Complex Wave Motions and Thermal Structure of the Oceans

C H A P T E R

3 Complex Wave Motions and Thermal Structure of the Oceans The water column structure of the sea oscillates continually. These oscillations cover a broad range of frequencies— a spectrum—with capillary waves in the high frequency band and planetary scale waves in the low frequency band. The in-between significant bands are small gravity waves, swell, storm surge, seiche, tsunami waves, Kelvin waves, and tides. Storm surge usually represents a meteorologically driven positive surge of water in the sea, manifested at the coast in the form of heavy flooding of coastal lands, often leading to major destruction and fatalities. Because the sea surface behaves as an inverted barometer, an atmospheric pressure drop (ie, depression) in a large region above the sea in association with storms leads to an upward surge of water in the form of a dome immediately below the atmospheric depression region (see Fig. 3.1). Because water is incompressible, to compensate for this positive surge (ie, elevated sea level) in one region, a negative surge (ie, depressed sea level) must necessarily occur in another neighboring region in the sea. Storms are associated with strong winds blowing over a large region. Apart from the inverted barometer effect, the intense winds accompanying the storm can also trigger positive and negative surges. In shallow waters, the gradient of the wind-induced surge is directly proportional to (wind stress/water depth). Such positive and negative surges trigger sea surface waves of differing frequencies and amplitudes. The high frequency (ie, short wavelength) waves attenuate fast and die down completely after traveling some short distance, but the low frequency (ie, large wavelength) waves travel over large distances in the form of swells. For example, the extreme southwestern coast of India (particularly the Kanyakumari-Thiruvananthapuram coastal belt) experiences large swells even during very calm atmospheric conditions because of the arrival of swells from as far away as the southern Indian Ocean (see Baba, 2005) and perhaps even from the Southern Ocean. Storm-induced positive surges are very common along several coasts. Negative coastal surges are also possible depending on the wind direction and the influence of the Coriolis force acting on the wind. Sea level oscillations of different kinds have always been a source of fascination (and sometimes terror!) to many observing minds. Perhaps the oscillations that are readily observed by the human eye and catch the immediate attention of an observer on the beach are the wind-driven gravity waves, commonly known as waves or wind waves (to distinguish from swell), which are classified in the category of high frequency oceanic waves. Interestingly, almost all oceans undergo continuous motions of several kinds, from the surface to the bottom, caused by wind-driven surface waves, ship wakes, tide, fronts and eddies, and different kinds of ocean currents, whose ranges and speeds vary from place to place depending on several factors, primarily topographic influences (see Joseph, 2013 for an account of major ocean currents and eddies). Some of these motions are a rich source of green energy that has begun to be tapped more vigorously in recent years in several parts of the world. Apart from the sea surface motions that are almost always present, large undersea earthquakes and volcanic eruptions generate destructive tsunami waves (see Joseph, 2011; Parker, 2012; Pugh and Woodworth, 2014) that last only for a few days and can be confined locally or may have a global extent. Violent atmospheric disturbances (eg, tornadoes and cyclones) give rise to storm surges. The ocean is one of the most important components of the climate system. It constitutes a large reservoir for heat and carbon and is thus a major player in controlling climate change (see Siedler et al., 2013). The thermal structure of different zones of the world oceans are often complex, exerting considerable influence on the spatial distribution of marine life and precipitation over land. Thermocline oscillations, the giant subsurface waves known as internal waves

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FIG. 3.1 Illustration of positive surge generated in the sea under the influence of an atmospheric storm and the accompanying atmospheric pressure depression.

(IWs), are important in terms of (1) the danger posed by them to offshore installations, and (2) their ecological and climatologic benefits in mixing the ocean.

3.1 RIDDLE OF THE BIBLICAL CROSSING OF THE RED SEA BY ISRAELITES LED BY MOSES The historic crossing of the Red Sea by the enslaved Israelites under the leadership of Moses, as described in Exodus, chapter 14, of the Old Testament, has traditionally been described as a miracle. Over the past 150 years, several researchers have attempted to provide a reasonable scientific explanation for this miraculous episode that is believed to have taken place about 1250 years before Christ. Based on such studies, it turns out that the biblical parting of the Red Sea that enabled the crossing of the fleeing Israelites and the drowning of the pursuing Egyptian Pharaoh and his entire army is an outcome of an interesting oceanographic phenomenon known as negative surge, an intense wave motion under a severe and persistent wind and its subsequent sudden relaxation. This happens very rarely. It would be interesting to skim through the biblical story and the relevant scientific studies that are considered to have finally shed light on this event of biblical and historical importance. Enjoy reading!!

3.1.1 The Biblical Story The book of Exodus, in chapter 14 of the Old Testament, contains a description of Moses and the Israelites making a dramatic escape from the pursuing Egyptian Pharaoh's army through the parted waters of the Red Sea. This crossing is an eventful episode believed to have taken place about 1250 years before Christ. This story is also mentioned in the

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Quran. It marks the point in Exodus at which the Israelites escape slavery in Egypt and enter a period of wandering in the wilderness in an effort to realize their dream of finding the land of Canaan, which, according to the Bible, God had promised to the Israelites. The Egyptian pharaoh agrees to their departure. Then quite unexpectedly he and his army pursue the Israelites on chariots, sparking terror in the fleeing population. But according to the biblical narrative, God protects the Israelites from the Egyptians by separating them with a pillar of fire by night and a pillar of cloud by day. Moses and the fleeing Israelites, however, become trapped between the Pharaoh's advancing chariots and a large body of water termed Red Sea. According to the Exodus event as narrated in the Old Testament, while the Egyptian army was closing in on the fleeing Israelites, in a divine miracle a mighty east wind blew all night, splitting the waters of the sea and leaving a passage of dry land with walls of water on both sides, as a result of which the Israelites were able to flee to the other shore. But when, according to the biblical narrative, the Egyptian Pharaoh's army attempted to pursue them in the morning across the Red Sea, Moses stretched out his hand at God's command over the Red Sea waters, which rushed back and drowned the soldiers. According to a narrative in the Song of the Sea, God cast the Egyptians into “tehomat,” the mythical abyss (see Soggin, 1999), which may perhaps be the Red Sea. Ancient Hebrew writers and modern Christians have generally viewed the Exodus episode of the Old Testament as a miracle and a mighty work of God. Free from scientific considerations, the writers merely focused on the Israelites' deliverance from certain death at the hands of Pharaoh and his army. According to the Exodus narrative, God sent the east wind at just the right time to part the Red Sea and to reveal a path of escape for Moses and his people. The miracle is in the timing and in the advance notice given to Moses. The Bible states that God used the natural agent of wind to deliver His chosen people. The ancient Hebrews believed that God was in charge of the natural world and all its forces. Modern Christians who believe in the existence and power of God see science as the study of God's creation.

3.1.2 In Search of Scientific Explanations to the Vexed Question The ancient flight of the Israelites from Egypt into Canaan and the unusual events associated with it are of considerable historical and biblical importance. It would, therefore, be of interest to examine whether events such as the Red Sea crossing can be explained in terms of natural phenomena. Furthermore, because of the significant advancement achieved in the realm of physical oceanography during the past 75 years, such an examination appears to be appropriate. Several investigators have made earnest attempts to unravel the mystery shrouded in the story of the biblical crossing, and it is entirely appropriate for science to study an event described as a miracle. That a reliable answer to the vexed question surrounding the parting of the Red Sea had finally been found at the end of a long search would be music to the ears of all those (believers and non-believers) who are curious to understand the wonderful mechanism behind the biblical story. Results of studies carried out in this line for over a century have been reported by several researchers such as Bartlett (1879), Tulloch (1896), Hellstrom (1950; a translation of Hellstrom's 1924 work), Dayan (1978), Har-el (1983, 1987), Goedicke (1992), Nof and Paldor (1992, 1994), Drews (2009), and Drews and Han (2010). This list of researchers who pursued studies relating to this legendary event is indicative of the scientific interest in this biblical topic, and such research is believed to have revealed the mechanisms responsible for the great miracle and to have located the site where it happened in the eastern Nile Delta. The purpose of such scientific investigations was never to prove or disprove that a crossing and exodus occurred but rather to examine whether or not a crossing phenomenon is plausible from a purely scientific point of view. Likewise, the investigating scientists' aim was never to address all the details of the biblical description, because some aspects of the Exodus appear to be mere spicy ingredients inserted by the Exodus narrator, probably with the willful intention of creating a mystical background to the entire story, and are clearly unscientific from a natural point of view (eg,, a pillar of fire and a pillar of cloud). The scientific investigations proceeded by merely accepting the biblical account as a possible “qualitative” description of an event which is said to have taken place in a distant past (about 1250 years before Christ). As indicated, attempts to explain the Red Sea crossing event in terms of natural phenomena including wind action can be traced as far back as the 19th century (see Bartlett, 1879). Because the Exodus was always a subject that intrigued scholars, it is reasonable to suppose that attempts to explain the crossing had been made even earlier, but their records probably did not survive because such attempts would at one time have been considered heresy. An excellent review is provided by Bartlett (1879), who noted that the tip of the Gulf of Suez could be the most likely site for the crossing. Bartlett further commented on the characteristics of the tides at Suez and inferred that the crossing was probably a

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result of a strong northwesterly wind. (Note that this wind direction is in disagreement with the east wind mentioned in the biblical narrative.) Furthermore, Bartlett identified a ridge (ie, an underwater hill) in the northern part of the gulf and argued that the crossing might have occurred along that ridge. Before addressing the scientific investigations carried out by various researchers to explain the riddle, it would be worthwhile to skim through some of the geometrical features of the Red Sea and the Gulf of Suez so that the arguments put forward by the researchers can be better understood and appreciated. The Red Sea is a seawater inlet of the Indian Ocean, lying between Africa and Asia. The connection to the ocean is in the south through the Bab el Mandeb strait and the Gulf of Aden. In the north, there is the Sinai Peninsula, the Gulf of Aqaba, and the Gulf of Suez (leading to the artificially constructed present-day Suez Canal, which of course was nonexistent during the historical Red Sea crossing episode). Morcos (1970) has noted that the Gulf of Suez is about 350 km long and 20–30 km wide. Its average depth is about 36 m and on its northern side the bottom slope is very gentle (2: 1000). The salinity of the gulf is higher (
40%) than that of most seas, primarily because of strong evaporation. Its typical temperature is about 22°C. Bartlett (1879) and Hellstrom (1950) attempted to explain the crossing purely in terms of natural phenomena. Wind setdown has been considered a plausible explanation because it is more closely related to the biblical description in terms of the strong wind prior to the event, the receding water, and the crossing in the midst of the sea. Note that wind setdown is a negative surge that occurs in shallow coastal areas when strong winds blow offshore. When wind stress acts for several hours on a body of water, the free water surface acquires a low-angle tilt. This tilt causes the water on the upwind side to recede from the original shoreline, leaving exposed mudflats on the bottom. As mentioned, wind setdown is directly opposite to positive surge and comparable in vertical displacement, although wind setdown is less well known because it usually poses no danger to lives and property. In simple language, wind setdown is the drop in water level caused by wind stress acting on the surface of a body of water for an extended period of time. As the wind blows, water recedes from the upwind shore and exposes terrain that was formerly underwater. Dayan (1978) made the first attempt to actually compute the necessary conditions for crossing the Red Sea. He used the well-known linear oceanic response to wind stress: ε¼ 

τL , gHρw

(3.1)

where ε is the sea level drop, τ the wind stress, L is the length of the gulf, H the water depth of the gulf, g the gravitational acceleration, and ρw the water density in the gulf. The dependence of the wind stress, τ, on the wind speed is given by: τ ¼ ρa CD w2 ,

(3.2)

where ρa is the density of air, CD is the drag coefficient (empirically defined), and w is the wind speed. Recognizing that the likelihood of a wind of, say, 20 m/s over the Gulf of Suez is very small and that it gives a linear wind setdown of merely 1.5 m, Dayan (1978) suggested that a combination of wind and extremely low tide could perhaps expose a submerged ridge. Using hydrographic maps from the period prior to the construction of the Suez Canal, Dayan identified a ridge where the crossing presumably occurred. He claimed to be the first to identify such a ridge, but in fact the same ridge was identified by Bartlett in 1879. In any case, Dayan (1978) resorted to employing a mechanism that involves a particular combination of wind and the low-tide phase of biweekly spring tides (during which the low tide is the lowest) to produce the desired sealevel drop. Because of this particular combination, the likelihood of such an event is rather small. The likelihood of Dayan's mechanism is no more than once in 30,000–60,000 years, implying that, although the mechanism is certainly possible, it is less appealing. Dayan did not publish his results in a scientific journal of natural sciences but rather in a scholarly journal of biblical studies (in Hebrew, later translated into Spanish) that is not usually accessible to natural scientists in the West. Consequently, Dayan's studies remained largely unknown for some time. It may be noted that the submerged ridge mentioned by Dayan (1978) and also by Har-el (1987) as a possible crossing site is fairly close to the center of the connected lakes that formed part of the Gulf of Suez. The idea that the wind could possibly be responsible for the crossing had already been qualitatively mentioned by Bartlett (1879) and Har-el (1983), but the quantitative scientific question of how such a process could actually occur had not been addressed. Doron Nof of the Florida State University and Nathan Paldor of the Hebrew University of Jerusalem held the view that it is quite possible that an event resembling the biblical story did occur. Nof and Paldor (1992) provided a possible scientific explanation for the Red Sea crossing by using an analysis of the possible oceanographic processes that the Red

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Sea can be subject to as a result of strong winds acting on it. With the aid of such an analysis, they argued that the Israelites' crossing and the Egyptians' drowning could have been a result of known natural phenomena. Further, they showed that although such events are not common, they are possible from a scientific point of view. Based on the application of fluid dynamical nonlinear theory (in contrast to the linear theory applied by Dayan (1978)), Nof and Paldor (1992) offered a possible explanation for the biblical crossing of the Gulf of Suez (see Fig. 3.2) in the Red Sea in terms of natural phenomena. They proposed an oceanographic explanation, the essence of which is a nonlinear process based on the unique geometry of the Gulf of Suez, ie, a long and narrow (ie, width/length ≪ 1) and shallow (ie, maximum depth/length ≪ 1) channel where the exodus presumably took place, which is connected to the deep Red Sea proper. Their wind setdown model of the Gulf of Suez represents the above-mentioned geometry (see Fig. 3.3). They showed that the oceanic response to wind action is governed by a nonlinear balance equation comprising the undisturbed water depth, the stress induced by the wind, the density of water, the free surface vertical displacement of water relative to the undisturbed depth, and the gravitational acceleration. Taking into account the wind stress, an exact nonlinear solution to this balance equation enabled Nof and Paldor (1992) to compute the water-receding distance from the coast and the associated sealevel drop for the Gulf of Suez geometry. In the absence of recorded evidence, they then argued that the ancient Gulf of Suez might have included a ridge (see Fig. 3.4) as testified by Dayan (1978) and Har-el (1987). Nof and Paldor (1992) showed that under such a scenario a northwesterly sustained wind blowing for 10–14 h with a speed of 20 m/s is sufficient to cause a sealevel drop of more than 2.5 m. Such a drop could potentially expose a swath of Red Sea bottom consisting of an underwater ridge, making a passage on “dry land” and

FIG. 3.2

The Gulf of Suez and the adjacent Red Sea Proper. Modified from: Nof, D., Paldor, N., 1994. Statistics of wind over the Red Sea with application to the Exodus question. J. Appl. Meteorol. 33, 1017–1025; © 1994 American Meteorological Society.

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y Gulf of Suez R B

tx

2W

A x

C

D L = 350 km Top view z f/2 x

tx H(x)

R

D = 70 m

L = 350 km

Cross-section

FIG. 3.3 Schematic diagram of the conceptual wind-crossing model. The gulf is taken to be a long and narrow channel with a linearly sloping bottom. Short thick arrows represent the wind stress at the surface. The variable H(x) is the undisturbed water depth, D the maximum depth of the linearly sloping bottom, 2w the width, L the length, and R the receding distance. From: Nof, D., Paldor, N., 1992. Are there oceanographic explanations for the Israelites' crossing of the Red Sea? Bull. Am. Meteorol. Soc. 73, 305–314; © 1992 American Meteorological Society.

Wind forces sea level down

Normal sea level

Underwater ridge exposed

FIG. 3.4

Schematic diagram of an assumed underwater ridge exposed by wind action. Owing to various geographical changes that have occurred during the last several thousand years (including the construction of the Suez Canal), it is currently difficult to identify such a ridge. However, Bartlett (1879) and later Dayan (1978) suggest that such ridges are common on the northern edge of the gulf. From: Nof, D., Paldor, N., 1994. Statistics of wind over the Red Sea with application to the Exodus question. J. Appl. Meteorol. 33, 1017–1025; © 1994 American Meteorological Society.

the Gulf of Suez crossing possible. Such an intense wind would also be associated with a receding of the shoreline (of the Gulf of Suez) of more than a kilometer from its original prewind position. These relatively high values of sealevel drop and the water-receding distance are a result of the unique geometry of the gulf (ie, its rather small width-to-length and depth-to-length ratios) and the nonlinearity of the governing equation. Nof and Paldor (1992) showed that a relatively simple oceanographic process of a northwesterly wind pushing the water offshore and the water then returning as a fast high-amplitude nonlinear gravity wave (as the wind relaxes within several minutes) could provoke floods of the entire receding zone within minutes, preventing any possible escape, thus explaining the “parting of the sea” episode in full measure. Nof and Paldor (1992) suggested that the crossing of the fleeing Israelites occurred while

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the water receded and that the drowning of the pursuing Egyptians was a result of the rapidly returning wave. It has been argued that the wind-crossing mechanism proposed by Nof and Paldor (1992) has much in common with the original biblical description because it involves pre-event winds (although with disagreement in wind direction), receding water, and a rapidly returning wave. Although the exodus story revolves around an easterly wind, Nof and Paldor (1992) employed a north-northwesterly wind in their computations. The basis for such an intentional departure from the biblical description of the wind direction is the following (see Nof and Paldor, 1992): because of the high mountain chains that run on both sides of the gulf, the wind is usually directed along the primary axis (ie, length) of the gulf. The dominant wind is from north-northwest (ie, north-northwesterly wind) during the entire year, although a south-southeasterly wind occasionally blows during the winter. For instance, during the spring and summer (the time of year when the crossing presumably occurred), the wind is from the north-northwest almost 70% of the time, and its average value is about 5 m/s (see Sharaf El Din, 1975). The maintenance of its direction and strength is assisted by the mountain chains on the two sides of the gulf, which are approximately parallel to the shoreline. The apparent discrepancy between Nof and Paldor's choice of a northwesterly wind and the biblical description of an east wind prior to the crossing has been attributed to the local variability of the wind. Nof and Paldor (1992) have, however, admitted that given the region's geography and, in particular, the relatively low mountains near the northern edge and the much higher mountains along most of the gulf, it is quite possible that in the relatively small area of crossing, the wind might have been indeed from the east, whereas over the much larger gulf the wind was from the northwest. The maximum monthly winds during the summer as given by the Comprehensive Ocean-Atmosphere Data Set (COADS) are about 8 m/s (see Woodruff et al., 1987). As a result of the wind stress acting for some time, a steady state is reached and the water recedes a distance R, given by the expression (Nof and Paldor, 1992):
 τx gρw D2 , (3.3) ln 1 + R¼ Lτx gρw ðD=LÞ2 where τx is the stress induced by the wind in the x-direction (ie, along the length of the gulf), D the maximum depth of the linearly sloping bottom, L the length of the Gulf, g the gravitational acceleration, and ρw the density of the gulf water. In the above expression, the gulf is assumed to be narrow and, therefore, the wind stress across the gulf is neglected. According to the theoretical calculations of Nof and Paldor (1992), the time, T, required for the wind to rffiffiffiffiffiffiffiffiffiffiffiffiffi LD . Note that in this expression, create the sea level setdown situation in the Gulf of Suez is given as, T
O τx =ρw “O” simply means “of the order of.” According to Nof and Paldor (1992), this corresponds to a Gulf of Suez sealevel setdown time of about 10 h for moderate winds and several hours for strong winds. This is in agreement with the biblical description of a “wind blowing the entire night” prior to the crossing. It has been suggested that a gradual relaxation of wind would cause a gradual return of the water to the prewind state, and this would present no threat to humans. However, if the wind relaxes or changes its direction abruptly (say, within several minutes), then the water returns as a fast nonlinear gravity wave or a “bore.” In this case, because the water behind the front of the nonlinear wave has larger depth than the water ahead, the back would travel faster than the front and the wave would ultimately pffiffiffiffiffi break. Even though the wave is highly nonlinear, its propagation speed is still given by gh, where h is the total depth of the water. This implies that for a wind setdown of 2.5 m (corresponding to a storm), the returning wave would travel at 5 m/s, so that the entire formerly exposed area would be flooded in just 4 min. Although Nof and Paldor (1992) showed that a passage on “dry land” and the Gulf of Suez crossing was theoretically possible in the presence of a north-northwesterly sustained wind of 20 m/s blowing for 10–14 h, there was no historical wind data available to unambiguously confirm the truth about the exodus story. Specifically, it has not been demonstrated how often a wind of 20 m/s lasting for 10–14 h is likely to blow over the Gulf of Suez in a northnorthwesterly direction. By and large, this is because the only reliable detailed wind dataset for the area was contained in an Israeli government report that was classified. Fortunately, portions of the wind data reports have been declassified subsequent to the theoretical investigations by Nof and Paldor (1992), making further studies possible. Now that detailed wind dataset for the area entered the public domain, Nof and Paldor (1994) have put the nonlinear theory on a firmer footing by providing a statistical analysis of the actual wind pattern in the area. The crossing presumably occurred in spring when the area is usually stormier than the rest of the year. Nof and Paldor (1994) used the Weibull distribution, the known duration of typical storms in the area, and direct measurements in the region for their research. The so-called Weibull distribution enables one to make predictions extending beyond the period for which data are available. The Weibull distribution is particularly important in handling wind data because it takes into account the fact that the winds are usually not isotropic but rather have preferred directions. The Weibull distribution is a general model that

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includes the Gaussian and Rayleigh distributions as special cases. On the basis of the Weibull distribution applied to long and reliable wind records in the part of the Indian Ocean adjacent to the Red Sea, ie, the northwestern Indian Ocean (see Pavia and O'Brien, 1986), and taking into account direct measurements along the Gulf of Suez, they argued that the likelihood of a storm sustaining winds of 20 m/s and lasting, as required, “the entire night” is roughly once every 2400 years. Nof and Paldor (1994) have suggested that the Red Sea crossing had been termed a “miracle” simply because the above likelihood period is exceedingly greater than the human life span, so that even if it occurred at a given time prior to the legendary crossing, it was not remembered by later generations. Nof and Paldor (1994) have indicated that their research on the vexed question of the legendary Red Sea crossing episode represents a departure from the traditional trend of physical oceanographic research, as their main aim was not limited to advancing physical oceanography per se. Rather they chose an unusual type of research that advances archeology, biblical history, and religion as well as physical oceanography because they view the role of science as an aid not only in advancing itself but also in advancing other avenues of human endeavor. As already indicated, previous researchers have suggested wind setdown as a possible hydrodynamic explanation for Moses's crossing of the Red Sea as described in Exodus 14. A study by Drews and Han (2010) analyzed the hydrodynamic mechanism proposed by earlier studies, focusing on the time needed to reach a steady-state solution. They performed a suite of model experiments to demonstrate a hydrodynamic mechanism that can cause an angular body of water to divide under wind stress. They utilized a modern ocean model, the three-dimensional Regional Ocean Modeling System (3-D ROMS), to investigate the interesting hydrodynamic event involving the phenomenon of wind setdown and performed a series of experiments for the idealized Suez basin with an underwater reef (see Fig. 3.4) as proposed by Nof and Paldor (1992; 1994). However, they used easterly wind forcing in accordance with the biblical narrative (unlike previous research, mentioned above, that employed northwesterly wind). According to Drews and Han (2010), geological and archaeological sources suggest that there once was a large coastal lagoon known as the Lake of Tanis into which the Pelusiac branch of the Nile flowed. The topography of this location resembles that of 1250 BC, when the Exodus is believed to have taken place. Based on this justification, Drews and Han (2010) chose this location to simulate a wind setdown event. Thus an important point to be noted is that Drews and Han's study location was different from that of the previous researchers, namely, across the Kedua Gap (30.98°N, 32.46°E) from West to East, a region about 3–4 km long and 5 km wide, which is in the proximity of the Lake of Tanis. Thus their study is based on a reconstruction of the likely locations and depths of Nile delta waterways, which have shifted considerably over time. Returning to the model experiments carried out by Drews and Han (2010), it may be noted that the ROMS is a modern Ocean General Circulation Model (OGCM) that can be configured to any ocean region ranging from local to basin scale (see Shchepetkin and McWilliams, 2005). ROMS implements a scheme for wetting and drying whereby the water's edge can advance to cover formerly dry land or recede and expose the underlying bathymetry. This scheme employs a critical depth, which is the minimum water depth that the ocean model will resolve. Coastal ocean modeling requires a rigorous transition from deep Ocean to shallow and intricate shorelines. While open-ocean dynamics can be represented using grid scales greater than 1 km, harbor and coastal features often require grid resolution on the order of 100 m. This range of scale presents a modeling and computational challenge. Unless the ocean model supports variation in the grid resolution (by nested grids or an unstructured grid), the modeler must increase the number of grid points to match the smallest feature in the domain. Following this strategy can easily increase the computational burden by many orders of magnitude, requiring a different class of computer on which to solve the problem. Based on this requirement, Drews (2009) used the “bluefire” supercomputer, available at the US National Center for Atmospheric Research (NCAR), for the simulation work. Bluefire supports higher-resolution modeling than a normal workstation can support, and this modeling is capable of revealing important features of wind setdown and storm surge in coastal areas. The computer simulation described demonstrates that a strong east wind, blowing overnight, could have pushed water back at a bend where an ancient river is believed to have merged with a coastal lagoon. With the water pushed back into both waterways, a land bridge would have opened at the bend, enabling the fleeing Israelites to walk across exposed mudflats to safety. As soon as the wind died down, the waters would have rushed back in. The simulations match fairly closely with the account in Exodus.

3.2 INVISIBLE COLOSSAL WAVES BELOW SEA SURFACE—INTERNAL WAVES In the oceans, there are more waves than meet the eye. Much below the whitecaps breaking on the sea surface, socalled internal gravity waves, or IWs, or alternately internal tides, may ripple through the interior of the oceans when favorable conditions for their generation occur. Although both the terms, IWs and internal tides, are used by different

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investigators to describe these waves, the former term is used in this book because unlike tides (which are of astronomical origin), IWs are not always of astronomical origin although strong tidal currents are often needed to trigger their generation. Furthermore, IWs are not generally coherent with the movements of the Moon and the Sun. Knowledge of IW movements are of increasing interest for underwater navigational precautions (see Osborne et al., 1978), offshore engineering applications (see Cai et al., 2003), and to oceanographers in general (see St. Laurent et al., 2012) because of their importance in ocean mixing. IWs, which are hidden below the sea surface and entirely between density layers within the interior of the oceans, are giant waves which can reach towering heights with profound effects on the Earth's climate and on ocean ecosystems. Typical speeds for IWs are around 1 m/s. They have been described as “the lumbering giants of the ocean.” They move fairly slowly relative to surface ocean waves but are very large in amplitude, carry a lot of energy, and can travel long distances.

3.2.1 Early History of Internal Wave Observations and Studies IWs travel along stratified layers of heavier, saltier than normal water as well as lighter, less salty water. They are ubiquitous in several regions and their existence has been known for well over a century. However, IWs have remained poorly understood because of the difficulty of observing them. Tracing the early history of IW study would therefore be of academic interest. Findings for oceanic IWs were reported in the early 1940s (see Munk, 1941; Ufford, 1947) and the early 1950s (see Shand, 1953) from the Gulf of California and the Georgia Strait. Much of the modern interest in large long nonlinear IWs and internal solitary waves (ISWs) in oceanography began in the 1960s to 1970s. Lee (1961) reported observations on IWs in shallow water, and LaFond (1962) reported some mechanisms of in situ measurements of IWs. Subsequently, IWs were observed in the mid-ocean (see Radok et al., 1967). The 1970s and the 1980s saw initiation of theoretical investigations and model studies of the IW phenomena (see Grimshaw, 1975; Garrett and Munk, 1975; Baines, 1982). Oceanic IWs are usually generated as tidal currents flow over a steep topography such as a ridge, a seamount, or a shelf break in the stratified ocean (see Baines, 2007). They are also found near islands, in straits, and along continental shelves. One such example is the region close to the Little Andaman Island and the Car Nicobar Island in the Andaman Sea (north of Sumatra in the Indian Ocean) where very strong oceanic IWs have been observed. Another example is the Strait of Messina separating the Italian Peninsula from the Island of Sicily (see Alpers and Salusti, 1983; Alpers et al., 1996). Yet another example of a region where eastward propagating oceanic IW packets (solitons) have been observed is the Strait of Gibralter, located between Spain and Morocco (see Watson and Robinson, 1990; Alpers and La Violette, 1993). Based on synthetic aperture radar (SAR) images taken from Space Shuttle “Endeavor” during a SAR mission over the continental shelf in the Atlantic Ocean, another location identified as sensitive to IW generation is off the coast of Namibia in East Africa (see Alpers, 1996). In the South China Sea and Yellow Sea, Hsu et al. (2000) identified IWs using SAR imagery. An important region where IWs break in the hidden depths of the ocean is along the 1600-milelong Hawaiian Ridge. Apart from large-amplitude IWs such as those just mentioned, weak IWs (amplitudes up to 2.5 cm) are also found in several parts of the World's Oceans.

3.2.2 Internal Wave-Induced Hazards Among several IW-induced hazards, one related to marine navigation is a curious phenomenon called dead water, first reported by the Norwegian oceanographer Fridtjof Nansen, in which a ship or boat may experience strong resistance to forward motion in apparently calm conditions. This occurs when the vessel is sailing on a layer of relatively fresh water whose depth is comparable to the ship's draft. This phenomenon is commonly encountered in fjords, where an interface is produced by a shallow layer of low-density freshwater from glacier runoff overlying relatively higher density salty oceanic water underneath. The ship produces a wake of IWs that dissipates a huge amount of energy. Thus if the interface on which the IW travels is very shallow, ships may find themselves in a situation in which most of the energy put into the propeller goes into driving the circular/elliptical particle motion of the IW at the interface, with the ship making little or no progress through the water. The passage of the ship through the highly stratified water layers (ie, low-density fresh water layer lying over a high-density salty water layer, thus forming a density interface) and the wakes shed by the passage of the ship together operate as triggering forces for the generation of IWs in this case. The peak-to-trough vertical distance (ie, wave height) of large IWs has been found to exceed a few hundred meters. For example, in the South China Sea, the observed vertical displacements of the ocean layers reach up to 200 m (see

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Ferrari and Wunsch, 2009; Alford et al., 2011). Large IWs have been found to pose hazards for manmade structures in the ocean, hindering underwater navigation (see Osborne et al., 1978) and causing grave danger to divers and offshore drilling operators. ISWs cause unusually strong underwater currents that may bring about large loads on deep-sea drilling rigs or platforms (see Cai et al., 2003). Osborne et al. (1978) observed IWs in the Andaman Sea, offshore Thailand, in water depths ranging from approximately 580 m to more than 1036 m during drilling by the drillship Discoverer 534. They conducted an engineering analysis on measured IW data and simultaneously measured drillship response. It was found that knowledge of IWs was required for designing production facilities in these deep waters. The oil companies working northwest of the Dongsha Islands in the South China Sea experienced equipment damage by ISWs during tanker operations and platform installations (see Ebbesmeyer et al., 1991; Bole et al., 1994). IW measurements are needed to characterize the energy of IWs at offshore drill sites. This information must be taken into account in the determination of strain on the drilling string; and in the case of standing platforms, IW measurements are required to determine the strain on the platform's supporting legs. Experiences from offshore engineering activities indicate that human lives and safety in the offshore environment depend on a clear understanding of IWs. Apart from causing danger to offshore operators and damage to offshore installations, evidence has been found to suggest that IWs trigger dangerous coastal seiches. For example, the east coast of Sri Lanka has been found to be particularly sensitive to the generation of seiches that are triggered by IWs generated in the Andaman Sea in the Indian Ocean. It was found that these seiches exhibit clear fortnightly and seasonal variations, having maximum amplitudes during March–April and October–November, during which stratification and IW activity are dominant (see Wijeratne et al., 2010). An investigation of global ocean ISWs has shown that westward propagating IWs could occur in the Bay of Bengal (in the Indian Ocean), originating from the Andaman Sea in the Indian Ocean (see Jackon, 2004). By means of a twolayer model simulation, it was shown by Wijeratne et al. (2010) that the seiche amplitudes can be excited by IW activity which travels over the Bay of Bengal from the Andaman Sea to Sri Lanka. It was also shown that occurrences of IWs in the Bay of Bengal are particularly likely during the months of April and November. Based on two-layer baroclinic model application with different stratification, Chapman and Giese (1990) concluded that the seiches are the result of IWs propagating onto the shelf. Seiches can sometimes be large enough to affect the coasts in a destructive manner (see Hibiya and Kajiura, 1982; Candela et al., 1999; Metzner et al., 2000). For example, a seiche event on Apr. 4, 2008, caused significant damage to fishing boats at Mirissa Harbor on the south coast of Sri Lanka (see Wijeratne et al., 2010). Among several other factors, locally or remotely generated IW activity in stratified water bodies has been found to be an important forcing mechanism of coastal seiches (see Giese et al., 1990; Chapman and Giese, 1990; Giese and Chapman, 1993). Strong fortnightly periodicity of the seiche amplitudes suggests that they are excited by the fortnightly forcing of the astronomical tides. Surprisingly, large seiches induced by IWs from distant sources have been noticed at some locations during neap tides, and the reason for occurrences of seiches during neap tides at such locations has been attributed to the IW's arrival time of approximately 1 week after their generation at distant sources (see Chapman and Giese, 1990). Giese et al. (1982, 1990) and Giese and Hollander (1987) proposed remotely generated IWs, which propagate to the Caribbean Coast as the seiche-generating mechanism in that area. Chapman and Giese (2001) have provided a similar argument for seiches in Puerto Princessa (Palawan Island, Philippines). They argued that IWs coming from the Sulu Sea, 450 km away, could be the source of seiches at Puerto Princessa, where maximum seiche activity occur 2–3 days after spring tides. Likewise, IWs, which are responsible for the generation of seiches on the east coast of Sri Lanka, originate at the distant ( 1200 km) Andaman Sea during spring tides and reach Sri Lanka during neap tide (see Wijeratne et al., 2010).

3.2.3 Similarities and Dissimilarities Between Surface Waves and Internal Waves Internal gravity waves, which are ubiquitous in the ocean, are the subsurface analogue of the familiar sea surface gravity waves that break on beaches. IWs, also called internal gravity waves, go by many other names depending upon the fluid stratification, generation mechanism, amplitude, and influence of external forces. If generated by flow over topography, they are called Lee waves. If generated in the ocean by tidal flow over submarine ridges or the continental shelf, they are called internal tides. IWs are usually distinguished from Rossby waves, which are influenced by the change of Coriolis frequency with latitude. Where low-density water overlies high-density water in the ocean, IWs propagate along the density boundary. They are especially common over the continental shelf regions of the world oceans and where brackish water overlies salt water at the outlet of large rivers. Note that associated with the continental shelf there exists a boundary line known as shelf break. The shelf break is the step between shallow seas (around

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continents and islands) and the deep ocean. It is the line at which tides usually start to generate IWs. While IWs of higher magnitudes will often break after crossing over the shelf break, smaller trains will proceed across the shelf unbroken (see Cairns, 1967; Winant, 1980). IWs are commonly found everywhere in the world and range in size from small-scale microstructures to waves with measured amplitudes as great as 200 m and more. Normally, IWs in shallow water are moderately sized, only a few meters or tens of meters high, being limited primarily by available energy sources, thermocline depth, and water depth. There is typically weak surface expression of the waves, aside from slick bands that can form over the trough of the waves. It would be interesting to draw some similarities and dissimilarities between the familiar sea surface waves and the not so familiar IWs in terms of their characteristics and generation mechanisms. Like the sea surface waves, which are waves at the interface of two media of different density (water and air), IWs are waves at the interface between water layers of different densities. The layers of different densities behave like completely different fluids. Thus if the water in the sea is stratified, there are water layers each with its own density, and these layers may start oscillating at their interfaces if energy sources are available to generate the waves. For example, current flow over uneven bathymetry, shear current flow, and atmospheric disturbances may cause IWs to form. These oscillations are often irregular. This type of oscillatory motion is called IWs. Associated with IWs are orbital motions of the water particles in the vertical plane, another similarity of IWs with surface gravity waves. IWs typically have much lower frequencies and higher amplitudes than surface gravity waves because the density differences (and therefore the restoring forces) within a fluid are usually much smaller. Similar to surface waves, IWs suffer modification as they approach the shore. As the ratio of wave amplitude to water depth becomes such that the wave “feels the bottom,” water at the base of the wave slows down due to friction with the seafloor. This causes the wave to become asymmetrical and the face of the wave to steepen, and finally the wave will break, propagating forward as an internal bore (see Defant, 1961; Cairns, 1967). The maximum amplitude of the IW normally is found near the average depth of the main thermocline. However, these waves cannot have amplitudes greater than the average thermocline depth and, just as surface waves, they are limited in height by water depth. Oceanic IWs are generated when the layered density interfaces are disturbed by some kind of obstacles and forcing. In the case of sea surface waves, the disturbance is usually caused by atmospheric pressure jumps or wind consistently blowing over the sea surface. However, oceanic IWs have a different story to tell. In the case of IWs, the disturbance at the density interfaces is usually caused by tidal flow pushing the layered water body over shallow underwater obstacles, eg, over shallow sills or shallow ridges (see Maxworthy, 1979; Helfrich et al., 1984; Lamb, 1994). As mentioned, IWs happen where the ocean is layered and a suitable forcing mechanism exists to trigger oscillations of these layers. Deep water is cold, dense, and salty, while shallower water is warmer, lighter, and fresher. The differences in density cause the various water layers of the ocean to behave like different fluids. IWs are generated in oceanic zones of density stratification as indicated. Typically, the largest IWs occur along the thermocline, which is a thermal boundary between the warmer surface waters and the cooler, deeper ocean waters. In general the seawater temperature decreases from the surface to the deepest levels, except in temperature inversion regions and high latitudes where the configuration can be more complex. There exists in most ocean areas (apart from polar and subpolar oceans) a zone where the rate of decrease of temperature is much larger compared with that above and below, hence the term thermocline. Because the thermocline separates the surface layers and the deep water of the ocean, it is often referred to as a boundary layer. Depending on the geographical location, the thermocline depth ranges from about 50 to 1000 m. A simplified view is to consider the thermocline as the separation zone between the mixed layer above (which is much influenced by atmospheric fluxes) and the deep ocean. In the tropics, the thermocline can be quite shallow on average, as in the eastern Pacific (50 m), or deeper, as in the western part (160–200 m). In extra-tropical regions, a permanent (or main) thermocline is found between 200 and 1000 m. However, the thermocline depth varies seasonally, especially in mid-latitude regions where a secondary and much shallower thermocline (above 50 m) occurs in summer. In high latitudes, a thermocline may appear only seasonally. The thermocline can also vary from one year to the next, as in the tropical Pacific where thermocline vertical displacements play a fundamental role during El Nino Southern Oscillation (ENSO). About 90% of the total volume of the ocean is found below the thermocline in the deep ocean. Fig. 3.5 illustrates a typical temperature-depth ocean water profile indicating the presence of a thermocline, comprised of layers of water where the temperature changes rapidly with depth. It can be seen that temperature decreases with increasing depth. Internal gravity waves are basically propagating disturbances (often nonlinear oscillations) of the ocean's density stratification (ie, thermocline). Fig. 3.6 shows an example of an IW traveling on the seasonal

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0

0°

Increasing temperature (°C) 4° 8° 12° 16°

20°

24°

500

1000

Thermocline

1500

2000 Increasing depth (m) 2500

3000

3500

4000

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FIG. 3.5 Illustration of temperature-depth ocean water profile indicating the presence of a thermocline, which comprises of layers of water where the temperature changes rapidly with depth. From https://en.wikipedia.org/wiki/Thermocline#/media/File:THERMOCLINE.png.

FIG. 3.6

Sketch of an internal wave propagating on the seasonal thermocline in the coastal ocean. Blue contours indicate isotherms (contours of constant temperature). The mixed layer is indicated by the greenish-blue region above the first isotherm, the thermocline by the crowding of isotherms at mid-depth. The slicks on the surface are produced by the convergence of water above the wave troughs in the mixed layer. Total water depth of the example shown is approximately 100 m. From: http://www.es.flinders.edu.au/
mattom/IntroOc/lecture10.html.

thermocline in coastal waters. Such waves typically have wave lengths of several tens of meters and periods of about 30 min. Convergence of surface particle movement above the wave troughs near the surface often collects floating matter (see Fig. 3.7) and makes the waves visible as slick marks. IWs are caused by the lower layer of the thermocline being forced against a shallow underwater obstacle, such as a ridge or sill, by tidal action. Generation of IWs as tides pass over a shelf break is quite usual, and in some regions IW bores are formed (see Rattray, 1960). The largest of the IWs are generated during spring tides, and those of sufficient magnitude break and progress across the shelf as bores (see Winant and Olson, 1976; Winant, 1980). These bores are evidenced by rapid, step-like changes in temperature and salinity with depth, the abrupt onset of upslope flows near the bottom, and packets of high frequency IWs following the fronts of the bores (see Shanks, 1995).

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FIG. 3.7 Oil slick from the Montara oil spill in the Timor Sea, September 2009. From: https://en.wikipedia.org/wiki/Oil_spill#/ media/File:Oil_Slick_in_the_Timor_Sea_September-2009.jpg. NASA Earth Observatory - http://earthobservatory.nasa.gov/images/imag erecords/40000/40254/timorsea_tmo_2009260_1.jpg.

The physics of IWs resembles that of surface gravity waves but with buoyancy rather than gravity providing their restoring force. The restoring force for waves is proportional to the product of gravity and the density difference between the two layers (the relative buoyancy). At IW interfaces, this difference is much smaller than the density difference between air and water (by several orders of magnitude). As a consequence, IWs can attain much larger amplitudes than surface gravity waves (amplitudes of tens to a few hundred meters instead of less than 10 m). It also takes longer for the restoring force to return particles to their average position. Furthermore, IWs travel much more slowly. They can travel thousands of kilometers from their sources before breaking (see Ray and Mitchum, 1996). Unlike normal sea surface waves, IWs in the ocean typically have wavelengths from hundreds of meters to tens of kilometers; wave periods from tens of minutes to several hours, even up to a half day (semidiurnal) or a full day (diurnal); and can last several hours. It is interesting to note that an IW may become confined to a finite region of altitude or depth as a result of varying stratification. Here the wave is said to be ducted or trapped, and a vertical standing wave may form, where the vertical component of group velocity approaches zero. A ducted IW mode may propagate horizontally, with parallel group and phase velocity vectors, analogous to propagation within a waveguide. When IWs become large and steep, they come under the category of nonlinear waves possessing trapped cores. Such waves have been predicted theoretically (see Derzho and Grimshaw, 1997) and observed in the laboratory (see Manasseh et al., 1998). Such IWs propagate in environments characterized by high shear and turbulence (see Scotti and Pineda, 2004). Pugh and Woodworth (2014) have explained the existence of IWs in a simple way by analogy with surface tidal waves. The propagation of waves on the sea surface depends on the density difference between air and water and on the gravitational acceleration; the air density is small enough to be ignored in the usual theoretical development. Waves may also propagate along density gradients within the ocean, at speeds that depend on the density differences. For IWs at the interface between two ocean layers, it is necessary to consider the density of both layers. With acceleration due to gravity g, the speed of propagation C of a long wave in a sea at the interface between two layers of density ρ1 and ρ2, the upper layer having a thickness D1, and the lower layer having a thickness D2, is given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ρ 2  ρ1 D1 D2 C¼ g (3.4) ρ2 D1 + D2 If the thickness of the bottom layer is much greater than that of the surface layer, then (D1 + D2)  D2 so that: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   ρ 2  ρ1 gð D 1 Þ C¼ ρ2

(3.5)

Eq. (3.5) is similar to the speed of a long surface wave but with the gravitational acceleration reduced by  ρ 2  ρ1 . (Note that the speed of propagation (c) of a long surface wave (eg, tide wave) is given by the expression: ρ p2ffiffiffiffiffiffiffi c ¼ gD, in which g is the gravitational acceleration and D is the local depth of the water body through which the long wave propagates.) According to Wijeratne et al. (2010), with g ¼ 9.81 m/s, for the east coast of Sri Lanka with D1  100 m, (ρ2  ρ1)  5 kg/m3 and ρ2  1020 kg/m3, the IW speed is calculated as C  2.2 m/s. The IW speed in the South China Sea has been reported by Ramp et al. (2010) to be roughly 3 m/s. Typical speeds for IWs are in the range 1–3 m/s, much smaller than for surface waves because of these smaller restoring forces. However, because

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they can have large amplitudes (as large as 200 m and more observed in the Luzon Strait (LS)), the currents associated with IWs can be considerable (see Arbic et al., 2012)—for example, roughly 3 m/s in the shallower waters of the Dongsha Island Plateau in the South China Sea (see Alford et al., 2015). Although the causative reason for generation of IWs is density difference of water layers as explained above, IWs are often triggered by sufficiently strong tidal forcing. For this reason, IWs are generated most strongly around the spring tides of new and full moon. However, the IWs are not usually coherent with the astronomical forcing (see Arbic et al., 2012). IWs can vary in size through the year as the density composition of the water varies with the seasons, but they can take many days to propagate across an ocean basin to where they are observed.

3.2.4 Peculiarities of Solitary Waves Solitons were first sighted in the month of Aug. 1834 by John Scott Russell, a Scottish Engineer, in the Union Canal near Edinburgh, Scotland, during his experiments in hydrodynamics (see Bullough, 1988; Darrigol, 2003; Craik, 2004). The generation of a soliton in this case was the result of an abrupt deceleration caused by the sudden stoppage of a fastmoving boat. In this case, the impulse launched a single smooth pulse of water “rolling on at a rate of some eight or nine miles an hour, preserving its original figure some 30 ft long and 1 to 1½ ft in height.” Based on painstaking experimentation, Russell viewed the solitary wave as a self-sufficient dynamic entity and described his “first chance interview with that singular and beautiful phenomenon” (which he then called the Wave of Translation) as “the happiest of his life” (see Darrigol, 2003). Any impulsive disturbance in a water body can give rise to the generation of solitons if conditions are supportive. A soliton, which forms an essential constituent of a large IW, is an extraordinarily smooth and well-defined heap of water in the form of a packet of high frequency nonlinear pulses. The packet of high-frequency waves that constitute the soliton hold themselves together “much like a mother duck keeping her brood collected when they try to stray.” Solitons are found in association with only very large IWs. The presence of solitons in such a large IW is due to the fact that this wave is generated following a large impulsive disturbance. The uniqueness of a soliton stems primarily from the fact that it is practically frictionless, does not break up, spread out, or lose strength. Unlike the plethora of familiar wind-generated water waves that we are accustomed to seeing, an IW can travel long distances without change of shape (ie, while preserving its original figure). A particularly interesting feature is that it preserves its unique wave-packet characteristic even after traveling through a maze of many other groups of oceanic waves on its way. Any linear wave that propagates through a dispersive medium such as the ocean, in which the phase and group velocities differ from each other, is bound to deform through dispersion. Amazingly, the wave pulses that constitute a soliton of even very large amplitudes can propagate without deformation. These very special attributes of water wave solitons are the result of a fine dynamic balance between dispersion (ie, the wave's tendency to spread out) and nonlinear effects. Soliton waves are nondissipative waves that occur at the boundary between two different tendencies of waves (ie, dispersion and nonlinearity). Low amplitude waves are linear waves that have a tendency to dissipate, as when we drop a pebble in a pond and observe the waves spreading out. Furthermore, the various frequencies that might comprise a low amplitude wave will gradually separate as a result of the different speeds at which they travel. Highamplitude waves, on the other hand, behave nonlinearly. They have a tendency to compress. However, just like low amplitude waves, the result is a rapid dissipation of energy and structure. Right at the boundary between these two tendencies we find soliton waves. It seems that at the boundary the nondissipative (or compressive) tendency of high-amplitude waves exactly cancels out the dissipative tendency of low amplitude waves. There is a strong relationship between the robustness of solitons and the degree of order or symmetry in the medium and boundary conditions that form their environment. What makes these waves so interesting is their robustness. Soliton theory was given a sound mathematical footing primarily by Korteweg and de Vries (see Korteweg and de Vries, 1895), although subsequent researchers have made excellent contributions to an understanding of the many mysterious facets of solitons (see Zabusky and Galvin, 1971; Hammack and Segur, 1974). Solitary waves can be described by a nonlinear wave equation (KdV equation, named after Kortweg and de Vries who originally derived it). Studies by Korteweg, de Vries, and many other researchers have shown that key to these very special characteristics of the soliton is a robust balance between the effects of dispersion and nonlinearity. Apart from their applications in the field of hydrodynamics and oceanography, the theory of solitons has found recent applications in biology, solid-state physics, optical communications, superconductive electronics, elementary-particle physics, solid-state physics, and cosmology. Although substantial theoretical work on solitary waves are available, only very limited experimental results have been published.

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3.2.5 Observations of Solitary Internal Wave Packets As deep ocean tides move water on and off the continental shelf, some of the tidal kinetic energy vertically perturbs the density structure in the water column, causing internal bores to form as trains of pulse-like density displacements which are commonly called solitons (see Ostrovsky and Stepanyants, 1989). During an experiment studying coastal IWs, Stanton and Ostrovsky (1998) observed extremely strong solitary IW packets over a 3-week period on a very shallow and strongly stratified pycnocline off northern Oregon. Note that a pycnocline is the cline or layer where the vertical density gradient (@ρ/@z) is greatest within a body of water. During periods of strongest tidal forcing, solitons were consistently observed on the leading edge of a semidiurnal internal tide bore, with pycnocline displacements up to 25 m downward from a 7 m initial depth in the first few solitons. The extreme nonlinearity of these IWs is believed to be unique in ocean observations. Stanton and Ostrovsky's study characterized these highly nonlinear Solitary Internal Waves (SIWs) and presented a second order KdV model that reproduced the form of the displacements and the small change in soliton width with amplitude predicted by this model. The area offshore from Tillamook to Oregon was found to be rich with strong IW surface signatures. The strength and extreme nonlinearity of the IW packets measured in the ocean profile observations made from research platform FLIP were a surprise to the investigators and appeared to be a “world record of nonlinearity” as characterized by a ratio of the maximum isotherm displacement to their initial depth. The objectives of Stanton and Ostrovsky's IW observations were to characterize the hydrodynamic nature of strong IWs propagating past FLIP, by determining their vertical displacements, particle velocities, and lateral displacements, and to determine their stability and contribution to upper ocean mixing. In Fig. 3.8, a 24 h period near the monthly maximum in tidal amplitude has been selected from the 3 weeks of observations to illustrate the internal tide characteristics. Two cycles of a steep-leading edged, smooth displacement of the thermocline, with an amplitude of approximately 10 m, was observed in the low frequency part of the time series. These two bore-like displacements could be interpreted as evidences of a strong, semidiurnal (
12 h period) internal tidal wave propagating past FLIP from a nearby, offshore generation site over the shelf break. Equally striking in Fig. 3.8 are the rapid vertical displacements (barely resolved at this time scale) extending down from the leading edge of the low frequency wave displacement by up to 25 m. Many studies have suggested nonlinear

Temperature (°C) 0

18.0 16.0

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–10 14.0 –20

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8.0 269

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269.6 269.4 Time (days)

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–10

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8.0 268.96

268.98 269 Time (days)

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FIG. 3.8 A color contour time series of temperature profiles from the surface to 35 m depth measured over a 1 day period. The 10°C span color contour scale is shown at the right side of the time series panel. The low frequency semidiurnal internal tide displacement can clearly be seen along the yellow isotherm. White areas indicate times with no data. From: Stanton, T. P., Ostrovsky, L.A., 1998. Observations of highly nonlinear internal solitons over the continental self. Geophys. Res. Lett. 25 (14), 2695–2698. This paper is not subject to US copyright. Published in 1998 by the American Geophysical Union. Paper number 98GL01772.

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mechanisms which cause the steep-edged internal tidal bore to disintegrate into packets of weakly to moderately nonlinear SIWs by dispersion (see Lamb, 1994). However, closer inspection of a 1.7 h profile time series expanded from the start of the first soliton displacement shown in the top plate of Fig. 3.8 shows the extreme nonlinearity of these SIWs, when the very shallow pycnocline depth of approximately 7 m seen at the start of the time series and the 15–25 m downward displacement amplitudes are considered. Comparisons between two theoretical forms for internal solitons, the KdV equation and the higher-order CombKdV model, suggest that the highly nonlinear solitons observed at this site are well represented by the CombKdV equation. Note that the next order model to the KdV model, the CombKdV model, accounts for a higher degree of nonlinearity (see Lee and Beardsley, 1974). Several measurements in the northern South China Sea (SCS) have shown that ISWs exist with amplitudes in excess of 100 m and phase speeds approaching 2.0 m/s, comparable to those in the Andaman Sea and Sulu Sea (see Apel et al., 2006), and that the generation of the ISWs is also associated with sharp topographic variations and strong tidal flows. Whereas the Hawaiian Ridge system dissipates only about 15% of the energy locally (see Klymak et al., 2006), the LS dissipates around 40% of the energy locally (see Alford et al., 2015). These results suggest that the LS is a more highly dissipative system than the Hawaiian Ridge system. Cai et al. (2012) found that there exists some difference in the ISWs between the SCS and other seas. First, almost all of the generated ISWs in the LS propagate westward to the deep basin of the northern SCS rather than eastward to the west Pacific; second, at the source site in the LS, besides strong tidal flows, the effect of a strong boundary current (the Kuroshio) makes the generation mechanism of the ISWs more complicated; finally, the generation mechanisms of the observed ISWs in the northern and western SCS were very different (see Ebbesmeyer et al., 1991; Zhao et al., 2004b; Zheng et al., 2007; Jackson, 2007). In the northern SCS, Cai et al. (2012) identified two types of ISWs: (1) a single-wave packet containing only one ISW with/without an oscillating tail, and (2) a multiple-wave packet composed of a group of rank-ordered ISWs. They found that, normally, the distance between two neighboring wave packets is about 10 km, each packet containing no more than ten rank-ordered solitons, with a packet width of 25 km. Liu et al. (2004) and Li et al. (2008) noticed that within a wave packet, the ISW separation distance (wavelength) appeared to be monotonically decreasing, front to rear, from 9.7 km to 500 m. Furthermore, near the Dongsha Islands, the westward propagating huge ISWs were often encountered, diffracted, and separated by the reef into the northern, the southern, and the middle parts. Several researchers (see Liu et al., 2004; Chao et al., 2008; Zhao et al., 2008) noticed that the northern and southern parts of the IWs in the SCS region continued to propagate westward, merged after passing the island, and interacted with each other, while the middle part was reflected eastward. Normally, the ISWs could occur in a whole year in the SCS, but there existed distinct seasonal variations. For example, Zheng et al. (2007) and Ho et al. (2009) found that the high monthly ISW occurrence frequencies are from April to August with a maximum frequency of 20–21.5% in June– July, and the low occurrence frequencies are from November to February of next year with a minimum frequency of 0.5–1.5% in January–February. It was found that the yearly distribution of ISW occurrence frequencies exhibit an interannual variability, implying the influence of longterm and large-scale processes in modifying the observed ISW occurrence. ISWs have been observed on the continental shelf (see Ostrovsky and Stepanyants, 1989), where the water column consists of a thin stratified upper layer overlaying a thicker well-mixed lower layer. On the California shelf, observations of ISWs have been documented by Howell and Brown (1985). In a set of moored array experiments conducted across the Chinese continental shelf and slope in support of the Asian Seas International Acoustics Experiment (ASIAEX), Ramp et al. (2004) observed highly nonlinear IWs (or solitons) which were generated near the Batan Islands in the LS and propagated 485 km across deep water to the observation region. Dubbed transbasin waves, to distinguish them from other, smaller nonlinear waves generated locally near the shelf break, these waves had amplitudes ranging from 29 to greater than 140 m and were among the largest such waves ever observed in the world's oceans till then. The IWs observed across the Chinese continental shelf and slope had several interesting features. For example, the waves arrived at the most offshore mooring in two clusters lasting 7–8 days each separated by 5 days when no waves were observed. Within each cluster, two types of waves arrived which have been named type-a and type-b. The type-a waves had greater amplitude than the type-b waves and arrived with remarkable regularity at the same time each day, 24 h apart. The type-b waves were weaker than the type-a waves, arrived an hour later each day, and generally consisted of a single soliton growing out of the center of the wave packet. Ramp et al.'s (2004) comparison with modeled barotropic tides from the generation region revealed that: (1) The two clusters were generated around the time of the spring tides in the LS; and (2) the type-a waves were generated on the strong side of the diurnal inequality while the type-b waves were generated on the weaker beat. The position of the

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Kuroshio intrusion into the LS was suspected to modulate the strength of the waves being produced. As the waves shoaled, the huge lead solitons first split into two solitons, then merged together into a broad region of thermocline depression at depths less than 120 m. Elevation waves sprang up behind them as they continued to propagate onshore. The elevation waves grew out of regions where the locally generated IW forced the main thermocline down near the bottom. The critical point α where the upper and lower layers were equal was found to be a good indicator of when the depression or elevation waves would form. It was further observed that critical point α was not a static point but rather varied in both space and time according to the presence or absence of the IWs and the incoming transbasin waves themselves. Liu et al. (2004) reported an interesting behavior of the ISW propagation in northeast of the Dongsha Islands (located on the southeast corner of the Dongsha plateau) in the SCS, in which its propagation direction rotated clockwise from 295 to 345 degrees, and the wave speed decreased from 1.8 to 0.72 m/s. Lien et al. (2005) found that the IW energy characteristics in the northeastern SCS are geographically distinct in four zones from the LS to the continental shelf. West of the LS, strong IWs are generated in the LS and propagate westward across the basin in a narrow beam, and the total IW energy (EIW) is 10 times that predicted by Garrett-Munk spectra (EGM). Near the Dongsha Islands, IWs are amplified by the shoaling continental slope and become nonlinear, and EIW ¼ 13  EGM. Chang et al. (2006) found that the presence of the Dongsha plateau directed the IW beam of approximately 100 km width to propagate onto the plateau and substantially convert to ISWs. ISWs disintegrate most of their energy before reaching the continental shelf. Furthermore, both depression and elevation ISWs are found to exist in this region. An interesting effect of formation of trapped cores and vertical displacement of water produced by IWs has been reported by Alford et al. (2015). The sequence of this event has been described thus: “Proceeding into the shallower waters of the Dongsha Island Plateau in the South China Sea at speeds of roughly 3 m/s, the ISWs begin to slow down (Ramp et al., 2010). Eventually, the wave-induced fluid velocities can exceed the wave speed, leading to the formation of trapped cores (Alford et al., 2010), wherein fluid is carried along with the wave. At this stage, the waves vertically displace water up to 170 m, nearly 40% of the local ocean depth, in only a few minutes and have wavelengths of only a few hundred meters.” Alford et al. (2015) observed the IWs to become convectively and shear-unstable, producing vertical overturns of up to 100 m within the core. Surprisingly, IW's effect on the surface of the ocean is quite weak, producing a rise of just inches that is virtually imperceptible on a turbulent sea; they are rarely noticeable on the sea surface unless one is looking down from space (see Ray and Mitchum, 1996). Recently conducted elaborate studies indicate that the giant IWs are potentially the key mechanism for transferring heat from the upper ocean to the depths (see Alford et al., 2015) and thus play an important role in climate change studies.

3.2.6 Internal Waves—Roles of Seafloor Obstacles and Currents IWs occur in oceanic regions which are layered in terms of water density structure and where a suitable forcing mechanism exists to trigger oscillations of these layers. It was realized since the beginning of IW studies that seafloor topography has an important role to play in their generation (see Baines, 1973). In subsequent studies, analyses of surface signatures of IWs detected in satellite images have also shown that energy in the IWs is radiated from localized areas of marked sudden depth changes and rough topography such as the Hawaiian Ridge in the Pacific Ocean. Other areas include the Mascarene Plateau in the Indian Ocean (see Da Silva et al., 2011), the Tuamotu Archipelago in the South Pacific, and near seamounts. There is also evidence for strong generation at straits that separate two areas of deep stratified seas; examples include the Straits of Gibraltar joining the Atlantic Ocean, the Straits of Messina in the Mediterranean Sea, and the LS in the SCS. The waves can propagate and can be identified over very long distances (see Zhao et al., 2012). Ridges and sills have been found to have a natural tendency to trigger the formation of thermocline. Seawater density layering akin to an open ocean thermocline occurs in the water body lying over shallow sills or shallow ridges because of the deep water (which is cold, dense, and salty) lying just outside of the ridges and because sills are forcibly brought underneath this shallower water body (which is warmer, lighter, and fresher) under the influence of large tidal forcing. When sufficiently strong tides drag the ocean water body over shallow barriers such as ridges or sills on the seafloor, the barrier performs two functions. It generates density layering and operates as a disturbing medium, both of which are necessary conditions for generation of IWs. As indicated, IWs occur in regions where the different density water layers are forced by the overall, depth-averaged, tidal currents to move up and down topographic features of the ocean bottom (such as at the continental shelf edge, deep ocean ridges, or ocean islands). As the tidal currents move up over a rise, they move the denser lower water up with them. On the other side, the denser water falls back to its normal

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level, generating IWs. There is significant dissipation of tidal energy by internal processes over rough topography in the open ocean (see Ray and Egbert, 2004; Garrett and Kunze, 2007). The observed occurrences of IWs immediately following spring tides (during which the tidal range is the largest) corroborates the notion that strong tidal forcing plays an important role in triggering IWs in the oceans. It has been found that, apart from ridges or sills on the seafloor, continental slopes also exert considerable influence in the intensification of IWs. For example, observations have indicated that when IWs shoal onto the continental slope of the SCS, the downward displacement of the ocean's layers associated with these solitary waves can exceed 150 m in 5 min (see Ramp et al., 2004). For a long time, confusion persisted regarding the nature of the mechanism of IW generation. It was suspected that IWs arise from seafloor obstacle-induced sharp hydraulic phenomena. For example, it was conjectured that the large-amplitude ISWs arise in the immediate vicinity of the LS via a hydraulic lee wave mechanism (see Guo and Chen, 2014). However, in a comprehensive set of field measurements, laboratory experiments, and model studies carried out in the LS with the aim of understanding the mechanisms of the generation, propagation, and demise of IWs on a basin scale, Alford et al. (2015) found that what emerges immediately to the west of the LS is a broad, energetic, spatially coherent, nearly sinusoidal IW at a combination of semidiurnal and diurnal frequencies. It was found that the IWs begin as sinusoidal disturbances rather than arising from sharp hydraulic phenomena. It was further noticed that the structure of the wave field is dominated by “mode 1” behavior (see Alford et al., 2011), this being the fundamental vertical mode of oscillation in which velocity in the upper few hundred meters of the ocean is in the opposite direction to, and oscillates out of phase with, the velocity in the deeper ocean. It was argued that the three-dimensional structure of the ridge system within the LS shapes the radiated semidiurnal and diurnal IWs differently because the horizontal wavelength of the former is half that of the latter. The baroclinic energy flux associated with propagating IWs is concentrated in beams that may extend thousands of kilometers from generation sites (see Simmons et al., 2004a,b). By far the largest and most powerful known IWs are those originating from the LS on the eastern margin of the SCS, located between the Philippines and Taiwan. Fig. 3.9 shows examples of beams of such IW energy fluxes in the LS region in the SCS. It was also found that the semidiurnal IW energy flux is strongest within a beam that emanates from the central section of the LS, while a broader beam of diurnal energy flux emanates from across the central and southern sections of the LS (see Fig. 3.9). These results and interpretations were furthermore supported by a rotating, large-scale laboratory experiment, using an accurate scale model of the LS (see Mercier et al., 2013). Apart from seafloor topographical effect as indicated by Baines (1973), Boyd et al. (1993) realized the role of strongly sheared water currents in the generation of high frequency IWs in the equatorial Pacific Ocean. Tidally generated IWs are usually highly nonlinear and occur often in wave packets, known as solitary waves. In the Strait of Messina, IWs are generated by a strong tidal current interacting with the shallow sill in its center (see Alpers and Salusti, 1983; Alpers et al., 1996). Many locations, such as the LS, generate these waves in a steady, predictable way as tides flow over submerged ridges and through narrow channels. A resulting 12-h periodicity is often clearly visible in remotely sensed IW data. In addition to the influence of tidal currents as mentioned, it was suspected that in the case of the IW field emanating from the LS, there could be an impact of the Kuroshio Current on the emergence of ISWs (see Guo and Chen, 2014). It may be noted that the Kuroshio Current is one of the oceans' foremost (in volume, extent, and velocity) western boundary currents, whose modeled lateral structure and velocities of over 0.5 m/s were confirmed by the detailed measurements reported by Alford et al. (2015) (see Fig. 3.10). Ramp et al. (2010) found that during Winter 2006, the Kuroshio Current was responsible for directing the semidiurnal IW beam and thus the solitary waves. Using new observations and numerical models, Alford et al. (2015) showed that when the Kuroshio western boundary current intrudes into the LS, the Kuroshio Current noticeably refracts the IW field emanating from the LS, profoundly affecting their subsequent propagation path. Alford et al. (2015) further found that the total energy flux in the northern section of the LS exhibits a clockwise pattern between the tall east and west ridges. This feature exists because the double-ridge structure creates a 100-km-scale resonant cavity for the
100-km wavelength semidiurnal IW (see Buijsman et al., 2014). The existence of this resonant cavity has been confirmed by observations of very high energy density but little energy flux between the two ridges, characteristic of a standing-wave pattern (see Alford et al., 2011). These and subsequent studies shed much light on the characteristics of IWs.

3.2.7 Measurement of Internal Waves and Internal Solitary Waves At the sea surface, the effects of IW-induced currents can be detected through high-resolution imagery obtained using remote sensors (eg, airborne cameras, satellite-borne SAR). Such images have provided a wealth of knowledge

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FIG. 3.9

Internal wave energy fluxes in the South China Sea. Semidiurnal (A) and diurnal (B) energy flux from the far-field model. (C) Energy flux along 21°N. Arrows in (C) indicate integrated energy fluxes (numbers on arrows are fluxes in kilowatts per meter) at 21°N in the semidiurnal and diurnal internal waves and in the solitary or nonlinear internal waves (NLIW). Flux values at 120°E are from the near-field model; flux and dissipation values at 115.19°E, 117.25°E and 117.895°E are from observations. (D) Bathymetry along 21°N. The processes of generation, breaking, propagation, steepening, and dissipation are shown schematically. From: Alford, M.H., Peacock, T., MacKinnon, J.A., Nash, J.D., Buijsman, M.C., Centuroni, L.R., Chao, S.-Y., Chang, M.-H., Farmer, D.M., Fringer, O.B., Fu, K.-H., Gallacher, P.C., Graber, H.C., Helfrich, K.R., Jachec, S.M., Jackson, C.R., Klymak, J.M., Ko, D.S., Jan, S., Johnston, T.M.S., Legg, S., Lee, I.-H., Lien, R.-C., Mercier, M.J., Moum, J.N., et al., 2015. The formation and fate of internal waves in the South China Sea. Nature 521, 65–69, http://dx.doi.org/10.1038/nature14399.

on the overall propagation pattern of IWs over a considerably large area, where they are seen in bands moving away from the generation areas. This feature provides a mechanism for identifying the generation areas of IWs. However, detailed quantitative measurements of IWs are obtained by means of in situ time-series profile measurements. Combined applications of both these methods have been found to yield valuable information on IWs and ISWs. For example, based on the analyses of both the satellite photographs and in situ observational data, several investigators (eg, Niwa and Hibiya, 2004; Ramp et al., 2004; Zhao et al., 2004b; Lien et al., 2005) inferred that the complicated bottom topography with several sill channels and sharp underwater ridges, strong tidal currents, and the Kuroshio intrusion favored the generation of IWs or ISWs in the LS in the SCS. Methods employed for in situ measurements and remote detection of IWs are summarized below. 3.2.7.1 In Situ Measurements Although IWs have been known for a long time, their measurement has been rather difficult. Interestingly, longterm temperature measurements have often showed subsurface water temperatures at tidal periods. Thus, time-series subsurface water temperature measurement has been identified as a means of measuring IWs. Oceanic IW activity is quantified and characterized by measuring the vertical oscillations in the thermocline by making time-series water

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Vertical displacement (m)

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22⬚ N

21⬚ N

50 0 −50 −100 −150 −200 01

Latitude 21.30⬚ N, longitude 118.60⬚ E 06

20⬚ N

19⬚ N

KS model Observed 0.5 m s−1

118⬚ E

(A)

119⬚ E

120⬚ E

121⬚ E

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(B)

11 Day in Feb. 2006

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50 0 −50 −100 −150 −200 01

(C)

Latitude 20.50⬚ N, longitude 119.00⬚ E 06

11 Day in Feb. 2011

16

FIG. 3.10

The Kuroshio Current and its influence on internal wave propagation in the Luzon Strait, South China Sea. (A) Observed (green) and modeled (gray) Kuroshio Current flow during Jun.–Aug. 2011 in the Luzon Strait region. The meshes are modeled phase lines of internal waves during Feb. 2006 (red) and Feb. 2011 (blue). (B, C) are measured wave displacement at the locations shown in (A). Waves were observed year-round at the southern station in 2011 (C), but not at the northern station in 2006 (B), when the Kuroshio Current deflected the internal wave paths southward (A, red). From: Alford, M.H., Peacock, T., MacKinnon, J.A., Nash, J.D., Buijsman, M.C., Centuroni, L.R., Chao, S.-Y., Chang, M.-H., Farmer, D.M., Fringer, O. B., Fu, K.-H., Gallacher, P.C., Graber, H.C., Helfrich, K.R., Jachec, S.M., Jackson, C.R., Klymak, J.M., Ko, D.S., Jan, S., Johnston, T.M.S., Legg, S., Lee, I.-H., Lien, R.-C., Mercier, M.J., Moum, J.N., et al., 2015. The formation and fate of internal waves in the South China Sea. Nature 521, 65–69, http://dx.doi.org/10.1038/ nature14399.

temperature profile measurements. Because the thermocline is a temperature boundary, a subsurface pressure wave recorder (which is one of the tools used for measurement of seasurface gravity waves) is not suitable for monitoring IWs. Note that the purpose of attaching pressure sensors on moorings is simply to provide depth reference to the various other measurements made from the moorings. IWs are traditionally measured by means of repeated bathythermograph tracings. However, taking into account the unusually large water current associated with large IWs, they can be measured also by means of a string of current meters. But these methods are very expensive. The most inexpensive and effective way to measure rapid, sweeping changes in the thermocline caused by IWs is with a string of fast-response temperature sensors (response time of 10 s or better). In fact, time-series fast-response temperature profile measurements, making a scan every minute for over a month or more for a fraction of the cost of repeated bathythermograph tracings or strings of current meters, represent a cost-effective breakthrough in IW monitoring capability. Fig. 3.11 provides a schematic representation of IW passage along a thermocline and the methods employed to measure the IW with the use of a vertical string of water temperature sensors deployed on a mooring line, which is held taut between the seafloor and a buoyant subsurface float. Alternatively, there are measurements made between an anchored ship and the seafloor. As part of the Oct. 1995 Coastal Ocean Probing Experiment (COPE), detailed measurements of the near-surface ocean and atmosphere structure were made from FLIP, a 108 m long research platform which was tri-moored in 140 m of water to provide a small horizontal cross section, stable platform from which the oceanographic, meteorological, and remote sensing instruments could be deployed. The stable platform provided by FLIP allowed detailed upper ocean current measurements to be made concurrently with continuous upper ocean profiling by the Loosetethered Microstructure Profiler (LMP), which was raised and then allowed to free-fall through the water column every 80 s by using a computer-controlled servo winch (see McPhee and Stanton, 1996). This rapid cycling provided 0.1 m vertical resolution temperature and salinity profiles from both upward and downward profiles every 40 s from the surface to a depth of 35 m. During the Internal Waves in Straits Experiment (IWISE) campaign, Alford et al. (2015) employed a series of approximately 30 densely spaced water temperature loggers and an Acoustic Doppler Current Profiler (ADCP) along the IW propagation path. The former gave the temperature (from which seawater density was computed) and the latter yielded water current velocity measurements in the upper ocean. Apart from these measurements, the moored

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FIG. 3.11 Schematic representation of internal wave passage along a thermocline and the methods employed to measure the internal wave with the use of a vertical string of water temperature sensors deployed on a mooring line held taut between (left) the seafloor and a buoyant subsurface float; and (right) an anchored ship and the seafloor.

profilers carried current meters and conductivity-temperature-depth (CTD) loggers, giving continuous, full water column measurements of density and velocity from a mooring. Carrying out both these measurements in an IW field represents a challenging task because moorings deployed in such a dynamic environment are prone to substantial knockdowns by the extreme currents associated with IW propagation. Knockdowns are minimized by highly taut designs and, in some instances, a low drag cylindrical float. Sandstrom et al. (1989) observed groups of solitary IWs and turbulence on the continental shelf off Nova Scotia using a towed CTD (BATFISH) and a 200 kHz echo sounder. The results of their studies indicate that a significant increase in acoustic backscatter and temperature gradient variance is observed in well-defined layers of approximately 5–15 m thickness and that these layers are generated by the passage of the group of IWs. Consideration of the nature of the IWs shows that the layers occur where shear instability is expected to be large and where the local Richardson number is smaller than the critical value required for the generation of turbulence. It was concluded that both acoustic backscatter and BATFISH temperature fine structure are associated with active turbulence generated by the IWs. An alternate moored measurement use bottom-mounted Pressure Inverted Echo Sounders (PIES), which measure bottom pressure and the round-trip bottom-top-bottom travel time of an acoustic pulse transmitted upward every few seconds. Because sound speed in water medium depends on water temperature, these signals are proportional to the mode-1 displacement of the thermocline (see Li et al., 2009). True mode-1 vertical displacements are computed from travel time using nearby moored in situ temperature measurements, and have an overall uncertainty of 4 m (Gregg, 1998). 3.2.7.2 Remote Measurements Because IWs are vertical oscillations of the thermocline, they propagate at the thermocline depths where the rate of decrease of temperature with increase of depth is the largest. These waves occur much below the sea surface, and they do not give rise to an elevation of the sea surface as familiar surface waves do. But they do give rise to successive cells of variable (horizontal) surface current as a result of the orbital motions of the water particles in the vertical plane. The water current velocity at the sea surface, resulting from such spatially adjacent cellular orbital motions, varies in magnitude and direction, thereby giving rise to convergent and divergent flow regimes at the sea surface. The IW pattern visible in remotely acquired imageries results from the strong surface expression generated on the sea surface as a result of the influence of the IWs on the sea surface current rather than spatial variability in the weak sea surface elevation. Sea surface disturbances caused by strong, near-surface IWs are widely seen in coastal regions as propagating bands of slick. Imageries of such surface slicks are typically manifestations of optical and radar backscatter properties above internal solitons. The scientific reasoning behind the feasibility of capturing the signatures of IWs on the sea surface can

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be explained in a more explicit manner as follows. When the crests of the IW pass below the sea surface, water in the surface layer essentially flows down the sides of the crest and pools into the area overlying the waves' troughs. Because surface water is diverging over the crests and converging over the troughs, biological surfactant materials and natural oils existing on the surface are alternately dispersed and concentrated in a wave pattern. These natural “slicks” alter the sea surface roughness pattern and thus exert considerable influence on the light reflection pattern of the sea surface. These surface slicks thus reveal the presence of the underlying IWs. The sea surface roughness pattern is especially obvious in areas of the image where the Sun's reflection off the water, called sunglint, is brightest. Thus, IWs become visible in remotely acquired images because they create alternating rough and smooth regions on the sea surface that align with their own crest and trough. Sunlight reflects the smooth sections, appearing as white arcs, while the rough sections stay dark in photographically obtained images. ISWs in the ocean were first seen from space by Vance Brand on the Apollo-Soyuz mission in 1975. Subsequently, oceanographer-astronaut Paul and his coworkers sighted IW solitons propagating from the Strait of Gibraltar in Oct. 1984. The train of solitons seen in satellite imagery is triggered by the inflowing Atlantic water, which has been accelerated by its passage through the narrow strait and across the sill at the entrance to the strait. Kropfli et al. (1999) observed strong IWs in the form of soliton groups off the Oregon coast with in situ and remote sensors, including shore-based X band and Ka band Doppler radars and an airborne microwave radiometer operating at a carrier frequency of 37 GHz. Analysis of these observations showed that the horizontal spatial structure of the IW field depends on whether it is forced by a strong (spring) tide or weak (neap) tide. Note that the spatial separation between the IW-induced successive convergent and divergent flow regimes at the sea surface is dependent on the wavelength of the IW motion. The above mentioned IW-induced variable surface current (ie, convergent and divergent flow regimes at the sea surface) modulates the sea surface roughness (see Hughes, 1978; Alpers, 1985). As a result of this interaction, IWs manifest themselves at the sea surface as a pattern of successive calm and rough water. This is the reason why oceanic IWs become visible on radar images of the sea surface and, in some cases, also on satellite images acquired in the visible, ultraviolet or infrared wavelength bands. From space, the appearance of the surface manifestation of the IWs (ie, variations of the small-scale sea surface roughness) is enhanced due to reflected sunlight, or sunglint, aimed back at the space station or a satellite, making the waves visible to an astronaut's camera or the satellite's sensor (see Hughes, 1978; Alpers, 1985). Calm, smooth waters reflect more light directly back to the satellite, resulting in a bright, pale stripe along the length of the IW. The rough waters in the trough scatter light in all directions, forming a dark line. Surface signatures of huge ISW packets with one to six solitons in a packet near the Dongsha Islands in the SCS were first observed by a satellite photograph in May 1973 (see Fett and Rabe, 1977). IWs are one of the earliest ocean phenomena detected on satellite images. Depression, elevation, broadening ISWs, and the continuous evolution process from depression to elevation ISWs can be discerned according to the different signatures in SAR images, in which a depression ISW packet can be identified as a bright band followed immediately by a dark band, while an elevation ISW packet can be identified as a dark band followed immediately by a bright band. In fact, since the 1970s, satellite imagery has provided an effective tool for detecting ISWs in the northern SCS (see Fett and Rabe, 1977). Recently, the European Remote Sensing (ERS) satellites 1/2 SAR, RADARSAT ScanSAR, SPOT, Landsat, IRS and NOAA AVHRR, Satellite Pour l'Observation de la Terre (SPOT) 1–5, and Envisat images from both optical and microwave sensors, including satellite ocean color products from the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) and the Moderate Resolution Imaging Spectroradiometer (MODIS), etc., have been employed widely to the study of ISWs in the SCS (see Liu et al., 1998, 2004; Hsu et al., 2000; Hsu and Liu, 2000; Zheng et al., 2001, 2007; Zhao et al., 2004a,b, 2008; Chao et al., 2008; Li et al., 2008; Du et al., 2008; Su et al., 2008; Ho et al., 2009). Fig. 3.12(top) shows IWs in the Sulu Sea (between the Philippines and Malaysia). In this true-color MODIS image acquired on Apr. 8, 2003, the slicks created by IWs appear as dark bands in the center of the image. In the Sulu Sea, sunglint highlights delicate curving lines of IWs moving to the northeast toward Palawan Island. The sunglint stretches diagonally across the image, turning the more turbulent SCS a diffuse gray and the calmer waters within the Sulu Archipelago a more vivid silver. In some spots, the sunglint does not appear at all and allows the natural vibrant blues and greens of the water to peek through. Fig. 3.12(bottom) shows IWs in the Tsushima Strait around Tsushima Island in Korea. On Jul. 4, 2000, the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) on NASA's Terra satellite captured this image of the area around Tsushima Island. In this false-color image (enhanced with shortwave- and near-infrared light), land surfaces appear bright red and clouds appear white. Of more interest, however, are the silvery blue waves that appear to be rippling the ocean's surface. Many of the waves appear to propagate toward the north. In the lower right part of the image, the waves push northward in a series of concentric arcs. To the left, another, fainter series of arcs pushes in nearly the opposite direction.

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FIG. 3.12 (Top) Internal waves in the Sulu Sea (From: NASA; http://earthobservatory.nasa.gov/IOTD/view.php?id¼3586; Image courtesy Jacques Descloitres, MODIS Land Rapid Response Team at NASA-GSFC).; (Bottom) Internal waves in the Tsushima Strait around Tsushima Island in Korea observed on Dec. 21, 2006 (From: NASA. http://earthobservatory.nasa.gov/IOTD/view.php?id¼7230; NASA image created by Jesse Allen, Earth Observatory, using data provided courtesy of the NASA/GSFC/METI/ERSDAC/JAROS, and the U.S./Japan ASTER Science Team).

Fig. 3.13(top) shows a photograph of IWs in San Francisco Bay. The photograph was taken using a digital camera from the International Space Station and was provided by the Earth Sciences and Image Analysis Laboratory at Johnson Space Center. Sets of IWs traveling east impinge on the coastline south of San Francisco. At the same time, fresher bay water flows out from the bay beneath the Golden Gate Bridge, creating a large plume traveling westward. Fig. 3.13 (bottom) shows a photograph of IW trains north of the Caribbean island of Trinidad, as featured by NASA's Earth Observatory in space. This photograph was taken by a crew member on the International Space Station. From space, the appearance of the waves is enhanced due to reflected sunlight, or sunglint, aimed back at the space station, making the waves visible to an astronaut's camera. The most prominent waves can be seen in the upper left of the photograph, moving in from the northwest due to tidal flow toward Trinidad. Another set can be seen moving in from the northeast, likely created at the edge of the continental shelf, where the seafloor abruptly drops off. A plume of milky

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FIG. 3.13 (Top) Internal waves in the San Francisco Bay. Courtesy of the Earth Science and Remote Sensing Unit, NASA Johnson Space Center; http:// earthobservatory.nasa.gov/IOTD/view.php?id=2474; (Bottom) Photograph of IW trains north of the Caribbean island of Trinidad, as featured by NASA's Earth Observatory in space. From: NASA; http://earthobservatory.nasa.gov/IOTD/view.php?id=80337.

sediment can also be seen moving to the northwest in the photograph. The sediment is carried by the Equatorial Current (also known as the Guyana Current), which flows from east to west, starting in Africa, and is driven toward the Caribbean by strong easterly winds. Fig. 3.14 shows IWs caused by tidal flow through the Strait of Gibraltar and made visible by sea surface roughnessenhanced sunlight backscatter. Note that the Strait of Gibraltar is a narrow strait that connects the Atlantic Ocean to the Mediterranean Sea and separates Gibraltar and Peninsular Spain in Europe from Morocco and Ceuta (Spain) in Africa. An interesting observation worthy of note in all the above images is the curved shape of the bands representing the IWs. Frequently, satellite images of IWs show up as bands of semicircular shape. The curved shape arises because of different propagation speeds of the IWs at different locations due to nonuniform depths D1 of the upper layer of the two stratified density layers, which results in the generation of IWs (see Eq. 3.5). Some bands are nearly straight, indicating uniform propagation speed in these regions (because of uniform depth of the upper layer of the two stratified density layers).

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FIG. 3.14 Internal waves (marked with arrows) caused by tidal flow through the Strait of Gibraltar and made visible by sea surface roughnessenhanced sunlight backscatter. From: Earth Observations Laboratory, Johnson Space Center. Arrows added and image rotated by Howcheng; http:// earthobservatory.nasa.gov/Newsroom/NewImages/images.php3?img_id¼16581.

3.2.8 International Efforts in the Study of Internal Waves ISWs, discussed earlier, are waves that more or less retain their shape during their propagation in a shallow, stratified layer by the balance between the nonlinear effects of steepening and dispersive effects (see Osborne and Burch, 1980). They travel below the pycnocline, causing unusually strong underwater currents that may bring about large loads on the deep-sea drilling rigs or platforms (see Cai et al., 2003). In the early decades of IW studies, IW measurements had been drawn from too narrow a slice of the region, resulting in conflicting results, rather like the fable of blind men describing a giant elephant. The first major field program of research on IW generation, known as Hawaii Ocean Mixing Experiment (HOME), was conducted by University of Washington (UW) researchers for the National Science Foundation (see Rudnick et al., 2003). It took place off the coast of Hawaii in 1999 and concluded in 2002. This was the first direct measurement of the energy flux of IWs along the Hawaiian Ridge. In the years since, scientists have come to a greater appreciation of the significance of these giant waves in the mixing of ocean water—and therefore in global climate. The most powerful IWs and ISWs discovered thus far are in the SCS. Numerous multinational efforts have been made over the years to gain a better understanding of the complexities of these strange waves. For example, from Apr. 2005 to Jun. 2006, the United States conducted the Windy Islands Soliton Experiment (WISE) and Taiwan the Variability Around the Northern South China Sea (VANS) experiment to obtain yearlong observations of ISWs from the LS to the northeast of the Dongsha Islands in the SCS (see Cai et al., 2012). In 2011, IWISE, mentioned above, was conducted to obtain the first comprehensive in situ data set from the LS, which in combination with high-resolution, three-dimensional numerical modeling was expected to support a cradle-to-grave picture of the life cycle of the world's largest known oceanic IWs. IWISE is the most substantial internal wave field program since HOME. Studies under the IWISE program revealed how most of the tidal energy from a complex ocean ridge system can be radiated away as nearly sinusoidal waves, with a small fraction dissipated locally. A pilot study was performed in the summer of 2010 to determine the feasibility of operating at specified locations (see Alford et al., 2011), and the full field program was executed throughout the summer of 2011. The IWISE program is broader in scope, however, because the

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radiated IW is nonlinear and therefore subject to additional steepening and wave-breaking processes. This international research effort was a rare undertaking in this field and the latest such field study on IWs on this scale. The new study looked at IWs that were stronger than usual and went significantly further in determining not just how the waves originated but how their energy dissipated. The IWISE program resulted in a first-order energy budget and an understanding of the key physical processes occurring in the LS that led to the formation of the world's strongest known IWs. Detailed new field studies, laboratory experiments, and simulations of the largest known IWs in the Earth's oceans—phenomena that play a key role in mixing ocean waters, greatly affecting ocean temperatures—provide a comprehensive new view of how these colossal, invisible waves are born, spread, and die off. As part of this comprehensive study program, Thomas Peacock, an associate professor of mechanical engineering at MIT, has teamed up with researchers from the Ecole Centrale de Lyon, the Ecole Normale Superieure de Lyon, and the University of Grenoble Alpes, all in France, as well as the Woods Hole Oceanographic Institution in the US, to carry out the largest laboratory experiment ever to study IWs. The team's largescale laboratory experiments on the generation of such waves used a detailed topographic model of the LS's seafloor, mounted in a rotating tank, 50 ft in diameter, in Grenoble, France, the largest such facility in the world. The experiments showed that these waves are generated by the entire ridge system on that area of seafloor and not in a localized hotspot within the ridge. The new observations resolved a longstanding technical question about how IWs propagate. Do the towering waves start out full strength at their point of origin or do they continue to build as they spread from that site.? Many attempts to answer this question have produced contradictory results over the years. This new research, which involved placing several long mooring lines from the seafloor to buoys at the surface, with instruments at intervals all along the lines, has decisively resolved several questions, including the observation that IWs grow larger as they propagate. According to the researchers, from 25 institutions in five countries, who carried out this comprehensive program, the results, a synthesis of the results from both programs mentioned above, published in the prestigious journal Nature (see Alford et al., 2015) contributed to a massive advance in our understanding of how these waves are generated and dissipated. They could add significantly to the improvement of global climate models.

3.2.9 Climatologic and Ecological Benefits of Internal Waves IWs can reach considerable heights (up to a few hundred meters) and generate powerful turbulence. They can also travel vast distances and can play a key role in the mixing of ocean waters, thus helping drive warm surface waters downward and drawing heat from the atmosphere. For example, the energy from the deep IWs that break in the hidden depths of the ocean along the 1600-mile-long Hawaiian Ridge is believed to help stir ocean waters, even those quite distant from the ridge. The ISWs become highly turbulent, leading to strong vertical mixing. Because of their size and behavior, such as strong vertical and horizontal currents, and the turbulent mixing caused by their breaking, IWs affect a panoply of ocean processes. They carry nutrients from ocean depths to the surface to enrich marine organisms; enhance sediment and pollutant transport; and affect acoustic transmission, ie, propagation of sound waves (see Williams et al., 2001). Bogucki et al. (1997) reported the observation of ISWs propagating upstream along a strongly stratified bottom layer on the California shelf, where an increased concentration of particulates in the water column was found to accompany the passage of these ISW packets. The leading ISW gave rise to reversed flow in an 8-m layer above the bottom. It has been suggested that the upstream propagating ISWs were generated by resonant flow over bottom topography. 3.2.9.1 Climatologic Benefits In the late 1990s, scientists began to hypothesize that the energy causing mixing in the oceans may be generated in places where tides draw deeper waters around and across rough seafloor features such as ridges. Tides traveling toward the Hawaiian Ridge collide almost directly into the ridge chain. During IW measurements at the Hawaiian Ridge under the mixing experiment program known as HOME, it was found that where the seafloor is the roughest, the mixing rates are considerably more intense than places without such topography. Furthermore, in terms of the energy flux of such IWs, some places turned out to be hot spots in the mixing experiment, with the energy of the most dramatic IWs measured at 60 kW/m at the French Frigate Shoals, 900 km NW of the Hawaiian Islands (see Ocean News & Technol., 2002; 8(2), p. 19). While the Hawaiian Ridge supports strong IW activity, the strong tidal forcing and ridge geometry at the LS result in not only some of the strongest IWs in the world's oceans but also the largest so far documented. As they propagate

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west from the LS, they steepen dramatically, producing distinctive solitary wave fronts evident in sun glint and SAR images from satellites (see Alford et al., 2015). Ferrari and Wunsch (2009) and Alford et al. (2011) found that the depthintegrated turbulent dissipation levels associated with the IWs in the SCS, where the observed vertical displacements of the ocean layers reach up to 500 m, with static instabilities >200 m high, approach 20 W/m2, exceeding open ocean values by a factor of 1000–10,000. Alford et al. (2015) noted that the associated vertical mixing of quantities such as temperature and salinity is orders of magnitude greater than open ocean values and that it probably plays a key role in setting large-scale circulation patterns throughout the SCS (Qu et al., 2006). Model results (see Alford et al., 2015) suggest that the ISWs can carry up to 73% of the generated long wave energy. It was found that the ocean conditions at the LS site are similar to those of other coastal sites, which suggest that this phenomenon may be common. It was further observed that IWs are a significant factor in the mixing of ocean waters, combining warmer surface waters with cold, deep waters, a process that is essential to understanding the dynamics of global climate. Deep-sea dissipation of IWs has been considered to play a crucial role in the ocean's global redistribution of heat and momentum (see Ferrari and Wunsch, 2009). It has been found that numerical climate models are sensitive to the effects of IWs (see Simmons et al., 2004a,b; Melet, et al., 2013), and therefore climatologists are faced with the challenge of gaining a proper understanding of these waves to effectively include them in climate models. A major challenge is to improve our understanding of IW generation, propagation, steepening, and dissipation, so that the role of IWs can be more accurately incorporated into climate models. It may be noted that IWs, by helping carry heat from the ocean's surface to its depths, function as an important parameter in modeling climate (see Alford et al., 2015). 3.2.9.2 Ecological Benefits In the initial decades of IW studies, the primary focus was to understand their influence on biological productivity, particularly fisheries. In parallel, their physics were investigated by physical oceanographers, and their inputs became beneficial to biological oceanographers and fisheries scientists. While both surface waters and those at depth tend to have relatively low primary productivity, thermoclines are often associated with chlorophyll-maximum layers (see Mann and Lazier, 1991). In fact, the importance of oceanic IWs in marine biological studies stems primarily from the fact that these waves are oscillations of the thermoclines, which are often associated with chlorophyll-maximum layers. Because IWs are associated with such oscillations, they have the potential to transfer the phytoplankton-rich waters downward, coupling benthic and pelagic systems (see Haury et al., 1979, 1983). Witman et al. (1993) found that areas affected by IWs show higher growth rates of suspension feeding ascidians and bryozoans, likely due to the periodic influx of high phytoplankton concentrations. They found pulsed phytoplankton supply to the rocky subtidal zone under the influence of IWs. It is believed that periodic depression of the thermocline (caused by its oscillations during IW events) and associated downwelling may also play an important role in the vertical transport of planktonic larvae. Arrival of cool deep water (associated with internal bores) into warm, shallower waters corresponds with drastic increases in phytoplankton and zooplankton concentrations and changes in plankter species abundances. For example, Leichter et al. (1998) noticed that breaking IWs on a Florida (US) coral reef is akin to a plankton pump at work. The chlorophyll-maximum layers in the oscillating thermocline in the IW attract large aggregations of mobile zooplankton (see Mann and Lazier, 1991) that internal bores subsequently push inshore. Whereas many taxa can be almost absent in warm surface waters, they are plentiful in these internal bores (see Leichter et al., 1998). An important influence of IWs on marine biology is their special ability to transport nutrients and plankton from deeper depths. It has been found that large, steep IWs containing trapped, reverse-oscillating cores can transport parcels of water shoreward (see Scotti and Pineda, 2004). The conditions favorable to the generation of IWs containing trapped, reverse-oscillating cores are likely to suspend sediment along the bottom as well as plankton and nutrients found along the benthos in deeper water. Apart from marine biological and fisheries interests, IWs have been found to play a significant role in sustaining coral-reef ecosystems, which are considered vulnerable to climate change and to other environmental effects. For example, in the SCS, an IW-induced mixing process and the supply of nutrients from the ocean depths have been found to help sustain an extensive coral-reef system. Apart from this beneficial ecological implication, it has been found that IWs are key to other ecosystems as well. For example, they are considered to play an important role in the cross-shelf transport of planktonic larvae between coastal and offshore environments (see Botsford et al., 1994). Some marine creatures essentially “surf” IWs to move in toward shore for feeding or breeding. Furthermore, they have been found to play an interesting role of piloting whale populations that forage in their wakes (see Moore and Lien, 2007). Mixing of warm surface waters and cold deep waters in the thermocline is an important component in helping to force nutrients up from the deep, where they can be used by tiny plants at the sea surface that are the

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foundation of the ocean's food web. The strong vertical mixing associated with ISWs in the SCS has been postulated to contribute to the high biological productivity in the vicinity of the nearby Dongsha coral atoll and to nourishing the coral reefs (see Wang et al., 2007). The process of IW mixing is enhanced in submarine canyons (well-studied examples include the Monterey Bay Canyon off California and the Hudson Canyon off the eastern United States). In these canyons, the waves are funneled and concentrated to give strong flows along the canyons' axis, converging at the upper end where they meet the continental shelf. The vertical internal movements can be important for bringing deep-water nutrient to the surface layers, both along continental margins and on the coasts of ocean islands, to nourish plankton, coral reefs, and seaweed.

3.3 INTRICATE TIDAL MOTIONS IN TOPOGRAPHICALLY COMPLEX WATER BODIES An observer who spends a day on a beach witnesses a low-frequency rhythmic rising and lowering of sea level known as “tidal oscillation” or simply “tide.” Tides have exerted such a great influence on the lives of coastal dwellers and beach visitors that the word itself may be found sprinkled throughout the poetry and folklore of many maritime regions. Indeed, so awe-inspiring are tides that they have inspired the creation of several myths. Meanwhile, progress in science has slowly and steadily begun to unravel the hidden secrets of these wonderful tidal rhythms. But prior to this progress and despite many direct and indirect influences of tidal rhythms on mankind, little was known about what caused them. This led to many strange notions and mythologies among ancient observers. Over time, the theory of tidal rhythms has received a sound mathematical footing from physicists, astronomers, and mathematicians including: Sir Isaac Newton (1642–1727), David Bernoulli (1700–1782), Marquis de Laplace (1749–1827), and Lord Kelvin (1824–1907). Investigations by these celebrated men of science have revealed that the origin of tidal rhythm is primarily astronomical with secondary bathymetric and topographical influences. To Newton we owe the artifice of the equilibrium tide, one that would exist in the absence of inertia on a world that is fully covered by water of infinite depth. In 1778, Laplace established the fluid dynamic equations of motion under gravity on a rotating sphere and described the tideproducing force as the residual differential force acting on a fluid particle in the ocean after the force at the center of the Earth is subtracted. It is now known that tidal rhythms are in reality produced by a combination of several forces, namely; • • • •

Force of gravitational attraction of the Moon and the Sun on Earth Centrifugal force produced by the revolution of the Earth-Moon system about their common center-of-mass Centrifugal force produced by the revolution of Earth around the Sun Rotation of the Earth about its own axis

Thus tides are forced oscillations generated by the attractions of the Moon and Sun, and they have the same periods as the motion of the Sun and Moon relative to the Earth. Every 14th day at full moon or new moon, the attraction forces of the Sun and Moon reinforce one another. These conditions give rise to “spring tide” during which the tidal range (ie, the difference between successive high water and low water elevations) is the largest in a fortnight. Interesting scientific explanations on tidal oscillations may be found in Young (1823), Airy (1845), Darwin (1898), Harris (1907), Doodson (1927), Doodson and Warburg (1941), Munk and Cartwright (1966), Garrett and Munk (1971), Pugh (1987), and Pugh and Woodworth (2014).

3.3.1 General Characteristics of Tidal Oscillations Sea level oscillations on an approximately twice daily (semidiurnal) or daily (diurnal) basis are a worldwide phenomenon routinely observed at continental coasts and shores of islands. At most coastal or island locations, the interval between successive high waters (or successive low waters) is about 12 h and 25 min, which is half the time of the Moon's apparent revolution around the Earth. This type of tidal oscillation is called semidiurnal tide. The fact that different oceans, basins, gulfs, and straits can respond to the different periods of the tide-generating forces allows a variety of tidal patterns to be developed. In some water bodies, the interval between successive high waters (or successive low waters) is about 24 h. This type of tidal oscillation is called diurnal tide. The East China Sea is an example of such a water body, suppressing semidiurnal tides and selectively promoting diurnal tides. In fact, mixed and diurnal tides predominate in parts of the Pacific. The most regular rhythmic motions in the sea level in several parts of the world's oceans have periods in the range of approximately 12–24 h.

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A parameter known as tidal form factor (F), defined by the ratio of the two main diurnal and semidiurnal constituents (K1, O1, M2, and S2) provides a quantitative measure of the general characteristic of tidal oscillations at a given location (ie, whether the tides are semidiurnal, diurnal, or mixed). F¼

ðHK1 + HO1 Þ ðHM2 + HS2 Þ

Tidal regimes where F is below 0.25 are normally said to be semidiurnal, F between 0.25 and 1.5 is said to be mixed and mainly semidiurnal, F between 1.50 and 3 is mixed and mainly diurnal, while F greater than 3 is diurnal (see Pugh and Woodworth, 2014). A map providing a quantitative measure of F in the world's oceans is shown in Fig. 3.15. Assuming that the Earth were fully covered by water (this, for simplicity of explanation), the full rotation of the solid Earth on its own axis (once in 24 h), combined with the “heaping” action resulting from the horizontal flow of water toward two diametrically opposite regions on the Earth representing the positions of the maximum gravitational attraction and centrifugal forces of the combined lunar and solar influences, makes each point on the ocean surface pass through two maximum levels (tidal bulges) and two minimum levels (tidal depressions) for each daily rotation of Earth. This causes two high waters and two low waters in a day in a given location, giving rise to the semidiurnal tidal rhythm. This is not applicable at amphidromic points, whose dynamics are entirely different. While the equilibrium high tides are generated as a result of the heaping action, the equilibrium low tides are created by a compensating maximum withdrawal of water from the null force regions around the Earth, which lies midway between these two tidal humps. The Moon, in making a full revolution around the Earth once each month, passes from a position of maximum angular distance north of the equatorial plane of the Earth to a position of maximum angular distance south of the Earth's equatorial plane and vice versa. The effect of this angular distance of the Moon relative to the equatorial plane of the Earth (ie, declination) is to produce inequalities in the heights of successive high waters (and successive low waters) at a given point because this point passes below the two tidal bulges as a result of the rotation of the Earth about its own axis. A point P which experiences a higher high tidal level (HHT) will experience a lower high tidal level (LHT) approximately 12 h later when the Earth's rotation on its own axis has brought this point to a diametrically opposite point P0 where the height of the ellipsoidal tidal bulge of water is lesser than that at P. This causes the successive tidal ranges (ie, the difference in levels between two successive high- and low-waters) to be different. This inequality in the range of two successive tides in a day is called the diurnal inequality of the tide. Diurnal inequalities (see Fig. 3.16) vary markedly with seasons. The strongest diurnal inequality is possible when spring tides occur during the solstices when both celestial bodies are near their maximum declination and acting together. For example, in June and December, the maximum diurnal inequalities of the year are observed because of the occurrence of the maximum solar declinations during these months. However, in March and September, solar Tidal form factor 0°

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FIG. 3.15 Map providing a quantitative measure of the tidal form factor in the world's oceans. The amplitudes used to make this map were taken from the Technical University of Denmark DTU-10 global tide model (see Cheng and Andersen, 2010). Courtesy of Dr. Philip Woodworth; Formerly, Leader, PSMSL.

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Regular variation of tidal range (observed at Ratnagiri, west coast of India) between two spring tides (which experience maximum tidal ranges) and two neap tides (which experience minimum tidal range) in a month. The data has been acquired from the real/near-real time reporting internet-accessible (http://inet.nio.org) sea level station network designed and established by the Instrumentation Division of CSIR-NIO in India (graph prepared by Prakash Mehra).

declination becomes zero (Equinox), causing the minimum diurnal inequalities of the year. From a purely theoretical point of view, the diurnal inequality arising from solar declination must vanish on the equator. However, lunar declination can give rise to diurnal inequality even on Equinox. Similarly, the diurnal inequality in any location must be zero when both the Moon and the Sun are on the equatorial plane of the Earth. However, as a result of the varying depths and boundaries, the real ocean responses to the tidal forcing are too complicated for these ideal situations to occur in practice. The interference of the solar tidal forces with the lunar tidal forces (which, due to the relative proximity between Earth and Moon, are about 2.2 times stronger than the solar tidal forces) causes the regular variation of the tidal range between spring tide (maximum tidal range) and neap tide (minimum tidal range). Spring tides occur when the Earth, Moon, and Sun are in line, which is at new moon and full moon. The synodic period from new moon to new moon is 29.5 days, and the time from one spring tide to the next is 14.8 days. Neap tides occur when the positions of the Moon and the Sun relative to the Earth are at right angles to each other, which is at half moon. During these times, the tide-generating forces caused by the Moon and the Sun counteract each other so that the resulting tide-generating force is reduced. Thus the observed high tides are lower than average, and the low tides are higher than average. Such tides of diminished range are called neap tides. There are two spring tides and two neap tides in a month, and one spring tide is usually larger than the other (see Fig. 3.16) because of the elliptical orbit of the Moon's motion around Earth. Within a lunar synodic period, the two sets of spring tides are usually of different amplitudes. This difference is due to the varying distance of the Moon from the Earth as a result of the elliptical path of the Moon around the Earth. At lunar perigee (the nearest approach of the Moon to the Earth), which occurs once each month, the lunar tide-raising force will be larger and therefore the spring tide during this period is the largest. Approximately 2 weeks later, when the Moon is farthest from the Earth (ie, at apogee), the lunar tide-raising force will be smaller, and the spring tide during this period is smaller than that at lunar perigee. One complete cycle from perigee to perigee takes 27.6 days. At perigee, the tidal forces are approximately 15% larger than the mean tidal forces. At apogee, the lunar forces are approximately 15% less than the mean tidal forces. Variations in the distance between the Earth and the Moon from a minimum to a maximum in 13.8 days cause fortnightly modulations in the tidal amplitudes. Very high spring tides occur when the Moon and Sun are overhead at the equator of the Earth, near the Mar. 21 and Sep. 23 equinoxes, when day and night are of equal length. The combinations of astronomical forcing that define spring and neap cycles and diurnal inequalities are further modified by local bathymetry and shoreline boundary influences. All of these factors combine to produce tidal envelopes that vary from location to location. The result is site-specific tidal signatures, which can be classified as semidiurnal, diurnal, or mixed. In the Sun-Earth system, the Earth follows an elliptical path relative to the Sun. When the Earth is closest to the Sun (perihelion), about Jan. 2 of each year, the solar contribution in the tidal pattern at a place will be enhanced. However, when the Earth is farthest from the Sun (aphelion), around Jul. 2, the solar contribution in the tidal pattern at a place will be reduced. Variation of the distance between the Earth and the Sun from a minimum to a maximum in 6 months causes a yearly modulation in the tidal amplitudes. Among several interesting characteristics of tidal oscillations, the spring tide enjoys a special status in a biblical sense—namely, the Easter spring. The high spring tides of Easter weekend are inevitable because Easter Sunday is defined in the western Christian calendar in terms of the lunar cycle and the date of the spring equinox. Easter is fixed as the first Sunday after the full moon that happens on, or next after, the Mar. 21 spring equinox. It may fall on any one of the 35 days from Mar. 22 to Apr. 25 (see Pugh and Woodworth, 2014). Thus Easter Sunday varies from year to year, depending on the time of spring tide after the Mar. 21 equinox.

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FIG. 3.17 Tidal oscillations during a year observed at Rat-

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In addition to semidiurnal and diurnal tides, there are also long-period tides. Surprisingly, periods of approximately 2 weeks (fortnightly) to approximately 19 years (Metonic cycle) exist. Tidal oscillations are periodic in a complex manner as mentioned because they are related to the complex motions of the Earth, the Moon, and the Sun. A fourth group of tides, not of direct astronomic origin, consists of the shallow-water tides that result from the mutual interaction of the semidiurnal and diurnal tides and several other constituents of the astronomical tides having specific periods. Tidal oscillations of all such periods are superimposed on the observed sea level oscillations. This allweather, worldwide, rhythmic rising and lowering of sea level is known as tidal oscillation or simply as tide. Fig. 3.17 shows tidal oscillations during a year observed at Ratnagiri sea level station on the west coast of India.

3.3.2 Topographical Influences on Tidal Range and Tidal Pattern In the open ocean, the tidal range (ie, difference in sea level elevation between successive high water and low water) is generally on the order of tens of centimeters up to a meter. Likewise, in some semi-enclosed seas (eg, the Mediterranean Sea and the Baltic Sea), the tidal range for both semidiurnal and diurnal tides is quite small. A detailed analysis of the characteristics of tides in several partially enclosed seas has been reported by Pugh and Woodworth (2014). Despite relatively small tidal amplitudes, the tidal currents in many of these semi-enclosed seas are very strong (2–3 m/s). The reasons for this have been found through thorough analyses as reported by Tsimplis et al. (1995) and Tsimplis (1997). Although astronomically induced tidal variations are in general less than a meter in range in the open ocean, when these tidal crests and troughs traverse into shallow waters, against landmasses and into confining channels, noticeable variations occur in the tidal range. In some other water bodies, particularly in bays and adjacent seas, the tidal range could be exceedingly large because their geometrical shape (bathymetry and topography) may favor tidal amplification. (This is called geometrical amplification.) It is known that the tidal range may be locally amplified further by several factors including the proximity of the period of natural resonant oscillation of the water body in the bay and estuary and the tidal period (see Clancy, 1968). Broad, funnel-shaped estuaries (with the broader portion open to the sea) cause exceptionally large tidal ranges in their head region, which is relatively much narrower and shallower than the mouth region that faces the sea. The Gulf of Kachchh and the Gulf of Khambhat in NW India are just two of several examples where such tidal amplification occurs, as the tide propagates up the gulf because of its funnel shape, with the mouth of the funnel opening to the sea. Fig. 3.18 shows the tidal pattern at the mouth and head of the Gulf of Kachchh on the west coast of India, illustrating the amplification. Geometrical amplification, as in the examples mentioned, takes place as the same volume of seawater that enters the mouth of the gulf under continuous tidal forcing is constrained to traverse up the estuary along progressively diminishing vertical cross-sectional area of the gulf, thereby compelling the water mass at a progressively increasing distance from the gulf mouth to spring up in order to conserve mass, momentum, and energy. However, frictional forces play an important role in influencing the quantum of amplification that really occurs as the tide propagates along the gulf towards its head region. The largest known tides occur in the Bay of Fundy (see Fig. 3.19) where spring tidal ranges up to 15 m have been measured. The reverse is true (ie, tidal range diminishes) in a broad water body if it opens to the sea via a narrow inlet. One of several examples to illustrate such a situation (geometrical attenuation) is the Chilika Lake in the Odisha State of India (see Fig. 3.20). Chilika Lake is the largest brackish water lake in Asia. It is approximately 65 km long, its width varying from 20 km in the middle to 7 km at the ends. The lake is connected to the Bay of Bengal through a 25-km long narrow inlet channel. In this case, the narrow tail of the water body opens to the sea. As a result, the tidal range in some locations within the lake diminishes up to one-seventh of that observed in the adjacent open sea. This is an example of geometrical attenuation. There are several water bodies of complex geometrical shape (eg, the Kochi backwaters shown in Fig. 3.21) in which the tidal range progressively decreases from their mouth to their head as a result of geometrical attenuation (see Fig. 3.22). The increasing prominence of the Msf tide towards the head region, resulting from

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FIG. 3.18 (Top) Partial map of India, indicat-

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ing the location of the Gulf of Kachchh. (Middle) Gulf of Kachchh on the north-eastern Arabian Sea (From Babu, M.T., Vethamony, P., Desa, E., 2005. Modelling tide-driven currents and residual eddies in the Gulf of Kachchh and their seasonal variability: a marine environmental planning perspective. Ecol. Model. 184, 299–312; Elsevier. © 2004 Elsevier B.V. All rights reserved). (Bottom) Measured tides at the mouth (Okha) and the head (Navlakhi) of the Gulf of Kutchch in India, illustrating the amplification of tides as it propagates up the gulf. Amplification occurs as a result of geometrical amplification and resonance (bottom figure prepared by PrakashMehra, CSIR-NIO, Goa, based on measurements carried out by CSIR-NIO).

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shallow water processes, is notable. In some gulfs and bays, tidal oscillations are triggered by the tides of the open ocean as a forced standing wave to which the Coriolis force adds a swinging cross-component, and both oscillations together result in a rotating tide. If an estuary is long, narrow, and shallows rapidly upstream, the rising tide (ie, flood tide) tends to be steepened when it meets the river water. This happens because as a progressive wave enters shallow water, its speed decreases.

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FIG. 3.19

171

Bay of Fundy where the largest known tides are found in the world. From: Google.

FIG. 3.20 Chilika Lake in the Odisha State of India, in which the water body opens to the sea via a narrow inlet resulting in geometrical attenuation of tidal range in the lake, which is 61 km long. The maximum width is 15 km. From: Google.

Because the trough is shallower than the crest, its retardation is greater, resulting in a steepening of the wave front. Therefore, in many tidal rivers, the duration of the rise is considerably less than the duration of the fall.

3.3.3 Wall of Tumbling and Foaming Water Waves—Tidal Bore A tidal bore (or simply bore, mentioned above) is also known as an aegir, mascaret, or pororoca in different countries. It is a sharp rise in free-surface water elevation propagating upstream in an estuarine system or narrow bay during a flood tide (ie, as the tidal flow turns to rising) against the direction of the river's or bay's current during spring tide. A tidal bore is literally a wall of tumbling and foaming water waves roaring up an estuary with high speed and thundering noise. The origin of the word “bore” is believed to derive from the Icelandic “bara” (billow), indicating a potentially dangerous phenomenon—ie, a tidal bore with a breaking roller (see Coates, 2007). In fluid dynamical term, a bore is, as mentioned earlier, a fast nonlinear gravity wave. Because the water behind the front of the nonlinear wave has larger depth than the water ahead, the back travels faster than the front and the wave ultimately breaks (see Stoker, 1957; Lighthill, 1978). A low barometric pressure and a positive surge on top of a spring tide are most effective for producing big bores. A tidal bore is in fact the most spectacular form of tidal distortion, seen

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FIG. 3.21 Map of Kochin backwaters. From: Joseph, A., Balachandran, K.K., Mehra, P., Prabhudesai, R.G., Kumar, V., Agarvadekar, Y., Revichandran, C., Dabholkar, N., 2009. Amplified Msf tides at Kochi backwaters on the southwest coast of India. Curr. Sci. 97 (6), 776–784.

in only a few estuaries. Most field occurrences showed well-defined undulations behind the leading wave, often described as an undular bore process (see Koch and Chanson, 2008). A tidal bore is a typical hydraulic phenomenon, frequently observed in some shallow estuaries during spring tide when the tidal ranges are appreciably large (exceeding 4–6 m) and the flood tide is confined to a narrow funnel-shaped estuary. As described above, this amplifies the tide wave in a process, discussed earlier, called geometrical amplification. In such a situation, the steepened flood tide becomes near-vertical in shape at the crest and the tide roars up the estuary as a wall of tumbling and foaming water producing a tidal bore. Simply stated, the bore is more or less the front of a tide wave where the water surface changes abruptly and surges forward, often as a group of steep waves. It may be noted that the bore height is not the same as the tidal height; in fact, the bore rides on the tide. So the bore height is actually the surplus height above the instantaneous tidal height. 3.3.3.1 Story of Tidal Bores There are numerous visual accounts of tidal bores. Most occurrences show well-defined undulations behind the leading wave—that is, an undular bore process (see Fig. 3.23). The other type of bore is the breaking bore that is rarely seen except at a few geometrically special locations. During the passage of a bore through a water body of varying depths, an undular bore is usually found in the deep water regions, while a breaking front is observed in the shallower waters near the bank and on dry flats. There have been instances in which the undular bore disappeared briefly on the channel centerline, possibly because of a deeper water hole, although the breaking bores were clearly seen elsewhere moving upstream (see Chanson, 2005). The basic requirement for the generation of a bore is a large tidal range, but this alone is not sufficient as the estuary geomorphology has a strong influence. The main characteristic of a bore is the very rapid rise in water level as its front advances past an observer. A bore can be several meters high and travels at a nonuniform speed, usually in the range of 18–28 km/h. The bore generates a powerful noise that is likened to the sound of a stampeding cavalcade. It is well

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FIG. 3.22 Progressive decrease of tidal range and increasing dominance of Msf tide in the Kochi backwaters in India, from its mouth to its head, as a result of geometrical attenuation. From: Joseph, A., Balachandran, K.K., Mehra, P., Prabhudesai, R.G., Kumar, V., Agarvadekar, Y., Revichandran, C., Dabholkar, N., 2009. Amplified Msf tides at Kochi backwaters on the southwest coast of India. Curr. Sci. 97 (6), 776–784.

known that tidal bores induce strong turbulent mixing in estuaries and river mouths. The effects of a tidal bore may be felt along considerable distances when the bore travels far upstream. According to Koch and Chanson (2008), the presently existing (though transient in character) most famous tidal bores are probably those of the Seine River (France) and Qiantang River (China). Another majestic bore is the “pororoca” of the Amazon estuary. Smaller tidal bores occur on the Severn estuary near Gloucester, United Kingdom; on the Garonne and Dordogne estuaries, France; at Turnagain Arm and Knik Arm, Cook Inlet (Alaska); in the Bay of Fundy, at Petitcodiac and Truro (Canada); on the Styx and Daly estuaries (Australia); and at Batang Lupar (Malaysia). The Messina Straits (Sicily), Gironde (France) and the Hooghly River (India) also exhibit occasional tidal bores. The Hooghly bore may propagate more than 80 km flowing past the port of Calcutta. Tidal bores have a notorious history of springing surprises on boatmen and navigators. For instance, the often sudden, irregular, and unpredictable nature of the bore renders it highly dangerous to boatmen navigating on such rivers. The anchored boats in the river, being struck by a rush of water, may swing around violently. The unanchored boats may move rapidly upstream, often unguided and out of control, and strike other boats or the riverbanks. The fascinating eyewitness account by Captain Moore of his dreadful experiences in the Tsien-tang-kiang river in China while he was surveying the river, commanding Her Majesty's survey ship Rambler is classic (see Darwin, 1898). Tidal bores have been dreaded even by notorious men of war and courageous navigators. For example, the tidal bore on the Indus estuary, now in Pakistan, mistreated the fleet of Alexander the Great in 325 BC (or 326 BC) and wiped out his fleet (see Arrian, 1976). The available accounts of this episode are detailed and provide accurate descriptions of

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FIG. 3.23 (Top) Transverse profile of the Selune River tidal bore (France) on Sep. 19, 2008. Note the “wavy” transverse profile of the undular tidal bore caused by the presence of shoals and bars (From: Chanson, H., 2011. Current knowledge in tidal bores and their environmental, ecological and cultural impacts. Environ. Fluid Mech. 11, 77–98, http://dx.doi.org/10.1007/s10652-009-9160-5). (Middle) Breaking bore in the Baie du Mont St Michel (France) on Oct. 19, 2008, at the Pointe du Grouin du Sud (From: Chanson, H., 2011. Current knowledge in tidal bores and their environmental, ecological and cultural impacts. Environ. Fluid Mech. 11, 77–98, http://dx.doi.org/10.1007/s10652-009-9160-5). (Bottom) Undular tidal bore of the Dordogne estuary at Port de Saint Pardon on Jul. 22, 2008, evening. Note the first wave crest breaking on the jetty in the foreground and the surfer riding on the third wave (From: Chanson, H., 2009. The rumble sound generated by a tidal bore event in the Baie du Mont Saint Michel. J. Acoust. Soc. Am. 125 (6), 3561–3568. http://dx.doi.org/10.1121/1.3124781; © 2009 Acoustical Society of America).

a tidal bore process (see Koch and Chanson, 2008). Likewise, during his expedition in the Qiantang estuary mouth in China, Captain Moore almost lost his survey ship on Sep. 20, 1888, when he inadvertently anchored in the Qiantang estuary (see Moore, 1888). Studies reported by Chanson (2011) indicate that tidal bores can cause major damage to river banks and create navigational hazards in tidal bore-affected estuaries. For example, tidal bores have adversely affected shipping and navigation in Mexico (Colorado estuary), Papua New Guinea (Fly and Bamu estuaries), Malaysia (Benak at Batang Lupar), and India (Hooghly estuary) (see Sykes, 1937). Notably, both the Seine and Qiantang estuary tidal bores have been notorious for their temperamental behavior. In modern times, the Qiantang River banks were overtopped by a tidal bore, and several dozen drownings are reported each year. Additional tragic occurrences of drowning in tidal bores

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and in “whelps” (a train of well-defined and quasiperiodic waves called undulations that follow a tidal bore) include the numerous human losses in the Colorado estuary (Mexico), Bamu and Fly estuaries (Papua New Guinea), and Seine estuary (France). 3.3.3.2 Thundering Noise of Tidal Bores A tidal bore propagates upstream in certain topographically and geometrically complex estuaries with a speed equal to that of a moving hydraulic jump. The upper mass of water may overtake the lower mass of water, and the crest eventually falls forward as the flood tide continues its advance. In some areas, the bore may be a startling and destructive force. In general, it is only at spring tide, and sometimes with certain meteorological conditions (wind and atmospheric pressure changes), that the phenomenon is striking. This means that the generation of a bore is the result of a very delicate balance between various conflicting forces. A tidal bore creates a powerful roar whose source includes the turbulence in the bore front and the whelps, entrained air bubbles in the bore roller, sediment erosion beneath the bore front and of the banks, scouring of shoals and bars, and impacts with obstacles. During a calm night, the murmurs of an approaching bore can be heard from afar and within minutes pass on with a thundering roar. The noise generated during the sudden surge of the rushing tidal bore has been likened to “the thundering noise of a hundred horses hooves stampeding.” Captain Moore heard the first murmur of the bore 1 h before it reached his Pandora ship in the Qiantang estuary (see Moore, 1888). In the Baie du Mont Saint Michel, the tidal bore is often heard 25–30 min before the bore reaches the spectators or listeners. In the Severn River, the bore may be heard more than 20 min ahead during the night. Thorne (1986) and Mason et al. (2007) found that in tidal channels and under waves, the sediment motion by bed load induces some particle collisions that transmit an acoustic pulse to the water with characteristic frequencies between 1.5 and 400 kHz. While the sediment motion is not the dominant cause of the tidal bore rumble noise, it nevertheless plays a secondary role in generating tidal bore rumble. Despite numerous anecdotal observations, the first quantitative acoustic measurements of a tidal bore event were recorded only recently in the Baie du Mont Saint Michel in France (see Chanson, 2009a,b). Detailed analysis of the recorded acoustic data suggested that the air bubbles entrapped in the eddies of the tidal bore roller are acoustically active and play the dominant role in the rumble sound generation. Analysis of the powerful and quite violent sound (acoustic signal) records taken during the passage of the tidal bore and the associated bore-whelps and flood flow indicated that for each power spectrum density function, the observed dominant frequency ranged from 74 to 131 Hz (see Chanson, 2009a,b). These values corresponded to a low pitch sound or rumble within the entire audible range of sounds for a human ear. Chanson (2009a,b) found that the sounds generated by a tidal bore event consist of three distinct periods: (1) when the sound amplitude increases with the approaching bore front; (2) as the tidal bore passes the observer/listener where the impact of the bore on the bank, rocks, or jetty generates powerful noise; and (3) during the upstream propagation of the bore when the flood flow motion causes additional loud noise. According to Chanson's studies, the dominant frequency during the bore-breaking process is around 76–77 Hz. The low-pitch rumble of the breaking bore has been found to have a dominant frequency comparable to the collective oscillations of bubble clouds (see Prosperetti, 1988, for the relevant mathematical equation). This suggests that air entrapment in the bore roller is likely to play a major role in the acoustic signature of the bore. The bore rumble is heard far away because its low frequencies can travel over long distances. Chanson (2011) has provided an elegant summary of a series of scientific investigations carried out by several researchers that lead to a greater understanding of the rumble noise that is invariably associated with tidal bores. Some detailed measurements show that the sounds generated by a breaking bore have a low pitch comparable to the sounds generated by bass drums and locomotive trains, and the dominant source of the rumble noise is the collective oscillations of the bubble clouds entrained in the tidal bore roller. Undoubtedly, a tidal bore is a beautiful, natural wonder and a fascinating geophysical phenomenon for surfers and kayakers as well as for estuarine populations and tourists. 3.3.3.3 Geomorphological Consequences of Tidal Bores A tidal bore induces a very strong shock-type mixing in its propagation path, and classical mixing theories do not account for this. Chanson (2011) has indicated that some bed erosion may take place during tidal bore passage, and the eroded materials as well as other scalars placed into suspension are advected with the whelps and secondary wave motions behind the tidal bore front. This is consistent with the very strong turbulent mixing, observed visually in the tidal bore-affected estuaries, which is associated with the accretion and deposition of sediment materials in the upper

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estuarine zone. Turbulent velocity measurements have suggested some energetic turbulent events beneath and after the tidal bore front. These are best seen by some sudden and rapid fluctuations of the transverse and vertical velocity data, while some recent numerical modeling highlights the production of large turbulent eddies beneath the bore front and their upstream advection behind the bore (see Furuyama and Chanson, 2008; Lubin et al., 2010). These vortical structures remain next to the bed as the bore propagates upstream, and the presence of these persisting turbulent structures indicate that a great amount of sediment materials are placed in suspension and advect upstream. Both physical and numerical modeling studies show some large transverse and vertical velocity fluctuations, implying the existence of transient secondary currents behind the bore front. The evidence of turbulence “patches” encompasses both undular and breaking bore conditions, and some simple considerations imply that the vorticity production rate is proportional to (Fr1  1)3, where Fr1 is the tidal bore Froude number, defined later (see Hornung et al., 1995). The vorticity “clouds” behind a tidal bore are a feature of tidal bores that are linked with some secondary current motion and are enhanced by the natural, non-prismatic channel bathymetry. The macro-scale turbulence is advected behind the bore front, contributing to the energetic turbulent velocity fluctuation periods observed in the Daly River with some surface “clockwise and counterclockwise rotating eddies” about 20 min after the tidal bore passage (see Wolanski et al., 2004). 3.3.3.4 Generation and Propagation Mechanisms of Tidal Bores Tidal bores have long attracted the attention of several curious hydraulic engineers and applied mathematicians, and during the past couple of centuries several attempts have been made to understand the inside story of this fascinating natural phenomenon. For example, in his milestone paper, Adhemar Jean Claude Barre de Saint Venant (1797–1886) applied his famous equations to the tidal bore of the Seine River (see Barre de Saint Venant, 1871). Likewise, Dennis Howell Peregrine (1938–2007) observed the Severn River bore many times, and his work on tidal bores was inspired by this majestic bore (see Peregrine, 1966). Other major contributions made in the study of tidal bores in the pre-21st century era include, in chronological order, the works of Bazin (1865), de Saint Venant (1871), Boussinesq (1877), Benjamin and Lighthill (1954), and Peregrine (1966). A tidal bore is simply a shock characterized by a sudden change in velocity and pressure fields (see Lighthill, 1978). The shock is followed by a highly turbulent flow motion with significant fluctuations of all velocity components. Observational data sets gathered by Chanson (2011) showed some large turbulent stresses and turbulent stress fluctuations beneath the tidal bore and ensuing whelp motion. Measurements carried out by Chanson (2011) indicated that the Reynolds stress magnitudes were significantly larger than those in the initially steady flow prior to the tidal bore. The turbulent stresses, also called Reynolds stresses, are the stresses in the water causing the random turbulent fluctuations in fluid momentum. They characterize a transport effect resulting from turbulent motion induced by velocity fluctuations with its subsequent increase of momentum exchange and of mixing. Visual observations of tidal bores have highlighted the turbulent nature of the advancing waters. A tidal bore induces a strong turbulent mixing in the estuarine zone, and the effects may be felt along considerable distances. Quantitatively, the levels of turbulent stresses were found to have been one to two orders of magnitude larger than the critical threshold for sediment motion in terms of both bed load and suspension. Field studies by several researchers (eg, Chen et al., 1990; Greb and Archer, 2007) demonstrated that the arrival of the tidal bore is always associated with some intense mixing and with the upstream advection of the suspended material. Donnelly and Chanson (2005) and Koch and Chanson (2008) found that in undular tidal bores (ie, tidal bores whose leading wave is followed by a train of welldeveloped undulations called whelps), the sediment suspension is further sustained by the strong wave motion of the whelps for relatively long periods after the bore passage. This facilitates the upstream advection of the solid matter within the flood flow behind the bore. The evidence of turbulent mixing induced by tidal bores is plentiful and sometimes challenging. The upstream advection of a “cloud” of turbulence and vorticity behind the bore is characterized by very turbid and murky waters and extends for a considerable distance. In fluid dynamic theory, the bore front is described as a discontinuity of the water depth and of the velocity and pressure fields: ie, it is a hydrodynamic shock. Many standard shock-capturing methods that have been well developed in aerodynamics have been found to be applicable to solve shallow water equations, as was done by, for example, Marshall and Mendez (1981), Glaister (1988), Toro (1992), Yang and Hsu (1993), and Hu et al. (1998). Since the 1980s, numerical simulations of the tidal bore on the Qiantang estuary have been extensively conducted by using the shockfitting (Zhao, 1985; Tan et al., 1995) and shock-capturing methods (Su et al., 2001). All of these studies provide a deeper understanding of the complicated characteristics of the tidal bore. Despite such continued efforts, many of its secrets remained unknown, and a passion for unraveling them continued unabated. In recent years, significant contributions have been made by a handful of researchers, and their studies have confirmed, unsurprisingly, that the dynamics in the generation and propagation of the tidal bore is rather complex.

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The driving process involved in its generation is primarily the large tidal amplitude in a shallow estuarine environment. After the formation of the bore, there is an abrupt rise in water depth at the tidal bore front, and the flow singularity may be analyzed as a hydraulic jump in translation (see Liggett, 1994; Chanson, 2004). Fluid dynamical principles demand that the flow properties immediately upstream and downstream of the tidal bore front must satisfy continuity and momentum principles (see Rayleigh, 1908; Henderson, 1966). The inception and development of a tidal bore may be predicted using the Saint Venant equations and the method of characteristics (see Henderson, 1966; Chanson, 2004). With reference to Fig. 3.24, the classical mathematical expression relating the parameters governing tidal bore propagation is given as (for details see Chanson, 2011): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  d2 1 1 + 8 Fr21  1 (3.6) ¼ d1 2 where Fr1 is the tidal bore Froude number defined as: V1 + U Fr1 ¼ pffiffiffiffiffiffiffi gd1

(3.7)

In the above expression, g is the acceleration due to Earth's gravity. Despite its apparent simplicity, Eq. (3.6) can explain the occurrence, or disappearance, and the strengthening or weakening of a tidal bore, as well as its changes in shape and appearance during its propagation. The strength of a bore-laden tide wave is often expressed in terms of its Froude number (Fr1). It may be noted that Fr1 is always greater than unity, and the term (Fr1  1) is a measure of the strength of the bore. If Fr1 is less than unity, the tide wave cannot become a tidal bore. For Fr1 < 1, the tide wave propagates upstream as a gentle surface slope, and there is no discontinuity in flow depth, hence no tidal bore. A tidal bore does not occur during an estuarine flood when the initial water depth d1 is large, nor when the tidal range is small (eg, neap tides). Interestingly, the shape of the tidal bore is directly linked with its Froude number. For example, an undular tidal bore (ie, a tidal bore whose leading wave is followed by whelps) is observed for a bore Froude number between 1 and 1.5–1.8 (see Koch and Chanson, 2008; Chanson, 2009a,b). For larger Froude numbers, a breaking bore takes place. The majority of tidal bore occurrences have an undular shape: (see Peregrine, 1966; Lewis, 1972). There have been instances in which more than 30 whelps, each 2–3 m high with 20–30 m between crests and extending behind the horizon with an estimated visibility of 20 nautical miles, have been witnessed (see Murphy, 1983). Immediately behind the bore front, the wave train presents a pseudo-periodic, undular profile, although the observations also show the development of semichaotic patterns with increasing time. Based on field observations, Chanson (2011) has also indicated the long-lasting effects of the whelps' wave motion, sometimes more than 20–30 min after the tidal bore passage. This aspect is well known to surfers and kayakers (those surfing with an Eskimo canoe of light wood covered with seal skin) who may have difficulty returning ashore after surfing. The breaking bores are observed for Fr1 > 1.5–1.8. Such an elevated fluid dynamical condition implies that a powerful tidal bore process can take place only during “king tide” conditions and low estuarine water levels—eg, the Seine Estuary and Qiantang Estuary bores during equinox spring tides in September (see Chanson, 2011). The bore front pffiffiffiffiffiffiffi celerity U is proportional to gd2 where d2 is the water depth immediately behind the bore front. Hence the bore speed is related to the rate of rise of the sea level and to the tidal range. The tidal bores are stronger, and advance faster, during spring tide conditions and rarely occur during neap tides. Furthermore, the tidal bores are better seen during the dry season and low river flow periods when the initial water depth d1 is small.

Downstream

d2 V2

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FIG. 3.24 Definition sketch of a tidal bore propagating

Upstream

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z

upstream U: propagation speed of bore wave front; d1 and d2 are respectively the flow depths immediately before and after the tidal bore passage; V1 and V2 are respectively the flow velocity corresponding to the flow depths d1 and d2 (positive downstream towards the river mouth). From: Chanson, H., 2011. Current knowledge in tidal bores and their environmental, ecological and cultural impacts. Environ. Fluid Mech. 11, 77–98, http://dx.doi.org/10.1007/ s10652-009-9160-5.

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Eq. (3.6) shows further that the bore shape and appearance may change rapidly in response to the estuarine bathymetry. In regions of deeper water (d1 large), the bore may disappear when the local Froude number is less than unity, while it may strengthen in regions of shallow waters, shoals, and bars (d1 small). In a given estuary section, the tidal bore may have a breaking bore appearance next to the bank in a region of shallow waters and an undular shape in a deeper section of the river channel. The transverse shape of the bore is also related to river bed topography. The existence of tidal bores is based upon a fragile hydrodynamic balance (between the tidal amplitude, the freshwater level and velocity, and the river channel bathymetry), which may be easily disturbed by changes in boundary conditions and freshwater inflow (eg, Malandain, 1988). Based on some simple theoretical considerations, Chanson (2011) found that this balance may be easily disturbed by some changes in the boundary conditions and freshwater runoff. In this context, it may be noted that a number of tidal bores disappeared because of river training, dredging, and damming. These interventions led not only to the loss of several bores but also had adverse effects on estuarine ecosystems. The tidal bore of the Seine Estuary (France) no longer exists after extensive training works and dredging; the Colorado Estuary bore (Mexico) is drastically smaller after some dredging as well as the damming of the river. Although the fluvial traffic gained in safety in both cases, the ecology of the estuarine zones suffered. The tidal bore of the Petitcodiac Estuary (Canada) almost disappeared after construction of an upstream barrage that resulted in the elimination of several native fish species such as the American shad, Atlantic salmon, Atlantic tomcod, striped bass, and sturgeon (see Locke et al., 2003). The proposed construction of the Severn Barrage in the UK is likely to be a major threat to one of the best documented tidal bores: the Severn Estuary bore. 3.3.3.5 Qiantang Bore—Most Spectacular Bore in the World The tidal bore at the Qiantang estuary in China (see Fig. 3.25) is presently the most spectacular in the world because of its height and its variety in appearance and shape. It resembles a wall of water 1–3 m high (the maximum height reaches about 4 m), traveling upstream at the speed of 20–30 km/h (see Pan et al., 2007). Dai and Zhou (1987) have provided general information about this tidal bore, including its formation, variation in appearance and shape, destructive capability, and historical evolution. Forming an intimate part of the Hangzhou Bay in the East China Sea, the bore exhibits various forms during its propagation, such as the λ-shape intersection of two bores from different directions, reflected surges from seawalls, almost vertical uplifting water sheets with foaming fronts, glittering and shining undular jumps, and so forth. The bore usually emits a thundering sound. Because of its uniqueness, it is called the “Qiantang Bore” and has become a famous scenic spot in China. Nonetheless, its destructive power could well spell disaster. Compared with other numerical schemes, one of the great advantages in the use of the Godunov-type scheme for the study of tidal bores lies in the fact that the model can be successfully used to simulate the flow with high velocity around the tidal bore as well as the bore itself. Pan et al. (2007) developed a mathematical model using the Godunovtype scheme combined with wet/dry techniques. Their model has been successfully applied to simulate the formation, evolution, propagation, and dissipation of the Qiantang Bore. Comparison between the computed results and the

FIG. 3.25 Qiantang Bore at Yanguan. From: Pan, C.-H., Lin, B.-Y., Mao, X.-Z., 2007. Case study: numerical modeling of the tidal bore on the Qiantang River, China. J. Hydraul. Eng., 130–138, http://dx.doi.org/10.1061/(ASCE)0733-9429(2007) 133:2(130). © ASCE/February 2007.

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observed data is reported to have shown good agreement. In addition, the model was able to replicate some of the fascinating bore scenery mentioned above during its propagation. It has been claimed that the model provides a useful tool to study macro-scale characteristics of the tidal bore. The Hangzhou Bay possesses a unique funnel shape (see Fig. 3.26). Its mouth is about 100 km wide and narrows upstream to 20 km wide at the head of the bay (Ganpu), 89 km away from the mouth. With rapid narrowing, the tidal range is increased by up to 75% at Ganpu, and the observed maximum height is 9 m. At the Qiantang estuary, the water depth decreases appreciably upstream of Ganpu due to large sand bars along the estuary. This shallowness results in an enhanced shallow-water effect, such that the tide wave is severely deformed and greatly contributes to the formation and development of the tidal bore. During the spring tide, the bore travels upstream along the river to a distance of more than 100 km. The Qiantang Bore is formed twice a day during spring tide during its flooding phase. The tidal bore is a local phenomenon and a special part of the tide wave. There exist complex flow structures within the tidal bore and in front of and behind it for a few minutes as the tidal level rises suddenly (see Pan et al., 1994). Because of the high speed at which the tidal bore advances ahead of the curious observer, it is as exciting to watch a tidal bore as a horse or a boat race. In view of the spectacular nature of the Qiantang Bore, it would be worthwhile to review the field observations carried out to obtain a quantitative assessment of the bore phenomenon in this geometrically complex estuary, and to shed light on its physics. It is worth remembering that because of the destructive capability of the bore, it can be difficult to fix measuring gadgets in the path of the tidal bore passage. This lacuna has resulted in scarcity of sufficient data sets to adequately study tidal bores. The large-scale observation of the tidal bore on the Qiantang River was organized in Sep. 2000 by a group of researchers, from tidal limit to the cross section of Jinshan in the Hangzhou Bay, with the deployment of about 20 water level gauges along the Qiantang River, recording the water level observations every 1–2 min during the passage of the bore. The computational region covered a distance of 72 km from Ganpu (the river width here is 20 km) to Changqian (the river width here is 2 km), as shown in Fig. 3.27. Based on the survey data collected during spring tide, the maximum tidal range was 7.72 m at Ganpu. Lin et al. (2002) analyzed the propagation characteristics of the tidal bore on the Qiantang estuary. The computed water elevation was compared with the observed data as shown in Fig. 3.28 where the circles denote the observed data and the solid line denotes the computed results. In the time series shown in Fig. 3.28, the bore is the steep front of the wave. Within the computational reach, the tidal bore is gradually formed and then grows up when the bore travels upstream. This phenomenon can be replicated clearly in the model as shown in Fig. 3.29, which presents water surface profiles along the main channel at different moments, every half an hour. When the flood tide front arrives at the downstream section where the Cao'e River flows into the Qiantang River, the water level jumps to several centimeters in height. This section can be considered to be the formation region of the tidal bore. After that, the height of the tidal bore increases gradually towards upstream. When the tidal bore reaches Yanguan Station, its height reaches the highest point. The bore height then decreases gradually because of dissipation. The height of the bore characterizes its strength and can be defined as the height of the steep front of the wave, which is different from the tidal range.

FIG. 3.26 Location of the Qiantang estuary. From: Pan, C.-H., Lin, B.-Y., Mao, X.-Z., 2007. Case study: numerical modeling of the tidal bore on the Qiantang River, China. J. Hydraul. Eng., 130–138, http://dx.doi.org/10.1061/(ASCE)0733-9429(2007)133:2 (130). © ASCE/February 2007.

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FIG. 3.27 Computational domain in the Qiantang estuary. From: Pan, C.-H., Lin, B.-Y., Mao, X.-Z., 2007. Case study: numerical modeling of the tidal bore on the Qiantang River, China. J. Hydraul. Eng., 130–138, http://dx.doi.org/10.1061/(ASCE)0733-9429(2007)133:2(130). © ASCE/February 2007.

The velocity evolution at the middle of Yanguan's section is shown in Fig. 3.30. The depth-averaged velocity during the ebb at this section ranges from 1 to 2.4 m/s, but it reaches 6 m/s when the bore passes by. The velocity jump may be greater than 7 m/s. The maximum flood velocity appears when the bore passes by. The maximum velocity at each point during one tidal cycle is shown in Fig. 3.31. Generally, the maximum velocity reaches 3–4 m/s on the open boundary at Ganpu, and then the velocity increases gradually upstream and reaches the extreme velocity around Daquekou, where the velocity is more than 6 m/s. Afterward the velocity decreases a little. This process accords with the height change of the tidal bore. As shown in Fig. 3.31, there is an area around Ershigongduan in which the velocity is much higher, more than 7 m/s, and the channel bends severely. On the opposite bank of Laoyancang, there is a similar situation. Surprisingly, the velocity near the convex bank was found to be much larger than that near the concave bank when the tidal bore passes by, and it reaches the largest one of 9.68 m/s near Ershigongduan. This phenomenon is different from the other curved flow in which high velocity usually happens near the concave bank in the river channel. It was found that large velocities appear in 9–40 min after the arrival of the tidal bore, and the durations are a little different at different locations. At the downstream Ganpu, the tide is somewhat like a standing wave, and the maximum flood velocity appears in about 2 h after low tide. When tide traverses upstream, the maximum velocity appears earlier and earlier. According to Pan et al. (2007), the sustaining time associated with high velocity is a very important factor. The sustaining periods are 8, 33, and 16 min when velocity is greater than 5.5 m/s at Dquekou, Yanguan, and Sigongduan, respectively. Especially at Yanguan, the sustaining period has been found to be more than half an hour when velocity is about 6 m/s, as shown in Fig. 3.32. According to the observations of Pan et al. (2007), the flood velocity increases quickly and reaches the highest velocity after 10–20 min. In practice, this is called high-velocity water. At the same favorable bore-watching spots, some beautiful forms of the bore, called bore scenery, can be seen, and the model was found to be able to replicate these forms. Near the section of Jianshan, the bore is divided into two fronts. In the reach from Jianshan to Daquekou, one front is almost parallel to the north bank, while another is almost perpendicular to the south bank; the two fronts intersect at the middle, which is called a crossed tidal bore or λ-shape bore. Fig. 3.33 shows the contours of tidal elevation when the tidal bore arrives at the Daquekou reach, the contours indicating clearly that the two fronts intersect at the reach. The area with dots represents the exposed tidal flat during the measurement period. After the bore passes through Daquekou, the two fronts combine into one. When the bore approaches Yanguan, even as it is still 15–20 km away, the bore can be heard about 1 h before arrival and can be seen as a distant silver line, traveling forward almost perpendicular to the seawall, which is called a thread-shape bore. The observed distant silver line (thread-shape bore) described above can be seen also in association with tsunami waves propagating along an estuary under suitable lighting conditions. For example, while sitting at an elevated shack

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FIG. 3.28

Comparison of tidal levels (A) at the mouth of the Cao'e River; (B) at Daquekou; and (C) at Yanguan. From: Pan, C.-H., Lin, B.Y., Mao, X.-Z., 2007. Case study: numerical modeling of the tidal bore on the Qiantang River, China. J. Hydraul. Eng., 130–138, http://dx.doi.org/10. 1061/(ASCE)0733-9429(2007)133:2(130). © ASCE/February 2007.

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at Siridao beach at a cove of the Zuari estuary in Goa (India) overlooking the mouth of the estuary just before noon, I (author of this book) had the wonderful opportunity to witness such a bright silver line fast approaching, while the Dec. 2004 Indian Ocean tsunami was unfolding in front of my eyes and propagating up the estuary. In my assessment, appearance of such a bright silver line was simply the result of sunlight reflection from the wave front. The sunlight was beaming toward the west and the tsunami wave was traveling toward the east, facilitating direct reflection of sunlight from the approaching tsunami wave front. Unlike ordinary wind waves that are random in character and consist of several constituent components of different amplitudes and frequencies traveling in different directions, the tsunami wave is nearly monochromatic, consisting of a single frequency component. It is therefore extraordinarily smooth in appearance when seen from far and locally unidirectional. For this reason, tsunami waves are capable of more efficient reflection of light, though its smoothness is totally lost when it breaks at shallow depths, such as a coast,

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FIG. 3.29

Water surface profiles along the main channel at different moments. From: Pan, C.-H., Lin, B.-Y., Mao, X.-Z., 2007. Case study: numerical modeling of the tidal bore on the Qiantang River, China. J. Hydraul. Eng., 130–138, http://dx.doi.org/10.1061/ (ASCE)0733-9429(2007)133:2(130). © ASCE/February 2007.

FIG. 3.30 Water flow velocity evolution at the middle of Yanguan's section. From: Pan, C.-H., Lin, B.-Y., Mao, X.-Z., 2007. Case study: numerical modeling of the tidal bore on the Qiantang River, China. J. Hydraul. Eng., 130–138, http://dx.doi.org/10.1061/(ASCE)0733-9429(2007)133:2(130). © ASCE/ February 2007.

FIG. 3.31 Computational contours of maximum velocity in Qiantang estuary, unit: m/s. From: Pan, C.-H., Lin, B.-Y., Mao, X.-Z., 2007. Case study: numerical modeling of the tidal bore on the Qiantang River, China. J. Hydraul. Eng., 130–138, http://dx.doi.org/10.1061/(ASCE)0733-9429(2007)133:2(130). © ASCE/February 2007.

3.3 INTRICATE TIDAL MOTIONS IN TOPOGRAPHICALLY COMPLEX WATER BODIES

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FIG. 3.32 Tidal level and velocity at 1 h after bore arrival. From: Pan, C.-H., Lin, B.-Y., Mao, X.-Z., 2007. Case study: numerical modeling of the tidal bore on the Qiantang River, China. J. Hydraul. Eng., 130–138, http://dx.doi.org/10.1061/(ASCE)0733-9429 (2007)133:2(130). © ASCE/February 2007.

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FIG. 3.33 Tidal level contours when the bore appears in Daquekou reach. From: Pan, C.-H., Lin, B.-Y., Mao, X.-Z., 2007. Case study: numerical modeling of the tidal bore on the Qiantang River, China. J. Hydraul. Eng., 130–138, http://dx.doi.org/10.1061/(ASCE)0733-9429(2007)133:2(130). © ASCE/February 2007.

or hits a wall. Such reflection can also occur during nights of a full moon if the light beam orientation is supportive of light reflection and the observers' location is supportive of reception of the reflected light emanating from the wave front. Thus I think that the observed distant silver line witnessed in a bore could be the result of reflected sunlight or moonlight emanating from the bore's wave front. Let us return to the Qiantang Bore. After the bore has passed through Yanguan to Laoyancang, where the channel turns to the south, the bore hits the seawall and reflects back, causing a violent upsurge, called a returned tidal bore. The returned bore causes the great difference of water level observed between the south and north banks, which may reach more than 3 m difference in height, as can be seen in Fig. 3.34. Because bores result from instabilities in the hydraulics of tidal flow, they are very sensitive to changes in estuarine morphology. Increasing the estuary depth by dredging and speeding the estuarine flow by building embankments reduce the chances of a bore developing. While there are reasons for supposing that the bore at Qiantang has become more spectacular over the centuries, in other rivers there have been deliberate attempts to reduce bores because of the

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FIG. 3.34 Tidal level contour when the bore appears in Laoyancang reach. From: Pan, C.-H., Lin, B.-Y., Mao, X.-Z., 2007. Case study: numerical modeling of the tidal bore on the Qiantang River, China. J. Hydraul. Eng., 130–138, http://dx.doi.org/10.1061/(ASCE)0733-9429(2007)133:2(130). © ASCE/February 2007.

damage they can cause to the banks of the estuary. For example, the pattern of silting and dredging at the entrance to the River Seine has reduced the size of the mascaret. Similarly, the burro, the bore in the Colorado estuary, has been progressively reduced by siltation and land drainage schemes (see Pugh and Woodworth, 2014).

3.3.4 Apparent Phase Reversal of Tide Wave Over Short Distance and Its Consequences

pffiffiffiffiffiffiffi The speed of propagation (c) of tide wave is given by the expression: c ¼ gD, in which g is the gravitational acceleration, and D is the local depth of the water body through which tide propagates. This implies that a decrease in water depth (shallowness of water body) reduces the speed of tidal propagation through such water bodies, giving rise to delay in tidal propagation commensurate with the shallowness. The delay in the tidal propagation caused by the morphology of the ocean basins and connected water bodies in some regions cause some peculiar effects. For example, tides appear to be out of phase on either side of the Valdes peninsula (see Fig. 3.35) in the Atlantic Ocean. Here, when it is high tide in the Nuevo Gulf, the tide is low in the San Jose Gulf, a few km away. The Valdes peninsula projects considerably into the Atlantic Ocean. At Valdes, the speed of propagation of tide wave from west to east is slowed down because of a much reduced water depth (D), caused by submarine elevation. Consequently, the tides are delayed a half period between San Jose Gulf and Nuevo Gulf. The tides being apparently out of phase, a level difference of up to 16 m results between the two Gulfs. Measurements carried out by the CSIR-National Institute of Oceanography (NIO) in India during Feb.–Apr. 2010 in the Sethusamudram channel, located between Sri Lanka and the southeast coast of India (see Fig. 3.36), reveal that an apparent phase reversal of tides occur over the northern and the southern boundaries of this canal. Tidal phases changing over short distances result in reversing tidal falls causing the gulf water body to behave like a seesaw, with alternate up-and-down motion of the water level elevation at these two boundaries of the gulf. The anticipated consequence of this seesaw behavior of the canal water body is to give rise to an enhanced tidal current. Tidal propagation time difference at various locations relative to Tuticorin is shown in Fig. 3.37. From the tidal propagation time difference at several locations relative to the Tuticorin station, it is evident that tidal propagation in this region happens from the Gulf of Mannar toward Palk Bay, and further toward the north. During this propagation, a delay of about 6 h occurs from the southern locations to the northern location (L8). Six hours and twelve minutes is the typical time delay between high water and low water in a topographically wide tidal regime (ie, 6 h and 12 min is the typical time delay between phase reversals). Thus the tidal propagation delay of 6 h over a short distance between these two regions (about 100 km apart) as a result of the shallowness of the gulf manifests as a phase reversal of tides over these two regions.

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FIG. 3.35 Location map of Valdes peninsula in the Atlantic Ocean, where apparent phase reversal (large phase differences) of tide occurs over either side of it over a short distance. Courtesy of NASA.

Some coasts respond to tides in other peculiar ways. For example, in the Lofoten islands of Norway, dangerous whirlpools appear twice a day during flood tide. The topographical influence of the ocean in modifying the tidal rhythms is indeed spectacular. Wherever tides propagate on either side of a land barrier and where there is a gap in the barrier, very strong tidal hydrodynamic currents occur. The Norwegian maelstrom is a well-known example; also the Pentlands Firth north of Scotland. San Jose Gulf was in fact suggested for a tidal power plant (Dr. David Pugh, Private communication). There are many other examples of tidal phases changing over short distances resulting in reversing tides, including some in Canada and in Pentlands Firth north of Scotland.

3.3.5 Ocean Basins Experiencing No Tidal Motions—Amphitrophic Zones In the real oceans, which are of nonuniform depth and bounded by continental shelves and land-sea interfaces, tide waves cannot propagate endlessly as progressive waves. They undergo reflection at sudden changes of depth and at the coastal boundaries. The reflected and incident waves combine together to produce the observed total wave (see Defant, 1961). In the open ocean, the nonuniformity of its depth causes some interesting effects on the tidal pattern. Continents, undersea mountain ranges (ridges), faults, and trenches produce a variety of partially connected or nearly closed oceanic basins. The interference between the two waves (incident and reflected waves) produces a fixed pattern of standing waves which have alternate nodes (positions where the amplitude is zero) and antinodes (positions where the amplitude is a maximum), each separated by a distance λ/4 where λ is the wavelength of the original progressive wave. Standing waves cannot transmit energy because they consist of two progressive waves of equal amplitude traveling in opposite directions. In situations where an ocean basin water body is of suitable dimensions to have a natural period of oscillation approaching one of the periods of the tide-generating forces, a state of resonance can be set up resulting in a series of standing oscillations. The natural period of oscillation of the water body is the time (T) taken for a wave to travel from one boundary and to return to the same boundary after reflection at the second boundary, and is given by the expression: 2L T ¼ pffiffiffiffiffiffiffi gD

(3.8)

pffiffiffiffiffiffiffi In this expression, gD is the speed of propagation of the tide wave, g is the gravitational acceleration, and D is the depth of water below the undisturbed sea level. L is the distance between the two boundaries mentioned above. Eq. (3.8) is known as “Merian's Formula” after its originator (see Proudman, 1953). Natural periods of large water basins may be estimated using this formula, but the observed periods will vary slightly because the depths are not uniform. The case of a standing wave oscillation on a rotating Earth is of special interest in studies of tide wave motion. Away from the boundary of the water basin, the tide waves can be represented by two Kelvin waves traveling in opposite directions and thus giving rise to the generation of standing waves. These standing oscillations are deflected in the

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FIG. 3.36 (Top) Sethusamudram channel, located between southeast India and Sri Lanka. (Bottom) Tidal propagation delay of about 6 h occurring over the northern and the southern boundaries of Sethusamudram channel is manifested as a phase reversal of tides over these two boundaries. Measurements carried out by the Indian CSIR-NIO team. Graph prepared by Prakash Mehra.

Propagation time difference (min)

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FIG. 3.37

Tidal propagation time difference at various locations relative to Tuticorin. Measurements carried out by the Indian CSIR-NIO team. Graph prepared by Prakash Mehra.

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horizontal by the Coriolis force, resulting from the gyroscopic spinning of the Earth about its own axis, to become amphidromic systems (also known as amphidromes), where the tide appears to move around a point. This point is known as the amphidromic point, where the tidal range is zero. In other words, amphidromic points are points at which the sea level remains unchanged by the tides. Thus amphidromic points occur because of the combined effects of the reflection and the resulting interference encountered by the tide wave within oceanic basins, seas, and bays and the Coriolis effect. Under the influence of the Coriolis force, an amphidromic system rotates about a nodal point (which is the amphidrome or amphidromic point). The term amphidrome derives from the Greek words “amphi” (around) and “dromos” (running), referring to the rotary tides running around them (see Cartwright, 2000). The theoretical ideas of progressive and standing waves, resonance, Kelvin waves, and amphidromic systems have effectively been applied to describe the dynamics of the observed tides in the oceans and shelf seas. The most obvious feature is the large number of amphidromes. Amphidromic systems create interesting wave patterns, as shown in Fig. 3.38. The sense of rotation of the tide wave around the amphidrome is anticlockwise in the northern hemisphere and clockwise in the southern hemisphere. The cotidal lines all radiate outwards from the amphidrome and the coamplitude lines form a set of nearly concentric circles with the center at the amphidrome, at which the amplitude is zero. The amplitude is greatest around the boundary of the basin. To be more specific, different harmonic constituents of the tide, such as M2, S2, etc., can exhibit their own separate amphidromes, having a point of zero amplitude. In most locations, M2 is the largest (semidiurnal) tidal constituent, with an amplitude of roughly half of the full tidal range. The tidal amplitude for that harmonic constituent progressively increases with distance from the amphidromic point. Fig. 3.39 shows the amphidromic system of the M2 constituent in the North Sea. Some of the known amphidromes are located at southeast of Madagascar, west of Perth, and between Africa and Antarctica. The islands of Madagascar (in the Indian Ocean) and New Zealand (in the Pacific

GOT99.2

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FIG. 3.38 Amphidromes in the oceans, shown in terms of M2 tidal constituent, the amplitude of which is indicated by color. The white lines are cotidal lines spaced at phase intervals of 30° (approximately 1 h). (Picture credit: R. Ray, TOPEX/Poseidon: Revealing Hidden Tidal Energy, GSFC, NASA). The amphidromic points are the dark blue areas where the lines come together. From: Credit to R. Ray, and NASA-GSFC, NASA-JPL, Scientific Visualization Studio, and Television Production NASA-TV/GSFC, http://en. wikipedia.org/wiki/Amphidromic_point.

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FIG. 3.39 Amphidromic system of the M2 constituent in the North Sea. The light-blue lines are lines of equal tidal phase for the vertical tide (surface elevation) along such a line, and the amphidromic points are denoted by 1, 2, and 3. From Wikimedia Commons, the free media repository; https://en. wikipedia.org/wiki/Amphidromic_point#/media/File:Amfidromieen.JPG.

Ocean) are amphidromic points in the sense that the tide goes around them (counterclockwise in both cases) in about 12 and a half hours, but the amplitude of the tides on their coasts is in some places large. It has been found that as a general rule the amphidromes conform to the expected behavior for Kelvin wave propagation, with anticlockwise rotation in the northern hemisphere and clockwise rotation in the southern hemisphere. However, exception to the general rule has been found in the real world and, therefore, Pugh and Woodworth (2014) have indicated that some caution is necessary regarding the actual sense of rotation of the tide wave in an amphidrome. For example, the M2 system west of Africa in the South Atlantic Ocean rotates anticlockwise (contradictory to the expected clockwise rotation), showing that other types of wave dynamics are also involved.

3.3.6 Implications of Coastal Tides and Tidal Bores Coastal tides and tidal bores have profound navigational, societal, and ecological implications. For example, they significantly affect the flow of water into and out of estuaries. Tidal streams in restricted channels too exhibit a regular pattern of ebb and flood. The tidal current that is directed toward land or runs up an estuary is known as a flood current because it floods the land. Conversely, the tidal current that is directed toward the sea or runs down an estuary is known as an ebb current. Coastal tidal oscillations are an important phenomenon accounting for a significant amount of the ocean's energy on the shelf. Their signals dominate the sea level and current spectra for frequencies of order 1 cycle per day (cpd) or greater. Conversely, their high energy implies that shelf currents can be strongly influenced by tidal oscillations, both through tidal rectification and tidal friction. Tidal currents also play an important role in mixing, material dispersion, and sediment transport. Information on the nature of tidal oscillations at a place is important in many ways. For example, it plays a significant role in the efficient use of nautical charts in shallow water bodies. To the mariner, the importance of tidal information increases very quickly as the land approaches, reaching critical level for deep-draft ships in shallow ports, straits, and channels. Navigators approaching a coast and steaming into harbor need information on the tidally influenced changing water depth in these areas in relation to the bottom of their ships. Tidal information pertaining to a given coastal water body, coupled with water depth information available from the nautical charts for this water body,

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helps the navigator to safely cruise over complex navigation channels. Moreover, it is advantageous for ships to sail in the direction of the tidal current flow, in consonance with the proverbial analogy of “swimming with the tide.” Coastal engineering work is also intensively concerned with the pattern and range of tidal oscillations. Coastal engineering activities such as the regulation of tidally influenced rivers (ie, estuaries), their damming, or the construction of harbors all require knowledge of the peculiarities of the local oscillations in tidal range and tidal currents. Artificial modifications of the topography and bathymetry of a tidally influenced water body may change the natural conditions to such a degree that unwanted or unexpected effects might arise. For instance, damming of an estuary may cause a flood current (ie, upstream current) of a very short duration and high speed, and consequently an ebb current (ie, downstream current) of long duration and low speed. In such a situation, soft-bedded estuaries cause transport of bottom materials landward during flood tide but do not return them into the sea during the succeeding ebb tide because of its low-speed ebb current. This sequence promotes gradual siltation, with the result that the depths of such estuaries decrease rapidly and ultimately render the waterways unsuitable for safe navigation. Information on the behavior of tidal oscillations in waterways that would result from artificial construction is therefore of great practical significance, and application of hydrodynamic-numerical models are often useful in such cases. Other engineering aspects related to the necessity for gathering information on tidal oscillations at various suitable locations are related to efforts to convert the energy associated with them into electric energy (see, eg, Booda, 1985; Gorlov, 2004), thus to harvest the much needed “green energy” from the oceans in the present era of global climate change, which is accelerated by the continued use of fossil fuels (see, eg, Johansson et al., 1993; Merry, 2005). Several maritime nations are traditionally at the mercy of tides. The study of tidal phenomena, with progress made in the development of tide-prediction techniques (see Schureman, 1958; Munk and Cartwright, 1966; Mays, 1987; Zuosheng et al., 1989), evolved primarily because of its great importance in improving the safety of navigation through shallow water bodies. The mariner determines the depth of water under his keel at a given time by algebraically adding the predicted tidal height to the depths shown on the nautical chart. Thus knowledge of tides and tidal currents serve important purposes. Tides have traditionally played an important role in the area of maritime navigation. In recent years, electronic nautical charts and electronic navigation have gained much importance (see Joseph, 2002). Millions of people live along the sea coasts, benefitting from rich agricultural land, easy transport connections for industry, fishing, and tourism. General knowledge of the periodic rise and fall of the sea level (read tide) is important to the daily lives of such coastal communities. The marshy areas that are flooded by tides provide a highly suitable habitat for a variety of fauna and generally contain a very rich and diverse range of species that provide delicacies to the coastal communities. From a recreational point of view, tidal pools provide a safer natural swimming place. Ocean currents that change under the influence of tidal oscillations exert strong direct or indirect influences on the distribution and behavior of some organisms. For example, the patchiness of phytoplankton in the vicinity of river mouths is coupled with the periodicity of the tidal cycle. Phytoplankton aggregations occur at frontal zones that form during flood tide when river- and tidal-flows are in opposition (see Dustan and Pinckney, 1989). Furthermore, migration patterns of some types of fish are considered to be related to tidal streams. Economists can readily identify immediate cash-yielding assets such as fisheries, wood, or agricultural grazing. They also acknowledge the value of ecosystem services such as nurseries, water purification, coastal protection from storms and waves, and even carbon sequestration, which involve no financial transactions but are important for the health of coastal populations, the ocean, and, ultimately, Planet Earth (see Pugh and Woodworth, 2014). Tidal patterns are important for plants and animals in the intertidal zones. The subject has spawned many classic accounts (eg, Carson, 1955; Ricketts et al., 1939; Stephenson and Stephenson, 1972; Lewis, 1964). The intertidal area between highest and lowest water levels generally shows high biological productivity and contains a very rich and diverse range of species. The most common intertidal species are crabs, periwinkles, barnacles, and mussels. These intertidal species often develop in intense clusters to create microhabitats in which more moisture is retained during exposure at low tide. Valuable seaweeds are often found in select intertidal areas. The many reasons for this high level of productivity include the regular availability of nutrients during each tidal cycle in water that is shallow enough for photosynthesis. However, it is also a region of great environmental stress. The potential for high productivity can also be realized by those species that have highly adapted survival mechanisms. Different coastal conditions require different mechanisms. Lunar periodicity plays an important role in the occurrence of crustaceans. For example, Chatterji et al. (1994) found lunar periodicity exhibiting a significant influence on the occurrence of edible crabs (Portunus pelagicus, Charybdis cruciata, and Portunus sanguinolentus). They found that whereas high density of these crabs was recorded in the trawl catches during full moon and new moon, low value was found during intermediate phase. Rock pools in the intertidal regions also provide special biological niches. Individual species have different physical and physiological mechanisms for coping with changes of exposure, temperature, higher light intensity, salinity, and

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other variables such as pH and the partial pressures of oxygen and carbon dioxide. Many species shelter in small shallow intertidal rock pools during periods of low tide, preferring the possible extremes of salinity, temperature, and of oxygen depletion to total exposure to air. Mangrove swamps, a diverse group of salt-tolerant plants that thrive in marshy beaches, support rich ecosystems, which contain terrestrial species in the branches of the mangroves and marine species within the underlying roots and sediments. Mobile animals such as crabs are particularly well adapted. The interactions between tidal bores and mankind are complicated. Some bores are dangerous and have had a sinister reputation (eg, the Qiantang River bore in China, the Seine River bore in UK, and the Bamu and Fly River bores) while also hindering local development and transportation. But tidal bores are also major tourist attractions, as in Canada, China, France, and the UK. Several tidal bores are regularly surfed by kayakers and surfers in Brazil, France, and the UK (see Chanson, 2011). The surfers aim to cover the longest distance and to ride the maximum duration. In terms of the community's interests, the study of tidal bores is clearly justified by their impact on a range of socioeconomic resources. This impact may be direct as in the net upstream advection of sediment materials, or it may be indirect as in the role of tidal bores in the reproduction and breeding of native fish species. Some animals are more sensitive to tidal bore sounds than humans (see Warfield, 1973; Fay, 1988). When a bore closes in, its rumbling noise disorientes some species. In the Baie du Mont Saint Michel, sheep have been outrun and drowned by the tidal bore. In Alaska, deer have tried unsuccessfully to outrun the bore (see Molchan-Douthit, 1998). In each case, the animals panicked with the deafening noise, although they could outrun the bore front. As described, a key feature of a tidal bore is its rumble noise that can be heard from far away. Tidal bores do have a significant effect on the natural channels and their ecology. Bores are usually associated with a massive mixing of the estuarine waters that stirs the organic matter and creates some rich fishing grounds. Its occurrence is essential to many ecological processes and to the survival of unique ecosystems. The tidal bore is also an integral part of the cultural heritage of many regions: for instance, the Qiantang River bore in China, the Severn River bore in the UK, and the Dordogne River in France. Chanson (2011) has addressed in some detail the environmental, ecological, and cultural impacts of tidal bores. According to his studies, tidal bores have a significant impact on ecosystems. For example, tidal bore-affected estuaries are the natural habitat, as well as the feeding zone and breeding grounds, of several forms of wildlife. The evidence connects both scientific and anecdotal observations. In Brazil, the bore sets organic matter into suspension and the estuarine zone is the feeding grounds of piranhas. In Alaska and in France, several birds of prey fish behind the tidal bore front and next to the banks: eg, bald eagles in Alaska and buzzards in France. Visual observations in Alaska and France showed a number of fish being ejected above the bore roller by the flow turbulence. For example, Chanson (2011) saw this several times in the Dordogne River. In Alaska, eagles catch fish jumping off and projected upward above the tidal bore roller. Several large predators feed immediately behind the tidal bore during its upstream progression. These include beluga whales in Alaska (at Turnagain Inlet), seals in the Baie du Mont Saint Michel, sharks in Queensland (the Styx River and Broadsound), and crocodiles in northern Australia and Malaysia (the Daly and Batang Lupar Rivers).

3.4 SUBSURFACE WARM WATER LAYER SANDWICHED BETWEEN COLDER WATERS ABOVE AND BELOW—THERMAL INVERSION The normal vertical distribution of seawater temperature consists of a warmer surface mixed layer and colder subsurface layers, with a thermocline sandwiched between these two distinct temperature layer zones (see Fig. 3.5). During the summer, less dense warm water floats on top of denser cold deeper water, with a thermocline separating them. A thermocline (large temperature gradient) means a specific layer of a water column where the temperature drops more rapidly than the layers above or below. Thus, the bottom layer of the thermocline is rather cold and dense, and the upper layer is rather warm and not as dense. The warm layer is called the epilimnion and the cold layer is called the hypolimnion. Because the warm water is exposed to the sun during the day, a stable system exists and very little mixing of warm water and cold water occurs, particularly in calm weather when the sea surface wave activity is rather weak. In some specific regions, when the sea surface temperature (SST) drops, the normal vertical profile of temperature, which features a warmer surface mixed layer and colder subsurface layer, is altered to form a subsurface warmer layer, a phenomenon known as temperature inversion. As winter approaches, the temperature of the surface water will drop and cooling dominates over the other air-sea heat transfer processes. Under normal conditions, when the surface water cools, it becomes denser and starts sinking. This process is called convective mixing. However, in regions where the

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surface salinity is less compared to subsurface water, the colder surface water still becomes lighter and continues to stay in the surface without mixing. Under this condition, the subsurface water is warmer compared to the layers above and below. Thus when the sea surface temperature drops, the normal distribution of temperature (ie, a warmer surface mixed layer and colder subsurface layers) is altered to form a subsurface warmer layer sandwiched between the colder waters above and below. This sandwiched warmer layer below the sea surface is referred to as the inversion layer (see Fig. 3.40). The process called temperature inversion is also called thermal inversion.

3.4.1 Prominent Thermal Inversion Zones in the World Oceans Thermal inversion is a phenomenon that occurs in the atmosphere as well as in the oceans. Oceanic thermal inversions have been reported from several specific regions of the world's oceans. In the Pacific Ocean, temperature inversions have been noticed in several regions such as in the western equatorial Pacific Ocean, in the coastal waters of the Yellow Sea (YS), and in the East China Sea (ECS) in winter (see Mao and Qiu, 1964; Lan et al., 1993; Lan, 1997). This also occurred in the frontal zone between the warm Kuroshio current and the cold Oyashio current near Japan (see Nagata, 1968, 1979), in the western Canadian Basin, and in the North Pacific subarctic ocean (see Ueno and Yasuda, 2000, 2001). Likewise, temperature inversion is a common phenomenon in the areas near the southeastern Chinese coast, west and south of the Korean Peninsula, and north and east of the Shandong Peninsula during October–May. The inversion is remarkable and has obvious temporal and spatial variabilities in both magnitude and coverage, with higher probabilities in regions near the southeastern Chinese coast (up to about 60%) and in the region north and east of the Shandong Peninsula (40%–50%) than in the region west and south of the

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Korean Peninsula (15–20%). Hao et al. (2010) noticed that the inversion gradually becomes stronger, deeper, and thinner from winter to spring and deepens and thickens from north to south. Nagata (1968, 1979) studied temperature inversions in the seas adjacent to Japan using bathythermograph (BT) observations and found that many of the temperature inversions occur in the mixed water region between the Oyashio and Kuroshio fronts. So the existence of inversions in the region, where cold Oyashio meets warm Kuroshio, is quite possible throughout the year. In the sea east of Honshu, the Kuroshio front forms the southern boundary of the area where temperature inversions are abundant. Though the occurrence frequency of the temperature inversion layers is very low in the sea south of Honshu, the path of the Kuroshio influences its regional distribution in this region also, and the high occurrence area shifts offshore when the large cold water mass is present off Enshu-nada. Nagata (1979) found that the magnitude of inversion thickness has a clear tendency to increase from south to north in the sea east of Honshu, reflecting the higher occurrence frequency of large-scale thick inversion layers in the northern part under the influence of the subarctic water mass. In the Atlantic Ocean, strong temperature inversions have been found in the northwestern tropical Atlantic, north of the Amazon mouth, and east of the Puerto Rico Islands. It has been found that there exist deep and strong colocalized vertical temperature inversions that exceed 0.6°C (see de Boyer Montegut et al., 2007a). Vertical temperature inversions in the northwestern tropical Atlantic are generally stronger and deeper but strongest in autumn. The barrier layer (BL) system in the northwestern tropical Atlantic is associated with significant inversions of the vertical temperature gradient. A unique BL system in terms of persistence, extension, and associated subsurface temperature maximum (ie, thermal inversion) is present seasonally in this region. Mignot et al. (2012) noted that subsurface temperature maxima of up to 1°C can be found in the northwestern tropical Atlantic, extending over a large region. In the Indian Ocean thermal inversion phenomena have been reported from the Arabian Sea (see Thadathil and Gosh, 1992), the Lakshadweep Sea (part of the Arabian Sea; Gopalakrishna et al., 2005), and the Bay of Bengal (see Thadathil et al., 2002). Surface layer temperature inversion in the southeastern Arabian Sea during winter (ie, the northeast monsoon period) has been studied by Thadathil and Gosh (1992) using BT data collected from 1132 stations. They found that while the inversion in this region is a stable seasonal feature, its occurrence is limited to the coastal waters and exists up to 15°N (see Fig. 3.41). It was found that the distribution of temperature depicts a very prominent patch of inversion at a depth of 30–50 m and has a horizontal extent of roughly 400 km. Thadathil and Gosh (1992) found that the inversions found in the southeastern Arabian Sea possess a strong localized character and are confined to the upper few tens of meters. The inversion layer was found to have a thickness varying from 10 to 80 m and a gradient of 0.0–1.2°C (see Fig. 3.42). In a further study, Gopalakrishna et al. (2005) observed thermal inversions in the Lakshadweep Sea. The expendable bathythermograph (XBT) profiles used in their study showed inversions of
20 m thickness, with a temperature increase of about 0.5°C from SST, typically occurring below 20–30 m depth. These inversions first occur during November–December and intensify in strength and spread spatially through January–February, disappearing by March (see Shankar et al., 2004). A modeling study of Durand et al. (2004) shows a similar life cycle but with a shift in phase, with the strongest inversions in the model climatology appearing in December–January and in observations appearing in January–February. The interesting phenomenon of spatially organized temperature inversion occurring in specified regions of the Bay of Bengal (BoB) in the Indian Ocean has been a subject of frequent studies by several researchers since the early 1980s (see Rao and Sastry, 1981; Martin et al., 1981; Rao et al., 1983, 1987; Suryanarayana et al., 1993; Shetye et al., 1996). In the course of these studies it has been found that strong temperature inversions occur during the winter season (November–February). The two major regions of temperature inversion in the BoB are the western and the northeastern bay. Thadathil et al. (2002) found inversions of a large temperature difference (of the order of 1.6–2.4°C) and a thin layer of thickness (10–20 m) in the BoB located adjacent to major fresh water inputs from the Ganges, Brahmaputra, Irrawaddy, Krishna, and Godavari rivers.

3.4.2 Temperature Inversion Formation Mechanisms The evolution of temperature inversion in the different regions of the world's oceans is governed by different dynamics. The controlling factors leading to the generation of temperature inversion include surface heat flux, freshwater flux, horizontal advection, and wind stress. The phenomenon is usually found in oceanic regions where the thermal structure is complicated and isotherms are rugged. Temperature inversion is embodied as a layer where temperature increases with depth and mainly appears in a cold season. Surface cooling is considered to be the primary reason for formation of temperature inversion. As seawater density depends not only on temperature but also on

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FIG. 3.41 Frequency of occurrence of temperature inversion in the southwest Indian coastal waters of the Arabian Sea in the Indian Ocean. The proportions of the number of bathythermograph (BT) traces containing temperature inversions larger than 0.2°C calculated for each one degree square grid are shown by the black area of each circle. Small circles indicate that the total number of analyzed traces is less than 5, while big circles show that the total number is between 5 and 10. From: Thadathil, P., Gosh, A.K., 1992. Surface layer temperature inversion in the Arabian Sea during winter. J. Oceanogr. 48, 293–304.

FIG. 3.42

Time series plot of temperature profiles with observed inversion found in the southeastern Arabian Sea in Jan. 1984. From: Thadathil, P., Gosh, A.K., 1992. Surface layer temperature inversion in the Arabian Sea during winter. J. Oceanogr. 48, 293–304.

salinity, temperature inversion is stable only if salinity within a temperature inversion layer increases rapidly with depth to compensate the temperature effect on density. Such a halocline is usually associated with freshwater flux in the surface layer. Thus temperature inversion is usually observed to coincide with the halocline, where higher salinity in the subsurface layer compensates stability loss due to lower SST and maintains the stable temperature inversion. Such a halocline is usually associated with significant freshwater flux from river runoff and precipitation. In addition, horizontal advection contributes dynamically to maintain the inversion. When cold and fresh water is advected over warm and saline water, temperature inversion forms and maintains. In fact, both the river plume and coastal currents are strongly affected by the local wind. Therefore, a combined effect of surface heat, freshwater flux, advection, and wind stress determines the strength and residence time of the inversion layer (see Hao et al., 2010).

3. COMPLEX WAVE MOTIONS AND THERMAL STRUCTURE OF THE OCEANS

FIG. 3.43 Typical temperature and salinity profiles showing the definition of the layer thickness (ΔD), the temperature difference (ΔT), the starting depth (Ds) and the salinity difference (ΔS) for an inversion layer. From: Thadathil, P., Gopalakrishna, V.V., Muraleedharan, P.M., Reddy, G.V., Araligidad, N., Shenoy, S., 2002. Surface layer temperature inversion in the Bay of Bengal. Deep Sea Res. I 49, 1801–1818. © 2002 Elsevier Science Ltd.

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The inverted vertical profile of temperature found in association with temperature inversion is caused by a peculiar heat balance. With reference to Fig. 3.43, the major parameters associated with a temperature profile with inversion are: layer thickness (△D), temperature gradient (△T), starting depth of inversion (Ds), and salinity gradient (△S). Thadathil et al. (2002) found that temperature inversions with thinner ΔD (varying from 10 to 20 m) are located in the coastal waters of the BoB, where ΔT is found to be large (1.6–2.4°C). Inversions with thicker ΔD (40–50 m), on the other hand, are located in the open ocean, where ΔT is comparatively smaller than it is in the coastal waters. Furthermore, Ds follows nearly the same distribution pattern as that of ΔD, with smaller values of Ds (10–20 m) in the coastal waters and larger Ds (30–50 m) in the open ocean region south of 16°N. The vertical stability of temperature inversion layers in the ocean depends upon the degree to which the increase in temperature with depth is compensated by a corresponding rise in salinity. Fedorov (1969) examined the ΔS-ΔT relationship of temperature inversion layers in the Atlantic Ocean and the Sea of Okhotsk (in the Pacific Ocean) and reported that the ΔS-ΔT relationship serves as a measure of their vertical stability. Neglecting the adiabatic temperature change in the surface layer, the static stability (E) of the inversion layer can be expressed as (Fedorov, 1969):
      1 @ρ @T @ρ @S (3.9) + m1 E¼ ρ @T @Z @S @Z where ρ, S, T, and Z are potential density, salinity, temperature, and the vertical coordinate, respectively. Considering that ð1=ρÞð@ρ=@T Þ, the coefficient of thermal expansion (α), ð1=ρÞð@ρ=@SÞ, and the coefficient of saline contraction (β) are constants, Eq. (3.9) can be simplified to:     @T @S (3.10) +β m1 E¼α @Z @Z Z vary substantially within the inversion layer, a better estimate would be the mean stability Because the stability may EðzÞdz > 0. Considering this mean stability, Eq. (3.10) could be rewritten as: defined as E ¼ ð1=ΔDÞ ΔD     α ΔD ΔS ¼ ΔT + E (3.11) β β

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FIG. 3.44 Scatter plot of ΔT and ΔS for the observed temperature inversion in the Bay of Bengal. From: Thadathil, P., Gopalakrishna, V.V., Muraleedharan, P.M., Reddy, G.V., Araligidad, N., Shenoy, S., 2002. Surface layer temperature inversion in the Bay of Bengal. Deep Sea Res. I 49, 1801–1818. © 2002 Elsevier Science Ltd.

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Eq. (3.11) represents the linear ΔS-ΔT relationship of inversion layers suggested by Fedorov (1969). It may be noted that in a complex oceanic environment, the ΔS-ΔT relation may often be not exactly linear but scattered due to several extraneous influences. For example, Thadathil et al. (2002) noticed that the observed inversion in the BoB in the Indian Ocean exhibits a scattered ΔS-ΔT linear relationship (see Fig. 3.44). Such a scattered ΔS-ΔT linear relationship may be unsuitable for representing the stability of inversions in some regions. For example, Thadathil et al. (2002) encountered such unsuitability with regard to the inversions occurring in the BoB. They therefore made direct use of Eq. (3.11), which is based on the static stability, to compute the mean and extreme values of stability by using the mean and extreme values of ΔS and ΔT. Based on such a computational scheme, the mean stability in the BoB was found to be 3600  108 m1 (with maximum and minimum values of 6105  108 and 1200  108 m1). 3.4.2.1 Freshwater Influx Temperature inversion in the oceans is usually observed to coincide with the halocline, where higher salinity in the subsurface layer compensates stability loss due to lower SST and maintains the stable temperature inversion. Such a halocline is usually associated with significant freshwater flux from river runoff and precipitation. There are large river runoffs in some oceanic regions. For example, Martin et al. (1981) reported that large river runoff from five major rivers (Ganges, Brahmaputra, Irrawaddy, Krishna, and Godavari) and excess precipitation (P) over evaporation (E) are significant sources of fresh water to the BoB in the Indian Ocean (see also Harenduprakash and Mitra, 1988; Prasad, 1997). Lukas and Lindstrom (1991) found that because of the supply of freshwater by river runoff in summer and fall, a sharp haline stratification is formed near the surface of the northern BoB, resulting in the formation of a BL and temperature inversion. In terms of the roles played by freshwater flux, surface heat flux, and advection, Thadathil et al. (2002) found that fresh water flux leads the occurrence process in association with surface heat flux and advection. In their studies, the leading role of fresh water flux in the generation and sustenance of temperature inversion in the BoB has been corroborated from the observation that the two occurrence regions of inversion (the western and northeastern Bay) have proximity to the two low salinity zones (with salinity values about 28–29%) that are located adjacent to major fresh water inputs from the Ganges, Brahmaputra, Irrawaddy, Krishna, and Godavari rivers. Thadathil et al. (2002) found that inversions of a large temperature difference (of the order of 1.6–2.4°C) are confined to the fresh water induced seasonal halocline of the surface layer. In another example pertaining to freshwater influx, a significant part of the freshwater input to the world's oceans takes place in the western tropical Atlantic, in particular, with the river outflow from South America. The main source of seasonal freshwater is coming from river outflow. Two of the strongest rivers (the Amazon and the Orinoco) in the

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FIG. 3.45

Map of a portion of South American coastal region indicating two of the strongest rivers (the Amazon and the Orinoco) in the World, which contribute immensely to the observed pronounced and specific Sea Surface Salinity (SSS) seasonal Cycle in this region of the Atlantic Ocean. The Figure also provides a schematic depiction of circulation in the western tropical Atlantic Ocean showing the North Brazil Current (NBC) retroflecting into the North Equatorial Countercurrent near 6°N. From: Fratantoni, D.M., Richardson P.L. (2006). The evolution and demise of north brazil current rings, J. Phys. Oceanogr., 36, 1241–1264.

world (see Fig. 3.45) outflow in this region, giving rise to a pronounced and specific Sea Surface Salinity (SSS) seasonal cycle. The Amazon River has the biggest flow in the world, 0.2 Sv (note that 1 Sv ¼ 106 m3/s) and is responsible for a large part of the low sea surface salinity (SSS) in the west tropical Atlantic ocean (see Masson and Delecluse, 2001). With reference to the northwestern tropical Atlantic, in a study using sensitivity experiments with a coupled climate model, Balaguru (2011) suggested an important role of surface salinity for the creation of strong vertical temperature inversions in this region. It has been argued that the presence of intense subsurface temperature maxima in the northwestern tropical Atlantic is indeed due to an early (in terms of seasonality) capping of surface mixing by freshwater input, while the mixed layer is sufficiently shallow to allow significant penetration of radiative heat flux. In summer, the BL is relatively shallow and thin, but subsurface temperature maxima are intense. The latter develop as a result of the specific seasonality of the freshwater discharge in this area. Because of the strong and shallow salinity gradient associated with the Amazon freshwater, an important part of the solar radiation is trapped in the BL and creates an inversion of the vertical gradient of temperature (up to 1°C in the northwestern tropical Atlantic, extending over a large region). Chen et al. (2006) analyzed the development of temperature inversion (subsurface warm water) in the ECS in the fall using two observational data sets and the results of a one-dimensional numerical model. It was found that during October, temperature inversion is developed in the ECS and remains until the end of November. The analysis showed that it is the Changjiang River Diluted Water (CRDW) that maintains the vertical structure of water column with a subsurface maximum temperature in this region. In fall, the diluted water flows southward in the ECS, leading to a lower salinity layer in the upper ocean. The existence of this lower salinity layer limits the depth of the vertical mixing induced by the loss of heat on the sea surface in fall and, therefore, maintains the higher temperature of the subsurface water. In winter, when the density's vertical gradient of the low salinity layer cannot balance the negative surface buoyancy flux, the temperature inversion vanishes. However, this situation does not occur outside the region where the CRDW extends because the stratification becomes unstable while the sea surface loses heat and the convective overturn readjusts the whole water column. Numerical results of a one-dimensional model reveal that the temperature inversion appears if the impact of the CRDW is included in the model; otherwise, the temperature inversion does not occur, thereby emphasizing the crucial role of freshwater influx in the generation of temperature inversion in the ECS. In another example, studies carried out by Hao et al. (2010) revealed that the occurrence frequency of about 15% west and south off the Korean Peninsula is the lowest among the three regions (areas near the southeastern Chinese coast, west and south of the Korean Peninsula, and north and east of the Shandong Peninsula), probably due to less river runoff and higher SSS that can hardly sustain the stability of the inversion. The inversion lasts for the longest period in the region near the southeastern Chinese coast (October–May) sustained by the Taiwan Warm Current (TWC) carrying the subsurface saline water, while evolution of the inversion in the region west and south of the Korean Peninsula is mainly controlled by the Yellow Sea Warm Current (YSWC).

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3.4.2.2 Sea Surface Cooling Sea surface cooling exerts considerable influence in the generation of temperature inversion in several oceanic regions. According to Chen et al. (2006), when the SST decreases in fall in the ECS due to heat loss, the existence of the low salinity upper layer will restrict the vertical mixing so that the subsurface water can keep its temperature higher than the SST. Hence the temperature inversion occurs. In summer, the BL still exists, but the temperature inversion cannot be observed because the surface water has a higher temperature than the underlying water. In the BoB, though, the surface heat loss in the coastal waters of the western and northern bay alone is sufficient to cool the surface waters (10–20 m) by
2°C over a period of 1 month. Thadathil et al. (2002) found that the advection of cold fresher water from the head of the bay to the south and southwest helps to retain the cooled water at the surface for causing inversion. Ueno and Yasuda (2000, 2001) suggested that two main processes are responsible for the formation of temperature inversion in the North Pacific Subarctic Ocean. First, in the winter season, abundant precipitation coupled with strong cooling and wind mixing lead to the formation of cold and low salinity waters (LSW) overriding the warmer and more saline waters. This leads to the formation of thermal inversion there during winter with a subsurface maximum vertical-temperature profile (mesothermal). It may be noted that differential horizontal advections of the waters with different temperatures and salinities create the mesothermal structure. Second, when surface warming occurs in the spring, subsurface minimum vertical-temperature profile (dichothermal) structures is formed between the sea surface and the warm and saline waters (see Uda, 1963; Roden, 1995; Ueno and Yasuda, 2000). The Miami Isopycnic Coordinate Ocean Model (MICOM) supports the thermal inversion mechanism for this region (see Endoh et al., 2004). 3.4.2.3 Horizontal Advection Horizontal advection involves primarily two driving forces: thermohaline driving and wind driving. These mechanisms are addressed below. 3.4.2.3.1 THERMOHALINE DRIVING

Thermohaline driving is an important factor in the development of temperature inversion in some specific oceanic regions. Certain regions of the Indian Ocean testify this observation. The northern Indian Ocean, spread out on either side of the Indian peninsula, has the distinction of having two different surface water masses: high salinity (36 ppt) water in the Arabian Sea and LSW (33 ppt) in the BoB. The former forms by the excess of evaporation in the central Arabian Sea, and the latter due to excess of precipitation and large run off into the BoB. During the northeast monsoon (November–January), the LSW moves south and then enters the Arabian Sea along two branches (see Wyrtky, 1971). One moves west, and the other north along the west coast of India. The causative factor for the temperature inversion in the Arabian Sea was identified to be the winter-time surface advection of cold, less saline water (with BoB origin) over the warm saline Arabian Sea water along the west coast of India. Most interestingly, Thadathil and Gosh (1992) found that the magnitude of the thermohaline forcing is approximately three times larger than the wind forcing, implying that the thermohaline driving would overwhelm wind driving. Likewise, Guan (1999, 2000) revealed a close relationship between the temperature inversion in winter and the warm current in the deep-bottom layer. Furthermore, Chen et al. (2006) found that the Yangtze River Diluted Water (YRDW) is important to maintain the temperature inversion in the ECS in the fall. In the Lakshadweep Sea (8–13°N and 72–76°E), LSW are transported from the BoB by the EICC and the Winter Monsoon Current (WMC). In response to this intrusion of LSW, a BL (haline stratification embedded within the near-surface isothermal layer) is formed in the LS leading to the buildup of a miniwarm pool (see Rao and Sivakumar, 1999; Shenoi et al., 1999; Masson et al., 2005). The XBT profiles collected along a repeat triangular transect in the Lakshadweep Sea (LS) have shown westward propagating (see Shankar et al., 2004; Durand et al., 2004) nearsurface thermal inversions (see Thadathil and Gosh, 1992) during November–February coinciding with the arrival of LSW from the BoB. Occurrence of the inversions in the LS is primarily attributed to the intrusion of relatively cooler and low salinity waters of the BoB origin entering into the LS and spreading over relatively warm and high saline waters. To achieve a pictorial insight of the thermohaline process causing the temperature inversion process that occurs in the BoB, Thadathil et al. (2002) constructed two composites of climatological surface currents in the bay for the months of December and February (Fig. 3.46) based on all the available literature. The composites of surface advection were then superimposed on the distribution maps of surface heat budget and occurrence of inversion. One of the telling features of the surface advection was found to be that the southward flowing East India Coastal Current (EICC), which

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FIG. 3.46

Maps showing regions of temperature inversion and a schematic composite of climatological surface currents in the Bay of Bengal superimposed on the distribution of the climatological, monthly mean net heat flux field (W/m2). The upper panel shows the surface currents, net heat flux, and regions of temperature inversion for December. The lower panel shows the same information for February. Red arrows indicate surface currents that transport warm, saline surface waters, and blue arrows show surface currents that transport cold, fresh water. From: Thadathil, P., Gopalakrishna, V.V., Muraleedharan, P.M., Reddy, G.V., Araligidad, N., Shenoy, S., 2002. Surface layer temperature inversion in the Bay of Bengal. Deep Sea Res. I 49, 1801–1818. © 2002 Elsevier Science Ltd.

exists in the coastal waters of the western bay during December, is replaced by a northward current in February. The EICC brings less saline and cold water from the head of the bay to the southwest bay, where it advects over warm saline water, thereby promoting temperature inversion in this region along with the surface heat loss. During later winter (February), the northward flowing coastal currents bring warm saline water to this region, favoring the disappearance of inversion. In the north eastern bay, the less saline cold water from the head of the bay and Irrawaddy basin is advected to the occurrence region of inversion during all winter months, causing sustained inversion here. 3.4.2.3.2 WIND DRIVING

In several instances, Ekman transport has been found to exert a substantial influence on temperature inversion's spreading and movement. For example, analysis of temperature inversion in the China seas by Hao et al. (2010) showed that advection of Yangtze and Yellow river freshwater in the surface layer, and that of ocean origin saline water in the subsurface layer, maintain stable stratification. It was suspected that the evaporation/excessive precipitation flux makes little contribution to maintaining the stable inversion. Advection of surface fresh water by winddriven coastal currents was found to result in the expansion of two inversion regions in the China Sea. The inversion

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was found to last for the longest period in one region (October–May) sustained by the TWC carrying the subsurface saline water, while evolution of the inversion in another region is mainly controlled by the YSWC. Based on the analysis by Hao et al. (2010) of a large set of hydrographic data archived from 1930 through 2001 (319,029 profiles), advection of the wind-driven surface coastal current and Ekman transport exert a substantial influence on the inversion's southward spreading from 32 to 28°N during October–May and offshore movement in April and May. Relative to advection, sea surface cooling, and thermohaline driving, the role of wind driving in the generation of temperature inversion is rather poor. Nevertheless, Thadathil et al. (2002) found that scattered inversions of a transient nature (with temperature differences of <0.5°C and lifetimes of the order of a few days) occur in the BoB on days of high winds and less solar radiation, especially during the summertime Indian monsoon period. Likewise ECS region, from which temperature inversion has been found to occur in winter (see Mao and Qiu, 1964), is distinguished by the indirect influence of monsoonal winds in terms of their role in generating a strongly stratified surface layer and seasonal variation of the circulation (see Guo et al., 2004). Based on the regional distribution of the temperature inversion in the YS and the ECS, Ding (1994) as well as Ding and Lan (1995) pointed out that the potential generating mechanisms are sea surface cooling, wind-driven currents, and warm advection. Thus, wind driving does exert a clear influence, though minor or indirect, in the generation of temperature inversion.

3.4.3 Climatic Impact of Oceanic Temperature Inversion Several studies have illustrated the climatic impact of temperature inversions in the Indian and Pacific Oceans (see Smyth et al., 1996; Vialard and Delecluse, 1998; Durand et al., 2004). In another study, Balaguru (2011) suggested the potential impact of oceanic temperature inversion in terms of tropical cyclone predictability and a possible link between BLs and temperature inversions in the tropical Atlantic. The BL system of the northwestern tropical Atlantic is located on the path of the surface branch of the Atlantic Meridional Overturning Circulation (AMOC). Therefore, its formation and seasonality could also be linked to remote oceanic conditions and influence the whole Atlantic climate through oceanic conditions. This location is thus unique in terms of climatic variability and climatic impact (see Mignot et al., 2012). Studies carried out by Han et al. (2001), Durand et al. (2004), de Boyer Montegut et al. (2007a,b), Wang et al. (2012), Girishkumar et al. (2013), and Akhil et al. (2014) indicate that temperature inversions can have a significant impact on the heat budget of the surface mixed layer because the mixed layer can be warmed by vertical heat advection, entrainment, or vertical diffusion at its base. In a model simulation, Durand et al. (2004) have shown that the heat trapped within the thermal inversions makes a significant contribution in raising the SST of the South Eastern Arabian Sea (SEAS) by 1.1°C during November–March. The notion that warm subsurface water is diffused to the surface is consistent with the studies of de Boyer Montegut et al. (2007a,b), who conducted an online calculation of the mixed layer heat budget and showed warming due to vertical diffusion in the northern BoB. Girishkumar et al. (2013) found a significant influence of temperature inversions on the mixed layer heat budget based on observations at the interior of the BoB. Observations near the coast possibly exhibit a stronger effect of temperature inversions than they found. Nagura et al. (2015) found that the region with the largest SST variability is located just off Bangladesh, which is over or near the shallow continental shelf. SST variability in the BoB region can have a significant impact on atmospheric circulation. Apart from surface meteorological impacts, environmental fluctuations can have important impacts on society because the northern BoB is surrounded by densely populated areas. For example, earlier studies demonstrated that SST off Bangladesh was associated with cholera incidence in Dhaka, the capital and largest city in Bangladesh (see Lobitz et al., 2000; Constatin de Magney et al., 2008; Hashizume et al., 2011). If interannual SST variability off Bangladesh is successfully predicted, it will be a great benefit to the local community. Nagura et al. (2015) found that the generation mechanisms of SST anomalies in a region of the BoB off Bangladesh are largely affected by the temperature inversion in fall and winter, which are characterized by cool water trapped in the surface mixed layer and warm water below. It was found that temperature inversions also play an essential role in the generation of SST anomalies off Bangladesh on interannual time scales. It may be noted that positive SST anomalies result in the enhancement of evaporation. Nagura et al. (2015) have summarized the mechanisms for the generation of a positive SST anomaly as follows: in fall and winter, when temperature inversions are clearest, wind anomalies in the bay turn southerward. The resulting Ekman pumping anomalies enhance upwelling of subsurface warm water to the surface. Simultaneously, the mixed layer off Bangladesh becomes deeper than normal, which entrains more subsurface warm water to the surface. Furthermore, the thicker-than-normal mixed layer increases the heat capacity of the mixed layer and thus reduces the

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severity of atmospheric cooling. Additionally, vertical diffusion at the bottom of the mixed layer anomalously warms the surface mixed layer. Thus, warming due to these oceanic processes overcomes anomalous atmospheric cooling and leads to the growth of positive SST anomalies off Bangladesh. The results in the studies carried out by Nagura et al. (2015) have revealed that haline stratification and temperature inversions together have a substantial role in the generation of SST variability off Bangladesh. They found that the SST variability is trapped near the coast, where the river runoff causes a clear temperature inversion. These researchers found that the amplitude of SST variability is about 5°C on the seasonal time scale and 0.4°C on interannual time scales.

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