Water transport in wake waves from high-speed vessels

Journal of Marine Systems 88 (2011) 74–81

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Journal of Marine Systems j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j m a r s y s

Water transport in wake waves from high-speed vessels☆ Tarmo Soomere a,⁎, Kevin E. Parnell b, Ira Didenkulova a a b

Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia School of Earth and Environmental Sciences, James Cook University, Townsville, Queensland, 4811, Australia

a r t i c l e

i n f o

Available online 2 March 2011 Keywords: Vessel waves Wave measurement Nonlinear waves Transport processes Coastal erosion Baltic Sea

a b s t r a c t A new mechanism producing onshore transport of substantial amounts of water remote from the fairway in wake waves generated by high-speed vessels is described based on high-resolution water surface profiling in Tallinn Bay, the Baltic Sea. In addition to water transported by precursor solitons, an elevation event that arrives in remote areas well after the precursors is able to carry several times as much water as the solitons and the other wave disturbances put together. When it arrives, just before the largest vessel waves, its interaction with the leading waves of wakes may produce water level set-up under groups of high ship waves. Its characteristic position at the study site, just before the highest wave group, suggests that it may be a forced disturbance created by the ship motion, the nature of which could be an almost degenerate undular bore. The backflow of this water potentially contributes to fast removal of sediment from non-equilibrium beaches by forming strong offshore-directed flow during the latter phase of the wake-wave event. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The specific properties of certain types of waves excited by vessels have been the subject of research since, at least, the first description of “great waves of translation”, or equivalently shallow-water solitons, by John Scott Russell (Russell, 1844). Although vessel wake waves (called vessel wakes below) add energy to the environment wherever they occur, the relative impact of these wakes depends significantly on the nature of the coastal environment that they reach. Their contribution is obviously negligible in deep sea areas where ship wakes are mostly linear, disperse relatively fast (Lamb, 1997; Lighthill, 1978) and almost never contribute significantly to overall wave activity. On the other hand, the damaging potential of different wake components has been widely recognized in archipelago areas, narrow straits and inland waterways where vessel wakes may contribute substantially to the overall hydrodynamic activity (Madekivi, 1993; Bourne, 2000; Schoellhamer, 1996; Neuman et al., 2001, among others), jeopardize the safety of people and their property, and may also seriously affect the coastal environment (Parnell and Kofoed-Hansen, 2001; Soomere, 2006; Parnell et al., 2007). Impacts of vessel wakes are frequently observed in low-energy environments. For example, in the Marlborough Sounds, New Zealand, the sudden change in the wave regime caused by the introduction of

☆ Based on presentation to the 41st International Liège Colloquium on Ocean Dynamics “Science based management of the coastal waters” hosted in University of Liège, Belgium, 4–9 May 2009. ⁎ Corresponding author. Tel.: +372 6204176, +372 5028921; fax: +372 6204151. E-mail addresses: [email protected] (T. Soomere), [email protected] (K.E. Parnell). 0924-7963/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmarsys.2011.02.011

high-speed craft (HSC) caused initial rapid and significant accretion. This process continued in many places for the duration of HSC operation (Parnell and Kofoed-Hansen, 2001). The beaches have not returned to pre-HSC morphologies since the ferries were required to travel more slowly in late 2000 (Parnell et al., 2007). The reason for almost irreversible changes in this environment is that natural energy levels are not sufficient to move larger gravel-sized sediment in supratidal berms created by vessel wakes. The new geomorphic features are now essentially relict and quite stable. Other significant adverse effects were noted in numerous locations (Parnell and Kofoed-Hansen, 2001; Soomere, 2007) soon after the introduction of either high-speed passenger ferries or large and fast HSC capable of carrying passengers and vehicles, with service speeds approaching 40 knots. As the periods of vessel-generated waves generally increase with the vessel's speed, HSC may produce long waves, the like of which are extremely seldom found in some environments. High leading waves that are practically non-dispersive compact entities carrying a massive amount of energy with shoaling causing violent plunging breakers far from the ship lane and a long time after the ship has passed (Hamer, 1999), highly monochromatic packets of relatively short waves (Brown et al., 1989; Soomere and Rannat, 2003), solitary and cnoidal wave trains ahead of the ship (Neuman et al., 2001), almost steady bores (Gourlay, 2001) or associated depression areas (Garel et al., 2008), all qualitatively different from the usual constituents of the linear (Kelvin) wake, frequently appear at certain speeds. Such effects typically become evident when the sailing regime of vessels contains extensive sections in which the depth Froude number Fh (the ratio of the ship's pffiffiffiffiffiffi speed V and the maximum phase speed of linear water waves gh for the given depth h where g is the acceleration due to gravity) exceeds 0.6

T. Soomere et al. / Journal of Marine Systems 88 (2011) 74–81

and the Kelvin wake starts to widen owing to effects of a finite depth (Sorensen, 1973). These nonlinear features of wake waves are usually the strongest pffiffiffiffiffiffi for ships sailing at so-called transcritical speeds when Fh = V = gh≈1 (Soomere, 2007). A new type of impact of vessel wakes upon usually accreting beaches was recently discovered during studies into vessel wake properties in Tallinn Bay, the Baltic Sea. A medium-energy beach on Aegna Island at the entrance to this bay revealed a clearly discernible impact of waves on the shoreline. The annual mean significant wave height near this beach is about 0.43 m (that is, about a half of the typical values for the open Baltic Sea) and wind waves over 1.2 m occur with a probability of about 2% (Kelpšaitė et al., 2009). Although single vessel waves reach heights of up to 1.5–1.7 m, with periods of ~ 10–13 s (Parnell et al., 2008), vessel waves contribute b10% to the total wave energy at this site. The beach profiles, however, demonstrate considerable variability over relatively short periods of time, with levels varying by over 0.5 m (Soomere et al., 2009). The most significant feature of this beach is that higher beach levels are associated with periods of high wind waves that usually feed this coastal section with sediment, whereas the beach loses sediment during periods of low wind waves. The rate of this loss is especially large during the initial stage of beach evolution after storm-induced accretion events. For example, wakes from four early morning ships reduced the beach volume by 0.75 m3/m. During the day, with low wind–wave-energy conditions, the beach volume further reduced, by 0.9 m3/m by the following morning (Soomere et al., 2009). Most of the sediment was removed to offshore of the typical depth of closure for this coastal section. As the sediment contained a substantial amount of gravel and limestone pieces with typical diameter of a few cm, such stripping of the beach was only possible with unusually strong offshore water velocities. Strong offshore velocities have been shown to frequently occur in situations where wave groups or waves of specific type propagate over nearshore areas of varying depth. Such offshore velocities regularly exist in the wash zone. The propagation of waves of finite amplitude to the coast is accompanied by onshore mass and momentum transport (Starr, 1947). Longuet-Higgins and Stewart (1962) demonstrated that under groups of relatively large non-breaking waves the water surface is depressed and the water is pushed offshore. This phenomenon (water level set-down) may occur quite far from the seaward border of the surf zone. Therefore, already well before the breaking point, groups of large wind waves usually create a local water level set-down while in the surf zone the release of the momentum flux from breaking waves leads to the piling up of water (set-up) in the nearshore (Dean and Dalrymple, 1991, p. 287; Dean and Walton, 2009) and an accompanying near-bottom offshore flow. In the context of vessel wakes, the presence of wave set-up or setdown in the sense of a steady-state (time-averaged) tilt of the water level off- or onshore from the breaking point has been generally discounted due to the short duration of wake events (Parnell and Kofoed-Hansen, 2001). There is, however, extensive evidence of high surges caused by certain components of vessel wakes in remote areas (Parnell et al., 2007). Various near-field flow disturbances created by a moving ship that can effectively carry substantial amounts of water both on- and offshore (Forsman, 2001; Gourlay, 2001; Balzerek and Kozlovski, 2007) have not been generally believed to extend for a distance of many kilometers. Consequently, it is usually thought that precursor solitons are the major agents carrying considerable amounts of water in the onshore direction (Soomere, 2007). It is well known that solitary waves of elevation may carry substantial water masses to the coast and cause effective beach erosion (that is especially significant due to large offshore velocities, Kobayashi and Lawrence, 2004). This study describes evidence of another possible mechanism contributing to onshore water transport and implicitly enhancing strong offshore-directed sediment flow. This mechanism is apparently

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connected with the specific far-field structure of the wakes from highspeed vessels. We demonstrate that a clearly identifiable water surface elevation event arrives before the highest waves of these wakes. Its superposition on the high ship waves produces water level set-up even well before the breaking point. This effect usually takes place before the highest wave crests arrive and becomes apparent with the use of a careful averaging procedure of the properly de-meaned time series of water level measurements. Although the magnitude of the related water level elevations is only about 1 cm, the described feature is able to carry substantial quantities of water towards the coast. The backflow of this water potentially contributes to the fast removal of sediment from non-equilibrium beaches. The paper is organized as follows. The description of the study site and experiment techniques is presented in Section 2. The methods used for the analysis of the raw data series of water surface elevation and for identification of wave set-up, set-down and local water level elevations are discussed in Section 3. The analysis of water transport caused by different parts of the wake is presented in Section 4. Finally, potential consequences of the intense water transport in terms of cross-shore sediment transport are discussed. 2. Study site and methods The analysis is based on results of an experiment undertaken in June and July 2008 on Aegna Island (located at the entrance of Tallinn Bay, the Baltic Sea, Fig. 1). Since about the year 2000, Tallinn Bay has been one of the few places in the world where high-speed ferries operate at service speeds close to the shoreline and/or at relatively shallow depths. The fleet has comprised a variety of vessel types including high-speed twin hull light hydrofoils (operating speeds ~ 60 km/h), conventional ferries (~ 30 km/h), and large mostly conventional but strong powered ships operating at ~50 km/h. The ‘classic’ high-speed craft (HSC, relatively lightweight fast ships capable of carrying vehicles, operating at ~ 55–65 km/h) were represented by two sister monohulls (both called SuperSeaCat), and two twin hull sister vessels (Nordic Jet and Baltic Jet, Parnell et al.,

Fig. 1. The Baltic Sea, and the location of Tallinn Bay.

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2008). At the time of the experiment, the fleet consisted of the vessel types listed above with the exception of the hydrofoils. The total number of departures of large high-speed vessels from Tallinn to Helsinki was 22–25 per day in summer 2008. These ships frequently operated at transcritical speeds in the inner parts of Tallinn Bay (Fig. 2; Parnell et al., 2008; Torsvik et al., 2009a). Significant effects of vessel wakes in the Tallinn Bay area have been reported in the recent past (Soomere, 2005; Erm and Soomere, 2006; Soomere et al., 2009). A partial reason for the relatively large impact of vessel waves is that the entire Baltic Sea is almost non-tidal: the amplitude of tides in most of the basin is 1–2 cm (Leppäranta and Myrberg, 2009). The water level variations in the Tallinn Bay area (historical extremes from − 0.95 m to 1.52 m with respect to the long-term mean) are driven primarily by weather systems. Over the summer months with intense ship traffic, the sea level variations are usually within 0.2–0.3 m and thus the hydrodynamic impact of both wind waves and ship wakes is confined to a narrow belt along the coast. The measurement site was located on the SW coast of Aegna, at a small mixed gravel-sand beach immediately West of a small jetty (Fig. 2, 59°34′19″N, 24°45′24″E) at about 2.7 km from the sailing line of outgoing vessels, at the closest point (Parnell et al., 2008). The site is fully open to the South and therefore receives wakes from vessels sailing from Tallinn to Helsinki, approaching from the South without substantial decay (Torsvik and Soomere, 2008). The location of the island, the geometry of the entire Tallinn Bay area and the angular

distribution of the predominant winds are such that effectively, no wave energy reaches the SW coast of Aegna from the East. The most influential waves come from the West, or NW. The littoral drift, therefore, is from West to East (Fig. 3). Along the shoreline to the West of the study site there are evident sediment deficit and coastal erosion (Kask et al., 2003). Long-term accretion has occurred only in a short section immediately adjacent to the jetty, where the beach is much wider than along the rest of the SW coast. The deposit consists of a relatively thin coating of finer sediment (including limestone pieces with a diameter of up to 5 cm), with cobbles and boulders (~0.2–0.5 m in diameter) permanently visible. The properties of the incoming waves were established from a time series of water surface elevations collected at a frequency of 5 Hz with the use of a high-resolution echosounder (LOG_aLevel® device, General Acoustics) mounted on a heavy tripod in about 2.7 m water depth, ~ 100 m offshore from the study site and 60 m from the southern end of the jetty (59°34.259′N, 24°45.363′E). The measurement range of the sensor is 0.5–10 m with a resolution for single measurements of ±1 mm. The typical time scale of factors potentially affecting the reading of the device caused by changes in the local air density (such as temperature, humidity, barometric pressure, and salinity) are usually much longer (a few hours) than the duration of a single wake (15–20 min). According to the manufacturer, such

Fig. 2. Tallinn Bay with the recorded sailing line and depth Froude number of SuperSeaCat on 29–30 June 2008. The eastern track is used by outgoing ships and the western track by incoming ships. High values of the depth Froude number occur near Paljassaare Peninsula and in several locations along the eastern track. A simulation using COULWAVE shows wake-waves at a single point. Depth contours are at 10 m intervals. Calculations and image by T. Torsvik (Parnell et al., 2008 courtesy of Estonian Journal of Engineering).

Fig. 3. (a) The study site on the western side of Aegna jetty. The triangle denotes the wave measurement location and the star at the landwards end of the jetty, the observation site of coastal profiles. Large arrows show the predominant wave propagation directions for wind and ship waves. Small arrows indicate the direction of sediment transport; (b) aerial photo of the study site showing the depths, isobaths, location of coastal recording equipment (A1–A3) and tripod (A4) for water surface time series. Figures by A. Kask (Parnell et al., 2008; courtesy of Estonian Journal of Engineering).

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changes in atmospheric conditions are accounted for internally using sound velocity compensation and the system needs no calibration. The nearshore at the study site, including the western side of the jetty, is such that any reflection of wave energy from the coast back to the tripod location is highly unlikely. Immediately onshore from the tripod location (A4 on Fig. 3b) there is a belt of large boulders that effectively damps any hydrodynamic activity. The surface elevation data were collected almost continuously over 30 days during the period from 21 June to 20 July 2008. The methods used for the data analysis and various properties of ship wakes and their impact on the coast are described in (Parnell et al., 2008; Didenkulova et al., 2009a; Kelpšaitė et al., 2009; Kelpšaitė and Soomere, 2009; Kurennoy et al., 2009a, 2009b; Soomere et al., 2009; Torsvik et al., 2009a; Torsvik and Soomere, 2009). A major challenge was the separation of certain components of wakes because of the occasional presence of relatively rough wind-seas and partial overlapping of wakes from different ships. While it was almost always possible to identify the long-wave components of vessel wakes, the shorter components were sometimes completely masked by wind waves. In the current analysis we rely on records from days with the lowest wind-wave activity (4–6 July) when the anthropogenic contribution to the wave field almost totally dominated (Parnell et al., 2008). 3. Water level, set-up and set-down in vessel wakes Groups of relatively large, weakly nonlinear waves usually create local water level set-down in water of finite depth. Its magnitude ηdown under a train of progressive waves in the framework of nonlinear second-order theory (Dean and Dalrymple, 1991, p. 287) is 2

ηdown = −

a k ; 2 sinh 2kh

ð1Þ

where a is the local wave amplitude, k is the wave number and h is the local water depth. To a first approximation, the local water level set-down can be estimated by assuming that the leading shipinduced waves are long waves. In this case sinh 2kh ≈ 2kh and ˜down = −a2 = ð4hÞ. This approximation underestimates the magniη tude of set-down by about 7% for waves with periods of 10 s and by about 25% for waves with periods of 5 s. The water level set-down calculated from Eq. (1) is almost negligible for waves with heights below 0.1 m and should reach values about 2–2.5 cm for 1 m high ship waves (Fig. 4) and may be up to 6–7 cm for the highest waves (1.5–1.7 m) observed at Aegna.

Surface elevation, mm

Surface elevation

600

40

400 200

Average amplitude

Smoothed water level

20 0

0 -200

60

Double-smoothed water level

-20

Mean water level, mm

80

800

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Surprisingly, the observed mean water level behaves differently in wakes generated by fast vessels. A slow but systematic increase in water level starts a few minutes (typically 2–3 min, for largest wakes 5–6 min) before the largest waves arrive. This increase reaches values of about 1 cm at the time when the group of largest waves arrives (Fig. 4). When the height of vessel waves reaches about 40 cm (amplitude about 20 cm), the mean water level starts to decrease, roughly following the above-described expected behaviour of the local water level in groups of large waves. This decrease continues until the highest waves have passed. The observed and theoretically predicted mean water levels almost coincide in the later stage of the wake when the wave heights have decreased to the level of about 1/2 of the highest waves. Such behaviour of water level in ship wakes suggests that there is a specific ship-induced feature that exists at very large distances from the sailing line. It is an elevation event that has a relatively low height but extremely long duration. Although it becomes evident in many records of ship wakes and always arrives before the highest waves at the study site, it is not possible to distinguish from our one-point measurements whether or not it is attached to the group of large waves or propagates as an independent entity. An important implication from its presence is that propagation of wakes from high-speed vessels is accompanied by transport of substantial quantities of water. Separation of these elevation events from the average water level and identification of the resulting water transport by ship wakes is an intricate issue. The deviations of the instantaneous water level in these events are quite small. Also, they are frequently masked by the changes to water level caused by external factors, the typical time scale at times being as short as about 5 min (that is, shorter than the duration of the entire wake). A central problem for the adequate analysis of these events is a proper reconstruction of the reference mean water level. The typical range of sea level variations in Tallinn Bay was about 0.1 m/day and in one case reached about 0.3 m/day during the experimental period. Data measured in Tallinn Harbour (Vanasadam) at the bayhead of Tallinn Bay about 12 km from the measurement site (Fig. 5) show that changes over time periods of about 10 h dominated the sea level variations, but oscillations with periods down to a few tens of minutes and with amplitudes up to 5 cm are also important. Some water level changes with a typical time scale of about 5 min frequently played a role. Many of these fluctuations are apparently caused by lowamplitude seiches in Tallinn Bay. Therefore, it is necessary to estimate the reference sea level separately for time periods of 10–20 min, which are shorter than the typical durations of wakes. Note that this “reference” level is to some extent affected by wave set-up or setdown in longer wave groups and therefore does not necessarily exactly coincide with the “theoretical” mean water level at the site that would have been measured in the absence of waves. To a first approximation, the mean water level for each instant was estimated as the running average over a 15 minute long section of the record. The original time series of water surface elevation was then

Theoretical set-down

-400

-40 11:21

11:24

11:27

11:30

Time on 05 July 2008 Fig. 4. De-meaned time series of water surface elevation in a wake of SuperStar on 05 July 2008, smoothed water level over 3 min, double-smoothed water level over 3 min, average wave amplitude over 30-s sections (dashed line) and expected water level setdown according to Eq. (1). Note the different scales for the record of surface elevation time series (left-hand axis) and for the smoothed quantities and set-down values (right-hand axis).

Fig. 5. Water level variations at Tallinn Harbour (Vanasadam) from 26 June to 22 July 2008 measured with an accuracy of 1 cm. Vertical grid lines correspond to 00:00 on the respective day. Data courtesy of the Marine Systems Institute at Tallinn University of Technology.

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de-meaned by point-wise subtraction of the average. This method results in a much better representation of the mean water level compared, for example, with a time series of measurements smoothed by means of a high-order digital low-pass filter with a cutoff period of 10 or 15 min. The appearance of sea level variations at Tallinn Harbour (Fig. 5) suggests that a large part of the variation of the reference level results from high-frequency components with periods below 10 min that at times cause quite rapid (albeit small amplitude) variations of the water level. A convenient way to visualize short-time changes in the average water level is obtained by using the integral of the de-meaned time series η(t) of water surface position t2

Iðt Þ = ∫ηðt Þ dt:

ð2Þ

t1

For an infinite wave train over a water body of constant depth, the variations in this quantity are limited by the amplitude of the waves, and its values ideally fluctuate around the zero mean. If the results of realistic measurements are integrated, there may be additional excursions of I(t) owing to the accumulation of measurement and rounding errors. These excursions are, however, quite small based on a few simulations of synthetic water surface time series with spectral properties coinciding with the measured series but with exactly zero mean value. The basic advantage of using this method is its ability to highlight the presence of systematic, albeit small, deviations of the water level from the reference level. An increase (decrease) in I(t) within a certain time interval reflects a temporary rise (fall) of the water level. The resulting relatively large integral variations are frequently better able to be detected on a noisy background than small deviations in the average water level. The magnitude of the excursion implicitly characterizes the duration of local set-up or set-down events, whereas the slope of I(t) provides information about the magnitude of the set-up or set-down. An example of the changes through time of I(t) in low wind wave conditions demonstrates that I(t) well reflects even quite small deviations of the water level (Fig. 6). Its excursion around the zero mean is relatively smooth and rather limited in amplitude. In particular, the presence of small sea-level variations with periods about 10–15 min and with an amplitude of a few millimeters that are hardly distinguishable in the smoothed time series of echosounder data is highlighted. In the case of (temporary) relatively large absolute values of I(t), the temporal variations of this quantity are much smaller than the relevant wave amplitudes and it therefore performs a smoothing of the signal. On the contrary, if I(t) oscillates around zero, its variation mostly matches the wave amplitudes and additional smoothing is necessary to adequately identify the magnitude of the water level variations. A spectral filter (a Matlab low-pass elliptic filter of 5th order with at most 1 dB passband ripple, 40 dB stopband attenuation, ±10% width of the cutoff band, and 20 s cutoff period) was used to remove such

Fig. 6. De-meaned water level variations at Aegna (rapid oscillations) and the integral I(t) (solid line) between 02:00–03:00 on 05 July 2008. Note the different scales for the surface elevation time series for the integral I(t).

oscillations. The procedure effectively damped the highest components of the wakes with periods of 8–13 s but did not affect low-frequency parts of the signal such as the precursor solitons. Note that in order to achieve sensible results it is important to perform the operation of integration first and filtering thereafter. For example, filtering may create a substantial phase shift of waves in groups (Parnell et al., 2008; Kurennoy et al., 2009a). Also, performing integration first allows damping, to some extent, oscillations of the water surface data in sections where the instantaneous values of I(t) are relatively large. The analysis of the water level time series with the use of the described technique reveals several new features of the structure of the wakes from fast ferries (Fig. 7). First, such wakes frequently contain a clearly discernible low-frequency signal of relatively low amplitude that arrives several minutes (about 2–3 min for relatively low wakes, and up to 7–9 min for the largest wakes) before the highest waves. The part of the wake from 11:17–11:20 in Fig. 7 contains three wave crests approaching 30 s apart and almost certainly is an ensemble of precursor solitons. The number of crests varies between 3–4 and 10 for different wakes. Their amplitude is usually quite low and normally does not exceed 0.1 m. Second, there is a clear gradual increase in the water level after the precursor solitons have arrived. This increase starts well before the large leading waves of the wake arrive and lasts for several minutes. The presence of such an elevation event suggests that a certain amount of water is being carried to the measurement site. A large part of it obviously is not connected with mass transport induced by the oscillating part of the wake. 4. Water transport The propagation of an infinite train of finite-amplitude waves generates transport of water in the surface layer due to differences in the water velocity around the crest and in the wave troughs (Starr, 1947; Dean and Dalrymple, 1991). To a first approximation, the total mean flux (flow of mass) in the water column is (Starr, 1947) M=

E 1 ρga2 = ; cf 2 cf

ð3Þ

where cf is the wave celerity (phase velocity). The depth-averaged velocity is u ¯ = M/ρh and the total volume of water carried by waves through a measurement site during a certain time interval [t1, t2] per each meter of wave crests is thus t2

Vw = ∫uh dt = t1

t

1 2 ga2 dt: ∫ 2 t cf

ð4Þ

1

The group of large and long waves at 11:25–11:28 (Fig. 4) has periods of about 10 s (and thus approximately constant k ≈ 0.12) and celerity cf ≈ 5.1 m/s at the site with a depth of about 2.7 m. The

Fig. 7. De-meaned water level variations at Aegna (rapid background oscillations), the integral I(t) expressed in terms of water transport Ve(t) ≈ cg maxI(t) given by Eq. (7) below and smoothed values (solid line) of water transport for a wake from a large highspeed ferry Superstar on 05 July 2008.

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amplitudes are about 0.15 m at the beginning and end of the group and reach about 0.5 m at its centre. The average of squared amplitudes is about 0.25 m and thus Vw ≈ 11 m3/m over the entire duration of the group of longest waves. The other parts of the wake cause much smaller water transport. This estimate greatly exaggerates the wakeinduced transport as it is based on the assumption of the presence of an infinite wave train. In reality, the group of high waves is likely to be too short to create any substantial onshore water transport as discussed above. Even if it brings water to the coast, the excess water will be distributed over the entire set-up area in the surf zone. The resulting maximum local water level setup at the shoreline is roughly 1/6 of the average wave height (Dean and Dalrymple, 1991), that is, b10 cm. This mechanism of water transport by large waves obviously cannot be applied to estimate the amount of water possibly carried during the above-discussed elevation event. An extension of the integral I(t) is a suitable measure to characterize this transport. The excess volume of water VS in the area of elevation at a time instant t0 is equal to the net change in the water level   VS = ∫ η′ ðx; y; t0 Þ−ηðx; y; t0 Þ dxdy

ð5Þ

S

over the area S where this elevation exists. Here η ′ is the instantaneous position of the water surface in the sea area containing ship-induced waves, η is the reference water level and(x, y) are spatial coordinates. To a first approximation, we assume that this elevation pattern is a long-crested feature and stationary in a suitably chosen moving co-ordinate system. In this case, the integral in Eq. (5) should be independent of the particular time instant t = t0. The measurement site is fairly remote from the ship lane and the waves have already been traveling 20–30 min by the time they reach the measurement site (Parnell et al., 2008). It is therefore natural to assume that the elevation event mostly moves together with the group of the largest waves, that is, it propagates with the group velocity of these waves. Another justification of this assumption is that no similar event has been identified in the vicinity of precursor solitons. Consequently, this feature apparently propagates more slowly than solitons. Based on these assumptions, a reasonable estimate of the crosssectional area Vˆ = ∫½η′ ðx; t0 Þ−ηðx; t0 Þ dx of the excess of the water volume within the elevation area can be obtained by integrating the water surface elevation time series at a fixed point over the entire duration of the elevation event. The assumption that the elevation event moves with the group velocity of large waves cg means that cg serves as the local Jacobian for the relevant change of variables. The resulting estimate for the volume of water transported through the measurement site per each meter of the ship wave crest is t

2  0  Ve = ∫ η ðx; t0 Þ−ηðx; t0 Þ dx≈∫ηðt Þcg dt;

ð6Þ

t1

where η(t) is the de-meaned position of the water surface, and t1 and t2 are the start and end of the elevation event. The integral in Eq. (6) differs from the similar integral in Eq. (2) only by the presence of the group velocity cg which insignificantly varies for the longest and highest ship waves. Therefore, t2

Ve ðt Þ = ∫ηðt Þcg dt≈cg max I ðt Þ;

ð7Þ

t1

pffiffiffiffiffiffi where cg max = gh is the maximum group velocity for linear waves for the given depth. In practice, the start and end of the elevation event can be relatively easily distinguished from the change through time of the integrals I(t) or Ve (Fig. 7). The start roughly corresponds to the

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section in the record in which the values of, for example, Ve, start to gradually increase (about 11:22.5 in Fig. 7). The end occurs at the instant when the values of Ve start to decrease (at 11:25.5 for the wake in Fig. 7). The cross-sectional area of the excess water mass is roughly the amount of increase in Ve during the event. For the wake in Fig. 7, this cross-section is about 6 m2. Consequently, about 6 m3 of water is transported towards the coast during these three minutes per each meter of wave crest. This volume is clearly larger than the volume transported by the group of high waves. This is also much larger than similar transport by the precursor solitons, the magnitude of which usually is below 1 m3/m (Fig. 7). These obtained values should be interpreted as an estimate of the lower bound of the magnitude of the water transport in such events. Most probably, a large part of such elevation events is masked not only by the presence of high vessel waves but also by the presence of much larger local water level set-down described by Eq. (1). It is not clear how the elevation events and the set-down regions are interconnected. In all our recordings where such events are identifiable (about 20 cases), the elevation arrives a few minutes before the group of largest waves and always partially overlaps with this group. Although this constellation might be interpreted as an intrinsic feature of the structure of the wakes from high-speed vessels, strictly speaking, this might equally well be simple coincidence for the particular study site. Based on the described one-point data, unfortunately, it is not possible to establish whether or not these two features are interconnected and/or are supporting each other. 5. Discussion The frequent presence of solitons and the highly nonlinear shape of the largest vessel waves (Soomere et al., 2005; Didenkulova et al., 2009a; Kurennoy et al., 2009a, 2009b) suggest that there is substantial net transport of water from the vicinity of the sailing line in wakes excited by ships sailing at transcritical speeds. It is well known that at these speeds, the ship motion is accompanied by an ever-lengthening depression area around the ship (Grimshaw and Smyth, 1986; Torsvik et al., 2009b) that may also extend to a considerable distance from the fairway (Forsman, 2001; Balzerek and Kozlovski, 2007). This feature causes the phenomenon of squat — a draw-down of the water surface around the ship (Constantine, 1961), which is particularly strong at transcritical speeds (Gourlay, 2006). It is natural to associate the development of the depression with the water transport by solitons and high nonlinear waves. The above analysis has highlighted the presence of another, previously unknown mechanism of water transport in wakes from high-speed vessels. A major mass flux connected with the elevation event arrives in remote areas well after the precursors and usually just before the group of the highest ship waves. It can carry several times as much water as the precursor solitons and the wave-like disturbances put together. Its characteristic position in the wake suggests that it may be a specific disturbance created by the ship motion, the nature of which could be an almost degenerate undular bore. To some extent, this line of thinking is confirmed by the crosssection of wave profiles in the numerical experiments of Torsvik et al. (2009b) where a similar structure is distinguishable; yet the elevation seems to be connected with the ensemble of precursor solitons. There are also theoretical arguments in favour of the existence of such structures in HSC wakes. For example, Grimshaw and Smyth (1986) study surface disturbances occurring in a framework identical to the ship motion at a critical speed. The situation is then governed by the forced Korteweg–de Vries (KdV) equation. Since neither the beginning nor the end of the depression area is a valid solution of the relevant equation, the depression should begin and end with a modulated wave train and thus resemble an undular bore. Another possible explanation is that this feature may be a forced disturbance directly driven by the ship motion. In that case it is simple

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coincidence that it arrives more or less simultaneously with the group of the largest waves. This concept would, however, extend the direct influence of ship motion to the pressure field in the marine environment to, at least, several kilometers from the fairway. There is some supporting evidence to this explanation in recent research into a forced Kadomtsev–Petviashvili equation for steady transcritical flows showing that an extremely long and long-crested elevation feature may extend very far from the obstacle (Esler et al., 2007). The observed phenomenon of water level set-up under a group of large vessel waves is interesting in itself as it is not in accordance with linear and second-order wave theory. Similar set-up events have been observed quite infrequently: for some single wave groups in very shallow water (Toffoli et al., 2007) and, remarkably, when the famous Draupner wave hit the oil platform on 01 January 1995 (Walker et al., 2004). The local increase in the nearshore water level may substantially affect the behaviour of the largest vessel wakes. In particular, it may considerably increase their runup height and consequently lead to significant penetration of a few large wave crests inland. This effect may be the reason why during experiments at Aegna, an area of sand deposition was found more inland than positions normally reached by breaking vessel wakes, although the offshore records indicated no extraordinary wave heights (Didenkulova et al., 2009a). The estimate for runup of periodic breaking waves (Hunt, 1959), which is used for designing coastal structures, underpredicts the runup of ship wave groups by 50–60%. On the other hand, estimates for runup of solitary waves (Li and Raichlen, 2003) show good agreement with the runup heights measured at Aegna (Didenkulova et al., 2009a). The resulting effect of the seaward sediment transport caused by fast ferries (Soomere et al., 2009) is similar to that established by Kobayashi and Lawrence (2004) in the sense that it is connected with the transport of water to the coast in the run-up process of positive solitary waves near the shoreline. The feature described above may change our understanding of the nature of the impact of transcritical vessel wakes in the nearshore. The major source of related problems in channels and shallow areas usually is that the different wake components may create extremely strong impulse loads that at times exceed similar loads caused by natural factors (Schoellhamer, 1996). The presence of large quantities of excess water may considerably modify the structure of wake-induced currents in the nearshore. What happens with this water depends on the nature of the coast. The elevation event has an extremely long spatial extension (equivalent to a wavelength in the order of a couple of kilometers). On a relatively steep, reflecting coast, this disturbance probably will not break but will be reflected and thus locally increases the nearshore water level for some time. The increase is evidently limited to a few centimeters. In some cases, for example in gradually narrowing bays and channels, the increase in elevation may be much larger. Its behaviour may be much more complicated on non-reflecting coasts that are able to concentrate large amounts of water in the nearshore area (Didenkulova et al., 2009b; Didenkulova and Pelinovsky, 2009). The moving elevation event will then behave as a mini-tsunami. In particular, it may develop a bore-like structure and penetrate much farther inland than might be expected based on its height as was observed during the catastrophic 2004 Indian Ocean tsunami, when the tsunami wave reached Sri Lanka (Liu et al., 2005). This behaviour may explain unexpectedly high surges observed at a distance of many kilometers from the ship lane in the Marlbourough Sounds, New Zealand (Parnell et al., 2007) and several remote effects of ship traffic in rivers (Neuman et al., 2001), narrow straits (Garel et al., 2008) and archipelago areas (Fagerholm et al., 1991) that can almost certainly not be caused by free waves. The major consequence is that the situation becomes very much different from the case of regular shoaling waves (that also transport water to the coast in the surf zone but result in more or less stationary

water level setup). When this water mass is released back from the vicinity of coastline, the result may be similar to backflow of a tsunami wave. Such a strong offshore flow may easily strip large sections of coasts of finer sediments (Soomere et al., 2009). Acknowledgements This study was supported by the Marie Curie RTN SEAMOCS (MRTNCT-2005-019374), TK project CENS-CMA (MC-TK-013909), Estonian Science Foundation (grant nos. 7000, 7413 and 8870), the Estonian Ministry of Education and Research (grants SF0140077s08 and SF0140007s11), and EEA grant EMP41. We gratefully acknowledge the contribution of the entire staff of the Wave Engineering Laboratory, Institute of Cybernetics in preparing for and undertaking the experiment at Aegna. Water level data in Tallinn Harbour were kindly provided by T. Kõuts and J. Elken, Marine Systems Institute at Tallinn University of Technology. References Balzerek, H., Kozlovski, J., 2007. Ship-induced riverbank and harbour damage. Hydro Int. 2–3 (September). Bourne, J., 2000. Louisiana's vanishing wetlands: going, going. Science 289 (5486), 1860–1863. Brown, E.D., Buchsbaum, S.B., Hall, R.E., Penhune, J.P., Schmitt, K.F., Watson, K.M., Wyatt, D.C., 1989. Observations of a nonlinear solitary wave packet in the Kelvin wake of a ship. J. Fluid Mech. 204, 263–293. Constantine, T., 1961. On the movement of ships in restricted waterways. J. Fluid Mech. 9, 247–256. Dean, R.G., Dalrymple, R.A., 1991. Water Wave Mechanics for Engineers and Scientists. World Scientific, Singapore. 353 pp. Dean, R.G., Walton, T.L., 2009. Wave setup. In: Kim, Y.C. (Ed.), Handbook of Coastal and Ocean Engineering. World Scientific, Singapore, pp. 1–23. Didenkulova, I., Pelinovsky, E., 2009. Non-dispersive traveling waves in inclined shallow water channels. Phys. Lett. A 373, 3883–3887. Didenkulova, I., Pelinovsky, E., Soomere, T., 2009a. Long surface wave dynamics along a convex bottom. J. Geophys. Res. Oceans 114, C07006. doi:10.1029/2008JC005027. Didenkulova, I., Parnell, K.E., Soomere, T., Pelinovsky, E., Kurennoy, D., 2009b. Shoaling and runup of long waves induced by high-speed ferries in Tallinn Bay. J. Coast. Res. I (Special Issue 56), 491–495. Erm, A., Soomere, T., 2006. The impact of fast ferry traffic on underwater optics and sediment resuspension. Oceanologia 48 (S), 283–301. Esler, J.G., Rump, O.J., Johnson, E.R., 2007. Non-dispersive and weakly dispersive singlelayer flow over an axisymmetric obstacle: the equivalent aerofoil formulation. J. Fluid Mech. 574, 209–237. Fagerholm, H.P., Rönnberg, O., Östman, M., Paavilainen, J., 1991. Remote sensing assessing artificial disturbance of the thermocline by ships in archipelagos of the Baltic Sea with a note on some biological consequences. 11th Annual International Geoscience and Remote Sensing Symposium. Helsinki, vol. 2, pp. 377–380. Forsman, B., 2001. From bow to beach. SSPA Highlights No 3, pp. 4–5. Garel, E., Fernández, L.L., Collins, M., 2008. Sediment resuspension events induced by the wake wash of deep-draft vessels. Geo-Mar. Lett. 28, 205–211. Gourlay, T.P., 2001. The supercritical bore produced by a high-speed ship in a channel. J. Fluid Mech. 434, 399–409. Gourlay, T.P., 2006. A simple method for predicting the maximum squat of a high-speed displacement ship. Mar. Technol. 43, 146–151. Grimshaw, R.H.J., Smyth, N., 1986. Resonant flow of a stratified flow over topography. J. Fluid Mech. 169, 429–464. Hamer, M., 1999. Solitary killers. New Sci. 163 (2201), 18–19. Hunt, J.A., 1959. Design of seawalls and breakwaters. J. Waterw. Port Coast. Div. 85, 123–152. Kask, J., Talpas, A., Kask, A., Schwarzer, K., 2003. Geological setting of areas endangered by waves generated by fast ferries in Tallinn Bay. Proc. Estonian Acad. Sci. Eng. 9, 185–208. Kelpšaitė, L., Soomere, T., 2009. Vessel-wave induced potential longshore sediment transport at Aegna Island, Tallinn Bay. Estonian J. Eng. 15, 168–181. Kelpšaitė, L., Parnell, K.E., Soomere, T., 2009. Energy pollution: the relative influence of wind-wave and vessel-wake energy in Tallinn Bay, the Baltic Sea. J. Coast. Res. I (Special Issue 56), 812–816. Kobayashi, N., Lawrence, A.R., 2004. Cross-shore sediment transport under breaking solitary waves. J. Geophys. Res. Oceans 109, CO3047. Kurennoy, D., Didenkulova, I., Soomere, T., 2009a. Crest–trough asymmetry of waves generated by high-speed ferries. Estonian J. Eng. 15, 182–195. Kurennoy, D., Soomere, T., Parnell, K.E., 2009b. Variability in the properties of wakes generated by high-speed ferries. J. Coast. Res. I (Special Issue 56), 519–523. Lamb, H., 1997. Hydrodynamics, 6th edition. Cambridge University Press. 738 pp. Leppäranta, M., Myrberg, K., 2009. Physical Oceanography of the Baltic Sea. Springer Praxis. Li, Y., Raichlen, F., 2003. Energy balance model for breaking solitary wave runup. J. Waterw. Port Coast. Ocean Eng. 129, 47–59. Lighthill, J., 1978. Waves in Fluids. Cambridge University Press.

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