Changes in energy of surface gravity waves in the Agulhas Current

Deep-Sea Research, 1976, Vol.23,pp.509to 518.Pergamon

Press. Printed in Great Britain.

Changes in energy of surface gravity waves in the Agulhas Current E. H. SCHUMANN* (Received 26 November 1914; accepted 7 August 1975)

Abstract-On a cruise of the R.V. Meiring Naud&, wave recordsweretaken in and out of the Agulhas Current. These have been analysed using the standard methods of spectral analysis and the energy increase in a frequency band from 0.067 to 0.02 Hz determined. Rough agreement with theory is found up to maximum current speeds of 1.6 m SY’. 1.

INTRODUCTION

THE CHANGESin

the characteristic of a wave group in a variable current regime has attracted considerable interest over recent years. In view of the dramatic increases in wave height and steepness which may occur in a strong current, and the consequent adverse effect on shipping, this is not surprising. However, the majority of investigations have been of a theoretical nature and the literature remains remarkably poor in the experimental verification, or otherwise, of the theory. The reasons for this are probably the limited areas in the ocean where suitable conditions exist, and the considerable experimental effort that has to go into such an investigation. This paper reports the result of a limited investigation, and rough agreement with theory is found. LONGUET-HIGGINS and STEWART (196 1, 1964) established the concept of ‘radiation stress’ as being the mean value of the flux of momentum across a plane in the presence of waves minus the mean flux in the absence of waves. This radiation stress (a force per unit length) due to the waves can interact with a rate of strain due to some other flow such as a current. Energy transfer will then be to or from the waves. This concept was extended by WHITHAM (1962) and later PHILLIPS (1969). WHITHAM (1965,

1967) provided a new technique for investigating the properties of nonlinear dispersive wave systems using a Lagrangian. CRAPPER (1972) used this technique in considering basically the same situations as LONGUET-HIGGINS and STEWART (1961, 1964); it

was found that the rates of growth of large waves in an adverse current were less than that predicted by the earlier theory. KENYON (1971) used the ray equations to consider the refraction of waves in an opposing current. Generally the waves will be bent towards the higher current velocity, though the group and phase velocity vectors follow different paths. It may also be possible for particular waves to be trapped along the peak of a current. In order to introduce the analysis, a brief description will be given of some previous results of linear small amplitude wave behaviour (see e.g. PHILLIPS, 1969). Let a component of a general wave field be specified by C(x, t) = a(x, t)e ix(x*rf,

(1.1)

where a(x, t) is the amplitude and x(x, t) the phase function; in general, both are functions of space x and time t. The wave number k and radian frequency n can then be defined by k=VX n = -

ax/at.

1

(1.2)

I

These two expressions can be combined to give the conservation of phase, g

+Tpl=o.

*National Research Institute for Oceanology, Box 17001, Congella 4013, South Africa.

509

(1.3) P.O.

510

E.H. SCHUMANN

The wave number here is the same as the usual definition of 2~t/3., where 3. is the wavelength. However, in a region where a current is flowing, the radian frequency n is not the intrinsic frequency of the waves. It is the frequency which is actually observed, and thus includes the convective effect of the current. If ~ denotes the intrinsic frequency, under steady conditions (Vn -----0) it is found that n ~-k'U(x,t)+tr

=%.

(1.4)

% is a constant, the subscript indicating a state where the current, U, is zero. For deep water waves, G

~/(g/k),

c .....

k

(1.5)

where g is the acceleration due to gravity and c is the phase speed. Combining equations (1.4) and (1.5) with waves and currents along the same line, c

c0--½+½

(

1+

4U~t

C-o~"

(1.6)

This result shows that the linear theory does not allow an adverse current -- U > co/4. At the critical point where the second term on the right-hand side of (1.6) vanishes, c = co/2, and U C

This means that the convection velocity is equal and opposite to the local group velocity cg ---- c/2, and energy can no longer be propagated against the stream. It should be noted that the non-linear analysis of Crapper allows -- U > co/4; however, in practice, it will generally be the case that the wave steepness will have increased to such an extent that the waves will break around this point. HUGHES and STEWART (1961) studied the interaction of a wave train with a stable Couette

shear flow, but were unaware of the full effects of capillarity. This was remedied by LONGUETHIGGINS and STEWART (1964). SUGIMORI (1973) investigated the dispersion of the directional spectrum of short gravity waves in the Kuroshio Current using the hologram method. The maximum wavelengths analysed were about 5 m. Suitable measurements of longer waves in the open ocean present many difficulties. In the first place, it should be possible to determine the energy spectra of the same wave group both outside and within a strong current; the stronger the current the wider the experimental range. Then all the other factors which may play a part must be monitored; at this stage the surface wind stress and the horizontal as well as the vertical current structure would probably be sufficient. Finally, of course, the wave direction must be determined, and possible effects of bottom topography eliminated. Obviously the correct choice of a suitable region will facilitate matters considerably. Thus it is a great advantage to take measurements in an area of strong current shear, so that the measurement points are close together and there can be greater surety that in fact it is basically the same wave group that is being recorded. This is an added advantage with a shipborne wave recorder, since then not much time will be wasted in steaming from one point to the next. The Agulhas Current off the east coast of South Africa has, on most occasions, a suitably strong gradient on its inner edge. Figure 1 shows such a situation with 10-m current vectors measured on one day of the cruise M N 71/16 of the CSIR R.V. Meiring Naudd. It is seen that over a horizontal distance of less than 10 km from the central line of ship stations to the outer line, the current speed increased by about 1 m s -1. Thus pairs of shipborne wave records in and out of the strong flow could be taken within an hour of each other along the central and outer lines of stations. The wave records were all made while the ship was drifting freely on station, sometimes with a drift speed in excess of 1 m s -1. Consequently, a correction had to be made to any wave frequency measured (see e.g. CARTWRIGHT,1961). Thus if v

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Fig. 1. Ten-metre current vectors measured on 23rd June, 1971. The bottom topography is also shown (1 fathom - 1.83 m) and the inset map shows the location of the Richards Bay area.

is the radian frequency recorded, with the true frequency being ~, -- v + vkcos+.

(1.6)

Here v is the ship's drift and + is the angle between the wave and drift directions. The bathymetry is also shown in Fig. 1 and indicates that in all cases the depth at the central and outer stations was greater than 50 m. All the wave spectra analysed showed energy peaks at wave periods less than 12 seconds, corresponding to deep water wavelengths, X, less than 220 m. For about 5 % accuracy in small amplitude wave theory, it is sufficient to take a definition of deep water as a depth greater than ~,/4, (see e.g. KINSMAN, 1965, p. 131). Consequently, all the waves considered in this paper will be taken to be propagating in deep water•

511

EXPERIMENTAL T E C H N I Q U E S

All the wave records used in the analysis were taken by a National Institute of Oceanography (now Institute of Oceanographic Sciences) shipborne wave recorder Mk II mounted on a vertical section of the hull of the R.V. Meiring Naudd. The wave heights are determined by summing the pressure due to the waves and the double integral of the acceleration. CARTWRIGHT (1961) investigated the response of such a wave recorder and found that the effective depth was 2 or 3 times the actual hydrostatic depth. DARBYSHIRE (1961) confirmed Cartwright's conclusions, and, in addition, found that the alignment of the ship was important. In the present analysis the wave recorder had not been accurately calibrated prior to the cruise; however, this is not important as only the relative variations in wave energy will be considered. However, the effect of the alignment of the ship is unknown, and therefore only those ship stations will be considered in the final analysis where the ship drift, current, and wave direction bore approximately the same relationship to each other. The techniques of spectral analysis used follow the methods outlined in JENKINS and WATTS (1968) for stationary, ergodic time series. The wave records were all of 15-rain duration and were digitized at 1-second intervals• The response of the shipborne wave recorder was such that little energy was allowed through for periods less than about 4 seconds, and thus the adverse effects of aliasing were minimal. The high pass filter mentioned by JENKINS and WATTS (1968, p. 300) was adapted to remove periods greater than about 20 seconds. The lag was chosen to be 50, and using the Tukey spectral window this gave about 52 degrees of freedom. Eighty per cent confidence intervals were calculated• The accurate determination of wave direction has been a problem ever since the analysis of waves took on a more quantitative aspect. This is especially so in a confused sea when there may be two or more wave groups interacting. On the other hand, on a calm day with a single swell running, the direction of propagation may be accurately determined by eye. Unfortunately, no

512

E.H. SCHUMANN

sophisticated methods were available for the cruise M N 71/16, and so the only available information on wave direction was that estimated by the ship's officers of the R.V. Meiring Naud#. While this may on occasion have been prone to considerable error, it will be seen in Section 3 that the changes calculated in the wave field agree well with the changes reported by the ship's officers. Consequently, it is felt that a good degree of reliance can be placed on these estimates. The details of the equipment and methods for measuring wind, currents and depth can be found from STAVROPOULOS (1971) and PEARCE (1973). Briefly, the wind was measured at a height of 13 m above sea level, while currents relative to the ship were measured by a Savonius rotor and magnetic compass on an underwater instrument. The depths at which these measurements were taken were about 10, 20, 30, 40, 50, 75 and 100 m, or to within 10 m of the ocean floor. Absolute ourrent speed and direction were then computed by adding the ship's drift vector while on station.

18 June, 1971

19 June, 1971

This latter parameter was determined from distances measured to fixed shore stations. Generally the current speed could be taken to be correct to about 10%, while direction was within about 15°. 3. W I N D A N D W A V E V A R I A T I O N S In this section the wind and wave variations over the 7 days of the cruise will be discussed. Figure 2 shows the wind speed and direction, and the ship's officers' estimates of the dominant wave direction and period on each of the stations. Some station numbers are also given, indicating that these are not synoptic pictures, but that time advances with station number. On the average about 1 hour elapsed between similar measurements on consecutive stations. It is obvious that the conditions were by no means steady throughout the 7 days, and that there appear to have been at least five major changes in the wave field over this time. These changes should also occur in the spectra

20 June, 1971

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Wind

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Period,

s

noi

//-

Fig. 2. Wind and wave vectors on the 7 days of the cruise MN 71/16. Some station numbers are also shown, indicating the route taken each day by the ship.

Changes in energy of surface gravity waves in the Agulhas Current

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MN 71/16. The numbers above refer to the central ship's stations where the wave records were taken. analysed if the ship's officers' estimates are to be trusted. Ideally, wave measurements should have been made several times a day at one particular point under constant current conditions in order to determine the spectral variations. However, this was not done, and so the next best thing is to consider the wave field variations at points which approximate the ideal situation. Generally, the central line of stations had similar currents on any particular day, and even though the spatial separation of Stas. A2 and E2 was some 48 km, a reasonable analysis can be made by considering the spectra at these stations. Figure 3 is a plot of equal energy contours on a frequency-time co-ordinate system using the stations indicated. Using Figs. 2 and 3 it is now possible to follow the development of the various wave fields during the seven days of the cruise. In this context it should be remembered that if a wave group was generated in some region distant from the area of measurement, a shift of the peak of the energy spectrum towards higher frequencies can be expected with time. This is because of the greater speed of the longer waves which travel ahead of the shorter waves and so are registered first (see e.g. MUNK, MILLER, SNODaRASS and BARBER, 1963). However, for local generation, as the waves become more fully developed a shift of the energy peak to lower frequencies occurs (see e.g. PHILLIPS, 1969).

From Fig. 3 it can be seen that conditions were fairly variable on the first day, with perhaps two different wave systems registered. The first major change in wave direction noted by the ship's officers occurred after Sta. 19 on the second day (19 June--see Fig. 2) and this change is echoed in Fig. 3. However, by the 20th June on both Figs. 2 and 3 the situation was steadier, with both wind and wave coming from roughly a southerly direction. Through most of 21 June the wave direction continued from the south, though the wind backed to oppose the waves. On the last station of the day (number 68) a shorter period wave was reported as the dominant swell. The existence of this group is corroborated in Fig. 3. For most of 22 June wind and wave were northerly, though at the end, with the wind veering, it appears that a new wave group entered the region from the south. This is confirmed in Fig. 3 on the following day, where there is a tendency for the peak of the energy spectrum to move to higher frequencies. The wind backed again on the 23rd and by the middle of the 24th June it appears that another change had occurred in the wave field. The cruise was terminated at this point in order to aid a fishing vessel in distress. With the agreement found between the ship's officers' estimates of the wave field and the calculated variation, it seems reasonable to accept

514

E..H. SCn~MANN

the estimates as giving valid measurements of wave direction. This is especially so in those circumstances when basically only one wave group was propagating through the region. This seems to have been the case during 20 and 21 June (Stas. 42 to 64); from the end of the 22nd and then the 23rd (Stas. 86 to 106) and to a lesser extent the first day (18 June, Stas. 2 to 15). In all these cases the wave and current directions were roughly in opposition. 4.

ANALYSIS OF THE W A V E - C U R R E N T INTERACTION

The theory discussed in Section 5 indicates that an energy increase can be expected at all frequencies when a wave group meets an opposing current. Ideally, by considering narrow frequency ranges, it should have been possible to determine the frequency dependence of the energy increase. However, with the limited resolution available it was not feasible to do this, and so only one frequencyrange coveringwave periods from 5 to 15 second was considered. The frequency corrections of Section 1 were applied [equations (1.4) and (1.6)] so that it was the same initial wave group which was being investigated. Generally, the resulting frequency range covered a width of about four bandwidths in the final spectrum. Some 25 points fell into this region for calculating the integrated energy. In order to determine the current influencing the waves at a point, account had to be taken of the vertical structure. The vertical dependence of a gravity wave was taken to be of the form exp(-- 2nd/~.), where L is the wavelength and d the depth. Taking an average period of about 9 seconds, it was found that the wave influence at 30 m is about ~ of that at l0 m. Using these proportions for the measured current vectors at the above depths, the effective current could be calculated. According to the theory of LONGUET-HIGGINS and STEWART (1961, 1964) if wave and current are not directly opposing, a tensor transformation involving the rates of strain of the mean flow can be performed to determine actual energy variations. Since the current field was not known

accurately enough, this could not be done, and the only comparison made in Section 5 is for those situations where wave and current are directly opposing. However, with the vagaries inherent in the direction measuring of both wave and current, it was felt that a latitude of about 30 ° could reasonably be tolerated. Consequently, this degree of variation was allowed, with the current vector being resolved in the opposing wave direction. In order to compare the energy increases for the various pairs of central and outer stations, it was necessary to find some basis for comparison. Generally, this can be taken to be tile fractional increase in energy above a zero point energy, Eo, where the opposing current, U, is zero. Figure 3 shows that E 0 was by no means constant, and so the central station in each pair (with a smaller current) was used to determine the local Eo, from whence the fractional increase above this level could be found for the outer station. It was not deemed advisable to assume that energy at the central station was E0 as a first approximation, since the current, though smaller than that at the outer station, was by no means negligible in the majority of cases. Theory indicates that it is reasonable to assume a linear dependence of energy increase for small opposing currents (see e.g. Fig. 5). Thus for currents less than 0.6 m s -1 the following dependence was assumed (waves and currents in same direction)

/r(~, U) E0

-

1 -7(~)U.

(4.1)

The coefficient ~/is a function of frequency, but once it was known, E0 could be determined from the current and energy measurement at the inner station of the current-wave pairs considered. Hence the energy increase, E/Eo, could be found for the outer station. On the first day (18 June), the inner boundary of the Agulhas Current was generally beyond the outer line of stations. Consequently, at four pairs of central and outer stations the current was less than 0.6 m s -1 and equation (4.1) could be used

Changes in energy of surface gravity waves in the Agulhas Current

515

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GI

5--

o "-

4

3--

i

I

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05

I

1.0 Opposing c u r r e n t ,

15

t

20

rn s - I

Fig. 4. Fractional energy increases calculated for waves with periods between 5 and 14 seconds. Pairs of ship stations in and out of the Agulhas Current were used, E0 being determined from the inner station with 7 = 1.0 s m-1 in equation (4.1). to find 7. In the four cases (Stas. 2, 3; 4, 5; 10, 11 and 14, 15) the value of 7 was found to be 1.31, 0.97, 2.30 and 2.06 s m -t, respectively. Referring back to Fig. 3, it can be seen that in the first two cases the spectral peak lay near 0.1 Hz, while in the latter two cases a spectral peak near 0.16 Hz had developed. This form of variation in the value of 3' is in accordance with theory, with smaller values occurring at lower frequencies. Further inspection of Fig. 3 shows that all the later spectral peaks of interest occurred at about 0.1 Hz, and so, guided by the values calculated above, as a first estimate the value of 7 was chosen as: 7 = 1.0 s m -1.

(4.2)

Using this value of 7 at the inner station of each of the pairs, from equation (4.1) the value of E o could be found. Hence the fractional increase in wave energy, E/Eo, could be found at the outer station. The results of these calculations have been plotted in Fig. 4. Unfortunately, no wave recordings were made

on the inner Stas. 45 and 96. However, a glance a Fig. 3 shows that conditions were reasonably steady at that time and so an approximation to the zero current energy E0 was made by taking the mean of the values found at the two neighbouring inner stations. Thus a total of 15 points could be plotted. Also indicated in Fig. 4 are approximate 80% confidence intervals. These have been determined by integrating along the 80% confidence intervals for each energy spectrum, and between the same frequency changes as the above energy integrals. Then if El, A~ and E, A represent the integrated energy and upper energy difference at the inner and outer stations, respectively, then the upper confidence interval, At, was calculated using the following formula for combination of errors:

~x~. = V(E?A~ + E'A?)IE?.

(4.3)

Similarly, the lower energy bound could also be determined.

516

E.H. SCHUMANN

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Opposing curren't, m s -~ Fractional energy increases f o r waves with 5- and 15-second periods using equations (5.1) and (5.2). Experimental points are also shown; these were determined using Y = 0"4 s m --1 in equation (4.1). 5.

DISCUSSION

E

c

(5.2) The results shown in Fig. 4 show that a Eo - - c + 2 U " dramatic increase in wave energy can result if a wave group encounters an opposing current. It is These two results have been plotted in Fig. 5 for necessary to determine how this increase cominitial wave periods of 5 and 15 seconds. pares with the theories mentioned in Section 1. It can be seen that the above theory confirms Only the radiation stress theory of LON6UETthat the assumption of a linear increase in wave HIC,GINS and STEWART (1961, 1964) will be energy with current speed is a reasonable one for considered here. Although the non-linear theory speeds less than about 0.6 m s -1. However, the of WmTHAM (1965, 1967) and CRAPPER (1972) value found for Y may have been too high, and indicates a smaller wave amplitude increase in from Fig. 5 an average value between the 5- and strong currents, and a dependence on the initial 15-second limits would be wave steepness, in the range considered here the two theories do not differ appreciably. ) , - - 0-4 s m -1. Two situations were investigated by LONGUETHIGGINS and STEWART. In both a wave group propagated into a region where there was an Also shown on Fig. 5 are the experimental results increasing, directly opposing current. In the first of 4, however using this lower value of 7. case, continuity was satisfied by upwelling from As expected, the lower current values of 18 below, and in the second by lateral spreading. June show a greater increase in wave energy than The results obtained f o r the fractional increase in that predicted by theory. The reasons for this are wave energy in the two cases were not clear, though perhaps the fact that the wave field was changing fairly rapidly (Fig. 3) may have played its part. The results for the other current E Co~ (5.1) pairs, taken under steadier conditions, show a Eo -- c(c + 2U) spread within the theoretical bounds of upwelling from below. However, the tendency is for the and experimental points to lie near the upper bound,

Changes in energy of surface gravity waves in the Agulhas Current

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Fig. 6. Refraction of 5- and 15-second waves initially propagating in the x direction. A current shear increasing the opposing current by 1 m s-~ in 10 km exists in the y direction. and several in fact are above the limit imposed by the theory assuming lateral spreading. A factor which may have been responsible for a greater energy increase than that predicted by the above theory is the refraction of waves into the core of the Agulhas Current. Unfortunately, the point measurements made of the current are not sufficient to determine its character, so it is not possible to calculate the effects of refraction. It may be thought that the ship's officers should have observed refraction if it did take place to any marked degree; however, Fig. 6 shows that this is not necessarily the case. Using the results of KENYON (1971), refraction of 5- and 15-second waves initially propagating along the x axis is shown. A current shear in the y direction has been assumed which increases an opposing current U by 1 m s -x over 10 km. As expected, the 5-second wave is refracted much more than the 15-second wave. However, an observer on a ship will notice the wave crests, and so see the phase velocity direction, and even for a 5-second wave the angular difference between the wave at current speeds of ~ 0.4 and -- 1.4 m s -x is less than about 20 °. This will not be readily distinguished. In conclusion, it can be stated that rough agreement has been found with theory, but that a much more precisely controlled experiment is

needed before finer detail can be examined. For instance, it is not known to what extent local winds affected the results, or the effect that meanders in the Agulhas Current may have had. With a slight angle between wave and current direction, waves in the inshore region may already have passed through the strong current. Though linear theory shows that the waves should have reverted to their pre-current characteristics, shorter waves may have broken with the resultant dissipation and perhaps transfer of energy. Finally, the effects of refraction should be included. Acknowledgements--The author is indebted to Mr. F. P. ANDEaSON for first suggesting this analysis, and to Dr. T. F. W. HARRIS for helpful discussion. Thanks also to all those who participated on the cruise of the R.V. Meiring Naudd. REFERENCES

CARTWRIGHT D. E. (1961) The use of directional spectra in studying the output of a wave recorder on a moving ship. Conference on Ocean Wave Spectra, Easton, Maryland, May 1--4, 1961. CRAPPER G. D. (1972) Nonlinear gravity waves on steady non-uniform currents. 52, 713-724. DARBYSSIRE M. (1961) A method of calibration of ship-borne wave recorders. 14, 5%63.

Mechanics,

graphischeZeitschrift,

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E.H. ScauMAr,rN

HuGI-mSB. A. and R. W. STEWART(1961) Interaction between gravity waves and a shear flow. Journal of Fluid Mechanics, 10, 385-400. JENKINS G. M. and D. G. WATrS (1968) Spectral analysis and its application, Holden-Day, 513 pp. KENVON, K. E. (1971) Wave refraction in ocean current. Deep-Sea Research, 18, 1023-1034. KINSMANB. (1965) Wind waves--their generation and propagation on the ocean surface, Prentice-Hall, 636 pp. LONGUET-HIGGINSM. S. and R. W. STEWART(1961) The changes in amplitude of short gravity waves on steady non-uniform currents. Journal of Fluid Mechanics, 10, 529-549. LON~tmT-HI~INS M. S. and R. W. STEWART(1964) Radiation stresses in water waves; a physical discussion with applications. Deep-Sea Research, 11, 529-562. Mur~ W. H., G. R. MmLr~, F. E. SNOOORASSand N. F. BARaER (1963) Directional recording of swell from distant storms. Philosophical Transactions of the Royal Society, A, 255, 505-584.

PEARCE A. F. (1973) Description of results of the twelve routine cruises by the R.V. Melting Naudd off Richards Bay. May, 1970 to March, 1972. C.S.I.R. Contract Report C FIS 37. Durban, South Africa. PHILLIPS O. M. (1969) The dynamics of the upper ocean, Cambridge University Press, 261 pp. STAVROPOULOSC. C. (1971) Data acquisition on the R.V. Meiring Naudd. Electronics and Instrumentation, May 1971, 11-15. SUGIMOrtI Y. (1973) Dispersion of the directional spectrum of short gravity waves in the Kuroshio Current. Deep-Sea Research, 20, 747-756. WmTHAM G. B. (1962) Mass, momentum and energy flux in water waves. Journal of Fluid Mechanics, 12, 135-147. WHITHAMG. B. (1965) A general approach to linear and non-linear dispersive waves using a Lagrangian. Journal of Fluid Mechanics, 22, 273-283. WHITHAM G. B. (1967) Non-linear dispersion of water waves. Journal of Fluid Mechanics, 27, 399-412.