The generation of longshore currents on a beach is investigated theoretically, using the concept of radiation stress to describe the flux of momentum associated with the incoming waves. Outside the surf zone, the longshore gradient of the radiation stress is always zero, and the flow is therefore driven only inside the surf zone...

Bowen, Anthony J.

English

Yale University

      P.O. BOX 208118 | NEW HAVEN CT 06520-8118 USA | PEABODY.YALE. EDU   JOURNAL OF MARINE RESEARCH The Journal of Marine Research, one of the oldest journals in American marine science, published important peer-reviewed original research on a broad array of topics in physical, biological, and chemical oceanography vital to the academic oceanographic community in the long and rich tradition of the Sears Foundation for Marine Research at Yale University.  An archive of all issues from 1937 to 2021 (Volume 1–79) are available through EliScholar,  a digital platform for scholarly publishing provided by Yale University Library at  https://elischolar.library.yale.edu/.  Requests for permission to clear rights for use of this content should be directed to the authors, their estates, or other representatives. The Journal of Marine Research has no contact information beyond the affiliations listed in the published articles. We ask that you provide attribution to the Journal of Marine Research.  Yale University provides access to these materials for educational and research purposes only. Copyright or other proprietary rights to content contained in this document may be held by individuals or entities other than, or in addition to, Yale University. You are solely responsible for determining the ownership of the copyright, and for obtaining permission for your intended use. Yale University makes no warranty that your distribution, reproduction, or other use of these materials will not infringe the rights of third parties.  This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. https://creativecommons.org/licenses/by-nc-sa/4.0/   The Generation of Longshore Currents on a Plane Beach 1 Anthony J. Bowen Tne University of Liverpool Tidal Institute and Observatory Birkenl,ead, Cl,eshire, England ABSTRACT The generation of longshore currents on a beach is investigated theoretically, using the concept of radiation stress to describe the flux of momentum associated with the incoming waves. Outside the surf zone, the longshore gradient of the radiation stress is always zero, and the flow is therefore driven only inside the surf zone. The exact form of the longshore current is dependent on assumptions about both the behavior of the waves in the surf zone and the form of the viscous forces. However, with reasonable assumptions, the theory leads to a complete description of the velocity field as a function of the distance from the shore. The model is intended to provide a mathematical framework in which the various possible assumptions can be tested by comparing them with reliable experimental data. One of the laboratory experiments of Galvin and Eagleson (1965) is used to show the relevance of this approach. I. Introduction. "The empirically modified, momentum-flux theory now accepted as the best prediction is based on an untenable assumption and sup-ported by inappropriate data" (Galvin 1967). This devastating but highly relevant statement stresses three important points: (i) existing knowledge about the theory of the generation of longshore currents is poor; (ii) the experi-mental data available, particularly the field data, are usually inadequate to test the relevance of theoretical models; (iii) the most promising approach to the problem has been that based on the concept of momentum flux (Putnam et al. 1949, Inman and Quinn 1951, Eagleson 1965). The concept of radiation stress, defined as the excess flow of momentum· due to the presence of waves (Longuet-Higgins and Stewart 1964), expresses the momentum input of waves in a fairly simple form. In particular, this concept allows the momentum flux due to waves to be defined everywhere in the nearshore region; existing theories tend to define the momentum input due to waves at only the line of the breakers. The gradient of the radiation r. Accepted for publication and submitted to press 28 January 1969. 206 Bowen: Generation of Longshore Currents 207 stress may therefore be defined, subject to suitable assumptions about the wave field, for the whole nearshore region; the equations of motion may then be solved to obtain the velocity field. 2. The Equations of Motion. In order to formulate a tractable flow problem, some general assumptions about the flow conditions are necessary. Following Arthur (1962), it is assumed that: (i) the currents are steady, hence there are no time-dependent terms; (ii) the water is homogeneous and incompressible, and the density, e, is constant; (iii) the pressure, p, is hydrostatic, and p = eg (ij-z), where rj is the difference between the mean-water level in the presence of waves and the still-water level; (iv) the Coriolis force may be neglected; (v) the currents are sufficiently small so that their ir1teractions with the waves are negligible. Laboratory observations suggest that (i), (ii), and (iv) must be good assump-tions; (iii) is the usual assumption of shallow-water theory; and (v) is a neces-sary first step in the solution of the equations. The equations of motion and continuity are then similar to the shallow-water equations of Stoker (1957) (see Fig. 1), c'Ju c'Ju c'Jrj u-+v- = -g- +Rz+i-z, c'Jx c'Jy c'Jx (1) c'Jv c'Jv c'Jrj u-+v- = -g- +R11 +-r: 11 , c'lx c'Jy c'Jy :)u(ij+h)] + :iv(ij+h)] = o; (3) here u(x,y) and v(x,y) are the depth mean velocities, Rz and R11 are frictional terms, and -,;z and -,;11 are the stress terms due to the radiation stress tensor, S. Then l (4) The tradional assumption for investigations into the mechanics of longshore currents is that longshore conditions are uniform. Then all y-derivatives are zero, and, using (3), u(x,y) = o. The solution will be of the form v(x,y) = v(x), ij(x,y) = ij(x), with h h(x). . Now, for most of the possible forms of Rz, Rz = o 1f u = o. Then the equations of motion reduce to 208 max still-water set-up line Journal of Marine Research breaker line wave crest ~------J'i'\. _ still-water level _ -x I 'tti~--, I h I I I I :Xs "'IE xb--Figure 1 . Schematic plan and section of the nearshore region. <Jij ___ /JSxx g /Jx = - e(ii+h) ~' R = + I /JSyx y e(ii+h) Jx' (5) (6) where (5) is the equation for the change in the mean water level due to the presence of waves derived by Longuet-Higgins and Stewart (1964) and (6) provides the solution for the longshore current. A current exists if /JS11x//Jx * o; its precise structure must depend on the assumed form of the viscous term, Ry. 3. The Form of Radiation Stress. Longuet-Higgins and Stewart derived the radiation stress tensor, S, in terms of the direction of wave propagation, Bowen: Generation of Longshore Currents 209 r, and the normal to the direction of propagation, s (Fig. 1 ), as a function of the wave energy E, the wave number k, and the depth h, where (Srr S,,) ( 2kh I o ) S = = E sinh 2 kh + 2 kh ; S,r S,, o sinh 2kh in shallow water, kh -+ o, S = E ( 3/ 2 0 ) . If rx is the angle of incidence of 0 1/2 the waves, then this tensor can be re-expressed in terms of x, y, and rx by using the usual tensor-transformation rules. Then Szz = cos2 ix S,, +sin2 ixS,,. In shallow water, Szz = E/2 · (3 cos2 ix+ sin2 ix) = E/2 • (3 - 2 sin2 ix). (7) Then Szz has a maximum value of 3E/2 when the waves approach normal to the beach, ix = o . Similarly, S11z = -sin IX cos IX (Srr-S,a). (8) When ix = o, then S11z = o and there is no longshore current, as expected. From (6), R 11 =- (_1 h)dd[sinixcosix(Srr-S,,)], (!'T}+ X . (9) and, in shallow water, Ry= - e(r/+n);[Esin ix cos ix]. (10) The regions inside and outside the surf zone must now be considered sepa-rately. Outside the surf zone, the effect of the set-down is small-17 (( h and ij + J, x tan f3. For this region, it is reasonable to assume that energy is conserved and that the rate of transmission of energy is therefore constant; then (11) where C is the phase velocity of the waves, Cn is the group velocity, and the subscript h denotes the conditions at the breaker line. The factor cos ix takes account of the change in the length of the wave crests due to refraction in this special case where h = h(x). The wave refraction is determined by C sin ixb = Cb sin ix. (12) 210 Journal of Marine Research [27,2 Then, from (11), En sin a. cos a.= E,, n,, sin a.,, cos a.,,, where n = 1/2 (1 + 2kh/sinh 2kh). Now, outside the surf zone, by substituting the values of Srr and S,, in (9), we obtain 1 d t . ( 2 kh I kh )] R-v = - eh dx stn a. cos a.E sinh 2kh + 2 - sinh 2kh ' 1 d [ . ( 2kh )] = - 2ehdx ESma.cosa. I+ sinh 2kh ' = - elh ![En sin a. cos a.]; but En sin a. cos a.= E,, nb sin a.,, cos ex,,= constant; therefore, R 11 = o. To the present order of calculations, there is no resultant gradient of stress outside the surf zone; therefore there is no driving term for a longshore current. A similar result was obtained by Bowen ( I 967) when he investigated the prob-lem of the generation of nearshore circulation cells. In that case also, the stresses that tended to produce circulation existed inside the surf zone only. Outside the surf zone the wave motion is assumed to be irrotational; therefore it cannot provide driving terms for currents of nonzero vorticity. Inside the surf zone, the set-up ij cannot be neglected. Following Longuet-Higgins and Stewart (1964) and Bowen et al. (1968), suppose that in this region the wave height is given by H = y(ij +h) and that the variation in a. across the surf zone can be neglected as a first approximation; then, using (7), the shallow-water approximation for the surf zone, S,u; = / 6egy2 (ij+h?(3-2sin2 cx,,) as E=iegH2• Then, from (5), or dij dh - = - K - = - K tan /J K = K(a. ) dx dx ' "' where /J and y are constant. If a new horizontal co-ordinate, x, is introduced, with its origin at the point of maximum set-up x = -x6 , then (15) and ij +h = mx, where m = (1 - K) tan /J. Then, from (6), Bowen: Generation of Longshore Currents R - - I d [ I i (- h)i . ] y- e(ij+h)dx gegy 'Y}+ smoc11COSOC11, R I . d y = - - gyi sm oc11 cos oc11 - (ij + h), 4 dx R I . y = - - gy2 sm oc11 cos oc11 m. 4 211 (16) A forcing function that tends to produce a longshore Bow therefore does exist in the surf zone. Although slightly different assumptions might be made about the behavior of the waves in the surf zone, it is clear that, in general, there must be a forcing function that is zero only when oc,, = o. 4. The Form of the Viscous Terms. The precise structure of the longshore current must depend on the assumption about the dissapative term, R11 • As-sumptions using only bottom friction of the form V Ry~--- or iJ + h vlvl -ij+h lead to velocity distributions that are discontinuous at the breaker line; as-sumptions using the horizontal eddy viscosity, .A11 , of the form ({iv {iv) d2 v Ry= .A,,, <5x2 + <5yi = .A,,, dx2 give an additional constant of integration that allows the velocity to be made zero at the breaker line; however, the velocity gradient will be discontinuous. The laboratory experiments of Galvin and Eagleson (1965) suggest that the velocity distribution should be of the form shown in Fig. 2. The longshore velocity should be zero at the point of maximum set-up (x = o), should reach some maximum value in the surf zone, and should decrease seaward. However, the Bow is not confined to the surf zone; the regime outside the surf zone, where no driving forces exist, must be purely diffusive, involving both fric-tional terms. Therefore, outside the surf zone the equation of motion must be of the form d2 v cv Ry =A11 dx2 - ij+h = o, and in this region ij ( ( h d2 v cv Ry = .A.,,, dx" - x tan /J = 0 · 212 V 25 B 208 Journal of Marine Research -Xs-0 --x. 0 .5 1.0 1.5 e , 16 0 , 9 e . 4 Figure 2 . The velocity distribution in the longehore current, when K = 0.4; the solid lines show the theoretical distribution for three values of 8; the circles show for each case the best fit of the Galvin and Eagleson points from their experiment 11.9. Similarly, inside the surf zone, from ( I 5) and ( I 6 ), d2 v cv l . R 11 = A11 d ::2 - = - - g y2 m sm ab cos ab . x mx 4 5. The Yelocity Distribution. The solution for the velocity distribution out-side the surf zone, subject to the boundary condition v- o as x- oo, can be expressed in terms of modified Bessel functions: v(x) = P).(x)•I• K, [J.(x)•I•] where ).2 _ 4 C • - A11 tan fJ' then dv P).2 - = - - K 0 [J.(x)•f•]. dx 2 Similarly, inside the surf zone, the solution subject to the boundary condition v = o at x = o is given by where v(x) = QA(x)''2I,[A(x)'' 2J +BA 2;, (20) I . m2 B = - gy2 sm ab cos ab - ; 4 cA2 then Bowen: Generation of Longshore Currents 213 The constants of integration, P and Q, are determined by the condition that v(x) and dv(x)/dx must be continuous at the breaker line; the breaker position is defined by (21) and the depth at the breaker point by hb = xb tan fJ = mxb = (1 - K) tan {Jib. (22) Let The velocity field outside the surf zone is given by ( X )''" [( X )''"] v(x) = P 0 xb K1 0 xb ; (23) the rate of decrease in v as a function of x/xb gives an estimate of 0; since hb and tan fJ are known or easily measured, an estimate of cf A" can be obtained. Now, equating the expressions for v(x) and dv/dx at the breakers, } so that P and Q depend on both 0 and K. The appropriate value of K depends on the distance between the still-water line (x = o) and the position of max-imum set-up, x = -x8 , as from (22), xb tan fJ = (1 -K) tan fJ xb = (1 - K) tan fJ (xb + x8), so that x 8 K xb = I -K· (25) It is necessary to introduce specific ideas about the particular problem in order to define the value of K and the range of values of 0 that are pertinent. As an example, experiment Il.9 (values at 5 m) of Galvin and Eagleson was considered. The run-up is of order of xb, ab~ 5°, y ~ 1.2 +, from (14). x~o+ 214 Journal of Marine Research The velocity profiles, calculated for 0 = 4, 9, and 16, are shown in Fig. 2. The magnitude of the longshore velocity in the surf zone increases with increasing 0; since the value of 0 is a function of the width of the surf zone, xb, the high velocities are associated with wide surf zones. This is physically reasonable, for it suggests that the intensity of the longshore currents is corre-lated with the breaker height, since the breaker height effectively determines the width of the surf zone. The rate of decay in the velocity outside the surf zone also increases with increasing 0, so that, although the shape of the velocity distribution inside the surf zone are similar, the complete profiles are noticeably different. Since all the parameters needed to calculate the value of the constant B were not measured in experiment II.9, the absolute magnitude of the theore-tical velocity is not known. To compare the shape of the measured distribution with that of the three theoretical curves, 0 = 4, 9, and 16, the measured values of velocity have been plotted on three different vertical scales so that, in each case, the measured and theoretical values at the break point (x = xb) are equal. In all cases, the fit inside the surf zone is good. Outside the surf zone, the observed rate of decay in velocity is greater than that of the models, 0 = 4, 0 = 9, but it is less than that of 0 = 16, suggesting that the observed velocity distribution would be well reproduced by a model 0 -:= 12. As in this experiment, xb = 33 cm, tan /J = 0.1, then c/A11 = 0.01 cm-•. By using the matching condition for the magnitude of the velocity, a value for B can be obtained, and from (20) the value of c can be calculated; then c = 0.20 cm/sec; therefore A11 = 20 cm2/sec. In this case, a relatively simple model, with A 11 and c taken as constants, seems to reproduce the observed situation very well. However, if we look at the experiments of Galvin and Eagleson, in which the longshore velocity is high (30-60 cm/sec), the velocity maximum seems to occur closer to the shore than the present model can allow. It is possible that in such cases assumption (iv) is no longer valid and that the interaction between the waves and the longshore current has to be considered. REFERENCES ARTHUR, R. S. 1962. A note on the dynamics of rip currents. J. geophys. Res., 67: 2777-2779. BoWEN,A.J. 1967. Rip currents. Ph.D. Thesis, University of California, San Diego. u5 pp. BOWEN, A. J., D. L. INMAN, and V. P. SIMMONS 1968. Wave 'set-down' and set-up. J. geophys. Res., 73: 2569-2577. EAGLESON, P. S. 1965. Theoretical study of longshore currents on a plane beach. Techn. Rep. M.I.T . Hydrodynamic Lab., 82; 3 r pp. Bowen: Generation of Longshore Currents 215 GALVIN, C. J. 1967. Longshore current velocity, a review of theory and data. Rev. Geophysics, 5 (3): 287-304. GALVIN, C. J., and P. S. EAGLESON 1965. Experimental study of longshore currents on a plane beach. Techn. Memo U.S. Army coastal Eng. Res. Ctr., 10; 80 pp. INMAN, D. L ., and W. H. QUINN 1951. Currents in the surf zone. Proc. 2nd Conf. coastal Engng., Council on Wave Research: 24-36. LoNGUET-HIGGINS, M . s., and R. W. STEWART 1964. Radiation stress in water waves, a physical discussion with applications. Deep-sea Res., II: 529-563. PUTNAM, J. A., W. H. MUNK , and M.A . TRAYLOR 1949. The prediction of longshore currents. Trans. Amer. geophys. Un., 30: 337-345. STOKER, J. J. 1957. Water waves. lnterscience, New York. 567 pp. 

The generation of longshore currents on a plane beach

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The generation of longshore currents on a plane beach

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