longshore current' A comparison of observations and

Transcription

1 Nonlinear evolution of shear instabilities of the longshore current' A comparison of observations and computations H. Tuba Ozkan-Haller Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor James T. Kirby Center for Applied Coastal Research, University of Delaware, Newark Abstract. The time dependent nearshore circulation field during 3 days of the SUPERDUCK field experiment is simulated. We consider the generation of nearshore currents due to obliquely incident breaking waves, damping effects due to bottom friction, and diffusion effects due to lateral momentum mixing caused by turbulence and depth-varying current velocities. Because of uncertainties in the friction and lateral mixing coefficients, numerical simulations are carried out for a realistic range of values for these coefficients. The resulting shear instabilities of the longshore current exhibit unsteady longshore progressive vortices with timescales of O(100 s) and length scales of O(100 m) and longer. The time dependent flow involves the strengthening, weakening, and interaction of vortices. Vortex pairs are frequently shed offshore. During this process, locally strong offshore directed currents are generated. We find that a stronger mean current and faster and more energetic vortex structures result as the friction coefficient is decreased. However, the longshore length scales of the resulting flow structures are not altered significantly. An increase in the mixing coefficient causes relatively small variations in the propagation speeds. However, the resulting flow structures are less energetic with larger longshore length scales. Shear instabilities are found to induce significant horizontal momentum mixing in the surf zone and affect the cross-shore distribution of the mean longshore current. Mixing due to the presence of the instabilities is found to be dominant over mixing caused by more traditional mechanism such as turbulence. For values of the free parameters that reproduce the propagation speed of the observed motions, the frequency range within which shear instabilities are observed as well as the mean longshore current profile are predicted well. 1. Introduction Surf zone current measurements from field experiments on both plane and barred beaches show that a variety of low-frequency motions coexist in the surf zone. The existence of gravity motions such as edge waves, leaky waves, and surf beat that span the range of frequencies less than 0.05 Hz has been well established. Recently, Oltman-$hay et al. [1989] observed a meandering of the surf zone longshore current during the SUPERDUCK field experiment over timescales up to O(1000 s) and showed that these motions have much shorter wavelengt hs than edge waves at those frequencies. Bowen and Holman [1989], in a companion paper, Copyright 1999 by the American Geophysical Union. Paper number 1999JC J performed an analytic study and showed that a shear instability of the longshore current can reproduce the nondispersive character and meandering nature of the observed motions. Several other mechanisms have been proposed to explain the experimental observations by Oltman-$hay et al. [1989]. Shemet et al. [1991] suggested that oscillations in the longshore current may be linked to oscillations in the radiation stresses that are due to the long time evolution of a three-wave system, composed of a carrier wave and the two most unstable Benjamin-Felt sidebands. Tang and Dalrymple [1989] and Fowler and Dalrymple [1991], in turn, showed that wave trains incident at slightly different angles to the beach can generate rip currents that migrate in the longshore direction at slow timescales. More recently, Hallet et al. [1 9] L " : J suggested that offshore wave groups can directly force low-frequency motions that resemble the observations. 25,953

2 25,954 )ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS All of these approaches assume the observed oscillations to be forced phenomena, whereas the linear instability theory proposes a mechanism resulting in free oscillations. It should also be taken into account that forced oscillations could provide the necessary perturbations for the modes associated with the linear instability the- ory. The instability theory has, so far, been the most studied alternative for explaining the observations by Oltman-$hay et al. [1989]. Several investigators, whose work will be reviewed in the next section, have applied the instability theory to realistic current and bottom profiles and studied the effects of bottom friction. Much has been learned about the instability properties of longshore currents as well as the nature of the fully developed fluctuations. However, most studies analyzing finite amplitude behavior have been carried out using analytic bottom and longshore current profiles and show that the fully developed fluctuations can behave in a variety of ways ranging from equilibrated, smallamplitude fluctuations to energetic, random fluctuations involving strong vortices, depending on the values of the frictional coefficients [e.g., $1inn et al., 1998]. In this stu'dy we seek to find out which type of behavior described by $1inn et al. [1998] is most likely to explain the observations at SUPERDUCK. Furthermore, it is still unknown if fluctuations in the current velocities re- sulting from fully developed shear instabilities alone can account for the energy in the shear wave band observed during SUPERDUCK. In this study we attempt to answer these question by modeling the time dependent current climate in the surf zone for realistic situations utilizing the nonlinear shallow water equations with additional terms to account for forcing and damping mechanisms. The effects of the incident incident wave field are included through a radiation stress forcing term. Bottom friction and various lateral mixing mechanisms are also taken into account through a linear damping and a momentum diffusion term, respectively. The free parameters in the problem, a friction coefficient and a lateral mixing coefficient, are sources of uncertainty and control the nature of the resulting motions. We simulate 3 days from the SUPERDUCK field experiment for a range of realistic friction and lateral mixing coefficients. We seek to find out if a good representation of the data can be obtained for realistic values of the free parameters and to analyze the characteristics of the resulting motions. It had previously been established that shear instabilities cause lateral momentum mixing. In this paper we also seek ment and the field conditions present during the days chosen for the simulations is given in section 3. The model formulation emphasizing the submodels related to the incident wave forcing, lateral mixing, and bottom friction effects is discussed in section 4. We present simulations involving measured bottom bathymetry and wave heights from the SUPERDUCK field study in section 5 and discuss some flow properties related to the vortex interactions observed in the simulations in sec- tion 6. The findings are summarized and conclusions are stated in section Review of Previous Work 2.1. Linear Instability of Longshore Currents Nearshore currents and changes in the mean water level are generated when wind and swell waves break and decay in the surf zone [Longuet-Higgins and Stewart, 1964]. In the absence of longshore variations in the bathymetry, a stationary, long-crested wave field approaching the shore at an oblique angle generates a longshore directed current and a setup of the mean water level. Since the longshore current is only weakly dependent on depth, it can be approximated by a twodimensional flow. Two-dimensional flows in fluid me- chanics have often been observed to be unstable [see Drazin and Reid, 1982], and the stability characteristics of the surf zone longshore current were considered when Oltman-Shay et al. [1989] observed a meandering of the longshore current during the SUPERDUCK field experiment. Observations showed that the undulations propagated in the direction of the longshore current and did not satisfy gravity wave dynamics. Therefore alternate mechanisms were sought to explain the observations. Bowen and Holman [1989] performed an analytic study of an idealized longshore current profile over constant depth using the "rigid lid" assumption, which was shown to be appropriate for longshore currents with small Froude numbers [Falquds and Iranzo, 1994]. Utilizing an instability analysis, Bowen and Holman [1989] identified a mechanism well studied in larger-scale physical oceanography, but new to the nearshore, namely, a shear instability of the mean longshore current. The restoring mechanism for the resulting alongshore propagating motions, termed shear waves, is potential vorticity, where the background vorticity is supplied by the shear structure of the mean longshore current in analogy to the effect of the Earth's rotation in larger-scale applications. Since the pioneering work by Bowen and Holman [1989] and Oltman-Shay et al. [1989], several subsequent investigators applied the linear instability theory to more realistic current and bottom profiles utilizing analytic [Dodd and Thornton, 1990] or numerical to assess the importance of this lateral mixing in comparison to the contribution by more traditional mixing mechanisms. Preliminary results of this study are given by ( zkan and Kirby [1996] and ( zkan-haller and Kirby [1997b, 1998]. [Putrevu and Svendsen, 1992] means. It was observed We begin by reviewing previous work on the subject that the instability is stronger on barred beaches. of shear instabilities of the longshore current in sec- Doddet al. [1992] lifted a limitation of the original tion 2. A brief overview of the SUPERDUCK experi- shear instability theory by introducing bottom friction.

3 ZKAN TM A,, 'R ^,Tr wm ny: INSTABILITIES OF LONGSHORE CURRENTS They simulated the linear shear wave climate for several days of the SUPERDUCK experiment. Their approach was to analyze the linear instability characteristics of the measured longshore current profile. They found good agreement with the observed range of frequencies and propagation speeds of shear waves. Since good agreement could be obtained using a linear analysis, results from this study seemed to suggest that the observed fluctuations were weakly nonlinear, equilibrated shear instabilities, but no conclusive information about the final amplitudes of the shear instabilities could be deduced owing to the linearity assumption. Church et al. [1993] attempted to circumventhis limitation by performing a linear instability study simulating the DELILAH experiment and computing the spatial variations of the velocity components. The amplitudes of the velocity components were then calibrated by scaling the computed energy density of the oscillations to reproduce the observed energy density. They found that velocity fluctuations with the inferred amplitudes can cause significant lateral mixing in the surf zone Nonlinear Instability of Longshore Currents In order to study the disturbances as they reach fione carried out by Dodd et al. [1992](using longshore nite amplitude, a nonlinear analysis needed to be emcurrent measurements in the field) and Reniers et al. ployed. Analytical studies utilizing weakly nonlinear [1997](using longshore current measurements in the theories were carried out by Dodd and Thornton [1993] laboratory) correspond to analyzing this latter profile. and Feddersen [1998], who examined weakly unstable Therefore this observation provides an explanation for longshore currents. the fact that results of linear instability analyses of Falqu s et al. [1995] modeled the nonlinear shallow measured currents can produce good agreement for the water equations utilizing a rigid lid assumption and perrange of unstable wavenumbers and propagation speeds formed numerical experiments for a plane beach geom- [see Dodd et al., 1992; Reniers et al., 1997]; however, the etry incorporating bottom friction and lateral mixing. They found that the instabilities equilibrate with constant or modulated amplitudes and observed that the period of the disturbances increased with increasing amplitude. Soon thereafter, Allen et al. [1996] carried out a detailed numerical study of the effects of varying the bottom friction coei icient as well as the longshore width of the modeling domain. They found that high values of the friction factor corresponding to a weakly unstable longshore current result in finite amplitude disturbances with constant amplitudes. As the friction factor is decreased, the disturbances display modulated amplitudes, period-doubling bifurcations, and, eventually, chaotic behavior. When the width of the modeling domain is increased, the behavior is dominated by the transition of the motions to larger-scale, nonlinear, propagating disturbances. Allen et al. [19 96], furthermore, found that fully developed shear instabilities alter the mean longshore current profile significantly. In the presence of the finite amplitude disturbances the mean longshore current displays marginal stability even though the initial mean current was strongly unstable. This is an important finding since measured mean current profiles, in reality, correspond to the final mean current in the presence of the fluctuations and could display much different stability characteristics than the often unknown, but relevant, fluctuation-free initial state. Most recently, $1inn et al. [1998] examined the non- linear instability of the longshore current over a barred topography including bottom friction as the only damping mechanism. They found equilibrated shear waves for high values of bottom friction and irregular fluctuations for lower values of the frictional coei icient and showed that these instabilities cause substantial lateral mixing of momentum in the surf zone and alter the initial current profile significantly. During their analysis, $1inn et al. [1998] carried out linear instability analyses of the initial fluctuation-free current profile as well as the final mean longshore current profile in the presence of the fluctuations. It should be noted that the latter current profile would be measured in the field. The results for the two current profiles show that the range of unstable wavenumbers as well as the most unstable wavenumber agree approxi- mately for cases involving irregular fluctuations. However, the growth rates in the former current profile are an order of magnitude larger than in the latter. Stability analyses of measured current profiles such as the growth rates may be underpredicted. In general, a measured longshore current profile is likely to include the effects of mixing due to shear instabilities. Therefore it is likely that the measured current profile displays a weaker instability than the relevant fluctuation-free initial current. Analysis of this unknown initial state could produce the observed growth rates and finite amplitude behavior. $linn et al. [1998] also state that good reproduction of the propagation speeds by linear theory therefore does not necessarily imply a weakly nonlinear flow regime. 3. SUPERDUCK Field Experiment 3.1. General Description Field experiments were conducted at Duck, North Carolina, in the fall of The field site was characterized by a steep foreshore slope (typically l:10), a shore parallel bar formation farther offshore, and a gentler beach beyond [Crowson eta!., 1988]. The experiments incorporated an alongshore array of Marsh- McBirney bidirectional current meters located in the

4 25,956 (DZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS lg5 lg ", 5 E 220 z o, Cross-shore, m Figure 1. (a) Perspective and (b) plan view of nearshore bathymetry at SUPERDUCK on October 16 [from Doddet al., 1992]. The dots show the positions of the alongshore array of seven current meters. trough shoreward of the bar formation. The array state that, based on both observation and dynamical was designed to observe primarily low mode progres- modeling, longshore variations in the bar regime were sive edge waves JOltman-Shay et al., 1989]. Current not significant enough to induce rip currents. Therefore measurements were obtained with a sampling frequency of 2 Hz for 4 hours centered about low or high tide. The incident wave climate was monitored using bottommounted pressure gages at 8 m water depth. Bathymetry data were also collected over the region where the surf zone instruments were deployed. We concentrate on days during the SUPERDUCK experiment where the wave climate was stationary over a period of hours and could be described fairly well by a directional spectrum that entails a dominant peak. We excluded conditions where intersecting sea and swell are likely to have occurred, resulting in a bimodal spectrum. Doddet al. [1992] carry out linear instability calculations for the chosen days using measured SUPERDUCK bathymetry from one transect assuming that the bottom contours are straight and parallel. We also adopt this assumption in our nonlinear modeling effort. Current and wave measurements were also obtained using wave gages and three Marsh-McBirney current meters mounted on a sled. The sled was initially deployed offshore of the breaker line in a geographic area with the highest degree of straight-and-parallel contours [Whitford and Thornton, 1996]. The sled was then gradually pulled onshore, collecting 34 rain time series of Of interest to this study are 3 days during the experi- current measurements at several cross-shore locations ment when a strong meandering of the longshore current was observed. The chosen days, October 15, 16, and 18, are characterized by locally generated incident waves at about 15 ø to the beach with a root-mean-square (rms) wave height of about 1 m measured at the 8 m array and a peak period of about 5 s. These waves generated a southward longshore current with a peak of 1 ms. There is evidence that a bimodal incident wave climate existed on October 17 [see Whitford, 1988]; therefore we exclude this day from our computations. The nearshore bathymetry in the region where the including the point of maximum breaking, on top of the nearshore bar, and in the nearshore trough. The cross-shore transects at which data were collected were located at longshore distances of 900 m on October 15, 1160 m on October 16, and 1170 m on October 18 according to the coordinate system used during the experiment (see Figure 1). Bathymetry measurements were also available at those transects. These depth measurements are used in our numerical simulations, resulting in a cross-shore distance of the current meter array from the shoreline of 45 m for October 15 and 35 m for Ocinstruments were deployed is depicted in Figure 1 for tober 16 and 18. October 16 and is typical of the analyzed days. The coordinate system in Figure 1 is that used in the experiment. The dots show the positions of the alongshore array of seven current meters. Doddet al. [1992] Whitford and Thornton [1996] used the current measurements from the sled along with wind and wave measurements to determine an appropriate friction coefficient by examining the longshore momentum balance

5 OZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25, ,, f k f k , 0.03 (b), O. 025,- I I I i J,l I tj II , o o (lm) (lm) Figure 2. Frequency-cyclic-wavenumber spectra S(f,k) (m3s) for longshore velocity from measurements on October 15. The equation for the best fit dispersion line (dashed line) is computed using equations (a) (1) and (b) (2) and is noted above each plot. Contour levels plotted are (10, 30, 60, 100, 200, 400, 800). A value of 0.8 ms is used for the estimated shear wave propagation speed Cesto constructhe upper and lower cutoff lines (dash-dotted lines) of the shear wave energy. The zero-mode edge wave dispersion lines for a plane beach slope of 0.05 are also shown (thick solid lines). for the period of October 15 through October 18. They obtained values for the friction coefficient cf in the range < cf < It should be noted that the current measurements obtained from the sled were acquired sequentially. Therefore, as Doddet al. [1992] also point out, the values are representative of a distinct space and time location. Oltman-Shay et al. [1989] and later Doddet al. [1992] used measurements from sensors within the surf zone ar- ray to construct frequency-wavenumber spectra of longshore and cross-shore currents using a high-resolution iterative maximum likelihood estimator (IMLE) [Pawka, 1983]. A contour plot of the frequency-longshore- wavenumber spectra S(f, K) for the longshore velocities on October 15 is reproduced in Figure 2a. For more information about the application of the IMLE method to the SUPERDUCK data set and the accuracy of the resulting spectra, the reader is referred to Doddet al. [1992]. Note that, following Doddet al. [1992], the cyclic longshore wavenumber K is defined as (ll), where L is the longshore wavelength of the motions. The angular wavenumber k is defined as 2vrK = 2vrL. The dispersion lines for a zero-mode edge wave for an effective plane beach slope of 0.05 are also shown in Figure 2a. Shear waves are readily distinguished from edge waves since they lie well outside the region bordered by the zero-mode edge wave dispersion curves. On all 3 days, we observe that the range of frequencies f < Hz is dominated by shear wave energy. In contrast to edge waves, shear waves exhibit a nearly linear dispersion curve. Taking advantage of the nearly nondispersive character of the observed motions, we seek to identify a representative propagation speed of the shear waves Estimation of Shear Wave Phase Speed For the purpose of identifying the dispersion line, the energy in the shear wave band needs to be identified. Howd ctal. [1991] used an energy-partitioning method between edge waves and shear waves by assuming that all infi'agravity energy lies in the region bordered by the dispersion curves for the zero-mode edge waves traveling in the positive and negative directions. Energy outside of this region is then attributed to shear waves. Here a simpler approach is taken. First, an estimate of the shear wave propagation speed Cest is made by inspection. An upper cutoff line with a slope of 2Cest intercepting the origin is defined so that, on this line, fup -- 2CestK. The lower cutoff line is defined with slope Cest intercepting the wavenumber axis at 0.01 (m-1) so that, on this line, flow - Cest( -0.01). These lines are also depicted in the frequency-wavenumber plot shown in Figure 2 for an estimated speed Cest of 0.8 ms. All energy lying between the defined cutoff lines is attributed to shear waves. This method assures that all gravity wave energy is excluded from the estimates of shear wave motion (but not the opposite). In order to obtain an improved estimate for the propagation speed of the shear waves, a frequency f is defined for every wavenumber as the quotient of the first and zeroth moments of the spectrum in frequency such that _ fffupfs(f,k)df f(k) -,ow, (1) f"p$(f,k')df low where the lower and upper limits of integration are cho-

6 25,958 (SZKAN-HALLER AND KIRBY' INSTABILITIES OF LONGSHORE CURRENTS sen as the upper and lower cutoff lines defined in the preceding paragraph. If it is assumed that the motions are nondispersive, a straight line can then be fitted in the least squares sense through the points (f,k) by performing a weighted, first-order polynomial fit. The weight of each data point (f, K) is determined by the total energy in that wavenumber bin Sf(K) given by the integration of the shear wave energy over frequency for each wavenumber. The slope and intercept values of the obtained linear fit then define the propagation speed. The best fit dispersion line computed in this manner for October 15 is shown in Figure 2a, where the equation for the dispersion line is given at the top. An alternate approach to estimate the propagation speed is to define a wavenumberi' for each frequency f as the quotient of the first and zeroth moment of the spectrum in wavenumber. upk$(f, K)dK K(f) - low, (2) low where the lower and upper limits of integration are again chosen such that only shear wave energy is included. A linear fit through the points (f, K) weighted by the energy Sx(f) in each frequency bin then results in an estimate of the dispersion line. This dispersion line for October 15 is shown in Figure 2b. It is seen that this method results in the prediction of higher propagation speeds by about 10%. The intercept of the dispersion lines with the wavenumber axis also occurs at higher values. The first method will be used in the remainder of the paper. The method outlined here is also used in order to estimate the propagation speeds of oscillations arising in the computations. The sensitivity of the results to the chosen value of test is less pronounced in comparison to the 10% variation due to the chosen weighting method. This 10% variation in the prediction of the speeds should be kept in mind when interpreting the results. 4. Model Formulation Nearshore circulation can be modeled using the mass and momentum conservation equations that have been integrated over the incident wave timescale and depth. Effects of processes at the incident wave timescale enter the equations of motion through radiation stress gradient terms. This concept was introduced by Longnet- Higgins and Stewart [1964] and was applied to the prediction of steady, depth-uniform longshore currents and wave setup by several investigators including Bowen [1969], Longuet-Higgins [1970a,b], Thornton and Guza [1986] and Larson and Krauss [1991]. These equations can also be applied to a two-dimensional-horizontal (2- DH) domain to study steady or time dependent, depthuniform, nearshore circulation. Examples of such studies are Noda [1974], Birkemeier and Dalrymple [1975], Keely and Bowen [1977], Ebersole and Dalrymple [1979], and Wind and Vreugdenhill [1986] Governing Equations The governing equations are in the form of the shallow water equations with additional terms to account for forcing and damping effects [ud] + [vd] - 0 (3a) Ou Ou Ou Or, 0- - U xx + V yy - -g xx + x + rx - rbx (3b) Ov Ov Ov Or, Ot h- U x x h- V yy ----g yy h- 5y h- -y-- -by. (3C) Here r] is the incident-wave-averaged water surface elevation above the still water level, h is the water depth with respect to the still water level, d = (h + r) is the total water depth, and u and v are the depth-averaged current velocities in the x and y directions, respectively, where x points offshore and y points in the longshore direction Wave forcing. The parameters 5x and 5y represent incident wave forcing effects and are expressed using the radiation stress formulation by Longuet-Higgins and Stewart [1964]. For straight-and-parallel con- tours these terms reduce to x- pd I (OSxx) Ox ' Y = pd I (OSxy) Ox ' (4) We employ the assumptions of longshore uniform bathymetry and linear water wave theory for the description of the incident wave climate throughout the study. We utilize measured bathymetry from the sled transect as well as measured incident wave properties at the offshore boundary. The values of the root-mean-square wave height Hrms, the peak frequency fp, and the mean angle of incidence 0 at the 8 m array for October 15, 16, and 18 are given in Table 1. Using these values and neglecting wave-current interactions, we compute the incident wave transformation for the entire domain prior to any computations of the current field. The computed wave height variation for October 15 along with the bottom bathymetry are shown in Figure 3. Also shown are wave height estimates from sled measurements. The wave height variation is characterized by strong, isolated wave-breaking regions over the bar crest and very close to the shoreline and is typical of the analyzed days. Details of the computations leading to the wave height as a function of cross-shore distance and the dependence of the radiation stresses Sxx and Sxy on the wave height variation are given in Appendix A Momentum mixing. The parameters rx and r represent the effects of lateral mixing due to two

7 OZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25,959 Table 1. Offshore Wave Conditions for SUPERDUCK at 8 m Water Depth of Root-Mean Square Wave Height Hrms, Peak Frequency fp and Mean Angle of Incidence 0 Date in Time, Hrms fp 0 October LST m Hz deg mechanisms: turbulence and depth variation in the current velocities [Svendsen and Putrevu, 1994]. Only the most dominant mixing terms are included in the computations to reduce the necessary computational time. The arguments that reduce the mixing effects to the terms shown below are presented in Appendix B. It is noted that we have carried out simulations including all dispersive mixing terms utilizing the formulations given in (B3) and (B4) and found that the results with the dominant terms listed show minor differences when compared to the results obtained using the full formulation. 20 (0u) 10 (0v) (5a) r; = d0z 10 ( dv, (5b) where the viscosity v is parameterized as - (6) The energy dissipation due to wave breaking b is modeled using the parameterization by WhitfoM [1988] stated in (A3). A realistic range for the mixing coefficient M for SUPERDUCK is determined to be 0 < M < 0.5 (see Appendix B). A typical cross-shore variation of the viscosity v is shown in Figure 4 for M=0.25 on October Bottom friction. The parameters rb and rbv in (3b) and (3c), respectively, are modeled using linear damping terms where Tbx -- t Tby- V, (7) 2 - -cuo, (8) and u0 is the amplitude of the horizontal orbital velocity of the incident waves and is given by (A6). Discussion of the adequacy of this formulation is presented in Appendix C. A typical cross-shore variation of is shown in Figure 4 for cy= on October Model Domain and Solution Method The model domain extends from the still water shore- line to a certain distance offshore. A moving shoreline boundary is incorporated at the onshore boundary of the modeling domain. An absorbing boundary condition is incorporated at the offshore boundary to allow transient gravity motions to leave the domain of interest. A periodicity condition is imposed in the longshore direction Shoreline boundary. One of the first simulations of finite amplitude shear instabilities was carried out by Allen et al. [1996], who utilized the rigid lid assumption and modeled the instability of an analytic longshore current profile on a sloping beach. We repeated simulations for the same case including the effects of the free surface and found that errors from ne- glecting the wave induced setup as well as the shoreline run-up due to surface fluctuations associated with the instabilities are minor for this case [see(szkan-haller, 1997]. In addition, the resulting water surface fluctua- tions are found to be small (0(1 cm)). These findings suggest that the effects of the steady setup as well as the effect of the shoreline run-up due to the water surface elevation associated with the instabilities can be neglected in this case. Therefore, for the simulations shown here, the steady setup is neglected and a wall boundary is incorporated at the shoreline to decrease the necessary computational time. However, the time dependent movement of the free surface is included i O0 2OO x (m) Figure 3. Computed wave height decay along with wave height measurements from the sled (open circles) and measured bathymetry for October 15.

8 25,960 ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS, ' ' I I I I I M=0.25 I 2 X 10-3 I I I I I i, I cf = Figure 4. Computed energy dissipation due wave breaking, measured bathymetry, computed frictional term and viscosity for October 16. M is the mixing coefficient and cf is the friction coefficient Offshore boundary. The offshore boundary condition has been shown to perform favorably for gravity wave motions exiting the domain of interest [see.o Ozkan-Haller and Kirby, 1997a]. However, it should be noted that the offshore boundary is not transparent to vortices that are advected from the interior. In general, if the vortices are somewhat weakened by bottom friction when they reach the boundary, they continue to dissipate at the location where they encountered the offshore boundary. However, if the vortices are strong when they reach the offshore boundary, they are advected along the boundary but remain close to it until they dissipate. Care has to be taken to place the offshore boundary far away from the surf zone so that offshore advection velocities are small when the vor- tices reach the of[shore boundary. Therefore the offshore boundary is placed approximately six surf zone widths away from the shoreline at 550 m Lateral boundaries. Since periodicity is imposed in the longshore direction, the longshore width of the modeling domain needs to be long enough for a sufficient number of waves to be present in the domain so that reliable spectral estimates can be obtained. In the present effort the width of the domain is chosen to be a multiple of the expected length scale of the unstable motions. This length scale is determined as follows. Dodd et al. [1992] previously applied the linear instability theory to October of the SUPERDUCK experiment. They generated longshore current profiles by using a mixing coefficient M of unity, and by calibrating the friction coefficient so that the generated mean longshore current profile, in the absence of any instabilities, resembled the measured profile closely. They subsequently analyzed the linear stability characteristics of these profiles and obtained good agreement between the predicted unstable wavenumbers and the observed range of wavenumbers and stated that the most unstable wavenumber associated with M = 1 for each day defines a representative length scale for the observed motions. Therefore, in our modeling effort, the domain for each day is chosen to be 16 times the predicted most unstable wavelength so that a sufficient number of waves can exist in the modeling domain. The most unstable angular wavenumbers kmax for the 3 days of the SU- PERDUCK experiment are given in Table 2. The domain width for each day is defined by Ly = 16 x Lmax. where Lmax: (271'kmax). Table 2. Most Unstable Wavenum- bers kmax and Corresponding Longshore Length Scale Lmax -- 2zrkm x Date in Time kmax Lmax October LST radm m The domain length Ly x Lmax.

9 0ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25,961 The longshore domain lengths defined above for each day are used for all the simulations for that day, even though the values of the friction factor and mixing coefficient are varied from one simulation to the next. We remark that Dodd [1994] found that varying the friction factor does not significantly change the value of the wavenumber associated with the maximum growth rate. However, decreasing the mixing coefficient will result in a mean current profile with stronger cross-shore gradients; therefore the maximum growth rate will occur at a higher wavenumber. The scaling argument presented in Appendix B suggests that a mixing coefficient of unity is unrealistically high; therefore the resulting most unstable wavenumber is low and the length scale associated with it is large, resulting in a conservative estimate of the required domain length Initial conditions. Given an initial condi- tion in the water surface elevation and the velocities u and v, the governing equations are integrated in time. The calculations are initiated with the fluid at rest ex- cept for small perturbations in the longshore velocities given by 'v(x, y, t - 0) - max{f} f' (9) where e = 1 x 10-4 and the function f is given by ND (27rjy+27rc)j) ' (10) 3=1 f -- y cos Ly Here, ND is the number of times the most unstable wavelength fits into the modeling domain and is equal to 16 and ½j represents a random phase function between-1 and 1. This initial condition ensures that the resulting longshore current is perturbed at the most unstable wavelength as well as all longer wavelengths that can exist in the modeling domain. Nonlinear interaction mechanisms present in the model equations will further cause the generation of any superharmonics. The initial cross-shore velocities, surface elevations, and the radiation stress forcing terms are not perturbed Solution method. The time integration of the governing equations is carried out using an explicit third-order Adams-Bashforth scheme. Fourier and Chebyshev collocation schemes are used to compute spatial derivatives in the longshore and cross-shore directions, respectively. For more information about the solution method, the reader is referred to ( zkan- Hallet and Kirby [1997a], where the numerical approach and application of the model to several shoreline runup problems as well as to the generation of subharmonic edge waves and their growth to finite amplitude are documented. The resulting equation is given by ot + < > - <e >- < + <%>. (i In the above equation the longshore average of a variable Q(x, y, t) is defined by 1 0 Ly (Q)(z, t) - Q(z,, t)d. (12) The terms in the 1 momentum equation involving! derivatives drop out of the mean balance. The effect of slow time variations in the mean longshore current (v) can be excluded by further time averaging (11) over timescales much longer than the varia- tions in (v). The time average of a variable Q(x, y, t) is defined as -- Q( ' 1 t t )- (t- t Q( ' t) t' ( s) Performing the time average of (11), the balance (u Ov v>-( >- ( > (14) >+( can be identified. Note that periodicity and longshore uniformity of the bathymetry imply that (u>: 0. Utilizing the time and longshore-averaged velocities and the fluctuating components, we can define the current velocities - { > + ', - { > + ' ß (i ) The effect of the first term in (14) can then be identified as lateral momentum mixing due to the fluctuating components since > - (, - >. (16) The resulting mean balance given by (14) states that the incident wave forcing { ) is balanced by bottom friction {l vd) and lateral mixing caused by the shear instabilities well as lateral mixing caused by turbulence and depth variations in the current velocities. The mean kinetic energy density (or kinetic energy per unit depth) of the fluctuating velocity components can be defined as E' (z) - + ). (17) It provides a good measure of the cross-shore distribution and extent of the fluctuations due to the shear instabilities Momentum Balance In the presence of the fluctuating motions, the mean momentum balance in the longshore direction that leads to the generation of a longshore current can be obtained by longshore averaging the y momentum equation (3c). 5. Simulation of SUPERDUCK In this section we document simulations involving the SUPERDUCK field experiment. Simulations are carried out by varying the friction coefficient with incre-

10 . _ 25,962 ZKAN-HALLER AND KIRBY' INSTABILITIES OF LONGSHORE CURRENTS 2 (a) cf ,,M:0.2,5,,,,,, o (b) cf ,, o (c) c.f , M= ß I (d) Data i i i I I I I t (hrs) Figure 5. Time series of velocities u, v (solid lines) and (v) (dashed line) on October 16 at (x, y)-(35 m, Ly2) for M=0.25 and (a) 0.003, (b) , and (c) 0.004, and (d) time series of velocities u and v of data. ments of 5 x 10-4 in the range < cf < while keeping the mixing coefficient M fixed at Subsequently, simulations are carried out by varying the mixing coefficient M within the range 0 < M < 0.5 for a fixed friction coefficient Effects of Bottom Friction Results from simulations for October 16 are presented for friction coefficients cf of 0.003, , and Time series in the bar trough region for October 16 are shown in Figure 5. The velocity time series from the current meter located closest to the cross-shore tran- sect of the sled has been low-pass filtered at 0.01 Hz and is also shown in Figure 5. The computed time series show that the instabilities reach finite amplitude within about 30 min. For cy = the instabilities reach finite amplitude slightly sooner than for the other cases. The mean (longshore averaged) current generation and the spin-up of the instabilities occur simultaneously in each case. The mean current displays some time variability. For all three cases, short-timescale oscillations are observed during the first 30 rain after in- stabilities are initiated. Later oscillations with longer timescales are observed. The longshore current oscillations are very energetic and are negative at times. This behavior is not observed in the measured time series where longshore velocities are always larger than 0.25 ms. Also, the computed time series display an intermittent character, whereas the field data do not. The cross-shore velocities display periods of small energy fluctuations (indicating primarily longshore flow) followed by periods of high-amplitude fluctuations. Often near-zero longshore velocities occur at the same time as high cross-shore velocities (up to 0.3 ms), indicating purely offshore directed flow. Since model computations are carried out with high resolution in both space and time, a direct Fourier transform in space and time is used to obtain the frequency-longshore-wavenumber spectra of the computed time series. The last 2.7 hours of the simulations are used. The spectra are calculated using a cosine taper on the first and last 10% of the time series. The resulting spectra are smoothed by band averaging over

11 ß., ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25, f - lk - 0.O0039 cf=o.003. M=0: i.. ß. ß I I'. 5 : f k cf= ' i -- -M= ! ß! ß '... I:" ''7'" ß ' I ß I '' ' 'W0'015 ' ' ß... '... i....'... '... "'--, ß : ß - K0(1m) - K0(1m) f = 0.78k f = 0.83k cf = ' i Data......i.. ß ß. ß.. ß. ß 'W0'015 ' :,, ß ' I' ' ' 7 '1 "'-, : ' " K (Im) K (Im) Figure 6. Frequency-cyclic-wavenumber spectra $(f,k) (mas) for computed and measured longshore velocity at x =35 m on October 16. The values for Cest are 1 ms for c = 0.003, 0.8 ms for c = and 0.004, and 0.8 ms for data, and are used to constructhe upper and lower cutoff lines (dash-dotted line) of the shear wave energy. Contour levels plotted are (10, 30, 60, 100, 200, 400, 800). The equation for the best fit dispersion line (dashed line) is noted above each plot. eight frequencies and two wavenumbers. The spectra obtained using time series of longshore velocities are shown in Figure 6. The frequency-wavenumber spectra of the measured velocities obtained utilizing the IMLE method are also included for comparison. As the friction factor is decreased, the range of frequencies affected by the instabilities increases. The propagation speed of the motions also increases. The propagation speed seen in the data is best reproduced with c=0.0035, resulting in an error of less than 5%. The time- and longshore-averaged mean longshore current velocities for the three cases are shown in Figure 7a. The longshore averages are performed over the entire width of the domain, while the time averages are obtained using the last 2.7 hours of the computed time series. The three current profiles display similar gradients, but, as expected, lower friction results in a stronger current. The cross-shore distributions of the kinetic energy density associated with these cases (see Figure 7b) display similar shapes with two local maxima, shoreward and seaward of the location of the mean current maximum. Although the mean current ( ) displays a local maximum close to the shoreline, the perturbation energy in the surf zone decreases monotonically toward the shoreline. The differences in kinetic energy density for the three cases are least pronounced in the bar trough region and most pronounced'around the longshore current peak and farther offshore.

12 25,964 OZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS.5 I I [ I I [ I I [ [ i'o5 _ 1 I I I I I O , I I I I I rigur 7. rim- ]ongho (b) energy density (12){u'2 + v '2} for M: 0.25 and c = (solid lines), c: (dashed lines), and cl: (dash-dotted lines) on October 16. Mean velocities from sled data (open circles) are also shown in Figure 7a. The spatial structure of the flow can best be observed utilizing snapshots of the vorticity field defined by c v c u q - Ox Oy' (18) Such plots for c = and cf: are shown in Plate 1. For both cases, four regions of alternating positive (red) and negative (blue) vorticity exist. These regions can be linked to the two local maxima of the mean longshore current profile. Since the current peak located near the shoreline is very narrow, the first two layers of positive and negative vorticity are closely confined to the shoreline. In contrast, the outer two layers of vorticity display complicated behavior. Features with positive and negative potential vorticity are seen to pair up to form vortex pairs. Some vortex pairs have been shed offshore earlier and are now seen about 400 m offshore. An important difference between the two cases is that the vortices are more energetic for c: This is especially evident offshore since the vortices associated with cf are seen to be weaker there. This observation is also consistent with the earlier state- ment that the differences in the kinetic energy density for the two cases are especially pronounced offshore of the current peak. Plate 2 shows a sequence of snapshots of the vorticity field at 5 rain time intervals for the simulation with cy = Regions of concentrated positive and neg- ative vorticity are irregularly spaced in the longshore direction and display the character of vortex pairs. They propagate in the longshore direction at a rate of about 250 m in 5 min (0.85 ms). The vortex pairs in the surf zone tend to occur in clusters. There are areas of primarily longshore directed flow (see t, y 800 m or t + 5 rain, y m 200 m) that subsequently start undulating, possibly because they develop renewed instabilities. Vortices are often ejected out of the surf zone and advect several surf zone widths toward the offshore, where they tend to remain and weaken. During the shedding process, locally strong offshore directed currents are generated. The offshore vortices appear in the form of pairs (see t+ 5 min, (x,y) m (400, 500)m)or sin- gle positive vortices surrounded by a region of negative vorticity (see t + 10 min, (x, y) m (250, 1750) m). The results documented in this section show that the propagation speed of the instabilities, the strength of the mean current, as well as the energy level of the in- stabilities increase with decreasing frictional damping. We only show simulations for a narrow range of friction coefficients and find the value that reproduces the propagation speed seen in the data. The results for October 16 are representative of the characteristics on all 3 days. The resulting behavior for all simulated days falls in the category of flow termed "turbulent shear flow" by $linn et al. [1998], which is the most energetic type of flow they observe.

13 )ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25,965 25O0 c,=0.003 c,= OO looo -O.Ol - 5OO x (m) x (m) Plate 1. Contour plots of vorticity q (s-1) at t - 5 hours for c I = and and hi = 0.25 on October Effects of Lateral Mixing Next, simulations are carried out by fixing the friction coefficient at the value that best reproduces the propagation speed seen in data and varying the mixing coefficient M. The identified value for the friction coefficient c 7 is for October 15, for October 16, and for October 18. These values are within the limits that were identified as reasonable for SUPERDUCK. We now document simulations for October 18 for c7 = and M = 0, 0.25, and 0.5. Time series of the computed velocities as well as low-pass filtered data are shown in Figure 8. The time series clearly show that the spin-up time is longer for larger M, reflecting the dissipational nature of the mixing terms. The time series show fewer high-frequency oscillations as M is increased. It is especially evident in the crossshore velocities that the instabilities are less energetic for higher M. The generated mean longshore current has similar magnitude in all three cases. However, the initial increase in the mean longshore current (v for M = 0.5 at this cross-shore location is due largely to the eddy viscosity mixing effects since the longshore current is generated prior to the onset of the instabilities. Oscillations at the timescale of 300 s are evident in the time series for M = 0 and M = Underlying longer oscillation can also be observed. Only oscillations with longer timescales are seen for M = 0.5, suggesting that the high-frequency oscillations are damped out by the eddy viscosity mixing effects. We observe that near-zero longshore velocities occur for low values of M. Simultaneously, high cross-shore velocities are observed, suggesting that the flow in the bar trough region becomes offshore directed. Frequency-wavenumber spectra of the longshore velocities are shown for three cases (M=0, 0.25, 0.5) in Figure 9. The spectra of the longshore velocities for M = 0 and 0.25 do not show significant differences. The spectrum for M = 0.5 displays noticeably less en-

14 25,966 ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25OO t t+5 min t+ 10 min t+ 15 min o.o2 15oo oo ' ß ß ß ß Plate 2. Snapshots of contour plots of vorticity q (s- ) at 5 min intervals for cf and M=0.25 on October 16. ergy. It is observed that the propagation speeds vary by less than 10% for different values of M. The size of this variation is equivalent to the uncertainty in the estimates of the propagation speed from the spectra; therefore the propagation speed is not strongly dependent on the value of the mixing coefficient. Quantitative comparisons to data are made by constructing a frequency spectrum that only contains energy due to shear waves. For this purpose, the energy in the region bounded by the upper and lower cutoff lines shown on the two-dimensional spectra plots in Figure 9 is summed for each frequency bin. Comparison of the computed and measured frequency spectra (Figure 10) shows that the amount of energy in the range < f < Hz is predicted reasonably well for the longshore velocities with M: 0 and Energy at frequencies below this range is overpredicted whereas energy above this range is underpredicted. Turning to the spectra of the cross-shore velocities, we note that the case involving M = 0 produces results closest to the observations in the range < f < Hz. The cases involving higher values for M show underprediction. The time- and longshore-averaged mean longshore current is shown in Figure 11a. For all three cases a peak is predicted about 100 m offshore. The strength of the peak as well as the offshore shear of the current profiles for different M values are remarkably similar, although the instability climates that generated them are different. Velocity measurements from the sled in the bar and trough regions are reproduced well. The mean longshore current associated with M - 0 exhibits some oscillations. The appearance of these oscillations is suspected to be linked to the unrealistically high gradients associated with the shoreline jet. This effect is especially pronounced for M = 0 since no eddy viscosity mixing mechanism exists to smooth the shoreline jet. Examining Figure 1 lb we note that higher perturbation kinetic energies are observed throughout the modeling domain for lower M. Snapshots of the vorticity are depicted in Plate 3. For M = 0 we see that a relatively short longshore length scale is evident in the nearshore region. The features are energetic and shed vortex pairs offshore. For M = 0.25 the layer of negative vorticity is notably weaker. In addition, the length scales of the motions are longer. For M = 0.5 the features have long length scales and are primarily confined to the surf zone. In this case, only occasional vortex shedding occurs. It is evident that the increase in the amount of eddy viscosity mixing causes the amount of high-wavenumber oscillations to decrease. The result is a decrease in the number of

15 OZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25,967 M = 0 M = 0.25 M = OO 2O0O O.Ol looo - 5oo O z (m) z (m) z (m) Plate 3. Contour plots of vorticity q (s-1) at t - 5 hours for cf and W=O, 0.25, and 0.5 on October 18.

16 ß. 25,968 ZKAN-HALLER AND KIRBY' INSTABILITIES OF LONGSHORE CURRENTS 2 (a) c=0.003, M-0 i i i I i i i i i o (b) c,f=o.003, M=0.25 o -0.5 (c) cf =0.003, M=0.5 2, (d) Data 0.5 o o -0.5 o t (hrs) Figure 8. Time series of velocities u, v and (v) (dashed line) on October 18 at (x, y)=(35 m, Ly2) for c=0.003 and M- (a) 0, (b) 0.25, and (c) 0.5, and (d) time series of velocities u and v of data. features in the modeling domain; consequently, the apparent longshore length scales of the motions increases. The motions are also weaker, presumably owing to the dissipational effect of the mixing. The results documented in this section are consis- tent for all days that were simulated. We find that increases in the mixing coefficient cause relatively small variations in the propagation speeds, but the resulting motions have longer longshore length scales. The total energy in the fluctuations decreases for higher mixing coefficient. However, differences in fluctuation energy are less pronounced in the nearshore region, especially in the bar trough. In spite of that, for October 18 (and also October 15) the spectra obtained from time series in the bar trough region show a decrease in the energy content for higher Jkat this cross-shore location. However, for October 16 it is difficult to reach such a conclusion with information in the bar trough region alone. Snapshots of vorticity (see Plate 4) clearly show the different flow patterns that occur for different ]F values, indicating that the resulting instability climates are a function of M. The perturbation kinetic energy density (Figure 12b) also shows that the motions are more energetic in most of the domain for lowerf; however, the values of E are very similar in the bar trough region (x 35 m). Consistent with this observation, we note that the frequency-løngshøre-wavenumber spectra obtained from time series in the bar trough region are very similar (see Figure 13). So although the shear instability climates are very different in most of the domain (especially around and offshore of the current maximum), these differences are not easily observed in the bar trough region where field data are available, making it difficult to discriminate between the cases and deduce the "correct" value of the mixing coefficient by comparing measurements to computations. Finally, we find that the longshore current profile is not altered by the value of M, although the instability climate is very different. For October 16 the current profiles in the presence of the instabilities for the three Jk values and the current profile that would result in the absence of instabilities for Jk=0.5 are shown in Fig-

17 )ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25,969 f k ,, 0.03 cf M=O 5 cf M=0.25 f k I,, I I ß I I 1,1 I I',, '. i '! - 0 K (lm) - o o.02 K (lm) cf:o.003 M=0.5 - f k f k , 0.03.' 5 Data, '. o, - ß ' ' - '. % ' I 0.01 ] o o - o Figure 9. Frequency-cyclic-wavenumber spectra S(f,k) (m3s) for computed and measured longshore velocity at x =35 m on October 16. The value for Cest is 0.75 ms and is used to constructhe upper and lower cutoff lines (dash-dotted line) of the shear wave energy. Contour levels plotted are (10, 30, 60, 100, 200, 400, 800). The equation for the best fit dispersion line (dashed line) is noted above each plot. ure 12a. The profiles in the presence of the instabilities are remarkably similar for all _3values and predict measurements well. The current profile that would result in the absence of the instabilities is markedly different, suggesting that the instabilities induce momentum mixing. This suggestion will be analyzed further in the next section. The importance of the fully developed instabilities as a mixing mechanism in the surf zone in comparison to other more traditional mixing mechanisms will also be assessed. As a final comment, the reader is reminded that in this study we calibrate the friction coefficient by matching the measured and predicted propagation speeds of the instabilities. Agreement with sled data is not used in the calibration process since caution should be used in interpreting sled data to model agreement because the sled measurements were carried out sequentially and the mean currents computed using sled data and model data are based on very different averaging periods. However, the correct prediction of the magnitude and location of the current maximum in comparison to sled data constitutes a "blindfold" test of the chosen friction coefficient Momentum Mixing due to Instabilities It is instructive to analyze the mean momentum balance that leads to the generation of the mean longshore

18 , 25,970 ZKAN-HALLER AND KIRBY' INSTABILITIES OF LONGSHORE CURRENTS 102 c,, ,, 101 (b) ' o 10 o ' o 10 o r-, f (Hz) f (Hz) Figure 10. Frequency spectra of (a) longshore and (b) cross-shore velocities for data (thick solid lines) on October 18, c i and M - 0 (thin solid lines), M (dashed lines), M (dash-dotted lines) current profiles in order to asses the roles of momentum mixing due to the instabilities as well as the eddy viscosity term. We examine the mean momentum balance in the longshore direction for the three cases involving M=0, 0.25, and 0.5 for October 18 utilizing Figure 14. The incident wave forcing term is the same for all three cases. The contribution of the bottom friction is very similar for all three cases, confirming that the resulting mean longshore current profile for all the cases should also be similar. Differences in the mixing caused by the instabilities are pronounced, especially offshore of the bar trough. The eddy viscosity mixing term ( - ) makes no contribution for M = 0. Its contribution increases as M increases. The term is especially active around the shoreline jet, where the shear instabilities do not induce mixing, and around the longshore current peak. The mixing caused by the instabilities is assessed by the size of the term (u(ovox)). This term is much larger -- than mixing induce due to the ( - ) term. However, the contribution due to the shear instabilities decreases as the contribution of the ( - ) term increases. The contribution of the eddy viscosity term ( - ) in the absence of shear instabilities can be assessed by confining the longshore length scale of the domain to 1.5 j i i I i I i i o.5 _ O o.1 I I I I I I I I I i 0.08 (b) O Figure 11. Time- and longshore-averaged (a) longshore currents (5) and (b) perturbation kinetic energy density (12)(u 2 + v 2) for ci and M - 0 (solid lines), M (dashed lines), M (dash-dotted lines) on October 18. Mean velocities from sled data (open circles) are also shown in Figure 11a.

19 )ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25,971 ' I I I I I I I I I I o I i I I I I I I I I, I I I I I I I I I I 0.08 (b) _ -.o.o f - 0 I I I O Figure 12. Time- and longshore-averaged (a) longshore currents ( and (b) perturbation kinetic energy density (12){u 2 + v 2) for cf = and M = 0 (solid lines), M = 0.25 (dashed lines), M = 0.5 (dash-dotted lines) on October 16. The mean current in the absence of instabilities V (dotted line) for M and values from sled data (open circles) are also shown in Figure 12a. prevent the growth of the instabilities, effectively simulating the one-dimensional problem. The mean momentum balance resulting from such a simulation for M = 0.25 is shown in Figure 15 and reveals that in the absence of instabilities the only mixing process is due to the eddy viscosity term {r ). The magnitude of the mixing effect around the location of the current peak analytic current profile on a plane beach by Allen et al. is approximately 0.7x 10-3 ms 2. The mean longshore [1996], where longshore propagating, quasiperiodic vorcurrent profiles resulting from the simulations includ- tex structures with steady amplitudes result for chosen ing and neglecting shear instabilities are shown in Fig- friction and mixing coefficients ( = 0.006, M = 0). We ure 16a and are seen to be very different, confirming use this simple case as a starting point to analyze interthat mixing due to the instabilities is significant. These actions between vortices in the surf zone in some detail. two current profiles can be considered to be the profiles before and after the onset of the instabilities. Our expectation is that mixing due to the {r ) term, which is present in the absence of instabilities, will continue to act in the presence of them, while the mixing due to the instabilities is added to the system and alters the mean current profile further. However, we find that the mixing term (r} becomes less important when mixing due to the instabilities is considered. A comparison of the mixing terms {r}) in the absence and presence of the instabilities is shown in Figure 16b. The contributions are similar in the area of the shoreline jet where the instabilities are not active. However, the contribution is reduced significantly (by 50%) if the instabilities are included. 6. Discussion of Flow Properties The results of the simulations for the SUPERDUCK experiment suggesthat energetic vortices that propagate in the longshore direction exist and interact in a complicated fashion in the surf zone. Simpler vortex interactions were first observed for simulations using an We then investigate i the nature of the interactions the SUPERDUCK simulations is similar Simulations on a Plane Beach Allen et al. [1996] carried out simulations assuming a rigid lid and utilizing an analytic longshore current profile as a basic state on a sloping beach with slope The longshore current profile along with the water depth h, the wave height decay, and setup for a peak frequency of 0.13 Hz and a mean direction of 20 at 18 m water depth are shown in Figure 17. We repeated simulations for the case involving and M = 0 allowing for a moving surface and show results for this simple case in order to introduce tools to analyze the nature of the resulting flow struc-

20 ß, 25,972 OZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS f k ,,, 0.03 f k " c: M=0 1.'.' ;,' 5,,G, cf: M=0.25 ' - ß. '. ' -, 0 K (Im) 0-0 K (lm) ,0.015 ' f k ,, ' cf M=0.5 I i i.i I I ' I I 0.03 Data I 5,, f k K (lm) - 0 K (lm) Figure 13. Frequency-cyclic-wavenumber spectra S(f,k) (m3s) for computed and measured longshore velocity at x =35 m on October 16. The value for Cest is 0.8 ms and is used to constructhe upper and lower cutoff lines (dash-dotted line) of the shear wave energy. Contour levels plotted are (10, 30, 60, 100, 200, 400, 800). The equation for the best fit dispersion line (dashed line) is noted above each plot. tures and their interactions. We found that the evolu- tion of the flow is strongly dominated by subharmonic transitions, resulting in a reduction of the number of disturbances in the domain and flow structures that could be characterized as migrating rip currents. Snapshots of the vorticity pattern shown in Plate 5 detail one subharmonic transition. A layer of positive vorticity (red region) can be observed close to the shoreline, a layer of negative vorticity (blue region) exists farther offshore, and near-zero vorticity exists far of[- shore. Periodically spaced disturbances can be seen in the vorticity field at t = 7.8 hours. We follow two disturbances, which are located at about y = 1800 and 2250 m at t : 7.8 hours. In 6 min (t: 7.9 hours) the disturbances propagate in the y direction, so that they are now located at y = 2000 and 2400 m. The distance between them has decreased, suggesting that the weaker, second disturbance travels faster than the stronger, first disturbance. Another 12 min (t -- $.1 hours) bring the two fronts closer together; they are now located at : 2400 and 2700 m. In another 24 rain they are observed to have merged and one strong disturbance is visible at = 3400 m. At t = 8.1 hours another pairing process is initiated since the distance between two fronts at y = 900 and 1350 m decreases, so that at - $.5 hours they are located at y = 1700 and 2000 m. The trailing disturbance can again be observed to be the weaker one. The details of the vortex-pairing process can best be observed when examining the time evolution of the vor-

21 .. ::. )ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25,973 25OO M- 0 M M ' OO o.o x (m) x (m) x (m) Plate 4. Contour plots of vorticity q (s-1) at t - 5 hours for cf and M=0, 0.25, and 0.5 on October 16. ticity q as a function of longshore distance and time at a chosen cross-shore location. In Plate 6 we show a contour plot of q(x0, y, t), where x0 = 90 m. The red regions represent positive vorticity; the blue regions represent negative vorticity. Plate 6 (left) shows the time series of q(xo, y0, t), where y0: 1940 m. This y location is also marked on the contour plot with a thick line. The crests of the vorticity waves are observed to propagate in the +y direction, with milder slopes of the resulting blue line indicating faster propagation speeds given by dydt. At t = 7.5 hours all eight disturbances are observed to travel at the same speed of about 0.4 ms. Around t = 8 hours one of the disturbances is seen to speed up. The time series in Plate 6 (left) confirms that the amplitude of the faster trailing wave is lower than the amplitude of the wave in front of it. The smaller disturbance eventually catches up with the disturbance in front of it and collides with it at about y = 3000 m. The resulting disturbance continues to propagate at the speed of the slower wave. This pairing event was also observed in the snapshots of vorticity in Figure 5. This type of pairing occurs again around t = 8.75 hours and y = 2500 m. The time series in Plate 6 (left) shows that the height difference between the two disturbances is larger than in the first merger since these two waves are at a later stage in the pairing process. The initial stages of this pairing process were observed in Plate 5. The third pairing evident in Plate 6 occurs around t = 9.25 hours. The details of the process can best be observed in this case, even though they are also evident in the previous events. It can be seen that the faster trailing wave collides with the slower wave in front of it. The trailing disturbance gains energy and slows down to the speed of the wave in front of it. The wave in

22 25,974 OZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS x 10-3 g2 "% ',,,, ', i i i i.. I \ I \ I' i o-2 -_,- I I I I O i i I i I i I I I I I i (b) I I I i x 10-3 L i '\ I' -1 " o ß I i i I Figure 14. Mean longshore momentum balance for c and M= (a) 0, (b) 0.25, and (c) 0.5 on October 18; {u(ovo:r)) (thick solid lines),-{ y) (dash-dotted lines),-( ) (thin solid lines), (l vd)(thick dashed lines), and residual (thin dashed lines). the front, in turn, loses almost all of its amplitude but 6.2. Simulations for SUPERDUCK remains intact, continues to propagate at the higher As a result of the plane beach simulations discussed speed of the initially trailing wave, and propagates into above, we find quasiperiodic disturbances with moda region of negative vorticity. This weakened wave can ulated amplitudes that occasionally collide and merge be seen in the time series in Plate 6 (left) at t = 9.5 with each other. The collision process in that case alhours as a small "blip" in the negative vorticity region. ways results in a reduction of the number of waves in It dissipates before it reaches the next front of positive the domain. In this section we investigate if similar vorticity. We can conclude that the pairing occurs in interactions between vortices occur in the case of the the form of a collision where most of the energy is trans- barred beach for SUPERDUCK. ferred to the trailing wave, which subsequently travels In Plate 7, we show contour plots of the vorticity at a slower speed. A small phase shift is also introduced q(x0, y, t) for computations simulating October 16 with at the time of the collision, so that the lines represent- cy = and 3: Snapshots of vorticity at 5 ing the lower and higher speeds are slightly shifted. One rain intervals were shown for this case in Plate 2. The final collision initiated at the end of the time series, re- cross-shore location x0 is chosen to be 100 m and correducing the number of propagating disturbances to four. sponds to a position within the layer of negative vortic-

23 OZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25,975 X I I \\ L,' '._. I -2 1 I I I I I O Figure 15. Mean longshore momentum balance in the absence of shear instabilities for c and M on October 18; (=(O O ))(thick solid line),-(fu} (dash-dotted line),-( - ) (thin solid line), (pvd) (thick dashed line), and residual (thin dashed line). ity seaward of the longshore current peak. Plate 7 (left) displays a time series of the vorticity q(z0,.v0, t), where Y0 m 700 m. This longshore location is also marked on the contour plot with a thick black line. In comparison to similar plots obtained for the plane beach simulations, Plate 7 appears to display an unorganized character. However, the vortex interactions can still be discerned. We can see several energetic features with positive vorticity that propagate through the do- main. Smaller positive vorticity features are frequently observed to catch up with the large vortices. For example, the positive vortex located at y = 700 m and t = 4 hours (marked by arrow) undergoes a collision at about y = 1250 m, consequently speeds up, and loses energy. It encounters another vortex in front of it, interacts again, this time gaining energy and slowing down. A short time after this interaction the vortex abruptly disappears in the region of negative vorticity, possibly ' 0 5O 1 O0 15O X , i i i 0.5 (b) O Figure 16. (a) Time- and longshore-averaged longshore currents (7) and (b) contribution to the mean momentum balance of the mixing term -( - ) for c and M suppressing (solid lines) and including (dashed lines) shear instabilities on October 18. Mean velocities from sled data (open circles) are also shown in Figure 16a.

24 ß. 25,976 )ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS -2O O O I I I I I I I o.1 o O o i ' O Figure 17. (a) Bathymetry for ra , (b) longshore current profile, (c) wave height decay, and (d) setup of the mean water level for fp Hz and ø. because it is shed offshore. Many of the features identified for the plane beach simulations apply. The faster positive vortex is observed to catch up to the slower vortex in front of it. As the two vortices attempt to merge, the trailing vortex is observed to gain energy and slow down. The vortex in front loses almost all of its energy and speeds up to the original speed of the trailing vortex. A phase shift is introduced at the time of the collision since the lines representing the lower and higher speeds do not intersect. The weakened disturbance in front propagates into a region of negative vorticity and continues to lose energy. Owing to frequent collisions of the form described above, the pattern in Plate 7 displays strong propagating positive vortex features (such as the feature running from t-4 hours and y=0 m to t=5 hours and y=2000 m) with long "streaks" on their left sides, representing the faster vortices as they catch up, and small streaks on their right sides, representing the vortices in front that lose energy but gain speed as a result of the interaction. The faster vortices appear to originate from regions of negative vorticity. As an example, we can observe a positive vortex that undergoes a collision at about t =3.7 hours and y =700 m (marked by arrow in Plate 7). It subsequently weakens substantially as it propagates away into a negative vorticity region. However, it does not dissipate altogether, starts gaining energy again around y =1400 m, and continues on to interact further. It is suspected that the reemergence of such structures is related to the onset of meandering that was observed in the snapshots in Plate 2 in regions where the longshore velocity appeared to be longshore uniform for a finite distance. It is possible that the longshore uniform current in those regions experiences a renewed local instability. A final case involves a value for the mixing coefficient of _ Returning to the snapshots of vorticity for this case (see Plate 3), we observe that the behavior in this case is reminiscent of the simulations for a plane beach. In Plate 8, we see that positive vortices are localized and propagate in the longshore direction. The pattern is indeed similar to the pattern observed during the plane beach simulations shown in Plate 6. The interactions between the vortices are clearly visible. In contrast to the plane beach simulations, the vortices that are weakened as a result of the collisions do not dissipate in the regions of negative vorticity. Instead, they propagate through the regions of negative vorticity and interact with the next region of positive vorticity that they encounter. As a consequence, a reduction of the number of structures in the domain is not necessarily the outcome of such interactions. 7. Summary and Discussion In this study, shear instabilities of the longshore current have been numerically simulated. The effects of stationary incident wave forcing, bottom friction, and

25 ß )ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25,977 t=7.8 hrs t=7.9 hrs t=8.1 hrs t-8.5 hrs 350O 3OOO OO looo -O.Ol x (m) x (m) x (m) x (m) Plate 5. Snapshots of vorticity q (s-1) for a plane beach.

26 25,978 )ZKAN-HALLER AND KIRBY' INSTABILITIES OF LONGSHORE CURRENTS q (1s) y (m) Plate 6. Contour plot of vorticity q (S--1) as a fi nction of y and t at xo m on a plane beach. (left) Time series of q(xo, yo, t), where Yo m l I O.O5 0 O.05 q (1s) y (m) Plate?. Contour plot of vorticity q (S--1)&$& function of y and t at xo m on October 16; c and M (left) Time series of q(xo, yo, t), where Yo m.

27 ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25, q (1s) y (m) Plate 8. Contour plot of vorticity q (S--1) as a function of y and t at x m on October 18; cf and M (left) Time series of q(x0, y0, t), where y0-870 m. turbulent momentum mixing as well as the effects of the depth variations of the current velocities were included in a rudimentary fashion. We incorporated simplifying assumptions that reduced the effects of these processes to the mathematically simplest formulations and carried out numerical simulations to model the shear instability climate during 3 days of the SUPERDUCK field experzment. We have varied the friction coefficient within a nar- row range close to a value that reproduced the propagation speed of the shear waves seen in the data. In summary, we found that a stronger mean current, more energetic fluctuations in the velocities, faster propagation speeds, and more energetic vortex structures result as the friction coefficient is decreased. However, the longshore length scales of the resulting flow structures were similar for the different friction coefficients. For a realistic value of the friction coefficient the propagation speed of the shear instabilities agreed well with observations. The range of wavenumbers at which vorticity motions are observed is reproduced. Although the choice of the friction coefficient was based solely on the propagation speed of the shear instabilities, the predictions of the peak longshore current values also agreed with current measurements from the sled. This obser- vation lends further confidence to the estimation of the friction coefficient. Although the propagation speeds of the instabilities as well as the maximum mean longshore current are predicted with satisfactory agreement, the observed frequency spectra are not modeled very well. The computed shear instabilities appear to have enough energy to account for the energy present in the shear wave band in data; however, the distribution of the energy in the frequency spectrum is not reproduced. Frequency spectra comparisons of computations to observations showed that the energy content in the frequency range < f < Hz can be reproduced. However, energy at lower frequencies is substantially overpredicted, whereas energy at higher frequencies is underpredicted. The details of the time dependent behavior of the flow structures are analyzed in the last section. We found that vortex collisions of the type observed for plane beach simulations occur. However, the interactions do not always lead to the reduction of the number of disturbances in the domain. The time dependent nature of the flow involves the strengthening, weakening, collision, and interaction of the vortices. Vortices are frequently shed offshore. During this process the flow structures exhibit narrow, offshore directed jets that are strong and transient, similar in description to the "migrating and pulsating" rip currents that Tang and Dalrymple [1989] observed during the Nearshore Sediment Transport Study (NSTS) experiment at Leadbetter Beach in As was reviewed earlier, when linear instability theory was first proposed, investigators such as Dodd et al. [1992] and Reniers et al. [1997] proceeded by examining

28 25,980 OZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS the linear instability of current profiles that were measured in the field or in the laboratory. They obtained good predictions for the range of unstable wavenumbers, leading to the concept that the observed fluctuations may be due to weakly nonlinear disturbances. However, we show that for a friction coefficient that reproduces the shear wave propagation speed inferred from the data and for a realistic range of mixing coefficients, the resulting motions all have the characteristics of highly transient, nonlinear vorticity waves. These results suggest that the shear wave climate observed during SUPERDUCK may be of a highly nonlinear nature. The predicted motions cause momentum mixing in the surf zone and alter significantly the resulting longshore current profile. The momentum mixing due to the instabilities is especially pronounced around the longshore current peak. Mixing is also induced in the bar trough region, causing the generation of significant mean longshore current velocities even in the absence of other more traditional mixing mechanisms. In the presence of other mixing mechanisms, which we param- eterize using an eddy viscosity formulation, the mixing due to the instabilities decreases. We find that the total Compared to the fluctuation-free current profile I7, the resulting current profile including the effect of the finite amplitude instabilities ( ) displays a weaker peak along with a milder seaward shear. It is important to note that the measured mean longshore current profile includes the effects of momentum mixing due to the fluctuations. The linear stability characteristic of the measured current is therefore not necessarily descriptive of the shear instability climate that exists along with it. Rather, the stability characteristics of the often unknown, but relevant, fluctuation-free initial state should be considered. Examining the mean longshore momentum balance, we found that the mixing induced by the instabilities dominates over mixing due to the eddy viscosity terms for reasonable mixing coefficients. We also found that the presence of the shear instabilities and the associated momentum mixing tend to suppress momentum mixing due to the eddy viscosity terms, which include the effects of turbulence and depth variation in the current velocities. We note that the cross-shore distribution of the momentum mixing due to the instabilities is such that the location of the longshore current peak is not altered. For all simulations carried out for the SUPERDUCK experiment, the longshore current peak is located over the bar crest. It is important to note, however, that sled measurements during SUPERDUCK also indicate that the current maximum occurred on the bar crest. Appendix A' Short Wave Submodel The incident wave forcing terms x and y are functions of the radiation stress components as given in (4). The radiation stress components Sxx and Sxy are defined in terms of the orbital wave velocities and can be computed utilizing an appropriate water wave theory. In this study we use linear water wave theory to obtain the radiation stress components as S = E c-- a(cos20+l)- ], (Ala) Sxy = EC sin0cos0, c (Alb) where 0 is the mean angie of incidence. The wave energy amount of mixing in the surf zone is virtually a constant density E is expressed in terms of the root-mean-square for a fixed friction coefficient, so that a remarkably sim- wave height Hrms as )pgh; ms. The group veilar mean longshore current profile is generated, regardless of the value of the mixing coefficient. If the mixing coefficient is increased, the amount of mixing due to the instabilities decreases proportionally. An increase in M causes relatively small variations in the propagation speeds but an overall decrease in the total energy of the fluctuations. However, differences in the cases are less pronounced in the nearshore region, especially in the bar trough, although the overall characteristics of the motions associated with the different mixing coefl:icien[s are very different since an increase in the mixing coefficient results in longer, less energetic motions with locity c a and celerity c can be computed from the linear dispersion relationship. The transformation of the random waves as they approach the shore is simulated using the wave height transformation model of Thornton and Guza [1983]. In order to apply this model, we assume random waves with a narrow-banded spectrum and Rayleigh distributed wave heights. The applicability of these assumptions to the SUPERDUCK experiment has been addressed by Whirford [1988]. The random wave transformation model by Thornton and Guza [1983] involves utilizing the energy equation weaker vortices. for the incident wave motions. For a stationary incident wave field over straight-and-parallel bottom contours and in the absence of any wave-current interactions, this equation is given by OEc a cos 0 =eb. (A2) Here c a is the group velocity associated with the the peak frequency fp. The cross-shore variation of O is computed using Snell's law of refraction. The parameter eb in (A2) is an ensemble-averaged energy dissipation due to wave breaking defined by Thornton and Guza [1983]. The form for eo suggested by Whirford [1988] for the SUPERDUCK experiment is used. - C H 3 C h (A3)

29 where ZKAN-HALLER AND KIRBY- INSTABILITIES OF LONGSHORE CURRENTS 25,981 3x B 3 C = -- pgfp, (A4) C tanh 8 -h 0.99 (A5) The coefficients used for the wave height transformation model are B (indicating the intensity of wave breaking) and -= Finally, an estimation of the amplitude of the horizontal orbital velocity of the incident waves is made since it enters the definition of the bottom friction term introduced in section For Rayleigh distributed wave heights the amplitude of the horizontal orbital velocity of the incident waves can be expressed in terms of Hrm s [Thornton and Guza, 1983] as and is a function U0- Hrms (A6) of the cross-shore location. r t: I OS;. (B1) pd We utilize a common eddy viscosity parameterization given by ( Ou Ou ) (B2) to relate the Reynolds' stresses to the mean flow. Here For waves that are obliquely incident at small angles the indices a, represent the x,y directions and rethe volume flux due to the waves will occur mostly in peated indices are assumed to be summed. The pa- the x direction so that rameter vt is the turbulent eddy viscosity and (us, u ) = (u, v) are the current velocities. Q wx >> Q wy. (B5) Another important source of lateral mixing in the surf zone is analogous to the Taylor dispersion pro- Under this assumption, Dxx > Dxy > Dyy. As a first cess of dissolved matter in pipe flow [Taylor, 1954]. approximation we retain only terms with the coefficient This process was identified by Svendsen and Putrevu Dxx. Including both the turbulent momentum mixing [1994], who considered the case of a steady longshore terms and the dominant dispersive mixing terms, the current and parabolic undertow profile on a longshore mixing terms reduce to uniform beach. They found that the nonuniformity of the nearshore currents over depth leads to additional terms in the depth-averaged momentum equations. The extension of the theory to unsteady flow over arbitrary bathymetry was performed by Putrevu and Svendsen [1999]. Considering a depth-uniform return flow to com- pensate for the wave-induced volume flux Qw, Putrevu and Svendsen [1999] argue that the dominant effects of the additional terms is additional lateral momentum mixing, termed dispersive mixing r and given by r = 1 0 d Ds +Ds ß (B3) Therefore, in the remainder of the discussion, we consider only the effects due to the D Z terms. The omitted terms modify the radiation stress term and the convective acceleration term. In the general case, D Z is a function of the turbulent eddy viscosity and the depth variations of the currents. Since undertow profiles are strongly curved in the surf zone but not outside the surf zone, varies with cross-sh6re distance. The specification of D Z therefore requires knowledge about the local depth profiles of the velocities. The quasi-three-dimensional (Q3-D)approach has been used in recent studies to obtain information about the vertical variations of the currents without the complications of a full 3-D model Appendix B- Lateral Momentum Mixing Submodel [e.g., De Vriend and Stive, 1987; Sanchez-Arcilla et al., 1993; Svendsen and Putrevu, 1994; Van Dongeren et al., B.1. General Formulation 1995; Faria et al., 1996]. The quasi-3-d approach involves calculating the depth-averaged current velocities The parameters rx and r in (3b) and (3c)repre- using the depth-averaged momentum equations and utisent the effects of lateral momentum mixing in the surf lizing the results to compute the local depth variation zone. One source of mixing in the surf zone arises owing to gradients of turbulence-induced momentum fluxes (the depth-integrated Reynolds' stresses $ ). The turof the velocities from a separate profile model. Since such computations are beyond the scope of this study, we seek to include the effects of the D Z terms to leadbulence-induced stresses rx t and ry t in the momentum ing order by estimating the order of magnitude of the equations are given by coefficients D Z along with a reasonable variation in the cross-shore direction. Putrevu and Svendsen [1999] stated that for a depthuniform return flow the coefficient matrix D,Z is pro- portional to the wave-induced fluxes in the horizontal directions such that + 10 [ + Ov I, (B6a)

30 25,982 OZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS I 0 Id(yt+Dxx)OV ] ß (B6b) It is evident from (B6a) and (B6b) that the Dx terms act to reinforce the turbulent momentum mixing in certain directions. The coefficient of these combined terms is a composite eddy viscosity y = Yt + D. Svendsen hv h < yt < 0.005hv h. (B11) and Putrevu [1994] show that the value of the coefficient D (termed Dc by Svendsen and Putrevu [1994]) can Svendsen and Putrevu [1994] state that a crude estibe orders of magnitude larger than vt. Therefore we mate of the magnitude of Dxx is given by retain only the mixing terms that are premultiplied by 1 Q2 (yt + D ), effectively neglecting turbulent momentum _ w, (B12) Dxx = 2 t mixing unless it is reinforced by the dispersion process. The resulting mixing terms are stated in (54) and (5b). where Qwx is the wave volume flux in the cross-shore di- Since both yt as well as D are more pronounced in rection with typical values in the range [see, e.g., Svendthe surf zone, the combined eddy viscosity v will also sen et al., 1987] be more pronounced inside the surf zone. To obtain a reasonable cross-shore variation, it is convenient to parameterize the total horizontal eddy viscosity following Battjes [1975] as where M is an O(1) coefficient and is the ensembleaveraged energy dissipation due to wave breaking given in (A3). This relationship was originally formulated to account for the cross-shore variation of the turbulent eddy viscosity vt. Therefore the order of magnitude of M needs to be reevaluated to account for mixing due to turbulence as well as the Taylor dispersion process. B.2. Estimation of the Magnitude of 3 In order to estimate the order of magnitude of 3for field applications, we focus our attention to the inner surf zone and use linear shallow water theory to construct a simple expression for eb given by b - xx pgh2 c, (BS) where H is the wave height and c is the incident wave celerity. Assuming wave breaking in a saturated surf zone, we can use (Hh) -'7 and write F -( h ) V A, (B9) where hx is the bottom slope. An average bottom slope of 0.05 applies to field experiments at Duck, North Carolina. Furthermore, Thornton and Guza [1983] found good agreement with measured wave height variations in field applications for a value of 3' given by With these values the order of magnitude of y can be estimated to be y O.X4Mh h. (BX0) On the other hand, y is composed of the turbulent mixing coefficient yt and the Taylor dispersion coefficient Dxx. A typical value of the turbulent eddy viscosity coefficient yt derived from laboratory studies by Svendsen et al. [1987] is 0.01 hx. However, Ceorge et al. [1994] state that this estimate should be reduced by 12 to 14 for field applications. A reasonable range of values for Pt in the field is then 0.03 h h < Qwx < 0.1 h. (B13) Once again, we can use (Hh) = ff = 0.42 and get o.oosnn < < 0.0snn. (e4) The effective eddy viscosity is given by the sum of vt and Dxx. Using (Bll), (B12), and (B14), reasonable values for will be in the range 0.00sn <. < 0.0cTnn. (es) Comparing (B10) and (B15), the range of values of the mixing coefficient M is estimated to be 0.0 < < 0.48, (elc) when lateral mixing due to turbulence as well as the dispersion mechanism described by Svendsen and Putrevu [1994] are taken into account in a rudimentary hshion. Appendix C' Bottom Friction Submodel The simplest formulation for the bottom friction is given by linear damping terms in the momentum equations and is purposefully used so that we can identify the resulting motions in the mathematically simplest setting. This formulation is given by (7) and has so far been used by a number of investigators for linear [Dodd et al., 1992] and nonlinear [Allen et al., 1996; Slinn et al., 1998] shear instability computations. Such a damping term rby can be obtained for the longshore momentum balance by assuming that the incident waves approach the shore with a small angle of incidence and the maximum orbital velocity uo associated with the incident waves is much larger than the mean current [Longuet-Higgins, 19704,b]. A similar linear damping term r is also introduced in the cross-shore momentum balance. However, a rigorous derivation of th e frictional terms in the x direction using the assumptionstated above [see Dodd, 1994]

31 ZKAN-HALLER AND KIRBY: INSTABILITIES OF LONGSHORE CURRENTS 25,983 would result in an asymmetry in the damping terms due to an additional factor of 2 in the damping term rbx in the cross-shore balance. This would result in elevated damping of the cross-shore velocities. Dodd [1994] compared the damping formulation given in (7) to the formulation containing the asymmetry and found that the linear instability characteristics are altered only slightly. We also carried out simulations for the nonlinear behav- ior using both formulations and found that the characteristics of the motions are not altered. Differences are especially minor when the friction coefficient cf is relatively low, as is the case in the simulations presented here. It is noted that the above formulation for bottom friction is rudimentary. The assumptions regarding the strength of the current in relation to the wave orbital velocity as well as the incorporation of linear shallow water theory for the computation of u0 are highly restrictive and often unrealistic. Furthermore, the size of the friction coefficient cf is a major source of uncertainty. Therefore simulations are carried out for a range of friction coefficients. Acknowledgments. Thanks are due to Joan Oltman- Shay and Nick Dodd for providing the SUPERDUCK data as well as the software to obtain estimates for frequencylongshore-wavenumber spectra of the data. Thanks also go to Bill Birkemeier, Peter Howd, Ed Thornton, and the staff at the Army Corps Field Research Facility (FRF) at Duck, North Carolina, for their contributions to the field data acquisition and distribution. This research has been sponsored by the Office of Naval Research, Coastal Dynamics Program, under grant N References Allen, J.S., P.A. Newberger, and R.A. 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