Shore-Connected Ridges Under the Action. of Waves and Currents

Transcription

1 Under consideration for publication in J. Fluid Mech. 1 Shore-Connected Ridges Under the Action of Waves and Currents By E M I L Y M. L A N E 1, A N D J U A N M. R E S T R E P 2 1 Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA U.S.A. 2 Department of Mathematics and Department of Physics, University of Arizona, Tucson, AZ U.S.A. (Received 11 July 25) Up-current rotated, shore-connected ridges are found in various coastal areas around the world. An often-quoted conjecture is that these ridges form during storm conditions through free instabilities in the erodible bed. Under these conditions both waves and currents are expected to play a significant roles in the hydrodynamics. Although the existing models referenced here have included the effects of waves parametrically in their bottom friction terms and sediment equations, the dynamical effects of wave-current interaction have not been explored. In this paper we begin to rectify this by considering the effects of wave-current interaction on the bedform instabilities of a simple model. This brings up the possibility of unsteady longshore flows and questions about the roles of wave parameters, reference bottom topography and boundary conditions which we address here. We show that the flow is stable under the wave forcing however the waves do affect the bedform instability. Under various conditions waves may suppress the instability, alter the shape of the unstable modes or introduce new unstable modes. They also affect

2 2 E. M. Lane and J. M. Restrepo the celerity of the ridges. We also investigate the mechanisms whereby the waves affect the instability. 1. Introduction Up-current rotated, shore-connected ridges are found in certain coastal areas, such as off the east coast of North America (Duane et al. 1972; Swift et al. 1972), off the Dutch coast (van der Meene 1996) and off the Argentinian coast (Parker et al. 1982). These ridges form at angles of around 2-35 to the shore, with the seaward side rotated upstream with respect to the landward side. They form in water between 5-3 meters deep. The ridges are spaced 5-1 kilometers apart and can be several meters high. Most ridge evolution happens during large storms which drive long shore currents on the order of.5 meters per second. These storms occur several times a year in these locations, with durations from hours to days. In the case of the Dutch Coast, tidal currents are also significant but in the Mid-Atlantic Bight and off the coast of Argentina the role of tides is negligible. The ridges migrate downstream with a celerity of a few meters per year. The original supposition was that shore-connected ridges were sillstands left over on the strand plain from the return of the Holocene sea (McClennen & McMaster 1971). However, their orientation, with the down-shore end connected to the shore, does not fit with this conjecture unless the angle of the shoreline has changed significantly since then. Also, as these ridges are active under present day hydrodynamic conditions (Swift et al. 1978), they are not just a relic feature. Swift et al. (1973) hypothesized that the ridges are an erosional feature: storm driven flows scour out the landward side of the shoreconnected ridges; gradually the shoreline retreats, and as it does, the ridges migrate

3 Shore-Connected Ridges 3 down current and also off-shore. A third hypothesis is that they form as instabilities of the erodible bed under the influence of the local hydraulic regime. This idea that these ridges formed through a free instability was postulated by Trowbridge (1995), largely inspired by the work of Huthnance (1982) based on Off s 196 observations of current-aligned ridges. Using a simple model where the evolution of the bottom topography is related to the divergence of the flow, Trowbridge (1995) shows that the unstable eigenfunctions are up-current rotated ridges similar to those observed. Mass considerations mean that the long shore flow is deflected off shore over the ridge crests. This deflection causes the flow to converge which, in turn, causes deposition increasing the height of the ridges. Thus up-current rotated ridges can form from a smooth bottom. Trowbridge s model, however, has no distinguished length scale to explain the size and spacing of the ridges. Falques et al. (1998b,a) (henceforth denoted F98) extends Trowbridge s model by using two-dimensional shallow water equations for the evolution of the currents and adding a diffusive term to the bottom topography evolution equation. The diffusive term represents the proclivity of sand grains to move downslope. With these additions the model selects a distinguished length scale. The fastest growing mode, which often represents the length scale of the final feature, is chosen through a balance between the destabilizing effects of the convergence term and the stabilizing effects of the diffusion term. This model still has the problem, however, that the ratio between e-folding timescale and the celerity is unrealistic. The basic model was modified by Calvete et al. (21) to include both bed load and suspended load; The morphological effects of allowing for different grain sizes was investigated by Walgreen et al. (23). Wave effects were also included parametrically in the bottom friction term and the sediment dynamics equations by Calvete et al. (21).

4 4 E. M. Lane and J. M. Restrepo These additions give more realistic ratios between the e-folding and migration timescales. However, although these give quantitatively better solutions, qualitatively the results are no different from those of F98 (Compare Calvete et al. (21) Figure 3 with the top row of Figure 3 in this paper). Finite amplitude effects are also explored in Calvete et al. (1999, 22) with the overall conclusion that the outcome of the linear instability calculations were robust, at least qualitatively. Field observations (Duane et al. 1972; Swift et al. 1972; Parker et al. 1982) have shown that considerable ridge evolution occurs during storms. In such oceanic conditions both waves and currents are prominent in the hydrodynamics. Restrepo (21) developed a coupled system of equations for the waves, currents and erodible bed by depth averaging the wave-current model in McWilliams & Restrepo (1999) and deriving a mass conservation statement for the sediment consistent with the tracer evolution equation which takes into account waves and currents. Restrepo (21) considered the effect of waves on the linear instability results that utilized current-only models for the hydrodynamics. Furthermore, he raised questions on the validity of the adiabatic approximation which was exploited in F98 to suppress potential instabilities due to the hydrodynamics. However, the analysis of the wave effects on the shore-connected ridge model in was tentative, for two reasons: the linear wave dynamics interacted only weakly with the rotational flow and the bottom topography and secondly, resolution constraints were impractical from a computational standpoint. Here we take another look at how wave effects change the outcome of instability calculations involving current-only models for the shore connected ridge problem. We do so armed with a much improved wave/current interaction model:derived by revisiting the scaling, going to higher order and circumventing the small scale resolution problem. This gives a much more complete and generally applicable model for wave-current interaction

5 Shore-Connected Ridges 5 on the continental shelf. This model appears in McWilliams et al. (24) which will be referred to here as MRL4. With a dynamic model for the fully three-dimensional hydrodynamics and a suitable sediment evolution model it is now possible to explore the shore-connected ridge problem over time scales that encompass the variable forcing conditions. Moreover, our model is capable of handling the extreme temporal scales, encompassing the range from stormdriven flows to hydrodynamics over decades, and spatial scales of the order of tens of meters to tens of kilometers. Before we can consider the transient problem, however, there are a number of fundamental issues related to the assumptions in the shore-connected ridge model that need to be addressed. This study will show how the shore-connected coupled sediment-ocean dynamics model is modified when waves are taken into account. Specifically, we consider (1) how the linear instability results on the shore-connected ridge problem change when waves are taken into account; (2) how these changes manifest themselves when wave parameters are varied. Furthermore, our consideration of wave-current interaction raised other issues; this paper also addresses: (3) the role of the reference bottom topography; (4) the difference between fixing the perturbation at the shoreward boundary (Dirichlet boundary conditions) and imposing a no-flux (Neumann) boundary condition; (5) whether the adiabatic assumption is justified i.e., whether currents are steady especially under wave forcing and whether the solutions are significantly affected if the morphological and hydrodynamic timescales are not well separated; and finally (6) what mechanisms the waves use to affect the morphodynamics.

6 6 E. M. Lane and J. M. Restrepo 2. Governing Equations Shore-connected ridges form in finite depth water on the continental shelf. The peak of the wave spectrum has a wavelength of around 1 m giving µ = k H 1, where k is the wavenumber magnitude and H is the mean depth of the water column. The ridges represent changes in bottom topography elevation of meters over distances of kilometres. Thus their length scales are long compared with the typical length scale of the waves. The storm driven currents have speeds on the order of m/s in microtidal regions Calvete et al. (21) with comparable wave orbital speeds. This suggests that the asymptotic theory derived in MRL4 is valid in this setting. Our mass equation includes a surface elevation tendency term, thus long waves due to changes in the wavetrain may be excited. However, the long wave contributions, as well as multifrequency wave-spectrum wave trains will be considered at some future time in the context of the variable forcing problem, which is outside of the scope of this paper We consider a model domain that is semi-infinite in the x (cross-shore) direction and periodic in the y (longshore) direction. The model does not take into account the surf zone so that x = should not be considered the shore per se but rather the shorewardmost part of the domain Waves We assume monochromatic waves at the peak of the spectrum with wavenumber, k = (k cos(θ), k sin(θ)), amplitude A and frequency σ (see Figure 1). The group velocity of the waves is C g. Quiescent ocean level is Z = with the ocean bottom at Z = H. The

7 waves propagate according to the equations (see MRL4) and ( Time is t and = x, y Shore-Connected Ridges 7 k t + C kσ g k = sinh[2kh] H, σ t + C g σ =, A 2 + (C g A 2) =. t ) is the horizontal derivative. For convenience we rewrite the wavenumber equations in terms of the scalar wavenumber k and the angle of the waves θ, with respect to the x axis: ( k t + C σk g k = cos(θ) H ) sinh(2kh) x + sin(θ) H and y ( θ t + C σ g θ = sin(θ) H ) sinh(2kh) x cos(θ) H. y Note that the scalar wavenumber s evolution is dependent only upon the bottom topography s gradient parallel to the direction of the waves, whereas the wave-angle s evolution depends only upon the bottom topography gradient perpendicular to the wave direction. The wave quantities will be specified in the far field (x ) and we assume a radiating boundary condition at x =. This is commensurate with the waves passing through this domain and then being dissipated through breaking closer to shore in the surf zone Currents We derive shallow water equations using equations (9.2,9.9,9.12,9.15,9.17) from MRL4 section 9. We assume that vertical variations in velocity are negligible (see MRL4 section 12 for a general discussion and also Lentz (21) for the situation of the coast of North Carolina). We add surface and bottom forcing terms analogous to those in F98. This gives the horizontal momentum equation V t + V V + f ẐZ V + g ζ c = g G ẐZ V (χ c + f) + τ rv H + ζ c. (2.1)

8 8 E. M. Lane and J. M. Restrepo V = (U, V ) is the vertically averaged current, χ c = ẐZ V the vertical vorticity, ζ c is the mean sea elevation (averaged over the waves), V = g A 2 2σH k = (U, V ) is the vertically-averaged okes drift. This expression for the vertically-averaged okes drift corrects an obvious typographical error that appears in Restrepo (21). The Coriolis parameter is f, gravity is g. G is a Bernoulli head wave forcing term G = K() 1 ˆp ˆζ ρ Z () P. (2.2) G combines wave effects on the pressure and sea elevation as well as the ambient pressure (further details are found in MRL4). G can be evaluated as + A 4 k 3 ( 1 G = 2 sinh(2kh) 2 + tanh2 (kh) sinh 2 (kh) ) A 2 k tanh(kh)k V σ 9 16 sinh 6 (kh) sinh 2 (kh). (2.3) The most well known effect of the Bernoulli head term is the wave set-up in the near shore region. The vortex force term ẐZ V (χ c + f) captures the effect of vorticity transferred to the waves through the action of the rotational currents and from planetary rotation. Note that the wave effects on the currents are completely characterized by the vortex force and Bernoulli head terms. As discussed in Lane et al. (25) this is an alternative to the radiation stress parametrization. If wave effects which enter explicitly in the okes drift velocity V and the Bernoulli head G are suppressed, we then obtain the current equation given in F98. The wind stress is represented by τ/(h + ζ) and the bottom friction is r/(h + ζ). A quadratic bottom friction term could be used but Falques et al. (1998a) showed that it makes minimal difference to the results.

9 The fluid mass conservation equation is Shore-Connected Ridges 9 t (ζc + H) + ((H + ζ c )(V + V ) ) =. (2.4) We note that unlike Restrepo (21), here we include the tendency term H/ t in equation (2.4). The bottom topography changes over timescales that are appreciably slower that the time scales associated with changes in surface elevation, thus H/ t can be thought of as a perturbation to (2.4). Over short time scales, the surface elevation balances with the divergence of the vertically integrated mass flux. Over long times scales, however, the bottom topography changes. The sea surface elevation then adjusts slightly to take this into account. The sediment dynamics equation for H/ t is H t = Q, (2.5) where Q is the erodible bed sediment flux. We adopt a similar sediment flux model to that of F98. The flux is composed of a convergence term and a diffusion term. Because we are considering storm dominated conditions suspended sediment plays an important role. In this case the sediment is advected by both the waves and currents. Accordingly, the mean tracer flux is the sum of the current and the okes drift i.e., V + V, as shown by Restrepo (21). The diffusion term represents the tendency of sand form smooth structures. The coefficient of diffusion is proportional to the absolute value of the sum of the vertically averaged current velocity and okes drift. Thus, Q = α ( V + V γ V + V h ), α is a small parameter related to the ratio between the morphological and hydrodynamic timescales, α = H Hydrodynamic Timescale Morphological Timescale,

10 1 E. M. Lane and J. M. Restrepo where H is the representative water depth. This sediment flux equation corresponds to m = 1 in F98, a regime that they focused on. Although more complicated sediment dynamics models such as those postulated by Calvete et al. (21) give quantitatively better results they do not produce qualitatively different solutions. Because the primary focus of this paper is to evaluate how the addition of wave current interactions alters the dynamics we feel that it is better to start with a simpler sediment model where the wave effects can be more clearly seen than consider a more complicated model which may make it difficult to differentiate between the effects of the various components. Once the wave effects are more fully understood we can consider sediment models that include both suspended- and bed-loads and other effects. To ensure that there is no net flux of water out the shoreward side of the domain we assume that the Eulerian velocity balances the okes flow in the crossshore direction, i.e., U(x = ) + U (x = ) =. We have a no stress boundary on V at x =. We assume that all quantities are finite for x. 3. Reference ate We derive a steady state solution to the equations of hydrodynamics and sediment motion, consistent with F98, which includes a wave component. Our reference bottom topography is independent of y. It increases from a depth of H at x = to H as x. Our default bottom topography decreases linearly from H to H over a distance L and is flat thereafter. In section 4 we also consider an alternative bottom topography.

11 Shore-Connected Ridges 11 A H H Inf ONSHORE y L OFFSHORE k θ x Figure 1. Schematic diagram of quantities used in calculations (Not to Scale). Top figure: side view; bottom figure: plan view Waves We assume that the far field boundary condition for the shore-incident waves is steady and independent of y. As the reference bottom topography is also independent of y, the incoming reference wave varies only in x. The frequency σ, is constant throughout the domain. The scalar wavenumber k(x), is obtained from the water depth H(x) using the implicit relationship σ 2 = gk(x) tanh(k(x)h(x)). θ and A 2 are found by solving the following relationships θ x = σ tan(θ(x)) H C g sinh(k(x)h(x)) x, A 2 (x) = C g cos(θ ) C g (x) cos(θ(x)) A 2, where C g, θ and A 2 are the group velocity, angle, and amplitude-squared, respectively, of the incident waves in the far field. Because the vertically-averaged okes drift is a wave quantity, its reference state is also independent of y.

12 12 E. M. Lane and J. M. Restrepo 3.2. Currents We assume a steady wind stress acts only in the y direction and τ = (, τ y ). We also assume that the sea elevation includes a small linear long-shore variation representing a pressure gradient that drives the long-shore flow, ζ c = ξ(x) + sy. Consistent with the Mid-Atlantic Bight we take s 1 7 m/m (see F98). Over the length scales of interest the sea elevation only varies by a centimetre, at most, and so is small compared to the water depth. We use this to justify assuming H + ζ c H. Thus, the long-shore velocity is forced by wind stress and a pressure gradient. Generally τ y and s are oppositely signed so that the two forces act in unison. In fact, for most of this analysis, we assume that the wind stress is negligible and concentrate on flows driven by a pressure gradient. This is equivalent to a = 1 in F98 and is also the situation Restrepo (21) considered. Apart from the pressure gradient we assume that all other reference state current variables are independent of y. In the cross-shore direction we assume a simple anti- okes flow, i.e., U(x) = U (x). Given these assumptions, the momentum equation (2.1) in the long-shore (y) direction becomes which gives gs = τ y rv (x), H(x) V (x) = τ y gsh(x). r The sea elevation is determined by the cross-shore (x) component of (2.1), ( ) gξ + G + U 2 = f(v + V V ru ) + V + x 2 x H, where ξ(x) is chosen so as to balance these terms.

13 4. Linear ability Analysis The steady state is perturbed according to Shore-Connected Ridges 13 (U, V ) = ( U (x) + u, V (x) + v), ζ c = ζ c (x, y) + z, H = H(x) h. The bottom perturbation, h, also perturbs the waves: θ = θ(x) + o, k = k(x) + p, A 2 = A 2 (x) + q. In the same manner that the wavenumber may be diagnosed from the bottom topography, the wavenumber perturbation, p, is enslaved to the bottom perturbation. The linear relationship between the two is p = k H h, where k H = 2k 2 sinh(2kh) + 2kH. Linearizing the equations for the wave-angle perturbation, o, and the wave amplitude perturbation, q, gives o t = W ooo + W oh h q t = W qoo + W qq q + W qh h (4.1) where the five W-operators are given in Appendix A. In the absence of bottom perturbations (i.e., h = ), the wave perturbations, o and q, decay to zero. We may see that these perturbations are enslaved to h by considering the case where h = (i.e., there are no bottom perturbations) and showing that, in this case, the wave angle and amplitude perturbations are linearly decaying. When h = the equation for the wave angle perturbation is o t = W ooo. The first two terms in W oo o are merely advection terms and the perturbation is assumed to be zero out to sea and bounded for all y. The third and fourth terms are the only ones

14 14 E. M. Lane and J. M. Restrepo that could cause instability. However, given our reference state, sin(θ) and θ x always have the same sign and H x is always positive so that wave angle perturbations will always decay in the absence of bottom topography perturbations. As the wave angle perturbations decay when h =, we need only consider q t = W qqq when determining the stability of the wave amplitude in the absence of bottom topography perturbations. Again, we will ignore the second term which only represents advection in the y direction. This leaves q t = x [ C g cos(θ)q] = [ C x x g q ]. We are interested in whether the perturbation q grows or decays in time and thus will consider q = ˆqe λ t, which gives Renaming Q(x) = C x g ˆq this becomes as C x g <. We may solve for Q giving λˆq = [ C x x g ˆq ]. Q x = λq(x) Cg x = λ Q(x) Cg x, Q(x) = Q()eR x λ C x g dx. However lim x Q = as Cg x is bounded at infinity and lim x q =. This means that λ must be negative and hence q must decay with time. Thus, in the absence of bottom perturbations the wave perturbations decay to zero. So all wave perturbations are enslaved to the bottom perturbations. Furthermore, the time scale for wave evolution is shorter than the time scale for the bottom topography. Thus, we assume that these adjustments are essentially instantaneous and solve for o and q as linear functions of h by setting the left hand side of (4.1) to zero.

15 The linearized equations for the perturbations are Shore-Connected Ridges 15 u t = M uuu + M uv v + M uz z + M uh h + M uo o + M uq q, v t = M vuu + M vv v + M vz z + M vh h + M vo o + M vq q, z t h t = M zuu + M zv v + M zz z + M zh h + M zo o + M zq q, 1 h α t = M huu + M hv v + M hh h + M ho o + M hq q, (4.2) where the M-operators are given in Appendix A. (4.2) may be written in matrix form as where B n = Mn, (4.3) t 1 u 1 v B =, n =, (4.4) 1 1 ζ 1 α h and M(x, x, y ) is a differential operator in x and y. The dependence on o and q is suppressed here because they can be represented as linear functions of h, as mentioned above. If we assume n(x, y, t) = ˆn(x, K y )e λt+ikyy, for λ C, K y >, then we may rewrite (4.3) as the eigenvalue problem with K y as a parameter. Bλˆn = M(x, x, ik y)ˆn, (4.5) We assume that the perturbations decay to zero for sufficiently large x and use this condition in the eigenvalue problem (4.5). The boundary condition at the shore end for the velocity is the linearization of U() = U (). For the bottom topography either

16 16 E. M. Lane and J. M. Restrepo Dirichlet, h() = or Neumann h x= = boundary conditions are prescribed. Dirichlet x boundary conditions represent the situation where the bottom topography perturbation is zero at the shoreward boundary. Neumann boundary conditions do not fix the perturbation at x = but rather specify that the flux is zero at this boundary. Recalling that the water has depth H at x =, this allows for the possibility that the perturbation may not disappear at this point. F98 used the former of these two boundary conditions and unless stated otherwise so do we. We take the opportunity in this study, however, to investigate how the results are affected by this alternative boundary conditions as it is not obvious what physical mechanism would hold the perturbation at zero at the shoreward boundary. We solve (4.5) for eigenvalues λ(k y ), and eigenfunctions ˆn(x, K y ), using the method outlined in Appendix B. For a given K y, an eigenfunction solution represents a mode with a periodicity 2π K y in y with a growth rate given by the real part of λ, R(λ). R(λ) positive (negative) indicates a growing (decaying) solution. A positive (negative) imaginary part of λ, I(λ), indicates that the mode advects in the negative (positive) y direction with celerity I(λ) K y. The parameter α in B gives the ratio of the hydrodynamic time scale to the morphological time scale and is assumed to be very small. F98 exploits this disparity in time scales to justify approximating B by B = 1 (4.6) (Note that there is a factor of α difference between these eigenvalues and those of the full problem). This is what we call the adiabatic approximation. Using this approximation,

17 Shore-Connected Ridges 17 the problem collapses to an eigenvalue problem in one variable, h. It assumes that, in the absence of changes in bottom topography, the flow is stable and as the bottom slowly changes, the flow merely relaxes to a slightly perturbed flow to account for changes in the bottom topography. In the adiabatic approximation the eigenvalue problem represents a situation wherein this relaxation happens instantaneously when compared with the morphological time scale. However, if the equilibrium flow were to lose stability due to changes in the bottom topography this approximation would no longer be valid as the unstable eigenvalues associated with the flow would be suppressed by this simplification. Furthermore, it might be the case that for certain equilibrium flows and bottom topographies the adiabatic approximation is appropriate, but in others not. In what follows we will investigate the adiabatic approximation in connection with a particular equilibrium solution. We know from F98 that for lim α the solutions are morphologically similar to shore-connected ridges. Restrepo (21) suggests, on the other hand, that for α large the perturbation solutions for the bottom topography may not resemble shoreconnected ridges. Below we intend to determine, for the particular equilibrium solution in question whether waves are present or not, how large α can be before the solutions deviate significantly from those obtained by F98. We will use the linear stability analysis to investigate the following questions: Waves versus no-waves Using our equations, we investigate how the presence of waves affect the results obtained when currents alone compose the velocity field of the fluid. Wave parameters θ, A, σ We investigate how changes in the angle of incidence, amplitude and frequency of the waves manifest themselves in the topography of the erodible bed. Bottom topography

18 18 E. M. Lane and J. M. Restrepo We investigate how the slope of the equilibrium bottom topography affects the outcome. Our default bottom topography is described in section 3. We also look at the linear stability analysis when the bottom topography is described by H(x) = H + (H H )e x L. Boundary conditions We investigate how the numerical calculation of the eigenfunctions depends on the choice of boundary condition on h at x =. In particular, we compare when h is set to zero at x = (Dirichlet), as compared to when dh/dx is set to zero (Neumann). These represent no perturbation and no perturbation flux at the x = boundary respectively. Following F98, our default boundary condition is the Dirichlet condition. Adiabatic assumption We determine a range of parameter values for which the adiabatic assumption is justified for the particular equilibrium solution that leads to shore-connected ridges and see how dropping this assumption changes the outcome of the linear stability analysis. Wave mechanisms We compare models with different levels of feedback to attempt to isolate the mechanisms by which the waves affect the outcome of the calculations. We also characterize the convergence/divergence of the waves and currents, a phenomenon that is instrumental in the evolution of the bottom topography and the formation of shore-connected ridge structures Results and Analysis We will focus on a problem examined in detail in F98. When waves are suppressed the parameters are chosen to agree with those in F98. The flow is in the negative y direction (corresponding to a Southward-directed flow in the Northern hemisphere) The current

19 Shore-Connected Ridges 19 Table 1. Reference values for bottom topography, currents, and waves; values are consistent with those used in F98 and Restrepo (21). Unless otherwise stated these are the values that are used in all of the numerical calculations. Parameter Value Units Description H 15 (m) Depth at shore end H 2 (m) Depth out to sea L 12 (km) Horizontal length scale β (m/m) bottom slope = H H L f 1 4 (radians s 1 ) Coriolis parameter γ.8 (dimensionless) Coulomb potential s (m/m) long-shore sea elevation slope r (m/s) bottom friction τ y (m 2 s 2 ) Wind ress A.8 (m) Wave Amplitude θ σ π 18 (radians) angle of incidence π 5 (radians s 1 ) frequency α.1 (s/s) ratio of time scales speed is taken as 25 cm/s, similar in magnitude to values reported for the the Mid Atlantic Bight case. In what follows we fix the wave period to be 1 seconds. The reference bottom topography is a smoothed linear function. The reference bottom slope decreases 5 m over 12 km. In the computations in F98 (and in their subsequent papers) a piecewise linear representation for the bottom topography is used. However, the jump discontinuity in the derivative causes Gibb s oscillations in the spectral approximation of the eigenvalue problem. To circumvent this problem we use a hyperbolic tangent function for the derivative, so that the bottom topography is smooth. This considerably

20 2 E. M. Lane and J. M. Restrepo improves the numerical approximations without affecting the comparison of outcomes to those in the F98 study. Unless otherwise stated, we assume the parameters are as given in Table 1. Recall that < θ < π/2 means that the waves come toward the shore from the North and East direction as depicted in Figure 1. Figures 2 shows the growth rate as solid lines plotted against the non-dimensionalized alongshore wavenumber of the instability K y L, for various wave-angles θ. In all cases A =.8 m. In contrast, the growth rates for the case of no waves are shown as dashed lines, superimposed on all the wave cases. Generally, the waves tend to suppress the instability. This suppression increases in strength with higher K y. Thus, the magnitude of the instability falls off faster as K y increases when waves are present, compared with the no-waves case. For low K y, the instability is not appreciably affected by the addition of waves. Sometimes, it is slightly augmented by the waves, shifting the peak of the growth rate towards lower values of K y. The waves affect the growth rate of different modes differently. In the no-wave case the modes have a definite ranking with the fastest growing mode being independent of K y ; whereas with waves, one mode may be fastest growing for low values of K y while a different mode may be fastest growing for higher values of K y. The main effect of the waves on the celerity is to increase or decrease it proportional to the magnitude of the alongshore okes drift. In the absence of waves the celerity is the same as that given by F98. A wave with amplitude 8 cm and angle θ = π 8 increases the celerity by around 2 %. Other values of A and θ affect the celerity proportionally to A 2 sin(θ) (i.e., in the same way as they would affect the okes drift). The variation in the celerity of different modes increases with the addition of waves. Changes to the fre-

21 Shore-Connected Ridges x 1 7 Growth Rate R(λ) θ=π/ x 1 7 θ=π/8 1.5 θ=π/18 θ=π/36 θ= π/4 θ= π/8 θ= π/18 θ= π/ x x x K y L Growth Rate R(λ) x x x K y L Figure 2. Growth rate as a function of K yl for different angles of incidence of the waves. The dashed lines represent the outcome when the waves are suppressed. The waves have an amplitude A =.8 m. quency (and thus also changes to the wavenumber k), however, do not show a significant effect on the celerity. The growth rates also show that the modes that are the fastest growing depend upon the direction θ of the incoming waves. For θ negative, the first three unstable modes are the same as without waves. Without waves there is a fourth mode that is also slightly unstable, but this is damped out by the addition of the waves. For waves coming in closer to perpendicular to the shore, a fourth mode becomes unstable. This is a different mode

22 22 E. M. Lane and J. M. Restrepo No Waves θ=π/ θ=π/18 θ=π/ First Mode Second Mode Third Mode Figure 3. Largest three unstable modes for peak growth rate for the no-wave case and also for waves with angle of incidence θ = π, π, π. The wave-induced mode is the first mode for θ = π, second for θ = π, and third for θ = π. Wave amplitude is A =.8m. The shoreward end is the left side of the rectangles. The axes have dimensions of km. to the one previously mentioned which is suppressed by the waves. Henceforth, we refer to this mode as the wave-induced mode. This wave-induced mode first appears when θ is around π 12. The growth rate of this new mode is larger for waves with a larger positive angle of incidence, θ. At θ = π 18 its growth rate is only slightly smaller than the first original mode and for θ larger than this it is, in fact, the fastest growing mode. The shape of this wave-induced unstable mode can be seen in Figure 3. It is pushed up against the shoreline and is slightly up-current rotated, but is far closer to perpendicular with the shoreline than the other modes. When the waves are very small (say A =.1 m and below) they have little effect

23 Shore-Connected Ridges 23 No Waves θ= π/ θ= π/36 θ= π/ First Mode Second Mode Third Mode Figure 4. Largest three unstable modes for peak growth rate for the no-wave case and also waves with angle of incidence θ = π, π, π. Wave amplitude A =.8m on the outcome, as compared to the no-wave case. There is a small quantitative change, more pronounced in the slower growing unstable modes, but no qualitative changes. For larger amplitude waves, however, changes can be seen. Figure 5 shows how the amplitude of the waves affects the growth rate of the instability for waves coming in at an angle of θ = π 18 (close to the observed prevailing swell direction in the Middle Atlantic Bite). Figure 6 gives contour plots of the most unstable modes corresponding to some of the cases considered in Figure 5. Again, the wave-induced mode can be seen increasing in importance. The wave-induced mode is stable for A =.1 m, the fourth unstable mode in this case is the same as the fourth mode in the no-wave case. For larger values of A the wave-induced mode is unstable and its growth rate increases with increasing wave

24 24 E. M. Lane and J. M. Restrepo A =.1m A =.2m A =.4m A =.6m 1.5 x Growth Rate R(λ) x x x K y L Figure 5. Growth rate as a function of K yl and amplitude of the waves. The dashed lines represent the instabilities due to currents alone. The angle of incidence of the waves is fixed at θ = π 18 The wave amplitudes are A =.1,.2,.4 and.6 m A =.2m A =.4m A =.6m First Mode 1 2 Second Mode 1 2 Third Mode Figure 6. Largest three unstable modes for peak growth rate for waves with amplitudes A =.2,.4 and.6 m; in all cases θ = π. Units on the axes are km. 18 amplitude. For A =.2 m and.4 m it is the third largest unstable mode and for A =.6 m it is the second largest unstable mode. As the wave amplitude increases, the smoothing effect of the waves, which suppresses

25 Shore-Connected Ridges 25 the instability, starts to dominate. For A = 1.5 m there is no instability for large K y and far slower growing instability than the no-wave case for smaller values of K y. Note that for A = 1.5 m, the wave slope ɛ = k A is around.7 and thus the wave equations derived in MRL4 remain valid. Figure 6 shows the first three modes for the value of K y which gives the maximum growth rate of the fastest growing mode. These modes are shown for different values of negative θ, with the amplitude fixed at A =.8m. Figure 3 shows the wave-induced mode gaining precedence for larger positive values of θ. This mode is clearly different from the unstable modes in the no-wave case. Another striking feature about the unstable modes for waves with θ positive or only slightly negative, is the number of modes which are long narrow features forming at an acute angle with the shoreline, very similar to shore-connected ridges. Without the waves, the first mode has this feature but all the subsequent modes are patchy with the second mode having two layers of patches, the third, three and the fourth, four (see the top row of Figures 3 and 4). Figure 4 shows how these modes develop into the long narrow ridges. For θ = π 8, the modes are still obviously the same patchy modes as in the no-wave case. For θ = π 18, diagonally opposite patches begin to merge. When θ = π 36, there are still lobes on either end but the intermediary patches have merged. For θ positive, there is very little sign of the original patchiness left. The modes form long skinny ridges at a steep angle with the shore. Thus, the waves alter the unstable modes with smaller growth rates. This has no effect on the first original mode, however, as it always was linear, up-current rotated ridges. Nonlinear modelling of the sediment problem in Calvete et al. (1999), however, suggests that the lesser unstable modes may also be important in explaining the final bottom topography after the instability has

26 26 E. M. Lane and J. M. Restrepo saturated. Thus, this change in the shape of the mode due to waves may enhance the formation of shore-connected ridges. In the numerical calculations reported here we have set the wave period to ten seconds but periods of 15 and 2 seconds were also tried. We observed little qualitative change in the linear stability analysis for period variations within this range. The celerity was not significantly affected by changes in the period on the waves. The reference bottom topography also affects the instabilities that form. The reference standard is a smoothed linear bottom topography. The reference bottom slope decreases 5 m over 12 km. As this slope is decreased, the growth rate of the instabilities also decreases. Trowbridge (1995) and F98 report this for the no-wave case, but it is even more dramatic for the wave case. For a 5% reduction in bottom slope, the only instabilities not completely damped out are for waves coming in against the current (i.e., θ π 8 ). For comparison, the same reduction in the no-wave case reduces the growth rates of the first three unstable modes by around 5% and suppresses the fourth unstable mode. Thus, with waves in the model, the gradient of H has a far larger effect. An exponential bottom topography does not significantly change the instabilities compared with a linear bottom. The first mode from the no-wave case and the wave-induced mode are pushed slightly further inshore (where the bottom slope is higher). The patchy modes, while they become longer and skinnier with the addition of waves, still show bumps along their length and so are not as completely smoothed out by the waves as for the linear bottom. However, the overall features remain qualitatively similar to the case with the linear bottom. The boundary condition for h at x = does not affect the morphology of the instabilities significantly. There are quantitative differences in the growth rates and the shapes of the modes near x = change to take into account the different boundary conditions,

27 Shore-Connected Ridges 27 1 (a) (b) (c) (d) x (km) Figure 7. First (a) and second (b) fastest growing modes for Dirichlet boundary conditions on h. First (c) and second (d) fastest growing modes corresponding to using Neumann boundary conditions on h. θ = π, A =.8 m and KyL = 1. Dashed lines are the real part, dash-dotted 18 lines the imaginary part and solid lines are the absolute value. as can be seen in Figure 7. Besides this, the gross features of the instabilities remain unchanged. When we employ Dirichlet boundary conditions on the bottom topography perturbation this wave-induced mode has a boundary layer near x = where its value goes to zero. See Figure 7b. However, as can be seen from Figure 7d, with Neumann boundary conditions no boundary layer is generated and so this mode is right up against the shoreline. Because the alignment of this mode is closer to perpendicular to the shoreline, it is unlikely that this mode augments the formation of shore connected ridges. For θ π 15, which is the dominant direction of waves in the case of the Mid-Atlantic Bight, this wave-induced mode and the fastest growing mode from the no-wave case are almost equal in growth rate and so linear stability analysis does not tell us how the struggle for dominance between the two modes will play out. Numerical experiments to test the adiabatic approximation confirm that this tends to be a reasonable assumption, provided α is chosen to be sufficiently small: One characteristic of the adiabatic approximation is that the reduced problem has fewer eigenvalues

28 28 E. M. Lane and J. M. Restrepo and the only eigenvalues with positive real part in the reduced problem appear to be legitimate eigenvalues according to the tests given in Appendix B, whereas, in the full problem there are many, obviously spurious, numerical eigenvalues that also have positive real part. This is possibly because the adiabatic approximation cuts down the degrees of freedom of the problem. We considered the full problem using B as given by (4.4) rather than by (4.6). As long as α is sufficiently small (α 1 2 suffices which corresponds to the morphological timescale being around 15 times greater than the hydrodynamic timescale), there are only small quantitative differences between the solution with and without the adiabatic approximation, i.e., the full problem is only a perturbation of the reduced problem. For α larger the differences are still not drastic but gradually the growth rates and mode shapes change. Furthermore, when the bottom topography is held constant (i.e., h = ), the current solution is found to be linearly stable. Thus, the current perturbations are enslaved to the bottom topography. The addition of waves does not change this stability. This is because the contribution of the waves comes into the purely imaginary part of the spectrum of the operator. α does not have any significant affect on the celerity of the instability. 5. Wave Effects In order to better understand the effect that waves have on the bottom topography we ran several experiments: No forced current. We consider the effect of the waves when the wind stress and pressure gradient are both set to zero. Wave pathways. We investigate the different pathways by which the waves affect the erodible bed.

29 Shore-Connected Ridges 29 Convergence. We look at the convergence patterns of the waves and currents and correlate them to the bottom topography patterns No Forced Current We consider the effect of the waves when the wind stress τ and pressure gradient s are both set to zero. The reference long-shore current is thus zero and the cross-shore flow balances the cross-shore okes drift. The only net movement of particles is due to the long-shore okes drift. In this case, the real parts of all the eigenvalues are negative. There are no linear instabilities. For larger K y, R(λ) is more negative. The waves smooth away any perturbation that forms, smoothing higher frequencies faster. Some of this smoothing is due to the diffusive term. However, because there is no current, the effect of the diffusive term is reduced as the diffusion coefficient is proportional to the sum of the current and the okes drift. The real parts of the eigenvalues tend to be around zero for K y small. For larger values of K y they are larger and more negative. This concurs with the fact that, in the full problem, the waves suppress the instability more strongly for higher frequencies. If the waves are exerting a net smoothing effect on the bottom topography and that smoothing is more pronounced for higher frequencies then we would expect what we see in Figures 2 and 5. The fact that the waves by themselves only smooth the bottom topography suggests that the new instability that occurs for positive θ in the presence of waves is not simply a wave instability but must also be due to some interaction with the currents Wave Pathways As illustrated in Figure 8, there are three pathways by which the waves may affect the erodible bed. These are:

30 3 E. M. Lane and J. M. Restrepo U,V,ζ u, v, z 3 H h θ, k, A o, p, q Figure 8. Schematic diagram of how variables influence each other. (1), (2) and (3) are the three pathways by which the waves influence the erodible bottom corresponding to those described in the text. (a) The original addition of the equilibrium wave solution to the current and their effect on the perturbations of the current and the bottom topography. We refer to this as the primary wave effect. (b) The perturbation of the bottom topography affects the waves. This can cause convergences of the okes drift, which in turn affects the bottom topography. We call this the direct wave feedback. (c) The wave perturbation also feeds back into the bottom topography indirectly. The wave perturbation alters the current perturbation, which in turn alters the bottom topography perturbation. We call this the indirect wave feedback. In order to understand the effects of these three pathways, we run three numerical experiments where we shut off different pathways. In the first experiment we keep the primary wave effect, however, we assume that there is no feedback from the bottom topography to the waves, i.e., we retain pathway (1) but remove pathways (2) and (3). The second experiment allows feedback from the erodible bed to the waves and back, but only directly, via the convergence of the okes drift, i.e., we keep pathways (1) and (2) but

31 Shore-Connected Ridges 31 remove pathway (3). The only feedback in the third experiment is indirect, via the current perturbations, i.e., we keep pathways (1) and (3) but remove pathway (2). Note that, because it does not make sense to have a wave perturbation without first having a wave, we do not consider cases where we have wave feedback (pathways 2 and 3) without having the primary wave effect (pathway 1). The results from the first and second experiments are qualitatively quite similar. The first experiment is more numerically stable, however. For most values of θ, there is only one unstable mode. For large positive values of θ (θ π 8 ) the first experiment has a second unstable mode. The modes are only unstable for smaller values of K y. They are stable if K y is large enough (K y L > 15, say). The main unstable mode seems to be a cross between the wave-induced mode and the first original mode. Like the wave-induced mode, it is pushed right up against the shore, and when Dirichlet boundary conditions are used, a boundary layer forms. However, the shape of the mode and its angle with the shore (irrespective of wave-angle θ) is more reminiscent of the first original mode. The second mode is also pushed up against the shore. It is somewhat patchy with a second (smaller) row of humps and depressions further sea-ward. In the third experiment the wave-induced mode is not seen, however, the original unstable modes are seen. For positive θ, the patchy modes are drawn out into long skinny modes as in the full wave case but for θ = π 36 the patches are beginning to show through and for θ = π 36 the modes are basically the same as the no-wave case. Thus, with just the primary wave effect and either no feedback or only the direct feedback, the unstable mode is most similar to the wave-induced mode. With the indirect wave feedback we get the original modes which show some of the effects from the waves, but the wave-induced mode is not present. This is interesting, as it suggests that the wave-induced mode is caused by the addition of the waves, with the feedback between

32 32 E. M. Lane and J. M. Restrepo the bottom topography and the waves playing only a minor role in its final shape. The direct wave feedback via the okes drift has only a small quantitative effect. The way the waves influence the currents, however, seems to balance out some of the effect of the waves in the first place as the original modes are once more unstable when these effects are included. Inclusion of both the direct and indirect feedback pathways, i.e., the full model, produces both the wave-induced mode and the original modes Convergence The bottom topography evolution equation has a gradient of a flux with two components: a convergence term and a diffusion term. The diffusion term represents smoothing of the bottom. Moreover, it is physically significant as it sets a length scale on the ridges. Instabilities occurs due to the convergence term. Our model differs from that of F98 in that the bottom topography is affected by the convergence of the current plus the vertically integrated okes drift, (U + U ). In order to understand the way that the waves change the shapes of the existing modes and enhance the wave-induced mode we look at these convergences. In the linear stability analysis the convergence becomes [ ) du (u + x dh h + )] dv (v + y dh h, where we are assuming that the wave components are enslaved to h, i.e., du dh = U k k h + U θ θ h + U A 2 A 2 h U H. The quantities U / k, U / θ, U / A 2 and U / H are straightforward to calculate: U k = g A 2 2σ cos θ sin θ,