Inverse Modeling of One-Dimensional Setup and Alongshore Current in the Nearshore

Falk Feddersen Scripps Institution of Oceanography, La Jolla, California

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R. T. Guza Scripps Institution of Oceanography, La Jolla, California

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Steve Elgar Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

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Abstract

Inverse models are developed that use data and dynamics to estimate optimally the breaking-wave-driven setup and alongshore current, as well as the cross-shore forcing, alongshore forcing, and drag coefficient. The inverse models accurately reproduce these quantities in a synthetic barred-beach example. The method is applied to one case example each from the Duck94 and SandyDuck field experiments. Both inverse solutions pass consistency tests developed for the inverse method and have forcing corrections similar to a roller model and significant cross-shore variation of the drag coefficient. The inverse drag coefficient is related to the wave dissipation, a bulk measure of the turbulence source, but not to the bed roughness, consistent with the hypothesis that breaking-wave-generated turbulence increases the drag coefficient. Inverse solutions from a wider range of conditions are required to establish the generality of these results.

Corresponding author address: Dr. Falk Feddersen, Scripps Institution of Oceanography 0209, 9500 Gilman Dr., La Jolla, CA 92093- 0209. Email: falk@coast.ucsd.edu

Abstract

Inverse models are developed that use data and dynamics to estimate optimally the breaking-wave-driven setup and alongshore current, as well as the cross-shore forcing, alongshore forcing, and drag coefficient. The inverse models accurately reproduce these quantities in a synthetic barred-beach example. The method is applied to one case example each from the Duck94 and SandyDuck field experiments. Both inverse solutions pass consistency tests developed for the inverse method and have forcing corrections similar to a roller model and significant cross-shore variation of the drag coefficient. The inverse drag coefficient is related to the wave dissipation, a bulk measure of the turbulence source, but not to the bed roughness, consistent with the hypothesis that breaking-wave-generated turbulence increases the drag coefficient. Inverse solutions from a wider range of conditions are required to establish the generality of these results.

Corresponding author address: Dr. Falk Feddersen, Scripps Institution of Oceanography 0209, 9500 Gilman Dr., La Jolla, CA 92093- 0209. Email: falk@coast.ucsd.edu

1. Introduction

Models for breaking-wave-driven nearshore circulation often are based on the depth-integrated, time-averaged, and constant-density Navier–Stokes equations and are simplified by assuming that all variables are independent of the alongshore coordinate y and time (i.e., steady). The cross-shore momentum equation becomes a one-dimensional (1D) balance between the cross-shore pressure gradient and the total (wind plus wave) cross-shore forcing Fx (e.g., Longuet-Higgins and Stewart 1964):
i1520-0485-34-4-920-e1
where g is gravitational acceleration, h is the water depth, x is the cross-shore coordinate, and η is the time- averaged free surface elevation relative to mean sea level without waves (i.e., setup).
The alongshore momentum equation is a 1D balance between the alongshore forcing Fy, bottom stress, and lateral mixing (e.g., Longuet-Higgins 1970):
i1520-0485-34-4-920-e2
where υ is the mean (time and depth averaged) alongshore current. The second term in (2) is a common bottom stress representation (Longuet-Higgins 1970; Thornton and Guza 1986; Garcez-Faria et al. 1998; and many others), where cd is a nondimensional drag coefficient, 〈〉 represents a time average over many wave periods, |u| is the total instantaneous horizontal velocity vector, and υ is the instantaneous alongshore velocity. Mean and wave-orbital velocities contribute to 〈|u|υ〉. The third term in (2) represents lateral mixing processes (ν is an eddy viscosity) including shear dispersion (Svendsen and Putrevu 1994), shear waves (Slinn et al. 1998; Özkan-Haller and Kirby 1999), and small-scale turbulent mixing by breaking waves (Battjes 1975). The cross- and alongshore forcings are the sum of wind (τwindx and τwindy) and wave forcing and are given by
i1520-0485-34-4-920-eq1
where ρ is the water density and Sxx and Syx are components of the radiation stress tensor (Longuet-Higgins and Stewart 1964).

The 1D setup [(1)] and alongshore current [(2)] dynamics are applicable to many laboratory and field situations (e.g., Bowen et al. 1968; Battjes and Stive 1985; Thornton and Guza 1986; and many others). Although simple, except for the pressure gradient term, the functional forms of the terms in (1) and (2) are not known and must be parameterized for use in models. Linear theory relates the wave forcing to the root-mean-square (rms) wave height Hrms, mean wave angle θ, and mean frequency f, quantities predicted by bulk wave transformation models (e.g., Thornton and Guza 1983). However, linear-theory-based surf-zone wave-forcing parameterizations are not sufficiently accurate for detailed alongshore current modeling on a barred beach (Church and Thornton 1993; Reniers and Battjes 1997; Ruessink et al. 2001). An additional water column stress due to the aerated front face of a broken wave (wave roller) has been hypothesized to shift the wave forcing shoreward. This concept is applied in heuristic roller models (e.g., Stive and de Vriend 1994) based on towed wavefoil experiments (Duncan 1981). Inclusion of a roller model with tuned parameters results in improved agreement with observations on barred laboratory (η and υ; Reniers and Battjes 1997) and natural (υ; Ruessink et al. 2001) beaches.

The closure of 1D integrated alongshore momentum balances on cross-shore transects (Feddersen et al. 1998; Feddersen and Guza 2003) suggests that cd〈|u|υ〉 adequately represents the bottom stress. A spatially constant cd often has been used in models (Longuet-Higgins 1970; Thornton and Guza 1986; Özkan-Haller and Kirby 1999). However, within the surf zone cd is elevated relative to seaward of the surf zone (Feddersen et al. 1998). A drag coefficient proportional to h−1/3 (cd increases in shallower depths) improves 1D υ model predictions compared with a constant cd (Ruessink et al. 2001). The elevated surf zone or shallow-water cd has been hypothesized to result from increased bottom roughness (e.g., Garcez-Faria et al. 1998) or breaking- wave-generated turbulence (e.g., Church and Thornton 1993), but the spatial variation of cd is not understood.

The wave forcing, cd, and the Reynolds stress terms are difficult to estimate directly, and therefore the quality of their parameterizations is not known. Instead, parameterizations are accepted or rejected by the accuracy of the model predictions. Parameterizations often can be tuned so that model predictions match a limited dataset and thus, rarely are rejected. Here, an inverse method is developed (section 2) that uses the setup and alongshore current observations and dynamics to solve for parameterized quantities, namely, the cross- and alongshore forcing and the drag coefficient. Specification of the measurement error variances and parameterized forcing and drag coefficient error covariances is required. Inverse solutions not consistent with the specified measurement and parameterization errors are considered spurious and are rejected. The inverse method is tested with a synthetic barred-beach example with known forcing and cd (section 3), and works well given the number and quality of field observations typically available. The inverse method is applied to one case example each from the Duck94 and SandyDuck field experiments (section 4). The case example inverse results are discussed in the context of wave rollers and possible drag coefficient dependence on breaking-wave- generated turbulence and bed roughness (section 5). The results are summarized in section 6.

2. Inverse modeling

a. Prior model and prior solutions

The equation for the setup (1) is linearized (i.e., the still-water depth h is used instead of h + η) to simplify the inverse problem. Prior model solutions with η included and excluded in h are similar in water depths ≥ 0.3 m, where the case example observations were obtained. The parameterized cross- and alongshore forcings are denoted as the prior forcing F(pr)x and F(pr)y. The prior drag coefficient c(pr)d is constant. The offshore (x = L) prior boundary condition for the setup model (1) is η = 0. The alongshore current model (2) uses prior slip boundary conditions dυ/dx = 0 at the shoreline (x = 0) and offshore (x = L) boundaries. The quadratic velocity term in the bottom stress is parameterized with 〈|u|υ〉 = B(υ) = σTυ[1.162 + (υ/σT)2]1/2 (Feddersen et al. 2000), where σ2T is the wave-orbital velocity variance. With specified prior forcing, drag coefficient, and boundary conditions, (1) and (2) yield the prior setup η(pr) and alongshore current υ(pr).

b. Setup inverse modeling

Error in the setup dynamics fx(x), attributed to error in the prior wave forcing, is allowed on the right-hand side of the cross-shore momentum equation [(1)]. The inverse forcing is given by F(i)x = F(pr)xfx. The forcing error (or correction) fx is assumed to be a zero-mean continuous Gaussian random variable with covariance Cfx(x, x′) = E[fx(x)fx(x′)]. Similarly, zero-mean Gaussian error with prior variance σ2ηL is allowed in the prior setup boundary condition η(L) = 0. The M noisy η observations d(η)m (m = 1, … , M) consist of signal and measurement error so that
d(η)mηxme(η)m
where measurement errors e(η)m are considered zero- mean identical and independent Gaussian random variables with (prior) variance σ2ηd.
Inverse estimates of η and fx that incorporate dynamics and data are found by minimizing a cost function that is a combination of dynamical, boundary condition, and data errors (e.g., Bennett 1992):
i1520-0485-34-4-920-e3
where each component of (3) is weighted by its inverse covariance. The minimum of the cost function yields the inverse setup η(i) and forcing F(i)x. With each term interpreted as a Gaussian random variable, cost-function minimization corresponds to maximum likelihood estimation (appendix A), and the (statistical) consistency of the inverse solutions with the prior assumptions (i.e., covariances) can be tested (appendix B).
The inverse of Cfx(x, x′) is defined so that
i1520-0485-34-4-920-e4
where δ(x) is the Dirac delta function. The η adjoint λη is defined as the convolution of fx with C−1fx,
i1520-0485-34-4-920-e5
so that
i1520-0485-34-4-920-eq3
Setting the first variation of the cost function J[η] to zero yields the Euler–Lagrange equations for the minimum of J[η]:
i1520-0485-34-4-920-e6a
which are solved directly for the inverse solutions η(i) and f(i)x [or F(i)x].
At the minimum, the cost function J[η] is rewritten, after integrating by parts, as
i1520-0485-34-4-920-e7
where Jmin is the minimum of the cost function (3) found by solving (6), and η′ are deviations from the inverse solution. The curvature of the cost function at the minimum [C(i)η(x, x′)]−1 is interpreted as the inverse η covariance (appendix A). The prior η covariance C(pr)η(x, x′) is related to the forcing error covariance by removing the data term from J[η] [(3)] and integrating by parts; that is,
i1520-0485-34-4-920-e8
(neglecting boundary terms). The prior covariance gives the η uncertainty when no data are available. The inverse η covariance C(i)η(x, x′) is then given by
i1520-0485-34-4-920-e9
The addition of data reduces C(i)η, thus reducing the uncertainty of the inverse solutions.

c. Alongshore current inverse modeling

Analogous to the inverse setup model, error in the alongshore current dynamics fy(x) is allowed on the right-hand side of (2), and represents error in the forcing, bottom stress, and lateral mixing. Because the forcing is considered to have the largest uncertainty and with the drag coefficient solved for separately, fy is ascribed to forcing error. Corrections to lateral mixing are neglected. The inverse alongshore wave forcing F(i)y is given by F(i)y = F(pr)yfy, and the forcing error (or correction) fy is assumed to be a zero-mean Gaussian random variable with covariance Cfy(x, x′). Errors in the prior slip boundary conditions are assumed to be zero- mean Gaussian random variables with variance σ2υx0 and σ2υxL at x = 0 and x = L. The inverse method also allows for drag coefficient deviations from the prior c(pr)d, adjusting the drag coefficient to make the inverse υ consistent with the data. The cd error is considered to be a zero-mean Gaussian random variable with prior covariance C(pr)cd(x, x′). The N noisy alongshore current observations d(υ)m consist of signal and measurement error, given by
d(υ)mυxme(υ)m
where em is zero-mean Gaussian measurement error with prior variance σ2υd.
The cost function I[υ, cd] is defined as a combination of dynamical, boundary condition, drag coefficient, and data errors:
i1520-0485-34-4-920-e10
where each component of the cost function (10) is weighted by its inverse covariance. The minimum of the cost function yields the inverse solutions. With the interpretation of each term as a Gaussian random variable, minimization of the cost function corresponds to maximum likelihood estimation (appendix A) and allows for testing the consistency of the inverse solution.
Setting the first variation of I[υ, cd] with respect to υ and cd to zero leads to the Euler–Lagrange equations for the cost function minimum:
i1520-0485-34-4-920-e11a
where the υ adjoint λυ is defined similarly to the η adjoint (5). The set of Euler–Lagrange equations (11) are nonlinear, ordinary differential equations for the inverse solutions υ(i), f(i)y, and c(i)d.
At the minimum, after linearizing and integrating by parts, the cost function I[υ, cd] is rewritten as
i1520-0485-34-4-920-e12
where Imin is the minimum of I[υ, cd] found by solving (11), and υ′ and cd are deviations from the inverse solutions. The curvatures of I[υ, cd] at the minimum {e.g., [C(i)υ(x, x′)]−1 and [C(i)cd]−1} are interpreted as inverse covariances (appendix A). Similarly, the prior υ covariance C(pr)υ is found by taking the first two terms of (10), linearizing about the prior υ solution, and integrating by parts, resulting in
i1520-0485-34-4-920-e13
(neglecting boundary terms). The inverse υ covariance C(i)υ is given by
i1520-0485-34-4-920-e14
with c(pr)d replaced by c(i)d in C(pr)υ and υ(i) used in dB;cldυ. As with η, the addition of data reduces the inverse uncertainty. The inverse cd covariance,
i1520-0485-34-4-920-e15
also is reduced relative to the prior because the first term in (15) is positive definite. The υcd covariance Cυ,cd is not discussed.

d. Prior covariances

Specifying the prior covariances is nontrivial. The covariance form chosen is a homogeneous (i.e., only a function of xx′) bell-shaped covariance often used in objective mapping (e.g., Brethereton et al. 1976):
i1520-0485-34-4-920-e16
where γ is either fx, fy, or cd. The fx, fy, and cd variances (σ2fx, σ2fy, σ2cd) and decorrelation length scales (lfx, lfy, and lcd) must be specified. This covariance form is a significant simplification. In particular, it is unlikely that the true forcing error covariances are homogeneous. Nevertheless, the inverse solutions with (16) appear to work well, as demonstrated below. Nonhomogeneous covariance forms similar to (16), but with (for example) σfx proportional to F(pr)x, were implemented. Results were similar to those using the homogeneous form (16). The homogeneous form was used for simplicity, because the form of the true covariances is unknown. Note that the homogeneous forcing error covariances, once filtered by the η and υ dynamics, result in nonhomogeneous prior and inverse η [(9)] and υ [(14)] covariances, and nonhomogeneous inverse cd [(15)] covariance.

3. Test of the inverse method

The ability of the inverse method to solve for the forcing and drag coefficient is tested with synthetic data. A true cross- and alongshore wave forcing (based on rollers) and a cross-shore variable cd yield [through (1) and (2)] the true η(tr) and υ(tr). Prior (nonroller) forcing and constant cd similarly yield the prior η(pr) and υ(pr) and reflect the imperfect knowledge of the dynamics. The true values represent the dynamical information that the inverse method should reproduce, given the prior values, noisy data, and assumptions about the errors.

a. True and prior conditions

Barred-beach bathymetry h from Duck, North Carolina (Lippmann et al. 1999), is used with a domain extending from the shoreline (x = 0 m) to 300 m offshore (Fig. 1a). The bar crest is located at x = 80 m and has a half-width of 15 m. At the offshore boundary, the wave height Hrms = 1.2 m, the wave period is 10 s, and the wave angle is 15° relative to shore-normal. The waves are transformed shoreward (Thornton and Guza 1983) over the barred bathymetry (Fig. 1b), yielding the (without rollers) prior wave forcing F(pr)x and F(pr)y (dark dashed curves in Figs. 1c,d), and also the wave-orbital velocity variance σ2T. A roller model (Stive and de Vriend 1994; Reniers and Battjes 1997; Ruessink et al. 2001) is used to calculate the true wave forcings F(tr)x and F(tr)y (solid curves in Figs. 1c,d). Relative to the prior, the roller model displaces shoreward and reduces the magnitude of the forcing peaks. In addition to the wave forcing, a spatially constant alongshore wind forcing of 10−4 m2 s−2 (roughly corresponding to a 14-kt alongshore wind) is added to the prior and true alongshore forcing.

Following Church and Thornton (1993), the true drag coefficient c(tr)d depends on the wave dissipation with a background (zero wave dissipation) value of 0.0015 (Fig. 1e). Maxima of cd, just offshore of the bar crest and near the shoreline, occur where breaking-wave dissipation is maximum. This cd is hypothetical and is used only to test the inverse method. A cd that depends inversely on water depth [e.g., the Manning–Strickler equation used by Ruessink et al. (2001)] gives qualitatively similar cd variation. The spatially constant c(pr)d = 0.0025 (Fig. 1e, dashed line) best fits the prior υ to the data. The spatially constant eddy viscosity ν = 0.5 m2 s−1 was used to model υ at two different barred beaches (Ruessink et al. 2001) and lies midway within the range of ν (0.1–0.9 m2 s−1) suggested by Özkan- Haller and Kirby (1999). With this eddy viscosity, the modeled magnitude of lateral mixing is small relative to the forcing (Ruessink et al. 2001).

These inputs are used within the setup [(1)] and alongshore current [(2)] models to generate true and prior η and υ (Figs. 2a,b). The sharp increase in the η(tr) and the main υ(tr) peak are moved onshore from the prior locations due to the roller and (for υ(tr)) by the reduced c(tr)d in the bar trough. Differences between the prior and true setup are significant (5 cm) in the bar–trough region. The difference between the prior and true υ exhibits the classic barred-beach model–data difference when roller models are not included (e.g., Church and Thornton 1993). Noisy η and υ data [d(η)m and d(υ)m, asterisks in Figs. 2a,b] are generated at eight locations by adding to the true values zero-mean Gaussian noise with standard deviations (referred to as std dev) of 0.004 m and 0.05 m s−1, representative of setup measurement (Raubenheimer et al. 2001) and electromagnetic current meter (Feddersen and Guza 2003) error, respectively. The eight data locations are typical of the cross-shore instrumented transects at Duck during the Duck Experiment on Low-Frequency and Incident-Band Longshore and Across-Shore Hydrodynamics (DELILAH), Duck94, and SandyDuck field experiments.

b. Prior covariances

The prior covariances of the forcing, drag coefficient, boundary condition, and data errors also must be specified. The data errors σηd = 0.004 m and συd = 0.05 m s−1 are those used to create the synthetic data. The magnitude of the forcing error is constrained by the prior forcing magnitude. Because the prior forcing is believed to be qualitatively correct, the forcing errors σfx and σfy are chosen to be 18% of the bar crest prior |F(pr)x| and |F(pr)y| maxima. The prior forcing ±2 std dev (σfx and σfy) are consistent with the true forcing (light dashed curves in Figs. 1c,d); however, this consistency cannot be examined in general with unknown true forcing. The cd error σcd = 0.0007 is chosen such that the c(pr)d ± 2σcd spans the expected cd range (0.001–0.004). The prior c(pr)d ± 2σcd also mostly contains the true cd (Fig. 1e). Because the bathymetry strongly controls the wave properties, the forcing and cd error length scales are chosen to match the sandbar half-width (lfx = lfy = lcd = 15 m). Varying the length scales between 10 and 30 m does not change the inverse solutions significantly in this or subsequent sections. The η offshore boundary condition error σηL = 0.01 m. The υ boundary condition errors are συx0 = 0.05 s−1 and συxL = 0.01 s−1, allowing for typical boundary shear of 1 m s−1 over 20 m at x = 0 m and 0.2 m s−1 over 20 m at x = L. The prior η and υ boundary condition errors do not affect the inverse solutions significantly. The prior covariances C(pr)η [(8)] and C(pr)υ [(13)] are estimated using the forcing covariances (light-dashed curves in Figs. 2a,b). One measure of consistency in the forcing error covariances is that most of the η and υ data are within two std dev (defined as the square root of the covariance diagonal) of the prior η and υ. This test can be applied in real inverse situations.

c. Inverse solution

With all the ingredients, the inverse method yields the inverse setup η(i) (Fig. 3a), inverse alongshore current υ(i) (Fig. 4a), and their covariances C(i)η [(9)] and C(i)υ [(14)]. The inverse solutions agree well with the true η(tr) and υ(tr) and are significant improvements over the prior solutions (Figs. 2a,b). The rms differences between inverse solutions and data are 2.4 mm and 2.6 cm s−1 for η and υ, respectively, consistent (at the 95% level) with the prior data error variance (appendix B). Inverse solutions should pass this test if they are to be accepted. The addition of data significantly reduces the uncertainty in the inverse solutions (the inverse η and υ std dev are 20%–50% of the prior). Note that the inverse solutions ±2 std dev usually contain the true solutions and the data. In regions with instrument gaps much larger than the 15-m decorrelation length scale (140 < x < 200 m), the inverse uncertainty increases.

The ability of the inverse method to reproduce the cross- and alongshore forcing is examined by comparing the inverse forcing corrections [f(i)x and f(i)y] with the true forcing corrections f(tr)x [i.e., F(tr)xF(pr)x] and f(tr)y (Figs. 3b and 4b). The inverse f(i)x and f(i)y qualitatively reproduce f(tr)x and f(tr)y, and result in significant improvements over the prior forcings. The location and magnitude of the forcing correction peaks are similar, although f(tr)x is underpredicted around x = 110 m. Onshore of the last data point (x = 20 m), without information (data) for the inverse, f(i)x and f(i)y relax to zero. The cross- and alongshore inverse forcing corrections are consistent with their prior covariances (appendix B). The c(i)d is consistent with the prior cd covariance and qualitatively reproduces c(tr)d in the bar crest–trough region where the data are concentrated (Fig. 4c). Onshore of x = 20 m and offshore of x = 150 m, the c(i)d relaxes back to the prior cd = 0.0015 both because of the data sparseness and because the inverse method can adjust fy to match the data with less cost. In the bar-trough region, the c(i)d uncertainties are reduced 15%–30% relative to the prior (Fig. 4c), which is less than the reduction in the η and υ uncertainties. These results suggest that the inverse method is capable of solving for the unknown surf-zone cross-shore forcing, alongshore forcing, and drag coefficient given the number and quality of data typically available.

d. Choosing covariance parameters

With the chosen covariance parameters, the inverse solutions pass the consistency tests and reproduce the true solutions. However, with real observations the choice of prior covariances is important and not straightforward. The effect of varying covariance parameters on the inverse solutions is examined to provide guidelines for general application. The boundary condition variances, data variances, and covariance length scales are held fixed at values used previously while σfx, σfy, and σcd are varied. The misfit between the true and inverse solutions is characterized by the metric
i1520-0485-34-4-920-e17
The integral spans the bar-crest region (x = 20–200 m) where data are concentrated, L is the integration distance (180 m), and α is any inverse or prior solution [e.g., F(i)x].

For the η inverse, σfx is varied between (%σfx) 5% and 50% of the bar-crest prior |F(pr)x| maximum (18% was used in Fig. 3). The F(i)x and η(i) misfits are reduced with increasing σfx (Fig. 5), and for %σfx > 10%, the F(i)x and η(i) misfits are 33%–50% and 25%–33% (respectively) of their prior misfits. The largest %σfx result in the smallest misfit. However, the inverse solutions with %σfx > 24% (right-hand shaded area in Fig. 5) are inconsistent because of data overfit (inverse solution matches data more closely than warranted, given the observational error), and solutions with %σfx < 16% (left-hand shaded area in Fig. 5) result in f(i)x being inconsistent with the prior Cfx. Inconsistent solutions are rejected. For the consistent solutions (16% ≤ %σfx ≤ 24%), the F(i)x and η(i) misfits are small, near the minimum misfits for all %σfx. Within the %σfx window of consistent solutions, larger %σfx are the most accurate. The f(i)x of the consistent solutions are almost identical.

For the υ inverse, σfy is varied between 5% and 50% (%σfy) of the maximum bar crest F(pr)y and σcd is varied between 10% and 45% of the constant c(pr)d. The inverse results in improved F(i)y with misfit 68% to 23% of the F(pr)y misfit, depending on σcd (Fig. 6a). For fixed %σfy, an increased σcd results in a larger F(i)y misfit (denoted by the arrow in Fig. 6a) because the inverse adjusts c(i)d instead of the forcing to fit the data. With adjustment of both the forcing and cd error, the υ(i) misfit reduction is dramatic {χ[υ(i)] is 20% of the υ(pr) misfit}. The c(i)d misfit χ[c(i)d] (not shown) is reduced only about 15% of the prior χ[c(pr)d] because of the cross-shore lag between c(i)d and c(tr)d maxima (Fig. 4c). As with the η inverse, the largest values of %σfy have the smallest misfits, but fail at least one consistency test (pluses in Fig. 6). The inverse solutions (asterisks in Fig. 6) that pass all consistency tests have small χ[υ(i)]. Consistent solutions with larger %σfy and σcd have small F(i)y misfit. Within the %σfy and σcd window of consistent solutions, larger values of these parameters are best. All consistent f(i)y and c(i)d solutions exhibit the pattern shown in Fig. 4, only with varying amplitude (factor of 2) depending on %σfy and σcd.

4. Case examples

The inverse method is applied to observations from two field experiments at Duck: Duck94 (Elgar et al. 1997; Feddersen et al. 1998; Ruessink et al. 2001) and SandyDuck (Elgar et al. 2001; Raubenheimer et al. 2001; Feddersen and Guza 2003; Noyes et al. 2004). Bathymetries are smoothed with a 10-m cutoff wavelength to remove bedforms that dominate the variance in the 1–5-m wavelength band (Thornton et al. 1998). All wave, setup, and alongshore current observations are based on hourly averages. In both cases the bathymetry is alongshore uniform, and the mean alongshore currents are consistent with 1D dynamics (Feddersen et al. 1998; Ruessink et al. 2001; Feddersen and Guza 2003).

a. Duck94 example

During Duck94, there were no setup observations, so only the alongshore current inverse method is applied. Wave breaking occurs offshore of and on the crest (x = 110 m) of a well-developed sandbar (Figs. 7a,b). In the bar trough (40–80 m from the shoreline), the wave height remains constant. A tuned 1D wave model (without rollers) (e.g., Thornton and Guza 1983) accurately (rms error 2.2 cm) predicts the wave height evolution (solid curve in Fig. 7b). The wave model (initialized with offshore Hrms and Sxy estimated from an array of pressure sensors in 8-m water depth), together with τwindy observations, gives the prior F(pr)y (Fig. 7c). The constant c(pr)d = 0.0015, based on alongshore momentum balances (Feddersen et al. 1998), results in similar data and prior υ(pr) peak magnitudes (Fig. 7d). The prior %σfy is 20% (light dashed curves in Fig. 7c), slightly larger than the largest test-case %σfy yielding consistent solutions (section 3d). The σcd = 4.5 × 10−4, 30% of the prior c(pr)d. The length scales (lfy and lcd) are 20 m, approximately the bar half-width. The eddy viscosity ν = 0.5 m2 s−1 is that used by Ruessink et al. (2001) to model a larger dataset from which one of the case examples is drawn. Inverse solutions with ν ranging between 0.1 and 2 m2 s−1 were similar, with smoother inverse solutions for larger ν (not shown). For these ν, the magnitude of the lateral mixing term was small and did not qualitatively change the results. The prior F(pr)y and c(pr)d (and prior covariances) are used to calculate υ(pr) and its error bars (Fig. 7d). Typical of barred-beach υ model runs without rollers, the prior υ(pr) rms errors (0.28 m s−1) are substantial.

The υ inverse method (section 2b), applied with data uncertainty συd = 0.05 m s−1, yields solutions (Fig. 8) that pass the consistency tests and agree well with the υ data (rms error of 0.038 m s−1). The υ uncertainty is reduced significantly. The F(i)y is reduced offshore of and on the bar crest (x ≥ 110 m), and is increased toward the trough (60 < x < 90 m), consistent with the concept of a wave roller (Fig. 8b). The slightly negative F(i)y near x = 30 m indicates a reversal of forcing and, although consistent with the fy error covariance, seems physically unrealistic (no mechanism for reversal is known). This may be the result of data noise mapped into the forcing correction. The c(i)d increases just offshore of the bar crest and is reduced in the trough (Fig. 8c). The c(i)d error bars are reduced by 15%–25% relative to the prior in the crest–trough region where data are concentrated. The inverse forcing correction f(i)y is compared with the change in alongshore forcing [f(r)y] calculated from a roller model (Stive and de Vriend 1994; Reniers and Battjes 1997). The inverse and roller reduction in alongshore forcing on and offshore of the bar crest (110 ≤ x < 200 m) are quite similar (Fig. 8d), as is the increase in alongshore forcing in much of the bar trough (70 < x < 100 m). Within the %σfy and σcd window for consistent inverse solutions, the f(i)y and c(i)d have the same structure as in Figs. 8b and 8c but with amplitude varying by 33% and 25%, respectively.

b. SandyDuck example

The SandyDuck case example does not have a well- developed bar (Fig. 9a). There is a steep slope region for 25 < x < 100 m and a nearly constant depth terrace for 100 < x < 200 m. Large waves begin breaking offshore of the terrace, have approximately constant height over the terrace, and then dissipate rapidly farther onshore on the steep slope (asterisks in Fig. 9b). A tuned 1D wave model (without rollers) accurately (rms error of 4 cm) predicts the wave height evolution (Fig. 9b). The wave model and observed wind give the prior F(pr)x and F(pr)y (Figs. 9c,d). The prior c(pr)d = 0.0015 (not shown). The σfx and σfy are chosen as 23% of the F(pr)x and F(pr)y absolute maxima (light dashed curves in Figs. 9c,d), and σcd = 7 × 10−4. Large covariance parameters within the window of parameters giving consistent solutions are chosen (section 3d). Although there is no sandbar to set the length scales, lfx, lfy, and lcd are set at 20 m (the same as in Duck94), which is a length scale for significant depth or wave height variation. The eddy viscosity from the Duck94 case example is used. The prior values and covariances are used to calculate η(pr) and υ(pr) with error bars (Figs. 9e,f). The errors in the (nonroller) prior model predictions of setup (rms error of 1 cm) and the alongshore current (rms error of 0.2 m s−1) exceed those expected from instrument noise alone. The expected η data error (σηd = 0.003 m) is based on rms errors of setdown predictions (Raubenheimer et al. 2001), and συd = 0.05 m s−1.

Consistent inverse solutions for η(i) and υ(i) (Figs. 10a and 11a) agree well with the data (rms errors of 3.9 mm and 3.3 cm s−1, respectively) with significantly reduced uncertainties. Both the cross- and alongshore inverse forcing magnitudes are increased relative to the prior in the terrace region (100 < x < 200 m) and are reduced near x = 80 m (Figs. 10b and 11b). As with the Duck94 example, changes in the cross- and alongshore forcing are consistent with the roller concept. The inverse model makes F(i)x slightly positive, predicting setdown for 85 < x < 100 m to match the setup observations in this region, consistent with observed wave shoaling at x = 100 m (Fig. 9b). The c(i)d pattern (Fig. 11c) is similar to the Duck94 case example (Fig. 8c). In the terrace region, c(i)d is reduced relative to the prior and is increased on the steep slope region for 70 < x < 90 m (Fig. 9b). The c(i)d error bars are reduced 25%–30% where data are concentrated. The cross- and alongshore inverse forcing corrections [f(r)x and f(r)y] are compared with the change in forcing predicted by the same roller model used in the Duck94 case example. The f(i)x and f(r)x qualitatively agree for 50 < x < 250 m (Fig. 10c). Offshore of x = 250 m, where there are no data, f(i)x is near zero, whereas nonzero f(r)x is predicted. The f(i)y and f(r)y also qualitatively agree for 50 < x < 200 m, although there is a factor of 2 difference in the magnitude of the f(i)y and f(r)y minima at x = 80 m (Fig. 11d).

5. Discussion

Overall, agreement between both the Duck94 and SandyDuck inverse forcing corrections and the roller model is remarkable, particularly because f(i)x and f(i)y are independent. No tuning of either the roller model parameters or of the forcing or drag coefficient error covariances was made to maximize this agreement. While the inverse forcings tantalizingly suggest that the roller model accurately predicts the change in cross- and alongshore forcing, no endorsement of a particular roller model is intended here. Varying the covariance parameters by 25% does not change the inverse case example results significantly, although not all inverse solutions are consistent with the prior assumptions.

Inferences can be drawn from the inverse-derived cd. Assuming that the maximum (and minimum) c(i)d is a Gaussian random variable with variance given by its zero-lag inverse covariance, the probability that the maximum (and minimum) cd is increased (reduced) from the prior cd can be calculated to determine the statistical significance of c(i)d variation. From both the Duck94 and SandyDuck case examples (Figs. 8c and 11c), the probability is over 90% that the maximum (and minimum) cd is significantly increased (reduced) from the prior.

Hypotheses that cd depends either on the bed roughness krms (e.g., Garcez-Faria et al. 1998) or on breaking- wave-generated turbulence (e.g., Church and Thornton 1993) are examined. Wave dissipation, a measure of the breaking-wave-generated turbulence source, is calculated from the modeled wave energy flux gradients in the region where c(i)d differs from the prior cd. A relationship between wave dissipation and c(i)d is observed (Fig. 12a) in both case examples (correlations r = 0.64 and r = 0.90 for Duck94 and SandyDuck, respectively), which is consistent with the hypothesis that increases in wave dissipation result in increased cd. No explicit or implicit connection exists in the inverse method between cd and wave dissipation. Bed roughness krms was estimated with eight fixed altimeters (Duck94) (Feddersen et al. 2003) and a towed altimeter (SandyDuck) (Gallagher et al. 2003). For the Duck94 example, krms varies between 1 and 7 cm, but for the SandyDuck example, the bed was smooth (krms < 2 cm). No relationship (i.e., statistically significant correlation) between krms and c(i)d exists for either the Duck94 or SandyDuck case, nor does a relationship exist between krms/h and c(i)d (not shown). Although the fixed-altimeter-based krms have errors (Feddersen et al. 2003), the lack of a relationship suggests that bed roughness is not a primary factor in determining cd (e.g., Feddersen et al. 2003).

The two inverse realizations presented here are insufficient to draw conclusions regarding forcing or cd parameterizations. Many inverse realizations of the forcing correction and cd, spanning a wide range of conditions, would allow statistical testing of wave-forcing or cd hypotheses. Additional interpretations of the inverse solutions are possible. For example, the alongshore forcing error could be ascribed to tidal (e.g., Ruessink et al. 2001) or buoyancy (e.g., Lentz et al. 2003) forcing.

6. Summary

Uncertainties regarding wave-forcing and drag coefficient parameterizations in the nearshore have motivated development of an inverse method that combines dynamics and data to yield optimal estimates of the setup η and alongshore current υ, together with corrections to the cross-shore forcing, alongshore forcing, and the drag coefficient cd. The method also yields error bars (covariances) for the η, υ, and cd inverse solutions. Tests that determine the consistency of the inverse solutions with prior assumptions were presented. The inverse method was tested with a synthetic barred-beach example, and consistent inverse solutions reproduced well the specified true cross- and alongshore forcing and cd.

The method was applied to two case examples from field experiments yielding inverse solutions that passed the consistency tests. The independently estimated cross- and alongshore inverse forcing corrections were similar to the modeled effect of wave rollers. The significant cross-shore variation of the inverse-derived cd was related to variations in wave dissipation, but was not related to variation in the observed bed roughness. Although consistent with the hypothesis that breaking- wave-generated turbulence increases cd, the two examples are not sufficient to examine this relationship statistically. Additional field cases spanning a wide range of nearshore conditions are needed to test hypotheses about the wave forcing and drag coefficient.

Acknowledgments

This research was supported by ONR, NSF, NOPP, and WHOI. The Duck94 and SandyDuck instruments were constructed, deployed, and maintained by staff from the Integrative Oceanography Division of SIO. Britt Raubenheimer helped design and manage the SandyDuck field experiment and provided high-quality setup data. Tom Herbers provided wave observations, and Edie Gallagher provided SandyDuck bed roughness observations. The Field Research Facility, Coastal Engineering Research Center, Duck, North Carolina, provided excellent logistical support, the bathymetric surveys, and 8-m depth pressure array data. A summer course on inverse methods taught by A. F. Bennett served as a springboard for this work. Gene Terray provided important insight, and John Trowbridge, Dennis McGillicuddy, and two anonymous reviewers provided valuable feedback.

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APPENDIX A

Relating Cost-Function Weights to Covariances

Minimization of a cost function [e.g., (3)] is shown to be equivalent to maximum likelihood estimation of a continuous Gaussian random variable and that the weight in the cost function [e.g., C−1fx in (3)] represents the inverse covariance. The nondegenerate and continuous, symmetric, positive definite kernel C(x, x′) is decomposed into (cf. Courant and Hilbert 1953)
i1520-0485-34-4-920-eqa1
where the ĉl and gl(x) are the eigenvalues (real and >0) and (orthogonal) eigenfunctions of C(x, x′). Orthogonality is defined as
i1520-0485-34-4-920-eqa2
where L is the domain size and δlm is the Kronecker delta function. For the set of continuous functions, the gl(x) provide a basis set. If C(x, x′) were a function of xx′ only (i.e., homogeneous), then the gl are complex exponentials. The inverse of C(x, x′) is
i1520-0485-34-4-920-eqa3
The basis set also decomposes the random function f(x),
i1520-0485-34-4-920-ea1
Substituting the decompositions for f(x) and C(x, x′) into a cost-function-type integral results in
i1520-0485-34-4-920-ea2
The expression (A2) is the argument in the exponential for an infinite set of independent zero-mean Gaussian random variables l with probability density function
i1520-0485-34-4-920-eqa4
With the continuous random function f(x) decomposed into an infinite sum of independent zero-mean Gaussian random variables l, it is straightforward to demonstrate that E[f(x)] = 0:
i1520-0485-34-4-920-eqa5
thus, E[f(x)] = 0 as well. The covariance of the random function f(x), defined as E[f(x)f(x′)], is
i1520-0485-34-4-920-ea3
Expanding E[lm],
i1520-0485-34-4-920-eqa6
and substituting into (A3) yields
EfxfxCx,x
Thus, cost-function (weighted by an inverse covariance) minimization is equivalent to maximum likelihood estimation of a continuous random function.

APPENDIX B

Consistency Checks with Prior Assumptions

The hypothesis that the difference (δυ) between inverse υ (or also setup η) solutions and data
δυnυ(i)xnd(υ)n
are (N) samples from a zero-mean Gaussian random variable with prior data variance σ2υd is tested. If the inverse solution is not consistent with the prior data variance, it should be rejected. Consider first the sample variance var(δυ). If this hypothesis is true, then with 95% probability the true variance falls within
i1520-0485-34-4-920-eqb2
where χ2N−1(y) represents the location where the chi- squared cumulative distribution function (cdf) with N − 1 degrees of freedom equals the probability y. If this confidence interval does not contain the prior σ2υd, then it should be rejected. Similarly, the 95% confidence limits on the sample mean 〈δυ〉 are given by a Student's t distribution with N − 1 degrees of freedom. If the interval
i1520-0485-34-4-920-eqb3
[where tM(y) is the location where the Student's t cdf equals probability y] does not contain zero, then the inverse solution is not consistent with the zero-mean data error and also should be rejected.
Similar consistency checks are performed for the inverse solutions for the forcing and cd error. The continuous functions (e.g., fx) are decomposed into Fourier coefficients [e.g., (A1)] using the basis functions of their respective prior covariances. Each Fourier coefficient (l) is then a sample from a zero-mean Gaussian random variable with variance given by the prior covariance eigenvalue (i.e., ĉl). If these hypotheses are correct, then the statistics (summed over the number of data N)
i1520-0485-34-4-920-eqb4
are zero-mean Gaussian random variables with variance ΣNl=1 ĉl and a (N − 1 degrees of freedom) chi-squared random variable, respectively. The significance tests described for the data are applied to test whether the inverse forcing error or cd error are consistent with the prior assumptions. The sum is over N (instead of ∞) because with finite data only a finite amount of information (i.e., approximately the first N Fourier coefficients) is added.

Fig. 1.
Fig. 1.

Test-case conditions vs distance from the shoreline: (a) depth h, (b) wave height Hrms, (c) cross-shore forcing Fx, (d) alongshore forcing Fy, and (e) drag coefficient cd. In (c), (d), and (e), the dark-dashed and light-dashed curves represent the prior and ±1 std dev, and the solid curves represent the true results

Citation: Journal of Physical Oceanography 34, 4; 10.1175/1520-0485(2004)034<0920:IMOOSA>2.0.CO;2

Fig. 2.
Fig. 2.

Prior and true (a) setup η and (b) alongshore current υ vs distance from the shoreline. The dark- and light-dashed curves represent the prior and ±1 std dev. The solid curves represent the true results. The asterisks represent the (noisy) data

Citation: Journal of Physical Oceanography 34, 4; 10.1175/1520-0485(2004)034<0920:IMOOSA>2.0.CO;2

Fig. 3.
Fig. 3.

(a) Inverse (dashed), true (solid), and data (asterisks) η, and (b) inverse (dashed) and true (solid) cross-shore forcing correction fx vs distance from the shoreline. In (a), the dark- and light- dashed curves represent the inverse and ±1 std dev

Citation: Journal of Physical Oceanography 34, 4; 10.1175/1520-0485(2004)034<0920:IMOOSA>2.0.CO;2

Fig. 4.
Fig. 4.

(a) Inverse (dashed), true (solid), and data (asterisks) υ; (b) inverse (dashed) and true (solid) alongshore forcing correction fy; and (c) inverse (dashed) and true (solid) cd vs distance from the shoreline. In (a) and (c), the dark- and light-dashed curves represent the inverse and ±1 std dev

Citation: Journal of Physical Oceanography 34, 4; 10.1175/1520-0485(2004)034<0920:IMOOSA>2.0.CO;2

Fig. 5.
Fig. 5.

Misfits [(17)] vs %σfx (% of maximum prior forcing magnitude): (a) inverse (asterisks) and prior (dashed line) cross-shore forcing misfit χ(Fx), and (b) inverse (asterisks) and prior (dashed line) setup misfits χ(η). The shaded areas are regions where either the data fit (right region) or f(i)x (left region) is inconsistent with the prior covariances. The circled asterisk indicates the solution in Fig. 3

Citation: Journal of Physical Oceanography 34, 4; 10.1175/1520-0485(2004)034<0920:IMOOSA>2.0.CO;2

Fig. 6.
Fig. 6.

Misfits [(17)] for (a) inverse alongshore forcing χ[F(i)y] and (b) inverse alongshore current χ[υ(i)] vs %σfy. The asterisks represent consistent inverse solutions. The pluses represent inconsistent inverse solutions. The horizontal dashed curve in (a) represents the prior forcing uncertainty χ[F(pr)y] = 0.0022 m2 s−2. The prior alongshore current misfit χ[υ(pr)] = 0.40 m s−1 is not shown in (b)

Citation: Journal of Physical Oceanography 34, 4; 10.1175/1520-0485(2004)034<0920:IMOOSA>2.0.CO;2

Fig. 7.
Fig. 7.

Duck94 conditions (1700 EST 10 Oct 1994) vs distance from the shoreline: (a) depth h, (b) model (solid) and data (asterisks) wave height Hrms, (c) F(pr)y, and (d) prior (dashed) and data (asterisks) υ. In (c) and (d), the dark- and light-dashed curves represent the prior and ±1 std dev

Citation: Journal of Physical Oceanography 34, 4; 10.1175/1520-0485(2004)034<0920:IMOOSA>2.0.CO;2

Fig. 8.
Fig. 8.

Duck94 inverse solutions vs distance from the shoreline: (a) inverse (dashed) and data (asterisks) υ, (b) inverse (dashed) and prior (solid) Fy, (c) inverse cd, and (d) inverse (dashed) and roller (solid) fy. In (a) and (c), the dark- and light-dashed curves represent the inverse solution and ±1 std dev. The roller forcing correction f(r)y in (d) uses roller parameter β = 0.05 (Ruessink et al. 2001)

Citation: Journal of Physical Oceanography 34, 4; 10.1175/1520-0485(2004)034<0920:IMOOSA>2.0.CO;2

Fig. 9.
Fig. 9.

SandyDuck conditions (1600 EST 18 Oct 1997) vs distance from the shoreline: (a) depth h, (b) model (solid) and data (asterisks) wave height Hrms, (c) prior cross-shore forcing F(pt)x, (d) prior alongshore forcing F(pr)y, (e) prior (dashed) and data (asterisks) η, and (f) prior (dashed) and data (asterisks) υ. In (c)–(f), the dark- and light-dashed curves represent the prior and ±1 std dev

Citation: Journal of Physical Oceanography 34, 4; 10.1175/1520-0485(2004)034<0920:IMOOSA>2.0.CO;2

Fig. 10.
Fig. 10.

SandyDuck η inverse solutions vs distance from the shoreline: (a) inverse (dashed) and data (asterisks) setup η, (b) inverse (dashed) and prior (solid) Fx, and (c) inverse (dashed) and roller (solid) cross-shore forcing corrections fx. In (a) and (c), the dark- and light-dashed curves represent the inverse solution and ±1 std dev

Citation: Journal of Physical Oceanography 34, 4; 10.1175/1520-0485(2004)034<0920:IMOOSA>2.0.CO;2

Fig. 11.
Fig. 11.

SandyDuck υ inverse vs distance from the shoreline: (a) inverse (dashed) and data (asterisks) υ, (b) inverse (dashed) and prior (solid) Fy, (c) inverse (dashed) cd, and (d) inverse (dashed) and roller (solid) alongshore forcing correction fy. In (a) and (c), the dark- and light-dashed curves represent the inverse solution and ±1 std dev

Citation: Journal of Physical Oceanography 34, 4; 10.1175/1520-0485(2004)034<0920:IMOOSA>2.0.CO;2

Fig. 12.
Fig. 12.

(a) Duck94 (circles) and SandyDuck (asterisks) inverse cd vs modeled wave dissipation (Figs. 7b and 9b). Comparison is made for 50 < x < 250 m (Duck94) and 30 < x < 200 m (SandyDuck) averaged over the decorrelation length scale lcd = 20 m, yielding approximately independent cd estimates. Changing the comparison regions does not alter the results. (b) Duck94 (open circles) and SandyDuck (small filled circles) c(i)d vs bed roughness krms. The SandyDuck results span 35 < x < 250 m and are averaged over lcd = 20 m

Citation: Journal of Physical Oceanography 34, 4; 10.1175/1520-0485(2004)034<0920:IMOOSA>2.0.CO;2

Save
  • Battjes, J. A., 1975: Modeling of turbulence in the surfzone. Proc. Symp. on Modeling Techniques, San Fransisco, CA, ASCE, 1050–1061.

  • Battjes, J. A., and M. J. F. Stive, 1985: Calibration and verification of a dissipation model for random breaking waves. J. Geophys. Res, 90 , 91599167.

    • Search Google Scholar
    • Export Citation
  • Bennett, A. F., 1992: Inverse Methods in Physical Oceanography. Cambridge University Press, 346 pp.

  • Bowen, A. J., D. L. Inman, and V. P. Simmons, 1968: Wave “set- down” and set-up. J. Geophys. Res, 73 , 25692577.

  • Brethereton, F. P., R. E. Davis, and C. B. Fandry, 1976: A technique for objective analysis of oceanographic experiments applied to MODE-73. Deep-Sea Res, 23 , 559582.

    • Search Google Scholar
    • Export Citation
  • Church, J. C., and E. B. Thornton, 1993: Effects of breaking wave induced turbulence within a longshore current model. Coastal Eng, 20 , 128.

    • Search Google Scholar
    • Export Citation
  • Courant, R., and D. Hilbert, 1953: Methods of Mathematical Physics. Vol 1. Wiley-Interscience, 560 pp.

  • Duncan, J. H., 1981: An experimental investigation of breaking waves produced by a towed hydrofoil. Proc. Roy. Soc. London, 377A , 331348.

    • Search Google Scholar
    • Export Citation
  • Elgar, S., R. T. Guza, B. Raubenheimer, T. H. C. Herbers, and E. L. Gallagher, 1997: Spectral evolution of shoaling and breaking waves on a barred beach. J. Geophys. Res, 102 , 1579715805.

    • Search Google Scholar
    • Export Citation
  • Elgar, S., R. T. Guza, W. C. O'Reilly, B. Raubenheimer, and T. H. C. Herbers, 2001: Wave energy and direction observed near a pier. J. Waterw. Port Coastal Ocean Eng, 127 , 26.

    • Search Google Scholar
    • Export Citation
  • Feddersen, F., and R. T. Guza, 2003: Observations of nearshore circulation: Alongshore uniformity. J. Geophys. Res.,108, 3006, doi:10.1029/2001JC001293.

    • Search Google Scholar
    • Export Citation
  • Feddersen, F., R. T. Guza, S. Elgar, and T. H. C. Herbers, 1998: Alongshore momentum balances in the nearshore. J. Geophys. Res, 103 , 1566715676.

    • Search Google Scholar
    • Export Citation
  • Feddersen, F., R. T. Guza, S. Elgar, and T. H. C. Herbers, 2000: Velocity moments in alongshore bottom stress parameterizations. J. Geophys. Res, 105 , 86738686.

    • Search Google Scholar
    • Export Citation
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  • Fig. 1.

    Test-case conditions vs distance from the shoreline: (a) depth h, (b) wave height Hrms, (c) cross-shore forcing Fx, (d) alongshore forcing Fy, and (e) drag coefficient cd. In (c), (d), and (e), the dark-dashed and light-dashed curves represent the prior and ±1 std dev, and the solid curves represent the true results

  • Fig. 2.

    Prior and true (a) setup η and (b) alongshore current υ vs distance from the shoreline. The dark- and light-dashed curves represent the prior and ±1 std dev. The solid curves represent the true results. The asterisks represent the (noisy) data

  • Fig. 3.

    (a) Inverse (dashed), true (solid), and data (asterisks) η, and (b) inverse (dashed) and true (solid) cross-shore forcing correction fx vs distance from the shoreline. In (a), the dark- and light- dashed curves represent the inverse and ±1 std dev

  • Fig. 4.

    (a) Inverse (dashed), true (solid), and data (asterisks) υ; (b) inverse (dashed) and true (solid) alongshore forcing correction fy; and (c) inverse (dashed) and true (solid) cd vs distance from the shoreline. In (a) and (c), the dark- and light-dashed curves represent the inverse and ±1 std dev

  • Fig. 5.

    Misfits [(17)] vs %σfx (% of maximum prior forcing magnitude): (a) inverse (asterisks) and prior (dashed line) cross-shore forcing misfit χ(Fx), and (b) inverse (asterisks) and prior (dashed line) setup misfits χ(η). The shaded areas are regions where either the data fit (right region) or f(i)x (left region) is inconsistent with the prior covariances. The circled asterisk indicates the solution in Fig. 3

  • Fig. 6.

    Misfits [(17)] for (a) inverse alongshore forcing χ[F(i)y] and (b) inverse alongshore current χ[υ(i)] vs %σfy. The asterisks represent consistent inverse solutions. The pluses represent inconsistent inverse solutions. The horizontal dashed curve in (a) represents the prior forcing uncertainty χ[F(pr)y] = 0.0022 m2 s−2. The prior alongshore current misfit χ[υ(pr)] = 0.40 m s−1 is not shown in (b)

  • Fig. 7.

    Duck94 conditions (1700 EST 10 Oct 1994) vs distance from the shoreline: (a) depth h, (b) model (solid) and data (asterisks) wave height Hrms, (c) F(pr)y, and (d) prior (dashed) and data (asterisks) υ. In (c) and (d), the dark- and light-dashed curves represent the prior and ±1 std dev

  • Fig. 8.

    Duck94 inverse solutions vs distance from the shoreline: (a) inverse (dashed) and data (asterisks) υ, (b) inverse (dashed) and prior (solid) Fy, (c) inverse cd, and (d) inverse (dashed) and roller (solid) fy. In (a) and (c), the dark- and light-dashed curves represent the inverse solution and ±1 std dev. The roller forcing correction f(r)y in (d) uses roller parameter β = 0.05 (Ruessink et al. 2001)

  • Fig. 9.

    SandyDuck conditions (1600 EST 18 Oct 1997) vs distance from the shoreline: (a) depth h, (b) model (solid) and data (asterisks) wave height Hrms, (c) prior cross-shore forcing F(pt)x, (d) prior alongshore forcing F(pr)y, (e) prior (dashed) and data (asterisks) η, and (f) prior (dashed) and data (asterisks) υ. In (c)–(f), the dark- and light-dashed curves represent the prior and ±1 std dev

  • Fig. 10.

    SandyDuck η inverse solutions vs distance from the shoreline: (a) inverse (dashed) and data (asterisks) setup η, (b) inverse (dashed) and prior (solid) Fx, and (c) inverse (dashed) and roller (solid) cross-shore forcing corrections fx. In (a) and (c), the dark- and light-dashed curves represent the inverse solution and ±1 std dev

  • Fig. 11.

    SandyDuck υ inverse vs distance from the shoreline: (a) inverse (dashed) and data (asterisks) υ, (b) inverse (dashed) and prior (solid) Fy, (c) inverse (dashed) cd, and (d) inverse (dashed) and roller (solid) alongshore forcing correction fy. In (a) and (c), the dark- and light-dashed curves represent the inverse solution and ±1 std dev

  • Fig. 12.

    (a) Duck94 (circles) and SandyDuck (asterisks) inverse cd vs modeled wave dissipation (Figs. 7b and 9b). Comparison is made for 50 < x < 250 m (Duck94) and 30 < x < 200 m (SandyDuck) averaged over the decorrelation length scale lcd = 20 m, yielding approximately independent cd estimates. Changing the comparison regions does not alter the results. (b) Duck94 (open circles) and SandyDuck (small filled circles) c(i)d vs bed roughness krms. The SandyDuck results span 35 < x < 250 m and are averaged over lcd = 20 m

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