1. Introduction
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
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
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 +
b. Setup inverse modeling
c. Alongshore current inverse modeling
d. Prior covariances
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
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
Following Church and Thornton (1993), the true drag coefficient
These inputs are used within the setup [(1)] and alongshore current [(2)] models to generate true and prior
b. Prior covariances
The prior covariances of the forcing, drag coefficient, boundary condition, and data errors also must be specified. The data errors σ
c. Inverse solution
With all the ingredients, the inverse method yields the inverse setup
The ability of the inverse method to reproduce the cross- and alongshore forcing is examined by comparing the inverse forcing corrections [
d. Choosing covariance parameters
For the
For the
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
The
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
Consistent inverse solutions for
5. Discussion
Overall, agreement between both the Duck94 and SandyDuck inverse forcing corrections and the roller model is remarkable, particularly because
Inferences can be drawn from the inverse-derived cd. Assuming that the maximum (and minimum)
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
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
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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