### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Observations
- 3. Equilibrium Change Observations
- 4. Model
- 5. Results
- 6. Discussion
- 7. Conclusion
- Acknowledgments
- References
- Supporting Information

[1] Shoreline location and incident wave energy, observed for almost 5 years at Torrey Pines beach, show seasonal fluctuations characteristic of southern California beaches. The shoreline location, defined as the cross-shore position of the mean sea level contour, retreats by almost 40 m in response to energetic winter waves and gradually recovers during low-energy summer waves. Hourly estimates of incident wave energy and weekly to monthly surveys of the shoreline location are used to develop and calibrate an equilibrium-type shoreline change model. By hypothesis, the shoreline change rate depends on both the wave energy and the wave energy disequilibrium with the shoreline location. Using calibrated values of four model free parameters, observed and modeled shoreline location are well correlated at Torrey Pines and two additional survey sites. Model free parameters can be estimated with as little as 2 years of monthly observations or with about 5 years of ideally timed, biannual observations. Wave energy time series used to calibrate and test the model must resolve individual storms, and model performance is substantially degraded by using weekly to monthly averaged wave energy. Variations of free parameter values between sites may be associated with variations in sand grain size, sediment availability, and other factors. The model successfully reproduces shoreline location for time periods not used in tuning and can be used to predict beach response to past or hypothetical future wave climates. However, the model will fail when neglected geologic factors are important (e.g., underlying bedrock limits erosion or sand availability limits accretion).

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Observations
- 3. Equilibrium Change Observations
- 4. Model
- 5. Results
- 6. Discussion
- 7. Conclusion
- Acknowledgments
- References
- Supporting Information

[2] Sandy beaches erode and accrete in response to changing wave conditions. Models for wave-driven change in beach sand levels span a wide range from complex flux gradient models to simple bulk response models. Flux gradient models estimate changes in sand level using conservation of mass, with spatial gradients in time-averaged sediment flux balanced by erosion or accretion. At the detailed end of the flux-gradient-model spectrum are wave phase resolving, two-phase flow models that include both intergranular interactions and turbulent suspension in flux estimates [e.g., *Dong and Zhang*, 2002; *Hsu et al.*, 2004]. These computationally intensive models predict time-dependent fluid velocities and sediment fluxes in the wave boundary layer and require input time series of velocity (including wave orbital velocity) above the wave boundary layer at many grid points. Values of several model coefficients are often unknown because many of the small-scale processes included are not well understood. Other flux gradient models empirically relate wave-averaged low-order moments (e.g., variance and skewness) of velocity and acceleration to wave-induced seabed stresses and sediment transport. For example, skewness (or the third moment) of cross-shore velocity [*Bailard*, 1981] and cross-shore acceleration [*Drake and Calantoni*, 2001] time series have been used in morphologic change models (*Roelvink and Stive* [1989], *Gallagher et al.* [1998], *Hoefel and Elgar* [2003], and others). Similar to the more complex two-phase models, spatial gradients in the estimated sediment flux are balanced by erosion or accretion.

[3] Bulk response models are essentially phenomenological; observations of waves and beach change are used to validate and calibrate simple heuristic rules for beach change. Equilibrium profile models, one subset of bulk response models, suggest that a beach exposed to steady wave conditions will evolve toward a unique equilibrium beach profile. When this shape is reached, no further change occurs. Equilibrium shapes have been suggested, for example *h*(*x*) = *Ax*^{2/3}, where *h* is the water depth, *x* is the distance offshore, and *A* depends on sediment grain size (*Bruun* [1954], *Dean* [1977], and others). Alternative shapes for the equilibrium beach profile, with finite shoreline slope or the inclusion of offshore sandbars, have been proposed [e.g., *Larson and Kraus*, 1989; *Inman et al.*, 1993; *Özkan Haller and Brundidge*, 2007]. Equilibrium beach response concepts have been used to model the evolution of beach profiles [*Larson and Kraus*, 1989] and nourishment projects [*Dean*, 1991], interannual variations in the cross-shore location of the sandbar crest [*Plant et al.*, 1999], and shoreline response to sea level rise [*Dubois*, 1990], storm surges [*Kriebel and Dean*, 1993], and storms [*Miller and Dean*, 2004].

[4] *Wright and Short* [1984] developed a conceptual set of equilibrium beach states, including straight and alongshore variable sandbars, which depend on the value of the Dean parameter, Ω = *H*_{b}/*w*_{s}*T* [*Gourlay*, 1968; *Dean*, 1973], where *H*_{b} is the breaking wave height, *w*_{s} is the grain-size-dependent sediment fall velocity, and *T* is the wave period. However, beach morphologies do not respond instantaneously to the changing wave field, and correlations between observed beach state and instantaneous Ω were weak. *Morton et al.* [1995], *Lee et al.* [1998], *Anthony* [1998], and *Jiménez et al.* [2008] further emphasize the observed low correlation between instantaneous wave conditions and beach state, on beaches dominated by storms with intermittent and seasonal recovery periods. For example, in a few hours during storm spin-up and spin-down, the wave field can vary far more rapidly than the morphology evolves.

[6] *Wright et al.* [1985] further suggested that beaches progress toward an equilibrium state, which depends on the instantaneous disequilibrium of the wave field (Ω − Ω_{eq}) and the relative magnitude of the wave event (Ω or Ω^{2}), but they lacked sufficient observations to confirm this hypothesis. More recently, *Miller and Dean* [2004] developed a simple model relating shoreline change to the disequilibrium of the shoreline position, which depends on the wave conditions, water level (including storm surge, wave setup, and tides), and berm height. They formulate shoreline change as

where *y* is the shoreline position, *y*_{eq} the equilibrium shoreline position, and *k* the rate constant, which can have different values for erosion and accretion. We pursue the equilibrium concepts of *Wright et al.* [1985] and *Miller and Dean* [2004] using extensive observations of shoreline location and hourly estimates of the wave field, which resolve even short-lived storms.

[7] At Torrey Pines State Beach in southern California, the site of the present study, seasonal erosion and accretion patterns were quantified with empirical eigenfunctions of monthly cross-shore profiles [*Winant et al.*, 1975; *Aubrey*, 1979]. *Aubrey et al.* [1980] made statistical predictions of weekly profile eigenfunctions using a combination of the weekly averaged wave energy and the previous profile eigenfunction values as predictors including the effects of the instantaneous forcing and the antecedent beach state. Their suggestion that a longer data set and shorter wave-averaging interval would decrease the forecast error is confirmed by our results.

[8] Here, multiyear observations of shoreline position and incident waves at Torrey Pines beach (described in section 2) are used to qualitatively illustrate equilibrium beach change concepts (section 3). A simple equilibrium shoreline model is developed (section 4), which reproduces well the observed shoreline movement at Torrey Pines and two nearby survey sites (section 5). The effects of survey sampling frequency and duration on model performance, the strong relationship between the displacement of shoreline and other depth contours, and the general model applicability are discussed (section 6).

### 3. Equilibrium Change Observations

- Top of page
- Abstract
- 1. Introduction
- 2. Observations
- 3. Equilibrium Change Observations
- 4. Model
- 5. Results
- 6. Discussion
- 7. Conclusion
- Acknowledgments
- References
- Supporting Information

[13] Beach sand level changes and waves are first related qualitatively using the extensive multiyear observations at Torrey Pines beach. Monthly (or more frequent) subaerial surveys of two approximately 2-km reaches were separated into eight 500-m alongshore sections (T1–T8, south to north, Figure 2a). To facilitate integration with the full bathymetry surveys, msl position was determined along the predefined cross-shore transects spaced approximately every 100 m alongshore. Changes in msl location were calculated on each transect and then averaged within each 500-m alongshore section. The temporal mean msl position for each 500-m section (beach width range in Table 1) was removed, yielding time series of msl position fluctuations about the mean (Figure 2d, msl position). Mean hourly wave energy was obtained by averaging the incident spectral wave energy estimates (at the −10 m depth contour), spaced every 100 m alongshore, over each 500-m section (Figure 2d, wave energy).

[14] In all eight sections, msl position and wave energy show large seasonal cycles. The wave energy is typically low during summer, with episodic, high-energy winter storms. At all alongshore locations, the beach is most accreted (positive msl position) after continuous, low-energy summer waves, and the beach is most eroded (negative msl position) after episodic, high wave energy winter storms. Msl position and wave energy statistics vary relatively little in the alongshore between most 500-m sections. Section T3 is representative and is shown in Figures 2c, 2d, 4, 5, 8, 9, 11, 13, 14, and 15.

[15] Time series of instantaneous msl position *S* and average wave energy (where the overbar denotes the average between successive surveys) are only weakly correlated (*R*^{2} < 0.14 for all alongshore sections). However, consistent with equilibrium concepts, the msl change rate does depend on for a given initial *S* (Figure 3). Eroding and accreting waves are separated by the equilibrium energy _{eq} (calculated using data averaged between successive surveys), which causes no msl change for a particular initial msl position (black line at blue-red boundary, Figure 3). The equilibrium energy depends on the initial msl position; therefore, the shoreline response will vary for two events with the same wave energy but different initial msl position. For example, a moderate wave energy of about 0.05 m^{2} (*H*_{sig} = 0.9 m), which erodes an accreted beach (positive msl, Figure 3) can accrete an eroded beach (negative msl, Figure 3). Larger wave energy events are required to continue eroding an already eroded beach. The msl change rate appears to increase when wave energy is farther from the equilibrium wave energy (e.g., as the disequilibrium − _{eq}, the deviation from the solid line, increases (see color scale in Figure 3)).

### 4. Model

- Top of page
- Abstract
- 1. Introduction
- 2. Observations
- 3. Equilibrium Change Observations
- 4. Model
- 5. Results
- 6. Discussion
- 7. Conclusion
- Acknowledgments
- References
- Supporting Information

[16] The beach response at Torrey Pines (Figure 3) suggests a simple equilibrium-type model. Following the concepts of *Wright et al.* [1985], the instantaneous msl change rate is assumed proportional to both the instantaneous energy *E* and the instantaneous energy disequilibrium Δ*E* for the current msl position

where *C*^{±} are change rate coefficients for accretion (*C*^{+} for Δ*E* < 0) and erosion (*C*^{−} for Δ*E* > 0), and the energy disequilibrium is

[17] The equilibrium wave energy *E*_{eq} depends on the initial msl position *S*, and the sign of the msl change rate *dS*/*dt* is determined by the sign of the energy disequilibrium Δ*E*. The factor *E*^{1/2} prevents nonphysical changes in msl position when *E* is small and will be discussed further in section 6.1. For simplicity, we define the equilibrium wave energy as a linear function of the msl position

where *a* and *b* are the slope and y intercept, respectively (similar to the solid line (_{eq}) in Figure 3). That is, for a given msl position *S* there is an equilibrium wave energy *E*_{eq} that causes no change. Rearranging (4) yields the equilibrium msl position for a given wave energy

The equilibrium msl position would be obtained if the wave energy remained constant for an extended period of time, allowing the beach to equilibrate fully with the wave forcing.

[18] The model approaches equilibrium exponentially, as suggested by *Swart* [1974] and *Larson and Kraus* [1989] and similar to the equilibrium model of *Miller and Dean* [2004]. The model behavior is illustrated in the simple case when the time series of wave energy *E* is a step function, either increasing or decreasing to fixed level and remaining constant thereafter. In this case, the solution to (2)–(5) is

where *S*_{0} is the initial msl position, and for a fixed wave energy *E*, *S*_{eq} is the equilibrium msl position, which depends on *a* and *b*(5). With equal *C*^{+} and *C*^{−}, the e-folding scale [*aC*^{±}*E*^{1/2}]^{−1} shows faster adjustment to high-energy waves than to low-energy waves. Time scale estimates, based on the free parameters fit to the observations, are discussed below.

[20] Hourly wave energy estimates *E* resolve even rapidly varying wave conditions, and after the free parameters are determined, allow hourly updates of shoreline location (2). However, the many hour time steps (about 44,000 in 5 years) complicate the numerics of finding the best fit parameters in this nonlinear system. Two derivative-free techniques were used to solve for the four free parameters that minimize the root-mean-square error (RMSE) between the model and observations: simulated annealing [*Barth and Wunsch*, 1990] and surrogate management framework [*Booker et al.*, 1999; *Marsden et al.*, 2004]. Derivative-free methods are used because the present system has many local minima in the four-dimensional parameter space that can trap gradient methods. Simulated annealing and surrogate management framework (SMF) search the parameter space differently but yield similar results for test cases. SMF required significantly fewer cost function evaluations to minimize the RMSE, and SMF results are presented below.