# The Vertical Structure of Cross-Shore Currents from Wind-Induced Setup

*by*Guy Gelfenbaum, U.S. Geological Survey, St. Petersburg, United States,

**Abstract:**Most of the storm surge models presented in the literature are vertically averaged and calculate only the sea-surface elevation and mean flow. Whereas these models may be adequate for predicting storm surge heights for flooding purposes, they neglect the vertical structure of the flow and the boundary shear stress, which are both critical for predicting cross-shore sediment transport. The steady and horizontally uniform equations of motion are used here to compute the sea-surface slope, the vertical structure of the cross-shore currents, and the boundary shear stress in a shallow wind dominated environment. The steady state model developed here balances the pressure gradient and the stress divergence, resulting in sea-surface slope and associated pressure gradient in the opposite direction of the wind, thus inducing a reversal in the currents near the bed. The Reynolds stress is modeled with a depth-dependent turbulent diffusion coefficient so that both the boundary shear stress and the velocity field are calculated, avoiding the need to set a bottom drag coefficient. Input parameters for this model are simply the wind stress, the water depth, and z

_{0}, the bed roughness parameter. A sensitivity test of the model results to various values of z

_{0}indicates that large changes in z

_{0}cause only minor differences in the surface slope, and moderate differences in the velocity field and boundary shear stress. Given the sediment size distribution and the small scale morphology of the bed, a reasonable estimate of z

_{0}may be obtained and the above uncertainty will be nearly eliminated.

**Subject Headings:**Shear stress | Boundary shear | Storm surges | Reynolds stress | Shores | Slopes | Terrain models | Hydrologic models

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