The Vertical Structure of Low-Frequency Motions in the Nearshore. Part II: Theory

Field observations from a vertical stack of two-component current meters obtained from the 1994 Duck94 nearshore field experiment (presented in a companion paper by Lippmann, et al.) show significant vertical structure in energy, phase, and rotation of motions at low frequencies around 0.005Hz. Low-frequency motions are typically modeled in the surfzone with the shallow-water (depth averaged) momentum equations that do not allow for any vertical structure. Following work from the shelf tidal community (Prandle), this study shows that the observations are consistent with the depth-varying momentum equations including shear stresses induced by a bottom boundary layer described by a constant eddy viscosity nt and bottom friction given by a constant drag coefficient and depth-averaged velocity cdjWj. The bidirectional flow field is solved over arbitrary depth profiles varying only in the cross-shore direction h(x) in the presence of a vertically uniform mean alongshore current with cross-shore shear structure V(x). Analytic solutions are found to depend on nt, cd, h, ›V/›x, and the parameter p5 [i(kV2s1 ›V/›x)/nt] 1/2 5 il, where s and k are the radian frequency and alongshore wavenumber of the oscillating motion. Model behavior is explored by plotting solutions for a given parameter space as functions of the nondimensional depth H 5 lh and dimensionless friction parameter J5 ntl/cdjWj that combines the effects of bottom drag and vertical mixing. The behavioral changes in amplitude, phase shift, and rotational structure over the water column are qualitatively similar to those observed in the field.