Modelling bottom stress in depth-averaged flows

Abstract

The relationship between depth-averaged velocity and bottom stress for wind-driven flow in unstratified coastal waters is examined here. The adequacy of traditional linear and quadratic drag laws is addressed by comparison with a 2 1/2-D model. A 2 1/2-D model is one in which a simplified 1-D depth-resolving model (DRM) is used to provide an estimate of the relationship between the flow and bottom stress at each grid point of a depth-averaged model (DAM). Bottom stress information is passed from the DRM to the DAM in the form of drag tensor with two components: one which scales the flow and one which rotates it. This eliminates the problem of traditional drag laws requiring the flow and bottom stress to be collinear. In addition , the drag tensor field is updated periodically so that the relationship between the velocity and bottom stress can be time-dependent. However, simplifications in the 2 1/2-D model that render it computationally efficient also impose restrictions on the time-scale of resolvable processes. Basically, they must be much longer than the vertical diffusion time scale. Four progressively more complicated scenarios are investigated. The important factors governing the importance of bottom friction in each are found to be 1) non-dimensional surface Ekman depth, u.5/fh where u.s is the surface shear velocity, f is the Coriolis parameter and h is the water depth 2) the non-dimensional bottom roughness, zo/h where zo is the roughness length and 3) the angle between the wind stress and the shoreline. Each has significant influence on the drag law. The drag tensor magnitude, r, and the drag sensor angle, θ are functions of all three, while a drag tensor which scales with the square of the depth-averaged velocity has a magnitude, Cd, that only depends on zo/h. The choice of drag Jaw is found to significantly affect the response of a domain. Spin up times and phase relationships vary between models. In general, the 2 1/2-D model responds more quickly than either a constant r or constant Cd model. Steady-state responses are also affected. The two most significant results are that failure to account for θ in the drag law sometimes leads to substantial errors in estimating the sea surface height and to extremely poor resolution of cross-shore bottom stress. The latter implies that cross-shore near-bottom transport is essentially neglected by traditional DAMs.