FY 1989

Depth dependence of bottom stress and quadratic drag coefficient for barotropic pressure-driven currents

Mofjeld, H.O.

J. Phys. Oceanogr., 18(11), 1658–1669, doi: 10.1175/1520-0485(1988)018<1658:DDOBSA>2.0.CO;2 (1988)

A level 2½ turbulence closure model is used to investigate the dependence on water depth H of bottom stress τ_b and quadratic drag coefficient C_d for a steady barotropic pressure-driven current in unstratified water when the current is the primary source of turbulence. For spatially uniform pressure gradient and bottom roughness z₀ the magnitude |τ_b | increases from small values in shallow water to a maximum (at a depth ~0.004 U₀/f where U₀ is the geostrophic current speed derived from the pressure gradient and f is the Coriolis parameter) at which the dynamics changes from being depth-limited to being controlled by similarity scales. As the depth increases further, |τ_b| decreases to its deep-water value that is 15% to 19% less than the maximum. The angle θ of the bottom stress relative to the geostrophic direction decreases rapidly from 90° in very shallow water, reaching its deep-water value (~11°–21°) at a somewhat shallower depth than does |τ_b|. At the maximum stress θ is 8° larger than the deep-water angle. A set of computationally efficient formulas matched to the model results gives |τ_b| and θ for all combinations of U₀, H, f and bottom roughness z₀. Comparison with a variety of other models satisfying Rossby similarity over oceanographic ranges of parameters shows agreement of ~10% for |τ_b| and ~5° for θ.

The coefficient C_d of the quadratic drag law relating |τ_b| to the vertically averaged velocity is found to be approximated reasonably well by a formula from nonrotating channel theory in which the coefficient depends only on the ratio H/z₀. The direction of the bottom stress relative to the vertically averaged velocity is equal to the geostrophic veering angle (~11°–21°) in deep water and decreases to ~5° for a range of intermediate depths (~0.004–0.01 U₀/f) where it is relatively independent of external Rossby number U₀/fz₀; the angle becomes less in shallower water.