## Abstract

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 *z*_{0} the magnitude |τ_{b}| increases from small values in shallow water to a maximum (at a depth ∼0.004 *U*_{0}/*f* where *U*_{0} 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 *U*_{0}, *H*, *f* and bottom roughness *z*_{0}. 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*/*z*_{0}. 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 *U*_{0}/*f*) where it is relatively independent of external Rossby number *U*_{0}/*fz*_{0}; the angle becomes less in shallower water.