Depth Dependence of Bottom Stress and Quadratic Drag Coefficient for Barotropic Pressure-Driven Currents

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  • 1 NOAA/ERL/Pacific Marine Environmental Laboratory, Seattle, Washington
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Abstract

A level 2½ turbulence closure model is used to investigate the dependence on water depth H of bottom stress τb and quadratic drag coefficient Cd 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 |τb| 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, |τb| 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 |τb|. 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 |τb| 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 |τb| and ∼5° for θ.

The coefficient Cd of the quadratic drag law relating |τb| 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.

Abstract

A level 2½ turbulence closure model is used to investigate the dependence on water depth H of bottom stress τb and quadratic drag coefficient Cd 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 |τb| 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, |τb| 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 |τb|. 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 |τb| 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 |τb| and ∼5° for θ.

The coefficient Cd of the quadratic drag law relating |τb| 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.

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