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A Numerical Study of the Adjustment of a Narrow Stratified Current over a Sloping Bottom

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  • 1 Woods Hole Oceanographic Institution, Woods Hole, Massachusetts
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Abstract

The adjustment of a narrow, stratified, cyclonic along-isobath current over a uniformly sloping bottom and the coupling between the current and the bottom boundary layer that develops beneath are investigated using a primitive-equation numerical model. The current generates a bottom Ekman layer immediately downstream of its origin, with downslope transport everywhere beneath the current, carrying lighter water under heavier water to produce a vertically well-mixed bottom boundary layer. At the top of the boundary layer, Ekman suction on the shallow side and pumping on the deep side lead to density advection in the vertical, tilted interior isopycnals, and thermal-wind shear of the interior along-isobath velocity. Flow above the bottom boundary layer is nearly perfectly geostrophic and along isopycnals. Buoyancy advection in the bottom boundary layer continues to cause growth of the boundary layer downstream, with subsequent reduction in bottom stress, until the flow reaches a steady downstream equilibrium beyond which only gradual changes occur as a result of viscosity and mixing.

The numerical results are compared with the idealized model of this adjustment process previously proposed by Chapman and Lentz. The same basic dynamics dominate, and some of the scales and parameter dependencies predicted by the idealized model apply to the numerical results. For example, the distance to the downstream equilibrium decreases with increasing buoyancy frequency and/or bottom slope, and the equilibrium structure is nearly independent of the bottom friction coefficient. The equilibrium bottom boundary layer thickness and the interior along-isobath velocity just above the boundary layer closely obey the idealized model scales; that is, the boundary layer thickness decreases with increasing buoyancy frequency and is independent of bottom slope, and the overlying current decreases while its width increases as either the buoyancy frequency or bottom slope decreases. However, the interior vertical shear in the numerical model tends to decouple the overlying current from the bottom boundary layer, so the structure of the bottom boundary layer in the downstream equilibrium is different from the idealized model, and neither the current width nor the surface currents are as sensitive to parameter variations as the idealized model suggests. Finally, the along-isobath current is not geostrophic near the bottom of the bottom boundary layer, as assumed in the idealized model, so the bottom boundary layer is not fully arrested, that is, bottom stress never quite vanishes downstream, suggesting that a completely frictionless downstream equilibrium is unlikely to be achieved.

Corresponding author address: David C. Chapman, Woods Hole Oceanographic Institution, Woods Hole, MA 02543.

Email: dchapman@whoi.edu

Abstract

The adjustment of a narrow, stratified, cyclonic along-isobath current over a uniformly sloping bottom and the coupling between the current and the bottom boundary layer that develops beneath are investigated using a primitive-equation numerical model. The current generates a bottom Ekman layer immediately downstream of its origin, with downslope transport everywhere beneath the current, carrying lighter water under heavier water to produce a vertically well-mixed bottom boundary layer. At the top of the boundary layer, Ekman suction on the shallow side and pumping on the deep side lead to density advection in the vertical, tilted interior isopycnals, and thermal-wind shear of the interior along-isobath velocity. Flow above the bottom boundary layer is nearly perfectly geostrophic and along isopycnals. Buoyancy advection in the bottom boundary layer continues to cause growth of the boundary layer downstream, with subsequent reduction in bottom stress, until the flow reaches a steady downstream equilibrium beyond which only gradual changes occur as a result of viscosity and mixing.

The numerical results are compared with the idealized model of this adjustment process previously proposed by Chapman and Lentz. The same basic dynamics dominate, and some of the scales and parameter dependencies predicted by the idealized model apply to the numerical results. For example, the distance to the downstream equilibrium decreases with increasing buoyancy frequency and/or bottom slope, and the equilibrium structure is nearly independent of the bottom friction coefficient. The equilibrium bottom boundary layer thickness and the interior along-isobath velocity just above the boundary layer closely obey the idealized model scales; that is, the boundary layer thickness decreases with increasing buoyancy frequency and is independent of bottom slope, and the overlying current decreases while its width increases as either the buoyancy frequency or bottom slope decreases. However, the interior vertical shear in the numerical model tends to decouple the overlying current from the bottom boundary layer, so the structure of the bottom boundary layer in the downstream equilibrium is different from the idealized model, and neither the current width nor the surface currents are as sensitive to parameter variations as the idealized model suggests. Finally, the along-isobath current is not geostrophic near the bottom of the bottom boundary layer, as assumed in the idealized model, so the bottom boundary layer is not fully arrested, that is, bottom stress never quite vanishes downstream, suggesting that a completely frictionless downstream equilibrium is unlikely to be achieved.

Corresponding author address: David C. Chapman, Woods Hole Oceanographic Institution, Woods Hole, MA 02543.

Email: dchapman@whoi.edu

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