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Response of an Oceanic Bottom Boundary Layer on a Slope to Interior Flow. Part I: Time-Independent Interior Flow

Dave RamsdenDaleth Research, Victoria, British Columbia, Canada

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Abstract

The behavior of oceanic boundary layers on a sloping bottom in the presence of stratification is investigated by the method of direct numerical simulations. The Navier–Stokes equations are decomposed into mean and turbulent components with the mean equations involving a slope Burger number: S = N2sin2θ/f2, where N is the buoyancy frequency, θ is the bottom slope angle, and f is the Coriolis parameter. The influence of the turbulent fluctuations is parameterized as eddy coefficients of viscosity and diffusivity.

Two regimes are considered. The first is for cases where the eddy coefficients are constant in time but variable in space and the interior flow is allowed to adjust to the boundary mixing. The numerical results agree with analytic theory for two limiting cases. The first case is when the mixing extends into the fluid interior, and the second is when the mixing is limited to a slab region.

The second regime involves cases in which the eddy coefficients are determined from a gradient Richardson number hypothesis. The interior alongslope flow is fixed in time. Two cases are considered, and it is shown that the resultant Ekman layer shutdown from downwelling favorable flows can be parameterized in terms of the Burger number and a “stratification drag” number, similar to that used by Trowbridge and Lentz, D = CdN/f, where Cd is the bottom drag coefficient.

Cases favorable to downwelling flow are shown to shut down in a time proportional to S−3/2D−1f−1 for small S. Cases favorable to upwelling flow are also investigated. The results are shown to depend on the values of diffusivity in the well-mixed and far-field region. The results, though not conclusive, suggest upwelling favorable flows shut down in a time proportional to S−1D−1/2f−1.

Abstract

The behavior of oceanic boundary layers on a sloping bottom in the presence of stratification is investigated by the method of direct numerical simulations. The Navier–Stokes equations are decomposed into mean and turbulent components with the mean equations involving a slope Burger number: S = N2sin2θ/f2, where N is the buoyancy frequency, θ is the bottom slope angle, and f is the Coriolis parameter. The influence of the turbulent fluctuations is parameterized as eddy coefficients of viscosity and diffusivity.

Two regimes are considered. The first is for cases where the eddy coefficients are constant in time but variable in space and the interior flow is allowed to adjust to the boundary mixing. The numerical results agree with analytic theory for two limiting cases. The first case is when the mixing extends into the fluid interior, and the second is when the mixing is limited to a slab region.

The second regime involves cases in which the eddy coefficients are determined from a gradient Richardson number hypothesis. The interior alongslope flow is fixed in time. Two cases are considered, and it is shown that the resultant Ekman layer shutdown from downwelling favorable flows can be parameterized in terms of the Burger number and a “stratification drag” number, similar to that used by Trowbridge and Lentz, D = CdN/f, where Cd is the bottom drag coefficient.

Cases favorable to downwelling flow are shown to shut down in a time proportional to S−3/2D−1f−1 for small S. Cases favorable to upwelling flow are also investigated. The results are shown to depend on the values of diffusivity in the well-mixed and far-field region. The results, though not conclusive, suggest upwelling favorable flows shut down in a time proportional to S−1D−1/2f−1.

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