Barotropic flow over bottom topography— experiments and nonlinear theory

Abstract

Abstract Barotropic flow over finite amplitude two-wave bottom topography is investigated both experimentally and theoretically over a broad parameter range. In the experiments, the fluid is contained in a vertically oriented, rotating circular cylindrical annulus. It is forced into motion relative to the annulus by a differentially rotating, rigid, radially sloping lid in contact with the top surface of the fluid. The radial depth variation associated with the slope of the lid, and an equal and opposite slope of the bottom boundary, simulates the effect of the variation of the Coriolis parameter with latitude (β) in planetary atmospheres and in the ocean. The dimensionless parameters which control the fluid behavior are the Rossby number (ϵ), the Ekman number (E), the β parameter, the aspect ratio (δ), the ratio of the mean radius to the gap width (α) and the ratio of the topographic height to the mean fluid depth (η). The Rossby and Ekman numbers are varied over an order of magnitude by conducting experiments at different rotation rates of the annulus. Velocity measurements using photographs of tracer particles suspended in the fluid reveal the existence of a stationary, topographically forced wave superimposed on an azimuthal mean current. With successively larger rotation rates (i.e. lower ϵ and E) the wave amplitude increases and then levels off, the phase displacement of the wave upstream of the topography increases and the azimuthal mean velocity decreases and then levels off. Linear quasigeostophic theory accounts qualitatively, but not quantitatively, for the phase displacement, predicts the wave amplitude poorly and provides no basis for predicting the zonal mean velocity. Accordingly, we have solved the nonlinear, steady-state, quasigeostrophic barotrophic vorticity equation with both Ekman layer and internal dissipation using a spectral colocation method with Fourier representation in the azimuthal direction and Chebyshev polynomial representation in the radial direction. For boundary conditions at the side walls, we specified zero velocity. Side wall boundary layers then appear explicitly in the numerical solution. At the bottom and top of the fluid, we specified that the vertical velocity at the mean height of each boundary is the sum of two components—one forced by Ekman suction in the absence of topography and the other by the condition that there can be no flow normal to the rigid boundary. We justify this choice by the smallness of the Ekman number and of the radial and azimuthal slopes of the topography. We have found that the use of three Fourier components and seven Chebyshev polynomials is sufficient to account qualitatively for the experimental results, although small quantitative discrepancies suggest that further investigation of the neglect of effects originally considered to be small is needed.