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Removal of a dense bottom layer by a gravity current. (English) Zbl 1486.76029

Summary: We investigate the removal of a dense bottom layer by a gravity current, via Navier-Stokes simulations in the Boussinesq limit. The problem is governed by a dimensionless thickness parameter for the bottom layer, and by the ratio of the density differences between bottom layer, gravity current and ambient fluids. A quasisteady gravity current forms that propagates along the interface and displaces some of the dense bottom fluid, which accumulates ahead of the gravity current and forms an undular bore or a series of internal gravity waves. Depending on the ratio of the gravity current front velocity to the linear shallow-water wave velocity, we observe the existence of different regimes, characterized by small-amplitude waves or by a train of steep, nonlinear internal waves. We develop a semiempirical model that provides reasonable estimates of several important flow properties. We also formulate a more sophisticated, self-contained model based on the conservation principles for mass and vorticity that does not require empirical closure assumptions. This model is able to predict such quantities as the gravity current height and the internal wave or bore velocity as a function of the governing dimensionless parameters, generally to within approximately a 10% accuracy. An energy budget analysis provides information on the rates at which potential energy is converted into kinetic energy and then dissipated, and on the processes by which energy is transferred from the gravity current fluid to the dense and ambient fluids. We observe that the energy content of thicker and denser bottom layers grows more rapidly.

MSC:

76D50 Stratification effects in viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
76M20 Finite difference methods applied to problems in fluid mechanics
86A05 Hydrology, hydrography, oceanography
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