Supercritical rotating flow over topography

Abstract

The flow of a one-and-a-half layer Boussinesq fluid over an obstacle of nondimensional height M, relative to the lower layer depth, is investigated in the presence of rotation, the magnitude of which is measured by a nondimensional parameter B ͑inverse Burger number͒. The supercritical regime in which the Froude number F, the ratio of the flow speed to the interfacial gravity wave speed, is significantly greater than one is considered in the shallow water ͑small aspect ratio͒ limit. The linear drag exerted by the obstacle on the flow is shown to be M 2 / ͑F ͱ F 2 −1͒ ϫ f͑B / F 2 −1͒, where f is a function specific to each obstacle. Explicit expressions are given for several common obstacle shapes, and the results are checked against nonlinear flows simulated by a shock-capturing finite volume numerical scheme. For flows within the supercritical regime ͑F −1ӷ M 2/3 ͒ the linear drag result is found to remain accurate up to ͑at least͒ M Ϸ 0.7. The success of the linear drag theory can be explained because, in the supercritical regime, strong nonlinearities are displaced to the wake regions at the flanks of the obstacle. In the presence of weak rotation and for small obstacle height the development of the nonlinear wakes is governed by the Ostrovsky-Hunter ͑OH͒ equation. Across a wide range of parameter space the wake pattern is determined by a single parameter ␤ =3M / B ͱ F 2 − 1. Numerical solutions of the OH equation illustrate the dependence of flow patterns and wave breaking regions on ␤. Results are again verified by comparison with numerical solutions of the full nonlinear rotating shallow water equations.

The Reduced Ostrovsky Equation: Integrability and Breaking

“…As noted there, solving in characteristic space is particularly useful for investigating wave-breaking as the wave breaks when the order unity quantity φ passes smoothly through zero. Esler et al [3] show that characteristic integrations can be carried smoothly past breaking, where they agree closely with finite volume integrations which fit "equal area" shocks to the waves after breaking. The system (41) was integrated numerically with spectral accuracy by performing the integration in (42) in Fourier space and then normalizing in real space using the result that the trapezium rule is spectrally accurate for periodic functions.…”

“…This system is solved to spectral accuracy for an unbounded interval in Esler et al [3] (note that there is a misprint in their Equation (26)) by expanding φ as a series of Chebyshev polynomials in X with time-dependent coefficients.…”

“…From our discussion of integrability in section 2 and our numerical results in section 4, we expect breaking to occur only at the edge of this region. Alternative breaking criteria were obtained by Liu et al [19], who used a priori estimates to deduce that breaking occurs when either J > (3H/2) 3/2 or when J > 0, H > 3/4, where J = − I (u 0x ) 3 dx, H = ( I |u 0 | 2 dx) 1/2 and I is again the periodic domain with period 2L = 1. They also obtained an analogous criterion on the infinite line.…”

Trapped internal waves over topography: Non-Boussinesq effects, symmetry breaking and downstream recovery jumps

“…12 Steady states for subcritical and supercritical flows over topography (using the more traditional definition of criticality) have been previously studied in the context of shallow water theory. [31][32][33][34] One of these steady states includes a downstream recovery jump when the flow upstream of the topography is subcritical but transitions to supercritical as it passes over the topography (i.e., a finite amplitude topography effect). A form of the downstream recovery jump was also observed by Stastna et al 35 in rotating continuously stratified flows.…”