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Wave–vortex interaction in rotating shallow water. Part 1. One space dimension
- ALLEN C. KUO, LORENZO M. POLVANI
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- Journal:
- Journal of Fluid Mechanics / Volume 394 / 10 September 1999
- Published online by Cambridge University Press:
- 10 September 1999, pp. 1-27
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Using a physical space (i.e. non-modal) approach, we investigate interactions between fast inertio-gravity (IG) waves and slow balanced flows in a shallow rotating fluid. Specifically, we consider a train of IG waves impinging on a steady, exactly balanced vortex. For simplicity, the one-dimensional problem is studied first; the limitations of one-dimensionality are offset by the ability to define balance in an exact way. An asymptotic analysis of the problem in the small-amplitude limit is performed to demonstrate the existence of interactions. It is shown that these interactions are not confined to the modification of the wave field by the vortex but, more importantly, that the waves are able to alter in a non-trivial way the potential vorticity associated with that vortex. Interestingly, in this one-dimensional problem, once the waves have traversed the vortex region and have propagated away, the vortex exactly recovers its initial shape and thus bears no signature of the interaction. Furthermore, we prove this last result in the case of arbitrary vortex and wave amplitudes. Numerical integrations of the full one-dimensional shallow-water equations in strongly nonlinear regimes are also performed: they confirm that time-dependent interactions exist and increase with wave amplitude, while at the final state the vortex bears no sign of the interaction. In addition, they reveal that cyclonic vortices interact more strongly with the wave field than anticyclonic ones.
Nonlinear Rossby adjustment in a channel
- K. R. HELFRICH, ALLEN C. KUO, L. J. PRATT
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- Journal:
- Journal of Fluid Mechanics / Volume 390 / 10 July 1999
- Published online by Cambridge University Press:
- 10 July 1999, pp. 187-222
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The Rossby adjustment problem for a homogeneous fluid in a channel is solved for large values of the initial depth discontinuity. We begin by analysing the classical dam break problem in which the depth on one side of the discontinuity is zero. An approximate solution for this case can be constructed by assuming semigeostrophic dynamics and using the method of characteristics. This theory is supplemented by numerical solutions to the full shallow water equations. The development of the flow and the final, equilibrium volume transport are governed by the ratio of the Rossby radius of deformation to the channel width, the only non-dimensional parameter. After the dam is destroyed the rotating fluid spills down the dry section of the channel forming a rarefying intrusion which, for northern hemisphere rotation, is banked against the right-hand wall (facing downstream). As the channel width is increased the speed of the leading edge (along the right-hand wall) exceeds the intrusion speed for the non-rotating case, reaching the limiting value of 3.80 times the linear Kelvin wave speed in the upstream basin. On the left side of the channel fluid separates from the sidewall at a point whose speed decreases to zero as the channel width approaches infinity. Numerical computations of the evolving flow show good agreement with the semigeostrophic theory for widths less than about a deformation radius. For larger widths cross-channel accelerations, absent in the semigeostrophic approximation, reduce the agreement. The final equilibrium transport down the channel is determined from the semigeostrophic theory and found to depart from the non-rotating result for channels widths greater than about one deformation radius. Rotation limits the transport to a constant maximum value for channel widths greater than about four deformation radii.
The case in which the initial fluid depth downstream of the dam is non-zero is then examined numerically. The leading rarefying intrusion is now replaced by a Kelvin shock, or bore, whose speed is substantially less than the zero-depth intrusion speed. The shock is either straight across the channel or attached only to the right-hand wall depending on the channel width and the additional parameter, the initial depth difference. The shock speeds and amplitudes on the right-hand wall, for fixed downstream depth, increase above the non-rotating values with increasing channel width. However, rotation reduces the speed of a shock of given amplitude below the non-rotating case. We also find evidence of resonant generation of Poincaré waves by the bore. Shock characteristics are compared to theories of rotating shocks and qualitative agreement is found except for the change in potential vorticity across the shock, which is very sensitive to the model dissipation. Behind the leading shock the flow evolves in much the same way as described by linear theory except for the generation of strongly nonlinear transverse oscillations and rapid advection down the right-hand channel wall of fluid originally upstream of the dam. Final steady-state transports decrease from the zero upstream depth case as the initial depth difference is decreased.