Stable and unstable shear modes of rotating parallel flows in shallow water

Abstract

This article considers the instabilities of rotating, shallow-water, shear flows on an equatorial β-plane. Because of the free surface, the motion is horizontally divergent and the energy density is cubic in the field variables (i.e. in standard notation the kinetic energy density is ½ h(u2 + v2)). Marinone & Ripa (1984) observed that as a consequence of this the wave energy is no longer positive definite (there is a cross-term Uh′u′). A wave with negative wave energy can grow by transferring energy to the mean flow. Of course total (mean plus wave) energy is conserved in this process. Further, when the basic state has constant potential vorticity, we show that there are no exchanges of energy and momentum between a growing wave and the mean flow. Consequently when the basic state has no potential vorticity gradients an unstable wave has zero wave energy and the mean flow is modified so that its energy is unchanged. This result strikingly shows that energy and momentum exchanges between a growing wave and the mean flow are not generally characteristic of, or essential to, instability.A useful conceptual tool in understanding these counterintuitive results is that of disturbance energy (or pseudoenergy) of a shear mode. This is the amount of energy in the fluid when the mode is excited minus the amount in the unperturbed medium. Equivalently, the disturbance energy is the sum of the wave energy and that in the modified mean flow. The disturbance momentum (or pseudomomentum) is defined analogously.For an unstable mode, which grows without external sources, the disturbance energy must be zero. On the other hand the wave energy may increase to plus infinity, remain zero, or decrease to minus infinity. Thus there is a tripartite classification of instabilities. We suggest that one common feature in all three cases is that the unstable shear mode is roughly a linear combination of resonating shear modes each of which would be stable if the other were somehow suppressed. The two resonating constituents must have opposite-signed disturbance energies in order that the unstable alliance has zero disturbance energy. The instability is a transfer of disturbance energy from the member with negative disturbance energy to the one with positive disturbance energy.

Potential vorticity dynamics in the framework of disk shallow-water theory

“…Instability can be best rationalized in terms of the interaction of the two individual edgewaves propagating along each defect of the symmetric step profile. This picture is consistent with the Hayashi-Young criterion for wave instability (Hayashi & Young 1987), which can be summarized as follows: instability of two waves separated by an evanescent layer occurs when the two waves (i) propagate opposite to each other; (ii) have almost the same Doppler-shifted frequency; and (iii) can interact with one another. Interaction in this case means that the edgewave at one defect contributes to the perturbation velocity profile at the other defect.…”

Nonlinear adjustment of a localized layer of buoyant, uniform potential vorticity fluid against a vertical wall

“…The strong anisotropy inherent in the initial condition and resulting flow may also be a factor (cf. Hayasha and Young, 1987). It would also be interesting to consider initial conditions that promote strong gradients in wave breaking which then give rise to potential vorticity changes and thus a more direct and faster interaction.…”

“…There are no potential vorticity gradients for the inertia-gravity advect and hence no interaction. This has been discussed by Hayasha and Young (1987) for wave-mean flow interactions in unstable semigeostrophic flows. This implies that the problem here may be a special case.…”

Potential vorticity dynamics in the framework of disk shallow-water theory

“…When the laboratory measured wavespeeds of both waves are nearly equal, instability may manifest itself, i.e. when the Hayashi-Young criterion is satisfied (Hayashi & Young 1987). Instability exists when the Rossby waves are essentially counterpropagating along the flow (Heifetz et al 1999) because it is only when the local propagating tendency of the Rossby wave is against the mean flow that the Hayashi-Young criteria of instability can be met.…”