Energetics of the instability of a depth independent equatorial jet

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Abstract
Negative energy perturbations may be linked to the instability of fronts and gravity currents, for instance when the latter have uniform potential vorticity (in the f-plane). The equatorial Gaussian jet ũ(y) = ũ(0) exp (−y1/L2), in a one-layer, reduced-gravity, β-plane model may also be unstable to negative energy perturbations, particularly when the meridional gradient of potential vorticity does not change sign. Thus, if the semi-width of the equatorial jet equals the deformation radius, L=R, the potential and planetary vorticities are proportional, and the northward gradient of the former is therefore positive everywhere (this flow has the structure of a steady Kelvin mode). A westward jet in this class is found to be unstable to varicose, negative-energy perturbations: a necessary condition for instability is that the flow be supercritical at the equator. The net energy transfer is from the perturbations, whose amplitude increases, to the mean flow, even though potential energy exchange is in the reverse direction. These energy transfers are of similar magnitude, but have signs opposite to those of the classical barotropic case, which is discussed next. (Total pseudo-energy, an order wave-amplitude squared conserved quantity, vanishes for a growing perturbation.) Narrower jets, L=0.4R, are unstable if the gradient of potential vorticity changes sign. The energy of the perturbation is positive. Most of the energy transfer is from the mean flow's kinetic energy to the perturbation's kinetic energy. This case is like the classical barotropic one, because divergence effects are negligible