On Symmetric Stability and Instability of Zonal Mean Flows Near the Equator

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Abstract
Obserations of longitudinally-averaged zonal flows in the atmosphere and ocean tend to display north–south symmetry about the equator, with a characteristic wind maximum or minimum, and therefore little horizontal wind shear locally near the equator. It is shown that this configuration is required for balanced flow on a sphere to be inertially stable. If dissipation can be neglected, any horizontal wind shear at the equator will cause inertial instability to develop. effectively eliminating the horizontal shear. It follows that the potential vorticity (q) must vanish at the equator for the symmetric circulation to be stable; and that it must increase to the north and decrease to the south. Balanced cross-equatorial flow can occur only if there is a north–south gradient in the torque or the diabatic heating at the equator. These conclusions are obtained under the assumption of a balanced zonal flow; i.e., acceleration and dissipation are explicitly neglected in the meridional momentum equation. The characteristics of the equatorial symmetric instability that develops if the mean flow is horizontally sheared at the equator are investigated. The analysis with Rayleigh friction and Newtonian cooling (i.e., scale-independent dissipation) extends We treatment of inviscid instability on the equatorial beta-plane by Dunkerton (1981). Quadratic as well as linear shear is treated, thereby enabling application to tropical jets. The instability is confined to the region in which the vertical component of absolute vorticity is of opposite sign to the local Coriolis parameter, i.e., where the square of the inertial frequency f(gz;+) is negative. The mode of greatest instability is a moridional overturning with a single cell in the horizontal dimension which tends to mix angular momentum, thereby eliminating the horizontal gradient of angular momentum at the equator. With the scale-independent parameterization of mechanical and thermal dissipation, the mixing occurs most readily at the smallest vertical scales and the gravest (n=0) meridional mode. When the low curvature is much less than β, the maximum growth rate for symmetric instability is approximately one-half the magnitude of the relative vorticity of the mean flow at the equator minus the mechanical dissipation rate. Hence the horizontal shear at the equator must exceed twice the Rayleigh friction coefficient for instability. Thermal dissipation does not affect the instability criterion. Many recent studies have been undertaken which investigate the effect of mean zonal flow on tropical waves and instabilities. A consequence of the present analysis is that a stationary. non-dissipative basic mate flow with horizontal shear at the equator is not an appropriate basic state for neutrally propagating waves because it is not a stable solution of the symmetric governing equations. The instability tends to eliminate the latitudinal shear of the zonal flow at the equator if mechanical dissipation is not 100 great. While Dunkerton (1981) focused on the ramifications for the middle atmosphere, this study applies the results primarily to tropospheric and oceanic circulations.

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