Equatorial Inertial Instability: Effects of Vertical Finite Differencing and Radiative Transfer

Abstract

The effect of vertical differencing on equatorial inertial instability is studied and explicit results obtained for growth rates as a function of the vertical resolution. It is found that for a basic state independent of height, the form of the growing modes is the same as that without vertical discretization except that the vertical wavenumber is replaced by an effective vertical wavenumber in the differential equation for the horizontal structure. This effective vertical wavenumber is bounded above by a value that depends on the spacing of the model levels, which implies that growing modes only occur when the shear exceeds a certain value. The upper bound is crucially dependent on the form of the difference scheme. For a scheme in which horizontal velocities and geopotential are evaluated on full levels and temperature and vertical velocity are evaluated on half levels (the Charney?Phillips scheme) the upper bound on the effective vertical wavenumber is 2/δ in the Boussinesq limit, where δ is the spacing between the model levels. For a scheme in which the horizontal velocity, geopotential, and temperature are evaluated on full levels, and only the vertical velocity on half levels (the Lorenz scheme), there is no upper bound on the effective vertical wavenumber in the Boussinesq limit so that growing modes occur for any nonzero value of the shear. This is contrary to the expectation that there is a minimum critical shear for instability because the vertical resolution limits the vertical wavenumber. The effect of Newtonian cooling is also considered and an expression for the growth rate as a function of the cooling coefficient and the effective vertical wavenumber is found. It is found that provided the shear at the equator is nonzero, there are growing modes for all vertical wavenumbers, unlike the case without Newtonian cooling, where a mode grows only if its vertical wavenumber exceeds a critical value that depends on the shear. The consequences for numerical models with finite vertical resolution are discussed.