Phase-lagged wave-CISK

Abstract

The conventional formulation of the wave-CISK mechanism involves a quasi-equilibrium assumption whereby the instantaneous large-scale diabatic heating that is induced by cloud-scale motions is assumed to be related to the instantaneous distribution of the large-scale flow, and in particular to the low-level moisture convergence of the large-scale flow. Thus in this context the deep convective clouds of the tropical atmosphere are viewed as instantaneous processors of the large-scale moisture supply. Herein a conventional, inviscid wave-CISK model is developed and an examination undertaken of the properties of the analytically determined solutions. The quasi-equilibrium assumption is then discarded by incorporating into the CISK model's formulation a simple time-dependent model of the dynamics of a cumulus cloud ensemble. The latter model has a finite adjustment time (TR) to changes in the large-scale forcing field. Hence the apparent diabatic heat source experienced by the large-scale fields becomes a function of the time history of that same forcing field. Neutral solutions of the refined wave-CISK model are derived as functions of the diabatic heating amplitude and the adjustment time. These solutions illustrate the frequency-selective stabilizing effect of the hypothesized cloud-scale dynamics. Again, the marginally unstable waves of the refined model are shown to exhibit properties that overcome some of the debilitating features of conventional wave-CISK. These solutions also reproduce some of the observed features of tropical wave motions. In particular, for Tr ∼ 12 h and with a physically reasonable value of the cumulus diabatic heating amplitude, it is shown that all the inertia-gravity modes are stable, and that the most unstable Kelvin wave, mixed Rossby-gravity wave and Rossby (n = 1 and n = 2) waves have finite preferred spatial length scales corresponding respectively to zonal wavenumbers (m) of 2, 4, 6 and 8. The periods of these four waves correspond approximately to 10, 4/1/2 12/1/2 and 15 days but the growth rates are comparatively small. The agreement with tropical wave data is not unreasonable when allowance is made for the Doppler shift of the Rossby waves. Also the vertical structures of these preferred modes are different: the Kelvin and mixed Rossby-gravity waves are associated respectively with the shortest and longest vertical wavelengths, and the meridional wind field of the Rossby wave (n = 1) has a maximum in both the upper and lower troposphere. The degree of mutual compensation in the model solutions between diabatic heating due to cumulus clouds and cooling due to large-scale ascent is not as strong as that observed in the atmosphere.