Wang, B., 1987: The nature of CISK in a generalized continuous model. Journal of the Atmospheric Sciences, 44(10), 1411-1426.

Abstract: In spite of the fundamental difficulties in interpreting the growth of tropical storms, the basic idea of CISK remains valuable in understanding the instability resulting from the interaction between cumulus convection and large-scale flows. A generalized solution of a quasi-balanced continuous model, which can be applied to various types of vertical heating distibution and basic-state stratification, is derived and used to explore the behaviors fo the CISK mode.

In the absence of cumulus momentum mixing, the CISK solution exhibits, in general, a scale selection. However, two types of unbounded growth rate associated with different closure assumptions may exist. Both of them take place in a common situation that is characterized by local warming at the top of the moist convergence layer in the area of rising motion. In these circumstances, the direct coupling between the heating and the large-scale moisture supply through the divergent wind component dominates over the indirect coupling through the rotational component. It is suggested that, for a feasible Ekman CISK mechanism, the dominant feedback of the convective heating to the low-level moisture convergence must be of an indirect nature. In this feedback process, planetary vorticity and/or preexisting relative vorticity play an essential role in converting heating-induced divergent kinetic energy to rotational kinetic energy, thus accelerating the spin-up of a large-scale vortex.

The cumulus momentum mixing destabilizes short waves by enhancing cyclonic circulation at the top of the Ekman layer and by reducing the vertical extent of the temperature disturbance; meanwhile, it stabilizes long waves by weakening the anticyclonic circulation in the upper levels.