WAVE CISK OF EQUATORIAL WAVES AND THE VERTICAL-DISTRIBUTION OF CUMULUS HEATING

The dependence of Kelvin and Rossby wave CISK (conditional instability of the second kind);on the vertical distribution of cumulus heating i s examined by expanding the vertical heating profile into its Fourier series with Fourier coefficients f(1), f(2),..., f(N). In the standard analysis presented, N = 8 is used. The use of eight Fourier terms pro vides an adequate vertical resolution considering the current state of knowledge of the dependence of cumulus heating profiles on environmen tal conditions. The results of the analyses are illustrated in the sta bility diagrams in the parameter space of Fourier coefficients, showin g regions of stability and instability. These results show that all Ke lvin wave solutions are stable when the heating parameter epsilon is s maller than a critical value epsilon(c), the precise value of which de pends on how the Fourier coefficients, f(n), decrease with n. For mode rately large values of the heating parameter (say, for epsilon greater than or equal to 2), Kelvin wave solutions become unstable for suffic iently negative values of f(2). The authors found that the stability d iagrams of Rossby waves are identical to those of Kelvin waves. If the Fourier coefficients f(n) decrease rapidly with n, negative values of f(2) mean the heating profiles have a maximum in the upper tropospher e. An examination of the composition of the apparent heat source due t o cumulus clouds indicates that for moderately large amounts of total precipitation equatorial Kelvin waves and Rossby waves are more likely to be unstable through the wave-CISK mechanism if the clouds are very vigorous, so that the convergence of the eddy heat Aux has a maximum in the upper troposphere.