Mapes, B. E., 2000: Convective inhibition, subgridscale triggering, and stratiform instability in a toy tropical wave model. J. Atmos. Sci., 57, 1515-1535.
A toy model of large-scale deep convection variations is constructed around a radiative-convective equilibrium climate, with an observed mean sounding as its thermodynamic basic state.
Vertical structure is truncated at two modes, excited by convective (one-signed) and stratiform (two-signed) heating processes in tropical deep convection. Separate treatments of deep and shallow convection are justified by observations that deep convection is more variable. Deep convection intensity is assumed to be modulated by convective available potential energy (CAPE), while occurrence frequency is modulated by the ratio of convective inhibition (CIN) to "triggering energy" K, a scalar representing the intensity of subgrid-scale fluctuations. Deep convective downdrafts cool and dry the boundary layer but also increase K. Variations of K make the relationship between convection and thermodynamic variables (CAPE, CIN, θe) nonunique and amplify the deep convective response to temperature waves of small (~1°C) amplitude.
For a parameter set in which CAPE variations control convection, moist convective damping destroys all variability. When CIN/K variations have dominant importance (the "inhibition-controlled" regime), a mechanism termed "stratiform instability" generates large-scale waves. This mechanism involves lower-tropospheric cooling by stratiform precipitation, which preferentially occurs where the already cool lower troposphere favors deep convection, via smaller CIN. Stratiform instability has two subregimes, based on the relative importance of the two opposite effects of downdrafts: When boundary layer θe reduction (a local negative feedback) is stronger, small-scale waves with frequency based on the boundary layer recovery time are preferred. When the K-generation effect (positive feedback) is stronger, very large scales (low wavenumbers of the domain) develop. A mixture of these scales occurs for parameter choices based on observations. Model waves resemble observed waves, with a phase speed ~20 m s-1 (near the dry wave speed of the second internal mode), and a "cold boomerang" vertical temperature structure.
Although K exhibits "quasi-equilibrium" with other convection variables (correlations > 0.99), replacing the prognostic K equation with diagnostic equations based on these relationships can put the model into wildly different regimes, if small time lags indicative of causality are distorted. The response of model convection to climatological spatial anomalies of θe (proxy for SST) and K (proxy for orographic and coastal triggering) is considered. Higher SST tends broadly to favor convection under either CAPE-controlled or inhibition-controlled regimes, but there are dynamical embellishments in the inhibition-controlled regime. The Kelvin wave seems to be the preferred structure when the model is run on a uniform equatorial β plane.