Modes of Tropical Variability under Convective Adjustment and the Madden–Julian Oscillation. Part I: Analytical Theory

Abstract The interaction between the collective effects of cumulus convection and large-scale dynamics is examined using the Betts–Miller moist convective adjustment (MCA) parameterization in a linearized primitive equation model on an equatorial β plane. In Part I of this paper, an analytical approach to the eigenvalue problem is taken using perturbation expansions in the cumulus adjustment time, which is short compared to planetary dynamical time scales. The modes of tropical variability that arise under MCA are dominated by the presence of moist processes; some modes act to adjust the system rapidly toward a convectively adjusted state, while others evolve on time scales set by the large-scale dynamics subject to near-adjusted (quasi equilibrium) thermodynamical constraints. Of the latter, a single vertical mode stands out, which obeys special balances implied by the quasi-equilibrium constraints and is the only propagating deep convective mode. The propagation speed is determined by an internally defi...