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

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 defined gross moist stability. For the Kelvin meridional mode, the phase speed and vertical structure are highly suggestive of those of the Madden-Julian (MJ) oscillation. For the simple case considered here, which assumes a homogeneous, separable basic state and sufficiently large zonal scales, the modes of variability found under MCA are all stable under reasonable conditions, although a large subclass of modes (including the MJ mode) is only slowly decaying. This contrasts with many studies using Kuo-like convective parameterizations, which have conjectured that convective instability of the second kind (CISK) plays a role in maintaining planetary-scale tropical variability. The authors suggest that a terminology is needed by which to refer to convective interaction with dynamics (CID), without necessarily assuming that large-scale instability arises from this interaction. Under MCA, there is strong CID but not generally CISK. Instability of the MJ mode can occur through evaporation-wind feedback. This behavior under MCA provides a suggestive prototype for tropical motions evolving under quasi-equilibrium convective constraints.