The Coupling between Gravity Waves and Moisture: A Mechanistic Picture

Motivated by the need to understand the influence of convection on tropical variability, the interaction between gravity and Kelvin waves and moisture in a shallow water model is analyzed with an emphasis on physical interpretation. Convection is represented by a simple Betts-Miller type relaxation, and analytical solutions for the influence of moisture on wave speed and stability are obtained, both at the limit of a vanishing convective relaxation timescale (or ``strict quasi-equilibrium'' (SQE)) and for finite relaxation timescales.

We show that the moisture and convergence are necessarily in-phase at the SQE limit, and that this phase relationship necessarily changes when SQE is relaxed. As a natural consequence of the equivalence between SQE and the moisture-convergence phasing, larger scale waves are always closer to SQE than the smaller scale waves.

Our analysis separates the phase and amplitude of the travelling wave, a relaxation timescale dependent ``gross moist stability'' and equivalent depth are derived for both one-dimensional gravity waves and Kelvin waves. We develop a mechanistic picture of how the phasing between the height and the divergence fields changes in the presence of moisture, and its effect on the stability of gravity and Kelvin waves. In particular, we show that rotation constrains moist Kelvin waves to be unconditionally stable at the SQE limit in our system by fixing the phasing between the height and divergence fields via geostrophic balance.

Away from the SQE limit, we show that a relaxation-type convection scheme can only damp gravity and Kelvin waves, and conditions for instability preclude coherent propagation of all fields (at and away from the SQE limit). Furthermore, we show that the well known condition for instability, (H - LQ) < 0 (where H is the mean depth of the fluid, L is the latent heat of condensation and Q is the column integrated saturation specific humidity), is true only at the SQE limit and a more general relation (H - \gamma LQ < 0) holds away from SQE. We derive analytical relationships for \gamma for gravity and Kelvin waves and try to understand what it implies physically.