The Basic Equations under Weak Temperature Gradient Balance: Formulation, Scaling, and Types of Convectively Coupled Motions

Abstract

Abstract The weak temperature gradient (WTG) approximation is extended to the basic equations on a rotating plane. The circulation is decomposed into a diabatic component that satisfies WTG balance exactly and a deviation from this balance. Scale analysis of the decomposed basic equations reveals a spectrum of motions, including unbalanced inertio-gravity waves and several systems that are in approximate WTG balance. The balanced systems include equatorial moisture modes with features reminiscent of the MJO, off-equatorial moisture modes that resemble tropical depression disturbances, “mixed systems” in which temperature and moisture play comparable roles in their thermodynamics, and moist quasigeostrophic motions. In the balanced systems the deviation from WTG balance is quasi nondivergent, in nonlinear balance, and evolves in accordance to the vorticity equation. The evolution of the strictly balanced WTG circulation is in turn described by the divergence equation. WTG balance restricts the flow to evolve in the horizontal plane by making the isobars impermeable to vorticity and divergence, even in the presence of diabatically driven vertical motions. The vorticity and divergence equations form a closed system of equations when the irrotational circulation is in WTG balance and the nondivergent circulation is in nonlinear balance. The resulting “WTG equations” may elucidate how interactions between diabatic processes and the horizontal circulation shape slowly evolving tropical motions. Significance Statement Many gaps in our understanding of tropical weather systems still exist and there are still many opportunities to improve their forecasting. We seek to further our understanding of the tropics by extending a framework known as the “weak temperature gradient approximation” to all of the equations for atmospheric flow. Doing this reveals a variety of motions whose scales are similar to observed tropical weather systems. We also show that two equations describe the evolution of slow systems: one that describes tropical thunderstorms and one for the rotating horizontal winds. The two equations may help us understand the dynamics of slowly evolving tropical systems.