Representing feedbacks from large-scale dynamics in limited domain models of tropical convection: an assessment of current approaches

Simulating moist convection in a limited domain model while parameterizing feedbacks from the large-scale flow is an effective strategy to gain insights into the interaction between convection and large-scale dynamics. In the tropics, one can compute the large-scale vertical motion required to reduce horizontal density gradients, and apply the advective tendencies associated with this vertical motion to the limited domain model to represent the feedback from the large-scale flow. There are, broadly speaking, two approaches to compute the large-scale vertical motion in the literature. The first approach, often known as the Weak Temperature Gradient (WTG) approximation, takes the large-scale vertical motion to be what is required to relax the horizontally averaged temperature of the limited domain model back to the large-scale mean temperature profile over a certain timescale. This will be called the relaxation approach. The second approach computes the vertical velocity based on two-dimensional gravity wave equations of a single horizonal wavenumber, and will be called the gravity wave approach. The relaxation approach, when recast in the framework of the gravity wave approach, effectively assumes that gravity waves of different vertical wavenumbers propagate at the same speed. While such assumption is valid in the limit of fast gravity wave adjustment (appropriate for horizontal scales of thousands of kilometers or less), for planetary scale phenomena, the relaxation approach produces vertical velocity profiles that are too top-heavy, and can cause significant errors in simulated behaviors of the system. This is demonstrated through comparisons with idealized Walker cell experiments and interpreted through a simple model.