Reduction of the Tropical Atmospheric Dynamics Into Shallow-Water Analogs: A Formulation Analysis

The shallow-water analogue models for the tropical atmosphere are examined from a formulational point of view. The normal-mode approach provides a formal procedure to reduce the primitive equation system to a shallow-water analogue, although approaches based on vertical integrals of the primitive equation system may be more intuitively appealing. Under a general framework of the latter, classical models by Gill (1980, ) and Lindzen and Nigam (1987, ) are derived in a deductive manner, by elucidating their limitations, implications, as well physical processes assumed. Major advantage of shallow-water analogue models is that after a vertical integral, the determination of convective heating rate simply reduces to that of a precipitation rate. Consequently, the question of representing convection also almost reduces to that of precipitation. This fact leads to confusions in literature about distinction between large-scale precipitation and subgrid-scale convection. This framework further supports a popular notion of the moisture as a key variable for describing convection. By reviewing the existing formulations, it is shown that convection can be parameterized without moisture under the limit of the parcel-environment quasi-equilibrium. When a depth of the water is much shallower than a typical horizontal extent of the flow, change of the flow pattern with height becomes no longer important, and the flow in this water can be treated as if a flow on a thin paper sheet. This system, called the shallow water, can furthermore be used to understand the basics of the atmospheric flows. Especially, the shallow water becomes an attractive approach for developing the theories for the tropical atmospheric dynamics, in which a conventional simplification of the description with an approximate balance between the Coriolis and pressure forces, as satisfied well for the midlatitude synoptic flows, no longer applies. Those tropical formulations of the shallow-water system are reviewed. Derivations of major shallow-water analogue atmosphere models from the primitive equation system are reviewed The Gill model with strong surface friction is justified only for the flows strongly confined to the boundary layer Convection can be under the parcel-environment quasi-equilibrium described without moisture, although it is popular to invoke it