Quasi-balanced dynamics in the tropics

We propose a quasi-balance dynamical system of equations for slowly evolving, large-scale motions in the tropics. Unlike other balance schemes, equatorially trapped Kelvin waves are included. Following the lead of Gill, this system is based on the neglect of the meridional acceleration (the so-called long-wave approximation). Consistent budgets for kinetic energy and potential vorticity are derived, with those quantities appropriately redefined. Unlike the quasi-geostrophic system, vertical advection of potential vorticity as well as of potential temperature is relevant. A primary circulation is defined, involving the zonal wind, geopotential, and potential temperature, such that potential vorticity depends on latitudinal and vertical gradients of the primary circulation. A residual secondary circulation is governed by an elliptic equation for a streamfunction in the meridional plane. The invertibility principle is affected by the Kelvin waves with zero potential vorticity. The eigenfrequencies for a tropospheric first internal mode are obtained both for the primitive equations and the quasi-balanced system. Applications of this dynamical system are proposed for diagnostic analysis and for investigating the slow evolution of the large-scale flow in the tropics.