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Global weak solution of planetary geostrophic equations with inviscid geostrophic balance

A reformulation of the planetary geostrophic equations (PGEs) with the inviscid balance equation is proposed and the existence of global weak solutions is established, provided that the mechanical force satisfies an integral constraint. There is only one prognostic equation for the temperature field, and the velocity field is statically determined by the planetary geostrophic balance combined with the incompressibility condition. Furthermore, the velocity profile can be accurately represented as a function of the temperature gradient. In particular, the vertical velocity depends only on the first-order derivative of the temperature. As a result, the bound for the L∞ (0, t 1 ; L 2) ∩ L 2 (0, t 1 ; H 1) norm of the temperature field is sufficient to show the existence of the weak solution.