quasigeostrophic equations

The system of momentum and thermodynamic equations, derived through the quasigeostrophic approximation, that fulfill the requirements of quasigeostrophic theory.
At the momentum level and using the pseudoheight coordinate z = [1 - (p/p₀)^κ]c_pθ₀/g the equations are

in which x, y, and z form a Cartesian coordinate system with u_g and v_g the eastward and northward geostrophic wind components, u_ag and v_ag the eastward and northward ageostrophic wind components, w the vertical velocity in the pseudoheight coordinate, φ the geopotential, θ the potential temperature, g the acceleration of gravity, f the Coriolis parameter, ρ the density in z space, θ₀ the reference potential temperature, and N the Brunt–Väisälä frequency.