Geostrophic approximation in horizontally differential atmospheric rotation

Effects of horizontally differential atmospheric rotation are considered in geostrophic dynamics of planetary and stellar atmospheres. The Coriolis parameter defined by the angular velocity of a basic flow f and the latitudinal gradient of the angular velocity Γ are used in the present study. Nondimensional differential rotation factor Γ/f and Rossby number R _o determine whether the geostrophic approximation can be applied to differential rotations of planetary and stellar atmospheres, or not. When an eddy with small intrinsic phase velocity satisfies the condition of F _r ≤ 1 (F _r: Froude number) and L/a ≤ R _o ≪ 1 (L: eddy horizontal scale, a: planetary radius), for rigid-body rotation (Γ/f ≪ R _o ² or Γ/f ∼ R _o ²) and weakly differential rotation (Γ/f ∼ R _o ¹), the geostrophic approximation can be applied. However, for strongly differential rotation (Γ/f ∼ R _o ⁰), the geostrophic approximation cannot be applied, even when R _o is sufficiently small.