Approximated momentum equations \[ \begin{aligned} fv &= +\frac{1}{\rho_s r \cos\theta}\frac{\partial p}{\partial \phi}\\ fu &= -\frac{1}{\rho_s r}\frac{\partial p}{\partial \theta}\\ \rho g &= -\frac{\partial p}{\partial r} \end{aligned} \]

The parameter $f$ is the local component of the planetary vorticity normal to the earth’s surface and is called the Coriolis parameter $$ f = 2\Omega \sin\theta $$ With radial coordinate, $r_0$ is the distance from the earth’s center to its surface, $z = r - r_0$, $z \ll r_0$ $$ \begin{aligned} fv &= \frac{1}{\rho_s r_0 \cos\theta} \frac{\partial p}{\partial \phi}\\ fu &= -\frac{1}{\rho_s r_0}\frac{\partial p}{\partial \theta}\\ \rho g &= -\frac{\partial p}{\partial z} \end{aligned} $$

In this limiting form the momentum balance for the horizontal velocity reduces to a balance between the horizontal pressure gradient and the horizontal component of the Coriolis acceleration. The velocity derived from this relation is called the geostrophic velocity. The geostrophic approximation can be written in vector form as $$ \underbrace{\mathbf{u}_H}_{\substack{\text{horizontal} \\ \text{velocity vector}}} = \frac{1}{f\rho_s} \; \underbrace{\mathbf{k}}_{\substack{\text{unit vector perpendicular} \\ \text{to the surface of the sphere}}} \times \nabla p$$ while $$ \rho g = - \frac{\partial p}{\partial z} $$ is the hydrostatic approximation