Thermal wind equation

Mathematical formalism

Differentiating the geostrophic wind, ${\textstyle \mathbf {v} _{g}={\frac {1}{f}}\mathbf {k} \times \nabla _{p}\Phi }$ (where ${\displaystyle f}$ is the Coriolis parameter, ${\displaystyle \mathbf {k} }$ is the vertical unit vector, and the subscript "p" on the gradient operator denotes gradient on a constant pressure surface) with respect to pressure, and integrate from pressure level ${\displaystyle p_{0}}$ to ${\displaystyle p_{1}}$, we obtain the thermal wind equation: