The term thermal wind refers to steady or slowly time‐varying shear flow, driven by buoyancy forces due to lateral gradients in density and modified by planetary rotation through the Coriolis acceleration.

For the atmosphere, the perfect-gas law applies and therefore \[\frac{1}{\rho} (\nabla p)_p = -\frac{1}{T} (\nabla T)_p = -\frac{1}{\theta} (\nabla \theta)_p\] so that the increase of the wind with height can be naturally written directly in terms of the horizontal gradient of temperature or potential temperature in constant-pressure surfaces

In the case of the oceans, the ratio of the terms on the right-hand side of \(\boxed{f \frac{\partial v}{\partial z} = -\frac{g}{\rho_s r_0 \cos\theta} \frac{\partial \rho}{\partial \phi} - \frac{1}{\rho_s} \frac{\partial \rho_s}{\partial z} f v}\) is \[ \frac{fv \, \partial \rho_s/\partial z}{(g/r_0 \cos \theta)\, \partial p / \partial \phi} = O\!\left(\frac{fU/H}{fU/D}\right) = \frac{D}{H} \] where $H$ is the density scale height $$ H = \left(-\frac{1}{\rho_s}\frac{\partial \rho_s}{\partial z}\right)^{-1} $$ and $D$ is the depth scale of the motion. The ratio $D/H$ is therefore of the order $\Delta \rho_s / \rho_s$, i.e., the order of the proportional change of the density over the vertical scale of motion. This never exceeds an amount of $O(10^{-3})$, so that to an excellent approximation for the oceans the density gradients in \(\boxed{\frac{\partial v}{\partial z} = -\frac{g}{f \rho r_0 \cos\theta} \left(\frac{\partial \rho}{\partial \phi}\right)_p}\) and \(\boxed{\frac{\partial u}{\partial z} = \frac{g}{f \rho r_0} \left(\frac{\partial \rho}{\partial \theta}\right)_p}\) can be evaluated in surfaces of constant height.