Thermal wind

Remarks

The proof differentiates the geostrophic relations with respect to $p$ and exchanges the order of derivatives: $\partial u_g / \partial p = -f^{-1} \partial_y (\partial \Phi / \partial p) = (R / (f p)) \, \partial T / \partial y$. Integration in $\ln p$ from $\ln p_0$ to $\ln p_1$ gives the second formula.

So the vertical shear of the geostrophic wind is parallel to the isotherms of the layer, with cold air to the left in the northern hemisphere ($f > 0$). Example: in mid-latitudes of the northern hemisphere the temperature decreases northward, $\partial T / \partial y < 0$. Since $p$ decreases upward, $u_g$ increases with height, which explains the westerly jet streams. With $f = 10^{-4}$ s$^{-1}$, $R = 287$ J/(kg K), $\partial T / \partial y = -5$ K per $1000$ km and $p_0 / p_1 = 4$, the shear is $u_g(p_1) - u_g(p_0) \approx 287 \cdot 5 \cdot 10^{-6} \cdot 1.39 / 10^{-4} \approx 20$ m/s.

In a Boussinesq fluid with buoyancy $b$ and height $z$, the same argument gives $f \, \partial u_g / \partial z = -\partial b / \partial y$ and $f \, \partial v_g / \partial z = \partial b / \partial x$. The relation fails near the equator, where $f \to 0$ and geostrophic balance does not hold.

Going deeper

For a student

The thermal wind relation combines the two leading balances of large-scale flow: geostrophic balance in the horizontal and hydrostatic balance in the vertical. It links the wind field and the temperature field, so a forecaster can infer the vertical shear from a temperature map. A wind that turns clockwise with height in the northern hemisphere (veering) indicates warm advection, and a wind that turns anticlockwise (backing) indicates cold advection. In quasi-geostrophic theory the relation holds automatically, since the wind and the buoyancy both come from one stream function. A flow with vertical shear in thermal wind balance stores available potential energy that baroclinic instability can release.