(b) Physical Discussion of Stratification and Vertical Shear

The thermal-wind relation directly links the **vertical shear** of the horizontal geostrophic velocity ( $\partial u/\partial z, \partial v/\partial z$) to the **horizontal gradient** of the density ( $\partial \rho'/\partial x, \partial \rho'/\partial y$).

• **Vertical Shear (LHS):** $\partial u/\partial z$ or $\partial v/\partial z$ means the horizontal velocity changes with height. This implies that the flow at the surface is different from the flow at depth, resulting in a tilted column of fluid (vertical shear).
• **Horizontal Density Gradient (RHS):** $\partial \rho'/\partial x$ or $\partial \rho'/\partial y$ means the density (or temperature/salinity for the ocean, or temperature for the atmosphere) is horizontally non-uniform. This state is known as **baroclinic stratification** (where surfaces of constant pressure are not parallel to surfaces of constant density).

Physically, the thermal-wind balance states that:

A **geostrophic current** can only possess vertical shear if there is a **horizontal gradient in density** (a baroclinic field).

If $\frac{\partial \rho'}{\partial x} = \frac{\partial \rho'}{\partial y} = 0$, then $\frac{\partial u}{\partial z} = \frac{\partial v}{\partial z} = 0$. This implies that the flow is **barotropic** ($\rho' = 0$ or $\rho'$ is uniform horizontally), and the geostrophic velocity $\mathbf{u}_h$ is constant with depth.

In essence, the thermal wind relation describes how the horizontal pressure gradient, which drives the geostrophic flow, changes with height due to horizontal variations in the fluid's weight (density).