(a) Derivation of the Thermal-Wind Relation

The governing equations for steady, hydrostatic, geostrophic flow are:

$$-fv = -\frac{1}{\rho_0}\frac{\partial p}{\partial x} x$$ (39)

$$fu = -\frac{1}{\rho_0}\frac{\partial p}{\partial y} y$$ (40)

$$\frac{\partial p}{\partial z} = -\rho g$$ (41)

We use the Boussinesq approximation for density, $\rho = \rho_0 + \rho'(x,y,z)$, where $\rho_0$ is the constant reference density. Since $\rho_0$ is a constant, we can rewrite the density in the hydrostatic equation, $\frac{\partial p}{\partial z} = -(\rho_0 + \rho')g$.

The procedure to eliminate the pressure ($p$) is to take the vertical derivative of the geostrophic equations (Eq. and ) and substitute the expression for $\frac{\partial p}{\partial z}$ obtained from the hydrostatic equation (Eq. ).