The equations for the total flow are
Continuity equation $$ \nabla \cdot \mathbf{u} = 0 $$ Horizontal momentum equations
$ \begin{aligned} &\frac{D u}{D t}-f v=-\frac{1}{\rho_{0}}\frac{\partial p'}{\partial x} +\nu_H\!\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right) +\nu_V\!\left(\frac{\partial^2 u}{\partial z^2}\right)\\ &\frac{D v}{D t}+f u=-\frac{1}{\rho_{0}}\frac{\partial p'}{\partial y} +\nu_H\!\left(\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}\right) +\nu_V\!\left(\frac{\partial^2 v}{\partial z^2}\right)\\ &\frac{D w}{D t}=-\frac{1}{\rho_{0}}\frac{\partial p'}{\partial z} -\frac{g\rho'}{\rho_{0}} +\nu_H\!\left(\frac{\partial^2 w}{\partial x^2}+\frac{\partial^2 w}{\partial y^2}\right) +\nu_V\!\left(\frac{\partial^2 w}{\partial z^2}\right) \end{aligned} \;\xrightarrow[\text{frictionless}]{\text{negligible $w$}}\; \begin{aligned} \frac{D u}{D t}-f v&=-\frac{1}{\rho_{0}}\frac{\partial p'}{\partial x}\\ \frac{D v}{D t}+f u&=-\frac{1}{\rho_{0}}\frac{\partial p'}{\partial y} \end{aligned} $
Vertical hydrostatic equilibrium $$ \frac{d p}{d z} = - \rho g $$ Density equation $$ \frac{D \rho}{D t} \equiv \frac{\partial \rho}{\partial t} + \mathbf{u} \cdot \nabla \rho = 0 $$

The total flow is assumed to be composed of an eastward wind $U(z)$ in geostrophic equilibrium with the basic density structure $\bar{\rho}(y, z)$, plus perturbations $$ u = U(z) + u'(x,t), \quad v = v'(x,t), \quad w = w'(x,t), \quad \rho = \bar{\rho}(y,z) + \rho'(x,t), \quad p = \bar{p}(y,z) + p'(x,t) $$

The basic flow is in geostrophic and hydrostatic balance $$ f U = - \frac{1}{\rho_0} \frac{\partial \bar{p}}{\partial y}, \qquad 0 = -\frac{\partial \bar{p}}{\partial z} - \bar{\rho} g $$

Eliminating the pressure, we obtain the thermal wind relation $$ \frac{d U}{d z} = \frac{g}{\rho_0 f} \frac{\partial \bar{\rho}}{\partial y} $$ which requires the eastward flow to increase with height because $\partial \bar{\rho} / \partial y > 0$