\[ \frac{Du}{Dt} - fv = -\frac{1}{\rho_0} \frac{\partial p'}{\partial x} + \nu_H \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) + \nu_V \left( \frac{\partial^2 u}{\partial z^2} \right)\\ \frac{Dv}{Dt} + fu = -\frac{1}{\rho_0} \frac{\partial p'}{\partial y} + \nu_H \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) + \nu_V \left( \frac{\partial^2 v}{\partial z^2} \right)\\ \frac{Dw}{Dt} = -\frac{1}{\rho_0} \frac{\partial p'}{\partial z} - g \rho' + \nu_H \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} \right) + \nu_V \left( \frac{\partial^2 w}{\partial z^2} \right)\] These are the equations for primarily horizontal fluid motion within a thin layer on a locally-flat rotating Earth

Consider a continuously stratified medium and assume that the horizontal scale of motion is much larger than the vertical scale. The pressure distribution is therefore hydrostatic, and the linearized equations of motion are the incompressible flow continuity equation \( \nabla \cdot \mathbf{u} = 0 \) \[ \frac{\partial u}{\partial t} - fv = -\frac{1}{\rho_0} \frac{\partial p'}{\partial x}, \quad \frac{\partial v}{\partial t} + fu = -\frac{1}{\rho_0} \frac{\partial p'}{\partial y}, \quad 0 = -\frac{\partial p'}{\partial z} - g \rho' \] and the linearized density equation \(\frac{D\rho}{Dt} \equiv \frac{\partial \rho}{\partial t} + \mathbf{u} \cdot \nabla \rho = 0 \): \[ \frac{\partial \rho'}{\partial t} - \rho_0 \frac{N^2}{g} w = 0 \] where \( p' \) and \( \rho' \) represent perturbations of pressure and density from the state of rest