\( \because \beta \overline{v} = \text{curl}_z(\boldsymbol{\tau}_T - \boldsymbol{\tau}_B) \)
\( \text{The top stress is the wind stress \(\text{curl}_z(\boldsymbol{\tau}_T) = F_T(x, y)\) where \( F_T \) is a known forcing (wind stress curl)} \) \( \text{The bottom stress is parameterized by linear Rayleigh drag \(\boldsymbol{\tau}_B = r \, \overline{\mathbf{u}} \Rightarrow \text{curl}_z(\boldsymbol{\tau}_B) = r \, \overline{\zeta}\) } \)
\( \text{where \( \overline{\zeta} = \frac{\partial \overline{v}}{\partial x} - \frac{\partial \overline{u}}{\partial y} \) is the vertically integrated relative vorticity} \) \( \therefore \beta \overline{v} = \text{curl}_z(\boldsymbol{\tau}_T) - \text{curl}_z(\boldsymbol{\tau}_B) = F_T(x, y) - r \, \overline{\zeta} \Rightarrow \boxed{\beta \overline{v} = -r \, \overline{\zeta} + F_T(x, y)} \)

\( \because \text{Assuming the vertically integrated velocity field is incompressible \(\nabla \cdot \overline{\mathbf{u}} = \frac{\partial \overline{u}}{\partial x} + \frac{\partial \overline{v}}{\partial y} = 0\)}, \overline{u} = -\frac{\partial \psi}{\partial y}, \overline{v} = \frac{\partial \psi}{\partial x} \)
\( \therefore \overline{\zeta} = \frac{\partial \overline{v}}{\partial x} - \frac{\partial \overline{u}}{\partial y} = \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} = \nabla^2 \psi \)
\( \therefore \beta \frac{\partial \psi}{\partial x} = -r \nabla^2 \psi + F_T(x, y) \Rightarrow \boxed{r \nabla^2 \psi + \beta \frac{\partial \psi}{\partial x} = F_T(x, y)} \) ◀ ▶