Consider a square domain of side $a$ and rescale the variables by setting $$ x = a\hat{x}, \quad y = a\hat{y}, \quad \boldsymbol{\tau} = \tau_0 \hat{\boldsymbol{\tau}}, \quad \psi = \hat{\psi} \frac{\overbrace{\tau_0}^{\text{amplitude of wind stress}}}{\beta} $$ The hatted variables are nondimensional and $\mathcal{O}(1)$ quantities in the interior

Consider a square domain of side $a$ and rescale the variables by setting $$ x = a\hat{x}, \quad y = a\hat{y}, \quad \boldsymbol{\tau} = \tau_0 \hat{\boldsymbol{\tau}}, \quad \psi = \hat{\psi} \frac{\overbrace{\tau_0}^{\text{amplitude of wind stress}}}{\beta} $$ The hatted variables are nondimensional and $\mathcal{O}(1)$ quantities in the interior

$$ r \nabla^2 \psi + \beta \frac{\partial \psi}{\partial x} = \text{curl}_z \boldsymbol{\tau}_T \Rightarrow \frac{\partial \hat{\psi}}{\partial \hat{x}} + \underbrace{\epsilon_S}_{=\,r/(a\beta)\,\ll\,1} \nabla^2 \hat{\psi} = \text{curl}_z \hat{\boldsymbol{\tau}}_T $$

Over the interior of the domain, away from boundaries, the frictional term is small $$ \psi(x,y) = \underbrace{\psi_I(x,y)}_{\text{interior streamfunction}} + \underbrace{\phi(x,y)}_{\text{boundary-layer correction}} $$

Away from boundaries $\psi_I$ is presumed to dominate the flow, and this satisfies $$ \frac{\partial \psi_I}{\partial x} = \text{curl}_z \boldsymbol{\tau}_T \Rightarrow \underbrace{\psi_I(x,y)}_{\text{Sverdrup interior}} = \int_0^x \text{curl}_z \boldsymbol{\tau}(x',y)\, dx' + \underbrace{g(y)}_{\text{arbitrary function of integration}} $$

The corresponding velocities are $$ v_I = \text{curl}_z \boldsymbol{\tau}, \quad u_I = -\frac{\partial}{\partial y} \int_0^x \text{curl}_z \boldsymbol{\tau}(x',y)\, dx' - \frac{dg(y)}{dy} $$