In the nonlinear problem we seek solutions to $$ \frac{\partial \zeta}{\partial t} + J(\psi,\zeta) + \beta \frac{\partial \psi}{\partial x} = - r \nabla^2 \psi + \text{curl}_z \tau_T + \nu \nabla^2 \zeta $$ which we have written in dimensional form. In the Stommel problem we set $\nu = 0$ and in the Munk problem we set $r = 0$. In general, solutions will be time-dependent and turbulent and this will create motion on small scales, so that $\nu$ cannot be neglected. The 'steady nonlinear Stommel-Munk problem' is $$ J(\psi,\zeta) + \beta \frac{\partial \psi}{\partial x} = - r \nabla^2 \psi + \text{curl}_z \tau_T + \nu \nabla^2 \zeta $$

We can scale this by first supposing that the leading-order balance is Sverdrupian (i.e., $\beta \, \partial \psi/\partial x \sim \text{curl}_z \tau_T$), from which we obtain the scales $\Psi = |\tau|/\beta$ and $U = |\tau| / (\beta L)$. The 'steady nonlinear Stommel-Munk problem' may then be nondimensionalized to yield $$ R_\beta J(\hat{\psi}, \hat{\zeta}) + \frac{\partial \hat{\psi}}{\partial \hat{x}} = - \epsilon_S \nabla^2 \hat{\psi} + \text{curl}_z \hat{\tau}_T + \epsilon_M \nabla^2 \hat{\zeta} $$ where \( R_\beta = \frac{U}{\beta L^2} = \frac{|\tau|}{\beta^2 L^3} \) the $\beta$-Rossby number for this problem, is a measure of the nonlinearity. Evidently, the nonlinear term increases in importance with increasing wind stress and for a smaller domain