A compact scheme for the Munk boundary-layer equation in one dimension

In this paper, we introduce a two-scale compact finite difference scheme for the equation (MK-1D)$\left(\begin{array}{cc}& -\beta \frac{d}{dx}u+\varepsilon {\left(\frac{d}{dx}\right)}^{4}u=f,x\in (a,b)\\ & u\left(a\right)=u\left(b\right)=u^{\prime }\left(a\right)=u^{\prime }\left(b\right)=0.\end{array}\right)$This equation serves as a model for the nonlinear barotropic equation (NB) governing oceanic flows. (NB)${\partial }_t\Delta \psi +{\nabla }^{\perp }\psi .\nabla \Delta \psi +\beta {\partial }_x\psi =\frac{1}{H}{(\nabla \times \tau )}_v-\mu \Delta \psi +\varepsilon {\Delta }^2\psi ,$where $\psi (x,y,t)$ and $\tau$ are the streamfunction and the wind stress tensor, respectively. This equation encodes the western boundary layer problem (Ghil et al. 2008) for the potential vorticity $\psi$, which corresponds to the sharp contrast between the gyres flow in the oceanic circulation at mid-latitude and the strong western boundary currents. Numerical results for Equation (MK-1D) show that, with this two-scale scheme, high order accuracy is preserved for $u$ and $\left(\frac{d}{dx}\right)u$ both in the boundary layer and in the central zone of the domain. The test cases are taken from Chekroun et al. 2020.