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 −β[Formula presented]u+ɛ([Formula presented])⁴u=f,x∈(a,b)u(a)=u(b)=u^′(a)=u^′(b)=0.This equation serves as a model for the nonlinear barotropic equation (NB) governing oceanic flows. ∂_tΔψ+∇^⊥ψ.∇Δψ+β∂_xψ=[Formula presented](∇×τ)_v−μΔψ+ɛΔ²ψ,where ψ(x,y,t) and τ 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 ψ, 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 ([Formula presented])u both in the boundary layer and in the central zone of the domain. The test cases are taken from Chekroun et al. 2020.