Harmonic Extension for Multiscale Analysis and Modeling Near Boundaries, with an Ocean Application

Treatment of fields near domain boundaries is a long-standing problem in signal processing that has come into renewed focus following recent efforts in convolution-based multiscale coarse-graining and in machine-learned parameterizations due to ocean boundary artifacts. Here, we propose a general method for extending fields beyond the domain boundaries by solving a Laplace boundary-value problem. Construction of the harmonic extension is well-posed, including uniqueness, and is consistent with the boundary conditions by design. The formulation applies to irregular boundaries such as discretized coastlines. The harmonic extension is physically desirable since it has minimum spatial variability among all admissible extensions satisfying the boundary conditions. The method is simple to implement using well-established numerical approaches, and is broadly applicable to extending oceanic variables over land boundaries. Other applications include machine learning parametrization and subgrid modeling of wall-bounded flows and multiphase flows. We demonstrate the method by extending sea-surface temperature (SST) over land using fixed temperature (Dirichlet) and no-flux (Neumann) boundary conditions: the land-filled solution is smooth with SST values between the coastal minimum and maximum.