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

TL;DR

This work tackles the challenge of extending ocean fields onto land to enable multiscale analysis and machine-learning applications near coastlines. It introduces a harmonic-extension framework built on solving the Laplace boundary-value problem $\nabla^2 u_*(x)=0$ in the fill region with boundary data tied to the existing domain, producing a unique extension that minimizes the Dirichlet energy $E(v)=\int_{\mathcal{D}_{fill}}|\nabla v|^2\,dA$ and remains fully consistent with boundary conditions. The method generalizes to vector fields and curved geometries via the Bochner Laplacian or Stokes-type formulations, supports Dirichlet or Neumann boundary conditions, and can be implemented with standard diffusion-solvers and multigrid techniques, incurring modest cost relative to full models. Demonstrations on global SST data show smooth, coast-consistent land fills for both Dirichlet and Neumann BCs, with convergence and coastal differences well-controlled. Overall, the approach provides a physically grounded, computationally efficient tool for boundary-aware multiscale analysis and ML-driven parameterizations, and it naturally lends itself to immersed-boundary style coupling in prognostic settings.

Abstract

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.

Figures (5)

• Figure 1: Schematic illustration of replicate-padding. Schematic illustrating why replicate padding does not yield a unique solution for filling interior land regions (grey). Each yellow cell represents a land point to be filled, while arrows indicate replicate-padding of data from neighboring ocean points (blue). How far inland one needs to fill depends on the lengthscales $\ell$ being parameterized, with larger scales requiring deeper padding over land, up to distances of order $\ell/2$ from the boundary. Because these land cells receive conflicting values from different directions, replicate padding introduces ambiguity and non-uniqueness in the filled field.
• Figure 2: Harmonic extension of sea-surface temperature over land. Land values are computed here by enforcing a no-flux boundary condition ($\partial T/\partial n = 0$) along coastlines, yielding a smooth, physically consistent harmonic extension of the oceanic SST field across land. Top [A]: global SST land-filling using a Neumann (no-flux) boundary condition. Middle [B,C,D]: three representative zooms highlighting coastal smoothness and fidelity. Bottom [E,F,G]: selected cross-coast transects through filled regions, illustrating zero coast-normal gradients and continuity between ocean and land values.
• Figure 3: Comparison of boundary conditions. Harmonically-extended sea surface temperature using [A] Neumann BCs and [C] Dirichlet BCs. (B,D) Histogram of coastal-normal temperature differences for the solution in (A,C), respectively. (E) Difference between the Neumann and Dirichlet harmonic extensions (A-C). (F) Overlay of the two histograms in (B) and (D) for direct comparison. Note that the plotted Neumann distribution (orange) does not integrate to unity because of the large number of zero values, which are at $-\infty$ on the log-axis.
• Figure 4: Transect comparing Neumann and Dirichlet results The transect follows the 10$^\circ$S parallel from the southeastern Atlantic (off Angola) across southern Africa to the southwestern Indian Ocean (off Tanzania). Original (ocean) SST (black lines) together with harmonic extensions using Dirichlet (blue) and Neumann (orange) boundary conditions. Shaded vertical bands mark longitudes corresponding to land. Note that the local maxima at $\sim26.5^\circ\mathrm{E}$ and $\sim28.5^\circ\mathrm{E}$ are the imprints of Lake Upemba and Lake Mweru, respectively. The inset map indicates the transect with the dashed green line.
• Figure B1: Convergence of the RMS Laplacian For both the Dirichlet (blue) and Neumann (orange) boundary conditions, we compute the RMS Laplacian of the land-filled scalar throughout the iterative solver. Convergence is reached once the RMS value falls below a desired tolerance. For reference, the dashed black line in Fig. \ref{['fig:app:convergence_1']} shows the RMS Laplacian evaluated over water.