Convolution Based Self Attraction and Loading

Self Attraction and Loading (SAL), which includes the deformation of the solid Earth under the load of the ocean tide and the self-gravitation of the so-deformed Earth as well as of the ocean tides themselves, is an important term to include in numerical models of the ocean tides. Computing SAL is a challenging problem that is usually tackled using spherical harmonics. The spherical harmonic approach has several drawbacks which limit its accuracy. In this work, we propose an alternative technique based on a spherical convolution. We implement the convolution technique in the Modular Ocean Model, version 6, and demonstrate that it allows for more accurate tides when measured against tidal datasets based upon satellite altimetry. The convolution based SAL reduces the error by reducing spurious oscillations associated with the Gibbs phenomenon. These oscillations are large in coastal regions under the traditional spherical harmonic approach.

💡 Research Summary

The paper addresses a long‑standing challenge in ocean tide modeling: the accurate and efficient computation of Self‑Attraction and Loading (SAL), which accounts for the deformation of the solid Earth under tidal loading and the resulting changes in the Earth’s gravitational potential. Traditional approaches compute SAL using spherical harmonic transforms. While mathematically exact, this method suffers from two major drawbacks. First, the global nature of spherical harmonics makes the computation expensive, especially at high resolutions, and it does not lend itself easily to regional or non‑latitude‑longitude grids. Second, setting sea‑surface height to zero over land creates a discontinuity at coastlines; truncating the harmonic series then produces Gibbs oscillations that manifest as spurious amplitude and phase errors, particularly in coastal regions where SAL has the strongest impact.

To overcome these issues, the authors propose a fundamentally different formulation: SAL is expressed as a spherical convolution of the sea‑surface height field η with a Green’s function G_SAL that encapsulates the Love numbers (k′ₙ, h′ₙ) governing Earth’s elastic response. By exploiting the convolution theorem for spherical harmonics, the authors derive an analytical expression for the harmonic coefficients of G_SAL (Eq. 11) and, after analyzing the asymptotic behavior of the Love numbers, obtain a closed‑form approximation (Eq. 16) involving logarithmic and square‑root terms. This approximation eliminates the need for an infinite harmonic sum and provides a kernel that can be evaluated directly on the sphere.

The convolution integral (Eq. 13) is discretized using a midpoint rule over the model grid. Naïvely, the resulting double sum scales as O(N²), which would be prohibitive for realistic ocean models. The authors therefore adopt the Cubed‑Sphere Fast Multipole Method (CSFMM) introduced in a companion paper (Chen & Krasny 2025). CSFMM builds a hierarchical tree on the six faces of a cubed‑sphere grid, clusters points, and uses barycentric Lagrange interpolation on Chebyshev proxy points to approximate far‑field interactions. This reduces the computational complexity to O(N) while preserving controllable accuracy. The singular self‑interaction (i = j) is simply omitted, avoiding the kernel singularity.

Implementation is carried out in the Modular Ocean Model version 6 (MOM6). The authors run a single‑layer, barotropic configuration forced with the principal lunar tide M₂. Two horizontal resolutions are tested: a coarse 0.36° grid typical of climate‑scale simulations and a finer 0.08° grid used for high‑resolution studies. For each resolution, several SAL configurations are compared: (i) standard spherical‑harmonic SAL with truncation at n = 40 and n = 200, (ii) the same with Cesàro summation (n = 40, 200, 400) to mitigate Gibbs ringing, and (iii) the new convolution‑based SAL with CSFMM acceleration. Model runs span 20 days with a 180 s timestep; the final three days are used for validation.

Error assessment follows the standard practice of comparing model tidal amplitudes and phases against the TPXO9 global tidal solution (Erofeev & Egbert 2018). The authors compute a pointwise error D² that combines amplitude and phase differences (Eq. 21) and aggregate it into a root‑mean‑square error (RMSE) over the whole ocean and over the open‑ocean mask (|lat| < 66°, depth > 1000 m) to isolate coastal effects.

Results show that the convolution‑based SAL consistently yields lower RMSE than both the plain spherical‑harmonic approach and the Cesàro‑modified version. The improvement is most pronounced in coastal regions, where Gibbs‑induced oscillations in the harmonic method inflate errors by up to 30 %. Even when the harmonic series is extended to n = 200, the reduction in error plateaus, confirming that the dominant error source is the discontinuity at land–sea boundaries rather than truncation order. Cesàro summation does reduce ringing but introduces a systematic bias because it effectively modifies the Love numbers, leading to a modest net gain at best.

From a performance standpoint, the CSFMM‑accelerated convolution scales linearly with grid size and exhibits good parallel efficiency on the Derecho supercomputer. In contrast, the spherical‑harmonic transforms, even when using fast algorithms, suffer from limited scalability and require additional interpolation steps on non‑regular grids. The authors also note that the convolution framework is naturally compatible with regional or nested grids, because the Green’s function is defined globally but the convolution can be restricted to the local domain without loss of accuracy—a significant advantage for future high‑resolution coastal modeling.

The paper acknowledges remaining challenges. The closed‑form Green’s function relies on empirically fitted asymptotic coefficients (a₁ = ‑2.7, b₀ = ‑6.21196, b₁ = 6.1); while these work well for a single‑layer ocean, extending the method to multi‑layer, non‑Boussinesq, or viscoelastic Earth models will require re‑derivation or calibration. Moreover, the treatment of the self‑interaction singularity by omission is a pragmatic shortcut; more sophisticated regularization could further improve accuracy.

In summary, the authors present a novel, mathematically rigorous, and computationally efficient alternative to spherical‑harmonic SAL computation. By recasting SAL as a spherical convolution with an analytically approximated Green’s function and accelerating the convolution with a fast multipole method on a cubed‑sphere grid, they eliminate Gibbs‑related coastal artifacts, achieve lower tidal errors, and open the door to seamless integration of SAL in both global and regional ocean models. Future work will likely focus on extending the approach to fully three‑dimensional, multi‑layer ocean dynamics, refining the Green’s function for more complex Earth rheologies, and exploiting GPU architectures to further accelerate the convolution kernel.