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Asymptotically well-balanced geostrophic reconstruction finite volumes numerical schemes for the 2D rotating NLSWE in spherical coordinates

Alejandro González del Pino, Manuel Jesús Castro Díaz, Jorge Macías Sánchez

TL;DR

The paper addresses geostrophic balance in large-scale geophysical flows by developing two high-order finite-volume schemes for the 2D rotating NLSWE on the sphere that are asymptotically well-balanced for small $Ro$. The authors derive the geostrophic equilibrium in spherical coordinates, construct locally geostrophic reconstructions, and prove asymptotic well-balancing properties, enabling accurate small-perturbation dynamics. They validate the methods through a suite of tests—Cartesian stationary vortex, order convergence, spherical jet, geostrophic perturbation, and a meteotsunami case—demonstrating improved accuracy and stability, including realistic nearshore forecasting potential. The work demonstrates significant practical impact for meteotsunami forecasting and trans-oceanic wave modeling, facilitated by GPU acceleration for large-scale simulations.

Abstract

The dynamics of large-scale geophysical fluids is primarily governed by the balance between the Coriolis force and the pressure gradient. This phenomenon, known as geostrophic equilibrium, is the basis for the geostrophic model, which has proven to be extremely useful for understanding and forecasting large-scale atmospheric and oceanic dynamics. In the present work, we develop second- and third-order finite-volume numerical schemes applied to the 2D rotating shallow-water equations in spherical coordinates. These schemes are designed to preserve the geostrophic equilibrium in the limit as the Rossby number tends to zero. The final goal is to design reliable and efficient forecasting models for simulating meteotsunamis, long-wave events generated in the ocean by atmospheric pressure disturbances. These disturbances produce long waves of small amplitude that gradually amplify as they approach the coast. The numerical results for various analytical and real-world test cases underscore the importance of maintaining geostrophic equilibrium over time.

Asymptotically well-balanced geostrophic reconstruction finite volumes numerical schemes for the 2D rotating NLSWE in spherical coordinates

TL;DR

The paper addresses geostrophic balance in large-scale geophysical flows by developing two high-order finite-volume schemes for the 2D rotating NLSWE on the sphere that are asymptotically well-balanced for small . The authors derive the geostrophic equilibrium in spherical coordinates, construct locally geostrophic reconstructions, and prove asymptotic well-balancing properties, enabling accurate small-perturbation dynamics. They validate the methods through a suite of tests—Cartesian stationary vortex, order convergence, spherical jet, geostrophic perturbation, and a meteotsunami case—demonstrating improved accuracy and stability, including realistic nearshore forecasting potential. The work demonstrates significant practical impact for meteotsunami forecasting and trans-oceanic wave modeling, facilitated by GPU acceleration for large-scale simulations.

Abstract

The dynamics of large-scale geophysical fluids is primarily governed by the balance between the Coriolis force and the pressure gradient. This phenomenon, known as geostrophic equilibrium, is the basis for the geostrophic model, which has proven to be extremely useful for understanding and forecasting large-scale atmospheric and oceanic dynamics. In the present work, we develop second- and third-order finite-volume numerical schemes applied to the 2D rotating shallow-water equations in spherical coordinates. These schemes are designed to preserve the geostrophic equilibrium in the limit as the Rossby number tends to zero. The final goal is to design reliable and efficient forecasting models for simulating meteotsunamis, long-wave events generated in the ocean by atmospheric pressure disturbances. These disturbances produce long waves of small amplitude that gradually amplify as they approach the coast. The numerical results for various analytical and real-world test cases underscore the importance of maintaining geostrophic equilibrium over time.
Paper Structure (17 sections, 1 theorem, 56 equations, 9 figures, 2 tables)

This paper contains 17 sections, 1 theorem, 56 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. PDE system
  3. Numerical Scheme
  4. Geostrophic equilibrium and well-balanced reconstructions
  5. Geostrophic equilibrium in spherical coordinates
  6. Asymptotic well-balancing
  7. Well-balanced reconstructions
  8. Finding approximated local geostrophic equilibria for the sec-ond--order scheme
  9. Finding approximated local geostrophic equilibria for the third-order scheme
  10. Asymptotic well-balanced behavior
  11. Numerical results
  12. Stationary vortex on Cartesian coordinates
  13. Numerical order test
  14. Test for the spherical geostrophic equilibrium
  15. Test for a perturbation of the spherical geostrophic equilibrium
  16. ...and 2 more sections

Key Result

Theorem 4.3

In the case of a flat bottom $\nabla H=0$ and constant in space atmospheric pressure $\nabla p^{a}=0$, the numerical scheme (semidiscreteNumericalSchemeUsingQuadrature) along with the reconstruction functions defined in subsection WB_reconstructions is asymptotically well-balanced for the geostrophi then $w_i^{'}(0)=\mathcal{O}(\epsilon^2) \, \, \forall i$, that is, the approximated geostrophic eq

Figures (9)

  • Figure 1: Stencil for the reconstruction operators. Second-order scheme (left) and third-order scheme (right). These are the local numeration, where $\Omega_0$ is the cell volume $V_i$.
  • Figure 1: Evolution of water height errors in $L^{1}$-norm for the stationary vortex test case. Dashed lines represent the non-asymptotically well-balanced schemes, while solid lines correspond to the asymptotically well-balanced schemes, where the complete local geostrophic reconstruction is performed.
  • Figure 2: Initial conditions for the zonal flow test. Left: zonal velocity component, $u_{\theta}$, as defined in (\ref{['test_5_3_ux']}). Right: total fluid height, $h$, obtained from (SM2.2) using a quadrature rule.
  • Figure 3: Relative errors for the evolution of $h_{\sigma}$ in $L^{1}$-norm for the zonal jet flow test. Both asymptotic well-balanced schemes Geos-AWB2, Geos-AWB3 evidence better results in presence of a stationary solution.
  • Figure 4: Perturbation of the height field $h'$ as defined in (\ref{['test_perturbation_hprima_function']}).
  • ...and 4 more figures

Theorems & Definitions (4)

  • Definition 4.1
  • Remark 4.2
  • Theorem 4.3
  • Remark 4.4