Asymptotically Well-Balanced Geostrophic Reconstruction Finite Volume Numerical Schemes for the Two-Dimensional Rotating NLSWE in Spherical Coordinates

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 two-dimensional 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.