Stable gradient-wind balance at high Rossby numbers: A semi-implicit method applied to high-resolution Ocean satellite data

The 2022 launch of the SWOT satellite (Surface Water and Ocean Topography) enabled sea surface height observations at unprecedented high resolution of approximately 2 km. These measurements are used to generate sea surface current maps by applying the geostrophic balance equation. At these fine scales, intense small-scale eddies become visible. These eddies exhibit strong ageostrophic behavior driven by non-linear advection, with Rossby numbers larger than 1. Theoretical work indicates that geostrophic current estimates can overestimate or underestimate actual current velocities by approximately a factor of 2 for ageostrophic cyclones and anticyclones respectively. This makes solving the gradient-wind equation essential for accurate representation. Earlier efforts to address this challenge employed explicit iterative finite difference schemes, which are known to lose stability when Rossby numbers exceed 1. We present a novel approach using a semi-implicit finite difference method. Our method is first tested against analytical solutions in a simplified framework, then validated using a 1 km resolution primitive equation ocean model. We demonstrate the method's application to SWOT observations of an intense oceanic submesoscale cyclone.