Continuous Data Assimilation in Quasi-Geostrophic Turbulence

Yiguo Li1, Peng Zhan1,2, Edriss S. Titi3,4, Ibrahim Hoteit1
1Division of Physical Sciences and Engineering, King Abdullah University of Science and Technology, Saudi Arabia
2Department of Ocean Science and Engineering, Southern University of Science and Technology, China
3Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK
4Department of Mathematics, Texas A & M University, USA

The ocean exhibits near-inviscid properties, posing a significant challenge for numerical modeling because the eddy viscosity required for numerical stability often over-dissipates small-scale energy and lacks necessary backscatter. This results in a biased representation of dynamics within ocean models. Continuous Data Assimilation (CDA), a downscaling framework inspired by the theory of determining modes, has been mathematically established for several classes of geophysical fluid models, including Rayleigh-Bénard convection and planetary geostrophic equation, to yield solutions that converge exponentially to the observed state by prescribing the scales resolved by nodal observations to constrain the unobserved, finer scales. Here we quantify the spectral limit and temporal retention of unobserved scales constrained by CDA under temporally discrete observations. We implemented CDA in a forced-dissipative simulation of two-dimensional turbulence governed by barotropic quasi-geostrophic (QG) dynamics. This model serves as a simplified paradigm to study multiscale ocean dynamics. We utilize spectral diagnostics to investigate the influence of forcing scales, eddy viscosity, and the beta-effect on the spectral limit and the retention time of the constrained scales. We show that the viscous scale controls the spectral limit of constrained scales; while strong viscosity rapidly damps small-scale variance, the predictability of intermediate scales is enhanced due to their spectral separation from the viscous scale, which hinders the propagation of damping effects via nonlinear energy transfer. These results provide theoretical insights for CDA-based downscaling in regional ocean models, showing how high-resolution gridded data, including future SWOT products, can extend the predictability across scales.