A Data Assimilation Algorithm for the Subcritical Surface Quasi-Geostrophic Equation

• Published in: Advanced nonlinear studies Vol. 17; no. 1; pp. 167 - 192
• Main Authors: Jolly, Michael S., Martinez, Vincent R., Titi, Edriss S.
• Format: Journal Article
• Language: English
• Published: De Gruyter 01.02.2017
• Subjects: 34D06; 35Q30; 37C50; 76B75; Data Assimilation; Fractional Poincaré Inequalities; Nudging; Quasi-Geostrophic and SurfaceQuasi-Geostrophic Equation; Surface Measurements
• ISSN: 1536-1365; 2169-0375
• DOI: 10.1515/ans-2016-6019

In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this boundary, a scalar unknown (buoyancy or surface temperature of the fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging is done on this two-dimensional model, yet ultimately synchronizes the streamfunction of the three-dimensional flow. The main analytical difficulties are due to the presence of a nonlocal dissipative operator in the surface quasi-geostrophic equation. This is overcome by exploiting a suitable partition of unity, the modulus of continuity characterization of Sobolev space norms, and the Littlewood–Paley decomposition to ultimately establish various boundedness and approximation-of-identity properties for the observation operators.