A data assimilation algorithm for the subcritical surface quasi-geostrophic equation

2016/07/28 by Jolly, Michael S., Martinez, Vincent R., Titi, Edriss S. · 3 citations
#34D06 #35Q35 #35Q86 #37C50 #76B75 #93C20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1607.08574

Abstract

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.

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• Continuous data assimilation for 2D stochastic Navier-Stokes equations
• Determining Modes, State Reconstruction, and Intertwinement: Existence of Self-Synchronizing Intertwinements
• Data Assimilation in Large Eddy Simulation: Addressing Model-Observation Mismatch from Navier-Stokes Data

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