Data Assimilation to the Primitive Equations in $H^2$

TL;DR

This paper proves that data assimilation equations for the primitive equations in $H^2$ ensure exponential convergence to the true solution even with limited initial information, under certain conditions.

Contribution

It establishes the stability and convergence of data assimilation solutions for primitive equations in $H^2$, even with incomplete external force information.

Findings

DA solutions exponentially converge to the true solution with known external forces.

DA remains stable with dense spatial observations of unknown external forces.

The approach guarantees accuracy despite limited initial data.

Abstract

In this paper we prove that the solution to the primitive equations is predicted by the corresponding data assimilation(DA) equations in $H^2$. Although, the DA equation does not include the direct information about the base solution and its initial conditions, the solution to the DA equation exponentially convergence to the base(original) solution when the external forces are known even before they are observed. Additionally, when the external force is not completely known but its spatially dense observations are available, then the DA is stable, $i.e.$ the DA solution lies in a sufficiently small neighborhood of the base solution.