Continuous Data Assimilation for a 2D B\'enard Convection System through Horizontal Velocity Measurements Alone

TL;DR

This paper introduces a continuous data assimilation algorithm for the 2D Bénard convection system that uses only horizontal velocity measurements to accurately recover the system's state over time.

Contribution

The study presents a novel nudging-based data assimilation method that guarantees exponential convergence to the true solution using only horizontal velocity data.

Findings

Convergence to the true solution is exponential under proper parameter choices.

The method can estimate errors when observational data contains inaccuracies.

The algorithm effectively reconstructs the system's state with error-free data.

Abstract

In this paper we propose a continuous data assimilation (downscaling) algorithm for a two-dimensional B\'enard convection problem. Specifically we consider the two-dimensional Boussinesq system of a layer of incompressible fluid between two solid horizontal walls, with no-normal flow and stress free boundary condition on the walls, and fluid is heated from the bottom and cooled from the top. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the horizontal velocity. We show that under an appropriate choice of the nudging parameter and the size of the spatial coarse mesh observables, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed data on the horizontal component of the velocity. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.