A data assimilation algorithm: the paradigm of the 3D Leray-alpha model of turbulence

TL;DR

This paper demonstrates a data assimilation algorithm applied to the 3D Leray-alpha turbulence model, showing that partial coarse measurements of velocity components suffice to recover the full solution exponentially fast, advancing turbulence modeling.

Contribution

It introduces a new data assimilation paradigm for the 3D Leray-alpha model, proving that measurements of only two velocity components can ensure exponential convergence to the true solution.

Findings

Partial velocity measurements suffice for full solution recovery.

Exponential convergence of the algorithm to the true solution.

Extension of results to the 3D Planetary Geostrophic model.

Abstract

In this paper we survey the various implementations of a new data assimilation (downscaling) algorithm based on spatial coarse mesh measurements. As a paradigm, we demonstrate the application of this algorithm to the 3D Leray-$\alpha$ subgrid scale turbulence model. Most importantly, we use this paradigm to show that it is not always necessary that one has to collect coarse mesh measurements of all the state variables, that are involved in the underlying evolutionary system, in order to recover the corresponding exact reference solution. Specifically, we show that in the case of the 3D Leray$-\alpha$ model of turbulence the solutions of the algorithm, constructed using only coarse mesh observations of any two components of the three-dimensional velocity field, and without any information of the third component, converge, at an exponential rate in time, to the corresponding exact reference solution of the 3D Leray$-\alpha$ model. This study serves as an addendum to our recent work on abridged continuous data assimilation for the 2D Navier-Stokes equations. Notably, similar results have also been recently established for the 3D viscous Planetary Geostrophic circulation model in which we show that coarse mesh measurements of the temperature alone are sufficient for recovering, through our data assimilation algorithm, the full solution; viz. the three components of velocity vector field and the temperature. Consequently, this proves the Charney conjecture for the 3D Planetary Geostrophic model; namely, that the history of the large spatial scales of temperature is sufficient for determining all the other quantities (state variables) of the model.