A Data Assimilation Algorithm: the Paradigm of the 3D Leray-α Model of Turbulence

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-α 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-α 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-α 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. This paper is dedicated to the memory of Professor Abbas Bahri MSC Subject Classifications: 35Q30, 93C20, 37C50, 76B75, 34D06.

Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data

“…In [1], it was shown that the AOT data assimilation scheme for the 3D NS-α model (with any admissible initial data) has solutions which converge to solutions of the 3D NS-α (with given admissible initial data). A similar result was later proved in the context of the 3D-Leray-α model in [44]. We note that in both of these cases, the reconstruction of the NS-α solution is done via data that do not come from the NS equations themselves, but from the NS-α model.…”

“…We note that in both of these cases, the reconstruction of the NS-α solution is done via data that do not come from the NS equations themselves, but from the NS-α model. In this sense, the observations are not physical, since the they are generated from NS-α, which itself is not physical, but is used as a regularized version of Navier-Stokes; however, [1] and [44] were important steps in demonstrating that AOT-type data assimilation is not limited to 2D equations, but can be extended to 3D in certain contexts. That is to say, the barrier to extending theoretical results about the AOT method for 2D NS to 3D NS is not directly due to the dimensionality, but rather it is likely due to the nature of the equations in three dimensions, which is the same barrier standing in the way of resolving the major open problem of global existence for strong solutions of the 3D NS equations.…”

Continuous data assimilation algorithm for simplified Bardina model

“…In [18], a similar result for a α-regularization model of the 3D Navier-Stokes equations was obtained. The authors applied the data assimilation algorithm for the case of 3D Leray-α model (see [12]…”

Assimilation of Nearly Turbulent Rayleigh–Bénard Flow Through Vorticity or Local Circulation Measurements: A Computational Study

“…For certain models this approach can work with data in only a subset of the system state variables. This is proved for the 2D NSE in [16], and for the 3D Leray-α model of turbulence model in [20] for data collected for only one and two component(s) of velocity, respectively. Notably, the treatment of each of the above mentioned systems has its own subtleties, and the studies are motivated by specific scientific questions, as we will clarify below.…”