Title: A Data Assimilation Algorithm: The Paradigm of the 3D Leray-alpha Model of Turbulence

Abstract: In the first part of the talk I will discuss some of the basic theoretical and practical properties of a turbulence model known as Lagrangian-Averaged Navier-Stokes equations, also called Navier-Stokes-alpha model.  In the process, I’ll identify core issues that arise when one uses a PDE model to characterize a complex flow.   Similarly, there are common issues that arise when one uses a purely data driven modeling strategy, with no governing equations,  to characterize multi-scale highly nonlinear complex systems.   This motivates the need to properly merge the model equations derived from physical laws with the available observational measurements.  I will introduce a continuous data assimilation (downscaling) algorithm for the two-dimensional Navier-Stokes equations employing coarse mesh measurements of only one component of the velocity field. This algorithm can be implemented with a variety of finitely many observables: low Fourier modes, nodal values, finite volume averages, or finite elements.   I will 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 necessary 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. Notably, similar results have 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 is joint work with Aseel Farhat and Edriss S. Titi.