Title: A data assimilation algorithm: the paradigm of the 3D Leray-alpha 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-$\alpha$ 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.