# Continuous Data Assimilation for a 2D Bénard Convection System Through Horizontal Velocity Measurements Alone

@article{Farhat2017ContinuousDA,
title={Continuous Data Assimilation for a 2D B{\'e}nard Convection System Through Horizontal Velocity Measurements Alone},
author={Aseel Farhat and Evelyn Lunasin and Edriss S. Titi},
journal={Journal of Nonlinear Science},
year={2017},
volume={27},
pages={1065-1087}
}
• Published 29 January 2016
• Mathematics, Computer Science, Physics
• Journal of Nonlinear Science
In this paper we propose a continuous data assimilation (downscaling) algorithm for a two-dimensional Bénard 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 conditions on the walls, and the 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…
42 Citations

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## References

SHOWING 1-10 OF 32 REFERENCES
Continuous data assimilation for the 2D Bénard convection through velocity measurements alone
• Mathematics
• 2015
© 2015 Elsevier B.V. An algorithm for continuous data assimilation for the two-dimensional Benard convection problem is introduced and analyzed. It is inspired by the data assimilation algorithm
Data Assimilation algorithm for 3D B\'enard convection in porous media employing only temperature measurements
• Mathematics, Physics
• 2015
In this paper we propose a continuous data assimilation (downscaling) algorithm for the B\'enard convection in porous media using only coarse mesh measurements of the temperature. In this algorithm,
Downscaling the 2D Bénard convection equations using continuous data assimilation
• M. Altaf, +4 authors I. Hoteit
• Environmental Science, Mathematics
Computational Geosciences
• 2017
We consider a recently introduced continuous data assimilation (CDA) approach for downscaling a coarse resolution configuration of the 2D Bénard convection equations into a finer grid. In this CDA, a
A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier-Stokes Equations
• Mathematics
• 2015
We study the numerical performance of a continuous data assimilation (downscaling) algorithm, based on ideas from feedback control theory, in the context of the two-dimensional incompressible Navier–
A Discrete Data Assimilation Scheme for the Solutions of the Two-Dimensional Navier-Stokes Equations and Their Statistics
• Computer Science, Mathematics
SIAM J. Appl. Dyn. Syst.
• 2016
An asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements is obtained, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors.
A discrete data assimilation scheme for the solutions of the 2D Navier-Stokes equations and their statistics
• Mathematics
• 2016
Author(s): Foias, Ciprian; Mondaini, Cecilia F; Titi, Edriss S | Abstract: We adapt a previously introduced continuous in time data assimilation (downscaling) algorithm for the 2D Navier-Stokes
Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model
• Mathematics, Physics
• 2015
In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is
Abridged Continuous Data Assimilation for the 2D Navier–Stokes Equations Utilizing Measurements of Only One Component of the Velocity Field
• Mathematics
• 2015
We 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.
Continuous data assimilation for the three-dimensional Navier-Stokes-α model
• Mathematics, Computer Science
Asymptot. Anal.
• 2016
A continuous data assimilation algorithm for three-dimensional viscous hydrodynamic models that provides conditions on the finite-dimensional spatial resolution of the collected data sufficient to guarantee that the approximating solution, which is obtained from this collected data, converges to the unknown reference solution over time.
Continuous data assimilation for the three-dimensional Navier-Stokes-$\alpha$
• Mathematics, Physics
• 2014
Motivated by the presence of a finite number of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages for dissipative dynamical systems, we present a continuous