In this paper we propose a continuous data assimilation (downscaling)
algorithm for a two-dimensional B\'enard 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 condition on the walls, and 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 equation of the horizontal velocity. We show
that under an appropriate choice of the nudging parameter and the size of the
spatial coarse mesh observables, and under the assumption that the observed
data is error free, the solution of the proposed algorithm converges at an
exponential rate, asymptotically in time, to the unique exact unknown reference
solution of the original system, associated with the observed data on the
horizontal component of the velocity. Moreover, we note that in the case where
the observational measurements are not error free, one can estimate the error
between the solution of the algorithm and the exact reference solution of the
system in terms of the error in the measurements.Comment: arXiv admin note: text overlap with arXiv:1506.0867
