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We introduce a continuous (downscaling) data assimilation algorithm for the2D B\'enard convection problem using vorticity or local circulationmeasurements only. In this algorithm, a nudging term is added to the vorticityequation to constrain the model. Our numerical results indicate that theapproximate solution of the algorithm is converging to the unknown referencesolution (vorticity and temperature) corresponding to the measurements of the2D B\'enard convection problem when only spatial coarse-grain measurements ofvorticity are assimilated. Moreover, this convergence is realized using datawhich is much more coarse than the resolution needed to satisfy rigorousanalytical estimates.

Assimilation of nearly turbulent Rayleigh-B\'enard flow through vorticity or local circulation measurements: a computational study