Reconstructing Rayleigh--Bénard flows from temperature-only measurements using the Ensemble Kalman Filter

Reconstructing the full state of a multiscale chaotic system from observations of only a subset of its degrees of freedom is a central problem in data assimilation. Turbulent Rayleigh--Bénard convection offers a paradigmatic example, for which Charney conjectured that temperature measurements alone should suffice to determine the velocity field. Synchronization-based methods such as nudging and direct replacement nonetheless fail in chaotic regimes, because they require the measurement of the determining modes of the system, a set to which the velocity belongs but the temperature does not. Here we show that the Ensemble Kalman Filter (EnKF), equipped with a proper inflation scheme, removes this asymmetry by exploiting the flow-dependent cross-covariances between temperature and vorticity, and recovers the velocity from noisy and spatio-temporally sparse temperature data alone. We assess the filter in two chaotic regimes of two-dimensional bulk convection by means of a scale-by-scale spectral analysis. When the whole temperature field is observed, both fields synchronize with the true state at all resolved scales, with the temperature estimate becoming even more accurate than the measurements themselves for sufficiently frequent observations. Equispaced observation networks preserve this full-spectrum synchronization down to a threshold density, below which only the largest scales remain synchronized, as also happens when observations are confined to a subdomain. We conclude that Charney's conjecture is numerically verified in two-dimensional turbulent convection when the assimilation is performed using a properly calibrated EnKF.