Ensemble‐type Kalman filter algorithm conserving mass, total energy and enstrophy

For the numerical discretization schemes, the violation of the enstrophy conservation causes a systematic and unrealistic energy cascade towards the high wave numbers. The same also holds for the data assimilation scheme where the total energy, enstrophy and divergence could be strongly affected by the data assimilation settings. The same occurs to data assimilation schemes, where the total energy, enstrophy and divergence could be strongly affected. In this paper, we construct the an ensemble data assimilation algorithm that conserves mass, total energy and enstrophy. The algorithm uses the B-spline functions for localization and the sequential quadratic programming to impose these nonlinear constraints solve nonlinear constrained minimization problems. Experiments with selected constraints Idealized experiments are performed using a 2D shallow water model, with selected contraints derived from the nature run. It is found that all experiments exhibit comparable RMSE with a slight advantage for ones with the ones that include the conservation constraint on the globally integrated enstrophy. However, for the kinetic energy and enstrophy spectra , the in experiments with the enstrophy constraint are considerably closer to the true spectra, in particular at the smallest resolvable scales. Therefore, imposing the conservation of enstrophy within the data assimilation algorithm effectively avoids the spurious energy cascade of rotational part and thereby successfully suppresses the noise generated by the data assimilation algorithm. The 14-day deterministic free forecast, starting from the initial condition enforced by both total energy and enstrophy constraints, produces the best prediction. The same holds for the ensemble free forecasts.

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