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

• Published in: Quarterly journal of the Royal Meteorological Society Vol. 143; no. 708; pp. 2902 - 2914
• Main Authors: Zeng, Yuefei, Janjić, Tijana, Ruckstuhl, Yvonne, Verlaan, Martin
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 01.10.2017; Wiley Subscription Services, Inc
• Subjects: Algorithms; B‐spline; Conservation; conservation laws; Data; Data assimilation; Data collection; Divergence; Energy; Ensemble forecasting; Enstrophy; Experiments; Kalman filters; Kinetic energy; kinetic energy spectrum; Localization; Mathematical models; noise; Shallow water; Spectra; SQP; total energy
• ISSN: 0035-9009; 1477-870X
• DOI: 10.1002/qj.3142

For numerical discretization schemes, the violation of enstrophy conservation causes a systematic and unrealistic energy cascade towards high wave numbers. The same occurs in data assimilation schemes, where the total energy, enstrophy and divergence could be strongly affected. In this article, we construct an ensemble data assimilation algorithm that conserves mass, total energy and enstrophy. The algorithm uses B‐spline functions for localization and sequential quadratic programming to solve nonlinear constrained minimization problem. 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 root‐mean‐square errors, with a slight advantage for those that include the conservation constraint on the globally integrated enstrophy. However, the kinetic energy and enstrophy spectra in experiments with the enstrophy constraint are considerably closer to the true spectra, in particular at the smallest resolvable scales. Therefore, imposing conservation of enstrophy within the data assimilation algorithm effectively avoids the spurious energy cascade of the 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. Unphysical noise can be generated by assimilating data. We construct an ensemble‐type Kalman filter algorithm that enforces the conservation of mass, total energy and enstrophy as equality constraints. Experiments with the enstrophy constraints E_BSP_Es and E_BSP_EnEs (the latter additionally imposing total energy conservation) avoid the spurious energy cascade of the rotational part, thereby suppressing noise and improving the quality of a 14 day free forecast, compared with experiments without the enstrophy constraints E_BSP_NO and E_BSP_En.