Conservation laws and ensemble Kalman filter algorithms

数值上维持物理守恒定律一直被认为是数值天气预报（NWP）模式发展的重要因素，而与其分辨率无关。然而，在更广泛的数据同化方面，只是最近才开始协调一致地努力在数值上保持守恒定律并了解这样做的意义。 解决高度非线性动力学和物理的大气数值模式对适当的初始和边界条件非常敏感。因此，解决多尺度运动的数值预报模式和更高的时间/空间密度/分辨率的观测的数据同化需要重新评估和改进目前从较少的非线性应用继承的方法。能量和拟能 将讨论两个问题。 首先，质量守恒和保存的积极性已被证明在最近的工作由首席研究员和同事的数据同化算法的重要约束。 这两个约束已被纳入一个新的算法，二次规划集合卡尔曼滤波器和线性动态测试。 在这个项目中，该算法将扩展，实施和测试的非静力，对流允许COSMO-DE模式在理想化的设置同化雷达反射率。 作为第二个问题，将解决，它将研究如何数据同化算法，如集合卡尔曼滤波器和二次规划集合卡尔曼滤波器的影响，在理想化的非线性二维浅水模式实验的守恒特性，以及这些集合为基础的算法是否以及如何可以修改，以获得解决方案，保存角动量，能量和涡度拟能。 能量守恒和涡度拟能有望改善系统中的非线性能量级联。 还将研究对预测准确性的可能影响。

Maintaining physical conservation laws numerically has long been recognized as being important in the development of numerical weather prediction (NWP) models independent of their resolution. In the broader context of data assimilation, however, concerted efforts to maintain conservation laws numerically and to understand the significance of doing so have begun only recently. The numerical models of the atmosphere that resolve highly nonlinear dynamics and physics are very sensitive to proper initial and boundary conditions. Consequently, data assimilation for NWP models that resolve many scales of motion and for observations of higher temporal/spatial density/resolution requires re-evaluating and improving methodology that is currently inherited from less nonlinear applications.The principal objective of this project is to develop an ensemble-based data assimilation algorithm that replicates properties of nonlinear dynamical systems such as conservation of mass, angular momentum, energy and enstrophy. Two problems will be addressed. First, conservation of mass and preservation of positivity have been shown in recent work by the principal investigator and colleagues to be important constraints for data assimilation algorithms. These two constraints have been incorporated into a new algorithm, the quadratic programming ensemble Kalman filter and tested for linear dynamics. In this project, the algorithm will be extended, implemented and tested with the non-hydrostatic, convection permitting COSMO-DE model in an idealized setup for the assimilation of radar reflectivity. As the second problem that will be addressed, it will be examined how data assimilation algorithms such as the ensemble Kalman filter and the quadratic programming ensemble Kalman filter affect the conservation properties in idealized nonlinear 2d shallow water model experiments and whether and how these ensemble based algorithms can be modified to obtain solutions that conserve angular momentum, energy and enstrophy. The conservation of energy and enstrophy is expected to improve the nonlinear energy cascade in the system. The possible impact on the accuracy of prediction will be examined as well.