Conservation laws and ensemble Kalman filter algorithms

长期以来，在数值上保持物理守恒定律在数值天气预报(NWP)模式的发展中一直被认为是重要的，而不受其分辨率的影响。然而，在更广泛的数据同化背景下，在数字上维持守恒定律并理解这样做的意义的协调努力直到最近才开始。解决高度非线性动力学和物理问题的大气数值模式对适当的初始条件和边界条件非常敏感。因此，对于可分辨多个运动尺度的数值预报模式和更高时间/空间密度/分辨率的观测数据同化，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。本项目的主要目标是发展一种基于集合的数据同化算法，该算法复制了非线性动力系统的性质，如质量守恒、角动量守恒、能量和辐合。将解决两个问题。首先，质量守恒和正性守恒在主要研究人员和同事最近的工作中被证明是数据同化算法的重要限制因素。这两个约束被结合到一种新的算法--二次规划集成卡尔曼滤波中，并对其进行了线性动力学测试。在这个项目中，算法将在一个理想的雷达反射率同化装置中用非静力、对流允许的COSMO-DE模式进行扩展、实施和测试。作为将要解决的第二个问题，将研究数据同化算法，如集合卡尔曼滤波和二次规划集合卡尔曼滤波如何影响理想化的非线性二维浅水模式实验中的守恒性质，以及这些基于集合的算法是否以及如何被修改以获得守恒角动量、能量和拟能的解。能量守恒和拟能效应有望改善系统中的非线性能量级联。还将检查对预测准确性可能产生的影响。

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.