Abstract The properties of the expected analysis and forecast error covariance matrices are explored using a novel method based on the tangent linearization and adjoint of a 4-dimensional variational (4D-Var) data assimilation system. The method is applied to the mesoscale circulation that develops in the presence of a baroclinically unstable mid-latitude ocean temperature front using a series of paternal twin experiments that employ both strong and weak constraint 4D-Var. Adopting the traditional view of Empirical Orthogonal Functions (EOFs) of a covariance matrix as the semi-major axes of a multi-dimensional hyper-ellipsoid, variations in the volume of the analysis and forecast error hyper-ellipsoids are explored which provides information about the flow of probability through state-space. The complementary variations in the expected total variance of the covariance matrix are also investigated. Two different kinds of behavior are identified that are associated with either the demise or growth of baroclinic instabilities. In both cases, the volume of the hyper-ellipsoid decreases during the 4D-Var analysis cycle. During the subsequent forecasts, the volume of the forecast error hyper-ellipsoid initially continues to collapse under both scenarios. During this time, the hyper-ellipsoid becomes increasingly elongated along some of the semi-major axes as forecast errors grow in preferential directions. Growth in these directions is controlled by the most unstable error modes, and projection of forecast error on to the precursors of these modes has been shown previously to be characterized by upscale energy transfer and non-normal processes. For the case of the growing wave, the forecast error hyper-ellipsoid continues to collapse through to the end of the forecast period. However, for the decaying wave, the hyper-ellipsoid may undergo expansion at longer forecast lead times.

中文翻译：

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关于存在斜压不稳定性的海洋分析和预报误差协方差的行为

摘要 使用基于 4 维变分 (4D-Var) 数据同化系统的切线线性化和伴随的新方法，探索了预期分析和预测误差协方差矩阵的性质。使用一系列采用强约束和弱约束 4D-Var 的父系双胞胎实验，将该方法应用于在斜压不稳定的中纬度海洋温度前沿存在的情况下发展的中尺度环流。采用协方差矩阵的经验正交函数 (EOF) 的传统观点作为多维超椭球的半长轴，探索分析和预测误差超椭球的体积变化，提供有关通过状态空间的概率流。还研究了协方差矩阵的预期总方差的互补变化。确定了两种不同的行为，它们与斜压不稳定性的消亡或增长有关。在这两种情况下，超椭球的体积在 4D-Var 分析周期中都会减少。在随后的预测中，预测误差超椭球的体积最初在两种情况下都继续崩溃。在此期间，随着预测误差在优先方向上增长，超椭球体沿着某些半长轴变得越来越拉长。这些方向的增长由最不稳定的错误模式控制，之前已经表明预测误差对这些模式的前兆的预测具有高级能量转移和非正常过程的特征。对于增长波的情况，预测误差超椭球继续崩溃，直到预测期结束。然而，对于衰减波，超椭球体可能会在更长的预测提前期发生膨胀。