中文摘要：

从位涡反演的基本定义和性质出发讨论了它的发展，指出基于非线性平衡（Charney平衡）来进行位涡反演的局限性，要准确地反演中尺度系统还有待于发展高阶位涡反演算子。文中通过分析位涡反演算子的构造得出，高阶位涡反演算子的存在性问题最终归结为探讨高阶平衡流表达式，即高阶平衡方程的存在性问题。然而，平衡流在运动过程中对惯性重力波的自发辐射使得并不存在精确的平衡方程来刻画平衡流的运动。为此，文中引入了一个与位涡反演和平衡流密切相关的概念，即所谓的慢流形。从慢流形的定义出发讨论了与它相关的一系列问题，包括慢流形与位涡反演的关系、慢流形上平衡流对惯性重力波自发辐射的方式等，并给出了一种处理惯性重力波自发辐射的方法。在此基础上给出慢流形的新定义，推导出高阶的平衡方程组及与之相应的高阶位涡反演算子。最后，以2008年7月21—24日由低涡东移所引发的江淮地区大范围降水事件为例展示了高阶位涡反演算子的具体应用，包括对高阶反演算子的可信度检验以及它与满足非线性平衡的反演算子之间的优越性，分析高阶位涡反演得到的平衡流、垂直速度场以及非平衡流的演变特征等。

英文摘要：

Having investingated into the general concept of potential vorticity inversion （hereafter, referred to as PV inversion） and its properties as well as its recent progress, the authors point out that this technology has deficiency in meso-scale systems research and, the higher-order PV inversion operator needs to be developed for these issues. The PV inversion operator depends on how to proper es- tablish a balance equation to a great extent. However, the spontaneous emission of inertia-gravity waves makes the exact balanced flow be hardly to exist and is potentially related to the fuzziness of the so-called slow manifold. A series of problems with the slow manifold were introduced and discussed, including how to overcome the fuzzy property. From the basic characters of slow manifold, a new defi- nition was given and the corresponding higher-order balance equations were deduced. The characteristics of this higher-order balance e- quation and its application in the PV inversion for meso-scale research are investigated. Finally, the technology was used to the diagnos- tic study of the rainfall event in the Yangtzi River-Huaihe River area caused by a low vortex moving eastward between July 21 and 24 2008. The analysis results demonstrate that the higher-order PV inversion operator is better than the old one satisfying the non-linear balance equation. In addition, some meaningful conclusions were drawn based on the analyses of the balanced flow, vertical velocity field and non-balanced flow.