We discuss the application of ambit fields to the construction of stochastic vector fields in two dimensions that are divergence-free and statistically homogeneous and isotropic but are not invariant under the parity operation. These vector fields are derived from a stochastic stream function defined as a weighted integral with respect to a Levy basis. By construction, the stream function is homogeneous and isotropic and the corresponding vector field is, in addition, divergence-free. From a decomposition of the kernel in the Levy-based integral, necessary conditions for the violation of parity symmetry are inferred. In particular, we focus on such fields that allow for skewness of projected increments, which is one of the cornerstones of the Kraichnan–Leith–Bachelor theory of two-dimensional turbulence.

中文翻译：

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二维各向同性、均匀、无发散和偏斜矢量场的范围场建模

我们讨论了范围场在二维随机向量场的构建中的应用，这些场是无散度的、统计上均匀的和各向同性的，但在奇偶运算下不是不变的。这些向量场来自一个随机流函数，该函数被定义为一个关于 Levy 基的加权积分。通过构造，流函数是齐次和各向同性的，并且相应的向量场也是无散度的。从基于 Levy 积分的内核的分解，推断出违反奇偶对称性的必要条件。特别是，我们专注于允许投影增量偏斜的领域，这是二维湍流的 Kraichnan-Leith-Bachelor 理论的基石之一。