Variational principles for stochastic fluid dynamics.
Variational principles for stochastic fluid dynamics.
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DOI：
10.1098/rspa.2014.0963
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发表时间：
2015-04-08
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影响因子：
--
通讯作者：
Holm DD
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作者：
Holm DD
本文从随机变分原理出发，导出了流体动力学的随机偏微分方程。本文利用SVP的变分导出随机Stratonovich流体方程；写他们的Itô表示；然后研究这些随机流体模型的性质，并与相应的确定性流体模型进行比较。发现随机Stratonovich流体方程的循环性质与确定性理想流体模型的循环性质非常相似。与确定性理想流一样，在不可压缩的随机流中，沿随机Stratonovich路径的运动也保留了涡旋场线的螺旋度。然而，这些Stratonovich性质在等价的Itô表示中并不明显，因为它们被Stratonovich到Itô变换中出现的二次共变漂移项所掩盖。这一项是二次共变漂移项的几何推广，该项已经在Stratonovich 1966年的著名论文中发现。本文还推导了两个随机地球物理流体力学实例的运动方程；即Euler-Boussinesq近似和拟地向近似。
This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The paper proceeds by taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their Itô representation; and then investigating the properties of these stochastic fluid models in comparison with each other, and with the corresponding deterministic fluid models. The circulation properties of the stochastic Stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models. As with deterministic ideal flows, motion along the stochastic Stratonovich paths also preserves the helicity of the vortex field lines in incompressible stochastic flows. However, these Stratonovich properties are not apparent in the equivalent Itô representation, because they are disguised by the quadratic covariation drift term arising in the Stratonovich to Itô transformation. This term is a geometric generalization of the quadratic covariation drift term already found for scalar densities in Stratonovich's famous 1966 paper. The paper also derives motion equations for two examples of stochastic geophysical fluid dynamics; namely, the Euler–Boussinesq and quasi-geostropic approximations.