Variational Principles for Stochastic Fluid Dynamics
This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The Legendre transform of the Lagrangian formulation of these SPDEs yields their Lie-Poisson Hamiltonian form. The paper proceeds by: taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their It\^o 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 It\^o representation, because they are disguised by the quadratic covariation drift term arising in the Stratonovich to It\^o transformation. This term is a geometric generalisation 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 (SGFD); namely, the Euler-Boussinesq and quasigeostropic approximations.