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Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier-Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin-Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler-Poincar\'{e} and stochastic Navier-Stokes-Poincar\'{e} equations respectively. The stochastic Euler-Poincar\'{e} equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems., Comment: 26 pages

An exact relation is derived between scalar dissipation due to molecular diffusivity and the randomness of stochastic Lagrangian trajectories for flows without bounding walls. This ‘Lagrangian fluctuation–dissipation relation’ equates the scalar dissipation for either passive or active scalars to the variance of scalar inputs associated with initial scalar values and internal scalar sources, as these are sampled backward in time by the stochastic Lagrangian trajectories. As an important application, we reconsider the phenomenon of ‘Lagrangian spontaneous stochasticity’ or persistent non-determinism of Lagrangian particle trajectories in the limit of vanishing viscosity and diffusivity. Previous work on the Kraichnan (Phys. Fluids, 1968, vol. 11, pp. 945–953) model of turbulent scalar advection has shown that anomalous scalar dissipation is associated in that model with Lagrangian spontaneous stochasticity. There has been controversy, however, regarding the validity of this mechanism for scalars advected by an actual turbulent flow. We here completely resolve this controversy by exploiting the fluctuation–dissipation relation. For either a passive or an active scalar advected by any divergence-free velocity field, including solutions of the incompressible Navier–Stokes equation, and away from walls, we prove that anomalous scalar dissipation requires Lagrangian spontaneous stochasticity. For passive scalars, we prove furthermore that spontaneous stochasticity yields anomalous dissipation for suitable initial scalar fields, so that the two phenomena are there completely equivalent. These points are illustrated by numerical results from a database of homogeneous isotropic turbulence, which provide both additional support to the results and physical insight into the representation of diffusive effects by stochastic Lagrangian particle trajectories.