Variational principles for fluid dynamics on rough paths

In this paper, we introduce a new framework for parametrization schemes (PS) in GFD. Using the theory of controlled rough paths, we derive a class of rough geophysical fluid dynamics (RGFD) models as critical points of rough action functionals. These RGFD models characterize Lagrangian trajectories in fluid dynamics as geometric rough paths (GRP) on the manifold of diffeomorphic maps. Three constrained variational approaches are formulated for the derivation of these models. The first is the Clebsch formulation, in which the constraints are imposed as rough advection laws. The second is the Hamilton-Pontryagin formulation, in which the constraints are imposed as right-invariant rough vector fields. The third is the Euler--Poincar\'e formulation in which the variations are constrained. These variational principles lead directly to the Lie--Poisson Hamiltonian formulation of fluid dynamics on geometric rough paths. The GRP framework preserves the geometric structure of fluid dynamics obtained by using Lie group reduction to pass from Lagrangian to Eulerian variational principles, thereby yielding a rough formulation of the Kelvin circulation theorem. The rough-path variational approach includes non-Markovian perturbations of the Lagrangian fluid trajectories. In particular, memory effects can be introduced through this formulation through a judicious choice of the rough path (e.g. a realization of a fractional Brownian motion). In the special case when the rough path is a realization of a semimartingale, we recover the SGFD models in Holm (2015). However, by eliminating the need for stochastic variational tools, we retain a pathwise interpretation of the Lagrangian trajectories. In contrast, the Lagrangian trajectories in the stochastic framework are described by stochastic integrals which do not have a pathwise interpretation. Thus, the rough path formulation restores this property.