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.