  Models under location uncertainty are derived assuming that a component of
the velocity is uncorrelated in time. The material derivative is accordingly
modified to include an advection correction, inhomogeneous and anisotropic
diffusion terms and a multiplicative noise contribution. In this paper,
simplified geophysical dynamics are derived from a Boussinesq model under
location uncertainty. Invoking usual scaling approximations and a moderate
influence of the subgrid terms, stochastic formulations are obtained for the
stratified Quasi-Geostrophy (QG) and the Surface Quasi-Geostrophy (SQG) models.
Based on numerical simulations, benefits of the proposed stochastic formalism
are demonstrated. A single realization of models under location uncertainty can
restore small-scale structures. An ensemble of realizations further helps to
assess model error prediction and outperforms perturbed deterministic models by
one order of magnitude. Such a high uncertainty quantification skill is of
primary interests for assimilation ensemble methods. MATLAB code examples are
available online.
