Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model

Abstract

The stochastic variational approach for geophysical fluid dynamics was introduced by Holm (Proc Roy Soc A, 2015) as a framework for deriving stochastic parameterisations for unresolved scales. This paper applies the variational stochastic parameterisation in a two-layer quasi-geostrophic model for a β-plane channel flow configuration. We present a new method for estimating the stochastic forcing (used in the parameterisation) to approximate unresolved components using data from the high resolution deterministic simulation, and describe a procedure for computing physically-consistent initial conditions for the stochastic model. We also quantify uncertainty of coarse grid simulations relative to the fine grid ones in homogeneous (teamed with small-scale vortices) and heterogeneous (featuring horizontally elongated large-scale jets) flows, and analyse how the spread of stochastic solutions depends on different parameters of the model. The parameterisation is tested by comparing it with the true eddy-resolving solution that has reached some statistical equilibrium and the deterministic solution modelled on a low-resolution grid. The results show that the proposed parameterisation significantly depends on the resolution of the stochastic model and gives good ensemble performance for both homogeneous and heterogeneous flows, and the parameterisation lays solid foundations for data assimilation.

Circulation and Energy Theorem Preserving Stochastic Fluids

“…The above considerations motivate the utility of circulation-theorem preserving stochastic models as reduced descriptions of nonlinear dynamical systems which account for the advective transport effects of the small, rapid, unresolvable scales of fluid motion on the variability of computationally resolvable. See, respectively, [12,13] for computational investigations of the Navier-Stokes-Poincaré equations in two dimensions for regions with fixed boundaries and for a 2-layer quasi-geostrophic model. See also [30] for a recent review, and see [31] for discussions of stochastic fluid models with non-stationary statistics.…”

“…A key feature of their derivation is the decomposition of the Lagrangian flow map into fast and slow components. The fast motion of the Lagrangian trajectory is represented by a stochastic process, whose correlate statistics are to be calibrated from data as in [12,13]. The stochastic decomposition proposed in [37] was later derived using multi-time homogenization by Cotter et al [11].…”

“…Here, P is the dual for 1-forms of the standard Leray projection operator for vector fields, as discussed in Appendix A. The symbol • denotes the Stratonovich sense of the stochastic product, the collection {ξ (k) } k∈N contains fixed, deterministic divergence-free vector fields ξ (k) : Ω → R d , to be calibrated in practice from data, e.g., as in [12,13], and we define the operator…”

Spatial Covariance Modeling for Stochastic Subgrid‐Scale Parameterizations Using Dynamic Mode Decomposition

“…Subgrid‐scale eddies and bottom friction are modeled by biharmonic viscosity, while in the upper layer (i.e., i =1), large‐scale forcing is provided by a prescribed background‐flow U =0.6 as, for instance, in Cotter et al. (2018) and Jansen and Held (2014). The external forcing leads to the formation of a jet stream with nontrivial meridional structure whose location experiences meridional shifts—a prominent feature of the observed atmospheric jet stream (Feldstein, 1998; James & Dodd, 1996; Riehl et al., 1950).…”

Stochastic Variational Formulations of Fluid Wave–Current Interaction

“…The motivations and recent applications of the SALT approach for uncertainty quantification and data assimilation are also briefly discussed (Cotter et al. 2019a , b , c ).…”