Rotating shallow water flow under location uncertainty with a structure-preserving discretization

Abstract

<p>We introduce a new representation of the rotating shallow water equations based on a stochastic transport principle. The derivation relies on a decomposition of the fluid flow into a large-scale component and a noise term that models the unresolved small-scale flow. The total energy of such a random model is demonstrated to be preserved along time for any realization. Thus, we propose to combine a structure-preserving discretization of the underlying deterministic model with the discrete stochastic terms. This way, our method can directly be used in existing dynamical cores of global numerical weather prediction and climate models. For an inviscid test case on the f-plane we use a homogenous noise and illustrate that the spatial part of the stochastic scheme preserves the total energy of the system. Finally, using an inhomogenous noise, we show  that the proposed random model better captures the structure of a large-scale flow than a comparable deterministic model for a barotropically unstable jet on the sphere.</p>

Stochastic parametrization: An alternative to inflation in ensemble Kalman filters

“…For a stationary noise, the variance field is constant in time, and thus not related to the evolving large‐scale components. The ability to build a flow‐dependent noise enables us to improve probabilistic forecasting skills (Brecht et al ., 2021). For the SQG dynamics several noise parametrizations have been compared and assessed through different statistical proper scores (Resseguier et al ., 2020).…”

“…In Holm (2015), it is dedicated to Hamiltonian dynamical systems and defined from a circulation‐preserving constrained variational formulation, whereas the one in Mémin (2014) is based on Newtonian principles and built upon classical physical conservation laws. This latter scheme has been used as a fundamental tool to derive stochastic representations of large‐scale geophysical dynamics (Bauer et al ., 2020a; 2020b; Brecht et al ., 2021; Resseguier et al ., 2017a; 2017b; 2017c) or to define large‐eddy simulation models of turbulent flows (Chandramouli et al ., 2020; Harouna and Mémin, 2017).…”

Hamiltonian formulation of the stochastic surface wave problem

“…Physically, we set ourselves hence in context with waves longer than few centimetres and shorter than few kilometres. The LU momentum equations are given by [1,6,27]…”

Stochastic data-driven parameterization of unresolved mesoscale eddies

“…From the system (2.9) a diverse set of approximated models under LU can be obtained through nondimensionalization and asymptotic approach with proper scaling. However, the noise introduces an additional degree of freedom that must be appropriately accounted for (Brecht et al, 2021;Bauer et al, 2020b;Resseguier et al, 2017b,c). The horizontal components of the quadratic variation are first scaled as a h ∼ U L, where U and L are typical velocity and length scales, and the factor is proportional to the ratio between the eddy kinetic energy (EKE) and the mean kinetic energy (MKE) and to the ratio between the small-scale correlation time and the large-scale advection time.…”

“…Unsurprisingly, stationarity/timevarying and homogeneity/inhomogeneity characteristics of the unresolved flow component have strong influences on the numerical results. For instance, Bauer et al (2020a) illustrates that the noise inhomogeneity induces a structuration of the large-scale flow reminiscent to the action of the vortex force associated to surface wave-induced Stokes drift; Bauer et al (2020b) shows that the introduction of inhomogeneous noise into the barotropic quasi-geostrophic (QG) model enables us to reproduce accurately, on a coarse mesh, the high order statistics of the eddy-resolving data; a stochastic shallow water model preserving the resolved total energy has been proposed by Brecht et al (2021) and the results show that the stochastic parameterization provides a good trade-off between model error representation and ensemble spread.…”