Data driven stochastic primitive equations with dynamic modes decomposition

The full numerical resolution of planetary flows, with the complex interdependence of mesoscale and sub-mesoscale dynamics that characterize such large scale circulation, is beyond reach with nowadays technology. When performing a numerical simulation of the ocean or the atmosphere, great care must be put in the choice of the parametrization of all those scales that are too small to be efficiently resolved.
This work investigates the benefits of a stochastic decomposition of the Lagrangian trajectory into a smooth-in-time large scale velocity and a random fast-evolving uncorrelated part, ideally accounting for mesoscale and submesoscale processes. This approach, named Location Uncertainty (LU) [1], is built upon a stochastic version of the Reynolds Transport Theorem allowing us to cast the classical physical conservation laws into this scale-separated framework. This framework has been proven to be successful in several large-scale models for ocean dynamics [2,3,4,5].
The derivation and implementation (within the community model NEMO, https://www.nemo-ocean.eu) of the hydrostatic primitive equations in this stochastic framework has been outlined in [6] and it is tested in this work with a novel data-driven approach based on dynamical mode decomposition [7]. The flow prediction in an idealized double-gyre configuration is shown to be improved by this stochastic contribution.

[1], E. Mémin Fluid flow dynamics under location uncertainty,(2014), Geophysical & Astrophysical Fluid Dynamics, 108, 2, 119–146.
[1] W. Bauer, P. Chandramouli, B. Chapron, L. Li, and E. Mémin. Deciphering the
role of small-scale inhomogeneity on geophysical flow structuration: a stochastic approach. Journal of Physical Oceanography, 50(4):983-1003, 2020.
[2] W. Bauer, P. Chandramouli, L. Li, and E. Mémin. Stochastic representation of
mesoscale eddy effects in coarse-resolution barotropic models. Ocean Modelling, 151:101646, 2020.
[3] Rüdiger Brecht, Long Li, Werner Bauer and Etienne Mémin. Rotating Shallow
Water Flow Under Location Uncertainty With a Structure-Preserving Discretization. Journal of Advances in Modeling Earth Systems, 13, 2021MS002492.
[5] V. Resseguier, L. Li, G. Jouan, P. Dérian, E. Mémin, B. Chapron, (2021), New trends in ensemble forecast strategy: uncertainty quantification for coarse-grid computational fluid dynamics, Archives of Computational Methods in Engineering.

[6] F.L. Tucciarone, E. Mémin, L. Li, (2022), Primitive Equations Under Location Uncertainty: Analytical Description and Model Development, Stochastic Transport in Upper Ocean Dynamics, Springer.

[7] L. Li, E. Mémin, G. Tissot,  Stochastic Parameterization with Dynamic Mode Decomposition, Stochastic Transport in Upper Ocean Dynamics, Springer.