Data-driven and data-agnostic stochastic parametrization of unresolved processes

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 therole 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 ofmesoscale eddy effects in coarse-resolution barotropic models. Ocean Modelling, 151:101646, 2020.[3] Rüdiger Brecht, Long Li, Werner Bauer and Etienne Mémin. Rotating ShallowWater 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.

<p>Planetary flows and large scale circulation systems are characterised by an interaction between scales that range over several orders of magnitude, with contributions given by mesoscale and submesoscale dynamics. Resolving numerically  such interactions for realistic configuration is, however, far beyond reach. Any large-scale simulation must then rely on parameterizations of the effects of the small scales on  the large scales. In this work, a stochastic parameterization is proposed based on a decomposition of the flow in terms of a smooth-in-time large-scale contribution and a random fast-evolving uncorrelated small-scale part  accounting for  mesoscales and submesoscales unresolved eddies. This  approach, termed modelling under location uncertainty (LU) [4], relies on a stochastic version of Reynolds Transport Theorem to cast physically meaningful conservation principles in this scale-separated framework. Such a scheme has been successfully applied to several large-scale models of the  ocean dynamics [1, 2, 3, 5]. Here a LU version of the  hydrostatic primitive equations is  implemented within the  NEMO community code (<span role="link"><span role="link">https://www.nemo-ocean.eu</span></span>) with a data-driven approach to establish the spatial correlation of the fast evolving scales. In comparison to a corresponding deterministic counterpart, this stochastic large-scale representation  is shown to improve, in terms of the eastward jet resolution and variabilities, the  flow prediction of an idealized wind forced double gyre circulation. The results are assessed through several statistical criterion as well as an energy transfer analysis [2,5]. <br>[1] W. Bauer, P. Chandramouli, B. Chapron, L. Li, and E. Mémin. Deciphering the<br>role of small-scale inhomogeneity on geophysical flow structuration: a stochastic approach.<br>Journal of Physical Oceanography, 50(4):<span role="link">983-1003, 2020</span>.<br>[2] W. Bauer, P. Chandramouli, L. Li, and E. Mémin. Stochastic representation of<br>mesoscale eddy effects in coarse-resolution barotropic models. Ocean Modelling, 151:101646,<br>2020.<br>[3] Rüdiger Brecht, Long Li, Werner Bauer and Etienne Mémin. Rotating Shallow<br>Water Flow Under Location Uncertainty With a Structure-Preserving Discretization. Journal of<br>Advances in Modeling Earth Systems, 13, 2021MS002492.<br>[4], E. Mémin Fluid flow dynamics under location uncertainty,(2014), Geophysical & Astrophysical Fluid Dynamics, 108, 2, 119–146.<br>[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.</p>

This study assesses the capability of Surface Quasi-Geostrophy (SQG) to reconstruct the three-dimensional (3D) dynamics in four critical areas of the Arctic Ocean: the Nordic, Barents, East Siberian, and Beaufort Seas. We first reconstruct the upper ocean dynamics from TOPAZ4 reanalysis of sea surface height (SSH), surface buoyancy (SSB), and surface velocities (SSV) and validate the results with the geostrophic and total TOPAZ4 velocities. The reconstruction of upper ocean dynamics using SSH fields is in high agreement with the geostrophic velocities, with correlation coefficients greater than 0.8 for the upper 400 m. SSH reconstructions outperform surface buoyancy reconstructions, even in places near freshwater inputs from river discharges, melting sea ice, and glaciers. Surface buoyancy fails due to the uncorrelation of SSB and subsurface potential vorticity (PV). Reconstruction from surface currents correlates to the total TOPAZ4 velocities with correlation coefficients greater than 0.6 up to 200 m. In the second part, we apply the SQG approach validated with the reanalysis outputs to satellite-derived sea level anomalies and validate the results against in-situ measurements. Due to lower water column stratification, the SQG approach’s performance is better in fall and winter than in spring and summer. Our results demonstrate that using surface information from SSH or surface velocities, combined with information on the stratification of the water column, it is possible to effectively reconstruct the upper ocean dynamics in the Arctic and Subarctic Seas up to 400 m. Future remote sensing missions in the Arctic Ocean, such as SWOT, Seastar, WaCM, CIMR, and CRISTAL, will produce enhanced SSH and surface velocity observations, allowing SQG schemes to characterize upper ocean 3D mesoscale dynamics up to 400 m with higher resolutions and lower uncertainties.

The present study investigates the reconstruction of the 3D dynamics of a turbulent mesoscale eddy field driven at a depth by a baroclinic instability of the Phillips type. It uses a high-resolution primitive equation simulation as a testbed. The method of reconstruction is based on potential vorticity principles and extends an earlier approach (Lapeyre and Klein, J Phys Oceanogr 36:165–176, 2006) to a regime where the signature of surface density anomalies on the dynamics is weak. The crux and the originality of the reconstruction lie in the estimation from sea surface height and surface density anomalies of the interior quasigeostrophic potential vorticity (PV) anomalies and its subsequent inversion. The estimation of PV anomalies relies on the vertical correlation between PV anomalies and on the knowledge on stratification and horizontal gradients of background PV. PV anomalies are accurately estimated over the first 500 m of the water column and over a wide range of wavenumbers. Density anomalies play a minor role in the PV estimation, though their omission leads to an overestimation of PV by a factor of less than 2 at scales of order 20 km and less. Inversion of the estimated PV leads to a geostrophic streamfunction which in turn provides reliable reconstructions of the relative vorticity and vertical velocity (via the omega equation).

The noise present in infrared satellite measurements of sea surface temperature (SST) hampers the use of surface quasi‐geostrophic (SQG) equations to diagnose ocean dynamics at high resolutions. Here we propose a methodology to reduce the contribution of noise when diagnosing surface vorticity, divergence, and vertical velocity from SST able to retain the dynamics at scales of a few kilometers. It is based on the use of denoising techniques with curvelets as basis functions and the application of a selective low‐pass filters to improve the reconstruction of surface upwelling/downwelling patterns. First, it is tested using direct numerical simulations of SQG turbulence and then applied to diagnose low‐frequency vertical velocity patterns from real MODIS (Moderate Resolution Imaging Spectroradiometer) images. The methodology here presented, which is not tied to the validity of SQG equations nor to the use of SST, is quite general and can be applied to a wide range of measurements and dynamical frameworks.

Global comparison of benthic nepheloid layers based on 52 years of nephelometer and transmissometer measurements

<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>

A one-dimensional turbulent Ekman layer model is used to simulate the upper-ocean dynamics in Santa Barbara Channel. Comparison of simulated and measured currents showed excellent agreement below subtidal frequencies.

The finite element method is one of the most widely used computational methods in engineering and science. It provides approximate solutions to boundary value problems. The quality of these solutions critically depends on the underlying discretization, the so-called mesh. To optimize the mesh, adaptive refinement methods have been proposed over the last years that can improve mesh quality over a series of iteration steps. Herein, we propose a novel deep learning architecture that can cut short the process of mesh optimization. This architecture exploits fundamental invariance and equivariance properties to keep the amount of training data modest. It can generate high-quality meshes for a given boundary value problem and a desired target approximation error in a direct, non-iterative way. We demonstrate the performance of our method by the application to standard two-dimensional linear-elastic elasticity problems. There, our method generates meshes that reduce the solution error by 22.6% (median) compared to uniform meshes with the same computational demand.

Anais do(a) XLII Ibero-Latin-American Congress on Computational Methods in Engineering

Due to their excellent mechanical performance and structural efficiency, curved tapered beams have been widely used in many engineering applications, such as, bridge structures, piping systems, biomedical devices, aerospace and aeronautical structures, etc. Their complex geometries pose challenges to the development of robust approaches for the modelling of their mechanical behaviour. Among the various approaches available in the literature for their analysis, those that are based on the finite element method have been the most successful, particularly due to their versatility. Nonetheless, when applied to Timoshenko based structural models, some of these finite element approaches are prone to shear locking when the beam elements become slender and to membrane locking when the curvature of the beam centroid curves increases [1]. The aim of the present contribution is to introduce a novel, simple and effective, finite element formulation for the analysis of two-dimensional Timoshenko curved tapered beams. This formulation relies on a complementary variational approach based on a set of approximations that satisfy in strong form all equilibrium conditions of the boundary-value problem [2], resulting thus in a formulation that is free from both shear and membrane locking phenomena. The effectiveness of the formulation is numerically demonstrated through its application to a circular clamped-clamped beam subjected to a mid-span concentrated load, and the obtained results are analysed and discussed. REFERENCES [1] H. Stolarski and T. Belytschko, “Shear and membrane locking in curved C0 elements”, Comput. Meth. Appl. Mech. Eng., Vol. 41, pp. 279–296, (1983). [2] H.A.F.A. Santos, “Complementary-energy methods for geometrically non-linear structural models: an overview and recent developments in the analysis of frames”, Archives of Computational Methods in Engineering, Vol. 18, (2011): 405.

Proceedings of the XXXVIII Iberian Latin American Congress on Computational Methods in Engineering

Agulhas Rings play a key role in the dynamics of the South Atlantic, especially through their interactions with the Walvis Ridge. However, the influence of mesoscale eddies, their interactions with bathymetry, and their interplay with internal waves on high frequency ocean dynamics remains unclear, especially in realistic setups. To address this issue, we combine high-resolution numerical simulations with telescopic grid refinement (achieving horizontal resolutions below 600 m over large regions of the South Atlantic) and observations from a dedicated field campaign, complemented by data from the recently launched SWOT mission. Our simulations, validated with mooring and Pressure Inverted Echo Sounder data from the Walvis Ridge region, show that tidal forcing and resolved submesoscale dynamics are critical to accurately capture high-frequency energy levels. Simulations without tides and submesoscale dynamics show significantly reduced energy at the high-frequency end of the spectrum. Using an eddy-tracking algorithm, we study the evolution of five anticyclones and three cyclones near the ridge. Hovmöller plots and bandpass-filtered fields show fluctuations at diurnal and semidiurnal frequencies, especially at the ridges and underneath large anticyclones. We provide new insights into mesoscale-bathymetry interactions and their role in shaping high-frequency ocean dynamics in the South Atlantic.

For the primitive equations of large-scale atmosphere and ocean dynamics, we study the problem of determining by means of a variational data assimilation algorithm initial conditions that generate strong solutions which minimize the distance to a given set of time-distributed observations. We suggest a modification of the adjoint algorithm whose novel elements is to use norms in the variational cost functional that reflects the H^1-regularity of strong solutions of the primitive equations. For such a cost functional, we prove the existence of minima and a first-order adjoint condition for strong solutions that provides the basis for computing these minima. We prove the local convergence of a gradient-based descent algorithm to optimal initial conditions using the second-order adjoint primitive equations. The algorithmic modifications due to the H^1-norms are straightforwardly to implement into a variational algorithm that employs the standard L^2-metrics.

Upper Ocean Dynamics and Horizontal Variability in Low Winds