Data driven stochastic primitive equations with dynamic modes decomposition

The simulation of planetary flows at all the scales that have a significant impact on the climate system is unachievable with nowadays computational resources. Large-scale simulations of the Ocean (as well as for Atmosphere) remains the primary tool of investigation while high resolution simulations can be obtained only for small geographical domains or short integration periods. The complex interdependence of mesoscale and sub-mesoscale dynamics, however, is lost in state-of-the-art simulations when performed at scales that are too large to capture these phenomena. Most of the modeling challenges arise from the representation of these effects in a parameterized manner. This work investigates the so called Location Uncertainty (LU) framework [1,2], that provides a solid theoretical background for the definition of a large-scale representation with an additional stochastic component representing the subgrid contribution, introducing new degrees of freedom to be exploited in the modeling of specific phenomena [2]. The model, that has been proven successful in several large-scale models for ocean dynamics [3,4,5,6], is implemented in the community ocean model NEMO (https://www.nemo-ocean.eu) in its hydrostatic primitive equation version, as outlined in [6] and already tested in [7]. An idealized double-gyre configuration is shown to be improved by the stochastic addition in both eddy permitting (~35km) and eddy resolving (~10km) regimes, under a variety of choices of the noise model including both data-driven and data-agnostic approaches. [1] E. Mémin Fluid flow dynamics under location uncertainty,(2014), Geophysical & Astrophysical Fluid Dynamics, 108, 2, 119–146.[2] G. Tissot, E. Mémin, Q. Jamet, (2023), Stochastic compressible Navier-Stokes equations under Location uncertainty, Stochastic Transport in Upper Ocean Dynamics, Springer. [3] 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.[4] 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] F.L. Tucciarone, E. Mémin, L. Li, (2023), Data driven stochastic primitive equations with dynamic modes 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>

Planetary flows are shaped by interactions at scales much smaller than the flows themselves, with mesoscale and sub–mesoscale eddies playing key roles in mixing, particle transport and tracer dispersion. To capture these effects, we introduce a stochastic formulation of the primitive equations within the Location Uncertainty (LU) framework. Derived from conservation principles via a stochastic Reynolds transport theorem, this approach decomposes velocity into a smooth–in–time large–scale component and a random–in–time field representing unresolved scales effects. To model the velocity noise term, we develop two data–driven methods based on Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) and extend this to hybrid approaches combining model– and data–driven constraints. Simulations show that the LU framework enhances gyre flow predictions, improving mixing, jet structure, and tracer transport while revealing the interplay between small– and large–scale dynamics.

To address the issue of low accuracy in the dynamic modal decomposition (DMD) method used for operational modal analysis (OMA) under noise conditions of aerospace structures, an enhanced identification approach is proposed in this paper, which integrates subspace orthogonal projection with DMD to better determine the modal properties of linear mechanical systems with noisy observations. Subspace orthogonal projection applied to the Hankelized matrix is utilized for denoising observation signals. Compact singular value decomposition (SVD) is employed on the projection matrix in order to acquire the optimal realization of system matrix. Subsequently, DMD is introduced to reduce the dimensionality of the system matrix and extract the dominant modal features. The effectiveness and practicality of the proposed method are confirmed through numerical and experimental examples. The proposed method demonstrates marginally improved identification accuracy in modal frequency and enhanced performance in damping ratios when compared to representative OMA methods under different white noise conditions.

The large-scale circulation (LSC) is the most fundamental turbulent coherent flow structure in Rayleigh-Bénard convection. Further, LSCs provide the foundation upon which superstructures, the largest observable features in convective systems, are formed. In confined cylindrical geometries with diameter-to-height aspect ratios of , LSC dynamics are known to be governed by a quasi-two-dimensional, coupled horizontal sloshing and torsional (ST) oscillatory mode. In contrast, in cylinders, a three-dimensional jump rope vortex (JRV) motion dominates the LSC dynamics. Here, we use dynamic mode decomposition (DMD) on direct numerical simulation data of liquid metal to show that both types of modes co-exist in and cylinders but with opposite dynamical importance. Furthermore, with this analysis, we demonstrate that ST oscillations originate from a tilted elliptical mean flow superposed with a symmetric higher-order mode, which is connected to the four rolls in the plane perpendicular to the LSC in tanks.

A physical stochastic parameterization is adopted in this work to account for the effects of the unresolved small-scale on the large-scale flow dynamics. This random model is based on a stochastic transport principle, which ensures a strong energy conservation. The dynamic mode decomposition (DMD) is performed on high-resolution data to learn a basis of the unresolved velocity field, on which the stochastic transport velocity is expressed. Time-harmonic property of DMD modes allows us to perform a clean separation between time-differentiable and time-decorrelated components. Such random scheme is assessed on a quasi-geostrophic (QG) model.

<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>

Microplastics (MPs) are pervasive contaminants, yet understanding their pathways and fate in the marine environment remains unclear. A key challenge is the lack of in-situ, complementary measurements linking MP quantification with oceanographic parameters, particularly concerning submesoscale processes and density fronts. Submesoscale dynamics, including filaments, eddies, and fronts, significantly influence the transport and accumulation of MPs by creating convergence zones and sharp density gradients. Density fronts serve as critical hotspots for MP aggregation, concentrating particles through upwelling and downwelling processes. Despite their importance, these interactions remain poorly studied, emphasizing the need for integrated approaches to directly measure the interplay between MPs and the physical processes that drive their distribution.This study addresses this gap by utilizing in-situ measurements collected with an autonomous surface vehicle (ASV) in the southern North Sea, simultaneously collecting water samples for MP analysis and key oceanographic data. The ASV simultaneously sampled air, sea surface microlayer, and underlying water for MP analysis. A weather station and conductivity, temperature, and depth (CTD) sensors were deployed on the ASV to further contextualize the distribution of MPs. Additionally, CTD profiles were obtained by an accompanying research vessel to investigate the influence of stratification and temporal dynamics on MP distribution. An acoustic Doppler current profiler measured water current velocities and flow direction.The measurements underscore the pivotal role of submesoscale fronts and filaments in shaping the accumulation and distribution of MP. Upwelling and downwelling processes at these fronts and filaments concentrated MP up to 30.48 µg MP L⁻¹, and distributed MPs vertically across depth profiles and horizontally across fronts. Wind direction was found to influence the presence of MP in the atmosphere, while wind speeds appeared to enhance heterogeneity in MP composition and concentration within the water.Submesoscale fronts and filaments are highlighted as key zones for MP accumulation, driven by the interplay of horizontal and vertical water flow linked to ageostrophic circulation. The data provide novel insights into their transport mechanisms in the marine environment.

Effect of a large-scale vortical structure on mixing and spreading of a shear layer is numerically investigated. Two strut configurations, namely, straight and tapered struts at two convective Mach numbers (Mc = 1.4 and 0.37) for two jet heights (0.6 and 1 mm) are investigated. The hydrogen jet is injected through a two-dimensional slot in oncoming coflow at Mach 2. An excellent agreement between simulated and experimental data is witnessed, whereas the instantaneous data reveal the presence of various large-scale structures in the flow field. From the instantaneous field, it becomes apparent that both the geometries have different vortical breakdown locations. It is also noticed that an early onset of vortex breakdown manifests itself into the mixing layer thickness enhancement, the effect of which is reflected in overall mixing characteristics. It becomes evident that the shear strength plays an important role in the near field mixing. The higher shear strength promotes the generation of large vortices. The analysis shows that the SS-0.6 case offers highest mixing efficiency being dominated by relatively large-scale structures. Eigenmodes obtained through Proper Orthogonal Decomposition (POD) confirm the presence of dominating structures and shed light into the series of events involved in vortex pairing/merging and breakdown. Dynamic modal decomposition also strengthens the observation made through the POD.

Recently, acoustic communication employing orbital angular momentum (OAM) opens another avenue for efficient data transmission in aquatic environments. Current topological charge (TC) detection of OAM beams relies on the orthogonality among different-order OAM beams. However, such strategy requires measurements of the complete azimuthal acoustic pressure, which inevitably reduces the efficiency and increases the bit error rate (BER). To address these challenges, this study proposes a modified dynamic modal decomposition (DMD) method by partially sampling the acoustic field for precise TC detection. Numerical simulations confirm the accuracy of this approach in extracting single or multiple TCs magnitudes within a partially sampled acoustic field. We theoretically compare the performance of the modified DMD approach with conventional orthogonal decoding method. Simulation results indicate that our modified DMD scheme exhibits lower BER under the same noise interference and is more robust to the array misalignment. This research introduces an efficient demodulation solution for acoustic OAM communication, offering potential benefits for simplifying receiver array design and enhancing long-distance underwater data transmission.

A one-year time series of Acoustic Doppler Current Profiler (ADCP) data was collected in Terra Nova Bay (TNB) polynya (Ross Sea, Antarctica) during 2000. Together with Automatic Weather Station (AWS) Eneide meteorological data and Special Sensor Microwave Imager (SSM/I) ice concentration data, ADCP data were analysed to investigate upper layer dynamics and variability due to atmospheric forcing. Empirical Orthogonal Function (EOF) analysis was performed to separate the surface variability caused by local forcing from the large-scale circulation component. In particular, the first mode represented the barotropic circulation while the second the stronger surface currents. The decrease in shelf water density from melting sea ice resulted in an off-shore density gradient producing a southern shift in the circulation. This result proved to be consistent with the in situ data acquired during February–April at 120 m depth. The observed variability of the surface currents was assessed with respect to the thermal wind equation and the steady Ekman model. Strong katabatic winds shifted the surface currents eastward with respect to the general north-eastern circulation. The wind stress acted as a relevant forcing for the surface large-scale circulation in TNB, but had negligible effects on the vertically integrated transport.

Oceanic submesoscale processes are ubiquitous in the North Pacific Subtropical Gyre (NPSG), where the biological carbon pump is generally ineffective. Due to difficulties in collecting continuous observations, however, it remains uncertain whether episodic submesoscale processes can drive significant changes in particulate organic carbon (POC) export into the mesopelagic ocean. Here we present observations from high-frequency Biogeochemical-Argo floats in the NPSG, which captured the enhanced POC export fluxes during the intensifying stages of a submesoscale front and a cyclonic eddy compared to their other life stages. A higher percentage of POC export flux was found to be transferred to the base of mesopelagic layer at the front compared to that at the intensifying eddy and the mean of previous studies (37% vs. ~10%), suggesting that the POC export efficiency was significantly strengthened by submesoscale dynamics. Such findings highlight the importance of submesoscale fronts for carbon export and sequestration in subtropical gyres.

Submesoscale processes, characterized by strong vertical velocities that generate sea surface temperature (SST) fronts as well as O(1) Rossby number (Ro), are critical to ocean mixing and biogeochemical transport, yet their observation is hampered by cost and spatial limitations. Hence, this study proposes an adaptive sampling framework for unmanned surface vehicles (USVs) that integrates Gaussian process regression (GPR) with submesoscale physical characteristics for efficient, targeted sampling. Three composite-kernel GPR models are developed to predict SST, zonal velocity U, and meridional velocity V, providing predictive fields to support adaptive path planning. A robust coupled gradient indicator (CGI) is further introduced to identify SST frontal zones, where the maximum CGI values are used to select candidate waypoints. Connecting these waypoints yields adaptive paths aligned with frontal structures, while a Ro threshold (0.5–2) automatically triggers spiral-intensive sampling to collect more useful data. Simulation results show that the planned paths effectively capture SST gradient and submesoscale dynamics. The final environment reconstruction achieved the desired accuracy after model retraining, and deployment analysis informs optimal platform deployment. Overall, the proposed framework couples environmental prediction, adaptive path planning, and intelligent sampling, offering an effective strategy for advancing the observation of submesoscale ocean processes.

<p>The interaction between large-scale tropical circulations and moist convection has been the focus of a number of studies. However, projections of how the large-scale tropical circulation may change under global warming remain uncertain because our understanding of this interaction is still limited.</p><p>Here, we use a cloud-resolving model (CRM) coupled with a supra-domain scale (SDS) parameterisation of the large-scale circulation to investigate how tropical circulations driven by sea-surface temperature (SST) gradients change in a future warmer climate. Two popular SDS parameterisation schemes are compared; the weak temperature gradient approximation and the damped-gravity-wave approximation. In both cases, the large-scale vertical velocity is related to the deviation of the simulated density profile from a reference profile taken from the same model run to radiative-convective equilibrium.</p><p>We examine how the large-scale vertical velocity profile varies with surface temperature for fixed background profile (relative SST) as well as how it varies with the surface temperature of the reference profile (background SST). The domain mean vertical velocity appears to be very top-heavy with the maximum vertical velocity becoming stronger at warmer surface temperatures. The results are understood using a simple model for the thermodynamic structure of a convecting atmosphere based on an entraining plume. The model uses a fixed entrainment rate and the relative humidity from the cloud-resolving model to predict a temperature profile. The vertical velocities calculated from these predicted temperature profiles is similar to the vertical velocity structures and their behaviour in a warmer climate that we see in the CRM simulations. The results provide insight into large scale vertical velocity structures simulated by SDS parameterisation schemes, providing a stepping stone to understanding the factors driving changes to the large-scale tropical circulation in a future warmer climate.</p>

Ocean surface current (or ocean flow) visualization plays a significant role in the understanding of dynamical processes of ocean. It has been a hot research topic in both computer science and oceanography. Ocean surface current is a turbulent flow field mixing of multi-scale ocean dynamics such as large-scale ocean circulations ( $$100\,\hbox {km}\sim$$ ), mesoscale eddies (10– $$100\,\hbox {km}$$ ), submesoscale processes (1– $$10\,\hbox {km}$$ ). Mesoscale eddies, which are strong but short-life movement relative to the large-scale ocean circulations, have great importance on the transportation of ocean water masses, momentum and energy. However, their detection and recognition, which are treated as the foundation of exploring their dynamical mechanisms, is still a challenging issue. For one thing, in the mixed ocean flow field, different ocean flows depended and influence with each other making existing methods difficult to identify among them. For another, mesoscale eddies are active signals on the ocean. They may change their forms and velocities at any time. This challenges existing works to deal with their boundary ambiguity and unremitting transitions. To solve these problems, this paper proposes a novel 2D and 3D ocean surface current visualization approach based on an amended Helmholtz–Hodge decomposition (HHD), which can be widely used for mesoscale eddy detection. In our method, HHD decomposes each mixed ocean flow field to two components: curl component and divergence component. Different ocean flows can be represented by these two components independently. In addition, to improve the performance of eddy identification and to reveal the 3D structure of ocean flows simultaneously, HHD transforms the 2D ocean flow field to 3D potential surfaces. Finally, comprehensive experiments are performed on both global and local ocean flow field (Black Sea and Mediterranean Sea) calculated from satellite maps of sea level anomaly to verify our method. Experimental results demonstrate the good effectiveness of our method.