arXiv subjects

Etienne Mémin

Publications and source records attributed to Etienne Mémin.

At least 19 recordsLinked to original sources

arXiv2026

Well-posedness of a stochastic Navier-Stokes system with dynamically coupled subgrid scales

We study a stochastic Navier--Stokes system in which the spatial modes of the transport noise evolve dynamically and are coupled to the resolved velocity. The model couples a Navier--Stokes stochastic PDE to an infinite family of linearized Navier-Stokes equations for the noise correlation modes. These latter equations may contain a hyperviscosity. We prove existence of martingale solutions in two dimensions for hyperviscosity exponent $s\geq1$, and pathwise uniqueness for $s>1$ under additional regularity of the initial noise modes. In three dimensions, we establish existence of martingale solutions for $s>3/2$. The proof combines Galerkin approximations and compactness with estimates adapted to the coupled drift and stochastic terms.

math.PR↗

arXiv2026

Coupled stochastic variational principles for multiscale surface gravity waves -- Part I: theoretical framework

This paper develops a comprehensive stochastic variational framework for surface gravity waves. Starting from Luke's variational principle for irrotational, incompressible free-surface flow, we introduce a decomposition of the velocity potential into a large-scale deterministic component and a regularized stochastic noise term representing unresolved scales. A path-wise variational principle yields the stochastic counterparts of the Laplace, Bernoulli, and kinematic boundary conditions. To close the system, a second variational principle in expectation is formulated, providing evolution equations for the noise correlation functions. The resulting coupled system preserves the Hamiltonian structure of the Zakharov--Craig--Sulem formulation. We further analyze explicit solutions via WKB approximation and show that the noise correlation functions satisfy a Hamilton-Jacobi equation with ray-tracing dynamics. This framework provides a rigorous foundation for reduced stochastic models of ocean waves and for the kinetic theory developed in the companion paper.

math-ph↗

arXiv2025

A Stochastic Ekman-Stokes Model for Coupled Ocean-Wave-Atmosphere Dynamics

Accurate representation of atmosphere-ocean boundary layers, including the interplay of turbulence, surface waves, and air-sea fluxes, remains a challenge in geophysical fluid dynamics, particularly for climate simulations. This study introduces a stochastic coupled Ekman-Stokes model (SCESM) developed within the physically consistent Location Uncertainty framework, explicitly incorporating random turbulent fluctuations and surface wave effects. The SCESM integrates established parameterizations for air-sea fluxes, turbulent viscosity, and Stokes drift, and its performance is rigorously assessed through ensemble simulations compared against observations from the LOTUS field experiment. A performance ranking analysis quantifies the impact of different model components, highlighting the critical role of explicit uncertainty representation in both oceanic and atmospheric dynamics for accurately capturing system variability. Among the tested configurations, the full model version -- including both Stokes drift and wave-induced mixing -- shows the best agreement with observations. Wave-induced mixing terms improve model performance, while wave-dependent surface roughness enhances air-sea fluxes but reduces the relative influence of wave-driven mixing. This fully coupled stochastic framework provides a foundation for advancing boundary layer parameterizations in large-scale climate models.

physics.ao-ph↗

arXiv2025

Variational principles for fully coupled stochastic fluid dynamics across scales

This work investigates variational frameworks for modeling stochastic dynamics in incompressible fluids, focusing on large-scale fluid behavior alongside small-scale stochastic processes. The authors aim to develop a coupled system of equations that captures both scales, using a variational principle formulated with Lagrangians defined on the full flow, and incorporating stochastic transport constraints. The approach smooths the noise term along time, leading to stochastic dynamics as a regularization parameter approaches zero. Initially, fixed noise terms are considered, resulting in a generalized stochastic Euler equation, which becomes problematic as the regularization parameter diminishes. The study then examines connections with existing stochastic frameworks and proposes a new variational principle that couples noise dynamics with large-scale fluid motion. This comprehensive framework provides a stochastic representation of large-scale dynamics while accounting for fine-scale components. Our main result is that the evolution of the small-scale velocity component is governed by a linearized Euler equation with random coefficients, influenced by large-scale transport, stretching, and pressure forcing.

physics.flu-dyn↗

arXiv2024

Quasi-geostrophic projection of rotating shallow-water equations

With a new unifying model for layered rotating shallow-water (RSW) and quasi-geostrophic (QG) equations, this paper sheds light on the relation between these two sets of equations. We propose here a new formulation of the quasi-geostrophic equations as a projection of the rotating shallow-water equations. This QG formulation uses the same prognostic variables as RSW, namely velocity and layer thickness, restoring the proximity of these two sets of equations. It allows direct access to the ageostrophic velocities hidden in geostrophic velocities resolved by the QG equations. It opens the path of studying the difference between QG and RSW using the same underlying numerical discretization. We illustrate this new possibility on a vortex shear instability and a double-gyre configuration.

physics.flu-dyn↗

arXiv2023

Ensemble forecasts in reproducing kernel Hilbert space family

A methodological framework for ensemble-based estimation and simulation of high dimensional dynamical systems such as the oceanic or atmospheric flows is proposed. To that end, the dynamical system is embedded in a family of reproducing kernel Hilbert spaces (RKHS) with kernel functions driven by the dynamics. In the RKHS family, the Koopman and Perron-Frobenius operators are unitary and uniformly continuous. This property warrants they can be expressed in exponential series of diagonalizable bounded evolution operators defined from their infinitesimal generators. Access to Lyapunov exponents and to exact ensemble based expressions of the tangent linear dynamics are directly available as well. The RKHS family enables us the devise of strikingly simple ensemble data assimilation methods for trajectory reconstructions in terms of constant-in-time linear combinations of trajectory samples. Such an embarrassingly simple strategy is made possible through a fully justified superposition principle ensuing from several fundamental theorems.

math-ph↗

arXiv2023

Linear wave solutions of a stochastic shallow water model

In this paper, we investigate the wave solutions of a stochastic rotating shallow water model. This approximate model provides an interesting simple description of the interplay between waves and random forcing ensuing either from the wind or coming as the feedback of the ocean on the atmosphere and leading in a very fast way to the selection of some wavelength. This interwoven, yet simple, mechanism explains the emergence of typical wavelength associated to near inertial waves. Ensemble-mean waves that are not in phase with the random forcing are damped at an exponential rate, whose magnitude depends on the random forcing variance. Geostrophic adjustment is also interpreted as a statistical homogenization process in which, in order to conserve potential vorticity, the small-scale component tends to align to the velocity fields to form a statistically homogeneous random field.

physics.flu-dyn↗

arXiv2023

Comparison of Stochastic Parametrization Schemes using Data Assimilation on Triad Models

In recent years, stochastic parametrizations have been ubiquitous in modelling uncertainty in fluid dynamics models. One source of model uncertainty comes from the coarse graining of the fine-scale data and is in common usage in computational simulations at coarser scales. In this paper, we look at two such stochastic parametrizations: the Stochastic Advection by Lie Transport (SALT) parametrization introduced by Holm and the Location Uncertainty (LU) parametrization introduced by Mémin. Whilst both parametrizations are available for full-scale models, we study their reduced order versions obtained by projecting them on a complex vector Fourier mode triad of eigenfunctions of the curl. Remarkably, these two parametrizations lead to the same reduced order model, which we term the helicity-preserving stochastic triad (HST). This reduced order model is then compared with an alternative model which preserves the energy of the system, and which is termed the energy preserving stochastic triad (EST). These low-dimensional models are ideal benchmark models for testing new Data Assimilation algorithms: they are easy to implement, exhibit diverse behaviours depending on the choice of the coefficients and come with natural physical properties such as the conservation of energy and helicity.

math.DS↗

arXiv2022

Input-output analysis of the stochastic Navier-Stokes equations: application to turbulent channel flow

Stochastic linear modelling proposed in Tissot, Mémin & Cavalieri (J. Fluid Mech., vol. 912, 2021, A51) is based on classical conservation laws subject to a stochastic transport. Once linearised around the mean flow and expressed in the Fourier domain, the model has proven its efficiency to predict the structure of the streaks of streamwise velocity in turbulent channel flows. It has been in particular demonstrated that the stochastic transport by unresolved incoherent turbulence allows to better reproduce the streaks through lift-up mechanism. In the present paper, we focus on the study of streamwise-elongated structures, energetic in the buffer and logarithmic layers. In the buffer layer, elongated streamwise vortices, named rolls, are seen to result from coherent wave-wave non-linear interactions, which have been neglected in the stochastic linear framework. We propose a way to account for the effect of these interactions in the stochastic model by introducing a stochastic forcing, which replace the missing non-linear terms. In addition, we propose an iterative strategy in order to ensure that the stochastic noise is decorrelated from the solution, as prescribed by the modelling hypotheses. We explore the prediction abilities of this more complete model in the buffer and logarithmic layers of channel flows at $Re_τ=180$, $Re_τ=550$ and $Re_τ=1000$. We show an improvement of predictions compared to resolvent analysis with eddy viscosity, especially in the logarithmic layer.

physics.flu-dyn↗

arXiv2022

Stochastic data-driven parameterization of unresolved mesoscale eddies

In this work, a stochastic representation based on a physical transport principle is proposed to account for mesoscale eddy effects on the large-scale oceanic circulation. This stochastic framework arises from a decomposition of the Lagrangian velocity into a smooth-in-time component and a highly oscillating noise term. One important characteristic of this random model is that it conserves the total energy of the resolved flow for any realization. Such an energy-preserving representation is successfully implemented in a well established multi-layered quasi-geostrophic dynamical core. The empirical spatial correlation of the unresolved noise is calibrated from the eddy-resolving simulation data. In particular, a stationary correction drift can be introduced in the noise through Girsanov transformation. This non intuitive term appears to be important in reproducing on a coarse mesh the eastwards jet of the wind-driven double-gyre circulation. In addition, a projection method has been proposed to constrain the noise living along the iso-surfaces of the vertical stratification. The resulting noise enables us to improve the intrinsic low-frequency variability of the large-scale current.

physics.geo-ph↗

arXiv2022

Analytical Properties for a Stochastic Rotating Shallow Water Model under Location Uncertainty

The rotating shallow water model is a simplification of oceanic and atmospheric general circulation models that are used in many applications such as surge prediction, tsunami tracking and ocean modelling. In this paper we introduce a class of rotating shallow water models which are stochastically perturbed in order to incorporate model uncertainty into the underlying system. The stochasticity is chosen in a judicious way, by following the principles of location uncertainty, as introduced in [Mémin, 2014]. We prove that the resulting equation is part of a class of stochastic partial differential equations that have unique maximal strong solutions. The methodology is based on the construction of an approximating sequence of models taking value in an appropriately chosen finite-dimensional Littlewood-Paley space. Finally, we show that a distinguished element of this class of stochastic partial differential equations has a global weak solution.

math.AP↗

arXiv2022

Modified (hyper-)viscosity for coarse-resolution ocean models

We present a simple parameterization for coarse-resolution ocean models. To replace computationally expensive high-resolution ocean models, we develop a computationally cheap parameterization for coarse-resolution models based solely on the modification of the viscosity term in advection equations. It is meant to reproduce the mean quantities like pressure, velocity, or vorticity computed from a high-resolution reference solution or using observations. We test this new parameterization on a double-gyre quasi-geostrophic model in the eddy-permitting regime. Our results show that the proposed scheme improves significantly the energy statistics and the intrinsic variability on the coarse mesh. This method shall serve as a deterministic basis model for coarse-resolution stochastic parameterizations in future works.

physics.flu-dyn↗

arXiv2022

Bridging Koopman Operator and time-series auto-correlation based Hilbert-Schmidt operator

Given a stationary continuous-time process $f(t)$, the Hilbert-Schmidt operator $A_τ$ can be defined for every finite $τ$\cite{Vautard1989SingularSA}. Let $λ_{τ,i}$ be the eigenvalues of $A_τ$ with descending order. In this article, a Hilbert space $\mathcal{H}_f$ and the (time-shift) continuous one-parameter semigroup of isometries $\mathcal{K}^s$ are defined. Let $\{v_i, i\in\mathbb{N}\}$ be the eigenvectors of $\mathcal{K}^s$ for all $s\geq 0$. Let $f = \displaystyle\sum_{i=1}^{\infty}a_iv_i + f^{\perp}$ be the orthogonal decomposition with descending $|a_i|$. We prove that $\displaystyle\lim_{τ\to\infty}λ_{τ,i} = |a_i|^2$. The continuous one-parameter semigroup $\{\mathcal{K}^s: s\geq 0\}$ is equivalent, almost surely, to the classical Koopman one-parameter semigroup defined on $L^2(X,ν)$, if the dynamical system is ergodic and has invariant measure $ν$ on the phase space $X$.

math.OC↗

arXiv2022

Level set based particle filter driven by optical flow: an application to track the salt boundary from X-ray CT time-series

Image-based computational fluid dynamics have long played an important role in leveraging knowledge and understanding of several physical phenomena. In particular, probabilistic computational methods have opened the way to modelling the complex dynamics of systems in purely random turbulent motion. In the field of structural geology, a better understanding of the deformation and stress state both within the salt and the surrounding rocks is of great interest to characterize all kinds of subsurface long-terms energy-storage systems. The objective of this research is to determine the non-linear deformation of the salt boundary over time using a parallelized, stochastic filtering approach from x-ray computed tomography (CT) image time series depicting the evolution of salt structures triggered by gravity and under differential loading. This work represents a first step towards bringing together physical modeling and advanced stochastic image processing methods where model uncertainty is taken into account.

cs.CV↗

arXiv2021

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

We introduce a physically relevant stochastic representation of the rotating shallow water equations. The derivation relies mainly on a stochastic transport principle and on a decomposition of the fluid flow into a large-scale component and a noise term that models the unresolved flow components. As for the classical (deterministic) system, this scheme, referred to as modelling under location uncertainty (LU), conserves the global energy of any realization and provides the possibility to generate an ensemble of physically relevant random simulations with a good trade-off between the model error representation and the ensemble's spread. To maintain numerically the energy conservation feature, we combine an energy (in space) preserving discretization of the underlying deterministic model with approximations of the stochastic terms that are based on standard finite volume/difference operators. The LU derivation, built from the very same conservation principles as the usual geophysical models, together with the numerical scheme proposed can be directly used in existing dynamical cores of global numerical weather prediction models. The capabilities of the proposed framework is demonstrated for an inviscid test case on the f-plane and for a barotropically unstable jet on the sphere.

physics.flu-dyn↗

arXiv2018

A model under location uncertainty to predict the mean velocity in wall bounded flows

To date no satisfying model exists to explain the mean velocity profile within the whole turbulent layer of canonical wall bounded flows. We propose a modification of the velocity profile expression that ensues from a recently proposed stochastic representation of fluid flows dynamics. This modeling, called modeling under location uncertainty introduces in a rigorous way a subgrid term generalizing the eddy-viscosity assumption and an eddy-induced advection term resulting from turbulence inhomogeneity. This latter term gives rise to a theoretically well-grounded model for the transitional zone between the viscous sublayer and the turbulent sublayer. An expression of the small-scale velocity component is also provided in the viscous zone. Numerical assessment of the results are provided for turbulent boundary layer flows, for pipe flows and channel flows at various Reynolds numbers.

physics.flu-dyn↗

arXiv2017

Stochastic modelling and diffusion modes for proper orthogonal decomposition models and small-scale flow analysis

We present here a new stochastic modelling in the constitution of fluid flow reduced-order models. This framework introduces a spatially inhomogeneous random field to represent the unresolved small-scale velocity component. Such a decomposition of the velocity in terms of a smooth large-scale velocity component and a rough, highly oscillating, component gives rise, without any supplementary assumption, to a large-scale flow dynamics that includes a modified advection term together with an inhomogeneous diffusion term. Both of those terms, related respectively to turbophoresis and mixing effects, depend on the variance of the unre-solved small-scale velocity component. They bring to the reduced system an explicit subgrid term enabling to take into account the action of the truncated modes. Besides, a decomposition of the variance tensor in terms of diffusion modes provides a meaningful statistical representation of the stationary or nonstationary structuration of the small-scale velocity and of its action on the resolved modes. This supplies a useful tool for turbulent fluid flows data analysis. We apply this methodology to circular cylinder wake flow at Reynolds numbers Re = 300 and Re = 3900, respectively. The finite dimensional models of the wake flows reveal the energy and the anisotropy distributions of the small-scale diffusion modes. These distributions identify critical regions where corrective advection effects as well as structured energy dissipation effects take place. In providing rigorously derived subgrid terms, the proposed approach yields accurate and robust temporal reconstruction of the low-dimensional models.

physics.flu-dyn↗

arXiv2017

Geophysical flows under location uncertainty, Part I Random transport and general models

A stochastic flow representation is considered with the Eulerian velocity decomposed between a smooth large scale component and a rough small-scale turbulent component. The latter is specified as a random field uncorrelated in time. Subsequently, the material derivative is modified and leads to a stochastic version of the material derivative to include a drift correction , an inhomogeneous and anisotropic diffusion, and a multiplicative noise. As derived, this stochastic transport exhibits a remarkable energy conservation property for any realizations. As demonstrated, this pivotal operator further provides elegant means to derive stochastic formulations of classical representations of geophysical flow dynamics.

physics.geo-ph↗