A consistent stochastic large-scale representation of the Navier-Stokes equations

In this paper we analyze the theoretical properties of a stochastic representation of the incompressible Navier-Stokes equations defined in the framework of the modeling under location uncertainty (LU). This setup built from a stochastic version of the Reynolds transport theorem incorporates a so-called transport noise and involves several specific additional features such as a large scale diffusion term, akin to classical subgrid models, and a modified advection term arising from the spatial inhomogeneity of the small-scale velocity components. This formalism has been numerically evaluated in a series of studies with a particular interest on geophysical flows approximations and data assimilation. In this work we focus more specifically on its theoretical analysis. We demonstrate, through classical arguments, the existence of martingale solutions for the stochastic Navier-Stokes equations in LU form. We show they are pathwise and unique for 2D flows. We then prove that if the noise intensity goes to zero, these solutions converge, up to a subsequence in dimension $3$, to a solution of the deterministic Navier-Stokes equation. similarly to the grid convergence property of well established large-eddies simulation strategies, this result allows us to give some guarantee on the interpretation of the LU Navier-Stokes equations as a consistent large-scale model of the deterministic Navier-Stokes equation

Ensemble forecasts in reproducing kernel Hilbert space family: dynamical systems in Wonderland

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 with kernel functions driven by the dynamics. This family is nicknamed Wonderland for its appealing properties. In Wonderland the Koopman and Perron-Frobenius operators are unitary and uniformly continuous. This property warrants they can be expressed in exponential series of diagonalizable bounded infinitesimal generators. Access to Lyapunov exponents and to exact ensemble based expressions of the tangent linear dynamics are directly available as well. Wonderland 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

Estimation conjointe de champs de vitesse denses et paramétriques pour des écoulements fluides

- Dans ce travail nous proposons une approche d'estimation dense couplée avec un modèle paramétrique non-linéaire. Ce modèle permet de décrire la forme du champ des vitesses apparentes de l'écoulement à l'aide d'entités cinématiques particulières telles que les vortex, puits ou sources. Il permet de rendre compte des composantes solénoïdale et irrotationnel du champ des vitesses tandis que le champ dense permet de décrire la composante laminaire de l'écoulement. L'estimation du champ dense et l'extraction des entités dynamiques sont effectuées de façon conjointe et coopérative

Viscosity contrasts in the Venus mantle from tidal deformations

The tidal deformations of a planet are often considered as markers of its inner structure. In this work, we use the tide excitations induced by the Sun on Venus for deciphering the nature of its internal layers. In using a Monte Carlo Random Exploration of the space of parameters describing the thickness, density and viscosity of 4 or 5 layer profiles, we were able to select models that can reproduce the observed mass, total moment of inertia, $k_2$ Love number and expected quality factor $Q$. Each model is assumed to have homogeneous layers with constant density, viscosity and rigidity. These models show significant contrasts in the viscosity between the upper mantle and the lower mantle. They also rather favor a S-free core and a slightly hotter lower mantle consistent with previous expectations.Comment: Accepted for publication in Planetary and Space Science

Decoding the origins of vertical land motions observed today at coasts

In recent decades, geodetic techniques have allowed detecting vertical land motions and sea-level changes of a few millimetres per year, based on measurements taken at the coast (tide gauges), on board of satellite platforms (satellite altimetry) or both (Global Navigation Satellite System). Here, contemporary vertical land motions are analysed from January 1993 to July 2013 at 849 globally distributed coastal sites. The vertical displacement of the coastal platform due to surface mass changes is modelled using elastic and viscoelastic Green's functions. Special attention is paid to the effects of glacial isostatic adjustment induced by past and present-day ice melting. Various rheological and loading parameters are explored to provide a set of scenarios that could explain the coastal observations of vertical land motions globally. In well-instrumented regions, predicted vertical land motions explain more than 80 per cent of the variance observed at scales larger than a few hundred kilometres. Residual vertical land motions show a strong local variability, especially in the vicinity of plate boundaries due to the earthquake cycle. Significant residual signals are also observed at scales of a few hundred kilometres over nine well-instrumented regions forming observation windows on unmodelled geophysical processes. This study highlights the potential of our multitechnique database to detect geodynamical processes, driven by anthropogenic influence, surface mass changes (surface loading and glacial isostatic adjustment) and tectonic activity (including the earthquake cycle, sediment and volcanic loading, as well as regional tectonic constraints). Future improvements should be aimed at densifying the instrumental network and at investigating more thoroughly the uncertainties associated with glacial isostatic adjustment models.This research benefited from financial support from the CNES (Centre National d’Etudes Spatiales, France) through the TOSCA committee fellowship and from the European Research Council within the framework of the SP2-Ideas Program ERC-2013-CoG, under ERC Grant agreement number 617588. GS is supported by a DiSPeA research grant (CUP H32I160000000005) and by Programma Nazionale di Ricerche in Antartide (PNRA 2013/B2.06, CUP D32I14000230005). AM was supported by an Australian Research Council Super Science Fellowship (FS110200045)

Convergence in measure under Finite Additivity

We investigate the possibility of replacing the topology of convergence in probability with convergence in $L^1$. A characterization of continuous linear functionals on the space of measurable functions is also obtained

Étude expérimentale de la turbulence dans une couche de mélange anisotherme

L'étude porte sur une couche de mélange plane horizontale générée par la rencontre de deux écoulements parallèles à vitesse et température différentes. Le mélange turbulent est analysé pour différentes conditions initiales en termes de gradients de vitesse et de température. On distingue en particulier des configurations en régime de stratification stable et instable sous l'effet des forces de flottabilité. L'analyse des corrélations entre les fluctuations de vitesse et de température s'appuie sur la technique expérimentale d'anémométrie à température de fil variable (PCTA), qui permet la mesure instantanée de la vitesse et de la température en un même point grâce à la variation périodique et par palier du coefficient de surchauffe du fil chaud utilisé. Un premier travail a consisté à étendre la technique PCTA à l'utilisation de fils croisés pour la mesure simultanée de la température et de deux composantes de la vitesse. Dans un premier temps, les statistiques en un point permettent d'identifier les caractéristiques de l'écoulement dans la région de similitude et d'y établir les équations de bilan pour l'énergie cinétique turbulente, l'intensité des fluctuations de température et les flux de quantité de mouvement et de chaleur. Il apparaît, vu les faibles nombres de Richardson en jeu (Rif<0,03), que les forces de flottabilité sont quasi-négligeables devant les moteurs principaux du mouvement. Pourtant, ce forçage thermique peu énergétique est suffisant, en configuration instable, pour augmenter significativement le taux d'expansion et la contrainte de cisaillement, ce qui correspond de fait à une augmentation de la production de turbulence. L'analyse des densités de probabilité jointes permet ensuite de mettre en évidence les mécanismes et évènements qui contribuent significativement aux flux transversaux de quantité de mouvement et de chaleur. Ces différentes contributions sont différenciées et quantifiées par une analyse en quadrants qui fait ressortir la prépondérance des mouvements d'entraînement et d'éjection. On examine enfin les statistiques en deux points associées aux incréments de vitesse et de température. Le comportement de ces incréments est étudié à travers leurs densités de probabilité et leurs coefficients de dissymétrie et d'aplatissement. Les exposants des fonctions de structure confirment l'intermittence plus grande de la température par rapport à celle de la vitesse. Les différents termes des équations de Kolmogorov et de Yaglom sont mesurés. L'équilibre de ces bilans par échelle permet de quantifier le terme qui intègre les différents forçages proposés dans la littérature.The turbulent mixing is studied in a plane mixing layer for a range of initial conditions applied in terms of velocity and temperature gradients between the two parallel inlet flows. A particular attention is paid to the effect of buoyancy forces, especially in the difference between the so-called stable and unstable configurations, in relation to the sign of the vertical temperature gradient applied. In this study, the novel experimental technique called PCTA, for Parameterizable Constant Temperature Anemometry, is used to enable the analysis of correlations between the velocity and temperature fluctuations. In a preliminary work, the PCTA technique, based on the implementation of repetitive multiple-overheat patterns to a hot wire, is extended and adapted for the instantaneous measurement of temperature and two components of velocity with X-wire probes. In a first stage, one point statistics are analysed. They provide a description of the flow features in the similarity region, where the balance equations for turbulent kinetic energy, temperature variance and the momentum and heat fluxes are established. Considering the low Richardson numbers at stake (Rif <0.03), the buoyancy forces appear logically to be quantitatively negligible compared to the main driving forces, but such a low energy forcing mechanism is in fact sufficient, in unstable configuration, to significantly increase the shear stress and the expansion rate of the mixing layer, both phenomena being associated to an enhanced production of turbulence. In a second stage, a joint probability density function analysis highlights the mechanisms and events that significantly contribute to the transverse momentum and heat fluxes. These contributions are differentiated and quantified through a quadrant analysis which emphasizes the dominance of the local movements of entrainment and ejection associated to the Kelvin-Helmholtz structures. Finally, the study focuses on the two points statistics associated with velocity and temperature increments. The behaviour of these increments is studied through their probability densities, examined along with the skewness and kurtosis coefficients. The structure function exponents confirm the stronger intermittency of temperature compared to that of the velocity. The different terms of the Kolmogorov and Yaglom equations are estimated. The balance of these scale budgets allows the quantification of the forcing term that has been introduced in the literature.RENNES1-Bibl. électronique (352382106) / SudocSudocFranceF