arXiv subjects

Corentin Herbert

Publications and source records attributed to Corentin Herbert.

29 records · Page 2Linked to original sources

arXiv2014

Mathematical and Physical Ideas for Climate Science

The climate is a forced and dissipative nonlinear system featuring non-trivial dynamics of a vast range of spatial and temporal scales. The understanding of the climate's structural and multiscale properties is crucial for the provision of a unifying picture of its dynamics and for the implementation of accurate and efficient numerical models. We present some recent developments at the intersection between climate science, mathematics, and physics, which may prove fruitful in the direction of constructing a more comprehensive account of climate dynamics. We describe the Nambu formulation of fluid dynamics, and the potential of such a theory for constructing sophisticated numerical models of geophysical fluids. Then, we focus on the statistical mechanics of quasi-equilibrium flows in a rotating environment, which seems crucial for constructing a robust theory of geophysical turbulence. We then discuss ideas and methods suited for approaching directly the non-equilibrium nature of the climate system. First, we describe some recent findings on the thermodynamics of climate and characterize its energy and entropy budgets, and discuss related methods for intercomparing climate models and for studying tipping points. These ideas can also create a common ground between geophysics and astrophysics by suggesting general tools for studying exoplanetary atmospheres. We conclude by focusing on non-equilibrium statistical mechanics, which allows for a unified framing of problems as different as the climate response to forcings, the effect of altering the boundary conditions or the coupling between geophysical flows, and the derivation of parametrizations for numerical models.

physics.ao-ph↗

arXiv2014

Nonlinear energy transfers and phase diagrams for geostrophically balanced rotating--stratified flows

Equilibrium statistical mechanics tools have been developed to obtain indications about the natural tendencies of nonlinear energy transfers in two-dimensional and quasi two-dimensional flows like rotating and stratified flows in geostrophic balance. In this article, we consider a simple model of such flows with a non-trivial vertical structure, namely two-layer quasi-geostrophic flows, which remain amenable to analytical study. We obtain the statistical equilibria of the system in the case of a linear vorticity-stream function relation, build the corresponding phase diagram, and discuss the most probable outcome of nonlinear energy transfers, both on the horizontal and on the vertical, in the presence of stratification and rotation.

physics.flu-dyn↗

arXiv2014

Restricted Equilibrium and the Energy Cascade in Rotating and Stratified Flows

Most of the turbulent flows appearing in nature (e.g. geophysical and astrophysical flows) are subjected to strong rotation and stratification. These effects break the symmetries of classical, homogenous isotropic turbulence. In doing so, they introduce a natural decomposition of phase space in terms of wave modes and potential vorticity modes. The appearance of a new time scale associated to the propagation of waves, in addition to the eddy turnover time, increases the complexity of the energy transfers between the various scales; nonlinearly interacting waves may dominate at some scales while balanced motion may prevail at others. In the end, it is difficult to predict \emph{a priori} if the energy cascades downscale as in homogeneous isotropic turbulence, upscale as expected from balanced dynamics, or follows yet another phenomenology. In this paper, we suggest a theoretical approach based on equilibrium statistical mechanics for the ideal system, inspired from the restricted partition function formalism introduced in metastability studies. In this framework, we show analytically that in the presence of rotation, when the dynamics is restricted to the slow modes, the equilibrium energy spectrum features an infrared divergence characteristic of an inverse cascade regime, whereas this is not the case for purely stratified flows.

physics.flu-dyn↗

arXiv2013

Restricted Partition Functions and Inverse Energy Cascades in Parity Symmetry Breaking flows

When the symmetries of homogenous isotropic turbulent flows are broken, different sets of modes with different physical roles emerge. In particular, choosing a forcing which puts more weight on one or the other of these sets may result in different statistics for the energy transfers. We use the general method of computing a partition function restricted to a portion of phase space to study analytically these different statistics. We illustrate this method in the case of parity symmetry breaking, measured by helicity. It is shown that when helicity is sign definite at all scales, an inverse cascade is expected for the energy. When sign-definiteness is lost, even for a small set of modes, this cascade disappears and there is a sharp phase transition to the standard helical equipartition spectra.

physics.flu-dyn↗

arXiv2013

Additional Invariants and Statistical Equilibria for the 2D Euler Equations on a spherical domain

The role of the domain geometry for the statistical mechanics of 2D Euler flows is investigated. It is shown that for a spherical domain, there exists invariant subspaces in phase space which yield additional angular momentum, energy and enstrophy invariants. The microcanonical measure taking into account these invariants is built and a mean-field, Robert-Sommeria-Miller theory is developed in the simple case of the energy-enstrophy measure. The variational problem is solved analytically and a partial energy condensation is obtained. The thermodynamic properties of the system are also discussed.

cond-mat.stat-mech↗

arXiv2013

Vertical Temperature Profiles at Maximum Entropy Production with a Net Exchange Radiative Formulation

Like any fluid heated from below, the atmosphere is subject to vertical instability which triggers convection. Convection occurs on small time and space scales, which makes it a challenging feature to include in climate models. Usually sub-grid parameterizations are required. Here, we develop an alternative view based on a global thermodynamic variational principle. We compute convective flux profiles and temperature profiles at steady-state in an implicit way, by maximizing the associated entropy production rate. Two settings are examined, corresponding respectively to the idealized case of a gray atmosphere, and a realistic case based on a Net Exchange Formulation radiative scheme. In the second case, we are also able to discuss the effect of variations of the atmospheric composition, like a doubling of the carbon dioxide concentration.

physics.ao-ph↗

arXiv2013

Predictive use of the Maximum Entropy Production principle for Past and Present Climates

In this paper, we show how the MEP hypothesis may be used to build simple climate models without representing explicitly the energy transport by the atmosphere. The purpose is twofold. First, we assess the performance of the MEP hypothesis by comparing a simple model with minimal input data to a complex, state-of-the-art General Circulation Model. Next, we show how to improve the realism of MEP climate models by including climate feedbacks, focusing on the case of the water-vapour feedback. We also discuss the dependence of the entropy production rate and predicted surface temperature on the resolution of the model.

physics.ao-ph↗

arXiv2012

Statistical mechanics of quasi-geostrophic flows on a rotating sphere

Statistical mechanics provides an elegant explanation to the appearance of coherent structures in two-dimensional inviscid turbulence: while the fine-grained vorticity field, described by the Euler equation, becomes more and more filamented through time, its dynamical evolution is constrained by some global conservation laws (energy, Casimir invariants). As a consequence, the coarse-grained vorticity field can be predicted through standard statistical mechanics arguments (relying on the Hamiltonian structure of the two-dimensional Euler flow), for any given set of the integral constraints. It has been suggested that the theory applies equally well to geophysical turbulence; specifically in the case of the quasi-geostrophic equations, with potential vorticity playing the role of the advected quantity. In this study, we demonstrate analytically that the Miller-Robert-Sommeria theory leads to non-trivial statistical equilibria for quasi-geostrophic flows on a rotating sphere, with or without bottom topography. We first consider flows without bottom topography and with an infinite Rossby deformation radius, with and without conservation of angular momentum. When the conservation of angular momentum is taken into account, we report a case of second order phase transition associated with spontaneous symmetry breaking. In a second step, we treat the general case of a flow with an arbitrary bottom topography and a finite Rossby deformation radius. Previous studies were restricted to flows in a planar domain with fixed or periodic boundary conditions with a beta-effect. In these different cases, we are able to classify the statistical equilibria for the large-scale flow through their sole macroscopic features. We build the phase diagrams of the system and discuss the relations of the various statistical ensembles.

cond-mat.stat-mech↗

arXiv2012

Phase transitions and marginal ensemble equivalence for freely evolving flows on a rotating sphere

The large-scale circulation of planetary atmospheres like that of the Earth is traditionally thought of in a dynamical framework. Here, we apply the statistical mechanics theory of turbulent flows to a simplified model of the global atmosphere, the quasi-geostrophic model, leading to non-trivial equilibria. Depending on a few global parameters, the structure of the flow may be either a solid-body rotation (zonal flow) or a dipole. A second order phase transition occurs between these two phases, with associated spontaneous symmetry-breaking in the dipole phase. This model allows us to go beyond the general theory of marginal ensemble equivalence through the notion of Goldstone modes.

physics.flu-dyn↗

arXiv2011

Entropy production and multiple equilibria: the case of the ice-albedo feedback

Nonlinear feedbacks in the Earth System provide mechanisms that can prove very useful in understanding complex dynamics with relatively simple concepts. For example, the temperature and the ice cover of the planet are linked in a positive feedback which gives birth to multiple equilibria for some values of the solar constant: fully ice-covered Earth, ice-free Earth and an intermediate unstable solution. In this study, we show an analogy between a classical dynamical system approach to this problem and a Maximum Entropy Production (MEP) principle view, and we suggest a glimpse on how to reconcile MEP with the time evolution of a variable. It enables us in particular to resolve the question of the stability of the entropy production maxima. We also compare the surface heat flux obtained with MEP and with the bulk-aerodynamic formula.

physics.ao-ph↗

arXiv2011

Present and Last Glacial Maximum climates as states of maximum entropy production

The Earth, like other planets with a relatively thick atmosphere, is not locally in radiative equilibrium and the transport of energy by the geophysical fluids (atmosphere and ocean) plays a fundamental role in determining its climate. Using simple energy-balance models, it was suggested a few decades ago that the meridional energy fluxes might follow a thermodynamic Maximum Entropy Production (MEP) principle. In the present study, we assess the MEP hypothesis in the framework of a minimal climate model based solely on a robust radiative scheme and the MEP principle, with no extra assumptions. Specifically, we show that by choosing an adequate radiative exchange formulation, the Net Exchange Formulation, a rigorous derivation of all the physical parameters can be performed. The MEP principle is also extended to surface energy fluxes, in addition to meridional energy fluxes. The climate model presented here is extremely fast, needs very little empirical data and does not rely on ad hoc parameterizations. We investigate its range of validity by comparing its performances for pre-industrial climate and Last Glacial Maximum climate with corresponding simulations with the IPSL coupled atmosphere-ocean General Circulation Model IPSL_CM4, finding reasonable agreement. Beyond the practical interest of this result for climate modelling, it supports the idea that, to a certain extent, climate can be characterized with macroscale features with no need to compute the underlying microscale dynamics.

physics.ao-ph↗