# Research

My research lies at the interface between statistical physics, fluid mechanics and climate dynamics.

I am interested in predicting, in a probabilistic sense, systems which are unpredictable in a deterministic sense. This is, broadly speaking, the purpose of statistical mechanics. A specificity of my work is to develop this approach for complex systems such as turbulent flows, and in particular, geophysical flows like the atmosphere and the oceans.

My work combines theoretical tools, from classical fluid mechanics, nonlinear dynamics and statistical physics, and numerical simulations (in particular, *Direct Numerical Simulations* for turbulent flows and *General Circulation Models* for idealized atmospheres).

# 2D and geophysical turbulence

The behavior of fluid flow confined to a two-dimensional surface is of interest for at least two reasons.

At a fundamental level, such flows have a radically different behavior from their 3D counterpart, because they spontaneously organize into coherent structures, such as large scale vortices or jets. This is at variance with 3D turbulence, which is essentially a terribly efficient dissipation mechanism, sucking energy out of any background mean-flow. From a physical point of view, the crucial point is that vortex tubes can be stretched in a 3D flow, while this mechanism is forbidden in 2D. This property is reflected in the mathematical structure of the equations, which admit additional conservation laws (in the absence of forcing and dissipation) in 2D. These invariants account for the inverse cascade and for the existence of quasi-stationary states. Many classical methods of theoretical physics, which fail in 3D turbulence, provide insightful approaches to characterize the statistics of 2D turbulent fields. For instance, the system has non-trivial statistical equilibrium states, and perturbative methods allow to close the hierarchy of moments in a self-consistent way.

In addition to the theoretical motivations, these flows are worth studying because they provide, under minor modifications, a surprisingly good model for flows encountered in nature, such as planetary atmospheres or oceans. Indeed, the combination of geometrical confinement, rotation and density stratification makes geophysical flows akin to 2D flows. Although imperfect, this analogy has spurred major developments in geophysical fluid dynamics, such as geostrophic turbulence, and continues doing so.

A 2018 article in Quanta magazine, reprinted on Wired.com, puts this line of research in the spotlight.

# Wave/vortex interactions in rotating-stratified flows

Both the atmosphere and the ocean are fully turbulent flows spanning a very broad range of length scales. The ocean for instance supports a thermohaline circulation at the planetary scale, on the order of ten thousand kilometers, western boundary currents such as the Gulf Stream and the Kuroshio, stretching over thousands of kilometers and surrounded by mesoscale rings of about 100 km diameter, as well as overturning and internal mixing occurring at scales as small as the centimeter. A fundamental question in geophysical fluid dynamics is to understand the energy pathway across this range of scales: how are large-scale circulations maintained? How do coherent structures such as mesoscale rings form? How are small-scale mixing and dissipation fed and how should we represent them in atmosphere and ocean models?

A major difficulty is that different processes dominate at different scales. This is at variance with more idealized turbulent flows, such as 3D homogeneous isotropic turbulence, for which the Kolmogorov theory predicts that large vortices break up into smaller vortices, and so on and so forth, down to the dissipation scale. In geophysical turbulence, the large scales of the flow are dominated by a balance between the Coriolis and the buoyancy forces and the pressure gradient. The behavior of these so-called 'geostrophic' vortices is very similar to that of a fluid on a two-dimensional surface: they tend to merge to form large-scale coherent structures. On the other hand, at very small scales, neither the Coriolis force nor the buoyancy force is felt any longer, and we expect the Kolmogorov theory of 3D turbulence to apply. In between, there exists a grey zone where the balanced motion coexists with much faster oscillatory motion referred to as inertia-gravity waves. The non-linearity of the equations leads not only to coupling between vortices of different size, as in geostrophic turbulence, but also between waves of different wavelength, and to interactions between vortices and waves.

My work on this topic aims at disentangling the role of vortices and waves in the energy transfer across the scales, using theoretical arguments and high-resolution numerical simulations. An important goal is to understand if, under certain circumstances, the range of scales can be divided into well-defined sub-ranges with universal energy spectrum and energy fluxes, like in the Kolmogorov theory.

See a slightly longer version of this text in the 2014 NCAR-ASP Annual Report

# Large deviations and rare event algorithms

Over the past decades, outstanding progress has been achieved in non-equilibrium statistical physics thanks to the theory of large deviations, which provides a natural generalization of the core concepts of equilibrium statistical mechanics: like entropy or free energy, large deviation rate functions encode all the relevant statistical information about an observable, such as its most probable values but also the probability of small and large fluctuations as well as phase transitions.
The main idea is to describe asymptotic regimes where the probability of some random variable (a *macroscopic observable*) takes an exponential form *exp(-I(x)/ε)*, where *ε* is a small parameter.
This framework is very general and can describe small-noise, large system or large time limits.

These new theoretical tools open up new prospects for understanding the statistics of turbulent flows.
One particular goal is to describe the effective dynamics of the large scales of geophysical flows, including for instance typical fluctations of a zonal jet or rare transitions between different flow configurations which could be connected to abrupt climate change.
For such complex systems, computing the large deviation rate function *I(x)* (or related quantities) is difficult.
Hence, we are also working on numerical algorithms to sample efficiently events which are so improbable that they are completely out-of-reach with standard Monte-Carlo methods.
Several algorithms exist, but the main underlying idea is to simulate an ensemble of trajectories with a controlled statistical bias to retain preferentially the ensemble members which are the best candidates for generating the rare event.

# Zonal jets in the atmosphere

The general circulation of the atmosphere consists in an overturning circulation, made up of three cells (the Hadley cell in the tropics, which is the strongest of the three, the Ferrel cell in the mid-latitudes, and a polar cell), and zonal jets at the boundaries between overturning cells: the sub-tropical jet and the polar front jet, also referred to as the *Jet Stream*.
These features play a major part in setting the meridional temperature distribution on the planet in the long-term climate, transporting roughly as much energy as the ocean towards the poles.
In addition, the fluctuations of the Jet Stream govern mid-latitude weather.

I am interested in the dynamics of zonal jets, both in idealized settings and in the atmosphere of the Earth. For instance, based on first principles, can we derive an effective equation describing the variability of the Jet Stream? For given external parameters, is there only one solution, or would a different structure for the general circulation be allowed? If so, what are the fluid mechanics principles maintaining the alternative circulation?

An example of a general circulation differing from that of the Earth is *equatorial superrotation*: it consists in a westerly jet at the Equator, carrying more angular momentum than solid-body rotation at the surface.
Superrotation is believed to be very general: it is observed on planetary atmospheres such as Venus, Jupiter, Saturn or Titan, as well as tidally-locked exoplanets like HD189733b or HD209458b.
It may also have played a part in past climates on Earth, or might occur in a warm future climate.
Recently, we have studied the nature of the transition to superrotation: is it smooth or abrupt?
In the second case, once we reach the tipping point, superrotation subsists even if we reverse the perturbation which made the conventional circulation unstable (*hysteresis* phenomenon).
Besides, spontaneous transitions between the two states may be possible, even before the tipping point is reached, due to atmospheric macroturbulence, thereby providing a mechanism for noise-driven abrupt climate change.
This is similar to the difference between first-order and second-order phase transitions in statistical physics.