Subject Panel: Mathematical Methods and ApplicationsSuggested title of dissertation: Reversals of zonal jets in 2D turbulenceDissertation supervisor: Dr. Vassilios Dallas

Description of proposal:

Freidlin–Wentzell theory and Eyring–Kramers lawRare transitions of atmosphere jets: numerics

Rare transitions of atmosphere jets: theory

Freidlin-Wentzell theoryTransition rates for non-gradient dynamicsSketch of the proof

Non-Equilibrium Phase Transition for the 2D Navier–StokesEq.The time series and PDF of the Order Parameter

Order parameter : z1 =R

dxdy exp(iy)w (x ,y).

For unidirectional flows |z1| ' 0, for dipoles |z1| ' 0.6�0.7

F. Bouchet and E. Simonnet, PRL, 2009.F. Bouchet CNRS–ENSL Large deviation theory and GFD.

Figure 1: Large scale vortices(top) and zonal jets (bottom)in 2D turbulence

The formation of large-scale coherent structures is widely ob-served in atmospheric and oceanic flows and ascribed to the nearlybi-dimensional nature of these flows. It is well known that the en-ergy of two-dimensional (2D) turbulent flows, is transferred fromthe forcing scale to larger scales due to the conservation of bothenergy and enstrophy by the inviscid dynamics [1]. In a squaredomain, the energy accumulates at the largest possible scale, thusgenerating coherent structures in the form of large-scale vortices(see Fig. 1 top). It has been also demonstrated that zonal jets (seeFig. 1 bottom) are spontaneously generated by changing the aspectratio of the domain [2]. Moreover, recently it was observed in lab-oratory experiments that the large-scale circulation generated byforcing a nearly 2D flow at small scales can display random flow re-versals in a confined geometry [3]. It is an open question, whetherzonal flows in rectangular domains confined in one direction canspontaneously generate zonal flows, whose mean flow directionsreverses randomly in time. The dynamics of zonal jets is an out-standing challenge to understand further planetary atmospheres,which has important implications to the Earth’s climate.

This project will involve numerical simulations of the incompressible 2D Navier-Stokes equation

∂tω + u ·∇ω = ν∇2ω + f

where u is the velocity field, ω = ez · ∇ × u is the vorticity, ν is the kinematic viscosity of thefluid and f is an external mechanical force driving the flow. The computations will be performedin rectangular domains with periodic boundary condition in x-direction and free-slip in the y-direction using a pseudospectral code that is designed to run on parallel computer clusters [4].

Possible avenues of investigation:The main focus of this project is to systematically vary the aspect ratio of the domain to study

a) the transition from large scale vortices to zonal flows and b) the reversals of zonal jets. Theresults will provide new insights into the non-linear, multi-scale and statistical dynamics of theseflows. This project will provide a unique experience on geophysical fluid dynamics, turbulencetheory, numerical methods and scientific computing. Important results will be prepared for publi-cation in a peer-reviewed journal.

Pre-requisite knowledge (listed as essential, recommended, useful)Essential: some familiarity with numerical methods for PDEs and a programming language

will be helpful.Useful: The analysis is going to be based on tools from turbulence theory, bifurcation theory

and statistical mechanics.

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Useful reading1. Tabeling P. Two-dimensional turbulence: a physicist approach. Phys. Rep. 362, 1–62

(2002)2. Bouchet F. and Simonnet E. Random Changes of Flow Topology in Two-Dimensional and

Geophysical Turbulence. Phys. Rev. Lett. 102, 094504 (2009)3. Fauve S., Herault J., Michel G. and Petrelis F. Instabilities on a turbulent background. J.

Stat. Mech., 064001 (2017)4. Gomez D. O., Mininni P. D. and Dmitrum P. Parallel simulations in turbulent MHD. Phys.

Scripta T116, 123–127 (2005)

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