* Christophe Gissinger (ENS Paris), Francois Petrelis (ENS Paris)*

The dynamics of planets, stars and galaxies is deeply linked to
nonlinear physics. The considerable progress of this discipline in
the last decades has dramatically changed our understanding of many
natural phenomena. This is partly due to the ubiquity of fluid
flows in the atmospheres and interiors of planets and stars. It is
now clear that geophysical and astrophysical fluid dynamics (GAFD)
differs from traditional fluid mechanics in that it deals with
turbulent, sometimes electrically conducting, and most often
rapidly rotating and highly stratified flows. These properties lead
to a multitude of nonlinear behaviors, and constitute the core of
this course.

**Chapter 1** :* Rotating flows. Coriolis force – geostrophic balance –
Proudman-Taylor theorem.*

In this chapter, we will study how the rapid rotation of planets
and stars profoundly modifies the nonlinear dynamics of flows,
generates new types of waves and produces the quasi-geostrophic force
balance at the basis of most theories of rapidly rotating flows.

** Chapter 2 **: *Electrically conducting fluids. Magnetohydrodynamics – dynamo instability -alpha and omega effects – chaotic field reversals*

The ubiquity of magnetic fields in the universe makes the study of
electrically conducting fluids (called magnetohydrodynamics)
essential. According to the dynamo theory, stellar and planetary
magnetic fields are in fact self-sustained by an instability of the
turbulent flow of plasma (in stars) or liquid iron (in planetary
interiors). This theory explains the extraordinary variability of
natural magnetic fields as well as the existence of hydromagnetic
waves controlling sunspot dynamics. The recent demonstration of this
effect in the laboratory, with the experimental observation of chaotic
and unpredictable reversals of the field polarity, similar to those
observed on Earth, is one of the greatest successes of nonlinear
physics in recent years. This will allow us to study dynamical models
of chaos relevant to astrophysical magnetic fields.

**Chapter 3 **:* Thermal convection. Rayleigh-Benard instability – pattern formation -Secondary instabilities.*

Beyond rotation and electrical conductivity, stratification, i.e. the
existence of flows with variable density, is also a crucial
aspect. The best example is thermal convection, where a thermal
gradient generates fluid motions which transport heat in planetary
atmospheres or in the interior of planets and stars. In this chapter,
we will focus on the occurrence of this instability and the
corresponding fluid dynamics.

**Chapter 4** : *Stably-stratified flows. Internal gravity waves, double-diffusion, baroclinicity*

In some regions of the ocean or in the radiative zone of stars, the
temperature and pressure distributions lead to stable density
gradients, so that no thermal convection is generated. But many
equally important phenomena are generated in these stably stratified
flows. Thus, the baroclinic instability, which describes the
destabilization of a flow in the presence of a density gradient along
surfaces of constant pressure, helps explain the existence of cyclones
and anticyclones in stratified regions subject to strong rotations,
the dynamics of the Jupiter bands, the shape of interstellar clouds,
etc. We will also study the dynamics of internal gravity waves, the
double-diffusion phenomenon in the ocean and the shear instabilities
occurring in stellar interiors.

** Chapter 5 ** : * Astrophysical turbulence. Angular momentum transport – accretion disks – rotation of radiative stars – Tayler-Spruit dynamo. *

Recent advances in observations and the implementation of new types of
telescopes are now challenging several theories of nonlinear
astrophysics. The emergence of new techniques of asteroseismology have
thus revealed the existence of an unexplained state of turbulence in
the stellar radiative zones. Similarly, recent observations from some
telescopes now question the very origin of turbulence in the accretion
disks around black holes. This last chapter will therefore combine
most of the notions seen in the previous chapters to focus on several
current theories of astrophysical fluid mechanics. We will study for
example the Tayler-Spruit theory, which allows to understand how
rotation, stratification and magnetic field simply combine to generate
a subcritical transition to turbulence and thus explain the rotation
profiles of stars. Another application is for example the transport of
angular momentum by a turbulent flow, which explains the accretion
mechanism around black holes.

** Prerequisites:**

Basic hydrodynamics (Eulerian or Lagrangian description, Navier-Stokes equation)

There are no prerequisites for electromagnetism.

**Bibliography:**

1. H.K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press

2. S. Chandraskhar, Hydrodynamic and Hydromagnetic Stability 1961, Clarendon Press, Oxford; reprinted by Dover Publications, Inc., 1981.

**Timing:**
The Course is offered in the first part of the M2 year.

It consists of 8 Lectures on Fridays from 1.45pm to 5.45pm at Sorbonne University, **room 14.24.305**

First lecture on **Friday 20th September**

**ECTS Credits:** 3

**Hours:** 32 hours.