Christophe Gissinger

Associate professor at Ecole Normale Superieure (ENS Paris)

Associate professor at Ecole Normale Superieure (ENS Paris)

DYNAMO THEORY

Dynamo is the mechanism by which magnetic fields of planets and stars are generated and amplified by the turbulent motions of the underlying electrically conducting fluid. We combine experiments, theory and numerical simulations to explore the exact mechanisms of magnetic field generation and its associated dynamics. Numerical modelling of the Earth core is a typical example of my research activities: for instance, we have studied how chaotic reversals of the Earth magnetic field can be generated in geodynamo simulations when the equatorial symmetry of the flow is broken by a heterogeneous heating at the core-mantle boundary. On the other side, experimental approach is very useful: I participated to the VKS experiment, the first experiment which reproduced dynamo field generation in a fully turbulent flow. In this experiment, we were able to reproduce field polarity reversals, which finally lead us to propose new models for the dynamics of astrophysical dynamos.

This picture shows the dipole
field generated at the core-mantle boundary in one of our
geodynamo simulations. In presence of a slightly
equatorially asymmetric heat flux, the dipole field
reverses its polarity chaotically. It can
also be shown that fluctuations of the electrical
conductivity (due to variations of temperature in the core
for instance) act as a new source for such astrophysical
magnetic fields.

*Gissinger et al, PRL 108,154502 (2012)*

*Petrelis et al, PRL (2016)*

Here is a schematic view of the
VKS (Von Karman Sodium) dynamo experiment. 50 liters of
liquid Sodium are put in motion by two counter-rotating
discs driven by 300kW motor power. When the discs rotate
sufficiently fast, a magnetic field with dipolar symmetry
is self generated, reaching approximately 50 Gauss. A
supercritical bifurcation is observed, with values in good
agreement with scalings predicted for turbulent fluid
dynamos.
*VKS Collaboration page*

When the discs in the VKS
experiment rotate at different speed, several dynamical
regimes are observed, including field reversals in which
the magnetic field randomly switches between two symmetric
solutions B and −B . Similarly to the Earth’s magnetic
field, the duration of the steady phases is widely
distributed, but is always much longer than the time
needed to switch polarity. The reversals result from a
strong interaction between the dipolar mode and a
quadrupolar structure.

*Gissinger, PRE 82,056302 (2010)*

To understand the surprising
results of the VKS experiment and to go beyond this
experiment, I performed various numerical simulations
aiming to reproduce laboratory dynamos. Among these
studies, we were able to emphasize the crucial role of
magnetic boundary conditions in the VKS dynamo, but also
the strong interaction between dipolar and quadrupolar
components of the magnetic field during field
reversals. The picture shows the structure of the magnetic
field obtained in one of these numerical models: by taking
into account the vortices created near the impellers, a
dipolar field similar to the VKS dynamo is generated. I
also work on other types of flows generating a dynamo
field, such as Taylor-Couette flow.

*Gissinger, EPL 87,39002 (2009)*

*Gissinger, Phys.Rev.Fluids 26, 044101 (2014)*

ELECTROMAGNETICALLY DRIVEN FLOWS

Plasmas and liquid metal can be put in motion by Lorentz forces, rather than driven by pressure gradients or mechanical impellers. This new situation can lead to very complex behaviors and fundamental questions. It is well known that a time varying magnetic field acting on an electrically conducting fluid causes motion within this fluid. Such electromagnetically induced flows are found in many applications, such as induction melting, magnetic levitation, or electromagnetic stirring. These types of induced flows also appear in nature, for instance in internal salty oceans of Jupiter’s moons. I try to study both theoretical and experimental aspects of these electromagnetically driven flows. In particular, we study the emergence of new MHD instabilities in these systems.

This represents a schematic view of our numerical
modeling of an Annular Electromagnetic Pump (EMP) similar to the ones
used in some secondary cooling system of fast breeder nuclear
reactor. For some parameters, a large scale instability is observed in
the fluid and leads to a dramatic drop of the flow rate developed by
the pump. This instability is driven by magnetic flux expulsion and
associated to magnetic field reconnection. Understanding and eventually
control of this behavior is a major industrial stake, since it would
allow the construction of large scale EMPs.

*Gissinger et al,
Phys.Fluids (2016)*

For sufficiently turbulent flows, the flow instability
occurring in electromagnetically driven flows yields a
surprising behavior : in a large region of the fluid
domain, the fluid moves backward compared to the
propagation of the driving electromagnetic wave,
eventually leading to a strongly chaotic and turbulent
states. The figure shows the axial velocity in the r-phi
plane. Note the strongly negative velocity close to the
inner cylinder. This behavior, which seems quite general
may explain some properties of large scale
electromagnetic pumps...

*Rodriguez-Imazio et al,
Phys. Fluids(2016)*

Again, since numerical simulations have strong limitations, we have also developed a laboratory experiment. A liquid metal (GaInSn) is confined between two concentric cylinders and subject to rotating magnetized discs, generating an electromagnetic forcing similar to the one used in electromagnetic pumps. Preliminary results indicate a driving of the fluid in good agreement with theoretical predictions. For some parameters, it is possible to observe unstable regimes, characterized by some intermittency of the flow.

CHAOS AND NON-LINEAR DYNAMICS

I also work on various non-linear problems, ranging from low-dimensional behavior in fluid dynamics to chaotic motions of mechanical systems. In general, one expects turbulent flows to show a very complex behavior, due to the infinite number of degrees of freedom. Yet, several turbulent systems, MHD or not, exhibit low-dimensional dynamics involving chaotic reversals between two symmetrical states. I work on deterministic or stochastic models aiming to understand such complex behaviors, using only a few modes in interaction.

For instance, the figure shows trajectories in phase
space (strange attractor) of a new deterministic 3-modes model
describing the chaotic inversions of the magnetic field of the
Earth. Compared to similar systems (like The Lorenz model for
instance), this new model produces chaotic reversals much
closer to the polarity inversions observed for Earth or VKS
laboratory dynamos. This is due to the particular type of
chaos ('crisis-intermittency') which is generated. Here,
reversals results from the interaction between a symmetrical
and an antisymmetrical magnetic mode, coupled through an
antisymmetric velocity field.

*Gissinger, EPJB
85,vol.4 (2012)*

*Gissinger et al, PRL
101, 144502 (2008)*

Here is a simple experimental study of the motion of
magnetized beads driven by a travelling wave magnetic
field. A bead moves on a flat surface under the effect
of a traveling magnetic wave created by 16 Neodymium
magnets disposed with a regular spacing along a circle
located on a rotating disc. For sufficiently large wave
speed, one observes a backward motion, in which the
sphere can move in the direction opposite to the driving
wave. We show that the transition to this new state is
strongly subcritical and can lead to chaotic motion of
the bead. These results can be understood in the framework
of a model based on the interplay among solid friction,
air resistance and magnetic torque.

*Gissinger, EPL
112,5 (2015)*

For some parameters, the counter-propagation of the
sphere submitted to a travelling magnetic field can be
one order of magnitude faster than the driving wave
speed. In the beginning of the movie, the magnetic
sphere stays above one magnet of the disc (red circles
on the rotating disc), thus achieving a forward
synchronous translation with no rotation. After a small
perturbation (here my finger), there is transition to a
backward translation associated to synchronous angular
rotation. Note that the backward motion, quite
counter-intuitive, is more than ten times faster than
the synchronous motion, due to the effect of wall friction !

*Gissinger, EPL
112,5 (2015)*

OTHER INTERESTS

I am also interested in other problems, ranging from astrophysical to laboratory scales. Below are some typical questions that picked my interest over the last years : is it possible to simply understand how the magnetic field of galaxies is generated ? What is the mechanism for the destabilization of accretions discs around black holes and stars? How efficient are shear layer instabilities for angular momentum transport ? I also have a strong interest for Taylor-Couette flow, in both spherical and cylindrical geometries...

The MagnetoRotational Instability (MRI) is a fluid
instability arising in MHD flows when the angular
velocity decreases outward and a magnetic field is
applied. It is currently the best candidate to explain
angular momentum transport in accretion disks around
stars and black holes. Indeed, the MRI explains how a
weak magnetic field can destabilize otherwise stable
Keplerian flows. Despite its simplicity, several
theoretical aspects of this instability are still poorly
known, like the mechanisms of saturation of the MRI or
the emergence of non-axisymmetric modes.
The figure shows the velocity structure obtained in a
numerical simulation aiming to observe the
magnetorotational instability in a cylindrical
Taylor-Couette geometry. I studied several aspects of
this problem during my postdoc at the Princeton
university, which currently works on the experimental
demonstration of this instability (visit the Princeton experiment webpage here).

*Gissinger et al, Phys.Fluids
24,074109 (2012)*

*Roach et al, Phys.Rev.Lett.
108,154502 (2012)*

Most of the galaxies have a coherent large scale
magnetic field. The flow of the interstellar medium presents all the
typical ingredients needed for dynamo action, i.e. differential
rotation and kinetic helicity. I worked on a simple model aiming to
explain the magnetic field of galaxies. This model is based on the
so-called alpha-omega dynamo, and involve the supernovae exploding
in the galaxy as a source of helical motion at small scale.
Numerical simulations of this simple model are able to reproduce
most of the characteristics of the galactic field, such as the
quadrupolar structure or the spiralling properties of the magnetic
field lines.

*Gissinger et al, M.N.R.A.S., 394 (2008)
*

This image reports numerical simulations of the flow of
an electrically conducting fluid in a spherical shell when a magnetic
field is applied. In this spherical Couette flow, either a Stewartson
layer or a Shercliff layer, accompanied by a radial jet, can be
generated depending on the rotation speeds and the magnetic-field
strength. These shear layers lead to various nonaxisymmetric
destabilizations of the flow through Kelvin-Helmoltz instability. The
two figures on the top show the velocity in the field, before (left)
and after (right) the instability. The bottom curve shows that strong
fluctuations and very efficient angular momentum transport are
associated with these destabilizations.

*Gissinger et al, Phys.Rev.E
84,026308 (2011)*