Table Of ContentTurbulence and
Nonlinear Dynamics
in MHD Flows
Proceedings of the Workshop on
Turbulence and Nonlinear Dynamics
in MHD Flows
Cargèse, France, July 4-8, 1988
Edited by
M. Meneguzzi
CERFACS
Toulouse, France
A. Pouquet
HAOINCAR
Boulder, U.S.A.
PL. Sulem
University of Nice
France
1989
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Library of Congress Cataloging-in-Publication Data
Workshop on Turbulence and Nonlinear Dynamics in MHD Flows (1988 :
Cargèse, France)
Turbulence and nonlinear dynamics in MHD flows : proceedings of
the Workshop on Turbulence and Nonlinear Dynamics in MHD Flows,
Cargèse, France, July 4-8, 1988 / edited by M. Meneguzzi, A.
Pouquet, P.L. Sulem.
p. cm. — (North-Holland delta series)
Includes indexes.
ISBN 0-444-87396-1
1. Magnetohydrodynamics—Congresses. 2. Solar wind—Congresses.
3. Turbulence—Congresses. 4. Dynamo theory (Cosmic physics)-
-Congresses. I. Meneguzzi, M. II. Pouquet, A. III. Sulem, P. L.
IV. Title. V. Series.
•C718.5.M36W67 1988
538' .6—dc20 89-1445
CIP
Printed in The Netherlands
ν
FOREWORD
Magnetic phenomena are widespread in the Universe, in the Sun, in stars
and in the interstellar medium where it is thought to be instrumental in
the process of star formation, and in galaxies. Closer to us, the Solar Wind
has been and continues to be the subject of several in situ observations,
thanks to the developmemt of satellite data. One such planned experiment
by the European Space Agency is CLUSTER which, with its tetrahedric
configuration, will yield data at points close enough that information on
the vorticity distribution, and hence the kinetic helicity, as well as the
electric current and magnetic helicity will hopefully be obtained.
In the meantime, data is being collected in laboratories in and around
Reversed Field Pinch fusion experiments. In such experiments, a fluid type
description in the framework of the MHD approximation can explain many
observations.
Also, the current progress in the design of fast computers with vector-
isation and parallelisation is procuring large amounts of data by modeling
the flows or by direct numerical simulations, sometimes in parameter domains
not accessible in the laboratory, as is the case with large magnetic Reynold
numbers. The flows are turbulent, and hence not fully understood because
of the strong coupling between an extended range of modes.
We felt that the time was ripe for bringing together scientists in the afore
mentioned related fields and thus provide a platform for discussions and
diffusion of knowledge from one branch of MHD to another. The response
to the Workshop was good, and we feel that cross fertilisation of ideas actually
took place. The main topics that were treated are magnetic fields in astro-
physical flows, slow and fast dynamos, MHD turbulence in space and labora
tory plasmas, exact solutions to MHD, topology and chaos in MHD, role of
helicity and velocity-magnetic field correlations and turbulent reconnection.
A short session was also devoted to non-magnetic flows.
Unfortunately, not enough time was available to discuss all aspects of these
fields, and some topics were sometimes deliberately left out. It was, for
example, a choice of the organizers to restrict Dynamo and Reconnection
sessions to a discussion of the effect of turbulence and strong nonlinearity.
vi Foreword
Many people have helped in organizing this Workshop, and we thank them
all. We wish particularly to mention the members of the Scientific Committee:
D. Biskamp (MPI, Garching, West Germany), H.K. Moffat (DAMTP,
Cambridge, GB), D. Montgomery (Dartmouth and LANL, USA) and A. Roux
(CNET-CRPE, France), as well as M.F. Hanseler, Administrator of the Centre
d'Etudes Scientifiques de Cargèse where the meeting took place. The help
of G. Thiery was greatly appreciated. The Workshop benefitted from the
framework provided by the Year on Nonlinear Phenomena. Additional
financial support was provided by the DRET, the CNRS, and the Observa
toire de Nice.
We hope this Workshop will stimulate many future collaborations of a
multidisciplinary nature.
M. Meneguzzi, A. Pouquet, P.L. Sulem
December 1988
Turbulence and Nonlinear Dynamics i MnHD Flows
M. Meneguzzi, A. Pouquet and P.L. Sulem (Editors)
© Elsevier Science Publisher sB.V. (North-Holland), 1989 1
STELLAR ROTATION, DYNAMO, ELECTROMAGNETIC BRAKING,
AGE AND LITHIUM BURNING
Evry SCHATZMAN
Miller Research Professor Berkeley.
On leave of absence from Nice Observatory.
Now at the Department of Stellar and Galactic Astrophysics (DASGAL),
Observatoire de Meudon, 92195 Meudon Principal Cedex, France.
ABSTRACT.
After an introduction describing the problem and the observational tests of
the theory, I present a consistant model of the dynamo mechanism in rotating
star. This provides for the electromagnetic braking a law
Ω ~ (1+t/tchar) in good agreement with the observations. This rests on
the hypothesis that the main contribution to the EM braking is due to the
magnetic field present in bipolar magnetic spots at the surface of the
stellar disk. The premain sequence EM braking provides an initial angular
velocity on arrival on the main sequence which is slightly smaller than the
angular velocity when the dynamo turns on. Starting the dynamo takes place
when the level at which the (αΩ) dynamo number becomes larger than one drops
below the ionization level of hydrogen. Before that time, the surface dynamo
mechanism would take place in a region of low ionization, where the magnetic
Reynods number is so small that dissipation overtakes the building of the
magnetic field.
Turbulent mixing with a turbulent diffusion coefficient proportional to
provides a consistant picture of the time and mass dépendance of the surface
abundance of Lithium. When the level of Li-burning is sufficiently far from
the bottom of the convective zone an asymptotic value of lithium abundance is
reached. This can explain the anomalous Li abundance of pop.II stars.
1. DYNAMOS IN STARS.
We shall review, in the following, some data on stars, which can be
explained in a consistent manner by the presence of a dynamo mechanism. In
most of the cases, there is no direct proof of the presence of magnetic
2 Ε. Schatzman
fields, but the consistency of the modelling can be considered as an indirect
proof.
A basic principle should always be kept in mind: a physical process has a
universal validity. In order to explain the solar properties it is necessary
to consider a number of physical processes. However, if theses processes are
at work in the Sun, they should be present also in similar stars. It is easy
to get rid of ad hoc hypothesis by applying them to other stars. Altogether,
what is suggested by the Sun can be tested on other stars, and, conversly,
understanding the stars can lead to a better understanding of the Sun.
We shall begin with a brief overview of stellar properties, and present the
facts which suggest that EM processes are at work. We shall mention how the
EM processes do act, before going into a description of the models.
2. BASIC DATA.
In this presentation of the astrophysical data, we shall be very brief,
refering the reader, for further information, to the bibliography. What we
are concerned with here, are stellar properties which are related to stellar
evolution; stellar rotation as a function of age, and surface abundances of
nuclearly processed elements. During the recent years, the improvement in the
signal to noise ratio has brought a large amount of information, which needs
to be interpreted in a consistent way.
2.1 Stellar evolution
There are strong arguments concerning star formation through contraction of
molecular clouds (Montmerle and Bertout 1986), but we are not concerned here
by the first steps of this evolution. However, it should be noticed that the
angular velocity of interstellar matter, in the solar neighborhood, provides
a very large angular momentum. For a molecular cloud with a density of 10^
protons cm"3, rotationnal breaking is reached for a density of 9.1018 H cm"3;
for a solar mass star, this would correspond for an homogeneous sphere, to a
radius of about 50 solar radii. These numbers are not supposed to correspond
to any real description of the early stages of star formation, but are given
to show that if there were no loss of angular momentum before reaching the
main sequence (see below), stars like the Sun should be rotating with
equatorial velocities corresponding to the rotationnal braking condition.
Two classical parameters, derived immediately from observations, are used
for representing stellar properties: luminosity and the so called effective
temperature, defined by 4πί*2σ T ff4 = L, σ being the Stefan constant. In a
e
bi-logarithmic scale we obtain the Hertzsprung-Russell diagram (Bouvier
ai*. 1986, Gilliland, 1986, d'Antona and Mazzitelli, 1984). The main feature
Stellar rotation 3
is the so called main sequence which corresponds to stars in the early phase
of stellar evolution. If we call £Q the energy available per unit mass of
hydrogen, X the hydrogen concentration, q a fraction of the order of (1/10),
qMXe
0
this corresponds to stars younger than t^s= — j ~ —. Such stars have burnt
only a small fraction of their hydrogen and are still close to a
convent ionnal state calle Zero age main sequence (ZAMS), which is defined by
stellar models of chemically homogeneous stars in a steady state. For the few
binary stars (about 30) for which the mass, luminosity and effective
temperature are known, the ageement with stellar models is reasonably good.
As far as we are concerned here we shall ignore the problems of the exact
fitting of stellar models with the observations. A one solar mass is close to
ZAMS as long as its age is less the 1010 years or thereabout.
Contracting stars move in the Hertzsprung Russell diagram along almost
vertical paths. The time scale of the contraction for a solar mass star is of
the order of 4.106 (1^ )"3/2 years. At the beginning, the contraction proceeds
5υΝ
so quickly that the probability of picking up a contracting star of a
luminosity larger than 102
LSUN ^s vanishingly small. In regions of star
formation, stars with L<100 LSTJN are observed. They all belong to a special
class of stars, the so called Τ Tauri stars, from the name of its prototype,
Τ Tauri, which belongs to the constellation of Taurus. The youngest ones (the
most luminous one) show a large velocity of rotation (100 km s_1 and more),
wheras the oldest ones (less than 10^ years old) rotates more slowly (20 to
50 km s-1) but still faster than main sequence stars (less than 20 km s"1) . Τ
Tau stars show emission lines, associated with a stellar wind and, in the
infrared, a strong emission which is due to the radiation of a disk of gas
and dust heated by the central star; a large fraction of these Τ Tau stars
seem to have recently lost their disk.
2.2. Stellar rotation and age
We shall summarize here the general properties concerning stellar
rotation and age (Benz et al 1984. Bouvier et al 1986, Duncan 1981, Mikkaden
1985, Noyés, 1984a,b, Simon et al.. 1985, Soderblom 1982, 1983, Stauffer e£
al. 1984, Vaughan et al.. 1980, Vogel et al. 1981).
Stars are born in clusters. In a cluster, one finds stars of different
masses. Stars with the largest masses evolve faster than stars with the
smallest masses. When a star has burnt about 10% of its hydrogen fuel, it
leaves the main sequence and evolves towards the giant branch. For stars with
a mass larger than the solar mass, this takes place after a characteristic
time, called nuclear time scale, which is of the order of 8.109 (Μ/MSTJN) "3
4 Ε. Schatzman
years. The Hertzsprung-Russell diagram of clusters show, on the high mass
side of the main sequence, a sudden turning point corresponding to the first
evolutionnary track towards the giant branch. From the luminosity and
effective temperature of the turning point, it is possible to obtain a
measure (which is model dependant) of the age of a cluster. The data are
available for several clusters for which a detailed analysis of the velocity
of rotation (from the line broadening by Doppler effect) and of the surface
abundance of lithium (a nuclearly processed element) has been carried.
The main results are the following:
(1) Along the main sequence, stars with a mass larger than about 1.4 MSUN
have a fast rotation (100 km s"1 or more), wheras stars with a smaller mass
rotate slowly (20 km s-1 or less) . Found by 0. Struve in 1950, this quasi-
discontinuity in the velocity of rotation has been confirmed by several
authors. Electromagnetic braking has been proposed by Schatzman (1959) in
order to explain the difference between fast and slow rotators. It is based
on the existence of a dynamo mechanism associated with the presence of a
surface convective zone for stars with M<1.4 MsUN* Over a very small interval
of mass, the surface convective zone recedes and disappears entirely.
(2) When comparing stars of different ages (Hyades, 8.10** years, UMa, 3.10**
years, Pleiades 70.106 years, α Per, 50.106 years), it appears clearly that
young stars, with M<1.4 MSUN rotate faster than old stars. A loss of angular
momentum takes place. The presence of spots at the surface of stars produces
a weak periodic effect associated with rotation, so that it is possible from
Doppler effect as well as from variability to obtain velocities of rotation.
Dating stellar ages give the possibility of plotting the period of rotation
as a function of age (Simon et al. 1985) . Without discussing here the
f
methods which have been used to estimate the ages of field stars (which do
not belong to clusters), it appears clearly that the stars with a mass of the
order of 1 MSUN experience rotationnal braking. One of the first problems
which we shall meet concerns the time dépendance of the velocity of rotation.
2.3 Internal rotation of the Sun
Helio-sismology provides a way of determining the internal velocity of
rotation of the Sun, Ω(Γ,Θ) from the splitting of the modes of oscillation.
For the time, being, the only reliable result concerns Q(r) . It appears
(Brown 1985, Brown et al. 1987, Duvall and Harvey 1984, Duvall e£ aj^ 1987,
Libbrecht 1986, Nemry 1986, Rhodes ei aj^ 1987) that the internal angular
velocity of the Sun, from the bottom of the convective zone (0.7 RSUN) to a
radius of about 0.15 RsUN* is almost constant, which means that the loss of
angular momentum of the Sun (with a surface equatorial velocity of 2 km s"1)
Stellar rotation 5
has been associated with a very efficient transport of angular momentum by
viscous shear flow. The quasi solid body rotation implies a coefficient of
viscosity of the order of (L^/tsuN) ~ 2.5.104 cm^ s"1. We shall see later
which constrain this implies.
2.4 Nuclearly processed elements.
Nuclearly processed elements can provide a very important test on the
internal motion of a star. We shall consider in the following surface
abundance of Lithium in stellar clusters and in the Sun and surface abundance
of the isotrope ^He in the Sun.
It is now know for many years, since the early work of Herbig (1965) that
the abundance of Lithium in main sequence stars decreases with time, and for
a cluster of a given age, decreases along the main sequence with the mass.
These two systematic effects (with some exceptions which we shall consider
later), suggest an explanation which implies a time dépendance (lithium
destruction with age) and a structural dépendance (mass dépendance of the
lithium abundance) . This last one is clearly connected to the distance of the
Lithium burning level to the bottom of the convective zone (Senatzman et al.
f
1981, Baglin et al., 1984) . We shall come back later to the quantitative
explanation of these observations. What we shall notice here, is that in OPer
and the Pleiades (two young clusters) there is an anomaly, which consists in
the presence of a higher abundance of Lithium in several fast rotators. This
anomaly raises a problem for the conservation of angular momentum and for
lithium destruction.
The isotope %e is produced inside the stars through the chain of pp
reactions. If there is a diffusion process which carries chemical elements
inside the Sun, calculations show that its efficiency is limited by the
surface abundance of %e. From the Schatzman-Maeder solutions (1981), it
appears (Schatzman 1984) that a turbulent diffusion mixing coefficient
DT ~ 30\faic ~ 300 cm2 S"l over about one half of the solar radius is an upper
limit, above which the predicted abundance of %e at the surface of the Sun
is incompatible with the observations.
3 MODEL OF TURBULENT DIFFUSION MIXING AND OF EM BRAKING
3.1 TDM and Li burning
We are concerned here by turbulent diffusion in the radiatively stable
region which extends from the bottom of the convective zone to the region of
Lithium burning. We can define the level of lithium burning by the condition
that the time scale of lithium burning is comparable to the age of the star.
In fact, if we consider the diffusion equation
6 Ε. Schatzman
where Κ (Τ) is the rate of destruction of Lithium, we have to satisfy to two
boundary conditions, one at the bottom of the region which we consider.
c—>0 as z—>-OO
the other at the boundary between the convective zone and the radiative zone.
where Η is a measure of the thickness of the convective zone. The problem is
to find the generation mechanism of the turbulent diffusion process and to
get an estimate of it.
J.P. Zahn (1988) has suggested the following picture
(a) stellar rotation generate a meridional circulation (Eddington Sweet
circulation);
(b) advection of angular momemtum by the circulation generates a differential
rotations;
(c) in the stellar stratified medium, the differential rotation generates a
2-D turbulence along equipotentials;
(d) Inertial forces exceed the coriolis force at small scales, and then the
2-D turbulence can feed a 3-D turbulence. This is possible as long as there
is no appreciable gradient of the chemical composition.
In laboratory experiments on rotating fluids, the transition from 3-D to 2-
D turbulence takes place for a Rossby number (u/2£2l)=0.2, where u and 1 are
respectively the r.m.s. velocity of the 3-D turbulence and its scale
(Hopfinger et al. 1982).
Zahn gives the energy available for production of the 3-D turbulence
(including the effects due to the presence of a surface where the direction
of the flow is reversed),
the efficiency of the conversion of thermal energy into mechanical motion.
2
Ω R -1
Η =
g
with
FRAD
the energy input in the turbulence.
