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Warwick Turbulence Symposium

Instabilities and Turbulence in MHD flows

Graduate summer school Abstracts
maths building

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Keith Moffatt
DAMTP, Cambridge
Introduction to Magnetohydrodynamics
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Faraday's Law of induction combined with Ohm's Law in a moving conducting fluid imply that magnetic field lines tend to be transported with the
fluid (like vortex lines in an ideal non-conducting fluid).  They also diffuse relative to the fluid.  Stretching of magnetic field lines can
lead to systematic field intensification despite the erosive effect of finite conductivity (dynamo action).  Left to itself however, a magnetic field will tend to relax to a magnetostatic state of minimum magnetic energy compatible with any imposed boundary conditions. In a perfectly conducting fluid, this relaxation process is also
constrained by the conserved topology of the magnetic field, with interesting consequences for the existence of 'knotted' equilibria.

These fundamental aspects of magnetohydrodynamics will be briefly reviewed in this introductory lecture.

Recommended reading:

Moffatt, H.K. (2000) Reflections on Magnetohydrodynamics. In: Perspectives in Fluid Dynamics (Eds Batchelor, Moffatt & Worster) CUP, 347-339.

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Andrew D. Gilbert
Exeter
Introduction to Dynamo Theory
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Abstract:

Dynamo theory concerns the generation of magnetic fields by fluid flows: when an electrically conducting fluid flows across a magnetic
field, a current may be driven. This current can amplify the original magnetic field, giving rise to a growing magnetic field, or dynamo
action. This process is responsible for the magnetic field of the Earth (the geodynamo), the Sun, many planets, stars and galaxies.
The dynamo effect has also been realised experimentally using the motion of liquid sodium, in Germany and Latvia,

We will review these processes of induction by moving electrical conductors, and survey the mathematical techniques used, including the
alpha-effect and methods based on dynamical systems. We will also give some discussion of nonlinear dynamos, and open issues in dynamo
theory.

Reading: Any of the books/articles below would be useful in giving an impression of dynamo theory and its applications. I don't recommend
reading them all!

The closest source would be, unsurprisingly:
--------------------------------------------

Gilbert, A.D. 2003
Dynamo theory. In: Handbook of Mathematical Fluid Dynamics,
volume 2 (ed. S. Friedlander and D. Serre), pages 355-441 (Elsevier).

Copyright does not allow me to put this review on the web, but I am
allowed to post a preprint version:

http://www.maths.ex.ac.uk/~adg/dynamo.ps.gz


The following books and reviews cover general dynamo theory:
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F. Krause & K.-H. Radler,
Mean-field magnetohydrodynamics and dynamo theory.
Pergamon Press (1980).

H.K. Moffatt, 
Magnetic field generation in electrically conducting fluids.
Cambridge University Press (1978).

E.N. Parker,
Cosmical magnetic fields.
Clarendon Press (1979).

P.H. Roberts, 
Fundamentals of dynamo theory.
In: Lectures on Solar and planetary dynamos
(ed. M.R.E. Proctor, A.D. Gilbert), pp. 1-58.
Cambridge University Press (1994).


Fast dynamos are reviewed in:
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B.J. Bayly,
Maps and dynamos.
In: Lectures on Solar and planetary dynamos
(ed. M.R.E. Proctor, A.D. Gilbert), pp. 305-329.
Cambridge University Press (1994).

S. Childress,
Fast dynamo theory.
In: Topological aspects of the dynamics of fluids and plasmas
(ed. H.K. Moffatt, G.M. Zaslavsky, P. Comte,  M. Tabor), pp. 111-147.
Kluwer Academic Publishers (1992).

S. Childress & A.D. Gilbert,
Stretch, twist, fold: the fast dynamo.
Lecture Notes in Physics: Monographs.
Springer--Verlag (1995).

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Anvar Shukurov
Newcastle University
Introduction to galactic dynamos
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Spiral galaxies are the largest objects in the Universe that have large-scale magnetic fields --- i.e., fields ordered at a scale comparable to the size of
the system. It is very plausible that these fields are the product of the mean-field dynamo action. The tenuous gas between stars also carries magnetic
fields at scales comparable to and smaller than the scale of interstellar turbulence. The fluctuation dynamo apparently contributes very significantly
to forming these magnetic fields.

Thus, spiral (and other) galaxies host a broad range of MHD phenomena including various forms of dynamo action. There are several features that make
galaxies, as MHD systems, different from stars, planets, accretion discs and other objects. Turbulent motions of the interstellar gas are driven
directly by energy injections from explosions of supernova stars, rather than by internal instabilities. Unlike most other natural dynamos, galaxies are
transparent to a broad range of electromagnetic waves, so that we know much about their internal kinematics, dynamics, and magnetic properties, and
theory can be compared with observations in great detail. Galactic discs are thin, which provides a natural small parameter to facilitate the development of
mean-field dynamo models. These features make galactic dynamo theory especially simple and attractive. This lecture will provide a review of observational properties and models of interstellar gas and its magnetic field, with emphasis on the origin and properties of galactic magnetic fields.

Recommended reading:

A. Shukurov, Introduction to galactic dynamos. In: Mathematical Aspects
of Natural Dynamos, EDP Sciences, 2006 (astro-ph/0411739).

L. M. Widrow, Origin of galactic and extragalactic magnetic fields, Rev. Mod.
Phys., Vol. 74, pp. 775-823, 2002.

A. Ruzmaikin, A. Shukurov & D. Sokoloff, Magnetic Fields of Galaxies (Kluwer,
Dordrecht, 1988).


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Arkady Tsinober
Imperial College London
Experimental observations of MHD turbulence with some emphasis on comparative aspects
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An overwiew of laboratory observations of/on turbulent MHD flows - starting ftom Hartmann and Lazarus (1937) - will be given. Some emphasis is made on comparative aspects, anistropy quasi-two-dimensional states and MHD as a means of studying general issues of fluid dynamics.


Based on reviews:

Tsinober, A. (1975a) MHD turbulence, Magnetohydrodynamics, 11, No.1, 5-17.
Tsinober, A. (1975b) The influence of the magnetic field on nonlinear hydrodynamic processes in liquid metals, pp. 314+104, The Doctor dissertation (West equiv. Habilitation),Riga, In Russian, available on internet 1469525355 tsinober-1973 150.pdf
Tsinober, A. (1990) MHD flow drag reduction, in D.M. Bushnell and J.N. Hefner, Viscous drag reduction in boundary layers,, Progr. Astronaut.Aeronaut., vol 123, pp. 327-249.
Moreau, R. Thess, A and Tsinober, A.. (2006) MHD Turbulence at Low Magnetic Reynolds Number: Current Status and Future Needs, in Magnetohydrodynamics: evolution of ideas and trends, Editors: S. Molokov, R. Moreau, H.K. Moffatt, Springer/Kluwer, in press.


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Alexander Schekochihin
DAMTP Canbridge
MHD turbulence in space and interstellar plasmas
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1. AA Schekochihin & SC Cowley,
Turbulence and magnetic fields in astrophysical plasmas,
in: Magnetohydrodynamics: Historical Evolution and Trends,
S. Molokov, R. Moreau, and H. K. Moffatt, Eds.
(Berlin: Springer, 2006), in press
available from http://www.damtp.cam.ac.uk/user/as629/mhdbook.pdf

2. AA Schekochihin, SC Cowley & W Dorland,
Interplanetary and interstellar plasma turbulence,
Plasma Phys. Control. Fusion, to be published (2006)
[invited talk for the 13th Int'l Congress on Plasma Physics, Kiev 2006]
available from http://www.damtp.cam.ac.uk/user/as629/kiev.pdf

3. AA Schekochihin & SC Cowley,
Turbulence, magnetic fields and plasma physics in clusters of galaxies,
Phys. Plasmas 13, 056501 (2006)
[invited talk for the 47th APS DPP Meeting, Denver 2005]
available from http://www.damtp.cam.ac.uk/user/as629/dpp05.pdf