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1. INTRODUCTION
Magnetohydrodynamics: MHD. Magneto- having to do with electro-magnetic fields;
hydro- having to do with fluids; dynamics- dealing with forces and the laws of motion.
Magnetohydrodynamics, or MHD, is the mathematical model for the low-frequency
interaction between electrically conducting fluids and electromagnetic fields.
So you will need to know about fluid dynamics, you will need to know about electromagnetism, and you will need to know some plasma physics.
An isotropic medium is one that has no preferred direction in space; it looks that
same and behaves in the same way in all directions. Hydrodynamics, which generally
deals with isotropic materials (fluids), is a very difficult subject. MHD is even more
difficult because a magnetic field identifies a preferred direction in space. A magnetized
fluid is said to be an anisotropic medium. Further, since there will be interaction between
the fluid and the magnetic field, some form of Maxwell’s must be solved simultaneously
with the dynamical equations for the fluid. You will see that this can be a formidable
task.
In this course, MHD will be treated as a topic in theoretical physics. We will make
some fundamental assumptions regarding the properties of the material and the time and
spatial scales of interest, and the rest will follow self-consistently (and hopefully
logically). Little or no experimental motivation will be given; but will rather be taken as
self-evident.
Mathematics is the language of theoretical physics. Specific topics that are essential
to the theoretical analysis of the dynamics of a magnetized fluid are

Vector and tensor analysis

ODEs

PDEs

Calculus of variations

Method of matched asymptotic expnsions
Short reviews of some of these topics will be provided at the appropriate times during the
presentation.
The validity of MHD as a mathematical model for magnetized plasmas has been
discussed in great detail elsewhere, particularly in the book by Freidberg. It cannot be
said better, so it will not be attempted here. Instead, from the beginning we adopt the
point of view that MHD describes the dynamics of a continuum fluid that is capable of
conducting an electric current, that this fluid can be characterized by a few parameters
such as mass density, velocity, and pressure, and that the material properties of this fluid
are independent of the physical size of the sample. As stated in Vol. 8 of Landau and
Lifschitz, “physical quantities are averaged over volumes that are ‘physically
infinitesimal’, ignoring the variations that result from the molecular structure of matter.”
That is, the material looks exactly the same no matter how finely it is subdivided. This is
our fundamental assumption.
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This assumption implies an ordering of length scales. Some important distances are

a 0 , the atomic radius

 , the “mean free path” between atomic collisions

 , the “physically infinitesimal” distance

L , the smallest relevant macroscopic distance to be considered (e.g.,
L ~ 1 / kmax , the shortest wavelength of interest)
Our fundamental assumption implies the ordering  ~ a0    L . Whenever    , L
the model breaks down, and it must be modified and further justified. This is the realm
of Extended MHD, and is beyond the scope of this course.
We will also assume an ordering of times scales. In particular, we will attempt to
describe only low frequency motions, those for which V 2 / c2  1 , where V is a
characteristic fluid velocity and c is the speed of light. With V   L , where  is a
characteristic frequency, we have  2  c 2 / L , or   1 /   L / c   c . The
characteristic time intervals for MHD are very much longer than the time it takes a light
wave to transit the macroscopic system.
We further assume that the smallest subdivision of the medium is in thermodynamic
equilibrium, which is provided by inter-atomic collisions. It can be characterized by a
temperature T . So, we also require that   1 /  c , where  c is the frequency of interatomic collisions.
The usual definition of a fluid is a substance that resists an applied compressive
stress, but continually deforms, or flows, under an applied shear stress, regardless of the
magnitude of the applied stress.
This behavior is illustrated in Figure 1, which shows the response of a fluid element
to an applied force. Figure 1a shows a compressive stress. The fluid generates a
restoring force that opposes the applied force; it can support compressive stresses. Figure
1b) shows a shearing stress. In this case the fluid generates no restoring force, and the
distortion will grow continually. The fluid cannot support a shearing stress.
We will see that the restoring force generated in response to an applied force leads to
wave propagation. Thus a fluid is commonly said to allow the propagation of
compressional waves (e.g., sound waves), but not shear waves.
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Now consider the sheared deformation of a fluid element permeated by a magnetic
field B . If the fluid is electrically conducting, we will see that the magnetic field is
deformed along with the fluid. This situation is sketched in Figure 2. The resulting
bending of the field lines produces a restoring force that opposes the applied stress.
Thus, an electrically conducting fluid can support the propagation of shear waves. This
result was considered so counter-intuitive and novel that at first it was not believed to be
true. Subsequent astronomical observations and experiments showed the existence of
these waves, and this led to the awarding in 1970 of the Nobel Prize in physics to Hannes
Alfvén, whose name is attached to these waves. This remains the only Nobel awarded in
plasma physics.
At this point it may be useful to define what we mean by a magnetic field line. The
magnetic induction, or magnetic field, is a vector function, denoted as B(x,t) , that
assigns a magnitude and direction to all points in space. (We will soon be more specific
about what we mean by the term vector.) Physically, this field is produced by electric
currents flowing somewhere in the universe. As with all vector fields, it is possible to
define a set of three-dimensional curves that are everywhere tangent to the vector B .
Sometimes these are called streamlines; here we call them field lines. The defining
equations of these curves are
dx dy dz dl



,
Bx By Bz B
(1.1)
where (x, y, z) are Cartesian coordinates, B  Bx2  By2  Bz2 is the magnitude of the
magnetic field (the magnetic field strength), and is the distance along the field line. The
trajectory of the field line x(l) , y(l) , and z(l) passing through a point (x0 , y0 , z0 ) can be
found by integrating the equations
dx Bx

dl
B
,
(1.2)
dy By

,
dl
B
(1.3)
and
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dz Bz

,
dl B
(1.4)
beginning from the point (x0 , y0 , z0 ) .
We take it as an experimental and observational fact that magnetic field lines either
close upon themselves, or fill space ergodically. (An ergodic trajectory is one that passes
arbitrarily close to all points in space without closing on itself.) In either case, the field
line has neither beginning nor end. If we consider an arbitrarily shaped surface S
enclosing a volume V , this means that as many field lines must leave the enclosed
volume as enter it. This is expressed mathematically as
—
 B  dS  0
(1.5)
,
S
or, using Gauss’ famous theorem,
   BdV  0
.
(1.6)
V
Since this must hold for every volume V , we must have
B  0 ,
(1.7)
which expresses the “endless” nature of field lines. This is one of the fundamental
equations of physics.
We will now proceed to derive
1. The equations for the dynamics of the fluid
2. The equations for the dynamics of the electromagnetic field
3. Some properties of the combined equations
These will require a familiarity with the language of scalars, vectors, tensors, and dyads.
This is reviewed briefly in the next section.
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