While doing some research and reading some of Hannes Alfven’s papers I came across
a paper that Hannes quoted, written by Heikkilla. Alfven was particularly disappointed that
years after Heikkila’s paper nothing had changed and the rather magical idea of frozen
in magnetic fields and magnetic reconnection were still being used to explain certain phenomena.
In order to understand why this is an issue, I felt it important to cover Heikkila’s
paper which covers the main problem with magnetohydrodynamics.
When we examine plasma in the Earth’s atmosphere, the Sun or in outer space the current trend
in Astrophysics is to describe these phenomena in terms of hydromagnetic-thermodynamics or
MHD concepts but this is often to the exclusion of more fundamental kinetic theory. This comes
largely because of the success of the fluid theory in describing the shape of the magnetopause
and the description of large scale magnetospheric convection. These successes have blinded many
astrophysicists to the limitations of the fluid theory.
If we were to define an ultimate description of a gas or plasma would have to include the
motions of all the individual particles, the external forces that each particle feels and
then the force that each particle then cause on the particles surrounding them. Realistically
this amount of detail is unattainable. What is useful to consider is a representative
particle in the gas, with the interaction between particles being represented by average
or statistical force fields.
It is there necessary to consider some kind of averaging over particles in order to get
macroscopic quantities. The most satisfying method makes use of phase space distribution
functions, which specify the numbers of particles with given positions and momenta. This statistical
kinetic theory generally provides an accurate physical picture, when it can be worked out.
Unfortunately, the kinetic theory is mathematically very difficult, and can only be fully worked
out for a few simple and idealised cases. It is therefore necessary to seek only partial
answers. One important method is to evaluate the velocity moments of the Boltzmann equation.
The moment equation reduces to the so-called hydrodynamic or fluid equations, which can
also be derived from first principles on the basis of fluid concepts. This helps simplify
the whole process but it must still be remembered that the fluid theory is only an approximation
to the more complete kinetic theory.
The kinetic description is more fundamental and only by means of it can more approximate
fluid theory be justified in detail. Approximations have to be made and can be justified in physical
theories but the use of an approximation for one purpose by no means implies that the neglected
process in the approximation can be neglected for all purposes. Any approximation must always
provide a source of suspicion when the relevant parameters are unknown. Regrettably, the limitations
of the fluid theory are too often forgotten.
So what are the limitations of the Fluid Theory?
There is a fundamental difference between a gas, fluid and a plasma. The properties
of an ideal gas are largely determined by close collisions, but those of plasma, results
from the long range Coulomb collisions. In a plasma each particle moves in an average
randomly changing field determined by many neighbouring particles. This difference is
so important that Stix made the following statement - “perhaps the most remarkable
thing about the two-fluid description of a plasma is its validity, for such a description
ignores the very nature of a plasma”.
So why is it then that the fluid theory has been so successful if it seems to ignore the
fundamental nature of plasmas. The fluid equations express the conservation of particles, of
momentum, and of energy, and these conservation relations are of fundamental importance. They
are sufficient for many purposes, such as for the specification of the shape of the
magnetopause.
The first problem that arises is in the choice of the equation of state. Here the difficulty
arises from the fact that groups of particles in a plasma do not generally move together,
and it is therefore not possible to specify a volume of plasma that is to be followed
in its motion. For the same reason, there is a serious difficulty in the choice of a
reference frame in which to describe the electrodynamics. Here we turn to the ‘frozen field’ concept.
Here a magnetic field line is identified by the plasma located on it. In such a description
the magnetic field lines are said to convect with the electric drift velocity V(e) = ExB/B^2.
This hydrodynamic description is a beautiful example of an approximation that is made for
a specific purpose, as was originally pointed out by Parker. In it, Maxwell’s and Newton’s
equations are first examined, and all the ‘nonessential field quantities’ are removed.
‘Only the field of principle interest, the magnetic field, remains. The dynamic of the
geomagnetic field is reduced to a discussion of the balance of forces.’
Parker’s word point directly at the reason why hydromagnetic theory cannot possibly be
adequate for the analysis of auroral particle energisation: it is not the magnetic field
that is the field of principle interest. There is no possibility of particle energisation
by a static magnetic field, since the Lorenz force on a particle is orthogonal to the velocity
vector. On the other hand, a significant induced electric field requires a rather rapid change
in the magnetic field. Such a rapid change is observed only during the expansive phase
of a substorm. However, there is plenty of evidence for the existence of a large scale
magnetospheric electric field at all times, which can provide the necessary energisation.
In order to understand this electric field it is necessary to keep track of charge separation
and electric currents. These quantities are explicitly left out of the hydromagnetic formulation.
It is therefore very apparent that the auroral particle problem is entirely different in
character from the magnetopause problem and yet people are still hell-bent on using the
same theory for both.
This had lead to a number of common fallacies that are worth discussing
A common factor that comes from the promiscuous application of fluid theory is a mistaken
identification of a mathematical sequence of calculations with the causal order of physical
phenomena.
As an example let consider the balance of forces between the solar wind and the geomagnetic
field. It has been possible to describe the magnetopause using magnetohydrodynamics in
quantitative terms. These include the magnitude of the discontinuity in the total magnetic
field. Having obtained this discontinuity it is then possible to calculate the strength
of the surface current that must flow. It is then stated that ‘the magnetic field
causes the current’. This is of course the reverse of what happens!
For a better understanding of the physics of the magnetopause it is necessary to turn
to a microscopic analysis, based on a consideration of particle trajectories. When this is done
you realise that the surface current is carried by the solar wind particles as they are deflected
by the magnetic field in the surface layer. It is the solar wind that drives the current
and that this is the direct cause of the discontinuity in the total magnetic field. Its magnitude
and location at equilibrium must of course reflect the balance of forces. Using this
picture there is no reason to throw away our previous concepts based on our experience
in the laboratory.
This way of looking at the problem is also in accord with the mathematical formulation
in which a vector field is said to be the result of sources which determine its divergence
and curl. Using Dungey’s notation we can now indicate the causal sequence as u->j->H
which is the usual relationship, and is the REVERSE of that proposed by Dungey.
On the basis of computational expedience, Dungey proceeds to reverse causal relationships
that have come to be accepted on the basis of considerable experience. On the surface,
the various equations including Maxwell’s would appear to be reversible but this is
a paradox which is true of any microscopic analysis. On a macroscopic scale relationships
are not reversible in general, as can best be understood by entropy considerations as
the problem at hand involves the dissipation of energy and therefore all doubts as to the
causal order can be resolved quite easily.
Thus we can probably agree that the energy flows from the solar wind into the magnetosphere,
and not vice versa. In particular, the main source of energy if the bulk motion of the
solar wind protons, and the ultimate recipient of this energy is the ensemble of particles
in the magnetotail. Within the magnetosphere it must be the electric field that is the
principal agent for the transfer of this energy. An electrostatic field must in turn arise
from the separation of charge as given by Poisson’s equation. Undoubtedly, such charge
separation in the magnetosphere involves currents.
The absurdity of frozen field convection
The identification of a magnetic field is somewhat arbitrary as only the magnitude and
direction at a point are specified by the fundamental relations. In the frozen field
description, the field lines and plasma are pictured as convecting together, as a fluid,
with the electric drift velocity v(E). In this drifting frame of reference, there is
no electric field, in conformity with the characteristic of a plasma that electric fields
are quickly shielded out by the redistribution of charge.
This above definition is to a large extent satisfactory for a cold plasma with large
external dimensions in a smoothly and slowly varying magnetic field. When particles of
higher energy or those with smaller external dimensions in a magnetic field with structure
are considered then the MHD model is no longer adequate. In the extreme case, this becomes
quite obvious, as it becomes permissible to ignore the electric field entirely, or at
least treat it as a minor perturbation, when trapped energetic particles in the radiation
belt are considered. For such particles in the magnetosphere, with energies exceeding
some 100keV, the major causes of drift are the gradient and the curvature of the magnetic
field lines. It turns out that for auroral particles with keV energies the electric and
magnetic drifts are of comparable importance. These will also depend strongly on the particles
energy, mass, the sign of the charge, and pitch angle. The particles do not travel together,
as they are assumed to do in fluid theory and it is essential that each particle be
followed in its motion separately. Failure to do so dooms any attempt to explain in detail
the process of auroral particle energisation and precipitation.
There is yet another line of thought which has taken hold. Here the magnetic field lines
are merged or annihilated during their convection across x-type neutral lines. The magnetic
field energy so annihilated is thought to be converted to particle kinetic energy. Again
it must be stressed that the only known energisation process by means of a magnetic field is through
the induced electric field, and this induced electric field is very small.
The MHD theory places undue importance on the magnetic field. When we consider the magnetospheric
problem in the Earth’s frame it becomes obvious that the magnetic field should be
considered as a passive element. The structure of the field is of great importance on the
motion of the particles but there is little or no work done by the magnetic field. The
work is done first of all by the flowing solar wind plasma in its interaction with the magnetosphere,
and subsequently by the magnetospheric electric field as particles cross the equipotential
surface.
The hydromagnetic approach leads us into a path that is seemingly difficult to reverse
out of for many astronomers. The only component of V relevant to their discussion is the ExB/B^2,
in the particle drift approach this term contributed nothing and the remaining terms alone are
thought to be relevant.
Any theory in which the basic processes are so distorted must surely be suspect. Many
will then come back with a statement like this: “This does not imply that microscopic
processes are unimportant or uninteresting, but rather that they are secondary..” These
types of statements are indicative of the reversed logic which is so characteristic
of this field.
Heikkilla ends the paper with the following statement:
“Let ys then use the fluid theory whenever we can, but let us not regard it as the fundamental
description. Let us be prepared to find that it can be deficient, and so let us use its
results with suspicion until they can be substantiated by a more basic theory of observation. Above
all, when the fluid theory fails let us neither force it nor castigate it, let us simply resign
ourselves to the use of the more basic and difficult particle-drift or kinetic theory.