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Old 1st February 2012, 08:10 PM   #6138
Tim Thompson
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Lightbulb Plasma Physics: "E orientation" or "B orientation"? II

In my previous post, Plasma Physics: "E Orientation" or "B Orientation"?, I presented the basic justification for the superiority of the "B" (magnetic field) orientation over the "E" (electric field) orientation when dealing with astrophysical plasmas. I want to complete that task in this post by presenting in more detail the physics involved. In this way we can go beyond any sense of relying solely on the assertion of individuals, even if their expertise is unquestioned, and have in place for ourselves an understanding for why the assertion is valid, based on sound physical principles. In that previous post I include a passage from the book Conversations on Electric and Magnetic Fields in the Cosmos by Eugene Parker, one of the foremost living plasma astrophysicists. In that comment he says, "As already noted, the difficulty is that there are no tractable dynamical equations for E and j". His "As already noted" refers to the passage I now quote first below. As before, any hilight emphasis is mine, but all other emphasis is carried over from the original.

Originally Posted by Eugene Parker, "Conversations on Electric and Magnetic Fields in the Cosmos", Introduction, page 2
It is here that a fundamental misunderstanding has become widely accepted, mistaking the electric current j and the electric current (sic) E (the E, j paradigm) (Parker 1996a) to be the fundamental physical entities. {Editors comment: clearly the second use of "current" for E is a typo where the word "field" is appropriate - TJT} Steady conditions often can be treated using the E, j paradigm, but the dynamics of time-dependent systems becomes difficult, if not impossible, because of the inability to express Newton's equation in terms of E and j in a tractable form. That is to say, E and j are proxies for B and v, but too remote from B and v to handle the momentum equation. So it is not possible to construct a workable set of dynamical field equations in terms of j and E from the equations of Newton and Maxwell. The generalized Ohm's law is often employed, but Ohm's law does not control the large scale dynamics. The tail does not wag the dog. This inadequacy has led to fantasy to complement the limited equations available in the E, j paradigm, attributing the leading dynamical role to an electric field E with unphysical properties. Magnetospheric physics has suffered severely from this misdirection, and we will come back to specific aspects of the misunderstanding at appropriate places in these conversations.

Here Parker makes a point that we have not emphasized in this thread, but will now. A plasma is a collection of charged particles and those particles have mass as well as charge, so they are not only affected by electric & magnetic fields as by Maxwell's equations, but they also have momentum and kinetic energy as by Newton's equations, and we have to respect all of the physics that counts, not just the parts we like. There is no overlap between the E, j paradigm and classical Newtonian mechanics; Newton's equations do not include either E or j. However, there is an overlap between the B, v paradigm and classical Newtonian mechanics; the velocity, v, of the particles shows up in both, which immediately connects the Newtonian energy & momentum to Maxwell's equations. Hence the obvious preference for the B, v paradigm: It makes the difference between tractable and intractable physics.

Parker says above that "we will come back to specific aspects of the misunderstanding ... ". I want to do that now so we can see exactly what is happening.

Originally Posted by Eugene Parker, "Conversations on Electric and Magnetic Fields in the Cosmos", chapter 7 "Moving Reference Frames", pages 69-70
Now, there has been an unfortunate practice in the (E, j) paradigm literature on magnetospheric physics to assert that E(r, t) plays an active dynamical role in driving the motions in the plasma. For instance, working in the frame of reference defined by Earth, there is an electric field Esw = -vsw X Bsw/c in the solar wind where the plasma velocity is vsw and the magnetic field is Bsw. It is sometimes asserted that Esw actively penetrates from the wind into the magnetosphere, thereby setting the plasma in the geotail into motion approaching the electron drift velocity

\bold v_D = c \frac{\bold B \times ( \bold v_s_w \times \bold B_s_w)}{B^2}

where now B is the geomagnetic field. Note that this motion can be in most any direction, depending on the relative orientation of B and Bsw. The antisolar streaming wind could just as well drive a solar-directed convection, contrary to the observed antisolar motion of the outer layers of the geomagnetic field. However, it is alleged that this is the basic cause of magnetospheric convection. A cross section of the geotail is sometimes drawn with the magnetotail between two wide parallel condenser plates, between which there is an implied potential difference in excess of 104 V in the direction to give the actual convection. But if the electric field were a driving force, in which frame of reference are we to use the electric field for computing the driving of the plasma? All those electric fields in the many different moving frames of reference are there, eagerly waiting to be exploited. There is, of course, no comparable magnetic field penetrating from the magnetosphere outward to affect the wind, so it appears that the wind suffers no gain or loss of momentum. On the other hand, if the calculation is carried out in the frame of the wind, then there is no electric field in the wind to drive magnetospheric convection, but rather an electric field +vsw X B/c in the magnetosphere that, by the same principle, actively penetrates into the solar wind, causing the electric drift velocity (vsw X B) X Bsw/B2 of the solar wind plasma relative to the solar wind. The principle of covariance is violated, and what ever happened to the conservation of momentum?

This author suggests that the electric field E' in the frame of the plasma is negligible, based on the considerations leading to eqn. (1.8). Another prejudice of the author is that Newton's equation of motion is the proper venue for the discussion of driven motions. Yet, when we come to Newton's momentum equation in chapter 8, we find no such driving effect introduced by electric fields. Indeed, it is obvious from eqn. (7.2) that the stresses, E2/8π, in the electric field are small O(v2/c2) compared to the stresses, B2/8π, in the magnetic field. That is to say, the stresses in the electric field are small to the same order as time dilation and Lorentz contraction, both of which we neglect to very good approximation. So the electric stresses are to be neglected. Vasyliunas (2001, 2005a, 2005b) has investigated in detail the dynamical role of E, showing how E is shaped by the plasma motions, rather than vice versa.

In contrast, in the E, j paradigm the force ħeE on an individual ion or electron looms large in the generalized Ohm's law and is considered to be an important dynamical effect. The net force on the plasma is very small, of course, because of the electrical neutrality of the plasma, so it is evident that ħeE has no role in the large-scale dynamics in spite of its prominence in the E, j paradigm. This is just one more illustration of how the E, j paradigm has such difficulty in coming to grips with the time-dependent dynamics of the plasma.

In this passage above the notation "O(v2/c2)" means "on the oder of v2/c2" and shows an approximate order of magnitude. Also note that the papers by Vasyliunas (retired director of the Max Planck Institute for Solar System Research) are particularly interesting and reinforce what Parker tells us quite nicely.

Parker's reference to equation 1.8 is this:

E'/B = c/4πσl = (10-4/l)(104/T)3/2

which equates the electric field in the moving reference frame of the plasma (E') and the magnetic field (B) (σ is the electrical conductivity). This is all found in Parker's Conversations on page 8. The upshot is that E'/B is likely never greater then 10-9, and since the stresses induced by the fields are proportional to (E'/B)2, the stress induced on the plasma by E' must be ~ 10-18 compared to the stress induced by B.

Parker's reference to equation 7.2 is this:

E(r, t) = [-v(r, t) X B(r, t)] / c

This is the electric field E(r, t) in the laboratory frame of reference when the electric field in the plasma frame of reference (E') is zero. It is found in Parker's Conversations at the bottom of page 68.

What it all boils down to is this: One cannot construct physically meaningful equations to describe the dynamics of a plasma in the "electric" paradigm favored by Mozina. One can do so only in the "magnetic" paradigm, increasingly favored by plasma physicists. This post and my previous post detail the physical reasoning behind this choice of "B" over "E". It is not as simple as Mozina claims, it's not simply a prejudice for one over the other, and it certainly is not "putting the magnetic cart before the electric horse", as Mozina has said it. The physical horse is in fact the magnetic field in almost all astrophysical cases. It is important to note the distinction that astrophysical plasmas and laboratory plasmas, despite both being plasmas, are not the same; astrophysical spatial scales cannot be reproduced in a laboratory, and that is significant. The relationship between electric and magnetic fields in the two plasmas are not the same, as a direct result of the difference in spatial scales, and that affects plasma dynamics. One should not blindly apply the paradigm of one unto the other. As is often the case in physics, each situation is like a position in a chess game: While there are general principles one can apply, each must be considered carefully on its own merits for its own proper solution.
The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it. -- Bertrand Russell
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