Plasma Physics for Dummies

[tusenfem]

I turned this thread into a sticky, and cleaned it up, so some of the really OT posts or non-info posts (also by myself) are gone.


Plasma Physics for Dummies

There is a lot of discussion about plasma physics on this board, and I thought it would be good to have at least the regular mainstream vision of plasma physics presented. This time plasma physics is presented not as short descriptions in the ATM section, where discussions seem to pop up most often, but in the general science section. I will assume that the persons reading this will have at least some knowledge of Maxwell’s equations, Newton’s laws and continuity equations, more about that later.
This first post is a little long, I will try to limit the next posts to 1 A4.

What is a plasma, and if so, is it a gas?
First of all we have to decide on the definition of a plasma. Although most of the readers will have an idea “it is a collection of charged particles with currents and magnetic fields and stuff,” the details are often stuffy, but the proof is not in the pudding but in the details.

First of all, a plasma is usually created (in the laboratory) from a gas in a plasma device. At one side of the plasma chamber will be a cathode and at the other side there will be an anode, and a strong voltage will be applied over the gas. The cathode can emit electrons, which can hit a gas atom (let’s start off with a simple atomic gas) and when the energy is strong enough one of the outer electrons of the atom can be knocked off. As there is a voltage difference in the room, the electron and ion will be accelerated; the electron more, because of its lighter mass. Now an avalanche effect can take place, by which more and more atoms will be ionized, thus creating in the plasma chamber a mixture of gas, ions and electrons. Indeed we might as well ionize all the gas atoms and what we will see is that the ions and electrons will fill up the whole chamber. So, there is a mixture of loose particles (charged in this case) that fills the whole room and has no specific shape (apart from the shape of the room) and thus we are dealing with a gas here, albeit a special kind. A very interesting paper on the creation of a plasma and associated electric fields is given in Physics Today, May 2007, pages19-21. (Time-resolved electric-field measurements probe plasma breakdown) You need a subscription to get the paper, but I have a copy of it.

A plasma is, amongst others, characterized by the following:
• A state of electric quasi-neutrality
• Behaviour is governed by collective effects due to long range electromagnetic interaction between the charged particles

’Quations! Get yasself some ‘quations!

Just to end this little introductory section, let us look at some of the necessary equations to do this little experiment of plasma physics:
Maxwell in differential form
1. div B = 0
2. div E = τ/ε₀
3. curl B = μ₀ J + c^-2 ∂E/∂t
4. curl E = - ∂B/∂t
Here div is the divergence of the vector (div A = ∂Ax/∂x + ∂Ay/∂y + ∂Az/∂z) and curl is the rotation of the vector ((curl A)_x = ∂Az/∂y - ∂Ay/∂z) where the rotation of the indices is clockwise to get the other two components.
τ is the charge density: τ = n_e q_e + n_i q_i, the sum of the electron density times the electron charge and the ion density times the ion charge. ε₀ is the electric permittivity of vacuum and μ₀ is the magnetic permeability of vacuum.

These equations tell us the following:
1. There are no magnetic “charges” called monopoles, magnetic field lines are closed loops
2. There are electric charges, and field lines go from positive to negative charges.
3. The spatial variation of the magnetic field can be generated by currents and by the time variation of a present electric field
4. The spatial variation of the electric field can be generated by the time variation of a present magnetic field.
These are the simplified versions of Maxwell’s equations, but we don’t need to complicate things any further for our purpose here. End of class.

What already end of class? Man you may have class but you ain’t got style.
This is the first step towards understanding plasmas. The next step will be to look at the characteristics of a plasma. What happens when we move the positive and negative cloud apart? What happens when we put 1 extra electron or ion in the plasma? And what does quasi-neutrality mean?

[iantresman]

Plasma Physics for Dummies

Thanks for that, all looks good go to me. The only comment I have relates to astronomy. On Earth, plasmas are quite rare, hence the requirement to create them in the laboratory. But above the atmosphere and throughout the Universe, plasmas are very very common, contributing to the ionosphere, the Sun and the stars, interplanetary, interstellar and intergalactic space.

[tusenfem]

Plasma Characteristics
To discuss some of the most basic plasma characteristics we have to do a little math. But before I scare you away, first a few words.

Plasma frequency
What happens when we have a plasma and take the two components, the positive ions and the negative electrons, as two separate entities and move them slightly apart? Well, naturally the electric force will start to act on the two parts, and when we let the two go, they will move back together again. But, as there is nothing to stop the motion when the two are together again, there will be an overshoot, and eventually the motion will be stopped again by the electric force. This continues in an oscillatory manner. Compare this with a pendulum, where you lift the weight to one side and let it go, it will swing past vertical and up again. The same way it works for the plasma components. The oscillation has a specific frequency, which we will now derive.

We will have to solve a few differential equations, which is usually done by looking at small disturbances of a balanced situation. This makes it possible that we linearize the equations. As the electrons are at least 1836 times lighter than the ions (unless we work with an electron-positron plasma) we can safely assume that the electrons do all the movement and the ions are a neutralizing background. So the electron density becomes n_e(x,t) = n_e0 + n_e1(x,t), where the last term is a small perturbation of the electron density in space and time.

We need the continuity equation which says that the time variation of a quantity can be described by the spatial variation in the flow of that quantity. In the case of the electron density this means: dn/dt + div( n v ) = 0. In the approximation of small disturbances, you can submit the above expression for the density, and assume that the equilibrium velocity is 0. We then find as first equation:

∂n_e1/∂t + n_e0 div( v_e1 ) = 0

Now we turn to Newton’s Law, F = m a, where F = dv/dt, because we expect the electrical force to work on the electrons:

m_e ∂v_e1/∂t = - e E₁

And with Maxwell’s equation we know how the electric field E is dependent on the charge density tau:

ε₀ div (E₁) = - e n_e1

Now we can combine these three equations to find:

{ ∂²/∂t² + (e² n_e0 / m_e eps₀) } n_e1 = 0

Which is a regular wave equation for which the solution is a harmonic oscillation with frequency

ω² = e² n_e0 / m_e ε₀

Which is called the (electron) plasma frequency. This frequency is often used in space physics to find the electron density. In a spectrogram of a plasma wave instrument this frequency this one jumps out. Now, does this mean that if I take one electron and move it that it also will start to oscillate at this frequency? One could think so, however, this is a collective phenomenon and not a single particle behaviour. In the next section we will see how a plasma reacts on the displacement of one particle.

[tusenfem]

No, this has nothing to do with Hollywood or with TVs!

Plasma Screening
In the previous installment we saw how the plasma can show a collective behaviour, oscillating around a certain equilibrium state at the plasma frequency. But what happens when we put an extra electron in the plasma, or move one electron from one location to another? Naturally, this will create a disturbance in the plasma. How does the plasma respond to this disturbance? In order to find out, we first have to consider (which has been neglected until now) that the plasma has a temperature T, where the electrons and the ions may well have different temperatures.

What happens in the plasma is caused by the mobility of the charged particles. When an electron gets placed somewhere in the plasma, the particles will start to re-adjust to the disturbance, the negative ones will move away a little and the positive ones close in a little, effectively screening off this charge disturbance. It was DeBye & Huckel who first came up with a calculation of this effect in 1923.

A charged particle population with a certain temperature can be described by the Maxwell-Boltzmann equation. Normally in the so-called distribution function of particles there is a description of how many particles there are at a certain place with a certain velocity. Integrating this distribution over all possible velocities gives the total density. Now we are dealing with charged particles, that can have an electric potential energy, so we have to change the distribution function and we find that we can describe the particles with:
f(x,v) = constant * exp{ - (kinetic energy + electrostatic energy) / thermal energy }
the kinetic energy is given by: 0.5 m v²
the electrostatic energy is given by: q φ
where: φ = q / 4 pi ε₀ r
the thermal energy is given by: k_B T

Now some math happens, which I will not put here (for a rainy Sunday to read up on), but integrating over velocity we find the density as a function of φ. This can be put into Maxwell’s equation div(E) = τ, where we have to consider that grad(φ) = E, and thus there is a second order differential equation, the so-called Laplace equation. The end result is that we find a length-scale in the plasma over which the influence of the charge disturbance is screened off. This is the DeBye length.

λ_D² = ε₀ k_B T / n₀ e²

So, what we find is that the mobility of the charged particles in a plasma and the electrostatic forces between the particles any disturbance of the equilibrium situation, a charge neutral plasma in the normal case, gets attacked and encapsulated by a sphere of radius λ_D, outside of which the plasma does not feel any influence of this disturbance. You could compare this to white blood cells in the human blood. Any strange particle coming into the blood stream gets encapsulated with the white cells, and the rest of the body does not notice it anymore (well, very simplified viewpoint).

This leads immediately to the explanation of a rather confusing term in plasma physics, and that is quasi-neutrality. Overall, when a plasma is created, either in the lab or in space, from a neutral gas, there will be equal amounts of electron charge and ion charge and the total sum over the whole cloud will be zero. However, this does not mean that variations in the net charge in small regions of the cloud cannot occur. We have just seen that the influence of a small local disturbance of the charge neutrality gets screened off. This means that deviations of charge neutrality of the plasma can take place on length scales on the order of the DeBye length. So, the plasma is neutral on the whole, but in small portions there can be a net charge
.
Just for laughs
NOTE: We have been talking about a neutral plasma. This does NOT mean that all the particles of the plasma are neutral (i.e. have no charge) because then we obviously would not have a plasma. A neutral plasma means that, take over a large volume the sum of negative and positive charges is zero.

[iantresman]

Plasma Screening

For those of us who are equationally challenged, I thought it be be useful to provided some examples of the plasma screening distance (Debye length) for some typical plasmas:

Plasma			Density	Electron 	Magnetic  Debye 
				temperature	field	  length
			ne(m3)		T(K)	B(T)	  λD(m)
Ionosphere		1012		103	10−5	   10−3
Magnetosphere		107		107	10−8	   102
Solar core		1032		107	--	   10−11
Solar wind		106		105	10−9	   10
Interstellar medium	105		104	10−10	   10
Intergalactic medium	1		106	--	  105

From which can be seen that shielding distance (Debye length) is a few millimeters in the ionosphere, about 10-metres in the Solar Wind (interplanetary medium) and interstellar medium, and 10-km in the intergalactic medium.

[tusenfem]

Thanks iantresman for the list of DeBye lengths.

The right-hand’s connected to the … electrical current? What the ...!
Well, that last part was no fun at all. I sat all Sunday trying to figure out the math for the DeBye length, and it was not even raining! How about something fun now. But this is science dude, there ain’t no fun in science! Or is there?

Magnetic Fields
We have not yet talked about magnetic fields, apart from mentioning them in the Maxwell equations. So, let’s start there, how are magnetic fields created in a plasma? Well, the easiest way is to have a big ol’ magnet plugged into the plasma, that’ll do it. But, in the beginning we had simplified versions of Maxwell’s equations, where the magnetization and the polarization had been neglected. So, we discard this option. Well, then the only thing that remains is a current and/or a time varying electric field.

Let’s start with the first one, electrical currents.

When for some reason or other (we will get to that later) a potential drop is created over a plasma, the result will be that currents will start to flow. Most of the current will be carried by the least massive particles, that means by the electrons. And because of some strange definition (basically because they used to think that in currents through wires the moving particles were positively charged), if the electrons move to the left, the current flows to the right.

So, what does Maxwell tell us about currents? He says that the rotation of the field (curl B) is proportional to the current density J. And now we can take this literally, the rotation or curl I mean.

Just a little side step:
There are two words for the same mathematical construct. The English language mostly uses “curl” whereas other languages seem to prefer to use “rot(ation)”. Naturally both word indicate the same that there is something turning around.

So, indeed, if we look at a current in a wire we find that the magnetic field consists of concentric circles of weaker and weaker strength as we move away from the current wire. So, where does the right hand come into play? If you curl your fingers, while keeping your thumb up, just like giving a thumbs-up sign, and your thumb is pointed in the direction of the current, then your fingers curl the same way as the magnetic field.

Gluon Glue?
Okay, but what does that do to the plasma? Now comes the interesting part. The charged particles interact with a magnetic field. To be specific, they get “picked up” by the magnetic field and start to gyrate around it. Ah, now we have to again do a little math.

The equation of motion for a charged particle in the presence of an electric and magnetic field is:

m dv/dt = q (E + v x B),

where the latter term means the cross-product of the velocity and the magnetic field vector. This is a vector perpendicular to both the velocity and the magnetic field with a size of vBsin(θ), where θ is the angle between the to vectors v and B. Simplifying things a little we will assume that there is no electric field, so E=0. That means that if a particle has a motion along the magnetic field, it will be unchanged. But as the only remaining force is perpendicular to the velocity, this means that the total velocity remains unchanged in size (not direction!!). Thus also the velocity perpendicular to the magnetic field remains constant in size. Assuming that the magnetic field is in the z-direction the equation of motion reduces to two simple equations:

∂²x/∂t² – (q B / m) ∂y/∂t = 0
∂²y/∂t² – (q B / m) ∂x/∂t = 0

which combine nicely to the equation for a harmonic oscillator and we find that the charged particle is running around in circles around the magnetic field with a specific frequency dependent on the charge q, the magnetic field B and the mass of the particle. This frequency, the gyro-frequency or cyclotron-frequency is given by:

ω_c = |q B / m|

where it should be noted that positive and negative particles gyrate in opposite directions. Which direction you ask? Well, the same as in the beginning, use the right-hand rule, but now with your thumb in the direction of the magnetic field. Curl your fingers again, and that will be the direction of the current. Positive particles gyrate in the direction of the current and negative particles gyrate in the opposite direction.

[tusenfem]

Okay, can we now finally start with something interesting? This stuff starts to bore me to pieces.
Sure! What would you like to know?
Well, how about frozen in fields, how can hot plasmas freeze anything anyway? Or maybe something more specific about currents in plasmas, something that we can “touch.”
Well, okay, but you will have to go through one little thingy still, and that is the time scales that are important in the plasma, because these rule how we can approach a plasma.
Okay, whatever, but make it quick, will you!

Time Scales
Time scales in plasma physics are very important. The reason for that is that for a short period something may well remain constant, whereas when you look at it longer you will see that it changes. To give an example that is easier to grasp we look at the Earth and at length scales. Now, when you work in the garden, or when you build a house, you can safely assume that the ground if flat, although you know that the Earth is a ball. So, the fact that the surface of the Earth is curved does not really matter when you look at scales of tens of meters. Similar in the garden, you plant a small tree. Now, coming back after a week, you will not notice any changes, but coming back after a year you will find that the tree has grown.

The first time scale we have is the time it takes for an ion to rotate around a magnetic field. This is the inverse of the cyclotron frequency. Looking at processes that are longer than this time scale allow us to neglect the gyration of the particles, they get averaged out. This makes that the plasma can be seen as a fluid (yeah, I know, above I said it was a gas and now it is a fluid!?!?). Indeed, this approach to plasma physics is called the MHD (MagnetoHydroDynamics) approach, invented by Hannes Alfvén and described in glorious details by Chandrasekhar in 1961 in his book Hydrodynamic and Hydromagnetic Stability.

The second time scale we have to look at is the diffusion time scale. Now, this is a bit more tricky, this deals with how fast magnetic fields can move through a plasma or through any conducting object in general. This time scale is dependent on the conductivity of the plasma.

τ_D = L² / η
η = (μ₀ σ)^-1

here L is a characteristic macroscopic length of the plasma you are looking at and sigma is the conductivity of the plasma (mmmm, we have not calculated conductivities yet). This time scale tells us how fast a magnetic field can move through a plasma, and this is essential for a very contentious topic (here on the board, but now and then also in the scientific community): the so-called "frozen in magnetic field."

So, this is a very small part of the basics of plasma physics, but it at least gives you a few of the important parameters that you have to keep in mind. To make it not too dry I will move to some more real-life problems. But first, question hour, I would like to have some input about what you would like to have explained/investigated.

The next part will be about frozen in fields and how to interpret this phenomenon invented by Hannes Alfvén and later sort-of rejected by him and his followers

[tusenfem]

Okay, let’s do some real plasma physics. Many of you have heard and/or talked about Frozen-In Magnetic Fields, but do you know what that means?

We have discussed above that the plasma particles are connected to the magnetic field, in such a way that they gyrate around the field. At the same time, when the magnetic field is moving, the plasma is moving along. However, we also know that a magnetic field can diffuse through a region with conductivity σ with a characteristic time scale τ_D. So, if the conductivity is infinite there will be no diffusion, if the conductivity is large, there will only be very slow diffusion. So, one can envision that, as there is a mutual attachment of the plasma and the magnetic field, if the one moves, the other will move along.

Let’s get into math mode for a minute. Everyone, I assume knows Ohm’s Law, connecting the current and the voltage and the conductivity, for example in a electrical circuit. To write it in quantities that are relevant in plasma physics Ohm’s Law takes the following (simplified) form:

J = σ ( E + v × B)

The current density is equal to the conductivity times the electric field plus the co-moving electric field. In the case of a collisionless plasma the conductivity σ can be so high that the only way to satisfy the equation above is:

E + v × B = 0

Now, here we see a very important equation in MHD and plasma physics (did we already discuss the differences between the two? I don’t think so, we’ll get back to that). First of all, take a plasma at rest, i.e. v = 0, which will mean that the electric field E = 0. This makes sense, as we are dealing with a collisionless plasma, and any stray electric field will quickly be negated by the motion of the particles in the plasma. In the case that v ≠ 0 we find that there has to be an electric field, and conversely, if there is an electric field then plasma must flow.

Now, one can prove that in a very high conductivity MHD fluid (or plasma), the magnetic flux can be frozen into the fluid (or plasma). Let’s take a surface S which is not parallel to the magnetic field lines. We can then calculate the magnetic flux Φ through this surface by:

Φ = ∫ B · dS
(if the two are perpendicular you just end up with Φ = **). Now the frozen-in flux conditions tell us that if we follow the fluid which was initially on surface S as it moves through the system, the flux will remain constant even as the surface changes its position and its shape. This can be shown very nicely mathematically, but I will leave that here (see e.g. Goedbloed and Poedts, Principles of Magnetohydrodynamics, pages 154 and 155 for a full mathematical derivation). This is an interesting result, and as Goedbloed and Poedts write, this gives more physical meaning to magnetic field lines than they ever had in Faraday’s time. This, by the way was first published by Alfvén in 1950 in Cosmical Electrodynamics page 81.

So, we have found a conservation law, for very high conducting plasmas the magnetic flux is conserved and moving along with the plasma. The field is frozen in. Oh brother! I can already feel the hot breath of adversaries in my neck, But Alfvén retracted his support for the frozen in field in his Nobel lecture!!. Did he?

Here is the lecture that he gave, in which he states something more differentiated, which can be seen in his Table 1: The first approach vs. the second approach. Now, we should be clear on the fact that Alfvén himself wrote a whole book on the first approach in which plasmas are ideal and homogeneous (the above mentioned one). Indeed, ideal plasmas do not exist, and therefore, we have to be careful with what we mean, and that is why I explained the time scales of plasmas first to you. It is always easier to get the physics right for the simplified model, and then move on to the next level and introduce complications.

Just to give an example, if the plasma is not perfectly conducting, but has a finite value for σ, what do we do then with this theorem of frozen-in field? Well, I would advise first to calculate the diffusion time, which is the time over which the magnetic field can significantly move through the conducting body or plasma. If the process that we are looking at is much longer than this diffusion time, then we cannot assume the frozen-in field to be applicable, however, if the process is shorter than the diffusion time scale, there is nothing wrong with having frozen-in fields.

To give an example of diffusion time scales in another object, consider the moon Europa around Jupiter, embedded in the Jovian magnetosphere. We know Europa is a conducting body, as we have observed an induced magnetic field around the moon. However, Europa has been so long inside of Jupiter’s magnetic field (a few billion years) that the non-time varying part of the magnetic field (basically in the direction of the rotation axis of Jupiter at Europa) has totally diffused into the moon. Only the time varying field (with a period of about 10 hours) changes too fast and cannot diffuse into the moon. It’s all about time scales.

[iantresman]

Indeed, Alfvén wrote that frozen-in magnetic fields in "plasma in interstellar space should be applied with some care":

I thought that the frozen-in concept was very good from a pedagogical point of view, and indeed it became very popular. In reality, however, it was not a good pedagogical concept but a dangerous "pseudopedagogical concept." By "pseudopedagogical" I mean a concept which makes you believe that you understand a phenomenon whereas in reality you have drastically misunderstood it.

I never believed in it 100 percent myself. This is evident from the chapter on "magnetic storms and aurora" in the same monograph. I followed the Birkeland-Störmer general approach but in order to make that applicable to the motion of low-energy particles in what is now called the magnetosphere it was necessary to introduce an approximate treatment (the "guiding-center" method) of the motion of charged particles. (As I have pointed out in [5, sec. III.1], I still believe that this is a very good method for obtaining an approximate survey of many situations and that it is a pity that it is not more generally used.) The conductivity of a plasma in the magnetosphere was not relevant.

Some years later, criticism by Cowling made me realize that there was a serious difficulty here. According to Spitzer's formula for conductivity, the conductivity in the magnetosphere was very high. Hence the frozen-in concept should be applicable and the magnetic field lines connecting the auroral zone with the equatorial zone should be frozen-in. At that time (1950) we already knew enough to understand that a frozen-in treatment of the magnetosphere was absurd. But I did not understand why the frozen-in concept was not applicable. It gave me a headache for some years.

In 1963, Fälthammar and I published the second edition of Cosmical Electrodynamics [12] together. He gave a much higher standard to the book and new results were introduced. One of them was that a non-isotropic plasma in a magnetic mirror field could produce a parallel electric field E|| . We analyzed the consequences of this in some detail, and demonstrated with a number of examples that in the presence of an E|| the frozen-in model broke down. On [12, p. 191] we wrote:

"In low density plasmas the concept of frozen-in lines of force is questionable. The concept of frozen-in lines of force may be useful in solar physics where we have to do with high- and medium-density plasmas, but may be grossly misleading if applied to the magnetosphere of the earth. To plasma in interstellar space it should be applied with some care."

Ref: Double layers and circuits in astrophysics, Alfven, Hannes, IEEE Transactions on Plasma Science (ISSN 0093-3813), vol. PS-14, Dec. 1986, p. 779-793 (also available online in full)

[tusenfem]

Frozen-in Revisited

Thanks for the quote from Alfvén, Ian. Indeed, as stated above, there are specifics to the MHD and frozen-in approach that have to be taken into account. First of all we have to be in the MHD regime, which means we are looking at plasmas on a timescale that is larger than the longest gyration time of the particles making up the plasma. Then, we throw away some terms in the generalized Ohm’s law, so we get the equation above for J, which means e.g. that we assume that there is no pressure gradient in the plasma, and also we are looking at length scales that are much larger than the largest Larmor radius. Are we allowed to do all this in typical space plasmas?

Marcel Goossens in his book An Introduction to Plasma Astrophysics and Magnetohydrodynamics gives an example, emphasizing that there is no such thing as “a set of typical values for v, L, n, B and T” in space plasmas, because they can be vastly different in different regions. He takes the following values, which are typical for e.g. the Earth’s magnetosphere:

v = 10⁵ m/s, L = 10⁶ m, n = 10⁷ m^-3
B = 10^-8 T and T = 10⁵ K

Looking at the discarded terms in the generalized Ohm’s law, which are Ohmic, Hall, battery and electron inertia terms with respect to the induction term, it is found that only the Hall term might play up in some cases with a relative weight of 0.1, the other terms are 10^-12, 10^-2 and 10^-5, respectively. This means that E = - v × B is a very good approach of the generalized Ohm’s law, when we take the limitations into account.

Now, Alfvén states his doubts on the frozen-in condition in the second volume of Cosmical Electrodynamics (1963) when space exploration was still starting up, and he was right about stating doubt on the general applicability of the frozen-in condition. Nowadays, we have dedicated magnetospheric missions, even multi-spacecraft, like Cluster. Now, we can measure magnetic fields, plasma data and electric fields in situ, and thus check whether or not the frozen-in condition is valid.

Rickard Lundin and co-workers published a paper in 2005 in Annales Gephysicae with the title Magnetospheric plasma boundaries: a test of the frozen-in magnetic field theorem. It goes too far to completely discuss the paper here in detail, and people interested can get a copy and read for themselves. However, I will give some of the conclusions they wrote down:

- Ideal MHD appears to apply on large scales in most parts of the exterior magnetosphere and I the “undisturbed” magnetosheath
- Departure from ideal MHD implies electromotive forcing
- Departure from ideal MHD in extended regions (several 1000 km) is observed near the magnetopause
- Departure from ideal MHD is also observed on smaller scales in magnetosheath plasma transients inside the magnetopause.
- Ideal MHD is certainly useful for transport applications where convection plays the dominant role for magnetized plasmas, like the cold plasma in the Earth’s magnetosphere and ionosphere.

This year (2007) Tony Lui and co-workers wrote a paper in JGR on Breakdown of the frozen-in condition in the Earth’s Magnetotail. This paper found that the ”… breakdown occurred in a low-density environment with moderate to high ion bulk flow and significant fluctuations in electric and magnetic fields. The breakdown can sometimes be seen simultaneously by cluster satellites separated in the X and Z coordinates by more than 1000 km apart, suggesting that the spatial extent of this breakdown region may exceed this dimension in both the X and Z directions. The examples in this paper show that during disturbed times E and –v × B do not agree.

However, Kunihiro Keika and co-workers have just submitted a paper to JGR where it is nicely shown that in the events that they studied, the induced electric field –v × B agrees very well with the measured electric field E. But in this case the flows v are not produced by reconnection (like in the Lui paper) but by a compression of the magnetosphere.

This shows, I think, that the frozen-in field approach does have its merits, when applied correctly. And that brings us back to Alfvén’s statement, in which he basically worries that this approach is blindly followed, without thinking. I hope I have shown that plasma(astro)physicists are well aware of the dangers of just assuming ideal MHD.

Know the limitations of your theory!

[tusenfem]

Waves in Plasmas

We have already seen the collective behaviour of plasmas, the oscillations at the plasma frequency. Naturally, there are more waves in a plasma than just this oscillation. We have already discussed the cyclotron motion of charged particles, and indeed, this will create waves at the cyclotron frequency in the plasma (we will not yet go into details as to why and how). Let’s first go to the basics, and back to Hannes Alfvén.

Let us assume we have a homogeneous plasma, i.e. constant density, no flows, constant pressure and the magnetic field is in the z-direction and no currents flow. Everything is quiet and nothing happens. No imagine we would disturb a magnetic field line, what will happen? Well, it is often said that magnetic field lines are like rubber bands, and if that is true, we would expect the magnetic field line to return back to its original state, but as a rubber band or a guitar string it will start to oscillate.

Time for some math again
Using the usual suspects in linearized form (which means that we have small disturbances of the starting equilibrium above and only use those terms that are linear in these small disturbances, which have subscript ₁) we find:

Continuity: ∂ ρ₁ / ∂t = - ρ₀ Div(v₁)
Eq. of Motion: ρ₀ ∂v₁/∂t = - grad p₁ + μ₁ ^-1 Curl(B₁) x B₀
Internal energy: ∂p₁/∂t = - γ p₀ Div(v₁)
Faraday: ∂B₁/∂t = Curl(v₁ x B₀)

Now, this is a full set of equations that can be put together with a little creativity. (This is homework stuff, make sure the above equations are correct, derive them yourself. Then you have to put them together to create the wave equation). If that is done correctly, we find the following expression:

ρ₀ ∂²v₁/∂t² = μ₀^-1 Curl(∂B₁/∂t) x B₀ = μ₀^-1 B₀ x Curl( Curl(B₀ x v₁))

Although maybe a little confusing, this is a standard form for a wave equation. The second time derivative is proportional to the second spatial derivative. A solution for such an equation can be found by assuming a plane-wave solution, which takes the form:

v₁(r,t) = v exp{ i ( k r – ω t) }

in that case the time derivative can be replaced by –i ω and the spatial derivative by i k. In the end, this leads to the following, that there are transverse waves, that travel along the magnetic field with a velocity:

v_A = B₀ / sqrt(μ₀ ρ₀)

This is the so-called Alfvén velocity. This was discovered by Alfvén in 1942, waves will travel in both directions along the magnetic field with the Alfvén velocity. These waves can exist because of the tension of the magnetic field lines (B²/ μ₀)

Naturally, there are also other waves possible in the plasma, we will ride those next time.

Questions?

[Nereid]

I do have some questions, but they're not, necessarily, directly connected with what's in this thread (so far).

I hope they won't be too off-topic; if so, say so, and I'll start a different thread.

First, how do plasmas react to electromagnetic radiation?

In particular (1), do the properties of real plasmas lead to a need to significantly adjust the Eddington limit?

In particular (2), do plasmas differ from neutral gases wrt the absorption and emission of electromagnetic radiation? In this I'm not interested in lines, only things like refraction and reddening (frequency dependent, broadband, absorption).

In particular (3), are plasmas completely opaque to electromagnetic radiation below the plasma frequency? What happens to the (EM below plasma frequency) energy absorbed by a plasma?

Second, in what significant ways do extremely dense plasmas behave differently from those of lower density than a good high school science lab vacuum? I'm thinking of white dwarfs and neutron stars, but the cores of high mass stars might also be interesting to examine.

[iantresman]

First, how do plasmas react to electromagnetic radiation?

Plasma may modify the wavelength-frequency relation, λf=c/n, where n is the refractive index of the plasma, which depends on the plasma wave frequency, magnetic field strength and orientation, plasma temperature, plasma constituency, and collision frequency.

Second, in what significant ways do extremely dense plasmas behave differently from those of lower density than a good high school science lab vacuum? I'm thinking of white dwarfs and neutron stars, but the cores of high mass stars might also be interesting to examine.

Hannes Alfvén and Carl-Gunne Fälthammar produced a table comparing high-density cosmic plasmas with low-density ones, and described the comparison with laboratory plasma. I hope this is the kind of comparison you are looking for.

[tusenfem]

I do have some questions, but they're not, necessarily, directly connected with what's in this thread (so far).

I hope they won't be too off-topic; if so, say so, and I'll start a different thread.

First, how do plasmas react to electromagnetic radiation?

In particular (1), do the properties of real plasmas lead to a need to significantly adjust the Eddington limit?

In particular (2), do plasmas differ from neutral gases wrt the absorption and emission of electromagnetic radiation? In this I'm not interested in lines, only things like refraction and reddening (frequency dependent, broadband, absorption).

In particular (3), are plasmas completely opaque to electromagnetic radiation below the plasma frequency? What happens to the (EM below plasma frequency) energy absorbed by a plasma?

Second, in what significant ways do extremely dense plasmas behave differently from those of lower density than a good high school science lab vacuum? I'm thinking of white dwarfs and neutron stars, but the cores of high mass stars might also be interesting to examine.

Dear Student!

Well, that's not nothing you are asking, but I will give it a try in the next few messages. Maybe not in the order that you asked them. We did waves, and so I will show how waves can propagate in a plasma (with a cut-off frequency) and how absorption and emission works. I guess I can group these two together, though it will be a tough ride.

Then I will take a look at the Eddington limit (it's like 15 years since I came across that one)

And then very dense plasmas (will have to look up in the literature a bit). But here the collisions in the plasma will become important in the dynamics as most likely the collision time will be shorter than the gyration time. This means collisionally dominated. I once did something with that in an accretion disk. Guess I will have to dig up my own paper.

Thanks for your attention.
See you in next class.

[tusenfem]

Waves in Plasmas

Waves in plasmas are as common as people on the beach on a nice summer day. We have already discussed two kinds of waves (at the plasma frequency, a collective oscillation of the plasma and the Alfvén waves, which are a result of the magnetic tension of the field lines). It is basically waves like the Alfvén waves that we are interested in. We found that the wave equation was given by:

ω = k v_A

and the propagation speed of these waves is the same for every frequency ω. However, in more general cases you will find that the wave equation looks like:

d² Ψ / dt² = v^-2 (d²/dt² – Ω²) Ψ

which has only non-trivial solutions when the part between brackets below is zero:

(ω² – Ω² - k² v²) Ψ = 0

And now we see that, when we calculate the velocity of the waves:

v = ω/k = +/- c ω / sqrt(ω² - Ω²)

is frequency dependent. This is called dispersion, waves with different frequencies travel at different velocities in a plasma (although this does not hold for pure MHD waves, which are higly anisotropic with respect to the magnetic field direction, but non-dispersive), and a good example of this effect is well known to people listening to AM radio and are whistler waves. When lightning strikes somewhere, you will hear a high tone that quickly falls down in frequency on the radio, like a whistle. The high frequencies reach you first and are then followed by the lower frequencies.

But notice that the dispersion relation that is found leads to an interesting phenomenon. If the frequency goes below Ω, the wave vector k becomes imaginary. Putting that into the plane wave formula, you find that the wave becomes evanescent, i.e. through the plasma it falls off exponentially:

exp{ i ( kr – ω t) } => exp( - K r) exp(- i ω t)

This will lead to damping of the wave in the medium and to total reflection of the wave, e.g. when light falls onto a metal, where the electron gas is acting as a plasma. The frequency Ω at which this happens is the plasma frequency in a regular “over the counter” plasma. Naturally, there can be other frequencies where this happens in the case of a multi-component plasma with a magnetic field.

So, in order to find out what happens, in this simple situation, one has to solve the equations of a plane wave traveling towards a boundary, and at the boundary there will be reflection and transmission, when we already know that the wave will be damped at the other side of the boundary. Now, of course this does not answer Neiried’s question yet, what happens to the damped wave? Where does the energy go? For that we have to go deeper into the theory of wave modes and find the relationship between the dielectric tensor epsilon and the dispersion tensor Lambda.

This will be discussed next time, as I am a little pressed for time at the moment.

[iantresman]

Waves in Plasmas

Many thanks for that. Others may be interested to read Hannes Alfvén's original paper in Nature, Existence of electromagnetic-hydrodynamic waves (1942). And a history of Alfvén waves can be read in "Biographical Memoirs: Hannes Alfvén" (page 653). I've also put together on my own Web site, an "Alfvén wave Timeline".

[tusenfem]

Okay, as you may have noticed, it has been a while since I wrote something. I left you all hanging at the damped wave, travelling through a plasma at a frequency below the plasma frequency. Apart from being very busy at work, there was also the problem of justifying my “gut feeling” about what happens with the wave.

It is obvious that the energy that is first in the wave mode (the so called Poynting flux, which has nothing to do with the flux pointing in some direction, but everything with John Henry Poynting ) is somehow dissipated. The Poynting flux is obtained by the cross product of the electric and magnetic fields of the waves. As the two are related this can be written (in vacuum) as:

S = (1/Z₀) E²

With Z₀ = sqrt(mu₀ / epsi₀), the “resistivity of vacuum”.
Now, “gut feelings” are sometimes good, when supported by some topical knowledge. In all I expected that the wave energy would be converted into heat, but how exactly does this happen?

As I wrote in the previous message, one has to turn back to the di-electric tensor, from which the dispersion tensor is found. The complete math is too much to write down here and I would like to point the math enthusiasts to Don Melrose’s Instabilities in space and laboratory plasmas (see here ) chapter 2.6 for a formal derivative on how the damping of the waves can be written in a form of electric dissipation through J • E, where J is the associated current density and E the electric field of the wave (see page 26).

So, in all the energy of the wave, when it enters a region where its frequency is lower than the plasma frequency gets converted into plasma thermal energy. Simplified you can look at it in the following way, the electrons (as they are lightest) slosh along with the wave and are thus energized. This is a very big simplification, which might make it look like Landau damping, in which the particles surf on the crests of the waves, taking the wave energy away, but that is another process. It is more comparable with Ohmic heating in a resistor.

Next time we try to look at the effects of a dense plasma, what happens when the collisions become important in a plasma?

[Nereid]

Well, the thread has been "dead" for a while, but that does not mean that the content is wrong or can be discarded. I just tried to make plasma physics as simple as I can, but from JREF I know that Robinson is not really gifted in math and physics.

But if there are any questions, then I am always willing to answer them.

I think Neiried still has some difficult questions about dusty plasmas and the eddington limit and stuff, that I never got to to answer.

Thanks very much Robinson, for reviving this thread!

Yes, I am still interested in several aspects of plasmas that tusenfem has not yet explained, starting with these:

Do the properties of real plasmas lead to a need to significantly adjust the Eddington limit?

Do plasmas differ from neutral gases wrt the absorption and emission of electromagnetic radiation? In this I'm not interested in lines, only things like refraction and reddening (frequency dependent, broadband, absorption) - this has been partially answered.

Are plasmas completely opaque to electromagnetic radiation below the plasma frequency? This is actually quite tricky, because real plasmas are not homogeneous, and my original question did not say anything about the source of any EMR. So, to make it a bit more concrete:

* suppose there is an efficient, compact radio (or microwave) transmitter in a homogeneous plasma comprised of fully ionised hydrogen

* suppose the transmitter makes EMR with a frequency that is one (or two) OOM below the plasma frequency

* suppose the plasma density and temperature are unchanged for the duration of our experiment

* suppose there is a sensitive receiver located a long way from the transmitter (>> the DeBye length, >> the wavelength of the transmitter's EMR, etc); perhaps this receiver is in a bubble of cold, non-ionised gas, perhaps this plasma-gas boundary needs to be considered?

In what significant ways do extremely dense plasmas behave differently from those of lower density than a good high school science lab vacuum? I'm thinking of white dwarfs and neutron stars, but the cores of high mass stars might also be interesting to examine.

[Gigabyte]

Thanks very much Robinson, for reviving this thread!

Hey! Don't blame me! Somebody was flogging it on another forum, I didn't know it was dead.

In what significant ways do extremely dense plasmas behave differently from those of lower density than a good high school science lab vacuum?

I have a lot of question, practical questions, about plasma, absorption lines, conductivity, frequency, and such. Most of the stuff I find while looking for answers is about fusion and the like.

Does anybody know the absorption spectra of hydrogen gas at different temperatures, levels of ionization?

[tusenfem]

Does anybody know the absorption spectra of hydrogen gas at different temperatures, levels of ionization?

I do hope you mean here at different levels of excitation, because hydrogen only has one electron to ionize.

A hydrogen atom has a lot of energy levels for its electron. Depending on the temperature of the gas there will be a distribution of excitation given by the Boltzmann excitation equation.

From here (could not find something else quickly here at home):

The Boltzmann equation tells us that, for a particular ionic species, the ground state (no excitation, C=0eV) is always the most populated and as the temperature rises the excited states become more and more likely to be populated. Lines arising from the ground level are generally the strongest lines observed and are called resonance lines.

So, basically if you know the temperature of the gas, you can find the excitation distribution of the gas and then theoretically find which absorption levels are present and at what strength. But it is rather complicated, you most likely need a large computer to calculate this.

Then again, you might mean, a hydrogen gas ionized to a certain degree, say 30% or so. Then you get a lot of other processes that may occur, e.g. free-free interactions. The best way of finding out all is by looking at Rybicky & Lightmann Radiative Processes in Astrophysics, which will give you all the gory details and more. (The book I have at work, not at home, so I cannot go into details here).

[tusenfem]

Do the properties of real plasmas lead to a need to significantly adjust the Eddington limit?

The Eddington limit is where the radiation pressure equals the gravitational force. I will have to look up the details, but basically it will be the photon collision with electrons and ions. I am not sure if "properties of real plasmas" (don't exactly know what you mean here) make any difference.

Do plasmas differ from neutral gases wrt the absorption and emission of electromagnetic radiation? In this I'm not interested in lines, only things like refraction and reddening (frequency dependent, broadband, absorption) - this has been partially answered.

There are various processes that happen in plasmas with respect to radiative transport. Like I mention in the reply to Robinson, all these processes are very well explained in R&L, but I will see if I can give a summary soon.

Are plasmas completely opaque to electromagnetic radiation below the plasma frequency? This is actually quite tricky, because real plasmas are not homogeneous, and my original question did not say anything about the source of any EMR. So, to make it a bit more concrete:

* suppose there is an efficient, compact radio (or microwave) transmitter in a homogeneous plasma comprised of fully ionised hydrogen

* suppose the transmitter makes EMR with a frequency that is one (or two) OOM below the plasma frequency

* suppose the plasma density and temperature are unchanged for the duration of our experiment

* suppose there is a sensitive receiver located a long way from the transmitter (>> the DeBye length, >> the wavelength of the transmitter's EMR, etc); perhaps this receiver is in a bubble of cold, non-ionised gas, perhaps this plasma-gas boundary needs to be considered?

Later, not here on the couch.

In what significant ways do extremely dense plasmas behave differently from those of lower density than a good high school science lab vacuum? I'm thinking of white dwarfs and neutron stars, but the cores of high mass stars might also be interesting to examine.

See above, this I cannot discuss easily here on the couch.

[DrRocket]

Well, the thread has been "dead" for a while, but that does not mean that the content is wrong or can be discarded. I just tried to make plasma physics as simple as I can, but from JREF I know that Robinson is not really gifted in math and physics.

But if there are any questions, then I am always willing to answer them.

I think Neiried still has some difficult questions about dusty plasmas and the eddington limit and stuff, that I never got to to answer.

I justy found this thread -- and thanks. The tutorial gives me more food for thought to add to the dE/dt thread. Please don't dumb down the math too far, that often leads to more confusion through less precision. Anyway there isn't anything too fancy here yet.

If you would care to elaborate I am still having a bit of trouble with the notion of magnetic diffusion time. The difficulty is that the derivations that I have seen that get to the diffusion equation (aka heat equation) arrive there after already assuming that dE/dt can be neglected and that E = -1/c curl B (Gaussian units) as in section 10.3 of Jackson's Classical Electrodynamics. This latter condition seems to come from assuming an infinite conductivity while simultaneously imposing some sort of unstated bound on current density.

[korjik]

I justy found this thread -- and thanks. The tutorial gives me more food for thought to add to the dE/dt thread. Please don't dumb down the math too far, that often leads to more confusion through less precision. Anyway there isn't anything too fancy here yet.

If you would care to elaborate I am still having a bit of trouble with the notion of magnetic diffusion time. The difficulty is that the derivations that I have seen that get to the diffusion equation (aka heat equation) arrive there after already assuming that dE/dt can be neglected and that E = -1/c curl B (Gaussian units) as in section 10.3 of Jackson's Classical Electrodynamics. This latter condition seems to come from assuming an infinite conductivity while simultaneously imposing some sort of unstated bound on current density.

I seem to remember that the simplest way to get a diffusion equation is to assume a stationary plasma with no current in it. That should bound J to zero. Dont quote me tho, I havent had lunch yet so I could be out to la la land (bloody diabetes )

[DrRocket]

I seem to remember that the simplest way to get a diffusion equation is to assume a stationary plasma with no current in it. That should bound J to zero. Dont quote me tho, I havent had lunch yet so I could be out to la la land (bloody diabetes )

An assumption like that ought to get to the diffusion equation, and rather quickly. What is done in Jackson, and other places is to use Ohm's law to drive J to zero by letting conductivity become large (infinite), but that only really holds if you can somehow bound J. What Jackson really wants is for J/sigma to go to zero as sigma tends to infinity. So physically you need a way for J/sigma to become small within what one can really expect for large but still finite sigma. Just letting sigma become large does not necessarily drive the ratio to zero (try laying a screwdriver across the terminals of a car battery to see what I mean).

I think the assumption of a stationary plasma would also severely limit the applicability of whatever one derives from it. Somewhere in the motivation for the question is the concept of frozen magnetic field lines, how one justifies that approximation and when the approximation is valid. Without some movement of charged particles (current, unless ion flow and electron flow are rather carefully balanced) I don't think the rest of the problem is going to be interesting.

[korjik]

An assumption like that ought to get to the diffusion equation, and rather quickly. What is done in Jackson, and other places is to use Ohm's law to drive J to zero by letting conductivity become large (infinite), but that only really holds if you can somehow bound J. What Jackson really wants is for J/sigma to go to zero as sigma tends to infinity. So physically you need a way for J/sigma to become small within what one can really expect for large but still finite sigma. Just letting sigma become large does not necessarily drive the ratio to zero (try laying a screwdriver across the terminals of a car battery to see what I mean).

I think the assumption of a stationary plasma would also severely limit the applicability of whatever one derives from it. Somewhere in the motivation for the question is the concept of frozen magnetic field lines, how one justifies that approximation and when the approximation is valid. Without some movement of charged particles (current, unless ion flow and electron flow are rather carefully balanced) I don't think the rest of the problem is going to be interesting.

J is not the internal movement of the plasma tho. If you have an isolated blob of plasma that dosent have any inflow or outflow from the system you are looking at, then you can treat J as zero. That dosent mean that you are considering the plasma as a static blob that doesnt evolve over time, just that you are considering it as isolated. In a magnetized plasma you will still have all sorts of ther stuff happening, all the gyromotions around the field lines, diffusion across the field lines, ect..

This has more applicability than you might think. Take a CME on the way to Earth. After it leaves the sun, you can treat it as a stationary plasma with a frozen in field (IIRC). The fact that it is travelling at several hundred kilometers per second isnt really relevant. Until it hits something, it acts as a stationary blob in its frame of reference.

As for the use in frozen in lines you have to remember that current implies a seperation of charges. In a natural system, especially an isolated one, you cannot sustain a large seperation. You arent generally going to have a large J, at least not for long. So when you compare the small J to a large sigma, you have a small E from Ohm's law.

So, let us take an isolated blob of magnetized ideal plasma. Ideal means that sigma=infinity, and isolated implies J is small. At this point your field is frozen.

Now make things less ideal. As long as the ratio J/sigma is still small, your field is mostly frozen. As long as you are looking at a process that is short duration, you can treat the field as frozen.

Look at the qualifiers. There are alot of restrictions on when you can use the frozen in condition. How you choose wether you should be using it can be more an art form than any real science. You just have to look at the conditions of your plasma and work out wether the approximation is accurate enough, or wether you have to go back and do things the hard way.

[DrRocket]

J is not the internal movement of the plasma tho. If you have an isolated blob of plasma that dosent have any inflow or outflow from the system you are looking at, then you can treat J as zero. That dosent mean that you are considering the plasma as a static blob that doesnt evolve over time, just that you are considering it as isolated. In a magnetized plasma you will still have all sorts of ther stuff happening, all the gyromotions around the field lines, diffusion across the field lines, ect..

So, isolated as used here means that any internal current flow produces no magnetic fields sufficient to materially affect either the phenomena of interest internal to the blob or the external currents that could in turn affect the blob itself. Correct ?

This has more applicability than you might think. Take a CME on the way to Earth. After it leaves the sun, you can treat it as a stationary plasma with a frozen in field (IIRC). The fact that it is travelling at several hundred kilometers per second isnt really relevant. Until it hits something, it acts as a stationary blob in its frame of reference.

I think that I understand and accept this. But I think that you are also saying that one does not have an isolated blob until it has left the sun. So that while the CME is forming on the sun you cannot consider it as isolated.

As for the use in frozen in lines you have to remember that current implies a seperation of charges. In a natural system, especially an isolated one, you cannot sustain a large seperation. You arent generally going to have a large J, at least not for long. So when you compare the small J to a large sigma, you have a small E from Ohm's law.

I don't follow this part. If you have electrons and ions that are mobile you can still get a current even if the mix of the two is effectively neutral. If the ions are moving north and the electrons are moving south then you still have a northerly current, but no macroscopic charge separation. This is essentially what goes on inside a solid conductor. It seems to me that in this case you can get a huge J, particularly if sigma is also large. All you need is an E field or a time-varying B field. It also seems to me that you could get large circulating currents in a "blob" that could generate large B fields and affect what is going on nearby the blob -- so that the blob is not isolated if there is something nearby that can be affected by the B fields.

So, let us take an isolated blob of magnetized ideal plasma. Ideal means that sigma=infinity, and isolated implies J is small. At this point your field is frozen.

Now make things less ideal. As long as the ratio J/sigma is still small, your field is mostly frozen. As long as you are looking at a process that is short duration, you can treat the field as frozen.

I follow this part, once you have a small J. In only a couple of steps you can from here get to a diffusion equation (aka heat equation) and from there justify the frozen field idea (as in Jackson or Landau and Lifshi tz).

Look at the qualifiers. There are alot of restrictions on when you can use the frozen in condition. How you choose wether you should be using it can be more an art form than any real science. You just have to look at the conditions of your plasma and work out wether the approximation is accurate enough, or wether you have to go back and do things the hard way.

That is what I am trying to understand -- the qualifiers. I think I can follow the science, but I don't have the experience for the art. But what you are telling me helps quite a bit.

I am still not clear on how to make sure that J is low. This is at the heart of what I don't understand yet.

I am also a bit confused by another point. This discussion makes it quite plausible that frozen magnetic field lines are appropriate in an isolated and relatively rarefied plasma like a CME. But maybe not so appropriate in a blob of plasma on the sun. On the other hand, Alfven in Cosmical Electrodynamics seems to think that frozen fields are OK in high density plasmas (like on the sun) but not in low density plasmas. (High density means the mean free path of electrons is much less than the Larmor radius and low density the opposite). This seems contrary to what I thought I was getting here. I think I am missing something important, and perhaps it is tied up with the issue of limiting J while sigma becomes large.

[tusenfem]

Sorry guys that I am not there, but several things happen at the same time. Next week I hope to have time again for detailed discussions. However, one little point about magnetic diffusion:

Take Maxwell equations (curl B and curl E) and Omh's law (J = sigma (E + v x B)) and then with a little math you can eliminate from these three equations everything, such that you only end up with magnetic field B, velocity v and conductivity sigma, which is the diffusion equation:

dB/dt = del x ( v x B ) + (c^2/ 4 pi sigma) del^2 B

here the only thing that is supposed it shat sigma is constant in space (so you can take it out of the derivative), there is no assumption that the current is zero or that J/sigma -> 0.

[Nereid]

[...]

here the only thing that is supposed it that sigma is constant in space (so you can take it out of the derivative), [...]

(bold added)

To what extent has this assumption been tested (is it testable?), for a CME (say)?

[DrRocket]

Sorry guys that I am not there, but several things happen at the same time. Next week I hope to have time again for detailed discussions. However, one little point about magnetic diffusion:

Take Maxwell equations (curl B and curl E) and Omh's law (J = sigma (E + v x B)) and then with a little math you can eliminate from these three equations everything, such that you only end up with magnetic field B, velocity v and conductivity sigma, which is the diffusion equation:

dB/dt = del x ( v x B ) + (c^2/ 4 pi sigma) del^2 B

here the only thing that is supposed it shat sigma is constant in space (so you can take it out of the derivative), there is no assumption that the current is zero or that J/sigma -> 0.

I think that to get to dB/dt = del x ( v x B ) + (c^2/ 4 pi sigma) del^2 B you start from Maxwell's equation del x B = 4 pi/c J + 1/c dE/dt then take the curl of that equation to yield

(-c/ sigma) del x J = 1/c del x dE/dt + (c^2/ 4 pi sigma) del^2 B

Also you take Maxwell's del x E = (-1/c) dB/dt and using Ohm's law and the spacial constancy of sigma get

dB/dt = del x (V x B) - (c /sigma) del x J

Now take the equation above for (-c/ sigma) del x J and you get

dB/dt = 1/c del x dE/dt + del x ( v x B ) + (c^2/ 4 pi sigma) del^2 B

The first term is then eliminated by the MHD assumption that the displacement current, 1/c dE/dt, can be neglected. So now we are back to the question as to why the displacement current can be safely neglected. So the problem that I am having is that the justification that has been discussed for neglecting the displacement current involves the notion of magnetic diffusion time, but the notion of magnetic diffusion itself seems to rest on the assumption that one can neglect the displacement current.

BTW, what I am used to seeing as the diffusion question is actually

dB/dt = (c^2/ 4 pi sigma) del^2 B

Which requires the low velocity assumption to eliminate the del x ( v x B ) term. I can see how to justify that assumption.

[korjik]

So, isolated as used here means that any internal current flow produces no magnetic fields sufficient to materially affect either the phenomena of interest internal to the blob or the external currents that could in turn affect the blob itself. Correct ?

Correct

However, chances are that what you are looking at in the plasma has everything to do with the internal currents and fields. EM waves and all that

I think that I understand and accept this. But I think that you are also saying that one does not have an isolated blob until it has left the sun. So that while the CME is forming on the sun you cannot consider it as isolated.

Probably not. Considering the amount of magnetic activity on the sun, I would be suprised if any frozen in condition lasted more than seconds.

I don't follow this part. If you have electrons and ions that are mobile you can still get a current even if the mix of the two is effectively neutral. If the ions are moving north and the electrons are moving south then you still have a northerly current, but no macroscopic charge separation. This is essentially what goes on inside a solid conductor. It seems to me that in this case you can get a huge J, particularly if sigma is also large. All you need is an E field or a time-varying B field. It also seems to me that you could get large circulating currents in a "blob" that could generate large B fields and affect what is going on nearby the blob -- so that the blob is not isolated if there is something nearby that can be affected by the B fields.

You have to think about the differences between a solid conductor carrying current and a plasma. For one, current dosent flow in a conductor at all without an EMF source. Second, plasma is a gas. It will leak out if given a chance. Third electric fields create truly immense forces.

The biggest thing about plasmas is that it is very hard to create a stable EMF across the plasma. The plasma is almost instantly going to move to counteract the EMF. The plasma will either conduct so much current as to discharge the EMF, leaving you with zero E, or the plasma will be pulled apart, to either side of the EMF, and you then only have background (vaccum or gas). So basically, yes you can have a huge J, but not for long.

Next, internal fields are treated differently. Yes, internal E and B and J do stuff. Lots of stuff. The thing is, frozen in condition is more of an external effect, not an internal. Properly done, the frozen in field is considered a background field that is constant, and then there are perterbations of that field caused by the internal effects of the plasma.

So basically, when working with a plasma, you are going to have an external (generally) fixed component to the field and an internal component to the field.

Lastly, if the isolated blob can affect other areas, it isnt isolated.

I follow this part, once you have a small J. In only a couple of steps you can from here get to a diffusion equation (aka heat equation) and from there justify the frozen field idea (as in Jackson or Landau and Lifshi tz).



That is what I am trying to understand -- the qualifiers. I think I can follow the science, but I don't have the experience for the art. But what you are telling me helps quite a bit.

I am still not clear on how to make sure that J is low. This is at the heart of what I don't understand yet.

I am also a bit confused by another point. This discussion makes it quite plausible that frozen magnetic field lines are appropriate in an isolated and relatively rarefied plasma like a CME. But maybe not so appropriate in a blob of plasma on the sun. On the other hand, Alfven in Cosmical Electrodynamics seems to think that frozen fields are OK in high density plasmas (like on the sun) but not in low density plasmas. (High density means the mean free path of electrons is much less than the Larmor radius and low density the opposite). This seems contrary to what I thought I was getting here. I think I am missing something important, and perhaps it is tied up with the issue of limiting J while sigma becomes large.

You have to look at the particulars of your specific plasma.