I came across a reference to a paper that should be of interest here. I think it shows that, as you said, one must be careful in applying the frozen flow assumption.
http://www.ann-geophys.net/23/2565/2...-2565-2005.pdf
I came across a reference to a paper that should be of interest here. I think it shows that, as you said, one must be careful in applying the frozen flow assumption.
http://www.ann-geophys.net/23/2565/2...-2565-2005.pdf
Indeed, and that is what most of the space physicists do, Rikkard had a nice publication there. I must say that in several of the papers published by my institute, where appropriate for the assumptions, similar calculations were made, with comparisons of the measured electric field and the calculated -vxB electric field.
Sorry to not have joined in this week, but too much stuff piled up here at work. Getting to a "real averaged" model of the Venusian magnetotail was a little more time consuming than I expected :-)
Have a nice weekend
M
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As Neiried asked this question a long time ago already, I will try to answer it first. I needed to look up some stuff, because the Eddington limit is not coming up often in magnetospheric physics, but anywhooooooo. Also I found that in the text books I had, there was only the general discussion of this phenomenon, which I will give in shorthand below.
Eddington published his model in 1921 in Zeitschrift der Physik (I am trying to get the original paper). His model for hydrostatic equilibrium for a star showed that there was a limit to the luminosity of the star. The equation for hydrostatic equilibrium is:
dp( r ) / dr = - G M( r ) ρ( r ) / r^{2}
which tells us that the change in pressure, when we move radially outward, is dependent on the gravitational interaction of the mass inside radius r and the density at radius r.
An interesting thing about radiation is that it can also produce pressure through e.g. scattering on ions and electrons. It would go too far to give a complete derivation here of the radiative transport but in the end one finds that the luminosity is given by:
L = - (4 π r^{2} c / κ ρ) dp_{r}/dr
here κ is the Rosseland mean opacity (which means averaged over angle and polarization) and we can write that p_{r}, the radiation pressure, is equal to (1-β) p, where p is the total pressure and β describes the part of the gas pressure, which often can be neglected. Putting these two equations together and rearranging a bit we get:
L = (4 π c G M / κ) (1 – β) < (4 π c G M / κ) = L_{Edd}
This is the Eddington luminosity, which states that there can be no hydrostatic equilibrium when this luminosity is exceeded, and e.g. in accretion processes the infall will be stopped by the radiation pressure. (BTW these days in text books it is usual to directly say “gravity and radiative force are equal, and often only the interaction of radiation with electrons is taken into account and then the Rosseland opacity is replaced by the Thompson scattering cross section.)
However, Neiried wanted to know if the fact that plasma is falling in would have a significant effect on this process. Actually, there is basically nothing in text books that shows a change in this luminosity, but there is some literature about it.
First of all we need to think about what can happen in a plasma, and the most logical idea is that there can be magnetic fields, which can upset e.g. the radiative transport equations. Indeed, that is what Canuto, Lodenquai & Ruderman (1971) did, when they discussed Thompson scattering in a strong magnetic field and what they basically found was that: in the presence of a strong magnetic field with Ω_{c} (the cyclotron frequency) exceeding the radiation frequency ω, then the opacity is reduced from the unmagnetized value by a factor of 2(ω/ Ω_{c})^{2}. (There are also effects when one has to take the ion interactions into account, which I will skip for now.)
Another way of influencing the Eddington luminosity is through the other equation, that of hydrostatic equilibrium. One can use the magnetic field to confine the plasma against the radiation pressure. The magnetic tension of the field lines then opposes the outward motion of the plasma driven by the radiation pressure. One can make an estimate of the minimum field that is necessary for this:
B > (2L/R^{2} c)^{1/2}
This is described for an application to soft gamma ray bursters in Katz (1996). This magnetic confinement basically changes the hydrostatic equilibrium equation, and higher luminosities are possible.
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hi
Can someone explain me polarization drift in a simple way?
anybody has some idea of two potential theory?
If I am not mistaken, but doing this now from the top of my head in a hurry, polarization drift happens because of the different propagation speeds for left- and right- handed polarized waves. Any wave form you can build up from a left- and a right-hand polarized wave (which means if you look at it coming at you it is rotating clockwise or counter-clockwise).
Now, because of the properties of a magnetized plasma, these two parts do not have the exact same propagation speed. This means if you have a linearly polarized wave, i.e. equal amounts of left and right, then after a certain period the one component is lagging or has changed its phase with respect to the other. When you add the two waves up with a detector, then you will find that the result does not have the same direction for the linear polarization, but it has rotated slightly.
"to potential theory" you would need to give some more info into how you mean this.
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I haven't kept up with the science and I hope that this is not a silly question:
Do you think the temperature of the of the electrons as well as the ions in an anisotropic plasma is best expressed as a 3 dimensional rank 2 tensor as opposed to a scalar?
The reason why I ask is because the temperature of a medium is related to the velocity of the particles. In an anisotropic (magnetized) plasma, the velocity of the electrons and the ions have a strong directional bias. I understand this to mean that the electron (or ion) temperature (in a small control volume) could have different readings when measured along the x, y, or z axis. In addition, the ion and electron temperatures could be different from each other. Or would it be better to express it as 2 vectors (one for the ions, and one for the electrons).
Thanks in advance for answering.
Last edited by adapa; 2009-Oct-14 at 01:41 AM.
Richard Fitzpatrick in his online notes "The Physics of Plasmas" in the section "Magnetic Drifts", he describes the...
:".. polarization drift by considering what happens when we suddenly impose an electric field on a particle at rest. The particle initially accelerates in the direction of the electric field, but is then deflected by the magnetic force. Thereafter, the particle undergoes conventional gyromotion .. Note that there is no deflection if the electric field is directed parallel to the magnetic field, so this argument only applies to perpendicular electric fields. .. Thus, when an electric field is suddenly switched on in a plasma, there is an initial polarization of the plasma medium caused, predominately, by a displacement of the ions in the direction of the field. If the electric field, in fact, varies continuously in time, then there is a slow drift due to the constantly changing polarization of the plasma medium."
Temperature in a magnetized plasma is indeed tensor like. When we assume that the magnetic field is in the z-direction of our coordinate system then the particles can flow unhindered along the z direction. However in the x and y direction the particles can only gyrate around the magnetic field (because of the Lorentz force). The total velocity of the particle is is now divided up in the parallel and perpendicular velocity, and this can be done in various ways, and thus a difference in those two temperatures is often measure in magnetized plasmas.
An interesting case is when particles with high parallel temperature (velocity) are flowing along the Earth's magnetic field to the magnetic poles. Then because of the adiabatic invariant, the perpendicular velocity (temperature) increases and thus you will find a different temperature ration for either ions or electrons near the poles than near the equator of the same field line.
Note that there are only 2 degrees of freedom in a magnetized plasma, and thus the temperarture tensor is a 3x3 tensor, but the xx and yy components are the same.
The temperature of electons and ions need not be the same, as different processes can act on electrons and ions, and electrons, because of the lower mass are much easier to accelerate. Therefore, you need two rank 2 tensor, one for electrons and one for ions. In the Earth's magnetotail both temperatures are different, but for the solar wind the two temperatures are often equal.
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The temperature of a plasma in space is measured by a plasma instrument. This instrument measures the composition and energy of the incoming ions (and electrons). These instruments scan around and in 4-8 seconds (for e.g. Cluster) or 3 minutes (for Venus Express) and obtain a 4π steradians distribution function. The second moment of the distribution function then gives the temperature of the plasma. Naturally, there is also a magnetometer on the spacecraft (except e.g. at Mars Express, which they regret something awful). The combination of knowing the magnetic field and the 3D distribution function can then be used to calculate the parallel and perpendicular components of the plasma temperature.
So, it's not like a fever thermometer that you just stick in somewhere, it is all based on knowing the particle distribution function.
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First, I would like to thank you for your straightforward answer to a curious layman such as myself. It helped to clarify quite a bit. Now I have even more questions:
1. Because there is primarily a parallel velocity and a perpendicular velocity, would using the metric tensor for a cylindrical manifold be more conducive to making calculations than using that for a flat manifold?
2. If the velocity of the particles is expressible as a rank 2 tensor, is it theoretically possible to calculate the eigenvectors for the tensor and then use these to diagonalize the original tensor into the canonical form? If the answer is yes, it seems like it would be useful in figuring out any resonant frequencies associated with the plasma and possibly even tell if it is stable or unstable (I think).
No problem, that's what this thread is for.
Well, basically you are right, there is a cylindrical symmetry because of the gyration of the particles. Given that, cylindrical coordinates would sound appropriate. So, why don't we do that? The explanation for that is that the EM and plasma differential equations get complicated, when not used in a Cartesian coordinate system. There are terms with r sinφ which enter into it, and that makes it rather tedious.
Yes, one can calculate the eigen vectors of the matrix. Also the temperature matrix is already diagonal, there are no cross terms, so one finds:
(Tpar 0 0;
0 Tperp1 0;
0 0 Tperp2)
Theorecitally Tperp1 and Tperp2 need not be the same, but in case of a magnetized plasma they are the same.
Furthermore, you are on the right track with eigenvalues and eigenvectors, but you are using the wrong tensor. What one does in plasma physics, with respect to waves and instabilities etc. is that one calculates the determinant of the dielectric tensor. Wherever that determinant goes to zero is where a resonance happens. A very good book on this topic is Swanson Plasma Waves.
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Thanks for the good info. If I understood you correctly, then it is essentially expressing the relative permittivity as a tensor as opposed to a scalar. To be honest, I would have never thought of that.
So, by finding the eigenvalues and eigenvectors of the dielectric tensor and then diagonalizing the matrix, one should be able to find the relative permittivities along the major axes. With that one can then calculate the electric fields which can then be used in a linear differential equation to calculate the resonant frequencies. Is that correct? If not, please feel free to nudge me in the right direction. Also, I imagine that with the dielectric tensor, there is a way to calculate refraction of these waves and (possibly even light) through the plasma.
I have another question about plasmas. This is based on my (hopefully correct) understanding that each electron in a neutral plasma will have the effect of its charge screened off by other charged particles. This means that its charge will have a negligible effect at distances much greater than the Debye length. Also, I imagine that the same thing must be simultaneously happening to each ion in the plasma.
Based on this, if all of the ions are of the same element, is it possible that there is a finite number of topological/geometrical arrangements of the electrons relative to the ions that will satisfy these conditions (even with the electrons moving at high speeds)?
In other words, do some types of plasma have some sort of stucture (even if it is a poorly defined structure)?
I realize that these last 2 questions are highly speculative, and that my idle speculations are likely to be way off. However, this is part of my natural learning process when dealing with subjects in which I have no real background.
Once more, thanks for your patience.
Well, the diagonalization usually is not done. The only important part for the resonances/instabilities is where the determinant of the tensor is equal to zero. Basically, before the tensor is created, the equations are already written in such a way that there is a split in parallel and perpendicular (so e.g. the xy xz, yx and yz components of the tensor are 0).
The rest goes a bit too far here to explain, as it would require lots of equations, and for that I should best point you to a book on plasma waves like the one I mentioned above. (If you want to really be kinky you could try Don Melrose's Instabilities in Space and Laboratory Plasmas)
Indeed, charges (be it pos or neg) put into a plasma get screened off. This is described somewhere at the beginning of this thread, the part about the DeBye length/sphere. So, the plasma will re-arrange itself in such a way that the electric field of the extra charge gets "blocked off."
Now, this calculation was done for a "cold" plasma, and basically for a zero temperature plasma, but it also holds for a plasma with a temperature. In the case of a temperature, the particles (ions and electrons) will have a velocity, so a "structure" as you are describing it will not be present. The only thing you can say it that statistically the particles have a distribution "screening off" the extra charge in the plasma.
Now, what you may have heard is about plasma crystals, but in that topic I am not really well versed as that deals with dusty plasmas, which is a whole large topic by itself.
I am looking for a reference about cooling down plasma, in which the plasma ions somehow get into a crystalline structure, but I cannot find it at the moment.
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Thanks for all the great info and the good references. I felt like I have learned quite a bit from this thread even if I didn't say so before. I still need to continue teaching myself the math but that's part of the fun.
Also, another question just popped into my head about plasmas:
How well understood are relativistic plasmas to current day science?
The reason why I ask is that ordinary plasma physics is more than difficult enough as is (Navier Stokes Equations, Electromagnetics, etc.). When we start adding in relativistic mass increase and time dilation, it seems that the math only gets more complicated. Also, it seems that apart from supernovae, there are not many opportunities to study this phenomenon. If there are, what would they be?
Maestro tusenfem, I'm posting this here, because when i got back from work last night I saw the following response:
,However, I am glad that you want to understand the really captivating topic of plasma physics. There is a "plasma physics for dummies" thread, giving an introduction for the physics/math apt members of this board.
so i've taken your advice and come over here to read and learn more...and how! Two words: you rock. I found your explanations of what plasma is and how it works some of the most lucid I have ever encountered, in large part because you have provided the equations and what's more, noted what the quantities in the equations represent. So I was ripping along, all ready to read about plasma collisions, where I hoped to find your descriptions and explanations of z and theta pinches, but then saw that the questions started coming in hot and heavy, you got busy, and poof, end of the road. It's a bit like reading a cliff-hanger novel, so if you have anything else somewhere I'd certainly like to read it. Thanks!
Further reading and research on z-pinches has uncovered some recent work by one A. Ciardi (Maître de Conférences - Lecturer Université Pierre et Marie Curie) who's homepage (http://amrel.obspm.fr/ciardi/index.html) links to some great articles about laboratory simulations of astropysical jets via z-pinch machines. Ciardi writes in a very accessible style, appears to be a thoughtful researcher and was a student/colleague (apparently) of Lebedev at the MAGPIE facility in the UK. Interesting reading. Another vein I've been following led me to an article by Richard F Post (http://www.21stcenturysciencetech.co..._on_fusion.pdf) that gives a brief history of LLNL, as well as the abandonment and recent "rediscovery" of axisymetric tandem mirror devices.
catching up slowly, I would like to ask more about temperature. I gather temperature in a plasma has both velocity and electric charge separation components and the velocities have separate components, parallel and perpendicular to the magnetic field lines and furthermore the electron temperature is or can be different from the ion temperature because of the mass difference. In a gas the atoms share temperature by collisions but I guess in a plasma the ions and electrons "share" temperature through charge forces although I see from the thread that emf does not last long. So..... is a plasma characterised by the mix of temperatures which oscillate within the plasma or is there still a net temperature at any point in the plasma which would correlate to the rate of heat radiation from that point?
As we learned above, plasma behaves like a gas, and thus it has an associated pressure. Also, when the plasma is magnetized, the gas pressure turns from a scalar into a tensor quantity, i.e. the pressure along the magnetic field has a value P_{par} and perpendicular to the magnetic field a value P_{perp} .
When the plasma is magnetized, and thus a magnetic field is present, Maxwell’s equations tell us that there is a pressure and a tension from the field. The total pressure of the magnetoplasma is then given by the sum of the magnetic and plasma pressure.
Now, let’s look at a very simple situation, a 1 dimensional current sheet. This was done in 1962 by Harris (E.G. Harris, On a plasma sheet separating regions of oppositely directed magnetic field, Nuovo Cimento, 23, 115, sorry cannot find an electronic link to the paper) who showed that this problem can be solved in a self-consistant way, either by an MHD or by a kinetic approach. We already know that the current sheet, between oppositely directed magnetic field regions will flow in a direction perpendicular to that magnetic field. In this case we will define tha the field is pointed in the x-direction and the current in the y-direction. What one finds is that the magnetic field strength and the plasma pressure is given by:
where h is a variable describing the thickness of the current sheet. And the interesting thing about this is, is that one can quickly show that the total pressure in this Harris current sheet is constant:
when one assumes that B_{0}^{2}/2μ_{0} = p_{0}. So crossing this current sheet from negative to positive z, one finds that the magnetic pressure first decreases until it is zero at z=0 and then increases again, whereas the plasma pressure is zero far from z=0 and maximum at z=0. This is the typical case of the Earth’s magnetotail, where exactly this situation takes place.
We can do some further math, calculating the the current through the curl of the magnetic field and that shows that:
And a current in a magnetic field creates a jxB force and the spatial varying plasma pressure also creates a force given by the gradient of the pressure. For those who like to twiddle with math, you can show that these two forces are in balance in this simple model:
Now, why go through this? First of all, because this current sheet is a very natural thing, as said above, it happens in the Earth’s magnetotail. Here is a paper of mine where I use this Harris model, where I also show that when this system gets disturbed, the magnetotail can start oscillating in a way that the magnetic pressure and plasma pressure keep on balancing out (more or less), if one increases the other decreases.
But this brings us to the already mentioned Willard Bennett who discussed Magnetically Self-Focussing Streams in 1934. Here we can consider a cylindrical plasma with a background magnetic field along the axis and a current flowing along the axis driven by some potential drop along the cylinder, which can be produced through various processes, but we don’t have to go into details of that.
A current along a magnetic field does not exert a force, as this is given by the cross-product, but we have to keep in mind that that current will also produce an azimuthal magnetic field around the cylinder, which is perpendicular to the current direction. And because this azimuthal magnetic field has circular field lines the jxB force is always inward all around the cylinder.
As we started with a cylindrical plasma, there is also a plasma pressure, and we can just use the perfect gas law here for simplicity: p = N k T for each species.
The knowledge that we have now, the magnetic field created by the current and the pressure of the plasma, we put together in the force balance equation, where we should note that there is a gradient on the right hand side and there is the jxB term with Curl(B)=μ_{0}j, so some vector calculus will have to be done and a bit of integration (which I will leave to the reader for a rainy Sunday afternoon), but in a simple plasma with singly ionized ions and electrons you arrive at the following force balance relation:
where I is the total current in the cylinder, N is the density and T_{e,i} are the electron and ion temperature and k is Boltzmann’s constant.
This tells us that there is a maximum current that can pass through the plasma without causing any problems, and it is given by the equation above. As soon as the current exceeds this limit, then the squeezing force becomes greater than the outward gas pressure of the plasma and there will be compression of the plasma.
Naturally, this is a very simple model, because we have not taken into account various things: there can be gradients in the plasma pressure, there can be multiple species, there can be heating of the plasma during the compression etc. etc. This general problem was attacked by E. A. Witalis in Plasma-physical aspects of charged-particle beams, where a generalized form for the Bennett relation, where changes in the kinetic, electric and magnetic energies are taken into account. I will not give the whole derivation here, you can find a very clear walk though in Anthony Peratt’s book “Physics of the plasma universe” (Chapter 2).
It is necessary to take these terms into account, because this squeezing (or more scientific pinching) of the plasma is used e.g. in tokamaks, where the plasma is held together by the pinching and can also be heated through the pinching.
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Just a quick post to make this thread more easily accessible and a request to "sticky" it, thus making it easier to find in the future. Also, this link to a site called Perspectives on Plasmas including a section on Plasma Astrophysics which has a nice table on space and astrophysical plasma parameters:
http://www.plasmas.org/space-astrophys.htm#research
And finally this part of the site where a number of online tutorials are linked: http://www.plasmas.org/plasma-physics.htm#tutorials
An interesting paper came out yesterday regarding the ionization of dense plasmas, which suggests that ionization may be greater than previously thought and that the Ecker and Kröll model might be a better fit of the observed data than the Stewart-Pyatt model (which is currently used in various simulation codes). The authors note this should be of interest to those working in astrophysics and cosmology, planetary science, and even inertial confinement fusion research.
There's a nice synopsis of the article here: http://physics.aps.org/articles/v5/88
Viewpoint: Extreme X Rays Probe Extreme Matter
Donald Umstadter, Department of Physics and Astronomy, University of Nebraska-Lincoln
Which also has a direct link to a free download of the actual paper:
http://physics.aps.org/featured-arti...ett.109.065002
Direct Measurements of the Ionization Potential Depression in a Dense Plasma
One other interesting aspect of this experiment is, as Umstader notes:
The XFEL work represents, in some sense, a cultural shift. In the past, these measurements were performed in a long-duration campaign by just a few individuals, while Ciricosta et al.’s experiment took more of a blitzkrieg approach: a team of 32 researchers from 10 institutions performed the experiment in a single week. A large collaboration is apparently required to plan and successfully execute an experiment within the short time span allotted to each group at the one-of-a-kind LCLS x-ray laser.
thanks for writing this tusenfem!