Talk:Magnetohydrodynamics

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1 Limits of MHD
2 Voltage at Waterloo Bridge in trivia
3 Magnetohydrodynamics, Electrohydrodynamics, and Electrokinetics
4 Image on the MHD page
5 Ideal MHD equations

[edit] Limits of MHD

"MHD is lacking when electric currents flows through these plasmas and produces filaments, double layers and plasma instabilities."

Aren't filaments essentially a product of the pinch effect, which is MHD? And there are certainly many fluid instabilities. You might be right about double layers. Can you (Iantresman) tell me in three sentences how they form? Art Carlson 06:20, 2005 Jun 8 (UTC)

I removed the offending sentence. Electric current flowing through plasmas, and the formation of "filaments ... and plasma instabilities" are explicitly part of magnetohydrodynamics. The formation of a double layer is an example of breakdown of MHD, but I don't believe that they are actually observed. zowie 19:24, 16 October 2005 (UTC)

I'm not sure I understand your reasoning. (1) What is "explicitly part of magnetohydrodynamics"? (2) If the formation of a double layer is an example of breakdown of MHD, then why can't we say so, as an example? (3) Likewise, aren't field-aligned currents a breakdown of MHD, in which case, why don't we say so. (4) You say that "don't believe that they are actually observed"... do you mean that double layers are not observed, and if so, where? --Iantresman 20:52, 16 October 2005 (UTC)

(1) Electrical currents of all types are allowed in magnetohydrodynamics. That is the whole and entire point of MHD - to deal with the coupling of Maxwell's Equations governing electrical phenomena and the Navier-Stokes Equations governing fluid dynamics. Filament formation and the various instabilities are well known magnetohydrodynamic phenomena; they are predicted by the MHD equations and hence are not examples of the breakdown of the theory.

(2) My bad - transient double layers are worth a mention, especially in the context of magnetic reconnection or electrojet formation. Sorry if I threw out a baby with that particular bathwater. (also, see (4) below).

(3) In the special case of quasi-static, low-beta plasmas (in which all non-magnetic forces are negligible) only field-aligned currents are allowed -- but in high-beta plasmas (in which the plasma gas pressure dominates over the magnetic pressure) non-field-aligned currents are also allowed. The important effect here is the Lorenz force, which is proportional to J x B (where J is the electric current and B is the magnetic field). In quasi-static, low-beta plasmas, J x B must be near zero everywhere, which means that J (if it exists) must be nearly parallel to B -- ie field-aligned. In other systems, the JxB force can be balanced by inertial forces ('mass times acceleration'; non-static) or by other forces (such as a gas pressure gradient or gravity; non-low-beta).

(4) Hmmm... When I typed that I was thinking of non-transient resistive regions in a plasma; but I had a look at some of the review articles in the double layer article (I think you may have linked to them; thanks!), and my mind is changed. I'm used to thinking of double layers as transient phenomena, perhaps related to anomalous resistivity in magnetic reconnection -- but I was surprised to read that relatively stable double layers can form in lab plasmas and are thought (by at least some) to form in astrophysical plasmas as well. So, I stand corrected here. Thanks!

Cheers, zowie 00:51, 17 October 2005 (UTC)

Is there a contradiction between (1) you say "Electrical currents of all types are allowed in magnetohydrodynamics.", but then in (3) ".. in low-beta plasmas .. only field-aligned currents are allowed". Does this imply that non-field-aligned currents are not allowed in low-beta plasmas, or not described. And if such currents are found, is that a breakdown?

Non-field-aligned currents are simply not force-free; they violate a particular further simplification of MHD. Hmmm... A good analogy: parasitic capacitance violates the assumptions of basic circuit design, because normally one assumes that the circuit traces are perfectly isolated. That doesn't mean Maxwell's Equations are wrong -- just that you can't use that particular simplifying assumption. zowie 15:11, 17 October 2005 (UTC)

I get the impression from Alfvén that certain currents in certain plasmas, do result in a breakdown of MHD?

The usual MHD equations predict the behavior of the fluid plasma for all finite configurations of electric current. Ideal MHD predicts the formation of ideal current sheets, which are not physical (ideal current sheets are infinitely thin and carry an infinite current density); but they are only singularities of the first kind (integrating across them gives a finite amount of current). Non-ideal MHD does not suffer from that problem, because the plasma resistivity limits the thickness of the sheet.

And that the electric field across a double layer resulting in beams and jets (electric currents?) are also a breakdown?

One normally assumes that the electric field is zero everywhere (ideal) or near-zero everywhere (non-ideal); double layers are not well described by the normal MHD equations. Of course one may augment them with some kind of anomalous resistivity term that describes double layer formation -- but that is outside the normal theory.

Do you have access to Alfvén's Cosmic Plasmas (1981)? --Iantresman 13:41, 17 October 2005 (UTC)

I do not have it in my office. I generally work with Priest's books ("Solar MHD" and "Magnetic reconnection").

I'll post (here in the discussion) what I mean by "the usual MHD equations" -- but I want to make sure I have a reference handy, so I don't muck it up by doing it from memory. That way at least we won't be talking past each other. zowie 15:11, 17 October 2005 (UTC)

Since double layers may generate quite large electric fields, (up to 10^4V in the auroral electrojets), can we say that MHD breaks down when significant non-field aligned electric currents are generated, for example, in in some double layers (ie. oblique double layers)? --Iantresman 15:55, 17 October 2005 (UTC)

The MHD equations are (from Priest's _Magnetic_Reconnection_):

Mass conservation:

$\frac{d\rho}{dt}=\frac{\partial\rho}{\partial t} + v\cdot y\nabla\rho = -\rho\nabla\cdot v$ (where

ρ

is the density and

v

is the plasma velocity)

Momentum conservation:

$\rho\frac{dv}{dt} = -\nabla p+ j\times B + \nabla\cdot S + F_g$ (where

p

the normal gas pressure,

j

is electric current,

S

is the viscous stress tensor [so $\nabla\cdot S$ is itself a vector], and

F g

is any externally applied volume force such as gravity) This is very similar to the Navier-Stokes momentum equation for fluids, (which itself is a fancy way of writing "ma=F"), except that it includes the Lorenz force $j\times B$ as one of the force terms.

Internal energy conservation:

$\rho\frac{de}{dt} + p\nabla\cdot v = \nabla\cdot(\kappa\cdot\nabla T) + (\eta_e\cdot j)\cdot j + Q_v - Q_r$ (where

e

is the internal energy per unit mass [something like temperature over the molecular mass],

κ

is the thermal conductivity tensor [indicating direction and strength of thermal flux, given a direction and strength of thermal gradient],

η e

is the electrical resistivity tensor [direction and strength of the electric field given a direction and strength of current flow], and

Q v

and

Q r

are the viscous (friction) heating and radiative loss terms, respectively).

Faraday's Law:

$\nabla\times E = -\frac{\partial B}{\partial t}$ (this is the electric induction equation)

Ampere's Law:

$\nabla\times B = \mu j$ (here,

μ

is the permeability of free space; it depends on the system of units you like)

Gauss's Law:

$\nabla\cdot B = 0$ (no magnetic monopoles)

Ohm's Law:

$E = \eta_e\cdot j - v\times B$ (Neglecting

η e

is what yields reconnection-free "ideal MHD")

Equation of State (ideal gas law)

$p = \rho \frac{k_B}{\mu_{av}} T$ (this is just the familiar PV = nRT, rewritten --

μ a v

is the average particle mass in the plasma, not to be confused with

μ

, the permeability of free space!)

It's usually more convenient to combine Faraday's Law, Ampere's Law, and Ohm's Law to eliminate the $j$ and $E$ terms -- then you get the magnetic induction equation:

$\frac{\partial B}{\partial t} = \nabla \times (v \times B) - \nabla \times ( (\frac{\eta_e}{\mu}) \nabla \times B)$ , which describes time evolution of the magnetic field in terms only of the velocity field and the resistivity. This makes it obvious why "ideal MHD" is so "ideal": turning off the resistivity completely eliminates one of two reasons why the magnetic field can change. The left-hand term describes the magnetic field being carried around by bulk motion; the right-hand term describes the decay of the magnetic field. Incidentally, keeping the

η e

inside that first curl operator is important for the physics you're interested in -- double-layer formation.

If the resistivity is constant across the whole plasma, then

η e

is just a constant and the right hand side gets even more simple -- $\nabla\times\nabla\times B$ is just $\nabla^2B$ , and you get the old, familiar RHS of the diffusion equation. That diffusion term is what prevents the formation of current sheet singularities in resistive MHD: in an very thin current sheet, $\nabla^2B$ , would be very large so $\frac{\partial B}{\partial t}$ would also be very large -- in the corrective direction.

If the resistivity varies in space, you can't commute it through the outer $\nabla$ , so you get additional terms that describe evolution of the magnetic field near an anomalously resistive area (such as a double layer or condensed region or whatever). The equations above can't predict that change in resistivity -- that is external to the system of equations. In that sense, the system "fails" near double layers: you have to put in more information about the resistivity of the plasma that isn't present in the original equations.

Anyway, the upshot of all of this stuff is that the physical equations don't fail simply because resistivity varies or there are field-aligned electric fields or whatever. Some idealizations may fail, but the fluid approach (informed by the underlyling particle kinematics) remains sound. I suppose it would be useful to list explicitly the assumptions most people use when working in this field ("assumptions of MHD") but I'm all out of steam for now. zowie 17:57, 17 October 2005 (UTC)

Thanks for all that, very impressive. Have you thought about transferring it to the MHD page proper? I can certainly recomend Alfvén's Cosmic Plasmas. As the man who devloped MHD, he will explain far better than I can, some of the points I am trying to convey, and you'd understand it far better than I can. His other book, co-authored with Carl-Gunne Fälthammar is Cosmical Electrodynamics (2nd Ed. 1963) is also recommended. --Iantresman 23:29, 17 October 2005 (UTC)

Well, not so impressive maybe -- but plenty tedious, which is why most folks prefer to wave their hands under simplifying assumptions! :-) You're probably right -- the equations and usual assumptions should be present. But I won't have time to look at it again for a few more days. Anyone else want to see this stuff in the article itself? zowie 23:58, 17 October 2005 (UTC)

There is a good reason that MHD is commonly treated with suspicion: it is often applied to plasmas where the assumptions of a locally thermal plasma do not hold. In particular, fusion, space and astrophysical plasmas are often so collisionless that the collisions can be ignored, and there is no reason for the plasma to be thermal. The question is usually not whether MHD can rigourously be applied (it cannot) but whether it still describes the qualitative physics, and approximates the quantitative physics (MHD is still usually pretty useful). For example, field aligned electric fields arise in collisionless plasmas due to kinetic effects, and it is usually difficult to even get the large scale physics correct with a fluid model (the Aurora is a case in point). Even large scale structures like the Earth's bow shock are only approximately resolved by MHD. Beams in collisionless plasmas can add a fair amount of pressure and move things around. But on the other hand, (fast) magnetic reconnection is now believed to be well modelled by Hall MHD (as long as you are interested in overall shape and timescales and not the details of particle acceleration).

So there probably needs to be a section on where MHD is valid, and why MHD is still useful even when not strictly valid.--Dashpool 17:09, 19 May 2006 (UTC)

Hmm, Dashpool -- do you have a good reference on Hall MHD and astrophysical magnetic reconnection? At least in the solar physics community I hear no clear consensus on how reconnection proceeds. People talk of anomalous resistivity and possible accelerating mechanisms to make the jump from Sweet-Parker type reconnection to faster modes like Petschek reconnection, but there's still an ongoing debate over things like why the energy scaling law appears linear in magnetic field strength (rather than quadratic). Pevtsov et al (2001?) observed the linear scaling law over a huge number of orders of magnitude -- something like 12 orders of magnitude in total magnetic flux, in astrophysical systems from solar bright points to quasars. Schrijver et al (2004?) did an interesting study where they examined forward models of coronal plasmas and tried to match the morphology of the observed corona. Longcope (several references over the last few years) seems to think, like me, that most of the energy is released NOT near the reconnection site but farther away. I've been modeling solar MHD systems using a simple ad-hoc anomalous resistivity that turns on when the current density gets too high -- but I don't think that anyone (yet) has a clear and unambiguous understanding of why and when such a resistivity should happen. zowie 15:24, 22 May 2006 (UTC)

You've probably already seen it, but can I suggest: Hall effects on magnetic relaxation, Turner, Leaf, IEEE Transactions on Plasma Science (ISSN 0093-3813), vol. PS-14, Dec. 1986, p. 849-857. --Iantresman 17:57, 22 May 2006 (UTC)

Probably I need to add some caveats and reduce the force of the statement I made earlier: even if I think an answer is emerging, there is still substantial controversy about reconnection mechanisms, and probably wikipedia isn't the place for picking a winner. Shay+Drake JGR 2001 is a good example of the school of thought I'm thinking about wrt Hall reconnection. I'm not sure that the energy scaling law you mentioned conclusively rules out fluid models of reconnection: Pevstov even mentions in his conclusion that he thinks models involving heating via Alfvén modes (mostly just MHD physics) might explain the power law dependence. Why do you think this scaling rules out reconnection models based on the Hall effect? I'm not sure that Hall MHD is inconsistent with energy release 'far from the reconnection region' either: geometrical effects are important. Also, the question is whether Hall MHD predicts global structures/timescales, rather than details. I think the best information on reconnection comes from plasmas which are well characterised, with in-situ measurements, and for these plasmas there is strong evidence for Hall MHD scaling. For example, the MRX (magnetic reconnection experiment) people presented results at the ICPP conference last week showing very good agreement with timescales expected from Hall MHD. Measurements from WIND in the magnetotail support the existence of quadrupole fields expected in Hall MHD reconnection (Oieroset, Nature 412, p414, 2001). The question of why and when anomalous resistivity arises is avoided in Hall MHD, because the (global) physics becomes insensitive to the resistivity mechanism. My view is that that reconnection in relatively simple scenarios is becoming fairly well understood, and that the problem is to understand the complex systems: but a Hall MHD simulation of a decent chunk of the corona would be difficult, and I don't know of any references on Hall MHD for solar plasmas. From the point of view of writing the article, maybe we could say "resistive MHD simulations usually find rates of reconnection far lower than those observed in systems with low collisionality, but there is evidence that the two-fluid MHD model can predict the rate of magnetic reconnection in many of these systems.". --Dashpool 16:39, 28 May 2006 (UTC)

that wording sounds fine to me. Thanks very much for the Shay & Drake reference (And thanks, Ian, for the Leaf reference as well). To clarify, I can't really opine over the role of the Hall effect on reconnection rates, since I haven't thought much about it -- I just wanted to find a place where I could read more about the physics... Kind regards, zowie 17:21, 28 May 2006 (UTC)

I have reorganised the Ideal MHD section to try to make it more logical and explain the various limitations clearly, and separate out the ideas about current sheets. Dashpool 14:56, 9 July 2006 (UTC) I have removed some material (the writeup on infinitely thin current sheets seemed spurious), and removed some of the references to individual physicists. We just need to explain the physics here, rather than the (in fact, uncontroversial) viewpoints of individuals.Dashpool 15:19, 9 July 2006 (UTC)

[edit] Voltage at Waterloo Bridge in trivia

Would someone do this calculation? I did it about forty years ago at school and remember the answer as one volt, but when I do it now I get a couple of orders of magnitude less. However I am now rusty on these things and they have changed to SI units, so someone should recheck my calculation. Assume that the water is flowing at one metre per second and the river is 200 metres wide. In London the earth’s magnetic field is 55,000 nanotesla and the dip angle is 67 degrees (it is zero at the equator and 90 degrees at the magnetic poles). You then have to invoke Fleming's right hand contortion to get the direction. JMcC 11:07, 20 June 2006 (UTC)

[edit] Magnetohydrodynamics, Electrohydrodynamics, and Electrokinetics

What's the difference? — Omegatron 20:22, 26 June 2006 (UTC)

To get things going:

"In more precise terms, EHD is the study of fluid flow behavior under the influence of electric fields. Electrohydrodynamics is, in essence, the mirror of electrokinetics, which is concerned with the thrust effect generated by these non-conducting media on the electric field generating apparatuses. As expected, the terms electrohydrodynamics and electrokinetics can be easily interchanged depending on the aspect used to describe the same system. For convenience, this section contains information concerning the hydrodynamic aspect of the apparatuses." [1]
"Electrohydrodynamics deals with fluid motion induced by electric fields." [2]
"Electrokinetics involves study of motion of liquid or particles under the action of electric field." [3]
"Magnetohydrodynamics (MHD). This is the theory of the macroscopic interaction of electrically conducting fluids with a magnetic field." [4]
"Magnetohydrodynamics (or MHD for short) is the study of the interaction between a magnetic field and a plasma treated as a continuous medium" [5]]
"[magnetohydrodynamics] , study of the motions of electrically conducting fluids and their interactions with magnetic fields." [6]

Ok, nevermind about MHD. I see the difference now. EHD and electrokinetics are not clear though. — Omegatron 23:29, 26 June 2006 (UTC)

I don't forget gravitoelectrodynamics [7]--Iantresman 23:46, 26 June 2006 (UTC)

That's not really related. These three are all involved with electrically conducting fluids. — Omegatron 23:56, 26 June 2006 (UTC)

[edit] Image on the MHD page

I think a picture without a scale and a legend (what is plotted?) is really not something that should be on wikipedia USferdinand 21:38, 16 January 2007 (UTC)

[edit] Ideal MHD equations

I think a form of the ideal MHD equations should be on the MHD page. The references to the single equation pages is confusing. In fact, the momentum equation page does not have the differential equation, and the Ampere's Law page only refers to the integrated form of the equation. I am fine with the Priest equations even though I prefer the divergenceless form of the MHD equations.

${\partial \rho \over \partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$

${\partial \rho \mathbf{u} \over \partial t} + \nabla \cdot (\rho \mathbf{u} \mathbf{u} + (p +{B^2 \over 8 \pi} )\mathbf{I} - {\mathbf{B}\mathbf{B} \over 4 \pi}) = \mathbf{F}$

${\partial \mathbf{B} \over \partial t} + \nabla \cdot (\mathbf{u} \mathbf{B}-\mathbf{B}\mathbf{u}) = 0$

where $ρ$ is density, t is time, B is the magnetic field, u is fluid velocity, and F is the forces by unit volume (such as $\rho \mathbf{g}$ for the gravity).

Also a reference to the divergenceless nature of B should be added. A entry on numerical MHD (8-wave formulation) is, in my opinion, needed. USferdinand 21:54, 16 January 2007 (UTC)