In solar physics, it is customary to use magnetic diffusivities which are orders of magnitude larger than kinetic values. But Parker (Space Sci. Rev. 144, 15 2009) points to fundamental difficulties with this concept: the back reaction of the Lorentz force on the plasma frozen to the field on scales above the real diffusion scale can stop the motion of fields through conducting plasma before it reaches the diffusion scale. Under what conditions might a "turbulent diffusivity" be a well grounded concept for highly conducting plasmas, like those in the Sun?
This question is related to the ability of magnetic field to reconnect in highly conducting fluids. Indeed, if the magnetic fields are frozen into fluids in accordance with the Alfven theorem, then there is no way that magnetic fields can diffuse at the rate given by turbulent diffusivities.
My answer to this question is based on the theoretical study in Lazarian & Vishniac 1999
http://adsabs.harvard.edu/abs/1999ApJ...517..700L
where we calculated the rates of magnetic reconnection in turbulent fluids and demonstrated that reconnection gets fast in the presence of turbulence. Our predictions were later numerically confirmed in
http://adsabs.harvard.edu/abs/2009ApJ...700...63K
and also related to the theoretical concept of Richarson diffusion in
http://adsabs.harvard.edu/abs/2011ApJ...743...51E
which for magnetic fluid was recently numerically confirmed in
http://adsabs.harvard.edu/abs/2013Natur.497..466E
All in all, these papers prove that magnetic field is not frozen in within turbulent (as opposed to laminar) conductive fluids and the concept of turbulent diffusivity is therefore applicable. The rates of such turbulent diffusivity are reduced for the subAlfvenic turbulence. I discuss some of these issues in my review dealing with star formation:
http://adsabs.harvard.edu/abs/2011arXiv1111.0694L
the updated version of which should be published in Space Science Reviews soon and is available upon a request.
Admittedly, I am more familiar with fusion plasmas. All the same, it seems to me that braiding of magnetic field lines should allow transport of both mass, energy and momentum on a much larger scale than predicted by kinetic treatment of collisions. We could expect such braiding to occur e.g. near separatrices, X-points and the like. Furthermore, if the Larmor radius is large enough then braiding is likely to make particle motion stochastic. This is particularly relevant for ions, because theirlarge mass raise both their Larmor radius and their contribution to viscosity. Turbulent diffusivities are therefore likely to play a relevant role locally at least, i.e. whenever the magnetic field is strongly non-uniform.
I totally agree that this is a key question about the solar corona and I myself am very interested in it. Well, there is evidence for higher diffusivity from observations (Nakariakov et al. 1999). On the theoretical side, it has been proposed that the cascade down to small scales eventually leads to dissipation through Alfven waves (van Ballegooijen et al. 2011). More can be said by scientists working on plasma turbulence.
I am glad that Alex has addressed this, as it is a question we discussed when he was visiting CfA over the summer. The application of reconnection diffusion I think addresses the issue in a robust way. There remains the question of testing the theory for large magnetic prandtl numbers (ratio of viscosity to magnetic diffusivity). When it is large (highly conducting viscous fluid) the pile up of small scale magnetic fields is most extreme. I don't know of any 3-D MHD turbulent simulations in the Prandtl number ~100 regime.
My answer is as follows. The turbulent diffusion decreases the spatial scale of magnetic field. In the simple case of the isotropic turbulence, we can meet two different situations. These cases have the equal probabilities. In one case the flow brings together the field lines which have the same direction. It produces the local increase of the field and it results to increase of the Lorenz force as well. Indeed, the Lorenz force can prevent the further motions locally. However, it is unlikely because there is the opposite situation when the motions brings together the field lines of the oposite polarities. In this case nothing prevent the field lines to reconnect. Thus, in these two opposite situations the second process plays the most important role. I'm fully agree with comment by Prof Lazarian in turbulent media the reconnection can go faster.
Very good question, and highly relevant to much of current research in solar physics. There is a lot of theoretical work on this that claims "yes" as an answer. But what you really want is an experimental confirmation. It is difficult to reproduce coronal plasma in the lab. There is some very recent work however that gives a strongly positive indication, not proof yet I would say. See: http://physics.aps.org/articles/v5/141. Note that the observed coupling between Alfven waves gets around Parker's objection (I think, not entirely sure yet).
The issue of high Prandtl number that Ed has mentioned is an important one. It is addressed in
http://arxiv.org/abs/1304.3133
where it is shown that the magnetic diffusion is marginally changed on the small scales by the Prandtl number being high (see logarithmic dependence in Eq. 8). At the scales less than that given by Eq. 8 the magnetic diffusion is suppressed.
My understanding of the original Sweet & Parker mechanism for reconnection is that the only body force that can drive the flow which advects the magnetic field is the magnetic force. For applications of this idea to solar flares, this constraint on the dynamics appears quite sound. By constraint, I mean that in the Sweet-Parker scenario there is no other force than the magnetic force that can drive the dynamics. In Sweet-Parker, the flow driven by the magnetic force is laminar, not chaotic neither turbulent.
On the other hand, Lazarian et al. have proposed that a turbulent flow can produce fast reconnection. Koval et al. have conducted a numerical test of this idea. They drive a turbulent flow localized around the reconnection layer. This flow is driven by an external body force, not the magnetic force. Can you Alex, tell me what external, non-magnetic body force is capable to be localized around the reconnection layer to drive turbulence there? I can't find any sound example.
"Turbulent diffusivity" is a well grounded concept for highly conducting plasmas if the typical scale of the turbulence is much shorter than the plasma scale ( diffusion scale). As example, I may refer to the ion-acoustic turbulence (e.g. V.Yu.Bychenkov, O.M.Gradov, and G.A.Chokparova, Generation of
quasistationary magnetic fields in a turbulent laser plasma. Sov. J. Plasma
Phys., 1984, v.10, N4, 430-433) which makes diffusion faster..
I realized I should made myself clear about the reasons I participate to this forum. I am genuinely interested in the problem that was posed. The reason is that I have worked on non-stationnary reconnection, called forced reconnection ( in the fusion litterature, the Taylor problem of forced magnetic reconnection, treated first by Hahm & Kulsrud for neutral sheets and more recently by Vekstein & myself at X-points ) . In forced, non steady reconnection there are 2 time scales involved. One is related to the formation of the current sheet ( an Alfven time) , the second is the dissipation time of the current sheet (a Sweet-Parker time for laminar fields). This is ordered in time in such a way that the current builds up first and then dissipates via reconnection. A major problem with steady fast dissipation is that it does not allow for the first phase which is the build-up of magnetic energy in the form of a current sheet.
Basically , turbulence does not allow for the reconnection to proceed because it does not allow for the accumulation of magnetic energy via the formation of the current sheet. Magnetic energy is dissipated at the same rate, at all time, than it is injected, in steady turbulence. Basically, for forced reconnection to proceed, we need a time lag. If turbulence is generated it needs to be so after the formation of the current sheet, not before, in order to allow the current sheet to exist. Turbulence can be triggered by currents. But in the MHD framework the type of turbulence ( I mean complex dynamics) triggered by unstable current sheets is far from being close to the Kolmogorov picture (and variants). Here I base my statement on numerical MHD simulations not having an external body force (of still unknown origin). So yes, I 'd like to understand the nature of the force producing the turbulence that broadens the current thickness to a size independent of the resistivity and that lags (comes after) the formation of the current sheet. Why not current driven electrostatic acoustic waves the former said ? This is an idea as old as Grisha Vekstein (fairly old) . There are also these calculations by Bellan that shows that for the range of beta I am interested in ( solar corona), Alfven waves are emitted first. Alex, can we make that idea works within your scenario for turbulent acceleration of reconnection rates by weak Alfvenic turbulence? What I understand about it is that even weak at the injection it becomes strong. Why I d prefer it to be current driven (instead of a fictious body force that lags the current sheet formation) is that it allows for the current sheet to build.
OK im realizing i am becoming to passionate, so if anyone is interested we can continue the conversation by any other means.
Best regards
Nicolas
Dear Bian, I have been impressed by your words 'independent of the resistivity '. I wonder if the critical size of physically relevant regions (e.g. near X-points) can be comparable either to ion Larmor size or to collisionless ion skin depth. In both cases MHD fails, and is to replaced by Hall MHD. The relevance of electrical resistivity to the estimate of the current thickness becomes questionable.
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