Understanding Ohm and Hall Diffusion in Cosmic Rays

[Pedro de la Torre]

TL;DR

I know that the diffusion of charged particles in a magnetic field is described as a diffusion tensor with 3 components: Diff parallel to the field, perpendicular and anti-symmetric diffusion. Is this last one that I do not understand.

Summary: I know that the diffusion of charged particles in a magnetic field is described as a diffusion tensor with 3 components: Diff parallel to the field, perpendicular and anti-symmetric diffusion. Is this last one that I do not understand.

Dear community,
I am PhD candidate in astroparticle physics and I study cosmic rays and their propagation in the Galaxy. To describe it we use a diffusive equation, with a diffusion coefficient which basically is a tensor. This tensor has the components of diff coef. parallel to the field, diff coef. perpendicular and anti-symmetric diffusion coefficient. The latter is called hall diffusion coefficient.

For me is not clear what can be the origin of this component of the tensor, but I have found that this Hall diffusion can also be described as a particle drift in a large-scale inhomogeneous magnetic field (e.g. Isenberg and Jokipii, 1979). This explanation is even more confuse for me since they also say that this diffusion is not important for small energies.

Checking more I found that this kind of diffusion is also used to describe movements of matter in accretion disks, and in that case they discern between Ohm and Hall diffusion.

So, my final question is, what does Hall diffusion is? and what is the difference with Ohm diffusion?

Thank you in advance

 

[Astronuc]

It is not unusual to express EM fields or mechanical behavior of fluids/solids with tensors.

https://quantummechanics.ucsd.edu/ph130a/130_notes/node451.html
https://ocw.mit.edu/resources/res-6-003-electromechanical-dynamics-spring-2009/part-ii/chap08_sec_emd.pdf
https://www.if.ufrgs.br/hadrons/espinosa2.pdf

So the treatment of weakly or highly ionized plasmas and the mechanical interactions due to magnetic and electric fields with tensors is expected.

B. P. Pandeyand Mark Wardle, "Hall magnetohydrodynamics of partially ionized plasmas," Mon. Not. R. Astron. Soc.385,2269–2278 (2008) - subscription or purchase required

Mark Wardle & Raquel Salmeron, "Hall diffusion and the magnetorotational instability in protoplanetary discs," Mon. Not. R. Astron. Soc.000, 1–20 (2011) - https://arxiv.org/pdf/1103.3562.pdf
Also - open access - https://academic.oup.com/mnras/article/422/4/2737/1046901

Wardle and Salmeron make the points: "
Ohm and ambipolar diffusion are stabilizing effects, whereas Hall diffusion may be destabilizing or stabilizing depending on whether the initial vertical magnetic field is parallel or antiparallel to the rotation axis, respectively.

Magnetic diffusion allows slippage between the field lines and the fluid, so that there is no longer a direct connection between the relative displacement of the fluid layers and the buckling of the magnetic field lines. One might expect that this effect should always reduce the instability because the displacement of the field would lag that of the fluid. However, this intuition is based on the limit of Ohm or ambipolar diffusion, in which the drift is in the direction of the magnetic stresses and tends to straighten up field lines. Hall diffusion, by contrast, creates a drift orthogonal to the tension forces and may, therefore, enhance or suppress the radial buckling depending on the situation."

See the discussion around Equation 13, with reference to "the inductive (I), Ohm (O), Hall (H) and ambipolar diffusion (A) contributions to the evolution of the magnetic field."

Does one have examples or equations for ambipolar, Hall and Ohm diffusion?