Expert: Philip A. Stahl - 4/6/2009

Question
Can you elaborate on the meaning and significance of the "Hall term"? Also when is it important and what does that mean in the context of "Hall MHD"?

Answer
Hello,

The "Hall term" arises in one electro-dynamic formulation of Ohm's law often used in solar work, and some other areas of astrophysics.

One usually starts with a motional emf arising from:

E = -v X B  +   (J X B)/ n_e e -  grad p_e/n_e e + nJ + f(J', vJ, Jv)

where E is the electric field intensity, v a fluid velocity, B the magnetic induction, p_e the electron pressure (grad is the gradient of it), e the electronic charge, n the resistivity, and J the current density. The last term which is a function f(J', vJ, Jv) is the electron inertial term and is often ignored if steady-state or quasi -equilibrium conditions are assumed.

Now, the *second term* on the right is what we call the "Hall term". It is important to understand that it cannot exist unless the J X B force exists. Thus, it is not going to be found in a force -free magnetic field (see other questions, answers on this) which dictates that:

J X B = 0

That is, one has J, B in the same direction say in a current-carrying coronal arch.

A simpler version of the preceding equation (which makes it easier to get at your next question of when it is important) can be found by recasting the above for the frame of the plasma such that now:

E' = E  +  v X B  +   (J X B)/ n_e e -  grad p_e/ n_e e  +  n J  


This can be simplified further is one assumes collisionless effects, and a dominant 2-fluid context so:

the ambipolar diffusion ( or polarization) term: grad p_e/ n_e e -> 0

the Ohmic resistance term n J  -> 0

Then:

E  +  v X B   =  - (J X B)/ n_e e    

So, the Hall term (RHS) will be important if it is roughly the same magnitude as the term on the LHS.

When will this occur?

Only when the characteristic length of the system is approximately on the order of the Hall scale defined by: L(H) = c V(A)/ V_o [2 pi f_i)

where c is the speed of light, V(A) is the Alfven velocity(google!), V(o) is a characteristic fluid velocity and 2 pi f_i is the ion plasma frequency.

"Hall MHD" means "Hall magneto-hydrodynamics" or the setting up of idealized MHD conditions via the Hall field, or Hall term.

Classification of differing fluid regimes is far beyond the scope of this answer, but you can find any number of good introductory plasma physics text that delve into details (e.g. 'Introduction to Plasma Physics' by Francis F. Chen).

The bottom line overview is that one proceeds by taking moments of the Boltzmann equation.

E.g. the Boltmann eqn. is:  @f/ @t + v*grad f + F/m*@f/@t = (@f/@t)_C

where @ denotes partial derivative, and (@f/@t)_C  is the time rate of change in f due to collisions.

The first moment, which yields a 'two-fluid' (e.g. electron-ion) medium is obtained by integrating the above eqn. with F = q/m (E + v X B). If one then assumes a sufficiently hot plasma so it's collisionless, the term on the RHS, (@f/@t)_C ->  0.

This is the Vlasov equation:

@f/@t + v*grad f + q/m (E + v X B)*@f/@t = 0

The 2nd moment is obtained by multiplying the original eqn. (Boltzmann) by mv then integrating it over dv.

Anyway, the progression by using this procedure is that one gets in succession:


Two -fluid theory (e.g. ions and electrons treated as a separate fluids)

!
!
!
V

One fluid theory (introducing low frequency, long wave length and quasi -neutral approximations, e.g. n_e ~ n_i
!
!
!
V

MHD Theory (proceeds from 1-fluid theory with further assumptions, simplifications)

Without belaboring too many more details, the Ideal MHD theory case revolves around a version of Ohms's law with J X B = 0 so can be written (see earlier generalized Ohm's law form):

J = oE = o(E  +  v X B)  where o is the conductivity


Hall MHD version leaves the Hall term in so again:

E  +  v X B   =  - (J X B)/ n_e e   

Now, in this "Hall MHD" regime, one does have field-freezing but now it is freezing of the magnetic field to the ELECTRON flow, not to the whole bulk velocity flow. In practical terms, what this means is that electron motion can be parceled out from the aggregate motion or simplified "one fluid" -type motion of standard MHD. It means, in a sense, one reaps the benefits of the more accurate two -fluid theory without having to incorporate all its complexities. One just needs to account for the Hall term, as opposed to neglecting it.

Hopefully this makes sense, but if not, feel free to ask follow-ups.