Fig.: Field-freezing. The plasma flow represents the motion relative to the existing (B)-field(say in a local solar field), and its "cutting" action on the field line leads to a change in magnetic flux and induced current. The conductivity of the plasma would be "infinite" given its high Reynolds number. The only way to avoid this unrealistic situation is for the plasma to be constrained to follow B-field lines rather than cross them. This is what is meant by field freezing. This continues the series for astronomy and space physics questions, with this one in the latter category:
Question: What exactly is the Hall term, and the Hall MHD term and how are they used - in what contexts?
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 + eta J + 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, eta, 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. (See diagram) Thus, it's not going to be found in a force -free magnetic field 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)/ ne e - grad p_e/ n_e e + (eta) 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 pe/ n_e e -> 0
and,the Ohmic resistance term: eta 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.
"Hall MHD" means "Hall magneto-hydrodynamics" or the setting up of idealized MHD conditions via the Hall field, or Hall term. ("Idealized MHD" means the magnetic field is frozen into the plasma). 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)/ Vo [(2 p f_i)]
(where c is the speed of light, V(A) is the Alfven velocity. V(o) is a characteristic fluid velocity and (2 p f_i ) is the ion plasma frequency.
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.
[*] Where f(J’, vJ, Jv) = m_e/ n_e e^2 dJ/ dt + Ñ × [Jv + vJ] ]
Question: What exactly is the Hall term, and the Hall MHD term and how are they used - in what contexts?
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 + eta J + 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, eta, 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. (See diagram) Thus, it's not going to be found in a force -free magnetic field 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)/ ne e - grad p_e/ n_e e + (eta) 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 pe/ n_e e -> 0
and,the Ohmic resistance term: eta 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.
"Hall MHD" means "Hall magneto-hydrodynamics" or the setting up of idealized MHD conditions via the Hall field, or Hall term. ("Idealized MHD" means the magnetic field is frozen into the plasma). 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)/ Vo [(2 p f_i)]
(where c is the speed of light, V(A) is the Alfven velocity. V(o) is a characteristic fluid velocity and (2 p f_i ) is the ion plasma frequency.
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.
[*] Where f(J’, vJ, Jv) = m_e/ n_e e^2 dJ/ dt + Ñ × [Jv + vJ] ]
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