Answer: In solar conditions, what we call 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 =
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, h the resistivity, and J the
current density. The last term, i.e. the function f(J', grad[vJ, Jv]) is the electron inertial term and is often
ignored if steady-state or quasi -equilibrium conditions are assumed.
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 (i.e. is non-zero). 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 earlier equation can be found by recasting it for the frame of the plasma such that:
E' = E + v X B + (J X B)/ ne e - grad pe/ ne e + h J
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 earlier equation can be found by recasting it for the frame of the plasma such that:
E' = E + v X B + (J X B)/ ne e - grad pe/ ne e + h J
This
can be simplified further if one assumes collisionless effects, and a dominant
2-fluid context so that:
grad
pe/ ne e ® 0
and, the Ohmic resistance term:
h J ®
0
Then: E + v X B = - (J X B)/ ne e
Then: E + v X B = - (J X B)/ ne e
So, the Hall term will be important if it is
roughly the same magnitude as the term on the left hand side of the above
equation. "Hall
MHD" means "Hall
magneto-hydrodynamics" or the setting up of idealized MHD conditions
via the Hall field, or using the Hall term.
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 VA/ Vo [(2 p fi)],
where c is the speed of light, VA is the Alfven velocity. Vo
is characteristic fluid velocity and (2 p fi
) is the ion plasma frequency.
The
Hall electric field intensity, E H , has been defined generically (Bray and
Loughhead, ibid.)[1]
, as:
E H = 1/nec [j
x B/m] =
(j B/ nec m) = j/ sH
Which
is typically ignored in solar flare applications. Sen and White (1972) were
among the first to show a possible role for it in conjunction with the
two-stream (Farley) instability and the triggering of a solar flare using a
“dynamo” model. They argued that a flare
trigger manifests when the electron drift velocity (vd) associated
with the Hall current density ( JH) exceeds the ion thermal velocity
(Vith). Their defined critical threshold value was: vd » 1 km s-1
Their
application depended upon using a sunspot model as depicted
below:
Thus,
the Hall (circuital) current density J H arises from the cross
product V(r) x H(z) (= B z/ m), in the directions
shown. A further point is that the radially directed electric field (E(r)) is anti-parallel to the velocity
direction, and directed toward
the umbral center. Their ansatz is consistent with the “three-finger rule” of electrodynamics, by which a conductor moving
across magnetic lines of force generates an E-field perpendicular to both the
velocity (v) and magnetic induction (B) such that: E = v X B.
In the Sen and White scheme, V(r) denotes a radially- directed (outward) convective-associated
velocity, typically of the order of 1 km s-1. Parker (1979 [2],
Fig. 2) later clarified that convective velocity is more plausibly inward
directed, ending in a “downdraft”
toward the umbral center and below it.
Based on a timed sequence of twenty-four photos in 6 minutes, and an
assumed “Evershed effect”, Schröter
(1962)[3]
inferred outward radial motions in penumbrae (for penumbral bright grains) from
1.0 km s-1 - 2.2 km s-1.
This result was contradicted by Tonjes and Wohl (1982)[4] who found a factor 10-15 reduced magnitude of motion. Sen and White (op. cit.) defined JH in esu-cgs units such that:
JH = ne ec E(r)/
H
= sH E(r).
Where s H is the Hall conductivity.
Based on an assumed H »
1000 G[5],
and ne = 10 11
cm-3 for electron number density, they obtained: s H = 1.44 x 10 9 statmho
Their Hall current density was therefore
computed:
JH » 5 x 10 6 statamps cm-2
From
this they computed a joule dissipation of:
JD = JH2/ seff = 1.7 x 10 4 ergs cm-3 s-1 ,
The order of physical magnitudes was more or less consistent with those measured, and the critical flare energy potentially released also made sense - arriving at the total energy dissipated as E(j) = V(f) x JD x 10 3 s = 2 x 10 32 ergs.
For a flare of duration » 10 3 s, this
meant a power output P(f) » 10 29
J s-1 or the minimum
expected for a moderate flare. However, this model is now mainly of historical interest only given most current sunspot magneto-hydrodynamics invoke force-free fields.
In addition, the model suffers from the problems peculiar to most dynamo sunspot models (e.g. Kan et al's (1983) in proposing significant unobserved motions in the photosphere:
In Kan et al’s (1983) dynamo flare model, it is also required that the dynamo action send currents to specific regions to provide a Lorentz force: (J ⊥ X B). This implies a current system which is non-force free in contradiction to numerous existing observations, and the fact that at the level of the photosphere –chromosphere, the plasma beta is much less than 1. For these and other reasons solar physicists tend to eschew the use of dynamo sunspot models.
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