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Coronal Heating: Basic 1-Fluid Plasma Equations
I’m going to begin by writing down some basic one-fluid plasma equations that provide a
framework for thinking about coronal heating processes. To derive these equations (which I’m
not doing here), we could first write down the equations for electrons and each ion species and
then sum them to get one-fluid plasma equations. Since the gas may be partially ionized, as it is
in the lower transition region and upper chromosphere, I’m going to include source and sink
terms in the mass conservation equation, since in the partially ionized region, the loss of the
ionized gas through recombination processes and its production through ionization processes
may not be negligible. An alternative approach to this is to write down the equations for each
species (electrons, ions, neutrals) separately, and that’s the way we actually address the
chromosphere-corona problem, but for what I want to illustrate here, that is too cumbersome.
For now, let’s assume we’re dealing with a hydrogen plasma (composed of electrons,
protons, and hydrogen atoms). Then the variables we’ll be working with are as follows:
n   n p  n e   number density
u   u p   flow velocity
p  p p  pe  n kB Tp  Te   pressure
q  heat flux density
J   n  u p  ue   electric current density
E  electric field
B  magnetic field
   e n p  ne   electric charge density
Here, T p and Te are the proton and electron temperatures, kB is the Boltzmann constant, and e
is the magnitude of the electron charge. Although I am using the term “magnetic field” for B ,
which is standard terminology in our area of research, many authors (such as Jackson or
Panofsky and Phillips) use the term “magnetic induction” for B , reserving the term magnetic
field for H .
Now I’ll write down the basic equations, beginning with the mass conservation equation,
where we represent the particle mass by m   m p  . Recall that I’m going to include a
source/sink term in this equation, and this implies the existence of related terms in the
momentum and energy equations, which I am not going to include at the present time.
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n m
   n m u   S
t
(1)
We will write the momentum and energy equations in this same general “conservation” form. In
the case of (1) this conservation form can be described in words by saying that the time rate of
change of mass in an infinitesimal volume (  nm  t ) is given by the rate at which mass is
flowing into that volume (   n m u  ) plus the net rate at which mass is being produced in that
volume ( S ). The same sort of verbal representation can be given to the following momentum
and energy equations.
In the momentum equation, I am going to write the pressure term as a force (i.e., the
pressure gradient force), which means I’m treating it as a momentum “source” (equivalent to the
mass source, S , in (1)). We could equally well treat it as a contributor to the momentum flux
density, just as we could the viscosity. (Together, the pressure gradient term and the viscous
term make up the divergence of the pressure tensor, which can be thought of as the divergence of
one part of the momentum flux density—the other part being the momentum flux density
associated with bulk flow.) Our momentum equation thus takes the form

GM
n m u     n m u u    p    π  J  B  n m 2 eˆ r  FR  Ff
t
r
(2)
where G is the gravitation constant, M the solar mass, π is the viscous stress tensor, FR is the
radiative force (not generally important for us), and Ff is the frictional force (owing to the
presence of hydrogen atoms). In principle, there is also a thermal force term, although this is not
as important for ion-neutral interactions as it is for ion-ion interactions in a multi-species
description. As noted earlier, there should also be a source term associated with the mass source
term, but we are not including that here. Finally, in the context of the earlier discussion, we
could have used a pressure tensor ( P ), in which case   P   p    π would appear as part of
the divergence term on the left side of (2), instead of as a source term (i.e., a force) on the right
side.
When the momentum equation (2) is derived, the electron and ion momentum equations
are added to each other. If we subtract these two equations instead, we obtain what is called the
generalized Ohm’s law:
me
n e2
 J


1
1
1
 uJ  Ju 
J J   E  u  B  η  J 
JB
 pe

ne
ne
ne


 t
(3)
where me is the electron mass, and η is the electrical resistivity tensor. Only the first three
terms on the right side of (3) are included in the ordinary Ohm’s law for a plasma, and for a
perfectly conducting plasma, only the first two terms. The terms on the left side of (3) are the
inertial terms and must be considered when rapid time variations or sharp spatial gradients are
present. The so-called Hall term is the fourth term on the right side. Note that the resistivity can
often be treated as a scalar (rather than a tensor); the tensor nature generally must be considered
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in a weakly ionized plasma, such as the E-region of the terrestrial ionosphere, and perhaps in
parts of the solar chromosphere (particularly, near the temperature minimum).
Now we turn to energy balance, which in conservation form can be written


 1
p 

2
    12 n m u 2 u 
p u  u π  q 
2nmu 


t 
 1 
 1


(4)
GM


J  E  u   n m 2 eˆ r  FR    u  uH   Ff
r


Here,  is the adiabatic index (i.e., the ratio of specific heats), which for three degrees of
freedom (as in our case) is 5 3 . Except for the J  E term, the source terms on the right side of
(4) describe the change in energy of our plasma owing to its direct acceleration (or deceleration).
The J  E term arises both from such an acceleration or deceleration (through the action of the
J  B force) and from heating or cooling, but in both cases, it is associated with the exchange of
energy between the electromagnetic field and the plasma. To understand this energy exchange
issue we must also consider Maxwell’s equations (of course, from a mathematical perspective,
we must consider them anyway, because with E and B as unknowns, we find ourselves with six
more unknowns than we have equations).
In SI units, Maxwell’s equations can be written


  B 0  J 

0 E
t


(5)
B
t
(6)
  0 E  
(7)
B  0
(8)
E  


Here,  0 is the electric permittivity of free space, and 0 is the magnetic permeability of free
space. Equations (7) and (8) are usually considered as initial conditions, and the sense of this
can be seen by taking the divergence of (5) and (6), which yields
 
  J     J  0



E
0

t 
t
(9)

B  0
t
(10)




-3-
Equation (10) tells us that   B does not vary with time, so that if it initially vanishes, then it
always vanishes. Similarly, (9) (which incidentally can also be derived by subtracting the
electron and ion continuity equations) describes the time evolution of   E or alternatively the
charge density. If we wish to follow the time evolution of the charge density, we can do this by
using (9), but it is not necessary to do this in order to solve our system of equations.
Speaking of our system of equations, we now have the 17 unknowns n , u , p , q , J , E ,
and B , and the 14 equations (1)-(6) (if we choose to follow the time evolution of  , this
becomes 18 unknowns and 15 equations, including (9)). [Actually, if we were to include π as
an unknown, that would give us nine more unknowns, but for now at least, we’ll ignore the
effects of viscosity. If we were to include them, then we would need a closure relation
equivalent to the one we are about to specify for q .] The additional three equations we need can
be provided by specifying a closure relation for q (closure in this case refers to closing the
moment equations, the first three of which we have written as (1), (2), and (4)), and for this
purpose we will use (for now) the classical expression for the heat flux:
q   T 5 2 T
(11)
where the thermal conduction coefficient,  , is a constant. Of course, (11) doesn’t quite solve
our problem, because we have introduced the new variable, T . To take care of this, we use the
ideal gas law to specify T in terms of n and p :
p  2 n kB T
(12)
where, for the time being, we are taking T  Te  Tp .
Now, let’s return to the subject of the J  E term in (4), which couples the
electromagnetic energy and the fluid energy. If we take the scalar product of (5) and E and then
make use of (6), we can show that


J  E     12 B 2 0  12  0 E 2     E  B 0 
t



(13)
So the J  E term is the negative of the sum of the time rate of change of the electromagnetic
energy density and the divergence of the Poynting vector, which measures the electromagnetic
energy flux density. If we had wanted to, we could have written (4) as conservation equation for
the total (fluid and field) energy by substituting (13) into (4) and moving the electromagnetic
terms to the left side of the equation. Sometimes, this is a more useful form of the energy
equation, but it is important to keep in mind that the two forms are completely equivalent and
simply provide different perspectives. Actually, people who forget about the equivalence of
these two perspectives tend to get into arguments about things like whether or not it is currents
that heat the corona.
Finally, let’s return to the subject of heating versus direct acceleration. The direct
acceleration component of the J  E term is just the work done by the J  B force, which is
-4-




u  J  B   J  u  B . So the heating component of the J  E term, which we will label QEM
is given by



QEM  J  E  u  J  B  J  E  u  B

(14)
One could substitute for E  u  B using the generalized Ohm’s law (3) to obtain a general form
for the electromagnetic heating term, but we will just note that in the commonly considered case
where Ohm’s law is written E  u  B   J our heating term takes the familiar form associated
with resistive heating:
QEM   J 2
(15)
I suppose that people who claim that all coronal heating is associated with currents have this
term in mind. Of course, as we can see from (14), if the heating involves a conversion of
electromagnetic energy into fluid energy, then it can always be said to be associated with electric
currents, even if the resistivity vanishes.
Anyway, I’ll stop for now. Obviously the above comments raise a variety of issues we’ll
have to deal with, and I hope what I have written begins to provide a framework we can use for
thinking about these issues.
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