The first step in our closure scheme is to approximate the actual collision
operator for Coulomb interactions by an operator which is strictly
*bilinear* in its arguments (see Sect. 3.3). Once this has been achieved,
the closure problem is formally of the type which can be solved using the
Chapman-Enskog method.

The electrons and ions collision times,
, are
written

respectively. Here, is the number density of particles, and is a quantity called the

(262) |

The basic forms of Eqs. (260) and (261) are not hard to understand.
From Eq. (231), we expect

(263) |

(264) |

The electron and ion fluid equations in a collisional plasma
take the form [see Eqs. (220)-(222)]:

and

respectively. Here, use has been made of the momentum conservation law (198). Equations (265)-(267) and (268)-(270) are called the

In the *unmagnetized* limit, which actually
corresponds to

(271) |

Here, is the net plasma current, and the

In the above, use has been made of the conservation law (205).

Let us examine each of the above collisional terms, one by one. The first
term on the right-hand side of Eq. (272) is a friction force due to the
relative motion of electrons and ions, and obviously controls the electrical
conductivity of the plasma. The form of this term is fairly easy to understand.
The electrons lose their ordered velocity
with respect to the ions,
,
in an electron collision time, , and consequently lose momentum
per electron (which is given to the ions) in this time.
This means that a frictional force
is exerted on the electrons.
An equal and opposite force is exerted on the ions. Note that, since the Coulomb
cross-section diminishes with increasing electron energy (*i.e.*,
),
the conductivity of the fast electrons in the distribution function
is higher than that of the slow electrons (since,
).
Hence, electrical current in plasmas is carried predominately by the
*fast* electrons. This effect has some important and interesting
consequences.

One immediate consequence is the second term on the right-hand side of Eq. (272),
which is called the *thermal force*. To understand the origin of
a frictional force proportional to minus the gradient of the electron temperature,
let us assume that the electron and ion fluids are at rest (*i.e.*,
). It follows that the number of electrons moving from left to right
(along the -axis, say) and from right to left per unit time is exactly the
same at a given point (coordinate , say) in the plasma. As a result
of electron-ion collisions, these fluxes experience frictional forces,
and , respectively, of order
,
where is the electron thermal velocity. In a completely homogeneous
plasma these forces balance exactly, and so there is zero net frictional force.
Suppose, however, that the electrons coming from the right are, on average, hotter
than those coming from the left. It follows that the frictional force
acting on the fast electrons coming from the right is less than
the force acting on the slow electrons coming from the left, since
increases with electron temperature. As a result, there is a net
frictional force acting to the left: *i.e.*, in the direction of .

Let us estimate the magnitude of the frictional force. At point , collisions
are experienced by electrons which have traversed distances of order a
mean-free-path,
. Thus, the electrons coming from the
right originate from regions in which the temperature is approximately
greater than the regions from which the electrons
coming from the left originate. Since the friction force is proportional to
, the net force
is of order

The term , specified by Eq. (273), represents the rate at which
energy is acquired by the ions due to collisions
with the electrons.
The most striking aspect of this term is
its *smallness*
(note that it is proportional to an inverse mass ratio,
). The smallness of is a direct consequence of the
fact that electrons are considerably lighter than ions. Consider the
limit in which the ion mass is infinite, and the ions are at rest on average:
*i.e.*, . In this case, collisions of electrons with ions
take place *without any* exchange of energy. The electron velocities
are randomized by the collisions, so that the energy associated
with their ordered velocity,
, is converted
into heat energy in the electron fluid [this is represented by the second term
on the extreme right-hand side of Eq. (274)]. However, the ion energy remains
unchanged. Let us now assume that the ratio is large, but finite, and
that . If , the ions and electrons are in thermal equilibrium, so
no heat is exchanged between them. However, if , heat
is transferred from the electrons to the ions. As is well known, when
a light particle collides with a heavy particle, the order of magnitude of the
transferred energy is given by the mass ratio , where is the
mass of the lighter particle. For example, the mean fractional energy transferred
in isotropic scattering is . Thus, we would expect the
energy per unit time transferred from the electrons to the ions to be roughly

(277) |

The term , specified by Eq. (274), represents the rate at
which energy is acquired by the electrons due to
collisions with the ions, and consists of three terms. Not surprisingly,
the first term is simply minus the rate at which energy is
acquired by the ions due to collisions with the
electrons. The second term represents the conversion
of the ordered motion of the electrons, relative to the ions, into random
motion (*i.e.*, heat) via collisions with the ions. Note that this
term is positive definite, indicating that the randomization of the electron
ordered motion gives rise to *irreversible* heat generation.
Incidentally, this
term is usually called the *ohmic heating* term. Finally, the third
term represents the work done against the thermal force. Note that this
term can be either positive or negative, depending on the direction of
the current flow relative to the electron temperature gradient. This
indicates that work done against the thermal force gives rise to *reversible*
heat generation. There is an analogous effect in metals called the *Thomson effect*.

The electron and ion heat flux densities are given by

respectively. The electron and ion

respectively.

It follows, by comparison with Eqs. (236)-(241), that the first term on the right-hand side of Eq. (278) and the expression on the right-hand side of Eq. (279) represent straightforward random-walk heat diffusion, with frequency , and step-length . Recall, that is the collision frequency, and is the mean-free-path. Note that the electron heat diffusivity is generally much greater than that of the ions, since , assuming that .

The second term on the right-hand side of Eq. (278) describes a convective
heat flux due to the motion of the electrons relative to the ions.
To understand the origin of this flux, we need to recall that
electric current in plasmas is carried predominately by the fast electrons
in the distribution function. Suppose that is non-zero. In the
coordinate system in which is zero, more fast electron move in the
direction of , and more slow electrons move in the opposite
direction. Although the electron fluxes are balanced in this frame of reference,
the energy fluxes are not (since a fast electron possesses more energy than a slow
electron), and heat flows in the direction of : *i.e.*, in
the opposite direction to the electric current. The net heat flux density is of
order : *i.e.*, there is no near cancellation of the fluxes
due to the fast and slow electrons. Like the thermal force, this effect
depends on collisions despite the fact that the expression for the convective
heat flux does not contain explicitly.

Finally, the electron and ion viscosity tensors take the form

respectively. Obviously, refers to a Cartesian component of the electron fluid velocity in Eq. (282) and the ion fluid velocity in Eq. (283). Here, the electron and ion

respectively. It follows, by comparison with Eqs. (235)-(241), that the above expressions correspond to straightforward random-walk diffusion of momentum, with frequency , and step-length . Again, the electron diffusivity exceeds the ion diffusivity by the square root of a mass ratio (assuming ). However, the ion viscosity exceeds the electron viscosity by the same factor (recall that ):

Let us now examine the *magnetized* limit,

(286) |

Here, the

Note that denotes a gradient parallel to the magnetic field, whereas denotes a gradient perpendicular to the magnetic field. Likewise, represents the component of the plasma current flowing parallel to the magnetic field, whereas represents the perpendicular component of the plasma current.

We expect the presence of a strong magnetic field to give rise to a
marked *anisotropy* in plasma properties between directions parallel
and perpendicular to , because of the completely different motions
of the constituent ions and electrons parallel and perpendicular to the field.
Thus, not surprisingly, we find that the electrical conductivity perpendicular
to the field is approximately half that parallel to the field [see Eqs. (287)
and (290)]. The thermal force is unchanged (relative to the unmagnetized case)
in the parallel direction, but is radically modified in the
perpendicular direction. In order to understand the origin
of the last term in Eq. (287), let us consider a situation in
which there is a strong magnetic field along the -axis, and an electron
temperature gradient along the -axis--see Fig. 5. The electrons gyrate
in the - plane in circles of radius
.
At a given point, coordinate , say, on the -axis, the electrons that
come from the right and the left have traversed distances of order .
Thus, the electrons from the right originate from regions where the
electron temperature is of order
greater than
the regions from which the electrons from the left originate. Since the
friction force is proportional to , an unbalanced friction force
arises, directed along the -axis--see Fig. 5. This direction
corresponds to the direction of
.
Note that there is
no friction force along the -axis, since the -directed fluxes are due
to electrons which originate from regions where .
By analogy with Eq. (276), the magnitude of the perpendicular
thermal force is

(291) |

In the magnetized limit, the electron and ion heat flux densities become

respectively. Here, the

(294) | |||

(295) |

Finally, the

(296) | |||

(297) |

The first two terms on the right-hand sides of Eqs. (292) and (293)
correspond to *diffusive* heat transport by the electron and ion
fluids, respectively. According to the first terms, the diffusive transport in
the direction parallel to the magnetic field is exactly the same as that in the
unmagnetized case: *i.e.*, it corresponds to
collision-induced random-walk diffusion
of the ions and electrons, with
frequency , and step-length . According to the
second terms, the diffusive transport in the direction perpendicular to the
magnetic field is *far smaller* than that in the parallel direction.
In fact, it is smaller by a factor , where is the
gyroradius, and the mean-free-path. Note, that the perpendicular
heat transport also corresponds to collision-induced random-walk diffusion
of charged particles,
but with frequency , and
step-length . Thus, it is the greatly reduced step-length in the
perpendicular direction, relative to the parallel direction, which ultimately
gives rise to the strong reduction in the perpendicular heat transport.
If , then the ion perpendicular heat diffusivity actually
*exceeds* that of the electrons by the square root of a mass ratio:
.

The third terms on the right-hand sides of Eqs. (292) and (293)
correspond to heat fluxes which are perpendicular to both the magnetic field
and the direction of the temperature gradient. In order to understand the
origin of these terms, let us consider the ion flux. Suppose that there
is a strong magnetic field along the -axis, and an ion temperature gradient
along the -axis--see Fig. 6. The ions gyrate in the - plane
in circles of radius
, where is the
ion thermal velocity. At a given point, coordinate , say, on the -axis,
the ions that come from the right and the left have traversed distances of
order . The ions from the right are clearly somewhat hotter than those
from the left. If the unidirectional particle fluxes, of order , are
balanced, then the unidirectional heat fluxes, of order , will
have an unbalanced component of fractional order
. As a result, there is a net heat flux in the -direction
(*i.e.*, the direction of
). The magnitude of
this flux is

(298) |

The fourth and fifth terms on the right-hand side of Eq. (292) correspond to
the convective component of the electron heat flux density, driven by
motion of the electrons relative to the ions. It is clear from the
fourth term that the convective flux parallel to the magnetic field is exactly the
same as in the unmagnetized case [see Eq. (278)]. However, according to the fifth term, the
convective flux is radically modified in the perpendicular direction.
Probably the easiest method of explaining the fifth
term is via an examination
of Eqs. (272), (278), (287), and (292). There is clearly a very close
connection between the electron thermal force and the convective heat flux.
In fact, starting from general principles of the thermodynamics of irreversible
processes, the so-called *Onsager principles*, it is possible to
demonstrate that an electron frictional force of the form
necessarily gives rise to an electron heat flux
of the form
, where the
subscript corresponds to a general Cartesian component, and
is a unit vector. Thus, the fifth term on the right-hand side of Eq. (292)
follows by *Onsager symmetry* from the third term on the right-hand
side of Eq. (287). This is one of many Onsager symmetries which
occur in plasma transport theory.

In order to describe the viscosity tensor in a magnetized plasma, it is
helpful to define the *rate-of-strain tensor*

(299) |

In a magnetized plasma, the viscosity tensor is best described as the
sum of *five* component tensors,

(300) |

(301) |

(302) |

(303) |

(304) |

(305) |

The tensor
describes what is known as *parallel viscosity*.
This is a viscosity which controls the variation along magnetic field-lines of the
velocity component parallel to field-lines.
The parallel
viscosity coefficients, and are specified in Eqs. (284)-(285).
Note that the parallel viscosity is unchanged from the unmagnetized case,
and is due to the collision-induced random-walk diffusion of particles,
with frequency , and step-length .

The tensors
and
describe what is known
as *perpendicular viscosity*. This is a viscosity
which controls the variation perpendicular to magnetic field-lines
of the velocity components perpendicular to field-lines. The perpendicular
viscosity coefficients are given by

(306) | |||

(307) |

Note that the perpendicular viscosity is

Finally, the tensors
and
describe what is known
as *gyroviscosity*. This is not really viscosity at all, since the
associated viscous stresses are always perpendicular to the velocity, implying that
there is no dissipation (*i.e.*, viscous heating) associated with
this effect. The gyroviscosity coefficients are given by

(308) | |||

(309) |

The origin of gyroviscosity is very similar to the origin of the cross thermal conductivity terms in Eqs. (292)-(293). Note that both cross thermal conductivity and gyroviscosity are