Boyd T.J.M., Sanderson J.J. The Physics of Plasmas

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12.6 Classical transport theory 487

On using the transformation equations (12.85) (ﬁnal column), (12.76) gives for

the evolution of ion and electron temperatures

∂T

∂t

u · ∇T

2ρT

∇ · u +



∂u

∂r

∇ · q

(12.92)

and

Zρ

∂T

∂t

Zρ



u −

Zeρ



· ∇T

2ZρT

∇ ·



u −

Zeρ





∂

∂r



−

Zeρ



∇ · q

(12.93)

Let us summarize what has been done in this section. The ion and electron

kinetic equations have been replaced by an inﬁnite series of moment equations.

The leading equations in this series have been obtained explicitly and describe

the evolution of the hydrodynamical (ρ,u, T

, T

) and electrodynamical (q, j)vari-

ables. However, this is not a closed set of equations for it contains the undetermined

quantities: 



,



, q

, R

, Q

and Q

. The system cannot be closed simply by

generating higher moment equations because the evolution equation for a moment

of order n automatically introduces a moment of order n +1, the evolution of which

demands yet another equation in the inﬁnite chain. Closure can be achieved only

by truncating on the basis of some appropriate physical approximation. This is the

subject of classical transport theory to which we now turn.

12.6 Classical transport theory

Plasma transport theory has much in common with the classical transport theory

of neutral gases and yet it is sufﬁciently different that it has remained an active

area of plasma research. The fullest and most rigorous account has been given by

Balescu (1988) who deals in turn with the regimes of classical and neoclassical

transport. Classical theory applies to plasmas in which the transport of matter,

momentum and energy is governed by the interaction of the particles through

binary collisions in the presence of slowly varying electric and magnetic ﬁelds;

the external magnetic ﬁeld is assumed to be straight, homogeneous and stationary.

Neoclassical theory applies to plasmas in curved, inhomogeneous ﬁelds. Phys-

ically, this is the only difference from classical theory but it turns out that the

global geometry of the ﬁeld, as opposed to its local value, plays a dominant role

in the transport so that a ﬂuid description remains valid even in the limit of very

weak collisions (long mean free path). Nevertheless, even in this limit, neoclassical

theory (a brief introduction to which was given in Section 8.3) remains a collisional

transport theory because collisions, though rare, play an essential role and no ac-

488 The classical theory of plasmas

count is taken of ‘collective’ interactions. When turbulent processes arising from

wave instabilities dominate we enter the regime of anomalous transport.

It is impossible to give here more than a brief introduction to classical transport

theory, the regime that is appropriate for the derivation of the MHD equations.

Our account is not much more than a pr

ecis of Balescu’s treatment. Truncation

and closure of the inﬁnite chain of moment equations is based on an analysis of

the time scales that characterize the plasma dynamics. In the MHD approximation

it is assumed that the hydrodynamic time scale τ

characterizes the electric and

magnetic ﬁelds E and B, as well as the hydrodynamic variables ρ,u, and T

;in

other words, ﬁeld ﬂuctuations in time and space arise from the plasma motion. One

immediate consequence of this, noted in Chapter 3, is that the electric ﬁeld is of

order (u/c) compared with the magnetic ﬁeld so that the displacement current may

be neglected and the Maxwell equations become

∇ × E =−

∂B

∂t

(12.94)

∇ × B = µ

j (12.95)

∇ · E =

(12.96)

∇ · B = 0 (12.97)

A second consequence is that the qE term in (12.89) may be neglected compared

with the j × B term since a dimensional analysis of (12.95) and (12.96) gives

|qE|

|j × B|

∼





(12.98)

This has the effect of removing the charge density entirely from the hydrodynam-

ical moment equations and makes (12.88) redundant. From (12.95) we see, in

fact, that this equation now reads ∇ · j ≈ 0,∂q/∂t ≈ 0, these approximations

being valid to order (u/c)

. Thus, in the MHD approximation q is determined by

(12.96) and j by (12.95). There are then two equations relating E and B, namely

(12.94) and the generalized Ohm’s law (12.91), and the hydrodynamic moment

equations (12.87), (12.89) (with q = 0), (12.92) and (12.93), for ρ,u, T

and T

respectively.

12.6.1 Closure of the moment equations

The physical approximation that enables us to achieve closure in classical transport

theory is that τ

is much longer than the collisional relaxation times τ

and τ

 τ

(α = i, e) (12.99)

12.6 Classical transport theory 489

It then follows that ions and electrons will, on the short time scales τ

, approach

local Maxwellian distributions

α0

(r, v, t) = n

(r, t)



2π T

(r, t)



3/2

exp



−

[v − u

(r, t)]

(r, t)



(12.100)

These functions, when substituted in the kinetic equations, give zero contributions

from the collision terms but are not solutions of the kinetic equations because

, u

and T

are not constants but vary with r and t .

It should be noted that the two-ﬂuid variables, n

, u

, T

, which characterize the

local Maxwellians in (12.100) vary on the slow time scale despite the presence of

collision terms in the evolution equations of u

and T

. That the equalization of ion

and electron temperatures takes place on the slow time scale is not surprising since

we have already observed that unlike particle collisions are inefﬁcient exchangers

of energy because of the disparity in the masses. The argument in the case of the

ﬂow velocities is best explained from (12.85). The ion ﬂow velocity is approxi-

mately the plasma ﬂow velocity and therefore slowly varying. Furthermore, the

difference between ion and electron ﬂow velocities is represented by the current j

which, in the MHD approximation, is deﬁned by (12.95) in terms of the magnetic

ﬁeld and, therefore, it also changes on the hydrodynamic time scale. Finally, as

discussed in the next section, short time scale variation arising from collision terms

is merely transient.

The distribution function is now written as

(r, v, t) = f

α0

[1 + χ

(

c;r, t)] (12.101)

where χ

measures the deviation of f

from f

α0

and for convenience we have

introduced the dimensionless velocity†

c =



(r, t)



1/2

[v − u

(r, t)] (12.102)

Standard procedure in transport theory is then to expand χ

in a series of orthogonal

polynomials in

c with coefﬁcients that are functions of r and t related to moments

of f

. There are two desirable features of any chosen procedure. One is to ﬁnd an

expansion which is rapidly convergent so that a truncated set of equations (for the

truncated set of moments) determines transport properties to the desired degree of

approximation. The second requirement is to choose a method of expansion that

is physically transparent. The singular achievement of Balescu’s approach is the

combination of both these features.

The tensorial Hermite polynomials are a natural choice for the expansion of

since they are orthogonal with respect to the weight function f

α0

. This choice

† Clearly,

c should have the label α but this is suppressed to avoid cluttering the notation.

490 The classical theory of plasmas

was used by Grad (1949) who developed the moment method of solution of the

kinetic equations. The method suffers a serious disadvantage, however, due to the

lack of a one-to-one correspondence between successive terms in the expansion

and the physical moments. The reason for this is that the expansion is in terms

of reducible Hermitian tensors so that there are, for example, contributions to the

vector moment from the contracted (reduced) third-order tensor. This also causes

slow convergence in this method. Balescu overcomes this disadvantage by ﬁrst

writing χ

in a representation which exhibits explicitly the possible anisotropies of

the distribution function, that is

(

c;r, t) = A

(

c;r, t) +˜c

(

c;r, t) +



˜c

−

˜c



(

c;r, t) +···

(12.103)

where A

is a scalar function, B

is a vector function and C

is a symmetric

traceless second-order tensor. Further terms (which we shall not need) involve

successively higher-order, irreducible tensors and tensor functions. Balescu then

expands the functions A

, B

, C

in series of irreducible tensorial Hermite poly-

nomials which maintain the Cartesian representations of the vector

c. This has

an advantage over the Chapman–Enskog method which takes (12.103) as a basic

ansatz but writes

c in spherical polar coordinates and expands χ

in an inﬁnite

series of Laguerre–Sonine polynomials and spherical harmonics.

The chief advantage of the representation of χ

in (12.103), however, lies in

the relationship between the terms in the mathematical series and the physical

quantities. In a transport theory that is linear in χ

it is clear that the heat ﬂux q

for example, being a vector moment of f

is completely determined by the vector

part of χ

, i.e. by B

. Likewise, 

is completely determined by C

. Furthermore,

choosing the exact values of n

(r, t), u

(r, t) and T

(r, t), as deﬁned by (12.55),

(12.56) and (12.63), to deﬁne the local equilibrium distribution (12.100) puts the

following constraints on χ



c f

α0

= 0



c f

α0

cχ

= 0



c f

α0

˜c

= 0







(12.104)

It follows that in linear theory only the vector and traceless second-order tensor

terms in (12.103) are required. Balescu shows, moreover, that the deeper physical

signiﬁcance of this lies in the fact that only these terms contribute to entropy

production in the linear regime. Thus, (12.103) is replaced by

(

c;r, t) =˜c

(

c;r, t) +



˜c

−

˜c



(

c;r, t) (12.105)

The expansions for B

and C

may be truncated at various levels with each suc-

12.6 Classical transport theory 491

cessive level adding three vector plus ﬁve (independent) tensor components so that

allowing for the ﬁve moments corresponding to n

, u

and T

the choices are for

5, 13, 21, 29,... moment approximations. Balescu shows that whereas there is a

considerable difference between the 13 moment (13M) approximation and the 21M

approximation no signiﬁcant improvement is provided by the 29M approximation.

There is not the space here to present the mathematical analysis of even the 13M

approximation. Rather we shall describe the procedure and present the main results.

12.6.2 Derivation of the transport equations

Balescu’s procedure leads to a natural classiﬁcation of the moments which arises

mathematically but is rooted in the physics. In the two-ﬂuid description of the

plasma the ﬁrst class contains the variables n

, u

and T

which Balescu calls

the plasmadynamical moments. The physical signiﬁcance of these moments is that

they determine the local equilibrium distribution. All the other moments in the

two-ﬂuid description (the complementary set) are called the non-plasmadynamical

moments and have the complementary property that they are identically zero in

the local equilibrium state; this follows from the orthogonality properties of the

Hermite polynomials. However, we shall not be directly concerned with the non-

plasmadynamical moments because our goal is the one-ﬂuid MHD description for

which we make the transformations





→



ρ = m

+ m

q =−en

+ Zen



(12.106)





→



ρu = m

+ m

j =−en

+ Zen



(12.107)

In this description the classiﬁcation is based on properties of the evolution

equations by means of which moments are ﬁrst separated into those which are

hydrodynamical and all others which are non-hydrodynamical. The class of hy-

drodynamical moments includes ρ,u, T

, T

whose evolution equations contain no

collision term or, in the case of the temperatures, one which is negligibly small.

The charge density q shares this property and could be added to this class but, as

we have seen already, it has no role in the MHD approximation and may be left

out of the discussion. The other electrodynamical moment j belongs to the non-

hydrodynamical class because its evolution equation (12.91) contains a signiﬁcant

collision term. This is also true of the electron and ion ﬂuxes deﬁned by



= n

(α = e, i) (12.108)

which are sometimes (e.g. in neoclassical theory) used instead of j. These mo-

ments share this property with the pressure tensors and heat ﬂuxes. Another shared

492 The classical theory of plasmas

property of these moments is that the evolution equations for 



, j,

and q

all contain a source term which is a function of the hydrodynamical moments.

Since these are the only non-hydrodynamical moments which have this property

and, furthermore, these are the only non-hydrodynamical moments appearing in

the evolution equations for hydrodynamical moments, it is clear that they play a

special role and the class of non-hydrodynamical moments is, therefore, further

subdivided into these privileged non-hydrodynamical moments and all others which

are non-privileged.

The physical signiﬁcance of this subdivision, as Balescu explains, is the one-

to-one correspondence between the solutions of the evolution equations for the

privileged moments and the transport equations of non-equilibrium thermodynam-

ics. The latter relate thermodynamic ﬂuxes, J

, and forces, X

, by a set of transport

equations



in which the elements of the transport matrix, L

, are the transport coefﬁ-

cients, having the Onsager symmetry L

= L

. Thus, the privileged non-

hydrodynamical moments may be identiﬁed with the thermodynamic ﬂuxes and the

source terms (involving the hydrodynamical moments) with the thermodynamic

forces. On the other hand, the solutions for the non-privileged moments have a

quite different structure and their main role is to improve the accuracy of the

transport coefﬁcients as one goes to higher moment approximations.

This classiﬁcation of the moments is highly signiﬁcant also for the next stage

of classical transport theory. Although closure of the set of moment equations

is achieved by adopting, say, the 21M approximation, there remains the task of

solving a sub-set of these equations so that most of the moments may be elim-

inated leaving a much smaller set of equations for the physically most interest-

ing variables. Speciﬁcally, the goal is to ﬁnd expressions for the privileged non-

hydrodynamical moments in terms of the hydrodynamical moments which can

then be substituted in the evolution equations of the latter thereby yielding the

set of MHD equations.

A subsidiary but necessary task is to express the collision terms as functions

of the moments. This is accomplished by ﬁrst substituting (12.105) in (12.101)

and then (12.101) in (12.77), (12.78) and the other collision terms arising in other

moment equations; the Landau collision integral (12.54) is used for (∂ f

/∂t)

.The

collision terms that we shall need explicitly are given by



+···



=−R

(12.109)

12.6 Classical transport theory 493

and

3Zm

− T

)

=−Q

− (u

− u

) · R

=−Q

j · R

(12.110)

where we have used (12.85). In the expression for R

successive terms arise from

progressively higher moment aproximations; the j term is there in the 5M approx-

imation, the q term in the 13M approximation, etc. In principle, we are discussing

the 21M approximation and so the next term in the bracket is included in Balescu’s

calculations. This can only be expressed in terms of a collision coefﬁcient and

the next Hermite coefﬁcient in the expansion of B

which we have not deﬁned. It

turns out, however, that we shall not need it explicitly for reasons that will become

apparent and so we do not write it out in (12.109). The point to notice about Q

the presence of the factor m

conﬁrming that transfer of thermal energy from

electrons to ions is a very slow process.

For what comes next, it should be noted that the collision terms in general are

all proportional to χ

, since f

α0

is Maxwellian and χ

= 0 in (12.101) would

make all collision terms vanish. In linear transport theory, solution of the moment

equations is carried out to ﬁrst order in the small parameter  = τ

/τ

. If all

moments are written in dimensionless variables then all terms have dimensions of

an inverse time which is either τ

or τ

. A self-consistent linear transport theory is

then obtained by assuming that χ

is at most of ﬁrst order in the small parameter

 = τ

/τ

. The hydrodynamical moments are taken to be of order zero with non-

hydrodynamical moments and collision terms at most of order one. No assumption

is made about Larmor frequencies which enter through the Lorentz force, so 

is treated as zero order.

Now if we examine the evolution equations (12.87), (12.89), (12.92) and (12.93)

for the hydrodynamical moments, we ﬁnd that these contain either no collision

terms or, in the case of T

and T

, collision terms which are either of order m

or 

and therefore negligible. The hydrodynamical moments consequently change

only on the slow time scale, as anticipated earlier.

The equations for the privileged non-hydrodynamical moments, on the other

hand, contain both slow and fast time scale terms with the leading order terms

showing that fast time scale collision terms are balanced by slow time scale terms

consisting of gradients of hydrodynamical moments and electromagnetic ﬁeld

terms. In (12.91), the only example of such an equation that we have explicitly

presented, the leading terms are the third, ﬁfth, sixth and seventh so that the lin-

494 The classical theory of plasmas

earized, generalized Ohm’s law is

Zeρ

(E + u × B) +

∇(ρT

) − j × B = R

(12.111)

It could be argued that the ∂j/∂t term should also be retained since, through the

collision term, j can vary on the fast time scale. This is quite true but Balescu shows

that fast time scale changes cause only transient behaviour which disappears after a

few collision times so that asymptotic results can be obtained by simply dropping

the time derivatives and solving the resulting algebraic equations. This transient

behaviour is not without physical interest for it is in the transition from transient

to asymptotic state that the system becomes Markovian. For a few collision times

the plasma maintains a dependence on its (unknown) initial state and its dynamical

behaviour is a function of the whole history of its motion up to that point. But for

t  τ

collisions have wiped out all knowledge of the initial state and any depen-

dence on past history extends no further back than about 7τ

which is negligible

in a consistent linear theory. There is a striking analogy here with the derivation of

the kinetic equation from the BBGKY hierarchy, with non-hydrodynamic moments

playing the role of the correlation function and hydrodynamic moments the role of

the one-particle distribution function; of course the collision time τ

has changed

its role from being the slow time scale (compared with τ

the duration of a collision)

to being the fast time scale compared with τ

Balescu’s transport equations expressing the privileged non-hydrodynamical

vector moments in terms of hydrodynamical moments and the electric and mag-

netic ﬁelds are

= σ

− α

∂T

/∂r

(12.112)

= α

− κ

∂T

/∂r

(12.113)

=−κ

∂T

/∂r

(12.114)

(12.115)

where the effective electric ﬁeld

E = E + u × B +

∇(n

) (12.116)

includes a thermoelectric component. The transport coefﬁcients are second-order

tensor quantities representing electrical conductivity, σ

, the thermoelectric co-

efﬁcients, α

, and thermal conductivity, κ

. They have just three independent

components and relative to Cartesian axes with B = B

z, take the form

L =





⊥

∧

−L

∧

⊥

00L







(12.117)

12.6 Classical transport theory 495

where L



, L

⊥

are the coefﬁcients parallel and perpendicular to B and L

∧

is the

only non-zero, off-diagonal element. This representation expresses the invariance

of these transport coefﬁcients under rotations of the axes in the plane perpendicular

to B. The traceless pressure tensors, 

, are already of second-order and so they

introduce a fourth-order viscosity tensor, µ

klmn

, through the relationship



=−µ

klmn

(12.118)

where



∂u

∂r

∂u

∂r

−

∇ · u



(12.119)

However, the same symmetries that reduce the second-order tensors to just three

independent elements mean that µ

klmn

has only ﬁve independent elements which

we label µ



,µ

. Balescu shows that (12.118) may then be written



=−

(µ



+ µ

)ν

−

(µ



− µ

)ν

+ µ



=−

(µ



− µ

)ν

−

(µ



+ µ

)ν

− µ



=−

− µ

= 



=−µ

+ µ

= 



=−µ

− µ

= 



=−µ













(12.120)

A further reduction arises from the relationships

(

) = µ

(2

), µ

(

) = µ

(2

)

discovered by Braginsky (1965) so that effectively we can again consider just three

coefﬁcients.

12.6.3 Classical transport coefﬁcients

A number of well-known effects arise from the non-diagonal elements of the

transport coefﬁcients. For example, a non-zero E

gives rise to a component of

current in the y direction, j

= σ

; this is the Hall current. Similarly, a

temperature gradient in the x direction produces a heat ﬂux in the y direction,

=−κ

∂T

/∂x, known as the Righi–Leduc effect.TheNernst effect is the

heat ﬂux q

= α

produced by an electric ﬁeld in the x direction, and

the Ettinghausen effect, produced by the same coefﬁcient, is the electric current,

=−α

∂T

/∂x, due to an electron temperature gradient in the x direction.

The transport coefﬁcients are, of course, dependent on the plasma parameters as

well as the magnetic ﬁeld. They can be written in terms of dimensionless functions

496 The classical theory of plasmas

of the atomic number Z and X

= 

as follows:





˜σ

(Z , X

)





1/2





˜α

(Z , X

)





˜κ

(Z , X

)

= n

˜µ

(Z , X

)(p =, 1, 2, 3, 4)











(12.121)

with

√

2π

3/2

1/2

3/2

ln 

√

2π

3/2

1/2

3/2

ln 











(12.122)

In (12.121) the dimensionless transport coefﬁcients (denoted by the tilde) are tab-

ulated functions of Z and X

the precise details of which depend upon the number

of moments taken in the approximation.

In the limit of zero magnetic ﬁeld, X

→ 0, we ﬁnd

⊥

= L



, L

∧

= 0 (12.123)

where L stands for σ, α or κ

,and

= µ



,µ

= µ

= 0 (12.124)

Thus, (12.112)–(12.118) reduce to

j = σ



E − α



∇T

(12.125)

= α



E − κ



∇T

(12.126)

=−κ



∇T

(12.127)



=−µ



(12.128)

We see that all the transport coefﬁcients have become scalars whose values are

given by the parallel coefﬁcients in this limit. The ion heat ﬂux is simply given by

Fourier’s law (12.127). The asymmetry between this equation and (12.126) for the

electron heat ﬂux, as between (12.113) and (12.114), arises from the mass ratio.

Only ion–ion collisions are effective for ion transport, ion–electron collision terms

being of order m

in comparison, so there is a decoupling of the ions.

The electron ﬂuxes, on the other hand, are subject to both electric and hydrody-

namic forces. If the latter are absent (∇T

= 0) we retrieve the simple Ohm’s law

j = σ



E (12.129)