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

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Ideal magnetohydrodynamics

4.1 Introduction

Ideal MHD is used to describe macroscopic behaviour across a wide range of

plasmas and in this chapter we consider some of the most important applications.

Being dissipationless the ideal MHD equations are conservative and this leads to

some powerful theorems and simple physical properties. We begin our discussion

by proving the most important theorem, due to Alfv

en (1951), that the magnetic

ﬁeld is ‘frozen’ into the plasma so that one carries the other along with it as it

moves. This kinematic effect arises entirely from the evolution equation for the

magnetic ﬁeld and represents the conservation of magnetic ﬂux through a ﬂuid

element. Of course, any ﬁnite resistivity allows some slippage between plasma

and ﬁeld lines but discussion of these effects entails non-ideal behaviour and is

postponed until the next chapter.

The concept of ﬁeld lines frozen into the plasma leads to very useful analogies

which aid our understanding of the physics of ideal MHD. It also suggests that one

might be able to contain a thermonuclear plasma by suitably conﬁgured magnetic

ﬁelds, although research has shown that this is no easily attainable goal. Further,

since the ideal MHD equations are so much more amenable to mathematical analy-

sis they can be used to investigate realistic geometries. The theory has thereby pro-

vided a useful and surprisingly accurate description of the macroscopic behaviour

of fusion plasmas showing why certain ﬁeld conﬁgurations are more favourable to

containment than others.

Notwithstanding the wide applicability of ideal MHD in space and laboratory

plasma physics a note of caution needs to be sounded over results derived from

it. Since the neglected dissipative terms are of higher differential order than the

non-dissipative terms, even a very small amount of dissipation can lead to solutions

which are signiﬁcantly different from those of ideal MHD. Mathematically, higher

differential order means that singular perturbation theory must be used to examine

78 Ideal magnetohydrodynamics

the effects of dissipative terms and the ‘ideal’ solution cannot be recovered by tak-

ing the dissipative solution to an appropriate limit. This is illustrated in Exercise 4.1

where it is shown in a particular example that as one approaches the ideal limit of

no dissipation the effect of the dissipative terms is restricted to a narrow sheath

in which steepening gradients compensate for vanishing dissipative coefﬁcients.

Generally, whilst an ideal MHD solution may be valid over most of a plasma

volume it cannot be entirely divorced from, and must be matched to, the non-ideal

solution at the boundary of such sheaths. Because of the steep gradients, often the

most interesting physics takes place in the sheath with dramatic consequences for

the whole plasma, as we shall discover in the next chapter.

4.2 Conservation relations

The basic physical properties of ideal MHD are related to the conservation of mass,

momentum, energy and magnetic ﬂux, so it is helpful to write the ﬂuid equations

in the form of conservation relations. The continuity equation

∂ρ

∂t

=−∇ · (ρu) (4.1)

is already in the required form, which is not surprising since it was derived by

expressing the conservation of mass in an arbitrary ﬁxed volume within the plasma.

The equation expressing the conservation of momentum is obtained by taking

the partial time derivative of ρu and using the continuity and momentum equations

for ∂ρ/∂t and ∂u/∂t to get

∂(ρu)

∂t

=−∇ · (ρuu) − ∇P +

(∇ × B) × B

Now, using the vector identity (∇×B)×B = (B·∇)B−∇(B

/2) and ∇·(BB) =

B · ∇B this becomes

∂(ρu)

∂t

=−∇ · (ρuu + PI −T) = ∇ ·



 (4.2)

say. In (4.2) I is the unit dyadic and

T =

BB −

2µ

is the Maxwell stress tensor, {ε

EE + BB/µ

−

(ε

+ B

/µ

)I}, in the non-

relativistic limit, ε

 1.

Similarly the equation of energy conservation follows from the partial time

derivative of the total energy density

U =

ρu

γ − 1

2µ

4.2 Conservation relations 79

comprising kinetic, internal, and magnetic energies, respectively. On using the

ideal MHD equations (Table 3.2) to evaluate the derivatives the result is

∂U

∂t

=−∇ · S (4.3)

where

S =



ρu

γ P

γ − 1



u +

B × (u × B)

is the total energy ﬂux (see Exercise 4.2). Each of these conservation equations

expresses the time rate of change of the conserved quantity, at any given point

in the ﬂuid, as the (negative) divergence of the corresponding ﬂux at that point.

Integrating these equations over an arbitrary volume and using Gauss’ theorem

gives the rate of change of the conserved quantity within the volume in terms of

the ﬂux through its surface.

It is less obvious that the evolution equation for the magnetic ﬁeld is a conser-

vation equation but, in what has become known as the frozen ﬂux theorem, Alfv

showed that a consequence of

∂B

∂t

= ∇ × (u ×B) (4.4)

is that the magnetic ﬂux, through any surface S bounded by a closed contour C

moving with the ﬂuid, is constant. From the two-dimensional extension of Leib-

nitz’s theorem (see (3.3) in Section 3.2) we have



B · dS =



∂B

∂t

· dS +



B · v

× dl (4.5)

so that in the case of a surface S whose boundary C is moving with the local ﬂow

velocity u, as illustrated in Fig. 4.1, we have



B · dS =



∂B

∂t

· dS +



B · u × dl (4.6)

where the ﬁrst term on the right-hand side of (4.6) represents the change of ﬂux due

to the time rate of change of B and the second term represents the change in the

surface area due to the movement of the bounding contour C (see Fig. 3.1(b)). Now

interchanging dot and cross in the triple scalar product and using Stokes’ theorem

to convert the line integral to a surface integral we see that



B · dS =





∂B

∂t

− ∇ × (u × B)



· dS = 0 (4.7)

on account of (4.4). Thus, the ﬂux through a surface S moving with the ﬂuid is

constant. Representing the ﬂux through the surface S by the totality of the ﬁeld

80 Ideal magnetohydrodynamics

u(r,t)

B(t)

Fig. 4.1. Magnetic ﬂux motion.

lines threading the loop C and bearing in mind that the theorem is true for arbitrary

C we see that the ﬁeld lines are constrained to move with C, i.e. with the ﬂuid.

Although not tangible physical quantities, ‘ﬁeld lines’ and ‘ﬂux tubes’ are help-

ful concepts for understanding the properties of magnetic ﬁelds. At every point,

whether in the ﬂuid or in the vacuum, ﬁeld lines follow the direction of the mag-

netic ﬁeld and are, therefore, deﬁned by the equations

A magnetic (or ﬂux) surface is one that is everywhere tangential to the ﬁeld, i.e.

the normal to the surface is everywhere perpendicular to B, as shown in Fig. 4.2(a).

Figure 4.2(b) shows an open-ended cylindrical magnetic surface which deﬁnes a

ﬂux tube and it is sometimes helpful to picture the density of the ﬁeld lines through

the tube as a representation of the ﬁeld strength; in other words, the number of ﬁeld

lines threading the tube is proportional to the ﬂux.

Clearly, the ﬂux through a magnetic surface is zero and by the frozen ﬂux

theorem must remain so if the surface moves with the ﬂuid. Let us now imagine

a long thin ﬂuid element which at any given moment lies along a ﬂux tube. As it

moves, its cylindrical surface remains a magnetic surface and the ﬂux through its

ends remains constant; hence the notion that the ﬁeld lines are ‘frozen’ into the

ﬂuid (see Fig. 4.2(c)).

4.2 Conservation relations 81

magnetic surface

flux tube fluid element

(a)

(b)

(c)

Fig. 4.2. Field lines and ﬂux tubes.

Next we consider the distortion of a ﬂux tube of initial cross-section dS

and

length l

as it moves with the ﬂuid. Since the ﬂux through the tube is constant a

motion which stretches the tube to length l > l

and narrows its cross-section to

dS < dS

will result in an increase in the ﬁeld strength since B

= B dS.As

the mass of the ﬂuid is also conserved, i.e. ρ

= ρ dSl, we may divide these

equations to get the result

ρl

(4.8)

For an incompressible ﬂuid (ρ = const.) this becomes

B = (B

)l (4.9)

i.e. the ﬁeld strength is proportional to the length of the ﬂux tube; these results were

ﬁrst noted by Wal

en (1946).

Flux conservation has a profound effect on the structure of the ﬁeld. A ﬂuid

element may change its shape but it does not break into separate pieces. The same

is true, therefore, of a ﬂux tube with the result that the topology of the ﬁeld cannot

change. This is a severe constraint since ﬁeld conﬁgurations in which a lower

energy state could be arrived at by line breaking and reconnection are both easy

to imagine and to realize in practice. Such changes, which are forbidden in ideal

MHD by ﬂux conservation, become possible with dissipation, however small, and

the consequences for plasma stability are dramatic, as we shall see in Chapter 5.

82 Ideal magnetohydrodynamics

4.3 Static equilibria

The fact that the ﬁeld lines are frozen into a perfectly conducting ﬂuid leads nat-

urally to the notion that by controlling the magnetic ﬁeld conﬁguration one might

be able to contain the ﬂuid. For the high temperature plasmas needed for fusion

reactions this is a vital matter since contact between plasma and wall is likely to

be deleterious to both. We turn, therefore, to the momentum equation to examine

the effect of the ﬁeld on the motion of the ﬂuid but, because we wish to examine

possible equilibrium conﬁgurations, we begin with the specially simple case in

which the ﬂuid is at rest.

For mathematical consistency this poses a problem for if u = 0 then R

= 0 and

ideal MHD is invalid! However, a dimensional analysis of the resistive MHD equa-

tions in Table 3.1 in this case shows that the pressure increases and the magnetic

ﬁeld diffuses on a time scale that is proportional to the plasma conductivity σ .The

requirement, therefore, is that σ should be sufﬁciently large that this diffusion time,

∼µ

σ L

, is greater than any other time of interest. We assume that such times

as will arise, e.g. the time for the growth of instabilities (see Sections 4.5–4.7) or

the period of plasma waves (see Section 4.8), are less than τ

. With this proviso

we may set u and all time derivatives equal to zero in the ideal MHD equations of

Table 3.2 to get

j × B = ∇P (4.10)

∇ · B = 0 (4.11)

j =

∇ × B (4.12)

If an equilibrium state is to be established the j×B force must balance the pressure

gradient and to investigate this it is convenient to use the static momentum equation

(4.10) in conservative form.

Deﬁning the total stress tensor

= [(P + B

/2µ

)δ

− B

/µ

] (4.13)

(4.2) becomes

∂T

∂r

= 0 (4.14)

The total stress tensor may be reduced to diagonal form by transformation to the

principal axes. The eigenvalues may be obtained from the secular equation

− δ

λ|=0

the solution being

= P + B

/2µ

= λ

,λ

= P − B

/2µ

4.3 Static equilibria 83

P +

= P*

)

Fig. 4.3. Total stress tensor relative to principal axes.

Thus, referred to the principal axes, T

takes the form





P + B

/2µ

0 P + B

/2µ

00P − B

/2µ





The principal axes are oriented so that the axis corresponding to λ

is parallel

to B and the axes corresponding to λ

,λ

are perpendicular to B. From this we

see that the stress caused by the magnetic ﬁeld amounts to a pressure B

/2µ

directions transverse to the ﬁeld and a tension B

/2µ

along the lines of force. In

other words, the total stress amounts to an isotropic pressure which is the sum of the

ﬂuid pressure and the magnetic pressure B

/2µ

, and a tension B

/µ

along the

lines of force. This is illustrated in Fig. 4.3. The ratio of ﬂuid pressure to magnetic

pressure, 2µ

P/B

, is an important parameter commonly denoted by β. In MHD,

it is often convenient to picture a tube of force behaving like an elastic string under

tension. Thus stretching the tube of force increases the tension, which means that

the ﬁeld is increased as explained in the previous section.

From the equilibrium condition (4.10) it follows that

B · ∇P = 0 (4.15)

84 Ideal magnetohydrodynamics

magnetic axis

P = 0

P = P

∇P

Fig. 4.4. Nested isobaric surfaces.

and

j · ∇P = 0 (4.16)

This means that both B and j lie on surfaces of constant pressure so that the cur-

rent ﬂows between magnetic surfaces. Supposing the constant pressure (isobaric)

surfaces to be closed, then, since (4.15) states that no magnetic ﬁeld line passes

through the surface, one may picture the surface as made up from a winding of ﬁeld

lines, i.e. the isobaric surfaces are also magnetic surfaces. Likewise, from (4.16)

the isobaric surfaces are made up from lines of current density; these lines will, in

general, intersect the ﬁeld lines. The cross-section in Fig. 4.4 shows a set of nested

surfaces on which the pressure increases in passing from the outside towards the

axis; the currents are such that j ×B points towards the axis. The implication here

is that a plasma may be contained entirely by the magnetic force, an arrangement

referred to as magnetic conﬁnement.

An important integral relationship, known as the virial theorem, follows from

(4.14) and shows that such magnetic conﬁnement cannot be achieved without the

aid of external currents. To demonstrate this we integrate the identity

∂

∂r

(

)

= r

∂T

∂r

+ T

over an arbitrary volume V , bounded by a surface S. Then, since ∂T

/∂r

= 0for

equilibrium, on using Gauss’ theorem we get



(3P + B

/2µ

)dV =



[(P + B

/2µ

)n · r − (B

/µ

)(r · b)(n · b)]dS

Now if the ﬁelds are due entirely to the plasma they must decrease as S →∞

at least as fast as a dipole ﬁeld (∝ r

−3

) so that the surface integral vanishes. The

volume integral, on the other hand, is positive deﬁnite and does not vanish, from

4.3 Static equilibria 85

which we conclude that we cannot set ∂T

/∂r

= 0 and there is no equilibrium

without an externally applied ﬁeld.

4.3.1 Cylindrical conﬁgurations

For simplicity let us ﬁrst consider the radial equilibrium of cylindrical plasmas. We

assume cylindrical symmetry so that all variables are independent of θ and z and

the lines of j and B lie on surfaces of constant r. Then

j =



0, −

(r)

(

(r)

)



(4.17)

and the radial component of (4.10) gives

[P + (B

+ B

)/2µ

] =−B

/µ

r (4.18)

which expresses the (radial force) balance between the total (gas and magnetic)

pressure gradient and the magnetic tension due to the curvature (if any) of the

magnetic ﬁeld. Of course, it is possible to remove magnetic curvature by choosing

= 0 and then the gas and magnetic pressure gradients must be oppositely

directed and in balance. In this case, setting B

= 0 in (4.18) and integrating,

we have

∗

≡ P + B

/2µ

= const. (4.19)

Such a ﬁeld may be produced by currents ﬂowing azimuthally; early devices de-

signed to contain plasma in this conﬁguration were known as theta-pinches (since

θ is used to denote the azimuthal coordinate). Azimuthal currents are induced by

discharging a current suddenly in a metal conductor enclosing the discharge tube.

The induced currents ﬂow in the opposite direction and an axial magnetic ﬁeld is

generated in the region between. The j×B force acts to push the plasma towards the

axis until the external magnetic pressure is balanced by the (total) internal pressure;

from (4.19)

P(r) + B

(r)/2µ

= B

/2µ

(4.20)

where B

is the external magnetic ﬁeld. Note that this means that the plasma acts

as a diamagnetic medium, B(r)<B

(see Section 2.2.1). In the absence of any

initial magnetic ﬁeld there is only the induced ﬁeld which remains entirely outside

the plasma since it cannot penetrate in ideal MHD and (4.20) reduces to

P = B

/2µ

(4.21)

i.e. the plasma is radially contained by the magnetic ﬁeld generated by the az-

imuthal current ﬂowing in its outer surface. If there is an internal magnetic ﬁeld

86 Ideal magnetohydrodynamics

Fig. 4.5. Z-pinch conﬁguration of axial current and azimuthal magnetic ﬁeld.

B(r), the current penetrates the plasma and (4.20) applies. It is customary to deﬁne

the plasma β with respect to the external magnetic ﬁeld, i.e. β(r) = 2µ

P(r)/B

so that from (4.20)

β(r) = 1 − (B(r)/B

)

which may take any value in the range 0 <β<1.

Another conﬁguration which has been important for studying plasma contain-

ment is the Z-pinch. Here the j, B lines of the theta-pinch are interchanged so

that with j now axial and B azimuthal, the j × B force is again directed towards

the axis. Consider a fully ionized plasma contained in a cylindrical discharge tube

with a current ﬂowing parallel to the axis of the tube, as shown in Fig. 4.5. Under

the action of the j × B force the plasma is squeezed or ‘pinched’ into a ﬁlament

along the axis of the tube. Since B

= 0 the static condition (4.18) may be written

=−

(rB) (4.22)

If the radius of the pinch is a, then multiplying (4.22) by r

and integrating gives



dr =−



(rB) d(rB)

i.e.

r=a

− 2



rPdr =−

2µ

(rB)

r=a

If we suppose that the plasma pressure vanishes at r = a the ﬁrst term vanishes

altogether and we get



rPdr =

2µ

(rB)

r=a

(4.23)