Chapter 6

MAGNETOHYDRODYNAMICS

6.1 Introduction

Magnetohydrodynamics is a branch of plasma physics dealing with dc or low frequency effects in fully ionized magnetized plasma. In this chapter we will study both “ideal” (i.e. zero resistance) and resistive plasma, introducing the concepts of stability and equilibrium, “frozen-in” magnetic fields and resistive diffusion. We will then describe the range of low-frequency plasma waves supported by the single fluid equations.

6.2 Ideal MHD – Force Balance

The ideal MHD equations are obtained by setting resistivity to zero: E + u×B =0 (6.1) ∂ ρ u × −∇p. ∂t = j B (6.2) In equilibrium, the time derivative is ignored and the force balance equation j×B = ∇p (6.3) describes the balance between plasma pressure force and Lorentz forces. Such a balance is shown schematically in Fig. 6.1. What are the consequences of this simple expression? The current required to balance the plasma pressure is found by taking the cross product with B.Then B×∇p B×∇n j kBTe kBTi . ⊥ = B2 =( + ) B2 (6.4) This is simply the diamagnetic current! The force balance says that j and B are perpendicular to ∇p. In other words, j and B must lie on surfaces of constant 136

∆ p u B Di uDe

ur

jD

Figure 6.1: In a cylindrical magnetized plasma column, the pressure gradient is supported by the diamagnetic current j. In time, however, the gradient is

dissipated through radial diffusion u⊥ = ur.

pressure. This situation is shown for a tokamak in Fig. 6.2. The current density and magnetic lines of force are constrained to lie in surfaces of constant plasma pressure. The angle between the current and field increases as the plama pressure increases.

Surfaces of constant pressure j and B lie on constant p surfaces

j jxB is radially inwards

B Helical lines of force

TOKAMAK

Figure 6.2: In a tokamak, the equilibrium current density and magnetic field lie in nested surfaces of constant pressure. 6.2 Ideal MHD – Force Balance 137

Equation (6.3) can also be resolved in the direction parallel to B to yield ∂p . ∂z =0 (6.5) Plasma pressure is constant along a field line. (Why is this?)

6.2.1 Magnetic pressure

Substituting Maxwell’s equation ∇×B = µ0j into Eq. (6.3) gives
 1 1 1 ∇p = (∇×B)×B = (B.∇)B − ∇B2 (6.6) µ0 µ0 2 or   B2 1 ∇ p + = (B.∇)B (6.7) 2µ0 µ0 In many case, B does not vary along B (for example in an axially magnetized cylinder) so that the right side vanishes. In this case, pressure balance gives B2 p + = constant (6.8) 2µ0 2 and B /2µ0 represents the magnetic pressure. Thus, if the plasma exhibits a pressure gardient, there is a corresponding gradient in the magnetic pressure that ensures the total pressure is constant in the plasma fluid. This is illus- trated in Fig. 6.3 which shows the balance between magnetic and pressure forces in a cylindrical plasma. Noting that the plasma is diamagnetic, an azimuthal current jθ flows that generates an axial field that suppresses the magnetic field in the region of plasma pressure. This is consistent with our earlier picture of diamagnetism. The ratio  nk T β B = 2 (6.9) B /2µ0 indicates the size of the diamagnetic effect. For tokamaks, β does not exceed a few percent.

6.2.2 Magnetic tension It is known that two wires carrying parallel currents attract as if the magnetic lines of force were under tension – see Fig. 6.4. Magnetic tension is described by the term ∂ ∇ B B ˆ B ˆ B ˆ ... (B. )B = x ∂x( xi + yj + zk)+ (6.10)

If the magnetic lines of force are straight and parallel then B = Bxˆi and (B.∇)B = 0. This term is only important if the magnetic lines of force are 138

Pressure Plasma region

Bz(r) 2 µ p +B /2 0 = constant

p(r)

r

Figure 6.3: The plasma thermal pressure gradient is exactly balanced by a radial variation in the magnetic pressure. This variation is generated by diamagnetic currents that flow azimuthally.

B lines of force

Parallel current elements

Figure 6.4: The magnetic lines of force for wires carrying parallel currents.

curved. To show this, consider the geometric construction shown in Fig. 6.5 and let Bˆ = B/ |B | be the unit vector in the direction of the field. It is clear that the term B.ˆ ∇ is equal to ∂/∂ where  is the coordinate along the line of force so that ∂ ˆ ˆ ∇ ˆ B . B. B = ∂ (6.11) In terms of infinitesimal quantities, we have

∆Bˆ Bˆ ( +∆) − Bˆ () = . (6.12) ∆ ∆ 6.2 Ideal MHD – Force Balance 139

With reference to Fig. 6.5, by construction we have

Bˆ ( +∆) − Bˆ ()=−nˆdθ (6.13) where nˆ is the unit normal to the field line, while ∆ = Rdθ. Substituting into Eq. (6.12) gives ˆ ∇ ˆ −nˆ B. B = R (6.14) so that the magnetic tension is inversely proportional to the radius of curvature of the magnetic field line. When the lines of force are curved, they behave as

n^ ^ B( l) dθ l l+∆l ^ Curved line B( l+∆l) of force R

dθ

Figure 6.5: The geometric interpretation of the magnetic tension due to curvature of lines of force.

2 though under a tension force B /µ0 per unit area. As shown above this produces a force that is perpendicular to B (parallel to nˆ) and inversely proportional to the radius of curvature. Thus the lines of force can be regarded as elastic cords 2 under tension B /µ0.

6.2.3 Plasma stability Consider an unmagnetized linear plasma pinch carrying current I as shown in Fig. 6.6. We have Bz =0andBθ = µ0I/(2πr). If the radius of the plasma channel in some region gets smaller, the magnetic field gets larger. This increases the magnetic pressure with the result that the channel shrinks further. Though the plasma pressure increases also, it spreads out along the column and cannot 140

Β θ = µ0I/2π r

2 µ Pressure B /2 0

B1 > B2

Figure 6.6: An unmagnetized linear pinch showing sausage instability.

balance the locally high magnetic pressure. This is an unstable equilibrium and the instability is known as the m =0sausage instability. If the column develops a kink, the increased pressure and tension on the high B side increases the bending (Fig. 6.7). This is known as the m =1kink instability.

2 µ Pressure B /2 0 High B

Low B

Figure 6.7: The kinking of a plasma column under magnetic forces.

Instabilities can be prevented by adding a longitudinal magnetic field Bz to 2 “stiffen” the plasma. The Bz /2µ0 pressure resists the m = 0 mode and the 6.3 Frozen-in Magnetic Fields 141

2 tension Bz /µ0 counters the bending. The m = 1 mode can also be stabilized by a conducting wall. As the plasma moves towards the wall, it squashes the trapped axial magnetic flux against the wall (the flux is trapped between the conducting wall and the conducting plasma). The increased magnetic pressure then acts to suppress the instability.

6.3 Frozen-in Magnetic Fields

We examine the behaviour of the plasma when the magnetic field changes with time. The latter generates an emf given by Faraday’s law ∂ ∇× − B . E = ∂t (6.15)

Taking the curl of the infinte conductivity Ohm’s law Eq. (6.1) and using Fara- day’s law then gives ∂ B ∇× × . ∂t = (u B) (6.16) This relation will allow us to study the behaviour of magnetic flux through a surface moving with the plasma at velocity u. With reference to Fig. 6.8, the magnetic flux through the surface S is  φS = B.ds. (6.17) S

This flux changes either through time variations in B or due to the surface moving with the plasma fluid element to a new position where B is different — the convective term. The first contribution is just  ∂φS ∂B = .ds. ∂t S ∂t

To evaluate the convective component, let d be a unit vector lying on the circumference of S. Due to the motion of the plasma fluid element, this element sweeps out an area dA = udt×d in time dt (see Fig. 6.8). The associated change in magnetic flux through this area element is dφ = B.dA. The total rate of change of flux is found by integrating along  around the circumference of S:  dφ = B.(u×d) dt  = (B×u).d  = ∇×(B×u).ds (6.18) 142

S u dt

B dl dl u dt θ

dA = u dt sin θ dl dA = u dt x dl

Figure 6.8: Left:The flux through surface S as it is convected with plasma velocity u remains constant in time. Right: showing the area element dA swept out by the plasma motion. Note that this area vanishes when u is parallel to the circumferential element d.

where we have applied Stokes’ theorem in the final step. If now we add the two components of the changing flux we obtain   DφS ∂B = −∇×(u×B) .ds =0. (6.19) Dt ∂t

In the last step we have used Eq. (6.16). The magnetic flux through S doesn’t change with time. This implies that the magnetic lines of force are “frozen” into the plasma and are transported with it. Thus, if the plasma expands, the field strength decreases (reminiscent of magnetic moment). If the flux did change, there would be an induced emf established to oppose the change (Lenz’s law). This would establish an E/B drift that would change the plasma shape to preserve the magnetic flux. If the plasma is resistive, currents can flow which can dissipate the magnetic energy as we shall soon see.

6.4 Resistive Diffusion

Ideal MHD assumes the plasma is a perfect conductor. Let us investigate the relationship between plasma and magnetic pressure when finite plasma resistivity allows the magnetic field to diffuse into the conducting fluid. The Ohm’s law is now E + u×B = ηj (6.20) 6.4 Resistive Diffusion 143 and Faraday’s law gives ∂B = −∇×(ηj − u×B) ∂t   η = −∇× ∇×B − u×B µ0 η = − ∇×∇×B + ∇×(u×B). (6.21) µ0

With the vector identity

∇×∇×B = ∇(∇.B) −∇2B = −∇2B (6.22) we finally obtain ∂B η = ∇2B + ∇×(u×B). (6.23) ∂t µ0 This is the finite resistivity version of Eq. (6.16). The right side now includes a diffusive term that accounts for resistive decay of the magnetic field energy. We make some order of magnitude comparisons of the two terms:

∆(uB) uB ∇×(u×B) ∼ ∼ ∆x L where L is the scale length for change of u or B by a factor of two or three. For the diffusive term we have η η ∂B/∂x η B ∇2B ∼ ∆( ) ∼ . 2 µ0 µ0 ∆x µ0 L

We now define the magnetic Reynold’s number RM as the ratio of the estimates of the convective and diffusive terms: uB µ L2 µ vL R 0 0 . M = L ηB = η (6.24)

For low resistivity plasma (η → 0), RM  1 and Eq. (6.23) reduces to Eq. (6.16) describing frozen-in flow. For higher resistivity, RM  1 the plasma behaves like an ordinary conductor (u×B = 0 ) and we obtain a diffusion equa- tion [compare with Eq. (3.23)]:

∂B η = ∇2B (6.25) ∂t µ0 which can be solved using separation of variables. As a rough estimate, we write ∂B η = 2 B (6.26) ∂t µ0L 144

giving B = B0 exp (−t/τR) (6.27) where µ L2 τ 0 R = η (6.28) is the characteristic time for magnetic penetration into the conducting fluid. As an example, consider a plasma squashed rapidly to half its diameter (in time t  τR). The field strength would increase by a factor of four (since RM  1 when u is large). On the other hand, when u is small, RM → 0andtheB field would relax diffusively to its initial value. In the case where B diffuses through a conducting plasma (u = 0), the moving B lines of force induce eddy currents that ohmically heat the plasma. This energy comes from the magnetic field. The induced current is µ0j = ∇×B ∼ B/L and the energy lost per unit volume by the field to heat the plasma in time τR is     2 2 2 2 B µ0L B ηj τR = η = . µ0L η µ0

This corresponds nicely with the initial magnetic field energy density. Thus τR is the time for the magnetic energy to be resistively dissipated into heat.

6.4.1 Some illustrations

Resistive diffusion Consider the magnetized current carrying plasma column shown in Fig. 6.1. Due to finite resistivity, the plasma will eventually diffuse across Bz to the walls with ambipolar velocity given by Eq. (5.62) η ∂p ur = − . B2 ∂r

2 Since the magnetic and plasma pressures must balance p + B /2µ0 = constant and with ur ∼ L/τ we obtain L η B2 ∼ 2 τ B 2µ0L or µ L2 τ ∼ 0 η and the characteristic time for diffusion of the plasma is just the resistive decay time of the diamagnetic field! 6.4 Resistive Diffusion 145 plasma stability As shown below, when the magnetic field is sheared, resistive reconnection of the magnetic lines can occur causing the appearance of magnetic “islands” in the magnetic topology. This phenomenon is very important in magnetic confine- ment devices (the island “short circuits” adjacent nested magnetic surfaces) and astrophysical phenomena such as solar flares etc

Magnetic "island"

Sheared magnetic lines of force "Reconnected" B-lines

Figure 6.9: Showing the process of magnetic reconnection.

Problems

Problem 6.1 Consider a cylindrically symmetric plasma column (∂/∂z =0,∂/∂θ = 0) under equilibrium conditions confined by a magnetic field. Verify that in cylindrical coordinates, the radial component of the MHD force balance equation (Eq. 9.7 in notes - ignore g) becomes dp(r) = Jθ(r)Bz(r) − Jz(r)Bθ(r) dr Using Maxwell’s equations, show that

dBz Jθ(r)=−(1/µ0) dr drBθ Jz(r)=(1/µ0r) dr Therefore obtain the following basic equation for the equilibrium of a plasma column with cylindrical symmetry   d B2 B2 1 B2 p + z + θ = − θ dr 2µ0 2µ0 µ0 r Give a physical interpretation of the various terms in this equation. 146

Problem 6.2 Consider a plasma in the form of a straight circular cyclinder with a helical magnetic field given by

B = Bθ(r)θˆ + Bz(r)zˆ.

Show that the force per unit volume associated with the inward magnetic pressure for this configuration is ∂ −∇ B2/ µ − B2 r / µ ⊥( 2 0)= rˆ ∂r[ ( ) 2 0] and the force per unit volume associated with the magnetic tension due to the cur- vature of the magnetic field lines is

2 B.∇B/µ0 = −rˆBθ (r)/(µ0r)

Problem 6.3 For an equilibrium θ-pinch produced by an azimuthal current jθ as

shown, determine expressions for the radial distributions of jθ(r) and p(r) in terms

of Bz(r). Draw a diagram illustrating these radial distributions for the special case

when Bz is constant.

Problem 6.4 A cylindrically symmetric plasma column in a uniform axial (solenoidal) B-field has the magnetic field strength ramped up slowly in time. Using the integral form of Faraday’s law show that the changing magnetic field induces an azimuthal electric field r ∂Bz Eθ = − 2 ∂t Write down an expression for the associated E/B drift. What is happening to the plasma? Show that the magnetic flux enclosed in any plasma annulus stays constant in time.