MAGNETOHYDRODYNAMICS

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 t = j B p. (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 j = B p B 2 =(k B T e + k B T i ) B n B 2. (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 u Di p u De B u r j D 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 = u r. 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 =0. (6.5) z Plasma pressure is constant along a field line. (Why is this?) 6.2.1 Magnetic pressure Substituting Maxwell s equation B = µ 0 j into Eq. (6.3) gives p = 1 ( B) B = 1 [ (B. )B 1 ] µ 0 µ 0 2 B2 (6.6) or ( ) p + B2 = 1 (B. )B (6.7) 2µ 0 µ 0 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 p + B2 2µ 0 = constant (6.8) and B 2 /2µ 0 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 illustrated 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 j θ 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 nkb T β = (6.9) B 2 /2µ 0 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 = B x x (B xî + B yĵ + B z ˆk)+... (6.10) If the magnetic lines of force are straight and parallel then B = B x î and (B. )B = 0. This term is only important if the magnetic lines of force are

138 Pressure Plasma region B z (r) p + B 2 /2µ 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 ˆB = B/ B be the unit vector in the direction of the field. It is clear that the term ˆB. is equal to / l where l is the coordinate along the line of force so that ˆB. ˆB = ˆB l. (6.11) In terms of infinitesimal quantities, we have ˆB l = ˆB(l + l) ˆB(l). (6.12) l

6.2 Ideal MHD Force Balance 139 With reference to Fig. 6.5, by construction we have ˆB(l + l) ˆB(l) = ˆndθ (6.13) where ˆn is the unit normal to the field line, while l = Rdθ. Substituting into Eq. (6.12) gives ˆB. ˆB = ˆn (6.14) R 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 Curved line of force l R ^ B( l) l+ l dθ ^ B( l+ l) dθ Figure 6.5: The geometric interpretation of the magnetic tension due to curvature of lines of force. though under a tension force B 2 /µ 0 per unit area. As shown above this produces a force that is perpendicular to B (parallel to ˆn) and inversely proportional to the radius of curvature. Thus the lines of force can be regarded as elastic cords under tension B 2 /µ 0. 6.2.3 Plasma stability Consider an unmagnetized linear plasma pinch carrying current I as shown in Fig. 6.6. We have B z =0andB θ = µ 0 I/(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 = µ 0 I/2π r Pressure B 2 /2µ 0 B 1 > B 2 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 = 0 sausage 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. Pressure B 2 /2µ 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 B z to stiffen the plasma. The Bz 2/2µ 0 pressure resists the m = 0 mode and the

6.3 Frozen-in Magnetic Fields 141 tension B 2 z /µ 0 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 E = t. (6.15) Taking the curl of the infinte conductivity Ohm s law Eq. (6.1) and using Faraday s law then gives = (u B). (6.16) t 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 φ S = 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 φ S t = S t.ds. To evaluate the convective component, let dl 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 dl 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 l around the circumference of S: dφ dt = = = B.(u dl) (B u).dl (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 dl. where we have applied Stokes theorem in the final step. If now we add the two components of the changing flux we obtain Dφ S Dt = [ ] t (u B).ds =0. (6.19) 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 t = (ηj u B) ( ) η = B u B µ 0 = η µ 0 B + (u B). (6.21) With the vector identity B = (.B) 2 B = 2 B (6.22) we finally obtain t = η 2 B + (u B). (6.23) µ 0 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: (u B) (ub) x ub 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 2 B η (/ x) µ 0 µ 0 x η µ 0 B L 2. We now define the magnetic Reynold s number R M as the ratio of the estimates of the convective and diffusive terms: R M = ub L µ 0 L 2 ηb = µ 0vL η. (6.24) For low resistivity plasma (η 0), R M 1 and Eq. (6.23) reduces to Eq. (6.16) describing frozen-in flow. For higher resistivity, R M 1 the plasma behaves like an ordinary conductor (u B = 0 ) and we obtain a diffusion equation [compare with Eq. (3.23)]: t = η µ 0 2 B (6.25) which can be solved using separation of variables. As a rough estimate, we write t = η µ 0 L 2 B (6.26)

144 giving B = B 0 exp ( t/τ R ) (6.27) where τ R = µ 0L 2 (6.28) η 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 τ R ). The field strength would increase by a factor of four (since R M 1 when u is large). On the other hand, when u is small, R M 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 µ 0 j = B B/L and the energy lost per unit volume by the field to heat the plasma in time τ R is ( ) 2 ( B ηj 2 µ0 L 2 ) τ R = η µ 0 L η = B2 µ 0. This corresponds nicely with the initial magnetic field energy density. Thus τ R 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 B z to the walls with ambipolar velocity given by Eq. (5.62) u r = η B 2 p r. Since the magnetic and plasma pressures must balance p + B 2 /2µ 0 = constant and with u r L/τ we obtain L τ η B 2 B 2 2µ 0 L or τ µ 0L 2 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 confinement 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) = J θ (r)b z (r) J z (r)b θ (r) dr Using Maxwell s equations, show that J θ (r) = (1/µ 0 ) db z dr J z (r) =(1/µ 0 r) drb θ dr Therefore obtain the following basic equation for the equilibrium of a plasma column with cylindrical symmetry d dr ( p + B2 z 2µ 0 + B2 θ 2µ 0 ) = 1 µ 0 B 2 θ 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 = B θ (r)ˆθ + B z (r)ẑ. Show that the force per unit volume associated with the inward magnetic pressure for this configuration is (B 2 /2µ 0 )= ˆr r [B2 (r)/2µ 0 ] and the force per unit volume associated with the magnetic tension due to the curvature of the magnetic field lines is B. B/µ 0 = ˆrB 2 θ(r)/(µ 0 r) Problem 6.3 For an equilibrium θ-pinch produced by an azimuthal current j θ as shown, determine expressions for the radial distributions of j θ (r) and p(r) in terms of B z (r). Draw a diagram illustrating these radial distributions for the special case when B z is constant. Problem 6.4 A cylindrically symmetric plasma column ina 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 E θ = r z 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.