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Chapter 6 MAGNETOHYDRODYNAMICS

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Chapter 6 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 × −∇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).
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