35. RESISTIVE INSTABILITIES: CLOSING REMARKS

Transcription

1 35. RESISTIVE INSTABILITIES: CLOSING REMARKS In Section 34 we presented a heuristic discussion of the tearing mode. The tearing mode is centered about x = 0, where F x that B 0 ( ) = k! B = 0. Note that this does not require ( ) = 0 ; it only requires that k be perpendicular to B there. Thus the tearing mode (and other resistive modes) can occur even when there is a large component of B in the z-direction. The scaling of the growth rate and resistive layer width with the Lundquist number and normalized wave number for the tearing mode are, for long wavelength, and!" A # $ %2/5 S %3/5 (35.1)! " # $3/5 S $2/5!!!. (35.2) The tearing mode derives its free energy from the configuration of the magnetic field. As is ideal MHD, there are other resistive instabilities that derive their free energy from different sources. For example, if we include gravity in the x-direction we can get an instability called the resistive g-mode. The growth rate and resistive layer width for this mode scale like and!" A # $ 2/3 S %1/3 G 2/3!!!, (35.3)! " # $1/3 S $1/3 G 1/6!!!, (35.4) where G =! A 2 A 1. The form of A 1 depends on the origin of the gravitational force. If it is from a true gravitational field, then A 1 ~! g " 0 d" 0 dx if it is from an accelerating frame of reference, then A 1!! 1 " 0 and, if it is from field line curvature, then A 1 ~! 1 " A 2!!!; (35.5) d ( dx " " 0 V 0 )!!!; (35.6) # a 2 & $ % 4R c ' ( d) 0 dx where R c is the radius of curvature of the field lines.!!!, (35.7) If we include a resistivity gradient d! / dx, the we can get an instability called the rippling mode, which scales like and!" A # $ 2/5 S %3/5!!!, (35.8) 1

2 ! " # $2/5 S $2/5!!!. (35.9) There are some resistive instabilities that do not satisfy the constant-! approximation. This can occur in a General Screw Pinch with a monotonically increasing q-profile, when the displacement has the form of the top hat trial function introduced in a previous Section. Then the perturbation must vary rapidly across the resistive layer centered about the singular surface. (This type of instability is thought to be responsible for the crash phase of tokamak sawtooth oscillations.) These non-constant-! modes can have a larger growth rate, i.e.,!" A ~ S #1/2, but still do not exceed the Sweet-Parker rate. Resistive instabilities can be wall stabilized. The magnetic field must remain parallel to any conducting boundary. Recall that the drive, or free energy, for the tearing mode comes from the outer region. If conducting boundaries are placed at x = ±L, they will inhibit the bending of the field lines in the outer region that accompanies the formation of magnetic islands. If they are placed close enough to the singular surface, an unstable mode with wave number k can be completely stabilized. This effect is obviously minimized at long wavelength. The transition from a state with no magnetic islands to one containing magnetic islands can be thought of as an example of neighboring equilibria. Consider two equilibrium states that satisfy the boundary conditions, one with magnetic islands and one without. Let the state without magnetic islands have energy W 1, and the state with magnetic islands have W 2, and let the system initially be in the state without islands. The system wants to be in the minimum energy state. If W 1 > W 2, the system would prefer to be in state 2, with islands. However, this transition is forbidden by the ideal MHD constraints on the invariance of the magnetic topology. In resistive MHD, such a transition is possible. This situation has been analyzed, and it turns out that the energy of such equilibria depend on the parameter "!. When "! < 0, then W 1 < W 2 and the state without islands is the preferred state; when "! > 0, then W 1 > W 2 and the state with islands is preferred. But "! > 0 is just the instability criterion for the tearing mode derived in Section 34, so the problems of stability to small perturbations and the energy of neighboring equilibria are equivalent in this regard. The mathematical theory of resistive instabilities relies on the method of matched asymptotic expansions. This is a method for obtaining an approximate solution to a boundary value problem with an ordinary differential equation whose highest derivative is multiplied by a small parameter. For example, consider the equation! d 2 u dx + ( 1 +!) du + u = 0!!!, (35.10) 2 dx ( ) = 0 and u( 1) = 1. One is tempted with! << 1, subject to the boundary conditions u 0 to set! = 0, and solve the simpler equation du + u = 0!!!. (35.11) dx 2

3 However, Equation (35.11) is of lower order than Equation (35.10), and therefore requires only a single boundary condition, whereas the solution of Equation (35.10) requires two boundary conditions. For Equation (35.11), the boundary condition corresponding to a non-trivial solution is u 1 imposed on u x way, if u x,! (35.11), then ( ) = 1. Therefore, the condition u( 0) = 0 ( ) cannot be satisfied by the solution of Equation (35.11). Put another ( ) is a solution Equation (35.10), and u( x, 0) is a solution of Equation lim u x,!!"0 ( )!# u( x,0)!!!. (35.12) This is an example of non-uniform convergence, and is illustrated in the figure. In Section (34), we showed that the perturbed velocity satisfies Equation (34.19), which can be rewritten as #! d 2 V x1 & " k 2 V dx 2 x1 $ % ' ( " k 2 2 B 0 V x1 = ik # ) "! B 0 ++ & )* o )* o $ % µ 0 ' ( B x1(0)!!!. (35.13) The small parameter! multiplies the highest derivative; if! = 0 we have ideal MHD. In light of Equation (35.12), we see that the solution of the resistive MHD equations in the limit! " 0 does not approach the solution of the ideal MHD equations. The ideal MHD equations require fewer boundary conditions. The tearing mode still exists in the limit! " 0, but it does not exist in ideal MHD. We now return to the boundary value problem given by Equation (35.10) and following. Since! << 1, we expect solutions of (35.10) and (35.11) to be almost equal except where u!! ~ 1 / " >> 1. From the figure, we see that this will occur in a small region near x = 0, where the solutions must rapidly diverge in order for u x,! the boundary condition at u 0 B 0 ( ) to satisfy ( ) = 0. We anticipate the region of disagreement to scale like!x ~ " #. Since this occurs near a boundary, the region!x is called a boundary layer. 3

4 We can obtain an approximate solution of Equation (35.10) by dividing the problem into two regions: 1. An outer region, x >! ", where u = u x,0 ( ) = 1. u 1,0 ( ), which satisfies! 2. An inner region, x <! ", where u = u x,! ( ) "! u "" + 1+! u + u = 0 with u 0,! These solution must be matched at x ~! ". ( ) = 0. u + u = 0 with ( ), which satisfies The solution in the outer region is u( x,0) = e 1! x. In the inner region, it is useful to rescale the independent variable as! = x / ". Then Equation (35.10) becomes 1 d 2 u 1! d" + 1 ( 2! 1+!) du 1 d" + u 1 = 0!!!, (35.14) If we now let! " 0 and integrate, we have du 1 d! + u 1 = C!!!, (35.15) where C is a constant of integration. The solution of Equation (35.15) that satisfies u 1 ( 0)! u( 0," ) = 0 is u 1 (!) = C( 1" e "! )!!!. (35.16) The key step in the method is to require that lim u 1 (!) = lim u x,0!"# x"0 ( )!!!. (35.17) This is often stated as the outer limit of the inner solution equals the inner limit of the outer solution. Applying Equation (35.17) results in C = e, so that u 1 ( x) = C( 1! e! x /" ). The approximate solution of the boundary value problem is then!u ( x,! ) = u( x,0) + u 1 ( x) " u( 0,0)!!!. (35.18) 4

5 We must subtract the common value of the two solutions at x = 0 and! " # to assure that the solution is continuous. The result is!u ( x,! ) = e( e " x " e " x /! )!!!. (35.19) Actually, in this example Equation (35.19) satisfies the differential equation (35.10) exactly. However,!u ( 1,! ) = 1 " e 1"1/! # 1, so that the boundary condition at x = 1 is not satisfied. It is therefore only an approximate solution of the boundary value problem. Nonetheless, lim!u 1,!!"0 ( ) = 1, so that the approximate solution converges to the actual solution in the proper limit. In this case it is possible to obtain an exact solution of the boundary value problem as u x,! ( ) / ( 1" e 1"1/! ). The difference between this exact solution and the ( ) = e e " x " e " x /! approximate solution of Equation (35.19) is sketched in the figure. Recall that in our analysis of the tearing mode, we found that the ideal MHD solution has a discontinuity "! in its first derivative at x = 0, and that this must be matched to the resistive solution in a thin resistive layer about x = 0. This layer is equivalent to the boundary layer in the above example (even though it doesn t occur at a boundary). Furth, Killeen, and Rosenbluth used the method of matched asymptotic expansions to solve this problem. The lower order ideal MHD MHD equations are valid in the outer region. The higher order resistive equations are valid in the inner region. In the inner region the equations are rescaled and solved. The solutions in the inner and outer regions are then matched by requiring B x1! " # " inner = "! outer. The ( ) inner = B x1 ( x " 0) outer and! mathematics required to solve the equations in the inner region is quite complicated, involving expansions in Hermite polynomials, etc. Nonetheless, the approach is as given here, and results in the scaling laws given by Equations (35.1) and (35.2). Of course, this can all be done in toroidal geometry. Then the criterion for instability becomes "! > "! C > 0, so that toroidal geometry is actually stabilizing. Finally, we comment on nonlinear effects on the growth of the tearing mode. In the linear regime the initial island width W, defined in a previous Section, is much smaller than the layer width!a, and the mode grows exponentially. However, it turns out that, when W! "a, there are nonlinear J! B forces that oppose the island growth, and these must compete with the drive of the linear instability. The result is that when the island 5

6 width becomes comparable to or larger than the width of the resistive layer, the island width grows as dw dt = 1.22! #"!!!, (35.19) or W ~!#" t ; exponential growth ceases and the island grows linearly in time with a rate proportional to the resistivity, or 1 / S. Equation (35.19) is called the Rutherford equation. Further analysis reveals that where "! W dw dt ( ) $ %W = 1.22! &' #" W ()!!!, (35.20) ( ) is now a function of W and! depends on equilibrium parameters. The ( ) # $W = 0, which can be solved for the island stops growing (saturates) when "! W saturated island width. And that s all for resistive instabilities! 6