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Interchange, Sausage and Kink Instabilities 85 Fig.8.3 Charge separation due to the di\ufb00erence in velocities of ions and electrons. The growth rate becomes \u3b3 \u2248 (a/R)1/2(vT/a) in this case. Analysis of interchange instability based on the linearlized equation of motion (8.32) with the acceralation term is described in (ref.[1]). For a perturbation with propagation vector k normal to the magnetic \ufb01eld B, i.e., (k ·B) = 0, another mechanism of charge separation may cause the same type of instability. When a plasma rotates with the velocity v\u3b8 = Er/B due to an inward radial electric \ufb01eld (\ufb01g.8.3), and if the rotation velocity of ions falls below that of electrons, the perturbation is unstable. Several possible mechanisms can retard ion rotation. The collision of ions and neutral particles delays the ion velocity and causes neutral drag instability. When the growth rate \u3b3 \u223c (gky)1/2 is not very large and the ion Larmor radius \u3c1i\u3a9 is large enough to satisfy (ky\u3c1i\u3a9) 2 > \u3b3 |\u3a9i| the perturbation is stabilized (ref.[3]). When the ion Larmor radius becomes large, the average perturbation electric \ufb01eld felt by the ions is di\ufb00erent that felt by the electrons, and the E ×B/B2 drift velocities of the ion and the electrons are di\ufb00erent. The charge separation thus induced has opposite phase from the charge separation due to acceleration and stabilizes the instabiltity. 8.1b Stability Criterion for Interchange Instability, Magnetic Well Let us assume that a magnetic line of force has \u201cgood\u201d curvature at one place B and \u2018bad\u201d curvature at another place A (\ufb01g.8.4). Then the directions of the centrifugal force at A and B are opposite, as is the charge separation. The charge can easily be short circuited along the magnetic lines of the force, so that the problem of stability has a di\ufb00erent aspect. Let us here consider perturbations in which the magnetic \ufb02ux of region 1 is interchanged with that of region 2 and the plasma in the region 2 is interchanged with the plasma in the region 1 (interchange perturbations, \ufb01g.8.4b). It is assumed that the plasma is low-beta so that the magnetic \ufb01eld is nearly identical to the vacuum \ufb01eld. Any deviation from the vacuum \ufb01eld is accompanied by an increase in the energy of the disturbed \ufb01eld. This is the consequence of Maxwell equation. It can be shown that the most dangerous perturbations are those which exchange equal magnetic \ufb02uxes, as follows. The energy of the magnetic \ufb01eld inside a magnetic tube is QM = \u222b dr B2 2\u3bc0 = \u222b dlS B2 2\u3bc0 (8.12) where l is length taken along a line of magnetic force and S is the cross section of the magnetic tube. As the magnetic \ufb02ux \u3a6 = B · S is constant, the energy is QM = \u3a62 2\u3bc0 \u222b dl S . The change \u3b4QM in the magnetic energy due to the interchange of the \ufb02uxes of regions 1 and 2 is \u3b4QM = 1 2\u3bc0 (( \u3a621 \u222b 2 dl S + \u3a622 \u222b 1 dl S ) \u2212 ( \u3a621 \u222b 1 dl S + \u3a622 \u222b 2 dl S )) . (8.13) If the exchanged \ufb02uxes \u3a61 and \u3a62 are the same, the energy change \u3b4QM is zero, so that perturba- tions resulting in \u3a61 = \u3a62 are the most dangerous. 85 86 8 Magnetohydrodynamaic Instabilities Fig.8.4 Charge separation in interchange instability. (a) The lower \ufb01gure shows the unstable part A and the stable part B along a magnetic line of force. The upper \ufb01gure shows the charge separation due to the acceleration along a \ufb02ute. (b) Cross section of the perturbed plasma. The kinetic energy Qp of a plasma of volume V is Qp = nTV \u3b3 \u2212 1 = pV \u3b3 \u2212 1 (8.14) where \u3b3 is the speci\ufb01c-heat ratio. As the perturbation is adiabatic, pV\u3b3 = const. is conserved during the interchange process. The change in the plasma energy is \u3b4Qp = 1 \u3b3 \u2212 1 ( p\u20322V2 \u2212 p1V1 + p\u20321V1 \u2212 p2V2 ) . where p\u20322 is the pressure after interchange from the region V1 to V2 and p\u20321 is the pressure after interchange from the region V2 to V1. Because of adiabaticity, we have p\u20322 = p1 ( V1 V2 )\u3b3 , p\u20321 = p2 ( V2 V1 )\u3b3 and \u3b4Qp becomes \u3b4Qp = 1 \u3b3 \u2212 1 ( p1 (V1 V2 )\u3b3 V2 \u2212 p1V1 + p2 (V2 V1 )\u3b3 V1 \u2212 p2V2 ) . (8.15) Setting p2 = p1 + \u3b4p, V2 = V1 + \u3b4V we can write \u3b4Qp as \u3b4Qp = \u3b4p\u3b4V + \u3b3p(\u3b4V) 2 V . (8.16) Since the stability condition is \u3b4Qp > 0, the su\ufb03cient condition is \u3b4p\u3b4V > 0. 86 8.1 Interchange, Sausage and Kink Instabilities 87 Fig.8.5 Speci\ufb01c volume of a toroidal \ufb01eld. Since the volume is V = \u222b dlS = \u3a6 \u222b dl B the stability condition for interchange instability is written as \u3b4p\u3b4 \u222b dl B > 0. Usually the pressure decreases outward (\u3b4p < 0), so that the stability condition is \u3b4 \u222b dl B < 0 (8.17) in the outward direction (ref.[4]). The integral is to be taken only over the plasma region. Let the volume inside a magnetic surface \u3c8 be V and the magnetic \ufb02ux in the toroidal direction \u3d5 inside the magnetic surface \u3c8 be \u3a6. We de\ufb01ne the speci\ufb01c volume U by U = dV d\u3a6 . (8.18) If the unit vector of the magnetic \ufb01eld B is denoted by b and the normal unit vector of the in\ufb01nitesimal cross-sectional area dS is denoted by n, then we have dV = \u222b \u2211 i (b · n)iSidl, d\u3a6 = \u2211 i (b · n)iBidSi. When the magnetic lines of force close upon a single circuit of the torus, the speci\ufb01c volume U is U = \u222e (\u2211 i (b · n)idSi ) dl\u2211 i (b · n)iBidSi = \u2211 i (b · n)iBidSi \u222e dl Bi\u2211 i (b · n)iBidSi . As the integral over l is carried out along a small tube of the magnetic \ufb01eld, \u2211 i (b · n)idSiBi is independent of l (conservation of magnetic \ufb02ux). As \u222e dl/Bi on the same magnetic surface is constant, U is reduced to U = \u222e dl B . When the lines of magnetic force close at N circuits, U is U = 1 N \u222b N dl B . (8.19) 87 88 8 Magnetohydrodynamaic Instabilities Fig.8.6 Sausage instability. When the lines of magnetic force are not closed, U is given by U = lim N\u2192\u221e 1 N \u222b N dl B . Therefore, U may be considered to be an average of 1/B. When U decreases outward, it means that the magnitude B of the magnetic \ufb01eld increases outward in an average sense, so that the plasma region is the so-called average minimum-B region. In other word, the stability condition for interchange instability is reduced to average minimum-B condition; dU d\u3a6 = d2V d\u3a62 < 0. (8.20) When the value of U on the magnetic axis and on the outermost magnetic surface are U0 and Ua respectively, we de\ufb01ne a magnetic well depth \u2212\u394U/U as \u2212\u394U U = U0 \u2212 Ua U0 . (8.21) 8.1c Sausage Instability Let us consider a cylindrical plasma with a sharp boundary. Only a longitudinal magnetic \ufb01eld Bz, exists inside the plasma region and only an azimuthal \ufb01eld H\u3b8 = Iz/2\u3c0r due to the plasma current Iz exists outside the plasma region. We examine an azimuthally symmetric perturbation which constricts the plasma like a sausage (\ufb01g.8.6). When the plasma radius a is changed by \u3b4a, conservation of magnetic \ufb02ux and the current in the plasma yields \u3b4Bz = \u2212Bz 2\u3b4a a , \u3b4B\u3b8 = \u2212B\u3b8 \u3b4a a . The longitudinal magnetic \ufb01eld inside the plasma acts against the perturbation, while the external azimuthal \ufb01eld destabilizes the perturbation. The di\ufb00erence \u3b4pm in the magnetic pressures is \u3b4pm = \u2212B 2 z \u3bc0 2\u3b4a a + B2\u3b8 \u3bc0 \u3b4a a . The plasma is stable if \u3b4pm > 0 for \u3b4a < 0, so that the stability condition is B2z > B2\u3b8 2 . (8.22) 88 8.1 Interchange, Sausage and Kink Instabilities 89 Fig.8.7 Kink instability. This type of instability is called sausage instability. 8.1d Kink Instability Let us consider a perturbation that kinks the plasma column as shown in \ufb01g.8.7. The con\ufb01gu- ration of the plasma is the same as that in the previous subsection (sharp boundary, an internal longitudinal \ufb01eld, an external azimuthal \ufb01eld). Denote the characteristic length of the kink by \u3bb and its radius of curvature by R. The longitudinal