Fundamentals of Plasma Physics
270 Chapter 9. MHD equilibria by assigning a magnetic ‘pressure’ proportional toB^2 acting in the direction perpendicular toB. B ...
9.5 Magnetic stress tensor 271 9.5 Magnetic stress tensor The existence of magnetic pressure and tension shows that the magnetic ...
272 Chapter 9. MHD equilibria term involving∇⊥B^2 portrays a magnetic force due to pressure gradients perpendicular to the magne ...
9.6 Flux preservation, energy minimization, and inductance 273 Using Ampere’s law, Eq.(9.18) can thus be rewritten as W= 1 2 ∫ V ...
274 Chapter 9. MHD equilibria 9.7 Static versus dynamic equilibria We define (i) a static equilibrium to be a time-independent s ...
9.8 Static equilibria 275 9.8 Static equilibria 9.8.1 Static equilibria in two dimensions: the Bennett pinch The simplest static ...
276 Chapter 9. MHD equilibria the existence in the plasma surface of an induced azimuthal current which creates a mag- netic fie ...
9.8 Static equilibria 277 a resounding ‘no’, is provided by a virial theorem due to Shafranov (1966). A virial is a suitably wei ...
278 Chapter 9. MHD equilibria Consider the virial expression ∇·(T·r) = ∑ jk ∂ ∂xj (Tjkxk) = (∇·T)·r+ ∑ jk Tjk ∂ ∂xj xk = TraceT ...
9.8 Static equilibria 279 then using Eq.(9.22) to determine a correspondingB(r)with associated currentJ(r);alter- natively one c ...
280 Chapter 9. MHD equilibria showing that the functional form of Eq.(9.34) is consistent with Ampere’s law ∮ B·dl=μ 0 I. The po ...
9.8 Static equilibria 281 which further implies that∇Imust be parallel to∇ψ.An arbitrary displacementdrresults in respective cha ...
282 Chapter 9. MHD equilibria The Grad-Shafranov equation is a non-linear equation inψ and, in general, cannot be solved analyti ...
9.8 Static equilibria 283 Figure 9.11: Contours of constantflux of Solov’ev solution to Grad-Shafranov equation. Even though it ...
284 Chapter 9. MHD equilibria separatrixand is given by the ellipse r^2 +4α^2 z^2 =2r^20. (9.56) On the magnetic axisψis a local ...
9.8 Static equilibria 285 field does not contribute to confinement ifI′=0.IfI′is finite, plasma diamagnetism or paramagnetism wi ...
286 Chapter 9. MHD equilibria 9.9 Dynamic equilibria:flows The basic mechanism driving MHDflows will first be discussed using th ...
9.9 Dynamic equilibria:flows 287 definition differs slightly from the conventional definition of vorticity (curl of velocity) be ...
288 Chapter 9. MHD equilibria z Ir,zconst. Figure 9.12: Current channelI(r,z). The magnetic force is therefore J×B= 1 2 π (∇I ...
9.9 Dynamic equilibria:flows 289 form ∇g+Q×∇φ=0. (9.78) which is the most general form of an axisymmetric partial differential e ...
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