Fundamentals of Plasma Physics
310 Chapter 10. Stability of static MHD equilibria result with the additional stipulation that the normal component of the veloc ...
10.3 The MHD energy principle 311 becauseǫis small the spatial dependence ofU 1 can be ignored when evaluating the time integral ...
312 Chapter 10. Stability of static MHD equilibria Multiplying Eq.(10.71) by the perturbed velocityξ ̇and integrating over the v ...
10.3 The MHD energy principle 313 which shows thatω^2 must be pure real. Negativeω^2 corresponds to instability. Equation (10.78 ...
314 Chapter 10. Stability of static MHD equilibria and δWP 1 = − 1 2 ∫ Vmf d^3 rξ·∇[ξ·∇P 0 +γP 0 ∇·ξ] = 1 2 ∫ Vmf d^3 r [ −ξ·∇(ξ ...
10.3 The MHD energy principle 315 and δWS′= 1 2 μ 0 ∫ Smf ds·ξ⊥(B 0 ·B 1 −μ 0 γP 0 ∇·ξ). (10.94) At this point it is useful to e ...
316 Chapter 10. Stability of static MHD equilibria stituting Eq.(10.97) gives δWF′ = 1 2 ∫ Vmf d^3 r γP 0 (∇·ξ ...
10.3 The MHD energy principle 317 and magnetofluid fields at the perturbed surface. If the equilibrium force balance is inte- gr ...
318 Chapter 10. Stability of static MHD equilibria the magnetofluid-vacuum interface remains. Since an element of surfacedspoint ...
10.5 Current-driven instabilities and helicity 319 10.4 Discussion of the energy principle The terms in the integrands ofδWF,δWi ...
320 Chapter 10. Stability of static MHD equilibria magnetic field reduces to B 1 ‖=−B 0 [2ξ⊥·κ+∇·ξ⊥]. (10.119) Thus, we may iden ...
10.6 Magnetic helicity 321 The question immediately arises whether this definition makes sense, i.e., is it reasonable to define ...
322 Chapter 10. Stability of static MHD equilibria Since bothB·ˆn=0andEt=0at the wall, this reduces to K= ∫ d^3 rA·B=const.; (10 ...
10.7 Qualitative description of free-boundary instabilities 323 whereλis spatially uniform. This equation is called the force-fr ...
324 Chapter 10. Stability of static MHD equilibria the pressure is assumed to be (i) uniform in the interior regionr<awhereai ...
10.7 Qualitative description of free-boundary instabilities 325 weaker azimuthal magnetic field stronger azimuthal magneticfield ...
326 Chapter 10. Stability of static MHD equilibria azimuthal field is weaker so that the convex bulge will tend to increase.This ...
10.8 Analysis of free-boundary instabilities 327 can be written as 0=−μ 0 ∇P 0 −∇ ( B 02 2 ) +B 0 ·∇B 0. (10.141) Because∇·B 0 = ...
328 Chapter 10. Stability of static MHD equilibria If a configuration is to be stable, any perturbation must generate a restorin ...
10.8 Analysis of free-boundary instabilities 329 so that, when expressed as a relationship between normalized fields, Eq.(10.154 ...
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