Fundamentals of Plasma Physics
330 Chapter 10. Stability of static MHD equilibria here the circumflex means the modified Bessel function is evaluated at the wa ...
10.8 Analysis of free-boundary instabilities 331 If we introduce the normalized pressure P ̄ 0 =^2 μ^0 P^0 B 02 vθ(a) (10.172) t ...
332 Chapter 10. Stability of static MHD equilibria Sausage modes Them=0modes are the sausage instabilities and here Eq. (10.174) ...
10.8 Analysis of free-boundary instabilities 333 so the stability condition becomes x^2 +(m+x)^2 >|m|=⇒stable. (10.185) Witho ...
334 Chapter 10. Stability of static MHD equilibria 10.9 Assignments Interchange instabilities and volume per unitflux- Another ...
10.9 Assignments 335 particular, indicate by shading regions whereF< 0 .Do this for bothm=0and for m=− 1 so as to show the re ...
11 Magnetic helicity interpreted and Woltjer-Taylor relaxation 11.1 Introduction The previous chapter introduced the concept of ...
11.2 Topological interpretation of magnetic helicity 337 BecauseBvanishes outside of the twoflux tubes, the helicity integral is ...
338 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxation Theflux tubes did not actually have to be thin: a fa ...
11.2 Topological interpretation of magnetic helicity 339 We next add another layer of ‘paint’ with more embeddedψflux, and also ...
340 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxation The twist of the magnetic field is defined as the nu ...
11.3 Woltjer-Taylor relaxation 341 11.2.3Conservation of magnetic helicity during magnetic reconnection Ideal MHD constrains mag ...
342 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxation (a) (b) (c) (d) (e) (f) Figure 11.3: (a) Two linked, ...
11.3 Woltjer-Taylor relaxation 343 The relaxation process involves minimizing the magnetic energy W= 1 2 μ 0 ∫ B^2 d^3 r (11.24) ...
344 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxation Minimization ofδW subject to the constraint thatKrem ...
11.4 Kinking and magnetic helicity 345 11.4 Kinking and magnetic helicity Magnetic helicity can be manifested in various forms a ...
346 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxation axis of helix helical flux tube , volume Vtu be link ...
11.4 Kinking and magnetic helicity 347 11.4.1KVtube,helicity content of theflux tube We now consider the helicity content of the ...
348 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxation axial segment of helical flux tube surface helical a ...
11.4 Kinking and magnetic helicity 349 Figure 11.6 shows a cross-section of this system. Theflux tube has minor radiusaand the h ...
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