Fundamentals of Plasma Physics
350 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxation two ends atφ=0andφ=2π.This definition implies thatψ= ...
11.4 Kinking and magnetic helicity 351 subribbon lying between flux tube outer surface and magnetic surface is flux through t ...
352 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxation and Ktwist= ∫ Vtube d^3 r ( ψ(Φ) 2 π ∇φ·Baxis+Aaxis· ...
11.4 Kinking and magnetic helicity 353 The three direction gradients∇θ,∇Φ,and∇φform an orthonormal coordinate system and an elem ...
354 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxation It should be noted that the ability forCto be contin ...
11.4 Kinking and magnetic helicity 355 Case where axis is helical andb>a Because the helix minor radiusbexceeds theflux tube ...
356 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxation ab a b flux tube surface flux tube surface a b b a ...
11.5 Assignments 357 A kink instability which starts withb=0and an initial twistTinitial=nwill grow until atb=athe twist vanishe ...
358 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxation property to argue that the rate of dissipation of ma ...
11.5 Assignments 359 (b) Cut the ribbon all the way along its length so that it becomes two interlinked ribbons of unequal width ...
12 Magnetic reconnection 12.1 Introduction Section 2.5.4 established the fundamental concept underlying ideal MHD, namely that m ...
12.2 Water-beading: an analogy to magnetic tearing and reconnection 361 long thin drop (a) (b) (c) contraction to single large d ...
362 Chapter 12. Magnetic reconnection width until circular. Because only modest frictional dragging of water across the substrat ...
12.3 Qualitative description of sheet current instability 363 which integrates to give By(x)=By(0)+ ∫x 0 μ 0 Jz(x′)dx′. (12.2) O ...
364 Chapter 12. Magnetic reconnection is weakest. The location of the weak spot can be deduced by examining Ohm’s law for a near ...
12.4 Semi-quantitative estimate of the tearing process 365 For|x|<<L,cosh(x/L)≃1+x^2 /L^2 while for|x|>>L,cosh(x/L)≃ ...
366 Chapter 12. Magnetic reconnection Equation (12.18) can be recast using Eqs.(12.15) and (12.19) as aninduction equation, − ∂A ...
12.4 Semi-quantitative estimate of the tearing process 367 Using Eq.(12.21) this can be written as ∂Ω 1 ∂t = 1 μ 0 ρ 0 ˆz· [ ∇Az ...
368 Chapter 12. Magnetic reconnection The stream-functionf 1 is a solution of the Poisson-like system Eq.(12.25) and Eq.(12.33), ...
12.4 Semi-quantitative estimate of the tearing process 369 product J 1 zB′x vorticity source J 1 z B′x x Figure 12.3: Product o ...
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