Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton m n ≈ 1 ι 0 = q 0 22.6 Then b r ( r 0 ,θ,φ) = b m,n sin mθ − nφ + β ( ) 22.7 with b m,n = 2 a m, n 2 + c m, n and tan( β) = b m ,n a m,n Next we define a new variable η = mθ − nφ + β 22.8 so that equation 22.4 becomes ( ) dx r 0 dθ = b sin η m ,n B θ (r 0 ) 22.9 We also obtain from equations 22.3 and 22.8 dη dθ = − n 0 ι + ' ι x + m 22.10 Substituting equation 22.9 into equation 2.10 gives us a non linear equation d 2 η = −n d ⎡ ⎢ dθ 2 dθ ⎣ 1 ⎛ 1− ι ' ⎞ ⎤ ⎜ x⎟ ⎥ = n r 0ι ' 2 ⎝ 0 ι ⎠ ⎦ 0 ι ι 0 ( ) ( ) b m,n sin η B θ r 0 = −Asin( η) 22.11 where A = −n r 0ι ' ι 0 2 b m ,n ( ) B θ r 0 22.12 We can take A > 0 for convenience. We see the behavior of the field lines is analogous to the motion of a particle in a periodic potential U = A (1-cos(η)), when we regard η and θ as position and time. Multiplying equation 22.11 by dη/dθ and integrating over θ gives 1 ⎛ dη ⎜ ⎞ ⎟ 2 ⎝ dθ ⎠ 2 = E − A(1 − cos( η)) 22.13 162
Magnetic fields and tokamak plasmas Alan Wootton with E = 1 ⎛ dη⎞ ⎜ ⎟ 2 ⎝ dθ ⎠ that 2 η= 0 a constant of integration analogous to a kinetic energy. Let k 2 = E 2A , so ⎛ dη⎞ ⎜ ⎟ ⎝ dθ ⎠ 2 ⎛ η = 4A k 2 − sin 2 ⎛ ⎞ ⎞ ⎜ ⎜ ⎟ ⎟ 22.14 ⎝ ⎝ 2 ⎠ ⎠ Two types of solution are possible, as shown in Figure 22.1. If E < 2A the field lines move periodically within a limit of η. Field lines with E > 2A are “passing”. _ dη dθ E>2A E=2A E
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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