Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton Often we will write an expression for the Fourier coefficients of the field at the resonant surface, b mn ( ) = µ 0 j mn r mn 2 . Note if we were working on a rectangular vessel we would measure r = r mn for example b η (see Figure 10.1): ⎛ r b η = b θ cos( θ )+ b r sin( θ) = b mn ⎞ mn ⎝ r ⎠ m +1 cos( m +1)θ + nφ − ω mn t) 17.15 i.e. we see that working with a rectangular contour the phase varies as (m+1)θ, instead of mθ when working with a circular contour. Now in toroidal geometry there are complications. We must consider that a) j mn produces a stronger field at the inner equator b) the perturbation are displaced because of a Shafranov shift c) the pitch of the field lines is no longer constant We will now derive the toroidal corrections for a case where we are considering surface current perturbations for our model equilibrium of section 6. We start with the field line equation for the total equilibrium fields, evaluated at a radius r = a p , the plasma minor radius: adθ = Rdφ 17.16 B θ B φ With R = R g + acos( θ) 17.17 B φ = B (R = R ) φ 0 g ⎛ 1 + a ⎞ ⎜ cos( θ ) ⎟ ⎝ R g ⎠ 17.18 B θa = µ I ⎛ 0 p 2πa 1 + a ⎞ ⎜ Λcos( θ) ⎟ 17.19 ⎝ R g ⎠ Λ = β I + l i −1 17.20 2 122
Magnetic fields and tokamak plasmas Alan Wootton Figure 17.2. Field lines around a torus, for the case q = 2. We can derive what is sometimes called q mhd , sometimes q I . This is the number of toroidal field revolutions a field line must be followed to make one poloidal revolution. Figure 17.2 shows the field lines for the case where q = 2. Generally we have q MHD = 1 2π aB φ 2π ∫ dθ 17.21 RB θ Keeping (a/R) 2 terms gives 0 q MHD = 2πa2 B φ 0 ⎡ ⎛ 1 + a ⎞ µ 0 I p R ⎢ ⎜ ⎟ g ⎝ R ⎣ g ⎠ ⎡ ⎛ = q circ 1 + a ⎞ ⎢ ⎜ ⎟ ⎣ ⎝ R g ⎠ 1 + 0.5 β I + l 2 ⎛ ⎜ ⎛ i ⎞ ⎞ ⎤ ⎟ ⎝ ⎝ 2⎠ ⎠ ⎥ ⎦ 2 1+ 0.5 β I + l 2 ⎛ ⎜ ⎛ i ⎞ ⎞ ⎤ ⎟ ⎝ ⎝ 2⎠ ⎠ ⎥ ⎦ 2 17.22 The toroidal angle covered when a field line is followed around a poloidal angle of θ is φ = aB ⎡ ⎛ ⎛ φ ∫ dθ = q circ 1+ a ⎞ ⎞ ⎜ RB θ ∫ ⎢ ⎜ ⎟ cos( θ ) ⎟ ⎝ ⎝ R ⎣ g ⎠ ⎠ ⎡ = q circ θ − a ⎤ ( 2 + Λ)sin( θ) ⎢ ⎣ R ⎥ g ⎦ −2 ⎛ ⎛ 1+ a ⎞ ⎞ ⎜ ⎜ ⎟ Λ cos( θ) ⎟ ⎝ ⎝ R g ⎠ ⎠ −1 ⎤ ⎥ dθ ⎦ 17.23 where we have written q circ for the value in a straight cylinder. In this straight cylinder we would have the toroidal angle φ covered in following a field line a given poloidal angle θ given by Equation 17.23 with a/R g = 0, i.e. φ = q circ θ. Now we see that, in transferring to toroidal geometry, we must replace the poloidal angle θ by θ*, where 123
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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