Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton θ * = φ = θ − a ⎛ β I + l i q MHD R ⎝ g 2 + 1 ⎞ ⎠ sin( θ) 17.24 i.e. the perturbing fields must have the form (for r mn ≈ a) ⎛ b θ = b mn ⎝ r mn r m +1 ⎞ ⎠ cos( mθ * + nφ − ω mn t) 17.25 Figure 17.3 illustrates the field line trajectories in (φ,θ) space. Here we have assumed that α ⎛ q MHD ( r) = q 0 + q a − q ⎞ 0 , with q ⎝ ⎠ 0 the value at r = 0, q a the value at r = a (the plasma edge). ( ) r a ⎛ This allows us to express r = a⎜ q (r) − q MHD 0⎞ ⎟ ⎝ q a − q 0 ⎠ 1 α . In the figure we show examples for a = 0.25 m, R g = 1 m, q 0 = 0.9, q a = 3.2, β I = 0.5, l i = 0.9. The solid lines are for q MHD = 3.2 (the plasma edge), and the broken lines for q MHD = 2. The shear in the q profile is apparent. φ θ Analysis techniques outside inside Figure 17.3. The trajectory of field lines in φ, θ space for q = 2 (broken lines) and q = 3.2 (solid lines). With a set of Mirnov coils spanning a poloidal cross section of a low beta, circular cross section tokamak, we can take a Fourier transform in θ to obtain the amplitude of each component cos(mθ+nφ-ω mn t). If we have a rectangular vessel, then we have shown that the relevant expression is cos(mθ+θ+nφ-ω mn t). This has been done both computationally, and using analog multiplexing. We should allow for the toroidal corrections discussed above when performing this Fourier analysis; so that θ is replaced by θ*. Figure 17.4 shows the placement of the coils 124
Magnetic fields and tokamak plasmas Alan Wootton around a circular vessel. Both poloidal and toroidal arrays are required to determine poloidal (m) and toroidal (n) mode numbers. Figure 17.4. The placement of Mirnov coils (magnetic pick-up coils) around a vessel Figure 17.5 shows an example of data obtained from PBX, a machine with a non circular vessel but operated with a circular plasma. Coils 1 through 5 are located on a vertical line at the outer equator, with coil 1 the lowest. Coils 6 through 10 are located on a vertical line at the inner equator, with coil 6 the highest. Coils 11 through 14 are elsewhere; but in particular coil 12 is at the outer equator (θ = 0) while coil 13 is at the inner equator (θ = π). Data is shown for a period of 10 ms. The scale in volts is shown on the right hand side. A peak first appears on coil 1, and then progresses to coil 5. An outside to inside asymmetry (coil 12 to coil 13) is evident. 125
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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