Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton Figure 17.7. The (m,n) space, with the limits imposed by q 0 = 1 and q a = 3.2. The darker shaded area represents the space considered in the analytic model described in the text. The measurements of the fluctuating fields are usually restricted to root means square (rms) fields, b rrms and b θrms , at r > a, the limiter radius. We would like to know how this is related to the field components at the resonant surface, so that we can calculate associated island widths. Typically we want to determine b rmn from a measured b θrms outside r mn . There is no unique transformation from b θrms to b rmn , so a model must be invoked. We proceed by assuming the fluctuating self generated field b θrms measured with magnetic probes at r > a is proportional to the required b rmn at r mn ≈ a. Evidence for such a model comes from correlation measurements between Langmuir probes and magnetic probes; the measured b θrms at r > a is apparently determined by plasma current fluctuations at r ≈ a, i.e. at the plasma edge. Outside the fluctuating current filaments (r > r mn ) the magnetic fields can be approximated by b r = ⎛ r ∑( b rmn ) ⎜ m ,n r r= r mn ⎝ mn ⎞ ⎟ ⎠ −(m +1) cos(mθ + nφ +δ mn ) 17.29 with φ the toroidal angle, θ the poloidal angle, and δ mn a random phase. This ignores toroidal effects, which introduce terms proportional to (m±1), and is strictly valid only for small n. If a conducting wall is present at r = b then the expression is modified by the factor [1-(r/b) 2m ]. For stationary ergodic turbulence the time and spatial averages are the same, so that the root mean square is found as a spatial average. Outside the singular surface we measure 128
Magnetic fields and tokamak plasmas Alan Wootton 2 ⎛ r ⎞ b rms = b r ,rms = b θ ,rms = 0.5∑ b rmn r= rmn ⎜ ⎟ 17.30 m ,n ⎝ ⎠ r mn −2(m +1) To go further and obtain a relationship between b θrms at r > a in terms of b rmn at r = r mn we must specify a given current density or safety factor profile q(r), the dependence of b rmn on m and n, locate all possible m and n pairs where q = m/n, specify upper and lower limits to either m or n. In general this must be done computationally. However, some progress can be made analytically. We can approximately solve Equation 17.30 for b θrms at r > a in terms of b rmn at r = r mn if we take b rmn independent of m and n over some range to be specified, and a q profile relevant to the 2 ⎛ r plasma edge, for example q = q ⎞ a . Then we can relate the resonance location r ⎝ a⎠ mn to a given q, r mn a = ⎛ ⎜ ⎝ 1 q ⎞ 2 ⎛ ⎟ m ⎞ = ⎜ ⎟ q a ⎠ ⎝ nq a ⎠ 1 2 17.31 With ρ = r/a defining the measurement position and q = m/n, Equation 17.30 is written as ⎛ ⎜ ⎝ b rms b rmn ⎞ ⎟ ⎠ 2 ρ ≥1 2(m+1) (m +1) ⎛ 1⎞ ⎛ m ⎞ = 0.5∑ ⎜ ⎟ ⎜ ⎟ 17.32 m ,n ⎝ ρ⎠ ⎝ nq a ⎠ ρ ≥ 1 is a constant, namely the location of the magnetic pick-up coils used to measure b rms . We now replace the summation by integrals. The space we are considering is shown in Figure 17.4. The first integral, ∫dn, is from m/q a to m, allowing all n values in between. The second integral, ∫dm, is from m 1 to m 2 , with m 2 >> m 1 . In Figure 17.4 m 1 = 6 and m 2 = 12. Unfortunately our chosen profile allows q ≤ 1 (when r mn /a ≤ 1/√q a ). To restrict n ≥ 1 we should choose m 1 ≥ q a ≈ 3. The result of the integration is b rms b rmn = ⎡ 1 ⎛ 1 4q a ln( ρ) ρ 2( m 1 +1) − 1 ⎢ ⎜ ⎢ ⎝ ⎢ 1 ⎛ ⎢ − ⎜ ⎣ ⎢ 2 ln( q a )+ 2 ln( ρ) ⎝ ( ) ρ 2( m 2 +1) ⎞ ⎟ ⎠ 1 ( m q 1 +1 ) a ρ 2 m 1 +1 ( ) − 1 ( m 2+1 q ) a ρ 2( m 2+1) ⎤ ⎥ ⎥ ⎞ ⎥ ⎟ ⎥ ⎠ ⎦ ⎥ 1 2 129
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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