Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton where the constants C 1 and C 2 are given by the boundary conditions (B . n) = 0 at the plasma surface (B r (a) = 0) and the continuity of tangential field at the plasma surface. At infinity ψ takes the form C 2 rcos(θ), i.e. this represents the part of ψ from the external sources. The first term on the RHS is the flux from a filament. Substituting 6.46 into 6.24 gives an expression for B θ (a), which is matched to the plasma solution equation 6.30 by taking C 1 ⎛ − C a 2 2 = ln 8R g ⎞ + 2Λ 6.47 ⎝ a ⎠ Now since B r = −1 rR g ∂ψ ∂θ the requirement B r(a) = 0 means that the coefficient of cos(θ) in 6.46 is zero at r = a. Therefore we obtain C 1 = a 2 ( Λ + 0.5) 6.48 ⎛ C 2 = −ln 8R g ⎞ + Λ − 0.5 6.49 ⎝ 2 ⎠ Thus we finally obtain an expression for the flux outside the plasma r > a: ψ = µ 0 R g I p 2π − µ 0 rI p 4π ⎛ ⎜ ⎛ ln 8R g ⎞ ⎝ ⎝ r ⎠ − 2 ⎞ ⎠ ⎡ ⎛ ⎜ 1 − a 2 ⎞ ⎛ ⎝ r 2 ⎠ Λ + 1 ⎞ ⎝ 2⎠ + ln ⎛ r ⎞ ⎤ ⎣ ⎢ ⎝ a⎠ ⎦ ⎥ cos θ The field components are given by ( ) 6.50 B θ = 1 R ∂ψ ∂r , B r = − 1 ∂ψ R g r ∂θ 6.51 i.e. B θ (r,θ) = − µ 0I p 2πr − µ 0I p ⎡ ⎛ 1 + a p 2 ⎞ ⎜ 2πR g ⎝ r 2 ⎠ ⎟ ⎛ Λ + 1 ⎞ ⎝ 2 ⎠ + ln ⎛ r ⎞ ⎤ ⎢ ⎜ ⎟ −1 ⎣ ⎝ a ⎥ p ⎠ cos θ ⎦ B r ( r,θ)= − µ 0I p ⎡ ⎛ 1 − a p 2 ⎞ ⎜ 4πR g ⎝ r 2 ⎠ ⎟ ⎛ Λ + 1 ⎛ ⎞ ⎝ 2 ⎠ + ln r ⎞ ⎤ ⎢ ⎜ ⎟ ⎣ ⎝ a ⎥ sin θ p ⎠ ⎦ ( ) 6.52 ( ) 6.53 66
Magnetic fields and tokamak plasmas Alan Wootton We can also separate the total flux ψ into external (ψ e ) and plasma (ψ p ) parts. At large r, the expression (ln(8R g /r) -2) really approximates an expression which goes to zero. Therefore at large r we have leaving ψ e = − µ I 0 p ⎛ ⎜ ⎛ ln 8R g⎞ 4π ⎝ a ⎠ + Λ − 1 ⎞ ⎝ 2⎠ r cos( θ) 6.54 ψ p = µ 0 R g I p 2π − µ 0rI p 4π ⎛ ⎜ ⎛ ln 8R g ⎞ ⎝ ⎝ r ⎠ − 2 ⎞ ⎠ ⎡ ⎛ −ln 8R g ⎞ ⎝ r ⎠ +1 − ⎛ a ⎞ ⎣ ⎢ ⎝ r ⎠ 2 ⎛ Λ + 1 ⎞ ⎤ ⎝ 2⎠ ⎦ ⎥ cos θ ( ) 6.55 i.e. the external field at the geometric center (R = R g ) has a component B z = 1 ∂ψ e R ∂R = − µ 0I p 4πR g ⎛ ⎜ ⎛ ln 8R g ⎞ ⎝ a ⎠ + Λ − 1 ⎞ ⎝ 2⎠ 6.56 This is the component of the field which must be produced by external conductors to maintain a tokamak equilibrium. Figure 6.12 shows contours of the poloidal flux for a plasma with geometric center 0.06 m outside the coordinate center, minor radius a p = 0.265 m, and Λ = 2. At the inner equator an X point appears, where ∂ψ/∂R = 0. At this point the external vertical field and the poloidal field from the plasma cancel, so that B z (= -B θ ) = 0. The flux surface through this point is called the separatrix. This X point approaches the plasma surface as β I increases. As this happens the surfaces are strongly distorted from circular, and the above equations starts break down. Figure 6.13 shows the poloidal flux ψ =Ψ/(2π) in the plane z = 0 for a plasma with minor radius a p = 0.2m, a major geometric radius R g = 1m, current I = 1/µ 0 , and Λ = 2. The external source (vertical field) and plasma components are shown separately. Figure 6.14 shows the minor radial (B r ) and poloidal (B θ ) field components for this plasma, on a circular contour concentric with the plasma minor radius. 67
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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