Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton 8. SOME FUNDAMENTAL RELATIONS Geometry In section 6 we derived an analytic expression for the flux outside a large aspect ratio (a/R g 0) is S φ , and the boundaries are S n and l. dV is the volume element on V, dS φ the area element on S φ , dS n is the area element on S n and dl the line element on l. Therefore dV = 2πRdS φ 8.1 dS n = 2πRdl 8.2 z dS n contour l φ plasma R dS φ Figure 8.1. Geometry Normal and tangential derivatives on l are ∂/∂n and ∂/∂τ. The positive orientation on l is that S φ lies on the RHS. Note this is DIFFERENT from what was assumed in section 6. The general question is: "How can we derive information on the magnetic field and plasma in V (or S φ ) from the fields measured on l" 78
Magnetic fields and tokamak plasmas Alan Wootton Field representation We shall try and be as general as possible. First we quickly re-derive the equilibrium equation. For the case of axisymmetry, we discussed in section 6 the description of the fields using the form where Ψ 2π = µ R I ⎛ ⎛ 0 l p ln 8R ⎞ ⎞ l ⎜ ⎜ ⎟ − 2⎟ − µ ⎡ 0ρI p 1− a 2 ⎛ ⎞ p ⎜ 2π ⎝ ⎝ ρ ⎠ ⎠ 4π ⎝ ρ 2 ⎠ ⎟ ⎛ Λ + 1 ⎞ ⎛ ⎜ ⎝ 2 ⎟ + ln ρ ⎞ ⎜ ⎟ − 2R ∆ ⎤ l g ⎢ ⎥ ⎣ ⎢ ⎠ ⎝ a p ⎠ ρ 2 ⎦ ⎥ cos( ω) 8.3 F = RB φ = µf 8.4 and ψ= RA φ From ∇xH = j, we can also write j = − 1 ⎛ µ L* ψ∇φ + ∇⎜ F ⎞ × ∇φ 8.5 ⎝ µ ⎠ where the operator L * is defined for axisymmetric scalar fields as ⎛ L * ψ = µR 2 ∇ • ⎜ ∇ψ ⎞ ⎝ µR 2 ⎠ 8.6 The operator L * satisfies the equation L * ψ = −µRj φ 8.7. For uniform permeability µ = µ 0 we have L * reducing to the operator ∆ * : ⎛ ∆ * ψ = R 2 ∇ • ∇ψ ⎞ ⎝ R 2 ⎠ = R ∂ ⎛ 1 ∂ψ ⎞ ∂R ⎝ R ∂R ⎠ + ∂2 ψ ∂z ∆ 8.8 2 That is, ∆ * is the operator on the LHS of the Grad Shafranov equation. We also introduce the operator L: Lψ = 1 µ ∇ • ( µ∇ψ) 8.9 79
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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