Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton symmetric f 0 = 1 f 0 = 0 g 0 = 0 ⎛ f 1 = x 1 + x ⎞ ⎜ ⎟ ⎝ 2R c ⎠ asymmetric g 0 = −1 ⎛ f 1 = z 1+ x ⎞ ⎜ ⎟ ⎝ R c ⎠ g 1 = z g 1 = − x ⎛ 1 + x ⎞ ⎜ ⎟ + z2 2 R c R c ⎝ 2R c ⎠ R c ⎛ f 2 = x 2 1 + x ⎞ ⎜ ⎟ ⎝ 2R c ⎠ 2 ⎛ − z 2 1 + x ⎞ ⎜ ⎟ ⎝ R c ⎠ 2 2 ⎡ ⎛ f 2 = 2xz 1+ x ⎞ ⎤ ⎢ ⎜ ⎟ − 4z3 ⎣ ⎝ 2R c ⎠ 3R c ⎦ ⎥ ⎛ 1+ x ⎞ ⎜ ⎝ R ⎟ ⎠ c g 2 = 2xz ⎛ 1+ x ⎞ ⎜ ⎟ − 2z3 2 g 2 = − x 2 ⎛ 1 + x ⎞ ⎜ ⎟ R c ⎝ 2R c ⎠ 3R c R c ⎝ 2R c ⎠ 2 − z2 R c ⎛ ⎜ 1+ 4x ⎝ R c 2 ⎞ ⎟ − 2z 4 3 ⎠ 3R c 10.2 + 2x2 R c 2 Because the Y m are sensitive only to currents flowing within the contour l (including vacuum vessel currents if the measurements are made outside this), either the total equilibrium fields, or just the plasma fields, can be used. The plasma fields can be calculated if external conductor currents are known. Using just the plasma fields alone may have advantages in terms of requiring fewer moments to accurately describe the data. We define the current center by setting Y 1 = 0. Using the symmetric set we obtain ⎡ ⎛ ∫ ⎢ ⎜ l ⎣ ⎢ ⎝ ( ξ − ∆ R + ξ − ∆ R) 2 2R c ⎞ ⎟ B τ + R l + ξ ⎤ ηB n ⎥ ⎠ R c ⎦ ⎥ dl ⎡ R − l + ξ ⎤ ∫ ⎢ ∆ z B n ⎥ dl = 0 10.3 l ⎣ R c ⎦ i.e. the current channel displacement with respect to the center of the contour l is ∆ R = ∆ R0 + ∆ R1 − ∆ 2 R0 10.4 2R l ∆ R0 = 1 ⎡ ⎤ ∫( ξB τ +ηB n ) dl µ 0 I p ⎣ ⎢ l ⎦ ⎥ 10.5 ∆ R1 = 1 ⎡ ⎛ ξ 2 B µ 0 I p 2R τ + ηξ ⎞ ⎤ ⎢ ∫ ⎜ B l R n ⎟ dl ⎥ 10.6 ⎣ l ⎝ l ⎠ ⎦ Ignoring the term ∆ R0 2 /(2R l ), then ∆ R is constructed from two integrals, namely ∫[ξ+(ξ 2 /2R l )]B τ dl and ∫[η+(ηξ/R l )]B n dl. The first integral is measured with a modified Rogowski coil whose winding density times cross sectional area varies as ξ +ξ 2 /(2R l ). The second integral is measured with a saddle coil whose width varies as η+ηξ/R l . Alternatively the integrals can be constructed from discrete local measurements. 88
Magnetic fields and tokamak plasmas Alan Wootton Consider as an example a circular contour of radius a l based on R = R l. Then ξ = a l cos(ω) and η = a l sin(ω). These coils measure the fields in the coordinate system (ρ,ω,φ) based on the vessel center, and B n = B ρ , B τ = B ω . Assuming a plasma with no vertical displacement, symmetric about z = 0, we can write B ω = B τ = µ I 0 ⎡ ⎤ 1 + ∑λ n cos( nω ) 2πa l ⎣ ⎢ n ⎦ ⎥ B ρ = B n = µ I 0 ⎡ ⎤ ∑ µ n sin(nω ) 2πa l ⎣ ⎢ n ⎦ ⎥ 10.7 10.8 Substituting these expressions into Equations 10.4 to 10.6 gives ∆ R ≈ a l ( 2 λ + µ 1 1)+ a 2 l ⎛ 1 + λ 2 4R l 2 + µ ⎞ ⎜ ⎟ 2 − a 2 l λ ⎝ ⎠ 1 + µ 1 8R l ( ) 2 10.9 i.e. for small displacements, (λ 1 +µ 1 ) 1) we have, ∆ R ≈ a l ( 2 λ + µ 1 1)+ a 2 l 10.10 4R l i.e. all that is needed to measure the displacement of a nearly circular plasma within a circular contour (λ n , µ n , n ≥ 2 = 0) is a modified Rogowski coil whose winding density varies as cos(ω), and a saddle coil whose width varies as sin(ω). This simple coil set gives the correct answer when the constant a l 2 /(4R l ) is allowed for. We already knew this. To allow for significant noncircularity the more general expression (Equations 10.4, 10.5 and 10.6) should be used. We can also derive equations to determine the vertical displacement of an arbitrarily shaped plasma, using the asymmetric components in Equation 10.2. Application to the large aspect ratio circular tokamak Let us apply these ideas to the circular equilibrium described in section 6, surrounded by a circular contour on which we have a sinusoidal area Rogowski coil and a cosinusoidal width saddle coil. The equations given in section 6 were in a coordinate system (r,θ,φ) based on the plasma geometric center; they were transformed into the vacuum vessel coordinate system in section 7, Equations 7.1, 7.2 and 7.3, allowing for a geometric shift ∆ g . The output from the integrated saddle coil with n w layers of width w(ω) = w 0 sin(ω) and integrator time constant τ int is 89
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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