Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton 2 πa p β I = 8π prdrdθ 2 µ 0 I p ∫ ∫ = 8π 2 prdr 2 µ 0 I p ∫ 0 0 = 8π 2 a ⎡ p pr 2 a p 2 − r 2 dp 0 µ 0 I p ∫ dr dr ⎤ ⎢ ⎥ = − 8π 2 a p r 2 dp 2 ⎣ 0 ⎦ µ 0 I p ∫ dr dr 0 a p 0 14.21 Here we have integrated by parts, with u = p, du = (dp/dr)dr, v = r 2 , dv = 2rdr. Next we use the approximate pressure balance equation (dp/dr = j θ B φ - j φ B θ , we will work in a RH coordinate system). We substitute for j θ and j φ in terms of B φ and B θ (from µ 0 j = ∇ × B), i.e. µ 0 j θ = − ∂B φ ∂r , µ 0 j φ = 1 r dp dr + d 2 ⎛ B ⎜ φ dr ⎝ 2µ 0 ( ) ∂r ∂ rB θ ⎞ ⎟ + B θ ⎠ µ 0 r , we obtain d ( dr rB θ )= 0 14.22 Substituting for dp/dr from equation. 14.22 into equation. 14.21 gives ( ) ( ) β I = 4π 2 a p 2 a ⎡ d B p φ r 2 dr µ 2 2 0 I p ∫ + 2B θ r d rB ⎤ θ ⎢ dr dr ∫ ⎣ dr ⎥ 0 0 ⎦ 1 ⎡ = a 2 2 p B ⎢ θa ⎣ a p ∫ 0 1 = 1 + a 2 2 p B θa ( ) dr d B φ 2 a p ∫ 0 d B φ 2 r 2 dr ( ) dr r 2 dr + ( rB θ ) 2 0 a p ⎤ ⎥ ⎦ 14.23 If (dB 2 φ /dr) > 0 then β I > 1, and if (dB 2 φ /dr) < 0 then β I < 1. Figure 14.1 shows this schematically. A reduced B φ (diamagnetic) inside the plasma is associated with β I > 1. The normal ohmic heating case, with β I ≈ 0.3 has an increased B φ inside the plasma (paramagnetic). The figures are not well drawn, because we must have that B φ R is constant on a flux surface. 108
Magnetic fields and tokamak plasmas Alan Wootton B B β I > 1 β I < 1 R Figure 14.1. The toroidal field with different values of β I . R Measurements coil output Figure 14.2. A simple loop to measure toroidal flux. To measure δΦ it is usual to use a wire wrapped around the vacuum vessel, as in Figure 14.2. The voltage output from the coil is integrated to give a signal proportional to the enclosed flux. For most cases δΦ is small, typically ~ 1mWb, which must be compared to a typical vacuum flux Φ v = 1Wb enclosed by the same loop. Therefore we have to use techniques which allow measurements to better than 1 part in 10 4 to get 10% accuracy in values of β I . This is done using compensating coils. 109
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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