Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton For a second example, consider the field lines and flux surfaces resulting from a quadrupole field applied to a single filamentary current, in straight geometry. In a straight system the field lines will lie in a surface defined by constant vector potential A (i.e. ψ constant, with R ⇒ ∞). The single (“plasma”) filament of current I zp lies at the origin of a rectalinear coordinate system (x,y,z) shown in Figure 1.16, where z will approximate the toroidal direction in a toroidal system. The vector potential at P is given by Equation 1.55 as A zp = − µ 0I zp 2π ln ( r ), with r the radius from the plasma filament to a point P. There are then four additional filaments, at a distance d from the plasma filament, with currents I zq alternatingly + (into the plane) and - (out of the plane). The i th additional filament (i = 1 to 4) has a vector potential A zqi = − µ 0I zq 2π ln ( r i ) in its own local coordinate system. Transforming to the coordinate system (r,θ) we obtain A z = −µ 0 I zp 2π ln( r) + µ I 0 zq 2π ( ) 1/2 ⎡ ln d 2 + r 2 − 2dr cos( θ) ⎢ ⎢ + ln d 2 + r 2 + 2drcos( θ ) ⎢ − ln d 2 + r 2 − 2drsin( θ) ⎢ ⎣ ⎢ − ln d 2 + r 2 + 2drsin( θ) ( ) 1/ 2 ( ) 1/2 ( ) 1/2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎥ 1.57 The resulting contours of constant A z are shown in Figures 1.17a through 1.17d for the ratio I zq /I zp = 1.0 to 2.0. In the examples d = 0.25 m. The equi-distant contours shown are different from frame to frame. Clearly approximately elliptic cross sections are obtained; in tokamaks elliptic surfaces are produced by applying quadrupole fields. A separatrix appears and defines the last closed vector potential surface (a straight circular system is the only one without a separatrix). We can derive an analytic expression for the shape of a given flux surface. By expanding Equation 1.57 to third order in r/d, we find A z = − µ 0 I zp 2π ln( r) − ⎛ r ⎝ d ⎞ ⎠ 2 µ 0 I zq cos( 2θ ) π 1.58 With I zq = 0 the surfaces are described by r = constant = a, i.e. a circle. If we assume the surfaces with I zq ≠ 0 are described by r = a + ∆ 2 cos( 2θ) 1.59 26
Magnetic fields and tokamak plasmas Alan Wootton Figure 1.17a. Vector potential contours with Figure 1.17a. Vector potential contours with I zq /I zp = 1. The quadrupole filaments are at I zq /I zp = 2. The quadrupole filaments are at ± 0.25 m. ± 0.25 m. then Equation 1.58 gives, after expanding in ∆/a, −A z = µ 0I zp 2π ln( a) + ⎛ ∆ ⎞ µ 0 I zp ⎝ a ⎠ cos 2θ 2π ( )+ a 2 ⎛ ⎞ µ 0 I zq cos( 2θ) ⎝ d ⎠ π ⎛ + a ⎝ d ⎞ ⎠ 2 ⎛ ∆ ⎞ ⎝ a ⎠ 2µ 0 I zq cos 2 ( 2θ ) For a/d
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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