Flux Coordinates and Magnetic Field Structure - University of ...

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80 4. Magnetic-Field-Structure-Related Concepts 'top view of torus' Fig. 4.16. Top view of a toroidal surface, cut by a / /major axis-) \ \/ =constant surface Here, d3R is an elementary volume element and Vis the volume enclosed by the flux surface. The second form follows from V. B = 0. In Fig. 4.16 we show a top view of the flux surface together with a c = constant (poloidal) surface. In particular, we choose the c = 0 + 2nk surface. This surface cuts the torus open, allowing us to consider it as a volume enclosed by the surfaces S,,,,,, SCzo and SS-,,,. If we now apply Gauss' theorem to the integral of (4.7.7) over the region of the cut-open torus, we obtain #(Beds= fJ L;B.dS+ ff cB.dS+ ff (B.dS . (4.7.8) S stor". S
82 4. Magnetic-Field-Structure-Related Concepts This equation states that there is no singularity in the gradient of the flux (label) for Vapproaching 0. Its physical meaning is easily understood. In the limit of V+ 0, we have that Ylor cc r2, where r is an average minor radius. Then VYtor = (dYl0,/dr)i allows us to conclude that VYt0,, evaluated at r = 0 is zero. (When taking Ytor,,,, the dependence of the area on r2 must be invoked to prove (4.7.15); when one wishes to prove (4.7.15) for Ypo,, one must make use of the fact that B, = 0 at the magnetic axis.) For every other surface label e(Y), we must make use of the property that: with the condition that de/dY is non-singular at the axis. 4.7.2 Open-Ended Systems In mirrors, similar flux labels as in toroidal systems are employed. The flux volume does not require further comment. Concerning magnetic fluxes, we are now limited to fluxes contained within the flux surface. By analogy to Ytor and !Piol in a torus, we have the axial flux Pax and the azimuthal flux YaZ, respectively. A scalar pressure p seems to be less convenient because the pressure is often anisotropic and not constant along a magnetic-field line. Depending on the choice of flux label, 1 of the (e, B, I) coordinate system points in the B direction or its reverse: i = f B. 4.8 The Rotational Transform in Toroidal Systems The total poloidal angle A8 traversed by a field line after one toroidal transit (A[ = 2n), is called the rotational transformation angle I (Greek small letter iota). The average rotational transformation angle is defined as (Morozov and Solov'ev 1966): Here I, represents the rotation angle resulting from the k-th toroidal transit. n is the product of the number of toroidal transits traced out by a single field line times the number of independent field lines on that flux surface (for an irrational surface, the latter is obviously one). The angle I of (4.8.1) is an average over the flux-surface and is by definition a flux-surface quantity. t : 4.8 The Rotational Transform in Toroidal Systems 83 It is more customary to work with the quantity called the rotational transform t (iota-bar) equals the average number of poloidal transits for one toroidal transit. (In the paper by Solov'ev and Shafranov (1970), the symbol p is used instead of t. Their Eqs. (1.7) and (1.8) should read m/n instead of n/m). A rational surface has t equal to a rational number; on an irrational surface, t is irrational. As mentioned above, in the tokamak literature, one prefers to work with the "toroidal winding number," which is the reciprocal of t: Because of its importance in stability calculations, q is called the safety factor. There is an equivalent and more useful expression for the (average) rotational transform t: The sign oft is chosen such that it is positive if the toroidal field and the poloidal field are in the positive [ and 8 directions, respectively (then B. Vl > 0, and B. V8 > 0). The dot represents the derivative with respect to the chosen flux- surface label e: We should note that for time dependent problems, t is defined by If we consider a functional relationship, Y;,, = Y~l(Ytor(R, t), t), where R is an arbitrary position vector, the chain rule of calculus leads to
Page 1 and 2: ~p~ger Senes in Computational Physi
Page 3 and 4: During the course of our research w
Page 5 and 6: X Contents b) Gradient ............
Page 7: 2 1. Introduction and Overview tens
Page 10 and 11: 8 2. Vector Algebra and Analysis in
Page 12 and 13: 12 2. Vector Algebra and Analysis i
Page 26 and 27: 3. Tensorial Objects 3.1 Introducti
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Page 56 and 57: 5. The Clebsch-Type Coordinate Syst
Page 58 and 59: 104 5. The Clebsch-Typ Coordinate S
Page 60 and 61: 108 5. The Clebsch-Type Coordinate
Page 62 and 63: 112 5. The Clebsch-Type Coordinate
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Page 84 and 85: 7. Conversion from Clebsch Coordina
Page 86 and 87: 8. Establishment of the Flux-Coordi
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Page 95 and 96: 178 111. Selected Topics In the thi
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182 10. "Proper" Toroidal Coordinat
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11. The Dynamic Equilibrium of an I
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190 11. The Dynamic Equilibrium of
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202 12. The Relationship Between 5
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206 13. Transformation Properties o
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212 14. Alternative Derivations of
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216 References Buck, R.C. (1965) Ad
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Subject Index Abraham 54 Absolute d
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224 Subject Index Conductivity (con
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228 Subject Index Equations, Hamilt
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232 Subject Index Lee 105, 164 Leig
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236 Subject Index Subject Index 237
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240 Subject Index Toroidal magnetic
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