J. Todoroki

National Institute for Fusion Science (NIFS-Repository)

§ 1. Calculating Rotational TransformFollowing Field Linewhere wm,n == ml - n.Differentiating eq.(5) with respect to ¢, we obtainTodoroki, J.dtJ- = l + i L..J cm,nw",.11 exp(i wm,n¢)'d¢ m,1I(6)where (rA (¢), ZA (¢)) stands for a closed curve containedinside the magnetic surface, e.g., the magnetic axis, therotational angle can be calculated by integrating the equationMagnetic coordinates are commonly used in the study ofplasma in toroidal magnetic confinement devices. Toconstruct the magnetic coordinates, Boozer's algorithm[ 1]following the magnetic field line is widely used. In suchcalculations the rotational transform is the most importantquantity of the magnetic surface. In this paper the efficientand accurate method calculating the rotational transform isdescribed.We consider the case in which the field line is traced inthe cylindrical coordinates (r, z, ¢). If the azimuthalcomponent of the magnetic field does not vanish, theequations for the magnetic field line can be written in theform(8)(7)(9)w(¢max) - w(-¢max)l = --="'------'-"=-.tPrnax(2 2)d w d tJ a a ¢d¢ = d¢ -J; exp - ¢max2(2 2)a d tJ a ¢ d¢l=- J -exp ---2 -,-J; -;N' d ¢ ¢max ¢maxThe same technique can be used to obtain the othersurface quantities, such as the specific volumewith the initial condition w(O) = O. The equation is solvedsimultaneously with the other equations following themagnetic line of force. The equations for the magnetic axisare also solved simultaneously, if required. After theintegration is finished, the rotational transform is obtainedby the formulawith a a number chosen to achieve a desired level ofaccuracy. In practice, a is about 3. In the actual calculation,we consider the following differential equationThe rotational transform l is the DC part of the function inthe right hand side of eq.(6). Hence by posing the Gaussianwindow in order to eliminate the end effects, we cancalculate using the relation(2)(1)dz rB_=_ltan tJ = z(¢) - ZA (¢)r(¢) - rA( ¢) ,If we accept the definition of the rotational angleFinally, it is worth noting that the method described inthis paper has no difficulty concerning the rational surface.(3)(2 2)a r a ¢ d¢U=- J -exp ---2 -.-J; B; ¢max ¢max(10)This equation is solved simultaneously with the equationsfor the field line.On the magnetic surface, the angle tJ is expressed interms of the magnetic coordinates eand ¢ as[1] A. Boozer, Phys. Fluids 25,520 (1982).tJ =e+ Lent,,, exp(i[me - n¢]). (4)m,nWe assume that the angle variable e is chosen so that themagnetic field line is straight in e -¢ plane. Since e = Ill>on a field line, the rotational angle behaves along the fieldline as(5)m,n178

Calculating Rotational Transform Following Field Line

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