Patent Number: 046684646
Section: description

EXAMPLE The above has been applied to a three-period, helical axis stellarator. This stellarator has a relatively large rotational transform which implies a relative small finite .beta. magnetic axis shift. The value of .chi. increases from slightly above 1.5 at the magnetic axis to about 1.7 at the edge. The axis shifts halfway to the outer flux surface at a .beta. of about 15%. The .delta..sub.nm.sup.v for a general vacuum field should decrease rapidly in amplitude as n or m increases. This is borne out for this heliac by a numerical Fourier decomposition of the vacuum field. The largest components correspond to (n,m)=(3,0), (0,1), and (3,1). Of course, n must be a multiple of 3 because of the periodicity. The .delta..sub.30 component corresponds to the field ripple on the magnetic axis. It is very nearly equal to 2r.sub.o /R=0.5, where r.sub.o is the radius of the helix formed by the magnetic axis and R is the major radius. The .delta..sub.01 term, which is due to the toroidal curvature, is about 2.5.rho./R, where .rho. is taken to be the circularized flux surface radius. The helical curvature gives the .sub.31 term, which has a value of about .delta..sub.31 =1.3.rho./r.sub.ch, where r.sub.ch is the helical radius of curvature of the magnetic axis, r.sub.ch =(1+k.sup.2 r.sub.o.sup.2)/(k.sup.2 r.sub.o 0), (2.pi./k is the periodicity length). The decrease of .delta..sub.nm.sup.v with increasing (n,m) implies that the most dangerous direct resonances are those with low n and m. Higher order resonances are due to coupling of the .delta..sub.nm.sup.v. The coupling is strongest for resonance with low n and m. So here again the low order resonances are the most dangerous. For this heliac, the most serious problem is posed by the n=3, m=2 resonance, which lies at the magnetic axis. (The transform at the axis is actually very slightly above 1.5. It is convenient for the following discussion to take .chi. there as 1.5 exactly, which has only a small effect on the results.) The radial m=2 component of the plasma field goes to zero at the axis; but d.chi./d.sub..rho. also vanishes there, so that the island width can nonetheless be finite. It is necessary to modify Eqs. (7) and (8) to take into account the vanishing d.chi./d.sub..rho.. Specializing to n=3, m=2, we estimate ##EQU5## where .DELTA..chi. is the change in .chi. across the minor radius, a. We obtain ##EQU6## This is the required modification of Eq. (8). The .delta..sub.31.sup.v and .delta..sub.01.sup.v components couple directly to give a nonlinear .delta..sub.32 component. The corresponding island width is equal to half the minor radius at .beta..sub.o .perspectiveto.0.017. The .delta..sub.31.sup.v and .delta..sub.01.sup.v Fourier components are intrinsic to the heliac vacuum field, and can be eliminated by the use of a helical equilibrium coil whose current is adjusted as a function of .beta. to suppress the resonant n=3, m=2 part of the equilibrium field. A few helical equilibrium coils would suffice to suppress the islands at the low order rational surfaces. In the general design of stellarator vacuum fields, we might have expected the requirement of good vacuum flux surfaces to suppress the resonant field amplitudes. Our calculation for the heliac reference design shows that the amplitudes of the direct resonances may nonetheless be unacceptably large. We conclude that it is necessary to incorporate the constraints on the resonant .delta..sub.nm.sup.v directly in the design procedures. Our application also shows that coupling of nonresonant components can give large islands, even for values of .beta. at which the axis shift is small relative to the minor radius. In summary, we have proposed the use of resonant coil systems, such as helical coils, carrying relatively small currents. The required current in the coils is determined by the plasma pressure, as given by equations (8) and (10). Initially the current is zero, to avoid the deleterious effect on the vacuum field. As the plasma pressure is raised, the current in the resonant coils must also be raised. The final current in these coils is typically 1% of that in the stellarator primary coils.