Unified theory of ripple transport in stellarators

Abstract

The distribution function for ripple-trapped particles has been found in a series form valid for all low collision frequencies, for standard and transport-optimized stellarators. The diffusion coefficient obtained with this distribution function shows excellent agreement with the results of Monte Carlo and Fokker-Planck computer codes, in the cases studied.PACS numbers: 52.25.Fi, 52.55.HeOver the past several years there has been renewed interest in the toroidal stellarator as a possible fusion reactor. Much of this interest has centered on the transport properties of a stellarator at the low collision frequencies expected in a fusion plasma, since particles trapped in the local ripple wells of the stellarator's magnetic field are expected to introduce a significant loss mechanism above and beyond that predicted by neoclassical theory for an axisymmetric tokamak. The theory of ripple transport for a "standard" stellarator at low collision frequency, v, was introduced 1 and developed 2 as a number of distinct results, each of which applies in different circumstances. In these works, regimes were identified in which transport coefficients scaled as v~l, v 1/2 , and v.Since then it has been demonstrated 3 that a certain class of stellarators exhibits enhanced confinement, and parts of the existing theory have been extended to treat the more general case of a multiple-helicity stellarator in the -1 • 4 5 v regime. In the present work a series solution in k 2 of the bounce-averaged kinetic equation for ripple-trapped particles is presented. Here, k 2 is a quantity which describes a particle's trapping state, with ripple-trapped particles satisfying 0<£ 2 <1. The proper choice of boundary conditions makes this solution valid for all collision frequencies and for all of the classes of stellarators discussed above. Diffusion coefficients obtained using this series solution show excellent agreement with those obtained numerically.The bounce-averaged kinetic equationis of primary interest in the study of transport due to particles entrapping and detrapping in the local wells of the stellarator field B=B 0 ll+e t (r)cose-€ h (r)(\ ~ crcos0)cos(/0+/?0)].(2)The symbols in Eqs. (1) and (2) are standard, except for the constant <r, introduced in Ref. 3. For a>0, Eq.(2) describes a stellarator with reduced transport rates in the v _1 regime (the standard form for B, which closely approximates most experimental situations, has cr=0). The usual means of solution of Eq. (l) begins with assumptions as to the relative size of the characteristic frequencies present in the problem. 1,2 The two frequencies of primary interest are the frequency with which particles are collisionally detrapped from a local well, v / efT == vV2f/ I , which appears through the collision operator, C(/), and the poloidal precessional frequency, #~ -Ci E = -E r /rB 0 .It is assumed that a large ambipolar radial electric field, E r , provides the dominant contribution to 9 through the ExB drift. Solutions obtained in this way have the disadvantage that they are...

Optimization of energy confinement in the 1/ν regime for stellarators

“…This is an advancement to the standard neoclassical theory (see, e.g. [27,33,34]) where simplifications to the magnetic field were made and only particles trapped within single wells were taken into account.…”

Tokamak ripple transport at arbitrary collision frequency

“…To obtain boundary conditions at k 2 = 1, we proceed as in Ref. [8]. To do so, we consider the orbits of 'transition' particles, which are alternately toroidally blocked (the 'tokamak banana') and ripple trapped during portions of their orbits.…”

“…In the case studied here, where ripple wells are localized on the midplane of the device, the series take a particularly simple form, and both the k 2 and 0 series truncate. If the ripple wells extend around the whole device, the solution resembles that in the stellarator case [8] and so we shall not consider this further.…”

“…This case was treated in Ref. [8] (for greater detail see also Ref. [16]), where it was initially assumed that f' = (X(k 2 ) cos 6 + Y(k 2 ) sin d)F^.…”

Trapped-Particle-Mediated Collisional Damping of Non-Axisymmetric Plasma Waves

“…Only a few theorists have tackled the subtle (ν/ω) 1/2 trapping scaling [8]. In stellarator magnetic fields a variety of TPM effects are thought to arise from helical ripples [9], but experimental tests are difficult.…”