Computation of the Spitzer function in stellarators and tokamaks with finite collisionality

The generalized Spitzer function, which determines the current drive efficiency in toka- maks and stellarators is modelled for finite plasma collisionality with help of the drift kinetic equation solver NEO-2 [1]. The effect of finite collisionality on the global ECCD efficiency in a tokamak is studied using results of the code NEO-2 as input to the ray tracing code TRAVIS [2]. As it is known [3], specific features of the generalized Spitzer function, which are absent in asymptotic (collisionless or highly collisional) regimes result in current drive from a symmetric microwave spectrum with respect to parallel wave numbers. Due to this effect the direction of the current may become independent of the microwave beam launch angle in advanced ECCD scenarii (O2 and X3) where due to relatively low optical depth a significant amount of power is absorbed by trapped particles.


Introduction
The generalized Spitzer function is an important part of current drive calculations within the adjoint approach [4] where it plays the role of current drive efciency in phase space.This function is well studied and is eectively computed in asymptotical regimes with high and low collisionality (short and long mean free path regimes).In these regimes, the dimension of the kinetic equation does not exceed two because the generalized Spitzer function has a parametric dependence on spatial coordinates (in the long mean free path regime it essentially depends on the integrals of motion in velocity space only).In regimes with nite plasma collisionality the dependence of this function on the position on the ux surface is not trivial anymore.Its computation on a given ux surface requires the solution of a 3D (4D) problem in tokamak (stellarator) geometry.This is a rather demanding task.Often it is performed using model simplications of the kinetic equation, in particular, of the collision model.In the present report, a parallel version of the kinetic equation solver NEO-2 [1] is used to solve the Spitzer problem in stellarator geometry.This code does not employ any simplifying assumptions for the collision operator or for the device geometry.Prior to the parallelization of the code, a sequential NEO-2 version was mainly used for computation of mono-energetic neoclassical transport coecients in stellarators [5] and for computation of the generalized Spitzer function in tokamaks [6], while computaa.e-mail: winfried.kernbichler@tugraz.at tions of this function for stellarators required rather long computing times.Specic features of the global ECCD eciency resulting from nite plasma collisionality are studied here with help of the ray tracing code TRAVIS [2] using the Spitzer function pre-computed by the code NEO-2.In contrast to asymptotical regimes where the Spitzer function is an odd function of parallel velocity, it does not have a distinct parity in nite collisionality regimes.This makes the generation of current by waves with symmetric spectra over parallel wave numbers [3] possible.As shown below, this eect may become dominant in ECCD scenarii with relatively weak absorption, such as O2 and X3, where a signicant power fraction is absorbed in the trapped particle region of velocity space.

ECCD computations in the adjoint approach
The standard method for calculation of ECCD generated current in tokamaks and stellarators is the adjoint approach where the solution to a linearized quasilinear kinetic equation written in terms of the integrals of motion in velocity space and ignoring the cross eld drift, is reduced to a generalized Spitzer problem.Here, f is the perturbation of the electron distribution function, h = B/B is a unit vector along the magnetic eld, is the parallel velocity, LCL is the linearized collision integral and is the quasilinear particle source in phase space resulting from the action of the quasilinear diusion operator LQL on the Maxwellian f M .Using the adjoint approach (see e.g., Ref. [4]), the ux surface averaged parallel current density is expressed through the adjoint generalized Spitzer function ḡ (current drive efciency) as follows, In the homogeneous magnetic eld or in the high colli- Due to the toroidal symmetry of a tokamak, the problem is periodic there with respect to ϕ s (see the lower plot in Fig. 1), and its solution (for a single eld period) can be eectively computed by a single processor unit.This periodicity is missing in a stellarator (see In Fig. 1 the generalized Spitzer function in a tokamak is shown together with its derivative over perpendicular velocity for a nite collisionality case, ν = 0.01 (ν * = 0.22), and for the long mean free path limit computed by the code SYNCH [8].Here ν = 4π 1/2 R/(3l c ) is the universal collisionality parameter and ν * = qνZ 2 eff A 3/2 π −1 is the conventional tokamak collisionality parameter, R, q, A and Z eff are major radius, safety factor, aspect ratio, and eective charge number, respectively.It can be seen that the Spitzer function is strictly antisymmetric only at the magnetic eld minimum and maximum points while for the points in between there is a signicant symmetric part connected with the combined action of the magnetic mirroring force and collisional detrapping.This part becomes small in the long mean free path case where it is localized in the narrow trapped-  drive by waves with symmetric spectra [3], which result in a symmetric source term Q RF (this is similar to bootstrap current generation by the symmetric source term −v g • ∇f M with v g being the cross-eld drift velocity).Since the sign of this part is determined by the sign of the parallel derivative of the magnetic eld module, the direction of the current is respectively determined by the location of the absorption zone in

ECCD in a tokamak
The impact of nite plasma collisionality on global ECCD eciency has been studied in tokamak geometry (see Fig. 3) using the ray-tracing code TRAVIS [2] and various models for the generalized Spitzer func-      asymptotical low and high collisionality limits where the symmetric part is small.As a result, the direction of the current may become independent of the microwave beam launch angle in advanced ECCD scenarii with relatively low single pass absorption such as O2 and X3.These scenarii are less ecient than standard scenarii (O1 and X2) but they might be necessary in high density plasmas where wave cut-o makes O1 and X2 resonances not accessible.

5 ) 2 , ( 6 )
sionality limit the function g is factorized to g = bλD where λ = p /p is the pitch parameter and D = D(p) is the classical Spitzer function[7].With the help of (2), the parallel current density is expressed via the momentum space ux density due to the waveinduced quasilinear diusion, Γ RF , and the deriva- tives of the adjoint Spitzer function, j b = e l c d 3 p ∂ḡ ∂p • Γ RF .(Within geometrical optics used for calculation of ECRH/ECCD, the quasilinear ux density is described in a local approximation.In this approximation Γ RF diers from zero in velocity space only at the resonance line where the (multiple) cyclotron resonance condition taking into account Doppler shift is fullled, ω = nω c + k v , where ω, ω c , n and k are wave frequency, relativistic cyclotron frequency, cyclotron harmonic index and parallel wave vector, respectively.For weakly relativistic electrons the largest component of the quasilinear ux density is over perpendicular momentum.Therefore, as follows from (5), the behavior of the derivative of ḡ over perpendicular momentum at the resonance curve is of main importance for ECCD. 3 Solution of the generalized Spitzer problem in toroidal geometry In general toroidal geometry described in terms of ux coordinates (s, ϑ, ϕ), the dimension of the generalized Spitzer problem (4) is four (ux surface label s plays the role of a parameter).Within a eld line integration technique used in the NEO-2 code this dimension is reduced by one because g is represented on the ux surface by its values on a single eld line, which is long enough to cover this ux surface densely.When presenting the dependence of the generalized Spitzer function on kinetic energy in the form of an expansion over the associated Laguerre polynomials of the order 3/2 (Sonine polynomials), g(r, p) = M m=0 g m (ϕ s , η)L (3/2) m u the problem is reduced to a set of coupled 2D equations for the expansion coecients g m (ϕ s , η), where ϕ s , η = p 2 ⊥ /(p 2 B) and u = p(2m e T e ) −1/2 = vv −1 T are the toroidal angle playing the role of a eld line parameter, the normalized perpendicular adiabatic invariant and the normalized momentum (normalized velocity) module, respectively.These equations are solved using a conservative nite dierence scheme on a 2D grid adapted over η in order to resolve the boundary layers η ≈ η b associated in the long mean free path regime with local magnetic eld maxima,

Fig. 2 )
Fig. 2) where it has to be solved for the whole eld line.Such a solution is obtained in terms of Green functions (propagators) introduced for separate eld line segments between the local maxima (ripples) with individual discretization over η.Propagators are then matched with each other using re-discretization.The computation of propagators has been parallelized recently what makes the evaluation of the Spitzer function in stellarators feasible on modern computer clusters.

Figure 1 .
Figure1.Generalized Spitzer function in a tokamak (upper plot) and its derivative over the normalized perpendicular velocity (middle plot) vs pitch parameter λ = p /p for the normalized velocity u = 2 at various spatial points.Colors of the curves are the same as colors of the markers showing the values of the upper phase space boundary ηmax = 1/B at respective observation points (lower plot).Solid and dashed curves correspond to the nite collisionality case and to the asymptotical low collisionality limit, respectively.

Figure 2 .
Figure 2. Derivative of the Spitzer function over the normalized perpendicular velocity in W7-X for u = 1 (upper plot) and u = 2 (lower plot).Colors and line styles are the same as in Fig. 1.

Figure 3 .
Figure 3. Ray trajectory (grey) in a tokamak for the toroidal launch angle 25 o .Upper plot -view from top, lower plot -ray projection to the toroidal cross-section ϕ = const (length units are SI).Cold resonance position is shown by the magenta line.
tion.In this example, parabolic density and temperature proles are used.The values of magnetic eld, central density and (homogeneous) Z eff are 5 T, 10 20 m −3 and 1.5, respectively.Two central temperature values, 1 keV and 2 keV resulted at the absorption region (shown with red color of the ray in Fig. 3) in plasma collisionalities ν * = 0.28 and ν * = 0.07, respectively.The microwave beam is launched o axis, in the horizontal plane, varying the toroidal launch angle β (angle between the central ray and the direction to the tokamak symmetry axis).Scans of the absorbed power fraction and total generated current over β are shown for the standard ECCD

Figure 4 .
Figure 4. O1 resonance: Absorbed power fraction (upper plot) and total generated toroidal current for central electron temperature 1 keV (middle plot) and 2 keV (lower plot) as functions of the toroidal launch angle β.High, low and realistic collisionality models for the generalized Spitzer function are indicated in the legend.

Figure 5 .
Figure 5.The same as in Fig. 4 for the X2 resonance.

Figure 6 .
Figure 6.The same as in Fig. 4 for the O2 resonance.

Figure 7 .
Figure 7.The same as in Fig. 4 for the X3 resonance.