Proceedings of ITC/ISHW2007

Development and application of the moments method transport
analysis to plasma flows in 3D configurations
D. A. Spong*, J. H. Harris*, A. S. Ware+, S. P. Hirshman*, L. A. Berry**Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 U.S.A.
+University of Montana-Missoula, Missoula, Montana 59812 U.S.A.

Shearing rates of neoclassical flows can attain levels that are relevant to turbulence suppression. Stellarators
offer several options for the control and generation of such flows. First, configuration selection and profile control
can lead to transport barrier formation, as has been demonstrated recently in the LHD (Large Helical device)experiment. Second, the nonlinear nature of the ambipolar electric field condition results in regimes where multiple
roots can exist. Near points where these roots merge/bifurcate, weak changes in plasma parameters can result inlarge increases in E × B shearing rates. Such regimes have been found particularly in the case of the QPS compact
quasi-poloidal stellarator and are analyzed here.
Keywords: moments method, E × B shear, sheared flow, neoclassical, ambipolar, momentum
conservation. stellarator.

1. Introduction
A moments transport method has been developed forstellarators [1] that predicts the three-dimensional variationof the neoclassical plasma flow velocity over the plasma
volume; in addition, the self-consistent ambipolar electricfield, particle and energy fluxes are calculated. In previous
applications of this model [2], we have examined theshearing rates of poloidal flow velocity components over arange of different stellarator devices, indicating asignificant dependence on magnetic configuration. These
velocity components include mixtures of perpendicular(E × B) and parallel (neoclassical, Pfirsch-Schlüter) flows.
While shearing of parallel velocities can be expected tohave an impact on finite k|| turbulence and nonlinear MHD,drift wave turbulence suppression is thought to be moststrongly influenced by shearing in the E × B flows. For
this reason, in this paper, the moments analysis isextended to calculate E × B shearing rates and to search
for regimes where this type of flow shearing can beenhanced.

2. Equations
A code (PENTA) has been developed that solves themoment equations derived in Ref. [ 3 ] for arbitrarystellarator configurations. The basic equations are
the parallel momentum balance
coupledwith
theperpendicular particle and energy transport equations.
author’s e-mail: spongda@ornl.gov#% M i + ! ii
%% !
ei
$

# Ui &&% (" ii! ie ( % Qi (mi ni( %U (Me + ! ee (' % e (%$Qe ('# *i &%(% qi / Ti (% * (% e (%$qe / Te ('# ni miNi%
2% " ii B%%
0%$==#N)% i%$ 00ne me" ee B 2# X i1 &%(0 & % Xi2 ((%Ne (' X e1 (%(%$ X e 2 ('& #U &(% i (( % Qi ((% (UNe ( % e (( %Q ($' e'#L+ % i%$ 0(1)# X i1 &%(0 & % Xi2 ((2)(%Le (' X e1 (%(%$ X e 2 ('These are given in equations (1) and (2) where
Ua = Bu||a , Qa =dT
21 dpad"
Bq||a , X a1 = !
! ea, X a2 = ! a5pa
na dsdsds# ii = !

$ ii % l11ii -l12ii ( %0' ii*!'iini mi '&-l 21l 22*) '&00 (*;2 *)# ei = !

% 1 0(Zni $ ee '*;3
ne $ ei ' !
0*'& 2*)# ie = !

# ee = !

%eeee( ' Z-l12
Z$ ee % l11'!
' ee*eene me '&-l 21 l 22*) ' ! 3 Z'& 2(3Z *2*
13 *2+Z*4 )!

%3(
Zni me ' 1 ! *2*;$ ei ''&0 0 *)and other quantities are as defined in Ref. [3]. The
neoclassical transport coefficients/viscosities Mi,e, Li,e, Ni,ehave been expressed in terms of velocity integrals overmono-energetic coefficients that are computed by theDKES code [ 4]. In order to provide a database for theseProceedings of ITC/ISHW2007
calculations, DKES is run for a range of flux surfaces,collisionalities and electric field values.

3. High E × B shearing regime in QPS
The QPS is a compact (<R>/<a> = 2.7), lowrotational transform (i = 0.15 to 0.3) configuration withapproximate quasi-poloidal symmetry and low effective
ripple (ε3/2 ~ 10-4 to 7×10-3). This form of quasi-symmetry
allows low effective levels of poloidal viscosity andhigher poloidal flow velocities than other configurations.
The combination of low transform and near poloidal
symmetry results in better alignment between theambiently-driven ⊥ flows (E × B and diamagnetic) andthe direction of minimum gradients in |B|, leading toFig. 2

Ambipolar electric field profiles for a range ofconfigurations.

Alldevices

arescaledto<B> = 1 Tesla and <a> = 0.32 m.

Fig 1(a)Elevation plot of Γion-Γelec vs. e<a>E r/kTe and<r>/<a> for n(0) = 7 × 1019 m -3.

Fig 1(b)Elevation plot of Γion-Γelec vs. e<a>Er/kTe and<r>/<a> for n(0) = 1020 m -3 (solid black contour
lines

representΓion-Γelec = 0.

author’s e-mail: spongda@ornl.gov

parameters

for

which

lowered magnetic pumping for such flows. This isreflected in smaller off-diagonal stress tensor components
and less conversion of ⊥ flows to || flows.
From the perspective of the ambipolar transportcondition (when properly coupled to neoclassical viscouseffects) these characteristics can lead to the existence ofadditional ambipolar roots beyond the usual ion andelectron solutions that are known for the case of transportmodels of simplified stellarators with limited helicities.
An example of this characteristic is shown in Fig. 1.
As indicated, for these parameters and profiles [hereTe(0) = Ti(0) = 0.6 keV], the two ambipolar contours
begin to merge/cross as the density is increased. This
leads to a region of very high E × B shearing that could
provide the basis for internal transport barrier formation.

Fig. 3E × B shearing rates for different configurations.

From limited surveys, such root merging, leading to highshearing rates has been found to be more prevalent inProceedings of ITC/ISHW2007
QPS than other configurations. In Fig. 2, ambipolarelectric field profiles are plotted for the same profiles andparameters for a range of devices. The associated E × B
shearing rates are plotted in Fig. 3. In order to suppress
drift turbulence, a simplified criterion is that such shearing
levels need to be similar to drift wave growth rates. As
analyzed in Ref. [5] for a variety of stellarators, typical
growth rates range from 0.5 to 1.1 × 105 sec-1 for QPS(albeit based on different parameters and profiles thanused here). The QPS shearing rates shown in Fig. 3
exceed these levels near the edge and <r>/<a> = 0.4 to 0.

4.

E × B shearing in LHD SDC regime

A recent experimental achievement of substantial
interest has been the generation of super dense core(SDC) plasmas in the LHD device [6] with low recycling
and high density gradient internal diffusion barriers. This
regime has been accessed by pellet injection andpotentially offers extrapolation to high density/lowtemperature ignition scenarios. In a previous analysis ofneoclassical flows in this regime, the moments model ofthis paper was applied to LHD, using an assumed
sequence of increasingly peaked density profiles. This
analysis has now been extended to examine a sequence ofdifferent equilibria with varying magnetic axis locationand flux surface shaping. The highest levels of E × B
shearing have been found either for equilibria with large
axis shifts or those with oblate flux surface shapes at theζ = 0 plane. An example of shearing rates in the E × B
velocity and contra-variant poloidal ion flow velocity are
shown in Fig. 4. Results from a sequence of profiles are
displayed here, starting with a broad density profile withn(0) = 6 × 1019 m-3 (red curve) and progressing to a very
peaked density profile with n(0) = 4.5 × 1020 m-3(magenta curve). While these shearing levels do not quite
reach levels that might be expected to form transportbarriers, they could provide a background level that could
supplement self-regulating turbulence-generated zonal
flows.

5. QPS flexibility at reduced magnetic field levels
If QPS is operated at reduced magnetic field levelsin the range of <B> = 0.3 Tesla, the planar toroidal
magnetic field coils have enough current capacity tointroduce QP-symmetry breaking effects. These effects
were analyzed previously in Ref. [2]. They have been
re-analyzed with the current version of the PENTA codewith respect to E × B shearing levels. As indicated in Fig.
5, variations in these coil currents can significantly
modify the effective ripple [7] coefficient.

Fig. 5 – Effective ripple coefficient for low field QPS operation(<B> = 0.3 T) as a function of planar toroidal fieldcurrent level.

Such variations in ripple levels lead to a range ofdifferent electric field levels, as are shown in Fig. 6.
These lead to the differing E × B shearing levels that are
shown in Fig. 7. As can be seen, in the <r>/<a> = 0.4 to0.5 range, the cases with the higher effective ripple leveltends to have lower shearing levels, while the cases withthe lower effective ripple have higher shearing levels.
This type of configuration change is thus expected toauthor’s e-mail: spongda@ornl.gov

Fig. 4

E × B shearing rates and poloidal flow shearing rates foran outward shifted LHD equilibrium with R 0 = 3.8 m.

Proceedings of ITC/ISHW2007
provide useful tests of the effects of different E × B
shearing levels.

References[1] D. Spong, Fusion Science and Tech. 50 (2006) 343.
[2] D. Spong, et al., Nuclear Fusion 47 (2007) 626.
[3] H. Sugama, S. Nishimura, Phys. of Plasmas 9 (2002) 4637.
[4] W. I. Van Rij, S. P. Hirshman, Phys. Fluids B1 (1989) 563.
[5] G. Rewoldt, et al., Phys. of Plasmas 12 (2005) 1047.
[6] N. Ohyabu, et al., Phys. Rev. Lett. 97 (2006) 055002-1.
[7] V. V. Nemov, et al., Phys. of Plasmas 6 (1999) 4622.

Fig. 6Ambipolar electric field profiles vs. radius andtoroidal planar coil current.

Fig. 7 E × B shearing levels vs. radius and toroidal planar coilcurrent.

author’s e-mail: spongda@ornl.gov