Page 6 - Suppressed ion-scale turbulence in a hot high-β plasma

ARTICLE
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13860
measurements. It is known that in tokamak/toroidal simulations the inclusion of proper sheath boundary conditions is paramount to the accuracy of a numerical SOL description and that this is a significant source of uncertainty. A fully self-consistent treatment of the dynamics parallel to the magnetic field would require sheath boundary conditions in the axial direction in the FRC SOL also. However simpler periodic boundary conditions are appro- priate in the axisymmetric FRC case within the local, electrostatic gyrokinetic approximation used here, as the relevant mode frequencies are substantially higher than the electron transit frequency. Periodic boundary conditions are used for the perturbed plasma density and electrostatic potential, and the particle motion in the toroidal direction and in the parallel direction at z 1⁄4 ±Lc. The axial boundary Lc is located outside the confined FRC region (Lc 1⁄4 2–4 m versus LFRCr1 m, where 2LFRC is the axial length of the (closed flux surface) FRC plasma). Even with electrostatic end plate biasing the sheath impedance is still moderately large as the parallel electron current is below the ion saturation current and very small compared to the thermal electron current (electron saturation current). The magnetic field lines are fixed (tied) in the electrostatic approximation. Radial electron temperature ðLTe Þ, ion temperature ðLTi Þ and density gradient scale lengths (Ln) are assumed equal ðZe 1⁄4Ln=LTe 1⁄4Zi 1⁄4Ln=LTi 1⁄41Þ here and in most of the GTC runs executed so far. Experimentally, R=Ln R=LTe within the available resolution of the electron temperature measurements via
40
multi-channel Thomson scattering , once SOL parallel transport
has established a connection to the axial boundary, as discussed
below in more detail. In the simulations, the FRC SOL is
close to collisionless with respect to ion-ion collisions, and
collisional with respect to electron-electron and electron-ion
calculated for a range of normalized density and temperature gradient scale lengths. For the SOL simulations, the normalization factor R/cs 1⁄4 2.51 10 6 s. The flux tube radius used in the simulation, at the axial midplane, is r/RsB1.3. It is remarkable that the growth rate spectrum (Fig. 6a) shows only unstable modes for 1.5rkyrsr20 with toroidal wavelengths of the order of and smaller than the ion Larmor radius. The prevalent poloidal mode numbers (along the magnetic field lines) are found to be m 1⁄4 0–3. Figure 6b shows the frequencies of the unstable linear modes versus normalized wavenumber, indicating propagation in the positive (electron diamagnetic) direction. For realistic driving gradients, the calculated frequencies are in between the ion transit frequency and the ion diamagnetic drift frequency. Benchmarking calculations performed by selectively removing gyro-averaging and the magnetic field gradient in the gyrokinetic solver indicate clearly that both FLR effects and rB are strongly stabilizing41. This is demonstrated in Fig. 7. Removing gyro- averaging (FLR effects) results in an increase of the normalized growth rate by almost an order of magnitude for realistic temperature and density gradients. Selectively removing rB, in contrast, affects the growth rate significantly only at low driving strength R/Ln.
The effect of including finite collisionality in the linear growth
rate calculations is demonstrated in Fig. 8, for a sample toroidal
wavenumber kyrs 1⁄4 4.1. The normalized electron collisionality is
collisions n 1⁄4 nii=ðvth=LcÞ 0:21, n 1⁄4 nee=ðvth=LcÞ 3:8 and iiee
collision rate, and ve =2Lc is the bounce frequency. Lc is the distance from the midplane to the simulation axial boundary (fieldline length 1⁄4 2Lc) in the SOL. Collisions are seen to have a stabilizing effect and reduce the normalized growth rate substantially. The frequency of the mode investigated here is also greatly reduced with increasing collision rate.
n 1⁄4n =ðvth=L Þ 8:6, where n , n and n are the electron- eieiec eeiieith
Quantitative comparisons with the measured turbulence wavenumber spectrum will require nonlinear gyrokinetic simulations. Nonlinear effects, including three-wave interaction, are expected to lead to forward cascading of the wavenumber spectrum towards viscous scales, and inverse cascading towards low toroidal wavenumbers where large-scale zonal flows may be excited via three-wave interaction. Hence the expected saturated spectrum can be broadened or shifted substantially compared to
electron, ion-ion and electron-ion collision frequencies, ve and vth are the electron and ion thermal velocities, and 2L is the field
ic line length in the SOL.
In the present simulation, with gyrokinetic ions and drift-kinetic electrons, linear normalized growth rates gR/cs are
1.0 0 0.5
0 4.8
3.6 2.4 1.2
k e
0.03 0.06 0.09 0.12
defined as n 1⁄4nei=ðvth=2LcÞ, where nei is the electron-ion ei the
a
SOL i= e =1
b
R/Ln=4.04 R/Ln=2.7 R/Ln=1.35
101 100
10–1 10–2
Figure 7 | Effect of finite Larmor radius and magnetic field gradient on stability. The figure shows the linear instability growth rate for the scrape-off layer for three cases, where Finite Larmor radius effects and the radial magnetic field gradient have been selectively removed in the gyrokinetic stability calculations. Other simulation parameters are as stated previously. Finite Larmor radius effects are found to reduce the scrape-off layer linear instability growth rate very substantially, by about an order of magnitude. The comparison shown in the figure has been obtained by artificially removing Finite Larmor radius terms in the gyrokinetic equations. In a separate set of calculations the magnetic field gradient rB has been neglected. The figure shows that the magnetic field gradient also has a stabilizing influence at low driving strength.
0 5 10 15 20 k s
4567 R/Ln,R/LTe
Figure 6 | Linear instability growth rate and frequency in the scrape-off layer. (a) Normalized linear instability growth rate from an electrostatic flux tube calculation using the Gyrokinetic Toroidal Code (GTC)20,21,35 of scrape-off layer (SOL) modes versus normalized toroidal wavenumber kyrs. The simulation flux tube radius (at the axial midplane) is r/RsB1.3. The ratio of density gradient scale length to electron and ion temperature scale lengths is unity ðZe 1⁄4 Ln=LTe 1⁄4 Zi 1⁄4 Ln=LTi 1⁄4 Z 1⁄4 1Þ. Results for different driving gradients are shown. With larger driving gradient, instability extends to lower normalized wavenumber. (b) Normalized frequency of scape-off layer (SOL) modes versus normalized toroidal wavenumber for the same simulation parameters.
6 NATURE COMMUNICATIONS | 7:13860 | DOI: 10.1038/ncomms13860 | www.nature.com/naturecommunications
k s=1.37
w/FLR, w/∇B w/o FLR, w/∇B w/FLR, w/o ∇B
R/cs
R/cs
R/cs