Page 6 - Cross-separatrix simulations of turbulent transport in the field-reversed configuration

Nucl. Fusion 59 (2019) 066018
C.K. Lau et al
Figure 5. The fluctuation spectrum corresponding to the post- saturation time from simulation Ä eφ = δne ä is plotted as the
measurements Ä δn ä is plotted as data points. The shaded region n0
indicates the standard deviation of the simulation spectrum over the time period for which this data represents. The deviation over the time period is smaller for unstable longer wavelengths in the SOL where the instability is driven.
6. Conclusions
In this paper, the simulation model has been extended to include nonlinear particle effects and interaction of multiple toroidal modes in a global geometry spanning both the core and SOL regions in the FRC confinement vessel. Previous local linear simulations found linear modes to be stable in the FRC core but unstable in the SOL [11, 12, 13]. Consistent with these past sim- ulations, linear simulations in the current work also finds insta- bility only in the SOL region. These results are in qualitative agreement with experimental measurements of density fluctua- tions in the C-2/C-2U FRC plasmas, which show core fluctua- tions to be lower in amplitude than SOL fluctuations [10].
As shown in figure 5, with the updated global model, the current work can now go beyond qualitative comparison and directly compare with experimental measurements. Nonlinear simulations show that instability saturates at levels compa- rable to experimentally measured amplitudes. An inverse spectral cascade in the SOL produces a toroidal wavenumber spectrum that decreases towards shorter wavelengths. In addition, the fluctuations spread from the SOL into the core at reduced amplitudes and shorter wavelengths, which pro- duces a core toroidal wavenumber spectrum that is much lower in amplitude at longer wavelengths. These simulated wavenumber spectra agree with the experimentally meas- ured spectra in several features [1]: lower amplitude in the core at longer wavelengths [2], higher amplitude in the SOL at longer wavelengths [3], decreasing trend towards shorter wavelengths in the SOL.
which tracks the energy moving in the direction across the field-lines due to the δ⃗E × ⃗B drift from the self-consistent elec- tric fields. Ion conductivity can then be found through Fick’s
law χs = Φq,s/ns∇Ts. In the simulations of the current work, the peak ion energy flux, Φq,i,max ∼ 6 (kW m−2 ), and conduc- tivity, χi,max ∼ 5 (m2 s−1 ), are found in the SOL. Under the adiabatic electron response model, particle flux across field- lines does not occur. However, test electron diagnostics (simu- lation particles which sample the self-consistent fields but do not contribute to them) have also been used to calculate an upper bound on electron transport quantities resulting in upper bound peak electron energy flux, Φq,e,max ∼ 7 (MW m−2 ) , and conductivity, χe,max ∼ 0.015 (m2 μs−1 ).
From the upper bounds found for electron conductivity, ‘worst case’ estimates of energy confinement times can be esti- mated for the core, τE,e 􏰂∼ 75 (μs), and SOL, τE,e 􏰂∼ 20 (μs) (using the relation χe ≈ a2/τE,e, where a → R at peak values in the respective regions) comparable to experimental measure-
Te n0
lines while the spectrum corresponding to the experimental
ments of τ
reach higher temperatures along with stronger magnetic field such that the motion of particles perpendicular to the field- lines should not change much. However, the motion of par- ticles along the field-lines will be faster as temperature rises, which can enhance the stabilizing effect of large k∥v∥ seen in past local simulations [13]. Interestingly, in the simulations using the test electron diagnostics, electron thermal transport was found to decrease with increasing electron temperature, possibly due to the effective averaging over fluctuations in the parallel direction due the faster motion along field-lines. More conclusive results on thermal transport will require simula- tions with electrons which contribute to the fields as the nature of the instability may change.
∼ 200 (μs). Future experiments are expected to
Φ ≡ ´ dv3 􏰁 1 mv2 − 3 T 􏰀 Ä⃗v · ψˆä δf , is calculated, q,s 22sδE×Bs
E,e
From these fluctuations, ion energy flux across field-lines,
Because long wavelength modes drive larger transport for the same amplitude of electric field, the wavenumber spectrum found in experiments and simulations is favourable for con- finement in the FRC core. The work of the paper has shown that this spectrum arises from SOL fluctuation spreading and not from inherent core instabilities. Because the core is stable to electrostatic drift-instabilities [13] due to the effects of typical FRC features (magnetic well, short electron field- line transit along, and large FLR), core turbulence in future devices or different equilibria is also likely to originate from outside instabilities, and the fluctuation spectrum is expected to remain favourable for confinement. Low-frequency insta- bilities of interest which may exist in the SOL include the uni- versal drift and ion-cyclotron electron drift modes suggested by Carlson [26] and the trapped electron mode seen in our previous local linear work.
Although the current work has extended previous simula- tion models, there are important physics present in experiment which have not been included. In previous local linear simula- tions, a drift-kinetic electron model was used and allowed for a trapped electron mode to arise in the SOL. Because these global FRC simulations are the first cross-separatrix FRC turbulence simulations that the authors know of, a simpler model using an adiabatic electron response was employed to begin. This may explain the discrepancy in the simulated SOL wavenumber spectrum, which shows slightly longer wave- lengths than experiments. In upcoming work, the electron model will be extended to drift-kinetic electrons to allow for
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