Page 2 - Cross-separatrix simulations of turbulent transport in the field-reversed configuration

Nucl. Fusion 59 (2019) 066018
simulations are comparable to the experimentally measured fluctuation threshold. On the other hand, the core is robustly stable due to the stabilizing FRC traits of short field-line con- nection lengths, radially increasing magnetic field strength, and the large finite Larmor radius (FLR) [15] of ions.
While there is also ongoing work towards understanding FRC transport with hybrid kinetic/fluid transport codes, namely Q1D [16] and Q2D [17], the present work is the first global nonlinear gyrokinetic transport study of turbulence in the FRC. We expand on the past linear physics simulations mentioned above [13] to push into the nonlinear kinetic simu- lations required for understanding turbulence-driven transport.
In this work, global linear simulations also find agreement with previous local linear simulations on core stability and SOL instability. Simulations of a single toroidal mode have demonstrated fluctuations spreading from the SOL region into the FRC core. Simulations of multiple toroidal modes, con- fined only to the SOL, show an inverse cascade in the fluc- tuation spectrum due to mode-mode coupling. Finally, global simulations of multiple toroidal modes including both the SOL and core regions find that the combined features of the inverse cascade and turbulence spreading lead to a fluctuation spectrum that is qualitatively comparable to the aforemen- tioned experimental results of Schmitz et al [10]
2. Simulation model
In this paper, simulations have been conducted with the tur- bulence code, ANC, in a C-2/C-2U-like magnetic geometry. ANC is a global particle-in-cell (PIC) code [18, 19], suit- able for simulation of electrostatic drift-wave turbulence. In this work, electrostatic perturbative δf simulations [20–24] are confined to a nonlocal domain spanning the confinement vessel region. The electrostatic Poisson equation is simplified by the Padé approximation
− ∇˜ 2 φˆ = ( 1 − ∇˜ 2 ) δ ˆn ⊥⊥
without collisions. Although neutral beam injection is used
C.K. Lau et al
These simulations are initialized with an FRC equilibrium [25] with density and ion temperature gradients and flat elec- tron temperature corresponding to table 1, with the distance from the machine axis to the separatrix at the outer midplane is Rs = 38 cm. Because ANC uses the perturbative δf model, these equilibrium profiles do not evolve within the simulation. The use of the flat electron temperature and the adiabatic elec- tron response serve to simplify the physics model as an initial step for the recently updated simulation model of this paper, which extends the simulation domain to include both the core and SOL regions. In upcoming work, the electron model will be extended to drift-kinetic electrons with a more realistic electron temperature profile.
In section 3, the simulation domain extends from the SOL to the core but only allows for a single toroidal mode (n = 20 is shown). In section 4, the simulation domain is confined to the SOL but allows for multiple toroidal modes (n = {5, 10, ... ,75, 80}). In section 5, the simulation domain extends from the SOL to the core and allows for multiple toroidal modes (n = {0, 5, 10, ..., 75}). The factors of 5 in the toroidal modes selected in sections 4 and 5 are due to the use of toroidal sym- metry for numerical reduction of the toroidal domain into a wedge 1/5 the size of the full toroidal domain. Simulations of a smaller toroidal wedge 1/10 the size of the full torus yield similar results in saturation levels and toroidal spectra.
3. Fluctuations spreading from SOL to core
With the equilibrium and model as described in section 2, linear simulations of a single toroidal mode show exponential growth of instability in the SOL with toroidal propagation in the ion diamagnetic direction. Comparison with local linear theory indicates that this is a slab-like ion temperature gra- dient (ITG) drift-wave instability [19].
In previous local, linear simulations [13], the SOL was found to be unstable while the core was stable. A variety of effects were studied, and core stability was found to be due to the stabilizing FRC traits of [1] short electron transit length [2], radially increasing magnetic field on the outboard side, and [3] strong FLR effects due to weak magnetic field. In the initial linear simulations of this section, nonlocal effects were numer- ically removed, effectively localizing the physics of the simula- tions. Consistent with past local simulations, there is no mode growth in the core due to the stabilizing effects mentioned, although the short electron transit effect is artificially enhanced by only using the electron adiabatic response in our equations.
In contrast to the past work, realistic nonlocal physics can be included through the Laplacian in the Poisson equation and through the gyroaveraging in the particle model. With these nonlocal effects included, drift-surfaces of different poloidal flux labels ψ are physically coupled. This introduces a radial wavenumber kψ into the wave dispersion, allowing for radial wave propagation such that a radial eigenmode structure forms across the SOL and core. Despite the physical cou- pling, in linear simulations, the amplitude of the eigenmode structure in the core is lower than in the SOL by more than
(where the φˆ is the normalized electrostatic potential, δˆn is the normalized perturbed charge density, and ∇˜ 2⊥ is the nor- malized Laplacian) and solved using the Portable, Extensible Toolkit for Scientific Computation (PETSc) via the Krylov method. Thermal ions are modeled with gyrokinetic deute- rium, time-advanced by Runge–Kutta 4th order, and elec-
trons are modeled with an adiabatic response Ä δne ≈ eφ ä, n0 Te
in the experiments to stabilize macro-instabilities and sustain the FRC, beam ions are not included in the current simula- tions. To preserve gyrokinetic validity, the magnetic field null region is excluded as shown in figure 1. Perturbed quanti- ties near the radial boundaries are smoothly set to zero to reduce boundary effects. In this simulation domain, periodic boundaries are used in the axial (Z) directions, neglecting parallel outflow effects. Details about the specifics of the code, including algorithm and benchmarks, will be published in a future paper.
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