Full radius linear and nonlinear gyrokinetic simulations for tokamaks and stellarators: zonal flows, appliedE×Bflows, trapped electrons and finite beta

Abstract

The aim of this paper is to report on recent advances made in global gyrokinetic simulations of ion temperature gradient (ITG) modes and other microinstabilities. The nonlinear development and saturation of ITG modes and the role of E × B zonal flows are studied with a global nonlinear δf formulation that retains parallel nonlinearity and thus allows for a check of the energy conservation property as a means of verifying the quality of the numerical simulation. Due to an optimized loading technique, the conservation property is satisfied with an unprecedented quality well into the nonlinear stage. The zonal component of the perturbation evolves to a quasi-steady state with regions of ITG suppression, strongly reduced radial energy flux and steepened effective temperature profiles alternating with regions of higher ITG mode amplitudes, larger radial energy flux and flattened effective temperature profiles. A semi-Lagrangian approach free of statistical noise is proposed as an alternative to the nonlinear δf formulation. An ASDEX-Upgrade experiment with an internal transport barrier is analysed with a global gyrokinetic code that includes trapped electron dynamics. The weakly destabilizing effect of trapped electron dynamics on ITG modes in an axisymmetric bumpy configuration modelling W7-X is shown in global linear simulations that retain the full electron dynamics. Finite β effects on microinstabilities are investigated with a linear global spectral electromagnetic gyrokinetic formulation. The radial global structure of electromagnetic modes shows a resonant behaviour with rational q values.

Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence

“…This identification might be useful when one tries to perform a long-time simulation with a slowly time varying Φ. [90][91][92][93][94] In passing, we remark that it's possible to formulate gyrokinetics in such a way that the polarization drift appears in the gyrokinetic Vlasov equation 95 and short scale fluctuations seems arbitrary, as noted in previous work. 65 It's our opinion that treating polarization physics as a shielding term 2 (i.e., polarization density) has been found to be more efficient for a wide range of numerical applications.…”

First principles based simulations of instabilities and turbulence

“…There is great interest in these phenomena, especially in the context of magnetic confinement fusion research, which has lasted for several decades and is still growing. The understanding of anomalous transport has greatly improved over the past years, thanks in large part to the role of large-scale numerical simulations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Predictions used for the design of the next generation of experimental fusion reactors have been largely made on extrapolations from a range of existing experiments worldwide and assumed semi-empirical scaling laws.…”

“…In a toroidal geometry they may be partly damped because of their coupling to m = 0 poloidal components, but there remains a totally undamped component. It has been shown in several simulations [1,3,5,6,9] that turbulence generates zonal flows nonlinearly, and that, in turn, the zonal flows reduce the unstable mode amplitudes by shearing the turbulent eddies. Thus, the system appears to be self-regulated, with the zonal flows playing the role of 'organizer' of the turbulence.…”

“…Clearly, numerical methods are a necessity. The past decade has seen several major developments in this area [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Such large-scale simulations have already given important physical results.…”

“…Such large-scale simulations have already given important physical results. To cite only a few: the importance of the zonal flows in determining the ITG turbulence level and the corresponding anomalous heat flux [9]; the Shafranov shift stabilization of trapped electron modes in the core of tokamak discharges with internal transport barriers [43]; the scaling of anomalous heat diffusivity with system size [1]; the importance of considering equilibrium distribution functions of the true constants of motion in order to avoid the spurious excitation of geodesic acoustic modes [5,44]; the importance of retaining the parallel nonlinearity for a correct zonal flow determination [6]; detailed comparisons with specific tokamak experiments have been made, e.g. with DIII-D [45] and Asdex-Upgrade [46]; the inclusion of non-adiabatic electron response and electromagnetic pertubations in linear [47] and nonlinear [2] global codes; the inclusion of realistic ideal MHD equilibrium geometry and truly global profiles in tokamak [50,46,6], and fully three-dimensional stellarator geometry [51,52]-the latter limited to electrostatic, linear perturbations.…”