Pullback scheme implementation in ORB5
Introduction
Electromagnetic effects are important in fusion plasmas. Alfvén waves, Magneto-Hydro-Dynamic (MHD) activity, electromagnetic modifications of the drift waves and the turbulent transport are well-known examples. In many cases, a combination of the global, electromagnetic and kinetic contributions is essential. Such complexity usually cannot be addressed analytically and calls for a numerical approach, certainly in realistic magnetic geometries under realistic fusion plasma conditions.
Global gyrokinetic particle-in-cell simulations represent such an approach. In this paper we focus on a particular code of this type, ORB5 [1]. This code has been intensively used for gyrokinetic turbulence studies, usually in the electrostatic regime [2]. The electromagnetic simulations have been inhibited by the so-called cancellation problem [3], [4]. In ORB5, this problem has been mitigated using the control variate approach [5]. This mitigation technique has been used for electromagnetic microturbulence simulations [6], and for the simulations of Toroidal Alfvén Eigenmodes [7], [8]. In this paper we describe an implementation in ORB5 of another mitigation scheme, the so-called pullback mitigation [9], [10], [11], [12]. This approach can be used in combination with the control variate scheme [5]. As a consequence, the code efficiency improves considerably. Previously, the pullback mitigation has been implemented in the EUTERPE code [10], [11], [12], [13], [14]. An alternative approach to the cancellation mitigation has been recently proposed for the GTC code [15], [16].
In our simulations, we consider the Toroidal Alfvén Eigenmodes [17], [18] destabilised by the fast particles [19], [20] and the internal kink instability [21] in tokamak geometry. To our knowledge, this is the first time the internal kink instability has been simulated using a global fully gyrokinetic particle-in-cell code in tokamak geometry at a realistic value of plasma . For simulations in straight tokamak, see Refs. [22], [23], [24].
The paper is organised as follows. In Section 2, the equations solved by ORB5 are presented. In Section 3, the discretisation used by the code is discussed. Simulations using the newly implemented schemes are presented in Section 4. Conclusions are made inSection 5.
Section snippets
Equations solved by ORB5
The global gyrokinetic particle-in-cell code ORB5 [1] solves the gyrokinetic Vlasov–Maxwell system of equations [25]. The species distribution function is split into the “background” control variate and the time-dependent deviation from the control variate so that . Here, the subscript indicates the particle species (bulk plasma ions and electrons, fast particles). The control variate is usually chosen to be a Maxwellian. The deviation from the control variate
Discretisation
The deviation of the distribution function from the control variate is discretised in the mixed variable [9], [10] with markers. This discretisation can formally be written as where is the number of markers, are the marker phase space coordinates and is the weight of a marker. The markers move along the gyrocenter orbits. The evolution of the marker weights is given by the gyrokinetic equation (1). An
Toroidal Alfvén eigenmode
For verification of the scheme newly implemented in ORB5, we consider the reference case of the international cross-code “ITPA-TAE” benchmark [19], [20]. In this benchmark, the Toroidal Alfvén Eigenmode with the toroidal mode number and the dominant poloidal mode numbers and has been considered in the linear regime. The mode has been studied in tokamak geometry with the small radius m, the large radius m, the magnetic field on the axis T, and the safety factor
Conclusions
The pullback scheme has been implemented in ORB5. The solver modifications needed for the pullback scheme implementation have been described. The new scheme has been verified using the ITPA-TAE benchmark [19], [20] both in the linear and nonlinear regimes. A considerable improvement of the code efficiency has been observed. Also, the efficiency of the pullback mitigation in ORB5 has been demonstrated using the internal kink mode in tokamak geometry. To our knowledge, internal kink mode
Acknowledgements
We acknowledge P. Helander, E. Sonnendrücker, F. Jenko, S. Günter, F. Zonca, and Ph. Lauber for their support. Numerical simulations were performed on the Marconi supercomputer within the framework of the OrbZONE and OrbFAST projects. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training program 2014–2018 under Grant Agreement No. 633053, for the CfP-AWP17-ENR-MPG-01 (2017/2018) project on “Nonlinear
References (31)
- et al.
Comput. Phys. Comm.
(2007) - et al.
J. Comput. Phys.
(2007) - et al.
Ann. Phys.
(1985) - et al.
Comput. Phys. Comm.
(2010) J. Comput. Phys.
(1987)- et al.
Phys. Plasmas
(2008) - et al.
Phys. Plasmas
(2001) - et al.
Phys. Plasmas
(2004) - et al.
Plasma Phys. Control. Fusion
(2011) - et al.
Phys. Plasmas
(2016)
Plasma Phys. Control. Fusion
Phys. Plasmas
Phys. Plasmas
Phys. Plasmas
Phys. Plasmas
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