# Agenda

# Tuesday 12 December 2017

### Geometric gyrokinetic reduction for strongly magnetised plasmas

#### Natalia Tronko (Max-Planck-Institut, Allemagne)

**Résumé**

Strongly magnetized plasmas, such as astrophysical and laboratory ones, represent complex

multi-scaled systems in space and time. Building up reduced models allows one to access physical

mechanisms in different regimes (e.g., turbulent, collisional) and geometry configurations.

Since more than three decades now, reduced kinetic models like gyrokinetics, resulting from

the elimination of the fast scales of motion associated with the particle rotation around magnetic

field lines, is the focus of intense research, both theoretical and numerical. Such an approach

allows a drastical reduction of computational time for numerical simulations. The gyrokinetic

reduction provides access to accurate prediction of long-scale processes such as transport which is

one of the main issues for fusion plasma confinement [1]. For astrophysical plasmas, the gyrokinetic

theory is also of interest [2]. Recently, gyrokinetic simulations have been used to access small scale

spectra, in order to fill the gaps in prediction of solar wind behavior when magnetohydrodynamics

approximations fail.

Building up solid theoretical basis following consistent derivation of the reduced equations is

an ultimate starting point for development of trustworthy numerical simulations. A systematic

derivation, which guarantees the energetic consistency of gyrokinetic models requires advanced

mathematical tools such as differential geometry (perturbative Lie-transformation techniques) as

well as advanced variational calculus on functional spaces.

The gyrokinetic dynamical reduction starts on the charged particle 6-dimensional phase space

with a series of gauge-transformations and Lie-transformations aiming to remove fast angle dependencies

from the single particle Lagrangian. At the second stage, fields and reduced particles

dynamics are coupled inside the specific variational principle.

The field-particles Lagrangian is the central object of the derivation, it contains information

about all the approximations and orderings, which will be naturally transferred to the equations

of motion and associated conserved quantitites. Furthermore, the Noether theorem can be applied

to the gyrokinetic Vlasov-Maxwell system issued from the variational formulation for derivation of

the associated conserved quantities. In particular, the energy invariant represents a special interest

for verification of quality of numerical simulations.

In this talk a systematic approach developed within the European enabling research project VeriGyro

for derivation of models implemented in major European gyrokinetic codes will be presented

[3]. An example of implementation of energy conservation law into the Particle-In-Cell gyrokinetic

code ORB5 will be considered for verification of quality of simulations [4].

1

References

[1] X. Garbet, Y. Idomura, L. Villard, and T. H. Watanabe. Gyrokinetic simulations of turbulent

transport. Nuclear Fusion, 50:043002, 2010.

[2] A. A. Schekochihin, S. C. Cowley, W. Dorland, G. W. Hammett, G. G. Howes, E. Quataert,

and T. Tatsuno. Astrophysical gyrokinetics:

weakly collisional plasmas. Astrophysical Journal Supplement, 182:310, 2009.

[3] N. Tronko, T. Bottino, A. Goerler, E. Sonnendr¨ucker, D. Told, and L. Villard. Verification

of gyrokinetic codes: theoretical background and applications. Physics of Plasmas, 24:056115,

2017.

[4] N. Tronko, A. Bottino, and E. Sonnendr¨ucker. Second order gyrokinetic theory for Particle-InCell

codes. Physics of Plasmas, 23:082505, 2016.