7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Local Momentum Balance in Electromagnetic Gyrokinetic Turbulence
H. Sugama1,21National Institute for Fusion Science2
Department of Advanced Energy, University of Tokyoe-mail (speaker): sugama@nifs.ac.jp
Gyrokinetic theory and simulation provide a powerfulframework to study microinstabilities and turbulent
processes in magnetized plasmas. The gyrokinetic
equations derived from the Lagrangian and/or
Hamiltonian [1,2] describing the gyrocenter motion
possess conservation properties, which are suitable forlong-time and global transport simulations. Variational
formulations based on the action integral of theLagrangian provide a useful and systematic means to
obtain governing equations and conservation laws ofenergy and momentum for considered systems.
Especially, the momentum balance equation satisfied by
the gyrokinetic model attracts our attention because it
determines plasma flow profiles, which are considered as
one of the key factors for improving plasma
confinement.
In Ref. [3], the Eulerian formulation, which is also
called the Euler–Poincaré reduction procedure, is applied
to derive the governing equations of the Vlasov–Poisson–Ampère (or Vlasov–Darwin) system and thoseof the drift kinetic system in the general spatial
coordinates, and the momentum balance equations forthese systems are obtained by using the invariance of theaction integrals for the systems under arbitrary spatial
coordinate transformations. The resultant momentum
balance equations contains the symmetric pressure tensor,
which is derived by taking the variational derivative of
the Lagrangian with respect to the metric tensor in themanner similar to that of the general theory of relativity.
This derivation is in contrast to the conventional method
using the spatial translation in which the asymmetriccanonical pressure tensor generally enters the momentum
balance equation.
In Ref. [4], the work in Ref. [3] is extended and theEulerian variational formulation of the gyrokinetic
system with electrostatic turbulence in general spatial
coordinates is presented to derive the local momentum
balance equation with the symmetric pressure tensor
including the effects of electrostatic turbulent
fluctuations on the momentum transport.
In the present study, the Eulerian formulation using thegeneral spatial coordinates is applied to derivation of thelocal momentum balance equation in the gyrokinetic
system with electromagnetic turbulence. The Lagrangian
describing the electromagnetic gyrokinetic turbulence ispresented in Ref. [5], where a novel second-order term is
included to correctly derive both equilibrium andperturbation parts of the gyrokinetic Ampère’s law. To
derive the governing equations for the gyrocenter
distribution functions and the turbulent electromagnetic
fields, the Lagrangian is written as 𝐿!"# = 𝐿!" + 𝐿#
where 𝐿!" and 𝐿# represent the gyrokinetic part(including the gyrocenter distribution functions) and theelectromagnetic field part, respectively. The symmetricpressure tensor is given by
𝛿𝐿!"#
Θ$% = 2𝛿𝑔$%
where 𝛿/𝛿𝑔$% represents the variational derivative withrespect to the metric tensor 𝑔$% . Here, Θ$% contains
the pressure tensor due to particles’ motion and theMaxwell electromagnetic tensor. Using the WKB
representation, the turbulent part of Θ$% is expressed in
the form comparable with the one obtained in the pastwork of the momentum transport. In addition, effects ofcollisions on the momentum balance is considered. It isalso shown that the Chew–Goldberger–Low (CGL)pressure tensor, which causes the neoclassical transport,
is also included in Θ$% .
References[1] H. Sugama, Phys. Plasmas 7, 466 (2000).
[2] A. J. Brizard and T. S. Hahm, Rev. Mod. Phys. 79,421 (2007).
[3] H. Sugama, M. Nunami, S, Satake, and T.-H.
Watanabe, Phys. Plasmas 25, 102506 (2018).
[4] H. Sugama, S. Matsuoka, M. Nunami, and S, Satake,
Phys. Plasmas 28, 022312 (2021).
[5] H. Sugama, S. Matsuoka, and M. Nunami, Phys.
Plasmas 29, 052509 (2022).