23P134-B
Higher Order Terms of the Guiding-Center Transformation and theGyrokinetic Quasi-Neutrality Condition
案内中心変換の高次項とジャイロ運動論的準中性条件
Naoaki Miyato1, Bruce D. Scott2 and Masatoshi Yagi1,3
宮戸直亮1, スコット・ブルース2，矢木雅敏1,3
1

Japan Atomic Energy Agency
2-116 Omotedate, Obuchi, Rokkasho, Kamikita-gun, Aomori 039-3212, Japan
原子力機構 〒039-3212 青森県上北郡六ヶ所村尾駮表舘2-166
2Max-Planck-Institut für Plasmaphysik
D-85748 Garching, Germany
マックス・プランクプラズマ物理研究所 ドイツ連邦共和国バイエルン州ガルヒンクD-85748
3Kyushu University
6-1 Kasuga, Fukuoka 816-8580 Japan
九州大学 〒816-8580 福岡県春日市春日公園6-1
The standard gyrokinetic model, which is originally formulated for perturbations with short wavelengthand small amplitude, is not always valid in the long wavelength regime. The reduced (gyrokinetic) Poisson
equation or the gyrokinetic quasi-neutrality condition in the standard model is no longer sufficient to obtainelectrostatic potential in the long wavelength regime. Since the polarization term including the electrostatic
potential goes to higher order, the other higher order terms which are not considered in the standard model
are needed to obtain the electrostatic potential. Taking into account a higher order displacement vector
associated with the guiding-center transformation, we find additional higher order terms coming fromnonuniformity of magnetic field in the gyrokinetic Poisson equation and the quasi-neutrality condition.

1. Introduction
Control of anomalous transport is one of very
important issues in magnetized fusion plasma
research. For this purpose, it is indispensable toclarify mechanism of formation of transportbarriers [1]. It is believed that drift wave typemicroturbulence driven by density and/ortemperature gradient is a principal cause of theanomalous transport [2]. Since spatiotemporalscale of the drift wave turbulence is small
compared to that of background profiles
including mean flow, it leads to separate
treatment of the drift wave turbulence andevolution of the background profiles. However,global simulation handling both the turbulence
and the profile evolution is necessary for ultimate
understanding of the formation of transportbarriers. The standard gyrokinetic model isformulated for investigation of the drift wave
turbulence, and perturbations with shortwavelength (∼ 1 ) and small amplitude⁄(≪ 1) (gyrokinetic ordering) are assumed in
the formulation of the gyrokinetic model [3-5].
Therefore, there is no guarantee that the standard
gyrokinetic model is valid in the long wavelength
regime as well. Although it was claimed that thestandard gyrokinetic model would be also valid in
the long wavelength by reinterpretation of thegyrokinetic ordering [6], it was reported later thatthe standard gyrokinetic model is not necessarily
sufficient in the long wavelength regime [7]. We
investigate this issue from a point of view ofpush-forward representation associated withphase space transformation [8,9] and show that
additional terms stemming from nonuniformity ofmagnetic field would be important in gyrokinetic
quasi-neutrality condition in the long wavelength
regime.
2. Gyrokinetic model
Modern formulation of the gyrokinetic model isbased on the phase space Lagrangian Lie-transform
perturbation method [10] and consists of two-stepphase space transformation from the particle phasespace to the gyro-center phase space [4-5]. In thefirst step the gyro-phase dependence of a gyrating
particle in equilibrium magnetic field is removed by
the phase space transformation from the particle
phase space to the guiding-center phase space(guiding-center transformation) [11]. Smallness
parameter of the guiding-center transformation is∼ ⁄ ≪ 1 whereis the Larmor radius of aparticle and L is the scale length of the magnetic
field. After the first step time-dependent
electromagnetic perturbations are introduced into
the system and the gyro-phase angle dependencereintroduced with the perturbations is removed by
the transformation from the guiding-center phasespace to the gyro-center phase space (gyro-centertransformation). Here smallness parameter is∼ ⁄ ≪ 1. As mentioned in Introduction, thestandard gyrokinetic model is originally constructed∼1 ,under the gyrokinetic ordering (⁄ ≪ 1). This condition can be interpreted as⁄ ≪ 1 which means that the E B drift
velocity is much slower than the thermal velocity.
The slow flow condition is also satisfied in the long⁄ ~1 ). This
wavelength regime (≪ 1,reinterpretation of the ordering seems to be noproblem in the phase space transformation for thesingle particle dynamics and in the gyrokinetic
Vlasov equation. The change of the ordering,
however, has an effect on the polarization term
includingin the gyrokinetic quasi-neutrality
condition (and the gyrokinetic Poisson equation)
1.

In the equation the ion particle density is expressedin terms of the gyro-center variables and it is called
the push-forward representation of particle density.
The representation depends on details of theguiding-center and gyro-center transformations:⋯.
The polarization term comes from the gyro-center
associated with thedisplacement vector. In the longgyro-center transformation at
wavelength regime the polarization term goes tohigher order than the standard gyrokinetic case.
Hence, we have to take into account the otherhigher order terms to obtain long wavelengthcomponent of the electrostatic potential. Then, wehave to consider another displacement vector
associated with the guiding-center transformation
as well [9, 12]. This piece is not considered in
the standard model.
3. Higher order displacement vector
The guiding-center transformation of particle
position x is written in general as1∙⋯,
2
whereis the nth-order vector field generating∙the guiding-center transformation and. Negative of 1 is the usual gyroradius

vector. We denote the second order piece as
12∙.

andare very
Since explicit forms ofcomplicated [11, 13], they are not shown here.
, we obtain additional terms in theConsideringpush-forward representation of particle density,
therefore, in the gyrokinetic quasi-neutrality
condition:
63

∙

,

where 〈∙〉 denotes gyro-phase average. The
is important for
gyro-phase angle average ofthe quasi-neutrality condition and is given by1∙∙
2
3log .
2
A similar result is found in Ref. [12] in which thefirst term is missing. Although the first term can beneglected in some cases, it may be important intokamaks with relatively large ripples of thetoroidal magnetic field. The second term iscommon to the result in Ref. [12], while thecoefficient in front of the last term is not. Effects of
the additional terms on electrostatic potential will
be investigated in future.
Acknowledgments
This work is partly supported by Grant-in-Aid forYoung Scientist (B) from Japan Society for thePromotion of Science (22760663).
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