22C03
Singularity analysis of two-dimensional two-fluid plasma
２次元２流体プラズマの特異性解析
Yuta Kaneko, Zensho Yoshida
金子雄太, 吉田善章
Graduate School of Frontier Sciences, The University of Tokyo
5-1-5, Kashiwanoha, Kashiwa-shi, Chiba 277-8561, Japan
東大 新領域 〒277-8561 千葉県柏市柏の葉5-1-5
An essential difference of the plasma theory from the neutral fluid mechanics is in that plasma models may
include a varity of "singular perturbations" determining scale hierarchies. Comparison of the ideal
magnetohudrodynamics (MHD) and the two-fluid MHD may be the best practice by whch we can explore
how a singular perturbation (electron inertia) determines an intrinsic (small) scale that is absent in thescale-invariant MHD system; in turn, how a sinlularity can emerge when an "obstacle" finite scale isremoved with neglecting the singular perturbation. Here we study the role of Casimir invariants thatcharacterize the "non-canonical" property of the determing symplectic geomerty; we define some different"sub-classes" of canonicalized self-contained mechanics of the reduced MHD, and report results ofnumerical analysis.

1. Introduction
Complex scale hierarchy is a fundamental
characteristic of plasmas.
Mathematically an
intrinsic scale is determined by a "singularperturbation" that is represented by a term includinga small scale parameter and a higher-orderdiffetential [1]. The ideal magnetohydrodynamics(MHD) is the base line that is free from any
intrinsic scale, i.e. scale invariant.
Includingvarious singular perturbations, we obtaincorresponding scales. Each intrinsic scale poses an
obstacle preventing creation of singularity; thisconjecture is still a tall order for rigorous proof(even for the conventional viscosity term that is alinear singular perturbation; the electron/ion inertia
terms are nonlinear singular perturbations that yielddispersive effects, instead of dissipation). The aimof present study is to analyze scale hierarchy as a"phase-space foliation" determined by "sub-classdynamics" immersed in the total phase space.
2. Non-canonical Hamiltonian dynamics
The plasma fluid models have some conservation
laws arising from either symmetries in theHamiltonian or a "topological defect" (kernel) of
the Poisson bracket [2]. The latter constants ofmotion is called Casimir invariants. A Hamiltoniansystem that has Casimir invariants is said"non-canonical." A Casimir invariant foliates thephase space, and the dynamics is constrained on alieaf of the Casimir invariant.---as far as the phasespace is of a finite dimension, this geometrical
picture applies properly. However, in an
infinite-dimension function space, the notion of

Casimir leaf must be more carefully defined; if theCasimir invariant is not a continuous functional,
even a bounded region on a Casimir leaf is notcompact ---a lief (level-set) of such a Casimir is not
such a smooth surface depict as a finite-dimension
maniforld. We are interested in a possible
pathological property created by a singularity of
non-canonical Poisson bracket.
3. Reduced magnetohydrodynamics
P.J. Morrison and R.D. Hazeltine proposed atransformation of RMHD into a canonical form [4].
We denote the electrostatic potential by , andthe flux function by.The vorticity isand the current is. Thenthe two-dimensional RMHD [3] are given by

where brackets are

Pegoraro et al. showed that the dynamicalsystem in the two-dimensional two-fluid
model that contained electron inertia wasrepresented as a non-canonical Hamiltonian
form [5,6,7].
Morrison has introduced the following newvariables.

andare new field variables
introduced here. This change of variables gives theperspective from higher degrees of freedom.
4. Construction of subclasses
Morrison suggested that the transformation ofthese variables is not unique. We found several
different kinds of transformation. According to
non-canonical RMHD there exist following
conserved Casimirs:

whereis an arbitrary function of
consider another transformation:. We canThe characteristic of this transformation is that
becomes zero. It is easy to check the following:,We also neglect
become.

dependence, then equations(12)

The prime denotes the derivative of a function withrespect to . These are the well-known wave
equations. We have observed the wave motion toproceed in the direction of y. This result isconsistent with the analytical solution.
We began with the initial conditions:

Now all variables are normalized. Fig.1 shows thevorticity of the fluid when
. It can be seen thatthe small vortex occurs continuously. We observe
gradually sharpening vorticity.

This transformation increases the degrees offreedom.
The following transformation maintains twovariables, which allows us writing down thecanonical Hamiltonian form.

RMHD becomes:

is the convective derivative,.
is equal to zero. We can analyze subclasses
without increasing the degrees of freedom.
5. Numerical experiments
In order to analyze RMHD systems, we built asimulation code. We performed preliminary
numerical experiments to verify whether the code iscorrect or not. The finite difference method wasused to develop code for the spatial second orderaccuracy. Time is advanced using third order
explicit Adams-Bashforts method. The boundary
conditions are periodic.
The analytical solution is compared to numericalsolution. We assume that,
whereandare the equilibrium
fields. We linearize the system

Fig.1. Vorticity
References[1] Z. Yoshida, Nonlinear Science ---the challenge ofcomplex systems (Springer-Verlag, Heidelberg,2009).
[2] 吉田善章,プラズマ・核融合学会誌, 86 (2010)
209-219 [解説].
[3] Strauss, H. R., Phys. Fluids, 19 (1976) 134.
[4] Morrison, P. J. and Hazeltine, R. D., Phys. Fluids, 27(1984) 886.
[5] Schep, T. J., Pegoraro, F. and Kuvshinov, B. N.,
Phys. Plasmas, 1 (1994) 2843.
[6] Kuvshinov, B. N., Pegoraro, F. and Schep, T. J.,
Phys. Lett. A, 191 (1994) 296.
[7] Kuvshinov, B. N., Lakhin V, Pegoraro F. and Schep
T. J., J. Plasma Phys., 59 (1998) 4.