Theory of action-angle variables for continuous spectrumin flowing plasma
流れをもつプラズマの連続スペクトルに対する作用・角変数理論
Makoto Hirota
廣田 真Japan Atomic Energy Agency, Naka, Ibaraki, 311-0193, Japan
日本原子力研究開発機構 〒 311-0193 茨城県那珂市向山 8-1

Development of linear and nonlinear mode coupling theory is essential for the stability analysis offlowing plasmas, where the action-angle representation of linear fluctuation is a fundamental issue inthe context of classical mechanics. Existence of continuous spectrum is however the greatest obstacle
originating from the infinite dimensionality of plasmas. This work has made it possible to performthe action-angle representation of the contiuous spectrum with the use of appropriate mathematicaltechniques. This formalism also clarifies the wave energy for each eigenmode and continuum mode,
where negative energy mode is generally responsible for onsets of various instabilities in flowing plasmas.
Resonance between an eigenmode and a continuum mode can be viewed as a mode coupling and the signsof their wave energies identify whether the coupling leads to an instability or not. This unified viewpoint
is applicable to various resonant phenomena including the Landau damping, the Alfvén resonance andthe critical layor instability in shear flow.

1. Introduction
Analysis of waves and stabilities in ﬂowing
plasmas is important towards the understanding of eﬃcient magnetic conﬁnement in fusion
devices and unresolved dynamics and transport in astrophysical plasmas. In the presenceof equilibrium ﬂow (e.g., rotation and shearﬂow), clear-cut stability criterion is hardly derived because the instability mechanism becomes highly involved. For instance, theDoppler shift of frequency causes both couplings and decouplings among various modes
depending on the ﬂow proﬁle. Such the complicated stability problem cannot be treatedby the energy principle [1] which has beensuccessful for the stability analysis of static
plasmas. According to classical mechanics
with ﬁnite degree of freedom, an instability
is triggered by resonant interaction between
two eigenmodes carrying opposite signs of perturbation energies (Krein’s theorem). On theother hand, since plasmas are inﬁnite dimensional systems, the eigenfrequency of linear
perturbation is often occupied by the continuous spectrum (which corresponds to a sumof inﬁnite number of singular eigenmodes, orsimply called the continuum mode). Although
it is well known that resonance with the con-tinuous spectrum strongly aﬀects linear stability, even the deﬁnition of wave energy for continuous spectrum has not been so clear except
for the quite limited cases [2], which seems
to prevent further rigorous developments oflinear and nonliner mode coupling theory in
ﬂowing plasmas.
In this work, we direct our attention togeneralizing the notion of “action-angle variables” to the continuum mode, in order toderive the energy and momentum carried byit. Based on this Hamiltonian description, we
have developed systematic methods for clarifying dynamical features (stabilities, eﬀects
of dissipation and nonlinear couplings etc.) of
ﬂuctuations in plasmas. Our achievements aresummarized in three parts as follows.

2. Action-angle representation of continuum mode
We have established a general method forderiving the action-angle variables of continuum modes in dissipationless plasmas bymeans of the Laplace transform and the hyperfunction theory [3]. This method can decompose the energy of linear perturbation (orthe variation H̃ of Hamiltonian) into the sumof wave energies for eigenmodes and contin-uum mode;∫∑H̃ =ωn µn + ωµ(ω)dω,nσwhere ω1 , ω2 , . . . ∈ C and an interval σ ⊂ R
are respectively assumed to be discrete andcontinuous spectra (i.e., mode frequencies). It
is the values µ1 , µ2 , . . . and the function µ(ω)
that signify the “action variables” for eachmode (the angle variables simply correspondto the phase angles). While they are equivalent to the so-called “wave action” in the wave
kinetic theory [4] when taking the eikonal (orshort wave-length) limit, our theory is widely
applicable to any eigenmode and continuum
mode for which the wave number is not always
well-deﬁned along the background nonuniformity. As an application of our method, wehave derived the action-angle variables of theAlfvén and sound continuous spectra in magnetohydrodynamics (MHD) and shown that
there exists continuum mode with negative energy [ωµ(ω) < 0] in ﬂowing plasma [5].
3. Stability of resonance between eigenmode and continuum mode
Once the energy of the continuum modeis clariﬁed, we can discriminate linear stability of resonant interaction between eigenmode and continuum mode (in analogy with
Krein’s theory for ﬁnite dimensional systems).
We have shown that the interaction leads to
the resonant damping (≃ Landau damping) of
the eigenmode if the continuum mode has thesame sign of energy as the eigenmode, whereas
it leads to the resonant growth (≃ inverse Landau damping) if the continuum mode has theopposite sign of energy [3, 6]. In addition, thenegative energy mode tends to be destabilized
by the eﬀect of dissipation. We have adapted
these theories to the stabilizing eﬀect of ﬂow
on the resistive wall mode [7], where the stableregime appears when the resonant damping
overcomes the dissipation-induced instability.
4. Adiabatic invariance of action variables
We have shown that the action variables of
eigenmode and continuum mode are invariant
when the mean ﬁelds are slowly varying compared with mode frequencies [6]. For example, the Landau damping can be understoodas a mode conversion from an eigenmode toa continuum mode, where the wave action of
the eigenmode∫µ0 is absorbed into that of continuum mode µ(ω)dω. The damping rate isclosely related to the spectral linewidth (equalto phase mixing rate) of the resultant continuum mode, which can be estimated by the invariance of wave-action without invoking theconventional analytic continuation of the dispersion relation [8]. This another viewpointof the Landau damping is generically valid forthe coupling between eigenmode and continuum mode in other dynamical systems (suchthe Alfvén resonance in MHD and the critiallayor instability in shear ﬂow).
Development of advanced theory of nonlinear mode coupling is also in progress basedon this action-angle formalism. Note thatthe conventional wave-kinetic theory and theweak turbulence theory deal with three-wave
resonance in wavenumber space (which is essentially resonance among plane waves). On
the contrary, our method can precisely discuss
the energy and momentum exchange among
the eigenmodes and the continuum modes,
which is clearly more appropriate for global(or long wave-length) ﬂuctuation when the inhomogeneity of mean ﬁeld is unignorable andplays a role of free energy source of ﬂuctuation.
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[4] R. Z. Sagdeev and A. A. Galeev, Nonlinear
Plasma Theory, Benjamin, New York (1969).
[5] M. Hirota and Y. Fukumoto, Phys. Plasmas 15,122101 (2008).
[6] M. Hirota and S. Tokuda, Phys. Plasmas 17,082109 (2010).
[7] M. Hirota, Plasma Sci. Technol. 11 409 (2009).
[8] L. Landau, J. Phys. (USSR) 10, 25 (1946).