J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

Action-angle representation of waves and instabilities in
ﬂowing plasmas
Makoto HIROTA
Japan Atomic Energy Agency, Naka, Ibaraki, 311-0193, Japan(Received: 27 October 2009 / Accepted: 17 February 2010)

Action-angle representation of linear ﬂuctuation in plasma is formally performed by taking the
Lagrange-Hamilton theory as the basis and invoking the recent spectral technique [Hirota and Fukumoto,
J. Math. Phys. 49 083101 (2008)], which is shown to be applicable equally to the magnetohydrodynamic(MHD) system and the Vlasov-Maxwell system. This formalism leads to a natural deﬁnition of the
wave energy (as well as the wave action) for each eigenmode and continuum mode. Negative energy
mode is generally responsible for onsets of various instabilities in ﬂowing plasmas. It is shown that the
canonical form of the linearized system, formulated by Hamilton’s principle of least action, facilitates theconsideration of the action-angle representation, and the spectral technique becomes more sophisticatedthan the previous work that based on the noncanonical form.
Keywords: action-angle variables, ﬂowing plasmas, magnetohydrodynamics, Vlasov-Maxwell equation,
continuous spectrum
gral transform akin to the Hilbert transform [18]. Recently, we have developed another new spectral technique [19], which enables us to evaluate wave energy(and also action-angle variables) for each eigenmodeand continuum mode in a uniﬁed manner by invoking
the Laplace transform. Our method has been applied
to the vortical continuum mode in shear ﬂow [19] andthe Alfvén and sound continua [20] in the MHD case.
The action-angle representation of linear ﬂuctuation
serves to provide a Hamiltonian interpretation of various resonant instabilities in ﬂowing plasmas. For instance, it is shown [19] that resonant coupling between
an eigenmode and a continuum mode having the same
sign of energy results in phase mixing (or continuum)
damping. In contrast, such resonance triggers an instability if their signs of energy are opposite. The
stabilization of the resistive wall mode by ﬂow can be
discussed in this general framework [21].
In this paper, we will rework our spectral
method [19] in a more sophisticated way by the use of
the Lagrangian approach to plasmas [22, 23, 24, 25].
We ﬁrst introduce Hamilton’s principle of least action
into linear perturbation, which enables us to write thelinearized system in the canonical form. This procedure will be demonstrated for the ideal MHD equationand the Vlasov-Maxwell equation as well, from which
one can expect the generalizations to other equations.
The canonical form is more intuitive and tractable
than the noncanonical form which we have employedin the previous work [19]. By constructing an appropriate Poincaré invariant, we will eﬀectively performthe action-angle representation even in the presence ofgrowing/damping eigenmodes and continuum modes,
which are not periodic in time by nature. The detailed calculation of wave energy for eigenmodes and1. Background
Stabilization or destabilization of plasma by ﬂow
is a key issue in recent fusion research and astrophysics. While its mechanism is drawing a lot of attention, theories for ﬂowing plasmas are quite limitedin comparison to static plasmas; especially, clear-cut
stability criteria are hard to obtain. For example, thevariational approaches[1, 2, 3, 4], which are nowadays
well-known such as the energy principle in the MHD
theory [5], the Arnold method [6, 7] and the energyCasimir method [8], are often fruitless for ﬂowing plasmas, i.e., they end up with extremely limited stability
conditions especially for 3D disturbances. In order togain a better understanding of various instabilities in
ﬂowing plasmas, it is informative to study the energyof eigenmode in the context of the stability theoryof Hamiltonian system [9, 10, 11, 12], in which negative energy mode is generally regarded as a sourceof instability. It is well known that negative energy
mode may occur in moving medium (i.e., ﬂow) [13],which is not always unstable but tend to be destabilized in combination with positive energy modes (reactive instability [9, 14, 15]) or by small dissipationeﬀect (dissipation-induced instability [16]).
However, such Hamiltonian theories are not yet
matured for the “continuum mode” that is well-known
to occur in ﬂuctuation of plasma. The existence ofcontinuum mode (or continuous spectrum) requires
careful mathematical treatments of singularities in the
eigenvalue problem. The wave energy for continuous
spectrum was ﬁrst successfully derived by Morrisonand Pﬁrsch [17] in the linear Vlasov-Poisson system,
where the corresponding action-angle representation
is rigorously performed by means of a singular inteauthor’s e-mail: hirota.makoto@jaea.go.jp463©2010 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchM. Hirota, Action-Angle Representation of Waves and Instabilities in Flowing Plasmas

Hamilton’s principle is therefore valid even for linear
dynamics. The linearized equation derived in this way
is especially called the Jacobi equation [6, 7].
As usual, the Legendre transform p̃ = ∂L(2) /∂ q̃˙
determines a quadratic Hamiltonian function H (2) inthe phase space (q̃, p̃) as far as (∂ 2 L/∂ q̇ 2 )−1 exists.
For later use, we introduce the following uniﬁed notations;
( )
(
)
q̃
0 1
ũ =
, J =
,
(7)
p̃
−1 0

continuum modes has been done for the MHD caseby assuming a slab geometry [20]. The present work
indicates that the same treatment is also valid for the
Vlasov-Maxwell system.

2. Hamilton’s principle of least action
for linearized systems
In this section, we discuss the canonical aspectof linear perturbation according to the LagrangeHamilton theory. In order to arrive at the canonicalform of a linearized dynamical system, it is essential
to note that the linearized system naturally inherits
the variational principle of least action. First, let us
simply review this fact by considering an action integral∫ t2
L(q, q̇, t)dt,
(1)
S=

and
H (2) (ũ)(
) ( 2 )−1 (
)
1
∂2L
∂ L
∂2L
p̃ −
q̃ ·
q̃
p̃ −
=
2
∂ q̇∂q∂ q̇ 2∂ q̇∂q
1
∂2L
(q̃)
(8)
− q̃ ·
2∂q 2
1(9)=: �ũ, Hũ� ,2t1

with a Lagrangian L for generalized coordinatesq(t) = (q1 , q2 , . . . , qn )(t) and velocities q̇ = dq/dt.
Hamilton’s principle of least action δS = 0 yields theEuler-Lagrange equation for q(t),
(
)
∂L
d ∂L
=
(2)
dt ∂ q̇∂qwhere we have naturally identiﬁed the phase space ũ
as a Hilbert space with an inner product �ũ1 , ũ2 � =q̃1 · q̃2 + p̃1 · p̃2 , and the linear operators J and H are
respectively anti-symmetric and symmetric by deﬁnition. Then, the Hamiltonian equation is shortly written as

Now, suppose that q(t) is any given solution (including steady one dq/dt = 0). We try searching a perturbed solution q � (t) = q(t) + q̃(t) in accordance with
Hamilton’s principle. By substituting the trial function q � (t) into S, we obtain the following expansion
with respect to the perturbation q̃(t),S =S (0) + S (1) + S (2) + S (3) + . . .

∂t ũ = Jor ∂t ũ = J Hũ.

(10)

It must be remarked that H (2) qualiﬁes as the perturbation (or wave) energy only when the basic solution
q(t) is an equilibrium state (dq/dt = 0). This is because the linear term H (1) of the series expansion of
the Hamiltonian H is not zero in general. It becomes
automatically zero only at an equilibrium state where∂H/∂q = ∂H/∂p = 0 is satisﬁed.
Let us derive the action-angle variables for an oscillatory eigenmode in the conventional manner. Suppose that the basic solution is an equilibrium state
and there is a single eigenmode,(3)

Since the basic solution q(t) is already extremum δS =0, we ﬁnd that S (1) = 0 holds automatically. If theamplitude of q̃(t) is suﬃciently small compared with
q(t), one may neglect the higher order terms S (n) , n ≥3, and obtain an action integral for the perturbationq̃(t) as∫ t2
(2)
L(2) dt
(4)
S =
t1

(
1 ˙ ∂2L ˙
∂2L ˙
q̃ ·
q̃
+
2
q̃
·
=
q̃
∂ q̇ 2∂q∂ q̇
t1 2
)
∂2L
q̃
dt,
+ q̃ ·∂q 2∂H (2)∂ ũũ(t) =ûe−iωt + c.c.,

∫ t2

(11)

with an eigenfrequency ω ∈ R and an eigenvectorû = (q̂, p̂). The complex conjugate (c.c.) is needed
to insure that ũ(t) is real. By introducing the angle
variable as θ(t) = ωt, the action variable is representedby(5)

where L(2) is thought to be a Lagrangian for the perturbation q̃(t) that is to be varied. Since S (2) is
quadratic, the extremum condition δS (2) = 0 with respect to δ q̃ yields the linearized Euler-Lagrange equation,()∂2L ˙ ∂2L
d ∂2L ˙∂2Lq̃=q̃. (6)q̃
+q̃ +dt ∂ q̇ 2∂ q̇∂q∂q∂ q̇
∂q 2

µ :=

�

1
p̃ · dq̃ =
2π

∫ 2π

∂ q̃
dθ
p̃ ·
∂θ
0
�
�
= ip̂ · q̂ − ip̂ · q̂ = û, iJ û .

(12)

It is easily veriﬁed that the energy of eigenmode is
H (2) = ωµ.

464

M. Hirota, Action-Angle Representation of Waves and Instabilities in Flowing Plasmas

where D2 /Dt2 = (∂t + v · ∇)(∂t + v · ∇) and note that∂t and v · ∇ do not commute for a time-dependent
ﬂow v. If the basic ﬁelds are an equilibrium state,
this equation reduces to the one derived by Frieman
and Rotenberg [26]. Together with the momentum
ﬁeld deﬁned by m̃ = ρDξ̃/Dt, (19) is written in the
canonical form (10) where
( )
(
)
−F
∇·v+v·∇
ξ̃
ũ =
, H=
,
−v · ∇
1/ρm̃
(20)

The derivation of the canonical equation (10) that
we have shown here can be applied to plasmas in asimilar way. While the conﬁguration space for thegeneralized coordinates is no longer ﬁnite-dimensionalq(t) ∈ Rn , the Lagrange-Hamilton theory for plasma
has been developed by several pioneering works. Here,
we brieﬂy summarize their results and show the canonical form explicitly.

2.1

Magnetohydrodynamics

The Lagrangian for the ideal MHD equations
was clariﬁed by Newcomb [23], where the generalized coordinates correspond to the ﬂuid particle orbits x(0) �→ x(t) that is mathematically understoodas a diﬀeomorphism group (the Lagrangian description of ﬂuid motion). The velocity ﬁeld v(x, t) in
the Eulerian coordinates is associated with the orbits
via ẋ(t) = v(x(t), t). The magnetic ﬁeld B(x, t), themass density ρ(x, t) and the speciﬁc entropy s(x, t) arefrozen to each inﬁnitesimal ﬂuid element convected by
the ﬂow x(0) �→ x(t).
Assume that (v, B, ρ, s) is a given basic solution of
the MHD equation. The above kinematical (or topological) constraints imply that the MHD ﬂuctuations
are expressed byṽ =∂t ξ̃ + (v · ∇)ξ̃ − (ξ̃ · ∇)v,

(14)s̃ = − ξ̃ · ∇s,(16)This agrees with the spatial integration of the wave
action density derived by Brizard [27].

2.2

(15)

Ẽ = − ∇φ̃ − ∂t Ã,

B̃ =∇ × Ã

where D/Dt = ∂t + v · ∇. The symmetric operator F
is deﬁned by+ ∇[(B 2 + ρc2s )∇ · ξ̃] − (∇ · ξ̃)∇p∗
(18)

D2 ξ̃
= F ξ̃,Dt2(25)

x� (t) = x(t) + ξ̃(x(t), v(t), t),

(26)

Dξ̃
(x(t), v(t), t).
Dt

(27)

v � (t) = v(t) +where p∗ = p(ρ, s) + B 2 /2 is the√ sum of kinetic and∂p/∂ρ is the soundmagnetic pressures and cs =speed. Hamilton’s principle leads to the equation of
motion,ρ(24)

in terms of the ﬂuctuations of the scalar φ̃(x, t) andvector Ã(x, t) potentials and the displacement vectorξ̃(x, v, t) observed in the phase space;F ξ̃ =(B · ∇)[(B · ∇)ξ̃] − (ξ̃ · ∇)∇p∗− ∇{B · [(B · ∇)ξ̃]},

Vlasov-Maxwell system

The Lagrangian theory for the Vlasov-Maxwell
equation was formulated by Low [22]. Here, we consider a plasma consisting of one species with particle
mass m and charge q (but the generalization to themulti-species case is straightforward). Let (f, E, B)
be a given solution, where f (x, v, t) is the distributionfunction, and E(x, t) and B(x, t) are respectively theelectric and magnetic ﬁelds. According to Low (seealso Galloway & Kim [25]), the linear perturbation isexpressed by
)
(
Dξ̃
˜
(23)
f = − ∇ · (f ξ̃) − ∇v · f
Dt

where the vector ﬁeld ξ̃(x, t) represents the displacement of the particle orbit, x� (t) = x(t) + ξ̃(x(t), t),observed in the Eulerian coordinates [26, 23].
The small-amplitude expansion of the MHD Lagrangian was performed by Dewar [24], which results
in
 �

�2
∫
�
�
1  � Dξ̃ �(17)ρ�
L(2) =
� + ξ̃ · F ξ̃  d3 x
� Dt �
2+ ∇(ξ̃ · ∇p∗ ) − (B · ∇)[(∇ · ξ̃)B](21)

As was shown in (12), the action variable for an
eigenmode ξ̃(t) = ξ̂e−iωt + c.c. is calculated as
∫ (
)
µ =i
ξ̂ · m̂ − m̂ · ξ̂ d3 x∫
(22)
=2 ξ̂ · ρ(ω + iv · ∇)ξ̂d3 x.

(13)B̃ =∇ × (ξ̃ × B),ρ̃ = − ∇ · (ρξ̃),and the inner product is∫�ũ1 , ũ2 � = (ξ̃1 · ξ̃2 + m̃1 · m̃2 )d3 x.

Here, one must regard the total derivative D/Dt asD∂q= + v · ∇ + (E + v × B) · ∇v .
Dt ∂t
m

(28)

The relation (23) originates from the fact that the
distribution function f is frozen to the phase space

(19)

465

M. Hirota, Action-Angle Representation of Waves and Instabilities in Flowing Plasmas

ﬂow (x, v)(0) �→ (x, v)(t). The condition ∇ · B̃ = 0
and Faraday’s law ∇× Ẽ +∂t B̃ = 0 have been alreadytaken into account in (24) and (25). By employing
the Coulomb gauge ∇ · Ã = 0, we can eliminate theelectrostatic potential φ̃ from the dynamical variables
since Gauss’s law relates φ̃ to ξ̃ through the Poisson
equation;
�0 ∆φ̃ = −∫q f˜d3 v =∫q∇ · (f ξ̃)d3 v,

be

Ỹ =

L

1
=
2

∫

3

d x

∫

[

+ 2q[(ξ̃ · ∇)A + Ã] ·

Dξ̃
Dt

]
∂ 2 (φ − v · A)˜˜− 2q ξ̃ · ∇(φ̃ − v · Ã)− q ξi ξj∂xi ∂xj
]
[
∫
1
1
3
2
2
d x �0 |∇φ̃ + ∂t Ã| −|∇ × Ã|
+
2
µ0
(30)
where µ0 is the vacuum permeability. In comparison
with the previous works [22, 25], note that φ̃ in L(2)must be substituted by the solution φ̃ of (29) and wehave a relation,∫|∇φ̃ + ∂t Ã|2 d3 x =∫(|∇φ̃|2 + |∂t Ã|2 )d3 x,(31)

since the functional spaces of gradient ﬁelds anddivergence-free ﬁelds are∫ orthogonal to each other.
t
The variation of S (2) = t12 L(2) dt with respect to δ ξ̃and δ Ã gives, respectively, the equation of motion andthe divergence-free part of Ampère’s law,m(35)

δH (2),δ ξ̃
δH (2)
δH (2), ∂t Ỹ = −,(36)∂t Ã =δ Ỹδ Ãwhich are shown explicitly in the Appendix.
For an eigenmode (ξ̃, Ã)(t) = (ξ̂, Â)e−iωt + c.c.,the action variable is calculated by
∫ ∫ (
)
µ =i
ξ̂ · m̂ − m̂ · ξ̂ d3 vd3 x
∫ (
)
(37)
Â · Ŷ − Ŷ · Â d3 x,+i∂t ξ̃ =��� Dξ̃ �2�
�
d vf m ��� Dt �3

δL(2)= �0 ∂t Ã,˙δ Ã

where the phase space ũ = (ξ̃, Ã, m̃, Ỹ ) is constrainedby ∇ · Ã = ∇ · Ỹ = 0. The Legendre transform
determines a proper Hamiltonian H (2) and canonicalequations,(29)

where �0 denotes the vacuum permittivity. In this way,
we regard the Coulomb gauge and the Poisson equation as kinematical constraints (which are imposedon the conﬁguration space to be varied in Hamilton’s
principle).
The Lagrangian for the ﬂuctuations ξ̃ and Ã isthen given by
(2)

]
[
Dξ̃
δL(2)
+ q(ξ̃ · ∇)A + q Ã ,
=f m
m̃ =
˙
Dt
δ ξ̃
(34)

δH (2),
δ m̃

∂t m̃ = −where m̂ and Ŷ can be eliminated by using the relations, (34) and (35) (e.g., Ŷ = −iω�0 Â).
For the purpose of ﬁnding the action-angle variables, this canonical formalism is quite beneﬁcial andintuitive. However, in contrast to the MHD case, theequations (32) is not necessarily required to be solvedin many cases, since we are usually interested in the
behavior of the scalar function f˜, not in the two vector ﬁelds ξ̃ and m̃. It is more economical to consider
the Lie-generated perturbation [27, 28, 29],q
(38)
f˜ =[g̃, f ] + Ã · ∇v f,mby solving the equation for the generating functiong̃(x, v, t),Dg̃
=q(φ̃ − v · Ã),
Dt
where the Lie bracket is deﬁned by(39)

1(∇f · ∇v g − ∇v f · ∇g)mq+ 2 (∇v f × ∇v g) · B,
(40)m
for any functions f, g of (x, v, t). One can conﬁrm
that f˜ generated by (38) solves the linearized VlasovMaxwell equation, and moreover ξ̃ and Dξ̃/Dt generated by[f, g] :=

D2 ξ̃
Dξ̃
×B
= q ξ̃ · ∇(E + v × B) + q
2Dt
Dt+q(Ẽ + v × B̃), (32)
[ ∫
]
1
2
3
˜
�0 ∂t Ã +∇ × B̃ = P q f vd v , (33)µ0where P denotes the projection onto the space ofdivergence-free vector ﬁelds. The gradient part ofAmpère’s law is automatically satisﬁed by the Poisson equation (29).
The corresponding momentum ﬁelds are found toξ̃ = [x, g̃] =
m

466

1
∇v g̃
m

(41)

q
Dξ̃
+ q Ã = [mv, g̃] = − ∇g̃ − B × ∇v g̃
Dt
m
(42)

M. Hirota, Action-Angle Representation of Waves and Instabilities in Flowing Plasmas

solve (32).
In terms of an eigenmode (g̃, Ã)(t)=(ĝ, Â)e−iωt + c.c., the action variable (37) isrewritten as
∫
∫ ∫
µ=i
f [ĝ, ĝ]d3 vd3 x + 2ω |Â|2 d3 x. (43)where ũ0 = ũ(0) denotes the initial data and Γ(σ)
represents a path of integration that encloses the spectrum σ ⊂ C counterclockwise. The operator (Ω−L)−1
is called the resolvent operator. By introducing a notation U(Ω) = (Ω − L)−1 ũ0 , the Laplace transform ofũ(t) corresponds to iU(Ω), which analytically depends
on Ω as long as Ω avoids the spectrum σ. The contour
integral in (46) plays the role of the inverse Laplacetransform.
We can prove that the spectrum σ is symmetricwith respect to both real and imaginary axes; σ = σ =−σ = −σ, which is a common property of Hamiltoniansystems [19]. [Proof: Since ũ(t) must be real, σ = −σholds. The similarity J −1 LJ = L∗ implies that σ =σ.]As a preparatory for the action-angle representation of ũ(t), we introduce a related Poincaré invariant(or the phase space volume enclosed by a family ofsolutions) [33]. First, let us decompose the spectrum
into σ = σ+ ∪ σ− such that σ+ (or σ− ) is inside theright (or left) half plane. According to the symmetryσ− = −σ+ , (46) is rewritten as

3. Spectral approach to action-angle
representation
We have seen that the linearized systems for theideal MHD equation and the Vlasov-Maxwell equation
are commonly written in the canonical form (10) witha judicious choice of variable ũ and its Hilbert space.
Accordingly, wave action of a single eigenmode has
been derived by simply applying the calculus of action
variable (12) to the inﬁnite-dimensional phase space ũ.
However, the ﬂuctuation ũ(t) of plasma generally contains not only eigenmodes but also continuum mode.
It is formally written as][∫∑−iωn t−iωtûn e+û(ω)edωũ(t) =
n+ c.c.,σc(44)

in the presence of semi-simple eigenvalues (or discrete
spectra) {ωn ∈ C|n = 1, 2, 3, . . . } and a continuous
spectrum σc ⊂ R lying on the real axis. The naive
formula (12) does not apply to continuum mode thatconsists of inﬁnite number of singular (or improper)
eigenmodes û(ω)e−iωt . If a singular eigenmode was directly substituted into (12), the action variable would
diverge since it is non-squire-integrable �û(ω)�2 = ∞.
In order to handle multiple eigenmodes and continuum modes in a uniﬁed manner, we use the following spectral approach. By just multiplying the imaginary unit i, the linear Hamiltonian equation (10) canlook like a Schrödinger equation,i∂t ũ =Lũ,�

1ũ(t) =2πi

Γ(σ+ )(Ω − L)−1 ũ0 e−iΩt dΩ + c.c.
(47)

We generate a family of solutions by replacing the initial data ũ0 by ũ0 e−iθ0 ,ũ(t, θ0 ) =

1
2πi

�

Γ(σ+ )(Ω − L)−1 ũ0 e−iΩt−iθ0 dΩ+ c.c.,(48)

which causes a uniform phase shift (+θ0 ) of all eigenmodes and continuum modes. Since this family ofsolutions is labeled by 0 < θ0 ≤ 2π and closed,ũ(t, θ0 ) = ũ(t, θ0 + 2π), we can deﬁne a Poincaré invariant as the ensemble average over θ0 ;(45)

where L = iJ H is a non-self-adjoint operator with respect to the inner product �◦, ◦� of the complex Hilbert
space (the overbar denotes complex conjugate). We
denote by σ ⊂ C the spectrum of L, which is the entirety of all eigenvalues and continuous spectra.
Note that the adjoint operator is given by L∗ =iHJ and its spectrum is the complex conjugate σ of
σ. The evolution equation (45) is said to possess a
pseudo-Hermitian structure [30, 31], since L is Hermitian (i.e., self-adjoint) with respect to an indeﬁnite
inner product �◦, J ◦�.
The solution of (45) is represented by the
Dunford-Taylor integral [32],�1(Ω − L)−1 ũ0 e−iΩt dΩ,(46)ũ(t) =
2πi Γ(σ)

S̃ :=

1
4π

∫ 2π �
0�
∂ ũ
, J ũ dθ0 .
∂θ0

(49)

It is easy to verify the invariance ∂ S̃/∂t = 0 by using(10), which is nothing but Liouville’s theorem for the
perturbation.
By substituting (47) into (49), the integration
with respect to θ0 results in��1(Ω� − L)−1 ũ0 e−iΩ� t dΩ� ,2πi Γ� (σ+ )
�
�
1
−1
−iΩt
(Ω − L) ũ0 edΩ . (50)iJ
2πi Γ(σ+ )S̃ =ReBy noting the identity J (Ω − L)−1 = (Ω − L∗ )−1 J ,467

M. Hirota, Action-Angle Representation of Waves and Instabilities in Flowing Plasmas

the Poincaré invariant is further reduced to��1�S̃ =RedΩdΩ(2πi)2 Γ� (σ+ )Γ(σ+ )���ũ0 , iJ (Ω� − L)−1 (Ω − L)−1 ũ0 ei(Ω −Ω)t ,�
1
D(Ω)dΩ,(51)=
2πi Γ(σ+ )where a function D : C → C is deﬁned by��
D(Ω) := ũ0 , iJ (Ω − L)−1 ũ0 ,is either exponentially growing or damping. Nevertheless, µn is naturally derived from the Poincaré invariant, and ωn µn indeed corresponds to the wave energy [19]. A little care needs to be paid to the factthat µn is complex when ωn is complex. Due to thesymmetry σ+ = σ+ , an eigenvalue ωn also belongs to
σ+ , and let ûn be the corresponding projection andµn be the ‘action variable’. Using the orthogonalityof the projection [32], we obtain

(52)

�
�
� �
µn = ûn , iJu ûn = ûn , iJu ûn = µn ,and we have used a property of the resolvent operator [32]. It should be remarked that the Poincaré invariant S̃ is deﬁned for general perturbations ũ(t) thatare not necessarily periodic in time. Namely, σ+ may
include complex eigenvalues (Im ωn �= 0) representing growing/damping eigenmodes as well as continuous spectrum which causes the phase mixing. Notethat S̃ deﬁned above recovers the action∫ variables for
2π
neutrally stable modes (ωn ∈ R), since 0 dθ in (12)∫ 2π
and 0 dθ0 in (49) become equivalent. Therefore, we
refer to S̃ as the wave action of ũ(t) in a wider sense.
In the last expression (51), the integration overthe phase angle 0 < θ0 ≤ 2π is eventually converted
into a contour integral in C enclosing the spectrum
σ+ . Note that if the integral path enclosed the whole
spectrum σ, it would always result in�1
D(Ω)dΩ = 0,(53)
2πi Γ(σ)

and hence the sum µn + µn of action variables for
growing and damping modes is always real. When ωn
is a real eigenvalue, there
is no �distinction between ωn�and ωn , and µn = ûn , iJ ûn ∈ R agrees with theprevious result (12).
As for the continuous spectrum σc ⊂ R on thereal axis, the path of integration is deformed into thetwo paths that run parallel to σc at the slightly upperand lower sides;
�1U(Ω)e−iΩt dΩ =2πi Γ(σc )∫i[U(ω + iε) − U(ω − iε)] e−iωt dω.
lim
ε→0 2π σc(57)Hence, it is reasonable to deﬁne a singular eigenfunction for ω ∈ σc byand mislead us into obtaining ‘zero wave action’. We
suggest that the correct wave action can be obtainedby counting only the contribution from σ+ , i.e. theright half of the spectrum. (While we have ignored
the spectrum on the imaginary axis so far, it would
also turn out to be zero wave action.)
Now, we are ready to perform the spectral decomposition of the wave action S̃. Since the complex function D(Ω) is analytic except for the spectrum σ, one
may analytically deform the integral path Γ(σ+ ) suchthat it consists of many closed paths that individuallyenclose each isolated singularity of D(Ω).
If there are semi-simple eigenvalues {ωn ∈ C|n =1, 2, 3, . . . }, the U(Ω) must have simple poles in the Ωplane,
U(Ω) =ûn+ ....
Ω − ωn(56)û(ω) :=i
[U(ω + i0) − U(ω − i0)] .
2π

(58)

This deﬁnition of û(ω) agrees with the Fourier transform of ũ(t) according to Sato’s hyperfunction theory [34] (see also the Appendix of Ref. [35]). Various
examples of singular eigenfunctions û(ω) are foundin literatures; see Van Kampen [36], Case [37, 38],
Sedláček [39] and Tataronis [40].
The wave action for the continuous spectrum isthen given as a function of ω;i[D(ω + i0) − D(ω − i0)]2π��= ũ0 , iJ û(ω) .

µ(ω) =(59)

Since the initial data ũ0 is usually non-singular, thisµ(ω) is well-deﬁned.
If the spectrum σ+ is composed of such semisimple discrete spectrum {ωn ∈ C : n = 1, 2, . . . }and a real continuous spectrum σc ⊂ R, the solution
is represented by (44) where the complex conjugate(c.c.) stems from the other spectrum σ− . The wave
action is decomposed into the action variables,∫∑µn +µ(ω)dω.
(60)
S̃ =

(54)

where ûn is the projection of ũ0 onto the eigenspace forωn . An integral path Γ(ωn ) surrounding ωn gives thewave action (or the action variable) for the eigenmode,
�
�
�
1
µn =
D(Ω)dΩ = ũ0 , iJ ûn .
(55)
2πi Γ(ωn )Strictly speaking, this is not the conventional action
variable when ωn is complex, for which the eigenmode

n

468

σc

M. Hirota, Action-Angle Representation of Waves and Instabilities in Flowing Plasmas

Similarly, the wave energy H (2) = (1/2)�∂t ũ, J ũ� becomes�1(2)ΩD(Ω)dΩ,(61)H =2πi Γ(σ+ )A. Canonical form of the linearized
Vlasov-Maxwell equation
The Hamiltonian function of the linearized system(or the wave energy) is
H (2) =
{
∫
∫
[
]�2
1 ��
�
d 3 x d3 v
�m̃ − qf (ξ̃ · ∇)A + Ã �
2mf
]
qf [
ξ̃ · (ξ̃ · ∇)∇(φ − v · A)− m̃ · Dξ̃ +2 }which is decomposed into
H (2) =∑n

ωn µn +∫

ωµ(ω)dω.

(62)

σc− qf ξ̃ · ∇(v · Ã)
]
[
∫
1
1
1
d3 x �0 |∇φ̃|2 + |Ỹ |2 ++|∇ × Ã|2 ,
2
�0
µ0
(63)

We remark that the derivation of the formulae,(55) and (59), becomes simpler than the previous
work [19] owing to the canonical formalism. The wave
actions of the vortical continuum [19] and the Alfvénand sound continua [20] have been derived from µ(ω)
in (59). Our methodology shown here is also applicable to the Vlasov-Maxwell system, and the calculationof the both wave energy and wave action will be muchfacilitated by standing on the present study.

where φ̃ is again regarded as the solution of (29) andD :=v · ∇ +q(E + v × B) · ∇v .
m

(64)

The canonical equations (36) are then written as

4. Summary∂t ξ̃ = − Dξ̃ + η̃,By tracing back to the Lagrange-Hamilton theory [23, 22], we have presented the canonical form of
the linearized systems for the cases of the MHD equation and the Vlasov-Maxwell equation. In the latter case, it is essential to impose the Coulomb gauge
as a kinematical constraint on the ﬂuctuation. The
action-angle variable for a neutrally stable eigenmodeis then easily obtainedby introducing the conven�
tional formula p · dq to the corresponding phasespace of the ﬂuctuation. As for aperiodic ﬂuctuation
stemming from continuum modes and exponentiallygrowing/damping eigenmodes, the notion of actionangle variables needs to be extended. We have naturally performed the action-angle representation forsuch general ﬂuctuation by invoking the spectral technique [19] and introducing the Poincaré invariant as
the ensemble average over the phase angle. The resultant expression (60) [or (62)] provides legitimate
wave action (or wave energy) for any eigenmodes andcontinuum modes.
Historically, the wave action density is wellstudied in the eikonal approximation [41], where thewave is locally assumed to satisfy the dispersion relation of uniform plasma. In contrast, our method corresponds to the spectral decomposition of the wave action and can deal with general ﬂuctuation in strongly
inhomogeneous media, in which the eikonal approximation is not always valid. Based on the present linear
analysis, we expect better understandings of quasilinear and weakly nonlinear behavior of plasmas.
Acknowledgments
The author is grateful to Zensho Yoshida, Yasuhide Fukumoto and Shinji Tokuda for useful comments and discussions.

[

(65)

∂t m̃ = − Dm̃ + qf η̃ × B + (η̃ · ∇)A]− (ξ̃ · ∇)∇(φ − v · A) − ∇(φ̃ − v · Ã) ,(66)

1∂t Ã = Ỹ ,
�0
}
{∫ [
]
3
q f η̃ − v∇ · (f ξ̃) d v∂t Ỹ =P−

1
∇ × (∇ × Ã),
µ0

(67)

(68)

where η̃ should be replaced by the following expression,
]
m̃
q [
−
(ξ̃ · ∇)A + Ã .
(69)
η̃ :=
mf
m

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