PHYSICS OF PLASMAS 12, 012107 (2005)

Resonance between continuous spectra: Secular behavior of Alfvén waves
in a flowing plasma
M. Hirota, T. Tatsuno,a) and Z. Yoshida
Graduate School of Frontier Sciences, University of Tokyo, Chiba, 277-8561, Japan(Received 21 July 2004; accepted 25 October 2004; published online 22 December 2004)
Conventional normal mode analysis often falls short in predicting a variety of transient phenomena
in a non-self-adjoint (non-Hermitian) system. Laplace transform is capable of capturing all possible
behavior in general systems. However, degenerate essential spectra require careful analysis. The
Alfvén wave in a flowing plasma is an example in which the coalescence of the Alfvén singularities
yields nonexponential growth of fluctuations. Invoking hyperfunction theory, rigorous expression of
the Laplace transform leads to an accurate estimate of the asymptotic behavior of resonant singularmodes. © 2005 American Institute of Physics. [DOI: 10.1063/1.1834591]
I. INTRODUCTION

evolution equation for the fluctuation parts ṽ, b̃, and p̃,Waves and instabilities in plasmas are far richer thanthose observed in fluids, solids, or various continuous media.
Dynamical representations of perturbations are beyond thescope of conventional normal mode analysis—a perturbation
u共x , t兲 may not assume the form of u共x , t兲 = e−i␻t␺共x兲; hence,
we cannot replace ⳵t with −i␻ in the determining evolution
equations.1 In analyzing a variety of transient phenomena inplasmas, we encounter two difficult problems that require
careful mathematical considerations. One is the general nonself-adjoint (non-Hermitian) property of plasmas. When plural branches of waves overlap in some ranges of frequencies,
these waves may interact through resonance (because themodes are not orthogonal), resulting in algebraic (secular)
amplification of the wave. The other is the existence of various continuous spectra (essential singularities in the dispersion relation). Resonance between continuous spectra is notas simple as those in point spectra (eigenvalues), because wemust analyze singular eigenfunctions (see Ref. 2 and Appendix A).
As is well known, Laplace transform is capable of capturing all possible behavior in general systems. However, theinverse Laplace transform (equivalent to the Dunford integral of the spectral theory) is not easy when multiple continuous spectra are degenerated. In the present paper, we
invoke the hyperfunction theory3,4 to provide a rigorous basis
for dealing with singular eigenfunctions (see Appendix B).
The theory reveals a natural relation between the Fouriertransform (characterizing the eigenfunctions) and theLaplace transform (defining the solution of an initial value
problem). We can derive an accurate estimate of theasymptotic behavior of resonant singular modes, which wasleft out in earlier theories.5–7
The subject of our analysis is the Alfvén waves governedby the ideal magnetohydrodynamics (MHD) equations. Linearizing the MHD equations around an equilibrium with velocity V, magnetic field B, and pressure P, we obtain ana)

Present address: Institute for Research in Electronics and Applied Physics,
University of Maryland, College Park, MD 20742-3511.

1070-664X/2005/12(1)/012107/11/$22.50

i⳵t f = Kf,

f = t共ṽb̃p̃兲,共1兲

where the generator K is a linear differential operator. We
will solve the initial value problem of (1) by assuming incompressibility.
Note that the non-self-adjointness discussed here is not
peculiar to flowing plasmas, since the well-known selfadjoint property of static plasmas is attributed to the
Lagrange representation. Introducing the Lagrange variable ˜␰
as ṽ = ⳵t˜␰ + V · ⵜ ˜␰ − ˜␰ · ⵜ V, (1) is reduced to⳵2t ˜␰ + 2V · ⵜ ⳵t˜␰ = F˜␰ ,共2兲

where F is a self-adjoint operator under appropriate boundary conditions.8,9 If the equilibrium is not flowing 共V ⬅ 0兲,
the evolution of ˜␰ is generated by only F. Due to Von Neumann’s theorem of the spectral resolution of the self-adjoint
operator,10 we can invoke the normal mode (spectral) analysis of F to generate the solution of (2). The MHD stability
analysis is, therefore, conventionally based on the dispersion
relation where time derivative 共i⳵t兲 is replaced by eigenvalue
共␻兲.
However, in deriving (2), we have assumed homogeneous initial conditions which satisfy
关p̃ + ˜␰ · ⵜ P + ␥ P ⵜ · ˜␰兴t=0 ⬅ 0,

共3兲关b̃ + ⵜ ⫻ 共B ⫻ ˜␰兲兴t=0 ⬅ 0,

共4兲

where ␥ denotes the specific heat ratio. Hence, the solutionof (1) is wider than that of (2), and K is no longer selfadjoint even for static plasmas. In this paper, we will showcomplicated algebraic growth of fluctuation by solving (1).
In Sec. II, we will formulate the evolution equation including the effect of shear flow. We consider an equilibriumwith slab geometry, which is inhomogeneous only in the x
direction. Assuming incompressibility, the fluctuations canbe represented by four variables f = t共w̃x j̃x ˜␷x b̃x兲. Writing
f 1 = t共w̃x j̃x兲 and f 2 = t共˜␷x b̃x兲, the evolution equation can becast in a block form,

12, 012107-1© 2005 American Institute of Physics

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012107-2

i⳵t

冉冊冉
f1

f2

Phys. Plasmas 12, 012107 (2005)

Hirota, Tatsuno, and Yoshida

=

K1 N
0

K2

冊冉 冊
f1

f2

,

共5兲

where K1, K2, and N are linear operators in 2 ⫻ 2 matrix
forms. As we will see later, the operator K2 has only realcontinuous spectra, ␴+c =兵␻a+共x兲 ; x 苸 D其 and ␴−c =兵␻a−共x兲 ; x
苸 D其, representing the (Doppler-shifted) Alfvén waves,
where ␻a+共x兲 and ␻a−共x兲 denote the local Alfvén frequencies
associated with the two propagation directions along magnetic field, and there is no unstable point spectrum undercertain conditions proposed by Barston5 and Stern.11 If thedomain D is bounded, i.e., the thickness of the slab is finite,
K2 is a bounded operator. It will be shown that the operator
K1 is a multiplication operator and has the same spectrum as
K2. Since these four continuous spectra are degenerating andthe generator has an off-diagonal element N, we can expect
algebraic growth of f 1 by analogy with the Jordan block inlinear algebra. However, the nilpotent among continuous
spectra is not mathematically resolved, and we therefore
must solve the initial value problem.
In Sec. III, the initial value problem will be solved byusing the Laplace transform. The major part of this section
will be devoted to solvingi⳵t f 2 = K2 f 2 .

共6兲

Barston,5 Sedláček,6 and Tataronis7 considered the mathematically equivalent problem without ambient flow, where
the continuous spectra do not receive the Doppler shift; ␴±c= 兵±␻a共x兲 ; x 苸 D其 with ␻a共x兲 : = ␻a+共x兲 = −␻a−共x兲. They assumed that ␻2a共x兲 ⬎ 0 for all x 苸 D, which implies that ␴+c and␴−c are disjoint. In this case, as Sedláček pointed out, theLaplace transform can be translated into the normal modes
analysis.
The normal modes analysis does not give a correct solution if there is a point xr that satisfies ␻a共xr兲 = 0. In the presence of this xr (so-called rational surface), the forward andbackward Alfvén continuous spectra (␴+c and ␴−c ) overlap.
Then we encounter a complicated singularity on the rationalsurface, which implies a resonance between the two continuous spectra at zero frequency in the subsystem (6). The hyperfunction theory enables analysis of this singularity. As aresult, we will find that the solution f 2 asymptotically
evolves into a standing wave that yields a magnetic island.
This phenomenon was omitted in the earlier works7,12 where
the Alfvén waves (except for zero frequency) receive thephase mixing damping. The standard analysis based on
Lagrange representations excludes this solution because of
the assumption (4).
The effect of flow becomes essential if the gradient of
the flow shear exceeds that of the magnetic shear. Our analysis will prove that the magnetic island cannot survive in sucha strong shear flow.
In the end of Sec. III, the equation for f 1 will be solved
by using the given f 2. We will observe an algebraic growth
localized on the rational surface, which can be understood asa resonance among the four Alfvén continuous spectra. The
physical explanation will be discussed in the summary.
II. GENERAL MODEL OF ALFVÉN WAVES

We formulate a system of equations that describes the
Alfvén waves in inhomogeneous ambient magnetic field andflow. The incompressible ideal MHD equations read as

⳵tv + 共v · ⵜ兲v = − ⵜ p +

冋

册

兩b兩2
1
+ 共b · ⵜ兲b ,
2 − ⵜ
2
a

共7兲

⳵tb + 共v · ⵜ兲b = 共b · ⵜ兲v,

共8兲

ⵜ · v = 0,

共9兲

ⵜ · b = 0,

where v, b, and p denote velocity, magnetic fields, and pressure, respectively. The mass density is assumed to be constant. Variables are in the standard Alfvén units. The scaling
parameter a denotes the Alfvén Mach number. In what follows, we choose the representative magnetic field so that a= 1.
We consider an equilibrium with slab geometry,
V共x兲 = 共0,Vy共x兲,Vz共x兲兲,

B共x兲 = 共0,By共x兲,Bz共x兲兲,and P共x兲 + 兩B共x兲兩 /2 = const.
2

共10兲

Since the equilibrium may have strong magnetic shear, we
do not invoke the reduced MHD approximation, but consider
all components of the vector field in the fluctuation partwhich depend on all coordinates. The essential independent
variable is, however, only x due to the homogeneity of theequilibrium in the y and z directions. We may introduce wave
numbers, ky and kz, and substitute the following expressions
into (7)–(9):v = V共x兲 + ṽ共x,t兲ei共kyy+kzz兲 ,共11兲b = B共x兲 + b̃共x,t兲ei共kyy+kzz兲 ,共12兲

p = P共x兲 + p̃共x,t兲ei共kyy+kzz兲 .

共13兲

For these fluctuations, the use of so-called normal velocity
and normal vorticity is more convenient (see Appendix C).
By introducing the x component of vorticity and current as,
respectively,w̃x = iky˜␷z − ikz˜␷yandj̃x = ikyb̃z − ikzb̃y ,共14兲

the four variables ˜␷x, w̃x, b̃x, and j̃x can reproduce all components of ṽ and b̃.
Now we linearize (7)–(9). By taking the divergence of(7), we obtain the linearized Poisson equation⌬p̃ + ⌬共B · b̃兲 = − 2ik · V⬘˜␷x + 2ik · B⬘b̃x ,共15兲

where k = 共0 , ky , kz兲 and a prime 共 兲 denotes the x derivative.
⬘
Using (15), we can eliminate p̃ from the system (i.e., p̃ is not
an independent variable). After some manipulations, the linearized equations are written in the matrix form of

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012107-3

冢 冣冢
w̃x

i⳵t

Phys. Plasmas 12, 012107 (2005)

Resonance between continuous spectra…

j̃x

⌬˜␷x

=

k·V

k·V

− 共B⬘ ⫻ k兲 · ex⌬00k · V − k · V ⬙⌬00

− k · B⌬−1

−k·B

b̃x

− k · B − 共V⬘ ⫻ k兲 · ex⌬−1

共B⬘ ⫻ k兲 · ex−1

共V⬘ ⫻ k兲 · ex

−1

− k · B⌬ + k · B⬙
k·V

冣冢 冣
w̃x
j̃x

⌬˜␷x

共16兲

.

b̃x

where ⌬ = ⳵2x − k2 with k = 兩k兩, and ex = 共1 , 0 , 0兲 is the unit vector. By considering a domain D 傺 R with a Dirichlet boundary−1
condition, the operator ⌬ is uniquely given as a convolution integral.
In terms of M± =˜␷x ⫿ b̃x and S± = w̃x ⫿ j̃x, (16) reads as

冢 冣冢
S+

i⳵t

S−

⌬M+

⌬M−

=

␻a+ 0
0 ␻a−

− ␫ +⌬−100⬘ ⳵ x⌬
␻a+ + ␻a+00

− ␻⬘a+⳵x⌬−1 − ␻⬙a+⌬−1− ␻⬘a−⳵x⌬ − ␻⬙a−⌬−1

␻a− + ␻⬘a−⳵x⌬−1

where
␻a±共x兲 = k · V共x兲 ± k · B共x兲and
␫±共x兲 = 关V⬘共x兲⫻ k兴 · ex ± 关B⬘共x兲 ⫻ k兴 · ex. If the ambient fields are uniform[V共x兲 ⬅ const and B共x兲 ⬅ const], the generator is reduced to adiagonal form, and therefore each variable oscillates independently with the frequency ␻a+ or ␻a−. The variables M±
and S± correspond to the Alfvén waves polarized in the two
directions perpendicular to B, and the subscripts + and −identify the propagation directions.
Inhomogeneity of the ambient fields causes interactions
between two polarized waves. The behavior of M± is notaffected by S±, while S± is forced by M±. In this paper, thelower two equations and the upper two equations in (17) [or(16)] will be referred to as master equations and slave equations, respectively, and we call M± (or ˜␷x and b̃x) mastervariables and S± (or w̃x and j̃x) slave variables.
If we consider normal modes such as ˜␷x共x , t兲 =˜␷x共x兲e−i␻t,the master equations are combined, by eliminating b̃x, into

冋

册

dũ
d
„␻ − ␻a+共x兲…„␻ − ␻a−共x兲…
− k2„␻ − ␻a+共x兲…„␻dxdx− ␻a−共x兲…ũ = 0,

共18兲

where ũ =˜␷x / 共␻ − k · V兲. This Sturmian equation becomes singular if ␻ 苸 ␴c = ␴+c 艛 ␴−c , where ␴+c and ␴−c are defined by
␴±c = 兵␻a±共x兲 ; x 苸 D其 and correspond to the (Doppler-shifted)
Alfvén continuous spectra. For a static equilibrium 关V共x兲⬅ 0兴 and any function B共x兲, Barston5 considered a mathematically equivalent problem to (18) and proved that there
is no spectrum in addition to ␴c. Even in the presence offlow, we can prove by applying Stern’s result,11 that exponentially growing or damping mode 共␻ 苸 R兲 does not exist if
兩k · V共x兲兩 艋 兩k · B共x兲兩

S−

0
−1

共19兲

is satisfied everywhere in a certain inertial frame. Even if(19) is violated, the system might be still free from the exponential growth mode. For example, if both k · V共x兲 andk · B共x兲 are linear functions of x, there is no drive for the
Kelvin–Helmholtz instability. We are interested in the continuous spectrum, and therefore consider only stable equilibria in the sense of the dispersion relation.

冣冢 冣
S+

− ␫−⌬−1

0

−1

⌬M+

共17兲

,

⌬M−

The slave equations have the same continuous spectra ␴+cand ␴−c due to the multiplication operator ␻a±共x兲 in (17).
Therefore, the evolution equation has four degenerate continuous spectra in total. Given that the mathematical structure of (17) is similar to that of Orr–Sommerfeld and Squire
equations13 in fluid dynamics, we can expect the algebraic
instability to be caused by the resonant energy transfer fromthe master equations to the slave one. For example, if wesubstitute the special configuration
V = 共0,0,0兲,

B = 共0,0,x兲,

ky ⫽ 0,kz = 0.

共20兲

the generator of (16) is reduced to a number matrix that iscalled nilpotent in the linear algebra (see Appendix A). Ourlinear system is, therefore, non-self-adjoint even for static
plasmas and the slave variables increase algebraically in proportion to t. This example is too simple because the Alfvén
wave does not propagate and the continuous spectra completely degenerate into a point 共␴c = ␴±c = 兵0其兲. We will discussa less trivial problem in the next section, where the four
Alfvén continuous spectra yield spatially and temporally
complicated behavior. Compared with our problem, the Orr–
Sommerfeld and Squire equations have only point spectra
due to the viscosity, and then the eigenvalue problem is applicable based on some general theories like Ref. 14. In oursystem, however, we must solve the initial value problem
directly because there is no theory for the non-self-adjoint
system with degenerate continuous spectra.
Before ending this section, we make a comment on theeffect of the compressibility. If we consider a compressible
plasma, we must include two other variables into the system,
viz., the pressure perturbation 共p̃兲 and the other componentof velocity perturbation (say iky˜␷y + ikz˜␷z). The six variables
are generally coupled and no longer decomposed into themaster and slave variables as in (17). The continuous spectrathat we found in the master and slave equations, then, appear
as the slow and Alfvén continuous spectra, respectively.
Since the frequency of the slow wave coincides with that of
the Alfvén wave on the rational surface, the degeneracy of
the four continuous spectra still exists, and hence, the special
configuration given by (20) will cause the algebraic growth

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012107-4

Phys. Plasmas 12, 012107 (2005)

Hirota, Tatsuno, and Yoshidaof ˜␷z even if we take into account the compressibility. Although the similar transient behavior is expected, the analysis of the compressible case is generally involved.

+ 2␤2x兴⳵xIII. ANALYSIS OF ALGEBRAIC BEHAVIOR

In the following analysis, we will solve the initial value
problem of (17) by assuming linear profiles of the ambient
magnetic field and flow. Let one rational surface exist in thedomain and the x coordinate be chosen so that the surface islocated on x = 0. In an appropriate inertial frame, we may
assume k · V共0兲 = 0 without loss of generality. Thus, we havek · V共x兲 = ␣x,

k · B共x兲 = ␤x

共␣, ␤ = const兲.

共21兲

In this coordinate system, we consider a finite domain x苸 关−L1 , L2兴 共L1 ⬎ 0 , L2 ⬎ 0兲 with the Dirichlet boundary condition
˜␷x共− L1,t兲 = ˜␷x共L2,t兲 = 0,b̃x共− L1,t兲 = b̃x共L2,t兲 = 0 共22兲for all t. Finally, we assume that the initial conditions areholomorphic functions on 关−L1 , L2兴. The Alfvén continuous
spectra discussed in the preceding section are represented by

␴+c = 兵␻ 苸 R; ␻ = 共␣ + ␤兲x,x 苸 关− L1,L2兴其,

共23兲

␴−c = 兵␻ 苸 R; ␻ = 共␣ − ␤兲x,x 苸 关− L1,L2兴其.

共24兲

We denote the spectrum of the evolution equation by ␴c= ␴+c 艛 ␴−c , for there is no other spectrum. This linear systemis stable with regard to the dispersion relation (all ␻ are real
numbers).
Because the spectrum ␴c is a bounded set, we can apply
the Laplace transform defined byL关ṽx共x,t兲兴: =冕

⬁ṽx共x,t兲ei⍀t dt,

共25兲

Ṽ
− 关2␣共⍀ − ␣x兲⍀ − ␣xṼ␺= ⌬␾ − ␤x⌬.
⍀ − ␣x⍀ − ␣x共27兲This equation becomes singular at 兵共x , ⍀兲 ; ⍀ − 共␣ ± ␤兲x = 0其,
which is related to the two continuous spectra ␴±c , but it isimportant to note that 兵共x , ⍀兲 ; ⍀ − ␣x = 0其 is not a singularityof (27) (see Appendix D). In general, we often encounter thisapparent singularity concurrent with the elimination ofvariables.15
Let us suppose that we have a solution of (27). By the
Cauchy–Kovalevsky theorem, the solution Ṽ共x , ⍀兲 must be aholomorphic function in
共x,⍀兲 苸 共关− L1,L2兴 ⫻ C兲 \ 兵共x,⍀兲;⍀ − 共␣ ± ␤兲x = 0其.
共28兲
Given that 关−L1 , L2兴 ⫻ 共C \ ␴c兲 is a subset of (28), thisṼ共x , ⍀兲 is holomorphic for all x 苸 关−L1 , L2兴 as far as ⍀
苸 C \ ␴c (resolvent set).
The relation between the Laplace transform and the Fourier transform is made clear in the hyperfunction theory (seeAppendix B). Since the spectrum ␴c is bounded, the inverse
Laplace transform is equivalent to the Dunford integral (orthe double Bromwich integral according to Sedláček6):
12␲

ṽx共x,t兲 = −冖

Ṽ共x,⍀兲e−i⍀t d⍀,

C共␴c兲

共29兲

where the integral path C共␴c兲 encircles the spectrum ␴c counterclockwise. By deforming C共␴c兲 into the vicinity of ␴c, we
may write

冕ṽˆx共x, ␻兲e−i␻t d␻ ,共30兲1
关Ṽ共x, ␻ + i0兲 − Ṽ共x, ␻ − i0兲兴.
ṽˆx共x, ␻兲 =
2␲

共31兲

ṽx共x,t兲 =0where ⍀ 苸 C must satisfy Im共⍀兲 ⬎ 0 and ␴c lies on the real
axis of the complex ⍀ plane.
By making use of the master-slave structure of (17) [or(16)], we can solve the master equations independent of theslave equations. In Secs. III A–III E, we will evaluate theasymptotic behavior of ṽx and b̃x instead of M±, since themaster equations of (16) are simpler than that of (17). Using
this result, the slave equations will be solved in Sec. III E.

冉 冊
冉 冊
冉 冊

关共⍀ − ␣x兲2 − ␤2x2兴⌬

␴c

where

This ṽˆx共x , ␻兲 on 关−L1 , L2兴 ⫻ ␴c corresponds to the Fouriertransform of ṽx共x , t兲 in a generalized sense, i.e., ṽˆx共x , ␻兲 is asingular eigenfunction defined by the hyperfunction theory.

B. Solutions near singular points
A. Fourier–Laplace analysis of master equations

In terms of Ṽ共x , ⍀兲 = L关ṽx共x , t兲兴 and B̃共x , ⍀兲
= L关b̃x共x , t兲兴, the master equations are transformed to共⍀ − ␣x兲⌬Ṽ = − ␤x⌬B̃ + ⌬␾ ,共⍀ − ␣x兲B̃ = − ␤xṼ + ␺ ,共26兲

where ␾共x兲 = iṽx共x , 0兲 and ␺共x兲 = ib̃x共x , 0兲 are holomorphic
functions on 关−L1 , L2兴.
By eliminating B̃, we obtain

The singularity of Ṽ共x , ⍀兲 can be investigated by the
Frobenius method.16 However, since (27) has complicatedinhomogeneous terms, we need more careful treatment toobtain the particular solution and understand its dependenceon ⍀.
We perform the series expansion of the equation in theneighborhood of two singular points, x = X+共⍀兲 and x= X−共⍀兲, which are defined as X±共⍀兲 = ⍀ / 共␣ ± ␤兲. In the following calculation, we solve both cases simultaneously, using the double signs x = X±. Let ␰ = 共x − X±兲 / r be a local variable that is scaled by

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012107-5

Phys. Plasmas 12, 012107 (2005)

Resonance between continuous spectra…

2␤⍀.
␤2 − ␣2

r = X+ − X− =共32兲In terms of this variable, (27) is transformed into

␰2⌬␰Ũ + ␰

2␰ ± 1
⳵␰Ũ = ⌽共x,⍀兲,
␰±1

共33兲

where ⌬␰ = ⳵2␰ − r2k2 andŨ共x,⍀兲 = r⌽共x,⍀兲 =Ṽ共x,⍀兲Ṽ共x,⍀兲=,⍀ − ␣x ␣共␥2 − ␰兲␰22
共␣ − ␤ 兲共␰ ± 1兲r

冋

冉
冉 冊

⫻ ⌬␰␾ + ␤共␥1 − ␰兲⌬␰

␥1 =

共34兲

冉 冊

1 ␣
⫿1 ,
2 ␤

␥2 =

␺
␣共␥2 − ␰兲

冊册

,

1 ␤
⫿1 .
2 ␣

共35兲

共36兲

In contrast to the normal modes analysis such as (18),
we have the complicated inhomogeneous terms ⌽共x , ⍀兲.
First, we solve the homogeneous equation of (33) bysetting ⌽共x , ⍀兲 ⬅ 0. According to the Frobenius method,16
the substitution of the series expansion Ũ共␰ ; ⍀兲⬁= 兺n=0Ũn共⍀兲␰n+␭ [where Ũ0共⍀兲 ⫽ 0] yields⬁m␰␭ 兺 兺 Ũn共⍀兲f m−n共n + ␭兲␰m = 0,

共37兲m=0 n=0

where
f 0共␭兲 = ␭2,

f 1共␭兲 = ± ␭,

f 2共␭兲 = − r2k2 − ␭,

共38兲

and

FIG. 1. Singular points X = X+ and X = X− in the complex X plane. The
Frobenius method develops the solutions in series near these points. The
circles of convergence are indicated by dotted lines, and the inner regionsare, respectively, denoted by ⌫+ and ⌫−. Wavy lines represent the branchcuts of the logarithmic singularities.

⌫±共⍀兲 = 兵X 苸 C;兩X − X±兩 ⬍ 兩r兩其.

The logarithmic function of complex variable, Log X, is defined on a Riemann surface 兵X 苸 C ; −␲ ⬍ arg X ⬍ ␲其, which
has a discontinuity on a real axis; we will use the formula
Log共x ± i0兲 = log兩x兩 ± ␲iY共− x兲

f j共␭兲 = − 共⫿1兲 ␭

for j 艌 3.

Ũr共␰ ;⍀兲 = 1 +and a singular solution冋冉

1 r 2k 2
Ũs共␰ ;⍀兲 = Ũr共␰ ;⍀兲Log ␰ + ⫿ ␰ −
−1
2 2

冉

冊

册

1 5r2k2+ 1 ␰3 + ¯ ,
␰ ⫿
3 36
2

冋

1
共2r␾2 − rk2␾0兲␣ − ␤22冉⫿ 2r␺2 − rk2␺0 +2␺1 2␺0+␥2 r␥22

共40兲

冊
共41兲

where we set Ũ0共⍀兲 ⬅ 1. The radius of convergence is foundto be 兩r兩 = 兩X+ − X−兩, which corresponds to the distance between the two singular points. Supposing that there is a complex plane X with Re共X兲 = x as shown in Fig. 1, we denote theinterior of the circle of convergence simply by

冊册

共44兲

.

Let us substitute a holomorphic function Ũ p共␰ ; ⍀兲⬁= 兺n=0Ũn共⍀兲␰n into (33). By comparing the coefficients ofn

␰ 共n = 0 , 1 , . . .兲, we obtainr 2k 2 2 r 2k 2 3␰ ⫿␰ + ¯,
418

共43兲

where Y共x兲 denotes the Heaviside function.
Next, we take into account the inhomogeneous term
⌽共x , ⍀兲, which includes the arbitrary initial conditions ␾ and␺. The particular solution (denoted by Ũ p) will be solved forall ⍀ 苸 C \ 兵0其, based on the fact that ⌽共x , ⍀兲 diverges at ⍀= 0 (or r = 0). Substituting arbitrary holomorphic functions⬁⬁
␾共x兲 = 兺n=0
␾nrn␰n and ␺共x兲 = 兺n=0
␺nrn␰n, we can also expand⬁⌽n共⍀兲␰n, where, for inthe inhomogeneous term as ⌽ = 兺n=1
stance,共39兲

We can determine 兵Ũn共⍀兲 ; n = 1 , 2 , . . .其 recursively by equating the coefficients of ␰m in (37) with zero. Since the indicial
equation f 0共␭兲 = 0 has a repeated root ␭ = 0, we obtain a regular solution

共x 苸 R兲,
3⌽1共⍀兲 = ±
j

共42兲

冢

f 1共0兲 f 0共1兲

0

⌽1

Ũ1

f 2共0兲 f 1共1兲 f 0共2兲Ũ2

f 3共0兲 f 2共1兲 f 1共2兲 f 0共3兲⯗冣冢 冣 冢 冣
Ũ0

0



Ũ3
⯗

=

⌽2

.

⌽3
⯗

共45兲

In these relations, we may fix Ũ0共⍀兲 ⬅ 0 to remove the indefiniteness of the homogeneous solution Ũr共␰ ; ⍀兲 from theparticular solution. Then we can determine 兵Ũn共⍀兲 ; n= 1 , 2 , . . .其 uniquely for any 兵⌽n共⍀兲 ; n = 1 , 2 , . . .其, due to thefact that the diagonal components f 0共1兲 , f 0共2兲 , . . . are not

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012107-6

Phys. Plasmas 12, 012107 (2005)

Hirota, Tatsuno, and Yoshida

zero and are different from each other. In other words, theparticular solution Ũ p共␰ ; ⍀兲 is found to be holomorphic at
␰ = 0 for any initial conditions. Although this Ũ p might besingular at ␰ = ␥2, we already know that ␰ = ␥2 is the apparent
singularity, and therefore Ṽ p = ␣共␥2 − ␰兲Ũ p must be holomorphic in ⌫±共⍀兲.
C. Singularity at ␻ « ␴c \ ˆ0‰

gions ⌫+共⍀兲 and ⌫−共⍀兲. For this purpose, we start our analysis with the solution near X+ and, from this point of view,observe the other singularity at X−. Noting the formula
Log共1 + X兲 = X −Using Ṽr,s,p共␰ ; ⍀兲 = ␣共␥2 − ␰兲Ũr,s,p共␰ ; ⍀兲, the general solution is represented by+ Ṽ p共␰ ;⍀兲in ⌫±共⍀兲.

The coefficients Cr±共⍀兲

and Cs±共⍀兲共46兲

Ṽ共L2,⍀兲 = 0,

共47兲

the coefficients Cr±共⍀兲 and Cs±共⍀兲 are uniquely determined,for ⍀ belongs to the resolvent set C \ ␴c.
Let us investigate the limits of Ṽ共x , ␻ + i0兲 and Ṽ共x , ␻− i0兲 for ␻ 苸 ␴c \ 兵0其 (the case ␻ = 0 will be discussed later).
In this case, the two singular points X+ and X− approach,respectively, x+ : = ␻ / 共␣ + ␤兲 and x− : = ␻ / 共␣ − ␤兲. Since Ṽs
has a branch cut in the complex X plane as shown in Fig. 1,
there is discontinuity due to the logarithmic term [see (43)],Ṽs共␰ ; ␻ + i0兲 ⫽ Ṽs共␰ ; ␻ − i0兲,

共48兲

which causes, in general,
Cr±共␻ + i0兲 ⫽ Cr±共␻ − i0兲,

Cs±共␻ + i0兲 ⫽ Cs±共␻ − i0兲.
共49兲Then the Fourier transform ṽˆx共x , ␻兲 given by (31) has singularities of log兩x − x±兩 and Y共x − x±兲. This result is qualitatively
the same as the conventional Alfvén singularity without flowwhich was studied by Barston,5 Hasegawa and Uberoi.12 The
inverse Laplace transform leads to phase-mixing damping⬀1 / t (see also Ref. 7).
Using this result, B̃共x , ⍀兲 is easily obtained byB̃共x,⍀兲 =− ␤xṼ共x,⍀兲 + ␺共x兲.
⍀ − ␣x共50兲

For the limit of ⍀ → ␻ 苸 ␴c \ 兵0其, the singularity of B̃共x , ⍀兲 isessentially the same as Ṽ共x , ⍀兲.
D. Singularity at ␻ = 0

Within the continuous spectra, the case of ␻ = 0 requires
more careful analysis. When ⍀ → 0, the two singular points
x = X+ and x = X− collide with each other on the rational surface x = 0, and the radius of convergence 兩r兩 becomes zero. In
this case, we must deal with the two singularities at once toobtain the solution Ṽ共x , ⍀兲 which is consistent in both re-

共51兲

冋

k2 2 k2 3
␰ − ␰ + ...
4
18

册

共52兲
共53兲

=1 + O共⍀兲,Ũs共␰ ;⍀兲 = Ũr共␰ ;⍀兲Log ␰ − Log 共1 + ␰兲冋

+ r2 −are determined by thefollowing consideration. For ⍀ 苸 C \ ␴c, the domain ofṼ共x , ⍀兲 can be extended to (28) by the analytic continuation.
By imposing the boundary condition
Ṽ共− L1,⍀兲 = 0,

共X 苸 C兲,

we can rewrite the homogeneous solutions in ⌫+共⍀兲 asŨr共␰ ;⍀兲 = 1 + r2

Ṽ共x,⍀兲 = Cr±共⍀兲Ṽr共␰ ;⍀兲 + Cs±共⍀兲Ṽs共␰ ;⍀兲

X2 X3
+
− ¯
2
3

=Log

k2 2 5k2 3␰ −␰ − ...
4
108

册

共54兲

x − X++ O共⍀兲,x − X−

共55兲

where ␰ = 共x − X+兲 / r, and O共⍀兲 represents the terms that converge to zero uniformly in ⌫+共⍀兲 for the limit of ⍀ → 0 (orr → 0). The function (55) describes how two logarithmic singularities collide in this limit.
At the same time, for ⍀ → 0, the inhomogeneous term
⌽共x , ⍀兲 diverges in proportion to 1 / ⍀ unless ␺0 = 0. By multiplying both sides of (33) by ⍀, we obtain␰2⌬␰共⍀Ũ兲 + ␰=−

2␰ ± 1
⳵␰共⍀Ũ兲
␰±1

␥1 − ␰
␰
␺0 + O共r兲.
␰ ± 1 ␣共␥2 − ␰兲3

共56兲

After some considerations, we find that␺0
+ O共r兲
␤共␥2 − ␰兲⍀Ũ p共␰ ;⍀兲 =共57兲

is a particular solution. It follows that⍀Ṽ p共␰ ;⍀兲 =␣
␺ + O共r兲 in ⌫±共⍀兲.
␤ 0

共58兲

From these results, we successfully extract the singularity of Ṽ共x , ⍀兲 in the neighborhood of ⍀ = 0. The generalsolution near x = X+ is represented by⍀Ṽ共x,⍀兲 = Cr+共⍀兲共⍀ − ␣x兲 + Cs+共⍀兲共⍀ − ␣x兲Log+

x − X+
x − X−

␣␺ + O共⍀兲 in ⌫+共⍀兲.
␤ 0

共59兲

Substituting this into (26), we get⍀B̃共x,⍀兲 = − Cr+共⍀兲␤x − Cs+共⍀兲␤x Log+ O共⍀兲 in ⌫+共⍀兲.

x − X++ ␺0
x − X−
共60兲

These Laplace-transformed variables Ṽ共x , ⍀兲 and B̃共x , ⍀兲have the same singularity and include 1 / ⍀ which must be

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012107-7

Phys. Plasmas 12, 012107 (2005)

Resonance between continuous spectra…

FIG. 2. In the case of ␤ ⬎ 兩␣兩 ⬎ 0, X+ and X− approach the x axis from,respectively, the upper and lower half-planes when ⍀ → + i0.

distinguished from an isolated pole in that it has the logarithmic singularities in the numerator. This complicated singularity at ⍀ = 0 is closely related to the magnitude relation of␣ and ␤.
If the magnetic shear is stronger than the flow shear
共兩␣兩 ⬍ 兩␤兩兲, the two singular points X+ and X− approach theorigin from opposite sides of the x = Re共X兲 axis for the limitof ⍀ → ± i0 (see Fig. 2). Let us assume ␤ ⬎ 0 (the case of␤ ⬍ 0 can be discussed in the same manner). The definition
X±共⍀兲 = ⍀ / 共␣ ± ␤兲 implies that the limit of ⍀ → ± i0 corresponds to X+ → ± i0 and X− → ⫿ i0. Then we get关⍀Ṽ共x,⍀兲兴⍀→±i0x ⫿ i0 ␣+ ␺0= − Cr+共±i0兲␣x − Cs+共±i0兲␣xLog

␤

x ± i0

FIG. 3. In the case of ␣ ⬎ ␤ ⬎ 0, X+ and X− approach the x axis from thesame direction when ⍀ → + i0.

g共x兲: =

冦sinh共− kL1 − kx兲
共− L1 艋 x 艋 0兲,
sinh共− kL1兲sinh共kL2 − kx兲
sinh共kL2兲

共0 ⬍ x 艋 L2兲.

Similarly, we obtain
关⍀B̃共x,⍀兲兴⍀→±i0 = ␺0g共x兲.

共66兲

By the inverse Laplace transform, the asymptotic behavior
共t → ⬁兲 is represented byṽx共x,t兲 →共61兲␣b̃ 共0,0兲g共x兲,
␤ x

共67兲

b̃x共x,t兲 → b̃x共0,0兲g共x兲,=− Cr+共±i0兲␣x ± Cs+共±i0兲␣x2␲iY共− x兲 +

␣
␺ .
␤ 0

共62兲

It is important to note that this equality is valid only in theinfinitely small region ⌫+共±i0兲. The whole solution in关−L1 , L2兴 will be attained by the analytic continuation whichis always possible in the domain (28). By multiplying theequations in (26) by ⍀ and taking the limit of ⍀ → 0, we
note that⌬关⍀Ṽ共x,⍀兲兴⍀→±i0 = 0in 关− L1,L2兴 \ 兵0其共63兲

must be satisfied except in the case of x = 0. The solutions of(63), a linear combination of ekx and e−kx, can be connectedwith the local solution (62) at x = 0, which determines
Cr+共±i0兲 and Cs+共±i0兲. Taking the boundary condition into
account, we obtain

where

共64兲

共68兲

where we used ␺0 = ib̃x共0 , 0兲. The convergence speed is characterized by ⬀1 / t according to the result in the preceding
section. We conclude that ṽx共x , t兲 and b̃x共x , t兲 converge tofunctions that have a derivative jump at x = 0. Especially, thevalue of b̃x共x , t兲 at x = 0 remains constant, which can be confirmed by viewing (26) only at x = 0. Even if we consider thecase of ␣ = 0 (no flow shear), the above result is applicable
and we obtain ṽx共x , t兲 → 0 instead of (67).
When the flow shear exceeds the magnetic shear 共兩␣兩⬎ 兩␤兩兲, the singularities of (59) and (60) change drastically.
Since the signs of Im共X+兲 and Im共X−兲 are the same, the two
singular points approach the origin from the same side of thex axis. For example, in the case of ␣ ⬎ ␤ ⬎ 0, both X+ and X−
approach from the upper half-plane for ⍀ → i0, and from thelower half-plane for ⍀ → −i0 (see Fig. 3). Then one of thetwo linearly independent solutions in (59) disappears due toLog␣关⍀Ṽ共x,⍀兲兴⍀→±i0 = ␺0g共x兲,␤共65兲

x − X+x ⫿ i0⬅0− → Logx−Xx ⫿ i0共⍀ → ± i0兲.

共69兲This situation must be avoided by the divergence of the coefficient Cs+共⍀兲 ⬀ 1 / ⍀, which recovers a linearly independent
solution. It follows that

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012107-8

Phys. Plasmas 12, 012107 (2005)

Hirota, Tatsuno, and Yoshida关⍀Ṽ共x,⍀兲兴⍀→±i0 = − Cr+共±i0兲␣x − C1共±i0兲␣关␹共x兲 − 1兴
+

␣
␺
␤ 0

in ⌫+共±i0兲,

共70兲

where we set C1共⍀兲 = rCs+共⍀兲 and␹共x兲: =

再S±共x,t兲 ⬃1 共x = 0兲,共71兲

0 共x ⫽ 0兲.

The continuation between the solution of (63) and (70) results in关⍀Ṽ共x,⍀兲兴⍀→±i0 =␣␺ ␹共x兲.
␤ 0

共72兲

Therefore, the inverse Laplace transform gives, for t → ⬁,ṽx共x,t兲 →␣b̃ 共0,0兲␹共x兲,
␤ x

共73兲

b̃x共x,t兲 → b̃x共0,0兲␹共x兲.

共74兲

We conclude that ṽx共x , t兲 and b̃x共x , t兲 converge to zero almost
everywhere.
E. Algebraic growths of slave variables

Finally, the slave equations are solved by using the resultof Ṽ共x , ⍀兲. The Laplace transform of M± = ṽx ⫿ b̃x is givenby− 共␣ ⫿ ␤兲共x − X⫿兲Ṽ共x,⍀兲 ⫿ ␺共x兲
. 共75兲
L关M±兴共x,⍀兲 =⍀ − ␣xWe note that either of the two logarithmic singularities,
Log共x − X+兲 or Log共x − X−兲, of Ṽ共x , ⍀兲 is smoothed by themultiplication of 共x − X⫿兲. Plugging this into the Laplacetransformed slave equations, we obtain
L关S±兴 = −

冋

␫⫿共x兲
␺共x兲
Ṽ共x,⍀兲 ±⍀ − ␣x⍀ − 共␣ ± ␤兲x+i

S±共x,0兲⍀ − 共␣ ± ␤兲x

册

共76兲

.

The inverse Laplace transform of the last term on theright-hand side yields the solution
S±共x,0兲e−i共␣±␤兲xt ,共77兲

which shows the phase mixing due to the Alfvén continuum
and always exists even if the master variables are initiallyzero.
The first term in (76) represents the response to the master variables. The singularity of this term at ⍀ = ␻ 苸 ␴c \ 兵0其
does not yield algebraic growth of S±. However, there is astrong singularity at ⍀ = 0, for the substitution of (59) leads
toL关S±兴 = −冋册x − X+
␫⫿共x兲 +
␺0
Cr 共⍀兲 + Cs+共⍀兲Log
− +
⍀
x−X
␤共X± − x兲+ O共⍀兲 + i

S±共x,0兲⍀ − 共␣ ± ␤兲x

For ␫⫿共0兲 ⫽ 0 and ␺0 = ib̃x共0 , 0兲 ⫽ 0, the singularity of
1 / ⍀共X± − x兲 brings about a localized algebraic growth at x
= 0. The most dominant asymptotic behavior is, therefore,
estimated byin ⌫+共⍀兲.

共78兲

1 − e−i共␣±␤兲xt␫⫿共0兲b̃x共0,0兲.
␤x

共79兲

By going back to w̃x = 共S+ + S−兲 / 2 and j̃x = 共S− − S+兲 / 2, we
can conclude that w̃x and j̃x increase remarkably near x = 0.
Given that we supposed k · B共0兲 = 0, the parallel components,ṽ储 and b̃储, with respect to B increase in proportion to t, as
follows:ṽ储共0,t兲 = w̃x共0,t兲/ik ⬀ t,

共80兲b̃储共0,t兲 = j̃x共0,t兲/ik ⬀ t.

共81兲If there is no shear flow 共␣ = 0兲, the growths of Re共w̃x兲 andIm共j̃ x兲 vanish and only the parallel motion ṽ储 increases inproportion to t.
By viewing the generator of (17), we note that, among
the four Alfvén waves, M+ (or M−) acts on S− (or S+) that
propagates in the opposite direction. That is why the resonantinteraction occurs only at the zero frequency ␻ = 0, while thefour continuous spectra degenerate in the region ␴+c 艚 ␴−c .
IV. SUMMARY

The system of linearized equations governing the fluctuations in ideal MHD has a non-self-adjoint generator withessential singularities. The resonant interaction among thefour Alfvèn continuous spectra causes an algebraic instability
localized on the rational surface x = xr where k · B共xr兲 = 0. The
initial fluctuation of magnetic field on the rational surface,i.e., b̃x共xr , 0兲, plays an essential role, which has been assumed to be zero in the stability analysis of the Lagrange
displacement [see (4)].
The physical mechanism of this instability can be understood as follows. If we have an ambient magnetic shear andno flow, the fluctuation on the rational surface does not inflect the magnetic field line and remains temporally constantbecause no force acts on it. In the equation of motion, theelectromagnetic force produced by this fluctuation b̃x共xr , 0兲and the ambient current constantly accelerates the parallel
motion ṽ储共xr , t兲 with respect to B, or, equivalently, the ambient pressure gradient along the magnetic field accelerates it,which yields the algebraic growth, ṽ储共xr , t兲 ⬀ t. In order toavoid this algebraic instability, it is required that the condition 关B⬘共xr兲 ⫻ k兴 · ex = 0 must be satisfied for arbitrary wave
number k, which is equivalent to the condition

冉 冊

兩B兩2 ⬘
共xr兲 = − P⬘共xr兲 = 0.
2

共82兲

Therefore, the pressure gradient on the rational surface mustbe zero.
In the presence of flow, this algebraic behavior undergoes a Doppler shift, ṽ储共xr , t兲 ⬀ te−i␻rt, where ␻r = k · V共xr兲.
The advection of fluctuations by mean shear flow yields the

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012107-9

Phys. Plasmas 12, 012107 (2005)

Resonance between continuous spectra…

Laplace transform, complicated singularities far from simple
poles appear in the Dunford integral path. The translationfrom the initial value problem into the normal modes analysis is generally difficult because it requires the spectral resolution of a non-self-adjoint operator, and we must generalize
the notion of nilpotent for degenerate continuous spectra. In
our model, the algebraic behavior of the solution arose fromthe coalescence of Alfvén singularities. The rigorous spectral
theory for this non-self-adjoint system will be discussed elsewhere.
FIG. 4. Contour plot of b̃x共x , t兲eikyy at t = 40 with the parameters ␣ = 0.5, ␤= 1, ky = 1, and 关−L1 , L2兴 = 关−1 , 1兴. This is a typical numerical solution of theinitial value problem (16).

algebraic growth of b̃储共xr , t兲 ⬀ t in addition to ṽ储共xr , t兲 ⬀ t. The
algebraic instability we have shown can be more important
than that of Lau and Liu,17,18 for the growth of the latterinstability is transient and saturated in a finite time.
Asymptotic behavior of the master variables, ṽx共x , t兲 andb̃x共x , t兲, changes discontinuously when the flow shear 共␣兲
exceeds the magnetic shear 共␤兲. It is interesting to note thatfor large Bz共=const兲 and small kz, the master variables are,respectively, similar to the stream function and the flux function in the reduced MHD theory (see Appendix C). For weakflow shear 共兩␣兩 ⬍ 兩␤兩兲, the asymptotic behavior given by (67)and (68) represents the generation of a magnetic island as
shown in Fig. 4. On the other hand, for strong flow shear
共兩␣兩 ⬎ 兩␤兩兲, the asymptotic behavior makes the transition to(73) and (74), where the stretching effect of shear flow destroys the magnetic island as shown in Fig. 5. The ratio␤2 / ␣2 is called the magnetic Richardson number, which is
known as an important parameter also in the normal modes
analysis.11 In real plasmas, such a time evolution into a singular structure is avoided by the dissipation effect. Our workis, however, useful for understanding the universal behaviorof fluctuation; the generation of a magnetic island and thestretching effect of shear flow. We clarified that the structureof the magnetic island is closely related to the Alfvén continuous spectrum and is identified as the coalescence of two
Alfvén singularities.
From a mathematical point of view, if the initial value
problem of a non-self-adjoint system is solved by the

ACKNOWLEDGMENTS

This work was supported by Research Fellowships of theJapan Society for the Promotion of Science (JSPS) for Young
Scientists and Grant-in-Aid for Scientific Research from theJapanese Ministry of Education, Science and Culture No.
14102033.
APPENDIX A: NON-SELF-ADJOINTNESSOF EVOLUTION EQUATION

In general, let us write a linear system in the form of anevolution equation: i⳵t f = Kf where f denotes a variable andK a linear operator.
When the space of variables has a finite dimension n,
and K is a matrix, the solution is definitely given by thelinear algebra as follows. A matrix is called semisimple if it
can be transformed into a diagonal matrix,

P−1KP =

冢

␻1
␻2


␻n

冣

共A1兲

,

by using a set of eigenvectors P = 共v1v2 ¯ vn兲, where␻1 , ␻2 , . . . , ␻n may include repeated eigenvalues. A selfadjoint matrix is always semisimple. For a semisimple generator K, the solution is represented by a linear combinationof the exponential behavior v je−i␻ jt.
A general matrix does not assume the diagonalized form
(A1), while it can be always cast into a Jordan canonicalform,19

P−1KP =

冢␻ 1I + N
␻ 2I + N
␻ mI + N

冣

,

共A2兲

where generally m 艋 n. The ␻ jI + N appearing on the diagonalline denotes a n j ⫻ n j, block,␻ jI + N =FIG. 5. Contour plot of b̃x共x , t兲eikyy at t = 40 with the parameters ␣ = 1, ␤
= 0.5, ky = 1, and 关−L1 , L2兴 = 关−1 , 1兴.

冢

␻j 1
 
␻j 1
␻j

冣

,

共A3兲

where I denotes unit matrix and N is called the nilpotent
matrix. In other words, a general matrix can be decomposed
into a semisimple part and a nilpotent part, and the diagonalization fails due to the latter. In the presence of the n j ⫻ n j

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012107-10

Phys. Plasmas 12, 012107 (2005)

Hirota, Tatsuno, and Yoshidablock, corresponding eigenvectors show algebraic behavior
such as tke−i␻ jt 共k = 0 , 1 , 2 , . . . , n j − 1兲. Physically, the algebraic growth of amplitudes implies the resonance of oscillators with a common frequency. A nilpotent matrix is, thus,
the mathematical representation of the resonant interactions
in degenerate (common eigenvalue) modes.
In the case of infinite dimensional (functional) space, theVon Neumann’s theorem10 enables spectral resolution of any
self-adjoint operator K, where we may observe a continuous
spectrum in addition to point spectra. However, we do nothave a general spectral theory for non-self-adjoint operatorsin functional space. The resonant interaction among degenerate continuous spectra yields much more spatially and temporally complicated behavior than degenerate point spectra.2
In plasma physics, inhomogeneity of equilibrium causes
a continuous spectrum (e.g., shear flow,20 magnetic shear,7
gradient of plasma density,5,6 and so on). Then, in the normal
modes analysis, the eigenvalue problem becomes a differential equation and the continuous spectrum is related to a singular point. We can formally obtain singular eigenfunctions
corresponding to the continuous spectrum (for example, Van
Kampen mode21). Although Case22 and Sedláček6 discussed
the equivalence of two approaches, the normal modes analysis and the initial value problem, the continuous spectrum
they considered seems to be semisimple. In this paper, we
solve an initial value problem by using the Laplace transform; the normal modes analysis is not well defined for degenerate continuous spectra that might conceal a nilpotent
part.

FIG. 6. Translation from (a) the Laplace transform (the initial value problem) into (b) the Fourier transform (the spectral resolution).

冕

f共x兲dx: = −f共x兲: = lim 关F共x + i⑀兲 − F共x − i⑀兲兴

共B1兲=F共x + i0兲 − F共x − i0兲,

共B2兲

⑀→+0

where F共z兲 is holomorphic everywhere in U except on D andcalled defining function. For example, the delta function ␦共x兲and the Heaviside function Y共x兲 are defined by F共z兲
= −1 / 共2␲iz兲 and F共z兲 = −Log共−z兲 / 共2␲i兲, respectively. The
derivative and integral of f共x兲 are calculated in terms of thedefining function,f ⬘共x兲: = F⬘共x + i0兲 − F⬘共x − i0兲,

共B3兲

共B4兲

F共z兲dz,where the integral path C共D兲 encircles D counterclockwise.
Now we consider a general evolution equation i⳵t f = Kf
as in Appendix A. For simplicity, let K has a bounded continuous spectrum ␴c, on the real axis. The solution is represented by the Dunford integral (or the inverse Laplace transform) which encircles ␴c [see Fig. 6(a)],APPENDIX B: HYPERFUNCTION THEORY

Generalized functions (such as delta functions) are defined by two approaches; Schwartz’s distribution theory and
Sato’s hyperfunction theory. We invoke the latter in this paper because it clarifies the relation between the Laplacetransform and the Fourier transform,3 which enables us toanalyze the continuous spectrum rigorously.
Let B共D兲 be the set of all hyperfunctions on the domain
D 傺 R. We introduce a region U 傺 C in the complex planesuch that D 傺 U and denote the set of all holomorphic function on U by O共U兲. Each hyperfunction in B共D兲 is uniquely
defined by a element of O共U \ D兲 / O共U兲, i.e., B共D兲⯝ O共U \ D兲 / O共U兲. Formally, we may define a hyperfunction
f共x兲 苸 B共D兲 by taking the difference of the boundary values
of F共z兲 苸 O共U \ D兲 like

冖

C共D兲

D

=−

冕

iR共⍀兲f共0兲e−i⍀t d⍀,冖F̂共⍀兲e−i⍀t d⍀,

1
2␲

f共t兲 = −

1
2␲

C共␴c兲

⍀ 苸 C,

C共␴c兲

共B5兲

共B6兲

where R共⍀兲 = 共⍀ − K兲−1 denotes the resolvent operator andF̂共⍀兲 = iR共⍀兲f共0兲 the Laplace transform of f共t兲. By regarding F̂共⍀兲 / 2␲ as a defining function, we obtain a hyperfunction f̂共␻兲 苸 B共␴c兲 which satisfies
f共t兲 =

冕␴cf̂共␻兲e−i␻t d␻ .

共B7兲In other words, we deformed C共␴c兲 into the vicinity of ␴c[Fig. 6(b)]. The integrand f̂共␻兲 is the Fourier transform off共t兲 and is expected to represent singular eigenfunction corresponding to the continuous spectrum. Since the spectral
theory is not established if K is non-self-adjoint, we mustderive f̂共␻兲 rigorously as stated above and the eigenvalue
problem 共␻ − K兲f̂共␻兲 = 0 is not valid.
APPENDIX C: NORMAL VELOCITY AND NORMALVORTICITY

The normal velocity ṽx and the normal vorticity w̃x arecommonly used in fluid mechanics to represent three dimensional incompressible fluctuations with only one independent

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012107-11

Phys. Plasmas 12, 012107 (2005)

Resonance between continuous spectra…

variable x.13 The definition of w̃x = i共kyṽz − kzṽy兲 and the condition ⵜ · ṽ = 0 allow us to calculate the other velocity components,ṽy = iky⳵xṽx + kzw̃x,k2ṽz = ikz⳵xṽx − kyw̃x
.
k2

共C1兲

These variables ṽx and w̃x recover the Clebsch representationin the following special cases. If kz = 0, we obtain1

ⵜ ṽx ⫻ ez + w̃xez
ṽ =
,
iky

共C2兲

and if ky = 0,ṽ = −

ⵜ ṽx ⫻ ey + w̃xey
,
ikz

共C3兲

where ṽx parallels the stream function.
Since the magnetic field is also incompressible, the same
representation is applicable to b̃.
APPENDIX D: APPARENT SINGULARITY

If we eliminate Ṽ from (26), we obtain关共⍀ − ␣x兲2 − ␤2x2兴⌬

冉 冊

冉 冊

B̃
B̃
− 关2␣共⍀ − ␣x兲 + 2␤2x兴⳵x
␤x
␤x=− ⌬␾ + 共⍀ − ␣x兲⌬

冉 冊
␺
␤x

obtain Ṽ, which must also be holomorphic at 兵共x , ⍀兲 ; ⍀− ␣x = 0其. It follows that the condition ⍀ − ␣x = 0 in (27) does
not yield singularity, which is so called the apparent
singularity.16 We can confirm this fact by applying the Frobenius method to (27) near x = ⍀ / ␣.
In the same manner, we can find that the point x = 0 is
also an apparent singularity of (D1).

共D1兲

instead of (27). In this equation, the point 兵共x , ⍀兲 ; ⍀ − ␣x
= 0其 is not singular, and as a consequence the solution B̃ must
be holomorphic there. By substituting this B̃ into (26), we

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