c 2003 Cambridge University Press
J. Plasma Physics (2003), vol. 69, part 5, pp. 397–412. 

397

DOI: 10.1017/S0022377803002241 Printed in the United Kingdom

Degenerate continuous spectra producing
localized secular instability – an example
in a non-neutral plasma
M. H I R O T A, T. T A T S U N O and Z. Y O S H I D A
Graduate School of Frontier Sciences, University of Tokyo, Tokyo 113-0033, Japan(hirota@plasma.q.t.u-tokyo.ac.jp)(Received 3 June 2002 and in revised form 30 November 2002)

Abstract. Fluctuations in ambient shear ﬂow exhibit interesting transient phenomena. Shear ﬂow produces not only Kelvin–Helmholtz modes (global exponential instabilities represented by point spectra) but also local algebraic instabilities
associated with multiple continuous spectra. Since the generating operator is nonHermitian, the orthogonality of eigenmodes is broken, and unresolvable mode couplings (resonances) bring about secular behavior (algebraic instability). We analyze
electrostatic ﬂuctuations in a magnetized non-neutral (single species) plasma where
the electrostatic potential parallels the stream function. This secular behavior isreproduced by solving the initial value problem with a renormalization method.

1. Introduction
Transient phenomena in a non-Hermitian system are far richer than those in a
Hermitian system. The notion of a ‘mode’ is no longer essential when we cannot
reduce the dynamical system to a complete set of independent (orthogonal) integrable dynamics of modes. The aim of this paper is to explore, with invoking
an appropriate renormalization technique, the time-asymptotic behavior of secular(algebraic) modes associated with degenerate (frequency overlapping) continuous
spectra. A shear ﬂow brings about non-Hermitian properties, and also produces anessential singularity generating a continuous spectrum [1]. We consider a simple
model of a magnetized non-neutral plasma, where the self-electric ﬁeld drives a
ﬂow in the direction perpendicular to the magnetic ﬁeld. A spatially localized
electrostatic instability [2] is the subject of our analysis.
Generating an ‘exponential function’ of an operator is the central problem of
linear theories. The behavior of such an exponential function, however, can be very
different from the elementary exponential function of numbers. The phase mixing
damping (well-known as the Landau damping of Langmuir waves [3, 4]) is oneof the most striking observations of complex behavior of exponential functionsin inﬁnite-dimensional function spaces. We must include continuous spectra, in
addition to point spectra (eigenvalues), to complete the spectral resolution of ageneral Hermitian operator, which describes damping (not necessarily exponential)
even though all spectra of a Hermitian operator are, by deﬁnition, real. The general
spectral resolution theory in function spaces (the Von Neumann theorem), however,applies only for Hermitian operators. The exponential function of a non-Hermitian

398

M. Hirota, T. Tatsuno and Z. Yoshida

operator may exhibit much more complicated evolution, because couplings of nonorthogonal modes occur [5–7].
A simple example of ﬁnite-dimensional dynamics reveals the essential nature ofsuch mode couplings. A linear map H (which may be non-Hermitian) on a ﬁnitedimensional vector space can be cast into a Jordan canonical form; by a regularmap P , we can transformP −1 HP = J1 +̇ J2 +̇ · · · +̇ Jν ,(1)where +̇ denotes the direct sum of linear maps, and each Jj is a Jordan block
corresponding to an eigenvalue λj ((λj I − Jj ) is a nilpotent map of class Nj , i.e.
(λj I − Jj )Nj = 0), which is represented by the Jordan matrix of dimension Nj ,


λj 1
0
0


 0 ... ... 0 .
(2)
Jj = 


..
..

.
. 1 
0
0 λj
When H is a normal map, all Nj are unity. Then, H can be diagonalized, and allmodes (eigenvectors) are decoupled. A Jordan block of dimension  2 represents‘unresolvable’ interactions among modes. Writingt2 (λj I − H)2−itH−itλj it(λj I−H)
−itλj
+ · · · , (3)=ee=e
I + it(λj I − H) −e2
we ﬁnd that e−itH acting on the generalized eigenspace (root vectors) belonging to
λj includes factors
e−itλj , te−itλj , . . . , tNj −1 e−itλj .
−itH

(4)

can describe ‘instabilities’
Therefore, even if every eigenvalue λj is real, e(growth of oscillations). The algebraic growth of amplitudes (the factors tp ) is called‘secularity’, which represents ‘resonant’ interactions of oscillators with a common
frequency (degenerate eigenvalue). This ‘resonance’ must be distinguished fromthe so-called ‘wave–particle resonance’ (Cherenkov resonance) which results in theLandau (phase mixing) damping. The latter is also related to continuous spectra,
while it is different from the process discussed here – wave–wave resonance among
overlapping continuous spectra.
The analysis is much more complicated when frequency overlapping occurs incontinuous spectra. As mentioned above, phase mixing due to continuous spectra
brings about complex transient phenomena that are strongly dependent upon initialconditions. The present study focuses on the degeneracy of spectra between pointand continuum, or continuum and continuum which has never been treated so far.
Our analysis will show an interesting relation between the order of secularity andthe number of degenerate (frequency-overlapping) spectra.
In Sec. 2, we begin by reviewing the general mathematical structure of continuous spectra associated with an ambient shear ﬂow in a system of various waves.
The convective derivatives produce essential singularity in the resolvent operator.
Inhomogeneity of the ﬂow also enables energy exchanges between ﬂuctuations andthe ﬂow. This interaction does not allow a Hamiltonian representation – the energy
is not conserved in the ﬂuctuation parts of the ﬁelds. In Sec. 3, we introduce a simple
model of a shear-ﬂow system. A single species (non-neutral) plasma produces a shear

Degenerate continuous spectra399

ﬂow to be conﬁned in a magnetic ﬁeld. The so-called diocotron instability is anelectrostatic analogue of the Kelvin–Helmholtz instability. When this exponential
instability is stable, the corresponding eigenvalue (point spectrum) occurs in theregion of the continuous spectrum (representing the shear ﬂow) on the real axis.
This ‘frequency overlapping’ causes degenerate spectra, and secular behavior stems
from there. However, this secular growth saturates because of the mixing effect. In
Sec. 4, we introduce another continuous spectrum caused by parallel (with respect
to the magnetic ﬁeld) motion of the plasma. The amplitude of the resonant ‘local’mode increases algebraically. We will solve the initial value problem and show thatan appropriate renormalization technique can predict the secular behavior of thelocal mode. We will compare the analytical result with our previous numericalsimulation [2]. In Appendix A, we will conﬁrm the validity of the renormalization
method by comparing it with the conventional Laplace transform approach.

2. Non-Hermitian property of shear flow
In this section, we review the effects of shear ﬂow and their mathematical representations. The simplest model of shear ﬂow is the vortex dynamics in inviscid
incompressible ﬂuids,(5)∂t w − ∇ × (v × w) = 0,
where v is the velocity and w = ∇ × v is the vorticity. In a two-dimensional space,
we may write v = ∇ψ(x, y) × ∇z, and rewrite (5) in the form of a Liouville equation,∂t w + {ψ, w} = 0,

(6)where w = −∆ψ is the z-component of w, and {ψ, w} = (∂y ψ)(∂x w) − (∂x ψ)(∂y w).
By decomposing the equilibrium (capital letters) and ﬂuctuation (tildes) parts, thelinearized equation becomes∂t w̃ + {Ψ, w̃} + {ψ̃, W } = 0.

(7)

In a stratiﬁed shear ﬂow V = (0, Vy (x), 0), we assume w̃(x, y, t) = w̃(x, t)e−iky y(ky is a good quantum number). Then, (7) reduces into the Rayleigh equation [8]−i∂t w̃ − ky Vy (x)w̃ − ky Vy (x)Gx w̃ = 0,(8)where, considering the whole space (without boundary),
Gx w̃ := −∆−1 w̃ =

1
2ky

∞
−∞

e−ky |x−ξ| w̃(ξ) dξ.

(9)

The third term on the left-hand side of (8) is the origin of the non-Hermitian
property of the problem. If Vy (x) changes the sign (i.e. the ﬂow Vy (x) has inﬂection
points), the generator of (8) cannot be a Hermitian operator in any deﬁnition ofHilbert space. Complex eigenvalues may occur, representing the Kelvin–Helmholtz(KH) instabilities. The second term on the left-hand side of (8), the convective
derivative, produces a continuous spectrum [1] (replacing ∂t by iω, the term ω −kVy (x) yields an essential singularity in the formal dispersion relation). The phasemixing effect due to the continuous spectrum, in addition to the non-Hermitian
property, causes a rather complex evolution of the ﬂuctuations. Possible instabilities
associated with the continuous spectrum (Sec. 1) are in marked contrast with the
KH instabilities – they are spatially localized, continuously changing the shape,
and temporally algebraic.

400

M. Hirota, T. Tatsuno and Z. Yoshida

A close cousin of the two-dimensional incompressible ﬂow are the electrostatic
perturbations in single-species plasma conﬁned by a homogeneous magnetic ﬁeld
B. The E × B drift velocity in the plane perpendicular to B can be written in theform −∇φ × B/B 2 . The Poisson equation implies that the vorticity −∆φ parallels
the density of the non-neutral plasma.
In a more general system, other physical variables may couple with the vorticity.
Let ξ be a state vector that is, in a plasma, a combination of the vorticity w, pressurep, density ρ, magnetic ﬁeld B and so on. We decompose ξ into the equilibrium Ξ andﬂuctuation ξ̃ parts. The transport equations (conservation laws) contain convective
derivatives [(v · ∇)ξ]. The linearized evolution equations, in two-dimensional spaces,
can be cast in the form of∂t ξ̃ + {Ψ, ξ̃} + {ψ̃, Ξ} = A ξ̃,(10)where A is a linear operator (including Ξ). The non-diagonal components of A
describe the couplings among different variables. If y is an ignorable coordinate ofΞ, (10) translates (using a good quantum number ky ) into−i∂t ξ̃ − ky Vy (x)ξ̃ − ky Ξ (x)Gx w̃ = −i A ξ̃.

(11)The operator [−i∂t − ky Vy (x)] acting on all variables yields multiple (overlapped)
continuous spectra. The aim of this paper is to analyze the secular behavior of
perturbations stemming from this overlapped continuous spectra.

3. Rayleigh equation and phase-mixing effect
As the simplest example of the two-dimensional, shear-ﬂow problem, we consider
the Rayleigh equation (8) describing electrostatic ﬂuctuations in a non-neutral(single species) plasma. In this model, we can observe the resonance between the
point spectrum and the continuous spectrum.
Let B = (0, 0, Bz ) be a homogeneous magnetic ﬁeld, which conﬁnes an electron
plasma within slab geometry. We normalize the variables by choosing the representative density, magnetic ﬁeld and length. We consider a simple ﬂat top density
N (x) =

1 |x|  1
0 |x| > 1,

(12)

which produces ambient shear ﬂow−1 x < −1(13)x−1 < x < 1
1
1 < x.
Let us Fourier-transform ﬂuctuations with respect to y and z. If kz 苷 0, the parallel (with respect to B) plasma oscillation couples with the perpendicular vortex
dynamics [2]. The determining equations are
Vy (x) =i∂t ñ + ky Vy ñ − ky Vy φ̃ + N kz ṽz = 0,i∂t ṽz + ky Vy ṽz =φ̃ = −Gx ñ := −

1
2k

∞
−∞

kz
φ̃,
s2

e−k|x−ξ| ñ(ξ) dξ,

(14)
(15)
(16)

401

Degenerate continuous spectra

ky2 + kz2 and s := ωp /ωc (with ωp = plasma frequency and ωc =where k :=cyclotron frequency). We assume s Ⰶ 1 (implying low density).
If kz = 0, (14) and (15) are decoupled. The KH instability predicted by thestandard Rayleigh equation (14) is the so-called diocotron instability [9, 10]. In
this section, we revisit the Rayleigh equation (assuming kz = 0) to highlight the
non-Hermitian property. Oblique modes (kz 苷 0) will be discussed in Sec. 4.
By (13), we observe that
Vy (x) = δ(x + 1) − δ(x − 1).

(17)

Therefore, the perturbed density (vorticity) must include delta functions,ñ(x, t) = a(t)δ(x + 1) + b(t)δ(x − 1) + f (x, t),(18)

where f (x, t) is the smooth part of the ﬂuctuation, while the delta functions represent the ‘surface waves’. Substituting (18) into the Rayleigh equation (14) (withkz = 0), we obtain  k x

0
0
y
f (x)
f (x)

1 −2ky  a .
−i∂t  a  =  ky G−1 − 12 (2ky − 1)
(19)

2e
1 −2ky1bb−ky G1−2e
2 (2ky − 1)
The coupling of a and b determines the two-point spectrum [8–11],
ω± = ± 12(2ky − 1)2 − e−4ky .

(20)

By equating the right-hand side of (20) to zero, we obtain a critical wave number
kc  0.639. For ky < kc , the frequencies ω± assume pure imaginary numbers,
representing the KH instability. The corresponding eigenfunctions are
 


0
fω± (x).
 aω±  = 
e−2ky
(21)
bω±
(2ky − 1) + 2ω±On the other hand, the continuous part f (x, t) receives phase mixing that is represented by the continuous spectrum σc = {ω ∈ R; −ky < ω < ky }. The corresponding
singular eigenfunction is formally written asfω (x) = δ(ω − ky x),
aω
bω



1
=
(ω − ω+ )(ω − ω− )
ω − 12 (2ky − 1)
1 −(ky +ω)
e

 2ky
.
×

 1
−(ky −ω)−e2ky− 21 e−2ky

1 −2ky2e
ω + 12 (2ky − 1)





(22)

A ﬂat top density ∂N/∂x = 0, as assumed in this paper, yields the real point
spectra (20) for ky  kc . Then, we fail to derive the singular eigenfunction withthe frequency ω = ω+ or ω = ω− in the expression of (22), and have to ﬁnd
a generalized singular eigenfunction (nilpotent), which constructs a Jordan block.
This Jordan block represents the ‘resonance’ in the overlapping spectra. Using (21),402

M. Hirota, T. Tatsuno and Z. Yoshida

the generalized eigenfunctions are calculated asfˆω± = ĉω± δ(ω± − ky x),

ĉω±
âω±




1 −(ky +ω± )
e
−
 2ky

=
1 −(ky −ω± )e2ky

ω± + 12 (2ky − 1)
1 −2ky
2e

−1





aω±
,
bω±

(23)

b̂ω± = 0,
where we have chosen b̂ω± = 0 for simplicity. Note that one may add (21) with themultiplication of an arbitrary constant to the expression (23).
In the analogy of linear algebra, one may expect a secular behavior of the KH
mode (21), such as teiω± t (see (4)). However, we do not observe secular behavior –an essential difference of inﬁnite dimensional Hilbert space from linear algebra.
This is due to the phase mixing in the continuous spectrum. Let us demonstrate
this fact by solving the initial value problem.
Since the evolution of f is independent of a and b, we readily obtain the generalsolution of f ,(24)
f (x, t) = f (x, 0)eiky xt .
The general solutions of a and b are given by the Duhamel formula,



a(t)
eiω+ t0−1 a(0)=TTb(t)0eiω− tb(0)eiω+ t0+T0eiω− tt0e−iω+ tiky (G−1 f )(t )
−1
×
dt ,
T


−iω− t
(G
f
)(t
)
−ik
0
e
y
1
0

(25)

where


aω+ aω−.
(26)
T=bω+ bω−If ω± ∈ R (stable KH modes), the ﬁrst term of the right-hand side of (25) describes
harmonic oscillation. The second term needs careful treatment.
We observe that∞1
(Gx f )(t) =e−ky |x−ξ| f (ξ, 0)eiky ξt dξ,(27)
2ky −∞∞=−∞

ˇ iξ̌t dξ,ˇg(x, ξ)e(28)

where ξˇ := ky ξ and1 −|ky x−ξ̌| ˇef (ξ/ky , 0).
(29)
2ky2
ˇ If |g(x, ξ)|ˇ is integrable
Equation (28) is the Fourier transform (ξˇ → t) of g(x, ξ).
ˇwith respect to ξ, the Riemann–Lebesgue theorem shows that limt→±∞ Gx f = 0,
implying the phase mixing damping.
ˇ :=g(x, ξ)403

Degenerate continuous spectra

Because the integrants G±1 f (t ) are decreasing functions of t , they do not include
stationary oscillations eiω± t . Therefore, the second term of (25) does not diverge
at the limit of t → ∞. We, thus, ﬁnd that the resonance between the continuous
spectrum and the point spectrum does not result in an algebraic instability – thephase mixing overcomes the resonant ampliﬁcation.
In the present work, we assume a ﬂat top density. If the equilibrium density has
a gradient, the real point spectra convert either to real continuous spectra or to
unstable (Im(ω) > 0) complex point spectra, depending on the sign of the density
gradient [9, 12]. Briggs et al. [12] have pointed out that a positive density gradientgives rise to the exponential growth of the diocotron wave, i.e. a complex point
spectrum is created. This exponential instability is distinguished from the present
secular behavior caused by frequency overlapping among the real point spectrum
and continuous spectrum. On the other hand, the point spectrum disappears in
the presence of a negative density gradient. Potential ﬂuctuations undergo phasemixing and damping.

4. Coupled continuous spectra and algebraic instabilities
In Sec. 3, we have considered the Rayleigh equation (14) governing the perturbations in the plane perpendicular to the ambient magnetic ﬁeld. In this section,
we consider the parallel motion (15) and the coupling between the parallel andperpendicular modes. If kz = 0 (decoupled), (15) becomesi∂t ṽz + ky Vy ṽz = 0,(30)which has a continuous spectrum σc = {ω ∈ R; −ky < ω < ky } due to the shear
ﬂow Vy (x) = x (−1 < x < 1). The corresponding singular eigenfunction isṽzω (x) = δ(ω − ky x).

(31)This continuous spectrum overlaps with that of perpendicular motion.
Since a ﬁnite kz yields a coupling of (14) and (15), the resonance of two continuous
spectra occurs. We rewrite (14)–(16) in matrix form,




f (x)
f (x)


∂ ṽz (x)
 = A ṽz (x) ,
−i 
(32)
 a 
∂t  a bbwhere the generator is


ky x
 kz
 2 Gx sA := 
 ky G−1−ky G1

kz

0

0



kz −k|x−1| kz −k|x+1|

e
e
2

2ks
2ks2

(33)
ky −2k ky.
e0− (2k − 1)

2k
2k


ky
ky −2k(2k − 1)
0
− e
2k
2k
Spectral analysis [13] shows that exponential instabilities (ω with an imaginarypart) are removed if kz 苷 0 and s2 is sufﬁciently small. However, numerical solutionsof the initial value problem show that oscillations grow in proportion to t near the‘resonant surfaces’ (x = µ± := ω± /ky ) when ky Ⰷ kz ∼ O(s) [2] (see Sec. 4.3).
ky x

404

M. Hirota, T. Tatsuno and Z. Yoshida

Since the phase mixing diminishes the magnitude of the convolution integral
Gx f , the ﬁrst approximation one may consider is to neglect the Gx f terms in thegenerator A [2]. Then, (32) gives a (3 × 3) Jordan block. This approximation yields
parabolical growth (∝t2 ) in the amplitude of f (µ+ ), which does not agree with thenumerical result (∝t).
This discrepancy is due to the neglect of Gx f . Because limt→∞ f (x, t) is singular,
we need a careful treatment of the integral terms Gx f . In the following subsections,
we shall solve the initial value problem and study the asymptotic behavior invoking
a renormalization method.
4.1. Renormalized perturbation theory (multiple-scale analysis)
We assume = kz = O(s2 ) is a small parameter, and consider two time scales,τ = t,

T = t.

(34)

The independent variables are expanded asf (x, t) =∞nfn (x, τ, T ),

(35)

n

ṽzn (x, τ, T ),

(36)

n

an (τ, T ),

(37)

n

bn (τ, T ).

(38)

n=0

ṽz (x, t) =

∞

n=0

a(t) =∞
n=0b(t) =∞
n=0

Using the relation
∂
∂
∂
=
+
,
∂t
∂τ
∂T
(32) reads

(39)








f (x)
f (x)
f (x)





∂ 
ṽz (x) = A ṽz (x) + i ∂ ṽz (x).
−i
(40)





a
a
a 
∂τ
∂T
b
b
b
Substituting (35)–(38) into (40), we can solve these equations successively in theorder of 0 , 1 , 2 , . . . .
To simplify the analysis, we consider a speciﬁc initial condition
f (x, 0) = 0,ṽz (x, 0) = 0,a(0) = e−2k ,where κ+ is a constant to be determined later in (46).
f0 obeys∂f0= ky xf0 ,−i∂τwhich yields f0 (x, τ, T ) ≡ 0. Then, a0 and b0 satisfy


∂ a0
ky
−2k + 1 e−2k
−i
=
−e−2k
2k − 1∂τ b0
2kb(0) = κ+ ,

(41)

(42)

a0
b0


.

(43)

405

Degenerate continuous spectra
The solution isa0 (τ, T ) = A(T )e−2k eiω+ τ ,b0 (τ, T ) = A(T )κ+ eiω+ τ ,

(44)

where
ω± := ±ky
2k

(2k − 1)2 − e−4k ,κ± := (2k − 1) ±(45)

(2k − 1)2 − e−4k .

(46)

The amplitude A(T ) is a certain function satisfying A(0) = 1. The slow evolutionof A(T ) will be determined later.
The short-term equation of ṽz0 is
−i

∂ṽz0
= ky xṽz0 + A(T )q(x)eiω+ τ ,
∂τ

(47)

which gives
1 − e−iU (x)τ
,
U (x)

(48)


kz  −k(x+1) −2kee+ e−k(1−x) κ+ ,
2ks2

(49)

ṽz0 (x, τ, T ) = A(T )q(x)eiω+ τ
whereq(x) :=

U (x) := ω+ − ky x.

(50)

At the resonant surface (x = µ+ := ω+ /ky ), ṽz0 is secular (∝τ ). Using ṽz0 , we may
determine the short-term behavior of f1 by−i∂f1= ky xf1 + ṽz0 ,∂τwhich yields


iω+ τ

f1 (x, τ, T ) = A(T )q(x)e

1 − e−iU (x)τ

(51)



U (x) − iτ e−iU (x)τ
.
U (x)

(52)

This f1 is also secular (∝τ 2 ) at the resonant surface (x = µ+ ). As mentioned above,the estimate of the power of the secularity disagrees with the numerical observation.
The slow variable A(T ) will amend the long-term behavior.
We proceed to the equations of a1 and b1 :



∂ a1
ky −2k + 1 e−2k
a1
−i
=
b1
2k − 1∂τ b1
2k −e−2k∂ a0
G−1 f1
+ ky
+i
.
(53)
−G1 f1∂T b0
By the transformation,
â1
b̂1


:=

e−2k
κ+

e−2k
κ−

−1


a1
.
b1

Equation (53) is diagonalized as




∂ â1
∂
â1
0
iω+
h (τ )−=+ A(T ) a0iω−hb (τ )b̂1
∂τ b̂1∂T(54)
A(T )eiω+ τ
,
0

(55)

406

M. Hirota, T. Tatsuno and Z. Yoshidawhereha (τ )hb (τ )


:=

e−2k
κ+

iky
A(T )

e−2k
κ−

−1


G−1 f1.
−G1 f1(56)

Since f1 ∝ A(T ), ha and hb are independent of T . The equation of â1 reads∂â1dA iω+ τ= iω+ â1 + A(T )ha (τ ) −
e
.
∂τ
dT
If ha (τ ) includes an oscillation with frequency ω+ , i.e.
ha (τ ) = σeiω+ τ + · · ·(σ = const),

(57)

(58)

â1 becomes a secular solution. We may remove this divergence by imposing arenormalization group equation
dA
= 0,
(59)
σA(T ) −dT
which gives
A(T ) = eσT .
(60)
We have derived the renormalized solutionâ0 = eσT eiω+ τ ,(61)

σt iω+ t

(62)

e

=e

,

σkz t iω+ t

e
.
(63)
=e
If the real part of σkz is negative, this solution represents the damping of â.
4.2. Fourier expansion
Let us proceed to estimate the ‘Fourier coefﬁcient’ σ deﬁned by
1
σ := limt →∞ tBy (52) and (56), we obtainha (t) = −iCeiω+ t

×

1
−1

t

ha (t)e−iω+ t dt.

(64)

0

(2e−2k + κ− e−2kξ + κ+ e2kξ )U (ξ) − ite−iU (ξ)t
dξ,
U (ξ)
kz ky e−2k.
C := 2 2
4k s (κ+ − κ− )1 − e−iU (ξ)t



(65)
(66)

We may writeha (t)e−iω+ t = iC
=C

1
−1

t

(2e−2k + κ− e−2kξ + κ+ e2kξ )

2e−2k
ky

ζe−iU (ξ)ζ dζ dξ,0tei(−ω+ +ky )ζ − ei(−ω+ −ky )ζ dζtζ

e−2k+i(−ω+ +ky )ζ − e2k+i(−ω+ −ky )ζ
dζ−2k + iky ζζ

e2k+i(−ω+ +ky )ζ − e−2k+i(−ω+ −ky )ζ
dζ.
2k + iky ζ0+ iCκ−
0t+ iCκ+
0

(67)

407

Degenerate continuous spectra
Using the relations∞0∞cos(βx)π −|α||β|edx =,α2 + x22|α|

we observe, for large t,
t

Im

eiβx
dx
x
α + ix00

x sin(βx)πβ −|α||β|edx =α2 + x22|β|(∀α, β ∈ R)
(68)

sin(βt)
−
+α
β

α
β
+
|α| |β|

π −|α||β|
e
.
2

(69)

Using (69) in (67) yields, for large t,Re(ha (t)e−iω+ t )  C

1sin(−ω+ + ky )t(2e−2k + κ− e−2k + κ+ e2k )ky(−ω+ + ky )−C1sin(−ω+ − ky )t(2e−2k + κ− e2k + κ+ e−2k )ky(−ω+ − ky )

+C


2kπ 
κ− e−(2k/ky )ω+ − κ+ e(2k/ky )ω+ .
ky |ky |(70)

Substituting (70) into (64), we ﬁnd that the third term on the right-hand side of(70) yields the real part of σ:Re(σ) =

kz πe−2kκ− e−(2k/ky )ω+ − κ+ e(2k/ky )ω+ .
2
2ks |ky |(κ+ − κ− )(71)

In the stable regime of the KH modes (ky > kc ), (71) is simpliﬁed as
Re(σ)  −kz π
2ky2 s2 e

(<0).

(72)

We have derived the damping rate of â (see (63)):ν = −Re(σ)kz ,

(73)

kz2 π
.
2ky2 s2 e(74)â(t) = e(iω+ −ν)t .

(75)

The asymptotic form of â is

4.3. Comparison with numerical solution
In this subsection, we compare the analytical result with numerical simulation. We
use the trapezoidal rule for spatial integration. Because of the continuous spectrum,a solution receives the mixing effect (see (24)). f (x, t) becomes strongly oscillatory
as t increases. The wave number with respect to x is of order ky t. For the range oft < T , we choose the mesh number per unit length (Nmesh ) to satisfy Nmesh Ⰷ ky T .
For numerical example, let us take ky = 1, kz = 10−3 and s = 10−2 . Then, (74)yields ν  0.005 78. This analytical estimate is compared with numerical simulation.
By substituting the initial condition (41), we observe the secularity at x = µ+ (seeFig. 1). The time evolution of a(t) shows damping which seems to be exponential(Fig. 2(a)). In Fig. 3, we compare the analytical ν (given by (74)) and the numericallyestimated damping rate (ky is the parameter).

408

M. Hirota, T. Tatsuno and Z. Yoshida400

t =100t =0~vz(x,t)

200
0

–200
– 400
–1

f (x,t)

25
20
15
10
5

– 0.5

0
x

0.5

1

0
x

0.5

1

t =100
t =0

0
–5
–10
–15
–20
–25
–1

– 0.5

Figure 1. Proﬁles of ṽz (x, t) and f (x, t) at t = 100. The parameters are ky = 1,
kz = 10−3 , s = 10−2 , and the initial condition is given by (41). Near the resonant surface(x = µ+ = 0.495), large oscillations appear.

By substituting A(T ) = e−νt into (47) and (51), we obtaine(iU (x)−ν)t − 1,
iU (x) − ν(iU (x)−ν)t(iU (x) − ν) + t
iky xt 1 − ef (x, t)  f1 = kz q(x)e.
iU (x) − νṽz (x, t)  ṽz0 = iq(x)eiky xt

(76)

(77)

We thus predict the saturation of ṽz (µ+ , t) and the growth of f (µ+ , t) in proportion
to t (see the dashed lines in Figs 2(b) and 2(c)). The remaining small error may beeliminated if we analyze the slow variations of ṽz0 and f1 .

5. Summary
In a non-Hermitian system, degenerate spectra (frequency overlapping) may cause
nilpotent, resulting in secular ampliﬁcation of even real frequency waves. If ω is adegenerate point spectrum (eigenvalue) of nullity n, the corresponding Jordan blockof dimension (n + 1) produces oscillations in the form of eiωt , teiωt , . . . , tn eiωt . Thiselementary result does not apply when we consider degenerate ‘continuous’ spectra.
We chose a single-species (non-neutral) plasma as a physical model for investigation.

409

Degenerate continuous spectra(a) 0.15
v =0.00578

0.1

a(t)

0.05
0
– 0.05– 0.1– 0.150(b) 1000

100

200

300
t

400

500

600

200

300
t

400

500

600

300
t

400

500

600

v =0.00578~
vz(µ+ ,t)

500
0
–500
–1000
0

f (µ+ ,t)

(c)

500
400
300
200
100
0
–100
–200
–300
–400
–500

100

v =0.00578

0

100

200

Figure 2. Time evolution of (a) the surface density ﬂuctuation a(t), (b) the parallel electron
velocity ṽz (x, t) at x = µ+ (resonant surface) and (c) the inner density ﬂuctuation f (x, t)
at x = µ+ . The parameters are ky = 1, kz = 10−3 , s = 10−2 , and the initial conditionis given by (41). Dashed lines denote the amplitudes of the analytical solutions (75)–(77),respectively.

The self-electric ﬁeld yields an ambient shear ﬂow that brings about continuous spectra (representing the convection of ﬂuctuations) as well as non-Hermitian
properties (interaction of ﬂuctuations and the ambient ﬂow). If a point spectrum
ω overlaps with other m continuous spectra (i.e. formal nullity is n = m + 1), we
obtain secularity of the form of eiωt , teiωt , . . . , tm eiωt . The diminution of the power

410

M. Hirota, T. Tatsuno and Z. Yoshida
0.012

Numerical Results
v=k z2Bπ/2k y2 s 2e

0.01
0.008
v 0.006
0.004
0.002
0

0.8

1

1.2 1.4 1.6 1.8
ky2

2.2 2.4 2.6

Figure 3. Damping rate of the surface wave for kz = 0.001, s = 0.01. The dotted line
denotes the plot of (74), and points are estimated by the ﬁtting of numerical results(Fig. 2(a)).

of t is due to the phase mixing in the continuous spectra. The resonance (frequency
overlapping) of a point spectrum and multiple continuous spectra is the cause of
secular behavior. This process may occur in many other shear-ﬂow systems.
The present analysis predicts an algebraic instability in a KH (diocotron) stablenon-neutral plasma. The instability stems from the coupling of perpendicular (withrespect to the magnetic ﬁeld) vortical motion and parallel plasma oscillation. The
ﬂuctuation is localized at the surface where the frequency (real) of the stablediocotron waves resonates with the ﬂow continuum.
Acknowledgements
This work is partially supported by a Grant-in-Aid from the Japanese Ministryof Education, Culture, Sports, Science and Technology, No. 14102033 andNo. 12780353.

Appendix A. Comparison with the Laplace transform approach
An alternative method to analyze the initial value problem is the use of the Laplacetransform. For the model of Sec. 4, we can construct a Green function to solve thetransformed equation. Let us write∞φ̃(x, t)e−pt dt.

(A 1)

ikz ṽz (x, 0)
kz2
if (x, 0)+.
φ̃p = −
s2 (ip + ky x)2
ip + ky x (ip + ky x)2(A 2)

φ̃p (x) =0

Equation (32) translates as
φ̃p − k 2 φ̃p +The boundary conditions read−(ip − ky )φ̃p (−1) + [ky + k(ip − ky )]φ̃p (−1) = ia(0),(A 3)−(ip + ky )φ̃p (1) + [ky − k(ip + ky )]φ̃p (1) = −ib(0),(A 4)where f (x, 0), ṽz (x, 0), a(0) and b(0) denote the initial values of the corresponding
variables. We assume the same initial values as (41). Then, the general solution of

Degenerate continuous spectra

411(A 2) is
1

1

(A 5)
φ̃p (x) = C1 χ 2 I−η (χ) + C2 χ 2 Iη (χ),
where χ := k(ip+ky x)/ky , η := 1/4 − (kz /sky )2 and I±η (χ) denotes the modiﬁed
Bessel function. Substituting (A 5) into the boundary conditions (A 3) and (A 4), we
can determine the coefﬁcients C1 and C2 :



1
C1
ia(0)
ϕ11 (p, η) ϕ12 (p, η)
,
(A 6)
=
C2
ky D(p, η) ϕ21 (p, η) ϕ22 (p, η) −ib(0)
where




1
∂  1
χ 2 Iη (χ) + (1 − χ)χ 2 Iη (χ)ϕ11 (p, η) = −χ
,
∂χ
x=1



1
∂  1
2
2
ϕ12 (p, η) = χχ Iη (χ) − (1 + χ)χ Iη (χ)
,
∂χ
x=−1

(A 7)
(A 8)

ϕ21 (p, η) = −ϕ11 (p, −η),(A 9)

ϕ22 (p, η) = −ϕ12 (p, −η),(A 10)

D(p, η) = ϕ11 (p, η)ϕ22 (p, η) − ϕ12 (p, η)ϕ21 (p, η).

(A 11)

If kz = 0 (i.e. η = 1/2), we ﬁnd D(p± , 1/2) = 0, where p = p± (= iω± ) are thepoles of φ̃p representing the KH modes. A small kz (we write η = 1/2 + δη) moves
the poles; p = p± + δp± . To calculate δp± , we need the analytical continuationof D(p, η) into the left-half plane of complex p, because D(p, η) is a multivalued
function when η 苷 1/2. We approximate D(p, η) by
D(p, η)  D(p± , 1/2) + δp±∂D∂D(p± , 1/2) + δη
(p± , 1/2).
∂p∂η

(A 12)

By setting D(p, η) = 0, we obtain∂D∂η (p± , 1/2)

δp± = −δη ∂D∂p (p± , 1/2).

(A 13)

After some manipulations, we ﬁnd, for ky Ⰷ kz ,Re(δp± )  −kz2 π.
2ky2 s2 e(A 14)

Therefore, we ﬁnd that both KH modes damp in the presence of a small kz . The
damping rate agrees with (74).
Since the modiﬁed Bessel function I±η (χ) is described by χ±η multiplied by1
an analytic function, the term with the strongest singularity in φ̃p (x) is χ 2 −η =(ip + ky x)−δη . By operating the Laplacian (∆ = ∂ 2 /∂x2 − k 2 ) on (A 5), the Laplacetransform of the density perturbation (ñp (x) = ∆φ̃p (x) may include a singularityof the form of (ip+ky x)−2−δη , the x-dependence of which represents the continuous
spectrum. The inverse Laplace transform, then, yields secular behavior [14]:ñ(x, t) ∼ t1+δη eiky xt(t → ∞).

(A 15)

412

M. Hirota, T. Tatsuno and Z. Yoshida

All other contributions from lower-order singularities yield slower growth that canbe neglected at large t. In (77), the small correction δη for the power of t was also
ignored.

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