PHYSICS OF PLASMASVOLUME 8, NUMBER 2

FEBRUARY 2001

Transient phenomena and secularity of linear interchange instabilities
with shear flows in homogeneous magnetic field plasmas
T. Tatsuno, F. Volponi, and Z. Yoshida
Graduate School of Frontier Sciences, The University of Tokyo, Tokyo 113-0033, Japan

共Received 14 August 2000; accepted 27 September 2000兲
Transient and secular behaviors of interchange fluctuations are analyzed in an ambient shear flow by
invoking Kelvin’s method of shearing modes. Because of its non-Hermitian property, complex
transient phenomena can occur in a shear flow system. The combined effect of shear flow mixing
and Alfvén wave propagation overcomes the instability driving force at sufficiently large time, anddamps all fluctuations of the magnetic flux. On the other hand, electrostatic perturbations can bedestabilized for sufficiently strong interchange drive. The time asymptotic behavior in each case isalgebraic 共nonexponential兲. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1336532兴

I. INTRODUCTION

interchange instabilities and its non-Hermitian mathematicalbackground, deriving the time asymptotic behavior by meansof Kelvin’s method.
Recently, Kelvin’s method has been applied to a varietyof linear shear flow problems.11–17 For neutral and magnetized fluids, many new fascinating phenomena were discovered; exchanges of energy between background flows andperturbation fields,13 shear flow induced coupling between
sound waves and internal waves and the excitation of beatwave,14 the asymptotic persistence due to the periodic energy
transfer for two-dimensional shear flows,15 and the emissionof magnetosonic waves by the stationary vortex perturbations.16 These results show that the modes, which are independent for static fluids, are no longer independent and thecoupling of these modes induces the observed energy transfer in the presence of the shear flow. The authors have also
applied this method to investigate the basic properties ofkink-type instabilities in the presence of a background shear
flow and obtained the result that the shear flow mixing always overcomes the kink driving at sufficiently large time.17
In this paper, we will first revisit Kelvin’s method fromthe viewpoint of the characteristics method 共Sec. II兲. We will
review the spectral theory focusing on the general mathematical concept of eigenmode in order to gain a better understanding of Kelvin’s method. In Sec. III, we will formulate the equations for the interchange instabilities. In Sec. IV,
we will derive the ordinary differential equation 共ODE兲 intime for the evolution of the amplitude of the interchange
instabilities by applying the analysis of shearing modes. In
Sec. V, by drawing the analogy with Newton’s equation it
will be shown that the solution to the above-mentioned ODE
for the flux function exhibits an asymptotic damped behavior
without any threshold of instability drive. We will also consider the electrostatic perturbations in Sec. VI. The solutionof the derived ODE for the stream function shows theasymptotic growth or decay of algebraic type depending on
the magnitude of instability drive. We will summarize theresult in Sec. VII. Moreover in the Appendix, we will also
show the difficulties encountered by including the magnetic
shear in the present formulation.

It is widely accepted that a shear flow yields stabilizing
effects on various fluctuations through convective deformations of disturbances.1,2 However, rigorous treatment of theshear flow effects encounters a fatal difficulty arising fromthe non-Hermitian 共non-self-adjoint兲 properties of the problem. We may not consider well-defined ‘‘modes’’ and corresponding ‘‘time constants.’’ The standard normal mode approach breaks down, and the theory may fail to give correct
predictions of evolution even if perturbation fields remain in
the linear regime. The discrepancies between the theory andthe experiment on the stability limit of neutral fluids arereviewed in Ref. 3. The aim of this work is to establish asolid foundation for the analysis of shear flow systems. We
apply Kelvin’s method of shearing modes.4 This scheme,
previously called the nonmodal approach, actually consistsin the combination of two methods which have been widelyused in solving wave equations; the modal and the characteristics methods.
Much work has been done on instability problems withshear flows by means of the modal approach. It is implicitly
assumed in the application of the modal scheme that the
motion can be decomposed into a set of independent normal
modes with certain time constants.5 As is well known, thismethod is effective in solving problems involving Hermitian
operators, however, when applying it to non-Hermitian systems, we may overlook the secular and transient behaviors.
On the other hand, the characteristics method has been usedin the context of rapid distortion theory for analyzing thefluid turbulence6 and in the eikonal representation of the ballooning mode stability.7 If we can treat the non-Hermitianpart of the whole operator as a singular perturbation to a
Hermitian operator,8,9 we may be able to construct the theory
in the framework of the perturbation theory for theoperator.10 But, unfortunately, the convergence of the perturbative series seems to be very ambiguous in case of the shear
flows due to the secularity of their time evolutions. Thus, athorough mathematical treatment of the non-Hermitian properties of shear flow systems has not been accomplished sofar. In this paper, we have analyzed the shear flow effect on
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400

Phys. Plasmas, Vol. 8, No. 2, February 2001

Tatsuno, Volponi, and Yoshida

II. NON-HERMITIAN PROPERTY OF SHEAR FLOWSYSTEMS

Before formulating the interchange instability equations,let us describe a rough sketch of the problem and explain themathematical tool to analyze the non-Hermitian dynamics.
As is well known, the force operator governing the linear
dynamics of static magnetohydrodynamic 共MHD兲 plasmas in
Hermitian,18 and therefore the perturbation fields can be decomposed into a set of orthogonal eigenmodes which show
purely exponential 共unstable兲 or purely oscillating 共stable兲
evolutions. A nontriviality stems from the Alfvénic andacoustic continuous spectra; the phase mixing damping occurs. This behavior, however, is totally within the frameworkof the well-known theory of Hermitian operators.19
In the case where ambient shear flow exists, however,
the operator becomes non-Hermitian and resolution in termsof orthogonal eigenmodes fails. From a dynamical point ofview, the system experiences evolutions of a complex type.
In the following sections, we will show examples of suchkind of ‘‘non-Hermitian’’ dynamics where transient phenomena and secular evolutions play a dominant role. Similar
evolutions are found in the case of non-Hermitian operators
in finite dimensional vector spaces.20 It has to be stressed
how the application of the traditional modal paradigm tonon-Hermitian systems, which assumes exponential evolution of the perturbation fields, hinders the possibility ofcatching these rich variety of transient and algebraic phenomena. In this section, we will discuss Kelvin’s method andshow its suitability to the analysis of shear flow nonHermitian systems. We will revisit it from the viewpoint of
the characteristics method showing that it represents a generalization of the modal approach.
Unlike matrices, clear classification of differential operators becomes extremely difficult due to their infinite dimensionality of the spaces they act on, and also to their unboundedness. The linearized dynamics of fluid systems in thepresence of sheared flow is governed by a general equationof the following type:⳵ t u⫹v•“u⫽Au,

共1兲

where A denotes a Hermitian differential operator 共timeindependent兲 defined in a Hilbert space V, v is the stationary
mean flow, and u(苸V) denotes a perturbation field.
It is the convective derivative, v•“, that introduces thenon-Hermitian property into problem 共1兲 and prevents thepossibility of representing the dynamics of the systems interms of the orthogonal and complete set of eigenfunctions.
This is a well-known difficulty in the stability analysis ofneutral fluids, such as Couette or Poiseille flows, where thepredictions obtained by means of the modal methods do notmatch the experiments.3
In the case of a spatially inhomogeneous stationary flowv, Eq. 共1兲 becomes non-Hermitian and a straightforward
spectral resolution is not effective. However, Kelvin’s
method permits to resolve, for some classes of mean flows,
the evolution of the system 共1兲 into new types of modes bymeans of which both transient and secular asymptotic behav-

iors are effectively described. Let us now explain the mathematical foundations of this scheme.
As mentioned in Sec. II, Kelvin’s method consists in the
combined application of two methods which have been extensively used in the analysis of wave equations. Preciselythe ‘‘Lagrangian’’ part of Eq. 共1兲, ⳵ t ⫹v•“, is solved bymeans of the characteristics method and the ‘‘Hermitian’’part A by means of the standard spectral resolution.
The characteristics method is applied to solve the characteristic ODE associated to the Lagrangian derivative moving along the characteristic curve of the ambient motion,
which is given by
dx
⫽v,
dt

x共 0 兲 ⫽ ␰.

共2兲

By inverting the modes, which are expressed in Lagrangian
coordinates as ␸ (k, ␰), they will be represented in Euleriancoordinates as
˜␸ 共 t;k,x兲 ⫽ ␸ 共 k, ␰共 t;x兲兲 ,共3兲

where ␰(t;x) denotes the inverse of x(t; ␰). The existence of
the inverse mapping x(t)哫 ␰ is guaranteed in the case ofincompressible mean flows. Due to Eq. 共3兲, ˜␸ (t;k,x) satisfies the characteristic equation

⳵ t ˜␸ 共 t;k,x兲 ⫹v•“ ˜␸ 共 t;k,x兲 ⫽0.

共4兲The essential condition for the applicability of Kelvin’s
method consists in the constraint for the function ˜␸ (t;k,x) toform the complete set of eigenfunctions of the operator A. If
such a set of eigenfunctions exists, we can decompose theperturbation field u by means ofu⫽冕û k 共 t 兲 ˜␸ 共 t;k,x兲 dk.

共5兲

We notice that due to Eq. 共3兲 the eigenvalues of A become
time dependent. The new eigenvalue problem for A reads
A˜␸ 共 t;k,x兲 ⫽␭ k 共 t 兲 ˜␸ 共 t;k,x兲 .

共6兲

Plugging Eq. 共5兲 into Eq. 共1兲 and exploiting Eqs. 共4兲 and
共6兲, we have冕 ⳵

␸ 共 t;k,x兲 dk⫽
关 t û k 共 t 兲兴 ˜冕û k 共 t 兲 ␭ k 共 t 兲 ˜␸ 共 t;k,x兲 dk.
共7兲

Due to the orthogonality of the modes ˜␸ (t;k,x), the evolution of û k is governed by the equationdû 共 t 兲 ⫽␭ k 共 t 兲 û k 共 t 兲 .
dt k

共8兲

If ˜␸ (t;k,x) do not satisfy both conditions given by characteristic equation 共4兲 and eigenequation 共6兲, Eq. 共7兲 will haveadditional terms which represent the complicated mode coupling and thus the applicability of Kelvin’s method is compromised.
Due to the time dependence present in the eigenvalues␭ k (t), the evolution of û k (t) will not exhibit a simple exponential dependence as in the Hermitian case, but more complicated behaviors, which are characteristic of non-Hermitiansystems. By analyzing this ODE, the motion of each mode

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Phys. Plasmas, Vol. 8, No. 2, February 2001

Transient phenomena and secularity of linear interchange . . .

can be classified, and the time asymptotic behavior can also
be shown. The following sections will be devoted to thederivation of ODE 共8兲 and the discussion of the behavior ofits solution for interchange instabilities in plasmas with shear
flow.
III. FORMULATION OF INTERCHANGE INSTABILITIESInterchange instabilities have been analyzed for static
plasmas by many authors.21–23 In the case of static 共stationary ambient flow v0 ⫽0兲 plasmas, the ideal MHD equations
can be reduced into a simple partial differential equation of
the form18

⳵ 2t ␰⫽F␰,共9兲

where ␰ is the displacement vector and F is the force operator which is Hermitian 共self-adjoint兲 when the plasma is surrounded by an ideal conducting wall. In order to analyze thestability of the system, we can apply the spectral method andrepresent the dynamics in terms of a superposition of harmonic oscillations of modes. Another method of analyzing
the stability of the static plasmas is to apply the energy
principle21 which is a variational approach based on the Hermitian property of the force operator F. These methods showthat the interchange modes have spatially localized structuresnear the marginal stability22 except when p ⬘ ⯝0 on the rational surface.23
It is remarkably difficult to estimate the exact linear stability of the system in the presence of a stationary shear flow,
since, as seen in the preceding sections, the dynamics become non-Hermitian and both the spectral and the variational
methods lose their mathematical foundations. Dispersion relations have been studied in many publications,1,5,8 however,as discussed in Sec. I, the evolution of a non-Hermitian system cannot be reconstructed from the formal dispersion relation, because we do not have a spectral theory. Since theproper asymptotic behavior of interchange instabilities are
not clearly shown yet, we will first analyze simplified systems focusing on the non-Hermitian property of the system.
In this section, we will derive the equations for stationary
flowing plasmas. Specifically we will investigate the effectof shear flows on interchange instabilities of plasma under
the influence of homogeneous magnetic field.
In the presence of gravitational force, the ideal MHD
equations read as␳dv⫽j⫻B⫺“ p⫹ ␳ g,
dt

共10兲

d␳
⫹ ␳ “•v⫽0,
dt

共11兲

⳵B
⫺“⫻ 共 v⫻B兲 ⫽0,
⳵t

共12兲

“•v⫽0,

共13兲

where ␳, B, and g are the density, magnetic field, and gravitational constant vector, and d/dt⫽ ⳵ t ⫹v•“ denotes the Lagrangian derivative. Here we assume the incompressibility ofthe velocity field v, instead of using the equation of state.

401

The ambient fields 共denoted by the subscript 0兲 mustsatisfy

␳ 0 v0 •“v0 ⫽j0 ⫻B0 ⫺“p 0 ⫹ ␳ 0 g.

共14兲

If we consider a parallel stationary shear flow of the formv0 ⫽(0,v 0y (x),0), straight homogeneous magnetic field B0⫽(0,B y ,B z ), and gravitational force acting in the positive x
direction, the convective derivative gives no contribution to
the stationary state and Eq. 共14兲 is reduced to“p 0 ⫽ ␳ 0 g.

共15兲

The above equation denotes that the pressure gradient is balanced by the gravitational force in the x direction. This is thesame condition which holds for static neutral fluids.
The perturbed magnetic and velocity fields are assumed
to be two dimensional in the x – y plane, and thus we canintroduce the poloidal flux function and stream function,
B1⬜ ⫽“ ␺ ⫻ez ,v1⬜ ⫽“ ␾ ⫻ez ,共16兲

where the subscript 1 denotes the perturbed quantities, ⬜
expresses the direction perpendicular to the dominant magnetic field directed along the z axis, and ez denotes the unit
vector in the z direction. Using these representations, we caneliminate the pressure from governing equations.
Taking the curl of the equation of motion and projecting
it along ez , we obtain␮ 0 ␳ 0 关共 ⳵ t ⫹ v 0y ⳵ y 兲 “⬜2 ␾ ⫺ v ⬙0y ⳵ y ␾ 兴⫽B0 •“ 共 “⬜2 ␺ 兲 ⫹ ␮ 0 ⳵ y ␳ 1 g,

共17兲

where “⬜2 ⫽ ⳵ 2 / ⳵ x 2 ⫹ ⳵ 2 / ⳵ y 2 . In deriving Eq. 共17兲 we haveused the Boussinesq approximation which consists in the neglect of the spatial variation of the stationary state density inthe inertial term of the equation of motion, but not in thecontinuity equation since it is the driving term for the interchange instability. Physically it is valid provided that thevariability in the density is due to variations in the temperature of only moderate amounts.24 The component of the flow
perpendicular to the ambient magnetic field can be considered consistently coming from the E⫻B drift, taking into
account the ideal Ohm’s law. It is noted that, if we neglect
the effect of the magnetic field, we recover the Rayleigh’s
equation for Kelvin–Helmholtz instability.25
The density fluctuation can be expressed as
共 ⳵ t ⫹ v 0y ⳵ y 兲 ␳ 1 ⫽⫺ ␳ 0⬘ ⳵ y ␾ ,共18兲

where the prime denotes the derivative with respect to x.
Now ␳ 0⬘ is considered as a constant which introduces a destabilizing force. The induction equation is the same as in theordinary reduced MHD equations26 and under the above assumptions on the stationary fields reads as
共 ⳵ t ⫹ v 0y ⳵ y 兲 ␺ ⫽B0 •“ ␾ .

共19兲Equations 共17兲–共19兲 constitute a closed system of equations. We can see that the static system ( v 0y ⫽0) governedby these equations shows Hermitian property, and the convective derivative ( v 0y ⫽0) brings the non-Hermitian prop-

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402

Phys. Plasmas, Vol. 8, No. 2, February 2001

Tatsuno, Volponi, and Yoshidaerty into our system. We will investigate the effect of theshear flow on the interchange instabilities in the following
sections.

⫽⫺

IV. DERIVATION OF ORDINARY DIFFERENTIAL
EQUATION

In this section, we derive the ODE for the amplitude ofKelvin’s modes, given in Eq. 共8兲, in the case of interchange
instabilities of plasmas. Let us first consider the electromagnetic case where B0 •“⫽0. From Eqs. 共18兲 and 共19兲, wehave⫺1␾ ⫽⫺ ⳵ ⫺1
共 ⳵ t ⫹ v 0y ⳵ y 兲 ␳ 1 ⫽ 共 B0 •“ 兲 ⫺1 共 ⳵ t ⫹ v 0y ⳵ y 兲 ␺ .
y ␳ 0⬘

共20兲

Since we have assumed the mean velocity v 0y ⫽ v 0y (x) andthe homogeneous ambient field B0 ⫽(0,B y ,B z ), the operator⳵ t ⫹ v 0y ⳵ y commutes with both ⳵ ⫺1
and (B0 •“) ⫺1 . Thusyacting on both sides of Eq. 共20兲 with the operator ( ⳵ t⫹ v 0y ⳵ y ) ⫺1 gives␳ 1 ⫽⫺ ␳ 0⬘ ⳵ y 共 B0 •“ 兲

⫺1

␺.

冋

d
d ␺ˆ
共 k̃ x 共 t 兲 2 ⫹k 2y 兲
dt
dt

共21兲

册

␳ ⬘0 gF2
ˆ,␺
共 k̃ x 共 t 兲 2 ⫹k 2y 兲 ␺ˆ ⫺k 2y
␮ 0␳ 0␳0

共26兲

where F⫽k•B0 ⫽k y B 0y ⫹k z B 0z , and we have dropped thesubscript k for simplicity. We notice that in the absence of
shear flow ( ␴ ⫽0) the usual interchange instability equation
for static equilibrium can be obtained.
Our procedure can be readily shown to coincide with thetraditional formulation of Kelvin’s method consisting in thecoordinate transform (t,x,y,z)哫(T, ␰ , ␩ , ␨ ) defined by
T⫽t,

␰ ⫽x,

␩ ⫽y⫺ ␴ tx,

␨ ⫽z,

共27兲

and the Fourier transform with respect to the new coordinatesû 共 k ␰ ,k ␩ ,k ␨ ;T 兲
⫽

冕冕冕

⫹⬁

⫺⬁

u 共 ␰ , ␩ , ␨ ;T 兲 e i 共 k ␰ ␰ ⫹k ␩ ␩ ⫹k ␨ ␨ 兲 d ␰ d ␩ d ␨ .

共28兲

共22兲

Normalizing the time t by the poloidal Alfvén time ␶ A⫽a 冑␮ 0 ␳ 0 /F, we can rewrite Eq. 共26兲 in dimensionless formas

Substituting Eqs. 共20兲 and 共22兲 into Eq. 共17兲, and acting with
B0 •“ on both sides, we obtain

␶2
d ␺ˆ
d
2
2 ˆ
2 A ˆ
2
2
⫽⫺ 共 k̃ x 共 t 兲 ⫹k y 兲 ␺ ⫹k y 2 ␺ , 共29兲
共 k̃ x 共 t 兲 ⫹k y 兲
dt
dt
␶G

From Eq. 共19兲,“⬜2 ␾ ⫽“⬜2 共 B0 •“ 兲 ⫺1 共 ⳵ t ⫹ v 0y ⳵ y 兲 ␺ .

共 ⳵ t ⫹ v 0y ⳵ y 兲 “⬜2 共 ⳵ t ⫹ v 0y ⳵ y 兲 ␺␳ ⬘0 g 2
共 B0 •“ 兲 2 2⫽“⬜ ␺ ⫺⳵ ␺.
␮ 0␳ 0␳0 y

共23兲

Since the operator on the right-hand side is Hermitian,
we can decompose the flux function ␺ by means of the shearing eigenmodes␺ 共 x,t 兲 ⫽冕␺ ␸ˆ k 共 t 兲 ˜ 共 t;k,x兲 dk,

共24兲

where each eigenmode can be expressed by the sinusoidal
function in our simplified case˜␸ 共 t;k,x兲 ⫽exp关 ik x x⫹ik y 共 y⫺ v 0y t 兲 ⫹ik z z 兴⫽exp关 ik̃ x 共 t 兲 x⫹ik y y⫹ik z z 兴 .

共25兲Here the mean flow is assumed to be v 0y (x)⫽ ␴ x andk̃ x (t)⫽k x ⫺k y ␴ t. It is explicitly shown that the wave number in the flow shear direction is linearly increasing withtime due to the distorting effect of the mean flow. Sincecontinuous variation of k̃ x (t) prevents from imposing theboundary condition in the bounded domain, we will concentrate on the analysis of localized perturbations by considering the infinite domain. Note that ˜␸ are the eigenfunctions of
the right-hand side of Eq. 共23兲, and also satisfy the characteristic equation 共4兲. It should be noted that the presence of
the Laplacian operator on the left-hand side of Eq. 共23兲 does
not hinder the application of Kelvin’s method since themodes ˜␸ are as well eigenfunctions of the Laplacian “⬜2 .
Thus, the time evolution equation for the amplitude ␺ˆ kcan be written as冋册

where the wave vectors are normalized by the characteristic2⫽⫺ ␳ 0 / ␳ 0⬘ g. Further we can rewrite
length scale a and ␶ G
Eq. 共29兲 in the formd ␺ˆ
d 2 ␺ˆ
⫹ 关 1⫺S 共 t 兲兴 ␺ˆ ⫽0,
2 ⫹␮共 t 兲
dt
dt

共30兲

where

␮ 共 t 兲 ⫽⫺2 ␴ k y k̃ x 共 t 兲k̃ x 共 t 兲 2 ⫹k 2y,

S共 t 兲⫽k 2y Gk̃ x 共 t 兲 2 ⫹k 2y,2and G⫽ ␶ A2 / ␶ G. Drawing an analogy with Newton’s equation, ␮ (t) represents the frictional term and S(t) the interchange drive term. Equation 共30兲 is the correspondent of Eq.
共8兲. As we have mentioned in Sec. II, the time evolution for
the amplitude of each eigenmode is no longer a simple exponential function. The behavior of ␺ˆ will be discussed in
the following sections.

V. ASYMPTOTIC AND TRANSIENT BEHAVIOR OF
EACH MODE

In the absence of a density gradient or shear flow, ␮ (t)⫽S(t)⫽0 in Eq. 共30兲 and we have a pure oscillation representing the Alfvén wave. If we include the density gradient,then S(t)⫽0 and we obtain the interchange instability for
negative ␳ 0⬘ . Since a homogeneous magnetic field is assumed in this paper, we have no stabilizing effect of themagnetic shear. The operator is Hermitian in these two cases,
therefore we have the simple exponential evolution with time
constants for each mode.

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Phys. Plasmas, Vol. 8, No. 2, February 2001

Transient phenomena and secularity of linear interchange . . .

403

When we include the shear flow, we have ␮ (t)⫽0 and
we can draw an analogy with the dynamics of a damped
oscillator with time dependent frictional coefficient ␮ (t).
When time goes, ␮ (t) becomes always positive, whichmeans a formal dissipation, and therefore the oscillation energy of the Alfvén wave 关 (d ␺ˆ /dt) 2 ⫹ ␺ˆ 2 兴 /2 decreases monotonically. In the following sections we will describe both theasymptotic and transient behaviors of the amplitudes ␺ˆ .
A. Asymptotic behavior

In order to study the time asymptotic behavior, we assume tⰇk x / ␴ k y ,1/␴ . In this time asymptotic limit we obtainthe following ODE:冉冊G/ ␴ 22 dd2
ˆˆ ⫹ 1⫺ 2 ␺ˆ ⫽0,
␺
2 ␺⫹
dt
t dt
t

共31兲

2
denotes the magnitude of the instability
where G⫽ ␶ A2 / ␶ G
drive term. In the absence of the instability drive G, the time
asymptotic behavior of the solution of Eq. 共31兲 is expressedas1␺ˆ ⬃ sin t,t

共32兲

which coincides with the result of Koppel27 which considered a time dependent nonperturbative state. Since Eq. 共31兲is the spherical Bessel equation, its general solution for G⫽0 is expressed by

␺ˆ ⫽

1

冑t

共 C 1 J ␯ 共 t 兲 ⫹C 2 Y ␯ 共 t 兲兲 ,共33兲

where J ␯ and Y ␯ denote the Bessel functions, and ␯⫽(G/ ␴ 2 ⫹1/4) 1/2. Therefore the time asymptotic behavior of
the mode is expressed generally as

冉

冊

␲␯
1
⫹␦ ,
␺ˆ ⬃ sin t⫺
t
2

共34兲

where ␦ denotes a constant phase depending on the initialvalues. Therefore the mode oscillates with amplitude ␺ˆ decaying with the inverse power of time. While the x component of the perturbation magnetic field b̂ x is proportional to␺, the y component b̂ y is proportional to k̃ x (t) ␺ˆ . Thus b̂ ytends to the pure oscillatory behavior

冉

冊

␲␯
⫹␦ ,
b̂ y ⬃sin t⫺
2

共35兲

as k̃ x (t) increases with proportion to time 共see Fig. 1兲. It
should be noted that there is no threshold value for the stabilization of the interchange instability, since we obtain thesame spherical Bessel equation 共31兲 for all modes. All modes
evolve as in Eq. 共31兲 independently of the values of wave
numbers k.
The final amplitude of each mode depends sensitively on
the parameters. As the shear parameter increases, the final
amplitude of b̂ y tends to be larger as is also shown by Chagelishvili et al.,12 while the mixing damping effect on b̂ x increases. The numerical integration of Eq. 共30兲 is shown in

FIG. 1. Direct numerical integration of Eq. 共30兲 for each mode. The parameters are as follows: k x ⫽10, k y ⫽1, k z ⫽0, ␴ ⫽0.2, G⫽1, and initial perturbations ␺ˆ ⫽0 and d ␺ˆ /dt⫽1.0 at t⫽0.

Fig. 1. It should be noted that the instability drive G asymptotically has the only effect to shift the phase of the oscillations as can be seen in Eqs. 共34兲 and 共35兲, and it does notaffect the principal time dependence. The combined effect of
the Alfvén wave propagation and shear flow mixing always
overcomes the interchange drive and the oscillations of themagnetic flux asymptotically decay with proportion to theinverse power of time.
B. Transient behavior

In this section, we analyze the transient behavior of eachmode. Since an analytic expression is not available, we discuss the transients by qualitatively analyzing the ODE 共30兲.
In the absence of the instability drive, we haveddt

冋冉 冊 册d ␺ˆ
dt

2⫹ ␺ˆ 2 ⫽⫺ ␮ 共 t 兲

冉 冊
d ␺ˆ
dt

2

,

where

␮ 共 t 兲 ⫽⫺2 ␴ k y k̃ x 共 t 兲k̃ x 共 t 兲 2 ⫹k 2y,

k̃ x 共 t 兲 ⫽k x ⫺ ␴ k y t.

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共36兲

404

Phys. Plasmas, Vol. 8, No. 2, February 2001

Tatsuno, Volponi, and Yoshida

Therefore, the frictional coefficient ␮ (t) acts as a damping
force for ␮ ⬎0. Since the sign of the denominator in ␮ (t) isalways positive, the behavior will be determined by that of
the numerator. The numerator can be expressed as 2 ␴ 2 k 2y t
⫺2 ␴ k y k x and according to its initial value we can single out
two classes of the transients.
When the product ␴ k y k x is negative, the frictional coefficient ␮ (t) is always positive from the beginning, therefore
the shear flow acts as a damping force at any time and themode shows simple damped behavior. On the other hand, if
the product ␴ k y k x is positive, the frictional coefficient ␮ (t)is initially negative and changes its sign at the instant t*⫽k x / ␴ k y . Therefore the mode experiences an initial amplification lasting until the time t , which is even faster than it*
would be in the presence of the only interchange drive. Thistransient behavior can also be seen in Fig. 1, where the initial
amplification lasts until the turning point t ⫽50 followed by*
the asymptotic decaying phase.
We have observed by numerical integration that the amplitude can be amplified to values of 1030 times larger than
the initial one. From a physical point of view, such huge
amplifications may break down the linearity of the perturbations and may lead to a nonlinear stage. This case is beyond
the scope of the linear theory and no sure conclusion can be
drawn from Kelvin’s method. Such huge amplifications areexperienced by modes with large t and G.

*

VI. ELECTROSTATIC INSTABILITY

When the wave vector is perpendicular to the ambient
magnetic field, the formulation for the flux function 共23兲
fails. For this ‘‘electrostatic limit,’’ we discuss the evolutionof the stream function ␾. The governing equations are Eqs.
共17兲 and 共18兲, since the flux freezing equation can be decoupled due to the fact that B0 •“⫽0. In the case of electrostatic
perturbations, drift wave may be destabilized, however, wehave dropped the drift wave branch here in order to concentrate our attention on the single fluid MHD model. Applying⳵ t ⫹ v 0y ⳵ y to both sides of Eq. 共17兲 and substituting it into
Eq. 共18兲, we have
共 ⳵ t ⫹ v 0y ⳵ y 兲 2 “⬜2 ␾ ⫽⫺

␳ ⬘0 g 2
⳵ ␾,
␳0 y

共37兲

for a linear shear flow. We represent ␾ in terms of the shearing modes given in Eq. 共25兲,␾ 共 x,t 兲 ⫽冕␾ ␸ˆ k 共 t 兲 ˆ 共 t;k,x兲 dk.

共38兲

By substituting Eq. 共38兲 into Eq. 共37兲, the following ODE is
obtained:d22 ˆˆ 兴 ⫽k 2y ␥ G␾,关共 k̃ x 共 t 兲 2 ⫹k 2y 兲 ␾
dt 2

共39兲

2
⫺2
⫽⫺ ␳ 0⬘ g/ ␳ 0 (⫽ ␶ G) denotes the characteristic
where ␥ G
growth rate of the interchange instability. Here again we
have dropped the subscript k for the sake of simplicity. In
order to investigate the time asymptotic behavior of eachmode, we assume tⰇk x /k y ␴ and tⰇ1/␴ , then Eq. 共39兲 becomes

d2
4 d
2⫺ ␣
ˆ
ˆ⫹ 2 ␾
ˆ ⫽0,
␾
2 ␾⫹
dt
t dt
t

共40兲

2
/ ␴ 2 denotes the ratio of the strength of thewhere ␣ ⫽ ␥ Ginterchange destabilizing effect and flow shear stabilizing
one. Note that this ODE is not dependent on the wave numbers k. The solution of Eq. 共40兲 reads␾ˆ ⫽C 1 t m ⫹ ⫹C 2 t m ⫺ ,

共41兲

where
m ⫾⫽

⫺3⫾ 冑1⫹4 ␣
.
2

共42兲

The time asymptotic behavior is therefore determined by thelarger index m ⫹ . Thus we can state the condition for theˆ asboundedness of ␾␣ ⭐2⇒⫺

1 ␳ ⬘0 g⭐ ␴ 2.
2 ␳0

共43兲

ˆ has been improved
The condition for the boundedness of ␾with respect to the static case ( ␳ 0⬘ ⭓0) due to the mixing
effect of the shear flow. The electrostatic perturbation can belinearly unstable while the electromagnetic one is completely
stabilized. The direct numerical integration of the ODE 共39兲is illustrated in Fig. 2. The transient behavior can be observed until the time t ⫽5, and the asymptotic behavior fol*lows. The asymptotic behavior is shown to be algebraic withthe power m ⫹ as pointed out by means of the analytic treatment.
We notice that the stability condition is not well definedˆ
here. In fact by imposing the boundedness of v̂ y ⫽ik̃ x (t) ␾⬃t 1⫹m ⫹ , the same condition ␳ 0⬘ ⭓0 as the static case is obtained. If we consider the other fields which are represented
by higher derivatives, e.g., the vortex perturbations, morestrict conditions for their boundedness will be recovered.
Since the mixing effect of the shear flow distorts the structure of the perturbations into smaller scales, the fields characterized by the higher derivatives will have stronger secularities. Unlike the static case where the evolution of the
fields can be expressed in a common exponential form, they
exhibit different evolutions with respect to each other inshear flow systems. This effect could be a pathological problem of describing the stability condition for shear flow systems.
VII. SUMMARY

Kelvin’s method of shearing modes is interpreted as a
combination of modal and characteristic methods for the
analysis of a non-Hermitian system. A shear flow distorts
each Fourier mode, resulting in a change of the wave number, which represents the stretching effect of the shear flow.
By means of this method, we have first analyzed theincompressible electromagnetic perturbations in the presenceof an interchange drive and obtained the ordinary differential
equation 共30兲 for the amplitude of the modes ␺ˆ k . All modes
show asymptotic decay proportional to the inverse power oftime 共nonexponential兲 without any threshold value. This

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Phys. Plasmas, Vol. 8, No. 2, February 2001

Transient phenomena and secularity of linear interchange . . .

405

type modes 关Eq. 共32兲 in Ref. 17兴 are mathematically equivalent. Of course these two modes may have spatially different
structures, at least this is the case for static equilibria. But
this fact means that they have no difference in time evolution, and we can say that these terms have the same effect inthe sense that they enlarge the spectrum to unstable eigenvalues. This equivalence stems from the assumption of a
spatially homogeneous magnetic field. The possibility of including the magnetic field inhomogeneity is investigated in
the Appendix.
We have also investigated the time evolution for electrostatic (k•B0 ⫽0) perturbations, which do not excite the
Alfvén wave, since they do not bend the magnetic field line
during their growth. The flow shear has been shown to havea stabilizing effect also on electrostatic disturbances, however, the phase mixing effect alone cannot completely stabilize the interchange instabilities. The condition for theˆ k can be expressed inboundedness of the mode amplitudes ␾
Eq. 共43兲 by means of a ratio of instability strength to shear
parameter of the mean flow. We have shown that the time
evolution of these unstable modes is again of algebraic type.
Notice that the conditions for the boundedness of different
observatives do not coincide. The discrepancies originate
from the fact that, in shear flow systems, different fields experience algebraic evolutions characterized by different powers of time, while the evolutions for any fields are expressedin a common exponential form for static systems.

FIG. 2. Direct numerical integration of Eq. 共39兲 for different ␣. The paramˆ ⫽0 andeters are as follows: k x ⫽10, k y ⫽2, ␴ ⫽1 and initial perturbations ␾ˆd ␾ /dt⫽1.0 at t⫽0. The amplitude of the stream function in case of ␣
⫽3.3 shows the algebraic growth corresponding to m ⫹ ⯝0.35.

means that the interchange instabilities are always damped
away at sufficiently large time due to the combined effect of
the Alfvén wave propagation and distortion of modes bymeans of the background shear flow; i.e., phase mixing effect. However, the transient behavior is not common for allmodes. Depending on the initial wave number, some of themshow transient amplifications which are even faster than theywould be in the presence of the only interchange drive.
These amplifications are so conspicuous that they may leadto the break down of the linearity of the perturbation fields.
It should be noted that, since our treatment considers thecase of parallel linear shear flow, Kelvin–Helmholtz instabilities, which originate from the second-order spatial derivative of the background shear flow,24,25 are beyond the scopeof the present theory. From a mathematical point of view, we
stress that the Kelvin–Helmholtz instability is a problem involving purely non-Hermitian operators in the sense that theoperator A of Eq. 共1兲 itself becomes non-Hermitian andtherefore the method developed in Sec. II cannot be applied.
This is a well-known instability in fluid dynamics whose
rigorous mathematical treatment presents highly nontrivial
difficulties.
We note that the ODE which gives the evolution of theamplitudes of the interchange modes 共30兲 and that of kink-

ACKNOWLEDGMENTS

The authors thank Professor Wakatani, Professor Mahajan, Professor Berezhiani, and Dr. Furukawa for valuable
discussions and suggestions. This work is supported in partby a Grant-in-Aid from the Japanese Ministry of Education,
Science, and Culture, Nos. 09308011 and 12780353.

APPENDIX: INHOMOGENEOUS MAGNETIC FIELD

In order to consider the effect of the magnetic shear, let
us consider the three-dimensional MHD equation for theevolution of the perturbation fields, which can be written in
Cartesian coordinates as

␳0

冉

冊

冉

冊

⳵ v1
⳵ v0
B0 •“bB0 •b⫹v0 •“v1 ⫹ v 1x⫽⫺“ p 0 ⫹
,
⳵t
⳵x
␮0
␮0
共A1兲

⳵b
⳵ v0
⫹v0 •“b⫽B0 •“v1 ⫹b x
,
⳵t
⳵x

共A2兲

where b denotes the perturbation magnetic field and v0⫽(0,␴ x,0). Assuming B0 ⫽(0,B 0y (x),B 0z (x)), we cantransform the coordinate as (x,y,z)哫(x, ␩ , ␨ ) with ␨ along
the local ambient magnetic field line and ␩ perpendicular tox and ␨. In this coordinate, we have the stationary flow expressed as (0,v 0 ␩ (x), v 0 ␨ (x)). Here, the spatial dependenceof the velocity components are

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406

Phys. Plasmas, Vol. 8, No. 2, February 2001

v 0␩⫽

1
B ␴ x,
B 0 0zv 0␨⫽Tatsuno, Volponi, and Yoshida1
B ␴ x.
B 0 0y共A3兲

If the magnetic fields are homogeneous, the coordinate transform is also spatially homogeneous and these velocity fields
are still linear functions with respect to x. Writing the above
equations by components in the new coordinates, we have

␳ 共 ⳵ t u⫹ v 0 ␩ ⳵ ␩ u⫹ v 0 ␨ ⳵ ␨ u 兲 ⫽冉␳ ⳵ tv ⫹ v 0␩⳵ ␩v ⫹ v 0␨⳵ ␨v ⫹
⫽

冉

冉

B 0z
␴u
B0

冊

冊

B 0⳵ ␨b ␩
B 0b ␨⫺ ⳵ ␩ p⫹,␮0␮0

␳ ⳵ t w⫹ v 0 ␩ ⳵ ␩ w⫹ v 0 ␨ ⳵ ␨ w⫹
⫽

冉

冉

冊

B 0⳵ ␨b x
B 0b ␨⫺ ⳵ x p⫹
,
␮0
␮0

B 0y
␴u
B0

冊

共A4兲

共A5兲

冊

B 0⳵ ␨b ␨
B 0b ␨⫺ ⳵ ␨ p⫹
,
␮0
␮0

共A6兲

⳵ t b x ⫹ v 0 ␩ ⳵ ␩ b x ⫹ v 0 ␨ ⳵ ␨ b x ⫽B 0 ⳵ ␨ u,

共A7兲

⳵ t b ␩ ⫹ v 0 ␩ ⳵ ␩ b ␩ ⫹ v 0 ␨ ⳵ ␨ b ␩ ⫽B 0 ⳵ ␨ v ⫹

B 0z
␴bx ,
B0

共A8兲

⳵ t b ␨ ⫹ v 0 ␩ ⳵ ␩ b ␨ ⫹ v 0 ␨ ⳵ ␨ b ␨ ⫽B 0 ⳵ ␨ w⫹

B 0y
␴bx ,
B0

共A9兲

and we can derive for the evolution of the amplitude b̂ x , thesame equation as previously obtained 关Eq. 共30兲兴 for ␺ˆ .
If we include the magnetic shear in the stationary condition, the above coordinate transform becomes spatially inhomogeneous. As can be seen from Eqs. 共A3兲, this transform
introduces a nonlinear spatial dependence of the backgroundshear flow profile even if this is assumed to be linear in

Cartesian coordinates. This fact shows that the introductionof the magnetic shear is essentially equivalent to that of thenonlinear shear flow profile.
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1

2

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