J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)173–175
000–000

Algebraic Behavior of Fluctuations Attributed to Non-Hermitian
Property of the Linearized MHD Equation
HIROTA Makoto and YOSHIDA Zensho
University of Tokyo, Tokyo, 113-0033, Japan(Received: 9 December 2003 / Accepted: 3 February 2004)

Abstract
Algebraic behavior of three-dimensional fluctuations is analyzed for a static equilibrium of ideal magnetohydrodynamics (MHD). Without the reduction of variables to the Lagrange displacement, the linearized ideal MHD
equations constitute a non-Hermitian (non-selfadjoint) system. Degeneracy of Alfvén continuous spectra causes a
resonant interaction between eigenmodes. Solution of the initial value problem shows a temporally algebraic andspatially localized growth even if there is no exponentially unstable mode.
Keywords:
linear stability, ideal magnetohydrodynamics, Alfvén continuous spectrum, algebraic behavior,
non-Hermitian operator

1. Introduction

growth (∝ t) of w̃x due to the resonance between eigenmodes,
which is analogous to the Jordan canonical form of the nonHermitian matrix in linear algebra. This algebraic growth isexpected to reach nonlinear regime and trigger turbulence.
However, the spectral resolution of non-Hermitianoperators is mathematically unsolved, and therefore we mustsolve the initial value problem directly. Especially in ideal
fluids and plasmas, the continuous spectrum reflects theinfinite dimensional property of functional space whichcannot be understood as a simple extension of linear algebra[2]. It is a mathematically nontrivial problem what happens
if the resonance between singular eigenmodes occurs.
In this short paper, we will demonstrate an algebraic
behavior which occurs in ideal magnetohydrodynamics(MHD) with slab geometry. It will be produced by degeneracyof Alfvén continuous spectra. The algebraic growth islocalized on the rational surface and may ubiquitously occur
even if no exponential instability exists.

The algebraic instability has been attracting a lot ofinterest in the linear stability analysis of various fluids andplasmas (non-neutral plasma, galactic phenomena,atmospheric dynamics and so on). Since the growth of theperturbation field is temporally algebraic (∝ t α ), thisinstability can not be found by the conventional normal modes
analysis or the dispersion relation which assumes theexponential behavior (e–iω t with eigenvalue ω) of solutions andthe existence of a complete set of eigenfunctions.
If we write linearized equations in the form of anevolution equationi∂ t f = Kf ,(1)

the spectral resolution of the linear operator K is guaranteedby the Von Neumann theorem so far as K is Hermitian(selfadjoint). Then the solution f can be completely
decomposed into eigenfunctions (or eigenmodes) which may
be singular if K has the continuous spectrum. The algebraic
behavior stems from the non-Hermitian property of K. A
typical example is the Orr-Sommerfeld and Squire equations[1] which are known as the incompressible viscid fluid
equations linearized around laminar shear flow. Its
mathematical structure is written like⎛ w˜ x ⎞ ⎛⎜K sqi∂ t ⎜⎟=⎝ ∆ v˜ x⎠ ⎜⎝ 0Kc ⎞⎟ ⎛ w˜ x ⎞,K os ⎟⎠ ⎜⎝ ∆ v˜ x⎟⎠

2. Non-Hermicity of ideal plasma
In general, stability of plasma is analyzed by solving thelinearized MHD equations (simultaneous partial differentialequations), which is represented by an evolution equation for
the fluctuation part of velocity (ṽ), magnetic field (b̃) andpressure ( p̃). In many cases, these perturbation fields in thevicinity of the equilibrium are reduced to a displacement
vector ( ) and the linearized equation is written as ∂t2 =H where H is known as a Hermitian operator [3]. Since
the spectral resolution of Hermitian operator is(2)where ṽx and w̃x , respectively, denote the x component ofvelocity and vorticity perturbation. If the spectrum of Ksq
coincides with that of Kos (degeneracy), we observe linear
Corresponding author’s e-mail: hirota@plasma.q.t.u-tokyo.ac.jp1173©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchHirota M. et al., Algebraic Behavior of Fluctuations Attributed to Non-Hermitian Property of the Linearized MHD Equation

If we consider normal modes such as ṽx (x,t) = ṽx (x)e–iω t,(12) and (13) are combined into, by eliminating b̃x ,mathematically possible, the MHD stability analysis isconventionally based on the dispersion relation which
determines if the imaginary part of ω is positive or not.
However, the special initial conditions, p̃⏐t=0 = 0 and b̃⏐t=0= 0, are assumed in the above reduction process, and thespectrum of the original system (1) for the seven variablesf = (ṽ, b̃, p̃) does not perfectly coincide with that of H.
In the following analysis, let us assume the incompressibility for simplicity and consider a slab equilibrium,

B = (0, By ( x ), Bz ( x )), V = (0, 0, 0),(3)P( x )+ | B( x ) |2 /2 = const.

(4)

[((5)

i( k y + k z )b˜ ( x , t ) e y z ,(6)i( k y y + k z z )p˜ ( x , t ) e,

σ c± = {ω ∈ ; ω = ±ω a ( x )}.

j˜ x = ik y b˜z − ikz b˜y ,(9)

We assume that the profile of the ambient magnetic fieldis linear and the rational surface is located on x = 0;

ω a ( x ) = k ⋅ B( x ) = x.

(10)i∂ t j˜ x = −ω a ( x ) w˜ x − ι( x ) v˜ x ,(11)i∂ t ∆ v˜ x = −ω a ( x )∆ b˜x + ω a″ ( x ) b˜x ,(12)i∂ t b˜ x = −ω a ( x ) v˜ x ,(13)v˜ x ( − L1 , t ) = v˜ x ( L2 , t ) = 0,(18)b˜ x ( − L1 , t ) = b˜ x ( L2 , t ) = 0.

(19)

We denote the spectrum by σc = σc+ σc– which are nothing
but σc+ = [–L1, L2] and σc– = [–L2, L1]. Using the Laplacetransformation defined by∞L[v˜ x ( x, t )] = ∫0 v˜ x ( x, t )eiΩt dt, (Ω ∈ )(20)

we obtain, instead of (15),[]∂ x (Ω 2 − x 2 )∂ x V˜ − k 2 (Ω 2 − x 2 )V˜

where ∆ = ∂x2 – k2 is Laplacian operator, a prime (′) denotes x
derivative, ωa (x) = k·B(x) denotes local Alfvén frequency and– b̃x) and
we put ι(x) = [B′(x) × k]·ex . In terms of M± = ∆(ṽx +–˜S± = w̃x + j x , these equations are rewritten in the matrix formof ⎛ S ⎞⎜⎜i∂ t ⎜⎜⎜⎜⎜⎝(17)

In this coordinate system, we consider a finite domainx ∈ [–L 1, L 2] (L 1 > 0, L 2 > 0) with a Dirichlet boundary
condition

as independent variables. Due to the incompressibility, thepressure fluctuation p̃ is given by the Poisson equation andthe linearized system can be described by the four
independent variables, ṽx , w̃x , b̃x and j˜x as follows.

i∂ t w˜ x = −ω a ( x ) j˜x + ι( x ) b˜ x ,(16)3. Analysis of algebraic behavior

where we made use of the wavenumber ky and kz . We will
also use k = (0, ky , kz) and k = ⏐k⏐ for simple notation. Letus introduce

(8)

(15)

Furthermore, the upper two equations in (14) have the same
continuous spectra due to the multiplication operator ±ωa (x).
As a result, the evolution equation has four degeneratecontinuous spectra in total. Since the variables S± are forced
by M± like the Orr-Sommerfeld and Squire equations, wecan expect an algebraic growth of S± due to the resonancebetween the continuous spectra. But, since we do not havemathematical theory to describe it, we will solve the initialvalue problem in the next section.

(7)w˜ x = ik y v˜ z − ikz v˜ y ,)which is well known as the eigenvalue problem of the Alfvén
wave (the stream function replaces ṽ x in the case of twodimensional fluctuation). Barston [4] considered this problem
and proved that there is no spectrum in addition to the Alfvén
continuous spectra σc+ and σc–, where

Taking strong magnetic shear into account, we do not adopt
the reduced MHD approximation, but consider “threedimensional” fluctuations, i.e.,i( k y + k z )v˜ ( x , t )e y z ,] ()∂ x ω 2 − ω a2 ∂ x v˜ x − k 2 ω 2 − ω a2 v˜ x = 0,= iΩ v˜ x ( x , 0) − ix∆ b˜ x ( x , 0),(21)where Ṽ(x,Ω) = L[ṽx (x,t)]. If the initial conditions ṽx (x,0) andb̃x (x,0) are regular functions, this solution Ṽ(x,Ω) becomessingular at {(x,Ω); x = ±Ω}. For Ω = ω ∈ σc\{0}, there are
two singular points x = ±ω, and the Frobenius method gives
the well known logarithmic singularities, log(x ± Ω). The
inverse Laplace transformation of these singularities shows
the phase mixing damping in proportion to 1/t [4,5].
On the other hand, for Ω = 0, the two singular points
overlap at the rational surface x = 0. After some carefulconsiderations, we can write the general solution around x =Ω = 0 as+ ⎟S− ⎟⎟M + ⎟⎟M − ⎟⎟⎠⎛ω a 0⎞ ⎛ S+ ⎞0ι∆ −1⎜⎟⎜ 0 −ω
⎟⎜S ⎟0−ι∆ −1−a⎟.
⎟⎜=⎜⎜ 00
ω a + ω a′ ∂ x ∆ −1 (ω a′ ∂x + ω a″ )∆ −1⎟ ⎜⎜ M + ⎟⎟⎟⎜⎟⎜0 −(ω a′ ∂x + ω a″ )∆ −1 −ωa − ω a′ ∂ x ∆ −1 ⎠ ⎜⎝ M − ⎟⎠⎝ 0(14)2174Hirota M. et al., Algebraic Behavior of Fluctuations Attributed to Non-Hermitian Property of the Linearized MHD Equationx−ΩV˜ ( x , Ω ) = C1 (Ω ) + C2 (Ω )log+ O(Ω ),x+Ω– x) brings about a localized algebraic
singularity of 1/Ω(Ω +growth at x = 0 as far as ι(0) ≠ 0 and b̃x (0,0) ≠ 0, which canbe written like(22)where C1(Ω) and C2(Ω) are coefficients determined by theouter solution and the boundary condition (so-calledcontinuation data), and O(Ω) denotes the terms that vanish
when Ω → 0. For the limit of Ω → ±i0, we obtain

V˜ ( x , ± i0) = C1 ( ± i0) ⫿C2 ( ± i0)2 πiY( − x ),S± ~ ±ι(0) b˜ x (0, 0) 1 − exwhere Y(x) denotes the Heaviside function. In the domain[–L1, L2] except for x = 0, Ṽ(x, ±i0) must satisfy (21) with Ω= ±i0, i.e.,[]

By considering incompressible three-dimensional
fluctuations, the linearized ideal MHD equation has four
Alfvén continuous spectra. They degenerate with each otherand the resonant interaction occurs only at zero frequency due
to the non-Hermitian property of the evolution equation.
Although the spectral theory for non-Hermitian operator has
not been established, the initial value problem indicates theexistence of algebraic growth which is localized on therational surface.
On the rational surface, the initial perturbation ofmagnetic field receives no force since it does not bend thefield line. Then, the field line remains inclined with respect
to the ambient pressure contour, which causes a constantacceleration of parallel plasma motion; ṽ|| ∝ t. This physicalmechanism of the algebraic instability is not included in thestability analysis of the Lagrange displacement which assumesb̃x (x,0) ≡ 0.
In more realistic situations, the viscosity and theresistivity have a considerable effect on the fine structure that
we showed. However, our algebraic instability induces thegrowth of perturbation energy in small scale, which can
rapidly grow depending on the initial condition, and may berelated to the development of turbulence in inhomogeneous
magnetic field. The effect of shear flow will be discussed
elsewhere.

Solving this equation and continuating the solution with (23)result in(25)⎧ sinh( − kL1 − kx ) x ∈[ −L , 0]1
⎪ sinh( − kL )1⎪.
g( x ) = ⎨⎪ sinh(kL2 − kx ) x ∈[0, L ]2⎪⎩ sinh(kL2 )(26)

whereFrom (13), B̃(x,Ω) = L[b̃x (x,t)] is given by− xV˜ ( x , Ω ) + ib˜ x ( x , 0)
B˜ ( x , Ω ) =.
Ω(27)

By the inverse Laplace transformation, we can finally obtain
the asymptotic behavior (t → ∞) asv˜ x ( x , t ) → 0,(28)b˜ x ( x , t ) → b˜ x (0, 0)g( x ).

(29)

This convergence speed is characterized by 1/t.
The behavior of the remaining variables, w̃ x and j˜x , areeasily estimated by using the above result. First, we obtain,by definition,∆ −1L[M ± ] =(Ω ± x )V˜ ( x , Ω ) ib˜ x ( x , 0).
ΩReferences[1] P.J. Schmid and D.S. Henningson, Stability andTransition in Shear Flows (Springer-Verlag, New York,2001).
[2] M. Hirota, T. Tatsuno and Z. Yoshida, J. Plasma Phys.
69, 397 (2003).
[3] J.P. Freidberg, Ideal Magnetohydrodynamics (PlenumPress, New York, 1987).
[4] E.M. Barston, Ann. Phys. 29, 282 (1964).
[5] J. Tataronis and W. Grossmann, Z. Phys. 261, 203(1973).

(30)

By substituting this expression, the Laplace transformation ofthe upper two equations in (14) leads to˜

L[S± ] = ± ι( x ) ⎡⎢V˜ ( x, Ω ) ± ib x ( x, 0) ⎤⎥ + iS± ( x, 0) .
Ω ⎣

Ω

x ⎦Ω

x(32)

4. Summary(24)b˜ ( x , 0) − b˜ x (0, 0)g( x )V˜ ( x , ± i0) = i x,x.

By calculating w̃ x = (S+ + S–)/2 and j˜x = (S– – S+)/2, we canfind that Im(w̃ x) and Re( j˜x) increase in proportion to timearound x = 0. Since we supposed k·B(0) = 0, Im(w̃ x)
corresponds to the parallel component of ṽ with respect to B;ṽ|| (0,t) ∝ t.

(23)

∂ x x 2 ∂ x V˜ − k 2 x 2 V˜ = ix∆ b˜ x ( x , 0).

ixt

(31)

The last term in the right hand side yields an solution which–shows a phase mixing like S±(x,0)e+ixt. On the other hand, the3175