Chapter 1
Introduction
One dimensional linear spectral properties in single ﬂuid magnetohydrodynamics(MHD) were explored in this thesis by considering plasmas with and without mass
ﬂows. Electromagnetic ﬁeld equations in vacuum are linear systems. However, byintroducing magnetoﬂuids in the system, MHD equations become highly nonlinear due to coupling between the electromagnetic and ﬂuid properties of plasmas. In
such nonlinear systems, a lot of fascinating collective phenomena will appear; chaotic
motions, self-organizations, nonlinear waves, and so on. It should be stressed thateven the linear problems, which describe the dynamics of small amplitude motion
around an equilibrium state, also contain much more interesting behavior than asimple system of point mass, or that governed by the linear Schrödinger equation.
The inﬁnite dimensionality of the generator induces continuous spectra, and moreover, as well as neutral ﬂuids, the generator in case of ﬁnite shear ﬂow equilibrium
becomes non-Hermitian (non-selfadjoint), which is still an unresolved problem inmodern mathematics.
In addition to the above mentioned signiﬁcance from mathematical physics, spectral analysis of MHD is also related to an energy research for future human lives;
it is magnetic fusion. A fundamental requirement for realizing a fusion reactor is
to conﬁne a high density and high temperature1 plasma for suﬃciently long time.
In order to obtain such a plasma, magnetic conﬁnement systems have been studied,which basically utilize cyclotron motion of charged particle in a toroidal magnetic
ﬁeld. For the sustainment of plasma as a whole, at least the linear stability of theconﬁned plasma is indispensable in the time scale of the plasma dynamics. Studies
of such stability properties are accomplished by the spectral analysis.
1

Actually, the highest temperature was achieved at JT-60U in 1998 at more than five hundred
million degrees of centigrade, which is considered to be a few tens times higher than the coretemperature of the Sun [13].

12Chapter 1: Introduction

The practical magnetic conﬁnement conﬁgurations for aiming at fusion reactors
are mainly toroidal. Thus, multi-dimensional analyses of MHD stability are required.
Multi-dimensional spectral properties for static plasmas are recently discussed inmany literatures, e.g. for 2D [56, 150] and 3D [59]. However, the multi-dimensional
stability analysis is much more diﬃcult for the magnetized plasmas with shear ﬂows.
In this thesis, one dimensional spectral properties are discussed rather deeply including discrete and continuous spectra in non-Hermitian operators. First, we willtreat point spectra in Chap. 3 and continuous spectra in Chap. 4 for Hermitian operators (static plasmas). Although there are still many unresolved problems of MHD
stability in static equilibria, recently MHD stability of equilibria with plasma ﬂows
are investigated intensively, because of the spontaneous formation of zonal ﬂows fromelectrostatic turbulence [82]. Another reason is the rotational stabilization of MHD
instabilities, e.g. resistive wall modes [122]. Recently, a conﬁnement system which
actively uses the ﬂow shear for the improved conﬁnement is even proposed [101].
Thus, the latter half of the thesis is devoted to non-Hermitian problems of sheared
plasma ﬂow. Point spectra and continuous spectra are discussed in Chap. 5 and inChap. 6, respectively. Finally, the resonance between point spectra and continuous
spectra is mathematically investigated in Chap. 7.
Let us describe the main results in each chapter with an emphasis on physics.
We will ﬁrstly introduce the MHD system of equations in Chap. 2. Usually, preMaxwell equations for electromagnetic ﬁeld, which comes from the neglect of thedisplacement current, are used in ideal MHD. We will discuss in detail why andhow the displacement current will be omitted. We will also introduce the spectral
method for the analysis of linearized MHD equations. The boundary condition
is well known to aﬀect the spectra of diﬀerential equations. Here, it is explicitly
shown that the ‘norm’ (or scalar product) also plays an essential role for especially
Hermiticity of the operator. The construction of general solution is guaranteed if
the generating operator is shown to be Hermitian under any norm. Mathematically,boundary condition and norm are considered in the deﬁnition of the Hilbert spacein which the formal diﬀerential operators are embedded. After all description of themathematical background, we will summarize the spectra of incompressible MHD
equations, on which the rest part of the thesis is based.
In Chap. 3, some unusual properties of the unstable point spectra are investigated[133, 45, 88, 47]. The Mercier (Suydam) criterion for localized interchange modes
in stellarators does not predict instability in cases where global modes are unstable. One case is non-resonant pressure driven instabilities with low mode numbers,
which become unstable even if the mode resonant surface does not exist inside theplasma column. The other case is interchange instabilities when the pressure gradi-Chapter 1: Introduction3

ent vanishes at the mode resonant surface. If the pressure becomes ﬂat in a narrow
region around the mode resonant surface, high mode number instabilities may beeliminated and the beta (ratio of plasma pressure to magnetic pressure) limit at theparticular resonant surface increases. Also radial mode structure at nearly marginal
beta changes signiﬁcantly. Resonant unstable modes have a sequence of eigenvalues which converges to the edge of the Alfvén continuum, and the corresponding
eigenfunctions tend to localize around the resonant surface. However, non-resonant
unstable modes show the global structure and have a step like structure aroundthe resonant surface even in the limit of marginal beta value. Property of the nonresonant mode and transition from the resonant to the non-resonant one are clariﬁed
with a cylindrical plasma model for a low shear stellarator with a magnetic hill.
In Chap. 4, Alfvén continuous spectrum is picked up [132]. The MHD wave isstudied when two steep density gradient regions exist at surfaces of slab plasma. Insuch a case, it is shown that the surface Alfvén wave has two branches with nearly thesame damping rates, since the steep density gradients are located closely each other.
However, for the sharp boundary plasma, the surface Alfvén wave does not damp. As
the density proﬁle is relaxed, the damping rates become larger, pass via extremum,
and again they become small when the scale length of the density gradients becomes
extremely large. These damping rates seem consistent with behavior of magnetic
ﬂuctuations observed in the Heliotron-E pellet injection experiment.
As an example of non-Hermitian operators, we will ﬁrst analyze the eﬀect of
the outer resistive conducting wall and the rigid ﬂow of the plasma on externalkink instabilities in Chap. 5 [153]. Rigid plasma ﬂow itself will not introduce nonHermiticity, however, the ﬁnite diﬀerence of the velocity between the resistive walland the plasma will be regarded as a kind of discontinuous shear of the ﬂow. We willﬁrstly show the physical mechanism of Kelvin-Helmholtz instability in the neutral
ﬂuids, then show the correspondence of external kink mode in the cylindrical plasma
with a surrounding resistive wall and a rigid plasma ﬂow to the Kelvin-Helmholtz
instability. It is shown that the resistive wall does not only aﬀect to push back theplasma surface wave, but also pull to destabilize the external kink modes.
In Chap. 6, transient and secular behavior of interchange ﬂuctuations is analyzed
in an ambient shear ﬂow by invoking Kelvin’s method of shearing modes [143, 131].
Because of its non-Hermiticity, complex transient phenomena can occur in a shearﬂow system. It is shown that, for each mode, the combined eﬀect of shear ﬂow
stretching and Alfvén wave propagation overcomes the instability driving force at
suﬃciently long time, and damps ﬂuctuations of the magnetic ﬂux for all wave
numbers. On the other hand, electrostatic perturbations can be destabilized for
suﬃciently strong interchange drive. The time asymptotic behavior in both cases is4

Chapter 1: Introduction

algebraic (non-exponential).
In Chap. 7, we will construct an exact mathematical formulation of neutral ﬂuids
which contains the resonance of point spectra (diocotron modes) and continuous
ones (entropy waves) [152]. It is easily understood by the ﬁnite dimensional spectral
theory that the simple resonance among point spectra will cause secular growth ofthe modes. However, by introducing a new deﬁnition of the Hilbert space, it isshown that the resonance, which contains the energy ﬂow from continuous spectra
to point ones, may not cause secular behavior of the perturbed quantities. By
considering another model, e.g. parallel dynamics, it is also shown in Ref. [84] thatsuch secular behavior will be realized due to the inclusion of the resonance between
two diﬀerent continuous spectra. It means that, even if the system has no unstableeigenvalue [91], growth of the algebraic type is possible, which may cause linear
instability of the system. It is pointed out in Appendix C that such kind of resonance(between continuum and point spectra) is also found in the kinetic treatment of theelectrostatic oscillations (plasma oscillations).