
The followings are an abstract of my recent
research. For more details, refer to
publications.

Flowing Plasma and Hall Effect
Flows in a plasma are attractive objects both for their physical and
mathematical contents. Recently plasma flows are considered to play
essential roles in nuclear fusions (Hmode, shear flow stabilization
etc) as well as space plasmas (Jets, CME etc). Magnetohydrodynamics
(MHD) is widely used as the macroscopic theory of electrically
conducting fluids, providing a theoretical framework for describing
both laboratory and astrophysical plasmas. The MHD description is
seemingly valid for a static and macroscopic system. In contrast to
the name of magnetohydrodynamics, however, the MHD model captures only
a rather small aspect of plasma flows. In fact, most MHD studies of
plasmas deal with magnetostatic configurations. This is not only
because of a convenience but because powerful mathematical methods
have been developed for flowless MHD plasmas, for example, the
GradShafranov equation for equilibrium, the energy principle for
linear stability theory and so on. Conversely, there are many
difficulties in investigating flowing plasmas. One of the
difficulties is singularities created by the plasma flow. For
instance, considering the equilibrium of flowing plasmas, the type of
equation changes alternatively between elliptic and hyperbolic
depending on the poloidal Alfven Mach number. The existence of
hyperbolic regimes implies a shock formation, and it is not clear
whether the solution exists in the bounded domain (in more than
2dimension). Assuming the incompressible flow, the equation becomes
elliptic, however it contains a singularity in the 2 dimensional
system. It is caused by the fact that the MHD does not allow the
perpendicular poloidal flow to the poloidal magnetic field. In the
area of linear stability analysis of MHD, it is generally not possible
to give necessary and sufficient condition for stability of flowing
plasmas because the generator of flow dynamics becomes nonHermitian
(nonselfadjoint) operator, in which system the spectral
decomposition is not assured. Unlike the static case, the spectral
analysis (dispersion relation) of the generator to find mode
frequencies of perturbations does not lead to a complete understanding
of stability of flowing plasmas. Furthermore, the MHD equation, which
does not have an intrinsic scale length, is not capable of simulating
the dynamics associated with the smaller scale than the system size.
The small scale component may play an important role in various plasma
phenomena, such as coronal heating, magnetic reconnection, Hmode
confinement.
In order to describe magnetofluid plasma dynamics in more detail, one
can employ the twofluid effect. The twofluid model of a plasma
describes the coupling between the magnetic and the fluid aspects of
the plasmas, and then helps us understand a variety of structures
generated in plasmas. Neglecting the electron inertia, the twofluid
model is written by the Hall MHD. The Hall MHD is defined to be
standard onefluid MHD plus the Hall (current) term in the induction
equation (Ohm's law), therefore the Hall term is the principal term
distinguishing the twofluid model from the onefluid model. The Hall
term, usually assumed to be small, is expressed by a high spatial
derivative term (i.e., mathematically a singular perturbation of MHD)
that introduces a short characteristic length scale (the ion skin
depth) to the otherwise scaleless MHD. It becomes possible, thus, to
have equilibria in which related physical quantities can vary on
vastly different length scales.
Equilibrium of Flowing Plasma
The MHD restriction that the plasma flow can not deviate from the
magnetic field in the poloidal section produces the singularity in the
equilibrium equation in 2dimensional system. The singular
perturbation of Hall effect can remove this difficulty. The Hall term
allow a perpendicular flow and the equilibrium equation can be
expressed by the coupled elliptic partial differential equations for
the magnetic flux and the stream function without any singularities.
The equation can be solved in a bounded domain. The figure below is
an example of solution called "double Beltrami field," which is
considered as the relaxed state of Hall MHD.
Equilibrium of double Beltrami field; flux function and pressure.

Hierarchy of relaxed state

Lyapunov Stability of Flowing Plasmas
Stability of a plasma with a shear flow constitutes a very challenging
problem because the interaction between the perturbations and the
ambient flow cannot generally be cast in an appropriate Hamiltonian
form. The linear operator of flow dynamics become ``nonHermitian''
due to the fact the energy (conjugate to ``time'') corresponds to the
frequency of perturbations, and the frequency of perturbations in a
flow may assume complex values, i.e., the standard notion of energy
(Hamiltonian) does not pertain and the energy may cease to be the
basic determinant of the stability of the flow (it is even difficult
to define an appropriate energy for perturbations). In such a system,
The standard spectral analysis cannot provide a complete understanding
of stability in a nonHermitian system, and then, the stability
analysis of flowing plasmas, then, requires a wider framework. The
notion of a Lyapunov function is a natural extension of Hamiltonian.
We can show a general abstract theorem for the stability analysis of a
special class of flows, which is called Beltrami flow. In connection
with a variational principle characterizing the Beltrami flow, we have
a constant of motion that bounds the energy of perturbations and it
leads to a sufficient (Lyapunov) stability condition. This stability
condition suppresses any instability including nonexponential
(algebraic) growth due to nonHermitian generator; it also prohibits
nonlinear evolution. The key to prove is the coerciveness of the
constant of motion in the topology of the energy norm. The theory is
applied to Beltrami flows in MHD and the stability condition is
obtained as the right figure shows.

Lyapunov Stability condition of Beltrami flow.

However, this method fails to analyze the stability of double Beltrami
flows in twofluid MHD because the singular perturbation term (mainly
expressed by the Hall term) destroys the coerciveness of the constant
of motion. The stability analysis of the double Beltrami flows
requires a certain stronger (more coercive) constant, which
corresponds to an enstrophy order constant. It is shown that,
although the enstrophy is not generally a constant because of a
vortexstretching effect, an enstrophy order constant can be found for
two special classes of the double Beltrami flows, (I) Longitudinal
flow system with a spiral magnetic field, (II) Longitudinal magnetic
field system with a spiral flow. In the above two cases, a Lyapunov
function that bounds the energy of possible perturbations can be
constructed.
Hall current and Alfven wave
In MHD, the Alfven wave (the dominant low frequency mode of a
magnetized plasma) displays a continuous spectrum associated with
singular eigenfunctions.
However, in a more realistic treatment of the plasma, some singular
perturbation effects are expected to convert the continuous to the
point spectrum. Such qualitative changes in spectrum are relevant to
many plasma phenomena and lead interesting physical
phenomena. For example, adding the finite
resistivity leads the tearing mode instability represented by an
imaginary frequency point spectrum occurring through a singular
perturbation of the edge of the Alfven continuous
spectrum. Electron inertia or other kinetic effects
are also well known to resolve the Alfven singularity by leading to a
singular perturbation represented by a higher order derivative term to
the mode equation. Since the Hall term is added
to MHD as a singular perturbation, it may be expected to remove the
Alfven singularity of MHD. It is shown that the coupling of the
Hall current with the sound wave induces higher (fourth) order
derivative in the Alfven mode equation, and by resolving the
singularity replaces the MHD continuum by a discrete spectrum.
Hall effect on relaxation process of flowing plasmas
The Hall effect on the nonlinear dynamics of a flowing plasma has been
studied by comparing the magnetohydrodynamics (MHD) equations and the
Hall MHD equations. Numerical simulations of both systems show that
the turbulence brings about dissipation of the magnetic and kinetic
(flow) energies, and selforganization of large scale structures
occurs. However, the perpendicular flow to the magnetic field is
generated more effectively and the kinetic energy dissipates much
faster in the Hall MHD system. The enhanced energy dissipation is
primarily due to production of small scale fluctuations, which proves
the creation of scale hierarchy by the singular perturbation of the
Hall effect.
Presentation (pdf),
Simulation results 1,
Simulation results 2
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