Research Area
Lake Toya, Hokkaido

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 (H-mode, 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 flow-less MHD plasmas, for example, the Grad-Shafranov 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 2-dimension). 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 non-Hermitian (non-self-adjoint) 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, H-mode confinement.

In order to describe magneto-fluid plasma dynamics in more detail, one can employ the two-fluid effect. The two-fluid 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 two-fluid model is written by the Hall MHD. The Hall MHD is defined to be standard one-fluid MHD plus the Hall (current) term in the induction equation (Ohm's law), therefore the Hall term is the principal term distinguishing the two-fluid model from the one-fluid 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 scale-less 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 2-dimensional 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

Equilibrium of double Beltrami field; flux function and pressure.

Hierarchy of relaxed state

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 ``non-Hermitian'' 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 non-Hermitian 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 non-exponential (algebraic) growth due to non-Hermitian 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 two-fluid 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 vortex-stretching 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 self-organization 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

(Top Page)