Mechanisms of plasma rotation effects on edge magnetohydrodynamic (MHD) stability are investigated numerically by introducing energies that are distinguished by physics. By comparing them, it is found that an edge-localized MHD mode is destabilized by the difference between the eigenmode frequency and the equilibrium toroidal rotation frequency, which is induced by rotation shear. In addition, this destabilizing effect is found to be effective in the shorter wavelength region. The effect of poloidal rotation on the edge MHD stability is also investigated. Under the assumption that the change in equilibrium by poloidal rotation is negligible, it is identified numerically that poloidal rotation can have both stabilizing and destabilizing effects on the edge MHD stability, which depends on the direction of poloidal rotation. A numerical analysis demonstrates that these effects of plasma rotation in both the toroidal and poloidal directions can play important roles in type-I edge-localized mode phenomena in JT-60U H-mode plasmas.

Effects of toroidal rotation and density profiles on the stability of an edge localized MHD mode are investigated numerically. From the numerical results we show that both the density gradient and the sheared rotation profile can destabilize the edge ballooning mode and the peeling–ballooning mode, and particularly, the sheared rotation can destabilize these modes effectively. To clarify the mechanisms of these destabilizing effects of density gradient and sheared rotation, we define some energies and distinguish them by physics. By comparing these energies, we clarify that the destabilization by the density gradient can be explained as the destabilizing effect of the centrifugal instability, and that by the sheared toroidal rotation is induced mainly by the difference between the eigenmode frequency and the toroidal rotation frequency of the plasma. Although the strong rotation shear also has a stabilizing effect on the MHD modes by changing the mode structure, the edge MHD mode first becomes unstable due to the appearance of the destabilizing effect before changing the mode structure.

The stability of stably stratified vortices is studied by local stability analysis. Three base flows that possess hyperbolic stagnation points are considered: the two-dimensional (2-D) Taylor–Green vortices, the Stuart vortices and the Lamb–Chaplygin dipole. It is shown that the elliptic instability is stabilized by stratification; it is completely stabilized for the 2-D Taylor–Green vortices, while it remains and merges into hyperbolic instability near the boundary or the heteroclinic streamlines connecting the hyperbolic stagnation points for the Stuart vortices and the Lamb–Chaplygin dipole. More importantly, a new instability caused by hyperbolic instability near the hyperbolic stagnation points and phase shift by the internal gravity waves is found; it is named the strato-hyperbolic instability; the underlying mechanism is parametric resonance as unstable band structures appear in contours of the growth rate. A simplified model explains the mechanism and the resonance curves. The growth rate of the strato-hyperbolic instability is comparable to that of the elliptic instability for the 2-D Taylor–Green vortices, while it is smaller for the Stuart vortices and the Lamb–Chaplygin dipole. For the Lamb–Chaplygin dipole, the tripolar instability is found to merge with the strato-hyperbolic instability as stratification becomes strong. The modal stability analysis is also performed for the 2-D Taylor–Green vortices. It is shown that global modes of the strato-hyperbolic instability exist; the structure of an unstable eigenmode is in good agreement with the results obtained by local stability analysis. The strato-hyperbolic mode becomes dominant depending on the parameter values.

Energy of waves (or eigenmodes) in ideal fluid and plasma is formulated in the noncanonical Hamiltonian context. By imposing the kinematical constraint on perturbations, the linearized Hamiltonian equation provides a formal definition of wave energy not only for eigenmodes corresponding to point spectra but also for singular ones corresponding to continuous spectrum. The latter becomes dominant when mean fields have inhomogeneity originating from shear or gradient of the fields. The energy of each wave is represented by the eigenfrequency multiplied by the wave action which is nothing but the action variable and, moreover, is associated with a derivative of suitably defined dispersion relation. The sign of the action variable is crucial to the occurrence of Hopf bifurcation in Hamiltonian systems of a finite degrees of freedom (Krein 1950). Krein's idea is extended to the case of coalescence between point and continuous spectra.

Linear stability of equilibrium states with flow is studied by means of the variational principle in Hall magnetohydrodynamics (MHD). The Lagrangian representation of the linearized Hall MHD equation is performed by considering special perturbations that preserves some constants of motion (the Casimir invariants). The resultant equation has a Hamiltonian structure which enables the variational principle. There is however some difficulties in showing the positive definiteness of the quadratic form in the presence of flow. The dynamically accessible variation is a more restricted class of perturbations which, by definition, preserves all the Casimir invariants. For such variations, the quadratic form (the second variation of Hamiltonian) can be positive definite. Some conditions for stability are derived by applying this variational principle to the double Beltrami equilibrium.

Action-angle variables corresponding to singular (or improper) eigenmodes are rigorously formulated for the Alfvén and slow (or cusp) continuous spectra of ideal magnetohydrodynamics. The perturbation energy is then transformed into the normal form, namely, the eigenfrequency multiplied by the action variable. It is shown that the Laplace transform approach expedites this action-angle formulation more efficiently than the existing ones devoted to other kinds of continuous spectra. The presence of flow that is either nonparallel to the magnetic field or supersonic at some places brings about singular eigenmodes with negative energy. The Alfvén and slow singular eigenmodes are neutrally stable even in the presence of any external potential fields, but may cause instability when coupled with another singular or nonsingular eigenmode with the opposite sign of energy.

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