In the H-mode[1] pedestal region, magnetohydrodynamic (MHD) activities, called edge localized modes (ELMs)[2], often occur. The ELMs are usually categorized in several types. Ideal (non-dissipative) MHD instabilities are considered to cause type-I ELM[2].
Plasmas in the pedestal region of tokamaks often rotate. It was theoretically found that toroidal rotation shear stabilizes ideal MHD high-n ballooning modes (n: toroidal mode number)[3-6]. We found numerically that the stabilization of the mode comes from the damping of the perturbation energy of high-n ballooning modes due to toroidal rotation shear[6]. However, it has not been studied how the toroidal rotation shear damps the perturbation energy.
For clarifying the mechanism of the damping, the ballooning equations[4], solved numerically in Ref.[6], are difficult to treat analytically, since they are coupled wave equations for two components of the displacement vector. Thus, for simplifying the equations, we considered the incompressible limit, since it was shown that the toroidal rotation shear stabilizes the ballooning modes in that limit[3,5]. However, the incompressible ballooning equations are also coupled wave equations. Thus, it is still difficult to treat analytically.
In the present paper, we propose a model equation for analyzing the mechanism of the damping,
ρ[|k|²∂²ξ/∂t²-2(k·∇Ω)∂ξ/∂t]=B²/μ₀B·∇(|k|²/B²B·∇ξ)+2/B²(B×k·κ)(B×k·∇p)ξ. (1)
Here, ρ is the mass density, k=∇ζ-q∇θ-(θ-θ_k+dΩ/dq t)∇q is the wave vector, ζ is the toroidal angle, θ is the poloidal angle, θ_k is the ballooning angle, q is the safety factor, ξ is the displacement perpendicular to the magnetic field and wave vector, Ω is the toroidal rotation frequency, B is the magnetic field, μ₀ is the vacuum permeability, κ is the magnetic curvature, p is the pressure. The inertia and line bending terms (the left-hand side and the first term of the right-hand side of Eq.(1), respectively) are the same with those of the perpendicular component of the incompressible ballooning equation. The potential term (the last term) is the same with that of the commonly-used incompressible ballooning equation for a static plasma.
We solved Eq.(1) numerically, and found that the perturbation energy damps similar to Ref.[6]. We will systematically summarize the property of Eq.(1); whether the solution of Eq.(1) is similar to or different from the original ballooning equations either qualitatively or quantitatively. We will also analyze Eq.(1) by using asymptotic perturbation methods.

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