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Author(s):
H. Sugama
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Title:
Damping of Toroidal Ion Temperature Gradient Modes
Date of publication:
Apr. 1999
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Key words:
toroidal ion temperature gradient mode, ballooning representation, analytic continuation, branch cut, normal mode, continuum mode
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Abstract:
The temporal evolution of linear toroidal ion temperature gradient (ITG) modes is studied based on a kinetic integral equation including an initial condition. It is shown how to evaluate the analytic continuation of the integral kernel as a function of a complex-valued frequency, which is useful for analytical and numerical calculations of the asymptotic damping behavior of the ITG mode. In the presence of the toroidal nabla B-curvature drift, the temporal dependence of the density and potential perturbations consists of normal modes and a continuum mode, which correspond to contributions from poles and from an integral along a branch cut, respectively, of the Laplace-transformed potential function of the complex-valued frequency. The normal modes have exponential time dependence with frequencies and growth rates determined by the dispersion relation while the continuum mode, which has a ballooning structure, shows a power law decay propto t^-2 in the asymptotic limit, where t is the time variable. Therefore, the continuum mode dominantly describes the long-time asymptotic behavior of the density and potential perturbations for the stable system where all normal modes have negative growth rates. By performing proper analytic continuation for the homogeneous version of the kinetic integral equation, dependences of the normal modes' growth rate, real frequency, and eigenfunction on eta_i (the ratio of the ion temperature gradient to the density gradient), k_ theta (the poloidal wavenumber), hat{s} (the magnetic shear parameter), and theta_k (the ballooning angle corresponding to the minimum radial wavenumber) are numerically obtained for both stable and unstable cases.
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