Abstract
A nonperturbative kinetic/magnetohydrodynamics eigenvalue code has been constructed for calculation of kinetic damping of shear Alfvén eigenmodes in general tokamak geometry with finite pressure. The model describes shear Alfvén waves with kinetic effects from thermal species including thermal ion finite Larmor radius effects and parallel electric field. An analytic formula for the radiative damping of reversed shear Alfvén eigenmodes is obtained for tokamak plasmas with reversed shear profile. Numerical calculations reveal the existence of multiple kinetic reversed shear Alfvén eigenmodes (KRSAEs). It is found that the damping rate of the KRSAEs scales linearly with the thermal ion gyroradius. The damping rates are larger for modes with more peaks in the radial structures. These results are consistent with analytic expectation. The KRSAEs found here can be used to interpret the RSAEs frequency sweeping down observed sometime in the tokamak experiments.
One of the authors (L. Yu) would like to thank Professor Liu Chen, Professor Jiaqi Dong, Professor Zhiwei Ma, and Professor Jiquan Li for helpful discussions. Part of the work was done while the author (L. Yu) was visiting Princeton Plasma Physics Laboratory (PPPL). The author (L. Yu) thanks the hospitality of PPPL. We also thank Dr. Nikolai Gorelenkov for useful discussions. This work is supported by the Chinese Scholarship Council, the U.S. Department of Energy under Contract No. DEA C0276Ch03073, the NNSF of China under Grant No. 10675102, and the National Basic Research Program under Grant No. 2008CB717806.
I. INTRODUCTION
II. REDUCED KINETIC MHD MODEL
III. NUMERICAL METHOD FOR THE EIGENMODE EQUATION
IV. CODE BENCHMARK
V. RESULTS
A. Radiative damping of RSAEs
B. Kinetic reversed shear Alfvén eigenmodes
VI. CONCLUSIONS
Key Topics
 Normal modes
 22.0
 Magnetohydrodynamics
 15.0
 Eigenvalues
 13.0
 Tokamaks
 13.0
 Tantalum
 11.0
Figures
The eigenfunction of GAEs: vs for the previous code (solid line), where the eigenvalue is ; in this code (open circles), the eigenvalue is , where is the minor radius and is the frequency in the center of tokamaks.
The eigenfunction of GAEs: vs for the previous code (solid line), where the eigenvalue is ; in this code (open circles), the eigenvalue is , where is the minor radius and is the frequency in the center of tokamaks.
The eigenfunction vs for constant density, pressure profile: , safety factor profile: (where , , , ), mode numbers: , inverse aspect ratio: . (a) ; (b) ; (c) ; and (d) .
The eigenfunction vs for constant density, pressure profile: , safety factor profile: (where , , , ), mode numbers: , inverse aspect ratio: . (a) ; (b) ; (c) ; and (d) .
The eigenvalue vs pressure gradient for the same parameters as in Fig. 2.
The eigenvalue vs pressure gradient for the same parameters as in Fig. 2.
(a) The safety factor ; (b) the pressure profile.
(a) The safety factor ; (b) the pressure profile.
The eigenfunction vs normalized radius for the parameters and profiles selected as constant density, profile: , parabolic pressure profile: , inverse aspect ratio: , mode numbers: . (a) for ; (b) for ; (c) for ; and (d) for .
The eigenfunction vs normalized radius for the parameters and profiles selected as constant density, profile: , parabolic pressure profile: , inverse aspect ratio: , mode numbers: . (a) for ; (b) for ; (c) for ; and (d) for .
The damping rate vs Larmor radius of thermal ion. The functions of damping rates are shown in two curves: solid and dashed ones by using this code and the previous code, respectively.
The damping rate vs Larmor radius of thermal ion. The functions of damping rates are shown in two curves: solid and dashed ones by using this code and the previous code, respectively.
The eigenfunctions of RSAEs for . (a) The eigenfunction for , ; (b) the eigenfunction for , . Here, is real part of the eigenvalue and is imaginary part of the eigenvalue.
The eigenfunctions of RSAEs for . (a) The eigenfunction for , ; (b) the eigenfunction for , . Here, is real part of the eigenvalue and is imaginary part of the eigenvalue.
(a) Damping rate vs normalized gyroradius for . The dash curve represents a fitting curve for electron damping and the solid one is total damping. (b) The logarithmic damping rate of radiative damping vs normalized inverse gyroradius . The solid curve is a linear fitting curve. (c) Damping rate vs normalized gyroradius for . The dash curve represents a fitting curve for electron damping and the solid one is total damping. (d) The logarithmic damping rate of radiative damping vs normalized inverse gyroradius . The solid curve is a linear fitting curve.
(a) Damping rate vs normalized gyroradius for . The dash curve represents a fitting curve for electron damping and the solid one is total damping. (b) The logarithmic damping rate of radiative damping vs normalized inverse gyroradius . The solid curve is a linear fitting curve. (c) Damping rate vs normalized gyroradius for . The dash curve represents a fitting curve for electron damping and the solid one is total damping. (d) The logarithmic damping rate of radiative damping vs normalized inverse gyroradius . The solid curve is a linear fitting curve.
(a) vs ; (b) vs .
(a) vs ; (b) vs .
(a) The eigenmode structure of KRSAEs for . (b) The eigenmode structure of KRSAEs for . (c) The real part of the eigenvalue vs radial number of grids for the cases of Fig. 10(a) (open squares) and Fig. 10(b) (open circles). (d) The imaginary part of the eigenvalue vs radial number of grids for the cases of Fig. 10(a) (open squares) and Fig. 10(b) (open circles).
(a) The eigenmode structure of KRSAEs for . (b) The eigenmode structure of KRSAEs for . (c) The real part of the eigenvalue vs radial number of grids for the cases of Fig. 10(a) (open squares) and Fig. 10(b) (open circles). (d) The imaginary part of the eigenvalue vs radial number of grids for the cases of Fig. 10(a) (open squares) and Fig. 10(b) (open circles).
Half width of a singlepeak eigenmode vs squared root of normalized ion Larmor radius for , .
Half width of a singlepeak eigenmode vs squared root of normalized ion Larmor radius for , .
Damping rate of a singlepeak eigenmode vs normalized ion Larmor radius for .
Damping rate of a singlepeak eigenmode vs normalized ion Larmor radius for .
(a) The eigenfunction of KRSAEs with two peaks for . (b) The eigenfunction of KRSAEs with four peaks for the same parameters in Fig. 13(a). (c) The damping rates vs normalized ion Larmor radius for KRSAEs with single peaks (open circles), two peaks (open squares), and four peaks (open triangles), respectively, for .
(a) The eigenfunction of KRSAEs with two peaks for . (b) The eigenfunction of KRSAEs with four peaks for the same parameters in Fig. 13(a). (c) The damping rates vs normalized ion Larmor radius for KRSAEs with single peaks (open circles), two peaks (open squares), and four peaks (open triangles), respectively, for .
The eigenmode structure of RSAEs for .
The eigenmode structure of RSAEs for .
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