(a) Dispersion relation for an , wave in a cylindrical plasma. The phase velocity is a strong function of radial position. (b) A hypothetical disturbance launched in the highlighted region. The pulse will rapidly disperse and shear.
Comparison of an optical fiber with a transmission gap for visible light and a plasma with a transmission gap for shear Alfvén waves. The fiber has a periodically modulated index of refraction in its core. The plasma has a variable magnetic field that results in periodic modulation of the index of refraction . The spatial period of the modulations is . Both systems have a propagation gap at the Bragg frequency and the width of the propagation gap is proportional to the amplitude of modulation of .
(a) Dispersion relation of two waves with toroidal mode number and poloidal harmonics of and in a plasma with a profile that increases monotonically with radius. Frequencies are plotted as positive for both signs of . The frequency gap increases with radius because increases, which increases the modulation of . The two waves are counterpropagating in the frequency gap. (b) Dispersion relation with (solid) and without (dashed) toroidal coupling of the waves. The and waves are also shown and intersect at larger radii. Based on Ref. 19.
Calculated Alfvén gap structure for an experimental stellarator equilibrium. The “T” (toroidicity), “E” (ellipticity), and “N” (noncircularity) induced gaps are labeled. In the region marked with the bold square, the gaps are caused by both “N” and “H” helicity-induced modulations. Adapted from Ref. 25.
Frequency (left) and mode structure (right) for three different types of Alfvén waves. The waves are excited by an antenna at the edge of the plasma. (a) A continuum wave has a very narrow mode structure that is strongly damped. (b) If the plasma profiles create an extremum in the Alfvén continuum, the antenna may excite a gap mode that is located near the extremum; this wave has (predominately) a single poloidal harmonic. (c) The antenna can also excite gap modes near the extrema created by mode coupling; in this case, the poloidal harmonics of the coupled waves appear. Adapted from Ref. 13.
Radial profile of electron temperature fluctuations as measured in the DIII-D tokamak. The profile is also shown. RSAEs with different toroidal mode numbers are located near . More globally extended TAEs are also observed. Adapted from Ref. 28.
Experimental data from the Joint European Torus (JET) tokamak showing the evolution of the center of the TAE gap (labeled ) and the amplitude of the magnetic field observed as an antenna sweeps its frequency across the gap. A normal mode is excited that exhibits weak damping (inferred from the quality factor of the resonance) and follows the evolution of the gap frequency in time. Adapted from Ref. 119.
Tomographic reconstructions of soft x-ray fluctuation data obtained on the Wendelstein 7-AS stellarator. The reconstruction on the left is of extremum type, a GAE with a single dominant poloidal harmonic. The reconstruction on the right shows the structure associated with two coupled harmonics ( and 6); the harmonics interfere constructively on the outside of the device and destructively on the inside. Adapted from Refs. 34 and 35.
Projection of the orbit of an deuterium beam ion in the DIII-D tokamak. (a) Elevation. The dashed lines represent the magnetic flux surfaces. The particle orbits poloidally with a frequency . (b) Detail of the beginning of the orbit. The rapid gyromotion, parallel drift along the flux surface, and vertical drift velocity are indicated. (c) Plan view of the orbit. The particle precesses toroidally with a frequency .
Classification of different orbit types for beam ions in the DIII-D tokamak vs magnetic moment and canonical angular momentum . The poloidal flux at the wall is , the particle energy is , and the magnetic field at the magnetic axis is . Particles that move outward from the magnetic axis move leftward on the axis. Four types of EP transport are illustrated. (Red rectangle) The wave can perturb the equilibrium , causing particles near a loss boundary to collide with the wall. (Green triangle) Particles that stay in phase with a mode throughout the plasma can convectively escape via the drift. (Blue diamond) Particles can diffuse as they receive velocity kicks associated with the many wave-particle resonances in the plasma. (Purple circles) If the EPs move outward, they can locally alter the EP gradient and destabilize a new wave that transports them further, where a new wave is destabilized, etc.
Calculated resonances for rf-accelerated tail ions in the JET tokamak in the energy/toroidal-angular-momentum plane. The toroidal and poloidal values of the resonances are labeled. The amplitude scale takes into account the probability of detection by a gamma-ray diagnostic. Adapted from Ref. 51.
Illustration of the dependence of EP drive (or damping) on the slope of the distribution function. (a) Classic Landau damping situation: for a monotonically decreasing distribution function there are more particles that gain energy from the wave than lose energy, so the wave damps. (b) The energy distribution typically is monotonically decreasing, so the wave damps. (c) The distribution function is usually peaked on axis. The toroidal angular momentum has the opposite dependence on radius than the flux function, so a peaked distribution function has a positive gradient and gives net energy to the wave.
(a) Experimental magnetics data from the JT-60U tokamak during beam injection showing bursts of Alfvén activity. Modeling shows that the smaller amplitude bursts are (c) TAEs with frequencies in the gap of the Alfvén continuum but the large amplitude modes with rapid frequency chirping are (b) EPMs with frequencies that can lie in the continuum. [In the original experimental papers, the TAEs were called “fast frequency-sweeping” modes and the EPMs were called “abrupt large-amplitude events (ALE).”] Adapted from Refs. 73 and 74.
Examples of two phenomena that are successfully explained by the Berk–Breizman model. (a) In the JET tokamak, modes that are driven unstable by rf-accelerated tail ions undergo frequency splitting. This behavior is predicted when the resonant particles are scattered quickly out of resonance (large ). Adapted from Ref. 85. (b) In the MAST spherical tokamak, modes that are driven unstable by neutral beam ions begin at the TAE frequency, then chirp upward and downward in frequency. This behavior is predicted for waves with low values of . From Ref. 87.
The radial fast-ion profile measured during Alfvén activity in the DIII-D tokamak is much flatter than classically predicted (dashed line). The data are from equilibrium reconstructions (solid line) and fast-ion D-alpha measurements (symbols). Adapted from Ref. 101.
Bicoherence of a reflectometer signal during beam injection into the NSTX spherical tokamak. The analysis indicates that TAE waves with frequencies between interact nonlinearly with a low-frequency mode. Adapted from Ref. 107.
Calculated mode structure that is consistent with experimental data during beam injection into the DIII-D tokamak. Very high toroidal mode numbers with a spatial scale that approaches the gyroradius of thermal ions are inferred. Adapted from Refs. 108 and 120.
Example of control of an EP-driven instability. (This instability is an interchange mode, not an Alfvén wave.) Energetic electrons in a dipole experiment drive the instability and cause frequency sweeping that is consistent with the Berk–Breizman model. The application of of rf power that increases the effective collisionality of the wave-trapped electrons alters the nonlinear dynamics, replacing the large bursts with relatively steady noise. Reused with permission from Ref. 114. Copyright © 2003 American Institute of Physics.
Nomenclature of shear Alfvén eigenmodes, listed in ascending (approximately) order of frequency. For coupling-type eigenmodes, the coupled poloidal or toroidal harmonics are given; for extremum-type eigenmodes, the source of the extremum is underlined. The citations are to the original theoretical and experimental papers.
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