M. S. Pekker

Theory of Alfvén eigenmodes in shear reversed plasmas

Plasma configurations with shear reversal are prone to the excitation of unusual Alfven eigenmodes by energetic particles. These modes exhibit a quasiperiodic pattern of predominantly upward frequency sweeping (Alfven cascades) as the safety factor q changes in time. This work presents a theory that employs two complementary mechanisms for establishing Alfven cascades: (1) a nonstandard adiabatic response of energetic particles with large orbits and (2) toroidal magnetohydrodynamic effects that are second-order in inverse aspect ratio. The developed theory explains the transition from Alfven cascades to the toroidicity induced Alfven eigenmodes (TAEs), including modifications of the TAEs themselves near the shear reversal point.

Plasma pressure effect on Alfvén cascade eigenmodes

Tokamak plasmas with reversed magnetic shear are prone to the excitation of Alfven cascade (AC) eigenmodes by energetic particles. These modes exhibit a quasiperiodic pattern of predominantly upward frequency sweeping. Observations also reveal that the AC spectral lines sometimes bend at low frequencies, which is a significant deviation from the shear Alfven wave dispersion relation. This paper shows that the underlying reasons for such bending are the finite pressure of the plasma and the geodesic curvature that precludes shear Alfven perturbations from being strictly incompressible. In addition to the geodesic effect, there are two other pressure effects on shear Alfven waves, which are the convection in the presence of an equilibrium pressure gradient and the toroidicity-induced coupling between shear Alfven waves and acoustic modes. An analytical treatment of the problem enables a parametric comparison of all three mechanisms. The key distinction between the geodesic compressibility and the acoustic c...

Spontaneous hole–clump pair creation

Numerical simulations and quantitative theoretical explanations are presented for the spontaneous formation of a hole–clump pair in phase space. The equilibrium is close to the linear threshold for instability and the destabilizing resonant kinetic drive is nearly balanced by either extrinsic dissipation or a second stabilizing resonant kinetic component. The hole and clump, each support a nonlinear wave where the trapping frequency of the particles is comparable to the kinetic linear growth rate from the destabilizing species alone. The power dissipated is balanced by energy extracted by trapped particles locked to the changing wave-phase velocities. With extrinsic dissipation, phase space structures always form just above the linear instability threshold. With a stabilizing kinetic component, an electrostatic interaction is considered with varying mass ratios of the stabilizing and destabilizing species together with collisional effects. With these input parameters, various nonlinear responses arise, on...

Critical nonlinear phenomena for kinetic instabilities near threshold

A universal integral equation has been derived and solved for the nonlinear evolution of collective modes driven by kinetic wave particle resonances just above the threshold for instability. The dominant nonlinearity stems from the dynamics of resonant particles that can be treated perturbatively near the marginal state of the system. With a resonant particle source and classical relaxation processes included, the new equation allows the determination of conditions for a soft nonlinear regime, where the saturation level is proportional to the increment above threshold, or a hard nonlinear regime, characterized by explosive behavior, where the saturation level is independent of the closeness to threshold. In the hard regime, rapid oscillations typically arise that lead to large frequency shifts in a fully developed nonlinear stage. The universality of the approach suggests that the theory applies to many types of resonant particle driven instabilities, and several specific cases, viz. energetic particle dr...

Numerical simulation of bump‐on‐tail instability with source and sink

A numerical procedure has been developed for the self‐consistent simulation of the nonlinear interaction of energetic particles with discrete collective modes in the presence of a particle source and dissipation. A bump‐on‐tail instability model is chosen for these simulations. The model presents a kinetic nonlinear treatment of the wave–particle interaction within a Hamiltonian formalism. A mapping technique has been used in this model in order to assess the long time behavior of the system. Depending on the parameter range, the model shows either a steady‐state mode saturation or quasiperiodic nonlinear bursts of the wave energy. It is demonstrated that the mode saturation level as well as the burst parameters scale with the drive in accordance with the analytical predictions. The threshold for the resonance overlap condition and particle global diffusion in the phase space are quantified. For the pulsating regime, it is shown that when γL≳0.16 ΔΩ, where γL is the linear growth rate for the unperturbed ...

Nonlinear response of driven systems in weak turbulence theory

A method is presented for predicting the saturation levels and particle transport in weakly unstable systems with a discrete number of modes. Conditions are established for either steady‐state or pulsating responses when several modes are excited for cases where there is and there is not resonance overlap. The conditions for saturation and the associated transport are discussed. Depending on parameters, the saturation level can be low, with only a small fraction of the available free energy released to waves and with no global transport, or the saturation level can be quite high, with almost complete conversion of free energy to wave energy coupled with rapid transport.

Simulation of Alfven-wave-resonant-particle interaction

New numerical simulations are presented on the self-consistent dynamics of energetic particles and a set of unstable discrete shear Alfven modes in a. tokamak. The code developed for these simulations has been previously tested in simulations of the bump-on-tail instability model. The code has a Hamiltonian structure for the mode-particle coupling, with the superimposed wave damping, particle source and classical relaxation processes. In the alpha-particle-Alfven-wave problem, we observe a transition from a single mode saturation to mode overlap and global quasi-linear diffusion, which is qualitatively similar to that observed in the bump-on-tail model. A considerable enhancement in the wave energy due to the resonance overlap is demonstrated. The effect of global diffusion on the energetic particle losses is also demonstrated

Evolution of toroidal Alfvén eigenmode instability in Tokamak Fusion Test Reactor

The nonlinear behavior of the toroidal Alfven eigenmode (TAE) driven unstable by energetic ions in the Tokamak Fusion Test Reactor (TFTR) [Phys. Plasmas 1, 1560 (1994)] is studied. The evolution of instabilities can take on several scenarios: a single mode or several modes can be driven unstable at the same time, the spectrum can be steady or pulsating, and there can be negligible or anomalous loss associated with the instability. This paper presents a comparison between experimental results and recently developed nonlinear theory. Many features observed in experiment are compatible with the consequences of the nonlinear theory. Examples include the structure of the saturated pulse that emerges from the onset of instability of a single mode, and the decrease, but persistence, of TAE signals when the applied rf power is reduced or shut off.

Alfven eigenmode induced energetic particle transport in JET

A Hamiltonian guiding centre particle following code has been developed to study the fast particle motion in the presence of arbitrary time dependent electromagnetic perturbations. In conjunction with an MHD stability code, this code was used to analyse TAE/KTAE induced alpha orbit diffusion and alpha losses in JET plasmas. Resonant alpha orbits are studied below and above the stochasticity thresholds, in the presence of single or several TAEs and KTAEs. Monte Carlo randomized ensembles of alpha particles in the presence of finite amplitude TAE/KTAEs are followed and estimates for the stochastic diffusion coefficients are obtained. Generalization of the method towards the self-consistent wave-particle evolution is described