Showing papers by "Herbert L Berk published in 1995"

Line broadened quasi-linear burst model [fusion plasma]

Abstract: A quasi-linear model is developed to produce realistic self-consistent saturation levels when modes do not overlap, and give self-consistent diffusion and wave evolution when modes do overlap. Both regimes give steady or pulsating behaviour in weakly driven systems with classical relaxation and background dissipation present. An avalanche response is demonstrated: wave momentum release caused by the overlap of closely spaced modes can produce mode overlap of more widely spaced modes (a domino effect) or the growth of modes which would be stable in systems unaffected by the diffusion of closely spaced modes. Detailed analysis and calculations are performed for the bump-on-tail instability, and extension of the method to more general problems is briefly discussed.

More on core‐localized toroidal Alfvén eigenmodes

Abstract: A novel type of ideal toroidal Alfven eigenmode, localized in the low‐shear core region of a tokamak plasma, is shown to exist, whose frequency is near the upper continuum of the toroidal Alfven gap. This mode converts to a kinetic‐type toroidal Alfven eigenmode above a critical threshold that depends on aspect ratio, pressure gradient, and shear. Opposite to the usual ideal toroidal Alfven eigenmode, this new mode is peaked in amplitude on the small‐major‐radius side of the plasma.

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

Abstract: 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 ...

Simulation of Alfven wave-resonant particle interaction

Abstract: 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

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

Abstract: This paper presents results of the simulations of the bump-on-tail instability with a weak source and sink. This problem has been posed as a paradigm for the important problem in controlled fusion, that of the unstable excitation of Alfven waves in a tokamak by resonant energetic alpha particles. The source of alpha particles is the controlled fusion reaction produced by the background plasma and the sink is the collisional transport processes that slow down or scatter the energetic particles. The mathematical techniques that are needed to address this applied problem can be demonstrated in the much simpler bump-on-tail problem, which is explained in this paper.

Alfven eigenmode induced energetic particle transport in JET

Abstract: 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

Orbit width scaling of TAE instability growth rate

Abstract: The growth rate of toroidal Alfven eigenmodes (TAEs) driven unstable by resonant coupling of energetic charged particles is evaluated in the `ballooning` limit over a wide range of parameters. All damping effects are ignored. Variations in orbit width, aspect ratio and the ratio of Alfven velocity to energetic particle `birth` velocity are explored. The relative contribution of passing and trapped particles, and finite Larmor radius effects, are also examined. The phase space location of resonant particles that interact strongly with the modes is described. The accuracy of the analytic results with respect to growth rate magnitude and parametric dependence is investigated by comparison with numerical results

Nonlinear response of driven systems in weak turbulence theory

Abstract: A method is presented for predicting the saturation levels and particle transport in weakly unstable systems where there are 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 achieving different levels of saturation are discussed. Depending on details, the saturation level can be quite low, where only a small fraction of the available free energy is released to waves, or the saturation level can be quite high, with almost a complete conversion of free energy to wave energy coupled with rapid transport.

Simulation of Alfven wave-resonant particle interaction

Abstract: 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. Our code developed for these simulations has been previously tested in the 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 the mode overlap and global quasilinear diffusion, which is qualitatively similar to that observed in the bump-on-tail model. We demonstrate a considerable enhancement in the wave energy due to the resonance overlap. We also demonstrate the effect of global diffusion on the energetic particle losses.

Nonlinear dynamics of a driven mode near marginal stability

Abstract: The nonlinear dynamics of a linearly unstable mode in a driven kinetic system is investigated to determine scaling of the saturated fields near the instability threshold. To leading order, this problem reduces to solving an integral equation with a temporally nonlocal cubic term. This equation can exhibit a self-similar solution that blows up in a finite time. When the blow-up occurs, higher nonlinearities become important and the mode saturates due to plateau formation arising from particle trapping in the wave. Otherwise, the simplified equation gives a regular solution that leads to a different saturation scaling reflecting the closeness to the instability threshold.