Self-consistent dynamics of nonlinear phase space structures
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This thesis investigates the self-consistent dynamics of nonlinear ”hole” and ”clump” phase space structures and the nonlinear modes supported by the structures in the presence of dissipation due to the background plasma. A system consisting of a single mode driven by a weakly destabilizing distribution function in a dissipative medium close to the threshold of linear instability exhibits explosive instability. This instability results in the formation of the phase space structures and the corresponding modes. The holes and clumps were expected to persist for an appropriate collisional time scale. However, for certain initial conditions Fokker-Planck calculations for the nonlinear system abruptly break down in the course of the calculation. We find that this is because an adiabatic description of phase space structures at zero collisionality does not necessarily lead to continual adiabatic frequency sweeping. For a class of initial distribution functions criteria are found that detervii mine whether adiabatic frequency sweeping will continue indefinitely or suddenly terminate. For certain other initial distribution functions that describe the predominantly deeply trapped particles, critical points may be encountered where the adiabatic analysis fails to yield a unique solution. Except for establishing boundary conditions, the contribution of passing particles is found to be unimportant in the dynamics of the phase space structures within the framework of the adiabatic description. We derived a self-consistent dispersion relation for the perturbed eigenmodes of the system and benchmarked the result with the dispersion relation obtained earlier in Ref.  and demonstrated their agreement. We analyzed this dispersion relation and demonstrated that the critical points of the adiabatic theory occur exactly where linear instability is triggered. Numerical runs were performed to test both the adiabatic theory and the instability analysis of a BGK (Bernstein-Greene-Kruskal) mode for a model problem where the distribution function of passing particles has zero slope with respect to the action variable. This problem has same essential features as the problem where the slope of the passing particle distribution function is constant. In particular, linear instability of the same nature is also predicted to arise whenever adiabatic analysis predicts termination of frequency sweeping. This procedure has the virtue of enabling a precise comparison of the theory with the simulations and indeed it does so until instability sets in. The model problem was also used to demonstrate the agreement between the numerical growth rate and the growth rate predicted in the instability analysis. Then a passing particle distribution function was used that has a constant slope with respect to the action variable, and it too showed agreement with the theory for the evolution of the adiabatic phase, for where the onset of the instability was predicted to occur, and for where the persistence of the phase space structures after the instability relaxation. Both cases showed that after the instability dies away, smaller phase space structures still persist and the frequency sweeping continues at a slower rate. The numerical simulations demonstrated the additional effect that several generations of the nonlinear phase space structures are often produced. The numerical data shows that the mode amplitude is reduced when there are neighboring modes. We considered two possible mechanisms that may account for such reduction of the primary mode amplitude. In one mechanism, the particle orbits remain regular and the mode amplitude reduction is caused by the accumulation (in the case of a clump) [or depletion in the case of a hole] of the passing particle distribution of one of the modes, because it is part of the trapped particle region of the other mode. The other mechanism is the chaotic erosion of trapped particles near the separatrix.