The resonant interaction of particles and waves is of fundamental interest in plasma phenomena. In linear theory, this resonant interaction  produces the well known  effect of Landau damping. In this case wave damping arises for a distribution function that  decreases with increasing energy if there are  particles that move at nearly the same velocity as the  velocity of the wave. If the distribution is inverted in energy a laser-like phenomena arises. Waves can then be  spontaneously excited  and grow by extracting energy from the background distribution function. The growth continues until the source of particle free-energy is depleted, and this process may cause turbulent diffusion of the energetic particles.
                In  the past this problem was tackled by assuming that there was a continuum spectrum  of waves present.  Then the  turbulence can be described by the  well developed quasi-linear theory (one of the initial pioneering works in this field was developed by Prof. Drummond). In recent years it was realized that the understanding of the dynamics a single wave is essential importance for the containment of charged fusion products  in a tokamak. The investigation of this problem has led to both new basic insights of the nonlinear  plasma-wave interaction and important applications to the understanding of phenomena in fusion related experiments.
               The simplest case to consider is when a destabilizing kinetic drive is just barely strong enough to overcome the damping due to background dissipation.  Important questions to address  are: to what level will the wave grow and will the wave amplitude be steady or pulsate.
               Together with Boris Breizman (Prof. Berk's close collaborator), a threshold nonlinear equation was derived [Phys. Rev. Letters, 1256, (1996)] that describes near-threshold phenomena. It was found that  effects of collisions on resonant particles are more important in the nonlinear regime than in the linear regime, and a collision rate, , needs to be introduced into the resonant particle response. Then the threshold equation  is applicable to nearly any kinetic system near its instability threshold. The equation has a cubic nonlinearity and is nonlocal in time. Specifically the mode amplitude,  satisfies nonlinear integral equation which is of the form:

Solutions to these equation have been obtained numerically and analytically. A sample of numerical solutions are shown below:

            If the collision frequency is high enough ( > 4.4), a steady solution is found as shown in figure (1). Observe that a steady solution is reached, which is at a level predicted by theory.
 
 

When  <  4.4, analysis shows that the steady solutions are unstable, and a pulsating solution develops as shown in figure (2).

As is further lowered the pulsations become more complicated as observed in figures (3) and (4).

              The various levels of complication are examples of  pitchfork bifurcation phenomena. The understanding and classification of the various bifurcations still remains to be performed. Experimental evidence exists that such bifurcations are arising in experiments with Alfven waves observed in tokamak experiments and studies of this effect are currently in progress (see below).
 

            Finally quite a dramatic event occurs if  is lowered even further. The level of saturation is no longer bounded, but  the threshold equation produces an unbounded solution as shown in figure (5).
 

           .

             The blow-up of this equation reflects the breakdown of the threshold equation. More accurate wave-particle dynamics is then needed to describe the system and the next step in the understanding was obtained after a graduate student, Nicholas Petviashvili, developed a particle simulation code that showed that the wave grows to a level that causes the resonant particles to be "trapped" by the wave at the   amplitude where the trapping frequency is comparable to the linear growth rate  . This saturation level was at a level that was roughly expected, but what was unexpected was the observation that  the wave amplitude persists for long times (many damping times) with significant frequency shifting, as is  shown in the figures (6) and (7) below (the dotted curve in figure (7) is the theory prediction).

 Persistence of mode amplitude intime  ( horizontal axisis the number of damping times ).
 
 

 Fig 7.

            It was realized that this persistence is due to the development of phase space structures that allow waves to continually shift in frequency by a large amount .  The plots below gives the evidence for the formation of phase space structures.
 
 

This  plot shows how the distribution function averaged over space, evolves in time. We clearly see a trough in the distribution function moving to higher velocity in time and a ridge in the distribution function moving to lower velocity.

Fig 9.

Fig 10.

At times 120 and 220, we see that the phase space structure continually change their velocity. The upper structure is a phase space"bubble" rising, and the lower structure  a phase space "clump" falling. The color code reflects the distribution weight of the particle, Note that the colorof the  upper and lower phase space structures at time 120, are the same as the color that the structure was born with. This indicates that particles are trapped in the phase space structure so that they move  synchronously with the sructures' motion. The colors only blend later in time, due to collisions that mix the trapped distribution of particles (or holes) with its surrounding area.

             These results are now being used to interpret data in various experiments of the controlled fusion program. For example consider figure (11) below, which was taken on the TFTR experiment in Princeton. Energetic particles are created by ion cyclotron heating, and through the particle wave resonance, excite Alfven waves.  On the left hand side we see the spectral intensity vs. time. On the right hand figure we see a numerical solution of the threshold equation discussed above. Both figures show an abrupt change in the wave amplitude that arises when the ion cyclotron heating is suddenly terminated. The steady mode amplitudes observed prior  and after the turn-off are interpreted as a consequence of the steady solutions manifested in figure (1) above. However, when ion cyclotron waves are present, the stochastic heating corresponds to the larger effective collision frequency felt by resonant particles than when the waves are absent. The theory predicts that the steady solution should lower  when heating is turned off. This effect is  is observed in experiment (figure 11a) and in the numerical solution (figure 11b).
 
 

 Theoretical fit of the threshold equation's solution for the mode amplitude of each spectral line.
 

            At yet smaller collision frequency the explosive behavior can evolve into  a phase space structure that sweeps in frequency. We presently evaluating several pieces of experimental data (again involving Alfven waves) that may be examples of such frequency sweeping. One case has been observed by Snipes on the CMOD experiment. The data is shown on figure (14). We see that a signal with a rapid frequency shift is measured as the frequency doubles in 10 ms. Work is in progress to determine if our theory is indeed explaining this effect.

Rapid frequency change observed of a sharp line in the Alfven wave frequency regime observed  in the CMOD experiment at MIT (Courtesy of Joseph Snipes)
 

               Phase space structures can  arise for other waves and in non-tokamak experiments. Impressive frequency sweeping effects (see figure 15 below) are observed on the Terrella plasma  dipole experiment at Columbia University.

Fig  15.

Rapidly changing frequencies observed of the Terrella experiment at Columbia University (Courtesy of Micheal Mauel)
 

               We are contining to analyze frequency bifurcations and sweeping phenomena that we expect to be generated by kinetically driven instabilities. We are interested in this phenomena in fusion laboratory experiments, space phenomena and in particle accelerators. Further, we intend to extend the investigation to determine how frequency sweeping may be important in more general turbulence theory. For example, what is the effect of frequency sweeping on anomalous particle transport? This is an area where we have been quite active in, and where there are many questions still to be answered.
 
 
 

               Many question still remain to be addressed in the research area described above. For further information feel free to contact  Prof. Berk   or  Dr. Breizman .