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Spectral contents of electron waves under strong Langmuir turbulence
All in all, Microsoft PowerPoint says an utmost wood. The process of normalizing the equation of motion to the action-angle coordinate system is described in many books on nonlinear dynamics including Refs. To arrive to the case of the nonlinear beating wave - ion interaction we reduce the above given equation to two waves. To simplify our analysis even further, we assume that the waves are of equal magnitude and have the same wave number.
The best we can do is to solve the equation numerically or seek an approximate solutions using perturbation theory. The solutions are then graphed as Poincare sections . The figures above show typical Poincare sections for nonlinear beating wave - ion interaction. The section on the right was obtained by solving the Hamiltonian analytically using 2nd order Lie transformations . You can see that both figures agree with each other pretty well The analytical solution does not predict the existence of the stochastic random region — indicated by random points in the upper part of the right phase diagram.
In the past few years of working on the theoretical model we were able to derive the sufficient and necessary conditions for ion acceleration by beating electrostatic waves using the phase diagram approach. Introducing collisions into the theoretical model complicates the model even further remember, so far we were ignoring collisions in our model. An alternative way to deal with collisions is to write a Monte Carlo code .
This approach is explained in the next section. Numerical solutions provide us with a useful way of analyzing ion dynamics during wave — ion interaction. However, as was mentioned in the previous section, analytical solutions are only valid for small wave amplitudes. Speaking more precisely, the perturbation analysis used to obtain analytical solutions is only valid for small perturbation strength.
In addition, analytical solutions will predict random stochastic motion. On the other hand, numerical solutions, are valid for all wave amplitudes and do show the rich variety of dynamical behavior displayed by ions during the interaction. Also, a numerical solver provides us with an easy way to do the parametric studies of the phenomena. For example, we can investigate how variation in the wave amplitude affect the solution.
The above plots of angle versus normalized Larmor radius at 3 different wave amplitudes show the results of such an investigation. We can see clearly that when the wave amplitude is increased, the region of stochastic motion reaches lower values of velocity, r. Since stochastic acceleration is much faster than regular, that means that as the wave amplitude is increased the ions will be heated more and more efficiently.
While 4th order Runge-Kutta method could be used to solve the equation of motion to obtain the Poincare sections, we use a more accurate method known as 4th order Simplectic Integration Algorithm SIA4 . Relying on the conservation of the autonomous Hamiltonian , this method is specifically tailored for these types of problems.
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