For my thesis research I have worked on the problem studying the scaling of unstable electrostatic waves in collisionless plasmas. From the dynamical systems point of view even the simplest examples of instabilities as described by Vlasov-Poisson system, have many unusual features related to the Hamiltonian character of the dynamics and the central role played by the neutral continuous spectrum in the appearance on the unstable modes. These novelties are not present solely in Vlasov theory; analogous features arise in simple models of unstable shear flows described by the Euler equation, populations of coupled oscillators and in stability calculations for certain classes of solitons. The analysis described below would, therefore be applicable to these systems as well.

Nonlinear scaling of Unstable Electrostatic Waves:

The basic model for a collisionless plasma describes the dynamics of each mobile species with a kinetic equation (the Vlasov equation), and these equations are coupled by the self-consistently determined electric field. An equilibrium solution for the model is a charge distribution which is homogeneous in space. Depending on stability of the equilibrium distribution ( i.e, whether it has any unstable eigenvalues or not) a perturbation to the equilibrium distribution grows or decays. When the equilibrium distribution does support an unstable mode, the electric field produced by the perturbation grows exponentially as $\exp(\gamma t)$, where $\gamma$ is the linear growth rate determined by the equilibria that was chosen.As the mode grows, nonlinear effects eventually become important and act to halt the instability. The nonlinear evolution of such an unstable electrostatic mode is a fundamental problem in plasma physics.

We analyze the problem perturbatively around an unstable equilibria. We simplify the problem by restricting the dynamics along the two-dimensional invariant unstable manifold.

Since the growth rates are small near the onset of the instability, nonlinear effects act to saturate the instability before the amplitudes grow appreciably. For this reason the nonlinear evolution of the unstable mode amplitude, near the onset of the linear instability can be described using an expansion in the amplitude A of the unstable modes.

 

\begin{displaymath} \frac{dA}{dt}= \gamma A + \sum_{j=1}^{\infty} p_j A \vert A\vert^{2j}\end{displaymath}

 

In the limit of weak instability $\gamma\rightarrow0^+$ the coefficients in the expansion pj become singular. These singularities can be absorbed by suitable rescaling transformations. These rescaling transformations provide information about the overall scaling of the nonlinear evolution [2].

Numerical calculations show that the scaling of saturation amplitude of the electric field in the case of fixed-ions ( $m_{ions}=\infty$) is $E\sim \gamma^2$. The general expectation was that this scaling should not change even when the ions are allowed to move. However amplitude equations analysis for the multiple-species Vlasov equation (i.e. with mobile ions) we find that the scaling of electric field changes in the presence of mobile ions [1]

\begin{displaymath}E \sim \gamma^{5/2}.\end{displaymath}

Further, from the singularity structure of the amplitude equation, the asymptotic features of the electric field and the charge distribution function can be determined. The asymptotic electric field is monochromatic (i.e. has only single Fourier component) at the wavelength of the unstable linear mode, with a nonlinear time dependence. The structure of the distribution function outside the resonant region is given by the linear eigenfunction but in the resonant region the distribution has a nonlinear dependence on the mode amplitude A [3]. This physical picture corresponds to the single wave model (SWM) originally proposed by O'Neil, Winfrey and Malmberg for the interaction of a cold weak beam with a plasma of fixed ions [5]. However the result we obtain is for a more general situation, with the only restriction on the distribution being that it supports one unstable mode.

The above physical picture can be used to construct a generalized single wave model for the electrostatic instabilities. The generalized SWM retains all the essential features of the full nonlinear problem - the model is Hamiltonian and predicts the correct scaling for the electric field [4]. The advantage of this reduced model is that it is easily amenable to numerical solution. We use a symplectic integration scheme to integrate the model and the scaling predictions of the amplitude equations analysis are being verified.

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References:
1.   J.D. Crawford and A. Jayaraman, Nonlinear saturation of an electrostatic wave: mobile ions modify trapping scaling, Phys. Rev. Lett. , 77 3549-3552 (1996).  [PRL96.ps.gz, PRL96.pdf]
2.   J.D. Crawford and A. Jayaraman, Amplitude equations for electrostatic waves: multiple species, J. Math. Phys.,  39   4546-4577 (1998).   [JMP-multiple.ps.gz, JMP-multiple.pdf]
3.   J.D. Crawford and A. Jayaraman, First principles justification of a Single Wave Model, Phys. of Plasmas 6 666-673 (1999).  [POP-swm.ps.gz, POP-swm.pdf]
4.  A. Jayaraman and J.D. Crawford, A Generalized Single Wave Model for electrostatic instabilities, in preparation.
5. T.M. O'Neil, J.H. Winfrey, and J.H. Malmberg, Nonlinear interaction of a small cold beam and a plasma, Phys.  Fluids, 14, (1971) 1204-1212.

 

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