1. Simulations of turbulent plasma heating by powerful electron beams Timofeev I.V., Terekhov A.V.

2. Formulation of the problem The aim of the study is to investigate nonlinear evolution of a beam-plasma system in the case of continuously injected electron beams. Both PIC and hybrid simulations are used to study long-time evolution of the beam-driven turbulence. Due to the dynamical range in space and time that should be resolved we are limited to the 1D geometry. We focus our attention on how the beam-plasma system evolves from regular dynamics to the steady-state turbulence and what nonlinear processes play the main role at the turbulent stage.

3. Hybrid model High frequency response In a uniform plasma without a beam:

4. Hybrid model Slow plasma dynamics

5. Beam relaxation in the system with open boundaries Dynamic stage Beam parameters: Plasma parameters:

6. Dynamic stage Wave energy exceeds thermal plasma energy Large amplitude wave is unstable to the short-wavelength perturbations

7. Dynamic stage Superthermal tail formation

8. Modulational instability Evolution of monochromatic Langmuir wave without beam effects

9. Modulational instability Dispersion relation in the limit Spectrum of unstable waves is in a good agreement with theoretical predictions.

10. Steady-state turbulence Hybrid and PIC simulations repro- duces similar results Modulational instability lead to thermalization of local oscillatory energy of plasma electrons Turbulence goes to the regime Nonlinear dissipation of beam- excited waves is produced by scattering off density fluctuations and forced wave collapse

11. Steady-state turbulence With the increase in electron temperature turbulence tends to operate in the regime In spite of intense plasma turbulence, beam interaction with the resonant wave remains regular with the coherent length allowing for beam trapping Under weak nonlinear dissipation pro- duced by plasma nonlinearities beam ab- sorbs some wave energy by itself and saturates the pumping power

12. Steady-state turbulence Saturation of pumping power Level of power saturation as a function of instability growth rate

13. Conclusion The problem of plasma heating by a continuously injected electron beam is studied using conventional PIC and simplified hybrid simulations. It is shown that the scenario of nonlinear evolution of the beam-plasma system consists of several typical stages: at the dynamic stage beam-excited Langmuir wave grows up to the energy significantly exceeding thermal plasma energy and drives short-wavelength modulational instability; the initial stage of this instability is adequately described by the simplified theory correctly accounting for kinetic effects of plasma electrons, and the nonlinear stage is governed by the strong electron nonlinearity, which provides efficient dissipation of unstable perturbations due to wave-breaking; strong dissipation results in the weak-pump turbulent regime W<nT, when the spectral energy transfer is determined by Langmuir waves scattering off density fluctuations and forced collapse;

14. with the increase in electron temperature turbulence tends to operate in the regime of the constant pump, when the saturation level of heating power is determined solely by the nonlinear interaction of beam particles with resonant waves and does not depend on the turbulence structure in the nonresonant part of the spectrum. Conclusion The hybrid model reproduces the main physical phenomena observed in PIC simulations and can be used to simulate turbulent plasma heating over macroscopic scales of real experiments.