1. Computationally efficient description of relativistic electron beam transport in dense plasma Oleg Polomarov*, Adam Sefkov**, Igor Kaganovich** and Gennady Shvets* *IFS, The University of Texas at Austin, TX, 78712; **PPPL, Princeton, NJ 08543

2. Abstract A reduced model of the Weibel instability and electron beam transport in dense plasma is developed. Beam electrons are modeled by macro-particles and the background plasma is represented by electron fluid. Conservation of generalized vorticity and quasineutrality of the plasma-beam system are used to simplify the governing equations. Our approach is motivated by the conditions of the FI scenario, where the beam density is likely to be much smaller than the plasma density and the beam energy is likely to be very high. For this case the growth rate of the Weibel instability is small, making the modeling of it by conventional PICs exceedingly time consuming. The present approach does not require resolving the plasma period and only resolves a plasma collisionless skin depth and is suitable for modeling a long-time behavior of beam-plasma interaction. An efficient code based on this reduced description is developed and benchmarked against the LSP PIC code. The dynamics of low and high current electron beams in dense plasma is simulated. Special emphasis is on peculiarities of its non-linear stages, such as filament formation and merger, saturation and post-saturation field and energy oscillations. Supported by DOE Fusion Science through grant DE-FG02-05ER54840.

3. Motivation To design an computationally efficient approach for modeling electromagnetic Weibel instability for the case of propagation of relativistic electron (ion) beams into dense background plasma Interesting features : pinching, filamentation, generation of strong magnetic field and its saturation Applications : Fusion - fast ignition, Accelerator physics - beam transport, Astrophysics - origin of cosmic strong electro-magnetic field.

4. Problem Setup Beam current v z >> v x, v y Plasma return current B x,y x z y EzEz In the quasi-neutral plasma the rising beam current produces encircling transverse magnetic field B x,y, which leads to the beam pinching and generates E z by the induction law, which produces a return plasma current and tries to stop and screen the beam current. Instability loop: Increase of the beam (filament) density -> larger beam (filament) current -> larger magnetic field -> larger pinching -> larger beam (filament) density The instability is collective, not resonant Relativistic beam with v z >> v x, v y propagates into dense background plasma n b < n p. The beam-plasma system tends to be quasi-neutral.

5. Existing computational approaches PIC L. O. Silva et al., Phys Plasmas 10, 1979 (2003), M. Honda et al., Phys Plasmas 7, 1302, (2000). Advances all plasma and beam particles Hybrid modeling. LSP (hybrid mode), T. Taguchi et al., Phys. Rew. Lett., 86, 5055 (2001). Solves hydrodynamic equation for background plasma Both computationally expensive

6. Main assumptions Plasma electrons are warm non-relativistic fluid Beam electrons are relativistic particles streaming along the z-direction Plasma ions are fixed neutralizing background, with n i =const “Plasma + beam” is quasi-neutral: n b +n p =n i L >> l D and t >> 1/w pe All quantities of interest are z independent

7. Conservation of generalized vorticity Assuming Combining the Faraday’s law and the equation of motion of fluid collisionless background plasma yields the conservation of generalized vorticity For initially quiescent plasma giving

8. Consequences of the vorticity conservation No need to solve the equation of motion for background plasma. Plasma velocity is related to magnetic field from conservation of vorticity and plasma density is from quasi-neutrality The only equations to be solved are equations for  (x,y) and B z (x,y) and equations of motions of the beam particles

9. Field equations Magnetic field: The axial projection of Ampere’s law produces the equation for axial vector potential Curl of the transverse part of Ampere’s law yields the equation for B z Note: Coefficients in the both equations depend only on the beam density and current. No time dependence - the equations are solved at each time step as the beam particles are evolved. For underdense beams the equations for B z can be neglected. Electric field: is expressed through vector potential and plasma fluid velocity which in turn is expressed through B z and beam density and current.

10. Beam equations Newton equations for “particles”: Note: Although the beam electrons phase space is {Vx,Vy,Vz,X,Y}, actual integration is done in {Vx,Vy,X,Y}.

11. Block-scheme of simulation Initialization: Given initial density and velocities of the beam, the Eq. for axial vector potential is solved. Cycles of time integration: a - > b - > a a) the Eqs. for beam particles are solved by PIC technique. The beam electrons momentae and positions are advanced in time dt using fields on a grid. The beam density and current are recalculated on the grid. b) the Eqs. for axial vec. potential and B z are solved by the MUDPACK multigrid solver for non-separable 2D elliptic equations. See: J. Adams, "MUDPACK: Multigrid Fortran Software for the Efficient Solution of Linear Elliptic Partial Differential Equations,“ Applied Math. and Comput. vol.34, Nov 1989, pp.113-146.

12. Time evolution of the underdence electron beam in ambient plasma. a) initial exponentially growing stage: Weibel (filamentation) instability of relativistic electron beam with diameter K p D=20. Simulation box is 256x256 (or 32 K p x 32 K p ), 2x10 6 particles. Peak beam density compression ~ 100 times.

13. b) Final stage of instability. Saturation Maximal pinching, Saturation.

15. Saturation of the beam-plasma energies The instability saturates when In agreement with of R. C. Davidson et al., Phys. Fluids 15, 317 (1972).

16. Initial rate of instability growth for warm beam Linearized Vlasov for the perturbed part of the distribution function under assumptions: leads to: Assuming “waterbag-like” distribution function Calculating z-Fourier component of the beam current note: no singularities in denominator as w is imaginary, substituting it into the linearized Fourier expanded Eq. for axial vector potential gives the dispersion relation

17. The dispersion relation Waterbag: Cold beam: For warm beam with Possible unstable solutions are only with pure imaginary frequency:

18. Typical dispersion behavior Maximal possible wave vector: Maximal possible transverse thermal velocity: Cold Waterbag Warm

19. Extracted magnetic energy n beam /n plasma =0.007 0.05 0.025 0.01

20. Plasma density 19 57 L ω pe /c 9.5 27.5 L ω pe /c initially n beam /n plasma =0.007 Initially n beam /n plasma =0.025 ω pe t = 3000 0.5 0.01 2 0.01 n beam /n plasma

21. Conclusion and further work Robust hybrid code for modeling Weibel instabilities is developed. Detailed studies of filamentation, tearing, instability saturation and non-linear stage of the beam dynamics will be continued.