aa r X i v : . [ phy s i c s . p l a s m - ph ] J a n Fast growing instabilities for non-parallel flows
A. Bret ∗ ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain (Dated: October 3, 2018)Unstable modes growing when two plasma shells cross over a background plasma at arbitraryangle θ , are investigated using a non-relativistic three cold fluids model. Parallel flows with θ = 0are slightly more unstable than anti-parallel ones with θ = π . The case θ = π/ θ = 0 one, but the fastest growing modes are oblique. While the most unstable wave vector varywith orientation, its growth rate slightly evolves and there is no such thing as a stable configuration.A number of exact results can be derived, especially for the θ = π/ PACS numbers: 52.35.Qz - 52.40.Mj
I. INTRODUCTION
Beam plasma instabilities are ubiquitous in physics and have been investigated for many decades [1]. The topiccurrently undergoes a renewed interest through the Fast Ignition Scenario for Inertial Fusion [2] or some scenarios ofGamma Ray Bursts production in astrophysics [3]. Indeed, Astrophysics offers a very wide range of unstable systemswhich can be magnetized, relativistic, homogenous or not. Counter-streams instabilities are usually studied assumingparallel streams. But why should real systems systematically fit this scheme? Admittedly, when an electron beamenters a plasma, a return current is prompted in opposite direction to neutralize it [4]. But two already current andcharge neutralized plasma shells could perfectly collide over a background plasma at an arbitrary angle. It seems theproblem of non-parallel streams has not be addressed so far, and the goal of this letter is to show that non negligiblegrowth rates can arise from non-parallel streams interactions as well.Let us consider two non-relativistic beams, both charge and current neutralized. For simplicity, we consider here twoelectron-proton beams where both species have equal densities so that charge and current neutrality are guarantiedregardless of the beams relative motion. The problem of the beams respective orientation is interesting only if thereis a background plasma. Otherwise, one just needs to consider the reference frame of one of the beams to cancel anyorientation parameter. We eventually come up with the setting pictured of Figure 1. Calculations are conducted inthe reference frame where the background plasma of electronic and protonic density N is at rest. In order to focuson the parameter θ , the two beams have equal protonic and electronic densities n , and both flow at the same velocity V along their respective direction. We now study the stability of harmonic perturbations ∝ exp( i k · r − iωt ) where FIG. 1: Scenario considered. ∗ Electronic address: [email protected]
FIG. 2: (Color Online) Growth rate map in terms of the reduced wave vector Z for α = 0 . β = 0 . θ = 0, π/ π . i = − k = ( k x , , k z ). Note that the all k unstable spectrum is evaluated in order to be able to spot the fastestgrowing modes for any given configuration. Finally, we neglect proton velocities perturbations in view of their muchlarger inertia. II. THREE FLUIDS MODEL
As a first approximation, we implement a cold three fluids model for the two beams and the plasma. Linearizingthe conservation and Euler equations for the three species, the dispersion equation is found evaluating the dispersiontensor in a standard way in terms of the dimensionless variables, α = nN , Z = k Vω p , β = Vc , x = ωω p , (1)where ω p = 4 πN e /m e is the background plasma frequency, e the electron charge and m the electron mass. Dueto the arbitrary orientation of the wave vector and of the two beams, the full dispersion tensor is too large to bereported here. Still, the dispersion equation remains polynomial and can easily be solved numerically. For Z x = 0,the zz component of the tensor give the growth rate of a two-stream like instability, as least for θ = 0 and π . Thecorresponding dispersion equation reads,1 − x − α ( x − Z z ) − α ( x − Z z cos θ ) = 0 . (2)The growth rate map in terms of Z for θ = 0, π/ π on Figure 2. The first and the last cases pertain towell-known systems as θ = 0 eventually comes down to one single beam of density 2 n interacting with the plasma,while θ = π corresponds to two counter-streams crossing over a background plasma. These situations have been wellstudied, and it is known that within the present non-relativistic regime [5], they are governed by the two-streaminstability which dispersion equation is precisely given by Eq. (2). The maximum growth rate δ M is in the dilutedbeam regime α ≪ δ M ( θ = 0) ∼ √ α / ω p , and δ M ( θ = π ) ∼ √ / α / ω p . (3)For θ = 0, the two beams act as one, and a resonant unstable mode is feed by the free energy of both at the sametime. For θ = π , resonant unstable modes can only travel with one single beam, yielding a smaller growth rate. III. MAXIMUM GROWTH RATE FOR θ = π/ Besides these extreme orientations of the two beams, Fig. 2 clearly displays some interesting features for the case θ = π/
2. One the one hand, the fastest growing mode is here found for Z z = Z x ∼ FIG. 3: Evolution of the maximum growth rate ( ω p units) for β = 0 .
1, in terms of the angle θ and for various beam to plasmadensity ratios α . hand, the maximum growth rate seems very close to the one reached for θ = 0, and some finer numerical evaluationshows that it is almost the same. Such equality can be demonstrated setting θ = π/ Z z = Z x . The dispersion equation for these modes reads, (cid:20) − x − α ( x − Z z ) (cid:21) P ( x ) = 0 , (4)where P ( x ) = ( x − x − Z z ) − α ( Z z + ( x − Z z ) ) − x − Z z ) Z z /β . (5)Since the first factor in Eq. (4) is strictly equal to the dispersion equation (2) for θ = 0, we here prove that themaximum growth rate for this branch is rigourously the same, and that it is reached for the very same Z z component.A closer look at P ( x ) shows that it yields another unstable mode. By setting Z z ∼ P ( x ) anddeveloping the result near x = 1, we find the growth rate of this second oblique unstable mode, δ M ∼ ω p β √ α. (6)For α and β lower than unity, this secondary growth rate remains smaller than δ M ( θ = 0). IV. CONCLUSION
The overall system is thus found exactly as unstable for θ = 0 as it is for θ = π/
2. Beyond this value of θ , themaximum growth rate must decrease to its final value δ M ( θ = π ), as given by Eq. (3). Figure 3 shows how δ M evolvesbetween θ = 0 and π , for β = 0 . α . As expected, δ M remains almostconstant between 0 and π/ δ M ( π ) for θ ∼
2. Because δ M is independent of β for θ = 0, π/ π , we can expect an overall weak dependance for other angles. This is confirmed through numericalcalculation as the three curved displayed on Fig. 3 are indistinguishable from their β = 0 .
01 counterparts.Ton conclude, it is important to emphasize that there is no such thing as a stable configuration. On the contrary,the system remains unstable regardless of the beams orientation, and the evolution of the maximum growth rate islimited. The most unstable wave vector is two-stream like for parallel and anti-parallel orientations, but turns obliquein the intermediate case to stay in phase with both beams at the same time. Noteworthy, relativistic and/or kineticeffects should force an oblique regime regardless of the beam orientation so that an evaluation of the whole unstablespectrum becomes mandatory.
V. ACKNOWLEDGEMENTS
This work has been achieved under projects FIS 2006-05389 of the Spanish Ministerio de Educaci´on y Ciencia andPAI-05-045 of the Consejer´ıa de Educaci´on y Ciencia de la Junta de Comunidades de Castilla-La Mancha. [1] D. Bohm and E. P. Gross, Phys. Rev. , 1851 & 1864 (1949).[2] M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J.Mason, Phys. Plasmas , 1626 (1994).[3] M. Medvedev and A. Loeb, Astrophysical Journal , 697 (1999).[4] D. A. Hammer and N. Rostoker, Phys. Fluids , 1831 (1970).[5] A. Bret, M.-C. Firpo, and C. Deutsch, Nuclear Instruments and Methods in Physics Research A , 427 (2005).[6] Y. B. Fa˘ınberg, V. D. Shapiro, and V. Shevchenko, Soviet Phys. JETP , 528 (1970).[7] F. Califano, R. Prandi, F. Pegoraro, and S. V. Bulanov, Phys. Rev. E58