a r X i v :p h y s i c s /0410188v 1 [p h y s i c s .p l a s m -p h ] 21 O c t 2004Impact of dissipative effects on the macroscopic evolution of a Vlasov systemL.Galeotti, F.CalifanoPhysics Dept.,University of Pisa and INFM,Pisa,Italy(Dated:February 2,2008)Numerical diffusion is introduced by any numerical scheme as soon as small scales fluctuations,generated during the dynamical evolution of a collisionless plasma,become comparable to the grid size.Here we investigate the role of numerical dissipation in collisionless plasma simulations by studying the non linear regime of the two stream instability.We show that the long time evolution of the Vlasov -Poisson system can be affected by the used algorithm.PACS numbers:52.65.Ff,52.35.Qz,52.35.Fp,52.35.MwMany space and laboratory plasmas can be considered as weakly collisional since the collisional frequency is smaller than all the other frequencies,as for example the plasma frequency.In other words,for these plasmas,the mean free path of the particles is (much)longer than all the other characteristic length scales of the plasma and,sometimes,even larger than the dimension of the plasma itself.At first approximations such plasmas can be considered as collisionless and their dynamics can be well represented using a Hamiltonian description.This approach is based on the idea that the dissipative scale (for example in numerical simulations the grid size)is much smaller than any macroscopic physical length scale of the system,so that dissipation has no feedback on the macroscopic asymptotic evolution of the system.Numerical simulations based on non collisional mod-els must necessarily face with the small scales genera-tion problem during the dynamical evolution of the sys-tem;indeed,when the typical length scales of the fluc-tuations become comparable to the grid size,numerical dissipation comes into play leading the system to vio-late the conservation constraints of Hamiltonian dynam-ics and to reconnect close isolines of the distribution func-tion (d.f.).This process,formally forbidden,is very well highlighted by the time evolutions of the system invari-ants N i =f i dxdv i =1,2,..and by the ”entropy”S =− f ln (f )dxdv (here f is the d.f.),showing sudden variations when closed vortices are formed in phase space as a consequence of particle trapping.In this paper,through numerical studies of a non-linear regime of a collisionless plasma,we discuss the role of ar-tificial dissipation introduced by a numerical scheme on the plasma dynamics,influencing the final Vlasov evolu-tion of the system even if the grid size is much shorter than any physical relevant scale length.The dynami-cal non linear evolution we chose for our numerical sim-ulations is the well-known two stream instability.In this case,dissipation allow for the formation of coher-ent macroscopic structures in phase space (vortices).Since the non-linear dynamics of the two stream in-stability is substantially driven by kinetic effects,es-pecially concerning the saturation phase where particle trapping play a crucial role,a kinetic approach is nec-essary.This can be done using Vlasov equation,which replaces Coulomb interactions between charged particles with a mean electromagnetic field.This field is deter-mined self-consistently trough the particle distribution function by Maxwell and Poisson equations.Since the two stream instability is driven by purely electrostatic mechanisms,we limit our study to the solution of the 1D-1V Vlasov -Poisson system of equations:∂f a∂x −m e∂x∂f a∂x 2= (f e −f p )dv ;E =−∂φ
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