THE VLASOV-POISSON SYSTEM
General Considerations
Plasma waves are generated by the charge separation that arises
when ions and electrons are displaced from their equilibrium
positions in a plasma. The dispersion properties and the stability of
these waves are important in many aspects of plasma physics and are
usually studied by kinetic theory methods. They tend to propagate
mainly
in collisionless
plasmas, since other wave phenomena are dominant in the presence of
collisions.
The kinetic description of plasma waves is inherently
nonlinear, since the time evolution of the distribution functions of
the plasma species
is affected by the self-consistent electric field, itself a function
of the charge density. Linearized models have been used by many
authors
to study the stability properties of plasma waves, most often by
analytical tools only.
Nonlinear studies have also been carried out,
but the difficulties in this case are overwhelming and one has to
resort to numerical techniques.
The effect of short range binary collisions on the stability
properties
of these spatially uniform Vlasov equilibria has also been studied,
both
analytically and numerically,
and the dampening effect of collisions has been shown.
Results
- Nonlinear plasma waves.
Numerical simulations of the one-dimensional Vlasov-Poisson system indicate that
the time-asymptotic behaviour of small perturbations to Vlasov
equilibria is often characterized by the presence of periodic
travelling structures in phase space and by electric fields that
exhibit standing-wave oscillations with constant amplitude (2). It has
been conjectured that these structures can be adequately described -
at least in first approximation - by using some sort of
combination of BGK modes, the nonlinear undamped travelling waves
discovered by Bernstein, Greene and Kruskal in 1957.
In (6) we present some numerical simulations showing that a
nonlinear superposition principle introduced recently by
Buchanan and Dorning gives a self-consistent plasma state with the
desired properties. In particular, we examine how these
nonlinear states evolve under the Vlasov dynamics and compare their
evolution with the time evolution of linear superpositions of the same
BGK modes.
In (1,3,4,5) we investigate numerically small amplitude undamped plasma
waves propagating near Maxwellian equilibria.
- L. Demeio, Numerical simulations of Vlasov plasmas,
Proc. NATO Adv. Res. Workshop, ``Physical
processes in hot cosmic plasmas", Vulcano, Italy,
May 29, June 2, 1989, 141 (1989).
- L. Demeio and P. F. Zweifel, Numerical simulations of
perturbed Vlasov equilibria, Physics of Fluids B, 2, 1252 (1990).
- L. Demeio, A numerical study about the existence of BGK
modes near a maxwellian equilibrium, Proc.
11th Intern. Transp. Theory Conf.,
Blacksburg, VA, U.S.A., May 1989 in ``Operator
Theory: Advances and Applications" 51, 109 (1991).
- L. Demeio and J. P. Holloway, Numerical simulations of BGK
modes, Journal of Plasma Physics, 46, 63 (1991).
- L. Demeio, Numerical simulations of BGK modes in Maxwellian
plasmas, Proc. 1st Symposium on Plasma Dynamics, Trieste,
Italy, June 26-28, 1991. Ed. M. Tessarotto, Consorzio di
Magnetofluidodinamica, Trieste University, Italy (1992).
- L. Demeio, A numerical study of linear and
nonlinear superpositions of BGK modes, mandato in
pubblicazione a Transport
Theory and Statistical Physics, Luglio 1999 (ICTT Atlanta
May 1999).
- Effect of short range binary collisions on the stability
properties of Vlasov equilibria.
A new algorithm has been introduced, which solves the Vlasov-Poisson
system in phase space with a BGK collision operator, both with constant
and velocity dependent collision frequency. The algorithm has been used to
examine the influence of binary collisions on the nonlinear behaviour of Vlasov
solutions. A linear stability analysis in presence of collisions,
described by the BGK collision operator with velocity dependent collision
frequency, has also been performed.
- L. Demeio, The inclusion of collisional effects in the
splitting scheme, Journal of Computational Physics, 99, 203 (1992).
- L. Demeio and G. Frosali,
Effects of short-range binary
collisions on the stability properties of longitudinal
plasma waves, Sommari del
III Congresso Nazionale della SIMAI,
Salice Terme, PV, 27-31 Maggio 1996, p. 526.
- L. Demeio and G. Frosali,
Effects of short-range binary
collisions on the stability properties of longitudinal
plasma waves, Rapporto Interno 7/1996 del Dip. di
Matematica ``V. Volterra", Universita' degli Studi di
Ancona.
- L. Demeio, Linear stability of the spatially
homogeneous equilibria of the Vlasov-Poisson system
with collisions, Reports on Mathematical Physics,
40, 455 (1997).
- L. Demeio, Collisional relaxation of
nonlinear plasma waves, IV International
Conference on Industrial and Applied Mathematics,
Edinburgh, Scotland, July 5-9, 1999, ICIAM99, p. 253.