Application of Picard iteration technique to self-consistent wave-particle interaction in plasmas

The validity of quasilinear theory to describe the weak warm beam-plasma instability is still an open issue. This work extends the analytical approach of the problem with a model where the beam is described as a set of particles while the waves are harmonic oscillators. An average over the initial wave phases is performed over the result of the third iterate of the Picard iteration technique applied to the equations of motion. This calculation shows that some of the results of the uniform particle density case remain correct even for a non uniform density. However, as for Langmuir wave amplitude evolution, there is a spontaneous emission by spatial inhomogeneities (turbulent eddies) on top of Landau growth or damping and of spontaneous emission by particles. The validity of quasilinear theory (QL) describing the weak warm beam-plasma instability has been a controversial topic for several decades (see the many references in the introduction of [1]). It involves the chaotic dynamics of self-consistent wave-particle dynamics, whose description sounds formidable for a Vlasovian description. This was an incentive to tackle this instability by generalizing [2, 3] a model originally introduced for the numerical simulation of the cold beam-plasma instability [4, 5]. This model describes the beam as a set of particles, while the waves are present as harmonic oscillators. A Langmuir wave with a phase velocity ω/k in a range of velocities where there are no resonant particles verifies the Bohm-Gross dispersion relation, and is equivalent to a harmonic oscillator. Wave-particle dynamics is described by the self-consistent Hamiltonian Hsc = N ∑ r=1 pr 2 + M ∑ j=1 ω j0I j − ε N ∑ r=1 M ∑ j=1 k−1 j β j √ 2I j cos(k jxr −θ j) (1) where ε = ωp[2mη/N] is the coupling parameter and β j = [∂εd(k j,ω j0)/∂ω ]−1/2, with ωp the plasma frequency, m the mass of particles set to unity, η the ratio of the tail to the bulk density, εd(k,ω) the bulk dielectric function, k j and ω j0 the wavenumber and pulsation of wave j. The conjugate variables for Hsc are (pr,xr) for the particles and (I j,θ j) for the waves. On top of the total energy Esc = Hsc, the total momentum Psc = ∑r=1 pr +∑ M j=1 k jI j is conserved. 39 EPS Conference & 16 Int. Congress on Plasma Physics P4.161