Vlasov multi-dimensional model dispersion relation

A hybrid model of the Vlasov equation in multiple spatial dimension $D>1$ [H. A. Rose and W. Daughton, Physics of Plasmas v. 18, 122109 (2011)], the Vlasov multi dimensional model (VMD), consists of standard Vlasov dynamics along a preferred direction, the $z$ direction, and $N$ flows. At each $z$ these flows are in the plane perpendicular to the $z$ axis. They satisfy Eulerian-type hydrodynamics with coupling by self-consistent electric and magnetic fields. Every solution of the VMD is an exact solution of the original Vlasov equation. We show convergence of the VMD Langmuir wave dispersion relation in thermal plasma to that of Vlasov-Landau as $N$ increases. Rotational symmetry about the $z$ axis in $3D$ of small perpendicular wavenumber Langmuir fluctuations is demonstrated for $N\geq 6$, with flows arranged uniformly over the azimuthal angle.