Contents

In this notes I compare results from 1D/1V simulations of
electrostatic shocks with those obtained from the multi-fluid,
5-moment model. The Poisson-bracket formulation of Vlasov-Poisson
equations is used. See *JE15* for
formulation of the equations, and basic benchmarks.

The basic setup for all problems is the same: two counter propagating beams of plasmas start interacting at \(t=0\). The beams have initial drift speed determined from their Mach number, \(M \equiv u/c_s\), where \(u\) is the drift velocity, and \(c_s \equiv \sqrt{T_e/m_i}\) is the sound speed. The domain size is \(500\) Debye lengths, with time measured in inverse plasma frequency \(1/\omega_{pe}\). For all problems, a DG/CG scheme with polynomial order 2 is used.

The multi-fluid simulations are setup in an identical way. However,
the full Maxwell equations are evolved instead of the Poisson
equation. A finite-volume scheme is used to solve the multi-fluid
equations. For benchmarks of the FV scheme, see *JE4* and other notes following that.

Note that for these problem the domain size, and hence the run-time, can be reduced by a 2X by using reflecting boundaries instead of two counter propagating beams. However, as of this writing, Gkeyll does not have reflecting BCs for the distribution function, and hence I am just using a bigger domain.

When two counter propagating, but otherwise identical, slabs of plasma collide, a potential well is created, leading to trapping of electrons and the formation of shocks from nonlinear steepening of wave fronts. However, this shock formation only happens to a critical Mach number of about \(M_c = 3\) [1]. The situation is different. however, when the plasma slabs have different temperatures and/or densities. In this case [2] any Mach number shock can be obtained, as long as the correct temperature and densities ratios are selected. The critical Mach number (above which an collisionless ES shock is not formed) is given by

\[M_c \approx \frac{3(Y+1)}{Y}\sqrt{\frac{\pi\Theta}{8}}\]

where \(Y = n_R/n_L\) and \(\Theta = T_{eR}/T_{eL}\) are the density and electron temperature ratios (ions assumed to be cold) of the slabs, respectively.

The potential well, besides trapping electrons, reflects ions back into the oncoming plasma and decelerates the ions in the interaction region. For higher Mach numbers (above the critical Mach number) there isn’t enough slowing down of the ions to create a shock, and the trapped particle region keeps growing with time (instead of nonlinear steepening leading to shock formation). For colder ions there are significant amplitude ion waves, which damp out in the warmer ion case.

In the following simulations, I use various mass ratios and electron/ion temperature ratios to simulate the formation of collisionless ES shocks. In addition, I compare the results with fluid simulations with identical parameters. Fluid models will always show shock formation, independent of the Mach number. However, the mechanism of shock formation is different (than in the kinetic case) and is a combination of charge separation and collisions (which keep the plasma Maxwellian). At some point it will be useful to perform simulations with some collision operator.

In this sequence of simulations, I use a hydrogen plasma,
\(m_e/m_i = 1/1836.2\), with \(T_e/T_i = 9.0\). I run
simulations with initial Mach numbers of \(1.5\) [*m1*], \(3.0\) [*m3*] and \(5.0\) [*m4*].

The distribution functions for the Mach \(1.5\) and Mach \(5.0\) at \(T\Omega_{pe} = 1000\) are shown below.

In the first case, the potential well traps the electrons and decelerates the ions in the interaction region, significantly slowing them down. The trapped electron component flattens the electron distribution function (leading to a “flat-top” distribution). The reflected ions seem to undergo some oscillations, which are, however, damped. The evolution is markedly different in the Mach \(5.0\) case. The ion momentum is so high that the potential well is unable to slow down the ions in the interaction region, and the reflected ions also have a very small increase in velocity. However, in each case, in the interaction region the electron distribution function takes a characteristic flat-top shape, as shown below.

I ran a set of two-fluid simulations (electrons and ions treated as separate fluids coupled via electromagnetic source terms, the field evolved with Maxwell equations) with the same parameters and initial conditions as used for the kinetic simulations. In general, the fluid results compare well (except for ion energy) with kinetic results for low Mach numbers. The ion energy outside the interaction region in the kinetic simulations is dominated by the high-energy reflected ions, leading to significant differences with the fluid results.

However, for higher Mach numbers the results are dramatically different. The fluid simulations always show a shock, independent of Mach number. The trapped particle effects are not captured in the fluid model, and collisions (missing in the kinetic model) force the distribution functions to be Maxwellian. Hence, the kinetic effects, which lead to the nonlinear physics (particle trapping and ion deceleration) in the interaction regions, are not correctly captured in the interaction region.

In the following plots, the kinetic and fluid results for density, momentum density (\(n u\)) and energy density (\(n u^2 + nv_{th}^2\)) are compared.

Kinetic simulations of collisionless electrostatic shocks confirm the lack of shocks above the critical Mach number. Comparisons with multi-fluid simulations show that the fluid model correctly predicts the shock location and profiles of moments (except for ion energy density) for low Mach number cases. However, the fluid results diverge significantly for high Mach numbers: shocks don’t form above a critical Mach number in the kinetic model, while continue to form in the fluid model. This indicate some caution in using fluid approximations to model collisionless shocks.

[1] | D. W. Forslund and C. R. Shonk “Formation and Structure
of Electrostatic Collisionless Shocks”. Phy. Review Letters,
25 (25), 1699. 1970. |

[2] | Sorasio, G., Marti, M., Fonseca, R., &
Silva, L. O. (2006). “Very High Mach-Number Electrostatic Shocks in
Collisionless Plasmas”. Phys. Review Letters, 96
(4), 045005. http://doi.org/10.1103/PhysRevLett.96.045005 |