High-Mach Number Relativistic Ion Acoustic

Shocks

J. Fahlen and W.B. MoriUniversity of California, Los Angeles

ShocksHigh mach number relativistic ion acoustic shocks are travelling discontinuities in electric potential, density, temperature and pressure.

High-intensity laser interactions can generate shocks and heat the electrons to relativistic temperatures.

High mach number shocks are those travelling at speeds greater than 1.6Ms.

Nonrelativistic theory with Boltzmann electrons predicts a critical mach number Mcr=1.6 indepentdent of electron temperature. The theory presented here predicts Mcr=3.1 for low temperatures decreasing to Mcr=2 for extremely relativistic temperatures.

Motivation

Intense lasers incident on thin metal foils have been shown in simulations to generate high mach number electrostatic shocks (L.O. Silva et al. PRL 92 015002 (2004)).

These shocks are characterized by a fast moving, large electric potential jump that can reflect ions and accelerate them to high energies.

Existing shock theories indicate that ions reflect at Mcr=1.6 or 3.1 (see below). However, relativistic electron temperatures require a modification to these theories.

Shock vs. Soliton

Critical Mach Number: The speed at which the structure begins to reflect ions.

1) 1 < M < Mcr: No or very few ions reflected, mostly soliton-like.2) M > Mcr: Many ions reflected, now a shock3) M>>Mcr: Shock doesn’t form, ions reflect off the wall throughout.

SolitonShock

Initial EquationsQuickTime™ and a

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Ion conservation ofenergy

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Ion continuity equation,Drop time derivative

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Electron Density?

Poisson Eq. Above requires the electron density. There are several choices:

1) Boltzmann equation

2) Use a trapped electron model from R.L. Morse (Phys. Fluids 8, 308 (1965)) and D.W. Forslund and C.R. Shonk (PRL 25 1699 (1970))

3) Trapped electron model extension for relativisitic electron temperatures.

Substitute these into Poisson equation and find the critical mach number, i.e., the speed at which ions reflect and a shock forms.

Electron Density?

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= eφBoltzmann: Sagdeev

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nen0

= 0.611 α 2φ2 + 2φ + eφ e−

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α 21+α 2p 2 −1 ⎛

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⎠ ⎟

α 2φ 2 +2φ

∞

∫ dp ⎡

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RelativisticTrapped Ele: €

α ≡v th /c

Boltzmann: Mcr=1.6

Simulations

Initial conditions: Uniform neutral plasma drifts to the right with finite electron temperature and Te/Ti=400. Right side boundary is reflecting. Electron temperature and drift speed are varied over many runs.

As plasma reflects, a sheath is formed which eventually becomes a shock if the conditions are correct.

Neutral Plasma

Uniform Drift

Reflecting wall

Simulation box

Simulations - Shock Formation

A) Soliton M=1.6B) Shock M=2.8C) No Shock, Initial Drift M = 2.5

Te=5MeV, Te/Ti=400

A

B

C

Ion Reflection Results

For a given shock speed, more ions will be reflected when the plasma conditions are such that the critical mach number Mcr is lower rather than higher. The lower Mcr is, generally the more ions that will be reflected.

Density vs. Potential

Dashed Line - Boltzmann equationDotted Line - Nonrelativistic Trapped Electron equation of stateSolid Line - Relativistic Trapped Electron Eq.

Density vs. Potential II

Dashed Line - Boltzmann equationDotted Line - Nonrelativistic Trapped Electron equation of stateSolid Line - Relativistic Trapped Electron Eq.

Shock Speed vs. Initial Drift

It is not clear how fast a shock will propagate given an intial temperature and drift speed. However, the points do fall on a fairly well defined line with a slope of a little less than 1.

Conclusion

New theory extending shock theory for relativistic electron termperatures was developed.

Simulation results are in qualitative agreement with theory.

Three different regimes seen in simulations:1) Soliton2) Shock3) No structure

More ions generally reflected for lower critical mach numbers.