Transcript High-Mach Number Relativistic Ion Acoustic Shocks

High-Mach Number
Relativistic Ion Acoustic
Shocks
J. Fahlen and W.B. Mori
University of California, Los Angeles
Shocks
High 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 shock
3) M>>Mcr: Shock doesn’t form, ions reflect off the wall throughout.
Soliton
Shock
Initial Equations
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Ion conservation of
energy
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Solve for u:
Ion continuity equation,
Drop time derivative
Poisson Eq.
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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?
Boltzmann:
ne
 e
n0

n
Relativistic
e
 0.611  2 2  2  e

Trapped Ele: n0



  vth /c
Sagdeev



dp


1 
 1 2 p 2 1


 2 
 e 
 2 2 2
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.
Uniform Drift
Neutral Plasma
Simulation box
Reflecting wall
Simulations - Shock Formation
B
A
Te=5MeV, Te/Ti=400
C
A) Soliton M=1.6
B) Shock M=2.8
C) No Shock, Initial Drift M = 2.5
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 equation
Dotted Line - Nonrelativistic Trapped Electron equation of state
Solid Line - Relativistic Trapped Electron Eq.
Density vs. Potential II
Dashed Line - Boltzmann equation
Dotted Line - Nonrelativistic Trapped Electron equation of state
Solid 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) Soliton
2) Shock
3) No structure
More ions generally reflected for lower critical mach numbers.