MHD instability

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MHD instability driven by fast electron
streaming Tony Bell University of
Oxford Rutherford Appleton Laboratory
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Fast electrons in high intensity laser-plasma
experiments behave like cosmic rays. In both
cases, highly energetic charged particles with a
large Larmor radius stream through a thermal
plasma which behaves as an MHD fluid. Cosmic
rays are confined close to the outer shock of
supernova remnants by an amplified tangled
magnetic field produced by an MHD instability
driven by the cosmic rays. The same MHD
instability may be driven by fast electrons
in laser-plasma experiments relevant to Fast
Ignition. At solid density the growth rate is
of the order of 1 psec. At lower densities as
in an ablated corona the instability grows more
rapidly. The MHD instability grows more slowly
than the Weibel instability, but it grows on a
larger scale and may be more effective in
inhibiting fast electron transport, especially
for Fast Ignition laser pulses lasting 10psec.
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MHD instability due to cosmic rays
Tycho 1572AD
Kepler 1604AD
Supernova remnants MHD instability produces
magnetic field Driven by cosmic ray streaming
Thin outer shock due to synchrotron cooling in
large B
SN1006
Cas A 1680AD
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MHD instability due to laser-produced fast
electrons
High density plasma
LASER High intensity (1019Wcm-2) Short pulse
(psec)
Fast (MeV) electrons
B1-100MG
Units of 10MG
Two instabilities driven by streaming fast
electrons

• Weibel instability
• Grows rapidly on fsec timescale
• Filaments currents on length scale c/wpe
• No ion motion in simplest form
• MHD instability
• Grows on psec timescale
• Starts small grows non-linearly by spatial
  expansion
• Ion motion an essential role

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MHD instability driven by fast electrons
Fast electrons Larmor radius gt instability
scalelength Detached from instability Uniform
constant electric current
Thermal electrons Larmor radius instability
scalelength Provide return current Frozen to
field linea (approx)
Fast e- current
Magnetic field frozen into thermal plasma
j
j x B
j
B
j x B
Current carried by thermal plasma
j x B force expands the spiral POSITIVE FEEDBACK
(INSTABILITY)
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Energetic particles coupled to MHD plasma
Lucek Bell (2000) applied to cosmic rays
Demonstrated dB/Bgtgt1
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Non-linear growth
3D MHD with fixed fast electron current
fast electron current
Slices through B - time sequence
Cavities and walls in B r
Non-linear growth expending spirals/cavities
Field lines wandering spirals
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Dispersion relation linearise for parallel k, B
jfast
Fluid acceleration
Maxwell
Ideal MHD Ohms law
Growth rate
Drift velocity of return current
Growth limited by field tension at short
wavelength
Ion Larmor frequency
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Growth rate (neglecting k2vA2)
B in 100MG
energy carried by fast electrons
in 1019 Wcm-2
k in mm-1
density in gm cm-3
fast electron energy in MeV
100
10
Growth rate (psec-1)
1
0.1
0.01
100
10
0.1
1
wavenumber k (mm-1)
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Comparison equations for MHD Weibel
instabilities
MHD Weibel
Fast electron momentum
a
r
a
a
Thermal electron momentum
a
r
Thermal ion momentum
Maxwell
Essence of MHD instability fast electrons are
relatively unmagnetised thermal electrons are
magnetised thermal ions provide inertia
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Expanding spiral filament (1D cylindrical MHD)
Magnetic field Bq
Density
Cavity in density and magnetic field
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Natural evolution towards beam in cavity
B
E0
R
particle beam (imposed)
E-uxB
E0
Beam does work generating turbulence beam energy
loss through electric field curl(E) produces
B focuses particles, evacuates cavity
Similar but different fast electrons slowed by
resistive electric field particles work against
collisional resistivity GENERIC GENERATION OF
MAG FIELD
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Time sequence magnitude of B - sections
through beam
Same results integrated across beam