A. E. Vladimirov, D. C. Ellison, A. M. Bykov
Strong nonresonant
amplification of magnetic
fields
in particle accelerating
shocks
Submitted to ApJL
In diffusive shock acceleration, the streaming of shockaccelerated particles may induce plasma instabilities.
A fast non-resonant instability (Bell 2004, MNRAS) may
efficiently amplify short-wavelength modes in fast shocks.
•We developed a fully nonlinear model* of DSA
based on Monte Carlo particle transport
•Magnetic turbulence, bulk flow, superthermal
particles derived consistently with each other
Amplified
MHD turbulence
W(x,k)
Shocked flow
u(x)
Accelerated
particles
f(x,p)
* Vladimirov, Ellison & Bykov, 2006. ApJ, v. 652, p.1246;
Vladimirov, Bykov & Ellison, 2008. ApJ, v. 688, p. 1084
Turbulence
Wavenumber, k
Particle mean free path, (p)
Turbulence spectrum, k·W(k)
Our model for particle propagation in strong turbulence
interpolates between different scattering regimes in
different particle energy ranges.
Particles
~(Wres)-1
~lcor
Momentum, p
~p2
k – wavenumber of turbulent harmonics
W(x,k)
– spectrum of turbulent fluctuations, (energy
per unit volume per unit ∆k).
Cascading
Dissipation
In du
this work we
W
 ignored
du  

u
  W  L Wcompression

  forclarity
kW

x
dxaffect the
xqualitatively
dx  k

(does not
new results)
Amplification
( corresponds
to Bell’s
instability)
Compression
(amplitude)
Compression
(wavelength)
We study the consequences of two
hypotheses:
A. No spectral
energy transfer
(i.e., suppressed
cascading),
=0
B. Fast
Kolmogorov
cascade,
 = W5/2k3/2ρ-1/2
Shock-generated turbulence with NO CASCADING
Trapping
Effective magnetic field
B = 1.1·10-3 G
~p2
Shocked plasma temperature
T = 2.2·107 K
•Without cascading, Bell’s instability forms
turbulence spectrum with several distinct peaks.
a
•The peaks occur due to the nonlinear connection
between particle transport and magnetic field
amplification.
•Without a cascade-induced dissipation, the plasma
in the precursor remains cold.
Shock-generated turbulence with KOLMOGOROV CASCADE
Resonant
scattering
Effective magnetic field
B = 1.5·10-4 G
~p2
Shocked plasma temperature
T = 4.4·107 K
•With fast cascading, Bell’s instability
forms a smooth, hard power law
turbulence spectrum
•The effective downstream magnetic
field turns out lower with cascading, as
well as the maximum particle energy
•Viscous
dissipation
of
small-scale
fluctuations in the process of cascading
induces a strong heating of the
backround plasma upstream.
Summary
• We studied magnetic field amplification in a nonlinear particle
accelerating shock dominated by Bell’s nonresonant shortwavelength instability
• If spectral energy transfer (cascading) is suppressed, turbulence
energy spectrum has several distinct peaks
• If cascading is efficient, the spectrum is smoothed out, and
significant heating increases the precursor temperature
With Cascading
Without Cascading
Discussion
• With better information about spectral energy transfer (
) we
can refine our predictions regarding the amount of MFA,
maximum particle energy Emax, heating and compression in
particle accelerating shocks (plasma simulations needed)
in a
strongly magnetized plasma with ongoing nonresonant magnetic field amplification, accounting for the interactions with streaming accelerated particles
• If peaks do occur, they define a potentially observable
spatial scale and an indirect measurement of Emax
• Peaks in the spectrum may help explain the rapid
variability of synchrotron X-ray emission*
• Observations
of
precursor
heating
may
provide
information about the character of spectral energy transfer
in the process of MFA
* Bykov, Uvarov & Ellison, 2008 (ApJ)
Q? A!
Plots from the paper (just in case)
The following sequence of slides shows
how the peaks are formed one by one in
the shock precursor.
(model A, no cascading)
Solution with NO CASCADING
Very far upstream…
Solution with NO CASCADING
Far upstream…
Resonance Turbulence
w/particles amplification
Solution with NO CASCADING
Upstream…
Solution with NO CASCADING
Particle trapping occured…
Solution with NO CASCADING
Second peak formed…
Solution with NO CASCADING
The story repeated…
Solution with NO CASCADING
And here is the result (downstream)…
The following sequence of slides shows
how the peaks are formed one by one in
the shock precursor.
(model B, Kolmogorov cascade)
Solution with KOLMOGOROV CASCADE
Far upstream…
Solution with KOLMOGOROV CASCADE
Amplification…
Solution with KOLMOGOROV CASCADE
Cascading forms a k-5/3 power law…
Solution with KOLMOGOROV CASCADE
Amplification continues in greater k…
Solution with KOLMOGOROV CASCADE
And a hard spectrum is formed downstream…