Strong nonresonant amplification of magnetic

fields in particle accelerating

shocks

A. E. Vladimirov, D. C. Ellison, A. M. Bykov

Submitted to ApJL

In diffusive shock acceleration, the streaming of shock-accelerated particles may induce plasma instabilities.

A fast non-resonant instability (Bell 2004, MNRAS) may efficiently amplify short-wavelength modes in fast shocks.

Shocked flowu(x)

Accelerated particles

f(x,p)

Amplified MHD turbulence

W(x,k)

•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

* Vladimirov, Ellison & Bykov, 2006. ApJ, v. 652, p.1246;

Vladimirov, Bykov & Ellison, 2008. ApJ, v. 688, p. 1084

~p2

~lcor

~(Wres)-1

Wavenumber, k

Tu

rbu

len

ce s

pectr

um

, k·W

(k)

Momentum, p

Part

icle

mean

fre

e p

ath

, (

p)

Turbulence Particles

Our model for particle propagation in strong turbulence interpolates between different scattering regimes in different particle energy ranges.

k – wavenumber of turbulent harmonics

W(x,k) – spectrum of turbulent fluctuations, (energy per unit volume per unit ∆k).

Wux

W L

dukW

x dx

duWdx

k

Amplification

( corresponds to Bell’s

instability)

Dissipation

Compression (amplitude)

Compression (wavelength)

Cascading

In this work we ignored compression for clarity

(does not affect the qualitatively new results)

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

Effective magnetic field B = 1.1·10-3 G

Shocked plasma temperature T = 2.2·107 K

~p2

Trapping

•Without cascading, Bell’s instability forms a turbulence spectrum with several distinct peaks.

•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

Effective magnetic field B = 1.5·10-4 G

Shocked plasma temperature T = 4.4·107 K

~p2

Resonant scattering

•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 short-wavelength 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

Without Cascading

With Cascading

Discussion• With better information about spectral energy transfer

(in a strongly magnetized plasma with ongoing nonresonant magnetic field amplification, accounting for the interactions with streaming accelerated

particles) we can refine our predictions regarding the amount of MFA, maximum particle energy Emax, heating and compression in particle accelerating shocks (plasma simulations needed)

• 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)

Very far upstream…

Solution with NO CASCADING

Far upstream…

Turbulence

amplification

Resonance

w/particles

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)…

Solution with NO CASCADING

The following sequence of slides shows how the peaks are formed one

by one in the shock precursor.

(model B, 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…

Solution with KOLMOGOROV CASCADE