Self-Similar Evolution of Cosmic Ray Modified Shocks

Nov. 1-3, [email protected]

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Self-Similar Evolution of Self-Similar Evolution of

Cosmic Ray Modified ShocksCosmic Ray Modified Shocks

Hyesung Kang

Pusan National University, KOREA


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- CR energy flux emerged from shocks FCR= (M) Fk

Thermal E

CR E

thermalization efficiency: (M)CR acceleration efficiency: (M)

1

Vs= u1

Egas

- kinetic energy flux through shocks

Fk = (1/2)Vs3

- net thermal energy flux generated at shocks

Fth = (3/2) [P2-P1u2

= (M) Fk

ECR


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- Astrophysical plasmas are composed of thermal particles and cosmic ray particles.

- turbulent velocities and B fields are ubiquitous in astrophysical plasmas.

- Interactions among these components are important in understanding the CR acceleration.

Astrophysical Plasma

thermal ions &

electrons

Cosmic ray ions & electrons tubluentmean

tot

BB

B

turbulentmean

tot


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scattering of particles in turbulent magnetic fields isotropization in local fluid frame transport can be treated as diffusion process

streaming CRs upstream of shocks excite large-amplitude Alfven waves amplify B field ( Lucek & Bell 2000)

upstreamdownstream

Interactions btw particles and fields: examples

) timecrossing timescatteringmean i.e. ,( || sh

g

V

r


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- Full plasma simulations: follow the individual particles and B fields, provide most complete picture, but computationally very expensive (see the next talk by Hoshino)

- Monte Carlo Simulations with a scattering model: steady-state only

particles scattered with a prescribed model assuming a steady-state shock structure

reproduces observed particle spectrum (Ellison, Baring 90s)

- Two-Fluid Simulations: solve for ECR + gasdynamics Eqns

computationally cheap and efficient, but strong dependence on closure parameters ( ) and injection rate (Drury, Dorfi, KJ 90s)

- Kinetic Simulations : solve for f(p) + gasdynamics Eqns

Berezkho et al. code: 1D spherical geometry, piston driven shock , applied to SNRs, renormalization of space variables with diffusion length i.e. : momentum dependent grid spacing

Kang & Jones code: CRASH (Cosmic Ray Amr SHock code)

1D plane-parallel and spherical grid comving with a shock

AMR technique, self-consistent thermal leakage injection model

Numerical Methods to study the Particle Acceleration

C ,

)()( ppx


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In kinetic simulations Instead of following individual particle trajectories and evolution of fields

diffusion approximation (isotropy in local fluid frame is required)

Diffusion-convection equation for f(p) = isotropic part

),()()(3

1)( ,, pxQ

x

f

xp

fpuU

x

fuU

t

f

jji

iw

iiwi

lines fieldmean

and normalshock btw angle

shock tonormalt coefficiendiffusion

sincos

speeddrift eAlfven wav 22

||

Bn

BnBnxx

wu

BB

nn

Bn

Geometry of an oblique shock

shock

Injection coefficient

x

So complex microphysics of interactions are described by macrophysical models for p & Q


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upstreamin disspation eAlfven wav todue termheating Gas :

speed.Alfven is 4/

,downstreamin 0 upstream,in is speed wavewhere

),(]),([)(3

1)(

r

P-W

B

uu

pxQx

fpx

xp

fpuu

xr

fuu

t

f

cA

A

wAw

ww

Diffusion-Convection Equation with Alfven wave drift + heating

streaming CRs

- Streaming CRs generate waves upstream

- Waves drift upstream with

- Waves dissipate energy and heat the gas.

- CRs are scattered and isotropized in the

wave frame rather than the fluid frame

instead of u smaller velocity jump

and less efficient acceleration

A

generate waves

A

UU11

upstream

Aw uuu

1

Pc


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Parallel (BnBn=0) vs. Perpendicular (BnBn=90) shock

Injection is less efficient, but the acceleration is faster at perpendicular shocks

Slide from Jokipii (2004): KAW3

diffusion

field-cross xx

diffusion

parallel

|| xx

FULL MHD

+ CR terms

Gasdynamics

+ CR terms


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U2U1

Shock front

particle

downstreamupstream

shock rest frame

Diffusive Shock Acceleration in quasi-parallel shocks

Alfven waves in a converging flow act as converging mirrors particles are scattered by waves cross the shock many times

“ Fermi first order process”

v~ sU

p

p energy gainat each crossing

Converging mirrors

Bmean field

cv


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B

prgB

3

1||

Parallel diffusion coefficient

For completely random field (scattering within one gyroradius, =1) “Bohm diffusion coefficient”

minimum value

1 and 3

1

3

1

p. f. m.

||||

||

g

g

r

r

particles diffuse on diffusion length scale ldiff = ||(p) / Us

so they cross the shock on diffusion time tdiff = ldiff / Us= ||(p) / Us2

smallest means shortest crossing time and fastest acceleration.Bohm diffusion with large B and large Us leads to fast acceleration. highest Emax for given shock size and age for parallel shocks

||

This is often considered as a valid assumption because of self-excited Alfven waves in the precursor of strong shocks.


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Thermal leakage injectionat quasi-parallel shocks:

due to small anisotropy in veloc

ity distribution in local fluid fra

me,

suprathermal particles in non-M

axwellian tail

leak upstream of shock B0

uniform field

self-generated

wave

leaking particles

Bw

compressed waves

hot thermalized plasma

unshocked gas

Suprathermal particles leak out of thermal pool into CR population.


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velocityflow downstream is where

)1(-exp

11)1(),(

2

1

d

Bd

B

B

ddB

d

B

dBesc

u

u

uu

uH

u

“Transparency function”: probability that particles at a given velocity can leak upstream.

e.g. esc = 1 for CRs

esc = 0 for thermal ptls

CRs

gas ptls B=0.3

B=0.25

Smaller B : stronger turbulence, difficult to cross the shock, less efficient injection

turbulent

fieldmean

)(

0

B

B

Mfcnuu

B

sd

th

d

So the injection rate is controlled by the shock Mach number and B


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02

42

1

2

1)(

3

4

),(]),([)(3

1)(

][)1()(

)()(

0)()(

0)(

p

dpppfcmP

pxQx

fpx

xp

fpuu

xr

fuu

t

f

LWx

uS

t

S

x

PuuPue

xt

e

PPuxt

ux

u

t

pc

ww

g

cgg

g

cg

g

Basic Equations for 1D plane- parallel CR shock

S = modified entropy = Pg/to follow adiabatic compression in the precursor

W= wave dissipation heating L= thermal energy loss due to injection

across the shock

outside the shock

ordinary gas dynamic Eq. + Pc terms


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CR modified shock: diffusive nature of CR pressure introduces some characteristics different from a gasdynamic shock.

- diffusion scales: td (p)= (p)/us2, ld (p)= (p)/us

wide range of scales in the problem: from pinj to pmax

numerically challenging ! not a simple jump across a shock total transition = precursor + subshock

- acceleration time scale: tacc(p) td(p)

instantaneous acceleration is not valid so time-dependent calculation is required

- particles experience different u depending on p due to the precursor velocity gradient + ld (p) f(x,p,t): NOT a simple power-law, but a concave curve

should be followed by diffusion-convection equation


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- For given shock parameters: Ms, us

the CR acceleration depends on the shock Mach number only.

So, for example, the evolution of CR modified shocks is “approximately”

similar

for two shocks with the same Ms but with different us,

if presented in units of

“Similarity” in the dynamic evolution of CR shocks

),(1

8)

)()((

32

2

2

2

1

1

21sd

s

sacc upt

M

M

u

p

u

p

uut

)(

1

82

2

ss

s

d

acc MfcnM

M

t

t

).( oft independen ith time,linearly w stretches structureshock

8

1)(

)(8 - max

2max

p

tuu

pl

u

pt s

sshock

s

- Thermal injection rate: depends on Ms and B

ssd u

p

u

pt

)( x,

)(d2


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Three runs with

(p) = 0.1 p2/(p2+1)1/2

(p) = 10-4 p

(p) = 10-6 p

at a same time t/to=10

(p) oft independen

ith t,linearly w stretches

structureshock the

8/

tul sshock

PCR,2 approaches time

asymptotic values for

t/to > 1.

1

but max p

At t=0, Ms=20 gasdynamic shock


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Numerical Tool:CRASH (Cosmic Ray AmrNumerical Tool:CRASH (Cosmic Ray Amr SHock ) Code SHock ) Code

Bohm type diffusion: for p >>1 - wide range of diffusion length scales to be resolved:

from thermal injection scale (pinj/mc=10-2) to outer scales for the highest p (~106)

1) Shock Tracking Method (Le Veque & Shyue 1995) - tracks the subshock as an exact discontinuity 2) Adaptive Mesh Refinement (Berger & Le Veque 1997) - refines region around the subshock with multi-level grids

Nrf=100

pp

pp

1)(

2

2

Kang et al. 2001

1024/

108 typically

level, grideach at

refinement twooffactor a

010

max,

xx

lg


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Time evolution of the M0 = 5 s

hock structure.At t=0, pure gasdynamic shock

with Pc=0 (red lines).

t=0

Kang, Jones & Gieseler 2002

-1D Plane parallel Shock DSA simulation

“CR modified shocks”- presusor + subshock- reduced Pg

- enhanced compression

precursor

No simple analytic shock jump condition

Need numerical simulations to calculate the CR acceleration efficiency

preshock postshock


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Evolution from a gasdynamic shock to a CR modified shock.1) initial states : a gasdynamic shock at x=0 at t=0

- T0= 104 K and us= (15 km/s) Ms , 10<Ms<80

- T0= 106 K and us= (150 km/s) Ms, 2<Ms<30

2) Thermal leakage injection :

- more turbulent B smaller B smaller injection

- pure injection model : f0 = 0

- power-law pre-existing CRs: fup(p) = f0 (p/pinj) -5

3) B field strength :

1)(

2

2

p

pp o4) Bohm type diffusion:

mcppp o of unitsin is where,)(or

BBB / where,25.02.0 0B

ISMfor 1~ ICM,for 1.0~

10 ,/

speeddrift wavedetermines ,4/ w

thB

A

EE

Bu


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Pcr,2 reaches to an asymptotic value,

The shock structure stretches linearly with t.

the shock flow becomes self-similar.

efficiency CR

constant

5.0

),()(

2,

30

t

tP

tV

txdxEt

c

s

CR

CR energy/shock kin. E.


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dtun

dppfpdxt

oo

CR

'

2 )(4)(

02t

01s

/

/

ration compressioshock

Pc,2 increases with Ms

but asymptotes to 50% of

shock ram pressure.

Fraction of injected CR particles is higher for higher Ms.


ea

4ln

)(ln

, ),( 4

pd

Gdq

dxppxfG

p

p

G p : non-linear concave curvature

q ~ 4.2 near pinj

q ~ 3.6 near pmax

f( xs, p)p4 at the shock

at t/to = 10


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CR acceleration efficiency vs. Ms for plane-parallel shocks-The CR acceleration efficiency is determined mainly by Ms . It increases with Ms, but it asymptotes to a limiting value of ~ 0.5 for Ms > 30.-larger ( larger A/cs): less efficient acceleration due to the wave drift in precursor

- larger weaker turbulence: more efficient injection, and less efficient acceleration- pre-existing CRs: higher injection: more CRs- these dependences are weak for strong shocks

tV

txdxEt

s

CR

305.0

),()(

Pre-exist Pc

B=0.25

B=0.2


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From DSA simulations using our CRASH code for parallel shocks:

-Thermal leakage injection rate is controlled mainly by Ms and the level of downstream turbulent B fields. (a fraction of = 10-4 - 10-3 of the incoming particles become CRs.)- The CR acceleration rate depends on Ms, because acc/td = fcn(Ms) in other words u/u = fcn(Ms)- The postshock CR pressure reaches a stable value after a balance between fresh in

jection/acceleration and advection/diffusion of the CR particles away from the shock is established.

-The shock structure broadens as lshock ~us t/8, linearly with time, independent of the diffusion coefficient.

So the evolution of CR shocks becomes approximately ``self-similar” in time.

It makes sense to define the CR energy ratio for the acceleration efficiency

- Ms increases with Ms, depends on B, EB/Eth (wave drift speed),

but it asymptotes to a limiting value of ~ 0.5 for Ms > 30.

SUMMAY


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Thank You !


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Plasma simulations at oblique shocks : Giacalone (2005a)

Injection rate weakly depends on Bn for fully turbulent fields.

~ 10 % reduction at perpendicular shocks

(B/B)2=1

The perpendicular shock accelerates particles to higher energies compared to the parallel shock at the same simulation time .

parallel

perp.


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Observational example:particle spectra in the Solar wind(Mewaldt et al 2001)

-Thermal+ CR populations-suprathermal particles leak out of thermal pool into CR population

CRgas


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initial Maxwellian

Concave curve

CR feedback effects

gas cooling (Pg decrease)

thermal leakage

power-law tail

concave curve at high Epower-law tail (CRs)

Particles diffuse on different ld(p) and feel different u,

so the slope depends on p.

f(p) ~ p-q

21

13)(

uu

upq

Evolution of CR distribution function in DSA simulation

f(p): number of particles in the momentum bin [p, p+dp], g(p) = p4 f(p)

injection momenta

thermal

g(p) = f(p)p4

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Self-Similar Evolution of Cosmic Ray Modified Shocks

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