Shock Acceleration at an Interplanetary Shock: A Focused Transport Approach

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1 Shock Acceleraton at an Interplanetary Shock: A Focused Transport Approach J. A. le Roux Insttute of Geophyscs & Planetary Physcs Unversty of Calforna at Rversde

2 . The Gudng center Knetc Equaton for f(x,p,p,t Transform poston x to gudng center poston x g (x = x g r g Transform transverse momentum to plasma frame p = p V E Smallness parameter δ = m/q << (small gyroradus, large gyrofrequency Keep terms to order δ o Average over gyrophase f t V g ( f x g dp * f p dp * f p = & % f t #! " c V where g dp = v b V, dp % % = qe = E # p v B p v B! B #! B mv p E db ", (! " V # b "! V E E

3 Transform (p, p (ε, M V V where g g dm % 0, d = f t = v b V & E qe V = v b V E, g V M q g M f ( x # B, # t g dm f M d * f M B! " B mb d ("! b! ( v b q B qb = & % f t #! " c Conservaton of magnetc moment Electrc feld drft Grad-B drft Curvature drft

4 To get focused transport equaton, assume n gudng center knetc equaton E = -U B, whereby V E = U Transform parallel momentum to movng frame (p = p mu Transform (p,p (p,. The Focused Transport Equaton for f(x,p,, t Convecton ( ( ( ( PUI j j j j Q f D p f p E b p q du b v x U b b x U f E b p q du b v x U b b x U x b v x f b v U t f! " # % & = ( ( *, ( ( *, Adabatc energy changes Parallel Dffuson Mrrorng/focusng

5 Advantages Same mechansms as standard cosmc-ray transport equaton No restrcton on ptch-angle ansotropy - njecton Descrbe both ( st order Ferm, and ( shock drft acceleraton wth or wthout scatterng Dsadvantages No cross-feld dffuson Gradent and curvature drft effect on spatal convecton gnored Magnetc moment conservaton at shocks

6 3. Results: (a Proton spectra n SW frame SW Kappa dstrbuton (κ = 3 /r 6 /km 3 (v(s f AU f(v v /km 3 (v(s f SW densty = C AU f(v v f(v v upstream v/u e downstream v/u e SW densty /r

7 SW Kappa dstrbuton (κ = 3 /r 0.3 AU AU 0.49 AU 6 _.._0.7 AU AU /km 3 (v(s f Shft n peak to lower energes f(v v -5.3 v/u e f(v v -3

8 (b Ansotropes n SW frame at AU downstream shock.00.0 (f 0.0 ξ=8% upstream (f ξ=5% kev MeV ξ=9% (f MeV

9 (c Spatal dstrbutons across shock kev (z f MeV 0 - MeV z(au 0 MeV

10 (d Tme dstrbutons at AU MeV t(f MeV MeV t(hours 00 kev

11 3. COMPARISON OF STRONG AND WEAK SHOCKS: (a Accelerated spectra 6 /km 3 (v(s f Strong shock (s= f(v v /km 3 (v(s f Weak shock (s= AU 0.7 AU f(v v -7. v/u e Spectrum harder than predcted by steady state DSA v/u e Spectrum softer than predcted by steady state DSA

12 (b Ansotropes Strong shock (s=4 Weak shock (s= (f 0 - (f kev kev Ptch-angle ansotropy larger for a strong shock

13 D parallel Alfven wave transport equaton at parallel shock solved I t ( U V ( sh z & U = ( % z ( k I k sh ( U V & % D kk A sh I z A #! I " I k k I ( I( k, k k #! " Convecton Convecton n wave number space- Compresson shortens wave length Compresson rases wave ampltude Wave generaton by SEP ansotropy Wave-wave nteracton forward cascade n wave number space n nertal range

14 Wave-wave nteracton Model (Zhou & Matthaeus, 990: D kk = k t s Kolmogorov cascade I(k k -5/3 t NL << t A (δu = δv A >> V A or (δb >> B o Krachnan cascade I(k k -3/ t A << t NL (δu = δv A << V A or (δb << B o t s = t NL = l! U t = t = s A l V A Comments Turbulence can be modeled as a nonlnear dffuson process n wavenumber space (D kk Ad-hoc turbulence model based on dmensonal analyss Contrary to weak turbulence theory assume that 3-wave couplng for Alfven waves s possble mply wave resonance broadenng momentum and energy conservaton n 3-wave couplng does not hold

15 Alfven wave Spectrum (Kolmogorov Cascade - (evcm (k I 0 6 downstream AU 0.5 AU 0.7 AU AU at shock at 0. AU k r e upstream Need C K > 8 for cascadng to work

16 Kolmogorov or Krachnan Cascade? Bastlle Day CME event Kallenbach, Bamert, le Roux et al., 008 Used Marty Lee (983 quas-lnear theory extended to nclude nonlnear wave cascadng va a dffuson term n wavenumber space. Appled theory to Bastlle Day and Halloween # CME events to calculate k mn for wave growth Fnd that Krachnan-type turbulence gves the correct k mn Kolmogorov-type turbulence overestmates k mn by a factor -4 Halloween CME event # Vasquez et al., 007 For cascade rates and proton heatng rates to agree n SW C K = needed nstead of the expected C K ~ Krachnan cascade rate s close to heatng rate at AU

17 A Weak Knetc Turbulence Theory Approach to Wave Cascadng? Wave Knetc Equaton Nonlnear wave-wave nteracton di( k = # ( k I( k % dk[ ak, k I( k I( k ak, k ak, k I( k I( k ]" (!( k!( k!( k Lnear wavepartcle nteracton Wave generaton Wave decay Wave-wave nteracton term a Boltzmann scatterng term When assume weak scatterng, term can be put n Fokker- Planck form so that nonlnear dffuson coeffcents for wave cascadng D kk can be derved

Turbulence. Lecture 21. Non-linear Dynamics. 30 s & 40 s Taylor s work on homogeneous turbulence Kolmogorov.

Turbulence. Lecture 21. Non-linear Dynamics. 30 s & 40 s Taylor s work on homogeneous turbulence Kolmogorov. Turbulence Lecture 1 Non-lnear Dynamcs Strong non-lnearty s a key feature of turbulence. 1. Unstable, chaotc behavor.. Strongly vortcal (vortex stretchng) 3 s & 4 s Taylor s work on homogeneous turbulence