Session 3: Radial diffusion, stochastic transport, and non-diffusive processes in radiation belts and the slot region

Solène Lejosne, Sasha Ukhorskiy

May 2015, Paris

Outline

3/ From dL*/dt to DLL (and reciprocally)

1/ What is diffusion?(D??)

2/ How can I trigger a variation of the 3rd adiabatic invariant?(dL*/dt??)

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

1. What is diffusion?

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

1. What is diffusion?

1.1. Random walk (1D)

Each particle goes to the left (-x) or to the right (+x)Probability (go to the right) = probability (go to the left)

once every τ seconds, moving a distance δ=ντ

⟨𝑥 ⟩=0 ⟨ 𝑥2 (𝑡 ) ⟩=𝑡 𝛿  2

𝜏 =2𝐷𝑡

𝐷=𝛿  2

2𝜏 >0 where <…> denotes an average over time or aggregate of similar particles

Galton machinehttps://www.youtube.com/watch?v=xDIyAOBa_yUhttps://www.youtube.com/watch?v=epq-dpMJIxs

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

1.2. Fick transport equation

+ Continuity equation:

𝑗= 12 (𝑛 (𝑥 )𝑣−𝑛 (𝑥+𝛿 )𝑣 )

Fick’s law (steady state):

𝜕𝑛𝜕𝑡 +

𝜕 𝑗𝜕 𝑥=0

𝜕𝑛𝜕𝑡 = 𝜕

𝜕𝑥 (𝐷 𝜕𝑛𝜕 𝑥 )

𝜕𝑛𝜕𝑡 =𝛁 ∙ (𝐷𝛁𝑛)

Diffusion equation:

“The flux goes from a region of high concentration to a region of low concentration, with a magnitude that is proportional to the concentration gradient”

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

1.3. Application to radiation belts: radial diffusion

𝜕𝑛𝜕𝑡 =𝛁 ∙ (𝐷𝛁𝑛) 𝜕 𝑓

𝜕𝑡 = 𝜕𝜕𝜙 (𝐷𝜙𝜙

𝜕 𝑓𝜕 𝜙 ) 𝜙∝𝐿∗−1

𝐿∗∝ 𝑟  /𝑅𝐸

𝜕 𝑓 (𝜙 )𝜕𝑡 =𝐿∗2 𝜕

𝜕𝐿∗ (𝐷𝐿𝐿

𝐿∗2𝜕 𝑓 (𝜙 )𝜕𝐿∗ )

𝜕 𝑓 (𝐿∗ )𝜕𝑡 = 𝜕

𝜕𝐿∗ (𝐷𝐿𝐿

𝐿∗2𝜕

𝜕𝐿∗ (𝐿∗2 𝑓 (𝐿∗) ))What are the underlying assumptions?

• Random walks along the φ axis• Small correlation values• Steady state

What are the consequences?

• Central role of the radial gradient

[Walt, 1994]

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

2. What triggers the variation of the third adiabatic invariant?

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

2.1. Describing the problem: single-particle tracking

0rB

At time t)( 0rB

?00 drr At time t+dt

[e.g., Roederer, 1970]

dtBqB

Mdt

2200

BEBBrVdr indD

It is possible to track the particle motion (i.e. no randomness there)

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

2.2. Describing the problem: guiding-contour tracking

0rB

At time t

)( 0rB

B

B

r0

B(r0+dr0)

r1

r2

B(r2+dr2)

B(r1+dr1)

)( 0rB

At time t+dt

Trigger == variations of the electromagnetic field that depend on magnetic local time (asymmetry)

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

2.3. Results: a general expression of dL*/dt exists

• In the most general case :

),()(),( tdtdt

dtd

qt

dtd D

00 rr

),()(),( 0 tdtdBt

dtdBCt

dtd

00 rr3

2 20

0rC

with

0)*,,,(*tLKM

dtdL

[Northrop, 1963][Cary and Brizard, 2009]

• In the equatorial case + in the absence of electrostatic potential:

[Lejosne et al., 2012]

[Lichtenberg et Lieberman, 1992]

and <…> means averaging over MLT

)()()( rqUrMBr m

)(),,(

2*),,(*

0

2

tdtdtr

dtd

kqRLtr

dtdL eD

• Consequence 1: => No bulk motion on average over MLT

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

2.4. Interpretation: competition τD and τc

),()(),( tdtdt

dtd

qt

dtd D

00 rr

Adiabatic

Transfo 1

Adiabatic

Transfo 2

)(),( tdtdt

dtd

0r )(),( tdtdt

dtd

0r

Conservation of M and L* symmetric acceleration by the curl of Eind [Fillius and Mc Ilwain, 1967]

Non Adiab

Transfo 1

Non Adiab

Transfo 2

Violation of L* : asymmetric acceleration by the curl of Eind

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

2.5. Conclusion: What triggers the variation of the third adiabatic invariant?

• Induced electric fields do not necessarily imply violation of L*.

• In fact, and work together so that the flux of particles initially on the same drift shell is conserved, on average.

• When the ELM field is symmetric, or equivalently, when the ELM field is varying slowly in comparison with the drift period , the (drift betatron) acceleration is reversible (L* is conserved).

• It is the degree of the asymmetry in the variations of the field that drives the intensity of the variations in L*.

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

3. From dL*/dt to DLL and reciprocally

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

3. From dL*/dt to DLL and reciprocally

tkRL

t

LD e

LL 2)2(*

2

* 2

20

242

?),*,,,( * tLKM

dtd

L

t

LLdt

dtdt )(),( **

)(1

*2

2

Ltt

4D Tracking

Integration over Δt

Drift Average

Operation, lots of

scenarios

Calculation of a DLL

Average over MLT

Average over x scenarios

Mysterious Δt (very short/very long/whatever)

),()( t

dtdt

dtd

qD

0r

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

dudttdtdu

dtdt

dtdu

dtd

tkqRLLKMD

tt

t

tt

tDe

LL

0

0

0

0

)()()()()(8

**),,( 22

0

24

dudttdtdBu

dtdBt

dtdBu

dtdBC

tkRLLKMD

tt

t

tt

te

LL

0

0

0

0

)()()()()(8

**),0,( 202

0

24

Ready-made general formula of a radial diffusion coefficient

In the equatorial case in the absence of electrostatic potentials:

3.1. Radial diffusion coefficients (from [Lejosne et al., 2012])

32

),(),( 2

0

00

rdltB

tBC

r

r

NOT “Ozeke”’s formula [Fei et al., 2006]

MORE GENERAL than Falthammar’s formula (pitch angle, energy, any ELM FIELD)

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

3.2. What has been done: Fӓlthammar’s work [1965]

1/ Equatorial case

2/ Simple 3D ELM model [Mead, 1964] (small perturbations)

B(r, π/2, φ,t) = k0/r3 + S(t) + A(t) r cos φ

3/ A(t)? realizations of a stochastic process, assumed stationary

4/ Scattering rate <(ΔL)2>/ Δt as Δt → ∞ (!) given by

(5/ If sudden impulses, H = const.Ω -2 => DLL energy independent)

102*),0,( LHLKMDLL

D2

0

cos)(~)(~4 dtAtAH

Power 8 ~by definition+ Power 2 bc B~ Ar= Power 10

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

3/ From dL*/dt to DLL (et vice et versa)

Liouville’s theorem

Fokker-Planck equation

some assumptions ….(+ no interest in phase variations)

4D + t

3D + t

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

3.2. What has been done: Sasha’s work

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

3.2. Discussion: maybe it is time to get over the diffusion framework

- DLL as a function of MLT??

Þ If we forget about MLT, it ~looks like diffusionÞ But what if the PSD was strongly dependent of MLT? (located injection/located loss)• Fokker-Planck framework: assumes phase-mixingÞ No radial diffusion if PSD is strongly phase-dependent

Þ thus, do radial gradients in PSD still matter?

- event specific DLL??

• Fokker-Planck framework: assumes stationary stateÞ If I wait long enough: overestimations compensate underestimations. on average = good jobÞ Resonance?

⟨𝑥 ⟩=0

?0),*,,,(*)*,,,(*0 tLKM

dtdLtLKM

dtdL

𝑗=−𝐷 𝜕𝑛𝜕 𝑥 ? ?

0)*,,,(*tLKM

dtdL

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

3/ From dL*/dt to DLL (et vice et versa)

Liouville’s theorem

Fokker-Planck diffusion equation

oversimplification A new compromise?

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

4. Take away messages

Quantifying radial diffusion is one of the oldest topics in radiation belt science.

Originally, radial diffusion coefficients have been introduced to characterize the dynamics of the radiation belts on average during long time intervals. ( DLL = constant = how much? // delta_t = infinity)

To this purpose, different approximations and assumptions have been made … … and have changed over time. (DLL = time-varying!)

Currently, the demand is : to understanding the dynamics of the radiation belts , ‘ideally to the point of predictability’.

Since radial diffusion is an average (over the drift phases, over different possible scenarios), the modeling of transport of radiation belt particles is smoothed.

=>> One should decide if the over simple diffusive framework is good enough or if something(?) better should be done to meet one’s needs.

Solène Lejosne, Tutorial S3, PARIS, MAY 2015

Radial diffusion ? we should do better than that.

4. Simple Take away message

Solène Lejosne, Tutorial S3, PARIS, MAY 2015