Download - Tutorial on radial diffusion
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