MotivationCoronal signatures

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Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
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Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
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the solar wind
Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
natural turbulence laboratory

• We aim to
• Compare turbulence in plasmas and neutral fluids
  so that we can learn more about the universal
  aspects of turbulent flows
• Specific understanding of plasma turbulence
  phenomenology and modeling of space weather
• Learn more about the conditions in the solar
  corona from which the solar wind originates.

High magnetic Reynolds number 105
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the solar wind
Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
natural turbulence laboratory

• We aim to
• Compare turbulence in plasmas and neutral fluids
  so that we can learn more about the universal
  aspects of turbulent flows
• Specific understanding of plasma turbulence
  phenomenology and modeling of space weather
• Learn more about the conditions in the solar
  corona from which the solar wind originates.

High magnetic Reynolds number 105
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the solar wind
Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
natural turbulence laboratory

• We aim to
• Compare turbulence in plasmas and neutral fluids
  so that we can learn more about the universal
  aspects of turbulent flows
• Specific understanding of plasma turbulence
  phenomenology and modeling of space weather
• Learn more about the conditions in the solar
  corona from which the solar wind originates.

High magnetic Reynolds number 105
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self-similarity
Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
Change scale from t to bt AND scale y to bHy
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self-similarity
Motivation Coronal signatures Scaling Introducin
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Change scale from t to bt AND scale x to bHx
If the statistics of bHy is the same as y then
process is statistically self-similar
Hurst exponent H
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Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
x(t,?) y(t?) - y(t)
?
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Motivation Coronal signatures Scaling Introducin
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x(t,?) y(t?) - y(t)
?
probability density function (pdf)
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pdf collapse
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pdf collapse
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moment scaling
Motivation Coronal signatures Scaling Introducin
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pdf scaling implies scaling of moments AND
monoscaling implies a linear increase in the
scaling of the moments with moment order p
?(p)Hp
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moment scaling
Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
pdf scaling implies scaling of moments AND
monoscaling implies a linear increase in the
scaling of the moments with moment order p
?(p)Hp
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Motivation Coronal signatures Scaling Introducin
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Lévy processes
Motivation Coronal signatures Scaling Introducin
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For Levy H1/?
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heavy-tails
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finite-size effects
Motivation Coronal signatures Scaling Introducin
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From extreme value theory (EVT)
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finite-size effects
Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
From extreme value theory (EVT)
gradient1/?
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Motivation Coronal signatures Scaling Introducin
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K. Kiyani, S. C. Chapman and B. Hnat, Phys. Rev.
E 74, 051122 (2006)
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multifractals
Motivation Coronal signatures Scaling Introducin
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non-linear
Why does this happen?
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multifractals
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non-linear
Why does this happen?
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digression
Nigel Goldenfeld. Roughness-induced criticality
in a turbulent flow. Phys. Rev. Lett. 96,
0445031-4 (2006)
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Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
real-world data
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solar cycle change
Motivation Coronal signatures Scaling Introducin
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• In situ data from solar wind monitors ACE WIND
  around the Sun-Earth L1 point (R. Lepping, K.
  Ogilvie)
• To study solar max (2000) and solar min
  (1996)

• Probe in detail the scaling of B2 - magnetic
  field energy density

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Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
Monofractal Lévy process
Multifractal p-model
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results
Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
2000 - Solar max
1996 - Solar min
?(p)
?(p)
Moment p
Moment p
H 0.44 0.02
K. Kiyani, S. C. Chapman, B. Hnat and R. M.
Nicol, Phys. Rev. Lett 98, 211101 (2007)
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Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
_at_ 1
1 year (2000 solar max) ACE 64 seconds calibrated
magnetometer data (R. Lepping)
2 hrs
5 mins
log10(Sp)
K. Kiyani, S. C. Chapman, B. Hnat and R. M.
Nicol, Phys. Rev. Lett 98, 211101 (2007)
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pdfs
Motivation Coronal signatures Scaling Introducin
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Functional form of pdf different at solar max and
min
Power law tail at solar maximum reminiscent of a
Lévy process
B. Hnat et. al. , Geophys. Res. Lett. 34, L15108
(2007)
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Lévy?
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H1/? for a Lévy process
Measured values H0.44 ? 1/?0.71
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Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
All epistemological value of the theory of
probability is based on this that large scale
random phenomena in their collective action
create strict, non-random regularity. (Gnedenko
and Kolmogorov, Limit Distributions for Sums of
Independent Random Variables)
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limit theorems
Motivation Coronal signatures Scaling Introducin
g memory Heavy-tails Summary
Central Limit Theorem (De Moivre, Laplace,
Lyapunov)
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limit theorems
Motivation Coronal signatures Scaling Introducin
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Central Limit Theorem (De Moivre, Laplace,
Lyapunov)
Generalized Central Limit Theorem (Lévy)
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Joseph effect
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Introduce memory via Mandelbrots notion of
Joseph (memory) and Noah (large events) effects
using
Linear fractional stable motion (lfsm)
Memory parameter d H - 1/?
tail parameter ?
dgt0 gt persistence dlt0 gt anti-persistence
self-similarity parameter H Hurst exponent
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lfsm
Motivation Coronal signatures Scaling Introducin
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fft based algorithm obtained from S. Stoev and M.
Taqqu (fractals 2004)
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finite-size scaling
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finite-size scaling
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?(p)
Moment p
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Motivation Coronal signatures Scaling Introducin
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2000 - Solar max
lfsm H0.44 ?1.4
?(p)
?(p)
Moment p
Moment p
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conclusions
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• We find a unique monofractal signature in the
  solar wind at 1 AU in the ecliptic at solar
  maximum.
• Use of a robust statistical estimator which
  operationally excludes a small percentage of
  poorly represented extreme events.
• Solar Max -- Complex topology larger events gt
  are we seeing remnants of solar activity?
• To obtain a more complete picture it is
  necessary to take into account non-Markovian
  effects

Possible signatures of the processes driving the
turbulence?

• Outlook - look to different spacecrafts to scan
  interplanetary solar wind to study the evolution
  of its turbulence. Check for detailed solar cycle
  dependence.

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references
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• Self-similar scaling of B2
• K. Kiyani, S. C. Chapman, B. Hnat and R. M.
  Nicol, Phys. Rev. Lett 98, 211101 (2007).
• B. Hnat, S. C. Chapman, K. Kiyani, G. Rowlands
  and N. W. Watkins, Geophys. Res. Lett. 34,
  L15108 (2007).
• Conditioning extreme events
• K. Kiyani, S. C. Chapman and B. Hnat, Phys. Rev.
  E 74, 051122 (2006).
• Applications to anomalous transport
• G. Zimbardo, Plasma Phys. Control. Fusion 47
  B755-B767 (2005).
• D. F. Escande and F. Sattin, PRL 99, 185005
  (2007).
• Reading on stable processes
• Samorodnitsky Taqqu, Stable Non-Gaussian
  Random Processes.

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Motivation Coronal signatures Scaling Introducin
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