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Cyprus 2011 complexity extreme bursts and volatility bunching in solar terrestrial physics

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Invited talk at Sigma Phi statistical physics meeting, Cyprus, 2011.

Invited talk at Sigma Phi statistical physics meeting, Cyprus, 2011.

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  • 1. Nick Watkins British Antarctic Survey, Cambridge, UK  2011, Tuesday, 12th July Complexity, extreme bursts, and volatility bunching in solar-terrestrial physics
  • 2. 1. BAS/Warwick research on multiscale complexity in Earth’s magnetosphere began ~15 years ago [e.g. Chapman et al, GRL, 1998; Watkins et al, GRL,1999; Freeman et al, PRE, 2000]. Within this my personal focus has been on self-similar and multifractal time series models. 2. For sigma phi audience & this workshop, highlight work in progress on i) temporal scaling of bursts above threshold in monofractal time series [Carbone & Stanley, PRE, 2004; Watkins et al, PRE, 2009], and ii) a multifractal feature, “volatility clustering”. Show that some space physics time series share this property, well known in some financial ones [see also Engle Nobel lecture, Mantegna & Stanley book; Rypdal & Rypdal, JGR, 2011]. Talk about a simple linear stochastic model, the Kesten process studied intensively by Sornette. Advocate use of this toy model for framing “null” hypotheses about volatility bunching. 3. Relevance goes beyond solar-terrestrial physics to broader issue of model choice and diagnostics for complex systems in complex environments which may be neither weakly coupled or slowly driven [Freeman & Watkins, Science, 2002].
  • 3. Thank many colleagues including: Tim Graves (Cambridge), Dan Credgington (Now UCL) , Sam Rosenberg (Now Barclays Capital), Christian Franzke (BAS), Bogdan Hnat (Warwick), Sandra Chapman (Warwick), Nicola Longden (BAS), Mervyn Freeman (BAS), Bobby Gramacy (Chicago) Watkins et al, Space Sci. Rev., 121, 271-284 (2005) Watkins et al, Phys. Rev. E 79, 041124 (2009a) Watkins et al, Phys. Rev. Lett. , 103, 039501 (2009b) Watkins et al, submitted to AGU Hyderabad Chapman Conference Proceedings Franzke et al, submitted Phil. Trans. Roy. Soc., arXiv:1101.5018
  • 4. “Standard model” of Solar Terrestrial Physics Solar wind Magnetosphere Ionosphere • Reconnection-driven plasma convection-”loading” • Magnetospheric substorms-”unloading”
  • 5. Convection (DP2) • Mass, momentum and energy input from reconnection at solar wind - magnetosphere interface. • Plasma circulation from day to night over poles and from night to day around flanks. • magnetic pole equator Sun flow solar wind magnetosphere
  • 6. Substorms (DP1) • Irregular, large-scale releases of energy in magnetotail -substorms. • Intense magnetic field- aligned currents accelerate particles to cause aurora. solar wind magnetosphere BANG!
  • 7. Multiscale magnetosphere ... Solar wind Magnetosphere Ionosphere Data Heavy tailed pdf of size of bursts above threshold in AE auroral index Tsurutani et al (1990) left, and Consolini (1997,98) above: drew attention to multiscale behaviour in 1D auroral time series Used a “burst” diagnostic derived from SOC. Reviews incl. Freeman & Watkins, 2002; Watkins et al, 2001. Averaged spectrogram of AE-”1/f” at low freq. “Burst”
  • 8. Often forgotten (or not realised) that Bak et al’s aim was to unify heavy tails in amplitude with “1/f” noise in time, via a physics-inspired model. The physical inspiration for SOC just happened to be from condensed matter, not from solar terrestrial physics ... & v 1.0 of the model didn’t produce 1/f noise in output ... Why an SOC approach?
  • 9. So a question, 1997-98, was … 22 April 2014 9 Does SOC apply to magnetospheric energy release events ? [Consolini 1997; Chapman et al, 1998; Uritsky & Pudovkin, 1998 ] ? Lui et al, GRL, 2000
  • 10. The joy of fractals ... • "It makes me so happy. To be at the beginning again, knowing almost nothing...a door like this has cracked open five or six times since we got up on our hind legs. Its the best possible time to be alive” – Tom Stoppard, Arcadia
  • 11. Scenario: “… the internal relaxations of the magnetosphere statistically follow power laws that have the same index independent of the overall level of activity, and that both the internal and global events are consistent with the behaviour of a finite size avalanche model. ... … The onset of local avalanches in the sandpile model can be physically related to the merging of coherent structures around Alfvenic resonances [Chang, 1998, 1999] or current disruption by kinetic instabilities [Lui, 1996] in the magnetotail”. To which might add multi site reconnection, made more explicit by Klimas et al, 2000 The SOC paradigm, led Lui, Chapman et al [GRL, 2000]; to study spatial “blobs” defined by thresholding in UVI images “… using the global auroral energy deposition as measure of the energy output of themagnetosphere”.
  • 12. ....spatial signals Integrated power in Polar UVI blobs exceeding a brightness threshold----subdivided into substorm, quiet time and pseudobreakup. Lui, NPG, 2002 Prediction in Chapman et al, GRL, 1998; Watkins et al GRL, 1999. Test in Lui, Chapman et al, GRL, 2000
  • 13. 22 April 2014 13 Lui, Chapman et al, GRL, 2000 Continuing Question: What would magnetotail exhibiting multisite reconnection [e.g. Daughton et al, 2011] look like in ionosphere ? Uritsky et al, JGR, 2002 (& Freeman & Watkins Science commentary), and their recent papers
  • 14. log s log P(T) log P() logT log  Poynting flux in solar wind plasma from NASA Wind Spacecraft at Earth-Sun L1 point Freeman, Watkins & Riley [PRE, 2000]. Dialogue on this topic is one of several directions research area has proceeded along post 2002 (c.f. Rypdal & Rypdal, JGR, 2010b). log P(s) size length waiting time Ambiguity: magnetosphere non- autonomous, what about driver ? Bursts seen in solar wind Poynting flux
  • 15. Dealing with ambiguity Difficulty of attributing complex astrophysical phenomena uniquely to SOC has led me to back up one stage, and to get interested in the known models for non-Gaussian, temporally correlated stochastic processes. Partly to try and see what physics was embodied in any given choice, partly for “calibration” of the measurement tools. [e.g. Watkins NPG, 2002; Watkins et al, SSR, 2005; PRL, 2009] In similar sense to Eliazar & Klafter’s work the models go beyond the CLT. They do not embody general “laws”, but map out a range of widely observed “tendencies”. We have become particularly interested in Mandelbrot’s models and their close relatives [e.g. Watkins et al, PRE, 2009].
  • 16. 4 “giant leaps” made by Mandelbrot between 1963 and 1974---”well known” but history is informative 1. BBM remarks heavy tailed fluctuations in 1963 in cotton prices---applies alpha-stable model & self-similarity idea 2. BBM hears about River Nile and “Hurst effect”. Initially (see his Selecta) believes this will also be explained by heavy tails. But when sees that fluctuations are ~ Gaussian applies self-similarity [Comptes Rendus,1965] in form of a long range dependent (lrd) model, the roots of fBm. BBM’s classic series of papers with Van Ness and Wallis (68-69) on fBm in maths & hydrology literatures. 3. BBM demonstrates a new self-similar model, fractional hyperbolic motion, in 1969 paper with Wallis on “robustness” [sic] of R/S. Combines 1 & 2 (heavy tails & lrd). 4. BBM becomes dissatisfied with purely self-similar models, develops multifractal cascade, initially in context of turbulence [JFM, 1974]. Later applications of multifractal models include finance.
  • 17. “Noah effect”- e.g. Lévy flights where  < 2 increases tail fatness =1 e.g. Hnat et al, NPG [2004] =2 “Levy flight”: applied to magnetometer data by Consolini Black line is AE differenced at ~ 15 minutes 1. BBM observes heavy tailed fluctuations in 1963 in cotton prices--- alpha-stable model , self-similarity idea
  • 18. “Joseph effect”-e.g. fractional Brownian (fBm) walk: steepness of log(psd) with log(f) increases with memory parameter d d=-1/2 d=0 S(f) ~ f-2(1+d) Fractional Brownian motion model: applied to AE by Takalo and Timonen, 1994 et seq. 2. BBM hears about River Nile and “Hurst effect”. Initially (see his Selecta) believes this will be explained by heavy tails, but when he sees that fluctuations are ~ Gaussian applies self-similarity [Comptes Rendus1965] in the form of a long range dependent (lrd) model, roots of fractional Brownian motion. BBM’s classic series of papers on fBm in mathematical & hydrological literature with Van Ness and Wallis in 1968-1969.
  • 19. 1 1 1 ( ) ( ) ( ) ( ) H H H H R X t C t s s dL s                     H = d+1/α: allows H “subdiffusive” (i.e. < ½) while α “superdiffusive” (i.e. <2). Memory kernel: Joseph effect α-stable jump: Noah effect LFSM of today is a stable successor to Mandelbrot’s model 3. BBM demonstrates a new self-similar model, fractional hyperbolic motion, in 1969 paper with Wallis on robustness of R/S. Combines effects 1 & 2 (heavy tails & lrd). Nowadays would use linear fractional stable motion---LFSM, applied in space plasmas by Watkins et al, 2005: NB H here is self-similarity exponent not identical to “Hurst” exponent except in Gaussian alpha=2 case
  • 20. 1D spreading exponent • Burst diagnostics previously proposed include 1D version of “spreading” exponent  used by Uritsky et al, GRL, 2001 [c.f. book by Marro & Dickman]. • Took ensemble time average as function of time  of activity AE of a curve after it has crossed a threshold at time t. N*() = <AE(t+  )> - L
  • 21. Modelling bursts Has potential wider application to prediction of “typical” burst size in fractal time series. Reported scaling of N*() as  to the 
  • 22. Brownian walkers
  • 23. Brownian “upstarts”
  • 24. Brownian “survivors”
  • 25. Surviving activity only Have repeated with fBm, LFSM,
  • 26. Noah Meneveau & Srinivasan p-model 4. BBM becomes dissatisfied with purely monofractal models, develops multifractal cascade, initially in context of turbulence, JFM,1974. Applied to fluctuations of AE index by Consolini et al, 1996. Later multifractal applications studied by Mandelbrot included finance in late 1990s. See also Ghashgaie et al, Nature, 1996 who used multifractal Castaing pdf, and interesting debate about alpha-stable versus multifractal models between them and Mantegna & Stanley, Nature, 1998.
  • 27. Noah Natural examples include ionospheric AE index (above), & ice cores (e.g. Davidsen and Griffin, PRE , 2009), . Rypdal & Rypdal, 2010-11 noted that effect not seen in monofractal models like LFSM 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 4 -600 -400 -200 0 200 400 600 increments,r First differences of AE index January-June 1979 -100 -80 -60 -40 -20 0 20 40 60 80 100 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 lagacf AE data: acf of returns -100 -80 -60 -40 -20 0 20 40 60 80 100 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 lag acf AE data: acf of squared returns First differenced AE data ACF of diff. AE ACF of (diff. Ae squared) Why did BBM become dissatisfied ? Partly his eyes told him to: One effect multifractals capture is “volatility clustering”
  • 28. Volatility clustering in AE 10 0 10 1 10 2 10 3 10 -3 10 -2 10 -1 10 0 lag acf AE data: acf of squared returns ACF of (diff. Ae squared) for 20 000 minutes after 1979 1st Jan Time lags up to 1000 minutes
  • 29. The Kesten process 0 2 4 6 8 10 12 x 10 5 -40 -30 -20 -10 0 10 20 30 40 -normalised Kesten process x(t)/(x) 0 2 4 6 8 10 12 x 10 5 -100 -80 -60 -40 -20 0 20 40 Walk y made from summing Kesten process x y(t)=(x) X(n+1) = A X(n) +B Where A and B both iid Normal, <A>,<B>=0, s d A =0.8, s d B =0.05 Parameter and distribution choices give wide range of behaviour. X Y= cum. sum of X Generalises X(n+1) = λ X(n) +ξ, the AR(1) Process, to case where correlation time varies.
  • 30. Multifractality -30 -20 -10 0 10 20 30 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Rescaling  y() by () ( y - < y>)/ ()P(y(t,)) =1 =10 =100 =1000 -1 0 1 2 3 4 5 6 -1 -0.5 0 0.5 1 1.5 2 2.5 3 (m) m Difference pdfs of walk y do not collapse, instead change shape. Curvature in function zeta (m), exponents of mth order structure functions versus m, indicates multifractality
  • 31. But “mild” volatility clustering ? 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 -2 10 0 10 2 10 4 FFT of Kesten process x RawPSD 0 10 20 30 40 50 60 70 80 90 100 -5000 0 5000 10000 ACF of the Kesten process x itself (NOT y) ACF()  10 0 10 1 10 2 10 3 10 1 10 2 10 3 Log log plot of ACF of the square of Kesten process x ACF()  Decay of ACF of square of X is slower than exponential, but finite ranged
  • 32. Conclusion • In 1D spreading exponents governed by H [Watkins et al., Chapman Conference Proceedings submitted, 2011]. Further generalisation to multifractals underway. • Volatility bunching, in sense of correlation of absolute values of time series, seen in auroral energy dissipation data [Watkins et al., op cit; Rypdal & Rypdal, JGR, 2011]. • Linear Kesten process shows “weak” volatility bunching
  • 33. The Bohr Atom • “The Bohr model of the atom ... was wrong, yet it turned out to be fruitful.” – Gene Stanley, Nature 2008 Rydberg formula