Towards an information theory Towards an information theory approach for monitoring the ionospheric approach for monitoring the ionospheric

convection dynamics convection dynamics I. CocoI. Coco(1)(1),, G. ConsoliniG. Consolini(1)(1), E. Amata, E. Amata(1)(1), M. F. Marcucci, M. F. Marcucci(1)(1), D. , D.

AmbrosinoAmbrosino(1)(1)

(1)(1) INAF – IFSI, Rome, ItalyINAF – IFSI, Rome, Italy

Contact: Contact: igino.coco@ifsi-roma.inaf.itigino.coco@ifsi-roma.inaf.it

SuperDARN 2011, Thayer School, NH, USA May 30 – June 3, 2011SuperDARN 2011, Thayer School, NH, USA May 30 – June 3, 2011

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Outline:Outline: A new approach on the study of the ionospheric electric potential at high latitudes is outlined, making use of Super Dual Auroral Radar Network (SuperDARN) convection velocity data.

Concepts of information theory are applied, for evaluating the degree of order/disorder in changes of the topology of ionospheric convection. This is done computing the Shannon’s entropy on the pseudo-occupation probability of spherical harmonics’ modes. A comparison among different IMF conditions is done, showing a good correlation of the Shannon’s entropy with the IMF Bz behaviour. The solar wind dynamic pressure, does not seem to influence the entropy too much.

Preliminary results concerning possible interhemispheric differences are also shown.

The combined effect of IMF Bz and By is also investigated over a 20-days data set.

Dynamical Complexity (Chang et al., 2006): “a phenomenon exhibited by a nonlinearly interacting dynamical system within which multitudes of different sizes of large scale coherent structures are formed, resulting in a global nonlinear stochastic behaviour for the dynamical systems, which is vastly different from that could be surmised from the original dynamical equations”.

• Complexity is the tendency of a non-equilibrium system to show a certain degree of spatio-temporally coherent features resulting from the competition of different basic spatial patterns playing the role of interacting sub-units.

• Complexity requires the occurrence of nonlinearities and the intertwining of order and disorder, and it is generally related to the emergence of self-organisation in open systems.

• Ionospheric convection can be seen as a complex interacting system, whose driving power comes from the interaction between the solar wind and the Earth’s magnetopause.

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IMF Bz < 0 Bz > 0

Bz > 0, By ~ Bz Bz ~ 0, By >>0

Turbolence and complexity Emergence of structures, ordered/disordered systems.

Are ionospheric convection patterns “complex” Are ionospheric convection patterns “complex” structures?structures?

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Let (xi,tj) be a spatio-temporal field which can be written as:

)()()(1

, ik

j

N

k

kji xtAtx

2

2

|)(||)(|)(tA

tAtp jj

k

k

where ’s are orthogonal functions and, the time fixed, A’s are their coefficients. A normalized probability function can be associated to each k-th eigenfunction as follows:

The Information Entropy formalism The Information Entropy formalism (1)(1)

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The Shannon’s entropy is defined as:

N

k

kks tptpktS1

)(ln)()(

The higher the value, the wider the spectrum of the accessible states, so that S(t) provides a measure of “disorder” (uncertainty).

The Shannon’s formalism can be applied to the ionospheric PCP pattern, bearing in mind it can be written in terms of spherical harmonics:

ml iml

m imlil tA

tAtp

,2

,

2,

|)(||)(|

)(

where Al,m (ti) are the coefficients computed by RST as a function of t i, time of the scan. As a first attempt of this analysis we sum over all m’s, taking only the main index l of the spherical harmonics into account.

So that the probability functions pl(ti) can be calculated as follows:

Note that Al,m (ti) are in general complex coefficients, and in the practice only the real part of the written above is taken into account.

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Let’s better characterize order/disorder transitions with the help of these two quantities:

max

)(StS

varying in the interval [0,1]. = 0 for S(t) = 0 (maximum order), = 1 for S(t) = Smax (maximum disorder).

)1(11 II Order Complexity Measure (Shiner et al, 1999). 11 will vanish at both equilibrium (=0) and complete desorder (=1), implying that complexity will increase in intermediate situations.

The Information Entropy formalism The Information Entropy formalism (2)(2)

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l

Pl

0 1 2 3 4 l

Pl

0 1 2 3 4

Most of the power is concentrated in few coefficients (one or two): the system is “ordered”, few physical states contribute to describe it.

All the states contribute with similar weights: the system is “disordered”, because different physical mechanisms are superposed and act simultaneously on the system.

S(t) 0, and 0 S(t) Smax, and 1

2002 - 12 – 22, 14 – 16 UT: almost steady IMF B2002 - 12 – 22, 14 – 16 UT: almost steady IMF Bzz > > 00

14:10 15:00 15:50

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2003 - 10 – 01, 19:30 – 21:30 UT: almost steady IMF B2003 - 10 – 01, 19:30 – 21:30 UT: almost steady IMF Bzz < 0< 0

19:30 20:30 21:30

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• Time series of 4th order polar cap potential coefficients have been obtained for the two periods.• From coefficients, Shannon Entropy S(t), ,11 have been calculated.

The result is very clear. The time series distributions of the negative IMF Bz and positive IMF Bz periods are neatly separate: convection during the negative IMF Bz time series tends to a state of “order”, while during positive IMF Bz time series a mixing of ordered and disordered states often occurs.

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2002 - 12 – 19, 06:00 – 12:00 UT: varying IMF B2002 - 12 – 19, 06:00 – 12:00 UT: varying IMF Bzz

Here complexity shows up: a transition order/desorder proceeds from Bz negative to Bz positive periods: complexity has a maximum where ordered and disordered states coexist.

IFSI – 19/05/2011IFSI – 19/05/2011

Time series of for 2002/12/19 06-12 UT. Note the qualitative correlation with IMF Bz: when Bz flips from positive to negative the convection tends to more ordered configurations, and vice versa.

= 0.5

IFSI – 19/05/2011IFSI – 19/05/2011

Ionospheric response to sudden changes of the Solar Wind Ionospheric response to sudden changes of the Solar Wind dynamic pressure.dynamic pressure.

Radar echo response to Sudden Increases of SW pressure during “quiet” geomagnetic conditions (low |AE|)

Cross-Polar Cap Potential response to Sudden Increases of SW pressure during “quiet” geomagnetic conditions (low |AE|)

N Hem

S Hem

73 cases

50 cases

31 cases

21 cases

Coco et al., Int. J. of Geophys., in Coco et al., Int. J. of Geophys., in presspress

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Normalized Shannon’s entropy for superposed epoch time series of ionospheric potential patterns across the occurrence of a SW pressure variation

“Quiet” event: |AE| < 200 nT througout the period

“Disturbed” event: |AE| > 400 nT througout the period

“I” event: Sudden Increase of SW pressure

“D” event: Sudden Decrease of SW pressure

The occurrence of a pressure variation does not seem to influence too much the convection patterns and the entropy. The average geomagnetic activity (AE index) seems to affect the entropy strength: lower values for “disturbed events” (0.2-0.3), higher values for “quiet events” (0.3-0.4).

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• The IMF Bz effects on convection patterns are statistically more important than the variations of the SW pressure: most of “disturbed events” occur during Bz < 0 periods (stronger coherence, lower entropy), and most of “quiet events” occur during Bz > 0 (higher complexity, higher entropy). • This is even more evident in the figure on the left: “quiet” events are further classified according to the IMF Bz sign: the change of entropy is closely related to the rotations of Bz which occur in coincidence with the pressure increases.• Bz+ Bz- : decrease of the entropy.• Bz- Bz+ : increase of the entropy.

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Study of an extended time interval: February Study of an extended time interval: February 20022002

• PCP coefficients have been computed for over 13600 2-min SuperDARN scans in Northern Hemisphere, during February 2002, a period characterized by very good radar coverage and a wide variety of IMF and solar wind conditions. • and 11 have been calculated and averaged in IMF [By,Bz] bins 1 X 1 nT wide, from -15 up to +20 nT for both By and Bz. Bins containing less than 10 scans have been discarded.

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Interhemispheric differences on entropy behaviourInterhemispheric differences on entropy behaviour

2002-12–19, 08:00 – 12:00 UT: Shannon entropy and IMF Bz in both

Hemispheres2002-12–19: Complexity index 11

vs norm. Entropy in both Hemispheres

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8:30-8:32 UT 8:40-8:42 UT 8:50-8:52 UT

BeforeBefore……

N = 0.15 N = 0.13 N = 0.14

s = 0.14 s = 0.18 s = 0.19

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9:14-9:16 UT9:04-9:06 UT 9:24-9:26 UTs = 0.19 s = 0.2s = 0.17

N = 0.56 N = 0.68 N = 0.64

• Summary and Conclusions:Summary and Conclusions: We studied the reconfiguration of ionospheric convection from the point of view of information theory and complex system physics, so far not applied to such an issue. Starting from the Polar Cap Potential coefficients, as obtained from SuperDARN convection velocity data, we derived the Shannon information entropy and the degree of complexity associated with the PCP structure on a global scale in different IMF and solar wind conditions.

The obtained results clearly evidenced a dynamical topological phase transition from a less ordered configuration to a more ordered one as a consequence of the IMF turning from northward to southward.

Furthermore, when |By|/Bz >> 1, a similar effect was found as a function of the IMF By intensity, so that both Bz and By may be regarded as acting as order parameters.

The observed decrease of disorder for southward IMF Bz and the reduction in complexity has to be related to the emergence of a large coherence in the PCP structure manifesting in a more simple two-cell structure. Conversely, the higher degree of disorder and complexity for northward IMF Bz conditions reflects the inherent multi-cell structure of ionospheric convection, which has to be associated with a reduced coherence in the large scale convection motions, giving rise to multiscale structures.Coco et al., submitted to Nonlin. Processes Geophys., 2011Coco et al., submitted to Nonlin. Processes Geophys., 2011

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Can Shannon entropy be used as a «quicklook» of the ionospheric convection?