On going and planned activities by LAC group: Processing, Visualization and Analysis of Spatio-temporal Dynamics Reinaldo R. Rosa reinaldo@lac.inpe.br.

On going and planned activities by LAC

group: Processing, Visualization and Analysis of Spatio-temporal Dynamics

Reinaldo R. Rosa reinaldo@lac.inpe.br

•Adriana P. Mattedi, Roberto A. Costa Junior, Erico L. Rempel.

•Cristiane P. Camilo, Márcia Rodrigues, Rogério C. Brito, Mariana Baroni

•F.M. Ramos, A Wilter Souza da Silva, A. Assireu, I. B. T. de Lima, N. Vijaykumar, A. Zanandrea, R. Sych, J. Pontes, H. Swinney,

• A. J. Preto, S. Stephany, J.Demisio da Silva, Maria Conceição Andrade,

NÚCLEO PARA SIMULAÇÃO E ANÁLISE DE SISTEMAS COMPLEXOS - NUSASC

LABORATÓRIO ASSOCIADO DE COMPUTAÇÃO

E MATEMÁTICA APLICADA

BDA WORKSHOP

04-05/9/2003, INPE

Related BDA Research at LAC:• Time Series Analysis of Solar Bursts and modelling

for emission mechanisms• Loop Tomography• Image Processing, Interface Data Base Softwares and

High Performance Computing• Neuronetworks for Solar Active Region Pattern

Recognition• Space Weather using Gradient Pattern Analysis and

Wavelets• 1997-2003: 08 papers, 05 masters and 02 pos-docs

Space Weather Data and Scientific Computing

processing, visualization, data base, data mining and analysis

Real time data: Time series (time and space) and Spectrograms from the solar-terrestrial plasma

environment + warning devices + making decision tools

Solar Data: Yohkoh, soft X-rays

Solar Active Loops:

X Ray and Radio Data:Nobeyama Radioheliograph

Space Weather Today:

“A Dinâmica de Padrões Espaço-Temporais é uma teoria fenomenológica que sistematiza as leis empíricas, sobre regimes não-lineares no domínio espaço-temporal, que ocorrem durante a formação e evolução de estruturas dinâmicas macroscópicas”

Falaremos aqui sobre uma nova teoria geométrica analítica conhecida como Análise de Padrões Gradientes ( Gradient Pattern Analysis - GPA) cuja principal propriedade é a sua extrema sensibilidade para detectar flutuações não-lineares no domínio espaço-temporal.

Spatio-Temporal Domain:

*Characterization of Spatio-Temporal Pattern Formation and Evolution in:

• Chaotic Coupled Map Lattices• Extended Difusion-Convection• Osmosedimentation• Reaction-Diffusion Systems: Amplitude Equations

and Protein Folding

• 3D-Turbulence (“Turbulator-Phase I”)• Granular Materials• Extended Nonlinear Plasmas (Solar Physics)

• Activity Porosity *nonequilibrium regimes

Nonequilibrium Regimes:• Common Source:

High Gradients in the Field Symmetry(T, v or C)

==> system is driven away from thermodynamical equilibrium (initial system state becomes unstable)==>

==> symmetry breaking of the geometry ==>==> spatial complex pattern formation

(Graham’s nonequilibrium potential)In this context: What types of main nonequilibrium regimes?

Spatio-Temporal Nonequilibrium Regimes:

• Structure Fragmentation and Coalescence of Ch

(“spatio-temporal velocity”)• Hysteresis and Dissipative regimes• “Spatio-temporal chaos” (activator-inhibitor

model dynamics)• Long range spatio-temporal correlations and

structures synchronization =>• Local and Global spatio-temporal patterns

stability rate (Ux,y,0 coherence and U0,0,t boundary influence)

• Pattern Relaxation: normal and abnormal

Gradient Pattern Analysis (GPA)(P/ entender e interpretar as causas da relaxão e da estabilidade==>

nova metodologia)

• Gt = [E(x,y,t]t • Gt is represented by 4 n.s. gradients moments:

g1(t) h1 ({(r1,1, 11) … , (rij, ij) … , (rkk kk)}t)

g2(t) h2 ({(r11), … , (rij), … , (rkk)}t)

g3(t) h3 ({(11), … , (ij), … , (kk)}t)

g4(t) h4 ({(z11) … , (zij) … , (zkk)}t)

As {r} and {} are compact groups, spatially distributed, they can be geometrically constructed as Haar-like measures ==> rotational and amplitude translation invariant

Gradient Pattern Analysis:

How to compute g1,g2, g3 e g4 ?

g1 (C - VA)/VA , C > VA

VA = amount of asymmetric vectors, (v1 + v2 0)C = amount of geometric correlation lines (Delaunay triangulation

TD(C,VA) )(Asymmetric Amplitude Fragmentation - AAF, Rosa et al.,

Int. J. Mod. Phys. C, 10(1)(1999):147.)

g2 (rij – rmn)2 /N ; g3 ( ij – mn)2 /N

g4 : Sz = - zi,j/z ln (zi,j/z) = Re(Sz) eiSz

| g4|=Re(Sz)=S(|z|) and (g4)=Im(Sz) Thus, |g4| and (g4) are invariant measurements of norm and

phase of the gradient entropy(Complex Entropic Form (CEF) by Ramos et al. Physica

A283(2000):171.)

Characterization of Asymmetric Fluctuations, Amplitude Dynamics and

Nonlinear Pattern Stability (Relaxation Regimes):

g1 x t

| g4| x t

g1 (t) x phase(g4 )(t)

g1 = 0 , g2 = 0 , g3 = 0 , |g4| = 0.20 , (g4)=0

g1 = (7 - 5) / 5 = 0.4, g2 = 0 , g3 = 0.20 , |g4| = 0.18, (g4)=0.20

Some Important Dynamical Properties of the Gradient Moments:

(1) Amplitude x Phase Dynamics

|g4|/ t < 0 ==> phase dynamics dominates and determine the relaxation (g1/ t > 0 => desordering vec. Norm)

2) Pattern Global Equilibrium (PGE) Conditions:

C1: |g4|/ t = 0 + g1/ t = 0 “Weak PGE”

C2: C1 e g4/ t = 0 “Strong PGE”

===> cluster around a characteristic point g2 ,g3)(More than 01 cluster ==> complex equilibrium regimes

mainly due to more than 01 dynamical constraint (ex. Boundary)Physica A 283:156(2000), Physica D 168:397(2002), PRL (2003)

Gradient Pattern Analysis of Relaxation Regimes

Systems:

• Oscillated Granular Layer: (> 104 0.15-0.18 mm bronze spheres - from CND Un.Texas)

• Knobloch Amplitude Equation (Simplification of Proctor’s Model)

tE = rE - (2E + 1)2E + . (|E|2 E),where E(x,y,t) is a measure of the vertically averaged

temperature perturbation due to fluid motion. When the parameter r > 0, the conductive solution E=0 is unstable.

Spatio-Temporal Relaxation: is it an universal regime?

Results(Considering Charac. Scales):

Concluding Remarks System Relaxation GPE Boundary InfluenceOscillons normal simple low Knobloch abnormal complex (3sR) high

• Abnormal relaxation comes from a strong Amplitude x Phase Dynamics where the amplitudes are very asymmetric in z (amplitude equations are not models for discrete oscillons)

• Gradient moments from active regions time series:Monitoring + forecasting + modelling validation

•

0,88

0,9

0,92

0,94

0,96

0,98

1

1 2 3 4 5 6 7 8

Seqüência1

g1 x time-step

Modelling validation:

B=f(t,x,y,z;d)