Cusp turbulence as revealed by POLAR magnetic Cusp turbulence as revealed by POLAR magnetic field datafield data
E. YordanovaE. Yordanova
Uppsala, November, Uppsala, November, 20052005
OutlineOutline
Cusp Cusp
Models of turbulenceModels of turbulence
Multifractal structure of cusp turbulenceMultifractal structure of cusp turbulence
Anisotropy in the cuspAnisotropy in the cusp
Cusp Cusp
Models of turbulenceModels of turbulence
Multifractal structure of cusp turbulenceMultifractal structure of cusp turbulence
Anisotropy in the cuspAnisotropy in the cusp
Uppsala, November, Uppsala, November, 20052005
CusCuspp
• depressed and irregular magnetic magnetic field
• magnetosheath plasma /high density and low energy/
• plasma of ionospheric origin
• the direction of IMF
• the tilt of the magnetic dipole
• the solar wind dynamic pressure
Uppsala, November, Uppsala, November, 20052005
Turbulent magnetic field in the Turbulent magnetic field in the cuspcusp
POLARPOLAR missionmission
Uppsala, November, Uppsala, November, 20052005
f < 10-2 Hzf < 102-103 Hz
Examples of power spectra Examples of power spectra of the magnetic field fluctuations, of the magnetic field fluctuations,
measured in the cusp (POLAR satellite)measured in the cusp (POLAR satellite)
By110497_1By110497_1 BB091096091096__55
Uppsala, November, Uppsala, November, 20052005
Magnetospheric cusp magnetic field (POLAR satellite)Magnetospheric cusp magnetic field (POLAR satellite)
cuspscuspsspiralsspirals
rampsramps
0 0
h
ng x P x x C x x
The singularity strength:
Hölder exponentHölder exponent hh((xx00)) - a measure of the
regularity of the function g at the point x0
- the statistical distribution of the singularity exponents h.
SSingularity spectrum Dingularity spectrum D((hh)) - a humped shape (hmin - strongest singularity;
hmax – weakest singularity)
Uppsala, November, Uppsala, November, 20052005
a maximum in the modulus of the wavelet transform
coefficients
Singularities
Modulus maximum of WT
‘any point (x0,a0) of the space-scale
half-plane which corresponds to the local maximum of the modulus of considered as a function of x’
Maxima linethe curve, connecting the modulus maxima
Singularity exponents a power law fit of the wavelet coefficients along the maxima line
Mallat and Zong (1992)Wavelet Transform Modulus Maxima Method (WTMM)Wavelet Transform Modulus Maxima Method (WTMM)
Wavelet Transform (WT)Wavelet Transform (WT) A tool for detecting the singularities
a - scale, b – translation or dilation, * - conjugated transforming function
dxa
bxxg
aabg
*1,
Uppsala, November, Uppsala, November, 20052005
Energy injection
Inertial range
Dissipation range
. . . . . . . . . . . . . . . .
Ric
hard
son
casc
ade
Ric
hard
son
casc
ade
Kolmogorov phenomenology (1941)Kolmogorov phenomenology (1941)
Self-similaritySelf-similarity in the inertial rangein the inertial range
LocalnessLocalness in the interaction in the interaction
Uppsala, November, Uppsala, November, 20052005
P model (Meneveau and Sreenivasan ‘87)P model (Meneveau and Sreenivasan ‘87)
Energy injection
Inertial range
Dissipation range
. . . . . . . . . . . . . . . .
Uppsala, November, Uppsala, November, 20052005
Calculation of the scaling properties of turbulenceCalculation of the scaling properties of turbulence
Structure functionsStructure functions of a measured fluctuating parameter g(x):
0
1~
Lq q
qS l g x l g x dx lL
!!fundamental quantity in classical theory of turbulence!fundamental quantity in classical theory of turbulence!
Singularity spectrumSingularity spectrum (Parisi and Frisch,1985)
q
qS l lLegendre Legendre transformtransform inf 1q qD h qh
D(h) D(h) - statistical distribution of - statistical distribution of the singularity exponentsthe singularity exponents hh
Uppsala, November, Uppsala, November, 20052005
Scaling law of the partition function along the maxima line:Scaling law of the partition function along the maxima line:
, ~ qZ q a a
Singularity spectrum D(h) of the WTMM function Singularity spectrum D(h) of the WTMM function (q):(q): infqD h qh q
Wavelet based partition function Wavelet based partition function ((Muzy, Bacry, Arneodo, 1991))::
'
, sup ' , 'q
la al L a
Z q a T g b a a
WTMM
L(a) - a set of all the maxima lines l existing at a scale a; bl(a) - the position, at a, of the maximum belonging to the line l
l={bl(a), a} is pointing towards a point bl(0) (when a goes to 0) which corresponds to a singularity of g
1q q
Relation between q and q
Uppsala, November, Uppsala, November, 20052005
Extended structure function modelsExtended structure function models (Tu et al. 1996, Marsch and Tu 1997)(Tu et al. 1996, Marsch and Tu 1997)
- scaling exponents for the Kolmogorov-like cascade:- scaling exponents for the Kolmogorov-like cascade:
/ 3/32 1 1
2/32/32 1 1
5 3log 1
2 2 3
1log 1
3
qqqq P P
P P
- scaling exponents for the Kraichnan-like cascade- scaling exponents for the Kraichnan-like cascade::
/ 4/ 42 1 1
1/ 21/ 22 1 1
3 2 log 14
1log 1
2
qqqq P P
P P
P-model
Uppsala, November, Uppsala, November, 20052005
Method: WTMM Constructing partition partition functionsfunctions – sums of the WT located in the modulus maxima (define the singularity)
Set of locations and strength of the singularities – singularity spectrumsingularity spectrum
MF fluctuations – singular behavior
The problemThe problem
Uppsala, November, Uppsala, November, 20052005
Fractional Fractional brownian signalbrownian signal
WTMM partition WTMM partition functionsfunctions
WTMM partition WTMM partition function exponentsfunction exponents
Singularity spectrum Singularity spectrum
Muzy, Bacry & Arneodo (1994)
Uppsala, November, Uppsala, November, 20052005
Devil’s staircase Devil’s staircase signalsignal
WTMM partition WTMM partition functionsfunctions
WTMM partition WTMM partition function exponentsfunction exponents
Singularity spectrum Singularity spectrum
Muzy, Bacry & Arneodo (1994)
Uppsala, November, Uppsala, November, 20052005
Comparison with models of turbulenceComparison with models of turbulence
partition function exponents (power law fit of wavelet coefficients
along maxima line)Non-linear behavior
Least-square fit of models of turbulence
Through numerical differentiation of the exponents curve singularity spectrum is derived (parabolic shape, typical for the non-linear systems)
Mean square deviation between numerical and theoretical spectra
Uppsala, November, Uppsala, November, 20052005
Probability distribution functions for different time delaysProbability distribution functions for different time delays
Data Data sampling sampling
frequency - frequency - 8.333 Hz8.333 Hz
Uppsala, November, Uppsala, November, 20052005
Kolmogorov – like turbulence
Bz < 0
p - model turbulence
Bz > 0
Results for 9 Oct 1996 caseResults for 9 Oct 1996 case
Uppsala, November, Uppsala, November, 20052005
Results for 11 Apr 1997 caseResults for 11 Apr 1997 case
Bx > 0
By < 0
Bz > 0
Kolmogorov – like Kolmogorov – like turbulenceturbulence p p - model - model
turbulenceturbulence
Uppsala, November, Uppsala, November, 20052005
HYDRA / POLAR
Uppsala, November, Uppsala, November, 20052005
1. Conclusions about the magnetic 1. Conclusions about the magnetic field intensityfield intensity
IMF Bz > 0 – p – model (fluid, fully developed)
IMF Bz < 0 - Kolmogorov- like (fluid, non fully developed)
Uppsala, June, 2005Uppsala, June, 2005
Uppsala, November, Uppsala, November, 20052005
B~90 nT
B~10 nT
BBzz
BBxyxy
SPCSPC
northnorth
BB5656
duskdusk
antisunantisun
(Bxy, Bz, B56)
(B1, B2, B0)
Anisotropy features of the magnetic field
Uppsala, November, Uppsala, November, 20052005
Power spectra in parallel and Power spectra in parallel and perpendicular directionsperpendicular directions
~ 1.62 ~ 2.41
f -5/3
~ 1.21
~ 1.93
~ 5
Uppsala, November, Uppsala, November, 20052005
Extended Self-Similarity AnalysisExtended Self-Similarity Analysis
PDF in parallel and perpendicular directionsPDF in parallel and perpendicular directions
= 6,12,24,48,96,192t
0
2 2 2
2 2 20 0
B B t B t
B B t B t
PSD - different scaling in parallel and perpendicular directions
ESS analysis – parallel fluctuations are characterized by monofractal nature; perpendicular - by a strong intermittent (multifractal) character
PDF – more intermittent character of the fluctuations in perpendicular direction then in parallel
Acknowledgements: E. Yordanova acknowledges the financial support provided through the European Community's Human Potential Programme under contract HPRN-CT-2001-00314, ‘Turbulent Boundary Layers’
Uppsala, November, Uppsala, November, 20052005
2. Conclusions about the anisotropy 2. Conclusions about the anisotropy in the cuspin the cusp
Uppsala, June, 2005Uppsala, June, 2005
V total
For 9 Oct 1996 case – V~100 km/sPOLAR speed is 2 km/s
For 11 Apr 1997 case – V~40 km/s
Taylor’s hypothesis
Structure functionStructure function (q) and (q) and (q)(q)
Uppsala, June, 2005Uppsala, June, 2005
Uppsala, June, 2005Uppsala, June, 2005
Power spectra of Power spectra of 11 April 1997 case11 April 1997 case
By110497_1By110497_1
By110497_By110497_22
By110497_By110497_33
-2.152.15(0.06 – 0.78 Hz)(0.06 – 0.78 Hz)