scaling of the geomagnetic indices and solar wind: a fokker-planck approach bogdan a. hnat...
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Scaling of the geomagnetic indices and solar wind: a Fokker-Planck
approach
Bogdan A. Hnat
Collaborators:S. C. Chapman and G. Rowlands; University of WarwickN. W. Watkins and M. P. Freeman; British Antarctic Survey
Complex magnetosphereGeneral features:
Non-equilibrium, non-linear system
Driven by variable solar wind
Dissipation via complex current systems
Specific issues:
Coupling to solar wind essential part of description
How to distinguish between features of the driver and the system?
Complex dynamics-simple statistics
Stochastic approach
Differenced quantities are easier to study
Fluctuations treated as stochastic variables
Scaling, derived directly from data, can simplify description
If mono-scaling found the Fokker-Planck approach can be used to model statistics
Scaling: basic concepts
Buildingblocks
tyτ+ty=δx Fluctuations: Scaling
Segments:
mm =l
3
Length depends on scale mm =L 3/4
Statistics depends on scale mζm
m τδx=τS
self-similarity αm=mζ
How to determine scaling
αsα δxτPτ=τδx,P
tyτ+ty=δx Fluctuations: Generalized Structure Functions: mζm
m τδx=τS
PDF rescaling:
Conditioning: consider events < 10 σ(τ)
Indices and solar wind dataSolar wind driver represented by Akasofu’s ε:
)/|arctan(|,7where),2/(sin)/( 042
002
zyE BBRllBv
2 years of WIND, ACE data used to compute ε
Geomagnetic indices: ~10 stations in auroral region AU-strength of E-ward electrojet AL-strength of W-ward electrojet Dates different than ε Indices definition-extreme events
Indices and solar wind: SF
03.037.001.043.0
02.032.0
)(
)(
AL
AU
02.039.001.035.0
02.026.0
)(
)(
AL
AU
Solar min-max comparison
Results: Scaling of δ(AL) nearly insensitive to solar cycle Scaling of δ(AU) follows the trend of δε All scaling exponent are different Constant conversion rate?- 06.0)( AU
Fokker-Planck equation for PDFt ( , ) ( ( ) ( , ) ( ) ( , ))P y t A y P y t B y P y t
Assume: /12
0/11
0 )(bx)B( and)()( xxaxA
Look for self-similar solutions: xxPP ss ,
ConclusionsScaling led to full characterise statistics of fluctuations up to 10σMono-scaling allowed direct comparison of solar wind driver and AE indicesCoupling between solar wind and different region of the magnetosphere can be studied
Applications:Sub-grid modelling of complex systemsConstrains on numerical models and theoretical predictionsPredictive power in statistical sense
Post seminar material
Conditioning plays important role in establishing scaling – why?