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?