MagnetosphericMagnetospheric DataData--BasedBased ConfigurationsConfigurations ObtainedObtained WithWith aa HighHigh--ResolutionResolution ModelModel ofof ExternalExternal FieldField SourcesSources

M. I. Sitnov

Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD

N. A. Tsyganenko

Universities Space Research Association and Laboratory, NASA Goddard Space Flight Center , Greenbelt, MD Motivation

• Present empirical geomagnetic field models are limited in their spatial resolution, because each magnetospheric current system is described by only a few “custom-made” mathematical modules. • Using formal general expansions (e.g., polynomial series) is not feasible because of complexity of the and non-uniform data coverage. • We propose a trade-off approach, using high-resolution expansions for the equatorial currents coupled with field-aligned currents, and demonstrate its feasibility. • Results of a new fitting code are described, including the effects of different activity levels, solar and IMF conditions, as well as phases. Magnetic field of an axisymmetric current disc

Ampere’s equation z 2 φφ Jφ ∂∂ρρρ1 ∂ Aφ ()ρρδA +=2 Jz()() ∂∂ ∂z y x

Solution [Tsyganenko, 1989]:

∞ A ()ρρ,expzCkkzJkkdk=− ()()() φ ∫0 1

Finite element form:

N Azφ ()ρρ,exp,=−∑ Cmm() kzJk10() m kkm m =+∆ m=0 Magnetic field of equatorial currents: I. Axisymmetric part

Vector potential

()m 22 11+ (mk−∆) AJkφ =−+=1 ()mmρ exp( kzD) , k m 0

Magnetic field components (D=const)

kzm 22ρ BJkkzDxmm=−+1 ()ρ exp( ) cosφ zD22+

kzm 22 BJkkzDymm=−+1 ()ρ exp( ) sinφ zD22+

22 Bzm=−+kJ0 () k mρ exp( k m z D) Magnetic field of equatorial currents: II. Introducing azimuthal asymmetry

Generic scalar potential for an cos mφ UkzJk=±exp()()m ρ infinitely thin current sheet sin mφ

Stern’s transformation [Stern,1987]: A =∇×∇ρ 2 ( U φ )

Vector potentialφ for a current sheet with finite thickness:

cos mφ 22 z '  Aee=−++ρ exp( kz D ρ Jmz() kρρ) J m() k  sin m  zD22+ 

Variable current sheet thickness:

ρ 22XY DD=+ D + Dexp 1 + 01ρρ22 2  2 + 020XY Sample basic modes of the equatorial current Shielding equatorial currents

sin mφ Generating scalar potential  Generating scalar potential UkzJkmn= sinh ()( n m n ρ ) finite elements cos mφ

myρ BmkzJkxnmn=− 2 sinφρ sinh ()( )

Shielding the magnetic field kxρρ m  −−n cosmkzJk sinhφρρ() ( ) Jk() , for the odd-parity elements nmn −1 mn  kn  of the model tail mx BmkzJk=ρρsin sinh ()( ) ynmn2 ρ φρ

kyn  m  −−cosmkzJkφ sinhφρ()nmn −1 ρρ ( ) Jk mn() ,  kn  ()( ) Bkmzn=− cos cosh kzJk nmn Shielding sample

Normal magnetic field Bn formed by the Normal magnetic field Bn formed by the equatorial o-type term with k=3 and m=4 corresponding shielding terms Modeling field-aligned currents

Currents corresponding to Main module deformed magnetic field

[Tsyganenko, 2002a] Undeformed conical current

Field-aligned currents Equatorial currents “Quadrupole” current

Coupling with equatorial currents:

“Quadrupole” current Axisymmetric current Partial Data used in this study

Our combined dataset comprised 5-min and 15-min avg B vectors, spanning the period 1994-2005 and including data from GOES 8, 9, 10, and 12, Imp-8, Polar, , and Cluster. Concurrent & IMF data were taken from OMNI database (1-hour resolution). Figures below are based on a quiet-time subset of the entire dataset.

Coverage of the modeling region by POLAR (red), GOES-12 (yellow), Cluster (magenta), Geotail (blue), IMP 8 (black).

Radial distribution of original data (solid line) and weighted data (dotted line) Kp binning: Field lines

78º Kp=0 Kp=2 76º

6.5 RE 7.0 RE (65º) (65º)

Kp=4 Kp=6-7+ 74º 70º

9.0 RE 14.5 RE (65º) (65º) Kp binning: External equatorial field

Kp=0 Kp=2

Kp=4 Kp=6-7+ Kp binning: Total equatorial field

Kp=0 Kp=2

Kp=4 Kp=6-7+ Different solar wind conditions

Closed magnetosphere

(IMF) (IMF) (IMF) Bz > 8 5.5 < Bz < 8 3 < Bz < 4

(IMF) (IMF) (IMF) -4 < Bz < -3 -8 < Bz < -5.5 Bz < -8

Open magnetosphere Comparison to MHD simulations

Closed magnetosphere Magnetic field fluctuations ±500nT Different storm phases: Equatorial field

Main phase Recovery phase Different storm phases: Equatorial currents

Main phase Recovery phase Tail current sheet structure: Tilt angle and twisting effects Model results VI: Field-aligned currents

1.5 < P < 2.5 nPa, -10nT < Bz <-6nT, |By| <10 nT

(tot) I1 = 2 MA (tot) I2 = 0.9 MA Penetrating magnetic field versus IMF Bz SummarySummary

A new approach to empirical modeling of the magnetosphere is presented, using a general high-resolution approximation for the field of equatorial and field-aligned currents.

The new model is fitted to one of the largest data sets, compiled using GOES 8, 9, 10, and 12, Imp-8, Polar, Geotail, and Cluster observations, to investigate variations of the spatial structure of the geomagnetic field for different activity levels, solar wind IMF conditions, and storm phases.

The most interesting new results include the spatial structure of the magnetic field in the transition region between the inner magnetosphere and the magnetotail flanks in the dawn/dusk sectors, characteristic features of the 'open''open' andand 'closed''closed' statesstates ofof thethe magnetospheremagnetosphere forfor southwardsouthward and northward IMF Bz, and the asymmetric inflation of the inner magnetosphere during space storms. SummarySummary (continued)(continued)

Effects derived from the data include:

• compressed field on the dayside that grows in magnitude with increasing solar wind pressure; • strongly eroded field in the subsolar region during the times of strong southward IMF, driving the storm main phase; • inner depression whose depth and dawn-dusk asymmetry dramatically increases during storm-time periods; • extended region of weak equatorial field in the near tail, getting somewhat larger at the tail's flanks, especially for strong northward IMF conditions

• strong correlation of the 'penetrating' δBz with the concurrent IMF Bz; • ring current enhancement both in morning and in evening sectors during the main storm phase with the hook-like shape of the current as a whole; • strong reduction of the dayside equatorial current during the main phase; • broad and azimuthally symmetric enhancement of the equatorial current during the recovery phase without any significant distinction between the ring and tail current systems. NextNext sixsix months:months:

Transform the original magnetic field data-fitting procedure with high resolution in space to a dynamical model similar to T02 or TS05 and ready for the external use. This will require the elaboration of ‘advanced binning’ procedures using elements of the nonlinear time series analysis, including time delay embedding and nearest neighbors.

Perform the analysis of the spatial structure of the magnetosphere for specific events on request from TR&T focused science team collaborators.