% ! Real%Time)Dynamics)of)Geomagne3c) Magne3c)Storms)and)Substorms)) E.)Spencer 1 ,)W.)Horton 2 ,)S.)Vadepu 1 ,)) ) P.)Srinivas 1 )S.)Patra 3) % % 1 Univ of South Alabama, Mobile, USA ) 2 Univ of Texas at Austin, USA 3 Univ of Oslo, Oslo, Norway 13 th% Interna+onal%Conference%on%Substorms% 25829%September%2017% %Portsmouth,%New%Hampshire,%USA % %%%
OUTLINE Brief Description of the WINDMI Model. Substorm trigger in the model. Periodic Substorms (Sawtooth Events) - Model behavior. Isolated Substorms - Parameter values on substorm trigger.
WINDMI Model Real-time low dimensional model for predicting substorms from ACE solar wind data Nonlinear ODE’s result from the application of conservation laws to global energy components of the system. Lumped nonlinear representation of magnetosphere. Kinetic effects are included in a simplified manner.
WINDMI Model WINDMI Current Systems
WINDMI Model The largest energy reservoirs in the magnetosphere-ionosphere system are: Plasma ring current energy W rc . Geotail lobe magnetic energy W m . The R1 FAC associated with the westward auroral electrojet. The FAC at the lower latitude closing on the partial ring current. Kinetic energy proportional to E × B perpendicular plasma flows. Kinetic energy K ∥ due to mass flows along magnetic field lines. Plasma Sheet thermal energy component p cps .
WINDMI Model Model input is V sw and Model outputs are I 1 and W rc L dI V sw ( t ) − V + M dI 1 = (1) dt dt C dV = I − I 1 − I ps − Σ V (2) dt Σ V 2 3 dp pVA eff − 3 p − u 0 pK 1 / 2 = Θ ( u ) − (3) ∥ 2 dt Ω cps Ω cps B tr L y 2 τ E dK ∥ K ∥ = I ps V − (4) dt τ ∥ dI 1 V − V I + M dI L I = (5) dt dt dV I C I = I 1 − I 2 − Σ I V I (6) dt dI 2 L 2 V I − ( R prc + R A 2 ) I 2 (7) = dt dW rc 2 + pVA eff − W rc R prc I 2 = (8) dt B tr L y τ rc
WINDMI Physical Parameters L 90 H Inductance of the lobe cavity surrounded by the geotail current I ( t ) . The nominal value is L = µ 0 A ℓ / L eff x in Henries where A ℓ is the lobe area and L eff x the effective length of the geotail solenoid. C 50000 F Capacitance of the central plasma sheet in Farads. The nominal value is C = ρ m L x L z / ( B 2 L y ) where ρ m is the mass density in kg / m 3 , L x L z is the meridional area of the plasma sheet, L y the dawn-to-dusk width of the central plasma sheet and B the magnetic field on the equatorial plane. Computations of C are given in horton1996. Σ 8 S Large gyroradius ρ i plasma sheet conductance from the quasineutral layer of height ( L z ρ i ) 1 / 2 about the equatorial sheet. The nominal value is Σ = 0 . 1 ( n e / B n )( ρ i / L z ) 1 / 2 . Computation of Σ is given in horton tajima. Ω cps 2 . 6 × Volume of the central plasma sheet that supports mean pressure p ( t ) , 10 24 m 3 initial estimate is 10 4 R 3 E . u 0 4 e 9 m − 1 kg − 1 / 2 Heat flux limit parameter for parallel thermal flux on open magnetic field − lines q ∥ = const × v ∥ p = u 0 ( K ∥ ) 1 / 2 p . The mean parallel flow velocity is ( K ∥ / ( ρ m Ω cps )) 1 / 2 . I c 1 . 78 × The critical current above which unloading occurs. 10 7 A ∆ I 1 . 25 × The rate of turn-on of the unloading function. 10 5 A 8 × 10 11 The geotail current driven by the plasma pressure p confined in the central α plasma sheet. Pressure balance between the lobe and the central plasma sheet gives B 2 ℓ / 2 µ 0 = p with 2 L x B ℓ = µ 0 I ps . This defines the coefficient α in I ps = α p 1 / 2 to be approximately α = 2 . 8 L x /µ 1 / 2 . 0 Table: WINDMI Nominal Parameters, estimated by physical considerations of the state and geometry of the nightside magnetosphere.
Model Input I Measured Solar Wind parameters from ACE. Two functions have been found to be most useful and reliable: Input Coupling Function 1 (Rectified vBs (kV)): V Bs sw = V 0 + v sw B IMF L eff s y Input Coupling Function 2 (in kV)(Newell 2007): V 0 + ν n v 4 / 3 sw B 2 / 3 sin 8 / 3 ( θ c / 2 ) V N = (9) sw T
Model Outputs Outputs of the WINDMI model The outputs are the AL and Dst indices, which are compared to ground measurements. The AL index from the model is obtained from the region 1 current I 1 The D st signal from the model is partly given by ring current energy W rc through the Dessler-Parker-Sckopke relation: W rc ( t ) Dst Wrc = − µ 0 B E R 3 2 π E
Substorm Mechanism and Control in WINDMI Model The trigger function controls when the substorm is initiated: Θ ( u ) = 1 � � I − I c �� 1 + tanh (10) 2 ∆ I The character (growth, expansion, recovery phases) is strongly controlled by the first three equations of the model: LdI V sw ( t ) − V + M dI 1 = (11) dt dt C dV = I − I 1 − I ps − Σ V (12) dt Σ V 2 3 dp pVA eff − 3 p − u 0 pK 1 / 2 = Θ ( u ) − (13) ∥ 2 dt Ω cps Ω cps B tr L y 2 τ E Parameters in red are tuned manually in this study.
October 4 2000 Sawteeth Event I ACE data between 3-7 October 2000
October 4 2000 Sawteeth Event II AL and DST Index Comparison Oct 3 − 7 2000 3000 Model ARV = 0.46 2000 Data COR = 0.75 − AL [nT] 1000 0 − 1000 0 − 50 Dst [nT] − 100 Model ARV dst = 0.57 Data − 150 − 200 0 50 100 Time [Hrs] See Spencer et. al. JGR 2009, Spencer et. al. JGR 2007 for details. Here I c = 10 . 5 MA , C = 105000 F. Using oxygen O2+ with number density 20e6 per cubic meter and magnetic field 18nT gives 107000 F.
Substorm And Pseudo-Breakup Data Set Kalmoni et. al. 2015JA021470 use a set of Substorm and Pseudo-Breakup events to study how the growth rate of auroral beads are related to possible instability mechanisms in the near-earth plasma sheet. We used the same set of events but studied the substorm energy and triggering conditions using solar wind and IMF as drivers. 17 events. Good solar wind data for 13 events. 9 events where triggering with the model is possible. Compared model substorm trigger time against auroral observations (Dotted vertical red lines in all figures). Ω = 10000 R 3 E (100 X 20 X 5) for all the results.
28 November 2005 10:08 am Solar Wind Input Switching Function 120 0.8 0.6 100 0.4 0.2 540 550 560 570 580 590 600 610 620 80 VSW [kV] Plasma Sheet Pressure 0.025 0.02 60 nPa 0.015 0.01 0.005 40 540 550 560 570 580 590 600 610 620 Geotail Current 20 4.5 [MA] 4 540 550 560 570 580 590 600 610 620 3.5 540 550 560 570 580 590 600 610 620 Crosstail E Field Westward Auroral Electrojet [mV/m] 2 1.5 1 0.8 0.5 0.7 540 550 560 570 580 590 600 610 620 Parallel Ion Velocity 0.6 [km/s] 250 I1 [MA] 0.5 200 0.4 150 540 550 560 570 580 590 600 610 620 0.3 Pressure Gradient Current 0.2 [MA] 3 2 0.1 1 540 550 560 570 580 590 600 610 620 540 550 560 570 580 590 600 610 620
02 October 2008 04:29 am Solar Wind Input Switching Function 200 0.8 0.6 180 0.4 160 0.2 140 240 250 260 270 280 290 VSW [kV] Plasma Sheet Pressure 120 0.03 100 nPa 0.02 80 0.01 60 240 250 260 270 280 290 40 Geotail Current 20 5 [MA] 4 240 250 260 270 280 290 3 240 250 260 270 280 290 Crosstail E Field 3 Westward Auroral Electrojet [mV/m] 2 1 1 0 0.8 240 250 260 270 280 290 Parallel Ion Velocity 0.6 300 [km/s] I1 [MA] 200 0.4 100 240 250 260 270 280 290 0.2 Pressure Gradient Current 4 0 [MA] 2 -0.2 240 250 260 270 280 290 240 250 260 270 280 290
15 March 2009 04:28 am Solar Wind Input Switching Function 0.8 0.6 100 0.4 0.2 80 240 250 260 270 280 290 VSW [kV] Plasma Sheet Pressure × 10 -3 14 60 12 nPa 10 8 6 4 40 2 240 250 260 270 280 290 Geotail Current 20 3.8 3.6 [MA] 3.4 3.2 240 250 260 270 280 290 3 2.8 240 250 260 270 280 290 Crosstail E Field Westward Auroral Electrojet 2 [mV/m] 1.5 1 0.7 0.5 240 250 260 270 280 290 0.6 Parallel Ion Velocity 220 [km/s] 200 0.5 I1 [MA] 180 160 140 0.4 120 240 250 260 270 280 290 0.3 Pressure Gradient Current 2.5 0.2 [MA] 2 1.5 0.1 1 240 250 260 270 280 290 240 250 260 270 280 290
07 March 2010 05:15 am Solar Wind Input Switching Function 180 0.8 0.6 160 0.4 0.2 140 300 310 320 330 340 350 120 VSW [kV] Plasma Sheet Pressure × 10 -3 100 15 nPa 10 80 5 60 300 310 320 330 340 350 40 Geotail Current 3.8 20 3.6 [MA] 3.4 3.2 300 310 320 330 340 350 3 2.8 2.6 300 310 320 330 340 350 Crosstail E Field Westward Auroral Electrojet [mV/m] 2 1 1 0 0.8 300 310 320 330 340 350 Parallel Ion Velocity 0.6 [km/s] I1 [MA] 200 100 0.4 300 310 320 330 340 350 Pressure Gradient Current 3 0.2 [MA] 2 0 1 300 310 320 330 340 350 300 310 320 330 340 350
Model Performance On Substorm Data Table: WINDMI Model Triggering Conditions And Associated Parameters Date Onset (Mdl. Onset) ( ∆ t ) Σ [S] I c [MA] C [F] 28/03/2008 05:36 (05:32) (-4) 10 3.7 8000 28/11/2005 10:08 (10:08) (0) 10 4.4 10000 22/02/2006 06:26 (06:36) (+10) 10 3.7 5000 07/03/2007 05:50 (05:47) (-3) 10 3.4 5000 02/10/2008 04:29 (04:23) (-6) 10 4.9 10000 03/01/2009 04:36 (04:24) (-12) 5 3.5 7000 24/02/2009 07:32 (07:26) (-5) 5 2.5 8000 15/03/2009 04:28 (04:24) (-4) 10 3.7 5000 07/03/2010 05:15 (05:25) (+10) 10 3.7 5000
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