deleted: 1 modeling of a substorm event using the rice...

jianyang RCME-Geotail20041029-growth_V2 8/28/2009 Frank Toffoletto� 8/28/09 4:57 PM

Modeling of a substorm event using the Rice Convection Model 1

with an equilibrium magnetic field: 1. Growth Phase 2

3

J. Yang1, F. R. Toffoletto1, R. A. Wolf1, G. M. Erickson2, S. Sazykin1 4 1Department of Physics and Astronomy, Rice University, Houston, Texas, USA. 5 2Solar Observatory, Prairie View A&M University, Prairie View, Texas, USA. 6

7

Abstract. 8

We present a simulation of an isolated substorm growth phase that occurred on Oct. 29, 2004, using 9

the Rice Convection Model coupled with a magneto-friction equilibrium solver. This model includes 10

the self-consistent feedback of both the ionospheric electric potential coupled with magnetospheric 11

convection and the magnetic field equilibrated with particle pressures. The modeling results are in 12

good agreements with Geotail observed magnetic field, plasma moments and estimated local flux tube 13

volume V and the entropy parameter PV5/3. We also present comparisons of the modeled magnetic 14

field and number density and temperature for <45keV particles and the differential flux for >50 keV 15

particles with multipoint observations at geosynchronous orbit. The model results confirm the global 16

view of a typical substorm growth phase, including stretching of the magnetic field, enhancement of 17

the cross tail current density, earthward motion of the plasma sheet, sharpening of the transition region 18

and dropouts of the energetic particle flux. The induction electric field calculated from the model is of 19

comparable magnitude to the moderate convective electric field, and thus significantly affects the 20

plasma drift. At the end of the growth phase, the model produces a very stretched magnetic field and 21

Bz minimum at ~-13 Re, which results in the plasma sheet from -7 to -17 Re mapping to an extremely 22

thin layer (~0.5 degree in latitude) on the ionosphere. These results suggests that magnetic field 23

mapping at the end of growth phase using a statistical model could lead to large errors . 24

25

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Frank Toffoletto� 8/28/09 4:57 PM

1

AGU Index Terms: 2

2712 Electric fields (2411) 3

2764 Plasma sheet 4

2760 Plasma convection (2463) 5

2790 Substorms 6

7514 Energetic particles (2114) 7

8

9

Keywords: 10

growth phase, force-balance, numerical modeling 11

12

Running title: 13

YANG ET AL.: RCM-E GROWTH PHASE 14

15

16

1. Introduction 17

A substorm is generally believed to be a loading-unloading process as the magnetosphere 18

responds to the solar wind [e.g., Akasofu, 1981; Baker, 1992]. Unlike the controversial mechanisms 19

related to substorm onset, the physical picture of substorm growth phase is generally well-accepted, 20

i.e., unbalanced dayside reconnection and tail reconnection leading to an increase of magnetic flux in 21

the lobe and the storage of magnetic energy in the tail, which also forces the plasma sheet to thin to 22

equilibrate the higher lobe field pressure. One of significant issues in substorm physics is that the 23

configuration near- and middle-Earth tail near the end of the growth phase may be critical to some 24

instabilities, which in turn may be closely related to the trigger mechanisms of substorm onset, e.g., 25

the current disruption [Lui, 1996] or the near-Earth-neutral-line [Baker et al., 1996]. It was suggested 26

by Lui et al. [1992] that a current density of 27~80 nA/m2 prior to the onset in the dipole-tail transition 27

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region may induce the cross-field current disruption. Although there is speculation that some threshold 1

condition would be met just prior to the fully developed unloading process [e.g., Koskinen et al., 1993; 2

Henderson et al., 2006], it is still uncertain what conditions differentiate growth phases that result in 3

pseudo breakup versus those that result in a substorm expansion phase. Accurate modeling of an 4

individual substorm growth phase may be able to provide a global view of the change of plasma sheet 5

and magnetic field configuration. 6

In global magnetospheric-ionosphere coupling, the magnetic field line mapping is an important 7

link the one-to-one correspondence of activities between the magnetotail and the ionosphere. However, 8

since there are large variations of substorms, statistical models often smooth out the highly stretched 9

feature of the magnetic field prior to substorm onset. Event-oriented magnetic field modeling [e.g., 10

Pulkkinen et al., 1991a, 1991b], basically modify the free parameters and/or mathematical 11

representations of module(s) in Tsyganenko models. Rather than choosing solar wind and 12

geomagnetic indices driven parameters, module parameters are set to best match multipoint magnetic 13

field observations. Kubyshkina et al. [1999, 2002] included particle pressure information to further 14

constrain the choice of free parameters. Since these event-oriented modeling applies almost all 15

available observations to fit their model and the number of parameters in the model is of the same 16

order of the number of simultaneous observations, it is hard to assess with other independent data-17

model comparisons. Furthermore, the modeling result is critically dependent on the choice of 18

mathematical representations. For example, the same substorm growth phase event modeling by 19

Pulkkinen et al. [1994] with symmetric ring current module and Kubyshkina et al. [1999] with 20

asymmetric ring current module gave quiet different cross tail current density. Therefore, an 21

alternative way of modeling is necessary that would be helpful to understanding the magnetic field 22

configuration prior to the onset. 23

Various models have been developed to represent a growth phase, including the force equilibrium 24

code [e.g., Lemon et al., 2003; Zaharia and Cheng, 2003]. Zaharia and Cheng [2003] compared the 25

equilibrium states by solving 3D force-balance in an Euler potential coordinates, to demonstrate the 26

field and current distribution for two different (quiet and active) initial conditions. Zaharia et al. [2006] 27

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Frank Toffoletto� 3/19/09 10:46 AMComment: It is likely you will get this person (or Pulikiinen) as a referee

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also coupled this equilibrium code with RAM model inside the geosynchronous orbit to test the 1

feedback of magnetic field in a geomagnetic storm. Lemon et al. [2003] calculated the force balance 2

state of the magnetotail, as well as inner magnetosphere, by modifying the ideal MHD equation with 3

an additional frictional dissipation term, which was descended from Hesse and Birn [1993]. This 4

Magneto-Friction code (MF) has been coupled to the Rice Convection Model (RCM), by assuming 5

adiabatic convection inside the RCM modeling region. The initial attempt to model a substorm was 6

done by Toffoletto et al. [2001] and Wolf et al. [2002]. The coupled code, (RCM-E) has also been used 7

to simulate an idealized storm, using a sophisticated depleted boundary condition [Lemon et al., 2004]. 8

However, there is neither realistic substorm nor realistic storm events that have been modeled with the 9

RCM-E. This paper, as well as the accompanying paper (The modeling of a substorm event using the 10

Rice Convection Model with an equilibrium magnetic field: 2. Expansion Phase) addresses this issue. 11

In this paper, Section 2 describes the modeled substorm event, Section 3 provides a short 12

description of the RCM-E model, followed by details of model setup in Section 4. We compare our 13

simulation results with multipoint observations in Section 5. The discussion is in Section 6 and we 14

conclude in Section 7 with a brief summary. 15

16

2. Event Overview 17

The modeled substorm event occurred on Oct. 29, 2004. Figure 1 shows that AE index and the 18

overview of solar wind parameters during the times 10:30UT to 12:30UT. The IMF Bz in GSM turned 19

south at around 03:45UT and remained southward with variations from -6nT to -1nT until around 20

12:00UT. The solar wind proton number density increased from around 10 cm-3 at 03:45UT to around 21

25 cm-3 at 12:00UT, and the flow velocity was relatively low at about 300 to 320 km/s. The AE index 22

increased from 0nT to about 100nT at 06:20 UT, which suggested that convection was enhanced after 23

that. The ground station at Dawson, CGM Latitude=65.76 and CGM longitude=273.66, recorded Pi2 24

pulsation at 11:22UT, so we refer 11:22UT as the substorm onset time. During this event, the Kp 25

index varied between 2 and 3, and the absolute value of Dst index was less than 10nT, suggesting that 26

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this was a non-storm substorm. The AE index increased sharply at 11:28UT, 5 minutes after the Pi2 1

pulsation at Dawson. 2

In the near-Earth-neutral-line (NENL) model, the substorm growth phase begins when the 3

IMF Bz turning southward for about one hour at the magnetopause [Baker et al., 1996], which 4

suggests that the growth in this event possibly began at ~04:45UT. Since we used Geotail observation 5

as input simulation, the simulation presented in this paper started at 10:47UT rather than 03:45UT. 6

Near this time, Geotail was moving to cross the plasma sheet from southern hemisphere to the 7

northern hemisphere, it entered the plasma sheet at time 10:46UT when ion β value exceeded 1.0 (not 8

shown here). We set out to model the substorm growth phase at 10:47UT with relaxed magnetic field 9

(the detailed initial condition and boundary condition setup is described in Section 4). 10

11

3. The RCM-E 12

RCM-E couples the Rice Convection Model (RCM) with the magneto-friction equilibrium (E) code 13

and self-consistently computes the plasma drifts paths, electric field and magnetic field within the 14

closed magnetic field line region. The RCM was developed to compute plasma’s adiabatic drift, 15

treating the plasma in the inner and middle magnetosphere as many fluids with assumptions of slow-16

flow and isotropic pressure distribution along the magnetic field line. The electric field is calculated 17

by solving the Vasyliunas equation [Vasyliunas, 1970] in the coupling of the magnetosphere and 18

ionosphere, as 19

(1) 20

Where Φ is the electric potential in non-corotating frame, is the field-line-integrated ionospheric 21

conductance tensor of both hemisphere, I is the magnetic field dip angle, Bi and Beq are magnetic field 22

magnitudes on the ionosphere and magnetospheric equatorial plane, is the unit vector of magnetic 23

field at the equatorial plane, and are gradient operators on the ionosphere and magnetosphere, 24

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P is plasma pressure and is the flux tube volume per unit magnetic flux. With the strong-1

elastic-pitch-angle scattering assumption, the energy invariant, defined as 2

(2) 3

is conserved along the particle drift path, where Ws is the particle kinetic energy. If we neglect the 4

losses and sources, the density invariant for specific energy invariant is also conserved along 5

drift path, i.e., 6

(3), 7

where is the bounce-averaged and gradient/curvature drifts, given as 8

(4). 9

The plasma moments, pressure P, number density N, temperature T could be computed as 10

(5) 11

(6) 12

(7) 13

In a typical RCM, the magnetic field is prescribed by using simple dipole field or sophisticated 14

empirical model, e.g. Tsyganenko models. Detailed description and application of the RCM can be 15

found in Wolf [1983] and Toffoletto et al. [2003]. 16

The magneto-friction equilibrium code uses the modified ideal MHD equations that includes a 17

frictional dissipation term, which evolves toward the equilibrium state of plasma pressure and 18

magnetic field. The frictional term provides dissipation in the momentum equation: 19

(8) 20

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where ρ is plasma density, is electric current, is magnetic field vector, is plasma velocity, P is 1

plasma pressure, is friction term and α is friction parameter. α is adjusted to optimize 2

convergence towards equilibrium: 3

(9) 4

During an RCM run, the magnetic field is held fixed, after it has run for a designated exchange time 5

(typically several minutes) , the RCM provides the plasma pressure to the equilibrium code, which in 6

turn returns the equilibrated magnetic field to the RCM. A detailed description of the RCM, the 7

equilibrium code and the coupling are described by Toffoletto et al. [2003] and Lemon et al. [2003]. 8

The current version of the RCM-E assumes zero-tilted Earth dipole field, which means that the 9

center of the neutral sheet always lies on the Earth’s magnetic equatorial plane. This would introduce 10

a systematic error in data-model comparison when the measurement is off the neutral sheet. To 11

circumvent this, we utilize an empirical current sheet model [Tsyganenko and Fairfield, 2004; 12

hereinafter referred to as TF2004] to estimate the distance from the center of the neutral sheet to the 13

Earth’s magnetic equatorial plane. We transform the satellite’s location in GSM coordinate system 14

(x,y,z) to GSW system (X,Y,Z), using the solar wind direction and the dipole field tile angle. We then 15

estimate the center of the local neutral sheet in GSW system, Zn as a function of solar wind parameter 16

and (X, Y).Then we treat the satellite as located in the RCM-E model (X,Y,Z-Zn) and also treat the 17

normal direction of the current sheet in GSW as the normal direction of equatorial plane in RCM-E 18

model. Therefore, A physical parameter S at (X,Y,Z-Zn) in our model should be compared with the in-19

situ physical parameter S’ at (x,y,z) in GSM measured by the satellite. The similar procedure is used in 20

an RCM simulation of a substorm event [Zhang et al., 2009]. Since the solar wind conditions during 21

this event was fairly steady, so we applied the averaged solar wind parameters from 10:47UT to 22

11:22UT to determine the coefficients G0, G1, S and RH in equation (1) and (2) in TF2004, i.e., Vx=-23

313 km/s, Vy=25.0 km/s, By=-2.07 nT, Bz=-4.28 nT, Psw=3.59 nPa. We arbitrarily set α=2.4 in order to 24

best fit our initial magnetic field and plasma moments to Geotail observations. 25

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4. Model setup 1

During the growth phase simulation, the RCM and the equilibrium code exchange the plasma 2

pressure and magnetic field every 5 minutes, from 10:47 to just before onset at 11:22. For simplicity, 3

the ionospheric Hall and Pederson conductance were assumed to be uniform and set to 5 S for each 4

hemisphere. To initiate the growth phase simulation, we started with the magnetic field T89 (Kp=3) 5

[Tsyganenko, 1989] and plasma pressure [Tsyganenko and Mukai, 2003; hereinafter referred to as 6

TM2003], which have been relaxed using equilibrium code [Lemon et al., 2003]. The solar wind 7

parameters to set up coefficients for driving the TM2003 model are the same as those used in TF2004 8

model. In principle, for the region outside 10 Re, the plasma moments are set according to TM2003 9

empirical model; for the region inside 10 Re, the plasma moments are tuned more arbitrarily to match 10

the Geotail observed plasma moments (Figure 3, third and fourth panels) but still within a reasonable 11

region in relevant statistical models. Since the TM2003 empirical model is only valid in the plasma 12

sheet outside 10 Re, the plasma pressure within 10 Re is set to power law P=A*RB, where R is the 13

distance to the center of the Earth in units of Re. The power law form of plasma pressure distribution 14

as a function of radial distance has been adapted by Borovsky et al. [1998] and partially adapted by 15

Spence and Kivelson [1993] in the statistical study and Zaharia and Cheng [2003] in initial condition 16

setup in a growth phase simulation. The coefficients A and B are set to fit the pressure at Geotail orbit 17

and the pressure at 10 Re as that specified by TM2003 model at midnight. The ion temperature Ti 18

outside 10 Re is given by TM2003 model, while the ion temperature inside 10 Re is set as 19

Ti=C*cos(R/20.0*π)+Ti_10 (10) 20

where, Ti_10 is the ion temperature at 10 Re at midnight given by TM2003, and C is the coefficient so 21

that Ti=9.0 keV at geosynchronous orbit. This gives a decreasing ion temperature as increasing radial 22

distance, consistent with statistical model, but a different form from the power law suggested by 23

Borovsky et al. [1998]. The ratio of ion temperature and electron temperature is given as 24

Ti/Te=4.0+2.0*tan-1(R-7.0) (11) 25

which gives the ratio of 3.23 at geosynchronous orbit and larger than 6.50 outside of 10 Re, which is 26

acceptable, because the ion-electron temperature ratio falls between 5.5 and 11 for most circumstances 27

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Frank Toffoletto� 3/19/09 10:46 AM




Comment: Why this function???

Comment: Again, why this function???

Comment: Why this function? How important is this choice?

Comment: Why??

Comment: Again why?

Comment: Is 3.23 chosen to match observations? If so, you should comment as such.

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in the plasma sheet [Baumjohann et al., 1989] and the ratio near geosynchronous orbit is usually 1

smaller. The initial ion number density and electron number density are calculated everywhere as 2

Ne=Ni=P/kB(Ti+Te), where kB is the Boltzmann constant. The numbers of the plasma pressure 3

(1.8nPa), number density (1.2cm-3) and ion temperature (9.0keV) at geosynchronous orbit are roughly 4

consistent with statistical result in Borovsky et al. [1998], which is about 1.6nPa, 1.0cm-3 and 10.0keV 5

respectively.Statistically, the plasma sheet particle distribution is a kappa distribution during 6

geomagnetically quiet times (AE<100nT) [Christon et al., 1989], in terms of energy invariant and 7

density invariant it is written as, 8

(12) 9

We set the initial plasma sheet particle distribution as a distribution outside 8 Re as well as on 10

the tailward boundary in the nightside, which is as suggested by Christon et al. [1989]. Arbitrarily, the 11

plasma distribution elsewhere is set as a distribution. The plasma moments on the boundary are 12

given by TM2003 model. 13

During the growth phase, the RCM simulation region is set to a D-shape region on the 14

equatorial plane, as shown in Figure 2, which is convenient for the setup of electric potential 15

distribution on the boundary. For simplicity, the electric potential drop is made to vary linearly along 16

the boundary, i.e., from V/2 at (X, Y)=(-17,-10) to V/4 at (0,-10) on the dawn side, from –V/2 at (-17, 17

10) to –V/4 at (0, 10) on the dusk side, from V/2 to –V/2 on the tail boundary, and from V/4 to 0 to –18

V/4 along the dayside boundary, respectively. Since the solar wind condition was fairly stable, we take 19

the averaged polar cap potential drop V=57kV through the entire substorm event, estimated using the 20

Boyle et al. [1997] formula. 21

22

5. Simulation results 23

5.1 Near-Earth magnetotail region 24

Deleted: 4/12/2009




Comment: Why this value of kappa?

Comment: Why this size for the region?

Comment: Entire or just the growth phase.

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The bottom three panels in Figure 3 show the comparison of ion moments (<40keV) between 1

observations (dotted lines) and the simulation (solid lines). Both ion pressure Pi, number density Ni 2

and temperature Ti agree well with Geotail observations. The gradual buildup of particle pressure 3

indicates earthward transport of plasma. Dotted lines in the top two panels show the equatorial entropy 4

parameter PV5/3 and flux tube volume V estimated from local Geotail observations [Wolf et al., 2006]; 5

solid lines show the simulation results. The estimation from Wolf et al. [2006] is based on the 6

modified solution of 2-D Grad-Shafranov equation near the central neutral sheet with calibrated 7

coefficients determined by empirical models. The error in this method can be as large as a factor of 8

2~3 statistically. Therefore, although the comparison between the estimation and the simulation in 9

both PV5/3 and V are less confident than moment comparison, the results are acceptable within the 10

given error range. 11

Figure 4 shows the magnetic field observed by Geotail (dashed lines) and modeled in the 12

RCM-E (solid lines). The discrepancy in By is because (1) (1) the current version of RCM-E is a 13

simplified zero-tilted model with initial configuration adapted from T89 model with no tilt; (2) the 14

IMF By penetration in real magnetosphere, which can not be captured in this RCM-E modeling. It is 15

clear that both the magnitude of Bz and |B| are decreasing throughout the growth phase, which is 16

attributed to two effects, the spacecraft was approaching the center of neutral sheet from south 17

hemisphere to north hemisphere and the magnetic field was stretching. Observationally, the plasma 18

ion beta value reached a peak value of 10.5 at around 11:23UT, indicating that the spacecraft crossed 19

at the center of the neutral sheet at that time; while the model shows that the crossing happened almost 20

consistently at 11:22 when Geotail was at the center of the estimated neutral sheet. The left plot in 21

Figure 14 shows that the magnetic field was stretching during this time, when Bz was decreasing from 22

7.8 to 2.8 nT at X=-9 Re. 23

Figure 5 plots the Y component of electric field in the observations (black) and the model 24

(blue). The modeled Ey is roughly duskward with the magnitude of ~2mV/m during the growth phase, 25

indicating that the net flux transport is earthward. The observed Ey shows short time scale wave-like 26

oscillations, which cannot be captured using RCM-E; however, on average, the modeled Ey is 27

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Comment: Isn’t that you carefully chose the boundary conditions? If so, it should be stated. If it was found by trial and error, some explanation of how it was done would be useful., Jian comment: No. the boundary in growth phase run is set according to TM2003. It is not tweaked.

Comment: I don’t see an explanation of how much trial and error was involved and how sensitive your restults were to your choices. Jian comment: no trial and error on the boundary condition.

jianyang 8/28/2009

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consistent with the observed values. The red and green lines show the local induction and convection 1

Ey. The induction electric field, calculated from the expression (13) 2

where is the local velocity of same grid point on ionosphere [Zhang et al., 2009a setup paper] is 3

dawnward with the magnitude of about -3mV/m, which is an evident effect of magnetic field 4

stretching. While the ~5mV/m convection electric field is always positive in Y direction, which is a 5

direct result of a 57 kV potential drop over the 20 Re wide in dawn-to-dusk direction. The simulation 6

results indicate that, at least in this case, the magnitudes of induction and convection electric field 7

were comparable. During adiabatic convection in the growth phase, the convection electric field 8

transport plasma earthward mainly by E×B drift; while the induction electric prevents plasma injection 9

from tail by stretching magnetic field. More detailed discussion will be given in section 6.1. 10

11

5.2 Geosynchronous orbit 12

Figure 6 shows the GOES-10 observed magnetic field (dashed lines) and the modeled results 13

(solid lines). During the growth phase, GOES-10 (MLT=UT-9) is at the post-midnight sector. The 14

magnetic inclination angle is decreasing from 80 to 75 degrees, indicating a moderate stretching at 15

geosynchronous orbit. The modeled magnitude of magnetic field at GOES-12 at dawnside shows 16

qualitative agreement with in-situ observations (not shown), however we are unable to adjust the 17

simulated magnetic field in this un-tilted model to the realistic GSM coordinate system because the 18

mentioned neutral sheet warping estimation technique described in Section 3 cannot be applied in this 19

sector far from midnight. 20

Figures 7 and 8 show the observed energetic particle fluxes from LANL-SOPA instruments 21

(right panels) and simulated results (left panels) for electrons and for protons respectively. The energy 22

ranges are from 50keV to 315keV for electrons and 50keV to 400keV for protons from black lines to 23

green lines. Five spacecraft, LANL-01A, LANL-02A, LANL-97A, 1990-095 and 1991-080 are 24

presented from top to bottom. Both the observations and simulation indicate the fluxes at LANL-97A 25

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Comment: Explain how the inductive field is computed

Comment: Inclination angle?

Comment: Should explain why here.

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are distinctly different from other four, which shows a typical energetic particle flux dropout as a 1

classical feature during growth phase [e.g., Baker and McPherron, 1990; Lopez et al., 1989]. The 2

mechanism concerning this dropout has been discussed by Sauvaud et al. [1996] as betatron 3

deceleration by calculating the particle trajectory with a time-dependent stretching magnetic field near 4

geosynchronous orbit. Baker and McPherron [1990] also suggested an important role of magnetic 5

field stretching or current thinning in the energetic particle flux dropout. Although the isotropic 6

pressure assumption in RCM is not necessary in these two above mentioned papers and the sense of 7

adiabatic convection with energy invariant is also different in our model from the classic first and 8

second adiabatic motion, Wolf et al. [2006] and Zhang et al. [2009] demonstrated the electric field 9

inducted by magnetic field changes can accelerate and decelerate particles in adiabatic convection. 10

Figure 9 shows the magnetic field configuration threading the three spacecraft, LANL-97A, 1991-080 11

and Geotail in the beginning (top plot) and the end (bottom plot) of the growth phase. During the 12

growth phase, the mapping of field through Geotail changed little because Geotail was close to the 13

equatorial plane; 1991-080, about 0.6~0.7 Re off the magnetic equator, was moving from pre-midnight 14

to post-midnight; however, LANL-97A, 1.6 Re away from the equator, was at about 21MLT and just 15

in the transition region. The effect of field stretching dramatically changed the field configuration 16

threading LANL-97A, ending up with an equatorial footprint moving from (X, Y)=(-5.4, 5.1) to (X, 17

Y)=(-9.7, 7.1). With the isotropic pressure assumption in the model, the increase of the flux tube 18

volume tends to de-energize particles. A similar scheme was shown in Figure 1 in Lopez et al. [1989]. 19

The <45keV plasma moments are compared in Figure 10 and Figure 11 for 1991-080 and LANL-97A, 20

which shows reasonably good qualitative agreement between the observation and simulation . 21

22

6. Discussion 23

6.1 Adiabatic convection with force-balanced magnetic field 24

In recent years several models have been developed compute equilibrium solutions of the 25

inner and middle magnetosphere [Lemon et al., 2003; Zaharia and Cheng, 2003] and for the 26

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magnetotail [Hesse and Birn, 1993]. These simulations did not attempt to reproduce the time-1

dependent evolution of magnetic field and plasma pressure as a result of adiabatic convection during 2

the growth phase, but provided a snapshot of force-balanced configuration based on empirical models. 3

Earthward convection plasma interacts with the magnetic field in such a way that the magnetic field 4

will stretch to a tail-like configuration. This magnetic field stretching will also result in an inductive 5

dusk-to-dawn electric field, this preventing plasma injection from the tail into the inner magnetosphere 6

[Toffoletto et al, 2001]. The results of the RCM-E simulation of the growth phase provide us a semi-7

global view of this field-plasma interaction during a growth phase. 8

Figure 12 shows (a) the entropy parameter PV5/3, (b) plasma pressure P, (c) ion temperature Ti 9

and (d) ion number density Ni along the X-axis at different times during the growth phase. The initial 10

condition at 1047UT in solid lines indicates the specified symmetric ring current with the peak 11

pressure at 3.8Re and the ion temperature as monotonically decreasing with radial distance. 12

Comparing the plasma pressure and ion number density at 1107UT and at the end of the growth phase 13

1122-εUT with the initial condition, both of these two parameters show considerable increase, which 14

is a sign of earthward moving plasma as a result of a 2 mV/m dawn-to-dusk electric field. In contrast, 15

the ion temperature shows a 20% decrease just before 1122 UT compared with initial condition at 16

around X=-8 Re. This is believed to be due to hot ions, that were initialized in the open drift paths and 17

moved westward to the dusk side and away from the midnight sector. As a result, the hot ions from 18

tail boundary could not penetrate to the inner edge of the plasma sheet. It is noted that the PV5/3 19

increases substantially outside 8Re, and displays a knee-like profile, which is a result of the local Bz 20

minimum (left plot of Figure 14). 21

We also did an RCM-E run with a but with a fixed magnetic field, i.e., all parameters are kept 22

as the same as the previous RCM-E simulation, but the magnetic field was held in the same 23

configuration as the initial field in this RCM-E simulation. In this case, the code was run without the 24

self-consistent magnetic field. The left plot of Figure 13 shows RCM-E result of the equatorial 25

pressure distribution as well as equipotential lines (every 5kV) at 11:22-ε UT. Two prominent 26

differences stand out in the constant magnetic field run shown in the right plot of Figure 13. First, a 27

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clearly asymmetric ring current has built up near X=-7 Re with the peak value of about 7 nPa. The 1

physical explanation of this difference is that the convection electric field ~5mV/m (green line in 2

Figure5) is strong enough to be able to transport plasma into the near-Earth region; while the induction 3

electric field acts in an opposite direction to prevent the plasma from moving earthward. The total 4

electric field about 2mV/m (blue line in Figure 5) in RCM-E modeling is rather weak so that the 5

particle ring current buildup is substantially suppressed. Second, as a result, there is a predominant 6

shielding effect associated with the partial ring current in the RCM-E without the self-consistent 7

magnetic field; while in the RCM-E modeling, there are weak region-2 currents to generate shielding 8

in the convective potential distribution. 9

10

6.2 Magnetic field mapping and cross tail current 11

An important problem in understanding the substorm growth phase is having accurate 12

information on the magnetic field mapping. As mentioned in Section 1, the mapping at the end of the 13

growth phase has a large uncertainty. Comparison of multipoint observations for this event simulation 14

provides more confidence in having a reasonably accurate view of mapping. The Bz component on the 15

equatorial plane as a function of X-axis is shown on the left of Figure 14. During adiabatic convection, 16

the convective dawn-to-dusk electric field transports plasma earthward and the resulting force-balance 17

magnetic field stretches to balance the RCM-computed pressure distribution. As a result, the magnetic 18

field at the end of growth phase has a more stretched configuration inside 12 Re (dotted line) than the 19

beginning of the simulation (solid line). A distinct Bz minimum forms with a value of about 1.3nT at -20

13 Re, which is consistent with earlier growth phase modeling [Erickson and Wolf, 19xx, Hau and 21

Wolf, 19xx] Other equilibrium models [e.g., Lemon et al. 2003; Zaharia and Cheng, 2003] have also 22

produced this kind of Bz minimum. Kubyshkina et al. [1999] and Kubyshkina et al. [2002] 23

reconstructed mathematical representations of the currents in modified the Tsyganenko model by 24

fitting multipoint observations and including the plasma pressure information, and obtained Bz 25

minimum with roughly the same magnitude and location as our result. We compare our theoretically 26

modeled highly stretched magnetic field against the Tsyganenko T89 model [Tsyganenko, 1989] in the 27

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Frank Toffoletto� 3/19/09 10:46 AMComment: This seems contradtictory, one would expect stronger shielding would suppress ring current buildup.

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left plot of Figure 14. The simulated Bz can be ~20nT smaller than the T89 model with Kp=3 from -6 1

to -8 Re, and several nT smaller outside -8 Re. 2

As shown in the right plot of Figure 14, the ionosphere-magnetosphere mapping is quite 3

sensitive. The plasma sheet from -7 to -17 Re is maps to 24.3 to 25.3 degree colatitudes in the 4

beginning of the run (solid line) and but is then confined to less than 0.5 degrees with equatorward 5

boundary moving to 26 degree on the ionosphere in the end of growth phase (dotted line). In contrast, 6

the T89 model gives a much thicker ionospheric footprint of the plasma sheet. To our knowledge, 7

there is no statistical empirical model with growth phase specification (e.g., Tsyganenko model) can 8

map the plasma sheet to this thin layer on the ionosphere. Although the substorm event is highly 9

variable case by case, our result is approximately consistent with the event-oriented modeling applied 10

to other substorm growth phases [Pulkkinen, 1991a; Kubyshkina et al., 1999 and Kubyshkina et al., 11

2002]. 12

The equatorial cross tail current density Jy is calculated from the Ampere’s Law 13

(14) 14

using the equilibrium magnetic field (with the dipole field subtracted out). The left plot of Figure 15 15

shows Jy along X-axis at times 10:47, 11:07 and 11:22-ε UT. The peak location moves earthward from 16

-14 to -10 Re and the peak value increases from less than 12 to 16 nA/m2. The first term in equation 17

(14) is the main contributor to the cross tail current, which suggests that the Z gradient of magnetic 18

field near the equator is significant especially when the plasma sheet thins. The location of Bz 19

minimum is about 2 Re tailward of the Jy peak, which is also observed in the equilibrium model 20

[Lemon et al., 2003] and event-oriented model [Kubyshkina et al., 2002]. 21

In the substorm current disruption (CD) model, Lui et al. [1992] summarized several features 22

before the CD onset, suggesting that plasma beta is as high as ~70, local current density is ~27-80 23

nA/m2. In this event, the plasma beta value peaks with the value of 340 near -8 Re (right plot of Figure 24

15) in the end of growth phase. However, Jy is still 40% smaller than the lower limit inferred by CD 25

model. Other equilibrium models [Lemon et al., 2003; Zaharia and Cheng, 2003] give an even smaller 26

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Frank Toffoletto� 4/12/09 4:31 PMComment: Even the TXX storm models?

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cross tail current densities (~10nA/m2). Although event-oriented modeling shows comparable 1

[Kubyshkina et al., 1999] or even stronger cross tail current densities [Pulkkinen et al., 1994] than CD 2

model inferred in some cases, the modeling itself is critically dependent on the choice of the 3

mathematical representation of the currents [Kubyshkina et al., 1999]. The results of this simulation 4

suggest that the cross tail current density cannot be enhanced to a current disruption threshold. 5

However we guess, with the higher grid resolution, we may be able to obtain stronger localized 6

current density, although we do not really know how large it can be in the modeling. An alternative 7

analysis was conducted using RCM-E simulation results of a ballooning stability test [Toffoletto et al., 8

2007] which suggests that the inner edge of the plasma sheet is MHD ballooning unstable during the 9

late stages of the growth phase could be ballooning unstable in the Fast MHD approximation where 10

short timescale motions along the fieldline are ignored [Crabtree and Chen, 2004]. The low frequency 11

modes due to the ballooning instability were observed in the near-Earth plasma sheet region [Roux et 12

al., 1991], which is considered related with the initiation of substorm expansion phase. Cheng and 13

Zaharia [2004] showed that the most unstable modes were located with strong cross-tail current 14

density, where the magnitude of the cross-tail current density was smaller than our results. Another 15

analysis based on OpenGGCM simulation results [Raeder et al., 2008] was presented by Zhu et al. 16

[2009], indicated that the instability was firstly initiated near the local Bz minimum region. In our 17

simulation, a similar local Bz minimum was obtained near X~-13 Re as a result of adiabatic convection. 18

It will be interesting to investigate the ballooning instability in the configuration of magnetic field and 19

plasma pressure at the end of this growth phase. 20

21

7. Summary 22

In this paper, we presented the first RCM-E simulation of a real substorm growth phase. This 23

event occurred on Oct. 29, 2004, during fairly stable solar wind conditions with an average IMF Bz at 24

about -4nT. The RCM-E is a coupled code of the Rice Convection Model and the magneto-friction 25

equilibrium code, calculating the plasma adiabatic convection with self-consistent electric field by 26

solving the Vasyliunas equation and force-balanced magnetic field. An empirical current sheet model 27

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Frank Toffoletto� 3/19/09 10:46 AMComment: Not sure we should say this unless we back it up with something concrete.

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is used to transform our simulation result in the zero-tilted model to the observational coordinate 1

system. Several features are obtained in the modeling. 2

1. After careful choice of boundary conditions, the results are compared to multipoint observations 3

with good agreement, including the Geotail (-9 Re) observed magnetic field and plasma moments; 4

geosynchronous orbit magnetic field, plasma moments and energetic particle flux. 5

2. The modeling confirms several classical growth phase pictures, i.e., magnetic field stretching, 6

local Bz minimum formation, cross tail current density enhancement, earthward motion of the 7

plasma sheet , sharpening if the tail-dipole transition region, dropout energetic particle fluxes at 8

geosynchronous. 9

3. The induction electric field significantly alters plasma convection pattern. The plasma and 10

magnetic field interacts in a consistent way that the earthward convection builds up the plasma 11

pressure in the near-Earth region; the increased plasma pressure balances the magnetic field by 12

stretching; this stretching induces the dusk-to-dawn electric field, which prevents the plasma from 13

further earthward convection. 14

4. The magnetic field mapping in the end of the growth phase is very sensitive that the plasma sheet 15

from -7 to -17 Re maps onto the ionosphere within half degree in latitude. 16

17

The accompanying paper presents the simulation of the expansion phase. 18

19

20

Acknowledgements: The Dst and Kp indices were provided by the World Data Center for 21

Geomagnetism, Kyoto. GOES data were obtained through SPIDR. ONMI data were obtained via 22

CDAWEB. Geotail magnetic field, electric field and plasma data were provided by T. Nagai, H. 23

Hayakawa and Y. Saito through DARTS at Institute of Space and Astronautical Science, JAXA in 24

Japan. The authors thank I.R. Mann and the CARISMA team for ground magnetometer data. 25

CARISMA is operated by the University of Alberta, funded by the Canadian Space Agency. We also 26

thank LANL MPA team and SOPA team for making the data available online. GRANTS NUMBER?? 27

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1

2

3

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Zaharia S., C. Z. Cheng, Near-Earth thin current sheets and Birkeland currents during substorm 2

growth phase, Geophys. Res. Lett., 30 (17), 1883, doi:10.1029/2003GL017456, 2003. 3

4

Zaharia, S., V. K. Jordanova, M. F. Thomsen, and G. D. Reeves (2006), Self-consistent modeling of 5

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Zhu, P., J. Raeder, K. Germaschewski, and C. C. Hegna (2009), Initiation of ballooning instability in 11

the near-Earth plasma sheet prior to the 23 March 2007 THEMIS substorm expansion onset, Ann. 12

Geophys., 27, 1129-1138. 13

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Figure 1. Overview of solar wind parameters shifted to the Earth magnetopause and AE index. From 7

top to bottom are AE index, solar wind raw pressure Psw, flow velocity V, proton number density Np, 8

magnetic field in GSM Bz (solid line) and By (dotted line). The vertical solid line indicates the 9

substorm onset time 11:22UT determined by ground magnetometers. 10

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Figure 2. The equatorial view of D-shape RCM simulation region for substorm growth phase. The sun 4

is to the left. The values near black spots on the boundary indicate the polar cap potentials, where 5

V=57kV. 6

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Figure 3. Comparison of Geotail observations (dotted line) and the modeling results (solid line). From 2

top to bottom are equatorial plasma entropy paramemter PV5/3, equatorial flux tube volume per unit 3

magnetic flux V, and local ion moments (<40keV), thermal pressure Pi, ion number density Ni and ion 4

temperature Ti. 5

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Figure 4. Comparison of Geotail observations (dashed line) and the modeling results (solid line). From 2

top to bottom are the X, Y, Z components of magnetic field, the magnitude of magnetic field and 3

inclination angle ( tan(θ)=Bz/Bx), respectively. The three horizontal dotted lines indicate Bx=0, By=0 4

and θ=90. All data are in GSM coordinate system. 5

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Figure 5. The calculated Y component of electric field based on Geotail observations (black line) and 2

the modeled result (induction electric field in red line, convective electric field in green line and total 3

electric field in blue line) in GSM coordinate system. 4

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1

Figure 6. Comparison of GOES-10 observed magnetic field (dashed line) and the modeled magnetic 2

field (solid line) in GSM coordinate system. The four plots from top to bottom indicate the X, Y, Z 3

components of magnetic field and the tilt angle ( ). 4

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Figure 7. The observed energetic electron differential flux (right) at geosynchronous orbit and the 4

simulated results (left). The black, dark blue, light blue, red and green lines indicate the 50-75, 75-105, 5

105-150, 150-225 and 225-315 keV energy ranges. Five panels from top to bottom are for LANL-01A, 6

LANL-02A, LANL-97A, 1990-095 and 1991-080, respectively. 7

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Figure 8. The observed energetic proton differential flux (right) at geosynchronous orbit and the 2

simulated results (left). The black, dark blue, light blue, red and green lines indicate the 50-75, 75-113, 3

113-170, 170-250 and 250-400 keV energy ranges. Five panels from top to bottom are for LANL-01A, 4

LANL-02A, LANL-97A, 1990-095 and 1991-080, respectively. 5

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Figure 9. The magnetic field line mappings threading 1991-080, LANL-97A and Geotail, labeled as 16

“080”, “97a” and “Geo” respectively. The colors show the plasma pressure on the equatorial plane. 17

Top and bottom plots are for T=10:47UT and T=11:22-εUT. 18

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Figure 10. The LANL-080 MPA data (dotted line) compared with the simulated results (solid line). 2

The proton temperature, electron temperature, proton number density and electron number density are 3

shown from top to bottom. 4

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Figure 11. The LANL-97a MPA data (dotted line) compared with the simulated results (solid line). 2

The proton temperature, electron temperature, proton number density and electron number density are 3

shown from top to bottom. 4

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Figure 12. The entropy parameter PV5/3, plasma pressure P, ion temperature Ti and ion number density 18

Ni along the X-axis for three different times. 19

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(a)

(d)

(b)

(c)

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Figure 13. The equatorial plasma pressure and the equipotential lines (every 5kV) for RCM-E (left) 16

and RCM-E without Friction code (right) at the end of growth phase. 17

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Figure 14. The magnetic field Bz along the X-axis (left) and the colatitudes in degree versus the 14

corresponding equatorial crossing point in the midnight meridian plane (right). 15

The solid, dotted, dashed, dotted-dashed lines represent the modeled field in the beginning of the 16

modeling (T=10:47), the modeled field in the end of the growth phase (T=11:22-ε), the T89 Kp=3 17

model, and the T89 Kp=6 model, respectively. 18

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Figure 15. (left) The Y component of electric current density (in unit of nA/m2) and (right) the plasma 12

beta along the X-axis for the times T=10:47 (solid line), T=11:07 (dashed line), T=11:22-ε (dotted-13

dashed line). 14

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