Magnetic flux in the magnetotail and polar cap during sawteeth, isolated substorms, and steady magnetospheric convection events

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2009ja014232, 2009 Magnetic flux in the magnetotail and polar cap during sawteeth, isolated substorms, and steady magnetospheric convection events Chao-Song Huang, 1 Anna D. DeJong, 2 and Xia Cai 3 Received 10 March 2009; revised 28 April 2009; accepted 14 May 2009; published 2 July [1] Different magnetospheric dynamic processes, such as sawtooth events, isolated substorms, and steady magnetospheric convection (SMC), can occur, depending on the solar wind condition. The purpose of this study is to calculate the magnetic flux in the magnetotail and in the polar cap during these magnetospheric modes and to establish a quantitative relation between the magnetic flux and the solar wind parameters. We use the Imager for Magnetopause-to-Aurora Global Exploration Far Ultraviolet Imager and Polar Ultraviolet Imager measurements to derive the magnetic flux in the polar cap and the Geotail measurements to derive the magnetic flux in the magnetotail. The average value of the magnetic flux at the sawtooth onset is 1 GWb in the polar cap and magnetotail, and the relative decrease of the magnetic flux from the maximum value at the sawtooth onset to the minimum value after the onset is %. The average magnetic flux in the polar cap at the onset of isolated substorms is 0.68 GWb and decreases by 26.5% after the expansion phase. The magnetic flux in the polar cap during SMC events varies between 0.3 and 0.8 GWb. The magnetic flux at the isolated substorm onset is at the upper limit of the magnetic flux of SMC events for the same merging electric field, and the magnetic flux at the sawtooth onset is always higher than that during isolated substorms and during SMC events. The magnetic flux in the magnetotail and polar cap during sawtooth events and isolated substorms increases gradually before the onset and then decreases rapidly after the onset, which is consistent with the traditional energy loading-unloading scenario. However, the maximum magnetic flux at the sawtooth or isolated substorm onset is not a constant but increases with the merging electric field and with the corrected Dst index. The results also provide reasonable explanation of the relatively constant period of sawtooth events. Citation: Huang, C.-S., A. D. DeJong, and X. Cai (2009), Magnetic flux in the magnetotail and polar cap during sawteeth, isolated substorms, and steady magnetospheric convection events, J. Geophys. Res., 114,, doi: /2009ja Introduction [2] Magnetospheric substorms are often described as an energy (magnetic flux) loading-unloading process [Russell and McPherron, 1973; Hones, 1984; Baker et al., 1996; Nagai et al., 1998]. During the growth phase of substorms with southward interplanetary magnetic field (IMF), magnetic reconnection between the interplanetary and terrestrial magnetic field lines occurs at the dayside magnetopause, and magnetic flux is transferred from the solar wind to the magnetotail. As magnetic flux builds up in the tail lobe, the plasma sheet thins to the scale of an ion gyroradius, and magnetic reconnection occurs in a near-earth neutral line (NENL) and results in the expansion onset of substorms. The magnetic flux stored in the magnetotail increases 1 Institute for Scientific Research, Boston College, Chestnut Hill, Massachusetts, USA. 2 Southwest Research Institute, San Antonio, Texas, USA. 3 Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA. Copyright 2009 by the American Geophysical Union /09/2009JA during the growth phase, reaches a maximum value at the onset, and is then released during the expansion phase. [3] Sawtooth events in the Earth s magnetosphere are global, large-amplitude oscillations of energetic plasma particle fluxes at geosynchronous orbit [Huang, 2002; Huang et al., 2003a, 2003b, 2004, 2005; Reeves et al., 2003; Lui et al., 2004; Henderson, 2004; Henderson et al., 2006a, 2006b; Pulkkinen et al., 2006, 2007]. Very similar variations of electron and proton fluxes are detected nearly simultaneously over a large local time range on the nightside. A sawtooth event is a series of individual tooth events, and each individual tooth has all well-known characteristics of magnetospheric substorms and represents one substorm. The substorm activity during sawtooth events shows periodic variations with a typical period of 3 h, and it is becoming widely accepted that sawtooth events are periodic substorms. Sawtooth events are driven by extremely strong solar wind and continuously southward IMF during magnetic storms, and the magnetospheric-ionospheric disturbances during sawtooth events are generally much larger than those during quiet time isolated substorms. Huang and Cai [2009] have conducted a statistical analysis of the 1of13

2 magnetotail total pressure and lobe magnetic field during sawtooth events and derived the quantitative relationship between the magnetotail total pressure/lobe magnetic field at the sawtooth onset and the solar wind driver. They suggest that the sawtooth onset occurs when the magnetotail reaches a critical state and that the critical state depends on the solar wind condition. [4] Nakai and Kamide [2003, 2004] studied the magnetospheric and solar wind conditions for isolated substorms. They found that the magnetotail magnetic field just prior to the substorm onset correlates with the Dst index and that the magnetotail total pressure increases with the solar wind pressure. Shukhtina et al. [2004] derived the quantitative relationship between the magnetotail and solar wind parameters during isolated substorms. Shukhtina et al. [2005] showed that the total magnetic flux in the magnetotail at the onset of isolated substorms is correlated with the merging electric field. [5] The magnetosphere can exhibit enhanced convective signatures for long periods of time without occurrence of substorms; such a state is termed the steady magnetospheric convection (SMC) [Pytte et al., 1978; Sergeev et al., 1996; O Brien et al., 2002; McPherron et al., 2005]. During the period of SMC, the dayside magnetic reconnection is balanced by the nightside magnetic reconnection in the distant tail. However, the magnetospheric convection is not completely steady, a new name, balanced reconnection intervals, has been proposed for the magnetospheric state driven by southward IMF but without substorms [DeJong et al., 2008]. We still use the conventional name SMC in this paper. [6] The total magnetic flux in the polar cap and magnetotail is an important parameter for the study of substorms and SMC events. The magnetic flux stored in the magnetotail varies during substorms and is relatively stable during SMC events. Caan et al. [1978] found that the tail lobe energy is increased before the substorm onset and then decreased after the onset. Petrinec and Russell [1996] estimated that the level of magnetic flux needed in the magnetotail for the occurrence of a triggered substorm is between 1.0 and 1.4 GWb. McPherron and Hsu [2002] conducted a statistical analysis of storm-time and quiet time substorms. They found that the average lobe magnetic field at the onset reaches a maximum value of 40 nt for storm-time substorms and 30 nt for quiet time substorms and that the percent change during storm-time and non-storm-time substorms is nearly the same. Milan et al. [2004, 2007, 2008] examined the variations of the magnetic flux in the polar cap during substorms and showed that the average open flux of the magnetosphere at the substorm onset is GWb. Shukhtina et al. [2005] found that the magnetic flux stored in the tail lobes during the substorm growth phase is proportional to the merging electric field. DeJong and Clauer [2005] and DeJong et al. [2007] analyzed the polar cap open magnetic flux and concluded that the polar cap magnetic flux during individual sawteeth is, on average, 150% as large as that during isolated substorms or during SMC events. [7] In this paper, we investigate the characteristics of the magnetic flux in the polar cap and magnetotail during sawteeth, isolated substorms, and SMC events. Our purpose is to find how the magnetic flux in the polar cap and in the magnetotail varies during different magnetospheric modes and how the maximum magnetic flux at the sawtooth/ substorm onset is related to the solar wind. The results will provide useful insight into the solar wind and magnetospheric conditions under which sawtooth events, isolated substorms, or SMC occur. 2. Observational Results 2.1. Sawtooth Events [8] We first show the sawtooth event on 18 April 2002 as example. Depicted in Figures 1a and 1b are the IMF B z component and solar wind dynamic pressure measured by the Advanced Composition Explorer (ACE) satellite located at 220 R E upstream. The solar wind data have been shifted to the Earth s bow shock nose with the minimum variance analysis technique developed by Weimer et al. [2003] and are plotted in the GSM coordinate. In all cases analyzed in this paper, the solar wind and IMF data have been shifted to the bow shock nose with the above technique. In the case of Figure 1, the IMF B z was continuously negative and stable for the entire day, and the solar wind pressure showed an enhancement around 0100 UT and became small after 0500 UT. [9] Figure 1c shows the proton flux measured by the LANL-02A geosynchronous satellite. The energy channels of the proton flux are 50 75, , , , and kev. The variations of the proton flux show a well-defined sawtooth-like shape. The gradual decrease of the proton flux occurs when the magnetotail becomes more stretched during the growth phase of substorms, and the sudden increase of the flux corresponds to the flux injection from the tail to the inner magnetosphere at the expansion onset. In this paper, we use the same criteria to identify sawtooth events as those of Cai et al. [2006]: At least one geosynchronous satellite is located around local noon (3 MLT hours from local noon) and one around local midnight (3 MLT hours from local midnight), and the plasma particle flux injection is observed globally. In order to distinguish the difference between sawtooth events and quiet time isolated substorms, the onset of the individual tooth (substorm) during sawtooth events is termed the sawtooth onset. The vertical dotted lines in Figure 1 denote the times of the flux injections (sawtooth onsets). [10] The magnetospheric parameters used in this paper were measured by the Geotail satellite. During the entire day of 18 April 2002, Geotail was located in the midtail, and the GSM coordinate of Geotail was between ( 22.19, 13.83, 12.49) and ( 29.32, 4.56, 8.55) R E. Figure 1d displays the equivalent lobe magnetic field measured by the Geotail satellite. The equivalent lobe magnetic field is defined as B 2 L /2m 0 = B 2 T /2m 0 + N i k(t i + T e ) and includes the contribution of the plasma pressure, where B T = (B 2 x + B 2 y + B 2 z ) 1/2 is the strength of the magnetospheric magnetic field. The corresponding total pressure is P T = B 2 T /2m 0 + N i k(t i + T e ). The magnetic pressure is much greater than the plasma pressure in the tail lobes, and the plasma pressure dominates the magnetic pressure at the center of the plasma sheet. When we use the equivalent lobe magnetic field (or the total pressure) for both the lobe and plasma sheet, we assume that the lobe magnetic pressure is balanced by the plasma pressure in the plasma sheet. Baumjohann et al. [1990] 2of13

3 Figure 1. (a f) Examples of sawtooth events. Figures 1a 1f show the interplanetary magnetic field (IMF) B z component, the solar wind dynamic pressure, the proton flux at geosynchronous orbit, the lobe magnetic field measured by the Geotail satellite between 22 and 30 R E, the radius of the tail lobe at the position of Geotail, and the magnetotail/polar cap magnetic flux on 18 April 2002, respectively. The vertical dotted lines denote the sawtooth onsets. made a statistical survey on the balance of total (thermal and magnetic) pressure in the plasma sheet and tail lobe. They found that lobe and plasma sheet pressure balance very well for a single case as well as statistically for a wide range of activities. Birn [1987] used the isotropic pressure profile in the magnetospheric model. Spence et al. [1989] used ISEE 2 data to determine the plasma sheet pressure versus distance in the midnight local time sector of the near-earth magnetotail plasma sheet. They also found that the pressure balance is valid for the radial distance of greater than 15 R E. Huang and Cai [2009] studied the magnetotail total pressure at the substorm onset of sawtooth events and showed that the plasma sheet pressure is balanced by the lobe magnetic pressure in the vertical direction. [11] The equivalent lobe magnetic field shows a very regular periodic variation with a period of 3 h. There are two processes that may contribute the observed variations of the lobe magnetic field. One process is the periodic generation of plasmoids in the midtail. When magnetic reconnection occurs in the midtail on the closed field lines of the central plasma sheet at the onset of a substorm, the reconnection creates a structure of closed magnetic loop which is termed the plasmoid. After the lobe reconnection begins, the plasmoid is released and travels tailward at high speeds. The magnetic field strength increases before each substorm onset, and the spike in the magnetic field strength after the onset represents the lobe magnetic field compressed by the plasmoid moving tailward through the satellite [Slavin et al., 1984; Taguchi et al., 1998; Huang, 2002; Huang et al., 2003b]. The other process is the flaring of the magnetotail. As the open flux content of the magnetosphere increases, it causes the magnetotail magnetopause to flare outward. The solar wind now strikes the magnetopause at a less-grazing angle, so the normal component of the ram pressure increases, raising the internal lobe magnetic pressure. Hence, changes in the lobe field strength are associated with the breathing in and out of the tail as the open flux content changes during the substorm cycle [Petrinec and Russell, 1996; Slavin et al., 2002; Milan et al., 2004, 2008]. [12] The total magnetic flux in the magnetotail is calculated as follows. The tail radius, R T, at the Geotail position is calculated with the empirical model of Shue et al. [1998] by using the shifted solar wind data as input and plotted in Figure 1e. The tail radius does not change much on this day because of the stable IMF B z and solar wind pressure. The magnetic flux in the magnetotail is defined as F T = pr T 2 B L /2. Because R T is, by definition, the radius of the magnetotail, the area pr T 2 is the cross section of the magnetotail including the tail lobes and plasma sheet. Therefore, F T = pr T 2 B L /2 represents the magnetic flux contained in a semicircular magnetotail cross section consisting of one tail lobe and half plasma sheet. We assume that the equivalent lobe magnetic field measured by Geotail represents the average value over the semicircular magnetotail cross section and use the product of this cross section and the measured B L to represent the magnetic flux contained in the semicircular area. [13] The magnetotail magnetic flux derived with the above method is plotted in Figure 1f. The variation of the 3of13

4 Figure 2. (a c) Superposed epoch analysis of sawtooth events. Figures 2a and 2b show the magnetic flux in the polar cap and in the magnetotail, respectively. Figure 2c shows the average values of the polar cap and magnetotail magnetic fluxes. The polar cap magnetic flux and the magnetotail magnetic flux are measured in different events. magnetotail magnetic flux is very similar to that of the lobe magnetic field. Superposed is the magnetic flux in the polar cap. The polar cap magnetic flux used in this paper is derived from the measurements of the Far Ultraviolet (FUV) Imager on board the Imager for Magnetopause-to-Aurora Global Exploration (IMAGE) satellite and the Ultraviolet Imager on board the Polar satellite, and the details of calculating the polar cap area from the IMAGE and Polar measurements and the polar magnetic field from the International Geomagnetic Reference Field (IGRF) are given by DeJong et al. [2007]. We calculated the magnetotail magnetic flux with different empirical models of the tail radius [Petrinec and Russell, 1996; Shue et al., 1997, 1998] and found that the output of Shue et al. s [1998] model agrees best with the measured polar cap magnetic flux. In fact, only Shue et al. s [1998] model is derived for extreme solar wind condition and expected to be consistent with sawtooth events that are driven by strong solar wind. As can be seen in Figure 1f, the magnetotail magnetic flux is in amazing agreement with the polar cap magnetic flux during the sawtooth event. [14] However, it is necessary to realize the problems that exist in the calculations of the open flux from point measurements in the tail lobe. As we already mentioned, one problem is that we cannot determine the size of the plasma sheet and the contribution from the closed flux of the plasma sheet. Another problem is whether the empirical model of the tail radius [Shue et al., 1998] is capable of estimating the variation of the tail lobe radius during the process of substorms. Milan et al. [2004] found that the magnetotail radius is not constant through the substorm cycle but shows departures from the prediction of Shue et al. s [1998] model because the open flux changes. We need other measurements to calibrate the open flux estimated from the empirical model. [15] The polar cap maps along the geomagnetic field lines to the tail lobe, and the mapped region does not include the plasma sheet. The magnetic flux in the polar cap is not the same as the magnetic flux contained in one tail lobe and half plasma sheet. Because we do not have the measurement of the sizes of the tail lobe and plasma sheet, what we can do is to calculate the magnetic flux in the magnetotail including 4of13

5 Figure 3. (a, b) The magnetic flux in the polar cap and in the magnetotail at the sawtooth onsets as a function of the merging electric field, respectively. The polar cap magnetic flux and the magnetotail magnetic flux are measured in different events. the tail lobes and plasma sheet. The measured polar cap magnetic flux provides the calibration for the magnetic flux in the tail lobe. Our results suggest that the amount of the magnetotail magnetic flux calculated with Shue et al. s [1998] tail radius model is approximately equal to the amount of the polar cap magnetic flux. [16] We searched the plasma particle flux data measured by the LANL geosynchronous satellites over 9 years ( ) and found 107 sawtooth events during which there are 431 individual teeth in total. We then searched the Geotail satellite data and found 20 sawtooth events with 50 individual teeth during which Geotail was in the midtail between 19 and 31 R E. DeJong et al. [2007] identified 13 sawtooth events with 29 individual teeth in the polar cap data. The Geotail satellite and the IMAGE/Polar satellite had simultaneous measurements in the sawtooth events on April Besides the 18 April case presented in Figure 1, the magnetotail magnetic flux is also in good agreement with the polar cap magnetic flux on 19 and 20 April. [17] Figures 2a and 2b show the superposed epoch of the magnetic flux in the polar cap during the 13 sawtooth events with 29 individual teeth identified by DeJong et al. [2007] and the magnetic flux in the magnetotail during 20 sawtooth events with 50 individual teeth identified from the LANL plasma particle flux data, respectively. The epoch time is chosen to be the time of the sawtooth onset. The magnetotail/polar cap magnetic flux varies between 0.4 and 1.4 GWb during the sawtooth events. The average values of the magnetic flux are plotted in Figure 2c. The average magnetotail magnetic flux starts to increase from 0.78 GWb at 78 min, reaches a maximum value of 0.98 GWb at the onset time, and decreases to a minimum value of 0.72 GWb at 70 min; the net (or relative) decrease of the magnetic flux after the onset is 0.26 GWb (or 26.5%). Similarly, the average polar cap magnetic flux also starts to increase from 0.78 GWb, reaches a maximum value of 1.04 GWb at the onset time, and then decreases to a minimum value of 0.79 GWb; the net (or relative) decrease after the onset is 0.25 GWb (or 24.0%). Note that the magnetotail magnetic flux and the polar cap magnetic flux are measured in different events. The small difference in the average values of the two data sets may be caused by the solar wind conditions and/or by the limited cases. The agreement between the absolute values and relative changes of the magnetic flux in the two data sets is very good. [18] The magnetotail magnetic flux immediately prior to the substorm onset may represent the magnetotail condition for the onset to occur, which corresponds to the maximum energy stored in the magnetotail. Our purpose is to find how the maximum magnetic flux in the magnetotail/polar cap at the sawtooth onset is related to the solar wind. It is generally believed that the dayside reconnection rate is controlled by the merging electric field, which is defined as E m = V SW (B y 2 + B z 2 ) 1/2 sin 2 (q/2) [Kan and Lee, 1979], where V SW is the solar wind velocity, B y and B z are the IMF components, and q is the IMF clock angle. For the angular dependence, some investigators used sin 3 (q/2) [Reiff and Luhmann, 1986; Shukhtina et al., 2004] or sin 4 (q/2) [Wygant et al., 1983]. In this paper, we use the original definition by Kan and Lee [1979]. We will study whether/how the angular dependence affects the correlation between the solar wind and magnetotail magnetic flux in the future. [19] Figure 3 shows the magnetic flux at the sawtooth onset as a function of the merging electric field. Considering that the magnetic flux varies with time, we have averaged the magnetic flux over 5 min prior to each onset and used the averaged value to represent the flux at the onset. The merging electric field is averaged over 60 min prior to the sawtooth onset; this 60-min interval presumably corresponds to the accumulation time of the flux in the tail for substorms [Bargatze et al., 1985; Shukhtina et al., 2004, 2005]. We also averaged the merging electric field over 30 min prior to each onset and found that there is no noticeable difference from the merging electric field averaged over 60 min. This is because the IMF is generally stable for many hours during sawtooth events, so the merging electric field does not change much over min. The solid lines in Figure 3 are derived from the 5of13

6 Figure 4. (a) The normalized lobe magnetic field and (b) the magnetotail magnetic flux at the sawtooth onsets as a function of the corrected Dst index, respectively. data regression with least squares fitting. The constant term in both regression lines is 0.80, and the slope is for the polar cap data and for the magnetotail data. The difference of the slope in the two data sets is only 5.5% and may be caused by the limited data number and the data scatter. Although the magnetotail and polar cap magnetic flux data are taken from different events, the agreement between the regression lines for the two data sets is excellent. We use the average value of the slopes from the two data sets to represent the slope for the magnetic flux in both the magnetotail and polar cap, and the empirical formula is written as F T;Saw ¼ F PC;Saw ¼ 0:80 þ 4: E m ; where F T,Saw and F PC,Saw represent the magnetotail magnetic flux and polar cap magnetic flux (in GWb) at the sawtooth onset, respectively, and E m is the merging electric field in mv m 1. The polar cap magnetic flux in Figure 3a covers a larger range of the merging electric field than that for the magnetotail magnetic flux in Figure 3b, so the average value of the polar cap magnetic flux is larger than that of the magnetotail magnetic flux, as seen in Figure 2c. [20] The sawtooth events often occur during magnetic storms. We show in Figure 4 how the lobe magnetic field and magnetotail magnetic flux are related to the strength of magnetic storms, in terms of the Dst index. Figure 4a shows the lobe magnetic field as a function of the corrected Dst index [O Brien and McPherron, 2000]. Similar to the method of Huang and Cai [2009], the lobe magnetic field is first normalized to the tail distance at 25 R E and then corrected by the solar wind pressure. The regression line derived by Huang and Cai [2009] from a larger data set is B LNC ¼ 26:8 7: Dst 0 ; where B LNC is the normalized, pressure-corrected lobe magnetic field in nt and Dst 0 is the corrected Dst index in nt. ð1þ ð2þ [21] Figure 4b shows the magnetotail magnetic flux at the sawtooth onset as a function of the corrected Dst index. We also conducted the same analysis for the polar cap magnetic flux; the polar cap data will be shown later. We calculated the regression lines for both the magnetotail and polar cap magnetic fluxes and found that there is only a small (<10%) difference between the two data sets. We take the average value of the two data sets and write the empirical formula of the magnetic flux in both the magnetotail and polar cap as F T;Saw ¼ F PC;Saw ¼ 0:78 2: Dst 0 : All events shown in Figure 4 occurred during magnetic storms, and the mean minimum Dst value of these storms is 85 nt. Note that Dst 0 is in general negative during magnetic storms, so the lobe magnetic field and the magnetotail/polar cap magnetic flux increase with the magnitude of the corrected Dst index. Equations (2) and (3) imply that the lobe magnetic field and the magnetotail/ polar cap magnetic flux must reach a larger value for the sawtooth onset to occur during stronger magnetic storm Isolated Substorms [22] We now examine the behaviors of isolated substorms and show two examples in Figure 5. The IMF B z, plotted in Figure 5a, was small on 11 March 1998, and the solar wind pressure, plotted in Figure 5b, was stable. The Geotail satellite was located at ( 20, 20, 10) R E during the period of interest. The lobe magnetic field measured by Geotail, depicted in Figure 5c, started to increase around 2200 UT and reached a maximum value at 2252 UT when the substorm onset occurred. We use the empirical formula of the tail radius at the isolated substorm onsets derived by Shukhtina et al. [2004] to calculate the area of the magnetotail and then multiply the area by the lobe magnetic field measured by Geotail. The magnetotail magnetic flux is presented in Figure 5d. Although the formula of Shukhtina et al. [2004] was derived only for the onset of isolated substorms, the variation of the magnetotail magnetic flux calculated with this formula coincides well with that of the ð3þ 6of13

7 Figure 5. (a h) Examples of the polar cap magnetic flux during isolated substorms. Figures 5a 5d show the IMF B z, the solar wind dynamic pressure, the tail lobe magnetic field measured by Geotail, and the magnetotail/polar cap magnetic flux on 11 March 1998, respectively. Figures 5e 5h are the same as Figures 5a 5d but for the event on 7 December The vertical dotted line denotes the substorm onset. polar cap magnetic flux during the entire substorm process including the growth and expansion phases. [23] Another substorm case, which occurred on 7 December 2000, is displayed in Figures 5e 5h. The Geotail satellite was located at ( 16, 9, 3) R E during the period of interest in this case. Figure 5h shows the magnetic flux in the polar cap and the magnetotail. The magnetotail magnetic flux calculated with the formula of Shukhtina et al. [2004] is in good agreement with the polar cap magnetic flux. We also tried other empirical formulas of the tail radius [Shue et al., 1997, 1998] which are derived from more general conditions, rather than limited to substorms alone. We found that the magnetotail magnetic flux calculated with Shue et al. s [1998] model is in reasonable agreement with the measured polar cap magnetic flux when the solar wind dynamic pressure is small (e.g., <2 npa) but becomes 50% higher than the polar cap magnetic flux when the solar wind dynamic pressure is greater than 4 npa. The magnetotail magnetic flux calculated with Shue et al. s [1997] model is always higher than that calculated with Shue et al. s [1998] model. The magnetotail magnetic flux calculated with the formula of Shukhtina et al. [2004] is the closest to the polar cap magnetic flux at the onset of the isolated substorms in our cases. [24] We have the measurements of the polar cap magnetic flux in 30 isolated substorms. Figure 6a shows the superposed epoch of the 30 substorms, and the gray line represents the average value. The maximum value of the average magnetic flux at the onset is 0.68 GWb, and the minimum value 68 min after the onset is 0.50 GWb. The relative decrease of the average magnetic flux after the expansion onset is 26.5%. [25] Figure 6b shows the relationship between the polar cap magnetic flux at the isolated substorm onsets and the merging electric field. Similar to the sawtooth onsets, the polar cap magnetic flux is averaged over 5 min prior to each onset, and the merging electric field is averaged over 60 min prior to the onset. The regression line can be written as F PC;Sub ¼ 0:62 þ 3: E m ; where F PC,Sub is the polar cap magnetic flux at the isolated substorm onset in GWb. The merging electric field used in equation (4) is averaged over 60 min prior to the onset. We ð4þ 7of13

8 Figure 6. (a) Superposed epoch analysis of the magnetic flux in the polar cap during isolated substorms, (b) the polar cap magnetic flux at the onset of isolated substorms as a function of the merging electric field, and (c) the polar cap magnetic flux at the onset of isolated substorms as a function of the corrected Dst index. also calculated the merging electric field averaged over 30 min prior to the isolated substorm onsets and found that the constant term and the slope, corresponding to those of equation (4), become 0.64 and , respectively. The difference between the 30-min and 60-min averaged merging electric field occurs because the IMF may have significant variations within 60 min during isolated substorms. [26] Shukhtina et al. [2005] derived a quantitative relationship between the magnetotail magnetic flux at the isolated substorm onset and the merging electric field. Both the constant term and slope of Shukhtina et al. s [2005] formula are larger than ours. Note that Shukhtina et al. s [2005] formula is derived from the magnetotail magnetic flux, while equation (4) is derived from the polar cap magnetic flux. We found that the magnetotail magnetic flux calculated with Shukhtina et al. s [2004] model of the tail radius is generally larger than the polar cap magnetic flux when the solar wind dynamic pressure is higher than 4 npa, which may cause the difference between the empirical formulas of the magnetic flux. [27] In Figure 6c, the polar cap magnetic flux at the onset of isolated substorms is plotted as a function of the corrected Dst index. The regression line is given by F PC;Sub ¼ 0:65 1: Dst 0 : The corrected Dst index is in general negative at the onset of isolated substorms. In our cases, Dst 0 is negative in 29 out of 30 substorms and is weakly positive only in one substorm. Hence the polar cap magnetic flux increases with the magnitude of the corrected Dst index. This property is the same as that for the sawtooth onsets presented in Figure SMC Events [28] Steady magnetospheric convection occurs when the IMF and solar wind pressure are relatively stable [Sergeev et al., 1996]. Figure 7 shows two SMC events during which simultaneous measurements of the magnetotail and polar cap magnetic fluxes are available. We use the empirical formula of the tail radius of Shukhtina et al. [2004], which ð5þ 8of13

9 Figure 7. (a h) Examples of steady magnetospheric convection (SMC) events. Figures 7a 7d show the IMF B z, the solar wind dynamic pressure, the tail lobe magnetic field measured by Geotail, and the magnetotail/polar cap magnetic flux on 3 February 1998, respectively. Figures 7e 7h are the same as Figures 7a 7d but for the event on 19 February is derived specifically for SMC events, to calculate the area of the magnetotail and then multiply the area by the lobe magnetic field measured by Geotail. We also tried the formula of the tail radius of Shue et al. [1998]. The magnetotail magnetic flux calculated with Shue et al. s [1998] formula is close to that calculated with Shukhtina et al. s [2004] formula when the solar wind pressure is 1 npa or smaller but becomes 10 30% higher when the solar wind pressure is in the range of 2 4 npa. Although we have plotted the magnetotail magnetic flux over the entire day for each case, only the interval with the polar cap magnetic flux measurement should be considered being SMC because this is the interval identified by DeJong et al. [2007]. It can be seen that the polar cap magnetic flux is close to, or smaller than, the magnetotail magnetic flux calculated with Shukhtina et al. s [2004] formula. [29] We have the measurements of the polar cap magnetic flux in 45 SMC events, covering 285 h in total. We averaged the polar cap magnetic flux and corresponding merging electric field over 3 min; there are 5700 data points in the 3-min averaged data in total. Figure 8a shows that the polar cap magnetic flux during SMC events, F PC,SMC, increases with the merging electric field, and the regression line is given by F PC;SMC ¼ 0:50 þ 3: E m : In Figure 8b, the polar cap magnetic flux during SMC events is plotted as a function of the corrected Dst index, and the regression line is F PC;SMC ¼ 0:51 2: Dst 0 : The polar cap magnetic flux increases with the strength of the magnetic storms because the corrected Dst index is generally negative during SMC events. [30] Finally, we compare the polar cap magnetic flux at different magnetospheric modes. Figure 9a shows the polar cap magnetic flux at the sawtooth onsets, at the isolated substorm onsets, and during the SMC events as a function of the merging electric field. In fact, Figure 9a is the combination of Figures 3a, 6b, and 8a. The polar cap magnetic flux at the isolated substorm onsets is located at the upper limit of the magnetic flux for SMC events, and the ð6þ ð7þ 9of13

10 Figure 8. The magnetic flux in the polar cap during SMC events as a function of (a) the merging electric field and (b) the corrected Dst index, respectively. polar cap magnetic flux at the sawtooth onsets is significantly higher than those for the isolated substorms and SMC events. Figure 9b shows the polar cap magnetic flux as a function of the corrected Dst index. The sawtooth events occur during strong magnetic disturbances, and the isolated substorms and SMC events occur during weak magnetic disturbances or quiet times. [31] Figure 9 provides important information on the solar wind condition and the corresponding magnetospheric mode. From the aspect of the solar wind driver, the isolated substorm onset occurs when the merging electric field is smaller than 4 mv m 1, the SMC may occur when the merging electric field is in the range 0 6 mv m 1, and the sawtooth onset occurs when the merging electric field is larger than 2 mv m 1. From the aspect of the magnetospheric mode, the primary magnetospheric dynamic process is the SMC if the magnetic flux in the polar cap is smaller than 0.5 GWb, the isolated substorm if the polar cap magnetic flux is GWb, and the sawtooth event if the polar cap magnetic flux reaches 0.7 GWb or higher, respectively. More magnetic flux (energy) can be stored in the magnetosphere during sawtooth events than during isolated substorms and SMC events. From the aspect of the strength of magnetic activity, sawtooth events occur mostly during moderate and intense magnetic storms with the corrected Dst index of less than 60 nt, and isolated substorms and SMC events occur during weak magnetic storms or during quiet times with the corrected Dst index of greater than 60 nt. In particular, only sawtooth events can exist when the merging electric field is larger than 6 mv m 1 or when the corrected Dst index is less than 60 nt. The observations suggest that sawtooth events are the only Figure 9. Comparison of the polar cap magnetic fluxes at sawtooth onsets, at isolated substorm onsets, and during SMC events. The magnetic flux is plotted as a function of (a) the merging electric field and (b) the corrected Dst index, respectively. 10 of 13

11 magnetospheric mode during strong solar wind driver and during intense magnetic storms. 3. Discussion [32] Magnetospheric substorms (including isolated substorms and sawtooth events) are often described as the energy loading-unloading process [Russell and McPherron, 1973; Hones, 1984; Baker et al., 1996; Nagai et al., 1998], and the onset occurs when the total energy (or magnetic flux) stored in the tail reaches a critical value. The observations presented in this paper support this scenario. As can be seen in Figures 2 and 6a, the magnetic flux in the polar cap/magnetotail increases during the growth phase, reaches a maximum value at the onset, and then decreases during the expansion phase. [33] The maximum content of the polar cap magnetic flux at the isolated substorm onsets presented in this paper varies between 0.55 and 0.85 GWb, with a mean value of 0.68 GWb. Petrinec and Russell [1996] estimated that the magnetic flux in the tail at substorm onsets is between 1.0 and 1.4 GWb. Shukhtina et al. [2005] found that the magnetotail magnetic flux at substorm onsets is between 0.8 and 1.4 GWb. The calculated magnetotail magnetic flux depends on the model of the tail radius and can be different from the polar cap magnetic flux. The following investigators analyzed the open magnetic flux in the polar cap. Milan et al. [2004] found that the maximum content of the polar cap magnetic flux is 0.7 and 0.65 GWb at two isolated substorm onsets. Milan et al. [2009] conducted epoch analysis of the polar cap open flux during substorms and showed that for the substorms with the largest open flux content the average value of the open flux at substorm onsets is 0.8 GWb. The average value of the maximum magnetic flux in the polar cap at the sawtooth onsets is 1.04 GWb in our cases. Hubert et al. [2008] found that the polar cap magnetic flux during a few sawtooth events is GWb. Our findings are consistent with the previous studies. [34] Figures 2 and 6 of this paper show that the total magnetic flux stored in the magnetotail is reduced by 24 26% after the expansion onset for both sawtooth events and isolated substorms. Caan et al. [1978] and McPherron and Hsu [2002] found that there is a reduction of 25 30% in the lobe magnetic field after substorm onsets. The relative decrease of the magnetic flux in our study is similar to the previous result. [35] The data set of the polar cap magnetic flux used in this paper is the same as that used by DeJong et al. [2007]. However, the method of calculating the average flux is different. In the study of DeJong et al. [2007], the maximum value of each flux near onset time (but not always at onset) is first identified, and then the maximum values of all cases are averaged. Similarly, the minimum value of each flux after the onset is identified, and the minimum values of all cases are averaged. In the present study, we first average the fluxes over all cases and then determine the maximum and minimum values of the average flux. Although the same data set for the polar cap magnetic flux is used in the two papers, the derived average values are somewhat different because of the different averaging methods. [36] In the scenario of substorms, the onset occurs when the energy (or magnetic flux) stored in the magnetotail reaches a maximum value. However, it is important to mention that the maximum energy at the onset varies with the solar wind but is not a constant for all substorms. Nakai and Kamide [2003, 2004] and Shukhtina et al. [2004, 2005] have found that the lobe magnetic field at the isolated substorm onset increases with the merging electric field and solar wind pressure. Huang and Cai [2009] have found that the magnetotail total pressure and lobe magnetic field at the sawtooth onset increase with the merging electric field and solar wind pressure. In this study, we have shown that the magnetic flux in the magnetotail and in the polar cap at the sawtooth/substorm onset increases with the merging electric field and with the corrected Dst index. [37] The increase of the maximum magnetotail magnetic flux at the substorm/sawtooth onset with the corrected Dst index is a very important property of the substorm dynamic process. As shown by Nakai and Kamide [2003, 2004], Huang and Cai [2009], and this paper, the magnetotail/polar cap magnetic flux at sawtooth/substorm onset increases with the storm strength (the magnitude of the corrected Dst index). The Dst index is a measure of the storm-time ring current. As discussed by Nakai and Kamide [2003] and by Milan et al. [2008], the enhanced ring current produces a significant B z component in the magnetotail and makes the field lines more dipolar. In the near-earth neutral line (NENL) model of substorms [Russell and McPherron, 1973; Hones, 1984; Baker et al., 1996], magnetic reconnection in the near tail occurs when the B z in the neutral current sheet is very small. The B z induced by the enhanced ring current will impede the formation of the NENL until the lobe magnetic field becomes stronger. The strong southward IMF can cause enhanced convection which has been considered to be a driving force of the ring current [e.g., Kozyra et al., 1998]. If so, the strong solar wind driving will cause significant enhanced ring current and thus the Dst index [Burton et al., 1975; O Brien and McPherron, 2000]. Therefore, the occurrence of the sawtooth/substorm onset requires larger lobe magnetic field/flux when the southward IMF is stronger. [38] A mysterious phenomenon in sawtooth events is the relatively constant (3 h) period of the sawtooth injections (substorm onsets) [Borovsky et al., 1993; Huang et al., 2003a, 2003b, 2004, 2005; Henderson et al., 2006a, 2006b]. J. E. Borovsky (personal communication, 2006) conducted a comprehensive analysis of the sawtooth period and solar wind parameters but did not find a clear trend of how the sawtooth period varies with the solar wind. The relation between the lobe magnetic field/flux at the sawtooth onset and the merging electric field revealed by Huang and Cai [2009] and in this paper can help solve this mystery. The rate of the magnetic flux transfer from the solar wind to the magnetotail is higher for stronger southward IMF. On the other hand, the total magnetic flux stored in the magnetotail must reach a larger value for the sawtooth onset to occur for stronger solar wind driver. The time interval for the magnetotail magnetic flux to reach the maximum value, which is approximately equal to the maximum magnetic flux divided by the flux transfer rate, will be relatively constant and does not significantly depend on the solar wind parameters. In this scenario, the strength 11 of 13

12 of the ring current and its effect on the magnetic reconnection in the NENL play a critical role in determining the period of the sawtooth oscillations. Huang et al. [2003a] suggest that magnetospheric substorms have an intrinsic period of 3 h and that the magnetosphere takes 3 h after a substorm onset to reach the critical state for the next onset. The findings of Huang and Cai [2009] and this paper support the interpretation of Huang et al. [2003a] and provide a reasonable explanation of the period of sawtooth events. 4. Conclusions [39] We have analyzed the magnetic flux in the polar cap and in the magnetotail during sawtooth events, isolated substorms, and SMC events. The polar cap area was measured by the Polar UVI Imager and the IMAGE FUV Imager. The magnetic field in the magnetotail was measured by the Geotail satellite, and the size of the tail was calculated with empirical models. The conclusions derived from this study are as follows: [40] 1. An epoch analysis of the magnetic flux in the polar cap and in the magnetotail during sawtooth events has been conducted. The average magnetic flux starts to increase 78 min prior to the sawtooth onset, reaches a maximum value at the onset time, and then decreases by 70 min to reach a minimum value. The maximum value of the magnetic flux at the sawtooth onset in the magnetotail and in the polar cap is 1 GWb, and the decrease of the average magnetic flux after the sawtooth onset is %. The magnetic flux in the magnetotail and in the polar cap at the sawtooth onset is not a constant but depends on the solar wind parameters; the magnetic flux increases with the merging electric field and with the strength of the ring current. [41] 2. During isolated substorms, the average magnetic flux in the polar cap at the onset is 0.68 GWb, and the minimum value of the average magnetic flux after the onset is 0.50 GWb. The decrease of the average magnetic flux from the maximum value at the onset to the minimum value after the onset is 0.18 GWb (or 26.5%). The magnetic flux in the polar cap at the isolated substorm onset is also not a constant but increases with the merging electric field and with the strength of the ring current. [42] 3. The steady magnetospheric convection, as identified from the polar cap magnetic flux, can last for up to 10 h during southward IMF. The magnetic flux in the polar cap during SMC events varies between 0.3 and 0.8 GWb and increases with the merging electric field and with the strength of the ring current. [43] 4. The magnetospheric mode depends on the solar wind and on the maximum magnetic flux in the polar cap/ magnetotail. Sawtooth events tend to occur at higher merging electrical field. Isolated substorms and SMC tend to occur at lower merging electrical field. In the range of 1 5mVm 1, it is possible to have three modes occurring with different thresholds of magnetic flux in the magnetotail (or polar cap). The polar cap magnetic flux at the isolated substorm onset is at the upper limit of the magnetic flux of SMC events for the same merging electric field. The polar cap magnetic flux at the sawtooth onset is always higher than that at the isolated substorm onset and during SMC events. Sawtooth events appear to be the only magnetospheric mode when the merging electric field is higher than 6mVm 1 and/or when the corrected Dst index is less than 60 nt. [44] 5. An important property of the magnetosphere during sawtooth events and isolated substorms is that the maximum magnetic flux in the magnetotail at the expansion onset is not a constant but depends on the solar wind. The magnetotail magnetic flux must reach a larger value for the onset to occur during stronger solar wind driver. This property provides a reasonable explanation of the relatively constant period of the sawtooth oscillations. [45] Acknowledgments. The work of C.S.H. at Boston College was supported by National Science Foundation award ATM We thank the IMAGE team and the Polar team for providing the image data. We thank the Geotail team for providing the Geotail magnetic field data (PI: T. Nagai) and plasma data (PI: Y. Saito) through DARTS at the Institute of Space and Astronautical Science, JAXA, in Japan. We also thank Los Alamos National Laboratory for providing the energetic plasma flux data measured by geosynchronous satellites and the NASA CDAWeb for providing access to the solar wind data. [46] Wolfgang Baumjohann thanks the reviewers for their assistance in evaluating this paper. References Baker, D. N., T. I. Pulkkinen, V. Angelopoulos, W. Baumjohann, and R. L. McPherron (1996), Neutral line model of substorms: Past results and present view, J. Geophys. Res., 101(A6), 12,975 13,010, doi: / 95JA Bargatze, L. F., D. N. Baker, R. L. McPherron, and E. W. Hones Jr. (1985), Magnetospheric impulse response for many levels of geomagnetic activity, J. Geophys. Res., 90(A7), , doi: / JA090iA07p Baumjohann, W., G. Paschmann, and H. Lühr (1990), Pressure balance between lobe and plasma sheet, Geophys. Res. Lett., 17(1), 45 48, doi: /gl017i001p Birn, J. (1987), Magnetotail equilibrium theory: The general three-dimensional solution, J. Geophys. Res., 92(A10), 11,101 11,108, doi: / JA092iA10p Borovsky, J. E., R. J. Nemzek, and R. D. Belian (1993), The occurrence rate of magnetospheric-substorm onsets: Random and periodic substorms, J. Geophys. Res., 98(A3), , doi: /92ja Burton, R. K., R. L. McPherron, and C. T. Russell (1975), An empirical relationship between interplanetary conditions and Dst, J. Geophys. Res., 80(31), , doi: /ja080i031p Caan, M. N., R. L. McPherron, and C. T. Russell (1978), The statistical magnetic signature of magnetospheric substorms, Planet. Space Sci., 26(3), , doi: / (78) Cai, X., C. R. Clauer, and A. J. Ridley (2006), Statistical analysis of ionospheric potential patterns for isolated substorms and sawtooth events, Ann. Geophys., 24, DeJong, A. D., and C. R. Clauer (2005), Polar UVI images to study steady magnetospheric convection events: Initial results, Geophys. Res. Lett., 32, L24101, doi: /2005gl DeJong, A. D., X. Cai, R. C. Clauer, and J. F. Spann (2007), Aurora and open magnetic flux during isolated substorms, sawteeth, and SMC events, Ann. Geophys., 25, DeJong, A. D., A. J. Ridley, and C. R. Clauer (2008), Balanced reconnection intervals: Four case studies, Ann. Geophys., 26, Henderson, M. G. (2004), The May 2 3, 1986 CDAW-9C interval: A sawtooth event, Geophys. Res. Lett., 31, L11804, doi: / 2004GL Henderson, M. G., G. D. Reeves, R. Skoug, M. T. Thomsen, M. H. Denton, S. B. Mende, T. J. Immel, P. C. Brandt, and H. J. Singer (2006a), Magnetospheric and auroral activity during the 18 April 2002 sawtooth event, J. Geophys. Res., 111, A01S90, doi: /2005ja Henderson, M. G., et al. (2006b), Substorms during the August 2000 sawtooth event, J. Geophys. Res., 111, A06206, doi: / 2005JA Hones, E. W., Jr. (1984), Plasma sheet behavior during substorms, in Magnetic Reconnection in Space and Laboratory Plasmas, Geophys. Monogr. Ser., vol. 30, edited by E. W. Horns Jr., pp , AGU, Washington, D. C. 12 of 13