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2 ARTICLE IN PRESS Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) Abstract Polar cap potential saturation: Observations, theory, and modeling Simon G. Shepherd Thayer School of Engineering, Dartmouth College, 8000 Cummings Hall, Hanover, NH , USA Received 26 April 2006; received in revised form 28 July 2006; accepted 31 July 2006 Available online 14 December 2006 The polar cap potential ðf PC Þ has long been considered an indicator for the amount of energy flowing into and through the magnetosphere ionosphere (M I) system. Studies have shown that F PC reaches an upper limit, or saturates, during geomagnetic storms where the solar wind electric field becomes increasingly large. Numerous observational, theoretical, and modeling studies of F PC have seemingly confirmed that saturation is real, however, many mechanisms for saturation have been suggested with no direct observational evidence confirming any single cause. This paper presents a review of F PC saturation from various perspectives and attempts to identify some of the outstanding issues in determining the causal mechanism of this important piece necessary to understanding the coupling between the solar wind and M I system. r 2006 Published by Elsevier Ltd. Keywords: Polar cap potential; Saturation; Sw M I coupling 1. Introduction The polar cap potential ðf PC Þ is referred to by several different names in the literature including the cross polar cap potential, transpolar potential, polar cap voltage, polar cap potential drop, and many variations of these. As is customary in most studies of F PC, and for the purposes of this review, we will define F PC simply as the difference between the extrema of the ionospheric electric convection potential. That is, F PC F max F min, (1) where F max and F min are the maximum and minimum values of the convection electric potential (F) Tel.: ; fax: address: simon@thayer.dartmouth.edu. in the high-latitude ionosphere. In the collisionless, high-latitude ionosphere, F is given by r? F ¼ ~E ¼ ~V ~B, (2) where ~E is the convection electric field in the rest frame of the Earth due to the flow of ionospheric plasma perpendicular to the magnetic field. Fig. 1 shows contours of the convection electric potential F derived from convection flows measured with the Super Dual Auroral Radar Network (SuperDARN) of ground-based HF backscatter radars located in the Earth s polar regions (Greenwald et al., 1995) (one technique used to determine global convection patterns and F PC ). For standard two-cell convection associated with southward interplanetary magnetic field (IMF), F PC indicates the strength of the dawn dusk electric field or equivalently the intensity and extent of plasma /$ - see front matter r 2006 Published by Elsevier Ltd. doi: /j.jastp

3 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) Fig. 1. Convection electric potential in the high-latitude ionosphere derived from SuperDARN HF radar observations of the circulating plasma. F PC is defined as the difference between the potential extrema labeled þ and. convection in the high-latitude ionosphere. To the degree that magnetic field lines are equipotentials, F PC is a measure of the amount of magnetic flux and plasma flowing through the magnetosphere. It is an instantaneous measure of the strength of convection in the magnetosphere ionosphere (M I) system. Because the process of magnetic reconnection is thought to be primarily responsible for magnetospheric convection, and the energy necessary for convection comes from the solar wind, F PC can be considered to be a measure of the degree to which the solar wind couples to the magnetosphere. In this view a portion of the dawn dusk solar wind electric field (E sw ¼ V x B z, where V is the solar wind speed and B z is the z-component of the IMF) along the reconnection line on the dayside magnetopause maps into the high-latitude ionosphere along equipotential field lines and is measured as F PC. The implication is that F PC can be completely determined from measurements of the upstream solar wind electric field. Early studies using measurements of electric fields or drifting plasma velocities from low-altitude satellites such as AE and S3 focused on determining empirical relationships between F PC and the observed solar wind and IMF (e.g., Reiff et al., 1981; Reiff and Luhmann, 1986; Doyle and Burke, 1983; Wygant et al., 1983). In these studies linear fits of F PC to various expression of the reconnection component of the solar wind electric field were performed. These forms, determined from empirical and theoretical evidence, included the solar wind electric field itself ðv x B z Þ, the so-called Akasofu Perrault parameter ð ¼ VB 2 T sin4 ðy=2þþ (Perrault and Akasofu, 1978), and an expression shown by Sonnerup (1974) to represent an upper limit of the reconnection electric field ðe r ¼ VB T sin 2 ðy=2þþ, among others. The latter is attributed to Petschek (1964, 1966); Nishida and Maezawa (1971) and is sometimes referred to as the Kan Lee electric field (E K2L ) in the literature (Kan and Lee, 1979). In these expressions B T is the transverse magnitude of the IMF ððb 2 y þ B2 z Þ1=2 Þ and y is the transverse IMF

4 236 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) clock angle ðcos 1 ðb z =B T ÞÞ, although in some instances the clock angle appears to be defined using the total IMF ðcos 1 ðb z =BÞÞ. For purely southward IMF, E r ¼ E sw, but because has units inconsistent with that of an electric field it is not possible to directly compare with E r or E sw. One example of these earlier studies by Doyle and Burke (1983) uses S3-2, S3-3, and AE data to show linear fits of F PC derived from these spacecraft data relative to hourly averaged values of E r obtained from the IMP class satellites (King, 1979). Fig. 2 shows the linear empirical relationships obtained by Doyle and Burke (1983) for the data from the three separate spacecraft. The solid curve represents the fit to the S3-2 data and is given by F PC ¼ 33:4 þ 24:0E r kv, (3) where E r is specified in mv m 1. The range of E r for this study was o5mvm 1 with the bulk of the data occurring below 2mVm 1. For reference, a typical value of E r for a nominal solar wind speed of 400 km s 1 and a purely southward IMF of B z ¼ 5nT is 2 mv m 1. During intense geomagnetic storms, however, the solar wind speed can exceed 1000 km s 1 and the IMF can exceed 50 nt in magnitude, corresponding to a much larger E r of 50 mv m 1 for purely southward IMF. During such strongly driven conditions linear empirical relationships, such as Eq. (3), predict values of F PC exceeding 1000 kv; more than four times larger than the maximum F PC ever observed (Hairston et al., 2005). Observations from the most recent solar cycle (23), which incorporate a larger range of E r, together with theory and advances in global magnetospheric modeling, suggest instead that F PC depends nonlinearly on E r and that it has an upper limit. That is, F PC saturates under strongly driven conditions. While there appears to be a strong consensus that F PC saturates, there is still some question as to the value of the saturated potential, and at what value of E sw does F PC begin to deviate from linear behavior. Most importantly, there remains debate as to what the exact processes are that cause F PC to saturate. By many measures F PC is an important parameter associated with ionospheric convection and is often used as a measure of the coupling between to the solar wind and the M I system. Models of the inner magnetosphere and the subauroral ionosphere thermosphere (Fuller-Rowell and Rees, 1980), for instance, use F PC as an input. Under strongly driven conditions the value of F PC predicted with linear models can differ significantly from the saturated value, causing dramatic variations in the model results (e.g., Liemohn et al., 2002). Determining the causes of F PC saturation is, therefore, an important part of understanding the complex coupling that occurs between the solar wind and the M I system. The purpose of this paper is to review F PC saturation from the observational, theoretical, and modeling perspectives and to discuss some of the outstanding issues. 2. Observations The techniques used to measure the value of F PC generally require that the convection electric field or velocity be measured over the entire high-latitude ionosphere. Strictly speaking, however, it is only necessary to obtain measurements in the region between the extrema of the convection potential, which is possible in certain situations using, for instance, ground-based radars (e.g., Shepherd and Ruohoniemi, 2000). In most situations, however, the convecting plasma or electric field is measured over only a portion of the high-latitude ionosphere and F PC is inferred directly from these data or from a fit to these data. The most common techniques used to measure or infer F PC include measuring the plasma drift or electric field along polar trajectories of low-altitude satellites such as Ogo 6, AE, S3, DE 2, or DMSP, which pass near or through the potential extrema Fig. 2. Examples of linear empirical relationships between F PC and E r. (Reprinted from Doyle and Burke (1983), Fig. 8, with permission from the American Geophysical Union.)

5 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) (Heppner, 1972); fitting backscattered line-of-sight ionospheric convection velocities measured with ground-based networks of radars to functional forms of the electrostatic convection potential (Ruohoniemi and Baker, 1998); and the assimilation and mapping of ground and satellite magnetometer data, and radar measurements, such as the assimilative mapping of ionospheric electrodynamics (AMIE) technique (Richmond and Kamide, 1988). Despite the focus on linear relationships between F PC and various forms of E sw, evidence of F PC saturation was noted in several of the early studies. For example, using AE and S3 data and hourly averaged solar wind and IMF data (Reiff et al., 1981) show that a linear trend generally exists between F PC and the Akasofu Perreault parameter. However, three data points at much larger values of clearly do not fit the trend, suggesting saturation of F PC. (Because does not have units consistent with that of an electric field, it is not possible to infer the size of E sw for these data from the figure.) The authors attribute the nonlinear behavior at large values of to a compression and amplification of the magnetic field in the magnetosheath. By correcting for the amplified magnetic field in, the linear trend is restored. Another study using electric field data from the S3-3 satellite and hourly averaged values of a modified version of the Akasofu Perreault parameter referred to as the I model (VB T sin 4 ðy=2þ), show that F PC exhibits a linear trend (albeit with significant scatter) over a range that extends up to 3mVm 1 (Wygant et al., 1983). However, as shown in Fig. 3, when the data are limited to values for which F PC remains steady for several hours prior, F PC appears to saturate at a level around 100 kv for values of the I model that exceed a surprisingly low 0:5mVm 1. According to this study the restriction of considering only intervals which are steady for several hours removes nonlinear ambiguities of the time response of the M I system to changes in the IMF and solar wind from the dependence of F PC on these parameters. It is stated that F PC is in a saturated state almost half of the time based on the data in this study. A summary of the observational studies using AE and S3 data together with hourly averaged solar wind and IMF data is presented by Reiff and Luhmann (1986). The authors state that the asymptotic value of F PC, a term indicating steadiness for a period of several hours, is linear with respect to VB sin n ðy=2þ, where n ¼ 3or4,as long as jbjp10 nt, otherwise a strong saturation effect occurs. This value of E sw corresponds to 4mVm 1 for a nominal solar wind speed of 400 km s 1 and purely southward IMF. It should be noted that only a few data points from these observations appear to exceed a value of 2mVm 1. Fig. 3. F PC apparently saturating to a value of about 100 kv for 40:5mVm 1. (Reprinted from Wygant et al. (1983), Fig. 7, with permission from the American Geophysical Union.)

6 238 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) Reiff and Luhmann (1986) attribute the saturation phenomenon to a mechanism proposed by Hill et al. (1976) (and following Rassbach et al., 1974) whereby F PC is limited in size due to a restriction on the size of the Birkeland currents such that they cannot cause significant changes in the magnetic field at the magnetopause. A second, independent mechanism suggested by Pudovkin et al. (1985) is mentioned, in which the length of the merging line is dependent on the strength of the solar wind and IMF. In another study, Boyle et al. (1997) use hourly averaged solar wind and IMF data from the IMP8 spacecraft and plasma flow data from the DMSP F8 and F9 satellites to determine an empirical relationship for F PC given by F PC ¼ 10 4 V 2 þ 11:7B sin 3 ðy=2þ kv, (4) where V is the solar wind velocity in km s 1, B is the magnitude of the IMF in nanoteslas, and y ¼ cos 1 ðb Z =BÞ GSM. DMSP passes are selected only for times when the IMF are steady for several hours in order to study the asymptotic nature of F PC -similar to Wygant et al. (1983). Although not explicitly stated, it can be inferred from the data that the maximum value of E sw was no bigger than 8:5mVm 1 and that the bulk of the data occurred below 4mVm 1 (see Fig. 4). Boyle et al. (1997) state that there is no evidence of saturation of F PC despite the prediction of Hill et al. (1976). It is not entirely clear why the result of this study contradicts that of Wygant et al. (1983). The range of E sw is sufficiently larger than the 0:5mVm 1 saturation threshold reported by Wygant et al. (1983), and both studies select data which are deemed to be steady for several hours. One difference between the two studies that may help explain the contradictory results is the manner in which steady is defined. Both studies use hourly averaged IMF, but Wygant et al. (1983) use only data for which F PC remains above or below a certain threshold for several hours, whereas Boyle et al. (1997) use only data for which each component of the IMF remains within a relatively coarse range for successive hour-long periods; two very different measures of steady. The other obvious difference is that different instruments on two different spacecraft were used. More recently, Russell et al. (2001) used data from five separate geomagnetic storms during to illustrate saturation of F PC derived Fig. 4. A linear fit to the given empirical relation for F PC in terms of the solar wind and IMF. (Reprinted from Boyle et al. (1997), Fig. 4a, with permission from the American Geophysical Union.) Fig. 5. An arctangent fit to F PC as a function of E sw. (Reprinted from Russell et al. (2001), Fig. 8 lower panel, with permission from the American Geophysical Union.) using the AMIE technique. For this study the AMIE model was run using 80 ground-based magnetometers, 2 4 DMSP satellites, 2 NOAA satellites, and 0 6 SuperDARN HF radars for the various storm periods. While E sw ranged up to 24 mv m 1, F PC never exceeded 200 kv, suggesting saturation. Using arctangent fits to F PC from each storm (shown in Fig. 5) and reanalyzing statistical data from Burke et al. (1999), the authors state that saturation occurs for values of E sw 43mVm 1.It should be noted that there is a great deal of spread

7 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) in the data for any given value of E sw and the level of E sw that corresponds to saturation is challenged in a comment by Liemohn and Ridley (2002). Using strict criteria to select periods of steady E r from the ACE spacecraft, Shepherd et al. (2002) show a clear nonlinear response in F PC derived from 2-min SuperDARN HF backscatter observations during In this study, steady periods of E r were selected in order to minimize uncertainties inherent in propagating observations from the L1 point to the ionosphere. These uncertainties are estimated to be 10 min (e.g., Ridley, 2000) and can lead to large errors in associating the appropriate upstream conditions with the ground observations, particularly during highly variable solar wind conditions. Fig. 6. Nonlinear dependence of F PC derived from 2-min SuperDARN data suggesting saturation of F PC for large values of E r. (Reproduced/modified from Shepherd et al. (2002), Fig. 4b, with permission from the American Geophysical Union.) Fig. 6 shows that the bulk of observations for this study occur for E r below 8mVm 1. Over this range the observed values of F PC deviate substantially from the empirical relationship determined by Boyle et al. (1997). For instance, an upper bound of 100 kv is observed at E r ¼ 5:5mVm 1 with the SuperDARN-derived F PC, while the formula in Eq. (4) predicts a value of 180 kv. While the range of E r does not appear to be sufficient to show the actual saturation value of F PC, the suggestion of saturation is clear. It should be noted that like all observational studies there is a very large spread (40 50 kv) in the values of F PC for a given value of E r, suggesting that some other processes not accounted for in E r influence the value of F PC. Finally, using a comprehensive set of DMSP observations during geomagnetic storms and superstorms from 1998 to 2003 for which E r ranges up to 40 mv m 1, Hairston et al. (2005) show that F PC saturates for large values of E r. The maximum observed F PC from this data set is 265 kv, although it should be noted that 50 kv error bars are not uncommon and a 100 kv spread is observed routinely at all E r. Fig. 7 shows that for values of E r above about 10 mv m 1, the value of F PC is relatively constant and falls within the range of kv, regardless of the value of E r. As is evident from the Hairston et al. (2005) study in which a total of 50 passes were obtained during the period that had sufficiently large values of E r, the conditions for which F PC saturates are somewhat uncommon (apart from the 0:5mVm 1 threshold suggested by Wygant et al., 1983). To gauge the rarity of these conditions, Fig. 7. DMSP-derived F PC from storms and superstorms between 1998 and 2003, showing F PC constant for E r above about 10 mv m 1. (Reproduced/modified from Hairston et al. (2005), Fig. 10, with permission from the American Geophysical Union.)

8 240 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) Fig. 8. Distributions of 5-min averages of E r, shown for solar maximum in black/purple (ACE 2003) and solar minimum in gray/red (WIND 1996), show the relative rarity of E r exceeding 10 mv m 1. E r was computed for a year of data near solar minimum (1996 using WIND) and solar maximum (2003 using ACE). The distribution of 5-min averages of E r is shown in Fig. 8. It can be seen that the number of hours for which E r exceeds 10 mv m 1 is o23 h near solar maximum and o5h near solar minimum. The events become even rarer for times in which E r exceeds 20 mv m 1 ; o8 h and o1 h, respectively. Note that although so-called level 2 data (data that has been verified by the instrument team and is deemed suitable for scientific studies) is used in Fig. 8, it is possible that during some very extreme storms data dropouts from some of the instruments caused the number of actual hours to be under represented. Regardless of the accuracy, the numbers in Fig. 8 are representative for space weather applications relying on these same data. The relative rarity of these extreme events illustrates one of the main difficulties facing observational studies of F PC saturation. While the studies presented here are by no means an exhaustive list, they are representative of the observational studies of F PC and saturation. The most recent studies, as well as some earlier studies, all support the notion that F PC depends nonlinearly on E sw and that for very large values of E sw the value of F PC saturates at some value below 300 kv. The value of E sw for which F PC saturates is given somewhere between 0:5mVm 1 for very steady periods to 10 mv m 1 during geomagnetic storms. 3. Theory Many theories have been suggested to explain saturation of F PC, or the lack thereof. They include, but are not limited to the following list. 1. F PC does not saturate; the observations somehow fail to capture the true value of F PC under strongly driven conditions. The argument for this reasoning stems at least partly from the fact that global magnetospheric MHD models (henceforth referred to simply as MHD models) initially predicted values of F PC 2 3 times larger than those observed (e.g., Raeder et al., 1998; Fedder et al., 1998 (Section 4). As is the case with any observation, there is always the possibility that the measurements are made outside the design range of the instrument. Each of the techniques presented in Section 2 has limitations to its application for accurately measuring F PC. Satellites are limited in their ability to measure the true value of F PC by the fact that they can only measure plasma drifts along the path of their orbit. If the path does not pass through both potential extrema, the observed value of F PC is considered a lower bound of the true value. In addition, polar orbiting satellites typically take min to traverse the polar cap. Any variations in the plasma convection during this transit time result in a nonzero value of (or offset in) F at the end of the pass. The offset can be removed by adding a constant drift along the satellite path

9 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) (Hairston and Heelis, 1990). In either case there is some uncertainty in the measured value of F PC. Keeping in mind that the occurrence of strongly driven conditions (e.g., E sw 420 mv m 1 ) are relatively rare, it is possible that other factors during these extreme events affect the measurements. Reiff et al. (1981) state that for two of the three S3-3 passed used in their study the electric field instrument saturated. The values of F PC for these cases, therefore, represent only a lower bound of the true value. The main limitations to using ground-based radars to measure F PC are the fact that Super- DARN radars require plasma density irregularities in order to detect a backscattered signal and measure the plasma velocity, and the fact that the radar viewing region is limited by the fixed geographic location of the radars. A technique described by Ruohoniemi and Baker (1998) is used to determine a global solution of F using the available backscattered data and a statistical model of F keyed to IMF. In the Shepherd et al. (2002) study, F PC was determined from these fitted patterns. As described in their paper, care was taken to select periods for which sufficient backscatter was present to adequately determine a solution of F. As described by Shepherd and Ruohoniemi (2000), during situations where suitable backscatter is present, the measured value of F PC is determined solely by the observations, and in some situations this can be satisfied with only two radars. In their current configuration, however, the SuperDARN radars are not able to make accurate measurements of F PC during large geomagnetic storms. Under these circumstances, the convection region expands equatorward to such an extent that the radars can no longer view the entire convection region. In this situation the measured value of F PC is an underestimate of the true value. As stated, the practical limit of the Shepherd et al. (2002) study is at about E r ¼ 8mVm 1. New radars are currently being built at mid-latitudes in order to expand the range over which F PC measurements can be made and at high latitudes to fill gaps in coverage in these areas. The AMIE technique is a powerful tool used to ingest or assimilate many types of data from both ground- and space-based instruments and produce estimates of several ionospheric parameters including F PC. Essential to this technique is the ionospheric conductance (the height-integrated conductivity) needed to convert ionospheric currents inferred from magnetometer observations to convection electric fields. It has been suggested that incorrect conductance estimates during these extreme conditions can lead to incorrect predictions of F PC. Russell et al. (2001) suggest this as a possibility in their study, but also point out the unlikeliness given that, in addition to magnetometers, measurements from DMSP and SuperDARN radar were also used as inputs to AMIE to determine F PC. 2. Ionospheric convection (and F PC ) saturates, whereas magnetospheric convection does not; another possibility suggested by Russell et al. (2001) that is based on observations of D st. Such a scenario would require parallel potential drops (exceeding 100 kv in some cases) between the magnetosphere and ionosphere. Observational evidence of such large parallel potential drops are unknown to this author. There are, however, reports of sufficiently large (50 60 kv) parallel potential drops in some studies involving MHD models (e.g., Merkine et al., 2003). It is generally believed that numerical resistivity plays a large role in the parallel potentials observed in these models (Merkin et al., 2005b). It should also be noted that the interpretation of the observations which supports the claim made by Russell et al. (2001), that magnetospheric convection does not saturate during these storms, is also disputed by Liemohn and Ridley (2002), who claim that the evidence shows that magnetospheric convection does saturate in this instance. 3. Mass loading of the magnetosphere. Winglee et al. (2002) use a multifluid global magnetospheric code to demonstrate that as the ratio of O þ to H þ increases in their simulations, the value of F PC decreases. The authors argue that ionospheric outflow of heavy ions ðo þ Þ populates the magnetosphere and provides a sink for the momentum transfer between the solar wind and magnetosphere. It is reasoned that as convection over the polar cap increases so too does outflow due to centrifugal acceleration of ionospheric plasma. The mass added to the magnetosphere acts to retard convection, thereby leading to saturation of convection and F PC. Whether these results apply to realistic outflow conditions is unclear. Verification of this mechanism is not possible with the more abundant single fluid global magnetospheric MHD codes. 4. The size of reconnection line on the dayside magnetopause and post-shock compression of IMF must be accounted for in the formula for F PC. Ridley (2005) suggests that a modification to the

10 242 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) formulation by Boyle et al. (1997), Eq. (4), which accounts for the post bow shock compression of the IMF and the length of the magnetopause reconnection line based on the magnetopause standoff distance removes much of the discrepancy between the predicted and observed values of F PC. In this study, 13 events taken from September 1999 to October 2001, in which E r exceeds 12 mv m 1 for at least some of the period, are used to compare values of F PC determined using the AMIE procedure with ground-based magnetometer input only to the values determined using a modified version of the empirical relation in Eq. (4) given by F PC ¼ð10 4 V 2 þ 11:7Bð1 e Ma=3 Þsin 3 ðy=2þþ r ms ; kv, ð5þ 9 where M a is the solar wind Alfve n Mach number and r ms is the standoff distance to the magnetopause in R E. Ridley (2005) argues that the length of the reconnection line on the dayside magnetopause needs to be included in the correct formulation of F PC. The author uses the factor r ms =9 in Eq. (5) to account for the size of the magnetopause (and thus the reconnection line length) relative to a nominal value. The magnetopause standoff distance r ms is estimated by requiring a pressure balance between the solar wind and the magnetosphere. No seasonal effects or distortions due to large IMF values are included. The nominal value of r ms is taken to be 9R E. A second modification suggested by Ridley (2005) is based on the observation that the solar wind Alfve n Mach number drops significantly during times when F PC is saturated. Following Reiff et al. (1981), it is argued that a better agreement is obtained by using the post-shock (magnetosheath) magnetic field strength in determining F PC. Because the magnetosheath field strength depends on the solar wind Alfve n Mach number, the formulation for F PC must take M a into account. By including the details of the bow shock, Ridley (2005) incorporates the dependence of M a as a decaying exponential function ðe Ma=3 Þ in Eq. (5) which is said to match the form of the ratio of the upstream and downstream magnetic field strength. The conclusion is that F PC, including saturation, can be explained by processes external to the M I system. 5. Region 1 current limited system. What has become known as the Hill Siscoe (H S) formulation of the transpolar potential is based on the idea suggested by Hill et al. (1976), that the strength of the region 1 current system is limited in such a way that it cannot produce significant changes in the magnetic field strength at the low-latitude magnetopause. Using theoretical arguments and modeling results from the ISM magnetospheric MHD code, the H S formulation is given by 57:6E r p 1=3 F PC ¼, (6) p 1=2 þ 0:0125xS P E r where p is the solar wind ram pressure and, following Siscoe et al. (2002), the IMF clock angle dependence of reconnection at the magnetopause has been taken as sin 2 ðy=2þ and x (a factor that depends on the geometry of the currents flowing into the ionosphere) is defined as 4:45 1:08 log S P based on results from ISM runs. As shown by Eq. (6), F PC can be determined solely from the upstream solar wind and IMF (E r and p) and S P, the ionospheric Pedersen conductance. Given the IMF and solar wind, the only free parameter in the H S model, as written in Eq. (6), is then S P. This fact led to several studies comparing observations of F PC with Eq. (6) in order to determine a best-fit S P, which is assumed to be uniform as in Siscoe et al. (2002). Using the data set from Shepherd et al. (2002), a best fit to the H S F PC is achieved by adding a constant offset potential of 17 kv and using a value of 23 S for S P in Eq. (6) (Shepherd et al., 2003). Such a large, uniform ionospheric Pedersen conductance is rather unrealistic, but necessary to minimize the differences between the value of F PC and the H S F PC over a range of E r up to 5mVm 1 for very steady periods. Although not explicitly stated, a value of 10 S for S P produces a reasonable fit between the F PC measured by DMSP and the H S F PC during storms and superstorms, which occur over a larger range of E r (Hairston et al., 2005). This value is perhaps more realistic, but still relatively high for a uniform hemispheric value of S P. In another study using DMSP data, Ober et al. (2003) demonstrate that F PC saturates at (or at least reaches) a higher value during solar minimum compared to the value at a given E r during solar maximum. During solar minimum, it is argued, the value of F 10:7 (10.7-cm solar radio flux) is lower and therefore, the corresponding value of S P is lower, thus leading to a higher saturation value as predicted by Siscoe et al. (2002).

11 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) Following Ober et al. (2003) we look to investigate the dependence of F PC on the ionospheric Pedersen conductance using 2-min SuperDARNderived values of F PC. Here, we use a different technique for identifying suitable periods than the previous SuperDARN studies; periods are selected which have low variance in E r over a 30 min window around the calculated lag time necessary to propagate satellite observations to the ionosphere. In addition, uncertainties are associated with the calculated values of F PC and only those values with low uncertainties are selected. Fig. 9a shows F PC for equinoctial months in 2004 and 2000, near the current solar minimum and maximum, respectively. Although the data set is limited due to the number of suitable periods satisfying the relatively restrictive criteria, a slight a b dependence on ionospheric Pedersen conductance is suggested by the data. Fig. 9b shows that the value of F PC averaged over 1 mv m 1 at each value of E r up to 8mVm 1 is somewhat larger for periods near solar minimum (shown in black/red for 2004) where F 10:7 is lower, than for those near solar maximum (shown in gray/blue for 2000). At best, the dependence is weak according to these data. In order to investigate a possible seasonal conductance dependence of F PC a similar comparison (not shown here) was performed on selected periods during summer and winter months from 1999 to 2003 in the northern hemisphere only. Ideally, a seasonal dependence would be determined by making simultaneous measurements of F PC in the winter and summer hemispheres, however, the spatial coverage of the SuperDARN radars is not yet sufficient in the southern hemisphere to identify periods of high confidence in both hemispheres simultaneously. Despite this fact, the results from the summer/ winter comparison show that in the summer hemisphere (higher conductance) F PC is very slightly larger than the corresponding values of F PC in the winter hemisphere (lower conductance) opposite to what is expected. In both comparisons, the dependence of F PC on conductance is weak. Of greater significance seems to be the relatively large variation that is observed in F PC at all values of E r, suggesting perhaps that ionospheric conductance is only a relatively minor factor in determining F PC. 4. Modeling Global magnetospheric MHD models have been used in many studies of magnetospheric convection and F PC. One clear advantage of using these models to study F PC, in particular, is that they provide a unique global perspective of the magnetosphere, its parameters, and the coupling to the solar wind and to the ionosphere. Also of importance is the fact that the relative rarity of the extreme conditions, which are somewhat of a hindrance in observational studies, is not a limiting factor in studying F PC saturation with these models it is possible to run numerous experiments with extreme solar wind and IMF conditions. Ultimately, however, modeling results must be verified against observations. Several studies using these models report values of F PC that are 2 3 times those observed under similar conditions (Raeder et al., 1998; Merkin et al., 2005a). Fig. 9. (a) F PC measured by SuperDARN at 2-min periods during equinoctial times near solar minimum (black/red) and solar maximum (gray/blue) showing a hint of F PC dependence on ionospheric conductance. (b) one mv m 1 averages (dots), standard deviations (error bars), and ranges (solid bars) of data shown in (a).

12 244 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) Siscoe et al. (2004) provide an excellent summary of recent studies of F PC saturation using MHD models, and states that all the models observe saturation. It is also stated that the various mechanisms put forth to explain F PC saturation are all essentially the result of a limitation of the size of the region 1 currents. In the saturated regime, according to the H S formulation, the magnetosphere acts as a constant current generator with a magnitude determined by the strength of the region 1 current system. In this scenario, the ionospheric Ohm s Law can be written as I 1 ¼ xs P F PC (7) and F PC depends inversely on S P. All single fluid magnetospheric MHD models seemingly show this inverse dependence. As reviewed by Siscoe et al. (2004), recent studies of F PC with MHD models give the following four possible mechanisms as the cause of F PC saturation under strongly driven conditions: 1. Region 1 currents become sufficiently large that they reduce the dipole field strength at the subsolar magnetopause, thus limiting reconnection, magnetospheric convection, and F PC (Hill et al., 1976). 2. Region 1 currents reconfigure, usurp, and eventually replace the Chapman Ferraro currents at the magnetopause and provide the ~J ~B force to hold of solar wind ram pressure (Siscoe et al., 2002). The limitation of the ram pressure to hold ISM off the solar wind thus limits the size of the region 1 currents and F PC in Eq. (7). 3. Magnetic fields due to the large region 1 currents cause a reduction in the magnetic field strength near the subsolar magnetopause, but combine with the dipole field at high latitudes to create shoulders in these regions. The result is the formation of a low-latitude dimple in the magnetosphere which forms a buffer between the magnetosheath plasma and reduces the effectiveness of reconnection (Raeder et al., 2001). Fig. 10 shows magnetospheric dimples formed in three different MHD models under increasingly large southward IMF (after Siscoe et al., 2004). 4. Enhanced ionospheric conductance during storm conditions causes the subsolar magnetosheath to thicken and the magnetopause to flare out on the flanks. The result is that magnetosheath plasma is more effectively slowed and diverted around the flared magnetosphere during these conditions. Plasma streamlines which intersect a given reconnection line on the dayside magnetopause, thus access a smaller portion of the solar wind electric field, thereby reducing F PC. It is argued that F PC saturation is a purely geometrical effect caused by the shape of the magnetopause (Merkine et al., 2003; Merkin et al., 2005b). Fig. 11 shows the magnetospheric flaring resulting from enhanced S P and the associated reduction in F PC. One possible cause for enhanced S P during geomagnetic storms is UCLA /UNH U Mich IMF Bz = -100 IMF Bz = -30 IMF Bz = Fig. 10. Magnetospheric dimples from three different MHD models under increasingly extreme IMF conditions. (Reprinted from Siscoe et al. (2002), Fig. 2, with permission from the American Geophysical Union.)

13 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) Fig. 11. Magnetopause flaring caused by enhanced S P which reduces F PC. (Reprinted from Merkin et al. (2005b), Fig. 4, with permission from the American Geophysical Union.) suggested by Merkin et al. (2005a) as being due to anomalous electron heating resulting from a Farley Buneman instability. 5. Discussion While nearly all studies of F PC seemingly agree that saturation occurs at large values of E r, there is still some disagreement as to value of E r (or E sw )at which the saturation process becomes dominant and, more importantly, the exact mechanism which is responsible for causing saturation. The fairly developed and tested H S formalism predicts an inverse dependence on S P (actually xs P, but according to Siscoe et al., 2002, x varies between 3 and 4). As discussed in Section 4, global magnetospheric single fluid MHD models and some observational evidence support this dependence. However, results shown in Fig. 9 indicate that such

14 246 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) a dependence may not be as important as previously suggested. It is clear, however, that S P plays a critical role in F PC and saturation in the MHD model simulations. As stated by Siscoe et al. (2004) the modeling results seem to support the idea that it is ultimately a limitation of the region 1 currents that is responsible for F PC saturation. According to the H S formulation the magnetosphere acts as a current generator in the saturation regime and F PC depends inversely on S P as given by Ohm s Law in Eq. (7). It is not entirely surprising, therefore, that magnetospheric MHD models show this dependence due to the fact that the field-aligned current system is primarily dominated by region 1 currents in many of these models. Realistic region 2 current systems are not typically represented in these models (e.g., Raeder et al., 1998). In a system dominated by region 1 currents, a reduction in F PC follows directly from Ohm s Law given a constant current (as is the case in the saturated regime according to the H S model). The majority of the field-aligned current, therefore, closes across the polar cap and F PC depends significantly on the value of S P. However, in the case where region 2 currents are also present, an alternate path for region 1 current closure is possible; through the inner magnetosphere. In this case, region 1 currents close at lower latitudes, thereby reducing the current that flows across the polar cap and the value of F PC. The paths across the polar cap and through the inner magnetosphere are in parallel and the value of S P could play little or no role in determining F PC depending on the more complicated coupling situation which includes the region 2 current system. There is some evidence that F PC is reduced in MHD models by including more realistic region 2 currents in the simulations. Using a two-way coupling between the block-adaptive-tree solarwind Roe-type upwind scheme (BATS-R-US) global magnetospheric MHD model and the Rice convection model (RCM) of the inner magnetosphere, Zeeuw et al. (2004) show that F PC is reduced by 8% from the value obtained from the uncoupled BATS-R-US code. Similar coupling between the Lyon Fedder Mobarry (LFM) MHD mode and the RCM shows a significant reduction in the value of F PC compared with the stand-alone LFM model (Toffoletto et al., 2005). The suggestion made here is that the addition of region 2 currents reduces the amount of current closure across the polar cap thereby reducing F PC and thus reducing the importance of S P in determining F PC. Other effects result from the inclusion of more realistic region 2 currents and more detailed studies are necessary to determine the true influence of S P on F PC. Of course the situation is much more complicated than simple wire circuit analogies allow. While progress is being made toward understanding F PC saturation, bringing closure to defining the responsible mechanisms will require comparisons of model predictions to global observations of the electrodynamic and mechanical coupling between the solar wind, magnetosphere, and ionosphere. Such observations, however, are quite rare as shown in Fig. 8. As networks such as SuperDARN, Iridium, ground-based magnetometers, and polar imagers expand and provide more reliable observations, the uncertainty in derived quantities such as F PC will be reduced, hopefully providing a clearer picture of F PC and the processes on which it depends. While it is true that F PC saturation during geomagnetic storms and superstorms is of great interest and it can be argued that such knowledge is critical to fully understanding the coupling between the solar wind and the M I system, the rarity of such events suggests that it is perhaps more fruitful to study this coupling during the more pedestrian solar wind and IMF conditions. As can be seen in every observational study of F PC, there is a great deal of variation in F PC at every value of E r that is yet unaccounted for. It is clear that a complete picture of the complicated coupling processes that give rise to F PC are not yet fully understood, even at modest values of solar wind and IMF. 6. Summary The polar cap potential ðf PC Þ is one of the primary indicators of the strength and the extent of plasma circulation in the ionosphere. It is a measure of the amount of magnetic flux and plasma flowing through the magnetosphere. As such it is used as a measure of the effectiveness of the coupling between the solar wind and the magnetosphere ionosphere system. Early studies of F PC using low-altitude satellites focused on determining linear relationships between F PC and various functional forms of the solar wind and IMF. Many of these studies consisted of data from a very limited range of solar wind electric field (E sw ). More recent studies show that F PC reaches an

15 ARTICLE IN PRESS S.G. Shepherd / Journal of Atmospheric and Solar-Terrestrial Physics 69 (2007) upper limit or saturates for increasingly large values of E sw. Observational studies using satellites, ground-based radars, and assimilative mapping techniques as well as modeling studies with global magnetospheric MHD codes all suggest that F PC saturates during strongly driven situations. There remain, however, some questions as to the exact mechanism which causes F PC saturation and at what value of E sw the mechanism becomes important. Some theoretical and modeling studies suggest that it is a limitation on the size of region 1 currents that is ultimately responsible for F PC saturation (Siscoe et al., 2004). Several mechanisms have been suggested for limiting the region 1 currents during strongly driven conditions, however, no clear, direct observational evidence exists to confirm one mechanism over another. In all these mechanisms the ionospheric Pedersen conductance (S P ) plays a key role in determining F PC in the global magnetospheric models. Some observational evidence supports this dependence, but other evidence (such as that presented here) suggests that the role may be less important. Inclusion of more realistic region 2 currents in the global MHD models by coupling to models of the inner magnetospheric ring current reduces the value of F PC, also suggesting that the role of S P may be less critical than the models dominated by region 1 currents suggest (e.g., Zeeuw et al., 2004). Other factors appear to contribute to F PC, including mass loading of the magnetosphere caused by ionospheric outflow of heavy ions (Winglee et al., 2002), post-shock compression of the magnetic field strength, the size of the magnetopause reconnection line (Ridley, 2005), and other internal or external processes which contribute to the relatively large variation in F PC that is observed. At least part of the difficulty in confirming predictions of the MHD models and other theories with observations stems from the fact that extreme events leading to F PC saturation are somewhat rare (particularly during solar minimum). As ground and space-based instrumentation expands, the number of observations of F PC in the saturated regime should increase. However, it may be that studying F PC during the rather more mundane solar wind and IMF conditions that are much more abundant, is more enlightening and will lead to a more complete understanding of F PC and ultimately the coupling between the solar wind and magnetosphere ionosphere system. Acknowledgments This work was possible by NSF grants ATM and ATM Operation of the Northern Hemisphere SuperDARN radars is supported by the national funding agencies of the US, Canada, the UK, and France. 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