Relationship between interplanetary parameters and the magnetopause reconnection rate quantified from observations of the expanding polar cap

[1] Many studies have attempted to quantify the coupling of energy from the solar wind into the magnetosphere. In this paper we parameterize the dependence of the magnetopause reconnection rate on interplanetary parameters from the OMNI data set. The reconnection rate is measured as the rate of expansion of the polar cap during periods when the nightside reconnection rate is thought to be low, determined from observations by the Imager for Magnetopause-to-Aurora Global Exploration (IMAGE) Far Ultraviolet (FUV) imager. Our fitting suggests that the reconnection rate is determined by the magnetic flux transport in the solar wind across a channel approximately 4 RE in width, with a small correction dependent on the solar wind speed, and a clock angle dependence. The reconnection rate is not found to be significantly dependent on the solar wind density. Comparison of the modeled reconnection rate with SuperDARN measurements of the cross-polar cap potential provides broad support for the magnitude of the predictions. In the course of the paper we discuss the relationship between the dayside reconnection rate and the cross-polar cap potential.


Introduction
[2] The ultimate driver of most dynamical phenomena within the Earth's magnetosphere is magnetic reconnection occurring between the interplanetary magnetic field (IMF) and terrestrial magnetic field lines, as first described by Dungey [1961].When the IMF is directed southwards closed magnetic field lines are converted to an open topology by reconnection.The open field lines are stretched antisunwards by the flow of the solar wind to form the magnetotail lobes.Eventually, reconnection in the neutral sheet of the tail acts to close open magnetic field lines, these newly closed field lines returning to the dayside to complete the cycle.Figure 1, modified from Milan [2009], shows the relationship between the open/closed topology of the magnetosphere and the mapping of open field lines into the polar ionospheres to form the polar caps, the dim regions encircled by the auroral ovals.Of the 8 GWb of magnetic flux associated with the Earth's dipole, that which is open is termed the polar cap flux, F PC .
[3] The rates of dayside and nightside reconnection, F D and F N , are defined as the magnetic flux per unit time converted from a closed topology to an open topology at the magnetopause or from open to closed in the magnetotail, respectively, measured in volts.(Open magnetic flux can be closed at the dayside by dual-lobe reconnection [Imber et al., 2006], but this is a rare occurrence.)The dayside rate is controlled by the ever-changing conditions in the upstream solar wind; the nightside rate is controlled by conditions in the magnetotail.In general, these rates are not instantaneously equal, so the open magnetic flux content of the magnetosphere varies as [e.g., Siscoe and Huang, 1985;Cowley and Lockwood, 1992;Milan et al., 2007, and references therein].Equation ( 1) is a statement of Faraday's law, requiring that the change of magnetic flux within the circuit described by the open/ closed field line boundary (OCB) is equal to the electric field integrated around the boundary, which is decomposed into the voltages associated with reconnection at the dayside and in the magnetotail.As described by Cowley and Lockwood [1992], the waxing and waning of F PC when there is a mismatch of the dayside and nightside reconnection rates leads to expansions or contractions of the polar caps, and an attendant equatorward and poleward motion of the auroral ovals, and these motions have a direct relationship to the excitation of magnetospheric/ionospheric convection.
[4] The "driver" of the magnetospheric system is F D which leads to the accumulation of open magnetic flux in the system and expansion of the polar caps, the substorm growth phase; F N can be thought of as the response of the magnetosphere to this driving, occurring to remove excess open flux, usually associated with substorm expansion phase [Lockwood and Cowley, 1992;Milan et al., 2003Milan et al., , 2007Milan et al., , 2008;;Gordeev et al., 2011].Integrated over a sufficient length of time (long with respect to the duration of the substorm cycle) the release of open flux in the magnetotail must balance its accumulation at the magnetopause, that is, 〈F N 〉 = 〈F D 〉, but on shorter timescales the variation in F PC can be as large as a factor of 4 or more, typically between 2.5 and 12% of the total flux [e.g., Milan et al., 2004].A crucial factor controlling substorm onset is thought to be the inflation of the magnetotail as open flux accumulates and the increase in stress exerted by the solar wind against the flared magnetopause [Milan et al., 2009].In this way, F D controls the variation of F N .Moreover, the time-history of F D and F N determine the large-scale structure of the magnetotail [Milan, 2004a].Clearly, allowing a method of predicting F D from upstream observations of the solar wind and IMF is important for understanding the dynamics and structure of the magnetosphere, and this is the aim of the present paper.
[5] In section 2 we briefly review previous attempts to parameterize solar wind-magnetosphere coupling.Section 3 discusses the relationship between dayside reconnection and the cross-polar cap potential.Section 4 outlines the methodology we use to measure the dayside reconnection rate and parameterize this by the upstream solar wind conditions.Section 5 discusses our results.Finally, section 6 concludes.

Solar Wind-Magnetosphere Energy Coupling
[6] There have been many attempts to parameterize magnetospheric activity in terms of upstream solar wind observations, so-called "coupling functions," both from a theoretical perspective and from more empirical approaches.The most well-known of the coupling functions is the ɛ parameter [Perreault and Akasofu, 1978] (in this form measured in Watts).In this function, V X is the speed of the solar wind, B is the magnitude of the IMF, such that where B X etc. are the GSM components of the IMF, and q is the clock-angle, the angle between the IMF vector projected into the GSM Y-Z plane and the Z axis, i.e., q = tan À1 (B Y , B Z ), and L 0 is an approximate measure of the linear dimension of the cross-section of the magnetopause presented to the solar wind.ɛ was postulated from an energetics argument, and can be thought of as the variation of the Poynting flux propagated through the magnetopause into the magnetosphere: B 2 /m 0 is proportional to the electromagnetic energy density in the solar wind, V X is the rate of transport of this energy toward the magnetosphere, and L 0 2 is an estimate of the cross-sectional area presented by the dayside magnetopause.The sin 4 1 2 q represents the increase of the reconnection-mediated coupling between the solar wind and magnetosphere from zero to a maximum as the IMF rotates from a parallel to antiparallel alignment with the northward-pointing magnetopause field.
[7] There have been many studies that use ɛ as an estimate of the energy input to the magnetosphere, to then subsequently determine the "partitioning" of this energy between different energy sinks, such as auroral precipitation power, Joule heating of the ionosphere, energization of the ring current, etc. [e.g., Akasofu, 1980;Guo et al., 2011].Many of these sinks are difficult to measure from first principles and instead geomagnetic indices, such as the auroral electrojet indices AE, AU and AL, the ring current indices SYM-H or D ST , or even planetary indices such as A m or K P are used as proxies [e.g., Scurry and Russell, 1991].Several statistical studies have correlated geomagnetic indices with upstream solar wind conditions to determine if a functional form can be found that improves on the ɛ parameter.Newell et al. [2007] provide an excellent introduction into the coupling functions that have been proposed in the past, and themselves propose a new coupling function, derived by correlation with 10 different measures of magnetospheric activity: Newell et al. [2007] describe dF MP /dt as the "rate magnetic flux is opened at the magnetopause," which makes it synonymous with our F D (equation ( 1)).We will return to discuss this function below.
[8] In the present study we parameterize the dayside coupling rate using a direct measure of the reconnection rate F D .It is not clear that correlation with geomagnetic indices can provide an adequate measure of F D ; certainly, geomagnetic indices tend to measure the response of the magnetosphere to previously accumulated open magnetic flux, that is, will respond more to increases in F N than F D .As mentioned above, on long timescales 〈F N 〉 = 〈F D 〉, and so correlative studies often resort to some degree of averaging, sometimes as long as several hours, to demonstrate a correlation between magnetospheric activity and interplanetary parameters.In this study we employ measurements of the expanding polar cap to quantify F D , and consequently minimal averaging is necessary.

Solar Wind-Magnetosphere Reconnection Rate Proxies
[9] In addition to studies which determine a coupling efficiency by correlating the interplanetary parameters with geomagnetic indices, other workers have recognized that F D , as a measure of magnetic flux transfer, should be directly related to phenomena such as magnetopause erosion [e.g., Holzer and Slavin, 1978] or the excitation of ionospheric convection [e.g., Reiff et al., 1981Reiff et al., , 1985]].Of these two effects, ionospheric convection is more straightforward to observe on a regular basis, and studies have used the cross-polar cap potential, or transpolar voltage, usually denoted as F PC , as a measure of this [e.g., Reiff et al., 1981Reiff et al., , 1985]].When undertaking correlative studies involving magnetic flux transfer phenomena it is necessary that the upstream coupling function has units of voltage, that is the solar wind motional electric field multiplied by a length scale, and a factor to account for the clock angle dependence of reconnection.
[10] The electric field represents the transport of magnetic flux per unit length transverse to the flow direction, given by V X B YZ , where B YZ 2 = B Y 2 + B Z 2 is the transverse component of the IMF.Three forms for the variation of the "reconnection electric field" (i.e., magnetic flux transport and clock angle dependence combined) that have been commonly used are [e.g., Burton et al., 1975;Holzer andSlavin, 1978, 1979], where B S is the southward component of the IMF (B S = |B Z | for B Z < 0, B S = 0 otherwise), [ Sonnerup, 1974;Kan and Lee., 1979], and [ Wygant et al., 1983].
[11] These reconnection electric fields are then converted to a reconnection voltage by multiplying by an effective length scale, L eff , which can be thought of as the width of the channel of magnetic flux in the upstream solar wind impinging on the bow shock that eventually reconnects at the magnetopause.Reiff et al. [1981] estimated that L eff was roughly equal to 0.1 to 0.2 of the width of the magnetosphere.More recently, Milan [2004aMilan [ , 2004b] ] argued that, based on equation (4), provides a reasonable parameterization of Milan et al. [2008] subsequently included an improved representation of the clock angle dependence of F D , suggesting, based on equation ( 5), that with L eff ≈ 2.75 R E .
[12] Of course, L eff may itself be dependent on the upstream conditions.For instance, it is possible to suppose that a high solar wind dynamic pressure could modify the propagation of the solar wind within the magnetosheath, changing the access of the IMF to the magnetopause, or changing the efficiency of reconnection by modifying the Alfvén velocity at the magnetopause.As mentioned above, Newell et al. [2007] proposed equation (3) which is a modification of equations ( 5) and ( 6), such that where B MP is the magnetic field strength at the magnetopause, which is itself proportional to V X due to the conditions of magnetopause pressure balance.This correction attempts to account for changes in reconnection efficiency depending on the relative strengths of the magnetic field on either side of the magnetopause.There is also a modification of the shear angle dependence of the reconnection efficiency, having an exponent somewhere between the 2 and 4 of equations ( 5) and (6).However, although Newell et al. [2007] define dF MP /dt as to be equivalent to our F D (see above), equation (3) does not have a normalizing factor and hence cannot give a predicted reconnection rate.

Measuring Dayside Reconnection Rate
3.1.The Relationship Between F D and F PC [13] As mentioned in the previous section, some studies have used the cross-polar cap potential, F PC , as a proxy for the dayside reconnection rate [e.g., Reiff et al., 1981Reiff et al., , 1985;;Grocott et al., 2009].There are two difficulties with this approach.The first difficulty is that a viscous interaction between the solar wind and the magnetosphere [e.g., Axford and Hines, 1961], if such a process is believed to occur, can excite convection without the occurrence of dayside reconnection.In some studies, once the variation in F PC associated with changes in the IMF have been accounted for, there remains a base-level of order 25 kV which is attributed to such a viscous interaction.Coupling functions which involve the solar wind number density or dynamic pressure are then invoked to parameterize this contribution [e.g., Newell et al., 2008].Lobe reconnection, occurring when IMF B Z > 0, can also excite ionospheric convection, independently of the occurrence of the low latitude magnetopause reconnection measured by F D .The second difficulty is that, irrespective of possible contributions from lobe reconnection or viscous processes, the relationship between F D and F PC is complicated, and we discuss this relationship in the rest of this section.
[14] Studies which correlate the cross-polar cap potential, F PC , with upstream conditions to determine the reconnection rate coupling function implicitly assume that F PC is directly controlled by F D .However, F PC is a measure of the antisunwards transport of open magnetic flux as it is created on the dayside and destroyed on the nightside.It is traditionally measured in two ways: (1) by spacecraft which traverse the polar cap approximately along the dawn-dusk meridian and integrate the measured electric field in the antisunwards plasma drift region [e.g., Reiff et al., 1981]; and (2) by radars (e.g., SuperDARN [see Chisham et al., 2007]) or magnetometers [e.g., Reiff et al., 1985] which determine the global pattern of plasma drift and define F PC as the difference between the maximum and minimum in the corresponding electrostatic potential pattern.To avoid confusion, we refer to these two measures as the dawn-dusk voltage F DD and the delta-potential DF, respectively.Milan [2004b] pointed out that in general these two estimates of F PC will not be equal and will not necessarily be instantaneously representative of changes in open magnetic flux in the magnetosphere.
[15] Figure 2 investigates the relationship between F D , F DD , and DF.  , 1985]), that is, where flow streamlines cross the OCB, the drift speed and OCB speed are equal.In our model, reconnection is assumed to occur at a uniform rate along a merging gap of specified width.In reality, the reconnection rate may be lower at the edges of the merging gap and extend to earlier and later local times.
[16] From these convection plots we can determine the cross-polar cap potential, both F DD and DF, and the variations of these are shown in Figure 2f also.F DD is straightforward to calculate as Lockwood [1991] reasoned that if the expanding/contracting polar cap remains circular, then the rate of magnetic flux transport across the dawn-dusk meridian is in Figure 2a, a spacecraft traversing the dawn-dusk meridian would measure a voltage difference of 25 kV between the edges of the polar cap.As discussed previously, averaged over a sufficient period 〈F N 〉 = 〈F D 〉 and hence 〈F DD 〉 represents the long-term average of the dayside reconnection rate.DF is different: as can be seen from Figures 2a-2d, the focii of the convection cells (the maximum and minimum of the electrostatic potential) form at the ends of the mostactive merging gap, and in general DF ≥ F DD .For example, in Figure 2a the voltage difference between the ends of the dayside merging gap is 39 kV.Only if F D = F N are DF and F DD equal.
[17] The exact value of DF depends on the length of the merging gap.If we consider the case in which F D > F N then DF varies with the angular width of the dayside merging gap, q D : DF → F D as q D → 0, and DF → F DD as q D → p.For the situation presented in Figure 2, in which we have used q D = q N = p/6, averaged over the duration of the modeled substorm 〈DF〉 is 140% of the value of 〈F DD 〉.Hence, we find that in general F DD and DF are not equal to each other and are not instantaneously equal to the dayside reconnection rate.Indeed, Milan [2004b] calculated that F DD can have an average of $25 kV during periods of northward IMF due to the decoupling of the cross-polar cap potential from the dayside driving by on-going nightside reconnection, and that this may account for the residual voltage often attributed to viscous processes.

Measuring F D From Changes in Polar Cap Size
[18] Two methods allow F D to be measured almost directly and instantaneously: (1) measure the ionospheric convection flow across the ionospheric projection of the dayside OCB (in the frame of reference of the moving boundary) [e.g., Baker et al., 1997;Milan et al., 2003;Hubert et al., 2006a;Chisham et al., 2008] ; (2) observe the rate of expansion of the polar cap assuming that no reconnection is occurring on the nightside (F N = 0), in which case, from equation (1), In principle, the first method works at all times, but relies on a reliable method of measuring the location of the OCB and simultaneous good measurements of the ionospheric convection flow.The second method requires only a measurement of the location of the OCB at all local times, from which F PC can be deduced, during periods of low nightside activity.Both methods have the advantage that they are not affected by any contributions from a viscous interaction.
[19] In this study we use the second method.This has previously been attempted by Lewis et al. [1998] and Yeoman et al. [2002], but those studies suffered from poor estimates of the size of the polar cap due to non-global observations of the OCB.In this study we determine the location of the OCB at all local times from observations of the poleward boundary of the auroral oval observed in global auroral imaging [e.g., Milan et al., 2003Milan et al., , 2007Milan et al., , 2008;;Boakes et al., 2008].The temporal resolution of the method depends on the length of time over which observations are necessary to accurately determine the rate of change of F PC .The method has the disadvantages that there is currently no reliable automated method to measure F PC from auroral images and that it is limited to periods when there is confidence that F N ≈ 0; both of these mean that it is not currently possible to perform a particularly large statistical analysis.

Methodology and Observations
[20] We employ observations of the auroras by the Far Ultraviolet (FUV) instrument [Mende et al., 2000a[Mende et al., , 2000b] ] onboard the Imager for Magnetopause-to-Aurora Exploration (IMAGE) spacecraft to estimate the location of the polar cap, and hence determine F PC , as described by Milan et al. [2003] and Boakes et al. [2008].We have selected intervals based on the following criteria.(1) Intervals begin with a prolonged period of northward IMF with low nightside activity as observed in the auroral images and the AU/AL indices, such that the polar cap remains of uniform size.
(2) There follows a southward rotation of the IMF vector, though nightside activity remains low, and the polar cap expands.(3) The interval ends once significant nightside activity commences, indicating the magnetospheric response (usually a substorm) to the preceding growth phase.The absence of nightside activity in criterion 2 is determined from two indictors: (a) a lack of significant auroral activity, specifically a substorm-type auroral expansion; (b) enhancements in AU and AL occur in roughly equal measure, indicating the enhancement of both eastward and westward electrojets in response to the southward turning of the IMF and attendant dayside-driven convection, but there is no significant substorm negative bay in AL.In these circumstances we assume that F N ≈ 0 during the growth phase, and the rate of expansion of the polar cap is a measure of the dayside reconnection rate (equation ( 11)).It was also necessary that suitable measurements of the interplanetary parameters are available for each interval.We have employed the OMNI data set of solar wind parameters [e.g., King and Papitashvili, 2005] (available from NASA's Space Physics Data Facility, http://omniweb.gsfc.nasa.gov).
[21] During the period May 2000 to April 2002 when suitable IMAGE/FUV observations were available, 26 intervals matched our selection criteria, totaling 49.2 h of observations.Table 1 summarizes these events, giving the date of the start of each interval and the UT of the start, t 1 , and end, t 2 , of the interval (if t 2 > 24 the end occurred on the following day).For some intervals there was an obvious discrepancy between the predicted time of arrival of solar wind features at the magnetopause in the OMNI data set and their manifestation in our observations (for instance solar wind shocks and the corresponding positive excursion in the SYM-H index), in which case we have included a correction, Dt, indicated in Table 1.
[22] Figure 3 presents two of these events, typical of the event list.Figures 3a-3d and Figures 3l-3o show noonmidnight and dawn-dusk keograms of auroral emission observed by the Wideband Imaging Camera (WIC), predominantly electron auroras, and Spectrographic Imager (SI12), sensitive to Doppler-shifted Lyman-a emission associated with proton precipitation.Dayglow obscures the noon sector auroral emissions in the WIC observations.Otherwise, the locations of the auroral oval are apparent.Figures 3e and 3p show the variation of F PC determined from the auroral images at a cadence of 123 s, using the poleward boundary of the auroral oval as a proxy for the OCB as described by Milan et al. [2003] and Boakes et al. [2008].Figures 3f-3i and Figures 3q-3t are the B Y and B Z components of the IMF, the solar wind speed V X and the solar wind number density, N. Figures 3j and 3k and Figures 3u and 3v show the AU and AL auroral electrojet indices and the SYM-H ring current index.In each example, IMF B Z > 0 at the beginning of the interval and F PC is approximately constant with time.There follows a southwards turning of the IMF, F PC grows and the auroras move to lower latitudes; the AU and AL indices increase in magnitude, but relatively symmetrically indicating enhancement of convection, but not the formation of a substorm electrojet.Eventually a substorm onset occurs, observed as an enhancement of the emission intensity in the nightside auroras and a negative excursion of AL.The vertical dashed lines indicate the times of substorm onsets identified by Frey et al. [2004].After substorm onset, F PC ceases growth, and even decreases if the nightside reconnection rate exceeds the dayside rate.We include in our analyses the periods encompassing the growth in F PC , up to the time of substorm onset.
[23] We note that in the second example of Figure 3, from 5 February 2002, that there is some indication of an auroral enhancement 10 min prior to the substorm, and an AL bay even 10 min before this.This suggests that there may be  some contamination by nightside reconnection during this or other intervals but we have tried to keep this to a minimum.We also note that a transpolar arc is present in the auroral observations from this interval: it can be seen most clearly in the dawn-dusk keograms, forming near 08:30 UT and eventually fading near 12:30 UT.Transpolar arcs are not uncommon in our observations, as one of our selection criteria was that intervals should begin with a prolonged period of northward IMF.As reported by Milan et al. [2005] and Goudarzi et al. [2008] (who studied this particular event in detail), the presence of a transpolar arc does not affect the expansion of the polar cap once the IMF turns southwards; we continue to use the poleward edge of the main auroral oval as a proxy for the OCB and our analysis proceeds as before.
[24] We endeavored to include intervals which had a wide range of values of B Y , B Z , V X and N to explore fully the parameter space.Figure 4 shows occurrence distributions of the values of IMF and solar wind parameters during the intervals employed in this study, showing good representation of all clock-angles and values of B YZ up to 12 nT, and combinations of V X and N that give a solar wind dynamic pressure up to 12 nPa.
[25] The aim of the study is to determine a functional form of the solar wind parameters that reproduces the observed expansion of the polar cap.We assume that this parameterization has the form where L, a, b, g, and d are to be determined by fitting to the observations.L is a constant of proportionality, with dimensions dependent on the values of the exponents a, b, and g, such that F D has units of voltage; if a = 0 and b = g = 1 then L has units of length, in which case we refer to it as L eff .
[26] Combinations of the parameters a, b, g, and d are tested to determine the best fit to the observations.For each combination, the analysis proceeds as follows.For each interval in Table 1 we integrate F D (equation ( 12)) to produce a model variation in polar cap flux F′ PC (t) running from time t 1 to t 2 : There will be an arbitrary offset between the observed time series F PC (t) and the model F′ PC (t) which is the open flux content of the magnetosphere at the start of the interval, i.e., F PC (t 1 ), which depends on the pre-history of solar windmagnetosphere coupling and nightside reconnection.To simplify comparison, for each interval we calculate the time series F PC (t) À 〈F PC (t)〉 and F′ PC (t) À 〈F′ PC (t)〉, which we call DF PC (t) and DF′ PC (t): that is, we are interested in the variation of F PC as reconnection proceeds, not its absolute value.We then pool the observed and modeled data from all 26 intervals and compute the Pearson correlation coefficient, r, between them to determine the goodness-of-fit.We vary a, b, g, and d to find the combination that maximizes r.As r is a measure of the linear correlation between variables, it is independent of the choice of L; L can be found as the coefficient of proportionality between F PC and F′ PC once the values of a, b, g, and d which maximize r have been determined.For illustration, Figure 5 shows the fits for two example choices of a, b, g, and d, comprising 26 superimposed curves of DF PC (t) and DF′ PC (t).Clearly the data are ordered better in Figure 5b than Figure 5a, the quality of the fit being quantified by the value of r.Maximizing r is similar to minimizing the mean square deviation between the observed and model variations in polar cap flux.
[27] We performed the fitting in two steps.First, a, b, g, and d were varied over the ranges À2 < a, b, g < 2 and 1 < d < 6 to determine that the maximization of r was wellbehaved, that is, there was a single maximum, which we found to be located near a = 0, b = 1.2, g = 1, d = 4.To demonstrate this, Figure 6 shows two slices through the parameter space.Figure 6a shows the a-b plane in which is contoured the maximum r in the other two dimensions, that is the g-d plane is effectively flattened; Figure 6b shows the equivalent plot for the g-d plane.The second maximization method involved an iterative gradient-following algorithm which stepped the parameters from an initial guess such that they progressed toward constantly increasing r. Figure 7 shows the increase in r as the parameters meandered from an arbitrary starting point toward the optimum fit.The values settle on a = 0, b = 4/3, g = 1, and d = 9/2, at which r = 0.972.With these exponents, the best fit value of L is 3.3 Â 10 5 m 2/3 s 1/3 .Our function is, then, If this is expressed in the form of the solar wind electric field applied over an effective length modulated by the solar wind velocity (see equation ( 8)), then it can be written as To demonstrate the overall fit of this function to the F PC data, Figure 8 presents a comparison with the model variation of F′ PC determined from equations ( 13) and ( 15) for each of the 26 intervals identified in Table 1.The black curves indicate the observed variation in F PC , the red curves show the predicted dayside reconnection voltage F D , and the dashed blue curves are F′ PC .In general we find a  reasonable agreement between the model predictions and observations, with values of F D varying between 0 and 200 kV for the range of interplanetary parameters shown in Figure 5.
[28] For completeness, we redid the above analysis substituting total IMF strength B for B YZ in equation ( 12) (see equation ( 2)).The best fit exponents were not significantly different from those given previously.However, r maximized at a value of only 0.945.The exponents corresponding to equation (2) give r = 0.899.We conclude that B YZ is the appropriate IMF component to use in the parameterization of F D .

Assumed Negligibility of Nightside Reconnection
[29] Our analysis proceeds on the assumption that nightside reconnection can be discounted in equation ( 1) for nonsubstorm intervals, such that we can use equation ( 11) to measure the dayside reconnection rate.Here we examine the validity of this assumption; specifically we examine the possibility that there is a low level of nightside reconnection and that it is proportional to the amount of open magnetic flux present in the magnetosphere, F PC .If this was indeed found to be the case, we would interpret it as reconnection occurring at a distant reconnection X-line, which would then be superseded by the formation of a near-Earth X-line at substorm onset.In this case, during non-substorm times, equation (1) would become where c would be a constant to be determined.This would have two major repercussions: (1) the polar cap would contract when there is no dayside reconnection, or if the polar cap remained of uniform size, even when the IMF was strongly northward, this would imply ongoing dayside reconnection at a rate F D that balanced the nightside reconnection rate of cF PC ; and (2) for a given dayside reconnection rate F D the growing polar cap flux would asymptote to a maximum value of though such asymptotic behavior is not observed.In addition, if such a nightside reconnection rate was present, but not taken into account in our analysis, our predicted reconnection rate would, on average, systematically overestimate the observed rate of polar cap expansion at low values of F PC and under-estimate it at high values of F PC , such that the relationship between DF PC (t) and DF′ PC (t) in Figure 5 would be nonlinear.However, there is no clear evidence for this in the figure.
[30] We are led to the conclusion that c, if nonzero, must be small.For all our events, at the start of each interval when the IMF is northward and F D is very close to zero, we see a polar cap of uniform size, not systematically contracting as would be suggested by equation ( 16).This is most apparent in events in Figures 8d, 8g, 8m, 8o, 8p, 8r, and 8z.We estimate that this puts an upper bound of 10 kV on the nightside reconnection rate, such that for F PC = 0.6 GW (e.g., events in Figures 8e, 8m, and 8z), c < 1.7 Â 10 À5 V Wb À1 .
[31] By the same argument, when the reconnection rate has been high such that F PC has grown close to F max (equation ( 17)), a sudden decrease in F D should be accompanied by a contraction of the polar cap.That no significant decreases in F PC are observed, except decreases associated with the substorms that mark the end of each interval, suggests again that c must be negligible for our purposes.

Comparison With Previous Coupling Parameters
[32] Correlating the expansion of the polar cap during substorm growth phases with interplanetary parameters shows that the dayside reconnection rate can be parameterized as the rate of magnetic flux transport in the solar wind, with a sin 9=2 1 2 q clock angle dependence and a small solar wind velocity correction, where the width of the solar wind channel that reconnects at the magnetopause is of order 4R E (see equation ( 15)).To examine the performance of this F D with respect to other coupling functions that have been proposed in the past, we use that same method described in section 4 to find the Pearson correlation coefficient and constant of proportionality (where appropriate) for a representative sample of coupling functions, presented in Table 2.For instance, the fits presented in Figures 5a and 5b are those corresponding to the exponents of equation ( 8) in Table 2, the Scurry and Russell [1991] coupling function (see below), with r = 0.885, and our best fit exponents (equation (1) in Table 2), r = 0.972.The functions have been tabulated in order of goodness of fit.Our new function equation (1) in Table 2 performs best.In second and third places are the Wygant et al. [1983] (equation (2) in Table 2) and Burton et al. [1975] (equation (3) in Table 2) functions (modified to give a reconnection voltage, not electric field).That these perform better than equation (4) in Table 2 proposed by Milan et al. [2008] indicates that a sin 2 1 2 q functional form Figure 7.The variation of a, b, g, and d from an arbitrary starting point, as an iterative, gradient-following algorithm is used to maximize the Pearson correlation coefficient r.
overestimates the reconnection rate for B Z > 0 nT.Newell et al. [2007] also recognized that equation (2) in Table 2 was a successful coupling function for many magnetospheric state variables.Our function only marginally outperforms equation (2) in Table 2; the improvement comes from the additional V X 1/3 term which leads to an increase in reconnection efficiency by 25% (from L eff = 3.6R E to 4.6R E ) as V X varies from 350 to 700 km s À1 .
[33] Many previous studies have determined coupling functions based on observations that are averaged over several hours.Our study correlates observations made at a 2 min cadence, and so study the prompt response of the Figure 8.The observed variation in F PC (black curve) for each of the 26 intervals studied, along with predicted F D (red curve) and the model variation F′ PC (dashed blue curve) for the best fit combination of a, b, g, and d. magnetosphere to reconnection.That we integrate F D before correlation with F PC introduces some level of temporal averaging, but we estimate that this is effectively of the order of 15 min.
[34] We conclude that care must be taken when inferring dayside reconnection rates from magnetospheric state variables.It is important that the duration over which averaging is applied to the observations is not too great, such that the prompt response of the magnetosphere can be studied.The choice of state variables employed in the correlation is important also.Newell et al. [2007] included F PC and the latitude of the cusp (which should be related to each other) in their analysis.However, as discussed in section 3.2, the reconnection rate is expected to be associated with changes in F PC , not its instantaneous value.Other studies which correlate with, for instance, F PC inevitably fold in contributions from any viscous interactions and the nightside reconnection rate.

Lack of Solar Wind Density Dependence
[35] Several previous studies have suggested coupling functions that depend on solar wind density, N, or pressure, p, for instance Scurry and Russell [1991]: (see also equation ( 8) in Table 2) or Vasyliunas et al. [1982]: (see also equation ( 6) in Table 2).Our study clearly suggests that the reconnection rate is not dependent on N, as any coupling function which involves N performs badly.One can ask why previous studies have found a dependence on N, and two possibilities present themselves.First, viscous processes could contribute to some of the measures of magnetospheric activity that have been used to determine previous coupling functions.Second, it is known that sudden enhancements in N, associated with solar wind shocks, can excite magnetotail reconnection and produce significant enhancements of auroral precipitation power [e.g., Boudouridis et al., 2003;Milan et al., 2004;Hubert et al., 2006b].However, this results from the sudden closure of magnetic flux which was opened prior to the shock, and the precipitation of particles trapped in the magnetosphere by solar wind coupling prior to the shock, and is not the result of an enhancement of dayside reconnection associated with elevated solar wind density.Hence, this mistaken relation may be caused by the use of the wrong proxy for reconnection rate.
[36] It has also been speculated that enhanced plasma densities at the dayside magnetopause, for instance fed from the inner magnetosphere via plasmaspheric drainage plumes [Borovsky and Denton, 2006], can choke the reconnection rate.Our results suggest that if such a mechanism does operate, it must do so at densities significantly higher than those supplied by the solar wind.

Comparison With SuperDARN Cross-Polar Cap Potential
[37] We sought an independent means of verifying the reconnection rates provided by equation ( 15).Despite the caveats expressed in section 3.1, we have compared SuperDARN [Chisham et al., 2007] cross-polar cap potential measurements, DF SD , with F D for the period 20-30 October 2001, a period previously studied by Milan [2009] and Milan et al. [2009].The results are presented in Figure 9a: the black and gray dots show the SuperDARN measurements and the red curve is our predicted F D .We find that there is a good general agreement between both traces, as long as a $25 kV offset is added to F D , which has been included in Figure 9a.
[38] The DF SD used here is a standard SuperDARN data product.SuperDARN expresses the electrostatic potential pattern as a spherical harmonic expansion, the coefficients of which are constrained by radar observations of plasma drift.To constrain the pattern in regions devoid of radar returns, the fit is seeded with an average convection pattern appropriate for the current interplanetary parameters [e.g., Ruohoniemi and Greenwald, 1996].The fitted convection pattern will be more accurate where significant numbers of radar observations have contributed to the fit, hence black dots in Figure 9a indicate measurements of DF SD with more than 400 inputs.There are three intervals when F D grossly over-estimates DF SD , on 21, 22, and 28 October.Milan et al. [2009] showed that these periods correspond to geomagnetic storm conditions, and it is known that SuperDARN underestimates the true convection potential at such times due to a reduction in backscatter occurrence and the expansion of the convection pattern to low latitudes.
[39] During non-storm periods, the correspondence is good, as demonstrated in Figure 9b which presents a scatterplot of DF SD (black points from Figure 9a) and F D , in  Scurry and Russell [1991] this case with no offset.The dashed line shows the line of equality.The SuperDARN measurements saturate above F D ≈ 75 kV for the reasons discussed above.Below F D ≈ 75 kV there is a linear relationship between the two parameters, with a constant of proportionality slightly above 1 (the red line superimposed on Figure 9b has a gradient of 1.4, corresponding to the relationship between F D and DF discussed in section 3.1).We do not expect an instantaneous correspondence between F D and DF, as discussed in section 3.1, but the relationship we find suggests that averaged over a sufficient length of time, the reconnection rate provided by equation ( 15) matches the magnetic flux transport measured in the convection pattern.
[40] There is, however, an offset of 25 kV.This could measure a contribution from a viscous interaction to the ionospheric convection.However, we feel that this is more likely due to the contribution of the model convection pattern in the SuperDARN fitting analysis.The model convection patterns will contain contributions from periods when nightside reconnection is active, which Milan [2004b] estimated would add on average $25 kV to the cross-polar cap potential even during periods when dayside reconnection is inactive.

Conclusion
[41] We have compared rates of expansion of the polar cap with upstream interplanetary parameters to provide an expression for dayside reconnection rate, equation (15).We find, contrary to several previous studies, that the reconnection rate does not depend significantly on the solar wind number density.The results are in general agreement with measurements of the cross-polar cap potential made by SuperDARN.
[42] The present study investigates the reconnection rate for a relatively small sample of data, and for ordinary interplanetary conditions.It is important to extend the parameter space over which observations are made to investigate the reconnection rate during extreme events, for instance to determine if polar cap saturation effects [e.g., Siscoe et al., 2002] are associated with changes in the dayside reconnection process.

Appendix A
[43] Here we present details of the expanding/contracting polar cap model used to produce Figure 2.This follows in large part in the procedure outlined by Freeman [2003], modified to include a nightside merging gap.The model is used to calculate the electrostatic potential pattern F(q, l) associated with ionospheric convection as the polar cap expands and contracts due to dayside and nightside reconnection, where q and l are azimuth and colatitude, respectively.The open flux content of the polar cap is F PC and changes in F PC occur due to dayside and nightside reconnection rates, F D and F N , according to For simplicity we assume a dipolar magnetic field with equatorial field strength B eq = 31,000 nT and we discount the differences in magnetic field strength between the Earth's surface and in the ionosphere (the auroral emissions of Figure 3 originate at an altitude of approximately 150 km), such that the radial component of the magnetic field at colatitude l is given by We assume that the polar cap is circular and centered on the geomagnetic pole, such that the magnetic colatitude of the open/closed field line boundary (OCB), l PC , is related to F PC by where B is the magnetic field vector and the integral is over the area of the polar cap, with ds as the surface element; R E is the radius of the Earth.We identify the polar cap boundary with the Region 1 field aligned current region, l R1 = l PC , and place the Region 2 FACs some distance equatorwards of this; in Figure 2 we have used an offset of 10 of latitude, l R2 = l PC + 10 .These current sheets are assumed to be thin, such that their latitudinal extent can be neglected.The dayside and nightside merging gaps are assumed to have an angular half-widths of q D and q N , and the azimuthal angle q varies between Àp and p, being 0 at the noon meridian.
[44] As the polar cap is expanding or contracting, the OCB has a velocity (positive equatorwards) of The ionospheric flow is equal to the boundary motion along "adiaroic" (non-reconnecting) sections of the OCB, with flow across the boundary at the merging gaps.We assume that flow is everywhere perpendicular to the boundary, so Then, as E ¼ ÀV Â B, the electric field around the adiaroic boundary is given by Along each of the merging gaps, each of length l D = 2q D R E sin l PC and l N = 2q N R E sin l PC , the electric field along the OCB in the frame of the moving boundary is F D /l D and F N /l N .In the frame of the Earth, these become and It is then straight-forward to integrate E ∥ to find the potential around the boundary, that is the potential at the region 1 current system, F R1 (q) = F(l = l R1 , q).Conversely, equatorward of the region 2 current system there is no convection and consequently F R2 (q) = F(l = l R2 , q) = 0.
[45] The electrostatic potential pattern F(l, q) can then be found by solving Laplace's equation r 2 F = 0, using the boundary conditions provided by F R1 and F R2 .This can be achieved by making the substitution x ¼ log e tan 1 2 l, after which the solutions in the regions poleward of R1 and between R1 and R2 are given by where B m are the coefficients of a truncated Fourier expansion of F R1 of order N.These coefficients can be determined analytically from the functional form of F R1 , or can be found by taking the Fourier transform of F R1 , e.g., Either way, as F R1 is odd the coefficients in the cosine terms in the Fourier expansion, A m = 0, and only the sine terms need be considered.In our modeling we have used a Fourier series where N = 20.Finally, the electric field and ionospheric drift vectors across the polar region can be found using E ¼ ÀrF and V ¼ E Â B=B 2 .
[46] Acknowledgments.[47] Philippa Browning thanks the reviewers for their assistance in evaluating this paper.

Figure 1 .
Figure1.A schematic diagram of the magnetosphere illustrating closed and open magnetic field lines by red and blue lines, respectively.F D quantifies the rate at which flux is opened at the dayside, while F N is the rate at which it is reclosed on the nightside.The rate of dayside reconnection is dependent on the solar wind speed V SW and the strength and orientation of the interplanetary magnetic field B embedded within it.The inset panel shows the relationship between the footprint of the open flux F PC and the size of the polar caps, the dim ionospheric regions encircled by the auroral ovals.
Figure2fpresents idealized time series of F D and F N , representing a period of southward IMF (F D > 0), during which a substorm occurs (F N > 0).Figure2eshows the variation in polar cap flux F PC in response to changing F D and F N , as dictated by equation (1).Figures2a-2dpresent snapshots of the ionospheric convection pattern associated with these synthesized time series [cf.Cowley and Lockwood, 1992] (calculated using the technique outlined byFreeman [2003], modified to add a contribution from nightside reconnection; see Appendix A for details).The red circle represents the open-closed field line boundary (OCB) or polar cap boundary.Those portions of the OCB shown dashed, known as the merging gaps, map to regions at the magnetopause and in the magnetotail where reconnection is ongoing.Black lines show contours of the electrostatic potential pattern associated with the convection (contours at 10 kV intervals) or, equivalently, streamlines of the ionospheric flow.Colored blobs numbered 1 to 10 are tracers in the flow: these can be thought of as parcels of ionospheric plasma entrained in the flow or, equivalently, the footprints of magnetic field lines participating in the associated magnetospheric convection.The blobs are colored green where the field lines are closed, blue where open.When either dayside or nightside reconnection is ongoing, flow crosses the OCB/merging gap as field lines change their open/closed topology and the polar cap expands or contracts.Away from the merging gaps, the OCB and ionospheric drift move together (are "adiaroic" [Siscoe and Huang

Figure 2 .
Figure 2. (a-d) Snap-shots of the electrostatic potential pattern associated with ionospheric convection in response to dayside and nightside reconnection rates shown in Figure 2f.In each panel, concentric gray circles and radial lines show geomagnetic latitudes in 10 steps and magnetic local time meridians.The red circle indicates the location of the open/closed field line boundary (OCB); reconnection is active on portions of the OCB shown dashed.The black lines are equipotential contours in steps of 10 kV, equivalent to plasma flow streamlines.Numbered blobs are tracers in the flow or, equally, the footprints of magnetic field lines participating in magnetospheric convection; these are colored blue and green where these field lines are open and closed, respectively.(e) The variation in open magnetic flux, F PC , in response to the reconnection rates shown in Figure 2f.(f) The modeled variations in dayside and nightside reconnection rates, F D (red) and F N (blue), respectively.Also shown are the corresponding dawn-dusk cross-polar cap potential, F DD , and the difference between the maximum and minimum in the potential pattern, DF.

Figure 3 .
Figure 3. Two typical examples of the intervals used in this study.(a-d, l-o) Keograms of the IMAGE/FUV observations from WIC and SI12: from the top these are the noon-midnight and dawn-dusk keograms from WIC, and the same for SI12.The color-coding shows auroral brightness, red being greatest, in arbitrary units.(e and p) The variation in open flux, F PC , derived from the auroral observations.(f and g, q and r) IMF B Y and B Z ; (h and i, s and t) solar wind speed and density, V X and N. (j and k, u and v) Geomagnetic indices AU, AL and SYM-H.The vertical dashed lines show the times on substorm onsets identified byFrey et al. [2004].

Figure 4 .
Figure 4.The range of IMF and solar wind parameters during the 26 intervals identified for study: (a) IMF B Y and B Z .Concentric circles and radial lines show values of B YZ in 10 nT steps and clock angle q in steps of 30 , respectively.(b) Solar wind speed and density, V X and N. Superimposed lines show the corresponding variation in solar wind pressure, P.

Figure 5 .
Figure 5.The relationship between the observed variation in open flux DF PC (t) and the modeled variation DF′ PC (t) (see text for details) for different exponents a, b, g, d indicated in the top left of each panel.Each line is for one of the 26 intervals studied.The Pearson coefficient of correlation, r, is indicated in the bottom right of each panel, as well as the coefficient of proportionality L which gives the best fit.

Figure 6 .
Figure 6.The variation of r in the (a, b, g, d) parameter space.(a) The maximum r for each value of a and b.(b) The maximum r for each value of g and d.

Figure 9 .
Figure 9. (a) A comparison of SuperDARN derived cross-polar cap potential, DF SD (black/gray curve), and predicted F D (red curve), for the period 20-30 October 2001.The DF SD curve is black where more than 400 radar data points contributed to the electrostatic potential pattern fit.The F D has had a 25 kV offset applied.(b) Scatterplot of F D and DF SD (black points from Figure 9a).The red line indicates a coefficient of proportionality of 1.4 and an offset of 25 kV.

Table 1 .
Summary of Events

Table 2 .
Representative Sample of Coupling Functions S.E.M. was supported by the Science and Technology Facilities Council (STFC), UK, grant ST/H002480/1.J.S.G. was supported by an STFC studentship.B.H. was supported by the Belgian National Fund for Scientific Research (FNRS).This work was supported by the PRODEX program of the European Space Agency (ESA).The IMAGE FUV data were supplied by the NASA Space Science Data Centre (NSSDC), and we are grateful to the PI of FUV, S. B. Mende of the University of California at Berkeley, for its use.The OMNI data were obtained from the GSFC/SPDF OMNIWeb interface at http://omniweb.gsfc.nasa.gov.