The rudiments of a theory of solar wind/magnetosphere coupling derived from first principles

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2007ja012646, 2008 The rudiments of a theory of solar wind/magnetosphere coupling derived from first principles Joseph E. Borovsky 1 Received 16 July 2007; revised 7 April 2008; accepted 22 April 2008; published 27 August [1] A formula that expresses the dayside reconnection rate in terms of upstream solar wind parameters is derived and tested. The derivation is based on the hypothesis that dayside reconnection is governed by local plasma parameters and that whatever controls those parameters controls the reconnection rate. The starting point of the derivation is the Cassak-Shay formula (from energy conservation principles), which expresses the dayside reconnection rate in terms of four parameters: the magnetic field strengths B m and B s in the magnetosphere and magnetosheath and the plasma mass densities r m and r s in the magnetosphere and magnetosheath. Using the Rankine-Hugoniot relations at the bow shock and an analysis of the magnetosheath flow, three of these parameters are expressed in terms of upstream solar wind parameters. These three expressions are then used in the Cassak-Shay formula to obtain the solar wind control function. The interpretation of the control function is that solar wind pressure largely sets the reconnection rate. The solar wind magnetic field enters into the control function because of a bow shock Mach number dependence. The onset of a plasmasphere effect occurs when r m >M A 0.87 r solarwind, wherein the magnetosphere begins to exert control over solar wind/magnetosphere coupling. Using the OMNI2 data set and seven geomagnetic indices, the solar wind control function is tested on its ability to describe the variance in the geomagnetic indices. The control function is found to be successful, statistically as good as the best solar wind driver function in the literature. This picture opens a new pathway to understanding and calculating solar wind/magnetosphere coupling. Citation: Borovsky, J. E. (2008), The rudiments of a theory of solar wind/magnetosphere coupling derived from first principles, J. Geophys. Res., 113,, doi: /2007ja Introduction: Reconnection is a Local Process [2] Dayside reconnection is largely believed to control solar wind/magnetosphere coupling [e.g., Dungey, 1961; Hones, 1984; Kamide and Slavin, 1986; Goertz et al., 1993; Russell, 2000; Cowley et al., 2003]. Reconnection is not the driver of the magnetosphere (the mechanism putting energy into the magnetosphere is not explored here), but it controls how much driving occurs. [3] The underlying hypothesis for this calculation is the fact that reconnection is a local process. Local means that the rate of reconnection is governed by the local plasma parameters near the reconnection site. The prevailing hypothesis is that external (boundary) conditions control the rate of reconnection [e.g., Axford, 1969; Vasyliunas, 1975]. In our hypothesis, external conditions can only modify the reconnection rate if they modify the local plasma parameters near the reconnection site. For two identical plasmas undergoing antiparallel reconnection, the familiar rule of thumb is that the plasma inflow velocity into the reconnection site is 0.1 v A, where v A is the Alfven speed in the 1 Space Science and Applications, Los Alamos National Laboratory, Los Alamos, New Mexico, USA. Copyright 2008 by the American Geophysical Union /08/2007JA plasma. This plasma inflow speed is a measure of the reconnection rate (the rate at which magnetic flux is carried into the reconnection site and reconnected). When the two reconnecting plasmas are not identical, then a hybrid Alfven speed describes the rate of reconnection, wherein the parameters of both plasmas near the reconnection site contribute to the hybrid Alfven speed. Specifically, these parameters are the magnetic field strengths and the mass densities of the two plasmas. [4] Controlling the values of these local plasma parameters controls the rate of reconnection. This must be true regardless of any driving of reconnection that is going on. Driving, if it occurs, will act to change these local plasma parameters to adjust the reconnection rate to match the driving (as has been demonstrated via a series of driven computer simulations [Birn et al., 2008]). This could occur via a pileup or depletion of plasma on one side or both sides of the reconnection site. Driving cannot directly force flux to reconnect, it must first adjust the local plasma parameters. Hence the temporal scenario is driving, then pileup, then increased reconnection. [5] As this article progresses, we will argue that dayside reconnection is not driven, in the sense that there is no pileup caused by the solar wind forcing the reconnection rate. Rather, the solar wind flow is diverted around the magnetosphere and some of the magnetic flux can be 1of14

2 captured from that flow via reconnection. The occurrence of reconnection does not significantly affect that flow. This driving versus no driving scenario may be paraphrased with the following culinary example: magnetic flux is not forced down the throat of the magnetosphere, rather the magnetosphere samples magnetic flux as it is carried past. [6] We will argue that the solar wind largely controls the rate of dayside reconnection, although the magnetospheric behavior also has some control. The solar wind control is not directly via the dawn-to-dusk electric field E y =v x B z of the solar wind, rather there is a more complicated control that involves the solar wind pressure and Mach number. [7] Using high-resolution three-dimensional MHD simulations of the solar wind interacting with the Earth s magnetosphere, Borovsky et al. [2008a] studied the parameters that govern the rate of reconnection between the magnetosphere and magnetosheath at the dayside magnetopause. That study was restricted to antiparallel merging at the nose of the magnetosphere, that is, the magnetic field of the solar wind was exactly southward and there was no tilt to the Earth s dipole. To ensure that fast (Petschek) reconnection [Petschek, 1964] occurred and to ensure that numerical effects at the grid were not controlling the reconnection process, a narrow spot of high resistivity was placed on the moving magnetopause. Provided that the resistivity of the spot is strong enough (see Appendix of Borovsky et al. 2008a), the reconnection rate is independent of the properties of the spot [Birn and Hesse, 2001, 2007; Birn et al., 2008]. Simulations with high Mach number wind and high-b magnetosheath and simulations with low Mach number wind and low-b magnetosheath both confirmed that the Cassak-Shay formula [Cassak and Shay, 2007] held for the reconnection rate R. For antiparallel merging at the dayside magnetopause, the Cassak-Shay formula is R K2m 1=2 o B 3=2 m B3=2 s ðb m r s þ B s r m Þ 1=2 ðb s þ B m Þ 1=2 ; ð1þ where B m is the z-component of the magnetic field strength in the magnetosphere just outside the reconnection site, B s is the z-component of the magnetic field strength in the magnetosheath just outside the reconnection site, r m is the plasma mass density in the magnetosphere just outside the reconnection site, r s is the plasma mass density in the magnetosheath just outside the reconnection site, and K 0.1 is a coefficient representing the geometry of the inflow and outflow to the reconnection site. (In Borovsky et al. [2008a] this constant was denoted C, but to avoid confusion with the shock compression ratio, the coefficient is here denoted K.) Theoretical arguments [Petschek, 1964; Parker, 1973] and numerical simulations [Birn et al., 2001] yield the value of 0.1 for K, independent of external boundary conditions (variable driving) [Birn and Hesse, 2007]. Expression (1) can be seen as equations (16), (17), and (18) of Cassak and Shay [2007], with K d/l being the reconnection aspect ratio. For identical plasmas, expression (1) reverts to the familiar R 0.1 v A B, stating that the magnetic flux B is carried into the reconnection site with the plasma inflow speed of 0.1 v A. [8] In testing the Cassak-Shay formula with simulations, the quantities B m, B s, r s, and r m were evaluated just outside the reconnection site. B and r on the magnetospheric side were evaluated where their smooth trends in the magnetosphere begin to steepen near the reconnection site, and likewise B and r on the magnetosheath side were evaluated where their smooth trends in the magnetosheath begin to steepen near the reconnection site. Hence the quantities B and r are evaluated in the magnetosheath outside of any boundary layer that forms on the magnetosphere. Hence, if there is a plasma depletion layer in the magnetosheath adjacent to the magnetopause [cf., Zwan and Wolf, 1976; Phan et al., 1994], the quantities B s and r s are taken in the magnetosheath outside of the depletion layer. [9] According to the Cassak-Shay formula (1), the rate of reconnection is directly governed by the four parameters B m, B s, r s, and r m near the reconnection site. Whatever controls these four parameters controls reconnection. Later in this article it will be shown that, through various processes, the solar wind largely controls these four parameters and hence largely regulates solar wind/magnetosphere coupling. [10] To generalize the Cassak-Shay formula to cases where the magnetic field of the solar wind is not directly southward, a clock-angle dependence on the orientation q of the solar wind magnetic field relative to the Earth s dipole is added following the logic of Sonnerup [1974] [see also Russell and Atkinson, 1973; Pritchett and Coroniti, 2004; Huba, 2005; Swisdak et al., 2005]. This clock-angle dependence is sin(q/2), where the magnetic field direction of the solar wind forms an angle q with respect to the northward magnetic field direction of the magnetosphere (with q = 0 being parallel [northward IMF] and q = 180 being antiparallel [southward IMF]). Using this, the Cassak-Shay formula for reconnection is written R K2m 1=2 0 sinðq=2þb 3=2 m B3=2 s ðb m r s þ B s r m Þ 1=2 ðb s þ B m Þ 1=2 : ð2þ [11] The reconnection rate is an inflow speed times a magnetic field strength: the reconnection rate R in the Cassak-Shay formula is the amount of magnetic flux reconnected per unit time per unit length of the reconnection line. Note, in the present formulation there is a fifth parameter that controls the reconnection rate: the clockangle q of the solar wind magnetic field. [12] In this article we will pursue a calculation of the solar wind control of the rate of reconnection on the dayside magnetopause. The solar wind control function will be derived by writing down the Cassak-Shay formula for the dayside reconnection site, and then expressing the formula s four parameters B m, B s, r s, and r m in terms of upstream solar wind parameters. [13] This article is organized as follows. In section 2 the role of the magnetosheath in solar wind/magnetospheric coupling is discussed. In section 3 the four local parameters that determine the dayside reconnection rate are expressed in terms of upstream solar wind parameters. In section 4 the solar wind coupling function is derived and interpreted. In 2of14

3 section 5 the solar wind coupling function is tested using the OMNI2 data set and six geomagnetic indices. The project is summarized in section 6 and future work is outlined in section The Role of the Shock and the Magnetosheath Flow Pattern [14] The unshocked solar wind does not directly make contact with the magnetosphere: the magnetosheath does. The magnetosheath bathes, squeezes, and reconnects with the magnetosphere and it is the magnetosheath that transfers energy, momentum, and mass to the magnetosphere. Furthermore, the properties of the magnetosheath plasma near the magnetopause largely govern the rate of reconnection between the magnetosphere and magnetosheath [Borovsky et al., 2008a; 2008b]. To understand solar wind/magnetosphere coupling, the role of the magnetosheath must be understood. [15] In studying the driving of the magnetosphere, upstream solar wind measurements are compared with magnetospheric measurements [cf., Kamide and Slavin, 1986; Hruska et al., 1993] without much consideration of the action of the magnetosheath. For such a study, an appropriate set of magnetosheath measurements would be difficult to obtain since the properties of the magnetosheath vary with spatial location. Fortunately, the properties of the magnetosheath are related to the properties of the upstream solar wind, which are easily obtained from several spacecraft data sets. [16] An understanding of the role of the bow shock produced magnetosheath is critical to a full understanding of the solar wind s interaction with the Earth. For example, the density and temperature of the hot magnetospheric plasma sheet are positively correlated with the density and velocity, respectively, of the upstream solar wind, with a few hours of time lag [cf., Borovsky et al., 1997, 1998; Terasawa et al., 1997; Denton and Borovsky, 2008]; this is because the magnetosheath is a predominant source of the plasma sheet and the density of the magnetosheath is linearly proportional to the density of the solar wind and the temperature of the magnetosheath is proportional to the square of the velocity of the solar wind. These are bow shock properties. Another example is that B y and B z in the magnetotail are positively correlated with B y and B z in the upstream solar wind [cf., Lui, 1984; Borovsky et al., 1998]; this is because B y and B z in the magnetosheath are linearly proportional to B y and B z in the upstream solar wind. Again, bow shock properties. Likewise, we speculate here that the temperature ratio T i /T e 6 of the magnetotail plasma sheet [Baumjohann et al., 1989; Baumjohann, 1993] has its origin withthe temperature ratiot i /T e 6ofthemagnetosheath.This temperature ratio is a property of collisionless shocks at supercritical Mach numbers [Schwartz et al., 1988]. [17] The role that the bow shock plays in solar wind/ magnetosphere coupling was investigated by Lopez et al. [2004] [see also Kataoka et al., 2005]. Using 3D MHD simulations of the solar wind-driven magnetosphere, Lopez et al. [2004] observed that changes in solar wind density could result in changes in the solar wind/magnetosphere coupling. Their explanation is that changes in the solar wind density can lead to changes in the solar wind Mach number, which can lead to changes is the compression ratio of the bow shock, which can lead to changes in the magnetic field strength in the magnetosheath. A stronger magnetosheath magnetic field applied to the magnetopause is deemed to increase the reconnection rate. A fuller interpretation of the Lopez et al. [2004] simulation results appears in a further report (B. Lavraud and J. E. Borovsky, 2008, The alteredsolar wind - magnetosphere interaction at low Mach numbers: Coronal mass ejections, submitted to Journal of Geophysical Research, 2008). [18] The plasma parameters on the magnetosheath side of the dayside reconnection site are determined ultimately by the upstream solar wind, but their values depend on the nature of the bow shock and on the flow pattern in the magnetosheath behind the shock. In the next section calculations of the four parameters B m, B s, r s, and r m will be made in terms of the upstream solar wind specifically accounting for how the solar wind sets the properties of the magnetosheath. 3. Determining the Four Parameters That Control Dayside Reconnection [19] For antiparallel merging at the subsolar point, the four parameters near the dayside reconnection site that govern and determine the rate of reconnection are B m (the magnetic field strength in the magnetosphere), B s (the magnetic field strength in the magnetosheath), r m (the plasma mass density in the magnetosphere), and r s (the plasma mass density in the magnetosheath). Along with the IMF clock-angle q, these are the four parameters on the right-hand side of the Cassak-Shay formula (1). Whatever controls the values of those four parameters, controls the rate of dayside reconnection. [20] In the following four subsections, the values of each of these four local parameters will be expressed in terms of upstream solar wind parameters. The reader is reminded that the four local parameters are evaluated just outside of the region of steep gradients near the reconnection site, i.e., at the outer edge of any boundary layer at the magnetopause The Magnetospheric Magnetic Field Strength B m [21] The first parameter that we will address is the magnetic field strength B m of the magnetosphere near the reconnection site. The field strength B m can be calculated in terms of upstream solar wind parameters by using pressure balance. The total pressure in the magnetosphere is written B 2 m /8p + P m, where P m is the pressure nk B T of the magnetospheric particle population (ions plus electrons) near the reconnection site. At the nose of the magnetosphere, this total pressure must balance the pressure of the solar wind; typically only the ram pressure r o v 2 o of the solar wind is accounted for [e.g., Schield, 1969; Sotirelis, 1996], but since we are including low Mach number effects the total pressure r o v 2 o + B 2 o /2m o + P o is taken where the subscript o denotes upstream (of the bow shock) solar wind. Note the P o is the sum of the ion and electron pressures in the solar wind. Using the definition of the magnetosonic speed C ms = (v 2 A + 5P/3r) 1/2 (e.g., equation (1.52) of Tidman and Krall [1971]) and defining the magnetosonic Mach number M ms =v o /C ms, this total 3of14

4 Figure 1. For six snapshots, the plasma b is plotted along the Sun-Earth line. Three curves are from Simulation A of Borovsky et al. [2008a] (M A = 9.8, 12.2, and 16.2) and three are from Simulation B of Borovsky et al. (M A = 1.9, 3.2, and 4.3). pressure is r o v 2 o ( M 2 ms ). Equating these two pressures and solving for B m yields B m 2m o r o v 2 o 2m op m 1=2 1 þ 0:5M 2 1=2: ð3þ [22] The magnetospheric plasma pressure P m on the dayside is a function of the solar wind number density and the solar wind velocity with several hours of time lag [Borovsky et al., 1998] and it is a function of the past history of geomagnetic activity. For the dayside magnetosphere, a parameterization of this particle pressure P m as a function of the solar wind has not yet been performed. Often, but not always, P m can be ignored in comparison with r o v o 2 in the dayside magnetosphere. Ignoring P m, expression (3) becomes B m ð2m o Þ 1=2 r 1=2 v o 1 þ 0:5M 2 1=2; ð4þ o which expresses the magnetic field strength in the magnetosphere near the reconnection site in terms of upstream solar wind parameters The Magnetosheath Magnetic Field Strength B s [23] Pressure balance must hold across the magnetopause between the magnetospheric plasma with total pressure B 2 m / 2m o +P m and the magnetosheath plasma with total pressure B 2 s /2m o + P s, where P s is the particle pressure of the magnetosheath plasma. The plasma b is defined as the ratio of the particle pressure to the magnetic field pressure [Krall and Trivelpiece, 1973], so the b value of the magnetosheath is written b s =P s /(B 2 s /2m o ). Using this b s expression to write P s as P s = b s (B 2 s /2m o ), the total pressure of the magnetosheath can then be written (B 2 s /2m o )(1+b s ). In the argument of section 3.1, the pressure in the magnetosphere at the nose is equal to the total pressure r o v 2 o ( M 2 ms ) of the solar ms ms wind, so with pressure balance across the reconnection site, the pressure (B 2 s /2m o )(1+b s ) of the magnetosheath at the nose is also equal to r o v 2 o ( M 2 ms ) of the solar wind. Equating the solar wind pressure r o v 2 o ( M 2 ms ) with the magnetosheath pressure (B 2 s /2m o )(1+b s ) and solving for B s yields B s ¼ ð2m o r o Þ 1=2 v o ð1 þ b s Þ 1=2 1 þ 0:5M 2 1=2; ms ð5þ where b s is the plasma b of the magnetosheath near the reconnection site. In Figure 1 the plasma b is plotted along the Sun-Earth line for several snapshots from global MHD simulations of the solar wind interacting with the magnetosphere. Three of the six curves are from the high Mach number simulation (Simulation A) of Borovsky et al. [2008a] (M A = 9.8, 12.2, and 16.2) and three are from the low Mach number simulation (Simulation B) of Borovsky et al. (M A = 1.9, 3.2, and 4.3). Note that in all cases, the plasma b decreases across the magnetosheath from the bow shock to the nose of the magnetosphere. For high Mach number cases, b decreases by about a factor of 4; for low Mach number cases b decreases by less than a factor of 4. Denoting the b value in the magnetosheath just behind the bow shock as b BS and the b value in the magnetosheath near the dayside reconnection site as b s, the ratio b BS /b s is plotted as a function of the Alfven Mach number M A in Figure 2; the values are obtained every 5 minutes for Simulations A (solid points) and B (hollow points). The data in Figure 2 are parameterized by b BS =b s ¼ 1:414 00:414M A for M A 2:55 ð6aþ b BS =b s ¼ 0:418M 0:18 A for M A 2:55 ð6bþ which is plotted as the two curves in Figure 2. 4of14

5 Figure 2. The ratio b s /b BS is plotted as a function of the Alfven Mach number M A, where b BS is the plasma b just behind the bow shock and b s is the plasma b in the magnetosheath near the nose of the magnetosphere. The values are obtained every 5 minutes from Simulations A (solid points) and B (hollow points). [24] From the Rankine-Hugoniot shock jump relations [e.g., Tidman and Krall, 1971; Burgess, 1995], the value b BS of the plasma b behind the bow shock can be expressed from upstream solar wind parameters; then using expression (6) the values b s of the magnetosheath near the reconnection site can be expressed in terms of upstream solar wind parameter from these b BS values. In the top set of black points in Figure 3, the Rankine- Hugoniot relations (assuming a quasi-perpendicular shock) are used to calculate b BS for all upstream solar wind conditions in the OMNI2 data set (182,703 hours of solar wind data). (The OMNI2 data set [King and Papitashvili, 2005] consists of hourly averaged solar wind values determined at the position of the Earth s magnetosphere.) To check this upper black curve, b BS is measured every 5 minutes in Simulation A (red points) and in Simulation B (blue points) and plotted onto Figure 3. The agreement is good. Expression (6) is then used to calculate b s from each b BS value for the OMNI2 data set and these b s values are plotted as the bottom set of black points in Figure 3. These b s points in Figure 3 are well fit by values is 9.8% and the mean error is 19.6%. Using expression (7) to replace b s in expression (5), expression (5) for the magnetic field strength in the magnetosheath near the reconnection site becomes B s ð2m o r o Þ 1=2 v o 1 þ 3: M 1:92 1=2; ð8þ which expresses the magnetic field in the magnetosheath outside the reconnection site in terms of upstream solar wind parameters. The reader is reminded that the Alfven Mach number M A is M A =(m o r o ) 1/2 v o /B o The Magnetosheath Mass Density r s [26] The mass density r s of the magnetosheath plasma near the reconnection site can be estimated using Figure 4. Here the plasma number density n is plotted as a function of X along the Sun-Earth line for six instants of time from Simulations A and B (as in Figure 1). As can be seen in Figure 4, the density jumps across the bow shock but then it remains approximately constant across the magnetosheath from the bow shock to the outer edge of the boundary layer on the magnetopause. Hence we can write A b s 3: M 1:92 A : ð7þ r s Cr o ; ð9þ [25] To double check this bottom black curve, b s is measured every 5 minutes in Simulation A (red points) and in Simulation B (blue points) and plotted onto Figure 3. As can be seen, the agreement is good: the median error between the formula b and the simulation-measured b where C is the compression ratio of the bow shock and r o is the mass density of the upstream solar wind. [27] The compression ratio C of the bow shock is parameterized in terms of upstream solar wind parameters using Figure 5, which was constructed as follows. For the 5of14

6 Figure 3. In the top set of black points the Rankine-Hugoniot relations are used to calculate b BS for all upstream solar wind conditions in the OMNI2 data set. Along with the top set of black points, b BS is measured every 5 minutes in Simulation A (red points) and in Simulation B (blue points) and plotted. From each black point in the upper set, b s is calculated from expression (6) and plotted as the bottom set of black points. Along with the bottom set of black points, b s is measured every 5 minutes in Simulation A (red points) and in Simulation B (blue points) and plotted. hourly averaged solar wind parameters of the OMNI2 data set (182,703 hours), the compression ratio C of the bow shock is calculated using the Rankine-Hugoniot MHD shock jump conditions. For simplicity, a quasi-perpendicular shock is assumed (e.g., equation (1.50) of Tidman and Krall [1971] or equation (5.13) of Burgess [1995]). For each of the 182,703 hours of data, C is calculated and plotted as a function of the Alfven Mach number of the solar wind in Figure 4. For six snapshots, the plasma number density n is plotted along the Sun-Earth line. Three curves are from Simulation A of Borovsky et al. [2008a] (M A = 9.8, 12.2, and 16.2) and three are from the Simulation B of Borovsky et al. (M A = 1.9, 3.2, and 4.3). From right to left, each curve is plotted from out in the solar wind upstream of the bow shock to the outer edge of the boundary layer on the magnetosphere. 6of14

7 Figure 5. For the hourly averaged solar wind parameters of the OMNI2 data set, the compression ratio C of the bow shock is calculated using the Rankine-Hugoniot shock jump conditions and plotted as a function of the Alfven Mach number of the solar wind. Figure 5 (black points). The points of Figure 5 are parameterized by the curve n o 1=6; C ½1=4Š 6 þ½1= ðþ1:38 log e ðm A ÞÞŠ 6 ð10þ which is plotted as the red curve in Figure 5. Using expression (10) to eliminate C in expression (9) yields n o 16ro r s ½1=4Š 6 þ½1= ð1 þ 1:38 log e ðm A ÞÞŠ 6 ; ð11þ which expresses the mass density in the magnetosheath outside of the reconnection site in terms of upstream solar wind parameters. [28] At high Mach numbers (M A greater than about 10) expression (11) becomes r s 4r o (where the bow shock compression ratio C! 4) and at low Mach numbers (M A less than about 4) expression (11) becomes r s ( log e (M A )) r o (where the compression ratio C! log e (M A )) The Magnetospheric Mass Density r m [29] The last parameter that we will discuss is the magnetospheric mass density r m near the reconnection site. The plasma density in the dayside magnetosphere changes with solar wind conditions and the level of geomagnetic activity. There are four populations of particles that contribute significantly to the mass density in the outer dayside magnetosphere: (1) hot plasma sheet plasma, (2) cold plasmaspheric plasma, (3) cold photoelectron-driven ionospheric outflows, and (4) warm auroral ionospheric outflows. These are discussed individually in the next four paragraphs. [30] Hot plasma sheet ions convect from the nightside magnetosphere around the dipole to the dayside, with precipitation losses to the loss cone (making proton aurora) and charge-exchange losses to the geocorona. Because of these losses, the dayside hot-plasma density is less than the nightside hot-plasma density. For quiet to normal levels of geomagnetic activity, the number density of hot ions in the dayside magnetosphere is typically 0.5 cm 3 or less [Korth et al., 1999; Denton et al., 2005], probably chiefly protons. At high levels of geomagnetic activity the number density of hot ions tends to be higher [Denton et al., 2006]. The density of this hot plasma is positively correlated with the density of the solar wind, with about a 15-hour time lag [Borovsky et al., 1998]. In the nightside magnetosphere at geosynchronous orbit the plasma sheet density n ps (in units of cm 3 ) can be approximated as n ps = 0.292n o 0.49, where n o is the number density of the solar wind upstream of the bow shock (in units of cm 3 ). A similar parameterization for the dayside hot-plasma density has not yet been performed. [31] High-density plasmaspheric plasma builds up in the inner magnetosphere, particularly when geomagnetic activity is low. When geomagnetic activity changes, some of this built-up plasma convects into the outer magnetosphere. If geomagnetic activity grows strong (storm levels), a large amount of this cold plasma will convect to the dayside reconnection line in the form of a drainage plume [Grebowsky, 1970; Spiro et al., 1981; Borovsky and Denton, 2008] that is a few R E wide in the equatorial plane. Because this plasma is cold (a few ev), it convects where the magnetic field convects, which is into the dayside reconnection line. Confirming this, cold plasmaspheric plasma has been seen in reconnection outflows at the magnetopause [Su et al., 2000, 2001a]. The density of drainage-plume plasma at the magnetopause has been measured at 200 cm 3 [Borovsky et al., 1997]. The effective mass of the plasmaspheric plasma is about 2.5 amu owing to the fractional oxygen and helium content of the plasma. [32] The photoelectron-driven outflow of cold ions from the ionosphere that results in the plasmasphere also acts in the outer dayside magnetosphere to add cold plasma. As measured by Gallagher et al. [1998], Thomsen et al. [1998], Lawrence et al. [1999] and Su et al. [2001b], early time refilling rates are approximately cm 3 hour 1 at the equator. For flux tubes exposed to the sunlit ionosphere for several hours, the mechanism tends to add a background cold plasma with a density of 1 cm 3 to the dayside magnetosphere. [33] During geomagnetic storms, the dayside magnetosphere is filled with warm (10s of ev) ions with distribution functions that are often magnetic field aligned. These warm ions probably originate from auroral-zone ionospheric outflows into the magnetosphere at all local times, followed by convection into the dayside magnetosphere to the dayside reconnection line. The number densities in the dayside magnetosphere are typically a few cm 3 (see Figures 2 and 3 of Lawrence et al. [1999] and Figure 5 of Su et al. [2001b]) if the ions are protons and 4 times that amount if the ions are oxygen. [34] The solar wind partially controls r m, with the solar wind density (with a time lag) and the level of solar wind driven geomagnetic activity both affecting r m. Almost certainly, the F10.7 flux of the Sun also plays a role. [35] In the next section expressions (3), (9), and (8) for B m, r s, and B s will be used in the Cassak-Shay formula (2) 7of14

8 to produce a solar wind control function for dayside reconnection at the nose. 4. The Solar Wind Control Function 4.1. Deriving the Solar Wind Control Function [36] The reconnection rate at the dayside magnetopause is governed by four local plasma parameters: B m, B s, r m, and r s. If these four parameters are known, then the reconnection rate can be calculated from the Cassak-Shay formula (expression (1)). In section 3, three of the four local parameters were algebraically expressed in terms of upstream solar wind parameters. Hence the dayside reconnection rate can be expressed as a function of upstream solar wind parameters and the mass density r m of the dayside magnetosphere near the reconnection site. [37] Writing down the Cassak-Shay formula (expression (2)), taking K = 0.1 [cf., Borovsky et al., 2008a] using expression (4) (which ignores the magnetospheric particle pressure P m ) to eliminate B m, using expression (5) to eliminate B s, using expression (9) to eliminate r s, and multiplying the numerator and denominator by (1 + b s ) 1/4 yields R ¼ 0:4m 1=2 o sinðq=2þr o v 2 o 1 þ 0:5M 2 ms ð 1 þ bs Þ 1=2 h i 1=2 h i 1=2 ð12þ Cr o þ ð1 þ b s Þ 1=2 r m ð1 þ b s Þ 1=2 þ1 where r o is the mass density of the solar wind upstream of the bow shock, v o is the velocity of the solar wind upstream of the bow shock, C is the compression ratio of the bow shock, b s is the plasma-b value of the magnetosheath plasma near the nose, and M ms is the magnetosonic Mach number of the solar wind. Expression (12) is supplemented with b s ¼ 3: M 1:92 A ð13aþ n o 1=6 C ¼ ½1=4Š 6 þ½1= ð1 þ 1:38 log e ðm A ÞÞŠ 6 ð13bþ M ms ¼ v o B 2 o =m 1=2 or o þ 5Po =3r o ð13cþ M A ¼ v o ðm o r o Þ 1=2 =B o ; ð13dþ i.e., expressions (7) and (10) plus the definitions of the magnetosonic and Alfven Mach numbers. [38] Expression (12) is the derived control function for solar wind/magnetosphere coupling. It is derived to hold over a wide range of Mach numbers. There are no adjustable parameters. Unlike other driver functions, it was derived without tuning to optimize correlation coefficients between magnetospheric measurements and solar wind measurements. [39] Evaluating the magnetosonic Mach number M ms in expression (13c) is often difficult, since solar wind electron temperatures (which enter into the solar wind particle pressure P o ) are typically not available. Fortunately, M A can usually be substituted for M ms in expression (12); the instances when M ms and M A of the solar wind are small to modest are almost always magnetic clouds, which have low b and M A M ms Interpreting the Control Function [40] The solar wind control function (right-hand side of expression (12)) has a total pressure of the solar wind in the numerator and a square root of a mass density in the denominator. Expression (12) is an Alfven speed (magnetic field in numerator and square root of mass density in the denominator) times a magnetic field strength. As seen in sections 3.1 and 3.2, the magnetic field strength is governed by the total pressure of the solar wind (cf., expressions (3) and (5)), hence the total pressure r o v 2 o ( M 2 ms )inthe numerator of expression (12). The (1 + b s ) 1/2 in the denominator of expression (12) carries the high Mach number reduction of B s owing to the particle pressure behind the bow shock. In essence, if a driver of reconnection is to be identified, it is the solar wind pressure which squeezes the magnetosheath and magnetosphere plasmas together. [41] Owing to the asymmetry between the magnetospheric plasma and the magnetosheath plasma, there is a complication to this simple pressure driver that brings the solar wind magnetic field strength B o into the control function. In the Cassak-Shay formula (expression (1)), the plasma with the slower Alfven speed governs the reconnection rate. (The slower of the two Alfven speeds presents a bottleneck to reconnection, varying the bottleneck varies the reconnection rate.) At low Mach numbers, the b of the magnetosheath is low [e.g., Farrugia et al., 1995; Borovsky and Denton, 2006b; Lavraud et al., 2007], as it is for the magnetosphere, and so B m B s. But at medium to high Mach numbers the bow shock produces a b s value that is non negligible, meaning that plasma pressure in the sheath contributes to balancing the magnetic pressure of the magnetosphere, so B s < B m. Now the magnetosheath parameters tend to govern the reconnection rate R. B s now depends on b s, which depends on M A, which depends on B o. The reader should note that it is not the case that B o of the solar wind is a controller of reconnection because the solar wind electric field is the product v o B o, rather v o and B o control (separately) the reconnection rate by setting the local plasma parameters through pressure and Mach number. [42] This last statement is supported by one of the conclusions of the study by Borovsky et al. [2008a], which was that the solar wind electric field v x B z does not drive dayside reconnection. The two major pieces of evidence found against the electric field being the driver are the following. (1) The measured rate of reconnection in southward-imf 3-D MHD simulations of the magnetosphere is not proportional to v x B z (cf., Figure 10 of Borovsky et al. [2008a]). (2) If reconnection is controlled by an upstream electric field, then plasma pileup at the nose of the magnetosphere is expected to adjust local plasma parameters such that the reconnection rate would match the driving; no evidence of plasma pileup at the nose of the magnetosphere was seen when southward IMF (reconnection on) and northward IMF (reconnection off) simulations were compared (cf., Figure 12 of Borovsky et al. [2008a]). 8of14

9 Figure 6. From Simulation B of Borovsky et al. [2008a], the solar wind parameters v o, n o, and B o and the magnetospheric plasma density near the nose reconnection site are plotted every 5 minutes. In the bottom panel, the measured reconnection rate (in units of nt km/s) is plotted in black, and the prediction of the reconnection rate from the control function (expression (12)) is plotted in red: the solid points use the measured plasma density of the magnetosphere and the hollow points take r m = 0 in the control function. [43] Note that the solar wind control function describes the plasmasphere effect [Borovsky and Steinberg, 2006; Borovsky and Denton, 2006a] wherein high-density magnetospheric plasma can mass load dayside reconnection and reduce solar wind/magnetospheric coupling. The control function describes this effect by means of the r m factor in the denominator of expression (12). As can be seen by inspecting the term [Cr o + (1 + b s ) 1/2 r m ] in the denominator, when r m (1+b s ) 1/2 Cr o, the magnetospheric mass density dominates and magnetospheric plasma mass loads dayside reconnection, reducing solar wind/magnetosphere coupling. To within 12% accuracy, this factor (1 + b s ) 1/2 C can be replaced by M A 0.87, giving the simpler criterion r m M A 0.87 r o for the plasmasphere effect to operate. When r m M A 0.87 r o, the magnetospheric plasma chokes dayside reconnection. [44] The plasmaspheric effect can be seen in the two panels of Figure 6, where key parameters and the reconnection rate are plotted as functions of time from Simulation BofBorovsky et al. [2008a]. In the top panel the upstream solar wind velocity v o, density r o (n o ), and magnetic field strength B o are plotted, along with the magnetospheric plasma density r m (n o ) measured in the simulation on the magnetospheric side of the reconnection site. In the bottom panel of Figure 6 the measured reconnection rate at the nose of the magnetosphere is plotted in black. As can be seen, the 9of14

10 reconnection rate drops at time t 150 minutes when the magnetospheric density increases (this is the plasmasphere effect). Also plotted in the bottom panel is the reconnection rate as calculated from the control function (expression (12) with expression (13)) using v o, r o, and B o from the simulation. The solid red points are the calculated reconnection rate using the plasmaspheric mass density determined in the simulation, and the hollow red points are the calculated reconnection rate taking r m = 0. By comparing the solid red points and the hollow red points it is seen that the control function captures the plasmasphere effect Limiting Cases [45] At very low Mach numbers, the compression ratio C log e (M A ) and the b value of the magnetosheath is b s 1. Using these, expression (12) becomes R / sinðq=2þr o v 2 o 1 þ 0:5M 2 ms ½ ro ð1 þ 1:38 log e ðm A ÞÞþr m Š 1=2 : ð14þ [46] For r m r o ( log e (M A )) (low-density magnetosphere), expression (14) becomes R / sinðq=2þr 1=2 o v 2 o 1 þ 0:5M 2 ms = 1 þ 1:38 loge ðm A ð ÞÞ 1=2 ð15þ and for r m r o ( log e (M A )) (high-density magnetosphere), expression (14) becomes R / sinðq=2 Þr 1=2 m r o v o 1 þ 0:5M 2 : ð16þ [47] The reader is reminded that r o v 2 o ( M 2 ms )isthe total pressure of the upstream solar wind; in these low Mach number expressions (particularly expression (14)) the pressure driving of dayside reconnection is most evident. [48] At very high Mach numbers, (1+0.5M 2 ms ) 1, the compression ratio C 4, and the b value of the magnetosheath is b s 1. Using these, expression (12) becomes h i 1=2: R / sinðq=2þr o v 2 o b3=4 s 4r o þ b 1=2 s / r m ð17þ [49] Using expression (7) which gives b s 0.032M A 1.92, and using M A =v o (m o r o ) 1/2 /B o, expression (17) becomes R / sinðq=2þr 0:28 o v 0:56 o B 1:44 o ms ½4r o þ 5:6MA 0:96 r m Š 1=2 : ð18þ [50] For r m 0.71 r o M A 0.96 (low-density magnetosphere), expression (18) becomes R / sinðq=2þr 0:22 o v 0:56 o B 1:44 o : ð19þ [51] For r m 0.71 r o M A 0.96 (high-density magnetosphere), this expression (18) becomes R / sinðq=2þr 0:52 o ro 0:5 v 1:04 o B 0:96 o : ð20þ [52] As can be seen in expressions (19) and (20), the solar wind control function has complicated dependences on v o and B o, the functional forms of which in fact depend on the density of the magnetosphere. The reader is encouraged to note again that v o and B o are not in the control function from the fact that the solar wind electric field is the product v o B o : they are there because the solar wind pressure and Mach numbers depend on v o and B o. 5. Testing the Solar Wind Control Function [53] The usefulness of the solar wind control function (expression (12) with expression (13)) is tested using the OMNI2 solar wind measurements and various geomagnetic indices. The OMNI2 data set [King and Papitashvili, 2005] consists of hourly averaged solar wind values determined at the position of the Earth s magnetosphere. The various geomagnetic indices will be taken as indicators of the dayside reconnection rate, since reconnection is thought to control the energy input into the magnetosphere [e.g., Dungey, 1961; Goertz et al., 1993; Russell, 2000; Cowley et al., 2003]. The OMNI2 data set is used for the years , and 1-hour averages of the various geomagnetic indices are used as available. In calculating the magnetosonic speed for the magnetosonic Mach number M ms,t e =T i is taken for the solar wind since measurements of the electron temperature T e are not available in OMNI2. For magnetic clouds, a temperature ratio T e /T i of somewhere from 2 [e.g., Newbury et al., 1998] to 7 [e.g., Skoug et al.,2000] might have been more appropriate. Measurements of T e could improve the use of the control function. [54] An example is shown in Figure 7, where the auroralelectrojet index AE is plotted as a function of the solar wind control function (expression (12)), with a 1-hour lag on the AE index. (Owing to the reaction time of the magnetosphere, a 1-hour-lagged AE index usually gives a better correlation with the solar wind than does a non-lagged index [McPherron et al., 1986; Borovsky and Funsten, 2003]). The 1-hour-lagged AE index is denoted here as AE 1. Since we have no information about the dayside magnetospheric mass density r m, we take r m = 0 in expression (12). The black points in Figure 7 are the individual hours of data. For this data set, the linear correlation coefficient between AE 1 and the control function is To show the trend underlying the scatter of points in Figure 7, a 300-point running average (boxcar average) of the data is plotted in red. This running average clearly shows a flattening of the AE1-versus-control function curve. This flattening at strong driving is familiar in the literature [e.g., Weimer et al., 1990; Nagatsuma, 2004] and has been attributed to polar cap saturation [Reiff and Luhmann, 1986; Siscoe et al., 2004] at low Mach numbers and high ionospheric conductivity (cf., Appendix of Borovsky and Denton [2006a]; Lavraud and Borovsky, submitted manuscript, 2008]; the strongest driving (magnetic clouds) tends to be associated with low Mach numbers. [55] In Table 1 the linear correlation coefficients between the solar wind coupling function (expression (12) with (7) and (10) and r m = 0) and various geomagnetic indices are listed. The indices used are AE 1 (the 1-hour-lagged total auroral electrojet index), AU 1 (the 1-hour-lagged eastward auroral electrojet index), -AL 1 (the 1-hour-lagged westward 10 of 14

11 Figure 7. The auroral-electrojet index AE (with a 1-hour lag) is plotted (black points) as a function of the solar wind control function (expression (12)). Every fifth point of the 158,580 hours of data is plotted. A 300-point running average of the black points is plotted in red. auroral electrojet index), PCI thule (the northern polar cap index with no time lag), MBI 1 (the 1-hour-lagged midnight boundary index), Kp (the planetary activity index with no time lag), and Dst 2 (the 2-hour-lagged disturbance storm time index). PCI is a measure of current flowing across the polar cap, AE, AU, and -AL are measures of the strength of magnetospheric convection. MBI and Kp are measures of the position of the inner edge of the electron plasma sheet (position of the low-latitude edge of the diffuse aurora). Dst is a measure of the strength of the ring current and partial ring current. [56] For comparison, this exercise is repeated on the same data for the solar wind driver functions -v o B south [Burton et al., 1975; Holzer and Slavin, 1982; Bargatze et al., 1986a] (first column), v o B o? sin 4 (q/2) [Wygant et al., 1983; Pudovkin et al., 1985] (second column) and v 4/3 o B 2/3 o? sin 8/3 (q/2) [Newell et al., 2007] (third column). Here B o? is the component of the solar wind magnetic field B o that is perpendicular to the Sun-Earth line. The functions - v o B south and v o B o? sin 4 (q/2) are representations of the solar wind electric field, with the 4th power of sin(q/2) being a tuning obtained from data fits [cf., Bargatze et al., 1986b; Scurry and Russell, 1991]. The Newell function v 4/3 o B 2/3 o? sin 8/3 (q/2) was constructed with some tuning of exponents to maximize correlations in data fits. Because it is a tuning, it may be biased toward the most statistically important solar wind parameters in the data set. As can be seen by comparing the columns of Table 1, the control function does a substantially better job of predicting the variances of the geomagnetic indices than do the standard drivers v o B south and v o B o? sin 2 (q/2), and it does a statistically equivalent job of predicting the variances as does the Newell function v 4/3 o B 2/3 o? sin 8/3 (q/2). The fractional differences in the seven correlation coefficients r between the control function and the Newell function 2 (r controll r Newell )/(r controll + r Newell ) vary from 3% to +7%. [57] The fact that the control function (expression (12) with expression (13) and r m = 0) does well at predicting the values of the geomagnetic indices is an indication that it is largely successful at predicting the rate of reconnection in the dayside magnetosphere. The reader is reminded that expression (12) is a first attempt at a derivation of a control function expressed in terms of upstream solar wind parameters, and there are obvious improvements to come. The control function tested here (1) contains no information about magnetospheric mass density (r m was taken to be zero), (2) does not attempt to calculate the reconnection rate anywhere except exactly at the nose, and (3) does not account for the differences between the total magnetic field B o in the solar wind and Table 1. Linear Correlations Between Three Solar Wind Driver Functions From the Literature Plus the Solar Wind Control Function Derived Here and Seven Geomagnetic Indices a vb south vb? sin 4 (q/2) Newell v 4/3 B? 2/3 sin 8/3 (q/2) Control Function AE AU AL PCI thule MBI Kp Dst a The control function is sin(q/2) r o v 2 o ( M 2 ms )(1+b s ) 1/2 [Cr o + (1 + b s ) 1/2 r m ] 1/2 [(1 + b s ) 1/2 +1] 1/2. For the correlations, 158,000 hours of solar wind measurements in the OMNI2 data set are utilized. 11 of 14

12 the perpendicular component B o?. Needed improvements will be itemized in section Summary [58] A formula that expresses the dayside reconnection rate in terms of upstream solar wind parameters was derived and tested. The philosophy behind the derivation is that the dayside reconnection rate is governed by local plasma properties and that whatever controls those properties controls reconnection. The starting point of this calculation was the Cassak-Shay formula [Cassak and Shay, 2007] applied to the magnetospheric and magnetosheath plasmas on ether side of the dayside reconnection site at the nose of the magnetosphere. The Cassak-Shay formula, derived from first principles, was found in a previous report to accurately describe the reconnection rate at the nose of the magnetosphere [Borovsky et al., 2008a]. According to the Cassak-Shay formula, four parameters govern the rate of reconnection: B m (the magnetic field strength in the magnetosphere), B s (the magnetic field strength in the magnetosheath), r m (the plasma mass density in the magnetosphere), and r s (the plasma mass density in the magnetosheath). An IMF clock-angle q dependence was added to the Cassak-Shay formula following the arguments of Sonnerup [1974]. [59] The dayside reconnection rate is believed to control the driver of the magnetosphere (whatever mechanism that is). The resulting control function (expression (12) with (13)) is written in terms of upstream solar wind parameters It has no adjustable parameters and its derivation did not involve any tuning to optimize results. [60] The findings of this project are summarized as follows. [61] 1. From an understanding of the physics of the bow shock and an analysis of the magnetosheath flow, three of the four local parameters governing dayside reconnection (B m, B s, and r m ) were successfully described in terms of upstream solar wind conditions. [62] 2. The fourth local parameter governing dayside reconnection (magnetospheric mass density) has not yet been parameterized in terms of upstream solar wind conditions. [63] 3. A solar wind control function (expression (12)) was derived from the Cassak-Shay formula. With the aid of expression (13), the control function is expressed in terms of v o,b o, r o, and P o of the solar wind plus the IMF clock-angle q. [64] 4. The physical interpretation of the control function is that if anything in the solar wind drives or controls reconnection, it is solar wind pressure, which sets the magnetic field strength at the dayside magnetopause. This is particularly evident for low Mach number solar wind. [65] 5. At modest and high Mach numbers, the coupling function becomes more strongly dependent upon the solar wind magnetic field strength B o. This B o dependence is not because the solar wind electric field is v o B o, rather it is because of bow shock Mach number effects. [66] 6. According to the control function, the solar wind has much of the control of dayside reconnection and therefore of solar wind/magnetosphere coupling. However, through the magnetospheric plasma mass density, the Earth s magnetosphere also can control dayside reconnection and hence control solar wind/magnetosphere coupling. When r m >M A 0.87 r o, the magnetosphere exerts control. This is a further confirmation of the plasmasphere effect. [67] 7. Using the OMNI2 data set and seven geomagnetic indices, the solar wind control function was tested on its ability to describe the variances of the geomagnetic indices. The control function was found to be successful, as good as the best solar wind driver function in the literature. 7. Deficiencies and Future Work [68] This calculation of the solar wind control function was based on the picture that (1) reconnection is governed by local plasma parameters and (2) whatever controls those parameters also controls the reconnection rate. This picture opens a new pathway to understanding and calculating solar wind/magnetosphere coupling. The older picture of solar wind/magnetosphere coupling was that the solar wind electric field drove dayside reconnection, the newer picture is that reconnection is not so much driven as it is controlled, and it is not the solar wind electric field that does the controlling. The older picture had the Earth exerting some control over solar wind/magnetosphere coupling via the ionospheric conductivity which could result in polar cap saturation (but see Borovsky et al. [2008b] for a refutation of that picture): this newer picture has the Earth exerting additional control over solar wind/magnetosphere coupling via the magnetospheric mass density (the plasmasphere effect ). [69] Tests of the derived solar wind control function were successful, indicating that this new picture has merit. The derivation here is a first attempt, in that only reconnection at the nose of the magnetosphere was considered and several approximations were used. Because of its success on the first attempt, future work to derive a more accurate coupling function should be valuable to improving our understanding of solar wind/magnetosphere coupling and increasing our ability to predict magnetospheric activity. [70] Obvious avenues to the derivation of a more accurate coupling function would entail following: [71] 1. Improve the Cassak-Shay formula by using a derivation that includes Ohmic heating in the energy conservation consideration. [72] 2. Try other hybrid Alfven speeds to see if a formula can be found that is more accurate than the Cassak-Shay formula at describing the rate of reconnection. Birn et al. [2008] found that the empirical formula R K2m 1=2 o B 3=2 m B3=2 m r 1=2 x ðb s þ B m Þ 1 ; ð21þ where r x is the mass density at the x line, is slightly more accurate than the Cassak-Shay formula (expression (1)) when tested against reconnection measurements in 2-D and 3-D MHD simulations. Using expression (21) for predictive purposes will require the extra step of parameterizing r x in terms of r m and r s. [73] 3. Run 3D MHD computer simulations of the Earth s magnetosphere to test the accuracy of the sin 1 (q/2) factor that was added onto the Cassak-Shay formula in a cavalier 12 of 14