Solar cycle dependence of solar wind energy coupling to the ...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116, A06302, doi:10.1029/2011JA016437, 2011

Solar cycle dependence of solar wind energy coupling to the thermosphere William J. Burke1 Received 3 January 2011; revised 10 February 2011; accepted 14 March 2011; published 1 June 2011.

[1] Solutions to the differential equations describing the behavior of driven‐dissipative systems are compared with measured exospheric temperatures (T∞) and provisional Dst indices acquired during 38 magnetic storms between mid‐2002 and 2008. The only storm selection criterion was the availability of solar wind and interplanetary magnetic field data to compute driving electric fields "VS. Globally averaged T∞ was inferred from measurements by accelerometers on the GRACE satellites. Statistical regression analyses indicate that the coupling coefficients for T∞, Dst, and their ratio are well represented as functions of 81 day averaged F10.7a. Using Dst as the driver, this functional relationship yielded reasonable estimates of the evolution of T∞ during the Halloween 2003 magnetic storm. Linear relations between T∞ and the total energy of the thermosphere (Eth) and between Dst and the energy of the ring current (ERC) allow estimates of the storm time energy partitioning. Empirical estimates of the energy coupling coefficients for the thermosphere (aE) and ring current (aERC) span the ranges 1.5–0.2 and 0.5–0.2 TW/mV/m, respectively. Outside of extreme solar minimum conditions, main phase increases in Eth exceed those of ERC. Citation: Burke, W. J. (2011), Solar cycle dependence of solar wind energy coupling to the thermosphere, J. Geophys. Res., 116, A06302, doi:10.1029/2011JA016437.

1. Introduction [2] This is the ninth in a series of papers exploring empirical relations between interplanetary forcing and global thermospheric responses during magnetic storms. The studies were designed to help improve models used to specify and/or predict atmospheric drag exerted on space objects in low‐Earth orbit (LEO) for collision avoidance [Wright, 2007]. The adopted electric field parameterization of interplanetary forcing includes storm time saturation of the cross‐ polar cap potential [Siscoe et al., 2002] and is rooted in measurements by plasma [McComas et al., 1998] and magnetic field [Smith et al., 1998] sensors on the Advanced Composition Explorer (ACE) satellite, near the first Lagrange point (L1). Thermospheric responses, in the form of increased globally averaged exospheric temperatures (T∞) and total energy (Eth) contents of the thermosphere [Burke, 2008], are estimated, via the J77 [Jacchia, 1977] model, from orbit‐averaged mass densities (r) measured by accelerometers on the polar‐orbiting Gravity Recovery and Climate Experiment (GRACE) satellites [Tapley et al., 2007]. [3] Burke et al. [2007a] compared thermospheric densities inferred from GRACE accelerometer measurements during the large magnetic storm of November 2004 with predictions of standard thermospheric models used within the

operational community. While the models reproduced most features of GRACE measurements before the disturbed period, errors exceeded 100% during the storm’s main and early recovery phases. However, traces of the magnetospheric electric field "VS [Burke, 2007] and −Dst tracked orbit‐averaged densities measured by GRACE. Burke [2007] approximated the Volland‐Stern electric field "VS in the equatorial plane of the magnetosphere as the polar cap potential FPC [Siscoe et al., 2002] divided by the width of the magnetosphere along the dawn‐dusk meridian 2RELY. The parameter LY represents the distance in Earth radii (RE) between the Earth’s center and the magnetopause at 18:00 and 06:00 LT. Burke [2008] examined consequences of a suggestion by Wilson et al. [2006] that the storm time thermosphere acts like a large thermodynamic system that on a global scale never strays far from diffusive equilibrium. This analysis demonstrated that if r is measured at known altitudes (h), the J77 model allows simple estimates of T∞, and that Eth is linearly related to T∞. Further analysis [Burke et al., 2009, 2010b] showed that (1) Eth has two independent but additive contributions from solar extreme ultraviolet (EUV) fluxes Eth UV and from the solar wind driving Eth SW and (2) during the magnetic storms of 2004 Eth SW responses to "VS variations mimicked those of driven‐dissipative systems. The differential equation governing the temporal (t) evolution of such systems has the form

1 Institute for Scientific Research, Boston College, Chestnut Hill, Massachusetts, USA.

dEthSW EthSW ¼ E "VS : dt E

ð1Þ

Copyright 2011 by the American Geophysical Union. 0148‐0227/11/2011JA016437

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[4] The coupling aE and relaxation t E coefficients were found empirically to be 1.58 TW/mV/m and 6.5 h, respectively [Burke et al., 2009]. Eth SW traces provided by numerical solutions of equation (1) closely resembled those derived from GRACE measurements during the large magnetic storms of July and November 2004. Independent confirmation of this approach was obtained while comparing aE"VS, the power input term in equation (1), with independently derived predictions of the Weimer [2005] Poynting flux model. [5] Bowman et al. [2008] pointed out that operational modelers use TC the global minimum exospheric temperature rather than Eth to estimate thermospheric densities and require a more reliable provider of driving parameters than ACE. Burke [2008] showed that within the J77 model, the globally averaged exospheric temperature T∞ ≈ 1.15 TC. Burke et al. [2010b] argued that the linear relationship Eth SW (J) = 8.78 • 1013 T∞ SW (°K) and the fact that a similar equation describes the storm time development of Dst [Burton et al., 1975] allow estimations of T∞ SW and TC, using Dst as a driver. Burke et al. [2007b] had previously demonstrated that "VS can be used to drive the Dst equation of Burton et al. [1975]. The governing equations take the form dT∞SW T∞SW ¼ T "VS dt T

ð2Þ

dDst Dst ¼ D "VS : dt D

ð3Þ

and

By eliminating "VS from (2) and (3), globally averaged T∞ SW can be determined numerically as a function of Dst: T∞ SW ðtnþ1 Þ ¼

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BURKE: SOLAR WIND ENERGY COUPLING

Dtn T∞ SW ðtn Þ 1 T T Dtn Dstðtn Þ ; Dstðtnþ1 Þ 1 þ D D

ð4Þ

where Dtn = tn+1 − tn. For Dst, Dtn = 1 h. To avoid operational reliance on ACE measurements, equations (3) and (4) use Dst rather than the pressure‐corrected Dst* introduced by Burton et al. [1975]. Equation (4) indicates that the temporal profile of Dst allows quantitative estimates of storm time T∞ SW(t) and thus, rth(h, t). Bowman et al. [2008] validated this conjecture by comparing Dst‐based predictions of T∞ SW(t) and rth(h, t) with values inferred from accelerometers on the GRACE and Challenging Minisatellite Payload (CHAMP) satellites during magnetic storms in 2004 and 2005 as well as with densities obtained from the drag exerted on 75 LEO reference satellites. The latter information is encapsulated in the High Accuracy Satellite Determination Model (HASDM) [Casali et al., 2002; Storz et al., 2002]. Using Dst as a driver, Bowman et al. [2008] demonstrated that predicted storm time density errors fell from >65% to about 15% with respect to HASDM predictions. Since Dst more accurately specifies the state of the thermosphere during magnetic storms than the currently used ap index, Bowman et al. [2008] recommended that it be adopted for operational use.

[6] To be useful for orbit tracking, Dst must be available in near real time. Recently, Burke et al. [2011] demonstrated a simple method for extracting accurate estimates of provisional Dst using horizontal magnetic perturbations (DBH) measured at times when spacecraft of the Defense Meteorological Satellite Program (DMSP) cross the dip equator. Data were restricted to years 2006 through 2008 when measurements from DMSP flights 16 and 17 were available. Dip equator crossing points are identified in DMSP data streams at times when the vertical component of the Earth’s field changes polarity. The correlation coefficient between provisional Dst and the DMSP‐based proxy was ∼0.96. However, when equation (4) was solved setting DBH ≈ Dst, computed T∞ SW traces systematically exceeded values inferred from GRACE measurements. This was no surprise. With a smaller database, Burke et al. [2010a] reached a similar conclusion studying high‐speed stream effects on T∞ SW with "VS as the driver. [7] Two contributing sources for the discrepancy were considered. First, during early studies, GRACE data for ∼180 days in 2004 were available at the Air Force Research Laboratory (AFRL). Derived densities were based on preflight calibrations conducted at the University of Texas (UTX). Sutton [2009] recalibrated the GRACE accelerometers in flight, and applied this calibration to all measurements acquired between launch in mid‐2002 and the end of 2008. Figure 9 of Burke et al. [2011] shows that during the common period in 2004 the UTX and Sutton calibrations were highly correlated. UTX calibrations produced systematically higher densities than did Sutton’s. However, the maximum T∞ SW difference estimated via the two calibration methods was ∼13%, significantly less than differences between observed and modeled T∞ SW during solar minimum storms. [8] The second possible explanation for the discrepancy is that the coupling coefficient aT varies over the solar cycle. Since aT / aE, physically this would indicate that the efficiency of energy coupling between the solar wind and the thermosphere also varies under different solar cycle conditions. In fact, electromagnetic energy (Poynting flux) at ultralow frequencies can only be transferred to the thermosphere via ion‐neutral collisions. Thus, aT should vary with the plasma content and consequent conductance of the ionosphere. Both parameters are higher near solar maximum than near solar minimum. Burke et al. [2011] pointed out that T∞ SW peaks near times of main‐phase minima in Dst when time derivatives in equations (2) and (3) vanish. Thus, T D T∞SW max : D T Dstmin

ð5Þ

T∞ SW and Dst are evaluated at their storm time extrema. Figure 10 of Burke et al. [2011] shows a scatterplot of aT/ aD as a function of time, determined by visual inspection of T∞ SW and Dst extrema during 13 magnetic storms between 2003 and 2006. Plotted aT/aD distributions are qualitatively consistent with the conjecture that the efficiency of coupling decreases significantly between solar maximum and minimum. [9] Recently, Weimer et al. [2011] specified the storm time variability of TC and its relation to the total electro-

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magnetic power (H(t)) into the ionosphere‐thermosphere by analyzing 4 years (2003–2006) of obit‐averaged values inferred from measurements of accelerometers on the CHAMP and GRACE satellites. H(t) was obtained by integrating Poynting fluxes, inferred from ACE measurements, over northern and southern high latitudes [Weimer, 2005]. They solved a numerical version of equation (2) for DTC (≈ T∞ SW /1.15) the storm time increases in TC above baselines due to solar EUV fluxes. Their driving term bH(t) should be proportional to aT "VS in equation (2). The main results of their analyses were that (1) TC inferred from CHAMP and GRACE measurements were different but highly correlated and (2) the recovery time coefficient t T was shorter during intense storms than in weak ones. They attribute the difference to rapid thermospheric cooling consequent to the more efficient production of NO molecules during large storms. No mention is made of how, with the same interplanetary magnetic field (IMF)–solar wind drivers, power input to the ionosphere, bH(t), might vary over the studied interval. [10] The objective of this report is to quantify the solar cycle variability of the coupling coefficients aT and aD based on an analysis of 38 magnetic storms spanning the period between mid‐2002 when GRACE was launched through the end of 2008. The only selection criterion was that plasma and field measurements needed to calculate "VS were available from ACE. Section 2 describes the methodology developed to support calculations and provides examples of its application. Section 3 shows that aT, aD and aT/aD are well represented as functions of the F10.7 index averaged over the preceding 81 days (F10.7a). Supporting evidence is presented by applying the solar cycle corrected aT/aD term in equation (4) to estimate T∞ SW during the Halloween Storm of 2003 and by comparing results with GRACE data. Section 4 (1) summarizes our empirically based conclusions, (2) considers the significance of the reported F10.7a dependence on the efficiency of solar wind energy coupling to the storm time thermosphere and ring current, and (3) comments on similarities/differences with results of Weimer et al. [2011].

2. Methodology [11] Data used in this study have three sources: (1) "VS(t) estimates are derived from hourly averaged IMF components and solar wind densities/speeds measured by sensors on ACE [Burke, 2007] (for application in equations (1)–(3), calculated "VS values are time stepped by 1 h to allow for transport between L1 and the dayside magnetopause); (2) orbit‐ averaged T∞ values are based on orbit‐averaged thermospheric mass densities measured by GRACE [Burke, 2008] (brief summaries of the methods used to calculate "VS and T∞ are given in Appendices A and B, respectively); and (3) provisional Dst indices were obtained from the University of Kyoto’s World Data Center. The analysis proceeded in six steps, as follows. [12] 1. All magnetic storms from mid‐2002 through 2008 were identified in Dst traces. [13] 2. Thirty‐eight storms for which ACE data allowed "VS (t) calculations were selected for analysis.

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[14] 3. Equations (2) and (3) were initially solved for the 38 storms using "VS (t) as the driver and with aT and aD equal to 55 (°K/h/mV/m) and −40 (nT/h/mV/m), respectively. These values were obtained in analyses of GRACE data during the 2004 storms for which UTX‐calibrated data were available at AFRL [Burke et al., 2009]. [15] 4. Orbit‐averaged values of Eth and T∞ were calculated using Sutton‐calibrated GRACE data for 2002 through 2008. Eth UV and T∞ UV were estimated empirically from prestorm values to obtain Eth SW = Eth − Eth UV and T∞ SW = T∞ − T∞ UV. Note that Eth UV and T∞ UV were treated as constants across the main and early recovery phases of storms. [16] 5. Modeled T∞ SW and Dst extrema were compared with observed storm time counterparts. Values of aT and aD needed to align modeled T∞ SW and Dst extrema with observations were then calculated. These calculations were facilitated by treating t T and t D as the constants established in 2004 storms, and using the self‐similarity of solutions to equations (2), (3), and (4). [17] 6. Scatterplots of corrected aT, aD and aT/aD as functions of F10.7 and its 81 day running average F10.7a were constructed and subjected to least square analyses. [18] Because of its centrality to calculations summarized below, it is useful to digress to demonstrate that solutions of equations (1)–(4) are self‐similar when the coupling coefficients change but the driving time series and relaxation time coefficients do not. That is, if a coupling coefficient changes from a1 to a2, while maintaining the same driving "VS (equations (1), (2), and (3)) or Dst (equation (4)) time series, then the calculated dependent variables Eth(t), T∞ SW(t) or Dst(t) are proportional to the ratio a1/a2. [19] Figure 1 shows examples of such calculations using drivers measured during the magnetic storm of Julian Day (JD) 348–350 (14–16 December) 2006. Plots in Figure 1a show two sets of solutions of equation (1) to measured "VS (t) inputs (dotted line) with aE arbitrarily set at 0.6 (dashed line) and 0.4 (solid line). Traces of modeled Eth, given in units of 1016 J, appear very similar in shape. Results of calculations based on "VS inputs for all of 2006 are presented in Figure 1b in the form of a scatterplot of Eth obtained with aE = 0.4 versus aE = 0.6. A least squares linear regression analysis shows an intercept that is close to zero, a slope that is equal to the ratio of the coupling coefficients, and the regression coefficient is R = 1. Plots in Figure 1c show Dst‐driven (dotted line) solutions to equation (3). The T∞ SW traces represent calculated responses with the coupling parameter aT /aD arbitrarily set at −1.75 (solid line) and −2.333 (dashed line). The linear regression analysis in Figure 1d compares T∞ SW solutions obtained during the 10 day interval centered on the storm period. Again, the intercept is ∼0, the slope equals the ratio of aT/aD, and R = 1. [20] The analysis presented below makes statistical comparisons between time series of T∞ SW, Dst, "VS, F10.7, F10.7a and coupling coefficients. While Dst and "VS are provided at a 1 h cadence, T∞ SW are given once per GRACE orbital period (∼95 min) and F10.7 once per day. Data streams were broken into several‐day segments centered on the 38 storm periods. Using an algorithm within the Kaleidagraph™ application, T∞ SW and F10.7 were interpolated to 1 h cadences. The interpolated T∞ SW data entries included estimates at

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Figure 1. Demonstration of self‐similarity of solutions for equations (1)–(4). (a) Traces show numerical solutions of equation (1) during a 2 day storm in December 2006. The "VS driver is represented by the dotted line; Eth solutions with aE = 0.6 and 0.4 (TW/mV/m) are represented by dashed and solid lines. (b) Correlation analysis of Eth obtained with two values of aE. (c) T∞ solutions of equation (3) obtained with Dst driver during the same 2 days with two aT/aD ratios. (d) Correlation analysis of T∞ obtained with two values of aT/aD. both 1 h intervals and at the times of GRACE entries. In all cases, the correlation coefficient between measured and interpolated T∞ SW values exceeded 0.99.

3. Data Presentation [21] Section 3 has two main segments. The first illustrates the methodology summarized above through detailed comparisons between measurements and solutions of equations (2), (3), and (4) at the time of the December 2006 magnetic storm. Two modeled solutions are presented with aT and aD set at values used by Burke et al. [2010a] and with solar cycle corrected values. The second segment provides statistical comparisons of the adjusted aT, aD and aT/aD values for 38 magnetic storms with F10.7a. As an independent test of the statistical results, aT/aD determined from F10.7a at the time of the Halloween 2003 storm was used to calculate T∞ SW using Dst as the driver. The Halloween storm was not included in the 38‐storm database. High fluxes of MeV solar protons penetrated SWEPAM’s shielding, making it difficult to specify the solar wind densities needed to calculate "VS [Gopalswamy et al., 2005; Skoug et al., 2004]. [22] Traces in Figure 2 illustrate steps used to solve equation (2) and compare the results with relevant GRACE measurements during the magnetic storm of December 2006. The trace in Figure 2 (bottom) shows hourly averaged "VS plotted as a function of universal time during Julian Days (JD) 348–350. "VS has a double‐peaked distribution, rising to a maximum of ∼1 mV/m early on JD 349. The trace in Figure 2 (middle) shows orbit‐averaged values of T∞ derived from GRACE measurements of r. The dotted line at

∼830° K approximates the baseline exospheric temperature T∞ UV. Figure 2 (top) contains plots of three parameters. The dotted line show GRACE‐based estimates of T∞ SW = T∞ − 830° K, interpolated to a 1 h cadence. Dashed and solid line traces are solutions of equation (2) with aT set at 55 and 30 (°K/h/mV/m), respectively. The latter value was found to replicate T∞ SW values derived from GRACE measurements during the storms of 2004. The revised value of aT = 30 (°K/ h/mV/m) yields good agreement between observed and modeled T∞ SW except near the time of the first rise in "VS late on JD 348. A linear least squares analysis of these data (not shown) yielded a correlation coefficient R = 0.9. [23] Data plotted in Figure 3 compare solutions of equation (3) using Dst (dashed and solid lines) as the dependent variable, with provisional Dst (dots). The bottom trace repeats the temporal evolution of "VS. The solid line trace, with an adjusted value of aD = −34 (nT/h/mV/m), is in close agreement with provisional Dst during the main and early recovery phases of the storm. Figure 4 (bottom) shows the provisional Dst driver for equation (4). Figure 4 (top) contains plots of measured T∞ SW (dots) and its computed values using aT/aD = −1.57 (black line) and aT/aD = −0.88 (solid line) (°K/nT), respectively. Note that the solar cycle adjusted coupling coefficient in equation (4), the ratio of adjusted values of aT = 30 (°K/h/mV/m) and aD = −34 (nT/ h/mV/m) shown in Figures 2 and 3 yield good agreement between the modeled T∞ SW and values inferred from GRACE measurements. [24] Procedures similar to those implemented in Figures 2 and 3 were performed for all 38 storms. The results are summarized in Table 1. From left to right columns of Table 1

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be performed on any of the 38 storms. However, "VS estimates were largely unavailable from 28 October through 1 November 2003, so the Halloween storm was not included in the 38‐case database and thus is a reasonable test candidate. Figure 6 (bottom) shows a trace of the Dst index across the storm period. The storm had two deep main phases with minima of −363 nT (303:00) and −401 nT (303:22). The assigned value of aT/aD = 1.33 resulted from substituting the prevailing value of F10.7a = 140 into the regression equation found in Figure 5c. The modeled and observed T∞ SW shown in Figure 6 (top) had similar values near and after the first main‐phase minimum. As seen in studies of the 2004 storms, the measured T∞ SW relaxed at a faster rate than the model based on Dst [Burke et al., 2009]. Recall that, like the present case, −Dst tracked orbit‐averaged storm time density variations through the main and early recovery phases. In the second main phase the model overestimated the maximum T∞ SW by ∼100° K but underpredicted the duration of the thermospheric heating. In this case, these represent off‐setting errors for approximating thermospheric drag effects on space objects.

4. Discussion

Figure 2. Observed and modeled determinations of T∞ SW during the December 2006 magnetic storm. (bottom) Values of "VS calculated using hourly averaged plasma and magnetic field measurements from ACE. (middle) Orbit‐averaged exospheric temperatures derived from GRACE accelerometer measurements (solid line) and the approximate contribution of T∞ UV (dashed line). (top) The dotted line shows measured T∞ SW, and the dashed and solid lines give the modeled T∞ SW estimated with aT values, in 55 and 30 (°K/h/mV/m), appropriate for solar maximum and minimum, respectively.

list the year, initial and final storm days, minimum Dst, approximate T∞ UV, adjusted coefficients aT, aD, aT/aD and F10.7a. After a period of trial and error, it became evident that the coupling coefficients were well organized by the square root of F10.7a. Figure 5 shows scatterplots of aT (Figure 5a), aD (Figure 5b), and aT/aD (Figure 5c) as functions of F1/2 10.7a; dashed lines indicate least squares quadratic fits to the three distributions. Analytic representations of these fitting lines and correlation coefficients are given in the three panels. Note that the correlation between aT and F1/2 10.7a is quite high, R = 0.94. Somewhat surprising was the substantial, albeit significantly lower, correlation (R = 0.78) between aD and F1/2 10.7a. Similar least squares analyses were performed using F10.7 values for the actual storm days. However, the correlations were not quite as good as those shown in Figure 5. Comment on the possible significance of F10.7a correlations is deferred to section 4. [25] It is useful to consider an independent test of the validity of procedures developed above. Since these procedures were designed to maximize fits, useful tests cannot

[26] Sections 1–3 outline and implement procedures designed to identify solar cycle variations of the coupling coefficients in equations (2) and (3). This was accomplished by solving the equations after applying coupling coefficients identified in previous studies of magnetic storms in 2004 based on orbit‐averaged densities derived from UTX calibrations of accelerometers on the GRACE satellites. The present analysis used recalibrated data from GRACE [Sutton, 2009] acquired during 38 magnetic storms between mid‐ 2002 through the end of 2008 for which plasma and IMF measurements needed to calculate "VS were available from ACE. The self‐similarity of solutions for the governing

Figure 3. Comparison of measured and modeled Dst during the December 2006 storm. (top) Plots show reported provisional Dst (dots) and solutions of equation (2) with aD values appropriate for solar maximum (dashed line) and minimum (solid line). (bottom) The "VS trace is provided for reference.

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ticles map to the inner magnetosphere. On the other hand, the correlation may simply reflect lower levels of interplanetary driving during solar minimum. From operational considerations, it is useful that the ratio aT/aD is highly correlated with F1/2 10.7a. Because F10.7a changes slowly, over the several‐day durations of magnetic storms aT/aD can be treated as having a nearly constant value. Application of equation (4) to the Halloween 2003 storm, using F10.7a to compute aT/aD, yielded a reasonable, though imperfect, approximation to the storm time history of T∞ SW inferred from GRACE accelerometer measurements. [27] We next consider some implications of these results for understanding the energetics of the storm time thermosphere and ring current. After observing an empirical relationship between Dst and the evolution of thermospheric densities during the November 2004 magnetic storm, Burke et al. [2007a] speculated that perhaps the storm time ring Figure 4. Comparison of measured and modeled T∞ SW during the December 2006 storm. (top) Plots show measured T∞ SW (dots) and solutions of equation (3) with aT/aD values of 1.57 and 0.88 (°K/nT) appropriate for solar maximum (dashed line) and minimum (solid line). (bottom) A trace of provisional Dst is provided for reference. differential equations allows adjustments of aT and aD to obtain good agreement between extrema in modeled and observed T∞ SW and Dst. The ratios of adjusted coefficients aT/aD were then calculated to support solutions of equation (4) in which Dst rather than "VS acts as the driving parameter. Adjusted aT coefficients listed in Table 1 were found to be well organized as functions of F 1/2 10.7a . Robinson and is proportional to the Vondrak [1984] argued that F1/2 10.7 ion‐electron production rate and the ionospheric conductance due to solar EUV fluxes. Using Chatanika incoherent radar backscatter measurements, they determined that the Pedersen and Hall conductances are well approximated by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑P = 0.88 · F10:7 Cos and ∑H = 1.5 · F10:7 Cos, respectively. The symbol c represents the local solar zenith angle. In the present case, estimating the effective global ionospheric conductance probably requires averaging of solar zenith angle contributions over the entire dayside. As such, the empirical results shown in Figure 5a are consistent with the conjecture that the observed decline in coupling efficiency during solar minimum reflects a reduction in the number of ion‐electron pairs needed to transfer energy and momentum to thermospheric neutrals. Although the correlation between F1/2 10.7a and aD is weaker than with aT, still its existence came as something of a surprise. The most thorough empirical investigations of the applicability of the Burton et al. [1975] equation were conducted by Temerin and Li [2002, 2006]. Although the studies covered the period 1995 to 2002, neither report mentions an explicit solar cycle dependence of Dst. It is difficult to determine whether the correlation in Figure 5b with F10.7a is real or simply fortuitous. If real, it may reflect a still underappreciated influence of ionospheric conductance on the distribution of electric potential in the global ionosphere [Nopper and Carovillano, 1978]. This in turn would reflect how electric fields that energize and transport ring current par-

Figure 5. Scatterplots of (a) aT, (b) aD, and (c) aT/aD as functions of F1/2 10.7a for 38 magnetic storms that occurred between mid‐2002 and 2008. Inserted equations and dashed lines indicate least squares quadratic fits of these quantities as functions of F1/2 10.7a.

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Table 1. Solar Cycle Corrected Coupling Coefficients Obtained for 38 Magnetic Storms From Mid‐2002 Through 2008 Year

Initial Day

Final Day

Dstmin (nT)

2002 2002 2002 2002 2002 2002 2003 2003 2003 2003 2004 2004 2004 2004 2004 2004 2005 2005 2005 2005 2005 2005 2006 2006 2006 2007 2007 2007 2007 2007 2007 2007 2008 2008 2008 2008 2008 2008

246 250 272 276 297 324 149 168 229 324 22 204 208 242 312 314 126 134 148 163 235 243 98 103 348 142 190 194 218 295 322 350 31 67 86 165 193 245

248 252 276 278 299 327 151 170 231 326 24 210 210 246 314 316 130 136 152 165 239 245 101 105 350 150 193 196 221 305 326 360 37 70 89 170 195 250

−105 −170 −183 −140 −90 −127 −119 −145 −163 −467 −149 −95 −197 −125 −368 −288 −127 −263 −135 −106 −215 −128 −80 −108 −144 −62 −38 −45 −33 −51 −69 −37 −41 −70 −41 −37 −39 −49

T∞

1025 1050 1000 1000 1040 1025 880 850 875 950 850 810 850 875 875 855 830 840 830 840 840 840 830 830 830 815 810 810 805 817 815 807 805 810 825 805 803 805

current acts as a reservoir from which the storm time thermosphere draws energy. The J77 model offered a means of testing and rejecting that conjecture. Burke [2008] showed that during the November 2004 storm Eth increased by about four times the energy accumulated in the ring current (ERC). Equations (1)–(4) and the statistical relationships found in Figure 5 allow a more general understanding of how solar wind energy is partitioned between the thermosphere and the ring current. Since the J77 model yields a linear relationship Eth SW (J) = 8.727 1013 T∞ SW (°K), the energy coupling coefficient in equation (1) can be written 1 0 1 J K B C B C E @hr mV A ¼ 8:727 1013 T @ hr mV A; m m

=

=

0

ð6Þ

expressed in more convenient power units for comparison with Weimer [2005]: 0

1 0 1 K K 87:27 B C B C T @ hr mVA 0:024 T @ hr mV A: ¼ 3600 m m ð7Þ

=

!

=

E

TW . mV m

UV

aT

aD

aT/aD

F10.7a

52 63 55 55 61 48 44 48 45 57 42 44 44 33 57 47 43 39 31 32 38 31 33 30 30 25 10 12 11 15 20 20 15 26 28 13 8 11

−36 −40 −40 −40 −36 −30 −31 −40 −38 −49 −29 −33 −37 −40 −44 −41 −32 −44 −31 −34 −37 −38 −40 −30 −34 −27 −19 −21 −19 −30 −21 −23 −23 −32 −24 −22 −21 −22

−1.444 −1.575 −1.375 −1.375 −1.694 −1.600 −1.419 −1.200 −1.184 −1.163 −1.448 −1.333 −1.189 −0.825 −1.296 −1.146 −1.344 −0.886 −1.000 −0.941 −1.027 −0.816 −0.825 −1.000 −0.882 −0.926 −0.526 −0.571 −0.579 −0.500 −0.952 −0.870 −0.652 −0.813 −1.167 −0.591 −0.381 −0.500

182 179 174 173 165 165 126 125 119 138 113 112 112 108 105 106 93 94 98 96 89 89 82 82 85 74 72 71 70 68 72 75 72 72 71 67 66 67

The regression equation in Figure 5a indicates that E

TW . mV m

! ¼ 6:15 þ 1:17

pffiffiffiffiffiffiffiffiffiffiffi F10:7a 0:045 F10:7a :

ð8Þ

Combining aT data listed in Table 1 with equation (7) indicates that aE ranged between 1.512 and 0.192 TW/ mV/m during the JD 250–252 2002 and JD 193–195 2008 storms, respectively. Equation (8) provides guidance that should not be interpreted too literally. It gives a least squares fit for coupling coefficients available in storms between mid‐2002 and the end of 2008 when F10.7a ranged between 182 and 68 flux units. By construction the energy coupling coefficient must be a positive definite quantity, aE ≥ 0. However, equation (8) has roots aE = 0 at F10.7a equal to 54 and 348 flux units and would be negative outside this range. The regression equation also yields a maximum aE = 1.45 TW/mV/m at F10.7a = 170 flux units. This is ∼8% less than the 1.58 TW/mV/m estimated by Burke et al. [2009] based on GRACE measurements during the July and November 2004 storms. During these storms calculated total power into the global thermosphere aE "VS was found

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Figure 6. (top) Comparison of measured T∞ SW (dots) with solutions of equation (3) (solid line) obtained by setting aT/aD = 1.33 (°K/nT) as determined from the prevailing F10.7a ≈ 140. (bottom) The Dst trace is provided for reference.

to be in substantial agreement with the Poynting flux predictions of Weimer [2005]. However, this estimate is within the 13% difference obtained for T∞ between UTX and Sutton [2009] calibrations of GRACE accelerometers [Burke et al., 2011]. [28] Energy provided to the storm time ring current can be estimated from the Dst index using the Dessler‐Parker‐ Sckopke (DPS) relation [Dessler and Parker, 1959; Sckopke, 1966; Carovillano and Siscoe, 1973]. While the accuracy of the DPS relation remains a subject of debate [cf. Liemohn, 2003, and references therein], Vasyliunas [2006a, 2006b] pointed out that the Burton et al. [1975] equation describes the progression of storm time Dst precisely because of the underlying relation between ERC and Dst captured by DPS. For the practical purposes of the present study, we use a simple formula found in the review by Stern [2005], ERC (J) ≈ 3.8 1013 Dst (nT). This information can be substituted into equation (3) to obtain an equation, analogous to (1) describing the evolution of ERC with a coupling coefficient aERC:

=

ERC

0 1 , ! nT TW C mV ¼ 0:0106 D B @ hr mV A: m m

ð9Þ

The dimensionless ratio aE/aERC ≈ −2.3 aT/aD can be substituted into the regression equation found in Figure 5c to obtain . E pffiffiffiffiffiffiffiffiffiffiffi ERC ¼ 8:98 þ 1:82 F10:7a 0:066 F10:7a :

ð10Þ

[29] Combining aD data listed in Table 1 with equation (9) indicates that estimated aERC ranged from a high of 0.52

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and a low of 0.2 TW/mV/m during the storms of JD 324–326 2003 and JD 218–221 2007, respectively. The relationship expressed in equation (10) indicates that at very low values of F10.7a (≤57 flux units) aE/aERC < 1, so that the ring current would extract energy from the storm time solar wind at a faster rate than the thermosphere. Conversely, as F10.7a approaches solar maximum values, power extraction favors the thermosphere by a factor of ∼3.6. [30] The present analysis and that of Weimer et al. [2011] should be regarded as complementary reflections on the application of equation (2) toward operational modeling of the storm time thermosphere. Where this study concentrated on the solar cycle dependence of driving terms, Weimer and coworkers focused on the relaxation terms. They provide plausible arguments that the production of NO molecules is more effective during solar maximum storms due to the higher particle‐energy inputs at auroral latitudes. Higher concentrations of NO in turn lead to faster relaxation times (t T) during solar maximum storms than during solar minimum storms. In the present study t T was assigned a constant value of 6.5 h, which is close to the solar maximum value determined by Weimer et al. [2011] through least squares analyses. This choice did not have the unintended consequence of creating a compensating, solar cycle dependence in aT. The opposite is probably true. In underestimating solar minimum t T, the present analysis may have slightly overestimated corresponding values of aT as F10.7a decreased.

5. Summary and Conclusions [31] This paper reports on thermospheric and magnetospheric responses to interplanetary driving during 38 magnetic storms between mid‐2002 and the end of 2008. The sole criterion for storm selection was the availability of ACE data required to calculate "VS. Using the self‐similarity of solutions to equations (2), (3), and (4) the analysis demonstrated that the coupling coefficients aT, aD and aT/aD are functions of the F10.7 index averaged over the previous 81 days. This indicates that the ionospheric conductance regulates the flow of energy to both the thermosphere and ring current. Except during extreme solar minimum the thermosphere’s energy budget exceeds that of the ring current. [32] A next logical step, that lies beyond the scope of this report, is to synthesize the approaches developed by Weimer et al. [2011] and here to quantify energy transferred from the interplanetary medium to the magnetosphere‐ionosphere‐ thermosphere system. Such a synthesis must go beyond the simple fix of parametrizing changes in t T and possibly t D over the solar cycle. Relationships between expressions for the driving terms aT "VS and bHJ [Weimer et al., 2011, equation (6)] should also be reviewed. The "VS driver for equation (2) was derived by Siscoe et al. [2002] to help explain saturated polar cap potentials FPC observed during the Bastille Day 2000 storm and was initially validated during the March 2001 storm [Ober et al., 2003]. Model predictions compared favorably with DMSP measurements of FPC during most large storms between 2000 through the end of 2006 [Burke et al., 2007b]. Similar comparisons of Siscoe‐Hill predictions of FPC with measurements have not yet been conducted for the smaller storms of solar minimum associated with high‐speed streams in the solar wind.

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[33] Electric and magnetic fields measured by sensors on the Dynamics Explorer 2 (DE‐2) satellite constitute the empirical basis for the Weimer [2005] model. Since DE‐2 was primarily a solar maximum mission questions about the model’s applicability during deep solar minima remain open. That is, given the same IMF/solar wind conditions, does the same Poynting flux energy enter the high‐latitude ionosphere‐thermosphere? The approach of Weimer et al. [2011] seems to assume that it does; results of the present study suggest otherwise. This indicates that a solar maximum to minimum comparison of Poynting fluxes into the high‐latitude ionosphere along the lines of Weimer [2005] should prove useful.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi potential term is FS (kV) = 1600 3 PSW ðnPaÞ/∑P(mho), where SP is the effective Pedersen conductance of the high‐latitude ionosphere. Burke [2007] found that setting SP ≈ 10 mho gave consistently good agreement between FPC predictions and values inferred from DMSP measurements.

Appendix B: Exospheric Temperature Calculations [37] By treating table entries to the J77 model as “real data,” Burke [2008] found that over the ranges 300 ≤ h ≤ 500 km in altitude, and 700 to 2000 °K in exospheric temperature, a quadratic relationship exists between thermospheric mass densities r(h) and the exospheric temper2 P ai(h)ri(h). T∞ is in degrees Kelvin, r in ature, T∞ = i¼0

Appendix A: Volland‐Stern Electric Fields [34] As formulated by Ejiri [1978] the Volland‐Stern (V‐S) model offers a simple, but limited, method for estimating the

kilograms per cubic meter, and h in kilometers. The coefficients ai(h) are also well represented by fifth‐order polynomials in h: 0

1 0 a0 ðhÞ 28:10 @ a1 ðhÞ A ¼ @ 4:733 1017 a2 ðhÞ 3:2695 1032 0

2:69 4:312 1015 4:620 1030

2:03 103 1:372 1013 2:618 1028

trajectories of ring current ions in the inner magnetosphere. One difficulty in applying the model concerned the connection between the interplanetary electric field (IEF) and its magnetospheric manifestations. Burke [2007] reformulated V‐S by combining it with the Siscoe‐Hill model of the polar cap potential (FPC) to show that in the absence of shielding, the electric field in the inner magnetosphere is given by "VS ≈ FPC/2RELY, where the denominator represents the width of the magnetosphere along the dawn‐dusk line. Empirical studies indicate that the dayside, equatorial magnetopause is nearly self‐similar in shape with LY ≈ 1.5 LX [Elsen and Winglee, 1997]. Force balance at the subsolar magnetopffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pause requires that LX ≈ 9.6/ 6 PSW ðnPaÞ and LY ≈ 14.4/ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 6 PSW ðnPaÞ. [35] Between 1980 and 2000 many investigators found that measured cross‐polar cap potentials linearly correlated with interplanetary parameters: FE(kV) = F0 + LG VSWBTSin2 2 . Here F0 ≈ ∼25 kV is a residual potential probably boundary layer, BT = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidue to the low‐latitude B2Y þ B2Z , and = Cos−1(BZ/BT) is the IMF clock angle in the GSM Y‐Z plane. If VSW BT is expressed in kV/RE (where 1 mV/m ≈ 6.4 kV/RE) then the regression slope LG ≈ 3.5 RE represents the width of a “gate” in the solar wind through which geoeffective streamlines (equipotentials) must pass to reach the dayside magnetopause. [36] Siscoe et al. [2002] suggested that during large magnetic storms Region 1 currents generate perturbation magnetic fields that alter the shape of the dayside magnetopause, thereby limiting access of solar wind streamlines and forcing the merging rate to saturate. Following the suggestion of Hill [1984], Siscoe et al. [2002] postulated that a “saturation” potential FS contributes to FPC via the relation, FPC = FE · FS/(FE + FS). The saturation

0 1:60 1010 7:456 1025

0 0 1:071 1023

1 1 1 BhC B 2C 0 B C A B h3 C 0 Bh C B 4C 6:237 1019 @h A h5

The standard procedure is to calculate a0, a1 and a2 for orbit‐averaged altitudes of GRACE. T∞ distributions are then obtained by substituting measured orbit‐averaged r(h) series into the quadratic equation. [38] Information contained in the J77 tables also allows estimations of thermal and gravitational potential energy density profiles for any value of T∞ and h. Burke [2008] numerically integrated these energy density profiles over the volume of the thermosphere for h ≥ 100 km to obtain the total energy content of the thermosphere Eth. Latitudinal integrations are accomplished by using orbit‐averaged densities from polar‐orbiting satellites like GRACE. Local time integrations were based on considerations of T∞ distributions in the J77 model which range between minimum and maximum values with a ratio T∞max/T∞min ≈ 1.3. Polar‐ orbiting spacecraft should sample globally averaged exospheric temperatures T∞ ≈ 1.15 · T∞min regardless of the local time of their orbital planes. Via numerical integrations with h ≥ 100 km, Burke [2008] showed that Eth is linearly related to T∞, Eth(J) = 5.365 · 1017 + 8.727 · 1013T∞ (°K). All of the regression coefficients obtained in these “statistical analyses” of J77 tables exceeded 0.999. [39] Acknowledgments. Research found in this report was supported by AFOSR task 2301SDA5. The author is grateful to two AFRL colleagues: L. C. Gentile for editing the manuscript and John O. Wise for providing orbit‐averaged GRACE data. Provisional Dst indices used in this paper were obtained from the University of Kyoto World Data Center. [40] Robert Lysak thanks Arrun Saunders and Daniel Weimer for their assistance in evaluating this paper.

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