Improved ionospheric electrodynamic models and application to calculating Joule heating rates

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110,, doi: /2004ja010884, 2005 Improved ionospheric electrodynamic models and application to calculating Joule heating rates D. R. Weimer Alliant Techsystems Mission Research Corporation, Nashua, New Hampshire, USA Received 4 November 2004; revised 28 January 2005; accepted 1 March 2005; published 18 May [1] Improved techniques have been developed for empirical modeling of the high-latitude electric potentials and magnetic field-aligned currents (FAC) as a function of the solar wind parameters. The FAC model is constructed using scalar magnetic Euler potentials and functions as a twin to the electric potential model. The improved models have more accurate field values plus more accurate boundary locations. Nonlinear saturation effects in the solar wind magnetosphere coupling are also better reproduced. The models are constructed using a hybrid technique, which has spherical harmonic functions only within a small area at the pole. At lower latitudes the potentials are constructed from multiple Fourier series functions of longitude at discrete latitudinal steps. It is shown that the magnetic (FAC) and electric potential models can be used together to calculate the total Poynting flux and Joule heating in the ionosphere. An additional model of the ionospheric conductivity is not required to obtain the ionospheric currents and Joule heating, as the conductivity variations as a function of the solar inclination are implicitly contained within the FAC model s data. The models outputs are shown for various input conditions and are also compared with satellite measurements. The calculations of the total Joule heating are compared with results obtained by the inversion of ground-based magnetometer measurements. Like their predecessors, these empirical models should continue to be useful research and forecast tools. Citation: Weimer, D. R. (2005), Improved ionospheric electrodynamic models and application to calculating Joule heating rates, J. Geophys. Res., 110,, doi: /2004ja Introduction [2] Empirical studies of the ionospheric electric potentials have had an important role in developing an understanding of the physical processes occurring in the coupled solar wind-magnetosphere-ionosphere system. These studies have steadily evolved from fixed statistical maps of the electric potential pattern [Heelis et al., 1982; Foster, 1983; Foster et al., 1986; Heppner and Maynard, 1987; Rich and Hairston, 1994; Ruohoniemi and Greenwald, 1996; Papitashvili et al., 1999] to more advanced computer models that can reproduce the statistical patterns for arbitrary solar wind conditions [Papitashvili et al., 1994; Ridley et al., 2000; Papitashvili and Rich, 2002]. These models can also be used in other research studies in topics ranging from the thermosphere to the ring currents and radiation belts [i.e., Angelopoulos et al., 2002; Liemohn et al., 2002] or for space weather predictions. [3] A similar but slower progress has occurred for the statistical mapping of the field-aligned currents (FAC) to and from the ionosphere, starting with the basic patterns of Iijima and Potemra [1976]. Only lately have more detailed FAC maps and models been produced [Waters et al., 2001; Papitashvili et al., 2001, 2002; Stauning, 2002]. Copyright 2005 by the American Geophysical Union /05/2004JA [4] This author s involvement with this progression started with statistical maps of the polar cap electric potentials for various groupings of interplanetary magnetic field (IMF) conditions, derived from measurements of the electric potential by the polar-orbiting DE 2 satellite. Data from 2900 passes, paired with simultaneous measurements of the IMF from IMP 8 and ISEE 3, were used [Weimer, 1995]. These maps were produced by least squares fits of the randomly placed data, using spherical harmonic (associated Legendre polynomial) coefficients. As the fits were confined to the polar region above 45 corrected geomagnetic latitude, this method is a variation of the spherical cap harmonic analysis (SCHA) technique [Haines, 1985]. [5] These results were then used to produce a more flexible model of the electric potentials, which could create a potential map for any arbitrary solar wind, IMF, and tilt angle conditions [Weimer, 1996]. This model had used a linear regression of the spherical harmonic coefficients in each IMF group to calculate how each coefficient varies with the solar wind conditions. This model was later improved [Weimer, 2001a] with the addition of a nonlinear response to the IMF magnitude and the inclusion of an optional parameter for varying the potential pattern according to the auroral electrojet AL index. Another improvement was the use of a regression formula for the colatitude of the lower boundary of the potential pattern. This formula 1of21

2 produced an explicitly defined boundary, which had the capability to expand beyond the previously fixed 45 boundary latitude for extreme southward IMF conditions. Having the potential pattern s boundary coincide with the SCHA boundary also eliminated the small harmonic oscillations present at the lowest latitudes in the 1996 version. [6] A similar empirical model was also developed for mapping the field-aligned current patterns in the polar ionosphere [Weimer, 2001b] on the basis of scalar magnetic Euler potentials derived from DE 2 satellite magnetometer measurements [Weimer, 2000]. A review of how these scalar magnetic potentials are used in the FAC model follows in section 6. The FAC patterns produced by this technique showed that the geometry and evolution of the current regions as a function of the IMF are much more complex than the canonical regions 1 and 2 defined by Iijima and Potemra [1976]. This FAC model could be considered a twin to the electric potential model as the calculations of the magnetic potentials are identical. [7] The purpose of this paper is to describe recent improvements made to the twin electric and magnetic potential/fac models, producing greater accuracy. Also described are how these two models can be used together to calculate the distribution of Joule heating in the ionosphere, which makes them even more useful. Variations in the total Joule heat energy dissipated in the ionosphere as a function of the IMF are also discussed. 2. Data Preparation [8] The electric and magnetic field measurements used to construct these empirical models have all been obtained from the Dynamics Explorer 2 satellite, which operated between August 1981 and March 1983 in a polar orbit at altitudes of km [Hoffman and Schmerling, 1981]. The electric field measurements are from the vector electric field instrument, a double floating probe, on DE 2[Maynard et al., 1981]. The magnetic field data were obtained by a triaxial fluxgate magnetometer [Farthing et al., 1981]. The electric potentials are obtained along the satellite path by an integration of the electric field component in the direction of motion. A similar procedure is used to obtain the magnetic potential along the path from the magnetometer data [Weimer, 2000]. Altitude adjusted corrected geomagnetic (AACGM) latitude and magnetic local time (MLT) are used in these models. These coordinates are described in Appendix A. [9] To improve these models, the intention had been to add measurements from other spacecraft to the database, but also it had been realized that improvements in the models basic algorithms were desirable. As the only way to test for improved algorithm performance is to use identical input data, the changes described here were made before adding additional data from other spacecraft. [10] Measurements of the solar wind and IMF conditions, simultaneous with each polar pass of DE 2, are also required to build the statistical models. These measurements have been from the IMP 8 and ISEE 3 spacecraft, using 40 min averages obtained from 5 min resolution data. More recently, it has been found that the propagation of the solar wind/imf from the point of measurement to the Earth may have highly variable timing delays due to tilting of the IMF phase fronts [Weimer et al., 2002]. In light of this discovery it was decided to reprocess all IMF measurements, using the modified minimum variance technique [Weimer et al., 2003] to obtain more accurate time delays. As the ISEE 3 spacecraft was often positioned very far (more than 100 R E ) from the Earth-Sun line, its data were expected to be more susceptible to variations in the timing. The new model was first developed using the old IMF values, and afterward, the model coefficients were recomputed using the updated IMF values. [11] The IMP 8 magnetometer data used for this project have a time resolution of 15 s, while the best resolution found in the archives for the ISEE 3 magnetometer data is 1 min. The solar wind velocity and density data have resolutions of 1 min and 168 s, respectively. At times where these plasma data are not available, average values are substituted as these data are neither as critical nor as variable as the magnetic field. [12] The tilted phase front computations were used to propagate the IMF/solar wind from both spacecraft to a fixed point at approximately the nose of the magnetopause at X =8R E GSE. Additional delays are expected in both the transmission of the IMF effects to the ionosphere and in the resulting ionospheric response time. To find the optimal delay to use, correlation tests were done between the propagated IMF and the peak electric potential drops measured on each DE 2 polar pass. The simultaneous AE indices, averaged over the 25 min duration of each polar pass, were also compared with the IMF. It was found that the best correlations were obtained by using an average of the propagated IMF values during the time intervals that start 55 min before the midpoint of each polar pass. The intervals end 10 min before the midpoints. This corresponds to an averaging interval of 45 min. The offset or delay of 10 min between the IMF measurement and the polar cap potential or AE index is in agreement with the range of delay times found by Ridley et al. [1998]. The 10 min delay includes the effects of a slower flow velocity between the bow shock and magnetopause. Correcting for a variable bow shock position was not done, as any such correction would be very small compared with the 45 min averaging interval. [13] As there are many time gaps in the IMF data, particularly from IMP 8, it was not required that IMF data be available throughout the entire 45 min period; otherwise, the majority of the satellite passes would be eliminated. The change in the timings that resulted from using the new IMF phase front orientations did cause some passes that previously had at least one corresponding IMF sample now to have none and therefore to be eliminated. The number of valid satellite passes in the final set is 2608 for the electric field measurements and 2403 for the magnetic field measurements. 3. Low-Latitude Boundary Fit [14] The electric and magnetic potential models have a boundary that defines the low-latitude edge of both the convection electric field and the magnetic perturbations due to the field-aligned currents. This boundary expands to lower latitudes as the IMF turns southward and contracts with northward IMF. In the 2001 versions of the two 2of21

3 models, which were created at different times, they each had their own set of boundary coefficients computed from their respective input data sets. The two boundaries were very similar (usually within 1 ) but not identical because of some differences in their input data. The new models now share a common low-latitude boundary that is computed prior to fitting the potential measurements in a separate step. [15] The data used to construct the boundary model are the points at both ends of every polar pass where the electric field or magnetic perturbations go to zero. These points were obtained by a manual inspection and marking of every pass in both the electric and magnetic field data sets. As these fields often have an asymptotic approach to zero, exactly where to place the marks is somewhat subjective. An explicit illustration of these boundaries in the magnetic field data is given by Weimer [2000, Figures 1 and 2]. The electric and magnetic potentials also start and end at zero at these endpoints. The electric and magnetic data were manually processed at different times and therefore have some differences in where the boundaries were marked on the same satellite pass over the pole. [16] The new boundary model consists of a circle with radius that varies as a function of the IMF and solar wind parameters. This circle, in AACGM coordinates, is offset from the AACGM pole toward 0 MLT. The variation in the boundary radius as a function of the controlling parameters and the offset distance were determined with a two-step process. The first step was to take each of the 2608 passes in the electric potential data and find 17 additional passes that had occurred under the most similar conditions. Eighteen of the most similar passes in the magnetometer data were also added, producing a total of 36. These passes have 72 endpoints where the electric or magnetic potentials start or end at zero, and these points have randomly located latitudes and magnetic local times. A simplex routine [Press et al., 1986] was then used to find the best fit of a circle to the locations of these endpoints. Both the offset distance of the center of the circle, on the 0 12 MLT meridian, and the radius of the circle are determined by this fit. The radius, offset, and the average IMF state vector (2608 sets) are saved for the second step. [17] The sorting process described above avoids the grouping of the passes into arbitrary, subjective bins to obtain a collection of points for an average fit. It should be mentioned how the most similar conditions for the pass groupings were determined. Initially, a normalized sixdimensional state vector is created for each pass, using the IMF components, solar wind velocity, AL index, and other parameters associated with the pass. The normalization maps the most common range of each component into a range of 1 to +1, but the more extreme cases are allowed magnitudes greater than 1. Then for each vector the distance to all other vectors is computed so that the smallest distances correspond to the most similar conditions. [18] The second step is to use the saved data to find the variation of the boundary as function of the IMF parameters. The radii ranged from 24 to 36, and the offsets were tightly clustered around a mean value of 4.2. Initially, it was assumed that both the radius and offset distance should change as a function of the solar wind, IMF, or tilt angle, but no significant correlations were found between the offset and these parameters. As a result the offset was set to a constant, using the mean value of 4.2. [19] The variation of the radius R as a function of the IMF is expressed by R ¼ X6 i¼0 B i W i ; where B i are the scalar coefficients and the parameters W i are W 0 ¼ 1; W 1 ¼ cos q; W 2 ¼ EðB T Þ; W 3 ¼ EðB T Þ cos q; W 4 ¼ V SW ; W 5 ¼ P SW ; W 6 ¼ AL; where q represents the IMF clock angle in the GSM Y-Z plane, V SW is the velocity of the solar wind, P SW is the dynamic pressure of the solar wind, and AL is the low auroral electrojet index. E(B T ) is an exponential function of the IMF s tangential magnitude in the Y-Z plane, defined by EB ð T Þ ¼ ½1 expðp 1 B T Þ ŠB p2 T : A least squares fit was used to solve for the coefficients B i in (1). The values of p 1 and p 2 are constants so that a simple linear fit could be used. However, the fit was repeated while the values of p 1 and p 2 were systematically varied until the values that produced the lowest error in the fit were found. The final values for the constants p 1 and p 2 in the boundary equation are 0.16 and Equation (1) was solved twice, with and without the term for the AL index, producing two sets of coefficients. The correlation coefficients between the 2608 R values in the database and the values obtained by equation (1) are and 0.918, with and without using AL. A few other parameters were initially used in solving for R, and they were dropped if they did not have a significant correlation with R. The dipole tilt angle is one such parameter that did not help to produce a lower error in the fit, so it is absent from (2). 4. Electric and Magnetic Potential Fitting [20] The previous versions of the electric/magnetic potential models have used spherical harmonic functions to map the potential variations as a function of AACGM latitude and MLT. One shortcoming of these previous versions is that they cannot reproduce the largest electric field values and acute reversals seen in the satellite measurements, even in the cases where the potential minimum and maximum values are the same as observed. The primary goal of the revised model algorithms has been to improve the accuracy of the potential gradients. [21] Some progress was made by using the same SCHA equations and boundary conditions described by Haines [1985] and in using different methods to derive the coefficients from the input data. Nevertheless, it was decided that spherical harmonics could not reproduce the sharp ð1þ ð2þ ð3þ 3of21

4 [23] Because of the offset of the magnetic pole from the geographic pole and the additional offset of the rings from the magnetic pole (Figure 1), not every satellite pass goes through the center region. As a result the number of satellite passes that cross each ring decreases closer to the pole. So that all rings have approximately the same number of data samples and consistent statistical fits, they are defined only for distances between R = 26/60 and R = 1, resulting in 34 steps in the region between the spherical harmonic cap and the outer boundary. This region contains the electric field reversals in the typical two-cell convection pattern as well as the primary auroral currents. [24] On each ring or annulus the potential is constructed from a Fourier series as a function of angular position around the ring: yðjþ ¼ X4 A m f m ðjþ; m¼0 ð4þ where j is the angle and f m (j) is given by Figure 1. Schematic diagram of potential models construction. The potentials are constructed with a spherical harmonic formula only within a small region near the pole, represented by the gray shading, combined with Fourier series formulas on a set of concentric rings (solid black lines). The radii of the pole region and rings expand in proportion to the radius of the low-latitude boundary, a function of the solar wind/interplanetary magnetic field (IMF) parameters. The center is offset from the altitude adjusted corrected geomagnetic (AACGM) pole, 4.2 toward midnight magnetic local time (MLT). Only a quarter of the total number of rings are illustrated for clarity. convection reversals. Simply increasing the degree and order of the polynomials does not work with the sparse and randomly scattered data as the solutions grow without bounds in the locations where no measurement is available to constrain the fits. [22] The newest version of the code is a hybrid, which uses spherical harmonics to represent the potentials only within a region around the offset pole, where the potentials have lower gradients. Figure 1 contains a graphical representation of the models construction, where the central, spherical harmonic region is represented by shading. Between this polar region and the low-latitude boundary the potentials are defined on a set of virtual, concentric rings, centered with the boundary, as illustrated in Figure 1. For clarity, only about one fourth of the actual number of rings are shown in Figure 1. The spacing between the rings is fixed at 1/60 of the radius of the outer boundary, rather than having a fixed step size in degrees, so that the radii of the rings are proportional to the size of the outer boundary. As a result the rings expand and contract along with the boundary in response to changes in the IMF. When the radius of the boundary is expanded to a size of 30, then the spacing between rings is 0.5. This technique allows the electric potential patterns to change in size, while maintaining the same relative shape. n¼0 f 0 ðjþ ¼ 1; f 1 ðjþ ¼ cosðjþ; f 2 ðjþ ¼ sinðjþ; f 3 ðjþ ¼ cosð2jþ; f 4 ðjþ ¼ sinð2jþ: Every Fourier coefficient A m in (4) has its own independent response function to the IMF, solar wind velocity, density, dipole tilt angle, and (optional) AL index variations. These coefficients are derived from the input parameters by another set of coefficients:! A m ¼ X4 X 5 C inm X i f n ðþ; q ð6þ i¼0 where q is the IMF clock angle in the GSM Y-Z plane and the function f n (q) is defined the same as in (5) but using this IMF angle rather than a longitudinal position. The parameters X i in (6) are X 0 ¼ 1; X 1 ¼ EB ð T V SW Þ; X 2 ¼ sinðþ; t X 3 ¼ sin 2 ðþ; t X 4 ¼ P SW ; X 5 ¼ AL: The function E(B T V SW ) is the same as in (3), except this time the independent variable is the solar wind electric field, obtained by multiplying the tangential magnetic field with the solar wind velocity and converted to units of mv m 1. Other terms are the sine of the dipole tilt angle, the sine of the angle squared, the dynamic pressure of the solar wind, and the AL index. [25] The function f n (q) used in (6) produces a variation due to the IMF clock angle, but the constant term X 0 should not have any clock angle variation. So for i = 0, only the one n = 0 value is used. Since the X 1 term contains the IMF, the summation for n goes from zero to 4 if i = 1. However, the IMF clock angle rotation does not affect the remaining terms as much, so only the n = 0 2 terms are used for i = ð5þ ð7þ 4of21

5 2 5. This scheme results in 18 C inm coefficients in (6) for each A m if the AL index is being used and 15 coefficients otherwise. As there are five Fourier coefficients A m in (4), the potentials on each ring are determined by a total of 90 coefficients. The total is 75 if AL is not used as then the summation for i in (6) only goes up to 4. [26] Equation (3) is used to obtain a response curve that rises quickly as the solar wind electric field increases but then levels off to a more gradual slope at higher magnitudes. The purpose is to reproduce the saturation effect in the polar cap electric potentials and currents [Hill et al., 1976; Siscoe et al., 2002, 2004], as is demonstrated in section 7. The values of p 1 and p 2 in (3) used for the two potential models are different from the pair used in the boundary model. Values of 1.33 and 0.47 are used for p 1 and p 2 in the electric potential model, and 1.76 and 0.70 are used in the magnetic potential model. These values were found to produce the overall best fit with the input data, where the fit errors are evaluated on every ring and are summed. The values of p 1 and p 2 may be revised in the future as more data are added at larger IMF magnitudes, leading to a better resolution of the saturation curve. [27] Before calculating the model coefficients the coordinates of each satellite pass need to be translated into normalized, offset coordinates. This translation starts by taking the IMF/solar wind parameters associated with the pass and using them in equation (1) to obtain the boundary radius for the given conditions. The corrected geomagnetic latitude and MLT positions along the path are then rotated into the coordinate system of the offset pole in Figure 1. A relative or normalized colatitude is assigned to each position, having a value of zero at the offset pole and one at the concentric outer boundary. [28] The electric (or magnetic) potential in the models is always zero on and beyond the outer boundary at R =1. Still, the normalized colatitude where the potentials on an individual pass reach the zero level may sometimes be above or below the statistical average for the given IMF conditions. If the potential measured on a pass reached zero inside the boundary, then the potential on the remainder of the path up to the boundary is padded with zeros. A pass that has a nonzero potential value on and below the boundary needs to be adjusted so that the potentials are consistent with an asymptotic approach to zero at the boundary. This adjustment is done by first locating the nearest potential minimum or maximum value along the orbit track. Next the potential curve between the location of the peak and the zero endpoint is compressed to fit into the space between the peak and the outer boundary, while keeping the same relative shape. This adjustment enables the final, best fit solution to have the correct shape between the potential maxima and the zero point. [29] After normalization of each pass the potentials and angular locations of the two intersections of the path with every ring are found. After all passes are processed, then for each ring the C inm coefficients are found by a least squares error fit of (4) and (6), using all potential values from every intersecting pass. To accomplish this fit, the solution is put into the form of a very large matrix equation (number of points times number of coefficients), which is solved by lower triangular upper triangular matrix decomposition ( LU decomposition ) [Press et al., 1986]. [30] The region poleward of the highest ring is modeled using spherical harmonics so that the potentials within the shaded area in Figure 1 are given by yðl; jþ ¼ X2 X 2 D lm P m l ðcos lþ gn m cos mj þ hm n sin mj : ð8þ l¼0 m¼0 is the associated Legendre polynomial, and l is a function of colatitude, varying from zero at the pole to p/2 at the most poleward ring. Large degree l and order m are not required in this region as the purpose is to smoothly bridge the polar gap with a continuous function. After the Fourier coefficients in (4) are determined for the most poleward ring, then the spherical harmonic coefficients, D lm, can be set so that the potential from (8) is exactly equal to that of (4) where the two regions are joined. This matching of the potentials is accomplished by making all sin(mj) and cos(mj) terms equivalent for all m. Using finite differences to calculate the derivative of these Fourier components at the boundary, the slopes of the potentials are also matched. This matching is simplified by the fact that all associated Legendre polynomials at the boundary (l = p/2) have either a zero value or zero slope. [31] This matching process can uniquely specify all of the spherical harmonic coefficients except two, D 20 and D 00,so these are solved for using a least squares error fit to the satellite measurements. Equation (6) is also used for the relationship between these associated Legendre coefficients and the IMF parameters, except that D lm is substituted for A m. Using all available measurements within the polar region, the corresponding set of C inlm coefficients is also obtained from a least squares error fit of the potentials using (8) and (6) and employing the same LU decomposition matrix technique. [32] To reconstruct a map of the electric or magnetic potentials for a given set of input conditions, the first step is to calculate the boundary radius from the solar wind and IMF parameters using (1). The coefficients A m on each ring and also the D 20 and D 00 coefficients for the center region are obtained from the same input parameters using (6). The individual A m coefficients are then smoothed in latitude to reduce statistical noise fluctuations. Next the boundary value and slope matching are used to set all spherical harmonic coefficients for the polar cap. D 20 and D 00 may be adjusted just slightly from their IMF-derived values so that the m = 0 terms match, while keeping their same proportions. [33] In order to obtain the potential at a given point, its corrected geomagnetic latitude and MLT coordinates are rotated into the offset pole coordinates. Then if the translated point lies within the center region, (8) is used to obtain the potential. Otherwise, the potential on the two nearest rings is calculated with (4), and the potential at points between the rings is obtained by interpolation. All points on or below the outer boundary have a potential of zero. P l m 5. Examples of Model Results [34] Figure 2 shows maps of the electric potential contours produced with the new method, using a fixed IMF 5of21

6 Figure 2. (a i) Polar cap electric potentials in the Northern Hemisphere, mapped as a function of AACGM latitude and MLT. Figures 2a 2d and 2f 2i show the patterns for eight different clock angle orientations of the IMF vector in the GSM Y-Z plane; the angle in degrees is indicated in the top left corner of each map. The IMF has a fixed magnitude of 5 nt, the solar wind velocity is 450 km s 1,the solar wind number density is 4 cm 3, and the dipole tilt angle is 0. Figure 2e shows the potential for zero IMF, with the same solar wind conditions. Minimum and maximum potential values are printed in the bottom left and right corners of each map, with locations indicated by the diamonds and pluses. magnitude of 5 nt in the GSM Y-Z plane at eight different clock angle orientations. The other parameters used are 450 km s 1 solar wind velocity, 4 cm 3 solar wind proton number density, and 0 tilt angle. The optional AL index was not used. The northward directed IMF (0 clock angle) pattern is shown in Figure 2b, and the southward IMF (180 clock angle) pattern is shown in Figure 2h. Figure 2e shows the electric potential pattern for zero IMF magnitude. This zero IMF map is similar to that obtained by Papitashvili and Rich [2002] for equinox conditions. 6of21

7 Figure 3. (a i) Magnetic Euler potentials in the Northern Hemisphere, mapped as a function of AACGM latitude and MLT. The format and the IMF/solar wind parameters are the same as in Figure 2. [35] The patterns shown here are for the Northern Hemisphere. To use the models in the Southern Hemisphere, the signs of both the IMF B Y component and the dipole tilt angle are reversed, as was done for the input data. Mixing observations from both hemispheres were necessary to obtain the best statistical sampling, particularly since the duty cycle of the instruments on the DE 2 satellite produced fewer measurements in the Southern Hemisphere. This reversal of the B Y component assumes a mirror symmetry where the potential pattern in the Northern Hemisphere with positive IMF B Y looks like the pattern in the Southern Hemisphere with negative B Y and vice versa. Overall, this symmetry holds true, but the South Atlantic anomaly may produce some minor differences. [36] The corresponding maps for the magnetic Euler potential and FAC density are shown in Figures 3 and 4. The total integrated sum of the current in one direction in units of MA is shown in the top right corner of each map in 7of21

8 Figure 4. (a i) Density of the magnetic field-aligned current flowing into the ionosphere in the Northern Hemisphere at an altitude of 110 km, mapped as a function of AACGM latitude and MLT. Positive (red) values are downward. The format and the IMF/solar wind parameters are the same as in Figure 2. These currents are derived from the potentials shown in Figure 3. The total integrated downward current is printed in the top right corner of each map. Figure 4. The densities are calculated for an altitude of 110 km. Field-aligned current maps resulting from this technique show an intricate evolution of the current systems as the IMF clock angle rotates. The basic region 1 and 2 currents are clearly visible, but where they overlap near 0 and 12 MLT the currents defy a simple region 0, 1, or 2 classification. The FAC map in Figure 4e may be compared with the zero IMF cases shown by Papitashvili et al. [2001, 2002]. [37] The potential patterns in Figures 2 and 3 generally look very much like those from all previous versions of the model, even though the techniques used to get from the 8of21

9 input data to the final product are very much different. Subtle differences exist as the potential minima and maxima often have larger magnitudes, and their derivatives (electric and magnetic fields) also have significantly larger magnitudes near the field reversals. The Harang Discontinuity [Maynard, 1974], the elongation of the negative in the MLT sector, is more pronounced in the new version. These are the desired results. Some comparisons of the electric potentials from the older and new models are given in sections 6 and 7. The FAC densities, obtained by finite differences, are also noticeably higher because of the larger second derivatives in the magnetic potential. [38] There are no noticeable changes in the potentials (Figures 2 and 3) at the locations where the spherical harmonic calculations at the pole join with the innermost ring. Because of the matching process mentioned above, the boundary between these regions is invisible. Sometimes there is a slight discontinuity in the second derivative of the potentials at this boundary, which is the main disadvantage to using this hybrid model technique. This jump may produce some artificial features in the FAC density plots right at this border (Figure 4). However, overall, the smooth bridging of the polar cap with the spherical harmonic function has minimized the FAC artifacts in this region. 6. Joule Heating [39] The combined electric and magnetic potential models can be used to calculate the Poynting flux flowing into the ionosphere, and the resulting Joule heating, due to the solar wind-magnetosphere-ionosphere interaction. This application greatly increases the utility of these two models. To demonstrate how the Joule heating is derived, first, it is necessary to review briefly how the magnetic potentials are used to map the FAC as a function of the IMF [Weimer, 2000, 2001b]. [40] The derivation uses a curvilinear coordinate system having orthogonal unit tangent vectors e 1, e 2, and e 3 and assumes that the field-aligned current is in the e 1 direction: J ¼ J k e 1 : If dipole coordinates are used, then e 1 is along field lines; in the polar cap it is approximately radial. The magnetic field produced by this current on the orthogonal surface is such that DB ¼ e 1 r S y; ð9þ ð10þ where r S is the surface gradient in the e 2 and e 3 directions and y is a toroidal scalar. The current through the surface and the magnetic field on the surface are related by m o J ¼rDB ¼rðe 1 r S yþ: ð11þ by Backus [1986]. It can be seen that the toroidal scalar y is the same as a scalar magnetic Euler potential [Stern, 1970] as the magnetic field lines on the surface are along lines of constant y. This y is not to be confused with the more conventional scalar magnetic potential, which has a three-dimensional gradient equal to the magnetic field vector. [41] To derive magnetic potentials from satellite magnetometer data, the first step is to subtract the Earth s internal field, using a definitive geomagnetic reference field model, from the measured field, producing delta-b. As (10) can be rewritten as e 1 DB ¼ r S y; ð13þ there is a similarity to electric fields, where the field E is related to the electric potential F by E ¼ r S F: ð14þ Satellite measurements of the electric field are integrated along the path of motion to obtain the potential as a function of location; by analogy the magnetic Euler potential y can be obtained by integrating this quantify e 1 DB, where e 1 is a unit vector in the upward, field-aligned direction. Once these magnetic potentials are derived from all satellite passes, they are treated the same way as the electric potentials. The FAC densities, as in Figure 4, are derived by equation (12). [42] To use the two models together to derive the Poynting flux above the ionosphere, S ¼ E DB=m o ; ð15þ the electric field is obtained from the gradient of the electric potential model (14), and from (10) the magnetic perturbations above the ionosphere are obtained from the gradient of the magnetic Euler potential. These magnetic perturbation vectors are in the direction of the equipotential lines in Figure 3. The direction of the Poynting flux is radial, parallel to the field-aligned current. [43] It is also possible to calculate the ionospheric Joule heating from the models directly, starting with the postulate that the height-integrated perpendicular current in the ionosphere is directly derivable from the gradient of the magnetic potential function J? ¼ r Sy m o : ð16þ The field-aligned current density into the ionosphere is related to this horizontal, height-integrated perpendicular current within the ionosphere according to the formula It follows that J k ¼r 2 S y=m o; ð12þ where r S 2 is the surface Laplacian on the two-dimensional spherical surface. These notations and equations are derived J k ¼r S J? : Substituting (16) into (17) results in r S y J k ¼r S ¼r 2 S m y=m o; o ð17þ ð18þ 9of21

10 Figure 5. (a i) Joule heating rate in the ionosphere in the Northern Hemisphere at an altitude of 110 km, mapped as a function of AACGM latitude and MLT. The format and the IMF/solar wind parameters are the same as in Figure 2. The heating rates are derived from the potentials shown in both Figures 2 and 3. The total integrated heating is printed in the top right corner of each map. which is identical to (12), giving credibility to the original hypothesis (16). The ionospheric current vectors calculated from (16) are perpendicular to the contour lines in Figure 3. These perpendicular currents need not be exactly parallel to the electric fields as there may be gradients in the ionospheric conductivity and neutral winds. The heightintegrated ionospheric Joule heating is calculated by taking the dot product of the electric field with the heightintegrated perpendicular current from (16). By using (10) and some algebraic manipulation it is found that this dot product is equivalent to the Poynting flux in (15): ˆrðE J? Þ ¼ ˆrðEr S y=m o ¼ E DB=m o ¼ S: Þ ¼ E ðˆr r S y=m o Þ ð19þ 10 of 21

11 This result is expected as in a steady state the Poynting flux into the ionosphere should be equal to the energy dissipated by the Joule heating, and throughout this paper the terms are used interchangeably. [44] Figure 5 shows maps of the Joule heating/poynting flux calculated by this procedure for the same solar wind and IMF parameters used in Figures 2 4. The total integrated sum of the heating over the polar cap is shown in the top right corner of each map. The orientation of the IMF has a strong influence on the total heating. For southward IMF the total ionospheric heating is roughly hundreds of gigawatts. It is very important to note that this calculation of the heating does not require the use of an ionospheric conductivity model. The conductivity distribution in the ionosphere for each IMF and tilt angle combination is implicitly contained within the statistical magnetic potential/fac model. In other words, if the dipole tilt angle increases to a summer orientation, then the magnetic perturbation, currents, and resulting Poynting flux/joule heating in the statistical model generally increase because of the higher ionospheric conductivity. [45] It is also possible to use the magnetic potential model alone for approximating the magnetic perturbations on the ground caused by ionospheric Hall currents without requiring a separate conductivity model. This technique, based on (16), will be the subject of a separate publication. Figure 6. Total Joule heat, magnetic potential drop, total field-aligned current, and electric potential drop graphed as a function of southward IMF magnitude at 450 km s 1 solar wind velocity, 4 cm 3 number density, and zero tilt. (top) Total Joule heating, (middle) the magnetic potential drop (solid line), and (bottom) the electric potential drop (solid line shows new version, and dotted and dashed lines show 1996 and 2001 versions). The dashed line in Figure 6 (middle) shows the total downward field-aligned current, using the axis scale on the right. 7. Model Response to Varying Input Parameters [46] Figure 6 shows how the new model calculations vary as a function of IMF magnitude for a southward orientation (negative B Z ) and a fixed solar wind velocity of 450 km s 1. The solar wind number density is 4 cm 3, the dipole tilt angle is zero, and the AL index is not used. Figure 6 (top) shows the total Joule heating integrated over the northern polar cap, and Figure 6 (middle) shows the variation of the difference between the largest and smallest values of the scalar magnetic potential (solid line). Figure 6 (middle, dashed line) shows the total FAC in one direction, integrated over the polar cap, using the axis scale on the right. Because of the complexity of the current patterns, there is no easy way to sum up the separate contributions of the different current regions as in places they connect with each other. Figure 6 (bottom, solid line) shows the polar cap electric potential drop, the difference between the largest and smallest (most negative) electric potentials. For comparison, the same quantity is shown as computed with the earlier 1996 and 2001 versions of the electric potential model, using the dotted and dashed lines. [47] The slope of the electric potential curve changes significantly at about B Z = 5 nt because of the saturation effect. This break in the curve is obtained by using equation (3) in (7) to control how the model coefficients vary as a function of the solar wind electric field. An analytical expression for the saturation curve was not used for generating the model as this presumes in advance which, if any, of the competing saturation theories is correct and would therefore bias the results. Equation (3) is flexible in that it could produce a straight line slope with no saturation (if that is what best matches the data) if the parameter p 1 has a 11 of 21

12 Figure 7. Total Joule heat, magnetic potential drop, total field-aligned current, and electric potential drop graphed as a function of dipole tilt angle, with IMF B Z = 5 nt, 450 km s 1 solar wind velocity, and 4 cm 3 number density. (top) Total Joule heating, (middle) the magnetic potential drop (solid line), and (bottom) the electric potential drop (solid line). The dashed line in Figure 7 (middle) shows the total downward field-aligned current, using the axis scale on the right, and the dotted and dashed lines in Figure 7 (bottom) show the absolute values of the minimum and maximum potentials. negative value with large magnitude and p 2 is equal to 1. Or if p 2 is less than 1, then a simple, exponential power law would result. If p 1 has a value that causes a break to occur, then the value of p 2 determines if the line is perfectly flat above the break (p 2 = 0) or continues in a straight line with a different slope (p 2 = 1). The result that was obtained by the best fit to the data is between these values, at p 2 = 0.47, so after the break in the slope the potential drop continues upward in a power law manner and reaches 250 kv at B Z = 50 nt. This value agrees very well with the potential drops reported by Hairston et al. [2005] from Defense Meteorological Satellite Program (DMSP) satellite measurements during the October November 2003 superstorms. The break in the curve near B Z = 5 nt, corresponding to an electric field of 2.25 mv m 1, causes this curve to closely resemble those shown by Ober et al. [2003], Siscoe et al. [2004], and Hairston et al. [2005]. [48] Interestingly, the magnetic potential curve and the associated total current (Figure 6, middle) show less of a break in their slopes at the point where the electric potential starts to saturate. The total Joule heating, from the two models combined, is a nearly straight line. One reason why the integrated Joule apparently does not saturate like the other two is because the low-latitude boundary of the models continues to expand to lower latitudes, thus increasing the total area for the integration. In addition, the total heating is approximately proportional to the square of the polar cap electric potential drop. [49] The comparison line for the 1996 version of the electric potential model (Figure 6, bottom, dotted line) is nearly straight as linear equations had been used. This linear fit did a reasonable job at matching the available DE 2 data, which were all obtained with an IMF magnitude of less than 15 nt, but the 1996 model does poorly with extrapolation to much greater IMF magnitudes. The 2001 version does better as a power law exponent of 0.6 was used with the IMF term in the model. The potential drop from the 2001 version meets and exceeds the value from the newest model at B Z = 36 nt, but at lower-magnitude B Z the electric potential drop in the newest version is generally higher by 20 kv. One reason why the newest version has larger potentials is that use of (3) allowed for a more accurate fit of the saturation curve. The other reason is because the smoothing of the peaks that resulted from the spherical harmonic fits was avoided by the evaluation of the electric potential on concentric rings. The close latitudinal spacing enabled the model to resolve the potential minimum and maximum points better. [50] Figure 7 shows how the results from the model calculations vary as a function of the dipole tilt angle for the Northern Hemisphere. The IMF has a fixed magnitude of 5 nt with a purely southward orientation, and the other parameters are the same as in Figure 6. The format of Figure 7 is also the same, except that in Figure 7 (bottom) the dotted and dashed lines now represent the absolute magnitudes of the potential minimum and maximum values, respectively, for the new model. The electric potential drop has the largest value near zero dipole tilt and decreases slightly for both negative (winter in the Northern Hemisphere) and positive tilt angles (summer). In contrast, the magnetic potential drop and total current (Figure 7, middle) both increase in going from negative to positive tilts. This 12 of 21

13 Figure 8. Total Joule heat, magnetic potential drop, total field-aligned current, and electric potential drop graphed as a function of IMF clock angle. The IMF has a fixed magnitude of 5 nt in the GSM Y-Z plane, the solar wind velocity is 450 km s 1, and the number density is 4 cm 3. The dipole tilt angle is (left) 20, (middle) 0, and (right) +20. The format within each column is the same as in Figure 7. increase confirms that seasonal variations in the ionospheric conductivity from solar EUV radiation are implicitly included in the magnetic potential/fac model. As a result the total Joule heating also increases, doubling in value as the tilt increases from 30 to +10. Above 10 the heating decreases because of the decreasing electric potential at the more extreme tilt angles. [51] This decreasing electric potential at extreme tilt angles results from the term in (7) having the sine of the tilt angle squared. This term was included after viewing scatterplots of some A m Fourier coefficients as a function of tilt angle, in which a nonlinear trend was evident. The total error in the fits was also reduced after this term was added. Adding this term did not cause the magnetic potential to decrease at the large tilt angle as this decreasing trend was not present in the magnetometer data. The physical explanation for why the electric potential decreases is not known. Speculating, the more extreme tilt angles might cause a decrease in the efficiency at which the electric field in the solar wind penetrates into the magnetosphere, or the rate at which magnetic flux is transferred. [52] Figure 8 shows the combined effects of the season (dipole tilt) and the orientation angle of the IMF vector in the GSM Y-Z plane. The three columns show results for dipole tilt angles of 20, 0, and +20, ordered from left to right. These graphs show the changes that occur as an IMF vector with a fixed 5 nt magnitude rotates from 0 (purely northward, or positive B Z ) to 180 (purely southward, or negative B Z ) and back to northward. Positive and negative IMF B Y orientations are at clock angles of 90 and 270. The effects of the IMF direction and season on the relative magnitudes of the negative and positive electric potential cells are readily apparent in Figure 8 (bottom, dotted and dashed lines). As the total Joule heating is approximately proportional to the square of the electric potential drop, the heating increases by a factor of 16 in going from north to south IMF orientations, while the polar cap electric potential drop only increases by a factor of 4. [53] Figure 9 shows what happens when the optional AL index components of the models are activated, in which case different sets of coefficients are used. This graph shows the effects of changing the AL index from 0 to 800 nt when the IMF is 5 nt with a northward orientation. A decrease in the AL index is often associated with a greater potential in the positive, dawn convection cell, while 13 of 21

14 potential pattern to selected groupings of satellite passes. As Figure 9 shows, the field-aligned currents and Joule heating also increase, as expected. [54] Figure 10 shows examples of the electric potential and Joule heating maps when the AL index that is input to the models is set at 500 nt with a northward +5 nt IMF, while keeping the other parameters the same as in Figures 2 and 5. A comparison with Figures 2b and 5b shows the effect of using this value of AL. Magnetospheric substorm activity, of which the AL index is one indicator, often occurs just after the IMF turns northward, and apparently, the substorms cause the normal four-cell +B Z pattern to be dominated by a resulting two-cell pattern. The electric potential pattern in Figure 10 is very similar to those derived by the assimilative mapping of ionospheric electrodynamics (AMIE) magnetometer inversion technique. Two examples are given by Knipp et al. [1989, Figure 5] at 1050 UT on 19 September 1984 and by Kamide et al. [1994, Figure 5] at 0735 UT on 25 October The 25 October example follows a northward turning of the IMF that occurred at 0720 UT, and only a two-cell pattern is seen. Using a negative AL index in the models when the IMF orientation is southward does not have nearly the same effect on the maps as it does when the IMF is northward as the two-cell pattern is already present. Figure 9. Total Joule heat, magnetic potential drop, total field-aligned current, and electric potential drop graphed as a function of absolute value of the AL index. The IMF is B Z = +5 nt, the solar wind velocity is 450 km s 1, and the number density is 4 cm 3. The format is the same as in Figure 7. simultaneously, the AU index and negative cell also increase in magnitude [Weimer et al., 1990]. The changes in the potentials shown in Figure 9 (bottom) agree with those found by Weimer [1999] using a fit of the whole electric 8. Comparison With Satellite Measurements [55] The instruments on the DMSP spacecraft provide an opportunity to compare the models predictions of electric and magnetic fields with measurements that were not part of the models input data. Figures show comparisons of the electrodynamic models with DMSP satellite measurements on four different polar passes. Figures 11a, 12a, 13a, and 14a show the model electric potential pattern, with the satellite path superimposed as a blue line. Figures 11b, 12b, 13b, and 14b show the electric potential graphs of the electric potential as a function of time, using a blue line for the values derived from the DMSP ion drift meter measurements and a red line from the newest model s calculations. For comparison, the values from the 2001 version of the potential model are superimposed with a green line. Figures 11c, 12c, 13c, and 14c show graphs of the corresponding electric field values, using the same bluered-green color representation. The input parameters, displayed in the subtitles of Figures 11 14, are from IMF measurements on the ACE spacecraft and time delayed to the Earth. [56] Figures 11d 11f, 12d 12f, 13d 13f, and 14d 14f show the maps of the magnetic Euler potentials and the magnetic perturbations perpendicular (B? ) and parallel (B k ) to the orbit path. The delta-b values from the model are again shown in red. The DMSP satellite magnetometer measurements, shown in blue, have had the background geomagnetic field subtracted prior to the receipt of these data. As the reference geomagnetic field and satellite position/attitude may not be perfect, there are often significant and variable offsets in these results, and the body-mounted magnetometer on F13 may also have interference from within the satellite. The residual offset values at the boundary locations were subtracted from the magnetometer data before drawing the blue lines. 14 of 21

15 Figure 10. (left) Polar cap electric potentials and (right) Joule heating in the Northern Hemisphere, showing the effects of using the models with the optional AL index parameter. The IMF is northward at +5 nt, and the AL index is 500 nt, which results in a substantial change in comparison with Figures 2b and 5b. [57] The first three examples show patterns for different IMF and tilt angle orientations. Figure 13 has a particularly large tilt angle, combined with an IMF in the +B Y direction, which results in a dominant negative cell. Figure 14 shows a pass during a major geomagnetic storm, where the magnitude of the IMF is more than double that of the largest IMF in the models input data. Note that the latitude scale of the maps has been changed to accommodate the greatly expanded potential pattern. The locations of the low-latitude boundary and electric field reversals are still reproduced very well with the newest model, but the older 2001 version (green line) had expanded too far to lower latitudes. The primary reason for the improved boundary fit is the use of the nonlinear equation (3). Figure 14 also shows how scaling the size of the entire potential pattern in proportion to the outer boundary works as the locations of the various features are fairly accurately predicted. [58] In these examples the improved performance of the new electric potential model in comparison with the 2001 version is apparent. In consideration of the possible magnetometer measurement errors these comparisons of the model predictions, based on solar wind values, and the satellite measurements are very good. Since the electric and magnetic fields from the two models are realistic, it can be concluded that the Poynting flux/joule heating calculations derived from them should also be producing reasonable results. There are many other cases where the electric potentials from the model match much better than the examples shown here, but often, the DMSP magnetometer data are not usable for inclusion in the comparison. 9. Comparison With Ground-Based Magnetometer Inversions [59] The values of the total hemispheric Joule heat dissipation obtained by the combined electric and magnetic potential models are similar to those obtained with the inversion of ground-based magnetometer measurements, such as the AMIE technique [Baumjohann and Kamide, 1984; Richmond, 1992; Chun et al., 1999, 2002]. However, at times the empirical models produce heating rates that are higher than those from AMIE. It should also be mentioned that Chun et al. [2002] have reported that the total integrated Joule heating, averaged over many AMIE patterns, increased from winter to equinox to summer. As they did not see evidence for the heating to decrease in the summer (i.e., large tilt angles), the results shown in Figure 7 are contradictory. The contradiction might be resolved by the fact that at all angles up to +24 this model s prediction of Joule heating is still larger than what it is at 0. [60] The best direct comparison between the two techniques is made by using a wide range of IMF conditions spanning a period of several days, as shown by Ballatore et al. [2000]. Figure 15 (bottom) shows the Y (green) and Z (blue) components of the IMF, measured on the Wind satellite, as a function of time during October Above the IMF are shown both the polar cap potential drop and the total integrated Joule heating in the Northern Hemisphere computed from this IMF with the new empirical models. The heating computed by the empirical models is drawn with the red line. A section of Ballatore et al. [2000, Figure 2], which contains a graph of the same quantity as computed with the AMIE technique, is superimposed in black. The results are nearly identical for the periods with moderate IMF, providing further proof that the method described in section 6 is accurately calculating the total ionospheric Joule heating (this agreement implies that the AMIE technique is also valid). However, during the period when the IMF magnitude exceeds 20 nt in the Z (southward) direction the empirical Joule heating model produces heating rates that are significantly higher than those from AMIE. 15 of 21

16 Figure 11. Comparison of model calculations with DMSP F13 satellite measurements on 23 September (a) Electric potential pattern is shown, with the satellite path superimposed. (b) Measured (blue) and model (red) electric potentials as a function of time are shown, as well as potentials from the 2001 version of the potential model (green). (c) Corresponding electric field values are shown, using the same blue-red-green color representation. (d) A map of the magnetic Euler potential and the measured (blue) and model (red) magnetic perturbations (e) parallel (delta-b k ) and (f) perpendicular (delta-b? to the orbit path are shown. The model input parameters, indicated in the subtitle, are from IMF measurements on the ACE spacecraft. [61] The most likely reason for the difference is that the AMIE technique needs to use an assumed model of ionospheric conductivity, whereas the empirical FAC model has an implicitly derived conductivity. Comparisons of the electric and magnetic fields from the models with DMSP measurements for extreme IMF cases show that the new versions of the models tend to err more on the low side, as in Figure 14 for IMF B Z = 33.6, rather than being too large. Also, both MHD and empirical models of the ionospheric electric potentials tend to produce magnitudes that are generally higher than those from the AMIE technique [Winglee et al., 1997]. Therefore it can be argued that the heating rates from the 2004 empirical models, which are greater than the AMIE results, are the more accurate. The 16 of 21

17 Figure 12. (a f) Comparison of model calculations with DMSP F13 satellite measurements on 26 January The format is the same as in Figure 11. total ionospheric heating may actually be higher as it has been proposed by Codrescu et al. [1995] that smallscale structure in the electric fields raises the total Joule heating. 10. Summary [62] Improved techniques have been developed for empirical modeling of the high-latitude electric and magnetic fields and the associated currents and Joule heating as a function of the solar wind parameters. An additional model of the ionospheric conductivity is not required to obtain the ionospheric currents and Joule heating, as the conductivity variations as a function of the solar inclination are implicitly contained within the models data. The improved models have more accurate field values plus more accurate boundary locations. Nonlinear saturation effects in the solar wind magnetosphere coupling are also better reproduced, but caution is advised when these models are used with input values far exceeding normal IMF conditions. The future addition of more data from cases with large IMF magnitudes may improve the accuracy of the saturation curve without requiring a further restructuring of the models techniques and code. [63] There are two aspects of the ionospheric electrodynamics that these empirical models currently do not handle. One problem is that of temporal resolution as the ionosphere may respond at different rates at different MLT locations 17 of 21

18 Figure 13. (a f) Comparison of model calculations with DMSP F15 satellite measurements on 22 June The format is the same as in Figure 11. when there are sudden IMF changes. Also, the models cannot show the changes in the ionosphere due to largemagnitude and sudden changes in the solar wind dynamic pressure [Boudouridis et al., 2003]. The pressure fronts may only cause momentary temporal effects, not well modeled by an average of data obtained from satellite orbits widely spaced in time and location. Another area of difficulty is the penetration electric field that occurs at subauroral latitudes during major geomagnetic storms [e.g., Foster and Rich, 1998; Wygant et al., 1998]. This penetration field is located below the normal outer boundary of these models. One way to handle to penetration fields, as suggested by Ridley and Liemohn [2002], may be to construct a separate model of the low-latitude storm time electric fields and potentials and superimpose them on the high-latitude model. [64] The prior versions of the electric potential model have been used in the past by numerous researchers for a variety of applications, such as showing where in the convection pattern a particular set of measurements has been obtained. Another use has been providing the electric fields to numerical simulations of the magnetosphere or thermosphere. The programs can also be used for predicting space weather effects in advance by using measurements of the IMF radioed to the Earth in real time. The more recently conceived capability of the combined models to calculate 18 of 21

19 Figure 14. (a f) Comparison of model calculations with DMSP F13 satellite measurements on 31 March The format is the same as in Figure 11 except the boundary of the maps is at a lower latitude. ionospheric Joule heating rates augments their utility even more, particularly since thermospheric heating currently is a topic of high interest in the NASA Living With a Star Program. These empirical models should continue to be useful research and forecast tools. Appendix A: AACGM Coordinates [65] Various techniques are used to compensate for irregularities in the Earth s internal magnetic field when translating measurements at various locations into a common geomagnetic frame of reference. One such system is the altitude adjusted corrected geomagnetic (AACGM) coordinates, which were originally known as the Polar Anglo- American Conjugate Experiment (PACE) coordinates [Baker and Wing, 1989] and were later renamed to AACGM. To calculate these coordinates for a given location, the final definitive geomagnetic reference field or the interim international geomagnetic reference field model that is appropriate for the given epoch is used to trace magnetic field lines. The lines are traced, numerically, from the specified point in space outward to the point where the line crosses the magnetic dipole equator. A pure dipole field is then used to follow a virtual magnetic field line back down from this point toward the Earth. For true corrected geomagnetic coordinates (CGM) [Gustafsson, 1984] the back- 19 of 21