Mode number calculations of ULF field-line resonances using ground magnetometers and THEMIS measurements

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, , doi: /2012ja018307, 2013 Mode number calculations of ULF field-line resonances using ground magnetometers and THEMIS measurements T. E. Sarris, 1,2 X. Li, 2 W. Liu, 3 E. Argyriadis, 1 A. Boudouridis, 4 and R. Ergun 2 Received 18 September 2012; revised 26 August 2013; accepted 2 October 2013; published 11 November [1] Using multiple pairs of International Monitor for Auroral Geomagnetic Effects ground magnetometers together with simultaneous measurements from two of the Time History of Events and Macroscale Interactions during Substorms (THEMIS) spacecraft constellation, when they were flying over the magnetometers in magnetic conjunction and in close azimuthal separation, we are able to calculate the phase differences of Ultra Low Frequency Field-Line Resonances, and, through that, their azimuthal mode number, wavelength and propagation characteristics. A cross-wavelet technique is applied, that exposes the times and frequencies of common power between time series from azimuthally aligned magnetometers in space or on the ground, yielding their relative phase. Using the amplitude ratio and phase differences between ground stations with similar longitudes that are separated in latitude, a correction to the mode number calculation is demonstrated, accounting for the phase differences that arise from the L shell separation of the THEMIS probes. Citation: Sarris, T. E., X. Li, W. Liu, E. Argyriadis, A. Boudouridis, and R. Ergun (2013), Mode number calculations of ULF field-line resonances using ground magnetometers and THEMIS measurements, J. Geophys. Res. Space Physics, 118, , doi: /2012ja Introduction [2] The Earth s magnetosphere often undergoes oscillations in the Ultra Low Frequency (ULF) range, roughly 1 mhz to 1 Hz (periods 1 to 1000 s), which are a fundamental response of the magnetosphere to various drivers: they can be externally driven by the solar wind and can be produced via the coupling of compressional magnetospheric pulsations to shear Alfvén waves [e.g., Lee and Lysak, 1989, and references therein], or they can be internally driven by plasma instabilities, such as the drift mirror [Hasegawa, 1969] and bounce resonance instability [Southwood et al., 1969]. ULF waves can also have significant effects, such as enhancing the radial diffusion of radiation belt electrons. They are classified as poloidal or toroidal mode pulsations: poloidal pulsations exhibit electric field oscillations that are polarized in the azimuthal direction and are generally characterized by small azimuthal scale sizes, or, equivalently, large azimuthal mode numbers m, whereas toroidal pulsations have electric field oscillations polarized in the radial direction, magnetic field oscillations in the azimuthal direction, and are characterized 1 Department of Electrical Engineering, Democritus University of Thrace, Xanthi, Greece. 2 Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado, USA. 3 School of Astronautics, Beihang University, Beijing, China. 4 Space Science Institute, Boulder, Colorado, USA. Corresponding author: T. E. Sarris, Department of Electrical Engineering, Democritus University of Thrace, Vasilissis Sofias 1, Xanthi 67100, Greece. (tsarris@ee.duth.gr) American Geophysical Union. All Rights Reserved /13/ /2012JA by large scale sizes in the azimuthal direction, or, equivalently, by a small azimuthal wave numbers m; determining the scale size and wave number of toroidal oscillations through multipoint measurements both in space and on the ground will be the topic of this paper. [3] Toroidal oscillations have been attributed to mechanisms such as the Kelvin-Helmholtz instability near the magnetopause, solar wind buffeting, or to sudden increases in the solar wind pressure (pressure pulses) at the front of the magnetopause. Such pulsations can be observed in space [e.g., Takahashi and McPherron, 1982; Anderson et al., 1990; Lessard et al., 1999; Takahashi et al., 2010; Denton et al., 2009; Sarris et al., 2007; Sarris et al., 2010], and, contrary to poloidal (large m) oscillations that are filtered by the ionosphere, they can also be observed by ground magnetometers [Walker et al., 1979; Waters et al., 1991, 1996; Menk et al., 1999; Chi et al., 2000; Dent et al., 2003; Sarris et al., 2009a, 2009b]. Through these and other studies, the spatial and frequency characteristics of Field-Line Resonances (FLRs) have been investigated. However, determining their azimuthal wavelength and corresponding mode number involves combining measurements from multiple stations and performing phase-difference calculations; such calculations have been performed using ground magnetometers (e.g., Olson and Rostoker, 1978; Sarris et al., 2009a, 2009b; Tan et al., 2011] and also HF radars [e.g., Ponomarenko et al., 2001]. In particular, Ponomarenko et al. [2001] compared the calculations of the azimuthal mode number between ground magnetometers and coincident HF radars; they found values of m = 3 5 and 12, respectively, and offered potential explanations for these discrepancies. [4] Mode number calculations using spacecraft measurements require that spacecraft are properly aligned in azimuth 6986

2 and at close enough separations, within one FLR wavelength, so as to be able to capture the correct phase difference, and are thus not as common. A different technique has been demonstrated by Zong et al. [2007], who used the phase differences observed in the fluxes of energetic electrons as they were modulated by ULF waves to calculate the mode number of low-m toroidal waves. [5] A rare occasion of the occurrence of a FLR event of particularly large-amplitude fluctuations that were simultaneously observed by both multiple International Monitor for Auroral Geomagnetic Effects (IMAGE) ground stations and multiple THEMIS satellites [Angelopoulos, 2008] occurred on 4 September 2007: during this event, an ideal azimuthal alignment between THEMIS satellites coincided with their flyby over several of the Scandinavian IMAGE ground magnetometers in close magnetic conjunction, meaning that the THEMIS satellites were passing in the vicinity of field lines mapping to the locations of the ground stations. The solar wind conditions on this day were rather quiet, namely Dst = 25 nt, solar wind ram pressure = 1.1 npa. In the following, we present correlations and comparisons between calculations of phase differences and mode numbers that were performed on the ground and in space: We first present an overview of the Cross-Wavelet Transform technique for calculating phase differences between adjacent time series; we then present observations of the FLR event by multiple azimuthally aligned IMAGE station pairs as well as between THEMIS satellites, and subsequently, we present the phase-difference calculations through which we are able to identify the distribution of the mode number, wavelength, and wave phase velocity in space. 2. The Cross-Wavelet Transform for Phase- Difference Calculations [6] In order to detect the phase differences (phase lag) between pairs of the time series from IMAGE ground magnetometers and from the two THEMIS satellites, we make use of a Cross-Wavelet Transform (XWT) technique. Cross-wavelet analysis between two time series is used increasingly in geophysical data analysis in order to examine if regions in time and frequency with large common power also have a consistent phase relationship [e.g., Grinsted et al., 2004], as it offers several advantages over classical methods such as the Fourier cross-spectral analysis, including a higher spectral accuracy in low frequencies, in particular for nonstationary signals and a more accurate detection of correlations between sinusoidal signals of the same wavelength, if these signals are phase shifted [e.g., Rioul and Vetterli, 1991]. [7] Toward implementing the XWT analysis, we first compute the Continuous-Wavelet Transform (CWT) of each data set, in a manner similar to the work of, e.g., Torrence and Compo [1998]. We note here that the computation is actually a discretized version of the CWT, with computations performed at a finite number of locations, as the original CWT is highly redundant and computationally expensive. We use a discretized version of the CWT over a set of scales (scale vector), s j, given by s j ¼ s 0 2 j:δj ; j ¼ 0; 1; 2; ; J (1) where s 0 defines the smallest scale of the wavelet. Here we use s 0 =2*δt, where δt is the sampling time of the time series. The parameter J defines the number of scales, which range from s 0 to s 0 2 j.δj, to give a total of (J + 1) scales; here we use J = 12/δj. The parameter δj defines the spacing between discrete scales; here we use δj = 1/12. [8] Among the different types of mother wavelets, we adopted the Morlet wavelet, because it provides the desired balance between time and frequency localization. One more attractive characteristic of this type of wavelet is that its scale is almost equal to the Fourier period. Given the practical equality of the wavelet scale with the Fourier period, we assume that the calculation of the Wavelet Transform is done for 12 octaves of frequencies, from zero to Nyquist frequency, with 1/12 suboctaves per each octave. [9] From the CWTs of the two data sets, we then compute the XWT. The technique used for the XWT calculations is similar to that described in Torrence and Combo [1998] and used by Torrence and Webster [1999] in a study involving time series of the El Niño-Southern Oscillation. In brief, the XWT of two time series x(t) and y(t) can be considered as a bivariate extension to wavelet transform and is defined as the product of the corresponding Wavelet Coefficients W n x (s) and W n y (s) at each scale and location W xy n ðþ¼w s x n ðþw s y n ðþ s (2) where W n y (s) * indicates the complex conjugate of the wavelet coefficient of y. The resulting XWT is a complex number, its modulus being the Cross-Wavelet Power that essentially reveals regions with higher or lower common power, whereas the argument of the XWT is the local relative phase between the two time series in the time-frequency plane. The XWT calculation produces plots (scalograms), which show the Wavelet and Cross-Wavelet Power Spectral Density (XWT), in units of nt 2 /Hz, with linear time and frequency axis; it also produces phase-difference plots, which show the calculated phase difference as a function of frequency and time. [10] In calculating the phase difference between the two time series of the narrow-band ULF waves, a threshold has been introduced in order to reveal only the frequencies and times of regions with high common power (cross correlation) between the two signals; the threshold has been set to 0.025, meaning that data of the XWT phase plot are displayed only at frequencies (scale) and times (location) when the corresponding XWT power is above 2.5% of the maximum value. Subsequently, we average the thresholded XWT phase plot over all computed frequencies so as to take a line plot of the average phase difference for every time bin; here each time bin is equal to the sampling time, which is 10 s for IMAGE magnetometers and 3 s for THEMIS magnetometers. The averaged-phase difference is then smoothed in order to clearly reveal any trends in phase differences; smoothing is performed by averaging the calculated phase differences with a window of 140 elements or 23.5 min for IMAGE and 7 min for THEMIS. Averaged-phase differences are subsequently used to calculate the azimuthal mode number (or azimuthal wave number) m as a function of time; m here is a measurement of the number of wave cycles which can occur around the Earth in the azimuthal plane and is defined by m ¼ Δφ= Δλ (3) where Δφ is the phase difference between two points on the earth and Δλ is the azimuthal or longitude separation between 6987

3 Figure 1. (a) Example of the Dynamic Power Spectrograms of ground magnetometer data from station AND. Two distinct regions of enhanced narrow-band ULF wave power can be identified (similarly in most IMAGE stations), one in the dawnside of the magnetosphere, marked with red lines, and the other in the duskside, marked with blue lines. These regions are marked in Figure 1c accordingly. (b) Pairs of IMAGE ground magnetometer stations across different latitudes are used to calculate the mode number, using the XWT analysis; grid lines show geographic latitude. (c) The location of each station of Figure 1b is mapped to the magnetic equatorial plane every 1 h (UT), by tracing the field-line passing through the ground station to the minimum-b plane, using the T96 magnetic field model. (d) Constant and consistent mode numbers are observed between ~03:00 and 07:30. these points. In the following, phase differences are calculated between the easternmost versus the westernmost locations of magnetometer pair; thus, phase differences and associated mode numbers that are found to be positive will correspond to westward-phase propagation. 3. Mode Number Calculations From IMAGE Ground Magnetometers [11] Magnetic field measurements from IMAGE ground magnetometers were used in order to identify FLR magnetic field signatures. The resolution of the data was 10 s, yielding a Nyquist frequency of 50 mhz. Magnetic field measurements had a distinct DC component and persistent power from DC to about 2 mhz; in order to obtain only the frequencies of interest and more clearly identify the narrow-band pulsations corresponding to the observed FLRs, we detrended the time series by removing a 45 min moving average. [12] Figure 1a shows an example of the spectra of the northward component of the magnetic field from the ground station at Andenes (AND), up to 50 mhz. In this figure, signatures of enhanced power in ULF pulsations appear to occur primarily between 03:00 06:30 and 09:30 15:30 UT, as marked by the red and blue lines, respectively; it is these pulsations that we associate with FLRs, as will be shown more clearly below in section 4, through THEMIS measurements. These time periods correspond to Magnetic Local Time (MLT) ranges of approximately 5:00 8:30 and 11:30 17:30, or the dawn and noon-to-dusk regions, and thus the FLRs are associated with flank oscillations, possibly caused by Kelvin-Helmholtz-like instabilities in the magnetosphere flanks. Similar signatures of dawnside and duskside pulsations were also observed at most IMAGE stations, across different latitudes and longitudes, mostly in the X (northward) component of the magnetic field; the IMAGE stations that are used in this analysis are marked in Figure 1b. [13] In order to associate the ground measurements of FLRs with FLR observations in the magnetosphere, we mapped the location of each of the ground stations marked in Figure 1b to the magnetic equatorial plane by tracing the field line passing through the ground station up to the location of the minimum magnetic field strength on that field line, 6988

4 Figure 2. (a) Phase differences are calculated at five station pairs of increasing azimuthal separation, to remove 2π ambiguity. (b) Power Spectral Density at station LOZ and (c) station AND are used to calculate (d) the Cross-Spectrogram Power, onto which a Power threshold has been applied. (e) Phase differences at all times and frequencies. (f) Only phase differences that have cross-power higher than the threshold used in Figure 2d are shown. (g) Calculations from all station pairs above show a consistent phase difference and mode number (m = 5 to 7) only between 03:00 and 07:30 UT. using the T96 magnetic field model [Tsyganenko, 1995]; these locations, traced from each ground station to the magnetosphereevery1h(ut),areshowninfigure1c.thenameof each station is marked accordingly at the mapped location at 00 UT. Each of the two stations that will be used in the following sections as a pair for phase-difference calculations is marked with a distinct color, and the stations of each pair are connected with a line, also plotted every hour in UT. Similar to Figure 1a, in this figure, we also marked with red and blue lines the local times corresponding to the dawn and dusk regions where pulsations and distinct frequency ranges are observed. [14] In calculating the phase difference between ULF waves as measured at two azimuthally aligned stations, a 6989

5 Table 1. IMAGE Station Geomagnetic Coordinates and Separations in Latitude and Longitude of All Paired Stations in Order of Increasing Longitudinal Separations Abbreviation Name Latitude Longitude Δlatitude Δlongitude AND Andenes TRO Tromsø KIL Kilpisjärvi MAS Masi IVA Ivalo LOZ Lovozero π ambiguity is involved, as the difference in phase in any frequency is only defined modulo 2π. In order to resolve this 2π ambiguity in the event presented herein, we calculate the phase difference between multiple pairs stations of increasing azimuthal separations, keeping one of the stations in each pair always the same; an example of such calculation is shown in Figure 2a, where stations at approximately similar latitudes are jointed in pairs of increasing azimuthal (longitudinal) separation, as follows: AND-TRO, AND-KIL, AND- MAS, AND-IVA, and AND-LOZ. The coordinates of all stations and the latitudinal and longitudinal separations of all paired stations are given in Table 1, in geomagnetic coordinates. Thus, a coherent signal at a particular frequency that is recorded at all stations should show a phase difference between station pairs that is linearly dependent upon the stations azimuthal separations, if the azimuthal separations are smaller than the wavelength of the sampled wave. [15] In Figure 2a, we show a map of six stations that are linked in five pairs of increasing azimuthal (longitudinal) separation, as described above. An example of the power as a function of time and frequency from the CWT of two of these stations (pair AND-LOZ, which have the largest separation in longitude) is shown in Figures 2b and 2c. In these figures, a high-pass filter has been applied to the magnetometer time series, to better identify the narrow-band ULF pulsations associated with FLRs from broadband, lowerfrequency pulsations below 5 mhz. After applying the Cross-Wavelet analysis that was described in the previous section, in Figure 2d, we plot the XWT Power between the two CWTs of stations LOZ and AND, also applying a threshold in power that allows only cross-power at times and frequencies with common high power at both stations to be depicted. In Figure 2e, we show the phase differences at all frequencies and at all times between these two ground stations, whereas in Figure 2f, we plot only the phase differences at the times and frequencies with high cross-power, as shown in Figure 2d. We subsequently average these phase differences across frequencies from 0 to 20 mhz for a given time, weighted by the cross-spectrogram power, and, finally, we calculate the mode number as a function of time by dividing the estimated average phase difference by the azimuthal separation between the two stations, as described in equation (3). In Figure 2g, we plot the average mode numbers from all four of the station pairs shown in Figure 2a. We can see that calculations from the various station pairs show a phase difference that is increasing linearly with azimuthal separation of the stations, and a consistent mode number that ranges between 5 and 7 only between 03:00 and 07:30 UT, whereas at other times the mode number is not consistent between station pairs, indicating that only during this time period is there a coherent ULF FLR wave that is observed at all ground magnetometers and also that, indeed, there is no 2π ambiguity involved in these calculations during this time and for this azimuthal separation of the stations. [16] A source of error in the calculation of the above mode number comes from the uncertainty in determining the phase difference, which, as described, was calculated from the average of phase differences at various frequencies, after applying a threshold in power, as shown in Figure 2f; as a measure of the error, we calculated 2σ or twice the square root of the variance of the averaged-phase differences. The resulting average error in the mode number calculation during the period of interest from 03:00 to 07:30 was found to be δm ~1.7. [17] After the period 03:00 and 07:30 UT, and in particular between 09:00 and 14:00, the dominantly blue and red colors (Figure 2f) indicate that the phase difference is close to 0 or 360, respectively, indicating that the mode number should be zero, corresponding to global pulsations, often termed as the breathing mode or global mode of the magnetosphere. We attribute this concurrent appearance of 0 and 360 to noise in the signals and uncertainties through the phasedifference calculation technique. Thus, the averaging procedure to find the mean phase difference does not yield any meaningful outcome during this time, as, for example, an average between 0 and 360 yields 180, which is an incorrect calculation. This could be resolved by a phase-shifting technique, defining the range from 180 to 180 instead of 0 to 360 and then taking the average, which would give a more meaningful average. In conclusion, the above calculations shown in Figure 2 can be interpreted correctly only when phase differences are linearly increasing with azimuthal separation and mode numbers are consistent between various station pairs, which are occurring only between 03:00 and 07:30 UT in our case. [18] Similar calculations to those presented above were performed for measurements from all station pairs shown in Figure 1b, where the ground station pairs are located at different geomagnetic latitudes, ranging from to As before, stations of a particular pair are aligned across similar latitudes. The geomagnetic coordinates and latitudinal-longitudinal separations of the IMAGE ground station pairs are shown in Table 2, in order of decreasing latitudes of the pairs. Table 2. IMAGE Station Geomagnetic Coordinates and Separations in Latitude and Longitude of All Paired Stations in Order of Decreasing Latitude of the Various Pairs Abbreviation Name Latitude Longitude Δlatitude Δlongitude TRO Tromsø KEV Kevo ABK Abisko IVA Ivalo KIR Kiruna SOD Sodankylä RVK Rørvi OUJ Oulujärvi DOB Dombås MEK Mekrijärvi UPS Uppsala NUR Nurmijärvi

6 [19] By tracing the field lines with foot-points at the stations of Table 2 to the magnetosphere, we map the locations of the station pairs to the plane of minimum magnetic field (B-min) in order to correlate the FLR measurements with measurements performed by THEMIS spacecraft (s/c) in the magnetosphere; the mapped locations are shown in Figure 1c, plotted every 1 h in UT, as marked; in the dawn region where the pulsations are observed, the ground stations map at a range of radial distances, from ~3 R E to ~7 R E.Ifthe observed ULF pulsations are persistent and of long duration, such as described in, e.g., Sarris et al. [2007, 2009a, 2009b], and provided that the ULF waves penetrate through the ionosphere, as seems to be the case in this particular event, this mapping allows us to scan the entire magnetosphere and estimate the mode number as a function of radial distance, local time, and/or time. [20] Following the procedure discussed in the preceding sections, we calculate the mode numbers at the marked station pairs; the estimated mode number from each pair is shown in Figure 1d. In this figure, we can see, as before, that consistent mode numbers are observed between approximately 03:00 and 07:30 UT across all sampled L shells. A deviation from the average mode number can be seen at the green line, corresponding to station pair RVK- OUJ, which shows phase differences corresponding to mode numbers lower than average before ~05:30 and higher than average after ~05:30. We attribute this to the fact that these two stations have the largest latitudinal separation, and second-largest separation in longitude, as can be seen from Table 2, leading to larger errors in mode number calculation. 4. Mode Number Calculations From THEMIS Spacecraft [21] In space, magnetic field measurements were obtained from two spacecraft of the THEMIS constellation, during an outbound pass of the five probes on 4 September 2007 from 04:00 to 10:00 UT. During this time, the two spacecraft remained azimuthally separated, with Δφ ranging from ~0.5 at 04:00 to 1.0 at 08:45; the spacecraft separation at 0400 UT was ~200 km in longitude and ~500 km in radial direction, and at 0845 UT it was ~900 km in longitude and ~300 km in radial direction. The components of the magnetic field vector were projected in a Mean Field-Aligned coordinate system in order to separate the ULF field variations perpendicular to as well as along the magnetic field direction. Thus, following common nomenclature, waves in the e ==, e r, and e φ directions are referred to as compressional, poloidal, and toroidal, respectively. During this event, most fluctuations in the azimuthal component of the magnetic field were observed, indicating toroidal fluctuations. Details of the observed fluctuations have been presented in a study by Sarris et al. [2009a, 2009b]; in this study, we present, instead, a reanalysis of these measurements based on the XWT technique, with a greatly enhanced time resolution, and a direct comparison, for the first time to our knowledge, of the mode number and azimuthal wavelengths between spacecraft and ground magnetometer measurements. [22] The close azimuthal separation between two of the THEMIS probes, D and E (referred to as TH-D and TH-E hereafter) during this event provide an ideal configuration to investigate the azimuthal mode number of the pulsations, through monitoring phase differences of the observed toroidal ULF waves, as they were captured in the time series of the azimuthal component of the magnetic field. In the mode number calculation, a similar analysis to that applied in the ground magnetometer measurements is followed, as described below. [23] In Figures 3a and 3b, the CWT of the time series of the azimuthal component of the magnetic field B φ is plotted from TH-E and TH-D, respectively, for frequencies up to 30 mhz. Fluctuations at ULF frequencies can be observed, together with their harmonics, whereas no magnetic field pulsations was observed in the field-aligned or radial directions; during the same time, fluctuations are observed at the same frequencies in the electric field instrument on board both THEMIS probes, primarily in the radial direction E r, indicating the prevalence of toroidal ULF pulsations. Calculations of the phase difference between E r and B φ show a phase difference of 90, indicating that these pulsations are Alfvén wave Field-Line Resonances (calculations not shown). Thus, in the spectra of the azimuthal components from both satellites narrow-band magnetic field oscillations appear shortly after 04:00 UT, lasting throughout the outbound orbit, until 10:00, when both spacecraft cross the magnetopause. The frequency of these pulsations is initially ~25 mhz, at ~04:00 UT, when the two THEMIS spacecraft are located at ~4 R E, and gradually drops to ~4 mhz, as the spacecraft move close to the magnetopause boundary. [24] In Figure 3c, the CWT of the azimuthal components from the two s/c is used to calculate the Cross-Wavelet Transform (XWT), as described in section 2. In this figure, a threshold in power has been applied, similar to the threshold applied in ground magnetometer measurements. Subsequently, in Figure 3d, the phase differences are plotted in a color scale as a function of time and frequency, derived from the imaginary part of the XWT calculations as described in section 2 above; in this figure, during the times and frequencies of the FLRs shown in the panel directly above a coherent blue color is prevalent, corresponding to low-phase differences close to zero. For the times and frequencies that have enhanced XWT power above the fixed threshold, the phase differences are plotted in Figure 3e; the two plots, Figures 3e and 3d are similar, both in content and in color scale. As before, it is these phase differences that are used in the mode number calculation. Finally, in Figure 3f, estimates of the wave mode number are calculated as a function of time, as derived from the average calculated phase difference of Figure 3e divided by the azimuthal separation between the two s/c. [25] From the calculations of the mode number as a function of time during the outbound pass of the two THEMIS s/c, we can see that the mode number is initially above 20, with large variations and measurement gaps; we note that, during this time, the azimuthal separation between the two spacecraft was rather small (~200 km in longitude and ~500 km in the radial direction), and estimates of the phase difference are expected to be less accurate. As the two spacecraft move outward, their azimuthal separation increases and their radial separation decreases, allowing a more accurate determination of the phase difference. After ~05:00 UT and until 07:00, the 6991

7 a. B CWT Spectrum, THE b. B CWT Spectrum, THD c. B XWT Spec., THE THD d. Phase Difference THE - THD e. Phase Difference THE - THD f. Mode Number, m Figure 3. (a) Continuous Wavelet Transforms (CWT) of the azimuthal component of the magnetic field, B φ at s/c THEMIS E and (b) THEMIS D are used to calculate (c) the Cross-Wavelet Transform (XWT) power, onto which a threshold in power has been applied. (d) Phase differences at all times and frequencies, based on the XWT calculations. (e) Only phase differences that have XWT power above the fixed threshold of Figure 3c are shown. (f) Estimates of the wave mode number as a function of time, based on the calculated phase difference and the azimuthal separation between the two s/c. calculated mode number drops to values of around 5, whereas after 07:00, the ULF pulsations are less obvious and overlap with low-frequency-broadband power, making the identification of phase differences less obvious. After 08:30, the two spacecraft start crossing a fluctuating magnetopause boundary, and estimates of the mode number are no longer possible. [26] A main source of error in calculating the phase differences and corresponding mode number from the THEMIS probes comes from the fact that the two probes have a small L shell separation, which leads to an additional phase difference; in the following section, we demonstrate using latitudinally separated ground magnetometer pairs, following the methodology described by Kawano et al. [2002], in order to evaluate the resonance width of the FLRs and to correct the phase difference and subsequently the mode number estimates. 5. Discussion [27] Based on the calculations of the mode number of the FLRs, as presented in Figures 2g and 3f above, we will derive the azimuthal wavelengths from the two separate measurements in space and on the ground and we will discuss the differences between the two measurement schemes. Thus, in Figure 4, we overplot the locations of the ground stations as mapped to the magnetic equatorial plane, together with the orbits of s/c TH-D and TH-E, marked in dark blue and lighter blue, respectively. Time along the orbit is also marked, every 1 h in UT. From this figure, we can see that TH-D and TH-E 6992

8 Figure 4. The azimuthal wavelength corresponding to the mode number calculated based on the XWT from each of the selected station pairs is plotted as thick black line at each station pair, centered around the time that narrow-band oscillations are observed. At the same time, THEMIS D and E crossed through the same region; THEMIS D and E orbits are plotted as light and dark blue lines, and their locations every 1 h in UT is marked with a cross of similar color. The azimuthal wavelength based on THEMIS calculations is shown in light blue lines. cross through the same region that the ground stations are monitoring at the same time, providing ideal in situ calculation of the azimuthal mode number that the ground stations are remotely sensing. The wavelength calculations of the FLR oscillations are done by translating the calculated mode number to an azimuthal extent equal to φ =2π/m, where m is the mode number, or to the corresponding arc length, from l = r φ. Thus, in Figure 4, we plot the extent of the azimuthal wavelength of the FLRs as derived from the ground measurement calculations in thick black lines, one line for each station pair, centered around the dawnside region where FLRs are observed; overplotted in light blue lines are the azimuthal wavelengths as derived from THEMIS measurements, with one line corresponding to the average m value every 15 min. Thus, calculated wavelengths range from 1.4 R E (inner) to 9.2 R E (outer), as calculated from THEMIS s/c (light blue lines), and from 3.2 R E (inner) to 10.3 R E (outer), as calculated from the IMAGE magnetometers (thick black lines). [28] Overall, through the above analysis, it was found that low-mode numbers, on the order of m = 4 to 7, are associated here with the particular FLR observed on this date; estimates from TH-D and TH-E measurements agree with estimates from ground measurements mostly at higher L shells, where we also note that there is sufficient azimuthal separation between the s/c to provide an accurate phase-difference estimate, whereas at lower L shells, the azimuthal separation is much smaller, thus providing less accurate estimates. [29] These results are in agreement with previous studies that associate low-mode numbers (m < 10) to field line guided Alfvén waves; these have been attributed to energy transfer from magnetospheric waveguide modes [e.g., Mann et al., 1998], compressional modes, cavity modes [e.g., Kivelson and Southwood, 1986], and the Kelvin- Helmholtz instability at the magnetopause [e.g., Pu and Kivelson, 1983]; the latter seems the most probable generation mechanism for these waves during this event, as there is no evidence for strong compressional modes, waveguide modes, or cavity modes during this event. In an MHD simulation by Claudepierre et al. [2008], which was driven by three different solar wind speeds, v x = 400 km/s, v x = 600 km/s, and v x = 800 km/s, it was found that magnetospheric waves at mhz had their peak power at m ~ 8; interestingly, this value did not vary significantly as the solar wind driving speed was varied. During the event studied herein, the solar wind velocity was ~600 km/s, with the rest of solar wind 6993

9 Figure 5. Amplitude ratio G and phase difference Δφ as a function of frequency and time during this event, from IMAGE ground stations IVA and SOD, which are aligned at similar longitudes but different latitudes. conditions being on the same order of magnitude to those of Claudepierre et al.; whether there is a solar wind, dependence needs to be confirmed through other case studies such as this one. [30] From the calculated mode numbers and the associated wavelengths, as plotted in Figure 4, and using the FLR frequencies as a function of radial distance as measured by the two THEMIS satellites during their outbound orbit, we subsequently calculated the azimuthal phase velocity of the FLRs at different locations in the magnetosphere, using υ ph = λ/τ = λω/ 2π. Thus, velocities range from 42 km/s to 85 km/s. [31] During this event, all five THEMIS satellites were flying on similar orbits (similar arguments of periapsis), but were on different phases along the same orbit; hence they monitored exactly the same region, even though at different times. Thus, combining measurements from other THEMIS s/c (not shown), we note that the FLRs are measured within the same region of interest by THEMIS B, about 1 h before THEMIS D and E cross through this region at ~04:30 to ~07:00. Furthermore, THEMIS A measurements indicate that the FLRs were present even during the satellite s inbound pass, at 00:00 to 02:00, and beyond 12:00. We therefore speculate that the FLRs were persistent throughout this day, but were only observed by the ground magnetometers and by THEMIS D and E in the local times marked in Figure 4; Hence, we speculate that the azimuthal extent of the observed FLRs is a spatial effect, potentially associated with the Kelvin-Helmholtz instability in the Earth flanks, rather than a temporal effect. [32] As mentioned above, a main source of error in the calculation of the mode number can be introduced by the potentially large transition in phase that appears across the resonance latitude of toroidal field-line resonances, if the two measurement points are situated at even slightly different L shells. Following the methodology employed by Pilipenko and Fedorov [1994] and Kawano et al. [2002], we used the IMAGE ground stations IVA and SOD, which are aligned at similar longitudes but different latitudes, in order to determine the amplitude ratio G and phase difference Δφ of the ground station pair as a function of frequency 6994

10 Figure 6. Magnetic latitude profile of the wave phase caused by the field-line resonance from the model of Pilipenko and Fedorov [1994], based on the measurements from ground stations IVA and SOD on 4 September 2007, at 05:00 UT. The model parameters used are: x R = 64.41, ε g = 1.26 (black solid line), and ε m = 0.36 (gray dashed line), where ε m includes the ionospheric correction discussed by Pilipenko and Fedorov (1994), assuming an ionospheric E layer height of h = 0.9. The relative locations of THEMIS probes D and E at 05:00 UT are also marked (blue lines), as mapped to the Earth along magnetic field lines. during this event; we selected the particular latitudinal pair because it maps closest to the locations of probes TH-D and TH-E at 05:00 UT, as shown in Figure 4 in the paper. The resulting ratio G( f ) between the Fourier spectra amplitudes from the two ground stations and the phase difference Δφ( f ) of the H components of the ULF waves are shown in Figure 5. Both the amplitude ratio and the phase-difference calculations show clear signals of a field-line resonance at a variable frequency, from 10 to 20 mhz during 02:00 to 08:00 UT, which is however consistent between the two panels. Furthermore, at 05:00 UT when the IVA-SOD pair maps closest to the locations of probes TH-D and TH-E, the FLR frequency calculated from the average between the amplitude ratio and the phase-difference methods from the IVA-SOD measurements is mhz, very close to the FLR frequency measured by the two THEMIS satellites, which is about 15 mhz. At that time, G =A IVA /A SOD is found to be 0.94 and Δφ = φ IVA φ SOD is [33] We subsequently use G and Δφ to calculate the resonance latitude x R and resonance width ε between the stations, using equations (6a) and (6b) of Kawano et al. [2002]. In their equations, x 1 corresponds here to the latitude of IVA, the northernmost of the two stations used in this analysis, and x 2 corresponds to the latitude of SOD. The resulting values are x R = and ε = From the estimated x R and ε, we can then find the relationship between the phase variation and latitude, similarly to the theoretical example illustrated in Kawano et al. [2002, Figure 1b]. For the particular measurements from ground stations IVA and SOD on 4 September 2007, at 05:00 UT, the curve of the latitude profile of the wave phase of the magnetic field caused by the fieldline resonance is plotted in Figure 6 (black solid line), based on the model of Pilipenko and Fedorov [1994]. The latitudes of THEMIS probes D and E at 05:00 UT are also marked in the figure (blue vertical lines), as mapped to the Earth along magnetic field lines. The parameters used to draw the latitude profile of the phase across the resonance latitude are listed in Table 3 below. [34] From the relative latitudinal locations of the two THEMIS probes with respect to the ground station locations and the curve shown in Figure 6, we subsequently calculate the expected phase difference, which arises from the L shell separation of the two probes; at the particular time when this methodology is tested (05:00 UT), the two probes were separated by 0.06 in L-shell. The latitudinal locations of the two probes are given in Table 3, as estimated by tracing field lines Table 3. Parameters Used in Phase-Correction Technique of Pilipenko and Fedorov [1994], Based on the Amplitude-Ratio and Phase- Difference Calculations Made at 05:00 UT From Measurements at Latitudinally-Aligned Ground Stations IVA and SOD Station Pair (x 1 x 2 ) Latitude Difference (x 1 x 2 ) f (mhz) Δφ (degree) G ( f ) x R ( f ) (degree) ε (degree) THEMIS Probes THEMIS Latitude Mapped to ground IVA-SOD TH-E TH-D

11 from their locations in space to the ground, using the T96 magnetic field model [Tsyganenko, 1995]. [35] Thus, from the model curve of Figure 6, the wave phase difference that is introduced by the small latitudinal separation of the two THEMIS probes is estimated to be 4.4. The remaining will be the longitudinal phase difference that we used above to calculate the mode numbers. This needs to be applied to the phase difference calculated through the cross-phase analysis, in order to isolate the phase difference only due to the wave propagation. [36] At 05:00 UT, the azimuthal separation between probes TH-E and TH-D is 0.8 and the cross-wavelet analysis yields a phase difference of 11 ; assuming that all this phase difference is attributed to the azimuthal separation leads to an estimated mode number of m = 13; however, when correcting for the L shell (or latitudinal) separation by subtracting the associated phase difference, calculated above to be 4.4, this leads to a phase difference of = 6.6, which is due to the longitudinal separation, or a mode number of m = 6.6 /0.8 = 8 instead of 13, much closer to the mode number m = 5 to 7 that is calculated at latitudes close to station IVA, as shown in Figure 2. [37] In the above calculation of the correction in the azimuthal wave number due to the latitudinal effect, we did not include the effects of the ionosphere. Pilipenko and Fedorov [1994] describe that the resonance width observed on the ground, ε g, will be smoother than that above the ionosphere in the magnetosphere, ε m, such that ε g = ε m + h, where h is the height of the ionospheric E layer. Assuming an E layer height h that corresponds to 0.9, the calculated ε g = 1.26 on the ground will translate to merely ~0.36 in the magnetosphere. Thus, in Figure 6, we also plot with a dashed gray line the curve of the latitude profile of the phase of the wave magnetic field based on the above resonance width. We observe that this corresponds to a sharper transition in phase across the resonance latitude, which will correspond to a larger change in phase at the latitude locations of two THEMIS probes; the required correction in the phase change caused by latitudinal separation of the THEMIS probes is found in this case to be 11.6 in the magnetosphere. This phase correction is substantially higher than the previous one and of similar magnitude to the azimuthal phase difference deduced from the Wavelet technique. It is therefore unknown what the effect of its application would be on the entire time period of the study. A more detailed study needs to be carried out, taking into account the particular characteristics of the ionosphere during the particular event, and also using a larger number of different latitudinally-aligned pairs of ground stations, in order to accurately determine the phase-difference correction that needs to be applied, as a function of both time and latitude (or L). 6. Summary and Conclusions [38] In this paper, we investigated the spatial characteristics of FLRs that appear at multiple stations in space and on the ground in the form of narrow-band ULF waves. Using simultaneous measurements from pairs of azimuthally (longitudinally) aligned ground stations that were located at various latitudes, we were able to compute phase differences of the FLRs, and, through that, their azimuthal mode number and azimuthal extent. Phase differences of the FLRs were calculated using a cross-wavelet analysis technique, which provides power, and also phase difference as a function of time and frequency; we subsequently calculated the phase difference of the FLRs by averaging only phase differences of time instances and frequencies that had power above a certain threshold in power, and calculated the corresponding mode number by dividing the average phase difference at every given time by the azimuthal separation of the magnetometers. The technique was also applied to a pair of azimuthally aligned THEMIS satellites, as they traversed the inner magnetosphere, and also to ground magnetometer pairs across a range of latitudes, from to 58.26, mapping to a range of radial distances in the equatorial plane of the magnetosphere, from ~3 R E to ~7 R E. The region that was scanned by the magnetic field lines originating from the selected ground magnetometers overlapped with the region that the THEMIS satellites traversed, providing a unique opportunity to compare and confirm the mode number calculations; more specifically, from UT = 04:00 to 07:00, TH-D and TH-E cross the region with L values between ~4 and ~8 and MLT between ~6 and ~8, while the ground stations scan the region between ~3.3 and ~6.4 and MLT between ~6 and ~9. We resolved the 2π ambiguity that is often associated with phasedifference measurements by performing similar measurements at a number of station pairs at close longitudes, but at gradually increasing latitudes, through which a linear correlation was observed between azimuthal separation and phase difference. The FLR wavelength corresponding to the calculated mode number from both the ground magnetometers and the THEMIS spacecraft closely matched the azimuthal extent of the entire region over which FLRs are observed. [39] In this study, we also applied the methodology described in Pilipenko and Fedorov [1994] and Kawano et al. [2002] in order to introduce a correction to the mode number calculation due to phase differences arising from the L shell separation of the THEMIS probes; in the applied methodology, we calculated the amplitude ratio and phase difference from a pair of ground stations with similar longitude that are separated in latitude, which closely map along field lines to the locations of probes TH-D and TH-E at a particular time. The calculated amplitude ratio and phase difference were then used to calculate the resonance latitude and resonance width, through which we calculated the wave phase difference that is introduced by the small latitudinal separation of the two THEMIS probes. When correcting the observed phase difference for the L shell (or latitudinal) separation by subtracting the associated phase difference, we get a better agreement of the estimated mode number between ground and space-based observations; however, when the ionospheric correction is also applied, this leads to a lesser agreement. Further investigation on this is needed, calculating this correction from multiple latitudinally-aligned station pairs and at different times, to confirm a consistent correction of the mode number across multiple L shells in the magnetosphere. [40] Acknowledgments. This study was supported by NASA grants (THEMIS, NNX10AQ48G and NNX12AG37G) and NSF grant ATM This research has also been cofinanced by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF) Research Funding Program: Thales. Investing in knowledge society through the European Social Fund. [41] Robert Lysak thanks Kazue Takahashi and an anonymous reviewer for their assistance in evaluating this paper. 6996

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