Radial diffusion of relativistic electrons into the radiation belt slot region during the 2003 Halloween geomagnetic storms

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2005ja011355, 2006 Radial diffusion of relativistic electrons into the radiation belt slot region during the 2003 Halloween geomagnetic storms T. M. Loto aniu, 1 I. R. Mann, 1 L. G. Ozeke, 1 A. A. Chan, 2 Z. C. Dent, 1 and D. K. Milling 1 Received 19 August 2005; revised 25 November 2005; accepted 18 January 2006; published 27 April [1] A study was undertaken to estimate the radial diffusion timescale, t LL, for relativistic electrons (2 6 MeV) to diffuse into the slot region due to drift-resonance with Pc5 ULF waves (2 10 mhz) on 29 October Large amplitude ULF waves were observed by ground-based magnetometer arrays to penetrate deep into the slot region (L 2 3) starting at 0600 UT and maximising (200 nt p-p) between UT. Around the same time, the SAMPEX PET instrument measured an over two orders of magnitude increase in relativistic (2 6 MeV) electron flux levels in 24 hours within the slot region. The ground-based D-component magnetic power spectral densities (PSD db ) for 29 October were estimated for six latitudinally spaced ground stations covering L for an observed ULF wave with central frequency 4 mhz. The PSD db values were used to calculate the in situ equatorial poloidal wave electric field power spectral densities (PSD dem ) using a standing Alfvén wave model. The radial diffusion coefficients, D LL, were estimated using the PSD dem values. The fastest t LL were 3 5 hours at L > 4, while t LL initially increased with decreasing L-value below L 4; peaking at L 3 with t LL hours with PSD dem estimated using a wave frequency bandwidth between Df = 1 mhz and Df = 2.5 mhz. The t LL over the L-range L were consistent with the timescales observed by SAMPEX for the increase in relativistic fluxes in the slot region on 29 October. The authors believe that this is the first example of the ULF wave driftresonance with relativistic electrons explaining a radiation belt slot region filling event. Citation: Loto aniu, T. M., I. R. Mann, L. G. Ozeke, A. A. Chan, Z. C. Dent, and D. K. Milling (2006), Radial diffusion of relativistic electrons into the radiation belt slot region during the 2003 Halloween geomagnetic storms, J. Geophys. Res., 111,, doi: /2005ja Introduction [2] Highly energetic (greater then a few hundred kev) electrons are often observed in the Earth s radiation belts. The kinetic energy of these electrons can approach, or be greater than, their rest mass and therefore they are often referred to as relativistic electrons. The electron radiation belts are best described as an inner belt which has relativistic particle flux maximum at L 1.5 and which is often co-located with a very stable relativistic proton belt (energies >10 MeV), and an outer radiation belt which can extend from L to geosynchronous and beyond, where relativistic electron intensity usually peaks at altitudes between L 4toL5. The two belts are separated by a region of enhanced losses centred at about L 2.5 called the slot region [Lyons and Williams, 1976]. 1 Department of Physics, University of Alberta, Edmonton, Alberta, Canada. 2 Rice Space Institute, Physics and Astronomy Department, Rice University, Houston, Texas, USA. Copyright 2006 by the American Geophysical Union /06/2005JA [3] The outer electron radiation belt is highly variable, with the outer belt s relativistic electron population sometimes varying by orders of magnitude over different timescales ranging from minutes to days [e.g., Blake et al., 1992; Li et al., 1997; Baker et al., 1994]. The central location of peak electron flux also varies, and even the slot region can sometimes be filled. However, the mechanisms responsible for creating, depleting and injecting this population of relativistic particles are not well understood [e.g., Friedel et al., 2002; Fung, 2004]. Although the dominant mechanisms are not well-established, a connection between enhancements in relativistic electrons and solar-terrestrial interactions has long been recognised, in particular observations often show relativistic enhancements associated with high solar wind speeds [e.g., Paulikas and Blake, 1976, 1979; Blake et al., 1997; Rostoker et al., 1998]. [4] In the magnetosphere a range of mechanisms have been suggested to explain the acceleration of relativistic electrons [e.g., see Friedel et al., 2002; O Brien et al., 2003]. The most popular involve ULF wave or VLF wave processes operating by violation of the third or first adiabatic invariant, respectively. A number of studies based on case studies [Rostoker et al., 1998; Baker et al., 1998a, 1998b; Green and Kivelson, 2001; Mathie and Mann, 2000] 1of15

2 coefficient (L 6 or L )[Falthammar, 1968; Schulz and Lanzerotti, 1974; Elkington et al., 2003] as well as the statistical decay of the ULF wave power towards lower L-values [e.g., Mathie and Mann, 2001]. During some storms however, particle transport can occur which fills the slot region. For example, during the March 1991 super storm shock acceleration resulting from the assumed impact of an interplanetary shock can explain the filling of the slot region and the creation of a new energetic inner belt [e.g., Li et al., 1993]. During the 2003 Halloween super storm, the slot region was also observed to be filled following storm onset [e.g., Baker et al., 2004]. In this paper the ULF wave power in the inner magnetosphere is examined during the first day of the 2003 Halloween storm. Specifically, the hypothesis that enhanced Pc5 ULF wave radial diffusion can explain the relativistic electron population of the slot region on 29 October 2003 is examined. Figure 1. Solar wind parameters from the ACE spacecraft during October From top to bottom: three components of the interplanetary magnetic field in GSM coordinates (Bx, By, Bz), the solar wind velocity (V sw ), solar wind proton number density (Np), dynamic pressure (Pdyn), and the Dst index. as well as others based on statistical analysis [O Brien et al., 2001, 2003] have suggested the importance of ULF waves in the acceleration process. Whilst studies such as Baker et al. [1998a] and Tan et al. [2004] suggest timescales for electron acceleration by ULF waves may be as short as a few hours during specific storms, statistical studies suggest that timescales of 2 3 days in the recovery phase are appropriate [Mathie and Mann, 2000; O Brien et al., 2001, 2003]. [5] While a range of theorised ULF wave acceleration mechanisms exist, including transit-time damping [Summers and Ma, 2000] and recirculation or magnetic pumping [Liu et al., 1999], much work has focused on enhanced Pc5 ULF wave radial diffusion [e.g., Elkington et al., 1999, 2003]. Statistically, radial diffusion is much faster at high- L values due to the implicated L-dependence of the radial diffusion 2. Observations 2.1. Solar Wind [6] Figure 1 shows the solar wind conditions during the period October Solar wind data are shown from the Solar Wind Electron Proton Alpha Monitor (SWEPAM) instrument [McComas et al., 1998] and the Magnetic Field Experiment (MAG) [Smith et al., 1998] onboard the Advanced Composition Explorer (ACE) spacecraft, which is in a halo orbit about the L1 Lagrangian point. The Level 2 data for the SWEPAM instrument is usually 64 s resolution; however, during the interval from 1241 UT on 28 October through 0051 UT on 31 October data were only available in a lower 33 minute time resolution. The density (Np) and dynamic (Pdyn) pressure between 0600 UT on 29 October to 0400 UT on 30 October, which are shown as data gaps, are considered unreliable [Skoug et al., 2004]. [7] In a previous study of this Halloween storm interval, Skoug et al. [2004] noted three shock events at ACE at 0131 UT on 28 October, 0558 UT on 29 October and 1619 UT on 30 October. The latter two shocks both had solar wind speeds exceeding 1500 km/s, reaching >1850 km/s on 29 October (note the SWEPAM measurements are considered unreliable above 1850 km/s). The total magnetic field (not shown) was enhanced after each shock, reaching 20 nt following the 28 October shock, 68 nt following the 29 October shock, and 40 nt around the time of the shock on 30 October. The magnetic field Bz-component (GSM) turned negative after each shock reaching 15 nt on 28 October, 68 nt on 29 October, and 35 nt on 30 October. [8] The hourly Dst index data taken from the OMNI Level 2 website ( shows three strong negative dips reaching 180 nt at 0900 UT on 29 October, < 360 nt around midnight on 29 October, and 400 nt at 2200 UT on 30 October. The first strong jdstj excursion starts immediately following the SSC, which occurred about 14 minutes after ACE observed the shock at 0558 UT on 29 October [Panasyuk et al., 2004]. The excursion towards the jdstj maximum at 2200 UT on 30 October began around UT, which is shortly after the shock seen at ACE at 1619 UT on 30 October. 2of15

3 Figure 2. The 12-hour averaged SAMPEX 2 6 MeV electron flux from day 280 to the end of 2003 derived from measurements using the Proton/Electron Telescope (PET) instrument. (a) Integrated flux levels over the bins L < 2.0 (thin line) and L = (thick line), with bin size dl = 0.2. (b) 6-hour time resolution flux over dipole L = , at dl = 0.2 resolution. [9] Very large geomagnetic storms can sometimes be associated with very large Np (>50/cm 3 ) and high Pdyn (>50 npa) [e.g., Skoug et al., 2004]. The measured proton densities and dynamic pressure in this super storm did not reach these extreme values, not exceeding 10 15/cm 3 and 20 npa, respectively. However, data gaps make it impossible to estimate the magnitude of these values on 29 October Radiation Belt Measurements [10] The Solar, Anomalous, and Magnetospheric Particle Explorer (SAMPEX) spacecraft was launched in 1992 into a low-altitude (600 km), high-inclination (82 ) orbit with an orbital period of about 95 minutes [Baker et al., 1993]. Figure 2 shows the 12-hour averaged, 6-hour resolution, relativistic electron (2 6 MeV) integrated (total) flux derived from measurements using the Proton/Electron Telescope (PET) [Cook et al., 1993] onboard SAMPEX, over the period day 280 (7 October) year 2003 to the end of that year. The top panel (Figure 2a) shows the 6-hour resolution averaged- L flux levels over the bins L < 2.0 (thin line) and L = (thick line), while in the bottom panel (Figure 2b) the fluxes for L = , calculated with a bin size dl = 0.2, are shown. The 12 hour beating or periodicity seen in the particle measurements occurs because of the asymmetry of the Earth s magnetic field, in particular the South Atlantic anomaly whose effect becomes significant at the low-altitude orbit of SAMPEX (M. Looper, personal communication, 2005). [11] Starting on day 301 (28 October) and based on 6-hour resolution data, the relativistic (2 6 MeV) electron flux increased across the outer zone (L 4) L-values from <10 cm 2 s 1 sr 1 to >10 3 cm 2 s 1 sr 1 in 24 hours. The beginning of this flux increase coincided with the arrival of the first shock early on 28 October. The 400nT Dst excursion around 24 UT on 30 October was accompanied by an abrupt flux decrease to 1 cm 2 s 1 sr 1 across the entire outer zone at L 4. The fluxes across L 4 subsequently recover over a three day period beginning on 1 November, peaking at 10 3 cm 2 s 1 sr 1 on day 308 (4 November) before dropping to 1cm 2 s 1 sr 1 on the same day (4 November) at the time of a rapid Dst decrease of about 100 nt. The 3 L < 4 region relativistic flux decrease started on day 302, decreasing from 4000 to 200 cm 2 s 1 sr 1 by day 305 (1 November). Fluxes subsequently rise again to reach their previous levels by day 308 (4 November). The flux levels remain high until 20 November (day 324) when SAMPEX observed a sudden drop in fluxes across all L-values when Dst reached 400 nt and where the outer belt flux levels changed from >10 4 cm 2 s 1 sr 1 to below 1cm 2 s 1 sr 1 in less than 24 hours. [12] In the region 2.5 < L <3, electron flux increased from 10 2 to 10 3 cm 2 s 1 sr 1 in about 24 hours starting on day 302 (29 October), peaked at cm 2 s 1 sr 1 several days later, before slowly decaying to 10 2 cm 2 s 1 sr 1 by day 325. In the slot region (L = ) on day 302, SAMPEX observed the fluxes rise from 1 cm 2 s 1 sr 1 to cm 2 s 1 sr 1 in 24 hours. From day 303 (30 October) to 307 (3 November) the fluxes in the slot region continued to increase from 10 3 to cm 2 s 1 sr 1, while the inner belt (L = ) flux levels rose from 20 to >10 3 cm 2 s 1 sr 1, creating a new sustained inner belt. Also during this period, any distinction in terms of a relativistic electron flux depletion between the outer belt (L > 2.5) and the inner belt disappeared; the lower edge of flux in the outer belt merging into the inner belt below L = 2.0 removing the usual feature of the slot. The L-value where the maximum flux occurs (L max ) moves from pre-29 October to 2.3 by 3 November. The usually separated outer belt only becomes distinguishable from the inner belt again beginning around day 316 (12 November) when the flux levels in the slot region decreased, reaching 200 cm 2 s 1 sr 1 by day 323 (19 November). Fluxes in the slot mostly disappear the following day, but then quickly recover again. The inner belt fluxes also drop by one order of magnitude during the period of the slot flux decrease on 20 November, before they also rise again after day 325 (21 November) Ground-Based Observations [13] Estimation of the radial diffusion timescales (t LL ) requires knowledge of the ULF waves power spectral densities (PSD). The ULF wave magnetic power spectral densities (PSD db ) were estimated from ground-based magnetometer observations. Time-series magnetograms from the CANOPUS/CARISMA [e.g., Samson et al., 1991], IMAGE [Luhr et al., 1998] and SAMNET [Yeoman et al., 1990] ground-based magnetometer arrays were surveyed for ULF waves during the Halloween 2003 geomagnetic storms. Note that the CANOPUS magnetometer array now operates as the CARISMA array (the Canadian Array for 3of15

4 Table 1. Ground Magnetometer Stations Used in This Study Station Array Code GeoLat GeoLon CGMLat CGMLon L Hartland SAMNET HAD York SAMNET YOR Tartu IMAGE TAR Crooktree SAMNET CRK Uppsala IMAGE/SAMNET UPS Lerwick SAMNET LER Dombas IMAGE DOB Pinawa CARISMA PINA Faroes SAMNET FAR Oulujarvi IMAGE/SAMNET OUJ Rorvik IMAGE RVK Island Lake CARISMA ISLL Hella SAMNET HLL Muonio IMAGE MUO Ivalo IMAGE IVA Kilpisjarvi IMAGE/SAMNET KIL Gillam CARISMA GILL Kevo IMAGE KEV Tromso IMAGE TRO Soroya IMAGE SOR Eskimo Point CARISMA ESKI Taloyoak CARISMA TALO Real-time InvestigationS of Magnetic Activity, see [14] Table 1 shows a list of the stations from the three arrays used while Figure 3 shows the locations of the stations. Together, the stations geographically cover around 180 in longitude and approximately 30 in latitude. Figures 4a and 4b show the H- and D-component unfiltered time-series from selected ground magnetometer stations, respectively. In the HDZ (geomagnetic) coordinate system the H-component points in the horizontal magnetic northsouth direction while the D-component points east-west, with eastward positive. The Z-component points vertically downward into the earth. The magnetograms shown in Figure 4 reveal multiple periods of very strong ULF wave activity that last for many hours during the interval from 28 October to the end of 31 October [15] Concentrating on 29 October, when filling of the slot region by relativistic electrons begins, an SSC observed at 0612 UT is immediately followed by ULF waves observed in both the H- and D-components across all L-values. The ULF waves reach maximum peak-to-peak (p-p) amplitudes in the interval UT at L 4 7on this day. Within this time interval and in the L 4 7 region the H-component (Figure 4a) p-p ULF wave amplitude is observed to peak as high as 1000 nt, while for the D-component (Figure 4b) p-p amplitudes reach >300 nt. Near and within the slot region (L = 2 3), the mid-latitude stations from the SAMNET and IMAGE arrays in this L-range observe ULF waves with unusually large p-p amplitudes for these latitudes of 200 nt in the D-component and a p-p maximum of 400 nt in the H-component. For example, at L = 2.3 (Hartland) up to 200 nt p-p amplitudes from UT could be seen in the D-component times-series and 400 nt maximum p-p amplitude were observed in the H-component magnetogram at the same station. The CARISMA magnetometer array unfortunately does not currently reach below L = 4. Similar ULF waves are observed again in this L-shell range on 30 October, but with much lower p-p amplitudes (50 nt) in both components. On 31 October, at least two very strong ULF wave pulsation wave trains were seen around UT and UT; p-p amplitudes >400 nt were seen in the latter wavepacket in both the H- and D-components. 3. Results 3.1. Relationship Between ULF Waves and MeV Electron Flux [16] Figure 5 shows a composite plot relating the dynamics of the radiation belt 2 6 MeV electrons as measured by SAMPEX and details of ULF wave power dynamics from Figure 3. The SAMNET, CANOPUS/CARISMA, and IMAGE ground-based magnetometer array station locations used in this study. 4of15

5 Figure 4. Unfiltered stacked magnetogram time-series from selected IMAGE, SAMNET and CANOPUS/ CARISMA ground-based magnetometer over the period October (a) H-component and (b) D-component. the European sector for October The groundbased magnetometer D-component time-series shown in Figures 5a and 5b are, respectively, the amplitude magnetogram and dynamic power spectrogram observed at Crooktree, which represent the typical temporal and power spectral characteristics observed on October by the mid-latitude ground-based magnetometers in the European sector. Figure 5c shows a dynamic spectrogram of the D-component integrated Pc5 ULF power in the 2 10 mhz band as a function of L-value for L = calculated using measurements from 11 latitudinally spaced IMAGE groundbased magnetometer stations (see Table 1) with a geographic longitudinal span of less than 20. This gives a global picture of the Pc5 ULF wave power over four days as a function of dipole- L. The IMAGE magnetometer array ground data were available at 10 seconds sample rate. The FFT and time-step lengths used to calculate the global dynamic ULF wave power map in Figure 5c were 2000 seconds and 1000 seconds, respectively. [17] The 2 6 MeV relativistic electron flux from the SAMPEX PET instrument for October 2003 is shown in Figures 5d and 5e. Again, the 12 hour periodicity due to the asymmetry of the geomagnetic field with respect to the SAMPEX orbit is clearly visible. The flux levels shown in Figure 5d are for L-value bins integrated over the range L = , L = , L = and L = , while discontinuities represent intervals with data gaps. 5 of 15

6 Figure 5. Combined SAMPEX observations of relativistic electron (2 6 MeV) fluxes and groundbased ULF wave observations for 28 October to the end of 31 October 2003 (see text for details). [18] The two dashed vertical red lines in Figure 5 represent the time interval UT on 29 October where enhanced ULF wave power can be seen across all L-values. The extremely high ULF power (red) regions on either side of the dashed vertical lines in Figure 5c correspond to intervals of bay activity, co-located with strongly negative Dst, and occurring during the storm main phase following the SSC at 0600 UT on 29 October. This extreme ULF power is seen again close to midnight on 30 October, following the arrival of the shock observed at ACE at 1619 UT, and resulting from the subsequent nightside activity. Interplanetary shocks can excite significant current systems in the magnetosphere which are represented as large amplitude magnetic bays in the ground-based magne- tometer time-series. In the spectral domain these bays can appear across a wide frequency range, including the lower end of the 2 10 mhz band, resulting in the very high 2 10 mhz ULF power measurements especially on the nightside during these strong storms as shown in Figure 5c. [19] The 12-hour periodicity in the SAMPEX flux data makes it difficult to determine the exact time of the onset of the flux increase in the slot region. However, Figure 5e does show clear evidence that the increase in the slot (L = ) flux begins on 29 October. The first evidence in Figure 5c is seen around UT during the period of ULF wave power enhancement bracketed by the vertical dashed lines between UT. Flux in the L = range (Figure 5d) continues to rise over the next two 6 of 15

7 days, a flux increase of over 2 orders of magnitude being seen by SAMPEX within 24 hours. Clearly there is a close correspondence between the penetration of the Pc5 (2 10 mhz) ULF wave power to low- L and the penetration of relativistic electrons into the slot (compare Figures 5c and 5e). In what follows the relationship between the ULF waves and the slot region electron flux enhancements on 29 October will be examined Drift Resonant Radial Diffusion [20] In order for ULF waves to accelerate electrons to relativistic energies the simplest scenario would be a drift resonance [e.g., Elkington et al., 2003] whereby, the phase speed of an azimuthally polarized wave electric field, E f, matches the azimuthal drift speed of an electron. This causing an inward motion which increases the electron energy through conservation of the first adiabatic invariant. As discussed by Elkington et al., [1999, 2003], the classical electric radial diffusion described by Falthammar [1968] and Schulz and Lanzerotti [1974] can be examined in the context where the electric field fluctuations are provided by Pc5 ULF waves. For an electron gradient-curvature drifting in the magnetosphere, the rate of relativistic energy change of the particle, dw/dt, is given by dw dt ¼ qe v d þ M g where q is the charge of the particle, v d is the guiding centre drift velocity, M is the relativistic first adiabatic invariant and g is the relativistic correction factor [Northrop, 1963]. Assuming a fundamental field-aligned mode guided poloidal ULF wave, and electrons with an equatorial pitch-angle of 90, the second term on the right-hand side of equation (1) can be neglected. For a symmetric dipole field, only the poloidal mode electric field, E f, will energise the particles. One can then estimate the rate of radial diffusion due to the poloidal mode electric fields in drift resonance with radiation belt electrons. [21] The symmetric drift resonance condition, w = mhw d i, where w is the angular wave frequency, hw d i is the electron bounce-averaged angular drift frequency and m is the azimuthal wavenumber, is supplemented by the asymmetric w =(m ±1)hw d i resonances in a compressed dipole [e.g., Elkington et al., 2003]. However, these additional asymmetric (m ± 1) resonances generate diffusion rates that are additive to the symmetric drift resonance. In this study, the rate of diffusion is calculated in a symmetric dipole neglecting possible effects from the (m ± 1) resonances. [22] The radial diffusion coefficient due to drift resonance with a poloidal mode electric field, E f, assuming a fundamental field-aligned ULF wave mode interaction with 90 pitch-angle particles moving in an uncompressed background dipole field [e.g., Brizard and Chan, 2004] is given by D LL ¼ L6 8B 2 E X PSD dem : m ð1þ ð2þ Here, B E is the equatorial magnetic field at the Earth s surface, R E is the Earth s radius, L is the McIlwain parameter [McIlwain, 1961] and PSD dem is the ULF wave azimuthal electric field power spectral density (PSD) evaluated at w = mhw d i. Inverting equation (2) gives an estimate of the timescale (t LL ) for an electron to diffuse 1 R E. In order to estimate t LL, the in situ equatorial poloidal ULF wave electric field PSD is required. Unfortunately no in situ equatorial electric field wave data were available for any significant time for this study. Instead, the in situ electric field ULF wave power in equation (2) was estimated by using a standing mode guided Alfvén wave model to map ground-based magnetometer Pc5 ULF power measurements into the magnetospheric equatorial plane. These methods are discussed below Ground-Based ULF Wave Power Spectral Density [23] In order to calculate D LL in equation (2), the ULF wave PSD is estimated using ground-based magnetometer data before mapping to the equator to give the equatorial ULF wave electric field (section 3.4). The D-component, which should represent the poloidal mode in the magnetosphere after an assumed Alfvén wave ionospheric polarisation rotation of 90 [e.g., Hughes and Southwood, 1976], is used for the power estimates. [24] The Falthammar [1965] computation of D LL used the one-sided power spectrum [e.g., Press et al., 1992] to define power in the discrete Fourier domain (DFT). This study uses the same convention to determine PSD. Given a discrete finite time-series of length N, the ULF wave magnetic PSD in the frequency and time domain may be represented respectively by and P N=2 k¼1 PSD dbf ¼ 2 Pf ð kþ Ndf PSD dbt ¼ 1 N ð3þ P N k¼1 db2 k : ð4þ Ndf Values of P(f k ) represent the wave power in the spectral domain at the kth DFT frequency, f k. Similarly, db k 2 is the wave power at the kth point along the amplitude time-series. In Fourier space df is the frequency resolution, while in the time domain Ndf is the time-series band-pass filter width. The summation in equation (3) is over the one-sided power spectrum, with the multiplication by 2 shown in equation (3) being required for negative frequencies. The units of equations (3) and (4) are nt 2 per Hertz and by Parseval s theorem both represent the PSD. Given the close relation of the two forms of the PSD it was decided to calculate D LL using PSD estimates in both the time and frequency domains over various limited frequency ranges close to w = mhw d i in the Pc5 band. [25] The ground ULF wave PSD in both the time and frequency domain were estimated for 29 October using magnetometer data from six ground-stations (HAD, YOR, CRK, UPS, DOB and FAR) from the IMAGE/SAMNET array (see Table 1). These stations are chosen to maximise 7of15

8 Figure 6. (a) The unfiltered magnetic D-component dynamic power spectrogram for the interval UT on day 302 (29 October) from the Crooktree ground station at L = 3. (b) The amplitude time-series at Crooktree, band-pass filtered with a filter width of Df = 2.5 mhz centred at 4.5 mhz. The horizontal dashed lines indicate the root mean squared (RMS) amplitude. (c) The static power spectrum for the same interval as Figure 6a. coverage from the slot region to mid-latitudes (L = ) and to minimised the longitudinal (<30 ) extent. In order to help explain how the two forms of PSD are calculated, the D-component data from the Crooktree station are shown in Figure 6. [26] The selection of intervals of dominant ULF wave power were based on visual inspection of the dynamic power spectrograms over the interval UT, which in the Crooktree case was found to be UT. The background magnetic field is removed in the amplitude time-series by high-pass filtering the selected time interval data with a 1 mhz cut-off. A Hanning window [e.g., Press et al., 1992] was then applied to the data before calculation of the Fast Fourier Transformation (FFT). The FFT values were squared and then normalised to account for the Hanning window, giving the power spectrum. Figure 6c shows the power spectrum from the Crooktree station D-component for the UT interval. The PSD dbf (equation (3)) is estimated by summing the P(f k ) elements over the number of discrete frequency bins within a chosen band-limited frequency interval, Df, and then dividing by this frequency width, Df. [27] For the time domain PSD dbt (equation (4)), the amplitude time-series were band-pass filtered across a finite bandwidth, Df, then the mean squares of the amplitude of the filtered time-series were calculated and divided by the finite frequency width, Df. Figure 6b shows the band-pass filtered D-component amplitude time-series. The dotted lines indicate the root mean squared (RMS) amplitude for the case of Df = 2.5 mhz. [28] The central wave frequency was estimated by taking a static FFT over the time interval of interest and selecting the frequency where peak power occurs, which for Crooktree was 4.5 mhz. The frequency resolution for the static FFTs was 0.2 mhz. The PSD values are clearly very sensitive to the choice of Df. Furthermore, equation (2) assumes a single frequency wave. However, it is obvious for example from Figure 6a that the spectral width of the time-series is spread over a range of frequencies. As a result of these considerations two Df values were used, a wide 2.5 mhz signal width, and a narrower width of 1 mhz. However, it was found that in the Df = 1 mhz case, when the filtered time-series are reconstructed, the time domain PSD drastically under-estimates the power due to the narrow band-pass filter. Therefore, the PSD estimates using Df = 1 mhz were calculated only in the frequency domain In Situ Equatorial ULF Wave Electric Field [29] Calculation of the radial diffusion coefficient in equation (2) requires knowledge of the in situ equatorial wave electric field PSD, PSD dem. The guided Alfvén wave model described by Ozeke and Mann [2004] was used to map the ground-based PSD db values into estimates of the in situ equatorial electric field PSD dem. The model assumed a finite thin-sheet ionosphere with a constant height integrated Pedersen conductivity of 10 S and a dipole magnetic field configuration. [30] On 29 October the ground-based H-component power was, on average, approximately one order of magnitude larger than the D-component. In order to estimate the azimuthal electric field E f at the magnetospheric equator the wave was assumed to be a dominantly toroidal Alfvén mode (H-component on the ground) but having a mixed mode polarisation which generates a small component with an azimuthal electric field, E f (D-component on the ground). Under this Alfvénic mapping assumption, there is no contribution from the compressional ULF magnetic component to the field mapping and the radial diffusion coefficient. Hence, it is reasonable to approximate the fieldaligned mapping of both the H- and D-component into radial and azimuthal equatorial electric fields, respectively, using the guided toroidal mode solutions of Ozeke and Mann [2004]. The assumption of toroidal modes for mapping the H- and D-component to equatorial electric fields has the additional pragmatic benefit that the dip-angle corrections of Allan and Knox [1979] can be applied for the toroidal mode at mid-latitudes. [31] The eigenmode solutions presented by Ozeke and Mann [2004] only show the relationship of the magnetic field at the thin-sheet ionosphere to the equatorial electric field. In order to map ground-magnetometer measurements of ULF power to values of PSD dem, it was also necessary to continue the solution from the thin-sheet ionosphere 8of15

9 Figure 7. The modeled ratio of equatorial ULF wave electric field (de m ) to the ground ULF wave magnetic field (db) over L = 2 4. The three lines represent results assuming different field-aligned plasma density (r) profiles. The values were calculated by solving a guided toroidal Alfvén mode wave equation (see text). through the atmosphere to the ground. The study by Hughes and Southwood [1976] concluded that Alfvénic signals decay according to P db P H exp ð k? hþ db I P where k? is the perpendicular wavenumber, h is the height from the ground to the ionospheric E region, db is the ground field and db I is the (toroidal) Alfvén wave magnitude at the top of the ionosphere. For, m 1atL 3 4, a latitudinal resonance width of 5 (0.4 R E in the equatorial dipole magnetosphere at L = 3) dominates over the azimuthal variation in determining k?. In this case k? 1/R E (360 /Dl) km 1 (Dl in degrees), k? 1/90 km 1 for Dl =5 and hence db/db I 0.6, assuming h = 110 km and S H /S P 2. Clearly, db/db I O(1) for m O(1) resonances with resonance width DL 0.4 at these L- values. In what follows, any magnetic amplitude difference resulting from mapping the wave amplitude from the ionosphere to the ground is neglected. [32] The guided Alfvén wave model requires knowledge of the profiles of equatorial plasma mass density, and its field-aligned dependence. In order to estimate the densities the cross-phase method was used [e.g., Waters et al., 1991; Loto aniu et al., 1999; Dent et al., 2003]. Field-line resonance (FLR) frequencies on 29 October were estimated for two pairs of adjacent magnetometer stations, one pair with mid-point latitude corresponding to L = 2.8 and the other L = 3.8. The resonance frequencies were determined using data windows of 50 minutes centred at 1000 UT. The FLR frequencies, 8 mhz at L = 2.8 and 4 mhz at L = 3.8 were linearly interpolated to give the densities over the region L = ð5þ 2 4. The interpolated values were then combined with the guided toroidal Alfvén wave equation assuming a dipole field [Radoski, 1967] and three different radial dependences for the field-aligned plasma mass density profiles (r) to give the modelled ratio of the in situ equatorial ULF wave electric field, de m,todb observed on the ground for L = 2 4. The results are shown in Figure 7. [33] The different field-aligned plasma density profiles had little affect on the ratios, consequently the red curve (r = r 3 ) was used for the PSD dem estimates. For the Crooktree example in Figure 6, at L = 3 the amplitude ratio for the red curve was 0.1 mv/m/nt, which in power would be 0.01 (mv/m/nt) 2. The resultant PSD db at L = 3 in the frequency domain assuming Df = 2.5 mhz centred on 4.5 mhz was nt 2 /Hz with the PSD dem (mv/m) 2 /Hz, which is equivalent to an E RMS in the equatorial magnetosphere of 3.6 mv/m over the 2 hour interval shown in Figure 6. Tables 2 and 3 include example estimates of PSD dem and E RMS using data from the Crooktree and PINA ground magnetometer stations, respectively Radial Diffusion Coefficients [34] Values of PSD dem are used in equation (2), assuming m = 2, to give values of D LL. The assumed azimuthal wave number is chosen based on estimates of the ULF wave phase change observed between longitudinally separated stations in the European and Canadian sectors on 29 October. The D-component magnetometer data from three station pairs covering a CGM latitude of (L ) were analysed and the phase changes corresponded to an m-number range of 1 < m 2. Hence, the assumption of m = 2 is reasonable. [35] Tables 2 and 3 show results for the Crooktree and PINA magnetometer stations, respectively. The t LL values are defined as 1/D LL, which when converted to units of hours represents the time for an electron to diffuse 1 R E from (1) L = 3 with initial energy 2.4 MeV (Table 2), (2) and from L = 4.2 with initial energy 3.5 MeV (Table 3), when interacting with a 4.5 mhz ULF wave in the fundamental field-aligned poloidal Alfvén mode. Results are shown based on both the ULF wave H- and D-components, as measured on the ground. However, in each case the power was assumed to be mapped into the poloidal, E f, mode in the equatorial magnetosphere. This is likely to be a much more realistic assumption for the D-component power. [36] The radial diffusion coefficient in equation (2) only works for the poloidal mode E f component of the in situ ULF wave. Hence, assuming a perfect ionospheric rotation of a purely Alfvénic poloidal mode in an uncompressed background dipole field would produce diffusion rates more appropriate for the D-component (E f in the magnetosphere) in Tables 2 and 3. Clearly, there is significantly more power in the H- than the D-component as seen on the ground. Table 2. Statistical Results for the Crooktree Station at L = 3.0 Component Time Interval, UT Domain PSD db, 10 5 nt 2 /Hz PSD dem, 10 3 (mv/m) 2 /Hz E RMS, mv/m t LL, a hours H 12:13:20 15:16:40 Time H 12:13:20 15:16:40 Spectral D 13:53:20 16:00:00 Time D 13:53:20 16:00:00 Spectral a Time taken for a 3.5 MeV electron to diffuse 1 R E from L = 3 due to interaction with 4.5 mhz ULF wave with m = 2. 9of15

10 Table 3. Statistical Results for the PINA Station at L = 4.2 Component Time Interval, UT Domain PSD db, 10 5 nt 2 /Hz PSD dem, 10 3 (mv/m) 2 /Hz E RMS, mv/m t LL, a hours H 12:13:20 15:16:40 Time H 12:13:20 15:16:40 Spectral D 13:53:20 16:00:00 Time D 13:53:20 16:00:00 Spectral a Time taken for a 2.4 MeV electron to diffuse 1 R E from L = 4.2 due to interaction with 4.5 mhz ULF wave with m = 2. Using our assumption of symmetric (uncompressed dipole, w = mhw d i) diffusion, a perfectly 90 polarisation rotated poloidal mode in the magnetosphere would be represented entirely by the D-component on the ground. However, any inhomogenities in ionospheric conductivity can allow poloidal mode Alfvénic wave power to be partially represented in the H-component on the ground. Therefore, storm time spatial inhomogenities in ionospheric conductivity could mean that some of the power observed in the H- component on the ground might be representative of poloidal mode power in the magnetosphere. Of course the reverse could be true, and the modes may not be purely Alfvénic. [37] Figures 8a and 8b show the PSD db and PSD dem on 29 October as a function of L-value, respectively. The PSD db values were estimated based on magnetic amplitude, db, data from six latitudinally space ground-stations assuming a dipole field (HAD, YOR, CRK, UPS, DOB and FAR, see section 3.3 and Table 1), while the PSD dem were calculated using db and a guided Alfvén wave model (see section 3.4). The solid line represents values calculated based on a ULF wave frequency spectral width of Df = 1 mhz, while the dashed and dotted lines represent estimations that used Df = 2.5 mhz and which were calculated in the time and frequency domain, respectively. All three PSD db curves follow similar trends decreasing with decreasing L-value and then levelling out below L 3. The PSD db calculation based on a ULF wave with Df = 1 mhz produced the largest PSD power of nt 2 /Hz at L = 4.3, while remaining at around nt 2 /Hz below L = 3.3. The PSD dem curves also follow similar trends to each other: relatively constant PSD dem (mv/m) 2 /Hz at higher- L until L 3.3, beyond which all curves increase with decreasing L reaching > (mv/m) 2 /Hz by L 2.3. [38] Figure 8c shows the estimated diffusion times (t LL ) for relativistic electrons on 29 October to diffuse 1 R E from different L-values due to interactions with a ULF wave of central wave frequency of 4 mhz with azimuthal wave number m = 2. The values were calculated using equation (2) and the PSD dem values in Figure 8b. The energy of electrons in drift resonance ranges from 2.4 MeV at L 4.3 to 4.8 MeV at L 2.3, assuming a dipole magnetic field configuration. The energy range is determined from the symmetric resonance condition, w = mhw d i, with hw d i proportional to energy; for example at L = 2.3 for m = 2 the electron energy would be 4.8 MeV given 4 mhz wave frequency. [39] All three curves in Figure 8c have t LL 3 5 hours at L 4.3, then increasing with decreasing L-value before reaching peak t LL at L 2 3. The values based on PSD defined for a bandwidth of Df = 2.5 mhz were independent of whether the time or spectral domain were used to estimate the PSD db. Both curves (dashed and dotted) follow very similar trends with peak t LL 24 hours at L 3.0, then decreasing to t LL hours near L 2.3. The t LL values for Df = 1 mhz in the frequency domain, with consequently larger estimates of PSD db, remains relatively constant below L 3.3 with t LL 12 hours. [40] Figure 9 shows a comparison of the time and frequency domain estimates of t LL based on Crooktree station (L = 3) data assuming a 3.5 MeV electron interaction with a 4.5 mhz ULF wave with m = 2, and calculated using different ULF wave frequency widths, Df. The triangles represent time domain estimates while the asterisks represent the spectral frequency domain estimates. The solid and dashed lines represent third order polynomial fits to the spectral and time domain values, respectively. The time domain estimates are not shown below 2 mhz due to difficulty in justifying the use of ever increasing normalisation factors in the computation of the power required to account for the decreases in signal power due to the time domain band-pass filter. The t LL values in general decrease with decreasing Df as expected since the PSD derived with these narrower band Df also increased. However, below Df 1 mhz for the frequency domain the t LL values approach a constant value of 12 hours suggesting that at Df 1 Figure 8. (a) and (b) The L-dependence of the ground ULF wave magnetic power spectral density (PSD db ) and the corresponding in situ equatorial electric ULF wave power spectral density (PSD dem ), respectively. The solid line represents values calculated based on a ULF wave frequency spectral width of Df = 1 mhz, while the dashed and dotted lines represent estimates that used Df = 2.5 mhz and were calculated in the time and frequency domain, respectively. (c) The L-dependence of t LL. 10 of 15

11 Figure 9. t LL calculated using ULF power estimated using different ULF wave signal frequency widths, Df. The triangles represent time domain estimation while the asterisks represent the spectral frequency domain estimations. The solid and dotted lines are third order polynomial fits to the spectral and time domain values, respectively. mhz the t LL values were close to being independent of the chosen frequency domain Df. 4. Discussion [41] The so-called Halloween storms period from 29 October to the end of 31 October 2003 was an interesting and unusual interval. The high solar wind speed (V sw 2000 km/s) observed by ACE on 29 October (Figure 1) along with the increase in relativistic electron flux levels observed by SAMPEX (Figures 2 and 5e) were consistent with previous studies showing strong correlations between high V sw and relativistic flux enhancements [e.g., William and Smith, 1965; William, 1966; Paulikas and Blake, 1979; Blake et al., 1997]. The high V sw, resulting in the shock impact on the Earth s magnetosphere at 0600 UT on 29 October, was likely due to the full-halo coronal mass ejection (CME) associated with a X17.2-class solar flare observed by SOHO on 28 October [e.g., Lopez et al., 2004]. The shock was followed by an SSC and the development of geomagnetic storms. Only one other recorded geomagnetic storm has been associated with such a high solar wind speed [Skoug et al., 2004]. The Dst measured during the four days was not however particularly unusual for large storms, as an example an event in March 2001 had a Dst of about 390 nt and the Bastille Day event in July 2000 reached 300 nt [e.g., Skoug et al., 2004]. [42] The large amplitude ULF waves observed by the ground-based magnetometer arrays starting at 0600 UT on 29 October and which peaked at 1000 nt p-p for the H-component (Figure 4a) and over 300 nt p-p in the D-component (Figure 4b) in the afternoon at L 4 7, along with the high solar wind speed, are consistent with the suggestion of enhanced Kelvin Helmholtz instability (KHI) along the flanks of the magnetosphere resulting in higher amplitude ULF waves [e.g., Rostoker et al., 1998; Mathie and Mann, 2001]. The unusually large amplitude ULF waves (200 nt p-p) observed at very low L-values (L <3)on 29 October have been suggested by Mann et al. [2004] as resulting from mass loading of the geomagnetic field-lines by cold heavy ions, pumped into the magnetosphere from the ionosphere perhaps through enhanced wave-particle interactions, lowering the local Alfvén continuum and allowing enhanced Pc5 ULF wave power to penetrate to lower L-values than normal. [43] Interpretation of the observations of relativistic electron (2 6 MeV) fluxes by the SAMPEX spacecraft PET instrument shown in Figure 2 were complicated by the low altitude of the orbit of the spacecraft and the fact that the Halloween interval was such an active period. The high magnetic activity on day 302 (29 October), means that the geomagnetic field would have been highly distorted. However, within the inner magnetospheric region and particularly inside the slot region (L = 2 3) the use of a dipole field model becomes more justified. Figure 2 shows a sudden increase in relativistic electron flux levels within the slot region starting on 29 October and lasting until a massive storm (Dst = 470 nt) on 20 November (day 324) caused a dramatic drop in the relativistic electron fluxes across all L- values. This suggests that the enhancements in the slot region seen after 20 November were not associated with the October storms. [44] A detailed examination of the flux increases from 31 October onwards is worthy of study but is not the primary subject of this paper. The current study was directed towards explaining the flux increases in the slot region on 29 October. The rest of the discussion be will focused on this flux enhancement. Before 29 October the D-component ULF wave power was restricted to above L = 4 (Figure 5c) and there were no detectable levels of relativistic electrons (2 6 MeV) by SAMPEX within the slot region L = (Figures 5d and 5e). Unusually high power (>10 3 nt 2 /Hz) in the 2 10 mhz bandwidth of the H- and D-components were first observed below L < 4 by the ground-based magnetometers on 29 October at around 0600 UT coinciding with the arrival of the shock, the triggering of the SSC and then the subsequent storm-time ring current enhancement. However, as previously mentioned (section 3.1), geomagnetic storms can result in large amplitude magnetic bays in ground-based magnetometer time-series. In the spectral domain these bays can include power in a range of frequency bins including the lower end of the 2 10 mhz band, which partly explains the high ULF power seen in the interval UT on 29 October. The power was also generally larger for the higher L-values across this 3.5 hour interval. These observations would suggest that possible radial diffusion of relativistic electrons on 29 October started immediately after the SSC in direct competition with the adiabatic Dst affect and was probably stronger at the higher L-values since in addition to the expected strong L- dependence of the radial diffusion coefficient, (e.g., L 6,or L 12 ) the ULF power was also enhanced at higher- L. [45] The region in between the two dashed vertical lines in Figure 5c, UT on 29 October, also shows enhanced ground-based Pc5 ULF power in the 2 10 mhz frequency band across the observed L-shell range of L 3 7. However, in this interval there are no large amplitude magnetic bays in the ground-based magnetometer timeseries (see Figure 5a), suggesting that all the power (2 10 mhz band) in this time interval was due to ULF waves. The increases in relativistic flux levels seen during the same interval in Figures 5d and 5e suggest that the slot region filling occurred around the same time. The enhanced levels of ULF wave power at low L-values continue right through 29 October until around midday on the following day. The ULF wave trains in the magnetic field H-component from 0930 UT onwards on 29 October follow a very similar trend 11 of 15

12 to the D-component except the average wave power spectral density (PSD) in the H-component is generally one order of magnitude higher. Accurately estimating the time lag between ULF wave power increases and relativistic electron flux enhancement in the slot is extremely difficult because of the 12 hour periodicity in the SAMPEX data due to the spacecraft s low-altitude orbit. Figure 5d was produced based on a 3 hour (2-orbit) flux averaging with a 1-orbit sliding window and the results suggest that the flux increases in the slot region start before noon on the 29 October. Although the exact time of apparent onset of this flux rise is heavily dependent on the averaging process it is clear that the flux increases in the slot region started on the morning-to-afternoon of 29 October exactly when enhanced ULF wave power was also seen at unusually low L-values (L < 3 4) by the CARISMA, SAMNET and IMAGE ground-based magnetometer arrays. [46] The wave frequency (3 7 mhz) of the dominant spectral power peaks over the UT interval was also unusually low for L 3. For example, at L = 2.8 the first harmonic of the standing Alfvén wave field line resonance (FLR) is usually around 15 mhz [e.g., Waters et al., 2001]. The low ULF wave frequency observed on 29 October could be due to the mass loading of the field-lines as suggested by Mann et al., [2004]. It should be noted that broadband Pc5 ULF wave power was seen beginning around 0600 UT on 29 October through to the following day meaning that radial diffusion could have been continuous at some level throughout that day and not just restricted to the interval UT. [47] The PSD db as a function of L-value curves for 29 October shown in Figure 8a show decreasing ULF wave power with decreasing L-value. As shown statistically by Mathie and Mann [2001], Pc5 ULF wave power in the morning sector in the L-range L = 3.75 to L = 6.79 increases with V sw, however the rise with V sw as well as the average power for fast solar wind speeds decreases with decreasing L. The ULF power on 29 October 2003 appears to follow this trend, however the penetration of enhanced Pc5 ULF power levels to low L is very pronounced. This may be explained by the KHI generating Pc5 FLRs along the magnetospheric flanks at high latitudes, and which decay in amplitude towards lower latitudes [e.g., Ansari and Fraser, 1986]. Alternatively, since the turning point for Pc5 fast mode waves in general moves to higher frequencies for lower- L, only fast mode waves in the band whose frequency is above the local turning point frequency will be able to propagate to that L, producing an integrated 2 10 mhz power level which would decay naturally with decreasing L-value. [48] The de m /db ratio shown in Figure 7 was obtained by numerically solving the guided Alfvén wave model discussed in Ozeke and Mann [2004] but in this case for the toroidal ULF wave mode. The important mode for symmetric radial diffusion in a dipole field is typically the poloidal mode [e.g., Elkington et al., 2003]. In mapping the ULF waves to the equator using the values in Figure 7 it was assumed that the wave was mixed mode such that the eigenmodes for the poloidal and toroidal modes were structurally similar along the field line, that the toroidal eigen-structure was a good approximation for the poloidal modes, and that no attenuation occurs between the thin sheet ionosphere and the ground, as discussed earlier in section 3.4. [49] In this study the diffusion rates have been estimated by using equatorial electric field power spectral density (PSD dem ) derived by numerically solving the guided Alfvén wave model and using as parameter inputs the combination of plasma mass density obtained by applying the FLR crossphase technique [e.g., Waters et al., 1991] and mapping ULF power data from ground-based magnetometers into the equatorial plane. Perry et al. [2005] modeled the effect of ULF wave electric and magnetic fields on the guiding centre trajectories of relativistic electrons by assuming a sinusoidal varying wave electric and magnetic field along the fieldlines. However, in reality the wave fields are not sinusoidal along the field-lines and using solutions to the Alfvén wave equation results in more realistic wave variations along the field-lines [e.g., see Ozeke and Mann, 2004]. [50] Li et al. [1993] successfully modelled the March 1991 storm relativistic electron injection that occurred with a timescale of minutes and which was believed to have been due to the propagation of an MHD shock, using a timedependent electric field pulse of amplitude 240 mv/m. Similarly, Li et al. [2003] used a 6 mv/m pulse to model the 26 August 1998 MHD shock driven event. Although the Halloween storm was a very strong event with V sw 2000 km/s, the flux increases in the slot region on 29 October occurred over several hours and hence shock acceleration would be too fast to explain the observed slot filling enhancements. [51] Interestingly, the PSD dem values in Figure 8b follow an opposite trend to the PSD db curves in Figure 8a, increasing with decreasing L-value below L 3.3 where all three curves reach > (mv/m) 2 /Hz at L 2.3. The solutions of the guided toroidal wave equation where obtained using the ionospheric boundary conditions 1/m o S P = de I /db I, corrected for toroidal mode dip-angle below L 4 [e.g., Allan and Knox [1979] and Ozeke and Mann [2004], where S P is the height integrated Pedersen conductivity (assumed fixed at 10 S in both hemispheres). Within this model, the ratio of the ionospheric wave electric to wave magnetic fields (de I /db I ) is approximately independent of L-value. However, the variation of de when it is mapped along the length of the field-line is a strong function of L-value (see the analytic solutions in Allan and Knox [1979] and Ozeke and Mann [2004]). Consequently, the ratio of the equatorial electric field to the ground magnetic field, d E m /db, can vary with L as shown in Figure 7. Therefore, increases in PSD dem with decreasing L-value such as those shown in Figure 8b are possible even if the level of PSD db decreases with decreasing L. In Figure 8a it can be seen that PSD db remains constant at nt 2 /Hz below L 3, whilst PSD dem still increased with decreasing L. Itis suggested that geomagnetic storms that show sustained levels of ground-based Pc5 ULF wave power which penetrate to lower L-values may produce relatively larger ULF wave electric field amplitude de m at these low L-values than at higher L-values, due to the L-dependence of de m /db. This would in turn contribute to enhancing the rate of radial diffusion at these lower L-values. [52] The ordinate in Figure 8c is defined as t LL = 1/D LL and represents the time (hours) for a relativistic electron at L to diffuse 1 R E due to drift resonance with a m = 12 of 15