Ultra-low-frequency (ULF) pulsations are critical in radial diffusion
processes of energetic particles, and the power spectral density (PSD) of
these fluctuations is an integral part of the radial diffusion coefficients
and of assimilative models of the radiation belts. Using simultaneous
measurements from two Geostationary Operational Environmental Satellites (GOES) geosynchronous satellites, three satellites of the Time
History of Events and Macroscale Interactions during Substorms (THEMIS)
spacecraft constellation and the two Van Allen probes during a 10-day period
of intense geomagnetic activity and ULF pulsations of October 2012, we
calculate the PSDs of ULF pulsations at different

Ultra-low-frequency (ULF) fluctuations in the Earth's magnetosphere have
frequencies in the millihertz (mHz) range, within the drift frequency of energetic
particles in the magnetosphere. For example, particles of energies 0.5, 1.0
and 5.0 MeV have drift frequencies

In radial diffusion models, the effects of ULF waves are described by the
diffusion coefficient,

In the following we focus on the radial diffusion effects of the
compressional magnetic field component of ULF waves. The diffusion
coefficient in this case is described by the following formulation:

In the following, we present an overview of some of the studies that have
shown a dependence of ULF wave power on

In this study we show results of compressional magnetic field PSD during a high-speed stream event, using measurements from the Van Allen probes, the THEMIS satellite constellation and GOES geosynchronous satellites. Furthermore, we use GOES geosynchronous measurements to estimate the distribution of power in the various wave numbers.

Three consecutive geomagnetic storms occurred in the first half of October
2012: on 1, 8 and 13 October. The first storm, as observed by the Van Allen probes and reported by Baker
et al. (2013), depleted the outer electron belt, and electron fluxes remained
low and constant until the second storm. During the second storm, and in
particular after 9 October 2012, the two RBSP satellites measured an
intense relativistic electron enhancement event: electrons at all energies
became enhanced, with lower-energy electrons being enhanced earlier and
penetrating further inward (Li et al., 2013). During the same event, the
development of peaks in electron phase space density were observed (Reeves
et al., 2013), which are considered evidence for local electron acceleration
in the heart of the outer radiation belt. The third storm was characterized
by a moderate drop of Dst to

The Van Allen Probes mission (Mauk et al., 2012), initially named the
Radiation Belt Storm Probes (RBSP) mission, was launched on 30 August 2012
and consists of two identical satellites, hereafter termed probes RBSP-A and
RBSP-B, which orbit the Earth with an apogee of

The THEMIS mission (Angelopoulos, 2008) originally consisted of five
nearly identical satellites, hereafter termed probes TH-A through TH-E,
which orbited the magnetosphere at different apogee; during the time of
interest of this study, three of the satellites remain in the magnetosphere,
while two have been moved into orbit near the Moon. The three remaining
satellites are probes TH-A, TH-D and TH-E, orbiting at nearly identical
orbits around Earth with apogees of 11.7

The Geostationary Operational Environmental Satellites provide
continuous monitoring from a geosynchronous orbit. Most commonly there is
one satellite over the east coast of the US at longitude 75

Magnetic field measurements from all the above platforms are rotated into a mean-field-aligned (MFA) coordinate system, which allows the magnetic field oscillations to be determined as toroidal (azimuthal), poloidal (radial) or compressional (parallel).

We use simultaneous measurements of broadband ULF waves by the above
measurement platforms (two Van Allen probes, three THEMIS and three GOES satellites)
in order to reconstruct the

In Fig. 1 we show the

Subsequently, after performing coordinate transformations from GSM
coordinates to the mean-field-aligned coordinate system, we calculated the
dynamic PSDs (power spectral densities as a function of time), which were
subsequently averaged within each time bin. The corresponding

The results are plotted in Fig. 2, where the power spectral density is
plotted in color in a frequency-vs.-

In the following we use cross-spectral and cross-phase calculations between
the time series of magnetic field measurements from azimuthally aligned
satellites to get the fraction of total power in each wave number. The
theoretical background for this technique has been discussed in greater
detail in Sarris (2014). There it was described that, for example, between
two measurement points that are separated in azimuth by

Calculations of the power per wave number using cross-spectrogram
calculations between GOES-14 and GOES-15, at an azimuthal separation of

For the calculation of phase differences between the two time series as a
function of time and at all frequencies a cross-wavelet transform (XWT)
technique is used (e.g., Grinsted et al., 2004; Sarris et al., 2013; Sarris,
2014). The results of this calculation are wavelet and cross-wavelet power
spectral density (XWT-PSD) in units of nT

For the period in October 2012 we use phase difference calculations between
the magnetometer measurements of GOES-13 and GOES-15 geosynchronous
spacecraft, hereafter referred to as G

In the lower panel of Fig. 3 a moving average of the calculated

In the above calculation a critical assumption is used, namely that there is
no spatial aliasing when calculating the phase differences of the waves
between the two satellites. Thus, under this assumption, a 60

During the same time period under investigation, measurements from another
geosynchronous satellite, GOES-14, were available. In order to identify if
there are waves of a multiple of

According to the technique used above, the azimuthal separation of

The

Data used in the above analyses are available at:

We thank the Van Allen Probes Science Team and B. Mauk for Van Allen Probes data, the THEMIS Science Team and Vasilis Angelopoulos for THEMIS data, and NOAA and Howard Singer for GOES magnetometer data. This study was supported by NASA grants (THEMIS, NNX15AF56G and NNX12AG37G) and NSF grant ATM 0842388. The topical editor, G. Balasis, thanks L. Ozeke and two anonymous referees for help in evaluating this paper.