Chorus-type whistler waves are one of the most intense electromagnetic waves
generated naturally in the magnetosphere. These waves have a substantial
impact on the radiation belt dynamics as they are thought to contribute to
electron acceleration and losses into the ionosphere through resonant
wave–particle interaction. Our study is devoted to the determination of
chorus wave power distribution on frequency in a wide range of magnetic
latitudes, from 0 to

The assessment of radiation belt dynamics is one of the most important
objectives of space weather programmes

To define the dynamics of radiation belts, the timescale for electron loss
and acceleration is calculated by numerical models

Realistic distributions of chorus wave power in the radiation belts should
thus be included in diffusion models to accurately reproduce the effects of
wave–particle interactions on energetic electrons

In this study, 10 years of Cluster data are analysed to determine the
parameters of field-aligned chorus spectral characteristics as a function of
latitude, in such a way that these parameters can be easily employed in
numerical models to calculate the resulting diffusion rates. The measured
statistics are in good agreement with our ray tracing simulations in a
realistic model of Earth's inner magnetosphere, that explains the observed
chorus spectral extent in terms of both wave growth/damping and cross-

In this section we first present the equatorial properties of chorus waves in
the inner magnetosphere, observed by different spacecraft since first space
missions, that are used to model the chorus “equatorial source region” in Sect.

Chorus waves typically appear as short (

In Earth's magnetosphere, chorus waves are usually observed in the day, dawn
and night sectors

Recently, an extensive study

However, in the off-equatorial magnetosphere the spectral extent of chorus
waves is still not well known nor understood. As stated in the previous
section, measurements of chorus spectral extent as a function of latitude
have already been presented in

Thus, in this section we process 10 years of magnetic and electric field
wave power measured in a frequency range from 8 Hz to 4 kHz by STAFF-SA
instrument onboard Cluster spacecraft

The parameters obtained from Cluster statistics shown above, that can be used
to calculate diffusion rates, are presented in Sect.

In this section we present how to model chorus spectral extent in the inner
magnetosphere, using the numerical model described in the Appendix B

Number of measurements (left panels) and amplitude in pT (right
panels) of chorus waves as a function of latitude obtained from 10 years of
STAFF-SA instrument measurements onboard Cluster spacecraft on the nightside
(top panels) and dayside (bottom panels) for

In order to reconstruct chorus frequency distribution statistics, we have
calculated numerous ray trajectories in the inner magnetosphere (in terms of

This numerical ray distribution is uniform in terms of

Each ray is then propagated throughout the magnetospheric model described in Appendix B and recorded at different latitudes in a certain

Power spectrum of chorus waves as a function of latitude obtained
from our simulations for MLT

The spectral extent of all rays is then reconstructed in Fig.

In this figure, the peak frequency (squares) simulated in the dayside
(MLT

In this paper we study the field-aligned spectral extent of chorus waves in
the radiation belts by means of 10 years (2001–2010) of Cluster measurements
and ray tracing simulations in a realistic model of Earth's inner
magnetosphere. As suggested by

Here 10 years of statistics recorded onboard Cluster spacecraft in the
dayside show a global decrease of the normalized frequency peak value with
increasing latitude, from

These numerical simulations allow us to interpret the causes of such peak
frequency variations observed on Cluster in terms of cross-

At low/mid latitudes (

Cluster statistics in the nightside presented in this paper show a chorus
peak frequency near the equator close (

Coefficients of the quartic function used to fit the peak frequency as a function of latitude in the dayside and nightside.

Pitch angle and energy diffusion coefficients as a function of
electron equatorial pitch angle for 1 MeV electrons interacting with
parallel whistler waves (wave parameters are the same as in

Equatorial distribution functions of a typical chorus event observed
on 15 July 2010 onboard Cluster C1 at

The behaviour of the chorus peak frequency as a function of latitude strongly
affects the dynamics of the outer radiation belt. As a matter of fact, the
distribution of mean wave frequency along field lines plays an important role
for electron resonant scattering by waves. For each value of electron energy
the most effective resonant interaction corresponds to a certain latitude
domain. For example, MeV electrons with small equatorial pitch angles
resonantly interact with whistler waves propagating parallel to the magnetic
field at latitudes

To follow the “standard” scheme

The chorus wave spectral power presented in this study can thus be used to
improve the accuracy of the diffusion coefficient calculations

Data coverage (MLT as a function of

Damping (blue) and growth (red) rate

The 10 years of Cluster statistics presented in this study are in agreement
with previous measurements of chorus spectral extent

Beside the electron resonant scattering by small-amplitude whistler waves,
the nonlinear electron interaction with high-amplitude waves

In this study we partially use a large data set, described in detail in

To simulate the chorus wave power distribution upon frequency along a
magnetic field line we use a 3-D ray-tracing code that includes a realistic
model of the Earth's inner magnetosphere, described in detail in

Density of suprathermal electrons has a small effect on rays' trajectories
since, according to the measured statistics used in this study, the values of
the two suprathermal electron populations (

The density of these two suprathermal electron populations in the inner magnetosphere
has then to be defined in order to include them into our model. The density value for each
population is taken from statistics at the equator (depending on

The growth/damping rate

The authors would like to thank Mikhail Rashev for the help in RAPID data calibrations. The work by O. Agapitov was performed under JHU/APL Contract No. 922613 (RBSP-EFW) and NASA NNX09AE41G-1/14 Contract. The work of A. Artemyev was partially supported by the grant MK-1781.2014.2.The article processing charges for this open-access publication were covered by the Max Planck Society. The topical editor C. Owen thanks the two anonymous referees for help in evaluating this paper.