VLF%20chorus%20emissions%20observed%20by%20CLUSTER%20satellites%20inside%20the%20generation%20region:%20%20comparison%20with%20the%20backward%20wave%20oscillator%20model%20%20E.%20E.%20Titova,%20B.%20V.%20Kozelov%20%20Polar%20Geophysical%20Institute,%20Apatity,%20Russia,%20V.Y.Trakhtengerts,%20A.%20G.%20Demekhov%20Institute%20

1
VLF chorus emissions observedby CLUSTER
satellites inside the generation region
comparison with the backward wave oscillator
model
E. E. Titova, B. V. Kozelov Polar Geophysical
Institute, Apatity, RussiaV.Y.Trakhtengerts, A.
G. DemekhovInstitute of Applied Physics, Nizhny
Novgorod, Russia O. Santolik, E. Macusova
Charles University, Prague, Czech Republic and
IAP/CAS, Prague, Czech Republic D. A. Gurnett,
J. S. Pickett University of Iowa, Iowa City, IA,
USA
2
Magnetospheric backward wave oscillator (BWO)
modelof VLF chorus generation predictions and
comparison with data

• Content
• Magnetospheric BWO property of the source region
  of magnetospheric chorus
• Field aligned scale of the source region
• Variations of chorus source location
• Influence of length of the BWO on chorus
  characteristics
• Magnetospheric BWO parameters of chorus elements
• Growth rate for BWO regime
• Frequency drift
• Saturation amplitude
• Conclusions

V. Y. Trakhtengerts, 1995,
1999
3
Field aligned scale of chorus source region
According to Trakhtengerts (1995), the
interaction length l of whistler waves and
energetic electrons can be written for the dipole
magnetic field as follows lBWO (R02L2/k)1/3
where R0 is the Earth's radius, L is the
geomagnetic shell, and k is the whistler wave
number. The backwardwave oscillator (BWO) regime
of the whistler cyclotron instability takes
place in a narrow near equatorial region
lBWO 103 km
LeDocq et al., 1998 Parrot et al., 2003
Santolik et al. 2003, 2005
4
Size and central position of the source region
from multipoint measurement of the Poynting flux
by the Cluster satellites
Santolik et al.
2003, 2005

• Poynting flux measurements show
• the size of the source region is a few
  thousands of km along the filed line
• strong variations of the central position of the
  chorus source region

5
Dynamical magnetic field model
Deviation of the observed magnetic field from
value modeled by Tsyganenko-96 model
For two currents and two positions of CLUSTER
satellites
p1,2 are known positions of two CLUSTER
satellites, I1 and I2 are two line currents, i1
and i2 are unit vectors of their directions, r1
and r2 are points at the line currents
The main characteristics of the magnetospheric
BWO effective length along the magnetic
filed line (LBWO) position at the
magnetic filed line (position of Bmin)
Kozelov, Demekhov, Titova, Trakhtengerts et al.,
2008
6
Estimation of effective BWO length
Introduce the relative magnetic-field
perturbation b(z)
The cyclotron resonance frequency mismatch is
The cyclotron resonance phase mismatch is
obtained using the known magnetic-field profile as
The experimentally derived BWO length LBWO is
then obtained as finding the distance between
such points z1 and z2 that
7
Results of modeling of the magnetospheric BWO
configurations on 31 March 2001 for a magnetic
field line at the CLUSTER-1 position.

• thin line (not seen) is the observed strength of
  the magnetic field, dashed line - calculated by
  Tsy-96, solid line - fitted by a model with 2
  additional currents.
• the modeled magnetic field along the magnetic
  field line.
• symbols mark the calculated positions of the
  magnetic field minimum, solid and dashed lines
  show, respectively, the smoothed evolution of
  this position and the CLUSTER orbit.
• evolution of the estimated length of the
  magnetospheric BWO.

Kozelov, Demekhov, Titova, Trakhtengerts et al.,
2008
8
Comparison of the minimum B location obtained
from local magnetic field modeling with the
center of the chorus source obtained from the VLF
STAFF data

• Solid line and symbols position of the BWO
  center (minimum-B point) estimated from the
  dynamical model of the local magnetic field red
  dashed line position of the VLF source from
  Santolik et al., 2005 long dashed lines
  satellite trajectories.
• Parallel component of the Poynting vector
  normalized by its standard deviation for
  CLUSTER-1 Santolik et al., 2005.

The obtained variation of the position of minimum
B along the field line qualitatively agrees well
with the variation of the chorus source location
previously obtained from the STAFF data Santolik
et al., 2005.
Kozelov, Demekhov, Titova, Trakhtengerts et al.,
2008
9
Results of simulations of nonlinear equations for
the magnetospheric BWO Demekhov and
Trakhtengerts, 2005 with two magnetic field
profiles corresponding to different time
intervals of April 18, 2002.

• the geomagnetic field dependences
• total Poynting flux in arbitrary units as a
  function of time and z-coordinate. The center of
  the source region corresponds to Stot  0.

The plots demonstrate that the center of the
chorus source region in the simulations remains
near the local minimum of the geomagnetic field.
10
The estimated threshold flux (cm-2s-1sr-1) for
the BWO generation in on-off intermittency
regime.

• The threshold electron flux Sthr for the BWO
  generation (Trakhtengerts et al., 2004)

vres is the parallel velocity of the resonant
electrons, for this event Ne 5 cm-3 we assume
for estimates that hstep 0.1 and v- vres,
A B C
36.4 17 5.6 Average chorus power, mV m-1 min-1
197 141 44 Average number of elements in minute
11
Estimation of growth rate of the BWO regime of
VLF chorus generation using frequency sweep rate
of chorus elements According of BWO model of
chorus generation (Trakhtengerts, 1999) the
frequency sweep rate df/dt at the exit from BWO
generation region can be written as df/dt (
?2BWO S1 ) 1.5 ? /(?H 2?) where ?
2BWO the growth rate of the absolute (BWO)
instability,
S1  0.3 V (d ?H/dz) characterizes the
magnetic field inhomogeneity effect. We
introduce the "reduced" frequency sweep rate G,
which is equal the growth rate of the BWO
regime for ? 2BWO gtgt S1. G 2 ? df/dt (? H 2
?)/ 1.5 ? ?2BWO
12
The frequency sweep rate of VLF chorus emissions
as inferred from Magion 5 satellite
Titova et al., 2003
G2 g 2 BWO df/dt (wH 2?)/ 1.5w The reduced sweep rate G2 as a function of L shell. The solid line shows the running average over 100 points of G2.
The mean of the "reduced" frequency sweep rate
G(L) varies within a small interval 100300 s1,
that is in a good agreement with estimates of
?2BWO from the BWO theory obtained from chorus
elements on MAGION 5 satellite as a function of
L shell. The chorus growth rate, estimated from
the frequency sweep rate, is in accord with that
inferred from the BWO generation mechanism
13
The growth rate of the absolute (BWO)
instability, the frequency sweep rates and the
chorus amplitudes
The growth rate of the absolute (BWO) instability
?2BWO ? BWO / O tr 32/ (3p)
where the trapping frequency ? tr is determined
by the expression O tr (k u ?H b)1/2 Here
b B /B L , B is the whistler wave magnetic
field amplitude, BL is the geomagnetic field, and
u is the electron velocity component across the
geomagnetic field. G2 df/dt (?H 2?)/
1.5? ? 2 BWO b the BWO model predicts an
increase in frequency sweep rate df/dt and G2
with chorus amplitude B (10 m c / e k
u)?2BWO
14
The frequency sweep rate of VLF chorus emissions
on CLUSTER satellite
The reduced sweep rate G2 as a function of chorus
amplitude for event CLUSTER 1, 18.04.2002
The reduced sweep rate G2 as a function of chorus
amplitude for event CLUSTER 1, 31.03.2001
The frequency sweep rate increases with chorus
amplitude, in accordance with the BWO model.
15
Summary of chorus parameters in the BWO model for
Cluster data 18.04.2002

• Basic parameters
• L 4. 4,
• cold plasma density Nc 2 cm -2 ,
• ?/?H 0.45 ,
• Wres me/2 (?H ?)2/k2 62 keV,
• the wavelength 26 km.
• The flux density of energetic
• electrons is assumed to be
• S 4108 cm2 s-1

The growth rate ? BWO Characteristic period of succession T gt T0 Frequency drift df/dt Wave amplitude B
? BWO p2/4T0 T0 lBWO (1/vg 1/vstep) 0.5 ? 2BWO (10mc / eku)? 2BWO
102 s-1 gt 0,025 s 0,7104 Hz s-1 100 pT
34-420 s-1 0,02-0,05 s 1,5 104 Hz s-1 100-300 pT
16

• We study the sweep rate of the emission frequency
  as a function of the cold plasma density in the
  equatorial plane and than we compare it with the
  prediction of BWO model.
• We investigate wave packets from detailed
  time-frequency spectrograms measured by the WBD
  instrument on board the Cluster spacecraft.
• The local electron densities during 15 processed
  time intervals were obtained from the WHISPER
  instrument.

17
December 22, 2001
t1 , f1 t2 , f2 f1 f2
(sweep rate)
An example of chorus elements measured on board
the Cluster satellites on December 22, 2001 by
the wideband (WBD) plasma wave instrument.
Spacecraft position is given on the bottom
UT-universal time ?m-magnetic dipole latitude
RE - Earth radius MLT-magnetic local time.
18
08.02.2005 (1151-1213 UT)
06.12.2003 (1430 - 1500 UT)
22.01.2002 (1740-1750 UT)
The electron density was about 11 cm-3 and the Kp
index was 2-.
The Kp index was 30 and the electron density was
about 10 cm-3
08.02.2001(ne-9 cm-3) df/dt10.61 kHz/s (N236)
and -9.64 kHz/s (N156) 22.12.2001(ne-10 cm-3)
df/dt no risers and -11.02 kHz/s
(N1036) 06.12.2003(ne-10 cm-3) df/dt 9.36
kHz/s (N5568) and no fallers
19
08.02.2005 (1151-1213 UT)
25.03.2002 (1356 - 1420 UT)
21.10. 2001 (2315 - 2335 UT)
The electron density was about 192 cm-3 and Kp
index was 8-
The Kp index was 20 and the electron density was
about 39 cm-3
12.4.2001 (ne- 27 cm-3) df/dt 5.97 kHz/s
(N351) and no fallers 25.03.2002 (ne- 39 cm-3)
df/dt 3.91 kHz/s and no fallers 21.10.2001
(ne- 192cm-3) df/dt 1.69 kHz/s (N972) and
-3.11 kHz/s
20
The sweep-rate estimate on base of BWO model of
Trakhtengerts et al., 2004 yields the following
scaling
df/dt C n -2/3
where n is the plasma density and C is a free
parameter.
21
Conclusions

• Within the framework of the BWO
  generation model, it is possible to explain the
  properties of the chorus emissions at frequencies
  below fH/2, observed by Cluster satellites, such
  as
• field aligned scale of the source region and the
  direction of the energy flux
• this motion of chorus source by deviation of the
  magnetic field minimum (the local magnetic
  equator)
• influence of effective length of the
  magnetospheric BWO on chorus characteristics
• chorus growth rate, estimated from the frequency
  sweep rate, is in accord with that inferred from
  the BWO generation mechanism.
• correlation between the frequency sweep rates and
  the chorus amplitudes.
• amplitude of chorus
• The theoretical scaling based on the BWO theory
  predict increasing sweep rate of chorus elements
  for decreasing cold plasma density as df/dt C
  n -2/3. The results observations on CLUSTER
  consistent with these predictions

22
Some principal questions remain unclear

• Relation between chorus and hiss emissions
• Formation of a step-like distortion at the
  electron distribution function
• Nonducted propagation
• Damping of chorus emission on fH/2