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

Magnetospheric backward wave oscillator (BWO) model of 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

According to Trakhtengerts (1995), the interaction length l of whistler waves and energetic electrons can be written for the dipole magnetic field as follows: l BWO = (R 0 2 L 2 /k) 1/3 where R 0 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: l BWO ~ 10 3 km LeDocq et al., 1998; Parrot et al., 2003; Santolik et al. 2003, 2005 Field aligned scale of chorus source region

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

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: p 1,2 are known positions of two CLUSTER satellites, I 1 and I 2 are two line currents, i 1 and i 2 are unit vectors of their directions, r 1 and r 2 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

The experimentally derived BWO length L BWO is then obtained as finding the distance between such points z 1 and z 2 that 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

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

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

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, 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 S tot = 0. The plots demonstrate that the center of the chorus source region in the simulations remains near the local minimum of the geomagnetic field.

The estimated threshold flux (cm -2 s -1 sr -1 ) for the BWO generation in “on-off” intermittency regime. The threshold electron flux S thr for the BWO generation (Trakhtengerts et al., 2004) ABC Average chorus power, mV m -1 min Average number of elements in minute v res is the parallel velocity of the resonant electrons, for this event N e = 5 cm -3 we assume for estimates that h step ≈ 0.1 and v ┴ ≈ v res,

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 = ( γ 2 BWO + S 1 ) 1.5 ω /(ω H + 2ω) where γ 2 BWO the growth rate of the absolute (BWO) instability, S 1 = 0.3 V || (d ω H /dz) characterizes the magnetic field inhomogeneity effect. W e introduce the "reduced" frequency sweep rate G, which is equal the growth rate of the BWO regime for γ 2 BWO >> S 1. G 2  df/dt (ω H + 2 ω)/ 1.5 ω = γ 2 BWO

G 2 =  2 BWO = df/dt (  H + 2  )/ 1.5  The reduced sweep rate G 2 as a function of L shell. The solid line shows the running average over 100 points of G 2. The mean of the "reduced" frequency sweep rate G(L) varies within a small interval 100–300 s –1, that is in a good agreement with estimates of γ 2 BWO 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 The frequency sweep rate of VLF chorus emissions as inferred from Magion 5 satellite Titova et al., 2003

The growth rate of the absolute (BWO) instability γ 2 BWO γ BWO / Ω tr ≈ 32/ (3π) where the trapping frequency Ώ tr is determined by the expression Ω tr = (k u ω H b) 1/2 Here b = B ~ /B L, B ~ is the whistler wave magnetic field amplitude, B L is the geomagnetic field, and u is the electron velocity component across the geomagnetic field. G 2 = df/dt (ω H + 2ω)/ 1.5ω = γ 2 BWO ~ b the BWO model predicts an increase in frequency sweep rate df/dt and G 2 with chorus amplitude B ~ = (10 m c / e k u)*γ 2 BWO The growth rate of the absolute (BWO) instability, the frequency sweep rates and the chorus amplitudes

The frequency sweep rate of VLF chorus emissions on CLUSTER satellite The frequency sweep rate increases with chorus amplitude, in accordance with the BWO model. The reduced sweep rate G 2 as a function of chorus amplitude for event CLUSTER 1, The reduced sweep rate G 2 as a function of chorus amplitude for event CLUSTER 1,

Summary of chorus parameters in the BWO model for Cluster data Basic parameters : L = 4. 4, cold plasma density N c ~ 2 cm -2, ω/ω H ≈ 0.45, W res = m e /2 (ω H – ω) 2 /k 2 = 62 keV, the wavelength 26 km. The flux density of energetic electrons is assumed to be S ~ 4*10 8 cm 2 s -1 The growth rate γ BWO Characteristic period of succession T > T 0 Frequency drift df/dt Wave amplitude B ~ γ BWO = π 2 /4T 0 T 0 = l BWO (1/v g +1/v step ) 0.5 γ 2 BWO (10mc / eku)*γ 2 BWO 10 2 s -1 > 0,025 s0,7*10 4 Hz s pT s -1 0,02-0,05 s1, Hz s pT

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.

December 22, 2001 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; λm-magnetic dipole latitude; R E - Earth radius; MLT- magnetic local time. [t 1, f 1 ] [t 2, f 2 ] f 1 ‹ f 2 (sweep rate)

( 11:51-12:13 UT) The electron density was about 11 cm −3 and the Kp index was (n e- =9 cm -3 ) df/dt:10.61 kHz/s (N=236) and kHz/s (N=156) (n e- =10 cm -3 ) df/dt: no risers and kHz/s (N=1036) (n e- =10 cm -3 ) df/dt: 9.36 kHz/s (N=5568) and no fallers (14: :00 UT) (17:40-17:50 UT) The Kp index was 3 0 and the electron density was about 10 cm −3

( 11:51-12:13 UT) The electron density was about 192 cm −3 and Kp index was 8 − (n e- = 27 cm -3 ) df/dt: 5.97 kHz/s (N=351) and no fallers (n e- = 39 cm -3 ) df/dt: 3.91 kHz/s and no fallers (n e- = 192cm -3 ) df/dt: 1.69 kHz/s (N=972) and kHz/s (13: :20 UT) (23: :35 UT) The Kp index was 2 0 and the electron density was about 39 cm −3

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.

Conclusions Within the framework of the BWO generation model, it is possible to explain the properties of the chorus emissions at frequencies below f H /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

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 f H /2