Properties of the backward wave oscillator model of VLFchorus generation inferred from CLUSTER measurements geomagnetic field. In the case of the BWO regime, the growth rate yBwo is related to the trapping frequency Ц г [1,2] as X vw o ^ ir ~3n /32 (2) Substituting Eq. (2) into Eq. (1) yields d f / d t = 0.3 y 2Bwo (3) According to the BWO model, in the case of propagation along the magnetic field the growth rate is given by [5,6] Ушо~ я/2 * ( q m - q ~Ш) / T 0 (4) In Eq. 4 T0 is the characteristic time scale of the BWO defined as T0= lBWO(l/V g + l/V * ) (5) where lBW0 ~ (B?L2/k)l/3 - the length of the chorus source region, V* = 2n (fH-J)/k is the parallel velocity of resonant electrons, Vg = 2 V*f/fH is the group velocity of the whistler-mode waves, L is the /.-shell, / and f n are the wave frequency and electron gyrofrequency, and q is the dimensionless parameter quantifying the excess of energetic electron density above the generation threshold. Note that parameter q is proportional to the height of the step [5] and the aim of this work is to determine and analyze this parameter for the magnetospheric BWO. Substituting Eq. (4) in Eq. (3) gives the following equation for q: (q 1/2- q ' m) 2 = 3 * (2/л)2 * df/dt * T 20 (6) As it follows from equations (5) and (6), to obtain the parameter q one needs simultaneous measurements of the frequencies and sweep rates of chorus elements, magnetic fields and plasma density in the source region. CLUSTER satellites provide all these data. 3. Analysis of the q parameter for the magnetospheric BWO using CLUSTER observations To analyze the sweep rate of the chorus elements on the CLUSTER satellites we used measurements of the WideBand Data (WBD) instrument [7], providing high- resolution measurements of one electric or one magnetic component. The WHISPER active sounder [8] onboard the CLUSTER spacecraft was used to measure the local electron densities ne. For 7 CLUSTER orbits the plasma density was measured with a good resolution (from 4s to 50s) during the time interval when chorus emissions were observed. For these CLUSTER orbits we obtained the frequency sweep rates for about 7000 elements and calculated the q parameter values for each chorus element in the source region near the equator (±5°). Figure 1 shows the values of the q parameter, depending on the cold plasma density for all chorus elements detected on the 7 Cluster orbits. Chorus elements with frequencies below and above one half of the electron gyrofrequency are plotted with black and grey colors, respectively. For calculation of the characteristic time scale of BWO T0 we use the lower frequency of chorus elements. Figure 1 shows that the parameter q varies in a wide range, but it does not show a definite dependence on the plasma density, and the q values are higher for the upper band chorus. 70 1 10 100 1000 Fig. 1. Parameter q for chorus emissions below (black diamond) and above (gray square) one half of the gyrofrequency for different plasma densities The mean values of the q parameter ( q m) for the intervals f / fH= 0.1 as a function of chorus frequency are given in Fig. 2. The vertical lines mark the standard deviation of the q values. As is seen in Fig. 2 the mean values o f q are higher for the upper band chorus: for the lower band, qm (f/fH< 0.5 ) ~ 8 and for the upper band, qm(f/fn> 0.5) =19. Fig. 2. The mean values of the parameter q in the interval 0,1 f / f H , depending on the chorus frequency normalized to the gyrofrequency. The values of the frequency sweep rate for the studied 7 orbits show no dependence on the frequency of chorus emissions. This means (see Eq. 6) that large q for the 91

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