N.G. Kleimenova et al. the distribution over the entire event. Let us demonstrate this deduction by numerical example. 6 December 2003, CLUSTER-3 Fig. 1. Distribution of the ^-parameter values obtained from chorus elements observed by CLUSTER spacecraft during the event on December 6 , 2003. The plasma density was Ne =10 cm '3. 3. Numerical model As in papers [6, 8], we consider the discrete model of electronic generator in BWO regime: A(t)=X(t-T0)[A(t-T0).A \t-T 0)] + 5, ( 2 ) where A(t ) is wave magnitude, T0 is characteristic time delay of feedback (it is a time step for the discrete model), 5 is a small constant which corresponds to the initial background wave, and X is the growth rate. For the magnetospheric BWO To ~ 0.1 s, and X can be obtained as [1, 2]: X=l+7c ( y slep /лов /(vgvslefl)Zi -л/2) = 1 + k 2 ( / ’ -l)/2 For example, X * 14 for q =12. The relationships between the model parameters and physical parameters of the magnetospheric BWO were discussed in [3, 6]. Let us consider X as a random variable: HO = C0+ C, ( 3 ) where C0 and Q are constants, and ^ is a random number. The simplest assumption that the random number ^( t ) have the uniform distribution in the range from 0 to 1 was considered in paper [6]. Now we consider a case of normal distribution of £ with a zero mean and unit variance. Then the constants C0and C, in equation (3) determine the mean value and variance ofX{t). Additionally, we assume that X(t) is non negative: X(t) = 0 for C0+ Ci £,(t) <0 4. Results The numerical model described by Eqs. (2)-(3) with parameters C0 = 6, Q = 12.0, and 5 = 10'8 demonstrates the generation of a sequence of peaks in A(t), see Figure 2. At a level Alhr= 3 the peaks are well distinguished. Let us consider the probability density functions (PDF) of the growth rate X for the entire simulation interval and separately for the moments just before exceeding the threshold value A,hr. The latter moments are actually the starting points of the chorus elements, so the corresponding parameter represent the BWO parameters deduced from the observed chorus elements. The PDFs are shown in Figure 3. One can see that the PDF for the X values over the entire simulation time occupies much smaller values than the PDF for the moments when Alhr is exceeded. More detail about the evolution of the model parameters near the moments of the threshold excess have been obtained by superimposed epoch method, see Figure 4. One can see that at these moments the wave amplitude is on average at least one order of magnitude higher than at other times, but the average growth rate is by 50% higher at than just one time step before. 12 10 8 0 0 1-104 2 -10 4 3 -104 VT0 Fig. 2. Dynamics of A(t) in the numerical model (2)-(3) with parameters C0= 6, Q = 12.0,, 5 = 10'8. C 0= 6.0, C 1= 1 2 . 0 80

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