Physics of auroral phenomena : proceedings of the 33rd Annual seminar, Apatity, 02 - 05 March, 2010 / [ed.: A.G. Yahnin, A. A. Mochalov]. - Апатиты : Издательство Кольского научного центра РАН, 2011. - 206 с. : ил.

А. V. Artemyev at al. Ax y = A e xP { ~ kl (* ~ x o " asin «О* ~ kl ( y - Уо Magnetic and electric fields can be found from Maxwell equations: S2(c/ou</) = ezrotA, E = c'dA /d t. Here к = {kx, ky) is the vector o f reversed spatial scale of a cloud, r0= {x0, y 0} is the vector of cloud position, a and со are magnitude and frequency o f cloud oscillations. We used a special distribution of amplitudes of magnetic clouds A0 and their positions to obtain the model with the structure function different from the Kolmogorov one well. It is known that some one­ dimensional maps of an interval at the real axis onto itself possess the property o f temporal intermittency [ Manneville, 1980]. To incorporate the one-dimensional map with intermittency into the model of magnetic clouds we use the following method. We divided the whole plane (x, y) into uniform cells with the size (A/,A/), then we place a single cloud with an amplitude A0 = into each of these cells, where the indices n and m define the numbering o f the cells along the x and у axis respectively. The coordinates of the clouds centres are given by r„m= {nAl + Д//2, m&l + Д//2}. The amplitudes f„m along the direction x are defined by the map f„+\m= F(f nm) where the map F is intermittent. The amplitudes of the maps are coupled by the discrete diffusion relation D(f„m _l -2f„m +/„m+l). Due to the fact that diffusion tends to make values neighbour in a space equal, the chaotic bursts and laminar phases (regions filled by f„m with similar values) will extend in both directions and will form clusters. The scale of the grid Al is chosen so that the inequality Д/|к<.| < 1 is satisfied (which means that each cloud is confined within its cell - see scheme in Fig. 1). We choose the map of the interval [0, 1] with the complex intermittent dynamics known as the Pomeau- Manneville map [. Manneville, 1980]: F(f n) = f„+ f/ (mod)l where f n e [0, 1], P > 1. It is known that this map possesses the three main dynamic regimes depending on the parameter Д When 1 < ft < 3/2 the dynamics is normal: the fluctuations of a random variable generated by the map are distributed by Gaussian law. When 3/2 < /? < 2 the dynamics is transitional-anomalous (non-gaussian) and when J3 > 2 the dynamics becomes strictly non- gaussian described by Levy statistics with the index \/{fi - 1). The latter regime corresponds to the map with intermittency. The given map generates one-dimensional series of intermittent turbulent bursts which are always positive. For our magnetic field model it is more natural for bursts to have both positive and negative values. This can be achieved by modifying the map F(f„) extending it to the negative values as follows, / „ | + | / / + l ) ( m o d 2 ) - l ■ a sin cot >1 ( 2 ) Figure 1. Scheme of magnetic clouds positions / „ +1 = ^ ( / „ ) = sgn (/n) f„ e [—1,1], P > 1 ( 4 ) The second component of our model is the ensemble of standing magnetostatic structures [Veltri et al., 1998; Zelenyi et al., 2008]: K a v e = В о ’2 : ( 1 + ( к л 1) ) “ < * * ( * * * + 9 o * ) ( 3 ) n Here 1 is the correlation vector, <p0n - initial phase has a random values for each wave, B 0W is the magnitude of waves. Comparison of model results with observations To compare our model with typical observations of turbulent magnetic field in the Earth magnetotail we construct the model magnetic field along the straight line in plane ( x ; y). Then for the obtained ID set the structure functions are calculated and power law exponent is plotted as a function of order p. Fig. 2(a) shows that model parameter /? controls the dependence on p: curvature of ^ increases with the increase o f Д Below we compare the model data with observation of turbulence in the Earth magnetotail [ Petrukovich, 2005]. Typical example of spacecraft (Interball-tail) data is shown in Fig. 2(b). Fig. 2(c) presents a model magnetic field. The model does not include a whole spectrum of the observed magnetic field (here we take into account only two orders in the spectrum), but even in such rather simple approximation the power law exponents from the model and observations have a similar behaviour (Fig. 2(d)). Therefore, one can conclude that the model constructed here is capable to reproduce main features of observation with good quality. 10