Physics of auroral phenomena : proceedings of the 40th annual seminar, Apatity, 13-17 March, 2017 / [ed. board: N. V. Semenova, A. G. Yahnin]. - Апатиты : Издательство Кольского научного центра РАН, 2017. - 143 с. : ил., табл.

V. Pilipenko et al. In the case of low level fluctuations, 8 « 1 , the approximate solution of (1) can be found by iteration method, developed in the theory of nonlinear stochastic mechanics [Dimentberg, 1980]. Beyond the small-5 approximation, this equation was numerically solved in [Coult et al., 2013]. We seek a solution in the following form: x ( 0 = *0( 0 + ^ ( 0 + S 2JC2(t) + ... (2) Substitution o f (2) into (1) and grouping of all the terms of the same 5 order provides an infinite system of equations in respect to Xi(t). In the О-approximation one obtains the classical formula for a driven linear oscillator x0(t) = b0sin(eot + <p0), Q 2 (3) yJ(Q.2- со2)2+ 4y2co2 = arctan - Cl1-сог 2yco Features of this solution are a resonant peak and phase reversal near the resonance to—»0. In the absence of background fluctuations (5=0), the peak amplitude b (Qw3X> and the semi-width of the spectral peak Лю are determined by the damping factor у of the system, or otherwise by the Q-factor Q = D.I 2y ■ In subsequent approximations (i=l,2,...) the fluctuation- induced correction to the solution can be found from a recurrence formula. Keeping terms up to 52 order, it follows that near the resonance co-»0 (x(t)) - b sin(o* + <p0 - cp x), _/(+>] h(m°o (4) 4 ya>[ Figure 2. Amplitude o f oscillations b(co) using A= 1, 0=1, y=0.01, and noise with spectrum 1 If The height of the peak for each curve is the effective Q factor. Here J (+> - ^ К ( т ) е * c o s [( o j ± co } ) r ] r f r , со2 = Q 2 - у 2 ■ о The additional phase shift (px=o{52) in (4) is of no importance to us. The relationship (4) predicts that the amplitude of the resonant oscillations should change under the influence of stochastic fluctuations o f O. When the spectral density Ф(со) of q(t) fluctuations is a non-growing function, these fluctuations cause a decrease of average amplitude of resonant oscillations. This deterioration of resonator properties is not related to the occurrence of anomalous resistivity or viscosity in a turbulent plasma. This decrease of the resonant response to an external monochromatic driving is caused by stochastic deviations o f eigenfrequency from exact resonance. We consider how the system response varies with the amplitude of the background fluctuations characterized by 5 (Fig. 2). We use the noise with power spectrum 1 If. Without fluctuations, the classic curve shows a sharp resonant peak near co=0. The addition of noise (5*0) reduces the height of the resonant peak and increases its width, whereas the effect is more pronounced for a larger 5. The suppression o f resonant oscillation depends essentially on the spectrum of fluctuations. For a model problem, we choose noise functions with power spectrum 1//“, where 0<a< l. In Fig. 3, we measure the effective Q factor for several values of spectral index a as a function of 5. Smaller values o f a result in less sensitivity of the Q factor to 5. For white noise (a=0). the effect is absent. The effect increases with an increase o f a. The deterioration of resonant properties may be quite substantial: fluctuations with 5=0.2 and a=1.0 decrease the Q- factor more than 3 times, from -5 0 to -15. 50, iC f ------- 4 5 ^ ' V !\ \ " X 35b \ ........ . . . \ \ 30 r \ . 4 \ 251 • \ X 20 p V . ч ..............N c . \ 15 j- a* 0.25 0 n = 0 5 .........r>*^........... 10*- V a = 0.75 ot= 1 0 5 ! 0.05 0.1 015 0 2 0 25 Figure 3. The dependence of the Q-factor of the resonator on 5 for A=l, 0=1, y=0.01, and various a. Discussion. The dayside magnetopause with a step-like jump in magnetic field strength and plasma density may be imagined as a stressed membrane with reflecting boundaries in the northern and southern ionosphere, which can be resonantly excited by magnetosheath turbulence. The MHD modeling of the magnetospheric response to impulsive solar wind dynamic 56