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14Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 2, pp. 289-296. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220209
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 65P20, 76W05, 85A30, 62M15
Spectral Properties of Low-order Dynamo Systems
P. Frick, R. Okatev, D. Sokoloff
The solar 11-year activity cycle is a famous manifestation of magnetic activity of celestial bodies. The physical nature of the solar cycle is believed to be large-scale magnetic field excitation in the form of a wave of a quasi-stationary magnetic field propagating from middle solar latitudes to the solar equator. The power spectrum of solar magnetic activity recorded in sunspot data and underlying solar dynamo action contains quite a stable oscillation known as the 11-year cycle as well as the continuous component and some additional weak peaks. We consider a low-order model for the solar dynamo. We show that in some range of governing parameters this model can reproduce spectra with pronounced dominating frequency and wide spectral peaks in the low-frequency region. The spectra obtained are qualitatively similar to the observed solar activity spectrum.
Keywords: solar activity, solar dynamo, low-order models, spectral properties
Received March 30, 2022 Accepted May 27, 2022
This study has been performed within the framework of RAS project AAAA-A19-119012290101-5. DS is gratefull for support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement No. 075-15-2019-1621 and BASIS foundation under grant 21-1-1-4-1.
Peter Frick frick@icmm.ru Roman Okatev okatev.r@icmm.ru
Institute of Continuous Media Mechanics ul. Akad. Korolyova 1, Perm, 614018 Russia Perm State University ul. Bukireva 15, Perm, 614068 Russia
Dmitry Sokoloff sokoloff.dd@gmail.com
Department of Physics, Lomonosov Moscow State University Moscow Center for Fundamental and Applied Mathematics GSP-1, Leninskie Gory, Moscow, 119991 Russia IZMIRAN
Kaluzhskoe sh. 4, Troitsk, Moscow, 108840 Russia
1. Introduction
The solar 11-year magnetic cycle is the most well-known manifestation of magnetic activity of celestial bodies. The physical nature of the solar cycle is large-scale magnetic field excitation in the form of a wave of a quasi-stationary magnetic field propagating from middle solar latitudes to the solar equator. It looks natural to expect that the basic features of the phenomenon should admit a description by several degrees of freedom and do not involve the whole variety of degrees of freedom responsible for all features of solar magnetism. Indeed, such an illustrative model was suggested as early as 1955 by E. Parker [1]. If desirable, this model can be presented in the form of a dynamical system truncating the Fourier decomposition of the original Parker equations to keep four terms only. It is noteworthy that such the simple model reproduces a variety of dynamic regimes, including stationary solutions, oscillatory regimes in the form of dynamo waves, vacillations, and chaotic regimes [2].
Dynamo waves provided by the low-order model in some range of governing parameters correspond in general to the actually observed dynamics of solar activity. The point, however, is that the variable solar activity is something more than an activity cycle and includes a number of variations which cover the spectral range from several months up to decades to form a power low continuous spectrum [3] with the 11-year cycle included in a form similar to a spectral line (eigensolution of governing equations). Continuous spectra are known for turbulent systems. The origin of continuous spectra was initially connected with excitation of many degrees of freedom (Landau scenario); however, later it became clear that strange attractors can produce such spectra using just a few degrees of freedom only.
Already in early studies of dynamical chaos, some attention was paid to the spectral properties of chaotic solutions [4-6]. Farmer et al. [7] noted that care must be taken in using power spectra to determine the onset of chaos, since a phase coherent strange attractor may look like a limit cycle, having a sharp peak in the power spectrum above the broad background. Frick [4] recognized that the famous Lorenz system demonstrates in some regimes continuous spectra with a sequence of weak peaks, each of which can be associated with typical fragments of the phase trajectory.
Our aim here is to show that, even in the framework of a low-order model, the occurring dynamo regimes require a more subtle classification than the division into periodic and chaotic regimes, since they demonstrate various structures of the spectral power decomposition. We discuss the spectral properties of a low-order Parker model and a possible analogy with the observed spectra of solar activity.
2. Low-order approximation of the Parker model
One of the simplest solar dynamo models was proposed by Parker[1]. In dimensionless form, the equations for the vector potential of the poloidal magnetic field A and the toroidal magnetic field B read
dA d2 A , ,
-9t=R°aB + m> {2A)
dB dA d2B , .
m=R^nl,m + (2/2)
where d is the latitude measured from the Northern pole, and Ra and are parameters characterizing the intensities of the a-effect and differential rotation, respectively. The time is measured
in units of the diffusion time where R is the convective zone radius and /3 is the turbulent diffusivity [2]. All quantities are averaged over the radial direction inside the convective shell. The model describes axisymmetric magnetic field generation and does not include in explicit form magnetic field transports in radial or/and longitudinal direction which are presented in more developed models.
Nonlinear dynamo saturation can be provided in the Parker model by the so-called «-quenching — a suppression of a by increasing the magnetic field. Then
a = a0 (1 - £2B2), (2.3)
where £ determines the amplitude of the saturated dynamo wave. For the sake of simplicity we choose a0 = cos 6.
We consider a Fourier decomposition of the equations and truncate them, keeping four variables, two for poloidal and two for toroidal components
A(6, t) = a1(t) sin 6 + a2(t)sin36, (2.4)
B(6, t) = b1(t) sin 26 + b2(t) sin 46. (2.5)
Substituting (2.4) and (2.5) into (2.1) and (2.2), we obtain a four-order approximation of Parker's dynamo [2]
cH = ^b1-a1-?^b1(bl + 2bl), (2.6)
R 3£2 R
¿2 = -f(h + b2) - 902 - "Y^&i + h) (6? + hh + (2.7)
R
b^-^ia,-^)-^, (2.8)
"UP , 2
3R.
b2 = -f-a2-16b2. (2.9)
The behavior of magnetic fields excited by Parker dynamo is determined by the quantity D = = RaRu known as dynamo number (however, strictly speaking, nonlinear stabilization depends on all parameters). The dynamical system (2.6)-(2.9) allows periodic as well as chaotic solutions (e.g., [2]). If D < 0, the periodic solution presenting a dynamo wave propagates equatorwards. For D > 0 the dynamo wave propagates polarwards. We concentrate here on the chaotic regime which appears for D > 0. The area of chaotic mode is shown in Fig. 1 on the parameter plane (Ra, Ru). For D < 0 we do not find a chaotic regime in the framework of the four-order model. We are not interested here in the very beginning of magnetic field evolution and present the results for the stage with initial dynamo action saturated by nonlinear terms. For this stage the character of temporal evolution does not depend on initial conditions and is determined by one governing parameter D only.
Numerical simulations were carried out using the fifth-order implicit Runge-Kutta method with an adaptive integration step size. We have verified that the step size choice does not affect the spectral characteristics. To perform the spectral analysis, we considered time series with a length of 50 000 units of time and sampling frequency vs = 1000. To compute the power spectral density (PSD), we used Welch's method with 214 points per segment.
Various toy dynamo models similar to some extent to the above low-dimensional dynamo are considered to explain the observed properties of astrophysical dynamos (e.g., [8, 9]). We note that there are models which use higher-order approximations of Eqs. (2.1)-(2.2), e.g., [10].
3000
2000-
1000-
3
0-
-3000
-1000 -
-2000 -
0.0
0.2
0.4
0.6
Fig. 1. Map of chaotic (shown in green) and substantially nonharmonic periodic solutions known as dynamo bursts which may mimic continuous spectrum (shown in gray) regimes in the plane (Ra, Ru). Black dots denote the regimes discussed below
Naturally, such models are more realistic than this simplest one. But in our work we prefer to consider a four-order system only to show that even the simplest model can reproduce spectra that are similar to the observable ones.
3. Spectral analysis of chaotic regimes
We consider the power spectral density (PSD) of the variable b1 which determines the behavior of a toroidal magnetic field, because this quantity seems to be most close to what directly comes from sunspot observations. The behavior of b2, a1, a2 is in general similar to that of b1. We show below PSD for several chaotic regimes of Eqs. (2.6)-(2.9) to demonstrate that two types of spectra are possible. Following the astronomical tradition, we present PSD spectra in terms of period lengths rather than frequencies.
First, we demonstrate a chaotic regime in which the spectrum of chaotic variations of the dynamo generated magnetic field is just continuous without dominating frequency. This kind of solution appears, i.e., at Ra = 0.25, Ru = 2453.0 and £ = 0.01 (Fig. 2). The figure presents fragments of time series for b1 with different time resolutions, which illustrate the structure of particular oscillations and the absence of long-term structures. The PSD indicates a maximum at T & 1, which evidently corresponds to the typical time of an individual oscillation; however, this peak is broad and only slightly elevated on the low-frequency (long periods) part of the spectrum.
The second kind of regimes is characterized by a much more pronounced dominating frequency (which again corresponds to the main dynamo cycle) and the appearance of additional peaks in the low-frequency part (long-term periodicity). Figure 3 presents results for a regime of this kind. Note that this regime appears at Ru = 2170.0 (all other parameters are the same as in Fig. 2), thus the corresponding points in the parameter plane are very close (see Fig. 1). The main peak at T & 0.73 is very narrow and the second one occurs at T & 5.63. Another example
760
50 0
-50 H
lA
n
780
800 t
820
840
820.0 822.5 825.0 827.5 830.0 t
Fig. 2. Simulated time series of the variable bx (above), its fragment (bottom, left) and its PSD (bottom, right) for Ra = 0.25, Ru = 2453.0, £ = 0.01
820.0 822.5 825.0 827.5 830.0 t
Fig. 4. Same as in Fig. 2 for Ru = 448.0
of a spectrum of this kind is shown below in Fig. 5 (right) and demonstrates even a more sharp main peak.
We note that the part of the spectrum in Fig. 3, on the left from the main peak, looks like several secondary peaks which can be considered as a result of nonlinear doubling of the main frequency. In order to confirm this interpretation, we show in Fig. 4 the time series and PSD for a run with complicated periodic behavior. We note a sequence of secondary peaks on the left from the main peak in the Fourier spectrum. It is natural to believe that the variability of the main solar cycle together with the moderate length of observational time series for solar magnetic activity would give here a continuous power spectrum and a broad spectral peak which corresponds to the main solar cycle. This explanation for the continuous component of the solar activity spectrum is suggested by [11] within the framework of a dynamo model based on partial differential equations as an explanation for mid-term solar activity variations (for a review, see [12]). We demonstrate here that four degrees of freedom are sufficient for this explanation.
4. Discussion and conclusions
Our study of spectral properties of a low-order solar dynamo model was motivated by the intention to find out whether a model of this kind could not only reproduce the basic types of dynamo regimes (namely, stationary, periodic, and chaotic regimes), but also come closer to modeling the spectra actually observed for solar and stellar activity. Solar observations significantly exceed all other available observations in quality and duration of observations and show that the pronounced quasi-periodic dynamo is characterized by a strong but rather wide spectral peak (which is due to permanent variations in the duration and intensity of individual cycles), which exists against the background of a continuous spectrum, including relatively weak, but noticeable local maxima on either side of the main peak. We demonstrate in Fig. 5 (left) the
103
10°
10-2
10°
101 102 T, years
101
T, years
Fig. 5. Power spectrum of solar activity variations as recorded in sunspot data [13] (left) and spectrum of b1 for Ra = 0.25, = 2060.0, £ = 0.01 (right)
PSD of solar activity, recorded in sunspot data from 1610 to the present day (we have used the database from [13]) for the range 0.5 < T < 500 years. In the same figure we show the PSD revealed in the explored four-order Parker-like model at Ra = 0.25, Ru = 2060.0, £ = 0.01 (this regime is also marked by a point on the parameter plane in Fig. 1). Comparing two spectra, we conclude that, while being not identical, they are, of course, reasonably similar.
Note that available stellar data include much shorter time series, but nevertheless allow one to recognize a variety of spectral portraits even being limited by old solar-type stars only [14]. We are not trying to look for analogs of various stellar spectra in our simulated PSD, but we note that among the latter there are also continuous spectra without pronounced maximum which may resemble power spectra for the stars like a fast rotating K2 dwarf V833 Tau [15]. Spectral details with time scales shorter than the main stellar activity cycle may be interesting in explaining solar activity details like 5.5-year oscillations, see, e.g., [16-18].
The general conclusion is that the effects of dynamical chaos as well as nonlinear frequency doubling can provide magnetic field variations with Fourier spectrum reasonably similar to those obtained for the solar activity spectrum as recorded in sunspot data. Such simulated spectra contain a pronounced spectral peak, corresponding to the main activity cycle, and a continuous part of the spectrum with longer time scale oscillations. This long-term part of the spectrum includes a local maximum at times that are an order of magnitude greater than the period of the main cycle. The corresponding solar cycle is known as the Gleissberg cycle (see, e.g., [19, 20]), which provides a peak at T & 100 years in the solar spectrum (Fig. 5). Other details of the solar spectrum with periods longer than the main activity cycle are also discussed (e.g., in [21]). The part of the simulated spectrum on the left from the spectral line contains a number of secondary lines which can mimic a continuous spectrum.
We stress that the chaotic regimes with an activity spectrum similar to the observed one are obtained for D > 0, i.e., for polarward activity waves, while the solar activity wave propagates equatorward. This means that we have failed to mimic all features of the solar activity spectrum within the framework of the low-dimensional model under consideration. It may hardly be expected that an obviously oversimplified dynamical system can simultaneously reproduce all features of such a complicated phenomenon as the solar cycle.
One more point is that moderate changes in governing dynamo parameters, e.g., in Ru, can result in substantial modifications of the magnetic activity spectrum.
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