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Small-scale Kazantsev-Kraichnan dynamo in a MHD shell approach

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Physica Scripta
Authors:
I Abushzada
I Abushzada
I Abushzada
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E Yushkov
E Yushkov
E Yushkov
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P. Frick at Institute of Contineous Media Mechanics
P. Frick
  • Institute of Contineous Media Mechanics
D Sokoloff
D Sokoloff
D Sokoloff
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Abstract and Figures

The small-scale magnetic energy generation in a turbulent velocity field is studied by two different approaches. One of them is based on the Kazantsev-Kraichnan model developed for turbulence with short-time velocity correlations, and the other uses the shell model of magnetohydrodynamic turbulence, describing the turbulent energy cascade on a finite number of spectral shells. We have found that the injection of weak magnetic field at the initial moment in both models leads to an exponential growth of magnetic energy and tried to determine whether these effects are of the same or different nature. The investigations have shown that the rates of growths and magnetic energy spectra in two approaches can be very much different, which can be attributed to the contradictions of the model assumptions and unknown correlation time. The discussion of these contradictions allows us to formulate a possible explanation, which is likely related to the fact that the small-scale magnetic field generation is under the influence of some spectral subrange, rather than the entire kinetic spectrum. Varying the correlation time of the velocity field and considering the spectral regions, we have determined the range of kinetic energy spectrum responsible for the small-scale dynamo generation.
Time evolution of total kinetic energy (upper panel) and total magnetic energy (lower panel). Different colors correspond to 1024 realizations of the turbulent cascade with slightly different initial distributions. The thick red line corresponding to the average behavior of all realizations shows the exponential growth of magnetic energy at the early stage of evolution, t ∈ [0, 4].
Time evolution of total kinetic energy (upper panel) and total magnetic energy (lower panel). Different colors correspond to 1024 realizations of the turbulent cascade with slightly different initial distributions. The thick red line corresponding to the average behavior of all realizations shows the exponential growth of magnetic energy at the early stage of evolution, t ∈ [0, 4].
… 
Averaged spectral densities of kinetic (upper panel) and magnetic (lower panel) energy at different time. Curves of different colors correspond to time indicated in the legends. It is clearly seen that the hydrodynamic energy practically does not change, while the magnetic energy increases exponentially with time. The wave number corresponding to the maximal magnetic energy will be referred below as the generation scale.
Averaged spectral densities of kinetic (upper panel) and magnetic (lower panel) energy at different time. Curves of different colors correspond to time indicated in the legends. It is clearly seen that the hydrodynamic energy practically does not change, while the magnetic energy increases exponentially with time. The wave number corresponding to the maximal magnetic energy will be referred below as the generation scale.
… 
Recovery of the magnetic field correlation function from the kinetic energy spectrum. Kinetic energy spectrum with indicated subranges, used for further consideration (a); the turbulent diffusion η(r) (b); potential of the Kazantsev eigenvalue problem (c); reconstructed correlation function F(r) (d) and its first(e) and second (f) derivatives.
Recovery of the magnetic field correlation function from the kinetic energy spectrum. Kinetic energy spectrum with indicated subranges, used for further consideration (a); the turbulent diffusion η(r) (b); potential of the Kazantsev eigenvalue problem (c); reconstructed correlation function F(r) (d) and its first(e) and second (f) derivatives.
… 
The correlation function M(r, t), reconstructed for different moments of time in the framework of the numerical solution based on the KK model (using the kinetic energy in the subrange 10⁰ < k < 10³) (a); the time evolution of the corresponding total magnetic energy (b), where the exponential growth rate is measured by the linear part of the black line. The spectral density of the magnetic energy, reconstructed for different time moments from the corresponding correlation functions (c). The generation scale is determined by the position of the maximum.
The correlation function M(r, t), reconstructed for different moments of time in the framework of the numerical solution based on the KK model (using the kinetic energy in the subrange 10⁰ < k < 10³) (a); the time evolution of the corresponding total magnetic energy (b), where the exponential growth rate is measured by the linear part of the black line. The spectral density of the magnetic energy, reconstructed for different time moments from the corresponding correlation functions (c). The generation scale is determined by the position of the maximum.
… 
The growth rate (a), (c) and generation scale (b), (d) versus the correlation, calculated within the framework of the KK model. Different colors correspond to different used subranges of kinetic energy spectrum. Both models show coinciding rates and scales of generation if KK model uses not the entire hydrodynamic spectrum, but only its large-scale part.
The growth rate (a), (c) and generation scale (b), (d) versus the correlation, calculated within the framework of the KK model. Different colors correspond to different used subranges of kinetic energy spectrum. Both models show coinciding rates and scales of generation if KK model uses not the entire hydrodynamic spectrum, but only its large-scale part.
… 
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Phys. Scr. 98 (2023)115966 https://doi.org/10.1088/1402-4896/ad0081
PAPER
Small-scale Kazantsev-Kraichnan dynamo in a MHD shell approach
I Abushzada
1
, E Yushkov
2,3,4
, P Frick
5
and D Sokoloff
2,6
1
Baku State University, Baku, Azerbaijan
2
Physics Department of Moscow State University, Moscow, Russia
3
Space Research Institute RAS, Moscow, Russia
4
Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
5
Institute of Continuous Media Mechanics UB RAS, Perm, Russia
6
Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation RAS, Troitsk, Russia
E-mail: yushkov@physics.msu.ru
Keywords: small-scale dynamo, MHD shell model, Kazantsev-Kraichnan approach, short-correlated turbulence
Abstract
The small-scale magnetic energy generation in a turbulent velocity field is studied by two different
approaches. One of them is based on the Kazantsev-Kraichnan model developed for turbulence with
short-time velocity correlations, and the other uses the shell model of magnetohydrodynamic
turbulence, describing the turbulent energy cascade on a finite number of spectral shells. We have
found that the injection of weak magnetic field at the initial moment in both models leads to an
exponential growth of magnetic energy and tried to determine whether these effects are of the same or
different nature. The investigations have shown that the rates of growths and magnetic energy spectra
in two approaches can be very much different, which can be attributed to the contradictions of the
model assumptions and unknown correlation time. The discussion of these contradictions allows us
to formulate a possible explanation, which is likely related to the fact that the small-scale magnetic
field generation is under the influence of some spectral subrange, rather than the entire kinetic
spectrum. Varying the correlation time of the velocity field and considering the spectral regions, we
have determined the range of kinetic energy spectrum responsible for the small-scale dynamo
generation.
1. Introduction
Amplification of the magnetic field in a flow of electrically conducting fluid was suggested in the first half of the
XX century as a suitable theoretical idea to explain the behavior of sunspots and the formation of the 11-year
solar cycle. In the magnetohydrodynamic (MHD)approximation, this theory is based on averaging the magnetic
induction equation, which makes it possible to relate the dynamics of the first and second magnetic field
statistical moments to the characteristics of the velocity field [1]. A natural simplification in such averaging is to
consider a statistically homogeneous in space and time velocity fields, in which the magnetic characteristics in a
certain situation can increase exponentially. The approach, which was developed in terms of the first statistical
moment, i.e. mean magnetic field, and used the ideas of E.Parker about the role of cyclonic motions [2]and the
mean-field electrodynamics introduced in 60th by M Steenbeck, F Krause and K-H Rädler [3]resulted after a
long standing revision and modification in a contemporary theory of solar cycle and solar dynamo, geodynamo
and dynamo in spiral galaxies [4]. In all these mean-field dynamo models, the generation of the field requires the
mirror asymmetric turbulence or convection and differential rotation of convective regions.
However, the dynamo excitation can exist without these special drivers, just by the developed turbulence,
producing a magnetic field on the turbulent motion scale. This process is known as the small-scale or turbulent
magnetic dynamo, see, e.g., [5].
The small-scale dynamo is an interesting phenomenon, which characterizes the transfer of the turbulent
kinetic energy to the energy of magnetic field localized on the scales smaller than the typical velocity correlation
length. A specific feature of such transfer is the ability to operate in a mirror-symmetric turbulence with zero
RECEIVED
22 August 2023
REVISED
27 September 2023
ACCEPTED FOR PUBLICATION
5 October 2023
PUBLISHED
23 October 2023
© 2023 IOP Publishing Ltd
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