Saturation of the turbulent dynamo

Editors' Suggestion

Saturation of the turbulent dynamo

J. Schober, D. R. G. Schleicher, C. Federrath, S. Bovino, and R. S. Klessen

Phys. Rev. E 92, 023010 – Published 6 August 2015

Abstract

The origin of strong magnetic fields in the Universe can be explained by amplifying weak seed fields via turbulent motions on small spatial scales and subsequently transporting the magnetic energy to larger scales. This process is known as the turbulent dynamo and depends on the properties of turbulence, i.e., on the hydrodynamical Reynolds number and the compressibility of the gas, and on the magnetic diffusivity. While we know the growth rate of the magnetic energy in the linear regime, the saturation level, i.e., the ratio of magnetic energy to turbulent kinetic energy that can be reached, is not known from analytical calculations. In this paper we present a scale-dependent saturation model based on an effective turbulent resistivity which is determined by the turnover time scale of turbulent eddies and the magnetic energy density. The magnetic resistivity increases compared to the Spitzer value and the effective scale on which the magnetic energy spectrum is at its maximum moves to larger spatial scales. This process ends when the peak reaches a characteristic wave number $k_{☆}$ which is determined by the critical magnetic Reynolds number. The saturation level of the dynamo also depends on the type of turbulence and differs for the limits of large and small magnetic Prandtl numbers Pm. With our model we find saturation levels between 43.8% and 1.3% for $Pm ≫ 1$ and between 2.43% and 0.135% for $Pm ≪ 1$ , where the higher values refer to incompressible turbulence and the lower ones to highly compressible turbulence.

Received 14 May 2015

DOI:https://doi.org/10.1103/PhysRevE.92.023010

Authors & Affiliations

J. Schober^*

Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Strasse 2, D-69120 Heidelberg, Germany and Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden

D. R. G. Schleicher

Departamento de Astronomía, Facultad Ciencias Físicas y Matemáticas, Universidad de Concepción, Avenida Esteban Iturra s/n Barrio Universitario, Casilla 160-C, Concepción, Chile

C. Federrath

Research School of Astronomy & Astrophysics, The Australian National University, Canberra, ACT 2611, Australia

S. Bovino

Hamburger Sternwarte, Gojenbergsweg 112, D-21029 Hamburg, Germany and Institut für Astrophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany

R. S. Klessen

Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Strasse 2, D-69120 Heidelberg, Germany

^*jschober@nordita.org

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 92, Iss. 2 — August 2015

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
An illustration of the amplification of magnetic fields and the resulting transport of magnetic energy to subviscous scales. Motions of the fluid on the viscous scale $ℓ_{ν}$ stretch and bend the magnetic field lines to scales $ℓ < ℓ_{ν}$ . This way antiparallel field lines are brought closer together and the magnetic energy eventually accumulates on the resistive scale $ℓ_{η}$ .
Reuse & Permissions
Figure 2
The spectra of turbulent kinetic and magnetic energy, $E (k)$ and $M (k)$ . The input parameters for this plot are the hydrodynamic Reynolds number $Re = 10^{6}$ , the magnetic Prandtl number $Pm = 10^{3}$ , and the slopes of the turbulence spectrum $ϑ = 1 / 3$ and $ϑ = 1 / 2$ describing Kolmogorov and Burgers turbulence. The green dashed line indicates the (initial) kinetic energy spectrum $E (k)$ . The gray lines, (a) and (b), show the magnetic energy spectrum $M_{k} (k)$ at different times in the kinematic phase. The two light blue lines, (c) and (d), give the spectrum in the first nonlinear phase $M_{nl}, 1 (k)$ , while the two dark blue lines, (e) and (f), give the one in the second nonlinear phase $M_{nl}, 2 (k)$ . The purple line, (g), represents the magnetic spectrum at saturation $M_{nl}, 2 (k_{☆})$ . Indicated as vertical dotted lines are the forcing scale $k_{L}$ , the peak scale at saturation $k_{☆}$ , the viscous scale $k_{ν}$ , and the initial resistive scale $k_{η}$ .
Reuse & Permissions
Figure 3
The saturation level of the turbulent dynamo as a function of the slope of the turbulence spectrum $ϑ$ . The upper boundaries of the bands are given by $m_{sat} / (ε_{0} - m_{sat})$ , while the lower boundaries are determined by $m_{sat} / ε_{0}$ . The upper purple band represents the limit of large magnetic Prandtl numbers Pm with the saturation levels given in Eqs. (41) and (42). The small Pm limit is illustrated by the lower gray band, where the boundaries are Eqs. (51) and (52).
Reuse & Permissions
Figure 4
The spectra of turbulent kinetic and magnetic energy, $E (k)$ and $M (k)$ . The input parameters for this plot are a hydrodynamic Reynolds number of $Re = 10^{6}$ and a magnetic Prandtl number of $Pm = 10^{- 3}$ . This figure is conceptually similar to Fig. 2.
Reuse & Permissions
Figure 5
The saturation level of the turbulent dynamo as a function of the Mach number $M$ . The crosses are the results of the numerical simulations from Federrath et al. [35] for solenoidally driven (purple crosses) and compressively driven turbulence (gray crosses). The hydrodynamical Reynolds number of these simulations is $Re \approx 1500$ and the magnetic Prandtl number is $Pm \approx 2$ . The black dashed-dotted line at low Mach numbers indicates the saturation level in simulations of Haugen et al. [34]. For comparison we include the predictions from our model by the blue shaded regions within the two limiting cases $m_{sat} / (ε_{0} - m_{sat})$ and $m_{sat} / ε_{0}$ [see Eqs. (41) and (42)] for different slopes of the turbulence spectrum $ϑ = 1 / 3$ and $ϑ = 1 / 2$ .
Reuse & Permissions
Figure 6
The saturation level of the turbulent dynamo as a function of the magnetic Prandtl number Pm. The purple crosses are the results of the numerical simulations from Federrath et al. [29]. In the analytical model the saturation level is independent of Pm in the limits $Pm ≪ 1$ and $Pm ≫ 1$ . We shade the region between the two limiting cases $m_{sat} / (ε_{0} - m_{sat})$ and $m_{sat} / ε_{0}$ for different slopes of the turbulence spectrum $ϑ = 0.4$ (dark blue dotted lines), $ϑ = 0.45$ (light blue dashed lines), and $ϑ = 0.5$ (gray solid lines).
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics