scholarly journals Turbulent viscosity and magnetic Prandtl number from simulations of isotropically forced turbulence

2020 ◽  
Vol 636 ◽  
pp. A93 ◽  
Author(s):  
P. J. Käpylä ◽  
M. Rheinhardt ◽  
A. Brandenburg ◽  
M. J. Käpylä
Keyword(s):  
Magnetic Fields ◽  
Prandtl Number ◽  
Large Scale ◽  
Turbulent Viscosity ◽  
Scale Dependence ◽  
Scale Separation ◽  
Separation Ratio ◽  
Magnetic Prandtl Number ◽  
Magnetic Diffusivity ◽  
Forced Turbulence

Context. Turbulent diffusion of large-scale flows and magnetic fields plays a major role in many astrophysical systems, such as stellar convection zones and accretion discs. Aims. Our goal is to compute turbulent viscosity and magnetic diffusivity which are relevant for diffusing large-scale flows and magnetic fields, respectively. We also aim to compute their ratio, which is the turbulent magnetic Prandtl number, Pmt, for isotropically forced homogeneous turbulence. Methods. We used simulations of forced turbulence in fully periodic cubes composed of isothermal gas with an imposed large-scale sinusoidal shear flow. Turbulent viscosity was computed either from the resulting Reynolds stress or from the decay rate of the large-scale flow. Turbulent magnetic diffusivity was computed using the test-field method for a microphysical magnetic Prandtl number of unity. The scale dependence of the coefficients was studied by varying the wavenumber of the imposed sinusoidal shear and test fields. Results. We find that turbulent viscosity and magnetic diffusivity are in general of the same order of magnitude. Furthermore, the turbulent viscosity depends on the fluid Reynolds number (Re) and scale separation ratio of turbulence. The scale dependence of the turbulent viscosity is found to be well approximated by a Lorentzian. These results are similar to those obtained earlier for the turbulent magnetic diffusivity. The results for the turbulent transport coefficients appear to converge at sufficiently high values of Re and the scale separation ratio. However, a weak trend is found even at the largest values of Re, suggesting that the turbulence is not in the fully developed regime. The turbulent magnetic Prandtl number converges to a value that is slightly below unity for large Re. For small Re we find values between 0.5 and 0.6 but the data are insufficient to draw conclusions regarding asymptotics. We demonstrate that our results are independent of the correlation time of the forcing function. Conclusions. The turbulent magnetic diffusivity is, in general, consistently higher than the turbulent viscosity, which is in qualitative agreement with analytic theories. However, the actual value of Pmt found from the simulations (≈0.9−0.95) at large Re and large scale separation ratio is higher than any of the analytic predictions (0.4−0.8).

scholarly journals Hydromagnetic dynamos at the low Ekman and magnetic Prandtl numbers

2016 ◽  
Vol 46 (3) ◽  
pp. 221-244 ◽  
Author(s):  
Ján Šimkanin
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Prandtl Number ◽  
Large Scale ◽  
Small Scale ◽  
Polar Regions ◽  
Magnetic Diffusion ◽  
Magnetic Prandtl Number ◽  
Prandtl Numbers ◽  
The Magnetic Field

Abstract Hydromagnetic dynamos are numerically investigated at low Prandtl, Ekman and magnetic Prandtl numbers using the PARODY dynamo code. In all the investigated cases, the generated magnetic fields are dominantly-dipolar. Convection is small-scale and columnar, while the magnetic field maintains its large-scale structure. In this study the generated magnetic field never becomes weak in the polar regions, neither at large magnetic Prandtl numbers (when the magnetic diffusion is weak), nor at low magnetic Prandtl numbers (when the magnetic diffusion is strong), which is a completely different situation to that observed in previous studies. As magnetic fields never become weak in the polar regions, then the magnetic field is always regenerated in the tangent cylinder. At both values of the magnetic Prandtl number, strong polar magnetic upwellings and weaker equatorial upwellings are observed. An occurrence of polar magnetic upwellings is coupled with a regenaration of magnetic fields inside the tangent cylinder and then with a not weakened intensity of magnetic fields in the polar regions. These new results indicate that inertia and viscosity are probably negligible at low Ekman numbers.

scholarly journals Turbulent magnetic Prandtl number and magnetic diffusivity quenching from simulations

2003 ◽  
Vol 411 (3) ◽  
pp. 321-327 ◽  
Author(s):  
T. A. Yousef ◽  
A. Brandenburg ◽  
G. Rüdiger
Keyword(s):  
Prandtl Number ◽  
Magnetic Prandtl Number ◽  
Magnetic Diffusivity

Investigation of Inverse-Turbulent-Prandtl Number with Four RNG k-ε Turbulence Models on Compressor Discharge Pipe of Bioenergy Micro Gas Turbine

2016 ◽  
Vol 819 ◽  
pp. 392-400 ◽  
Author(s):  
Ahmad Indra Siswantara ◽  
Budiarso ◽  
Steven Darmawan
Keyword(s):  
Prandtl Number ◽  
Turbulence Model ◽  
Large Scale ◽  
Swirling Flow ◽  
Turbulent Viscosity ◽  
Turbulence Models ◽  
Small Scale ◽  
Turbulent Prandtl Number ◽  
Curved Pipe ◽  
Molecular Viscosity

Inverse-Turbulent Prandtl number (α) is an important parameter in RNG k-ε turbulence models since it affects the ratio of molecular viscosity and turbulent viscosity. In curved pipe, this highly affects the model prediction to a large range eddy-scale flow. According to Yakhot & Orzag, the α range from 1-1.3929 has not been investigated in detail in curved pipe flow (Yakhot & Orszag, 1986) and specific Re. This paper varied inverse-turbulent Prandtl number α to 1-1.3 in RNG k-ε turbulence model on cylindrical curved pipe in order to obtain the optimum value of α to predict unfully-developed flow in the curve with curve ratio R/D of 1.607. Analysis was conducted numericaly with inlet specified Re of 40900 which was generated from the experiment at α 1, 1.1, 1.2, 1.3. Wall surface roughness is not considered in this paper. With assumption that thermal diffusivity is always dominant to turbulent viscosity, higher Inverse-turbulent Prandtl number represent domination of turbulent viscosity to molecular viscosity of the flow and predict to have more interaction between large scale eddy to small scale eddy as well. The results show the use of α = 1.3 has increased the turbulent kinetic energy by 7% and the turbulent dissipation by 5% compared to general inverse-turbulent Prandtl number of 1. The value difference shows that the use of higher α on RNG turbulence model described more interaction between eddies in secondary and swirling flow at pipe curve at Re = 40900.

scholarly journals Advection/diffusion of large scale magnetic field in accretion disks

2009 ◽  
Vol 16 (1) ◽  
pp. 77-81 ◽  
Author(s):  
R. V. E. Lovelace ◽  
G. S. Bisnovatyi-Kogan ◽  
D. M. Rothstein
Keyword(s):  
Magnetic Field ◽  
Large Scale ◽  
Turbulent Viscosity ◽  
Pressure Ratio ◽  
Rotational Instability ◽  
Small Scale ◽  
Magnetic Diffusivity ◽  
Large Scale Magnetic Field ◽  
Scale Field ◽  
Large Scale Field

Abstract. Activity of the nuclei of galaxies and stellar mass systems involving disk accretion to black holes is thought to be due to (1) a small-scale turbulent magnetic field in the disk (due to the magneto-rotational instability or MRI) which gives a large viscosity enhancing accretion, and (2) a large-scale magnetic field which gives rise to matter outflows and/or electromagnetic jets from the disk which also enhances accretion. An important problem with this picture is that the enhanced viscosity is accompanied by an enhanced magnetic diffusivity which acts to prevent the build up of a significant large-scale field. Recent work has pointed out that the disk's surface layers are non-turbulent and thus highly conducting (or non-diffusive) because the MRI is suppressed high in the disk where the magnetic and radiation pressures are larger than the thermal pressure. Here, we calculate the vertical (z) profiles of the stationary accretion flows (with radial and azimuthal components), and the profiles of the large-scale, magnetic field taking into account the turbulent viscosity and diffusivity due to the MRI and the fact that the turbulence vanishes at the surface of the disk. We derive a sixth-order differential equation for the radial flow velocity vr(z) which depends mainly on the midplane thermal to magnetic pressure ratio β>1 and the Prandtl number of the turbulence P=viscosity/diffusivity. Boundary conditions at the disk surface take into account a possible magnetic wind or jet and allow for a surface current in the highly conducting surface layer. The stationary solutions we find indicate that a weak (β>1) large-scale field does not diffuse away as suggested by earlier work.

Renormalization group analysis of the magnetohydrodynamic turbulence and dynamo

2021 ◽  
Vol 926 ◽  
Author(s):  
Krzysztof A. Mizerski
Keyword(s):  
Renormalization Group ◽  
Magnetic Fields ◽  
Large Scale ◽  
Magnetic Energy ◽  
Group Analysis ◽  
Small Scale ◽  
Mhd Turbulence ◽  
Dynamical Equations ◽  
Renormalization Group Analysis ◽  
Magnetic Diffusivity

The magnetohydrodynamic (MHD) turbulence appears in engineering laboratory flows and is a common phenomenon in natural systems, e.g. stellar and planetary interiors and atmospheres and the interstellar medium. The applications in engineering are particularly interesting due to the recent advancement of tokamak devices, reaching very high plasma temperatures, thus giving hope for the production of thermonuclear fusion power. In the case of astrophysical applications, perhaps the main feature of the MHD turbulence is its ability to generate and sustain large-scale and small-scale magnetic fields. However, a crucial effect of the MHD turbulence is also the enhancement of large-scale diffusion via interactions of small-scale pulsations, i.e. the generation of the so-called turbulent viscosity and turbulent magnetic diffusivity, which typically exceed by orders of magnitude their molecular counterparts. The enhanced resistivity plays an important role in the turbulent dynamo process. Estimates of the turbulent electromotive force (EMF), including the so-called $\alpha$ -effect responsible for amplification of the magnetic energy and the turbulent magnetic diffusion are desired. Here, we apply the renormalization group technique to extract the final expression for the turbulent EMF from the fully nonlinear dynamical equations (Navier–Stokes, induction equation). The simplified renormalized set of dynamical equations, including the equations for the means and fluctuations, is derived and the effective turbulent coefficients such as the viscosity, resistivity, the $\alpha$ -coefficient and the Lorentz-force coefficients are explicitly calculated. The results are also used to demonstrate the influence of magnetic fields on energy and helicity spectra of strongly turbulent flows, in particular the magnetic energy spectrum.

scholarly journals Circulation conservation and vortex breakup in magnetohydrodynamics at low magnetic Prandtl number

2018 ◽  
Vol 857 ◽  
pp. 38-60 ◽  
Author(s):  
D. G. Dritschel ◽  
P. H. Diamond ◽  
S. M. Tobias
Keyword(s):  
Magnetic Fields ◽  
Prandtl Number ◽  
Scaling Laws ◽  
Vortex Patch ◽  
Magnetic Prandtl Number ◽  
Weak Magnetic Fields ◽  
Dimensional Parameters ◽  
Hydrodynamical Problem ◽  
And Diffusion

In this paper we examine the role of weak magnetic fields in breaking Kelvin’s circulation theorem and in vortex breakup in two-dimensional magnetohydrodynamics for the physically important case of a fluid with low magnetic Prandtl number (low  $Pm$ ). We consider three canonical inviscid solutions for the purely hydrodynamical problem, namely a Gaussian vortex, a circular vortex patch and an elliptical vortex patch. We examine how magnetic fields lead to an initial loss of circulation $\unicode[STIX]{x1D6E4}$ and attempt to derive scaling laws for the loss of circulation as a function of field strength and diffusion as measured by two non-dimensional parameters. We show that for all cases the loss of circulation depends on the integrated effects of the Lorentz force, with the patch cases leading to significantly greater circulation loss. For the case of the elliptical vortex, the loss of circulation depends on the total area swept out by the rotating vortex, and so this leads to more efficient circulation loss than for a circular vortex.

scholarly journals Scaling of Turbulent Viscosity and Resistivity: Extracting a Scale-dependent Turbulent Magnetic Prandtl Number

2021 ◽  
Vol 917 (1) ◽  
pp. L3
Author(s):  
Xin Bian ◽  
Jessica K. Shang ◽  
Eric G. Blackman ◽  
Gilbert W. Collins ◽  
Hussein Aluie
Keyword(s):  
Prandtl Number ◽  
Turbulent Viscosity ◽  
Magnetic Prandtl Number

scholarly journals The onset of turbulent rotating dynamos at the low magnetic Prandtl number limit

2017 ◽  
Vol 822 ◽  
Author(s):  
Kannabiran Seshasayanan ◽  
Vassilios Dallas ◽  
Alexandros Alexakis
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Large Scale ◽  
Injection Rate ◽  
Magnetic Reynolds Number ◽  
Strong Decrease ◽  
New Paradigm ◽  
Turbulent Fluctuations ◽  
Rotation Rates ◽  
Magnetic Prandtl Number

We demonstrate that the critical magnetic Reynolds number $Rm_{c}$ for a turbulent non-helical dynamo in the limit of low magnetic Prandtl number $Pm$ (i.e. $Pm=Rm/Re\ll 1$) can be significantly reduced if the flow is subjected to global rotation. Even for moderate rotation rates the required energy injection rate can be reduced by a factor of more than $10^{3}$. This strong decrease in the onset is attributed to the transfer of energy to the large scales, forming a large-scale condensate, and the reduction in the turbulent fluctuations that cause the flow to have a much larger cutoff length scale than in a non-rotating flow of the same Reynolds number. The dynamo thus behaves as if it is driven just by the large scales that act as a laminar flow (i.e. it behaves as a high $Pm$ dynamo) even though the actual Reynolds number is much higher than the magnetic Reynolds number (i.e. low $Pm$). Our finding thus points to a new paradigm for the design of new experiments on liquid metal dynamos.

scholarly journals A multiscale dynamo model driven by quasi-geostrophic convection

2015 ◽  
Vol 780 ◽  
pp. 143-166 ◽  
Author(s):  
Michael A. Calkins ◽  
Keith Julien ◽  
Steven M. Tobias ◽  
Jonathan M. Aurnou
Keyword(s):  
Prandtl Number ◽  
Viscous Dissipation ◽  
Time Scale ◽  
Large Scale ◽  
Force Balance ◽  
Dynamo Model ◽  
Small Scale ◽  
Ohmic Dissipation ◽  
Evolution Time ◽  
Magnetic Prandtl Number

A convection-driven multiscale dynamo model is developed in the limit of low Rossby number for the plane layer geometry in which the gravity and rotation vectors are aligned. The small-scale fluctuating dynamics are described by a magnetically modified quasi-geostrophic equation set, and the large-scale mean dynamics are governed by a diagnostic thermal wind balance. The model utilizes three time scales that respectively characterize the convective time scale, the large-scale magnetic evolution time scale and the large-scale thermal evolution time scale. Distinct equations are derived for the cases of order one and low magnetic Prandtl number. It is shown that the low magnetic Prandtl number model is characterized by a magnetic to kinetic energy ratio that is asymptotically large, with ohmic dissipation dominating viscous dissipation on the large scale. For the order one magnetic Prandtl number model, the magnetic and kinetic energies are equipartitioned and both ohmic and viscous dissipation are weak on the large scales; large-scale ohmic dissipation occurs in thin magnetic boundary layers adjacent to the horizontal boundaries. For both magnetic Prandtl number cases the Elsasser number is small since the Lorentz force does not enter the leading order force balance. The new models can be considered fully nonlinear, generalized versions of the dynamo model originally developed by Childress & Soward (Phys. Rev. Lett., vol. 29, 1972, pp. 837–839), and provide a new theoretical framework for understanding the dynamics of convection-driven dynamos in regimes that are only just becoming accessible to direct numerical simulations.

scholarly journals An 84-μG Magnetic Field in a Galaxy at Z=0.692?

2008 ◽  
Vol 4 (S254) ◽  
pp. 95-96
Author(s):  
Arthur M. Wolfe ◽  
Regina A. Jorgenson ◽  
Timothy Robishaw ◽  
Carl Heiles ◽  
Jason X. Prochaska
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Faraday Rotation ◽  
Large Scale ◽  
Dynamo Model ◽  
Magnetic Field Generation ◽  
Interstellar Gas ◽  
Splitting Technique ◽  
Average Value ◽  
The Magnetic Field

AbstractThe magnetic field pervading our Galaxy is a crucial constituent of the interstellar medium: it mediates the dynamics of interstellar clouds, the energy density of cosmic rays, and the formation of stars (Beck 2005). The field associated with ionized interstellar gas has been determined through observations of pulsars in our Galaxy. Radio-frequency measurements of pulse dispersion and the rotation of the plane of linear polarization, i.e., Faraday rotation, yield an average value B ≈ 3 μG (Han et al. 2006). The possible detection of Faraday rotation of linearly polarized photons emitted by high-redshift quasars (Kronberg et al. 2008) suggests similar magnetic fields are present in foreground galaxies with redshifts z > 1. As Faraday rotation alone, however, determines neither the magnitude nor the redshift of the magnetic field, the strength of galactic magnetic fields at redshifts z > 0 remains uncertain.Here we report a measurement of a magnetic field of B ≈ 84 μG in a galaxy at z =0.692, using the same Zeeman-splitting technique that revealed an average value of B = 6 μG in the neutral interstellar gas of our Galaxy (Heiles et al. 2004). This is unexpected, as the leading theory of magnetic field generation, the mean-field dynamo model, predicts large-scale magnetic fields to be weaker in the past, rather than stronger (Parker 1970).The full text of this paper was published in Nature (Wolfe et al. 2008).

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