Temperature and Entropy in Ideal Magnetohydrodynamic Turbulence

2014 ◽  
Vol 136 (6) ◽  
Author(s):  
John V. Shebalin
Keyword(s):  
Dynamical System ◽  
Probability Density Function ◽  
Fourier Analysis ◽  
Numerical Experiments ◽  
Magnetic Energy ◽  
Fourier Mode ◽  
Mhd Turbulence ◽  
Magnetohydrodynamic Turbulence ◽  
Ideal Mhd ◽  
Ideal Magnetohydrodynamic

Fourier analysis of incompressible, homogeneous magnetohydrodynamic (MHD) turbulence produces a model dynamical system on which to perform numerical experiments. Statistical methods are used to understand the results of ideal (i.e., nondissipative) MHD turbulence simulations, with the goal of finding those aspects that survive the introduction of dissipation. This statistical mechanics is based on a Boltzmannlike probability density function containing three “inverse temperatures,” one associated with each of the three ideal invariants: energy, cross helicity, and magnetic helicity. However, these inverse temperatures are seen to be functions of a single parameter that may defined as the “temperature” in a statistical and thermodynamic sense: the average magnetic energy per Fourier mode. Here, we discuss temperature and entropy in ideal MHD turbulence and their use in understanding numerical experiments and physical observations.

Recent progress made in numerical experiments on magnetic reconnection and the related solar activities 

2021 ◽  
Author(s):  
Jun Lin ◽  
Jing Ye
Keyword(s):  
Magnetic Reconnection ◽  
Solar Flares ◽  
Numerical Experiments ◽  
Recent Progress ◽  
Magnetic Energy ◽  
Solar Physics ◽  
Plasma Heating ◽  
Mhd Turbulence ◽  
2D And 3D ◽  
Made In

<p>Magnetic reconnection plays a crucial role in the process of solar flares and coronal mass ejections, in which large amounts of magnetic energy (10^29-10^32 ergs) are converted into kinetic energy and thermal energy, even allowing for particle acceleration. On the platform of the Computational Solar Physics Laboratory of Yunnan Observatories, we have performed a series of numerical experiments on magnetic reconnection related to solar eruption events as well as numerical method developments both in 2D and 3D. In this talk, we will present some recent studies on the topic of plasma heating by reconnection, MHD turbulence, wave structures and complicate structures of CMEs, etc. Our numerical results have great potentials to explain and predict many related solar activities in the corona. </p>

scholarly journals On uniqueness of transfer rates in magnetohydrodynamic turbulence

2019 ◽  
Vol 85 (5) ◽  
Author(s):  
Franck Plunian ◽  
Rodion Stepanov ◽  
Mahendra Kumar Verma
Keyword(s):  
Kinetic Energy ◽  
Transfer Rate ◽  
Stokes Equations ◽  
Magnetic Energy ◽  
A Priori ◽  
Magnetic Helicity ◽  
Mhd Turbulence ◽  
Magnetohydrodynamic Turbulence ◽  
Transfer Rates ◽  
Induction Equation

In hydrodynamic and MHD (magnetohydrodynamic) turbulence, formal expressions for the transfer rates rely on integrals over wavenumber triads $(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})$ satisfying $\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q}=0$ . As an example $S_{E}^{uu}(\boldsymbol{k}\mid \boldsymbol{p},\boldsymbol{q})$ denotes the kinetic energy transfer rate to the mode $\boldsymbol{k}$ , from the two other modes in the triad, $\boldsymbol{p}$ and $\boldsymbol{q}$ . However as noted by Kraichnan (Phys. Rev., vol. 111, 1958, pp. 1747–1747), in $S_{E}^{uu}(\boldsymbol{k}\mid \boldsymbol{p},\boldsymbol{q})$ , what fraction of the energy transferred to the mode $\boldsymbol{k}$ originated from $\boldsymbol{p}$ and which from $\boldsymbol{q}$ is unknown. Such an expression is thus incongruent with the customary description of turbulence in terms of two-scale energy exchange. Notwithstanding this issue, Dar et al. (Physica D, vol. 157 (3), 2001, pp. 207–225) further decomposed these transfers into separate contributions from $\boldsymbol{p}$ -to- $\boldsymbol{k}$ and $\boldsymbol{q}$ -to- $\boldsymbol{k}$ , thus introducing the concept of mode-to-mode transfers that they applied to MHD turbulence. Doing so, they had to set aside additional transfers circulating within each triad, but failed to calculate them. In the present paper we explain how to derive the complete expressions of the mode-to-mode transfers, including the circulating transfers. We do it for kinetic energy and kinetic helicity in hydrodynamic turbulence, for kinetic energy, magnetic energy and magnetic helicity in MHD turbulence. We find that the degree of non-uniqueness of the energy transfers derived from the induction equation is a priori higher than the one derived from the Navier–Stokes equations. However, separating the contribution of magnetic advection from magnetic stretching, the energy mode-to-mode transfer rates involving the magnetic field become uniquely defined, in striking contrast to the hydrodynamic case. The magnetic helicity mode-to-mode transfer rate is also found to be uniquely defined, contrary to kinetic helicity in hydrodynamics. We find that shell-to-shell transfer rates have the same properties as mode-to-mode transfer rates. Finally calculating the fluxes, we show how the circulating transfers cancel in accordance with conservation laws.

Fully developed MHD turbulence near critical magnetic Reynolds number

1981 ◽  
Vol 104 ◽  
pp. 419-443 ◽  
Author(s):  
J. Léorat ◽  
A. Pouquet ◽  
U. Frisch
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Isotropic Turbulence ◽  
Magnetic Energy ◽  
Reynolds Numbers ◽  
Magnetic Reynolds Number ◽  
Mhd Equations ◽  
Liquid Sodium ◽  
Turbulent Heat ◽  
Mhd Turbulence

Liquid-sodium-cooled breeder reactors may soon be operating at magnetic Reynolds numbers RM where magnetic fields can be self-excited by a dynamo mechanism (as first suggested by Bevir 1973). Such flows have kinetic Reynolds numbers RV of the order of 107 and are therefore highly turbulent.This leads us to investigate the behaviour of MHD turbulence with high RV and low magnetic Prandtl numbers. We use the eddy-damped quasi-normal Markovian closure applied to the MHD equations. For simplicity we restrict ourselves to homogeneous and isotropic turbulence, but we do include helicity.We obtain a critical magnetic Reynolds number RMc of the order of a few tens (non-helical case) above which magnetic energy is present. RMc is practically independent of RV (in the range 40 to 106). RMc can be considerably decreased by the presence of helicity: when the overall size of the flow L is much larger than the integral scale l0, RMc can drop below unity as suggested by an α-effect argument. When L ≈ l0 the drop can still be substantial (factor of 6) when helicity is a maximum. We examine how the turbulence is modified when RM crosses RMc: presence of magnetic energy, decreased kinetic energy, steepening of kinetic-energy spectrum, etc.We make no attempt to obtain quantitative estimates for a breeder reactor, but discuss some of the possible consequences of exceeding RMc, such as decreased turbulent heat transport. More precise information may be obtained from numerical simulations and experiments (including some in the subcritical regime).

scholarly journals Multivalued Discrete Tomography Using Dynamical System That Describes Competition

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Takeshi Kojima ◽  
Tetsushi Ueta ◽  
Tetsuya Yoshinaga
Keyword(s):  
Dynamical System ◽  
Theoretical Analysis ◽  
Continuous Time ◽  
Numerical Experiments ◽  
Nonlinear Dynamical System ◽  
Reconstructed Image ◽  
Optimization Approach ◽  
Discrete Tomography ◽  
Nonlinear Dynamical ◽  
Time Optimization

Multivalued discrete tomography involves reconstructing images composed of three or more gray levels from projections. We propose a method based on the continuous-time optimization approach with a nonlinear dynamical system that effectively utilizes competition dynamics to solve the problem of multivalued discrete tomography. We perform theoretical analysis to understand how the system obtains the desired multivalued reconstructed image. Numerical experiments illustrate that the proposed method also works well when the number of pixels is comparatively high even if the exact labels are unknown.

Fourier Transforms in Probability, Random Variables and Stochastic Processes

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Fourier Analysis ◽  
Stochastic Processes ◽  
Probability Density ◽  
Density Function ◽  
Fourier Transforms ◽  
Random Variables ◽  
Cumulative Distribution ◽  
The Face

In this Chapter, we present application of Fourier analysis to probability, random variables and stochastic processes [1089, 1097, 1387, 1329]. Arandom variable, X, is the assignment of a number to the outcome of a random experiment. We can, for example, flip a coin and assign an outcome of a heads as X = 1 and a tails X = 0. Often the number is equated to the numerical outcome of the experiment, such as the number of dots on the face of a rolled die or the measurement of a voltage in a noisy circuit. The cumulative distribution function is defined by FX(x) = Pr[X ≤ x]. (4.1) The probability density function is the derivative fX(x) = d /dxFX(x). Our treatment of random variables focuses on use of Fourier analysis. Due to this viewpoint, the development we use is unconventional and begins immediately in the next section with discussion of properties of the probability density function.

scholarly journals MHD turbulence

2019 ◽  
Vol 5 (1) ◽  
Author(s):  
Andrey Beresnyak
Keyword(s):  
Numerical Experiments ◽  
Turbulence Models ◽  
Mhd Equations ◽  
Turbulence Theory ◽  
Current Status ◽  
Small Scale ◽  
Mhd Turbulence ◽  
Wave Modes

AbstractWe review the current status of research in MHD turbulence theory and numerical experiments and their applications to astrophysics and solar science. We introduce general tools for studying turbulence, basic turbulence models, MHD equations and their wave modes. Subsequently, we cover the theories and numerics of Alfvénic turbulence, imbalanced turbulence, small-scale dynamos and models and numerics for supersonic MHD turbulence.

scholarly journals Planet migration and magnetic torques

2015 ◽  
Vol 11 (A29A) ◽  
pp. 14-18
Author(s):  
A. Strugarek ◽  
A. S. Brun ◽  
S. P. Matt ◽  
V. Reville
Keyword(s):  
Numerical Experiments ◽  
Three Dimensional ◽  
Magnetic Star ◽  
Magnetic Topology ◽  
Global Models ◽  
Planet Migration ◽  
Order Of Magnitude ◽  
Ideal Magnetohydrodynamic ◽  
The Impact ◽  
The Ideal

AbstractThe possibility that magnetic torques may participate in close-in planet migration has recently been postulated. We develop three dimensional global models of magnetic star-planet interaction under the ideal magnetohydrodynamic (MHD) approximation to explore the impact of magnetic topology on the development of magnetic torques. We conduct twin numerical experiments in which only the magnetic topology of the interaction is altered. We find that magnetic torques can vary by roughly an order of magnitude when varying the magnetic topology from an aligned case to an anti-aligned case. Provided that the stellar magnetic field is strong enough, we find that magnetic migration time scales can be as fast as ~100 Myr. Hence, our model supports the idea that magnetic torques may participate in planet migration for some close-in star-planet systems.

scholarly journals On energetics and inertial-range scaling laws of two-dimensional magnetohydrodynamic turbulence

2012 ◽  
Vol 703 ◽  
pp. 238-254 ◽  
Author(s):  
Luke A. K. Blackbourn ◽  
Chuong V. Tran
Keyword(s):  
Energy Transfer ◽  
Kinetic Energy ◽  
Energy Flux ◽  
Scaling Laws ◽  
Magnetic Energy ◽  
Inertial Range ◽  
Two Dimensional ◽  
Magnetohydrodynamic Turbulence ◽  
Direct Energy ◽  
Energy Equipartition

AbstractWe study two-dimensional magnetohydrodynamic turbulence, with an emphasis on its energetics and inertial-range scaling laws. A detailed spectral analysis shows that dynamo triads (those converting kinetic into magnetic energy) are associated with a direct magnetic energy flux while anti-dynamo triads (those converting magnetic into kinetic energy) are associated with an inverse magnetic energy flux. As both dynamo and anti-dynamo interacting triads are integral parts of the direct energy transfer, the anti-dynamo inverse flux partially neutralizes the dynamo direct flux, arguably resulting in relatively weak direct energy transfer and giving rise to dynamo saturation. This result is consistent with a qualitative prediction of energy transfer reduction due to Alfvén wave effects by the Iroshnikov–Kraichnan theory (which was originally formulated for magnetohydrodynamic turbulence in three dimensions). We numerically confirm the correlation between dynamo action and direct magnetic energy flux and investigate the applicability of quantitative aspects of the Iroshnikov–Kraichnan theory to the present case, particularly its predictions of energy equipartition and ${k}^{\ensuremath{-} 3/ 2} $ spectra in the energy inertial range. It is found that for turbulence satisfying the Kraichnan condition of magnetic energy at large scales exceeding total energy in the inertial range, the kinetic energy spectrum, which is significantly shallower than ${k}^{\ensuremath{-} 3/ 2} $, is shallower than its magnetic counterpart. This result suggests no energy equipartition. The total energy spectrum appears to depend on the energy composition of the turbulence but is clearly shallower than ${k}^{\ensuremath{-} 3/ 2} $ for $r\approx 2$, even at moderate resolutions. Here $r\approx 2$ is the magnetic-to-kinetic energy ratio during the stage when the turbulence can be considered fully developed. The implication of the present findings is discussed in conjunction with further numerical results on the dependence of the energy dissipation rate on resolution.

scholarly journals Dust settling and rings in the outer regions of protoplanetary discs subject to ambipolar diffusion

2018 ◽  
Vol 617 ◽  
pp. A117 ◽  
Author(s):  
A. Riols ◽  
G. Lesur
Keyword(s):  
Large Scale ◽  
Diffusion Theory ◽  
Ambipolar Diffusion ◽  
Mhd Turbulence ◽  
Low Velocity ◽  
Zonal Flows ◽  
T Tauri ◽  
Ideal Mhd ◽  
Weakly Ionized ◽  
Protoplanetary Discs

Context. Magnetohydrodynamic (MHD) turbulence plays a crucial role in the dust dynamics of protoplanetary discs. It affects planet formation, vertical settling, and is one possible origin of the large scale axisymmetric structures, such as rings, recently imaged by ALMA and SPHERE. Among the variety of MHD processes in discs, the magnetorotational instability (MRI) has raised particular interest since it provides a source of turbulence and potentially organizes the flow into large scale structures. However, the weak ionization of discs prevents the MRI from being excited beyond 1 AU. Moreover, the low velocity dispersion observed in CO and strong sedimentation of millimetre dust measured in T-Tauri discs are in contradiction with predictions based on ideal MRI turbulence. Aims. In this paper, we study the effects of non-ideal MHD and magnetized winds on the dynamics and sedimentation of dust grains. We consider a weakly ionized plasma subject to ambipolar diffusion characterizing the disc outer regions (≫1 AU). Methods. To compute the dust and gas motions, we performed numerical MHD simulations in the stratified shearing box, using a modified version of the PLUTO code. We explored different grain sizes from micrometre to few centimetres and different disc vertical magnetizations with plasma beta ranging from 103 to 105. Results. Our simulations show that the mm-cm dust is contained vertically in a very thin layer, with typical heightscale ≲0.4 AU at R = 30 AU, compatible with recent ALMA observations. Horizontally, the grains are trapped within the pressure maxima (or zonal flows) induced by ambipolar diffusion, leading to the formation of dust rings. For micrometre grains and strong magnetization, we find that the dust layer has a size comparable to the disc heightscale H. In this regime, dust settling cannot be explained by a simple 1D diffusion theory but results from a large scale 2D circulation induced by both MHD winds and zonal flows. Conclusions. Our results suggest that non-ideal MHD effects and MHD winds associated with zonal flows play a major role in shaping the radial and vertical distribution of dust in protoplanetary discs. Leading to effective accretion efficiency α ≃ 10−3–10−1, non-ideal MHD models are also a promising avenue to reconcile the low turbulent activity measured in discs with their relatively high accretion rates.

scholarly journals Magnetic dynamo action in two-dimensional turbulent magneto-hydrodynamics

1977 ◽  
Vol 17 (2) ◽  
pp. 317-335 ◽  
Author(s):  
David Fyfe ◽  
Glenn Joyce ◽  
David Montgomery
Keyword(s):  
Initial Conditions ◽  
Magnetic Energy ◽  
Field Energy ◽  
Magnetic Excitation ◽  
External Forcing ◽  
Two Dimensional ◽  
Dynamo Action ◽  
Magnetohydrodynamic Turbulence ◽  
Long Wavelength ◽  
Magnetic Field Energy

Two-dimensional magnetohydrodynamic turbulence is explored by means of numerical simulation. Previous analytical theory, based on non-dissipative constants of the motion in a truncated Fourier representation, is verified by following the evolution of highly non-equilibrium initial conditions numerically. Dynamo action (conversion of a significant fraction of turbulent kinetic energy into long-wavelength magnetic field energy) is observed. It is conjectured that in the presence of dissipation and external forcing; a dual cascade will be observed for zero-helicity situations. Energy will cascade to higher wavenumbers simultaneously with a cascade of mean square vector potential to lower wavenumbers, leading to an omni-directional magnetic energy spectrum which varies as k-⅓ at lower wavenumbers, simultaneously with a build-up of magnetic excitation at the lowest wavenumber of the system. Equipartition of kinetic and magnetic energies is expected at the highest wavenumbers in the system.

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