Magnetohydrodynamic Turbulence
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Magnetohydrodynamic turbulence concerns the chaotic regimes of magnetofluid flow at high
Reynolds number In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
.
Magnetohydrodynamics In physics and engineering, magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is a model of electrically conducting fluids that treats all interpenetrating particle species together as a single Continuum ...
(MHD) deals with what is a quasi-neutral fluid with very high conductivity. The fluid approximation implies that the focus is on macro length-and-time scales which are much larger than the collision length and collision time respectively. Understanding MHD turbulence is fundamental because most of the visible matter in the universe is in the plasma state and this plasma is mainly turbulent.


Incompressible MHD equations

The incompressible MHD equations for constant mass density, \rho=1 , are : \begin \frac + \mathbf \cdot \nabla \mathbf & = -\nabla p + \mathbf \cdot \nabla \mathbf + \nu \nabla^2 \mathbf \\ pt \frac + \mathbf \cdot \nabla \mathbf & = \mathbf \cdot \nabla \mathbf + \eta \nabla^2 \mathbf \\ pt \nabla \cdot \mathbf & = 0 \\ pt\nabla \cdot \mathbf & = 0. \end where * represents the velocity, * represent the magnetic field, * represents the total pressure (thermal+magnetic) fields, * \nu is the
kinematic viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
and * \eta represents
magnetic diffusivity The magnetic diffusivity controls the rate of magnetic field diffusion. Since its role in the evolution equation for the magnetic field is analogous to that of the viscosity for the velocity field, some authors refer to it as the 'magnetic viscos ...
. The third equation is the incompressibility condition. In the above equation, the
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
is in Alfvén units (same as velocity units). The total magnetic field can be split into two parts: \mathbf = \mathbf + \mathbf (mean + fluctuations). The above equations in terms of Elsässer variables ( \mathbf^ = \mathbf \pm \mathbf ) are : \frac\mp\left(\mathbf _0\cdot\right) + \left(\cdot\right) = -p + \nu_+ \nabla^2 \mathbf^ + \nu_- \nabla^2 \mathbf^ where \nu_\pm = \frac(\nu \pm \eta) . Nonlinear interactions occur between the Alfvénic fluctuations z^ . The important nondimensional parameters for MHD are : \begin \text Re & = & U L /\nu \\ \text Re_M & = & U L /\eta \\ \text P_M & = & \nu / \eta. \end The
magnetic Prandtl number The Magnetic Prandtl number (Prm) is a dimensionless quantity occurring in magnetohydrodynamics which approximates the ratio of momentum diffusivity (viscosity) and magnetic diffusivity. It is defined as: :\mathrm_\mathrm = \frac = \frac = \frac ...
is an important property of the fluid. Liquid metals have small magnetic Prandtl numbers, for example, liquid sodium's P_M is around 10^ . But plasmas have large P_M . The Reynolds number is the ratio of the nonlinear term \mathbf \cdot \nabla \mathbf of the Navier–Stokes equation to the viscous term. While the magnetic Reynolds number is the ratio of the nonlinear term and the diffusive term of the induction equation. In many practical situations, the Reynolds number Re of the flow is quite large. For such flows typically the velocity and the magnetic fields are random. Such flows are called to exhibit MHD turbulence. Note that Re_M need not be large for MHD turbulence. Re_M plays an important role in dynamo (magnetic field generation) problem. The mean magnetic field plays an important role in MHD turbulence, for example it can make the turbulence anisotropic; suppress the turbulence by decreasing energy cascade etc. The earlier MHD turbulence models assumed isotropy of turbulence, while the later models have studied anisotropic aspects. In the following discussions will summarize these models. More discussions on MHD turbulence can be found in Biskamp, Verma. and Galtier.


Isotropic models

Iroshnikov and Kraichnan formulated the first phenomenological theory of MHD turbulence. They argued that in the presence of a strong mean magnetic field, z^+ and z^- wavepackets travel in opposite directions with the phase velocity of B_0, and interact weakly. The relevant time scale is Alfven time (B_0 k)^. As a results the energy spectra is : E^u(k) \approx E^b(k) \approx A (\Pi V_A)^ k^. where \Pi is the energy cascade rate. Later Dobrowolny et al. derived the following generalized formulas for the cascade rates of z^ variables: : \Pi^+ \approx \Pi^ \approx \tau^_k E^(k) E^(k) k^4 \approx E^(k) E^(k) k^3 / B_0 where \tau^ are the interaction time scales of z^ variables. Iroshnikov and Kraichnan's phenomenology follows once we choose \tau^ \approx 1/(k V_A) . Marsch chose the nonlinear time scale T_^ \approx (k z_k^)^ as the interaction time scale for the eddies and derived Kolmogorov-like energy spectrum for the Elsasser variables: : E^(k) = K^ (\Pi^)^ (\Pi^)^ k^ where \Pi^+ and \Pi^- are the energy cascade rates of z^+ and z^- respectively, and K^ are constants. Matthaeus and Zhou attempted to combine the above two time scales by postulating the interaction time to be the harmonic mean of Alfven time and nonlinear time. The main difference between the two competing phenomenologies (−3/2 and −5/3) is the chosen time scales for the interaction time. The main underlying assumption in that Iroshnikov and Kraichnan's phenomenology should work for strong mean magnetic field, whereas Marsh's phenomenology should work when the fluctuations dominate the mean magnetic field (strong turbulence). However, as we will discuss below, the solar wind observations and numerical simulations tend to favour −5/3 energy spectrum even when the mean magnetic field is stronger compared to the fluctuations. This issue was resolved by Verma using
renormalization Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
group analysis by showing that the Alfvénic fluctuations are affected by scale-dependent "local mean magnetic field". The local mean magnetic field scales as k^ , substitution of which in Dobrowolny's equation yields Kolmogorov's energy spectrum for MHD turbulence. Renormalization group analysis have been also performed for computing the renormalized viscosity and resistivity. It was shown that these diffusive quantities scale as k^ that again yields k^ energy spectra consistent with Kolmogorov-like model for MHD turbulence. The above renormalization group calculation has been performed for both zero and nonzero cross helicity. The above phenomenologies assume isotropic turbulence that is not the case in the presence of a mean magnetic field. The mean magnetic field typically suppresses the energy cascade along the direction of the mean magnetic field.


Anisotropic models

A mean magnetic field makes turbulence anisotropic. In the limit \delta z^ \ll B_0 , an analytical theory can be developed which asymptotically leads to a kinetic equation that describes the evolution of the energy spectrum over long timescales. This work was done by Galtier et al. who found the Kolmogorov-Zakharov spectrum : E(k) \sim (\Pi B_0)^ k_\parallel^ k_\perp^ where k_\parallel and k_ are components of the wavenumber parallel and perpendicular to the mean magnetic field. The above limit is called weak wave turbulence and the spectrum is an exact solution of MHD turbulence. Under the strong turbulence limit, \delta z^\pm \sim B_0 , Goldereich and Sridhar argue that k_\perp z_ \sim k_\parallel B_0 ("critical balanced state") which implies that : \begin E(k) & \propto k_\perp^; \\ ptk_\parallel & \propto k_\perp^ \end The above anisotropic phenomenology has been extended for large cross helicity MHD.


Solar wind observations

Solar wind plasma is in a turbulent state. Researchers have calculated the energy spectra of the solar wind plasma from the data collected from the spacecraft. The kinetic and magnetic energy spectra, as well as E^ are closer to k^ compared to k^ , thus favoring Kolmogorov-like phenomenology for MHD turbulence. The interplanetary and interstellar electron density fluctuations also provide a window for investigating MHD turbulence.


Numerical simulations

The theoretical models discussed above are tested using the high resolution direct numerical simulation (DNS). Number of recent simulations report the spectral indices to be closer to 5/3. There are others that report the spectral indices near 3/2. The regime of power law is typically less than a decade. Since 5/3 and 3/2 are quite close numerically, it is quite difficult to ascertain the validity of MHD turbulence models from the energy spectra. Energy fluxes \Pi^ can be more reliable quantities to validate MHD turbulence models. When E^+(k) \gg E^-(k) (high cross helicity fluid or imbalanced MHD) the energy flux predictions of Kraichnan and Iroshnikov model is very different from that of Kolmogorov-like model. It has been shown using DNS that the fluxes \Pi^ computed from the numerical simulations are in better agreement with Kolmogorov-like model compared to Kraichnan and Iroshnikov model. Anisotropic aspects of MHD turbulence have also been studied using numerical simulations. The predictions of Goldreich and Sridhar ( k_ \sim k_^ ) have been verified in many simulations. The wave turbulence regime which is more difficult to check carefully has also been verified.


Energy transfer

Energy transfer among various scales between the velocity and magnetic field is an important problem in MHD turbulence. These quantities have been computed both theoretically and numerically. These calculations show a significant energy transfer from the large scale velocity field to the large scale magnetic field. Also, the cascade of magnetic energy is typically forward. These results have critical bearing on dynamo problem. ---- There are many open challenges in this field that hopefully will be resolved in near future with the help of numerical simulations, theoretical modelling, experiments, and observations (e.g., solar wind).


See also

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Magnetohydrodynamics In physics and engineering, magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is a model of electrically conducting fluids that treats all interpenetrating particle species together as a single Continuum ...
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Turbulence In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...
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Alfvén wave In plasma physics, an Alfvén wave, named after Hannes Alfvén, is a type of plasma wave in which ions oscillate in response to a restoring force provided by an Magnetic tension force, effective tension on the magnetic field lines. Definition ...
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Solar dynamo The solar dynamo is a physical process that generates the Sun's magnetic field. It is explained with a variant of the dynamo theory. A naturally occurring electric generator in the Sun's interior produces electric currents and a magnetic field, ...
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Reynolds number In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
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Navier–Stokes equations The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
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Computational magnetohydrodynamics Computational magnetohydrodynamics (CMHD) is a rapidly developing branch of magnetohydrodynamics that uses numerical methods and algorithms to solve and analyze problems that involve electrically conducting fluids. Most of the methods used in CMHD ...
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Computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid dynamics, fluid flows. Computers are used to perform the calculations required ...
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Solar wind The solar wind is a stream of charged particles released from the Sun's outermost atmospheric layer, the Stellar corona, corona. This Plasma (physics), plasma mostly consists of electrons, protons and alpha particles with kinetic energy betwee ...
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Magnetic flow meter {{Short description, Device for measuring flow of a fluid A ''magnetic flow meter'' (mag meter, electromagnetic flow meter) is a transducer that measures fluid flow by the voltage induced across the liquid by its flow through a magnetic field. A ...
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Ionic liquid An ionic liquid (IL) is a salt (chemistry), salt in the liquid state at ambient conditions. In some contexts, the term has been restricted to salts whose melting point is below a specific temperature, such as . While ordinary liquids such as wate ...


References

{{reflist Magnetohydrodynamics Turbulence