magnetohydrodynamic turbulence

The incompressible MHD equations for constant mass density, $\backslash rho=1$, are

:$$

\begin{align}

\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} & = -\nabla p + \mathbf{B} \cdot \nabla \mathbf{B} + \nu \nabla^2 \mathbf{u} \\[5pt]

\frac{\partial \mathbf{B}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{B} & = \mathbf{B} \cdot \nabla \mathbf{u} +

\eta \nabla^2 \mathbf{B} \\[5pt]

\nabla \cdot \mathbf{u} & = 0 \\[5pt]

\nabla \cdot \mathbf{B} & = 0.

\end{align}

where

• {{math|u}} represents the velocity,
• {{math|B}} represent the magnetic field,
• {{math|p}} represents the total pressure (thermal+magnetic) fields,
• $\backslash nu$ is the kinematic viscosity and
• $\backslash eta$ represents magnetic diffusivity.

The third equation is the incompressibility condition. In the above equation, the magnetic field is in Alfvén units (same as velocity units).

The total magnetic field can be split into two parts: $\backslash mathbf\{B\}\; =\; \backslash mathbf\{B\_0\}\; +\; \backslash mathbf\{b\}$ (mean + fluctuations).

The above equations in terms of Elsässer variables ($\backslash mathbf\{z\}^\{\backslash pm\}\; =\; \backslash mathbf\{u\}\; \backslash pm\; \backslash mathbf\{b\}$) are

:$$

\frac{\partial {\mathbf{z}^{\pm}}}{\partial t}\mp\left(\mathbf {B}_0\cdot{\mathbf \nabla}\right){\mathbf z^{\pm}} + \left({\mathbf z^{\mp}}\cdot{\mathbf \nabla}\right){\mathbf z^{\pm}} = -{\mathbf \nabla}p

+ \nu_+ \nabla^2 \mathbf{z}^{\pm} + \nu_- \nabla^2 \mathbf{z}^{\mp}

where $\backslash nu\_\backslash pm\; =\; \backslash frac\{1\}\{2\}(\backslash nu\; \backslash pm\; \backslash eta)$. Nonlinear interactions occur between the Alfvénic fluctuations $$

z^{\mp} .

The important nondimensional parameters for MHD are

:$$

\begin{array}{lcl}

\text{Reynolds number } Re & = & U L /\nu \\

\text{Magnetic Reynolds number } Re_M & = & U L /\eta \\

\text{Magnetic Prandtl number } P_M & = & \nu / \eta.

\end{array}

The magnetic Prandtl number is an important property of the fluid. Liquid metals have small magnetic Prandtl numbers, for example, liquid sodium's $P\_M$ is around $10^\{-5\}$. But plasmas have large $P\_M$.

The Reynolds number is the ratio of the nonlinear term $\backslash mathbf\{u\}\; \backslash cdot\; \backslash nabla\; \backslash mathbf\{u\}$ 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,D. Biskamp (2003), Magnetohydrodynamical Turbulence, (Cambridge University Press, Cambridge.) Verma.{{cite journal | last=Verma | first=Mahendra K. | title=Statistical theory of magnetohydrodynamic turbulence: recent results | journal=Physics Reports | volume=401 | issue=5–6 | year=2004 | issn=0370-1573 | doi=10.1016/j.physrep.2004.07.007 | pages=229–380| arxiv=nlin/0404043 | bibcode=2004PhR...401..229V | s2cid=119352240 }} and Galtier.

IroshnikovP. S. Iroshnikov (1964), Turbulence of a Conducting Fluid in a Strong Magnetic Field, Soviet Astronomy, 7, 566. and Kraichnan{{cite journal | last=Kraichnan | first=Robert H. | title=Inertial-Range Spectrum of Hydromagnetic Turbulence | journal=Physics of Fluids | publisher=AIP Publishing | volume=8 | issue=7 | year=1965 | issn=0031-9171 | doi=10.1063/1.1761412 | page=1385| bibcode=1965PhFl....8.1385K }} 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 $(B\_0\; k)^\{-1\}$. As a results the energy spectra is

:{{anchor|eq:Kraichnan}}$$

E^u(k) \approx E^b(k) \approx A (\Pi V_A)^{1/2} k^{-3/2}.

where $\backslash Pi$ is the energy cascade rate.

Later Dobrowolny et al.{{cite journal | last1=Dobrowolny | first1=M. | last2=Mangeney | first2=A. | last3=Veltri | first3=P. | title=Fully Developed Anisotropic Hydromagnetic Turbulence in Interplanetary Space | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=45 | issue=2 | date=1980-07-14 | issn=0031-9007 | doi=10.1103/physrevlett.45.144 | pages=144–147| bibcode=1980PhRvL..45..144D }} derived the following generalized formulas for the cascade rates of $z^\{\backslash pm\}$ variables:

:{{anchor|eq:Dobrowolny}}$$

\Pi^+ \approx \Pi^{-} \approx \tau^{\pm}_k E^{+}(k) E^{-}(k) k^4 \approx E^{+}(k) E^{-}(k) k^3 / B_0

where $\backslash tau^\{\backslash pm\}$ are the interaction time scales of $z^\{\backslash pm\}$ variables.

Iroshnikov and Kraichnan's phenomenology follows once we choose $\backslash tau^\{\backslash pm\}\; \backslash approx\; 1/(k\; V\_A)$.

MarschE. Marsch (1990), Turbulence in the solar wind, in: G. Klare (Ed.), Reviews in Modern Astronomy, Springer, Berlin, p. 43. chose the nonlinear time scale $T\_\{NL\}^\{\backslash pm\}\; \backslash approx\; (k\; z\_k^\{\backslash mp\})^\{-1\}$ as the interaction time scale for the eddies and derived Kolmogorov-like energy spectrum for the Elsasser variables:

:{{anchor|eq:Kolm}}$$

E^{\pm}(k) = K^{\pm} (\Pi^{\pm})^{4/3} (\Pi^{\mp})^{-2/3} k^{-5/3}

where $\backslash Pi^+$ and $\backslash Pi^-$ are the energy cascade rates of $z^+$ and $z^-$ respectively, and $K^\{\backslash pm\}$ are constants.

Matthaeus and Zhou{{cite journal | last1=Matthaeus | first1=William H. | last2=Zhou | first2=Ye | title=Extended inertial range phenomenology of magnetohydrodynamic turbulence | journal=Physics of Fluids B: Plasma Physics | publisher=AIP Publishing | volume=1 | issue=9 | year=1989 | issn=0899-8221 | doi=10.1063/1.859110 | pages=1929–1931| bibcode=1989PhFlB...1.1929M }} 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{{cite journal | last=Verma | first=Mahendra K. | title=Mean magnetic field renormalization and Kolmogorov's energy spectrum in magnetohydrodynamic turbulence | journal=Physics of Plasmas | publisher=AIP Publishing | volume=6 | issue=5 | year=1999 | issn=1070-664X | doi=10.1063/1.873397 |arxiv=chao-dyn/9803021 | pages=1455–1460| bibcode=1999PhPl....6.1455V | s2cid=2218981 }} using renormalization group analysis by showing that the Alfvénic fluctuations are affected by scale-dependent "local mean magnetic field". The local mean magnetic field scales as $k^\{-1/3\}$, 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 $k^\{-4/3\}$ that again yields $k^\{-5/3\}$ 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.{{cite journal | last1=Shebalin | first1=John V. | last2=Matthaeus | first2=William H. | last3=Montgomery | first3=David | title=Anisotropy in MHD turbulence due to a mean magnetic field | journal=Journal of Plasma Physics | publisher=Cambridge University Press (CUP) | volume=29 | issue=3 | year=1983 | issn=0022-3778 | doi=10.1017/s0022377800000933 | pages=525–547| bibcode=1983JPlPh..29..525S | hdl=2060/19830004728 | s2cid=122509800 | hdl-access=free }}

A mean magnetic field makes turbulence anisotropic. In the limit

$\backslash delta\; z^\{\backslash pm\}\; \backslash ll\; B\_0$, 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.{{cite journal | last1=Galtier | first1=S. | last2=Nazarenko | first2=S. V. | last3=Newell | first3=A. C. | last4= Pouquet | first4=A. | title=A weak turbulence theory for incompressible magnetohydrodynamics | journal=Journal of Plasma Physics | publisher=Cambridge University Press (CUP) | volume=63 | issue=5 | year=2000 | issn=0022-3778 | doi=10.1017/s0022377899008284| arxiv=astro-ph/0008148 | pages=447–488| bibcode=2000JPlPh..63..447G | s2cid=15528846 | url=http://wrap.warwick.ac.uk/843/1/WRAP_Galtier_weak_turbulence.pdf }} who found the Kolmogorov-Zakharov spectrum

:$$

E(k) \sim (\Pi B_0)^{1/2} k_\parallel^{1/2} k_\perp^{-2}

where $k\_\backslash parallel$ and $k\_\{\backslash perp\}$ 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, $\backslash delta\; z^\backslash pm\; \backslash sim\; B\_0$, Goldereich and Sridhar{{cite journal | last1=Goldreich | first1=P. | last2=Sridhar | first2=S. | title=Toward a theory of interstellar turbulence. 2: Strong alfvenic turbulence | journal=The Astrophysical Journal | publisher=IOP Publishing | volume=438 | year=1995 | issn=0004-637X | doi=10.1086/175121 | page=763| bibcode=1995ApJ...438..763G | url=https://authors.library.caltech.edu/38003/1/1995ApJ___438__763G.pdf }} argue that $k\_\backslash perp\; z\_\{k\_\backslash perp\}\; \backslash sim\; k\_\backslash parallel\; B\_0$ ("critical balanced state") which implies that

:$$

\begin{align}

E(k) & \propto k_\perp^{-5/3}; \\[5pt]

k_\parallel & \propto k_\perp^{2/3}

\end{align}

The above anisotropic phenomenology has been extended for large cross helicity MHD.

{{main|Solar wind turbulence}}

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 $E^\{\backslash pm\}$ are closer to

$k^\{-5/3\}$ compared to $k^\{-3/2\}$, thus favoring Kolmogorov-like phenomenology for MHD

turbulence.{{cite journal | last1=Matthaeus | first1=William H. | last2=Goldstein | first2=Melvyn L. | title=Measurement of the rugged invariants of magnetohydrodynamic turbulence in the solar wind | journal=Journal of Geophysical Research | publisher=American Geophysical Union (AGU) | volume=87 | issue=A8 | year=1982 | issn=0148-0227 | doi=10.1029/ja087ia08p06011 | page=6011| bibcode=1982JGR....87.6011M | hdl=11603/30515 | hdl-access=free }}D. A. Roberts, M. L. Goldstein (1991), Turbulence and waves in the solar wind, Rev. Geophys., 29, 932. The interplanetary and interstellar electron density fluctuations also provide

a window for investigating MHD turbulence.

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.{{cite journal | last1=Müller | first1=Wolf-Christian | last2=Biskamp | first2=Dieter | title=Scaling Properties of Three-Dimensional Magnetohydrodynamic Turbulence | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=84 | issue=3 | date=2000-01-17 | issn=0031-9007 | doi=10.1103/physrevlett.84.475 | pages=475–478| pmid=11015942 |arxiv=physics/9906003| bibcode=2000PhRvL..84..475M | s2cid=43131956}} 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 $\backslash Pi^\{\backslash pm\}$ can be more reliable quantities to validate MHD turbulence models.

When $E^+(k)\; \backslash gg\; E^-(k)$

(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 $\backslash Pi^\{\backslash pm\}$ computed from the numerical simulations are in better agreement with Kolmogorov-like model compared to Kraichnan and Iroshnikov model.{{cite journal | last1=Verma | first1=M. K. | last2=Roberts | first2=D. A. | last3=Goldstein | first3=M. L. | last4=Ghosh | first4=S. | last5=Stribling | first5=W. T. | title=A numerical study of the nonlinear cascade of energy in magnetohydrodynamic turbulence | journal=Journal of Geophysical Research: Space Physics | publisher=American Geophysical Union (AGU) | volume=101 | issue=A10 | date=1996-10-01 | issn=0148-0227 | doi=10.1029/96ja01773 | pages=21619–21625| bibcode=1996JGR...10121619V | hdl=11603/30574 | hdl-access=free }}

Anisotropic aspects of MHD turbulence have also been studied using numerical simulations. The predictions of Goldreich and Sridhar ($k\_$

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).

• Magnetohydrodynamics
• Turbulence
• Alfvén wave
• Solar dynamo
• Reynolds number
• Navier–Stokes equations
• Computational magnetohydrodynamics
• Computational fluid dynamics
• Solar wind
• Magnetic flow meter
• Ionic liquid

{{reflist}}

Category:Magnetohydrodynamics

Category:Turbulence