MHD turbulence

• Dr. Peter Goldreich, Institute for Advanced Study

MHD turbulence describes turbulence in an electrically conducting, magnetized fluid (Biskamp 2003). Strictly speaking, MHD only applies to collision dominated fluids. However, it is often a useful guide to the behavior of magnetized plasmas even in the collisionless limit. Turbulence is a generic property of large scale fluid flows. Hydrodynamic (HD) turbulence is a familiar phenomenon. Flows of human dimensions commonly reach high Reynolds numbers; values in excess of \(10^4\) are achieved in the air we push aside when we walk and in the water we disturb when we swim. By contrast, the limited electrical conductivity of available fluids makes it difficult to excite flows with high magnetic Reynolds numbers in terrestrial laboratories.

Nature routinely produces MHD turbulence. Ionized gas pervades the regions between and within galaxies and inside stars. On large scales even modest velocities imply enormous hydrodynamic and magnetic Reynolds numbers. In the absence of stable stratification, these flows must be turbulent with frozen in magnetic fields. It is likely that most of the baryonic matter in the universe is in a state of MHD turbulence.

The evolution of a weak magnetic field in isotropic MHD turbulence poses an interesting problem. By stretching magnetic field lines, turbulence tends to increase magnetic energy, but by creating small scale structures, it enhances the rate of ohmic dissipation. Evidence from analytic models (Kazantsev 1968) and numerical simulations implies that the magnetic energy increases, but the level at which it saturates and the dependence on magnetic Prandtl number of the scale at which it peaks have yet to be firmly established. These issues fall within the purview of dynamo theory (Brandenburg & Subramanian 2005). Some of the recent experiments on dynamo are very exciting, and they provide important clues for understanding dynamo.

A short article cannot adequately cover all aspects of MHD turbulence. The choice made here is to focus on the inertial range of turbulent cascades in incompressible MHD. This is an active topic of current research in which analytical models and numerical simulations play leading roles. Guidance is provided by in situ measurements of velocity and magnetic field fluctuations in the solar wind (Goldstein, Roberts & Matthaeus 1995), and indirectly by scintillations of small angular diameter radio sources which reveal the spectra of interplanetary and interstellar electron density fluctuations (Rickett 1990). On the scales of interest, both interplanetary and interstellar plasmas are collisionless. Additional astrophysical implications of MHD turbulence are described in Cho, Lazarian & Vishniac (2003).

Incompressible MHD Turbulence

<label>sec:general</label>

Incompressible MHD is described by two solenoidal vector fields, the velocity, \(\mathbf v\), and the magnetic intensity, \(\mathbf B\). These must simultaneously satisfy the Navier Stokes and induction equations. The addition of the induction equation makes MHD turbulence a much richer problem than HD turbulence, offering a greater variety of solutions. One feature that greatly helps analyze MHD equations is the existence of linear MHD waves called Alfvén waves. In fact, the incompressible MHD equations given below are conveniently described in terms of Elsasser variables \( {\mathbf z^{\pm} = \mathbf u \pm \mathbf b} \), that are motivated by Alfvén waves: <math eq:motionzpm> \frac{\partial {\mathbf z^{\pm}}}{\partial t}-\left(\mathbf {V}_A\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},  

</math> with a constraint that \( \nabla \cdot {\mathbf z^{\pm}} = 0 \). Here \( \nu_{\pm} = \nu \pm \eta \), where \( \nu \) is the kinematic viscosity, and \( \eta \) is the magnetic diffusivity. \( V_A = B_0/\sqrt{4 \pi \rho} \) where \( {\mathbf B_0} \) is the mean magnetic field, and \( \rho \) is the density. \( p \) represent the total pressure that can be determined using

<math eq:solvep>

\nabla^2 p=- \partial_i \partial_j z^+_i z^-_j . </math>

Alfvén waves \( {\mathbf z^{\pm}} \) are the exact solution of the linearized MHD equations in the limit of vanishing viscosity and resistivity. Also note that the nonlinearity vanishes when either \( z^+ \) or \( z^- \) vanishes. The nonlinear term introduces interactions among these modes that leads to turbulence.

The Alfvén waves are of two types, referred to as shear and pseudo Alfvén waves. The latter is the incompressible vestige of the slow mode of compressible MHD. The displacement vector of a shear Alfvén wave is perpendicular to the plane defined by its wave vector, \({\mathbf k}\), and a uniform background magnetic field, \({\mathbf B}_0\), whereas that of a pseudo Alfvén wave lies in this plane. The two wave modes share the dispersion relation,

<math eq:disp>

\omega^2=\frac{({\mathbf k}\cdot{\mathbf B}_0)^2}{4\pi\rho}\equiv (k_\parallel V_A)^2\, , </math> and propagate with group velocity, \({\mathbf v}_A\), either parallel or antiparallel to \({\mathbf B}_0\) depending upon the sign of \(k_\parallel\).

In three dimensions, the conserved quantities of MHD under vanishing viscosity and resistivity are total energy

<math eq:conservE>

E=\frac{1}{2}\int d^3x\left(v^2 +b^2\right)=\frac{1}{4}\int d^3x\left((z^+)^2+ (z^-)^2\right) , </math> cross helicity

<math eq:conservH>

H=\int d^3x\,{\mathbf v}\cdot{\mathbf b}=\frac{1}{4}\int d^3x\,\left((z^+)^2- (z^-)^2\right)\, , </math> and magnetic helicity

<math eq:conservH>

H_M =\int d^3x\,{\mathbf a}\cdot{\mathbf b} , </math> where \( {\mathbf a} \) is the vector potential. This article focusses on energy and cross helicity.

At large Reynolds number, the magnetic and velocity field becomes turbulent. The mean magnetic field plays an important role in 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 these models can be found in the book by Biskamp (2003) and review article by Verma (2004). In the first part we will summarize the isotropic MHD turbulence models.

Isotropic MHD turbulence models

Iroshnikov (1964) and Kraichnan (1965) 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 the opposite directions with the phase velocity of \( V_A \), and interact weakly. The relevant time scale is Alfvén time \( (V_A k)^{-1} \); as a results the energy spectra is

<math eq:Kraichnan>

E^u(k) \approx E^b(k) \approx A (\Pi V_A)^{1/2} k^{-3/2}. </math> where \( \Pi \) is the energy cascade rate.

Later Dobrowolny et al. (1980) derived the following generalized formulas for the cascade rates of \( z^{\pm} \) variables:

<math eq:Dobrowolny>

\Pi^+ \approx \Pi^{-} \approx \tau^{\pm}_k E^{+}(k) E^{-}(k) k^4 \approx E^{+}(k) E^{-}(k) k^3 / V_A </math> where \( \tau^{\pm} \) are the interaction time scales of \( z^{\pm} \) variables, and \( \Pi^{\pm} \) are the energy cascade rates of \( z^{\pm} \). Iroshnikov, Kraichnan, and Dobrowolny et al. chose \( \tau^{\pm} \approx 1/(k V_A)\) and predicted the energy spectra of MHD turbulence to be proportional to \( k^{-3/2} \). In this article we refer to these phenomenology as KID's phenomenology in short.

Marsch (1990) chose the nonlinear time scale \( T_{NL}^{\pm} \approx (k z_k^{\mp})^{-1} \) as the interaction time scale for the eddies \( z_k^{\pm} \) and derived Kolmogorov-like energy spectrum for the Elsasser variables

<math eq:Marsch>

E^{\pm}(k) = K^{\pm} (\Pi^{\pm})^{4/3} (\Pi^{\mp})^{-2/3} k^{-5/3} </math> where \( K^{\pm} \) are constants. In Marsch's theory \( \Pi^+ \ne \Pi^- \) when \( E^+(k) \ne E^-(k) \) or the cross helicity is nonzero. In recent literature it is also referred to as imbalanced cascade. The occurrence of \( E^+ \ne E^- \) or \( \Pi^+ \ne \Pi^- \) is quite common in the solar wind plasma. Matthaeus and Zhou (1989) 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 is that KID's phenomenology should work for strong mean magnetic field (weak turbulence), 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 (1999) by showing that the Alfvenic fluctuations are affected by scale-dependent "local mean magnetic field". The local mean magnetic field scales as \( k^{-1/3} \), substitution of which in equation (<ref>eq:Kraichnan</ref>) yields Kolmogorov's energy spectrum for MHD turbulence. This analytic calculation was based on renormalization group calculations.

Verma (2001, 2004) also performed renormalization group analysis 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 calculations have been performed for both zero and nonzero cross helicity. When cross helicity is nonzero, the energy cascade rates \( \Pi^+ \) and \( \Pi^- \) are unequal; the scaling laws used in this theory is similar to those used by Marsch. The constants \( K^{\pm} \) have been calculated by field theoretic techniques (Verma 2004).

The mean magnetic field brings in anisotropu in MHD turbulence. In the following we will discuss anisotropic MHD turbulence.

Anisotropic MHD turbulence models

Shebalin, Matthaeus & Montgomery (1983) proved that the assumption of isotropy in the presence of a background magnetic field is inconsistent with the frequency and wave vector closure relations of resonant triads;

<math eq:closeom>

\omega_1+\omega_2 = \omega_3\, </math>

<math eq:closek>

{\mathbf k}_1+{\mathbf k}_2 = {\mathbf k}_3. </math> Their argument is as follows. Since \(\omega=v_A|k_\parallel|\), equation (<ref>eq:closeom</ref>) and the \(\parallel\) component of equation (<ref>eq:closek</ref>) yield

<math eq:closeomp>

|k_{1\parallel}|+|k_{2\parallel}| = |k_{3\parallel}|\, </math>

<math eq:closekp>

k_{1\parallel}+k_{2\parallel} = k_{3\parallel}\, </math> Because nonlinear interactions are restricted to waves which propagate in opposite directions, a nonzero 3-mode coupling coefficient requires that \(k_{1\parallel}\) and \(k_{2\parallel}\) have opposite signs. In this case, adding equations (<ref>eq:closeomp</ref>) and (<ref>eq:closekp</ref>) proves that either \(k_{1\parallel}\) or \(k_{2\parallel}\) must vanish. Since one incoming wave has zero frequency, 3-wave interactions do not transfer energy along \(k_\parallel\). More generally, numerical simulations with isotropic excitation on large scales produce small scale fluctuations that are elongated parallel to the local mean magnetic field (e.g., Cho & Vishniac 2000).

In the anisotropic turbulence the inertial range cascades are characterized by one-dimensional (1D) energy spectra, \(E(k_\perp)\), or equivalently, by velocity or magnetic field differences across separation \(\lambda\sim 1/k_\perp\). These are related by \(v_\lambda^2\sim b_\lambda^2\sim k_\perp E(k_\perp)\). The choice of \(k_\perp\) or \(\lambda\) as an independent variable is made because MHD cascades are anisotropic; gradients are steeper in directions perpendicular to the background magnetic field than they are parallel to it. Where needed, the parallel extent of a wave packet with perpendicular dimension \(\lambda\) is denoted by \(\Lambda(\lambda)\). A convenient measure of nonlinearity is the ratio of the nonlinear strain rate, \(v_\lambda/\lambda\), to the linear wave frequency, \(v_A/\Lambda\);

<math eq:defChi>

\chi_\lambda\equiv \frac{v_\lambda \Lambda}{v_A \lambda}\, . </math>

The concept of the background magnetic field in MHD turbulence requires elaboration. Nonlinear interactions involving fluctuations of scale \(\lambda\) are referred to a background field obtained by averaging the magnetic field over a scale a few times larger than \(\lambda\). This is referred to as the local mean magnetic field on scale \(\lambda\). Although somewhat ad hoc, this practice proves crucial for relating predictions of analytic models with results from numerical simulations (Cho & Vishniac 2000, Maron & Goldreich 2001, Cho, Vishniac & Lazarian 2002). It is not known whether there are examples of inertial range MHD turbulence that don't involve an effectively uniform background magnetic field.

In anisotropic energy cascade, the energy is transferred more rapidly along \(k_\perp\) than along \(k_\parallel\). Wave packets are distorted as they move along magnetic field lines perturbed by counter propagating waves.

Figure 1: Wave packet distortion through field-line wander. Left: Sample of field lines perturbed by downward-propagating waves. Right: Distortion of an originally circular bulls-eye pattern as it moves upward following these field lines. Reproduced with permission of the AAS from Maron & Goldreich (2001), ApJ, 554, 1175.

Field lines perturbed by unidirectional waves define a two dimensional

mapping between planes oriented perpendicular to the guide field and separated along it. Shear Alfvén waves are responsible for the mapping's shear and pseudo Alfvén waves for its dilatation. Once the anisotropy becomes pronounced, the shear exceeds the dilatation (Goldreich & Sridhar 1997). This accounts for the dominance of shear Alfvén waves in controlling the cascades of both shear and pseudo Alfvén waves. The latter play a passive role (Maron & Goldreich 2001).

The applicability of MHD turbulence to collisionless plasmas is made plausible by the dominance of shear Alfven waves (Schekochihin, Cowley & Dorland 2007). Unlike compressive MHD waves, shear Alfvén waves do not suffer linear damping due to interactions with resonant particles (Barnes 1966).

Weak anisotropic MHD turbulence

In weak MHD turbulence limit, the strong mean magnetic field dominates the fluctuations, and the Alfven waves interact very weakly. In isotropic situations, KID models predict the energy spectrum to be proportional to \( k^{-5/3} \). The mean magnetic field however changes the dynamics drastically. Ng and Bhattacharjee (1997) provided a phenomenological theory, while Galtier et al. (2000, 2002) studied this regime using kinetic equation. They showed that the energy cascade is perpendicular to the mean magnetic field. An important result of these calculations is that the three-wave interaction is nonzero in this regime. The energy spectrum in this regime is proportional to \( k_{\perp}^{-2} \). The nonlinearity parameter increases toward smaller scales, \(\chi_\lambda\propto \lambda^{-1/2}\); if not terminated by dissipation, the weak balanced cascade becomes strong.

When \( \Pi^+ \ne \Pi^- \) (nonzero cross helicity or imbalanced cascade), the scaling arguments merely constrain the sum of the indices of the individual 1D energy spectra to total 4. Kinetic equations derived from perturbation theory show that the 1D energy spectrum of the dominant waves is steeper than that of the subdominant ones; in the limit of infinite flux ratio, the indices are 3 and 1 (Galtier et al. 2000, 2002). Provided dissipation terminates the cascade before it becomes strong, the energy densities of the oppositely directed waves reach equality at the dissipation scale (Grappin, Leorat & Pouquet 1983, Lithwick & Goldreich 2003). This has the remarkable consequence that in the approach to a steady-state, there is feedback from the dissipation scale to the outer scale.

Strong anisotropic MHD turbulence

Strong cascades are of particular interest for applications in natural settings because weak ones strengthen with decreasing \(\lambda\). Analytical tools are of limited value when applied to strong cascades. Scaling arguments combined with numerical simulations have led to considerable progress but crucial issues remain unresolved.

An undamped weak cascade transitions to a strong one as \(\chi_\lambda\) approaches unity. Recall that for the weak balanced cascade \(\chi_\lambda\propto \lambda^{-1/2}\). The hypothesis of critical balance states that \(\chi_\lambda\) is of order unity throughout the inertial range of the strong cascade (Goldreich & Sridhar 1995). In other words, in a strong balanced cascade, MHD waves suffer order unity distortions on time scales comparable to their periods. Two arguments have been made for the saturation of \(\chi_\lambda\).

• 1) The frequency closure relation, equation (<ref>eq:closeom</ref>), cannot be satisfied to better than the cascade rate. Thus strong interactions must be accompanied by an increase in frequency and hence a decrease in the parallel scale \(\Lambda\).

• 2) If \(\chi_\lambda\) were larger than unity, then motions with perpendicular scale \(\lambda\) on planes separated by parallel scale \(\Lambda\) would proceed faster than Alfvén waves could couple them. Thus these distortions would proceed independently causing \(\Lambda\) to decrease.

Critical balance based on the cascade rate \(v_\lambda/\lambda\) leads to an anisotropic cascade in which excited modes are confined inside a cone bounded by \(k_\parallel\propto k_\perp^{2/3}\); this scaling was anticipated by Higdon (1984). The cone's opening angle, \(\theta(k_\perp)\propto k_\perp^{-1/3}\), defines a scale dependent anisotropy that increases with \(k_\perp\). The 1D energy spectrum mimics the Kolmogorov spectrum of incompressible HD turbulence, \(E(k_\perp)\propto k_\perp^{-5/3}\). Although the turbulence is strong in the sense that the MHD waves are critically damped, deep in the inertial range the energy in turbulent fluctuations is smaller than that in the background magnetic field by the factor \(\theta(k_\perp)^2\ll 1\).

The above arguments for anisotropic MHD turbulence have been generalized to nonzero cross helicity or imbalanced cascade by Lithwick, Goldreich & Sridhar (2006). The scaling laws used have resemblance to those used by Marsch (1990) and Verma (2001, 2003, 2004).

Numerical Simulations of 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 (e.g., Müller & Biskamp, 2000). There are others that report the spectral indices near 3/2 (e.g., Maron & Goldreich 2001). 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.

Verma et al. (1996) showed that the energy fluxes \( \Pi^{\pm} \) are more reliable quantities to validate MHD turbulence models. When \( E^+(k) \gg E^-(k) \) (imbalanced cascade or high cross helicity), the energy flux predictions of KID models (equation (<ref>eq:Dobrowolny</ref>)) is very different from that of Kolmogorov-like model (equation (<ref>eq:Marsch</ref>)). Verma et al. (1996) showed that the fluxes \( \Pi^{\pm} \) computed from the numerical simulations are in better agreement with Kolmogorov-like model compared to KID's model.

Anisotropic aspects of MHD turbulence have also been studied using numerical simulations. The critical balance predictions of Goldreich and Sridhar (\( k_{||} \sim k_{\perp}^{2/3} \)) has been verified in many simulations (Cho & Vishniac 2000, Maron and Goldreich 2001, Cho, Lazarian & Vishniac 2002). In another set of simulations, Mason, Cataneo & Boldyrev (2006) report dynamic alignment of velocity and magnetic field fluctuations in the inertial range, and \( k^{-3/2} \) energy spectra.

Solar wind observations

Solar wind plasma is in turbulent state. Researchers have calculated the energy spectra of the solar wind plasma from the data collected by the spacecrafts. The kinetic and magnetic energy spectra, as well as \( E^{\pm} \) are closer to \( k^{-5/3} \) compared to \( k^{-3/2} \), thus favoring Kolmogorov-like phenomenology for MHD turbulence (Matthaeus & Goldstein 1982, Goldstein, Roberts & Matthaeus 1995). The Kolmogorov-like spectrum for \( E^{\pm} \) have been observed even when cross helicity is very high (e.g., Marsch & Tu, 1990). There are recent attempts to test the validity of Boldyrev's (2005) predictions using solar wind observations.

The interplanetary and interstellar electron density fluctuations also provide a window for investigating MHD turbulence.

Energy transfers in MHD turbulence

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 (e.g., Dar et al. 2001, Verma 2004). 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. Most of the energy transfers have also been found to be local in wavenumber space consistent with Kolmogorov's theory of turbulence. These results have critical bearing on dynamo problem.

In summary, there are many open challenges in this field that hopefully will be resolved in near future. Many related active areas of research that have not been included in this short review are intermittency, dynamo, magnetic helicity, etc. These topics are/will be being covered by others in scholarpedia.

References

Beresnyak, A. & Lazarian, A. (2006), Polarization Intermittency and Its Influence on MHD Turbulence, Astrophysical Journal, 640, L175-L178. ApJ

Beresnyak, A. & Lazarian, A. (2008), Strong Imbalanced Turbulence, arXiv:0709.0554v2. astro-ph

Boldyrev, S. (2005), On the Spectrum of Magnetohydrodynamic Turbulence, Astrophysical Journal, 626, L37-L40, ApJ

Boldyrev, S. (2006), Spectrum of Magnetohydrodynamic Turbulence, Physical Review Letters, 96, 115002 (1-4). astro-ph

Chandran, B. D. G. (2008), Strong Anisotropic MHD Turbulence with Cross Helicity, arXiv:0801.4903. astro-ph

Cho, J. & Vishniac, E. T. (2000), The Anisotropy of Magnetohydrodynamic Alfvénic Turbulence, Astrophysical Journal, 539, 273-282. ApJ

Cho, J., Lazarian, A. & Vishniac, E. T. (2002), Simulations of Magnetohydrodynamic Turbulence in a Strongly Magnetized Medium, Astrophysical Journal, 564, 291-301. ApJ

Cho, J., Lazarian, A. & Vishniac, E. T. (2003), MHD Turbulence: Scaling Laws and Astrophysical Implications, Lecture Notes in Physics, 614, 56-98. ADS

Dar, G., Verma, M. K. & Eswaran, V. (2001), Energy transfer in two-dimensional magnetohydrodynamic turbulence: formalism and numerical results, Physica D, 157, 207.

Dobrowlny, M., Mangeney, A. & Veltri. P. (1980), Fully developed anisotropic hydromagnetic turbulence in interplanetary plasma, Phys. Rev. Lett. 45, 144.

Galtier, S., Nazarenko, S. V., Newell, A. C. & Pouquet, A. (2000), A weak turbulence theory for incompressible magnetohydrodynamics, Journal of Plasma Physics, 63, 447-488. astro-ph

Galtier, S., Nazarenko, S. V., Newell, A. C. & Pouquet, A. (2002), Anisotropic Turbulence of Shear-Alfvén Waves, Astrophysical Jounal, 564, L49-L52. ApJ

Goldreich, P. & Sridhar, S. (1995), Toward a theory of interstellar turbulence. 2: Strong Alfvénic turbulence, Astrophysical Journal, 438, 763-775. ApJ

Goldreich, P. & Sridhar, S. (1997), Magnetohydrodynamic Turbulence Revisited, Astrophysical Jounal, 485, 680-688.ApJ

Grappin, R., Leorat, J. & Pouquet, A. (1983), Dependence of MHD turbulence spectra on the velocity field-magnetic field correlation, Astronomy & Astrophysics, 126, 51-58. ADS

Iroshnikov, P. S. (1964), Turbulence of a Conducting Fluid in a Strong Magnetic Field, Soviet Astronomy, 7, 566-571. Translated from Astronomicheskii Zhurnal, 40, 742-750, (1963). ADS

Kraichnan, R. (1965), Inertial-Range Spectrum of Hydromagnetic Turbulence, Physics of Fluids, 8, 1385-1387. Sci

Lithwick, Y. & Goldreich, P. (2003), Imbalanced Weak Magnetohydrodynamic Turbulence, Astrophysical Jounal, 582, 1220–1240. ApJ

Lithwick, Y., Goldreich, P. & Sridhar, S. (2006), Imbalanced Strong MHD Turbulence, Astrophysical Jounal, 655, 269-274. ApJ

Maron, J. & Goldreich, P. (2001), Simulations of Incompressible Magnetohydrodynamic Turbulence, Astrophysical Journal, 554, 1175-1196. ApJ

Marsch, E. (1990), Turbulence in the solar wind, in Reviews in Modern Astronomy : G. Klare (Ed.), Springer, Berlin, p. 43.

Marsch, E. & Tu, C.-Y. (1990), On the radial evolution of mhd turbulence in the inner heliosphere, J. Geophys. Res., 95, 8211.

Mason, J., Cattaneo, F. & Boldyrev, S. (2008), Numerical measurements of the spectrum in magnetohydrodynamic turbulence, Physical Review E, 77, 036403 (1-4). astro-ph

Matthaeus, W. H. & Goldstein, M. L. (1982), Measurement of the rugged invariants of magnetohydrodynamic turbulence in the solar wind, J. Geophys. Res., 87, 6011.

Matthaeus, W. H. & Zhou, Y. (1989), Extended inertial range phenomenology of magnetohydrodynamic turbulence, Phys. Fluids B, 1, 1929.

Müller, W.-C. & Biskamp, D. (2000), Scaling properties of three-dimensional magnetohydrodynamic turbulence, Phys. Rev. Lett., 84, 475–478.

Müller, W.-C., Biskamp, D. & Grappin, R. (2003), Statistical anisotropy of magnetohydrodynamic turbulence, Physical Review E, 67, 066302 (1-4). Phys. Rev.

Ng, C. S. & Bhattacharjee, A. (1996), Interaction of Shear-Alfvén Wave Packets: Implication for Weak Magnetohydrodynamic Turbulence in Astrophysical Plasmas, Astrophysical Journal, 465, 845-854. ADS

Ng, C. S. & Bhattacharjee, A. (1997), Scaling of anisotropic spectra due to the weak interaction of shear-Alfvén wave packets, Physics of Plasmas, 4, 605-610.

Shebalin, J.V., Matthaeus, W.H. & Montgomery, D. (1983), Anisotropy in MHD turbulence due to a mean magnetic field, Journal of Plasma Physics, 29, 525-547. ADS

Verma, M. K., Roberts, D. A., Goldstein, M. L., Ghosh, S. & Stribling, W. T. (1996), A Numerical Study of Nonlinear Cascade of Energy in Magnetohydrodynamic Turbulence, J. of Geophys. Res., 101, 21619.

Verma, M. K. (1999), Mean magnetic field renormalization and Kolmogorov's energy spectrum in magnetohydrodynamic turbulence, Phys. Plasmas, 6, 1455.

Verma, M. K. (2001), Calculation of renormalized viscosity and resistivity in magnetohydrodynamic turbulence, Phys. Plasmas, 8, 3945.

Verma, M. K. (2004), Statistical theory of magnetohydrodynamic turbulence, Phys. Rep., 401, 229-380.

Recommended reading

Barnes, A. (1966), Collisionless Damping of Hydromagnetic Waves, Physics of Fluids, 9, 1483-1495.

Biskamp, D. (2003), Magnetohydrodynamical Turbulence, (Cambridge University Press, Cambridge).

Brandenburg, A. & Subramanian, K. (2005), Astrophysical magnetic fields and nonlinear dynamo theory, Physics Reports 417, 1-209. ADS

Goldstein, M. L., Roberts, D. A. & Matthaeus, W. H. (1995), Magnetohydrodynamic Turbulence In The Solar Wind, Annual Reviews of Astronomy and Astrophysics, 33, 283-326. ARA&A

Higdon, J.C. (1984), Density fluctuations in the interstellar medium: Evidence for anisotropic magnetogasdynamic turbulence. I - Model and astrophysical sites, 285, 109-123. ApJ

Kazantsev, A.P. (1968), Enhancement of a magnetic field by a conducting fluid, Soviet Physics (JETP), 26, 1031-1034.

Rickett, B. J. (1990), Radio propagation through the turbulent interstellar plasma, Annual Reviews of Astronomy and Astrophysics, 28, 561-605. ARA&A

Schekochihin, A. A., Cowley, S. C. & Dorland, W. (2007), Interplanetary and interstellar plasma turbulence, Plasma Physics and Controlled Fusion, 49, A195-A209. arXiv

See also

<review>

• The MHD equations have been rewritten in terms \( z^{\pm} \); this notation is more popular.
• The isotropic turbulence theories of Dobrowolny et al., Marsch, Verma have been discussed.
• The text rearranged a bit.. We divided the theories in two types: Isotropic and Anisotropic. The anisotropic theories get more prominence like in the original article.
• Imbalanced cascade corresponds to nonzero cross helicity. Cross helicity is commonly used in the community. Marsch, Marsch and Tu, Verma, and others had worked extensively on MHD turbulence for nonzero cross helicity. These work have been included. The text appropriately modified (very briefly)
• A separate sections on numerical results, solar wind, and energy transfers in MHD turbulence have been added. The intention was to highlight the main results in these areas.

</review>

_AUTOLINKER{1|stable|vector fields|averaging |linear wave|kinetic equation}

Sponsored by: Dr. Søren Bertil F. Dorch, University of Copenhagen and The Royal Library, Denmark, Copenhagen, Denmark

Reviewed by: Dr. Stanislav Boldyrev, Department of Physics, The University of Wisconsin, Madison

Reviewed by: Mahendra K. Verma, Indian Institute of Technology Kanpur, Kanpur, India

Accepted on: 2009-02-08 18:45:18 GMT