# MHD turbulence

Curator and Contributors

1.00 - Peter Goldreich

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).

A short article cannot adequately cover all aspects of MHD turbulence. Verma (2004) provides a more detailed review. 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

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. It might seem that the addition of the induction equation would make MHD turbulence a much harder problem than HD turbulence. It is certainly richer, offering a greater variety of solutions than HD turbulence. However, in some ways it is also simpler to analyze. Unlike HD turbulence, MHD turbulence has a weak limit which is amenable to perturbation theory. This is a consequence of the existence of linear MHD waves. In the incompressible limit, these 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, $\tag{1} \omega^2=\frac{({\mathbf k}\cdot{\mathbf B}_0)^2}{4\pi\rho}\equiv (k_\parallel v_A)^2\, ,$

and propagate with group velocity, $${\mathbf v}_A\ ,$$ either parallel or antiparallel to $${\mathbf B}_0$$ depending upon the sign of $$k_\parallel\ .$$

Three properties of incompressible MHD form the bulwark of current models of MHD turbulence. In the limit of vanishing viscosity and resistivity:

• Wavepackets which propagate in one direction along the magnetic field are exact solutions of the nonlinear equations of MHD.
• Collisions between oppositely directed wave packets cause distortions that give rise to the turbulent cascade.
• Wavepackets do not exchange energy when they collide.

The first and second items are immediate consequences of the equations of ideal MHD written in terms of Elsässer variables, $${\mathbf z^{\pm} = \mathbf v \pm \mathbf b} \ ,$$ where $${\mathbf b}$$ is the magnetic fluctuation about a uniform background field expressed in velocity units according to $${\mathbf b}\equiv \Delta{\mathbf B}/(4\pi\rho)^{1/2}\ .$$ $$\tag{2} \frac{\partial {\mathbf z^{\pm}}}{\partial t}\mp\left(\mathbf {v}_A\cdot{\mathbf \nabla}\right){\mathbf z^{\pm}} = -\left({\mathbf z^{\mp}}\cdot{\mathbf \nabla}\right){\mathbf z^{\pm}} -{\mathbf \nabla}p\, ,$$

with $\tag{3} \nabla^2 p=- \partial_i \partial_j z^+_i z^-_j\, .$

The third is a consequence of the separate conservation of energy $\tag{4} E=\frac{1}{2}\int d^3x\left(v^2 +b^2\right)=\frac{1}{4}\int d^3x\,\left[(z^+)^2+ (z^-)^2\right]\, ,$

and cross helicity $\tag{5} H=\int d^3x\,{\mathbf v}\cdot{\mathbf b}=\frac{1}{4}\int d^3x\,\left[(z^+)^2- (z^-)^2\right]\, .$

In this article inertial range cascades are characterized by one-dimensional (1D) energy spectra, $$E(k_\perp)\ ,$$ or equivalently, by rms 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, $$v_A/\Lambda\ ;$$ $\tag{6} \chi_\lambda\equiv \frac{v_\lambda \Lambda}{v_A \lambda}\, .$

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.

Iroshnikov (1963) and Kraichnan (1965) pioneered the application of the above properties together with the assumption of spatial isotropy, $$\Lambda\sim \lambda\ ,$$ to derive a 1D inertial range spectrum, $$E(k)\propto k^{-3/2}$$ or $$v_\lambda\propto \lambda^{1/4}\ ,$$ for MHD turbulence. They explicitly assumed the presence of a local mean magnetic field. IK's cascade is an example of weak turbulence because wave packets undergo multiple collisions before being significantly deformed. The nonlinearity of the IK cascade weakens toward smaller scales, $$\chi_\lambda\propto\lambda^{1/4}\ ,$$ so its inertial range could be arbitrary long. Only dissipation would terminate it. The IK model has since been superseded by others which do not assume spatial isotropy. Otherwise, the principles upon which it was based have endured.

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; $\tag{7} \omega_1+\omega_2 = \omega_3\,$

$\tag{8} {\mathbf k}_1+{\mathbf k}_2 = {\mathbf k}_3.$

Their argument is as follows. Since $$\omega=v_A|k_\parallel|\ ,$$ equation (7) and the $$\parallel$$ component of equation (8) yield $\tag{9} |k_{1\parallel}|+|k_{2\parallel}| = |k_{3\parallel}|\,$

$\tag{10} k_{1\parallel}+k_{2\parallel} = k_{3\parallel}\,$

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 (9) and (10) 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).

Current knowledge about steady-state turbulent MHD cascades is summarized below. These cascades are anisotropic; 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).

Cascades with equal fluxes of energy directed parallel and antiparallel to the ambient magnetic field are called balanced. They are simpler both to analyze and to describe than imbalanced cascades. Scaling arguments, which go a long way toward characterizing balanced cascades, are less potent when applied to imbalanced ones.

Weak MHD turbulence is well-characterized. Three wave interactions are dominant (Ng & Bhattacharjee 1996). There is no cascade along $$k_\parallel\ .$$ Individual waves survive for many collisions; this is why the turbulence is called weak. Weak cascades strengthen toward larger $$k_\perp\ .$$ In the absence of dissipation, they ultimately become strong.

The principal features of weak, balanced cascades follow from scaling arguments (Goldreich & Sridhar 1997, Ng & Bhattacharjee 1997). The 1D energy spectrum is a power law, $$E(k_\perp)\propto k_\perp^{-2}$$ or $$v_\lambda\propto \lambda^{1/2}\ .$$ The nonlinearity parameter increases toward smaller scales, $$\chi_\lambda\propto \lambda^{-1/2}\ ;$$ if not terminated by dissipation, the weak balanced cascade becomes strong.

Perturbation theory is required to determine the structure of weak imbalance cascades. Individual 1D energy spectra of oppositely directed waves are power laws in $$k_\perp\ .$$ 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 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 (7), 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 eddies with perpendicular scale $$\lambda$$ extend a distance $$\Lambda\propto \lambda^{2/3}$$ along the local mean magnetic field; the anisotropy increases with decreasing scale. This scaling was anticipated by Higdon (1984). 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 $$(\lambda/\Lambda)^2<<1\ .$$ The 1D energy spectrum mimics the Kolmogorov spectrum of incompressible HD turbulence, $$E(k_\perp)\propto k_\perp^{-5/3}\ .$$

Numerical simulations offer support for critical balance, scale dependent anisotropy, and the dominance of shear Alfvén waves (Cho & Vishniac 2000, Maron & Goldreich 2001, Cho, Lazarian & Vishniac 2002). However, in the presence of a strong guide field they yield energy spectra $$E(k_\perp)\propto k_\perp^{-\alpha}$$ with $$\alpha$$ closer to 3/2 than 5/3 (Maron & Goldreich 2001, Müller, Biskamp & Grappin 2003, Müller & Grappin 2005, Mason, Cattaneo & Boldyrev 2008). The flattening of the energy spectrum implies a weakening with decreasing scale of the nonlinear interactions that drive the casade. Boldyrev (2005, 2006) proposed that the nonlinear interactions weaken because of dynamic alignment, the progressive alignment or anti-alignment of velocity and magnetic field fluctuations. For $\tag{11} \delta=cos^{-1}\left(\frac{2|{\mathbf v}\cdot{\mathbf b}|}{v^2+b^2}\right)\ll 1\, ,$

the nonlinear interaction rate weakens by a factor $$\delta\ .$$ Since $\tag{12} \frac{2\left|{\mathbf v}\cdot{\mathbf b}\right|}{v^2+b^2}=\frac{\left| (z^+)^2- (z^-)^2\right|}{ (z^+)^2+ (z^-)^2}\, ,$

dynamic alignment is equivalent to a local imbalance that increases with decreasing scale. $$E(k_\perp)\propto k_\perp^{-3/2}\ ,$$ or equivalently, $$v_\lambda\propto \lambda^{1/4}$$ follows provided $$\delta\propto \lambda^{1/4}\ .$$ Dynamic alignment at about this level has been observed in a variety of numerical simulations (Beresnyak & Lazarian 2006, Mason, Cattaneo & Boldyrev 2008).

Strong imbalance turbulence is expected to be the dominant form of MHD turbulence in nature. Turbulence in the solar wind is both strong and imbalanced. A few models of strong imbalanced turbulence have been proposed (Lithwick, Goldreich & Sridhar 2006, Beresnyak & Lazarian 2008, Chandran 2008, Perez & Boldyrev 2009). It is too early to critically assess them given that critical issues involving strong balanced cascades remain unresolved. Nevertheless, four key questions are worth raising.

• Do the dominant and subdominant modes have the same spectral slopes?
• Does critical balance apply to either or both of the dominant and subdominant fluxes?
• Are the energy densities of the dominant and subdominant modes pinned at the dissipation scale?
• What role does dynamic alignment play?

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• Giovanni Gallavotti (2008) Fluctuations. Scholarpedia, 3(6):5893.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

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