Magnetohydrodynamic turbulence

Magnetohydrodynamic turbulence is irregular nonlinear motion of an electrically conducting fluid coupled to a magnetic field. Alfvén waves and magnetosonic waves give propagation times in addition to the nonlinear mixing time. A strong guide field distinguishes directions parallel and perpendicular to it, so the resulting cascades need not be isotropic.

Dimensional freedom of an MHD spectrum ( $E (k) = ϵ^{2/3} k^{- 5/3} Φ (v_{A} ϵ^{- 1/3} k^{1/3})$ )

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In a local isotropic hydrodynamic inertial-range model with only

ϵ

and

k

, dimensional analysis fixes the Kolmogorov 1941 theory spectrum. A mean magnetic field supplies the independent Alfvén speed, permitting the arbitrary dimensionless function in the displayed formula even before anisotropy adds parallel scales. Cascade-time assumptions such as weak decorrelated interactions or critical balance are therefore necessary to select an MHD spectrum.

Electron-magnetohydrodynamic cascade

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An electron-magnetohydrodynamic cascade transfers magnetic fluctuations through the nonlinear electron magnetohydrodynamics induction equation. Writing

b_{l} = δ B_{l} / B_{0}

, its perpendicular nonlinear interaction time is

τ_{nl} \sim l_{⊥}^{2} / (v_{A} d_{i} b_{l})

. The anisotropic whistler wave time is

τ_{w} \sim l_{∥} l_{⊥} / (v_{A} d_{i})

. Scaling predictions depend on locality, phase decorrelation and the assumed balance of these times.

Critically balanced electron-magnetohydrodynamic cascade ( $b_{l} \propto l_{⊥}^{2/3}, l_{∥} \propto l_{⊥}^{1/3}$ )

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Critical balance sets

τ_{nl} \sim τ_{w}

, hence

l_{∥} \sim l_{⊥} / b_{l}

. A local strong cascade with constant energy flux has

ϵ \sim v_{A}^{3} d_{i} b_{l}^{3} / l_{⊥}^{2}

, giving the displayed exponents and perpendicular magnetic energy spectrum

E_{B} (k_{⊥}) \propto k_{⊥}^{- 7/3}

. The anisotropy becomes stronger towards small perpendicular scales.

Weak electron-magnetohydrodynamic cascade ( $b_{l} \sim [ϵ l_{⊥}^{3} / (v_{A}^{3} d_{i} l_{∥})]^{1/4}$ )

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For

b_{l} l_{∥} / l_{⊥} ≪ 1

, the weak wave cascade estimate gives

τ_{cas} \sim l_{⊥}^{3} / (v_{A} d_{i} b_{l}^{2} l_{∥})

. Constant energy flux

ϵ \sim v_{A}^{2} b_{l}^{2} / τ_{cas}

yields the displayed amplitude. An additionally isotropic local cascade has

b_{l} \propto l^{1/2}

and shell-integrated magnetic energy spectrum

E_{B} (k) \propto k^{- 2}

. These statements are phenomenological consequences of the chosen locality and decorrelation assumptions.

Critical balance ( $τ_{nl} \sim τ_{wave}$ )

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Critical balance is a scale-dependent ordering in which the wave-propagation time and nonlinear interaction time are comparable. For strong Alfvénic turbulence,

k_{∥} v_{A} \sim k_{⊥} δ u_{⊥}

. For electron magnetohydrodynamics, the linear rate is

v_{A} d_{i} k_{∥} k_{⊥}

and the nonlinear rate is

v_{A} d_{i} k_{⊥}^{2} δ B / B_{0}

, so critical balance instead gives

δ B / B_{0} \sim k_{∥} / k_{⊥}

. This assumption is an estimate of decorrelation dynamics, not an exact equality at every point.

Alfvénic turbulence ( $z_{⊥}^{\pm}$ )

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Alfvénic turbulence is the turbulent interaction of fluctuations polarized primarily as Alfvén waves. In Elsässer variables

z^{\pm} = u_{⊥} \pm δ B_{⊥} / 4 π ρ_{0}

, each wave population is nonlinearly distorted by the oppositely propagating population. Equal populations give a balanced cascade; unequal populations give an imbalanced cascade. The absence of one population removes the leading nonlinear interaction in ideal reduced magnetohydrodynamics.

Weak Alfvénic cascade ( $E_{⊥} (k_{⊥}) \sim (ϵ v_{A} / l_{∥})^{1/2} k_{⊥}^{- 2}$ )

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For balanced anisotropic Alfvénic turbulence, suppose

τ_{A} = l_{∥} / v_{A} ≪ τ_{nl} = l_{⊥} / δz

and resonant interactions preserve the parallel scale during the perpendicular cascade. Random encounter accumulation gives

τ_{cas} = v_{A} l_{⊥}^{2} / (δ z^{2} l_{∥})

. Constant energy flux then gives

δz \propto l_{⊥}^{1/2}

and the displayed perpendicular turbulent energy spectrum. Unlike isotropic phenomenology, the fixed parallel scale is independent of the decreasing perpendicular scale.

Weak-to-strong transition of an Alfvénic cascade ( $l_{⊥, *} \sim L_{⊥} χ_{0}^{2}$ )

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In a weak Alfvénic cascade at fixed

l_{∥}

χ = τ_{A} / τ_{nl} \propto l_{⊥}^{- 1/2}

grows as the perpendicular scale shrinks. If its outer value is

χ_{0} ≪ 1

L_{⊥}

, it reaches order one at the displayed scale. Beyond it the weak-interaction ordering fails and a critical balance model may become appropriate.

Iroshnikov-Kraichnan spectrum ( $E (k) \sim (ϵ v_{A})^{1/2} k^{- 3/2}$ )

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The Iroshnikov-Kraichnan spectrum phenomenology assumes balanced, isotropic local Alfvénic turbulence with

δ z_{l} ≪ v_{A}

, decorrelated counterpropagating encounters,

τ_{A} = l / v_{A}

and

τ_{nl} = l / δ z_{l}

. A weak wave cascade takes

τ_{cas} \sim τ_{nl}^{2} / τ_{A} = v_{A} l / δ z_{l}^{2}

. Constant energy flux gives

δ z_{l}^{4} \sim ϵ v_{A} l

and the displayed shell-integrated turbulent energy spectrum. Isotropy is an additional assumption, not a consequence of a strong guide field.

Goldreich–Sridhar turbulence ( $E (k_{⊥}) \propto k_{⊥}^{- 5/3}$ )

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Goldreich–Sridhar turbulence combines critical balance, balanced counterpropagating Alfvén waves, a local perpendicular cascade, and constant energy flux per unit mass

E

. With

τ_{nl} \sim ℓ_{⊥} / δ u_{ℓ}

and

E \sim δ u_{ℓ}^{2} / τ_{nl}

, one obtains

δ u_{ℓ} \sim (E ℓ_{⊥})^{1/3}

. Defining the one-dimensional perpendicular turbulent energy spectrum by

δ u_{ℓ}^{2} \sim k_{⊥} E (k_{⊥})

gives

E (k_{⊥}) \sim E^{2/3} k_{⊥}^{- 5/3}

. Critical balance also gives

k_{∥} \sim E^{1/3} k_{⊥}^{2/3} / v_{A}

. This scaling argument neglects scale-dependent alignment and intermittency corrections, which are additional physical effects rather than consequences of the stated assumptions.

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Magnetohydrodynamic (MHD) turbulence is a complex field of study that combines aspects of fluid dynamics and magnetohydrodynamics, which is the behavior of electrically conducting fluids in the presence of a magnetic field. MHD turbulence is particularly relevant in astrophysical contexts, such as in the behavior of plasmas in stars, galaxies, and interstellar space, as well as in industrial processes involving liquid metals and other conducting fluids.

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