Chandrasekhar's invariant theory of isotropic magnetohydrodynamic (MHD) turbulence, Part II (Proc. R. Soc. Lond. A 207, 301, 1951), expresses the correlations of the total pressure with pairs of velocity or magnetic-field components in terms of second-order correlations by assuming that fourth-order moments factorize as for a jointly normal (Gaussian) distribution; its reference list includes Alfvén and Fermi, to whom is attributed the conjecture that the velocity and magnetic spectra are identical in stationary MHD turbulence. The open problems left by the paper's Gaussian closure and by the Alfvén–Fermi spectrum-equality conjecture cited in it are: whether real homogeneous isotropic MHD turbulence satisfies the Gaussian factorization and, if not, how the derived integral expressions for the pressure-correlation scalars are modified by nonzero fourth-order cumulants (including their dependence on the assumed decay of correlations at large separations); and what the correct relation between the velocity and magnetic spectra — the residual energy — in stationary MHD turbulence is.

Background

Magnetohydrodynamics (MHD) describes the motion of an electrically conducting, incompressible fluid coupled to a magnetic field through the induction and Lorentz-force terms of the equations of motion. When the turbulence in such a fluid is homogeneous (statistically invariant under translations) and isotropic (statistically invariant under rotations and reflections, with no imposed mean magnetic field), its statistical state is described by two-point correlation tensors between the velocity u\mathbf{u} at one point and the velocity u′\mathbf{u}', magnetic field h′\mathbf{h}', or pressure at another point a distance rr away. Building on Batchelor's MHD equations and Robertson's tensor-reduction method — the systematic reduction of isotropic two-point correlation tensors to a small set of scalar functions of rr — Chandrasekhar developed in 1951 an 'invariant theory' of isotropic MHD turbulence: the kinematic reduction of all two-point double and triple correlations to defining scalars, dynamical equations for those scalars, and Loitsiansky-type invariants — time-conserved integrals of correlation functions over all separations, named after Loitsiansky's hydrodynamic prototype I=∫0∞r4f(r) drI=\int_0^\infty r^4 f(r)\,dr for the longitudinal velocity correlation f(r)f(r), whose conservation requires sufficiently rapid decay of correlations as r→∞r\to\infty.

Part II of the theory (the source paper) extends it to the correlations of the total pressure P=p+12ρ∣h∣2P=p+\tfrac{1}{2}\rho|\mathbf{h}|^2 (hydrodynamic plus magnetic pressure) with two components of the velocity or of the magnetic field at another point. Fourth-order moments of the fields enter these correlations, and Chandrasekhar closes them with the hypothesis that fourth-order correlations are related to second-order correlations exactly as they would be for a jointly normal (Gaussian) distribution — the quasi-normal or zero-fourth-cumulant hypothesis, which Batchelor had used the same year for hydrodynamic pressure fluctuations. Under this hypothesis the scalars defining the pressure–velocity and pressure–magnetic correlation tensors (P1,P2,Π1,Π2P_1, P_2, \Pi_1, \Pi_2 in the paper's notation) and the pressure–pressure correlation PP′‾(r)\overline{P P'}(r) are expressed as integrals of the second-order scalars. The stationary state, in which an external agency supplies kinetic energy at a constant rate, raises a second question, tied to the conjecture attributed to Alfvén and Fermi — both cited in the paper — that the velocity and magnetic spectra are then identical (Q(r)=H(r)Q(r)=H(r) in the source notation), a condition under which the pressure–total-energy correlation would vanish.

In modern language, the second question is the residual-energy problem: the difference between the kinetic and magnetic energy spectra, ER=Ev−EbE^R=E^v-E^b, in MHD turbulence, routinely measured in direct numerical simulations and by spacecraft in the solar wind and modeled by closures such as the eddy-damped quasi-normal Markovian (EDQNM) approximation — a Markovianized, eddy-damped version of the quasi-normal closure in which fourth-order moments are still reduced to second-order ones. Neither the two-point fourth-order statistics nor the spectral partition has a closed, validated theory, so the source paper's questions survive in modern form.

Problem Statement

In homogeneous isotropic incompressible MHD turbulence, determine the two-point fourth-order statistical structure that replaces the Gaussian hypothesis of Chandrasekhar's invariant theory, and the resulting velocity–magnetic energy partition in the stationary state. Concretely: (1) determine whether real homogeneous isotropic MHD turbulence satisfies the quasi-normal (Gaussian) factorization of two-point fourth-order moments of the velocity and magnetic fields used in deriving the pressure-correlation expressions, and, if it does not, how the integral expressions satisfied by the scalars defining the pressure–velocity and pressure–magnetic correlation tensors (P1,P2,Π1,Π2P_1, P_2, \Pi_1, \Pi_2) and the pressure–pressure correlation PP′‾(r)\overline{P P'}(r), as well as the consequences derived under the assumed rapid decay of correlations as r→∞r\to\infty, are modified when the fourth-order cumulants are nonzero; (2) determine the correct relation between the velocity and magnetic spectra in stationary MHD turbulence, testing the Alfvén–Fermi conjecture of equality (Q(r)=H(r)Q(r)=H(r) in the source notation), under which the pressure–total-energy correlation would vanish — in modern terms, establish the sign, scaling, and parameter dependence of the residual energy ER=Ev−EbE^R=E^v-E^b in the statistically stationary state. Both components are linked through the same two-point fourth-order moment structure of the turbulence: part (1) fixes the explicit integral relations for the pressure-correlation scalars, and part (2) the derived velocity–magnetic spectral relation for the sign, scaling and parameter dependence of the residual energy.

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