Regime of validity and mode selection in Chandrasekhar's elementary theory of hydromagnetic turbulence: finite-dissipation analysis of the coupled cascade equations, closure justification, and correlation boundary conditions

Chandrasekhar's 1955 elementary theory of hydromagnetic (magnetohydrodynamic) turbulence extended Heisenberg's 1948 eddy-viscosity cascade to conducting fluids by introducing, alongside the eddy viscosity, an eddy resistivity and modeled cross-transfers between kinetic and magnetic energy. In the stationary, zero-viscosity, infinite-conductivity limit, the resulting cascade equations admit two distinct solution families: a velocity mode in which the ratio of magnetic to kinetic spectral energy tends to zero at high wavenumber, and a magnetic mode in which it tends to about 2.6, both with equipartition at the largest scales. Four questions jointly delimiting the theory's regime of validity were left open and remain so: (1) the analysis of the coupled equations with nonzero molecular viscosity and resistivity, and the mechanism that selects among the modes, including whether the double zero-dissipation limit is path dependent; (2) the justification of the phenomenological closures from the exact spectral-transfer terms of the magnetohydrodynamic equations, or their refutation; (3) the formulation of unambiguous boundary conditions for the coupled longitudinal correlation functions underlying the deductive theory, excluding solutions manufactured by reduction to differential form; and (4) the consequences for the interstellar magnetic field under realistic dissipation and forcing.

Background

Hydromagnetic turbulence, now called magnetohydrodynamic or MHD turbulence, is the turbulent flow of an electrically conducting fluid in which the velocity field and the magnetic field are dynamically coupled. A standard description uses the kinetic and magnetic energy spectrum functions $F(k)$ and $G(k)$, giving the energy contained in Fourier modes of wavenumber magnitude $k$. Heisenberg's 1948 elementary theory of ordinary turbulence modeled the nonlinear transfer of energy toward smaller eddies by an eddy viscosity $\nu(k)=\kappa\sqrt{F(k)/k^{3}}$, where $\kappa$ is a dimensionless constant: the smaller eddies act on the larger ones like an effective dissipation. Chandrasekhar's 1951 invariant-theory papers and his two 1955 hydromagnetic papers transplanted this program to MHD turbulence. The first 1955 paper (a deductive theory) derived exact equations for the two-point longitudinal correlation functions of the velocity and magnetic fields (the correlations, as functions of the separation between two measurement points, of the field components along the line joining the two points). The second 1955 paper (the source of this problem, Proc. R. Soc. Lond. A 233, 330) developed the elementary theory: it introduced an eddy resistivity $\lambda(k)=\kappa\sqrt{G(k)/k^{3}}$ for magnetic diffusion, modeled the cross-transfers between kinetic and magnetic energy, and obtained the coupled cascade equations (17) and (18) of that paper for $F(k)$ and $G(k)$.

The modeled cross-transfers are the factorized transition-probability densities

$$Q(v,k';v,k'')=\kappa\,F(k')\,k'^{2}\sqrt{\frac{F(k'')}{k''^{3}}},\qquad Q(h,k';v,k'')=\kappa\,G(k')\,k'^{2}\sqrt{\frac{G(k'')}{k''^{3}}},$$ $$Q(v,k';h,k'')=\kappa\,G(k')\,k'^{2}\sqrt{\frac{F(k'')}{k''^{3}}}+\kappa\,F(k')\,k'^{2}\sqrt{\frac{G(k'')}{k''^{3}}},$$

where $Q(v,k';v,k'')$ is the kinetic-to-kinetic transfer rate, $Q(h,k';v,k'')$ the magnetic-to-kinetic transfer rate, and $Q(v,k';h,k'')$ the kinetic-to-magnetic transfer rate. Here and below, symbols with an argument, $\nu(k)$ and $\lambda(k)$, are the $k$-dependent eddy coefficients, whereas the argument-free $\nu$ and $\lambda$ are the molecular kinematic viscosity and molecular magnetic diffusivity (electrical resistivity); with the latter as the direct-dissipation coefficients, the resulting coupled cascade equations (17) and (18) are

$$\frac{1}{2}\frac{\partial F(k)}{\partial t}=\kappa\sqrt{\frac{F(k)}{k^{3}}}\int_{0}^{k}F(k')\,k'^{2}\,\mathrm{d}k'+\kappa\sqrt{\frac{G(k)}{k^{3}}}\int_{0}^{k}G(k')\,k'^{2}\,\mathrm{d}k'-\kappa F(k)\,k^{2}\int_{k}^{\infty}\sqrt{\frac{F(k'')}{k''^{3}}}\,\mathrm{d}k''-\kappa G(k)\,k^{2}\int_{k}^{\infty}\sqrt{\frac{F(k'')}{k''^{3}}}\,\mathrm{d}k''-\kappa F(k)\,k^{2}\int_{k}^{\infty}\sqrt{\frac{G(k'')}{k''^{3}}}\,\mathrm{d}k''-\nu F(k)\,k^{2},$$ $$\frac{1}{2}\frac{\partial G(k)}{\partial t}=\kappa\sqrt{\frac{F(k)}{k^{3}}}\int_{0}^{k}G(k')\,k'^{2}\,\mathrm{d}k'+\kappa\sqrt{\frac{G(k)}{k^{3}}}\int_{0}^{k}F(k')\,k'^{2}\,\mathrm{d}k'-\kappa G(k)\,k^{2}\int_{k}^{\infty}\sqrt{\frac{G(k'')}{k''^{3}}}\,\mathrm{d}k''-\lambda G(k)\,k^{2},$$

In the stationary limit of zero molecular viscosity and infinite electrical conductivity, the system admits two distinct families of solutions. In the velocity mode the ratio $G/F$ of magnetic to kinetic spectral energy tends to zero as $k\to\infty$, while in the magnetic mode it tends to about 2.6 (the closed form $(3+\sqrt{5})/2\approx2.618$); in both modes the largest eddies ($k\to0$) are in equipartition between kinetic and magnetic energy, and both spectra are Kolmogorov-like, proportional to $k^{-5/3}$, in the inertial range (the range of scales large compared with the dissipation scales and small compared with the forcing scale). The paper also applied the theory to the interstellar magnetic field, where the two modes make different predictions for the field's strength and uniformity relative to the turbulent gas. Two idealizations bear the weight of these predictions: the phenomenological closures (the eddy resistivity and the modeled cross-transfers) were justified only by analogy with Heisenberg's eddy viscosity, and the entire mode analysis is performed in the strict dissipationless limit. Chandrasekhar's own closing discussion records the open questions analyzed here: the finite-dissipation analysis and the role of the order in which the limits $\nu\to0$ and $\lambda\to0$ are taken, the justification of the closure ansatz, the need for unambiguous boundary conditions for the correlation functions underlying the deductive equations, and the application to the interstellar field. Chandrasekhar returned to the energy-partition question in 1957 with a deductive, quasi-Gaussian closure on the correlation equations rather than the elementary cascade; he left that program's own limiting-procedure question (the role of the path by which $\nu,\lambda\to0$) unresolved. After criticisms by Kraichnan and others, Chandrasekhar left turbulence research, and the field's center of gravity moved to wave-based phenomenologies (Iroshnikov 1964; Kraichnan 1965), exact structure-function laws, anisotropic cascade models, and direct numerical simulation, so the elementary theory's open questions were never revisited within their own framework.

Problem Statement

The task is to delimit the regime of validity of Chandrasekhar's elementary theory of hydromagnetic turbulence, as posed by the four open questions in the closing discussion of 'Hydromagnetic turbulence II. An elementary theory' (1955). The theory models the spectral energy transfers of incompressible MHD turbulence by a Heisenberg-type eddy viscosity $\nu(k)=\kappa\sqrt{F(k)/k^{3}}$ acting on the kinetic spectrum $F(k)$, an eddy resistivity $\lambda(k)=\kappa\sqrt{G(k)/k^{3}}$ acting on the magnetic spectrum $G(k)$, and modeled cross-transfers converting kinetic to magnetic energy and vice versa; its central objects are the coupled cascade equations (17) and (18) of that paper for $F$ and $G$, and its central result is that in the stationary limit of zero molecular viscosity and infinite electrical conductivity the system admits exactly two solution families, a velocity mode with $G/F\to0$ as $k\to\infty$ and a magnetic mode with $G/F$ tending to about 2.6 (closed form $(3+\sqrt{5})/2\approx2.618$) as $k\to\infty$, both with equipartition at $k\to0$ and $k^{-5/3}$ inertial-range spectra. The problem is to resolve, without restricting to particular $\nu/\lambda$ regimes, geometries, or answer forms: (1) the stationary analysis of the coupled equations with nonzero molecular viscosity $\nu$ and resistivity $\lambda$, including the existence, persistence, and deformation of the two dissipationless modes at finite dissipation, and the mechanism selecting among them, with attention to whether the double limit $\nu\to0$, $\lambda\to0$ commutes or depends on the path, equivalently on the limiting ratio $\nu/\lambda$; (2) the justification of the phenomenological closures - the eddy resistivity ansatz and the modeled cross-transfer terms - from the exact nonlinear spectral-transfer terms of the MHD equations, or their refutation; (3) the formulation of complete and unambiguous boundary conditions for the coupled longitudinal correlation functions underlying the deductive theory, such that the stationary problem is well posed and selects only the physical solutions, excluding solutions manufactured by reduction from integral to differential form; and (4) the consequences of (1)-(3) for the theory's application to the interstellar magnetic field under realistic dissipation and forcing, where the two modes make distinguishable predictions. Any resolution must engage the cascade equations (17) and (18) as stated rather than a silently improved closure.

References

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