In 1951, S. Chandrasekhar initiated the statistical theory of density fluctuations in homogeneous isotropic turbulence of a compressible fluid. From the equation of continuity alone he derived an exact integral invariant of the two-point density covariance, and from the momentum equation he derived a closed equation of motion for that covariance. Obtaining the closed equation required the adiabatic pressure-density relation, a Gaussian (zero-fourth-cumulant) factorization of fourth-order correlations, and the removal of the viscous term, since retaining viscosity while neglecting heat conduction is thermodynamically inconsistent. The open problem is to construct a statistical theory of density fluctuations for compressible Navier-Stokes-Fourier turbulence that retains viscosity and heat conduction consistently, closes the correlation hierarchy by a controlled and realizable approximation instead of the unjustified Gaussian hypothesis, and thereby determines the evolution of the density covariance and its spectrum. Later literature has generalized and numerically confirmed the exact invariant (Jaupart and Chabrier 2021; Dumond et al. 2025), demonstrated that the Gaussian closure is unrealizable in the incompressible analogue (O'Brien and Francis 1962; Ogura 1963), and produced weak-compressibility analyses, exact isothermal relations, and direct numerical simulation characterizations, but no closed theory of the density covariance with consistent thermo-viscous physics exists. The problem remains open.

Background

In the statistical theory of turbulence, one describes a turbulent flow not by its instantaneous velocity field but by correlation functions between fluctuating quantities at two points separated by a distance rr. A flow is homogeneous and isotropic when these statistics are invariant under translations and rotations. For incompressible turbulence this program was well developed by 1951, including the discovery by Loitsiansky that a certain integral of the velocity correlation is conserved in time, which implies that the largest eddies retain memory of the initial conditions (the 'permanence of big eddies'). For a compressible fluid, whose density ρ\rho fluctuates about a mean ρˉ\bar{\rho}, no analogous theory existed. The fluctuation δρ=ρ−ρˉ\delta\rho=\rho-\bar{\rho} is described by its two-point covariance

ϖ(r,t)=δρ(x,t) δρ(x′,t)‾,r=∣x′−x∣,\varpi(r,t)=\overline{\delta\rho(\mathbf{x},t)\,\delta\rho(\mathbf{x}',t)},\qquad r=|\mathbf{x}'-\mathbf{x}|,

where the overbar denotes a statistical average.

Chandrasekhar (1951) addressed this gap. First, using only the equation of continuity (mass conservation), he derived an exact integral invariant,

∫0∞ϖ(r,t) r2 dr=constant,\int_0^\infty \varpi(r,t)\,r^2\,dr=\text{constant},

the compressible counterpart of the Loitsiansky-type invariant, implying that the largest scales of the density fluctuations are permanent features fixed by the initial state. Second, combining the continuity and momentum equations, he derived a closed equation of motion for ϖ\varpi. Achieving closure required three approximations that the paper itself acknowledges: (i) the adiabatic relation δp=c2 δρ\delta p=c^2\,\delta\rho between pressure and density fluctuations, where cc is the speed of sound — this excludes heat exchange; (ii) a Gaussian closure (zero-fourth-cumulant hypothesis), in which fourth-order correlations such as ρρ′uiuj‾\overline{\rho\rho'u_i u_j} are factored into products of second-order correlations exactly as for a jointly Gaussian distribution; and (iii) the neglect of viscosity, because keeping the viscous term while assuming adiabaticity — that is, keeping viscosity, which generates heat, while neglecting heat conductivity — is physically inconsistent. Within these approximations the closed equation is reported to yield, in the subsonic regime where the mean square turbulent velocity u2‾\overline{u^2} is small compared with c2c^2, wave-type oscillatory asymptotic behavior of ϖ\varpi at large rr; this prediction is not independently reproduced in the later literature. A companion paper in the same issue (Chandrasekhar 1951, gravitational instability of a turbulent medium) applied the density-fluctuation framework to star formation, the thread that leads to the modern astrophysical literature on supersonic turbulence.

The three approximations define what is missing. The physically complete model of a compressible Newtonian fluid is the Navier-Stokes-Fourier system, which retains viscosity μ\mu and heat conductivity κ\kappa together, so that dissipative heating is balanced by thermal diffusion and the entropy equation replaces the adiabatic relation. Writing closed moment equations for density correlations in this system faces the classical closure problem of turbulence: the hierarchy of correlation equations never terminates, and the specific Gaussian closure used in 1951 was later shown to be unrealizable in the incompressible analogue, where it predicts negative spectra (O'Brien and Francis 1962; Ogura 1963). Modern work provides strong adjacent results — direct numerical simulations (DNS, i.e. fully resolved numerical solution of the governing equations) characterizing thermodynamic fluctuations (Donzis and Jagannathan 2013), exact two-point relations for the special case of isothermal compressible turbulence (Galtier and Banerjee 2011), and rigorous Onsager-type results on dissipative anomalies (that is, results in the spirit of Onsager's conjecture that turbulent energy dissipation can remain finite as viscosity vanishes) in the compressible cascade (Eyink and Drivas 2018) — but not a closed statistical theory of the density covariance with consistent thermo-viscous physics.

Problem Statement

Construct a statistical theory of density fluctuations in homogeneous isotropic turbulence of a compressible Newtonian fluid, in the spirit of Chandrasekhar (1951), that removes the three approximations on which his closed equation of motion for the two-point density covariance ϖ(r,t)\varpi(r,t) rested. Specifically, the theory must: (a) be thermodynamically consistent — retain viscosity and heat conductivity together, replacing the adiabatic relation δp=c2 δρ\delta p=c^2\,\delta\rho by the full non-adiabatic energy (entropy) closure of the compressible Navier-Stokes-Fourier system, so that dissipative heating is balanced by thermal diffusion; (b) close the resulting hierarchy of density-velocity-temperature correlations without the unverified Gaussian (zero-fourth-cumulant) hypothesis, using a controlled, realizable closure (one that produces non-negative spectra and bounded variances), with the closure's regime of validity specified; and (c) determine the evolution of the density covariance (equivalently, of its Fourier transform, the spectrum of density fluctuations). A successful theory would naturally indicate the regimes in which the adiabatic, inviscid, Gaussian predictions of the 1951 theory hold and those in which they fail; this is a consequence one may read off the constructed theory, not a separate deliverable. These three components serve one objective — a consistent closed theory of density fluctuations in compressible turbulence: the closure (b) is what closes the equations derived under (a), and (c) is the resulting prediction whose comparison with simulation and experiment tests the whole. The initial covariance ϖ(r,0)\varpi(r,0) remains an external input, as in any initial-value statistical theory; the target is the law of evolution, not the initial state.

References

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