The 1950 final-period decay theory of statistically axisymmetric homogeneous turbulence (Chandrasekhar, Proc. R. Soc. Lond. A 203, 358) rests on two mathematical constructions whose physical status it did not settle: a special decoupled solution state, pictured as a superposition of an isotropic field and a non-interacting axisymmetric field (arising when the defining scalar Q1Q_1 is independent of the angular variable, under certain conditions on the third-order correlation scalars), and two radial fourth-moment integrals Λ1\Lambda_1 and Λ2\Lambda_2 of the initial data whose assumed existence fixes the late-time decay constants. The open problem is to determine whether the decoupled state can be occupied by real turbulence or is a kinematic curiosity, and for which classes of initial axisymmetric data the fourth-moment integrals exist and are genuine dynamical invariants of the decay, so that the asymptotic decay laws built on them are physically valid. The later literature (Batchelor–Proudman long-range correlations, Saffman's k2k^2 alternative, the modern axisymmetric Saffman decay theory, eddy-damped quasi-normal Markovian (EDQNM) closure studies, and direct numerical simulation (DNS) studies) does not settle either question: homogeneous axisymmetric turbulence itself is realizable in experiment and simulation, but the decoupled class is untouched, and the invariance of the Batchelor-type (k4k^4) fourth-moment integrals is unresolved even in the isotropic case.

Background

A turbulent flow is called homogeneous when its statistical properties are invariant under spatial translations, so that its structure is captured by two-point correlations such as the double velocity correlation ⟨ui(x)uj(x+r)⟩\langle u_i(\mathbf{x})u_j(\mathbf{x}+\mathbf{r})\rangle and the triple correlation ⟨uiujuk′⟩\langle u_i u_j u_k'\rangle (primed velocities evaluated at the second point), which obey the von Kármán–Howarth equation — the exact evolution equation linking the double and triple correlations of homogeneous turbulence through the Navier–Stokes nonlinearity and viscous diffusion. When the turbulence is also isotropic (statistics invariant under all rotations and reflections), the double correlation is determined by a single scalar function of the separation magnitude rr, and the classical decay theory of isotropic turbulence follows. Isotropy is an idealization: turbulence produced by grids, contractions, rotation, or stratification is anisotropic, and the simplest anisotropic case admitting a tractable exact formalism is axisymmetric turbulence — homogeneous turbulence whose statistics are invariant under rotations about one fixed preferred direction (the symmetry axis). Batchelor (1946) and Chandrasekhar (1950, Phil. Trans.) developed this formalism: the double correlation tensor is then determined by two scalar functions (denoted Q1(r,μ,t)Q_1(r,\mu,t) and Q2(r,μ,t)Q_2(r,\mu,t) in Chandrasekhar's notation), where rr is the separation magnitude and μ\mu is the cosine of the angle between the separation vector and the axis, and the triple correlation is likewise expressible through further defining scalars (denoted T1,T2,…T_1,T_2,\ldots in Chandrasekhar's notation, together with a contribution ω\omega from the vorticity correlation), which replace the single scalar of the isotropic case in the axisymmetric version of the von Kármán–Howarth equation.

The final period of decay is the late, low-Reynolds-number stage of freely decaying turbulence in which the nonlinear transfer between scales is negligible and the correlation equations reduce to linear diffusion equations; for isotropic turbulence, Batchelor & Townsend (1948) showed that the energy then follows power-law decay laws whose exponents and coefficients are fixed by the behavior of the initial energy spectrum at small wavenumber kk. For initial data in the so-called Batchelor (or big-eddy) class, in which the spectrum behaves as E(k→0)∼k4E(k\to 0)\sim k^4, the decay constant is the fourth radial moment of the initial correlation function — the quantity known as the Loitsyansky integral, i.e. the r4r^4-weighted integral of the two-point velocity correlation, whose conservation (when it exists) underlies the classical large-eddy decay predictions. The convergence of such fourth-moment integrals is not automatic: it requires the initial correlations to decay sufficiently fast at large separations, and the whole question of which initial data guarantee it was already delicate in 1950.

Chandrasekhar's decay paper (1950, Proc. R. Soc. A) extended the final-period analysis to axisymmetric turbulence. It obtained explicit late-time solutions for Q1Q_1 and Q2Q_2, in agreement with Batchelor's general big-eddy results, and its asymptotic formulae rest on two radial fourth-moment integrals of the initial defining scalars (denoted Λ1\Lambda_1 and Λ2\Lambda_2 in the source), whose existence is assumed rather than established. The paper also showed that the axisymmetric decay equations admit a remarkable special state: per the published abstract, "a state of turbulence exists which may be pictured as a superposition of two non-interacting fields which are isotropic and axisymmetric, respectively" — realized, in Chandrasekhar's formulation, when Q1Q_1 is independent of μ\mu under the third-order scalar conditions T1=T2=T6=0T_1=T_2=T_6=0, T4=T4(r,t)T_4=T_4(r,t), with T3+ωT_3+\omega and T5T_5 spatially uniform. Whether such a state is physically realizable, and whether the assumed invariants actually exist and constrain the decay of general axisymmetric data, are the two unresolved questions.

The later literature reshaped the landscape around these questions. Batchelor & Proudman (1956) showed for the isotropic case that pressure forces generically induce long-range triple correlations falling off as r−4r^{-4}, so that the standard proof that the fourth-moment integral is a dynamical invariant is invalidated, and results built on rapidly decaying cumulants are suspect. Saffman (1967) exhibited the alternative initial-data class with E(k→0)∼k2E(k\to0)\sim k^2 (eddies with finite linear impulse), for which the fourth-moment integral diverges and different (impulse-type) invariants hold. In the modern revival of the axisymmetric decay problem, Davidson, Okamoto & Kaneda (2012) and Davidson (2023) established the decay laws and invariants of axisymmetric turbulence for the k2k^2 class (u⊥2∼u∥2∼t−6/5u_\perp^2\sim u_\parallel^2\sim t^{-6/5}, with the integral scales — the characteristic large-eddy length scales — growing as t2/5t^{2/5}, where ⊥\perp and ∥\parallel mark directions relative to the symmetry axis), while the k4k^4 (Batchelor-type) axisymmetric branch — the home of Λ1\Lambda_1 and Λ2\Lambda_2 — remains treated only at the level of assumed constraints, its invariance unresolved even isotropically (Ishida, Davidson & Kaneda 2006; Davidson 2023).

Problem Statement

For homogeneous turbulence whose statistics are axisymmetric about a fixed direction, in the framework of the 1950 decay theory (double correlation defined by the scalars Q1(r,μ,t)Q_1(r,\mu,t) and Q2(r,μ,t)Q_2(r,\mu,t) with μ\mu the cosine of the angle between the separation vector and the symmetry axis; triple correlation expressed through the corresponding third-order defining scalars T1,…,T6T_1,\ldots,T_6 together with the vorticity-correlation contribution ω\omega, in Chandrasekhar's notation), determine the physical status of the two mathematical constructions on which the theory's final-period asymptotics rest:

  1. Decoupled superposition state. Establish whether the special statistical state shown to be compatible with the axisymmetric decay equations — the superposition of an isotropic field and a non-interacting axisymmetric field, realized when Q1Q_1 is independent of μ\mu under the third-order scalar conditions T1=T2=T6=0T_1=T_2=T_6=0, T4=T4(r,t)T_4=T_4(r,t), with T3+ωT_3+\omega and T5T_5 spatially uniform (as given by Chandrasekhar 1950) — can be occupied by real turbulence (that is, whether it is realizable, dynamically inaccessible or non-generic for Navier–Stokes evolution, or a kinematic curiosity only).

  2. Fourth-moment invariants. Determine for which classes of initial axisymmetric data the two radial fourth-moment integrals of the initial defining scalars, Λ1=−∫0∞q0(1)(r) r4 dr\Lambda_1=-\int_0^\infty q_0^{(1)}(r)\,r^4\,dr and Λ2=−∫0∞q0(2)(r) r4 dr\Lambda_2=-\int_0^\infty q_0^{(2)}(r)\,r^4\,dr (where q0(1)q_0^{(1)} and q0(2)q_0^{(2)} are the initial radial functions of the defining scalars, in the source's notation), exist (converge), and whether — for initial data in that class — they are genuine dynamical invariants of the decaying turbulence, including the nonlinear regime in which long-range, pressure-induced triple correlations may break their conservation, such that the late-time decay laws and invariant values predicted from them are physically valid.

The two parts share one objective: to settle the physical foundations of the 1950 decay theory for axisymmetric turbulence — which statistical states the theory's solution class can actually describe, and which integral quantities genuinely constrain the late-time decay of axisymmetric turbulence.

References

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