Stochastic identities for random isotropic fields

This letter presents new nontrivial stochastic identities for random isotropic second rank tensor fields. They can be considered as markers of statistical isotropy in turbulent flows of any nature. The case of axial symmetry is also considered. We confirm the validity of the identities using different direct numerical simulations of turbulent flows.

💡 Research Summary

The paper introduces a set of non‑trivial stochastic identities that must be satisfied by random second‑rank tensor fields when the underlying probability measure is invariant under rotations, i.e., when the field is statistically isotropic. Starting from a generic tensor A (which may represent velocity gradients, magnetic‑field gradients, or second derivatives of a scalar), the authors construct the positive‑definite quadratic form Γ = AᵀA and apply an LDU (lower‑diagonal‑upper) decomposition Γ = ZᵀD²Z. The diagonal matrix D = diag(D₁,…,D_d) is not composed of eigenvalues; instead each D_i² equals the ratio of successive leading principal minors of Γ, D_i² = s_i/s_{i‑1}.

Assuming the probability density ρ(A) is invariant under rotations in a given coordinate plane (or under the full orthogonal group O(d)), the authors define a correlator C(m₁,…,m_d) = ⟨∏i D_i^{m_i+1}⟩. They prove that C is symmetric under the exchange of any adjacent pair of exponents (a consequence of integrating over the SO(2) subgroup). Full rotational invariance then implies symmetry under any permutation of the indices. By setting the exponents to m_i = −i, they obtain the fundamental identity ⟨∏i D_i^{−i}⟩ = 1, which can be rewritten in terms of the principal minors as ⟨s{π(2)}^{−1} … s{π(d)}^{−1} s_d⟩ = 1 for every permutation π.

In three dimensions (d = 3) this yields five distinct identities, listed in Table 1. Each identity involves a dimensionless product of leading principal minors, for example ⟨s₁ s₂^{−1} s₃⟩ = 1, ⟨s₂^{−1} s₁ s₂⟩ = 1, etc. The authors emphasize that these identities are not mere consequences of permuting coordinate labels; they arise from the continuous rotational symmetry, and rotating the coordinate frame generates new, linearly independent families of identities. In general dimension d, there are (d! − 1) families, each with d(d − 1)/2 continuous parameters.

The paper also treats the case of axial (one‑axis) symmetry, where the probability measure is invariant only under rotations about a single direction. In this situation only a subset of the isotropic identities survives (those involving the symmetry axis). By rotating the coordinate system, the authors obtain identities that involve not only leading minors but also mixed 2‑rank minors m_{ij}. Table 2 collects the identities that arise for three possible choices of the symmetry axis, illustrating how each axial symmetry yields two one‑parameter families of stochastic identities.

To validate the theory, the authors analyze data from the Johns Hopkins Turbulence Database. They use two direct‑numerical‑simulation (DNS) datasets: (i) forced isotropic turbulence on a 1024³ periodic grid (Taylor‑scale Reynolds number R_λ ≈ 433) and (ii) a turbulent channel flow at friction Reynolds number R_τ ≈ 1000 on a 2048 × 512 × 1536 grid. From each dataset they compute the velocity‑gradient tensor A_{ij}=∂u_i/∂x_j, form Γ = AᵀA, and evaluate the left‑hand sides of the identities in Table 1 by spatial averaging (or temporal averaging when appropriate).

The results (Fig. 1) show that in the isotropic case all five averages converge to unity, confirming the identities. Convergence is faster for quantities with low powers of the gradient components in the denominator (e.g., ⟨s₂/s₁⟩) and slower for those with higher powers (e.g., ⟨s₃/(s₁ s₂)⟩). In the channel flow, averages taken at the centre plane also approach 1, supporting Kolmogorov’s hypothesis of local isotropy. Near the wall, most averages deviate substantially, but the axial‑symmetry identity ⟨s₁ s₂^{−1} s₃⟩ remains within 10 % of unity, indicating that the flow retains a stronger symmetry about the streamwise direction than about the other axes.

To test sensitivity to controlled anisotropy, the authors add a constant shear component δA_{12}=a to the isotropic dataset. The modified averages display systematic departures from 1 that are comparable in magnitude to those observed near the wall in the channel flow (Fig. 3). Some identities increase while others decrease, yet the identity associated with the y‑axis symmetry and the mixed‑minor identity ⟨s₂^{−1} s₃^{−1} s₂ s₃⟩ are the least affected. This demonstrates that the proposed stochastic identities are highly responsive to even modest violations of statistical isotropy.

The authors note that the identities also hold for tensors of the form B = f(A) A, where f(A) is a scalar function possibly introducing anisotropy, although real turbulent anisotropy is typically generated by large‑scale mean flows rather than such deterministic transformations.

In conclusion, the paper provides a mathematically rigorous set of stochastic constraints for random tensors in isotropic or axially symmetric turbulence. These constraints go beyond the traditional second‑order moment checks (e.g., ⟨δu_i δu_i⟩/3 = 1) by involving nonlinear combinations of principal minors, thereby probing higher‑order structure and intermittency. The identities can serve as robust diagnostics for isotropy in experiments and simulations, as additional constraints in turbulence modelling, and as a foundation for extending statistical symmetry analyses to more complex tensorial fields in magnetohydrodynamics, scalar transport, and related disciplines.