Euler Ensemble as Decaying Turbulence Attractor: Universality, Stability and Parity Classes

Abstract

We study local Lyapunov stability of the Euler ensemble in compact rescaled momentum-loop dynamics for freely decaying Navier-Stokes turbulence. The ensemble consists of exact self-similar finite-cutoff solutions whose momentum loops are equal-step polygons on a sphere; their planar representatives define the Euler ensemble. In two dimensions, after quotienting global rotation and the time-origin shift, the odd-\(N\) planar representatives have no local shape instability. The even-\(N\) ensemble contains an alternating unstable mode with \(\lambda=\cot^2(\pi p/q)>0\), and is excluded as a local attractor. Thus the odd Euler ensemble is the locally stable planar sector. For \(d>2\), the planar ensemble lies in a continuous manifold of equal-step spherical polygons. Transverse deformations along this manifold are exact tangent zero modes. Integrating over these unconstrained spherical modes in the continuum limit gives a singular Wilson-loop functional supported only on collapsed, globally rotated planar coordinate loops. These modes are therefore projected out of the admissible homogeneous isotropic ensemble. The physical stability problem is the decay of normal perturbations, represented by local edge-length defects. In the endpoint-local sector, the planar odd representative has exact defect spectrum \(\lambda_m=-\sec^2(\pi m/N)<0\). In the smooth angular continuum limit, the leading stability operator is the universal angular Laplacian. Nearby spherical zero modes do not modify this leading operator; their first effect appears only at order \(N^{-4}\), through higher-derivative corrections computed symbolically. Hence the leading local stability mechanism is universal and independent of spherical zero modes.