A directed probability current in the vorticity–strain alignment state space of homogeneous isotropic turbulence

Abstract

A stationary distribution can rest on balanced exchange between states, or it can conceal a steady circulation through them. Beneath the familiar occupancies of the vorticity–strain alignment in homogeneous isotropic turbulence, we find the second: a directed probability circulation. Counting transitions along Lagrangian trajectories between four coarse alignment states, the antisymmetric finite-lag edge current is almost entirely divergence-free, and its projection on the cycle E1 → E2 → E3 → E1 is positive and two orders of magnitude outside a reversible null that randomises only the direction of transitions. The circulation was preregistered with frozen sign and thresholds and replicates, at matched Kolmogorov-scaled lags, across two independently produced simulations with different codes, forcings and Reynolds numbers. Sign, replication and distance to the nulls are the load-bearing results; amplitudes are quoted with the spread across averaging windows, which exceeds the within-window bootstrap several-fold at short spans and closes to one per cent at long ones. Conditioning on the velocity-gradient invariants accounts for one third of it, and a causally evaluated realignment rate at the source point predicts the coming transition. Preprint, 16 October 2026. Code, English versions of the preregistrations and all result files: https://github.com/SRT-Compute/alignment-current-hit. The kinematic origin of the cycle orientation is analysed in the companion preprint https://doi.org/10.5281/zenodo.23417576.