Lagrangian Phase-Lag and Geometric Precedence in Turbulent Vortex Stretching

Abstract

This study investigates the causal timeline of vortex stretching in high-Reynolds-number turbulence ($Re_λ\approx 433$) using Lagrangian tracking in $1024^3$ direct numerical simulations. While classical theories often assume an instantaneous alignment between strain and vorticity, the present analysis identifies a systematic Lagrangian phase lag governing the onset of intense dissipation. By conditionally averaging the dynamics of fluid parcels, a distinct phase-space hysteresis is revealed. Trajectories are captured by the saddle-point topology of the pressure field ($λ_{min}^p < 0$) prior to experiencing peak enstrophy amplification. This temporal ordering ($τ> 0$) demonstrates that the pressure topology acts as a deterministic geometric precursor, organizing the flow structure before the bursting event occurs. The robustness of this mechanism is verified in magnetohydrodynamic (MHD) turbulence, where the Lorentz force is found to suppress the hysteresis loop, forcing a transition from causal precedence to simultaneous locking.

Figures (3)

• Figure 1: Geometric Interpretation of the Lagrangian Stability Analysis. The flow map $\varphi_t$ acts as a time-dependent coordinate transformation. Rather than tracking the vorticity vector $\boldsymbol{\omega}$ in a fixed frame, we track the deformation of the underlying metric tensor $C_t$ (blue ellipse). The "Geometric Locking" hypothesis tested in this paper posits that the Pressure Hessian $\nabla \nabla p$ determines the growth rate of the principal axis of $C_t$ (the unstable manifold) before the vorticity vector fully aligns with it.
• Figure 2: Geometric structure of 3D linear stagnation flow.(a) The Eulerian flow field $\bm{u}(\bm{x})$. Streamlines indicate compression in the horizontal $xy$-plane (blue inflow) and extension along the vertical $z$-axis (orange outflow). (b) Lagrangian material deformation. An initial unit sphere (gray ghost) deforms into a prolate ellipsoid (solid orange) due to volume-conserving strain. The vorticity vector $\bm{\omega}(t)$ aligns perfectly with the axis of maximum expansion (the unstable manifold), illustrating the geometric equivalence between material stretching and enstrophy production.
• Figure 3: Geometric hysteresis and temporal precedence of the pressure instability.(a) Phase-space evolution of the conditional mean trajectory in forced isotropic turbulence. The clockwise loop indicates that negative pressure curvature ($\lambda^p_{\min}$) intensifies prior to the peak in enstrophy production ($\sigma$). (b) MHD turbulence, exhibiting a collapsed loop due to magnetic suppression of the geometric phase lag. (c) Lagrangian cross-correlation $R(\tau)$, showing a distinct positive time lag ($\tau > 0$) for the isotropic case (blue) versus a centered response in MHD (red dashed). (d) Temporal evolution of normalized conditional means. The raw signal demonstrates the pressure curvature pulse (blue dashed) preceding the enstrophy amplification (black solid).