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Elongation of material lines and vortices by Euler flows on two-dimensional Riemannian manifolds

Koki Ryono, Keiichi Ishioka

TL;DR

The paper addresses how the curvature of a 2D flow domain governs Euler fluid dynamics on curved surfaces by deriving a second-order time derivative of the squared separation between nearby fluid particles, which includes a curvature contribution. It extends Haller's Lagrangian hyperbolic-domain framework to curved manifolds through the strain-acceleration tensor \mathsf{M} and the curvature term $-R(\bm{\xi},\bm{u})\bm{u}$, and validates the theory with a sphere and a curved torus. Key findings show that negative curvature accelerates material-line elongation and can promote vortex filamentation, while positive curvature tends to suppress it, with explicit computations and FTLE analyses illustrating these effects. The work provides a geometric, largely intrinsic approach to predicting mixing and coherent-structure behavior in 2D flows on curved domains and points to applications in geophysical and planetary contexts as well as directions for studying time-varying metrics and geodesic-flow connections.

Abstract

We study the influence of the differential geometry of the flow domain on the motion of fluids on two-dimensional Riemannian manifolds, particularly on the elongation of material lines and vortices. We derive a formula for the second order time derivative of the square of the distance between close fluid particles and show that a curvature term appears. The elongation of a material line is accelerated by negative curvature. The use of this expression extends Haller's definition of hyperbolic domains to flows on curved surfaces. The need to consider curvature effects is illustrated by three examples. The example of a curved two-dimensional torus implies that the filamentation of vortices can be triggered by negative curvature.

Elongation of material lines and vortices by Euler flows on two-dimensional Riemannian manifolds

TL;DR

The paper addresses how the curvature of a 2D flow domain governs Euler fluid dynamics on curved surfaces by deriving a second-order time derivative of the squared separation between nearby fluid particles, which includes a curvature contribution. It extends Haller's Lagrangian hyperbolic-domain framework to curved manifolds through the strain-acceleration tensor \mathsf{M} and the curvature term , and validates the theory with a sphere and a curved torus. Key findings show that negative curvature accelerates material-line elongation and can promote vortex filamentation, while positive curvature tends to suppress it, with explicit computations and FTLE analyses illustrating these effects. The work provides a geometric, largely intrinsic approach to predicting mixing and coherent-structure behavior in 2D flows on curved domains and points to applications in geophysical and planetary contexts as well as directions for studying time-varying metrics and geodesic-flow connections.

Abstract

We study the influence of the differential geometry of the flow domain on the motion of fluids on two-dimensional Riemannian manifolds, particularly on the elongation of material lines and vortices. We derive a formula for the second order time derivative of the square of the distance between close fluid particles and show that a curvature term appears. The elongation of a material line is accelerated by negative curvature. The use of this expression extends Haller's definition of hyperbolic domains to flows on curved surfaces. The need to consider curvature effects is illustrated by three examples. The example of a curved two-dimensional torus implies that the filamentation of vortices can be triggered by negative curvature.
Paper Structure (15 sections, 82 equations, 13 figures)

This paper contains 15 sections, 82 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Theory
  3. Euler equation on a Riemannian manifold
  4. Review of Haller's hyperbolic domain
  5. Elongation of the material lines and hyperbolic domain on general two-dimensional Riemannian manifolds
  6. Elongation of material lines
  7. Extension of Haller's hyperbolicity
  8. Examples
  9. Jet on a sphere
  10. Hyperbolic domains of vortices on a sphere
  11. Onset of the vorticity filamentation on a curved torus
  12. Discussion and conclusions
  13. The derivation of \ref{['xi_evolution_M']}
  14. Evaluation of the spectral coefficients of the Riemannian metric
  15. Explicit specification of the initial vorticity field on the curved torus

Figures (13)

  • Figure 1: The vorticity field $q$ defined by \ref{['sphere_q_def']}. The horizontal axis is longitude $\lambda$, and the vertical axis is sine latitude $\mu$.
  • Figure 2: Top panel: the hyperbolic domain (colored blue) of the vorticity field \ref{['sphere_q_def']}. Bottom panel: the same as the top panel, except that the curvature term of the strain acceleration tensor is neglected.
  • Figure 3: The (forward) finite-time Lyapunov index over the time interval from $t=0$ to $t=2$ for the flow given by the vorticity field defined by \ref{['sphere_q_def']}.
  • Figure 4: Top panel: total time spent by a particle starting from each point at $t=0$ in the hyperbolic domain up to $t=2$ for the flow field given by \ref{['sphere_q_def']}. Bottom panel: same as the top panel, except that the hyperbolic domain is computed by neglecting the curvature term of the strain acceleration tensor $\mathsf{M}$.
  • Figure 5: Total time spent by a particle starting from each point at $t=0$ in the strong hyperbolic domain up to $t=2$ for the flow field given by \ref{['sphere_q_def']}.
  • ...and 8 more figures