The early stage of the motion along the gradient of a concentrated vortex structure

TL;DR

The paper addresses the initial-stage dynamics of a highly concentrated vortex in a background vorticity field and proves, rigorously, that the vortex travels along the gradient of the background vorticity with a logarithmically enhanced acceleration as the blob radius tends to zero. It introduces a blob-wave model on a 2D torus and analyzes the barycenter motion to derive a precise second-derivative formula involving the gradient of the background field, then extends the framework to a quasi-2D setting by examining vortex filaments in a 3D shear flow with a mollified Biot–Savart kernel. The vortex-wave system on the rotating sphere is reviewed, including global existence, a blob-based convergence result, and a Lie–Poisson Zeitlin discretization that preserves key invariants, complemented by direct numerical simulations illustrating the coupled vortex–background dynamics. In the thin-domain limit, the same gradient-driven mechanism is shown to persist for vortex filaments, with a refined set of small-parameter conditions, demonstrating the robustness of gradient-driven aggregation across dimensional reductions. Overall, the work provides a rigorous first-principles account of gradient-driven aggregation in 2D and quasi-2D vortex dynamics, supported by numerics and culminating in explicit asymptotic displacement laws and open questions for fully 3D configurations.

Abstract

We give a rigorous mathematical result, supported by numerical simulations, of the aggregation of a concentrated vortex blob with an underlying non-constant vorticity field: the blob moves in the direction of the gradient of the field. It is a unique example of a Lagrangian explanation of aggregation of vortex structures of the same sign in 2D inviscid fluids. The result is also extended to almost vertical vortex filaments in a (possibly thin) three-dimensional domain.

Figures (4)

• Figure 1: Vertical coordinate trajectory $y=y(t):=\theta(0)-\theta(t)$ of the barycenter of a blob of radius $2\varepsilon=2/N$, with same energy of the blob in the previous picture and , for $1/N$ the smallest spatial scale resolved by the discrete model. The blob is surrounded by a zonal flow. The curve $y=y(t)$ is plotted in loglog scale and obtained by a DNS of \ref{['eq:vw_sphere_zeitlin']}, for $N=257$. The slope of the curve, after an initial short time, is around $\frac{3}{2}$, until we see a linear growth. We notice that increasing the radius of the blob produces a less sharp transition from the growth power law of $\frac{3}{2}$ and the linear one.
• Figure 2: Vertical coordinate trajectory $y=y(t):=\theta(0)-\theta(t)$ of the barycenter of a blob of radius $\varepsilon=1/N$, for $1/N$ the smallest spatial scale resolved by the discrete model. The blob is surrounded by a zonal flow. The curve $y=y(t)$ is plotted in loglog scale and obtained by a DNS of \ref{['eq:vw_sphere_zeitlin']}, for $N=257$. The slope of the curve, after an initial short time, is around $\frac{3}{2}$, until we see a linear growth.
• Figure 3: Vertical coordinate trajectory $y=y(t):=\theta(0)-\theta(t)$
• Figure 4: From the top-left to the bottom-right, time snapshots of the evolution of the vorticity field $\omega$ obtained by a DNS of \ref{['eq:vw_sphere']} for $N=501$, plotted in longitude-latitude coordinates $(\varphi,\theta)$ of the Mercator projection. The initial (unstable) equilibrium of 6 equatorial point-vortices placed at the vertices of a regular equatorial hexagon

Theorems & Definitions (15)

• Theorem 1
• proof
• Remark 2
• Remark 3
• Theorem 4
• Theorem 5
• Lemma 6
• proof
• Lemma 7
• proof