The Batchelor spectrum for a deterministically driven passive scalar

TL;DR

This paper demonstrates that in a deterministic setting with specific flow conditions, a passive scalar's long-term behavior aligns with Batchelor's law, marking a novel theoretical achievement.

Contribution

It provides the first deterministic example where Batchelor's law is rigorously established for a passive scalar under smooth, time-periodic forcing.

Findings

All smooth initial data are attracted to a limiting solution.

The limiting solution satisfies a form of Batchelor's law.

This is the first deterministic forcing example confirming Batchelor's law.

Abstract

We study the long-time behavior of a passive scalar transported by an incompressible flow in the presence of smooth, deterministic forcing. For a specific spatially Lipschitz and time-periodic velocity field, we prove that all sufficiently smooth initial data is attracted to a limiting solution that satisfies a cumulative form of Batchelor's law. To our knowledge, this provides the first example for which a version of Batchelor's law can be established with deterministic forcing.