Fast dynamo action on the 3-torus for pulsed-diffusions

Abstract

We study a pulsed-diffusion version of the kinematic dynamo equation on the three-dimensional torus, in which vector transport and resistive diffusion act alternately over unit time intervals. We provide a rigorous proof that the fast dynamo conjecture holds for this model. Our approach is genuinely perturbative in the magnetic diffusivity. We construct a time-periodic, divergence-free, and Lipschitz stretch-fold-shear velocity field, which generates a uniformly hyperbolic flow map. To analyze this system, we develop anisotropic Banach spaces specifically adapted to the map's dynamics, allowing us to recover favourable spectral properties for the associated dynamo operator. By characterizing the ideal dynamo operator in the strong-chaos limit, we prove that it admits an eigenvalue with modulus strictly greater than 1. Finally, we demonstrate that this instability persists under the singular perturbation of the heat semigroup for all sufficiently small values of the diffusivity, thereby establishing fast dynamo action.

Figures (5)

• Figure 1: The partition of $[0,1]^2$ into the regions $\mathcal{M}_1,\mathcal{M}_2,\mathcal{M}_3,\mathcal{M}_4$ and the sets $T_\alpha(\mathcal{M}_1), T_\alpha(\mathcal{M}_2), T_\alpha(\mathcal{M}_3), T_\alpha(\mathcal{M}_4)$, with $\alpha=16$.
• Figure 2: The sets $T_\alpha(\mathcal{M}_\ell)$ in $[0,1]^2$ with $\alpha=16$, with an admissible curve $W \in \Sigma$ and one highlighted subcurve $W_i \subset W\cap \overline{T_\alpha(\mathcal{M}_1)}$.
• Figure 3: Pre-processing of two admissible curves $W_1,W_2$: a small initial curve $U_{1,1}$ is cut from $W_1$ and a small final curve $U_{2,2}$ is cut from $W_2$. The base points of $W_1 \setminus U_{1,1}$ and $W_2 \setminus U_{2,2}$ lie in the same half-plane and on the same $\{ x+ \alpha y = q_{in}\}$ for some $q_{in} \in \mathbb{R}$ (indicated by the dashed strip lines); likewise for the endpoints of $W_1 \setminus U_{1,1}$ and $W_{2} \setminus U_{2,2}$ lie in the same half-plane and on the same $\{ x- \alpha y = q_{fin}\}$ for some $q_{fin} \in \mathbb{R}$.
• Figure 4: We represent $W_1, W_2$ as two black lines. We zoom near $y=\frac{1}{2}$. The two dashed lines are $\{ x+ \alpha y = q_{2, down}\}$ and $\{ x-\alpha y = q_{2, up}\}$ passing through $W_2 \cap \{ y = 1/2 \}$. The red segment is the unmatched curve $U_{1,1}\subset W_1$.
• Figure 5: We represent $T_\alpha^{-1} (W_1)$ and $T_\alpha^{-1} (W_2)$ as the two black lines in the region $\mathcal{M}_4$. In the proof these lines are cut in $\mathcal{O}(\alpha)$ curves of length $1$. The red segment $\tilde{U}_1$ is the initial unmatched curve of $T_\alpha ^{-1} (W_1)$, starting from $x=0$ until it reaches the same $y$-level as the curve $T_\alpha ^{-1} (W_2)$. The green segment $\tilde{U}_2$ is the final unmatched curve of $T_\alpha ^{-1} (W_2)$.

Theorems & Definitions (44)

• Theorem 1
• Remark 2.1
• Remark 2.2: On the strong-chaos limit
• Proposition 2.3
• Proposition 2.4: Keller--Liverani
• Remark 2.5
• Proposition 2.6: Lasota--Yorke inequality
• Theorem 2.7
• Proposition 2.8
• proof : Proof of Theorem \ref{['thm:fastpulsed']}