The Batchelor spectrum for a deterministically driven passive scalar

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.

Figures (5)

• Figure 1: Plots of $\mathcal{S}^-$ for $\alpha = 8$ and $\alpha = 8.5$ obtained from \ref{['eq:S-explicit']}. The green and orange lines show $T_2(\{x=0\})$ and $T_2(\{x=1/2\})$, respectively.
• Figure 2: A typical pair $(W_1,W_2)$ is shown on the left and some representative elements of the decomposition \ref{['eq:DecompTriple']} are colored on the right. Two pairs $(W_{1,j}, W_{2,j})$ are shown in pink. The segments in each pink pair have length on the order of $\alpha^{-1}$, while the distance between them is on the order of $\epsilon$. The vertical coordinates of the upper endpoints on the curves differ by roughly $\epsilon \alpha^{-1}$. Examples of the $U_{i,j}$ and $V_{i,j}$ are displayed in yellow and red, respectively.
• Figure 3: An example of $W_1$ and $W_2$ as in the statement of Lemma \ref{['lem:onestepkey']}. For the configuration shown, $T^{-1}W_1$ is contained in the right half of the fundamental domain and $T^{-1}W_2$ in the left half. The situation would be reversed if the point shared by $W_1$ and $W_2$ were instead on an orange spanning curve.
• Figure 4: A typical example of the pre-images $V_1 = T^{-1}(W_1)$ and $V_2 = T^{-1}(W_2)$ for $\alpha = 16$ and $W_1$ and $W_2$ configured as in Figure \ref{['fig:adjacent']}. Each $V_i$ consists of approximately $\alpha$ segments in $\Sigma$ that span $\mathbb{T}^2$ vertically, together with two segments that may not span. The parallelogram $P$ is shaded in pink and the remainder set $E = R_1 \setminus P$ is shaded in yellow. The angle $\theta$ satisfies $|\pi/2 - \theta| \approx \alpha^{-1}$ since the slope of each pink line is roughly $\alpha$.
• Figure 5: The singularity set $\mathcal{S}^+$ of $T$ when $\alpha = 8$.

Theorems & Definitions (78)

• Theorem 2.1
• Remark 2.1: Cumulative vs. Mode-wise
• Remark 2.2: $\log \alpha$ scaling in Batchelor's law
• Remark 2.3: Extensions to $\kappa > 0$
• Remark 2.4: Time-dependent discrete forcing
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Remark 2.5
• Remark 2.6