Mixing and enhanced dissipation in a time-translating shear flow

Abstract

Motivated in part by the work of Vanneste and Byatt-Smith, we study mixing and enhanced dissipation for the advection-diffusion equation with velocity field $\mathbf{u}(x,y,t)=(\sin(y-ct),0)$, a shear flow whose profile translates rigidly with speed $c$. This is a prototypical example of a flow whose critical points move in time. We quantify how the decay properties depend on the relation between translation speed $c$ and diffusivity $ν$. We first analyse the inviscid transport problem and establish time-averaged $H^{-1}$ mixing estimates for $t\lesssim c^{-1}$, yielding decay rates faster than stationary estimates. Building on these estimates, we prove enhanced dissipation for moderate translation speeds $c=c_0ν^\ell$ with $\ell\in(1/3,3/4)$. In this regime we obtain decay at rate $ν^{(1+2\ell)/5}$, which interpolates continuously between the sharp rates $ν^{1/2}$ for stationary shear flows with simple critical points and $ν^{1/3}$ for monotone flows. This quantifies how increasing translation speed progressively weakens the influence of the critical points. Comparing the inviscid mixing and enhanced dissipation timescales heuristically explains the lower endpoint $\ell=1/3$. For $c\gg 1$, we show that solutions remain close to those of the heat equation on fixed time intervals, such that the rapid translation averages out advection and weakens mixing. The mixing estimate relies on a refined stationary phase analysis exploiting cancellations generated by the motion of the critical points. The enhanced dissipation result requires an adaptation of the hypocoercivity framework for stationary shear flows to the non-autonomous setting. The translating flow prevents the commutator hierarchy from closing in the standard way, which we overcome by constructing an extended energy functional. The large-$c$ analysis exploits the averaging effect of rapid translations in this regime.

Figures (3)

• Figure 1: Illustration of the time–dependent shear $t \mapsto \sin(y-ct)$ at fixed spatial location $y$ for three different $c$. The shaded regions corresponds to the first sign–coherent intervals during which phase gradients accumulate and mixing strengthens. Once the shear changes sign, the dynamics begins to unwind previously generated gradients, explaining why the mixing estimate is naturally restricted to times of order $c^{-1}$.
• Figure 2: Snapshots of the inviscid solutions $\hat{\Theta}(k,y,t)$ first Fourier mode ($|k|=1$), with initial datum $\hat{\Theta}_0 = \cos(2y)$ and translation speed $c = 0.1$, at times $t = 0,\, \pi/2c,\, \pi/c,\, 2\pi/c$. The solution develops increasingly fine-scale oscillations within the time interval $[0, \pi/c]$, consistent with the mixing mechanism of Theorem \ref{['mix:thm:time-avg-mixing']}. After one full period ($t = 2\pi/c$), the solution returns to its initial profile.
• Figure 3: Numerically fitted exponent of $\nu$ in the dissipation time $t_\varepsilon$ (defined as the first time the norm drops below $\varepsilon = 0.1$) as a function of $\ell$, where $c = c_0\,\nu^\ell$. Exponents are extracted via log--log regression of $t_\varepsilon$ against $\nu$ over ten values logarithmically spaced in $\nu \in [3\times 10^{-7}, 10^{-2}]$. The solid line is the theoretical prediction $-(1+2\ell)/5$ from Theorem \ref{['hyp:thm:L2_phys']}, the dotted line the corresponding stationary exponent $-1/2$. The fitted exponents agree with the predicted curve to within an average relative error of $0.17\%$.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 3.1
• Proposition 3.2
• Lemma A.2.1
• proof
• Lemma A.2.2