平流扩散壳模型中最优混合的长时间行为

中文摘要 AI 辅助

我们采用壳模型框架研究平流扩散方程中最优混合的长时间行为，重点量化在拟涡能约束的搅拌作用下，以负Sobolev范数$H^{-1}$度量的标量方差衰减情况。我们分别使用时间局部（最大化瞬时混合速率）和时间全局（最大化指定终态时刻的混合程度）两种优化策略开展长时间数值计算。\n对于存在扩散（$κ>0$）的混合过程，数值结果表明标量长度尺度最终会受广义Batchelor尺度限制，与理论预测高度吻合。在此区域内，$H^{-1}$混合范数随时间呈指数衰减，且衰减速率与扩散系数$κ$无关。与纯平流情形（$κ=0$）相比，扩散显著提升了长时间混合速率；此外，进一步增大扩散系数可通过降低指数衰减的前因子来提高混合效率。\n受这些数值观测结果启发，我们推导了$H^{-1}$范数的新条件下界，对于所有$κ>0$，其指数衰减速率严格独立于扩散参数$κ$。我们还建立了标量方差增强耗散最大速率的条件上界，表明有效扩散时间尺度至少为$|\text{log}κ|$量级。

英文摘要

We investigate the long-time behavior of optimal mixing in an advection-diffusion equation using a shell model framework. Our focus is on quantifying the decay of the scalar variance, measured by the negative Sobolev norm $H^{-1}$, under enstrophy-constrained stirring. We perform long-time computations using both local-in-time (maximizing the instantaneous mixing rate) and global-in-time (maximizing mixedness at a prescribed final time) optimization strategies. For mixing with diffusion ($κ>0$), the numerical results show that the scalar length scale eventually becomes limited by a generalized Batchelor scale, in close agreement with theoretical predictions. In this regime, the $H^{-1}$ mix-norm decays exponentially in time with a decay rate that is independent of the diffusivity $κ$. Compared with the purely advective case ($κ= 0$), diffusion significantly enhances the long-time mixing rate; moreover, increasing diffusivity further improves mixing efficiency by reducing the prefactor of the exponential decay. Guided by these numerical observations, we derive new conditional lower bounds on the $H^{-1}$ norm whose exponential decay rates are strictly independent of the diffusivity parameter $κ$, for all $κ> 0$. We further establish conditional upper bounds on the maximal rate of enhanced dissipation of the scalar variance, showing that the effective diffusion time scale is at least of the order $|\logκ|$.

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35 pages, 7 figures