Maximum palinstrophy growth in 2D incompressible flows

In this study we investigate vortex structures which lead to the maximum possible growth of palinstrophy in two-dimensional incompressible flows on a periodic domain The issue of palinstrophy growth is related to a broader research program focusing on extreme amplification of vorticity-related quantities which may signal singularity formation in different flow models. Such extreme vortex flows are found systematically via numerical solution of suitable variational optimization problems. We identify several families of maximizing solutions parameterized by their palinstrophy, palinstrophy and energy and palinstrophy and enstrophy. Evidence is shown that some of these families saturate estimates for the instantaneous rate of growth of palinstrophy obtained using rigourous methods of mathematical analysis, thereby demonstrating that this analysis is in fact sharp. In the limit of small palinstrophies the optimal vortex structures are found analytically, whereas for large palinstrophies they exhibit a self-similar multipolar structure. It is also shown that the time evolution obtained using the instantaneously optimal states with fixed energy and palinstrophy as the initial data saturates the upper bound for the maximum growth of palinstrophy in finite time. Possible implications of this finding for the questions concerning extreme behaviour of flows are discussed.

在这项研究中，我们研究了在周期域上的二维不可压缩流中导致拟涡能最大可能增长的涡旋结构。拟涡能增长问题与一个更广泛的研究计划相关，该计划专注于涡度相关量的极端放大，这可能预示着不同流动模型中奇点的形成。通过对适当的变分优化问题进行数值求解，系统地找到了这种极端涡旋流。我们确定了几个由拟涡能、拟涡能与能量以及拟涡能与涡量参数化的最大化解族。有证据表明，这些解族中的一些使通过严格的数学分析方法得到的拟涡能瞬时增长率的估计达到饱和，从而证明这种分析实际上是精确的。在小拟涡能的极限情况下，通过解析方法找到了最优涡旋结构，而对于大拟涡能，它们呈现出自相似的多极结构。还表明，以固定能量和拟涡能的瞬时最优状态作为初始数据得到的时间演化使拟涡能在有限时间内的最大增长上限达到饱和。讨论了这一发现对有关流动极端行为问题的可能影响。