Shavit Michal, Falkovich Gregory
Weizmann Institute of Science, Rehovot 76100 Israel, Landau Institute for Theoretical Physics, Moscow 117240 Russia.
Phys Rev Lett. 2020 Sep 4;125(10):104501. doi: 10.1103/PhysRevLett.125.104501.
How weak is the weak turbulence? Here, we analyze turbulence of weakly interacting waves using the tools of information theory. It offers a unique perspective for comparing thermal equilibrium and turbulence. The mutual information between modes is stationary and small in thermal equilibrium, yet it is shown here to grow with time for weak turbulence in a finite box. We trace this growth to the concentration of probability on the resonance surfaces, which can go all the way to a singular measure. The surprising conclusion is that no matter how small is the nonlinearity and how close to Gaussian is the statistics of any single amplitude, a stationary phase-space measure is far from Gaussian, as manifested by a large relative entropy. This is a rare piece of good news for turbulence modeling: the resolved scales carry significant information about the unresolved scales. The mutual information between large and small scales is the information capacity of turbulent cascade, setting the limit on the representation of subgrid scales in turbulence modeling.
弱湍流有多弱？在这里，我们使用信息论工具分析弱相互作用波的湍流。它为比较热平衡和湍流提供了独特的视角。在热平衡中，模式之间的互信息是平稳且小的，但在此处表明，对于有限盒子中的弱湍流，它会随时间增长。我们将这种增长追溯到共振面上概率的集中，这可以一直延伸到奇异测度。令人惊讶的结论是，无论非线性多么小，任何单个振幅的统计多么接近高斯分布，平稳的相空间测度都远非高斯分布，这表现为相对熵很大。这对于湍流建模来说是一条难得的好消息：已解析尺度携带了关于未解析尺度的重要信息。大尺度和小尺度之间的互信息是湍流级联的信息容量，为湍流建模中亚网格尺度的表示设定了限制。
