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arXiv physics/0304076physics.flu-dyn
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局域波Turbulence的非平稳谱

Non-stationary Spectra of Local Wave Turbulence

Colm Connaughton, Alan C. Newell, Yves Pomeau

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revtex4, 19 pages, 10 figures
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摘要：

本文研究了弱湍流Kolmogorov-Zakharov（K-Z）谱在强局部相互作用极限下的演化，其中描述谱波作用密度随时间演化的动力学方程可以用偏微分方程近似。如果波作用最初在频率空间中是紧支撑的，那么它会被共振相互作用重新分布，产生通常的正级联和逆级联，导致K-Z谱的形成。这里的重点是直接级联。演化过程中形成了一个自相似的前沿，该前沿向右传播，在其尾流中留下一个准稳态。这个前沿是尖锐的，因为解在到达无穷远之前保持紧支撑。如果能量谱具有无限大的容量，则波前需要无限长的时间才能达到无限大的频率，并在其尾流中留下K-Z谱。另一方面，如果能谱具有有限的容量，则前缘在有限时间t* 内达到无穷大，并且尾流比K-Z谱更陡。在这种情况下，K-Z谱是在波前到达无穷远后从右边开始建立的。尾流中的解的斜率可以与波前的传播速度有关。结果表明，在有限容量的情况下，异常斜率对应于独特的前速度，确保前尖端包含一个有限的能量作为连接到无穷大。我们还介绍，第一次，在波动湍流的熵产生的概念，并显示它是如何演变的系统接近平稳的K-Z谱。

英文摘要：

The evolution of the Kolmogorov-Zakharov (K-Z) spectrum of weak turbulence is studied in the limit of strongly local interactions where the usual kinetic equation, describing the time evolution of the spectral wave-action density, can be approximated by a PDE. If the wave action is initially compactly supported in frequency space, it is then redistributed by resonant interactions producing the usual direct and inverse cascades, leading to the formation of the K-Z spectra. The emphasis here is on the direct cascade. The evolution proceeds by the formation of a self-similar front which propagates to the right leaving a quasi-stationary state in its wake. This front is sharp in the sense that the solution remains compactly supported until it reaches infinity. If the energy spectrum has infinite capacity, the front takes infinite time to reach infinite frequency and leaves the K-Z spectrum in its wake. On the other hand, if the energy spectrum has finite capacity, the front reaches infinity within a finite time, t*, and the wake is steeper than the K-Z spectrum. For this case, the K-Z spectrum is set up from the right after the front reaches infinity. The slope of the solution in the wake can be related to the speed of propagation of the front. It is shown that the anomalous slope in the finite capacity case corresponds to the unique front speed which ensures that the front tip contains a finite amount of energy as the connection to infinity is made. We also introduce, for the first time, the notion of entropy production in wave turbulence and show how it evolves as the system approaches the stationary K-Z spectrum.

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