Wave Turbulence in the Strongly Nonlinear Regime: Theory and Applications

波浪湍流是一种数学理论，旨在描述包含大量相互作用的波浪的波场的平均行为，例如可能发生在有风的日子的海洋表面上。不太明显的例子包括我们大气中的大尺度行星波（罗斯贝波），它在产生天气或密度波（漂移波）中起着重要作用，密度波在强磁化等离子体中传播，并在未来聚变反应堆的设计中提出了关键的工程挑战。一个优雅的数学理论存在预测的平均行为的波场在弱非线性限制。从本质上讲，这种弱非线性理论首先确定非相互作用（线性）波系统的行为，这在数学上是简单的，然后通过将波相互作用的影响视为对非相互作用情况的小修正来分析相互作用（非线性）波。然而，在许多应用中，波之间的相互作用足够强，以至于它们不能被视为小的校正。这项建议的目的是，首先，扩展理论，允许强非线性的情况下进行数学研究，其次，以确定在何种程度上这些新的理论结果是相关的应用。本课程将特别关注海浪和Rossby波的例子，并通过利用基本守恒定律（如能量守恒定律）对波场的约束来获得理论结果，即使在波浪相互作用很强的情况下，这些定律也仍然成立。已建立的流体动力学湍流理论，其中非线性总是很强的，将提供一些指示，如何发展强波湍流的类似理论，虽然有本质的区别。最重要的区别是存在一个弱非线性极限波湍流没有类似的经典湍流，并将提供，在某种程度上，一个起点的分析描述强波湍流。然而，计算机模拟将是必要的，以补充理论研究。将通过与感兴趣的大气动力学和波浪预报方面的研究人员专家建立新的合作来指导将结果应用于真实的波浪问题。

Wave turbulence is a mathematical theory which aims to describe the average behaviour of wave fields containing large numbers of interacting waves such as might occur, for example, on the surface of the ocean on a windy day. Less obvious examples include the large scale planetary waves (Rossby waves) in our atmosphere which play an important role in generating the weather or the density waves (drift waves) which propagate in strongly magnetised plasmas and present a key engineering challenge in the design of future fusion reactors. An elegant mathematical theory exists which predicts the average behaviour of wave fields in the weakly nonlinear limit. In essence, this weakly nonlinear theory works by first determining the behaviour of a system of non-interacting (linear) waves, which is mathematically straightforward, and then analysing interacting (nonlinear) waves by treating the effect of the wave interactions as a small correction to the non-interacting case. In many applications, however, the interactions between waves are sufficiently strong that they cannot be treated as a small correction. This proposal aims, firstly, to extend the theory to allow cases with strong nonlinearity to be studied mathematically and, secondly, to determine the extent to which these new theoretical results are relevant to applications. There will be particular focus on the ocean wave and Rossby wave examples.The theoretical results will be obtained by exploiting the constraints imposed on the wave field by fundamental conservation laws, such as conservation of energy, which remain true even when wave interactions are strong. The established theory of hydrodynamic turbulence, for which nonlinearity is always strong, will provide some indication of how to develop the analogous theory for strong wave turbulence although there are essential differences. The most important difference is the existence of a weakly nonlinear limit for wave turbulence which has no analogue for classical turbulence and will provide, to some extent, a starting point for an analytical description of strong wave turbulence. Nevertheless, computer simulations will be necessary to complement the theoretical study. The application of the results to real wave problems will be guided by the establishment of new collaborations with interested researchers expert in atmospheric dynamics and wave forecasting.