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，例如在大风天的海洋表面上可能发生的情况。不太明显的例子包括大气中的大规模行星波（罗斯比波），它在产生天气方面发挥着重要作用，或在强磁化等离子体中传播的密度波（漂移波），并在未来聚变反应堆的设计中提出了关键的工程挑战。存在一种优雅的数学理论，可以预测弱非线性极限下波场的平均行为。本质上，这种弱非线性理论的工作原理是首先确定非相互作用（线性）波系统的行为（这在数学上很简单），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，不能将其视为小修正。该提案的目的首先是扩展该理论，以允许对具有强非线性的情况进行数学研究，其次是确定这些新的理论结果与应用的相关程度。将特别关注海浪和罗斯贝波的例子。理论结果将通过利用基本守恒定律（例如能量守恒定律）对波场施加的约束来获得，即使在波相互作用很强时，这些守恒定律仍然成立。已建立的水动力湍流理论（其非线性总是很强）将为如何发展强波湍流的类似理论提供一些指示，尽管存在本质区别。最重要的区别是波湍流存在弱非线性极限，它与经典湍流没有类似物，并且在某种程度上为强波湍流的分析描述提供了起点。然而，计算机模拟对于补充理论研究是必要的。将通过与大气动力学和波浪预测方面感兴趣的研究人员建立新的合作来指导将结果应用于实际波浪问题。

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.