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波的波场的平均行为，例如，在刮风的日子里，在海洋表面可能发生的波浪。不太明显的例子包括我们大气中的大规模行星波（罗斯比波），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，是未来聚变反应堆设计中的一个关键工程挑战。存在一个优雅的数学理论来预测弱非线性极限下波场的平均行为。从本质上讲，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，这在数学上是直截了当的，然后通过将波相互作用的影响视为对非相互作用情况的一个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一个小的修正。本提案的目的是，首先，扩展理论，允许用数学方法研究具有强非线性的情况，其次，确定这些新的理论结果与应用相关的程度。将特别关注海浪和罗斯比波的例子。理论结果将通过利用基本守恒定律施加在波场上的约束来获得，例如能量守恒定律，即使在波的相互作用很强时，它仍然是正确的。对于非线性总是很强的水动力湍流理论，虽然存在本质的区别，但已建立的理论将为如何发展强波浪湍流的类似理论提供一些指示。最重要的区别是波浪湍流的弱非线性极限的存在，它与经典湍流没有相似之处，并将在某种程度上为强波浪湍流的解析描述提供一个起点。然而，计算机模拟将是必要的，以补充理论研究。将结果应用于实际的波浪问题将以与大气动力学和波浪预报方面有兴趣的研究人员建立新的合作关系为指导。

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.