Wave Turbulence in the Strongly Nonlinear Regime: Theory and Applications

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.

This proposal is about improving the fundamental understanding of the statistical properties of dispersive wave turbulence. Since we treat wave phenomena from a quite general perspective there are diverse potential applications. As already mentioned, the principal applications for this proposal are wave forecasting (ocean gravity waves), climate modelling (atmospheric internal and Rossby waves) and nuclear fusion (``drift'' waves in magnetised plasmas). These are applications of considerable current importance with many potential beneficiaries. Ocean wave forecasts are used daily by the shipping and fishery industries. The accurate prediction of storm surges and sea swell are of increasing importance to the insurance and tourism industries as well as the local and national authorities responsible for managing development and planning for extreme weather in populated coastal areas. Accurate climate models, and a quantification of their inherent uncertainty, are now among the key desiderata asked of the scientific community as the world's policy makers struggle to frame a collective response to the challenges posed by climate change. Controlled nuclear fusion offers the prospect of a relatively clean energy source for the future which has gained considerable momentum in recent years with the commencement of construction of ITER, the first controlled self-sustaining fusion device scheduled to complete in 2018. It is not claimed that fundamental research contained in this proposal will impact directly on these applications but basic science does contribute to these issues. In particular, fundamental understanding is essential to arrive at efficient and reliable parameterisations of wave systems. In computer simulations of systems such as a tokamaks or a the weather, wave turbulence mostly occurs at scales smaller than the resolution of the numerical model but nevertheless influences the large scales via turbulent feedback. Parameterisations allow the effects of the waves to be included without paying the computational cost of resolving them explicitly. This research will facilitate the development of physics-based parameterisations of wave turbulence which are more reliable than empirical ones. Such parameterisations also reduce the number of free parameters required in a numerical model. This is of importance in reducing the danger of statistical over-fitting in models with many parameters such as climate models. The proposal will ensure that the improved theoretical understanding resulting from the project has the possibility of impacting upon these downstream applications by establishing lines of communication to researchers who are closer to these applications. Researchers, from engineering, physics and dynamical meteorology, have been identified from the PI's existing circle of collaborators who, although they will not be actively involved in doing the proposed research, are willing to comment and advise on its implications in their areas of application. In addition, some funds are requested to allow the PI to establish new relationships with other end-users, particularly the UK Met Office, in order to learn what questions they feel are important for theorists to answer in order to improve the parameterisations in their models. The development of appropriate new collaborations as a result of this networking activity has been included as a specific objective of the project. Finally, an international workshop, ``Statistical wave problems in theory and practice'', will run at the end of the project in which the PI will attempt to bring together relevant individuals from this new network with appropriate theorists to attempt to bridge the gap.