Efficient numerical methods for wave-action transport and scattering

在大气和海洋中传播的波需要在用于天气和气候预测的数值模式中表示，以便捕捉它们对大气和海洋环流、海面状况和污染物传输的强烈影响。然而，这不能直接实现，因为典型的波长比即使是最高分辨率的数值模式的网格尺度要短得多。对短波长取平均值的简化数学模型提供了一个解决方案，但构成了一个主要的计算挑战。它描述了波作用密度在扩展的位置-波数相空间中的分布；因此，它需要求解多达6个类空维度的偏微分方程组。这超出了传统的离散化方法的能力范围。这个项目的目的是证明另一种方法的可行性，该方法基于波作用密度的动态低阶近似。这种方法将波作用扩展为几个变量函数的乘积的和，动态构造以在最小化误差的同时将动力学投影到低阶函数空间。该项目将制定一个基于低阶近似和分裂的算法，实施使用基于网格和光谱离散化的不同组合的两个版本，并针对光线跟踪算法(专门为捕获几个波包的动力学而设计)和针对基本流体方程的直接数值模拟进行测试。该方案的制定和实施将强调在超级计算机上的并行性和效率，并在ARCHER2上进行测试。该项目的主要目标是内波的模拟，重点放在湍流散射和非线性波-波相互作用的表示上。对海洋表面波的应用也将被考虑。

Waves propagating in the atmosphere and ocean need to be represented in the numerical models used for weather and climate prediction in order to capture the strong impact they have on the atmospheric and oceanic circulation, on the state of the sea surface, and on the transport of pollutants. This cannot be achieved directly, however, because the typical wavelengths are much shorter than the grid scales of even the highest resolution numerical models. A reduced mathematical model that averages over the short wavelengths offers a solution but poses a major computational challenge. It describes the distribution of wave-action density in an extended position-wavenumber phase space; hence, it requires solving a partial differential equation in up to 6 space-like dimensions. This is beyond the reach of traditional discretisation methods. This project aims at demonstrating the feasibility of an alternative approach, based on a dynamical low-rank approximation of the wave-action density. This approach expands the wave action as a sum of products of functions of a few variables, constructed on-the-fly to project the dynamics onto the space of low-rank functions while minimising an error. The project will formulate an algorithm based on low-rank approximation and splitting, implement two versions that use different combinations of grid-based and spectral discretisations, and test them against a ray-tracing algorithm (specifically designed to capture the dynamics of a few wavepackets) and against direct numerical simulations of the underlying fluid equations. The formulation and implementation will emphasise parallelisation and efficiency on supercomputers, with testing carried out on ARCHER2. The project primarily targets the modelling of internal waves, with a focus on the representation of their scattering by turbulence and of nonlinear wave-wave interactions. Applications to ocean surface waves will also be considered.