Variational principles for stochastic parameterisations in geophysical fluid dynamics

我们的建议的灵感来自于了解天气和气候的统计可变性的明确和现实的需要。动态天气预报源于19世纪中叶建立的力学和热力学决定论定律。随着20世纪下半叶数字计算机的出现，这些想法导致了可操作的数值天气预报(NWP)，此后不久，随着卫星观测的出现，数值实验探索了大气的一般环流。通过数值模拟进行的新型科学探索很快提出了大气动力学可预测性极限的问题，这是由于初始状态的不确定性、未解决的运动尺度以及数值输出对这些不确定性的极端敏感性。这种敏感度因蝴蝶效应而广为人知。对数值预报的可预测性丧失的认识促使人们在设计数值预报模拟器时采用随机方法进行研究。数值预报不能完全是确定性的，但也必须包含某种形式的随机性或噪声。一种将随机性、概率与决定论相结合的数值预报方法应运而生。20世纪90年代初S提出的并行处理方法和改进的业务预报系统，无论是在模拟器物理方法还是资料同化方法上，都导致了现在欧洲气象中心和英国气象局使用的现代业务随机动力预报系统产生的更可靠的预报。然而，仍然需要确定将随机动力学引入模拟器的最合适的方法，以便将数据同化与集合预报结合起来，并确定集合中足以保证所需可靠性的样本数量。目前的工作继续以极大的活力探索这些途径。该项目通过采用数据驱动的数学建模、兼容的数值和模型驱动的数据同化的综合方法，解决了随机动力学对数值预报的剩余挑战。该数学模型使用了一种将随机性引入地球物理流体动力学(GFD)的优化、系统的方法。该方法基于变分原理族的随机版本，其临界点产生了理想GFD在每一级近似下的确定运动方程的整个序列。近似水平是由不可近似变分原理的渐近展开得到的，该原理得到了旋转、分层、不可压缩流体的基本欧拉方程。将随机性引入变分原理，利用起伏示踪路径观测获得的数据的空间相关性进行分解。反过来，随机变分原理为携带这些示踪剂的流体沿其波动路径生成运动方程。对这些新的运动方程的数学研究将与数值模拟和数据同化方法相结合，旨在为数值预报、气候科学和其他高度不稳定的流体动力学应用创造一种具有重要意义的可实施的模拟方法。为此，我们采用贝叶斯观点将新开发的SPDEs与数据完全整合在一起，如图1所示，与其建模和模拟工作相结合。同样，数值算法将由数学分析提供信息。一旦开发和执行了数值模拟，随后的数据同化将通过粒子滤波方法产生模式当前状态的后验分布。

Our proposal is inspired by the clear and present need for understanding statistical variability of weather and climate. Dynamical weather prediction stems from the deterministic laws of mechanics and thermodynamics, established by the mid-19th century. With the advent of digital computers in the second half of the 20th century, these ideas led to operational Numerical Weather Prediction (NWP) and shortly thereafter, with the advent of satellite observations, to numerical experiments that explored the atmosphere's general circulation. The new type of scientific exploration via numerical simulations soon raised the issue of limits of predictability of atmospheric dynamics, due to uncertainty in the initial state, unresolved scales of motion, and the extreme sensitivity of the numerical output to these uncertainties. This sensitivity was famously popularised as the Butterfly Effect. The recognition of the loss of predictability for NWP summoned research into a stochastic approach in designing simulators for NWP. NWP cannot be entirely deterministic, but must also involve a form of randomness, or noise. A new approach to NWP arose, which coupled randomness and probability with determinism. Parallel processing methods in the early 1990's and improved operational forecasting systems, in both simulator physics and data assimilation methods, have led to more reliable forecasts produced by modern operational stochastic dynamic Ensemble Prediction Systems (EPS) now used at ECMWF, and the UK Met Office. Yet it still remains to determine the most appropriate way to introduce stochastic dynamics into the simulator, so as to couple data assimilation with ensemble forecasting and to determine the number of samples in the ensemble sufficient for a required reliability. Current work continues to explore these avenues with great vigour.This project addresses the remaining challenge of Stochastic Dynamics for NWP, by taking an integrated approach to data-driven mathematical modelling, compatible numerics and model-driven data assimilation. The mathematical modelling uses an optimal, systematic method of introducing stochasticity into Geophysical Fluid Dynamics (GFD). The method is based on a stochastic version of the family of variational principles whose critical points yield the entire sequence of deterministic equations of motion for ideal GFD at each level of approximation. The levels of approximation are obtained from asymptotic expansion of the unapproximated variational principle that yields the fundamental Euler equations for a rotating, stratified, incompressible fluid. Stochasticity is introduced into the variational principle by using resolved spatial correlations of data obtained from observations of fluctuating tracer paths. In turn, the stochastic variational principle generates the equations of motion for the fluid flow carrying these tracers along their fluctuating paths. The proposed mathematical research on these new equations of motion will be integrated with numerical simulations and data assimilation methods, aiming to create an implementable modelling approach of significance for the mathematical foundations of NWP, climate science, and other highly unstable fluid dynamics applications. For this, we adopt a Bayesian perspective in blending the newly developed SPDEs with data completely integrated with its modelling and simulation efforts with connections as shown in Figure 1. Likewise, the numerical algorithms will be informed by the mathematical analysis. Once the numerical simulations are developed and performed, the subsequent data assimilation will produce the posterior distribution of the current state of the model via particle filtering methods.