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Variational principles for stochastic parameterisations in geophysical fluid dynamics

地球物理流体动力学中随机参数化的变分原理

基本信息

批准号：
EP/N023781/1
负责人：
Darryl Holm
金额：
$ 98.6万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN023781%2F1
关键词：
Variational principles stochastic parameterisations geophysical

项目摘要

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.

我们的建议是受到明确和当前需要了解天气和气候的统计变异性的启发。动力学天气预报起源于世纪中期建立的力学和热力学的确定性定律。随着世纪后半叶数字计算机的出现，这些想法导致了业务数值天气预报（NWP），此后不久，随着卫星观测的出现，探索大气环流的数值实验。通过数值模拟进行的新型科学探索很快就提出了大气动力学可预测性的极限问题，这是由于初始状态的不确定性、运动的未解决尺度以及数值输出对这些不确定性的极端敏感性。这种敏感性被称为蝴蝶效应。认识到NWP的可预报性的损失召唤研究到一个随机的方法在设计模拟器NWP。NWP不能完全确定，但也必须涉及某种形式的随机性或噪声。一种新的数值预报方法出现了，它将随机性和概率与确定性结合起来。20世纪90年代早期的并行处理方法和改进的业务预报系统，在模拟器物理和数据同化方法中，已经导致了现代业务随机动态Enhancement预报系统（EPS）产生的更可靠的预报，现在ECMWF和英国气象局使用。然而，它仍然需要确定最适当的方式来引入随机动力学的模拟器，以便耦合数据同化与集合预报，并确定在集合中的样本的数量足以满足所需的可靠性。目前的工作继续以极大的活力探索这些途径。该项目通过对数据驱动的数学建模、兼容的数值和模型驱动的数据同化采取综合方法，解决了NWP随机动力学的剩余挑战。数学建模采用了一种最佳的，系统的方法，引入随机性的地球物理流体动力学（GFD）。该方法是基于一个随机版本的家庭变分原理，其临界点产生的整个序列的确定性运动方程的理想GFD在每个级别的近似。近似的水平是从渐近展开的不可近似的变分原理，产生的基本欧拉方程的旋转，分层，不可压缩流体。随机性引入到变分原理中，通过使用从波动示踪剂路径的观测获得的数据的解析空间相关性。反过来，随机变分原理产生的运动方程的流体流动携带这些示踪剂沿着其波动路径。对这些新的运动方程的拟议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的数学基础创造一种可实施的建模方法。为此，我们采用贝叶斯的观点，将新开发的SPDE与数据完全集成，并将其建模和模拟工作与图1所示的连接相结合。同样，数值算法也将通过数学分析得到信息。一旦数值模拟被开发和执行，随后的数据同化将通过粒子滤波方法产生模型当前状态的后验分布。