Multi-scale Mathematics applied to Parameterisation of Convection

为数值天气预报和气候建模设计的计算机模型必须使用有限网格来解析地球大气层。发生在比该网格更小的尺度上的物理过程，例如水平尺度为一公里或更小的大气对流环流，不能被明确解析。然而，这些对流环流在更大尺度上的累积效应必须用一种称为参数化法的技术来表示。建立一个精确的对流参数是模拟地球大气中的动量、水分和熵传输的关键。正如最近NERC的一个主要方案(“了解和代表尺度上的大气对流”)所强调的那样，当前的参数化方案和已解决的(大尺度)动力学之间的耦合是一个需要改进的高度优先事项。在支持对流的大气波的速度方面，不准确的耦合是明显的，导致对重要的大尺度过程，如热带的Madden-Julian振荡的模式表示不佳。我们的科学假设是，耦合误差的一个关键来源是在模式中实现参数化的当前方法中所作的假设的结果。目前，与(假设的)小尺度对流羽流相关的动量、质量和水汽通量简单地被直接添加到大尺度模式方程中。这里的假设是，大尺度流动的演变基本上就像(未解决的)小对流环流可以直接平均化一样。然而，对一个相关问题的简单数学处理表明，大尺度惯性重力波在可变环境中的传播速度实际上对存在小而有限的层结减少区域表现出非常强烈的敏感性，例如在对流羽流中出现的区域。换句话说，假设的简单平均过程是不正确的，需要更复杂的平均技术。该项目的目的是探索使用一种系统的和严格的数学方法--“多尺度数学”(MSM)--对对流羽流进行平均的新技术。MSM一直是水文学、结晶学和超材料科学等不同领域进步的基础，这些领域中的每一个都涉及到需要在小规模结构上进行系统平均的问题。学生将在地球物理流体动力学(Esler)、MSM(Smyshlyaev)和最先进的对流计算模型(Whitall，Met Office)专家的指导下工作。该方法将是系统化的，首先通过处理相对简单的数学问题来发展学生的直觉，同时进行计算方面的培训。接下来，将详细探讨一个相对简单的数值模式，在该模式中对流可以显式求解。其目的将是评估传统参数化的性能，并将其与基于MSM的新方法进行比较。最后，将评估切换到基于MSM的参数化对热带波速度的影响，并将评估使用MSM修改Met Office UnifiedModel中对流参数化实施的可行性。学生机会包括接受数学和气候科学前沿技能的培训，然后创造性地探索技能，以及发展最先进的大气模型方面的专业知识。这项研究可能会产生极大的影响，因为它可能会带来潜在的预测准确性，具有非常高的经济价值，以及改进气候预测的高社会效益。更广泛的好处包括气象局、气候模型社区和数学家之间的知识交流，导致将MSM技术引入一大类相关问题。

Computer models designed for numerical weather prediction and climate modelling necessarily resolve the Earth's atmosphere using a finite grid. Physical processes that occur on scales smaller than this grid, for example atmospheric convective circulations that have horizontal scales of a kilometre or less, cannot be resolved explicitly. Nevertheless, the cumulative effects of these convective circulations on the larger scale must be represented, using a technique called parameterisation. Formulating an accurate parameterisation of convection is key to modelling the transport of momentum, moisture and entropy in the Earth's atmosphere. As highlighted by a recent major NERC programme (`Understanding and representing atmospheric convection across scales') coupling between current parameterisation schemes and resolved (large-scale) dynamics is a high priority for improvement. Inaccurate coupling is evident in systematic biases in the speedsof those atmospheric waves that support convection, leading to poor model representation of important large-scale processes such as the Madden-Julian Oscillation in the tropics.Our scientific hypothesis is that a key source of coupling error is the result of assumptions made in the current method used to implement parameterisations in models. Currently, the momentum, mass and moisture fluxes associated with (hypothesised) small-scale convective plumes are simply added directly to the large-scale model equations. The assumption here is the large-scale flow evolves essentially as if the (unresolved) small-scale convective circulations can be directly averaged out. However, a simple mathematical treatment of a related problem reveals that the propagation speed of large-scale inertia-gravity waves, moving through a variable environment, in fact shows very strong sensitivity to the presence of small but finite regions of reduced stratification, such as occur in convective plumes. In other words, the simple averaging process assumed is not correct, and a more sophisticated averaging technique is required. The aim of the studentship is to explore new techniques for averaging across the convective plumes using a systematicand mathematically rigorous approach: `multi-scale mathematics' (MSM). MSM has been fundamental to advances in diverse fields such as hydrology, crystallography, and the science of meta-materials, each of which involves problems requiring systematic averaging across small-scale structure. The student will work under the guidance of experts in geophysical fluid dynamics (Esler), MSM (Smyshlyaev) and state-of-the- art computational modelling of convection (Whitall, Met Office). The approach will be systematic, first developing the students' intuition by working on relatively simple mathematical problems, while training proceeds in the computational aspects. Next, a relatively simple numerical model, in which convection can be explicitly resolved, will be explored in detail. The aim will be to evaluate the performance of a traditional parameterisation, and compare it to the new approach based on MSM. Finally, the impact of switching to an MSM-based parameterisation on tropical wave speeds will be estimated, and the feasibility of using MSM to modify the implementation of the convective parameterisation in the Met Office UnifiedModel will be evaluated. The studentship opportunity includes being trained in, and then creatively exploring, skills at the cutting edge of mathematics and climate science, as well as developing expertise in state-of-the-art atmospheric modelling. The research is potentially extremely high-impact, as it could result in gains potential forecast accuracy of very high economic value, in addition to the high societal benefit of improved climate forecasts. Wider benefits include knowledge exchange between the Met Office, climate modelling communities and mathematicians, leading to the introduction of MSM techniques to a wide class of related problems.