Multi-scale Mathematics applied to Parameterisation of Convection

为数值天气预报和气候模拟设计的计算机模型必须使用有限网格来解析地球大气。在小于这个网格的尺度上发生的物理过程，例如水平尺度为1公里或更小的大气对流环流，不能明确地解决。然而，必须使用一种称为参数化的技术来表示这些对流环流在更大尺度上的累积效应。对对流进行精确的参数化是模拟地球大气中动量、水分和熵的传输的关键。正如最近NERC的一个主要项目（“理解和表示跨尺度的大气对流”）所强调的那样，当前参数化方案与解决（大尺度）动力学之间的耦合是一个需要优先改进的问题。在支持对流的大气波的速度的系统偏差中，不准确的耦合是明显的，导致重要的大尺度过程（如热带的麦登-朱利安涛动）的模式表现不佳。我们的科学假设是，耦合误差的一个关键来源是在当前用于实现模型参数化的方法中所做假设的结果。目前，与（假设的）小规模对流羽流相关的动量、质量和水分通量只是简单地直接添加到大规模模型方程中。这里的假设是大尺度流的演变基本上就好像（未解析的）小尺度对流环流可以直接平均出来一样。然而，对一个相关问题的简单数学处理表明，大尺度惯性重力波的传播速度，在一个可变的环境中移动，实际上对小而有限的减少分层区域的存在表现出很强的敏感性，例如在对流羽流中。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。该研究的目的是探索利用系统和数学上严格的方法“多尺度数学”（MSM）平均对流羽流的新技术。MSM在不同领域的进步中起到了重要作用，如水文学、晶体学和超材料科学，这些领域都涉及到需要在小尺度结构上进行系统平均的问题。该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。这种方法将是系统化的，首先通过处理相对简单的数学问题来培养学生的直觉，同时在计算方面进行训练。接下来，将详细探讨一个相对简单的数值模型，其中对流可以明确地解决。目的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的影响，并评估使用MSM修改Met Office统一模式中对流参数化实现的可行性。学生的机会包括接受培训，然后创造性地探索数学和气候科学的前沿技能，以及发展最先进的大气建模专业知识。这项研究的潜在影响非常大，因为除了改进气候预测的高社会效益外，它还可能获得具有很高经济价值的潜在预测准确性。更广泛的好处包括气象局、气候建模社区和数学家之间的知识交流，导致将MSM技术引入广泛的相关问题。

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.