Topological Tools for Finding Vortex Rings in Cloud Formation

对气候的预测和建模提高了我们对大气现象、气候和日常天气的认识。这些是形成我们对长期气候趋势和气候变化可能影响的理解的关键。云在气候模型中扮演着至关重要的角色，根据其高度和组成，云可以使地球表面变冷或变暖。影响云发展的因素包括不同高度的大气条件，如温度、压力和湿度，更好的预测通常需要更多的变量。云形成的计算机模拟可以通过改变特定参数来检验不同假设的影响。目前人们感兴趣的一个假设是环形涡结构的作用。就像烟圈的普通例子一样，这些结构发生在以不同速度移动的空气团相互作用时，产生的环状结构可以在消散之前传播一段很长的距离。在云的形成过程中，漩涡环可以使云顶上升到更高的高度，那里的温度更低，云的形成更容易。但是，在数值云形成模型中包含涡环需要对涡环的性质和来源进行假设。虽然涡性在流体动力学中被很好地理解，但在模拟中检测涡环并不容易，并且已经开发了许多方法，例如q准则，这是一种二次计算，表明空间区域的涡状程度。在3-D中，使用可视化工具将数据呈现在屏幕上，这使得人类更容易在视觉上发现漩涡。尽管这是一种强大的方法，但它是人力密集型的，并且在更大的数据集上分解。因此，一种可靠的自动探测涡旋环的方法对于研究云的形成是非常重要的。幸运的是，数学有一些工具可以适应这项任务。特别地，我们可以看看一个物体的“属”——即它有把手的数量，一个环总是有一个把手，就像在咖啡杯上看到的那样。这种“属”是被称为拓扑学的数学分支所研究的基本性质之一，并且可以自动计算。然而，在气象学中使用这种方法并不容易，因为有几个参数必须正确选择。因此，该项目将研究可用于使用拓扑学自动检测涡旋环的参数，以便建立可靠的工具来分析云的形成。

Prediction and modelling of climate enhance our understanding of atmospheric phenomena, climate, and daily weather. These are key in forming our understanding of the long-term climate trends and the likely impacts of climate change. Clouds play a crucial role in climate models, and can either cool or warm the Earth's surface, depending on their altitude and composition. Factors affecting cloud development include atmospheric conditions at different altitudes such as temperature, pressure, and humidity, with better predictions typically requiring more variables. Computer simulations of cloud formation can examine the effects of different hypotheses by changing specific parameters. One hypothesis currently of interest is the role of structures known as ring vortices. Like the mundane example of a smoke ring, these structures occur when parcels of air moving at different speeds interact, generating ring shapes which can travel a significant distance before dissipating. In cloud formation, vortex rings can cause the cloud tops to rise to higher altitudes where temperatures are colder and cloud formation is easier. But including vortex rings in numerical cloud formation models requires developing hypotheses about the nature and source of vortex rings. While vorticity is well understood in fluid dynamics, detecting vortex rings in simulations is not easy, and a number of approaches have been developed, such as the Q-criterion, a secondary computation which indicates how vortex-like a region of space is. In 3-D, this makes the task of spotting vortices visually easier for the human, using visualisation tools to present the data on a screen. Although this is a powerful approach, it is human-intensive and breaks down on larger data sets. A reliable way to detect vortex rings automatically is therefore important for studying cloud formation. Fortunately, the mathematics has tools that can be adapted for this task. In particular, we can look at the "genus" of an object - i.e. the number of handles it has, and a ring always has one handle, as can be seen on a coffee cup. This "genus" is one of the fundamental properties studied in the branch of mathematics known as topology, and can be computed automatically.However, using this in meteorology is not easy, as there are several parameters that must be chosen correctly. This project will therefore study the parameters that can be used to detect vortex rings automatically using topology in order to build reliable tools for analysis of cloud formation.