Staircase Formation in Fluid Dynamical Systems

某些流体动力系统最有趣的特征之一是它们倾向于形成层或“阶梯”，在这种情况下，一个关键的物理量，如密度，呈现出一种有深度的阶梯结构，而不是像人们预期的那样分布得更平稳。这种现象发生在物理上看起来相当不同的系统中，因此一个令人感兴趣的研究问题是，在本质上，分层的基本物理在这些不同的系统中是否相同。了解楼梯的形成不仅具有内在的科学价值，而且对于了解湍流的传输也是至关重要的。分层系统和非分层系统具有非常不同的传输特性；因此，为了在大型海洋或大气模式中真实地将湍流传输参数化，了解这一过程是至关重要的。在大气和海洋的背景下，已知有两个特定的感兴趣的系统容易受到分层的影响。一种是双扩散对流；在海洋中，关键的量是热和盐，它们以非常不同的速度扩散--这个过程被称为温盐或热溶解对流。对流既可以由具有稳定热梯度的不稳定的溶质梯度驱动(在较暖的海洋中发生)，也可以由具有稳定的溶质梯度的不稳定的热梯度驱动(较冷的海洋中的情况)。在这两种情况下，密度都呈现出阶梯结构，似乎具有显著的弹性。有趣的是，类似的过程被认为在恒星核心中也很重要，在那里相互竞争的元素是热和成分梯度。在大气中起作用的另一个分层过程是在位涡中形成层(阶梯)，这表现为强急流的出现。这个过程不仅与我们自己的大气层有关，而且还与木星大气层的外层有关，木星大气层的特征是强烈的带状结构。这个项目将通过简化的模型来探索楼梯形成的整个性质--它们形成所必需的物理成分，它们最初形成的规模，以及随后的楼梯合并是如何发生的。模型方程至少在最初将是一个空间方向(例如，对于温盐对流的高度)和时间的非线性偏微分方程组。这些方程将比三维流体动力学的完整方程更简单，但仍将允许详细分析和理解提供分层所需的确切条件。该项目将结合解析法和渐近法，以及模型方程的数值解。

One of the most interesting features of certain fluid dynamical systems is their tendency to form layers or "staircases", in which a key physical quantity, such as the density, exhibits a staircase structure with depth, rather than being more smoothly distributed as one might expect. This phenomenon occurs in systems that appear to be rather different physically, and so a research question of some interest is whether, at heart, the underlying physics of layering is the same in these different systems. Understanding staircase formation is not only of intrinsic scientific interest but is vital to understanding turbulent transport. Layered and unlayered systems have very different transport properties; thus it is essential to understand the process in order that turbulent transport can be realistically parameterised in large oceanographic or atmospheric models.In the context of atmospheres and oceans there are two particular systems of interest that are known to be susceptible to layering. One is double-diffusive convection; in the oceans the key quantities are heat and salt, which diffuse at very different rates - the process is then known as thermohaline or thermosolutal convection. Convection can then be driven either by an unstable solutal gradient with a stable thermal gradient (which occurs in warmer oceans) or, alternatively, by an unstable thermal gradient with a stable solutal gradient (the situation in colder oceans). In both these cases, the density is observed to take on a staircase structure, which appears to be remarkably resilient. Interestingly, a similar process is believed to be of importance in stellar cores, where the competing elements are heat and a compositional gradient. The other layering process at work in the atmosphere is that of the formation of layers (staircases) in potential vorticity, which is manifested by the appearance of strong jets. This process is relevant not only for our own atmosphere, but also for the outer layers of Jupiter's atmosphere, which is characterised by a strongly banded structure.This project will explore, through simplified models, the entire nature of staircase formation - the physical ingredients necessary for their formation, the scale at which they initially form, and how subsequent staircase mergers occur. The model equations will, initially at least, be nonlinear partial differential equations in one spatial direction (height, for example, for the case of thermohaline convection) and time. These will be simpler than the full equations of three-dimensional fluid dynamics, but will nonetheless allow a detailed analysis and understanding of what exactly is needed to provide layering. The project will involve a combination of analytical and asymptotic approaches, together with numerical solutions of the model equations.