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”，其中一个关键的物理量，如密度，呈现出深度的阶梯结构，而不是像人们可能期望的那样更平滑地分布。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，在这些不同的系统中，分层的潜在物理原理是否相同。理解阶梯的形成不仅具有内在的科学意义，而且对理解湍流输运也至关重要。层状和非层状系统具有非常不同的输运性质；因此，为了能够在大型海洋学或大气模式中实际地参数化湍流输送，了解这一过程是至关重要的。在大气和海洋的背景下，已知有两个特别的系统易受分层影响。一种是双扩散对流；在海洋中，关键的量是热量和盐，它们以非常不同的速率扩散——这个过程被称为热盐对流或热溶质对流。因此，对流可以由不稳定的溶质梯度和稳定的热梯度驱动（在较暖的海洋中发生），或者由不稳定的热梯度和稳定的溶质梯度驱动（在较冷的海洋中）。在这两种情况下，密度被观察到采取楼梯结构，这似乎是非常有弹性的。有趣的是，一个类似的过程被认为在恒星核心中很重要，那里的竞争元素是热量和成分梯度。在大气中起作用的另一个分层过程是在位涡中形成的层（阶梯），这表现为强喷流的出现。这个过程不仅与我们自己的大气层有关，而且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。该项目将通过简化模型来探索楼梯形成的整个本质——它们形成所需的物理成分，它们最初形成的规模，以及随后的楼梯合并是如何发生的。至少在最初，模型方程将是一个空间方向（例如，热盐对流的高度）和时间上的非线性偏微分方程。这些将比三维流体动力学的完整方程更简单，但仍然可以详细分析和理解提供分层所需的具体内容。该项目将结合分析方法和渐近方法，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.