Collaborative Research: CMG--Anti-Turbulence, Horizontal Convection and Thermalence

水平对流是指由流体密度的空间梯度引起的流体板的水平运动。相当令人惊讶的是，理论论证表明，流体层顶部表面的不均匀加热或冷却对驱动环流是无效的。更具体地说，如果作用在流体上的唯一力是非均匀表面加热，并且如果运动粘度和热扩散率在其比值固定的情况下接近于零，那么可以证明，通过对流体的整个体积进行积分确定的单位质量的能量耗散趋于零。因为没有能量耗散意味着没有湍流，这有时被称为“反湍流定理”。这个结果是违反直觉的。事实上，人们一直认为，全球尺度海洋环流的主要特征可能是由两极到赤道的温度梯度来解释的，但抗湍流定理似乎暗示，表面的差异加热本身并不能驱动环流。这一建议的动机是需要理解的物理过程，说明反湍流定理和其他纯粹的数学约束水平对流。这些约束的含义是，最小尺度上的流体运动必须与更大尺度的运动紧密耦合，作为一个相互联系的多尺度系统的一部分。目标是将对水平对流的更好理解与海洋环流问题联系起来。该方法包括解析和数值技术的结合，如上界理论，利用连续方法计算平衡解，这些平衡的线性稳定性分析，多尺度渐近方法和直接数值模拟。这项工作将有助于改进海洋环流和气候的模型。它是由海洋科学部和数学科学部通过国家科学基金会项目“数学科学与地球科学研究合作”（CMG）共同支持的。

Horizontal convection refers to horizontal motions in a fluid slab caused by spatial gradients of fluid density. Rather surprisingly, theoretical arguments show that nonuniform heating or cooling at the top surface of a fluid layer is ineffective in driving the circulation. More specifically, if the only force acting on the fluid is nonuniform surface heating, and if the kinematic viscosity and thermal diffusivity are allowed to approach zero with their ratio fixed, then it can be shown that the energy dissipation per unit mass, determined by integrating over the whole volume of the fluid, goes to zero. Because no energy dissipation implies no turbulence, this is sometimes called the "anti-turbulence theorem." This result is counterintuitive. In fact, it has been thought that the main features of the global-scale ocean circulation might be accounted for by the pole-to-equator temperature gradient, but the anti-turbulence theorem seems to imply that differential heating of the surface cannot by itself drive the circulation. This proposal is motivated by the need to understand the physical processes that account for the anti-turbulence theorem and other purely mathematical constraints on horizontal convection. Implications of these constraints are that fluid motions on the smallest scales must be closely coupled to larger-scale motions as part of a linked, multiscale system. The goal is to relate the improved understanding of horizontal convection to the problem of ocean circulation. The approach consists of a combination of analytical and numerical techniques such as upper bound theory, numerical computation of equilibrium solutions using continuation methods, linear stability analysis of these equilibria, multiscale asymptotic methods, and direct numerical simulation. The work will contribute to improved modeling of the ocean circulation and hence climate. It is supported jointly by the Division of Ocean Sciences and the Division of Mathematical Sciences through the NSF Program, Research Collaborations between the Mathematical Sciences and the Geosciences (CMG).