标题： 统计与动力学性质在耦合的大气-海洋模型中的协变李雅普诺夫向量 - 多尺度效应、几何退化和误差动力学 显示英文标题

标题： Statistical and Dynamical Properties of Covariant Lyapunov Vectors in a Coupled Atmosphere-Ocean Model - Multiscale Effects, Geometric Degeneracy, and Error Dynamics

摘要： 我们使用协变李雅普诺夫向量（CLVs）的形式主义来研究一个简化的耦合大气-海洋模型，该形式主义将基于物理的扰动方向与增长/衰减率联系起来。 该模型是通过对两种流体的准地转方程进行严重截断得到的，并包括了它们动力学/热力学耦合的简单但具有物理意义的表示。 该模型有36个自由度，参数的选择使得观察到混沌行为。 发现有两个正的李雅普诺夫指数（LEs），十六个负的LEs，以及十八个接近零的LEs。 许多接近零的LEs的存在来自于两种流体特征时间尺度之间的巨大时间尺度分离，并导致由相应CLVs张成的切空间中非平凡的误差增长特性，这些CLVs在几何上非常退化。 这样的CLVs对应于两种不同的海洋/大气耦合模式。 而由正负LEs对应的CLVs张成的切空间则表现出非病理性的行为，并且可以为有限时间LEs构建稳健的大偏差定律，从而提供了一个通用模型，用于评估沿这些方向上的长期到超长尺度的可预测性。 令人有些惊讶的是，不稳定流形的切空间在大气和海洋分量上都有很强的投影。 我们的结果突显了在多尺度混沌动力系统中使用双曲性作为概念框架的困难，而部分双曲性框架似乎更合适，这可能表明了混沌假设的一种替代定义。 我们的结果表明，当研究耦合的大气-海洋模型时，需要对不同时间和空间尺度上的误差动力学进行准确分析，并仔细设置同化方案。 显示英文摘要

摘要： We study a simplified coupled atmosphere-ocean model using the formalism of covariant Lyapunov vectors (CLVs), which link physically-based directions of perturbations to growth/decay rates. The model is obtained via a severe truncation of quasi-geostrophic equations for the two fluids, and includes a simple yet physically meaningful representation of their dynamical/thermodynamical coupling. The model has 36 degrees of freedom, and the parameters are chosen so that a chaotic behaviour is observed. One finds two positive Lyapunov exponents (LEs), sixteen negative LEs, and eighteen near-zero LEs. The presence of many near-zero LEs results from the vast time-scale separation between the characteristic time scales of the two fluids, and leads to nontrivial error growth properties in the tangent space spanned by the corresponding CLVs, which are geometrically very degenerate. Such CLVs correspond to two different classes of ocean/atmosphere coupled modes. The tangent space spanned by the CLVs corresponding to the positive and negative LEs has, instead, a non-pathological behaviour, and one can construct robust large deviations laws for the finite time LEs, thus providing a universal model for assessing predictability on long to ultra-long scales along such directions. It is somewhat surprising to find that the tangent space of the unstable manifold has strong projection on both atmospheric and oceanic components. Our results underline the difficulties in using hyperbolicity as a conceptual framework for multiscale chaotic dynamical systems, whereas the framework of partial hyperbolicity seems better suited, possibly indicating an alternative definition for the chaotic hypothesis. Our results suggest the need for accurate analysis of error dynamics on different time scales and domains and for a careful set-up of assimilation schemes when looking at coupled atmosphere-ocean models.