Abstract The atmosphere is an exemplary case of uncertain system. The state of such systems is described by means of probability density functions which encompass uncertainty information. In this regard, the Liouville equation is the theoretical framework to predict the evolution of the state of uncertain systems. This study analyses the morphological characteristics of the time evolution of probability density functions for some low complexity geophysical systems by solving the Liouville equation in order to obtain tractable solutions which are otherwise unfeasible with currently available computational resources. The current and usual modest approach to overcome these obstacles and estimate the probability density function of the system in realistic weather and climate applications is the use of a discrete and small number of samples of the state of the system, evolved individually in a deterministic, perhaps sometimes stochastic, way. We investigate particular solutions of the shallow water equations and the barotropic model that allow to apply the Liouville formalism to explore its topological characteristics and interpret them in terms of the ensemble prediction system approach. We provide quantitative evidences of the high variability that solutions to Liouville equation may present, challenging currently accepted uses and interpretations of ensemble forecasts.

中文翻译：

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通过求解现实地球物理模型的刘维尔方程探索集合预报的极限

摘要 大气是不确定系统的一个典型例子。这种系统的状态通过包含不确定性信息的概率密度函数来描述。对此，刘维尔方程是预测不确定系统状态演化的理论框架。本研究通过求解 Liouville 方程，分析了一些低复杂度地球物理系统概率密度函数时间演化的形态特征，以获得可用现有计算资源无法实现的易处理解。在现实天气和气候应用中克服这些障碍和估计系统概率密度函数的当前和通常的适度方法是使用离散和少量的系统状态样本，以确定性的，也许是单独演化的有时随机，方式。我们研究了浅水方程和正压模型的特定解，这些解允许应用 Liouville 形式主义来探索其拓扑特征，并根据集合预测系统方法对其进行解释。我们提供了 Liouville 方程解可能呈现的高度可变性的定量证据，挑战了目前公认的集合预报的使用和解释。我们研究了浅水方程和正压模型的特定解，这些解允许应用 Liouville 形式主义来探索其拓扑特征，并根据集合预测系统方法对其进行解释。我们提供了 Liouville 方程解可能呈现的高度可变性的定量证据，挑战了目前公认的集合预报的使用和解释。我们研究了浅水方程和正压模型的特定解，这些解允许应用 Liouville 形式主义来探索其拓扑特征，并根据集合预测系统方法对其进行解释。我们提供了 Liouville 方程解可能呈现的高度可变性的定量证据，挑战了目前公认的集合预报的使用和解释。