For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in the past our planet flipped between these two states. The main physical mechanism responsible for such an instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attraction. In the weak noise limit, large deviation laws define the invariant measure, the statistics of escape times, and typical escape paths called instantons. By constructing the instantons empirically, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first-order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we put forward a new method for constructing Melancholia states from direct numerical simulations, black which provides a possible alternative with respect to the edge-tracking algorithm.

中文翻译：

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气候的全球稳定性特性：忧郁状态、不变测度和相变

对于入射太阳辐射的大范围值，地球至少具有两个吸引状态，它们对应于相互竞争的气候。温暖的气候类似于现在的气候；滚雪球气候以全球冰川和几乎无法支持生命形式的条件为特征。古气候证据表明，过去我们的星球在这两种状态之间翻转。造成这种不稳定性的主要物理机制是冰反照率反馈。在之前的工作中，我们定义了介于两种气候之间的忧郁症状态。这种状态嵌入在两个吸引力盆地之间的边界内，并且在低纬度地区具有广泛的冰川作用。这里，我们通过引入随机扰动作为对入射太阳辐射强度的调制来探索系统的全局稳定性特性。我们观察到相互竞争的吸引力盆地之间由噪声引起的转变。在弱噪声极限中，大偏差定律定义了不变测度、逃逸时间的统计以及称为瞬时子的典型逃逸路径。通过凭经验构建瞬时子，我们表明忧郁状态是噪声引起的转变的门户。在多稳态区域中，在零噪声极限下，该测量仅在竞争吸引子之一上得到支持。对于太阳辐照度的低（高）值，限制措施是雪球（温暖）气候。两种状态之间的转换对应于系统中的一阶相变。我们提出的框架似乎与复杂多稳态系统的研究具有普遍意义。最后，我们提出了一种从直接数值模拟中构建 Melancholia 状态的新方法，黑色为边缘跟踪算法提供了一种可能的替代方案。