论文标题
分析两层能量平衡模型：长时间行为和温室效应
Analysis of a two-layer energy balance model: long time behaviour and greenhouse effect
论文作者
论文摘要
我们研究了两层能量平衡模型，该模型允许在表面层和大气之间进行垂直交换。表面温度和大气温度的演化方程是通过红外辐射的发射耦合到一个水平的，该发射由另一层捕获，以及所有非辐射垂直能量交换的效果。因此，基本参数是大气的吸收性，表示为$ε_a$。 $ε_a$的值严重取决于温室气体：$ co_2 $的浓度增加，$ ch_4 $导致更不透明的氛围，$ε_a$的值更高。首先，我们证明，当且仅当$ε_a\ in（0，2）$中时，系统的解决方案的整体存在存在，并且如果$ε_a> 2 $，则会在有限的时间内发生爆炸。 （请注意，$ε_a$的物理范围是$（0，1] $。）接下来，我们解释了（0，2）$中$ε_a\ $ε_a\的长时间动态，我们证明所有解决方案都会融合到某个平衡点。最后，由物理上下文激励，我们研究了对循环的依赖，并在涉及的参数方面进行了尊重，并且我们证明了参数的温度，我们会互动，我们的差异是在涉及的参数方面，我们是在涉及的参数方面的依赖。 $ε_a$。
We study a two-layer energy balance model, that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being captured by the other layer, and the effect of all non radiative vertical exchanges of energy. Therefore, an essential parameter is the absorptivity of the atmosphere, denoted $ε_a$. The value of $ε_a$ depends critically on greenhouse gases: increasing concentrations of $CO_2$ and $CH_4$ lead to a more opaque atmosphere with higher values of $ε_a$. First we prove that global existence of solutions of the system holds if and only if $ε_a \in (0, 2)$, and blow up in finite time occurs if $ε_a > 2$. (Note that the physical range of values for $ε_a$ is $(0, 1]$.) Next, we explain the long time dynamics for $ε_a \in (0, 2)$, and we prove that all solutions converge to some equilibrium point. Finally, motivated by the physical context, we study the dependence of the equilibrium points with respect to the involved parameters, and we prove in particular that the surface temperature increases with respect to $ε_a$. This is the key mathematical manifestation of the greenhouse effect.
