Analysis of a two-layer energy balance model: Long time behavior and greenhouse effect.
Analysis of a two-layer energy balance model: Long time behavior and greenhouse effect.
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DOI：
10.1063/5.0136673
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发表时间：
2022-11
期刊：
影响因子：
2.9
通讯作者：
P. Cannarsa;V. Lucarini;P. Martinez;Cristina Urbani;J. Vancostenoble
中科院分区：
文献类型：
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作者：
P. Cannarsa;V. Lucarini;P. Martinez;Cristina Urbani;J. Vancostenoble
我们研究了一个两层能量平衡模型，该模型允许表层和大气之间的垂直交换。表面温度和大气温度的演化方程通过一个层的红外辐射发射（该发射被另一层部分捕获）以及所有非辐射垂直能量交换的影响来耦合。因此，一个重要参数是大气的吸收率，用εa表示。 εa 的值主要取决于温室气体：CO2 和 CH4 浓度的增加导致大气更加不透明，εa 值更高。首先，我们证明系统解的全局存在性当且仅当εa∈(0,2)时成立，并且如果εa>2则在有限时间内发生爆炸。 （请注意，εa 值的物理范围是 (0,1]。）接下来，我们解释 εaε(0,2) 的长期动力学，并证明所有解都收敛于某个平衡点。最后，在物理背景的推动下，我们研究了平衡点与所涉及参数的依赖性，特别证明了表面温度相对于 εa 单调增加。这是温室效应的关键数学表现。
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 partly 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 εa. The value of εa depends critically on greenhouse gases: increasing concentrations of CO2 and CH4 lead to a more opaque atmosphere with higher values of ϵa. First, we prove that global existence of solutions of the system holds if and only if εa∈(0,2) and blow up in finite time occurs if εa>2. (Note that the physical range of values for εa is (0,1].) Next, we explain the long time dynamics for εa∈(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 monotonically with respect to εa. This is the key mathematical manifestation of the greenhouse effect.