Ice clouds, so-called cirrus clouds, occur very frequently in the tropopause region. A special class are subvisible cirrus clouds with an optical depth lower than 0.03, associated with very low ice crystal number concentrations. The dominant pathway for the formation of these clouds is not known well. It is often assumed that heterogeneous nucleation on solid aerosol particles is the preferred mechanism although homogeneous freezing of aqueous solution droplets might be possible, since these clouds occur in the low-temperature regime iT/i?&?235?K. For investigating subvisible cirrus clouds as formed by homogeneous freezing we develop a reduced cloud model from first principles, which is close enough to complex models but is also simple enough for mathematical analysis. The model consists of a three-dimensional set of ordinary differential equations, and includes the relevant processes as ice nucleation, diffusional growth and sedimentation. We study the formation and evolution of subvisible cirrus clouds in the low-temperature regime as driven by slow vertical updraughts (0?&?iw/i?≤?0.?05?m?ssup?1/sup). The model is integrated numerically and also investigated by means of theory of dynamical systems. We found two qualitatively different states for the long-term behaviour of subvisible cirrus clouds. The first state is a stable focus; i.e. the solution of the differential equations performs damped oscillations and asymptotically reaches a constant value as an equilibrium state. The second state is a limit cycle in phase space; i.e. the solution asymptotically approaches a one-dimensional attractor with purely oscillatory behaviour. The transition between the states is characterised by a Hopf bifurcation and is determined by two parameters – vertical updraught velocity and temperature. In both cases, the properties of the simulated clouds agree reasonably well with simulations from a more detailed model, with former analytical studies, and with observations of subvisible cirrus, respectively. The reduced model can also provide qualitative interpretations of simulations with a complex and more detailed model at states close to bifurcation qualitatively. The results indicate that homogeneous nucleation is a possible formation pathway for subvisible cirrus clouds. The results motivate a minimal model for subvisible cirrus clouds (SVCs), which might be used in future work for the development of parameterisations for coarse large-scale models, representing structures of clouds.
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机译：冰云，即所谓的卷云，在对流层顶地区非常频繁地发生。一类特殊的是光学深度小于0.03的亚可见卷云，其冰晶数浓度非常低。这些云形成的主要途径尚不清楚。尽管水溶液液滴的均匀冻结是可能的，但通常认为固体气溶胶颗粒上的异相成核是优选的机制，因为这些云团在低温条件下发生，并且温度＜235。 K.为了研究由均质冻结形成的亚可见卷云，我们从第一原理开发了简化的云模型，该模型与复杂模型足够接近，但对于数学分析也足够简单。该模型由三组常微分方程组组成，并包括冰成核，扩散生长和沉降等相关过程。我们研究了由缓慢的垂直上升气流（0？<？ w ？≤？0.？05？m？s 引起的）在低温状态下亚可见卷云的形成和演化。 ？1 ）。该模型经过数值集成，并且还通过动力学系统理论进行了研究。对于亚可见卷云的长期行为，我们发现了两个性质不同的状态。第一个状态是稳定的焦点;即，微分方程的解执行阻尼振荡并且渐近地达到恒定值作为平衡状态。第二种状态是相空间中的极限环。即解决方案渐近逼近具有纯粹振荡行为的一维吸引子。状态之间的过渡以Hopf分叉为特征，并由两个参数确定-垂直上风速度和温度。在这两种情况下，模拟云的性质与更详细的模型的模拟，先前的分析研究以及对亚可见卷云的观测都相当吻合。简化的模型还可以在定性接近分叉的状态下使用复杂且更详细的模型对模拟进行定性解释。结果表明，均匀成核是亚可见卷云的可能形成途径。结果激发了亚可见卷云（SVC）的最小模型，该模型可在将来的工作中用于开发代表云结构的粗略大规模模型的参数化。
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