The formation and life-span of clouds as well as the associated unsteady processes concerning the micro-physics of the water phases they may contain are open questions in atmospheric physics.

We here use three-dimensional direct numerical simulation to analyse the temporal evolution of a small portion of the top of a cloud. The Eulerian description of the turbulent velocity, temperature and vapor fields is combined with the Lagrangian description of two different ensembles of cloud droplets, that is, with a monodisperse and a polydisperse size distribution. A shear-free turbulent mixing layer is used to model the background air flow of the cloud top. This flow is considered appropriate because clouds cannot stand the presence of shear, which inevitably destroys them quickly. Luke-warm clouds are generally found at an altitude of 1000-2000 meters, live for a few hours or up to 1-2 days, continuously change shape, and have typical dimensions of some hundreds of meters. The global time-scale of these changes is recognized as being of the order of 100 seconds (Shaw (2003), Warhaft (2009)). From the formation phase to the dying out phase, clouds live under a continuous sequence of transients that are slightly different one from the other.

In this study, we have tried to reduce the simplification level with respect to the real warm cloud situation as much as possible. We have included the same level of supersaturation of warm clouds, the same amount of liquid water content, and thus, the same numerical number of water droplets, and finally, a typical unstable perturbation of the density stratification and a typical kinetic energy cloud / clear air ratio (order of 10). We have considered an observation duration of the order of a few seconds (about 10 initial turnaround times). During this time, the kinetic energy decays throughout the system by 95%. It should be recalled that the kinetic energy inside the interfacial layer (the shear-free turbulent mixing layer that matches the cloud region to the ambient air region) also decays spatially, by nearly 85%. We observed, with respect to the cloud region, in the interfacial layer, a five times faster achievement of a common value of standard deviation for the probability density of both the monodisperse and poly-disperse populations. This acceleration of the dynamics is remarkable and is somewhat counterintuitive. It is closely correlated with the intermittency of the small scale of the air flow and of the supersaturation fluctuation. We give information on the size distribution of both the positive and negative droplet growth and on the drop size and the corresponding numerical concentration value of the distribution peak as time passes. Finally, we comment on the extension of the concept of the collision kernel for an unstable and inhomogeneous system in which turbulence decays faster than the time scales of the involved aqueous phases.

云的形成和寿命，以及与它们可能包含的水相的微观物理学有关的不稳定过程，是大气物理学中的未解决问题。

我们在这里使用三维直接数值模拟来分析一小部分云顶的时间演化。湍流速度，温度和蒸汽场的欧拉描述与云滴的两个不同集合的拉格朗日描述相结合，即具有单分散和多分散的粒度分布。使用无剪切湍流混合层来模拟云顶的背景气流。之所以认为这种流动是适当的，是因为云无法承受剪切力的作用，而剪切力不可避免地会很快破坏它们。温热云通常发现在1000-2000米的高度上，可以存活数小时或长达1-2天，不断变化形状，典型尺寸为几百米。这些变化的全球时间尺度被认为约为100秒（Shaw（2003），Warhaft（2009））。从形成阶段到濒临灭绝阶段，云团生活在一系列连续的瞬态序列之间，瞬态序列彼此之间略有不同。

在本研究中，我们尝试针对实际暖云情况尽可能降低简化程度。我们已经包括了相同水平的暖云过饱和度，相同量的液态水含量以及相同数量的水滴，最后还包括了密度分层的典型不稳定扰动和典型的动能云/晴空气比（10的数量级）。我们已经考虑了大约几秒钟的观察持续时间（大约10个初始周转时间）。在这段时间内，动能在整个系统中衰减了95％。应当记得，界面层（使云区域与周围空气区域匹配的无剪切湍流混合层）内部的动能在空间上也衰减了近85％。我们观察到 关于云区域，在界面层中，单分散和多分散总体的概率密度的标准偏差的公共值的获得速度快五倍。动态的这种加速是显着的，并且有点违反直觉。它与小规模气流和过饱和波动的间歇性密切相关。我们提供有关正负液滴生长的大小分布以及随着时间的流逝的液滴大小和分布峰的相应数值浓度值的信息。最后，我们评论了碰撞核的概念对于一个不稳定且不均匀的系统的扩展，在该系统中湍流的衰减快于所涉及的水相的时间尺度。在界面层中，单分散和多分散总体的概率密度的标准偏差的公共值的获得速度快五倍。动态的这种加速是显着的，并且有点违反直觉。它与小规模气流和过饱和波动的间歇性密切相关。我们提供有关正负液滴生长的大小分布以及随着时间的流逝的液滴大小和分布峰的相应数值浓度值的信息。最后，我们评论了碰撞核的概念对于一个不稳定且不均匀的系统的扩展，在该系统中湍流的衰减快于所涉及的水相的时间尺度。在界面层中，单分散和多分散总体的概率密度的标准偏差的公共值的获得速度快五倍。动态的这种加速是显着的，并且有点违反直觉。它与小规模气流和过饱和波动的间歇性密切相关。我们提供有关正负液滴生长的大小分布以及随着时间的流逝的液滴大小和分布峰的相应数值浓度值的信息。最后，我们评论了碰撞核的概念对于一个不稳定且不均匀的系统的扩展，在该系统中湍流的衰减快于所涉及的水相的时间尺度。单分散和多分散总体的概率密度的标准偏差的公共值的获得速度快五倍。动态的这种加速是显着的，并且有点违反直觉。它与小规模气流和过饱和波动的间歇性密切相关。我们提供有关正负液滴生长的大小分布以及随着时间的流逝的液滴大小和分布峰的相应数值浓度值的信息。最后，我们评论了碰撞核的概念对于一个不稳定且不均匀的系统的扩展，在该系统中湍流的衰减快于所涉及的水相的时间尺度。单分散和多分散总体的概率密度的标准偏差的公共值的获得速度快五倍。动态的这种加速是显着的，并且有点违反直觉。它与小规模气流和过饱和波动的间歇性密切相关。我们提供有关正负液滴生长的大小分布以及随着时间的流逝的液滴大小和分布峰的相应数值浓度值的信息。最后，我们评论了对于不稳定和不均匀系统的碰撞核概念的扩展，在该系统中湍流的衰减快于所涉及水相的时间尺度。