Analysis of surface wave formation of icicles and its application to the formation of stalactites

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We have proposed a new model for the surface formation of icicle, we also proposed some mathematically related works, and some experiments and some field works are done.For the problem of icicle, it was found that we cannot express the experimental fact (formation of surface wave with specific wave length) when we take into account the surface tension of water. Further consideration is necessary to find some physical factors which explain the experimental results. On the other hand, the similar model made by Ueno says that every experimental result can be explained. However, first, his boundary condition for the temperature field can not be accepted physically. Second, his result which suggests the motion of wave going upward is asserted to be proved by the trace of bubbles inside of icicles. But we cannot find the explicit motion of waves on the surface of icicle by hundreds video filming and the picture of formation of many crystals inside of icicles. So we have concluded that there is no reason to connect the wave motion and the trace of bubbles inside of icicles. And it should be said that this problem is still an open one. We will further continue the research on this point.Next we have considered the related mathematical problems. One is about the surface tension. We considered its value from the statistical mechanics view point, and next tried the problem of measurement of one from the form of pendant drop. Further the morphology of snow grains from the view point of Gaussian curvature is discussed. This is really related to the surface energy. On the other hand, To study the relation between the thermal problem and surface formation, phase field model is studied to compare to the usual theories.The field works to take the data on ice-stalagmites, stalactites, stalagmites and rim stones are done in 6 caves in Hokkaido, Iwate prefecture and Okinawa islands. These data are used to construct the new morphological theories for these objects.