Nakao-Kusune Sachi, Sakaue Takahiro, Nishimori Hiraku, Nakanishi Hiizu
Department of Physics, Kyushu University, Fukuoka 819-0395, Japan.
Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Kanagawa 252-5258, Japan.
Phys Rev E. 2020 Jan;101(1-1):012903. doi: 10.1103/PhysRevE.101.012903.
It has been observed that the direction in which a sand dune extends its crest line depends on seasonal variation of wind direction; when the variation is small, the crest line develops more or less perpendicularly to the mean wind direction to form a transverse dune with some undulation. In the case of bimodal wind with a large relative angle, however, the dune extends its crest along the mean wind direction and evolves into an almost straight longitudinal dune. Motivated by these observations, we investigate the dynamical stability of isolated dunes using the crest line model, where the dune dynamics is represented by its crest line motion. First, we extend the previous linear stability analysis under the unidirectional wind to the case with nonzero slant angle between the wind direction and the normal direction of the crest line, and show that the stability diagram does not depend on the slant angle. Second, we examine how the linear stability is affected by the seasonal changes of wind direction in the case of bimodal wind with equal strength and duration. For the transverse dune, we find that the stability is virtually the same with that for the unidirectional wind as long as the dune evolution during a season is small. On the other hand, in the case of the longitudinal dune, the dispersions of the growth rates for the perturbation are drastically different from those of the unidirectional wind, and we find that the largest growth rate is always located at k=0. This is because the growth of the perturbation with k≠0 is canceled by the alternating wind from opposite sides of the crest line even though it grows during each duration period of the bimodal wind. For a realistic parameter set, the system is in the wavy unstable regime of the stability diagram for the unidirectional wind, thus the straight transverse dune is unstable to develop undulation and eventually evolves into a string of barchans when the seasonal variation of wind direction is small, but the straight longitudinal dune is stabilized under the large variation of bimodal wind direction. We also perform numerical simulations on the crest line model, and find that the results are consistent with our linear analysis and the previous reports that show that the longitudinal dunes tend to have a straight ridge elongating over time.
据观察，沙丘延伸其脊线的方向取决于风向的季节性变化；当变化较小时，脊线大致垂直于平均风向发展，形成带有一些起伏的横向沙丘。然而，在相对角度较大的双峰风情况下，沙丘沿平均风向延伸其脊线，并演变成几乎笔直的纵向沙丘。受这些观察结果的启发，我们使用脊线模型研究孤立沙丘的动力学稳定性，其中沙丘动力学由其脊线运动表示。首先，我们将先前在单向风下的线性稳定性分析扩展到风向与脊线法线方向之间存在非零倾斜角的情况，并表明稳定性图不依赖于倾斜角。其次，我们研究了在强度和持续时间相等的双峰风情况下，风向的季节性变化如何影响线性稳定性。对于横向沙丘，我们发现只要一个季节内沙丘的演化较小，其稳定性与单向风的情况基本相同。另一方面，在纵向沙丘的情况下，扰动增长率的色散与单向风的情况有很大不同，并且我们发现最大增长率总是位于k = 0处。这是因为即使在双峰风的每个持续时间段内扰动都会增长，但k≠0的扰动增长会被来自脊线两侧的交替风抵消。对于实际参数集，该系统处于单向风稳定性图的波动不稳定区域，因此当风向的季节性变化较小时，笔直的横向沙丘不稳定，会发展出起伏，最终演变成一串新月形沙丘，但在双峰风方向的大变化下，笔直的纵向沙丘会稳定下来。我们还对脊线模型进行了数值模拟，发现结果与我们的线性分析以及先前的报告一致，这些报告表明纵向沙丘往往会随着时间的推移有一条笔直的脊线延伸。
