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Stabilization of straight longitudinal dune under bimodal wind with large directional variation

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 non-zero 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. Secondly, 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 $k \ne 0$ 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.