We develop a reduced complexity model for three-dimensional sand dunes, based on a simplified description of the longitudinal and lateral sand transport. The spatiotemporal evolution of a dune migrating over a nonerodible bed under unidirectional wind is reduced to the dynamics of its crest line, providing a simple framework for the investigation of three-dimensional dunes, such as barchan and transverse dunes. Within this model, we derive analytical solutions for barchan dunes and investigate the stability of a rectilinear transverse dune against lateral fluctuations. We show, in particular, that the latter is unstable only if the lateral transport on the dune slip face prevails over that on the upwind face. We also predict the wavelength and the characteristic time that control the subsequent evolution of an unstable transverse dune into a wavy ridge and the ultimate fragmentation into barchan dunes.

A bifurcation analysis of dune shape transition is made. By use of a reduced model of dune morphodynamics, the Dune Skeleton model, we elucidate the transition mechanism between different shapes of dunes under unidirectional wind. It was found that the decrease in the total amount of sand in the system and/or the lateral sand flow shifts the stable state from a straight transverse dune to a wavy transverse dune through a pitchfork bifurcation. A further decrease causes wavy transverse dunes to shift into barchans through a Hopf bifurcation. These bifurcation structures reveal the transition mechanism of dune shapes under unidirectional wind.

To analyze theoretically the stability of the shape and the migration process of transverse dunes and barchans, we propose a skeleton model of 3D dunes described with coupled dynamics of 2D cross-sections. First, 2D cross-sections of a 3D dune parallel to the wind direction are extracted as elements of a skeleton of the 3D dune, hence, the dynamics of each and interaction between them is considered. This model simply describes the essential dynamics of 3D dunes as a system of coupled ordinary differential equations. Using the model we study the stability of the shape of 3D transversal dunes and their deformation to barchans depending on the amount of available sand in the dune field, sand flow in parallel and perpendicular to wind direction.

We conduct numerical simulations based on a model of blowing snow to reveal the long-term properties and equilibrium state of aeolian particle transport from $10^{-5} \hspace{0.5 ex} \mathrm{m}$ to $10 \hspace{0.5 ex} \mathrm{m}$ above the flat surface. The numerical results are as follows. (i) Time-series data of particle transport are divided into development, relaxation, and equilibrium phases, which are formed by rapid wind response below $10 \hspace{0.5 ex} \mathrm{cm}$ and gradual wind response above $10 \hspace{0.5 ex} \mathrm{cm}$. (ii) The particle transport rate at equilibrium is expressed as a power function of friction velocity, and the index of 2.35 implies that most particles are transported by saltation. (iii) The friction velocity below $100 \hspace{0.5 ex} \mu\mathrm{m}$ remains roughly constant and lower than the fluid threshold at equilibrium. (iv) The mean particle speed above $300 \hspace{0.5 ex} \mu\mathrm{m}$ is less than the wind speed, whereas that below $300 \hspace{0.5 ex} \mu\mathrm{m}$ exceeds the wind speed because of descending particles. (v) The particle diameter increases with height in the saltation layer, and the relationship is expressed as a power function. Through comparisons with the previously reported random-flight model, we find a crucial problem that empirical splash functions cannot reproduce particle dynamics at a relatively high wind speed

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