Extended Data Fig. 4: Dispersion relation solutions over a wide range of planetary transport conditions.
From: Emergence of wind ripples controlled by mechanics of grain–bed impacts
Different planetary conditions are represented by changing the average hop length 〈ℓ〉. Positive values are shown as solid lines and negative values are dashed lines. The growth rate σ rescaled by the homogeneous vertical flux φ0 and the grain size d versus versus λ rescaled by the weighted average lag length \({\langle {\varepsilon }_{\theta }\rangle }_{n}={\langle \bar{\varepsilon }\cot \theta \rangle }_{n}/{\langle \cot \theta \rangle }_{n}\) shows that the fastest-growing wavelength in this model scales with \({\langle {\varepsilon }_{\theta }\rangle }_{n}\). The rescaled speed c versus λ rescaled by average hop length 〈ℓ〉 shows that the ripple propagation speed in this model is a unique function of λ/〈ℓ〉—and when evaluated at the fastest-growing wavelength \({\langle {\varepsilon }_{\theta }\rangle }_{n}\propto d\) the associated speed is c/φ0 = f(〈ℓ〉/d). These solutions were derived for α = 0, γn = 0.4, γε = 0.25, γθ = 0.25 and \({\bar{\varepsilon }}_{1d}/d=\cot {\theta }_{1d}=1\). These parameters represent typical conditions for saltation, ρp/ρf ≳ 100 (Extended Data Table 1). We can see that for λ ≈ 200d the ripple-antiripple transition (c = 0) occurs at a critical hop length 〈ℓ〉c ≈ 67d consistent with stationary ripple simulations observed for Titan (Fig. 4). To a good approximation the speeds evaluated at the fastest-growing wavelength scale as \(c/{\varphi }_{0}\propto {(\langle \ell \rangle /{\langle \ell \rangle }_{c})}^{{\gamma }_{c}}-1\), with γc ≈ 0.4 for the parameters used.
