Extended Data Fig. 1: Scale-free temporal fluctuations in the number of grains in transport.
From: Emergence of wind ripples controlled by mechanics of grain–bed impacts
(a-d) DEM simulations of transport on Earth showing a time series of the number of grains in transport per unit area n(t) rescaled by the average N ≈ 1/50d2. Panel (b) is the zoomed in box outlined in (a) and (c) is the black box in (b) and similarly (d) is the zoomed box of (c). Panel (e) shows the spectrum of fluctuations in n(t) of a given frequency \(| \hat{n}(f)|\) (that is, the square root of the power spectrum). The spectrum shows a power law decay as 1/f hinting at the critical sensitivity of the granular surface: With the wind shear stress at the bed buffered to the threshold of motion10,24, the bed is sensitive to perturbations. In this condition, when an impact ejects grains that then go on to eject other grains, and so on, this feedback can cascade into large bursts of transport (Supplementary Videos). These cascades cause fluctuations in transport across a range of scales. Highlighted by the solid markers in (e) are the frequency windows used in (a-d). Note that (b-d) fall in the scale-free region. The insets in (e) show the spectrum for Mars (red), Earth (blue), Titan (green) and Venus (gold) all showing similar scale-free \(| \hat{n}| \propto 1/f\) spectra. We note that the subtle peaks in the spectral density around \(f \approx 0.03\sqrt{g/d}\) and \(f \approx 0.4\sqrt{g/d}\) correspond to the characteristic transport time scales. The \(f \approx 0.03\sqrt{g/d}\) peak is set by the average travel time scale related to 〈ℓ〉 (therefore this peak occurs at smaller frequencies on Mars; see inset). And the \(f \approx 0.4\sqrt{g/d}\) peak is set by the average ejection time scale—the time it takes for grains to go from bed-bound to ’counted’ as a part of transport.
