River-bed sediments display two universal downstream trends: fining, in which particle size decreases; and rounding, where pebble shapes evolve toward ellipsoids. Rounding is known to result from transport-induced abrasion; however many researchers argue that the contribution of abrasion to downstream fining is negligible. This presents a paradox: downstream shape change indicates substantial abrasion, while size change apparently rules it out. Here we use laboratory experiments and numerical modeling to show quantitatively that pebble abrasion is a curvature-driven flow problem. As a consequence, abrasion occurs in two well-separated phases: first, pebble edges rapidly round without any change in axis dimensions until the shape becomes entirely convex; and second, axis dimensions are then slowly reduced while the particle remains convex. Explicit study of pebble shape evolution helps resolve the shape-size paradox by reconciling discrepancies between laboratory and field studies, and enhances our ability to decipher the transport history of a river rock.

The shape of fragments generated by the breakup of solids is central to a wide variety of problems ranging from the geomorphic evolution of boulders to the accumulation of space debris orbiting Earth. Although the statistics of the mass of fragments has been found to show a universal scaling behavior, the comprehensive characterization of fragment shapes still remained a fundamental challenge. We performed a thorough experimental study of the problem fragmenting various types of materials by slowly proceeding weathering and by rapid breakup due to explosion and hammering. We demonstrate that the shape of fragments obeys an astonishing universality having the same generic evolution with the fragment size irrespective of materials details and loading conditions. There exists a cutoff size below which fragments have an isotropic shape, however, as the size increases an exponential convergence is obtained to a unique elongated form. We show that a discrete stochastic model of fragmentation reproduces both the size and shape of fragments tuning only a single parameter which strengthens the general validity of the scaling laws. The dependence of the probability of the crack plan orientation on the linear extension of fragments proved to be essential for the shape selection mechanism.

Sediments moved by wind, rivers, and waves round in the same way due to geometry.

Striking shapes in nature have been documented to result from chemical precipitation -such as terraced hot springs and stromatolites -which often proceeds via surface-normal growth. Another studied class of objects is those whose shape evolves by physical abrasion -the primary example being river and beach pebbles -which results in shape-dependent surface erosion. While shapes may evolve in a self-similar manner, in neither growth nor erosion can a surface remain invariant. Here we investigate a rare and beautiful geophysical problem that combines both of these processes; the shape evolution of carbonate particles known as ooids. We hypothesize that mineral precipitation, and erosion due to wave-current transport, compete to give rise to novel and invariant geometric forms. We show that a planar (2D) mathematical model built on this premise predicts time-invariant (equilibrium) shapes that result from a balance between precipitation and abrasion. These model results produce nontrivial shapes that are consistent with mature ooids found in nature.Ooids are rounded, sand-sized particles of calcium carbonate that typically form by mineral precipitation in warm and shallow coastal waters. Their transport by waves and currents gives rise to spectacular shoals and white sand beaches, for example in the Bahamas 1,2 ( Fig. 1). Because ooids grow under a restricted range of conditions, they are increasingly being investigated for their potential to record environments of the geologic past [3][4][5] . A remarkable aspect of ooids is that-like trees-they record their own growth history, as accretion of carbonate occurs in concentric layers (Fig. 1). Unlike tree rings, however, the formation of concentric ooid layers is poorly understood [6][7][8] . Although aspects of ooid formation remain enigmatic, there is general agreement among modeling, laboratory and field studies on the qualitative picture. In the absence of collisions with other particles, isolated ooids form spheres with little to no layering 6,7 . This may occur in quiescent environments 9 , or when particles are small enough to be suspended by wave action 6 . If growth is restricted by neighboring particles, ooids can become non-spherical and eventually merge together 10 . The mechanism of growth is (possibly biologically-mediated) surface precipitation of calcium carbonate under super-saturated conditions 11,12 . The presence of ooid-sand waves in energetic environments shows that these particles are also transported as bed load 1,2 -i.e., by rolling, sliding and hopping along the sediment bed -a process that is known to result in abrasion 13,14 . Importantly, ooids transported in bed load often exhibit non-spherical shapes and concentric layering. An additional relevant observation is that layer thickness tends to decrease from the center outwards 15 , suggesting an increase in abrasion rate (relative to precipitation rate) as ooids grow.Patterns resulting from either growth by chemical precipitation 16,17 , or erosion by physical abrasion 13,18,19 , are well...