Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T11:51:50.116Z Has data issue: false hasContentIssue false
  • English
  • Français

Ooid growth: Uniqueness of time-invariant, smooth shapes in 2D

Part of: Equations of mathematical physics and other areas of application Geophysics (general) General theory in ordinary differential equations Geophysics

Published online by Cambridge University Press:  08 February 2019

ANDRÁS A. SIPOS Open the ORCID record for ANDRÁS A. SIPOS [Opens in a new window]
ANDRÁS A. SIPOS*
Affiliation:
MTA-BME Morphodynamics Research Group, Budapest, Hungary Department of Mechanics, Materials and Structures, Budapest University of Technology and Economics, Budapest, Hungary email: siposa@eik.bme.hu
Get access Rights & Permissions [Opens in a new window]

Abstract

Evolution of planar curves under a nonlocal geometric equation is investigated. It models the simultaneous contraction and growth of carbonate particles called ooids in geosciences. Using classical ODE results and a bijective mapping, we demonstrate that the steady parameters associated with the physical environment determine a unique, time-invariant, compact shape among smooth, convex curves embedded in ℝ2. It is also revealed that any time-invariant solution possesses D2 symmetry. The model predictions remarkably agree with ooid shapes observed in nature.

Keywords

Shape evolutionooid growthtime-invariant solutionnonlocal equation

MSC classification

Primary: 35Q86: PDEs in connection with geophysics
Secondary: 34A05: Explicit solutions and reductions 34A26: Geometric methods in differential equations 86A60: Geological problems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 1 , February 2020 , pp. 172 - 182
Copyright
© Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

The author acknowledges the support of the NKFIH grant K-119245 and grant BME FIKP-VÍZ by EMMI.

References

Bloore, F. J. (1977) The shape of pebbles. Math. Geol. 9, 113–122.CrossRefGoogle Scholar
Daskalopoulos, P., Hamilton, R. & Sesum, N. (2010) Classification of compact ancient solutions to the curve shortening flow. J. Differ. Geom. 84(3), 455–464.CrossRefGoogle Scholar
Domokos, G. & Gibbons, G. W. (2012) The evolution of pebble size and shape in space and time. Proc. Roy. Soc. A. 468, 3059–3079.CrossRefGoogle Scholar
Grayson, M. (1989) Shortening embedded curves. Ann. Math. 129, 71–111.CrossRefGoogle Scholar
Sipos, A. A., Domokos, G. & Jerolmack, D. J. (2018) Shape evolution of ooids: a geometric model. Sci. Rep. 8, 1–7, article number: 1758.Google ScholarPubMed
Trower, E. J., Lamb, M. P. & Fischer, W. W. (2017) Experimental evidence that ooid size reflects a dynamic equilibrium between rapid precipitation and abrasion rates. Earth Planet. Sci. Lett. 468, 112–118.CrossRefGoogle Scholar