Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-03T18:26:05.964Z Has data issue: false hasContentIssue false
  • English
  • Français

The hydrodynamic genesis of linear karren patterns

Published online by Cambridge University Press:  26 February 2021

M.B. Bertagni Open the ORCID record for M.B. Bertagni [Opens in a new window]  and
C. Camporeale Open the ORCID record for C. Camporeale [Opens in a new window]
M.B. Bertagni*
Affiliation:
The High Meadows Environmental Institute, Princeton University, Princeton, NJ08544, USA
C. Camporeale
Affiliation:
Department of Land, Infrastructure and Environmental Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10124Torino, Italy
*
†Email address for correspondence: matteobb@princeton.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In karst and alpine areas, the interactions between water and rocks give rise to a large variety of marvellous patterns. In this work, we provide a hydrodynamic model for the formation of dissolutional patterns made of parallel longitudinal channels, commonly referred to as linear karren forms. The model addresses a laminar film of water flowing on a rock that is dissolving. The results show that a transverse instability of the water–rock system leads to a longitudinal channelization responsible for the pattern formation. The instability arises because of a positive feedback within the channels between the higher water flow and the enhanced chemical dissolution. The spatial scales predicted by the linear stability analysis span different orders of magnitude depending on the Reynolds number. This may explain why similar patterns of different sizes are observed on natural rocks. Results also show that the rock solubility affects just the temporal scale of the instability and the rock inclination plays a minor role in the pattern formation. It is eventually discussed how rain is not strictly necessary for the appearance of linear karren patterns, but it may affect some of their features.

JFM classification

Interfacial Flows (free surface): Thin films Nonlinear Dynamical Systems: Pattern formation Phase change: Morphological instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , A34
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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.)

References

REFERENCES

Abramowitz, M., Stegun, I.A. & Romer, R.H. 1988 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. American Association of Physics Teachers.CrossRefGoogle Scholar
Bender, C.M. & Orszag, S.A. 2013 Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer Science and Business Media.Google Scholar
Berner, R.A., Lasaga, A.C. & Garrels, R.M. 1983 The carbonate-silicate geochemical cycle and its effect on atmospheric carbon dioxide over the past 100 million years. Am. J. Sci. 283, 641–683.CrossRefGoogle Scholar
Bertagni, M.B. & Camporeale, C. 2017 Nonlinear and subharmonic stability analysis in film-driven morphological patterns. Phys. Rev. E 96 (5), 053115.CrossRefGoogle ScholarPubMed
Bögli, A. 1960 Kalklösung und karrenbildung. Z. Geomorphol. (Suppl. band 2), 4–21.Google Scholar
Buhmann, D. & Dreybrodt, W. 1985 The kinetics of calcite dissolution and precipitation in geologically relevant situations of karst areas: 1. Open system. Chem. Geol. 48 (1–4), 189–211.CrossRefGoogle Scholar
Camporeale, C. 2015 Hydrodynamically locked morphogenesis in karst and ice flutings. J. Fluid Mech. 778, 89–119.CrossRefGoogle Scholar
Camporeale, C. 2017 An asymptotic approach to the crenulation instability. J. Fluid Mech. 826, 636–652.CrossRefGoogle Scholar
Camporeale, C. & Ridolfi, L. 2012 Hydrodynamic-driven stability analysis of morphological patterns on stalactites and implications for cave paleoflow reconstructions. Phys. Rev. Lett. 108 (23), 238501.CrossRefGoogle ScholarPubMed
Chang, H. & Demekhin, E.A. 2002 Complex Wave Dynamics on Thin Films. Elsevier.Google Scholar
Claudin, P., Durán, O. & Andreotti, B. 2017 Dissolution instability and roughening transition. J. Fluid Mech. 832, R2.CrossRefGoogle Scholar
Cohen, C., Berhanu, M., Derr, J. & du Pont, S.C. 2020 Buoyancy-driven dissolution of inclined blocks: erosion rate and pattern formation. Phys. Rev. Fluids 5 (5), 053802.CrossRefGoogle Scholar
Cross, M. & Hohenberg, P.C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (3), 851.CrossRefGoogle Scholar
Curl, R.L. 1966 Scallops and flutes. Trans. Cave Res. Group Great Brit. 7 (2).Google Scholar
Davis, E.J. 1973 Exact solutions for a class of heat and mass transfer problems. Can. J. Chem. Engng 51 (5), 562–572.CrossRefGoogle Scholar
Dreybrodt, W. 2012 Processes in Karst Systems: Physics, Chemistry, and Geology, vol. 4. Springer Science and Business Media.Google Scholar
Fairchild, I.J. & Baker, A. 2012 Speleothem Science: From Process to Past Environments, vol. 3. John Wiley and Sons.CrossRefGoogle Scholar
Ford, D. & Williams, P.D. 2013 Karst Hydrogeology and Geomorphology. John Wiley and Sons.Google Scholar
Ginés, A., Knez, M., Slabe, T. & Dreybrodt, W. 2009 Karst Rock Features. Karren Sculpturing, vol. 9. Založba ZRC.Google Scholar
Glew, J.R. & Ford, D.C. 1980 A simulation study of the development of rillenkarren. Earth Surf. Process. 5 (1), 25–36.CrossRefGoogle Scholar
Goldenfeld, N., Chan, P.Y. & Veysey, J. 2006 Dynamics of precipitation pattern formation at geothermal hot springs. Phys. Rev. Lett. 96 (25), 254501.CrossRefGoogle ScholarPubMed
Guérin, A., Derr, J., Du Pont, S.C. & Berhanu, M. 2020 Streamwise dissolution patterns created by a flowing water film. Phys. Rev. Lett. 125 (19), 194502.CrossRefGoogle ScholarPubMed
Hammer, Ø., Dysthe, D.K., Lelu, B., Lund, H., Meakin, P. & Jamtveit, B. 2008 Calcite precipitation instability under laminar, open-channel flow. Geochim. Cosmochim. Acta 72 (20), 5009–5021.CrossRefGoogle Scholar
Jamtveit, B. & Hammer, Ø. 2012 Sculpting of rocks by reactive fluids. Geochem. Perspect. 1 (3), 341–342.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M.G. 2011 Falling Liquid Films, vol. 176. Springer Science and Business Media.Google Scholar
Kaufmann, G. & Dreybrodt, W. 2007 Calcite dissolution kinetics in the system CaCO$_3$-H$_2$O-CO$_2$ at high undersaturation. Geochim. Cosmochim. Acta 71 (6), 1398–1410.CrossRefGoogle Scholar
Mac Huang, J., Moore, M.N.J. & Ristroph, L. 2015 Shape dynamics and scaling laws for a body dissolving in fluid flow. J. Fluid Mech. 765.Google Scholar
Meakin, P. & Jamtveit, B. 2009 Geological pattern formation by growth and dissolution in aqueous systems. Proc R. Soc Lond. A 466 (2115), 659–694.Google Scholar
Moore, M.N.J. 2017 Riemann-Hilbert problems for the shapes formed by bodies dissolving, melting, and eroding in fluid flows. Commun. Pure Appl. Maths 70 (9), 1810–1831.CrossRefGoogle Scholar
Morrow, L.C., King, J.R., Moroney, T.J. & McCue, S.W. 2019 Moving boundary problems for quasi-steady conduction limited melting. SIAM J. Appl. Maths 79 (5), 2107–2131.CrossRefGoogle Scholar
Morse, J.W. & Arvidson, R.S. 2002 The dissolution kinetics of major sedimentary carbonate minerals. Earth-Sci. Rev. 58 (1–2), 51–84.CrossRefGoogle Scholar
Mottershead, D. & Lucas, G. 2001 Field testing of Glew and Ford's model of solution flute evolution. Earth Surf. Process. Landf. 26 (8), 839–846.CrossRefGoogle Scholar
Nusselt, W. 1916 Die oberflachenkondesation des wasserdamffes the surface condensation of water. Z. Verein. Deutsch. Ing. 60, 541–546.Google Scholar
Plummer, L.N., Wigley, T.M.L. & Parkhurst, D.L. 1978 The kinetics of calcite dissolution in CO2-water systems at 5 degrees to 60 degrees C and 0.0 to 1.0 atm CO2. Am. J. Sci. 278 (2), 179–216.CrossRefGoogle Scholar
Polyanin, A.D., Kutepov, A.M., Kazenin, D.A. & Vyazmin, A.V. 2001 Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, vol. 14. CRC Press.CrossRefGoogle Scholar
Polyanin, A.D. & Nazaikinskii, V.E. 2015 Handbook of Linear Partial Differential Equations for Engineers and Scientists. CRC Press.CrossRefGoogle Scholar
Regnier, P., et al. . 2013 Anthropogenic perturbation of the carbon fluxes from land to ocean. Nat. Geosci. 6 (8), 597–607.CrossRefGoogle Scholar
Sauro, U. 2009 I Karren: le forme scolpite sulla roccia. Società Speleologica Italiana.Google Scholar
Short, M.B., Baygents, J.C. & Goldstein, R.E. 2005 Stalactite growth as a free-boundary problem. Phys. Fluids 17 (8), 083101.CrossRefGoogle Scholar
Slabe, T., Hada, A. & Knez, M. 2016 Laboratory modeling of karst phenomena and their rock relief on plaster: subsoil karren, rain flutes karren and caves. Acta Carsologica 45 (2).CrossRefGoogle Scholar
Vesipa, R., Camporeale, C. & Ridolfi, L. 2015 Thin-film-induced morphological instabilities over calcite surfaces. Proc R. Soc. Lond. A 471 (2176), 20150031.Google ScholarPubMed
Veysey, J. & Goldenfeld, N. 2008 Watching rocks grow. Nat. Phys. 4 (4), 310–313.CrossRefGoogle Scholar
Wykes, M.S.D., Mac Huang, J., Hajjar, G.A. & Ristroph, L. 2018 Self-sculpting of a dissolvable body due to gravitational convection. Phys. Rev. Fluids 3 (4), 043801.CrossRefGoogle Scholar