Accurate simulation of transient landscape evolution by eliminating numerical diffusion: the TTLEM 1.0 model

Earth Surface Dynamics Earth Surf. Dynam. 2196-632X 2017 5 1 Accurate simulation of transient landscape evolution by eliminating numerical diffusion: the TTLEM 1.0 model Benjamin Campforts https://orcid.org/0000-0001-5699-6714 Wolfgang Schwanghart https://orcid.org/0000-0001-6907-6474 Gerard Govers https://orcid.org/0000-0001-9884-4778

01 18 2017 47 66 Belgian Federal Science Policy Office http://dx.doi.org/10.13039/501100002749 P7/24: SOGLO https://creativecommons.org/licenses/by/3.0/ referee comment 1 to the article 10.5194/esurf-2016-39-RC1 author comment 1 to the article 10.5194/esurf-2016-39-AC1 referee comment 2 to the article 10.5194/esurf-2016-39-RC2 author comment 2 to the article 10.5194/esurf-2016-39-AC2 Video supplement 10.5446/19197 Video supplement 10.5446/19196 10.5194/esurf-2016-39 10.5194/esurf-5-47-2017 https://esurf.copernicus.org/articles/5/47/2017/ https://esurf.copernicus.org/articles/5/47/2017/esurf-5-47-2017.pdf 10.1029/JB092iB12p12857 Andrews, D. J. and Bucknam, R. C.: Fitting degradation of shoreline scarps by a nonlinear diffusion model, J. Geophys. Res., 92, 12857, https://doi.org/10.1029/JB092iB12p12857, 1987. 10.1073/pnas.1415443112 Baynes, E. R. C., Attal, M., Niedermann, S., Kirstein, L. A., Dugmore, A. J., and Naylor, M.: Erosion during extreme flood events dominates Holocene canyon evolution in northeast Iceland, P. Natl. Acad. Sci., 112, 2355–2360, https://doi.org/10.1073/pnas.1415443112, 2015. 10.1130/G36527.1 Blöthe, J. H., Korup, O., and Schwanghart, W.: Large landslides lie low: Excess topography in the Himalaya-Karakoram ranges, Geology, 43, 523–526, https://doi.org/10.1130/G36527.1, 2015. 10.1046/j.1365-2117.1997.00030.x Braun, J. and Sambridge, M.: Modelling landscape evolution on geological time scales: A new method based on irregular spatial discretization, Basin Res., 9, 27–52, https://doi.org/10.1046/j.1365-2117.1997.00030.x, 1997. 10.1016/j.geomorph.2012.10.008 Braun, J. and Willett, S. D.: A very efficient O(n), implicit and parallel method to solve the stream power equation governing fluvial incision and landscape evolution, Geomorphology, 180–181, 170–179, https://doi.org/10.1016/j.geomorph.2012.10.008, 2013. 10.1038/379505a0 Burbank, D. W., Leland, J., Fielding, E., Anderson, R. S., Brozovic, N., Reid, M. R., and Duncan, C.: Bedrock incision, rock uplift and threshold hillslopes in the northwestern Himalayas, Nature, 379, 505–510, https://doi.org/10.1038/379505a0, 1996. 10.1002/2014JF003376 Campforts, B. and Govers, G.: Keeping the edge: A numerical method that avoids knickpoint smearing when solving the stream power law, J. Geophys. Res.-Earth, 120, 1189–1205, https://doi.org/10.1002/2014JF003376, 2015. 10.1016/j.epsl.2016.01.017 Campforts, B., Vanacker, V., Vanderborght, J., Baken, S., Smolders, E., and Govers, G.: Simulating the mobility of meteoric 10Be in the landscape through a coupled soil-hillslope model (Be2D), Earth Planet. Sc. Lett., 439, 143–157, https://doi.org/10.1016/j.epsl.2016.01.017, 2016a. 10.5446/19197 Campforts, B., Schwanghart, W., and Govers, G.: TTLEM – TopoToolbox Landscape Evolution: Synthetic model run, https://doi.org/10.5446/19197, 2016b. 10.5446/19196 Campforts, B., Schwanghart, W., and Govers, G.: TTLEM – TopoToolbox Landscape Evolution: Spatially and temporally variable tectonic configuration, https://doi.org/10.5446/19196, 2016c. 10.5194/esurf-2-155-2014 Croissant, T. and Braun, J.: Constraining the stream power law: a novel approach combining a landscape evolution model and an inversion method, Earth Surf. Dynam., 2, 155–166, https://doi.org/10.5194/esurf-2-155-2014, 2014. 10.1086/626891 Culling, W. E. H.: Soil Creep and the Development of Hillside Slopes, J. Geol., 71, 127–161, https://doi.org/10.1086/626891, 1963. 10.1029/2011JF002095 DiBiase, R. A. and Whipple, K. X.: The influence of erosion thresholds and runoff variability on the relationships among topography, climate, and erosion rate, J. Geophys. Res.-Earth, 116, F04036, https://doi.org/10.1029/2011JF002095, 2011. 10.1016/j.epsl.2009.10.036 DiBiase, R. A., Whipple, K. X., Heimsath, A. M., and Ouimet, W. B.: Landscape form and millennial erosion rates in the San Gabriel Mountains, CA, Earth Planet. Sc. Lett., 289, 134–144, https://doi.org/10.1016/j.epsl.2009.10.036, 2010. 10.1029/135GM09 Dietrich, W. E., Bellugi, D. G., Sklar, L. S., Stock, J. D., Heimsath, A. M., and Roering, J. J.: Geomorphic Transport Laws for Predicting Landscape form and Dynamics, in: Prediction in Geomorphology, edited by: Wilcock, P. R. and Iverson, R. M., 103–132, American Geophysical Union, Washington, DC, USA, 2013. 10.1007/978-1-4419-6412-0 Durran, D. R.: Numerical Methods for Fluid Dynamics, Springer New York, NY, USA, 2010. 10.1130/L322.1 Gasparini, N. M. and Whipple, K. X.: Diagnosing climatic and tectonic controls on topography: Eastern flank of the northern Bolivian Andes, Lithosphere, 6, 230–250, https://doi.org/10.1130/L322.1, 2014. 10.1017/CBO9780511809101 Gerya, T.: Introduction to Numerical Geodynamic Modelling, Cambridge University Press, Cambridge, UK, 2010. 10.1130/G36702.1 Glotzbach, C.: Deriving rock uplift histories from data-driven inversion of river profiles, Geology, 43, 467–470, https://doi.org/10.1130/G36702.1, 2015. Godunov, S. K.: A Finite Difference Method for the Computation of Discontinuous Solutions of the Equations of Fluid Dynamics, Math. USSR-Sbornik, 47, 271–306, 1959. 10.1002/esp.3514 Goren, L., Willett, S. D., Herman, F., and Braun, J.: Coupled numerical-analytical approach to landscape evolution modeling, Earth Surf. Proc. Land., 39, 522–545, https://doi.org/10.1002/esp.3514, 2014. Gulliver, J. S.: Introduction to Chemical Transport in the Environment, Cambridge University Press, Cambridge, UK, 2007. 10.1016/0021-9991(83)90136-5 Harten, A.: High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49, 357–393, https://doi.org/10.1016/0021-9991(83)90136-5, 1983. 10.1029/2004JF000248 Herman, F. and Braun, J.: Fluvial response to horizontal shortening and glaciations: A study in the Southern Alps of New Zealand, J. Geophys. Res., 111, F01008, https://doi.org/10.1029/2004JF000248, 2006. 10.1029/2006TC002082 Hoke, G. D., Isacks, B. L., Jordan, T. E., Blanco, N., Tomlinson, A. J., and Ramezani, J.: Geomorphic evidence for post-10 Ma uplift of the western flank of the central Andes 18°30′–22°S, Tectonics, 26, TC5021, https://doi.org/10.1029/2006TC002082, 2007. 10.1029/94WR00757 Howard, A. D.: A detachment-limited model of drainage basin evolution, Water Resour. Res., 30, 2261–2285, https://doi.org/10.1029/94WR00757, 1994. 10.1130/0016-7606(1983)94<739:CCIB>2.0.CO;2 Howard, A. D. and Kerby, G.: Channel changes in badlands, Geol. Soc. Am. Bull., 94, 739–752, https://doi.org/10.1130/0016-7606(1983)94&lt;739:CCIB&gt;2.0.CO;2, 1983. 10.5194/gmd-8-1005-2015 Jaruga, A., Arabas, S., Jarecka, D., Pawlowska, H., Smolarkiewicz, P. K., and Waruszewski, M.: libmpdata+ + 1.0: a library of parallel MPDATA solvers for systems of generalised transport equations, Geosci. Model Dev., 8, 1005–1032, https://doi.org/10.5194/gmd-8-1005-2015, 2015. 10.1029/2008JF001086 Jungers, M. C., Bierman, P. R., Matmon, A., Nichols, K., Larsen, J., and Finkel, R.: Tracing hillslope sediment production and transport with in situ and meteoric 10Be, J. Geophys. Res., 114, 1–16, https://doi.org/10.1029/2008JF001086, 2009. 10.1130/0091-7613(2001)029<0415:QDRURV>2.0.CO;2 Kirby, E. and Whipple, K.: Quantifying differential rock-uplift rates via stream profile analysis, Geology, 29, 415–418, https://doi.org/10.1130/0091-7613(2001)029&lt;0415:QDRURV&gt;2.0.CO;2, 2001. 10.1130/G21959.1 Korup, O.: Rock-slope failure and the river long profile, Geology, 34, 45–48, https://doi.org/10.1130/G21959.1, 2006. 10.1002/esp.3462 Lague, D.: The stream power river incision model: evidence, theory and beyond, Earth Surf. Proc. Land., 39, 38–61, https://doi.org/10.1002/esp.3462, 2014. 10.1073/pnas.1312251111 Lamb, M. P., Mackey, B. H., and Farley, K. A.: Amphitheater-headed canyons formed by megaflooding at Malad Gorge, Idaho, P. Natl. Acad. Sci. USA, 111, 57–62, https://doi.org/10.1073/pnas.1312251111, 2014. 10.1038/ngeo1479 Larsen, I. J. and Montgomery, D. R.: Landslide erosion coupled to tectonics and river incision, Nat. Geosci., 5, 468–473, https://doi.org/10.1038/ngeo1479, 2012. 10.1002/cpa.3160130205 Lax, P. and Wendroff, B.: Systems of conservation laws, Commun. Pur. Appl. Math., 13, 217–237, https://doi.org/10.1002/cpa.3160130205, 1960. 10.1029/JB077i014p02460 Luke, J. C.: Mathematical models for landform evolution, J. Geophys. Res., 77, 2460–2464, https://doi.org/10.1029/JB077i014p02460, 1972. 10.1130/B30930.1 Mackey, B. H., Scheingross, J. S., Lamb, M. P., and Farley, K. A.: Knickpoint formation, rapid propagation, and landscape response following coastal cliff retreat at the last interglacial sea-level highstand: Kaua'i, Hawai'i, Geol. Soc. Am. Bull., 126, 925–942, https://doi.org/10.1130/B30930.1, 2014. 10.1002/2014JF003387 Moon, S., Shelef, E., and Hilley, G. E.: Recent topographic evolution and erosion of the deglaciated Washington Cascades inferred from a stochastic landscape evolution model, J. Geophys. Res.-Earth, 120, 856–876, https://doi.org/10.1002/2014JF003387, 2015. 10.1002/esp.3923 Mudd, S. M.: Detection of transience in eroding landscapes, Earth Surf. Proc. Land., 42, 24–41, https://doi.org/10.1002/esp.3923, 2016. 10.1002/2013JF002981 Mudd, S. M., Attal, M., Milodowski, D. T., Grieve, S. W. D., and Valters, D. A.: A statistical framework to quantify spatial variation in channel gradients using the integral method of channel profile analysis, J. Geophys. Res.-Earth, 119, 138–152, https://doi.org/10.1002/2013JF002981, 2014. 10.1007/b98874 Nocedal, J. and Wright, S. J.: Numerical Optimization, Springer, New York, USA, 1999. 10.1017/CBO9780511813849 Pelletier, J. D.: Quantitative Modeling of Earth Surface Processes, Cambridge University Press, Cambridge, UK, 2008. 10.1016/j.geomorph.2010.06.001 Pelletier, J. D.: Minimizing the grid-resolution dependence of flow-routing algorithms for geomorphic applications, Geomorphology, 122, 91–98, https://doi.org/10.1016/j.geomorph.2010.06.001, 2010. 10.1029/2010JF001801 Perron, J. T.: Numerical methods for nonlinear hillslope transport laws, J. Geophys. Res., 116, F02021, https://doi.org/10.1029/2010JF001801, 2011. 10.1016/j.earscirev.2015.02.002 Phillips, J. D., Schwanghart, W., and Heckmann, T.: Graph theory in the geosciences, Earth-Sci. Rev., 143, 147–160, https://doi.org/10.1016/j.earscirev.2015.02.002, 2015. 10.1029/1998WR900090 Roering, J. J., Kirchner, J. W., and Dietrich, W. E.: Evidence for nonlinear, diffusive sediment transport on hillslopes and implications for landscape morphology, Water Resour. Res., 35, 853–870, https://doi.org/10.1029/1998WR900090, 1999. 10.1002/jgrf.20031 Royden, L. and Perron, T. J.: Solutions of the stream power equation and application to the evolution of river longitudinal profiles, J. Geophys. Res.-Earth, 118, 497–518, https://doi.org/10.1002/jgrf.20031, 2013. 10.1016/j.envsoft.2009.12.002 Schwanghart, W. and Kuhn, N. J.: TopoToolbox: A set of Matlab functions for topographic analysis, Environ. Model. Softw., 25, 770–781, https://doi.org/10.1016/j.envsoft.2009.12.002, 2010. 10.5194/esurf-2-1-2014 Schwanghart, W. and Scherler, D.: Short Communication: TopoToolbox 2 – MATLAB-based software for topographic analysis and modeling in Earth surface sciences, Earth Surf. Dynam., 2, 1–7, https://doi.org/10.5194/esurf-2-1-2014, 2014. 10.1002/esp.3452 Schwanghart, W., Groom, G., Kuhn, N. J., and Heckrath, G.: Flow network derivation from a high resolution DEM in a low relief, agrarian landscape, Earth Surf. Proc. Land., 38, 1576–1586, https://doi.org/10.1002/esp.3452, 2013. Seidl, M. and Dietrich, W.: The problem of channel erosion into bedrock (Supplement), Catena, 23, 101–104, 1992. 10.1175/1520-0493(1983)111<0479:ASPDAS>2.0.CO;2 Smolarkiewicz, P. K.: A Simple Positive Definite Advection Scheme with Small Implicit Diffusion, Mon. Weather Rev., 111, 479–486, https://doi.org/10.1175/1520-0493(1983)111&lt;0479:ASPDAS&gt;2.0.CO;2, 1983. 10.1016/0021-9991(90)90105-A Smolarkiewicz, P. K. and Grabowski, W. W.: The multidimensional positive definite advection transport algorithm: nonoscillatory option, J. Comput. Phys., 86, 355–375, https://doi.org/10.1016/0021-9991(90)90105-A, 1990. 10.1029/2002WR001879 Soille, P., Vogt, J., and Colombo, R.: Carving and adaptive drainage enforcement of grid digital elevation models, Water Resour. Res., 39, 1366, https://doi.org/10.1029/2002WR001879, 2003. 10.1029/98JB02139 Stock, J. D. and Montgomery, D. R.: Geologic constraints on bedrock river incision using the stream power law, J. Geophys. Res., 104, 4983, https://doi.org/10.1029/98JB02139, 1999. 10.1007/b79761 Toro, E. F.: Riemann solvers and numerical methods for fluid dynamics – A Practical Introduction, Springer, New York, USA, 2009. 10.1029/2008JF001079 Valla, P. G., van der Beek, P. A., and Lague, D.: Fluvial incision into bedrock: Insights from morphometric analysis and numerical modeling of gorges incising glacial hanging valleys (Western Alps, France), J. Geophys. Res.-Earth, 115, F02010, https://doi.org/10.1029/2008JF001079, 2010. Valters, D.: Modelling Geomorphic Systems: Landscape Evolution, in: Geomorphological Techniques, vol. 12, edited by: Cook, S. J., Clarke, L. E., and Nield, J. M., 1–24, British Society for Geomorphology, London, UK, 2016. 10.1016/j.geomorph.2014.09.013 Vanacker, V., von Blanckenburg, F., Govers, G., Molina, A., Campforts, B., and Kubik, P. W.: Transient river response, captured by channel steepness and its concavity, Geomorphology, 228, 234–243, https://doi.org/10.1016/j.geomorph.2014.09.013, 2015. 10.1006/jcph.1997.5704 van Leer, B.: Towards the Ultimate Conservative Difference Scheme, J. Comput. Phys., 135, 229–248, https://doi.org/10.1006/jcph.1997.5704, 1997. 10.1002/jgrf.20121 West, N., Kirby, E., Bierman, P., Slingerland, R., Ma, L., Rood, D., and Brantley, S.: Regolith production and transport at the Susquehanna Shale Hills Critical Zone Observatory, Part 2: Insights from meteoric 10Be, J. Geophys. Res.-Earth, 118, 1877–1896, https://doi.org/10.1002/jgrf.20121, 2013. 10.1029/1999JB900120 Whipple, K. X. and Tucker, G. E.: Dynamics of the stream-power river incision model: Implications for height limits of mountain ranges, landscape response timescales, and research needs, J. Geophys. Res.-Sol. Ea., 104, 17661–17674, https://doi.org/10.1029/1999JB900120, 1999. 10.1111/j.1365-2117.2007.00337.x Whittaker, A. C., Cowie, P. A., Attal, M., Tucker, G. E., and Roberts, G. P.: Contrasting transient and steady-state rivers crossing active normal faults: New field observations from the central apennines, Italy, Basin Res., 19, 529–556, https://doi.org/10.1111/j.1365-2117.2007.00337.x, 2007. 10.1029/1999JB900248 Willett, S. D.: Orogeny and orography: The effects of erosion on the structure of mountain belts, J. Geophys. Res.-Sol. Ea., 104, 28957–28981, https://doi.org/10.1029/1999JB900248, 1999. 10.2475/ajs.301.4-5.455 Willett, S. D., Slingerland, R., and Hovius, N.: Uplift, shortening, and steady state topography in active mountain belts, Am. J. Sci., 301, 455–485, 2001. supplement file to the article 10.5194/esurf-5-47-2017-supplement https://esurf.copernicus.org/articles/5/47/2017/esurf-5-47-2017-supplement.zip