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# Topography of mountain belts controlled by rheology and surface processes

## Abstract

It is widely recognized that collisional mountain belt topography is generated by crustal thickening and lowered by river bedrock erosion, linking climate and tectonics1,2,3,4. However, whether surface processes or lithospheric strength control mountain belt height, shape and longevity remains uncertain. Additionally, how to reconcile high erosion rates in some active orogens with long-term survival of mountain belts for hundreds of millions of years remains enigmatic. Here we investigate mountain belt growth and decay using a new coupled surface process5,6 and mantle-scale tectonic model7. End-member models and the new non-dimensional Beaumont number, Bm, quantify how surface processes and tectonics control the topographic evolution of mountain belts, and enable the definition of three end-member types of growing orogens: type 1, non-steady state, strength controlled (Bm > 0.5); type 2, flux steady state8, strength controlled (Bm ≈ 0.4−0.5); and type 3, flux steady state, erosion controlled (Bm < 0.4). Our results indicate that tectonics dominate in Himalaya–Tibet and the Central Andes (both type 1), efficient surface processes balance high convergence rates in Taiwan (probably type 2) and surface processes dominate in the Southern Alps of New Zealand (type 3). Orogenic decay is determined by erosional efficiency and can be subdivided into two phases with variable isostatic rebound characteristics and associated timescales. The results presented here provide a unified framework explaining how surface processes and lithospheric strength control the height, shape, and longevity of mountain belts.

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• ### 3D geodynamic-geomorphologic modelling of deformation and exhumation at curved plate boundaries: Implications for the southern Alaskan plate corner

Scientific Reports Open Access 22 August 2022

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## Data availability

All data supporting the findings of this study are contained within the article and Supplementary Information.

## Code availability

Numerical models are computed with published methods and codes, described in the Methods and Supplementary Information. The code for longitudinality index calculations is available from the corresponding author on request.

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## Acknowledgements

S.G.W. and X.Y. acknowledge support from TOTAL through the COLORS project. C. Beaumont is thanked for constructive comments on an earlier version of the scaling analysis developed here and for proposing the introduction of the surface process Damköhler numbers.

## Author information

Authors

### Contributions

S.G.W., R.S.H., J.B. and X.Y. designed the experiments, discussed the results and implications, and wrote the article. With help from R.S.H., J.B. and X.Y., S.G.W. developed the coupling between the tectonic model and the surface process model. S.G.W. conducted the comparison to Nature, and ran and visualized the models.

### Corresponding author

Correspondence to Sebastian G. Wolf.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature thanks Kelin Whipple, Greg Houseman and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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## Extended data figures and tables

### Extended Data Fig. 1 Model setup with boundary conditions (a), initial landscape with surface process parameters (b), and material legend with properties (c).

a, The model is 1200 km wide, 600 km deep and has a uniform layered material distribution. A zoom into the continental lithosphere with corresponding yield-strength-envelope is shown as insert. Mountain building is modelled by applying a velocity boundary condition on both model sides in the lithosphere. Inflow is balanced by small distributed outflow of material in the sub-lithospheric mantle. The side and lower model boundaries have free slip conditions and the upper surface is free. The thermo-mechanical model is coupled to the surface process model FastScape, which starts out with a fluvial network with maximum 250 m elevation (b). The free surface corresponds to the average FastScape elevation. The initial temperature distribution in the continent corresponds to 1D-thermal steady state and the temperature in the sub-lithospheric mantle follows an adiabatic gradient of 0.4 °C. The side boundaries are insulated and the top and bottom of the model domain have fixed temperatures with respectively 0 °C and 1522 °C. c, Material legend shows colour, scaled flow law, and density of model materials. Blues for syn-contractional sediments alternate every 5 Myr. WQtz is the wet quartz flow law from Gleason and Tullis60; DMD is dry Maryland flow law from Mackwell et al.61; WOl is the wet olivine flow law from Karato and Wu62.

### Extended Data Fig. 2 Evolution of Bouyancy Forces.

a–c, Buoyancy force plots for three different timesteps with explanation in (g). The plots show the computed buoyancy force as black/grey lines and the sum of integrated overpressure $${\bar{P}}_{O}$$ plus Fint as red stippled lines. The stippled lines frame a typical range of values computed in the foreland crustal column in models 1 and 2, at several timesteps. On average, $${\bar{P}}_{O}\approx {F}_{int}$$. d–f, Viscosity field and temperature contours (red) at the end of the shortening phase (25 Myr).

### Extended Data Fig. 3 Supplementary growth-only models with variable fluvial erodibility.

a–i, Snapshots of supplementary models 19 after 25 Myr of model evolution. Each panel consists of: Zoom into the model domain showing material distribution and temperature contours of the thermo-mechanical tectonic model, map-view landscape from landscape evolution model FastScape, and swath elevation profile of the landscape. Swath profiles have the same scale in a-f, but a different scale in g-i. Top-row models do not reach flux steady state (Type 1); middle-row models reach steady state and are to first order strength-limited (Type 2); bottom-row models reach flux steady state and are erosion-limited (Type 3). Note: Non steady state models exhibit rivers flowing in longitudinal valleys in orogen core, and erosion-limited orogens do not form thrust sheets on the left side. H is the maximum mean elevation plotted as function of time in Extended Data Fig. 4b. Colours of model titles a-i correspond to colours in Extended Data Fig. 4.

### Extended Data Fig. 4 Time-dependent evolution of mountain width and height.

a, b, Evolution of mountain width and maximum mean elevation through time for the 9 growth-only models with different fluvial erodibility shown in Extended Data Fig. 3. The mountain width is calculated every 0.5 Myr between the two outermost points above 600 m, or above 15% of maximum mean height (black Type 3 models). Steps in width correspond to new outward-propagating thrusts.

### Extended Data Fig. 5 The influence of crustal rheology.

a, b, Evolution of mountain width and maximum mean elevation through time for three models with different fluvial erodibility and low crustal strength. Colours are the same in both plots. Mountain width is calculated every 0.5 Myr between the two outermost points which are above 600 m, or above 15% of maximum mean height (in MKf20 and MKf20Weak). Steps in width correspond to new outward-propagating thrusts. c–e, Snapshots of models MKf0.5Weak, MKf5Weak, and MKf20Weak after 25 Myr of model evolution. Each panel consists of: Zoom into the model domain showing material distribution and temperature contours of the thermo-mechanical tectonic model (see Extended Data Fig. 1 for material colours), map-view landscape from landscape evolution model FastScape, swath elevation profile of the landscape, and buoyancy force plot. The buoyancy force plot shows one earlier timestep (10 Myr) as a grey line and the sum of integrated overpressure $${\bar{P}}_{O}$$ plus Fint as red stippled lines. The stippled lines frame a typical range of measured values. On average $${\bar{P}}_{O}\approx {F}_{int}$$; $${\bar{P}}_{O}$$ and Fint are computed in the foreland crustal column in models MKf0.5Weak and MKf5Weak, at several timesteps.

### Extended Data Fig. 6 Influence of variable structural style (decoupled thick- and thin-skinned tectonics).

a, b, Evolution of mountain width and maximum mean elevation through time for three models with different fluvial erodibility and with a shallow crustal decoupling horizon (“salt" layer). Colours are the same in both plots. The mountain width is calculated every 0.5 Myr between the two outermost points above 600 m, or above 15% of maximum mean height (in MKf20 and MKf20Salt). Steps in width correspond to new outward-propagating thrusts. c–e, Snapshots of models MKf0.5Salt, MKf5Salt, and MKf20Salt after 25 Myr of model evolution. Each panel consists of: Zoom into the model domain showing material distribution and temperature contours of the thermo-mechanical tectonic model (see Extended Data Fig. 1 for material colours, purple is the weak layer), map-view landscape from landscape evolution model FastScape, swath elevation profile of the landscape, and buoyancy force plot. The buoyancy force plot shows one earlier timestep (10 Myr) as a grey line and the sum of integrated overpressure $${\bar{P}}_{O}$$ plus Fint as red stippled lines. The stippled lines frame a typical range of measured values. On average, $${\bar{P}}_{O}\approx {F}_{int}$$; $${\bar{P}}_{O}$$ and Fint are computed in the foreland crustal column in models MKf0.5Salt and MKf5Salt, at several timesteps.

### Extended Data Fig. 7 Analytical scaling relationship for decay phase II - effectively local isostatic rebound.

a–f, Elevation-time plots of FastScape-only models with low and high erodibility, variable orogen width and variable initial orogen height. Each sub-figure shows the evolution of topography and corresponding analytical solution of four models starting with the same width but with different initial heights. Uplift in these models is local-isostatic $$(U=(1-\rho {\prime} )\times \dot{e})$$, erosion follows the (extended) stream-power law with the same parameters as used in the coupled models. U is the uplift rate, $$\dot{e}$$ is the erosion rate, and $$\rho {\prime}$$ is the isostatic compensation factor (see Supplementary Information).  g, Shows evolution of maximum mean topography of the three models presented in the main text (Figure 1) with corresponding analytical solutions. The analytical solution is derived in the supplemental material. We see that a wider orogen and lower erodibility lead to slower decay; a lower initial starting height corresponds to a shift in time on the decay curve. The latter is displayed in (h). All models show good fit between analytical solution and evolution of topography.

### Extended Data Fig. 8 Non-dimensional Surface Processes Damköhler numbers and Beaumont number.

a, Theoretical box model that describes the mechanics of surface processes with fluxes between different hypothetical reservoirs. b, Definition of the four Surface Processes Damköhler numbers (DaSP) determining the mechanics and efficiency of surface processes. c, Definition of a new non-dimensional number, termed Beaumont-number (Bm) that determines the interaction between surface processes and tectonics. NTec is the non-dimensional number determining topographic growth, here the crustal Argand number Ar in case of collisional orogens, Ne is the uplift-erosion number.

### Extended Data Fig. 9 Longitudinality index in modelled orogens (a,b) and natural examples (d-f), steepness index of modelled orogens (h).

a, b, 2D longitudinality index (LI) plots of Models 1 and 2 showing the FastScape elevation as grey-shade in the background, the LI of each source point (A = 1 × 105 m2) of a river as colour coding, and the corresponding rivers as light-grey and transparent overlay. The black lines are the orogen boundaries corresponding to 300 m average elevation. c–f, DEM and (LI) plots for Taiwan (Tw), the Southern Alps of New Zealand (SANZ), Himalaya-Tibet (Him), and the Central Andes (And). The DEMs show elevation as colour coding, the manually picked orogen boundaries as black lines, and a stippled box outlining the LI plots. The LI plots show elevation as grey-shade in the background, each LI of a source point (A = 1 × 105 m2) of a river as colour coding. g, Box-and-whisker plots show the full LI datasets with boxed first and third quartile, whiskers expanding to the minimum and maximum of the datasets (+1.5 IQR), median as green line, mean as green dot, and outliers as grey circles. The grey, stippled line corresponds to a value of 1.5, which is roughly the maximum value of model 2. h, Swath profile of steepness indices of models 1 and 2 at the end of shortening and during orogenic decay. Bold line shows median, shaded area frames value range (Methods).

### Extended Data Fig. 10 Digital elevation models, swath profiles, and geological cross sections of the Southern Alps of New Zealand, Taiwan, Himalaya-Tibet and Central Andes.

The swath profiles are created in the vicinity of the cross sections, oranges are crust, greys are lithospheric mantle, and vc is the convergence velocity related to crustal thickening. a, Cross section modified from Little33 and Herman et al.35. b, Cross section modified from Brown et al.37 and Van Avendonk et al.40, lc is lower crust, oc is oceanic crust, so are syn-orogenic sediments, sr are pre-orogenic syn-rift sediments, LA is the Luzon Arc, LV is the longitudinal valley, orange color is crustal rocks. c, Cross section modified from Owens & Zandt63. Yellow color is lower crust, orange is crust from the pro-plate, light-orange is Tibetan retro-plate crust. The light-grey lithospheric mantle is possibly removed. d, Cross section modified from DeCelles et al.41. The additional swath profiles II and III show that the actively shortening Central Andes reach similar heights independent of orogen width.

## Supplementary information

### Supplementary Information

(1) An explanation of the supplementary model animations. (2) An extended methods section explaining the modelling basics and setup choices of the thermo-mechanical landscape evolution model. (3) A detailed description of the supplementary models. (4) The derivation of the scaling relationship between surface processes and tectonics during orogenic growth. (5) A comprehensive comparison between model inferences and the natural examples discussed in the text. (6) The derivation of the scaling relationship between surface processes and tectonics during orogenic decay.

### Supplementary Information

Legends for the Supplementary Videos.

### Supplementary Videos

A zip file containing 21 animations of the presented numerical models; see separate file for legends.

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Wolf, S.G., Huismans, R.S., Braun, J. et al. Topography of mountain belts controlled by rheology and surface processes. Nature 606, 516–521 (2022). https://doi.org/10.1038/s41586-022-04700-6

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• DOI: https://doi.org/10.1038/s41586-022-04700-6

• ### 3D geodynamic-geomorphologic modelling of deformation and exhumation at curved plate boundaries: Implications for the southern Alaskan plate corner

• Alexander Koptev
• Matthias Nettesheim
• Todd A. Ehlers

Scientific Reports (2022)

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