## 2012-02-15

### Steady-state topography during orogenesis: Erosion vs. Tectonics competition

I uploaded to Youtube a simple but interesting numerical model. It computes a constant tectonic uplift with the competing river erosion along a cross section, allowing enough time to reach two successive topographic steady states (one during uplift, another one after uplift):
Steady state topography development, and other parameters of the model.
Uplift occurs between x=-50 and x=+50 km. The dashed line indicates
the topographic profile that would develop in absence of erosion.
Equilibrium topography is reached at ~4 and at ~8 Myr.

This is calculated assuming constant uplift rate at the center (x=-50 to +50 km) and a 1D stream power law erosion model. Uplift rate is 1 mm/yr and stops at t=5 Myr. River erosion is proportional to slope and water discharge. Precipitation rate is constant over the entire profile. Calculations are performed under Linux with the program tAo (Garcia-Castellanos, 2007, EPSL). +info and software download here: https://sites.google.com/site/daniggcc/software/tao

As you can see, topographic growth goes on until a first steady state (with a maximum topography of ~3000 m) is reached before 5 Myr. If you look at the numbers, you'll see an equilibrium between erosion rates and uplift rates at that time. At 5 Myr uplift stops and then erosion leads to the new equilibrium: a flat topography (at 8 Myr).

Now, the question is: does steady-state topography exist in nature? And if it does, can we recognize it? In real Earth, neither climate nor tectonics are constant through time. The questions are probably too big for this small blog, but you can find some hints in this article by Willett and Brandon (2001).

Nevertheless, the notion of steady-state topography is useful to understand some basic principles of orogenesis, as Whipple (2009) showed in a very simple and elegant way. Consider these two end-member types of orogen:

 Evolution of for parameters (orogen width, erosion, topography, and rock uplift) for two simple models of orogenic growth. Left: fixed width orogen; Right: Self-similar growth. At t=0, the erosion coefficient is set to a double value (red) and to half of the reference value (green). Erosion is assumed proportional to elevation. The parameters are shown normalized. Redrawn from Whipple (2009).
Assume both orogens grow in response to the convergence of two tectonic plates, producing a constant tectonic flow Fa, and that they are eroded at a rate proportional to elevation. The red lines in the figures above correspond to a change to double erosion efficiency. With such a simple representation, it becomes clear that if erosion mechanisms become more efficient, both orogen types initially undergo an increase in erosion rate, but this will gradually decrease back to the initial erosion value (the one compensating the imposed tectonic flow, as in the animation above). The way the orogen returns to the original low erosion rate is by decreasing its elevation R.
One interesting thing is that, whereas for a fixed width, rock uplift rates return to normal after some time, the self-similar growth predicts a permanent increase in uplift rates.
And the other interesting conclusion is that the time response is controlled mainly by the erosion efficiency itself (within the approaches of the model, of course).

Simple models are generally more inspiring than the most complex ones.

References:

Whipple, K. (2009). The influence of climate on the tectonic evolution of mountain belts Nature Geoscience, 2 (2), 97-104 DOI: 10.1038/ngeo413

Willett, S., and Brandon, M. (2001). On steady states in mountain belts Geology, 30, 175-178