For the experts.

All the following equations and parameters are described in the Theory
page.

Intuitively, decreasing the uplift rate may be equivalent to increasing the
rainfall rate in terms of mountain height. We can evaluate the equivalence of
changing parameters by rewriting the equation in a ”non-dimensional” form. For
example, instead of having elevation z we will have z dividing by some constant
reference elevation, or a scaling factor, let’s say H, and this will give us
z_{*}, an elevation but without dimension. Its unit is ... an arbitrary unit of
elevation. We do that for all the parameters, replace them into the original
mass-balance equation and look at the resulting new equation, that will have some
universal meaning as all its parameters have an arbitrary unit of length, time
etc.

So we start with the initial mass balance equation (cf Theory page.)

Then, with m = 0.5 and n = 1 and using the scaling factors H for mountain
height, L for mountain width, P for effective precipitation rate (runoff) and U uplift
rate, H∕U for time, we obtain the non-dimensional form (_{*}) of the mass balance
equation 1 (Carretier et al., 2018):

where

N_{riv} = KU^{-1}P^{0.5}L^{-0.5}H

N_{depo} = ξ^{-1}P^{-1}

N_{hill} = κHU^{-1}L^{-1}

These numbers are called ”non-dimensional numbers” (Rmq. N_{depo} is G in Guerit
et al. (2019) - see the post about the scaling between mountain height and rainfall).
These numbers say that for two simulations 1 and 2 with different parameters but
such that their non-dimensional numbers are the same, then the evolution of the
mean non-dimensional erosion (ϵ_{1}∕U_{1} and ϵ_{2}∕U_{2}) over non-dimensional time
(t_{1} * U_{1}∕H_{1} and t_{2} * U_{2}∕H_{2}) or the final non-dimensional mountain height (z_{1}∕H_{1}
and z_{2}∕H_{2}) will be similar. Experts can read Whipple and Tucker (1999).

In particular, let’s imagine two simulations with K_{1} = 2K_{2} and U_{1} = U_{2}∕2. Then
non-dimensional numbers are the same in both cases and it can be predicted that the
evolution of the landscape will be similar in both cases. In other words, doubling the
erodibility or reducing the uplift rate by 2 have the same effect on the topography.
It is moreover a problem to interpret the different altitudes of mountains
in terms of tectonic uplift because their erodability is generally unknown.

However, this analysis does not predict everything. For example, the relative
effect of U and P is difficult to establish because the parameter chosen for the
altitude H, which is the final altitude of the mountain, itself depends on P and
U. Similarly, as explained in the post ”The scaling of mountain relief with
rainfall”, the relative weight of U and P depends on ζ, the transport length
parameter.

Suggestion for teaching:

From dimensionless numbers, predict the equivalence between some parameters and
verify this equivalence using the site simulations.

Re-write the non-dimensional equation with other scaling factors and discuss the
resulting non-dimensional numbers.

### References

Carretier, S., Y. Godderis, J. Martinez, M. Reich, and P. Martinod (2018), Colluvial deposits as a possible weathering reservoir in uplifting mountains, Earth Surface Dynamics, 6(1), 217–237, doi:10.5194/esurf-6-217-2018.

Guerit, L., X.-P. Yuan, S. Carretier, S. Bonnet, S. Rohais, J. Braun, and D. Rouby (2019), Fluvial landscape evolution controlled by the sediment deposition coefficient: Estimation from experimental and natural landscapes, Geology, 47(9), 853–856, doi:10.1130/G46356.1.

Whipple, K. X., and G. E. Tucker (1999), Dynamics of the stream-power incision model: implication for heigth limits of mountain ranges, landscape response timescales, and research needs, J. Geophys. Res., 104, 17,661–17,674.