Question
Ahnert (1976) proposed a model of mountain building (orogenesis), in which the orogeny-scale erasion rate [E) is proportional to the average elevation (eta, measured with respect to steady sea level) of the orogen, i, e. E = k eta, where k is a constant that embodies rock strength and climate. This is an example of negative feedback. Suppose initially peneplained continental lithosphere is uplifted at a rate U: your goal is to develop a spreadsheet solution to Ahnert's model: Delta_eta/Delta_t = U - Your spreadsheet should 'march forward in time' and predict the evolution of the mountain-range elevation, i.e. you want to find eta(t) for t >0. Note that Delta_tau = eta^new where eta^new and eta are the average elevations at the 'end' and 'beginning' of a time of duration Delta_t, respectively. To gain a better appreciation for the utility of a spreadsheet solution, integrate Eq. 1 manually, i.e., march forward in time for lie time steps, using Delta_t = 100 ka. I want you to compute (from Eq. 1) eta(t = 100 ka), eta(t=200 ka), eta(t=300 ka), eta(t=400 ka), and eta(t=500 ka). This exercise amounts to 'marching forward' in time for five equally spaced time steps: the solution at each 'new' time makes use of the solution from the previous ('old') time. Use U = 1 mm/a and k = 10^6 a^-1, present your results in a simple table.
Explanation / Answer
Since delta n/delta t=U-kn
where u=rate of uplift=1mm/a
k=10-6/a
U-kn1=1-10-6*100=1-10-6*100=1-0.0001=0.9999
U-kn2=1-10-6*200=1-0.0002=0.9998
U-kn3=1-10-6*300=1-0.0003=0.9997
U-kn4=1-10-6*400=1-0.0004=0.9996
U-kn5=1-10-6*500=1-0.0005=0.9995
