Using constant (non-uniform) N on fixed grid results in very minimal retreat

[glugeorge]

It reaches steady state almost instantly (had to reduce timestep by a lot), and is very close to the initial condition

[glugeorge]

x-axis: distance from divide, y-axis: effective pressure
N is indeed staying the same (these are N profiles along the channel at t = 0y to t = 25y. Zooming in on terminus:

Looks like when N is nonzero at the terminus the ice doesn't like to move? Could be because that means that shear stress is high in that region.

The steady state N that the ice cannot move past is 4.4267e+04 Pa. (When C = 0.1, perturbing from A = 2.9e-25 to A = 4.9e-25).

Another observation is that increasing C (making higher friction bed) results in a faster steady state + less retreat.

Next step will be to look term by term in the equations

[jkingslake]

interesting! So this is after you corrected for the $\sigma$ scaling of $N$ to keep $N$ constant everywhere (in terms of real, non-scaled space)?

What do color of the dots mean in the bottom plots?

(on a different topic, feel free to assign me to issues like this, so that I am notified and I have it in a list somewhere).

[glugeorge]

@jkingslake Yep, I kept N constant everywhere. The color dots (marking effective pressure at the new terminus) is just for me to discern timesteps - I just plotted the first 5 timesteps with r y g b k colors.

[jkingslake]

How large are the time steps?

How far for the 'dynamic $N$' grounding line move in the end?

[glugeorge]

Here, the time steps are 5 years each. The dynamic N grounding line moves 38 km whereas the static N grounding line moves 2km.

[jkingslake]

OK, very interesting. So its like we thought, the higher $N$ reduces the change. It would be interesting to know how much that depends on the characteristic shape of $N$ which we have been talking about (with the peak 10-20km back from the GL).

By the way, so axes labels would help with others keeping track of this (and us coming back to this in a year's time ;-)

Great work though! So cool to see some strange behavour we dont yet understand emerge from the model!

[glugeorge]

I've tried switching out the sliding law, such that now we're following a constant*u sliding law, and then I matched the initial steady state non-dimensional basal shear term with it to invert for this constant which I dubbed lambda. This constant will represent the static bed properties. Like the situation where I use the same sliding law but keep N static and constant over time, this model also retreats very minimally.
Here's what the retreat looked like with the same sliding law, constant N:

Here's what it looks like with the new power law, N absorbed into the constant:

Even less retreat!

[jkingslake]

Cool. I think this sounds really good, but just to make sure I know exactly what you are doing, you have the maths written out?

[glugeorge]

From the nondimensional equation for the ice stress balance, I set the old basal shear term equal to the new law: $\gamma N (\frac{u}{u+N^n})^{1/n} = \lambda u$. I'm claiming that any relevant non-dimensionalization gets absorbed into the $\lambda$. The u scale with A_s from the old law gives a scale of around 300 m/yr which I think is a valid scale to use, and As doesn't appear anywhere else so I'm fine with not creating an entire new law. With those two sides equal, I just plug in the steady state coupled solution for u and N and solve for $\lambda$. I then keep $\lambda$ static and equal and allow u to evolve.