Unexpected behavior when changing flow law exponent n

clairewcook

For Glen's flow law, I'd expect that an increase in the flow law exponent n, with everything else kept constant, would lead to faster flowing ice. This is not what I observe happening in models I've set up in ISSM. For instance, when I change n from 3 to 4, everything else kept constant, the surface velocity decreases by six orders of magnitude. I've set up an identical case using a different finite element model. The results for ISSM and the other model match for n=3, but do not for n=4. The other model gives, as expected, higher velocity for n=4.
Any suggestions as to why this is the case?

mathieumorlighem

Hi Claire,

the way ISSM is coded you should be able to pick any n that you want, there is nothing specific to n=3 or n=4. Now, whether you get a faster or slower flow really depends on many things: boundary conditions, floating vs grounded ice, surface slopes etc. Did you run exactly the same test case on your other model? Did you use the same flow equations (SSA or HO?)?

clairewcook

All the test cases I ran were exactly the same (boundary conditions, surface slopes, flow equation, etc). I only tested variations in the n parameter.

I have now tested this for a few different setups, including the ISMIP A model provided as an example. For all of them, just changing from n=3 to a higher n leads to lower surface velocity in ISSM.

mathieumorlighem

I did a few tests on my end and I find the same as you. I looked into a few things but it seems to be making sense (at least mathematically). Strain rates are generally small, and so the effective strain rates are on the order of 0.01 yr^-1 (3e-10 s^-1).

The ice viscosity is defined as B/(2* eps_e^((n-1)/n))
So if you increase n, you end up with a smaller numerator and the viscosity will get larger and larger.

For example, for B=1e8 (SI) we get:

n=3 -> viscosity = 1.1157e+14 Pa s
n =4 -> viscosity = 6.9363e+14 Pa s

and so the initial viscosity is higher with n=4, we get a slower speed and then we iterate until convergence.
Anyway, I don't see why the ice should be faster, but I am happy to investigate further.
Mathieu

clairewcook

I think that argument only works if you assume the strain rate doesn't vary with stress. When I'm thinking about ice flow equations conceptually, I think of things in terms of stress rather than strain rate. This makes more sense to me, at least in an ice flow context, where flow is driven by stress due to gravity, not an external applied strain.

Say you rewrite that viscosity equation in terms of effective stress using eps_e=Atau_eⁿ, where A=B^-n.
Then viscosity = 0.5(A*tau_e^n-1)^-1. For a given stress, if n increases, viscosity decreases. Hence, I would expect higher flow speeds for higher values of n.

mathieumorlighem

Hello Claire,

remember that it is not the stress that is conserved but its divergence:

\nabla \cdot \sigma' - \nabla P + \rho g = 0

For example, in 1D, using the SSA approximations, we get

d/dx (2 sigma'_xx) = d/dx (4 mu dvx/dx) = rho g ds/dx

and the right hand side is independent of n (it is the driving stress). Using the same equation, the effective strain rate for 1D SSA is simply |eps_xx|. Assuming that eps_xx>0 to simplify things, we get

d/dx( B eps_xx^{1/n -1} * eps_xx ) =d/dx( B eps_xx^1/n )= rho g ds/dx

and so, if ds/dx and B are constants and eps_xx(0) = 0 as a boundary condition, we get:

eps_xx = ( 1/(B) rho g ds/dx * x)ⁿ

again, B is large (_1e+8 SI), and surface slopes are small (e.g. ₀.01). So we are taking the power of a number <<1. As n increases from 3 to 4, we do expect eps_xx to get smaller and so v_x to be smaller too.

If you have a numerical experiment ready from another model, I am more than happy to look into it, but I don't see anything unexpected at this point. Feel free to email me (Mathieu.Morlighem@Dartmouth.edu) if you would like to follow up!
Cheers
Mathieu

rueckamp

Hi Mathieu, Claire,

I observe the same behaviour as Claire's simulations when setting n=4. When runnig ISMIP-HOM A (L=80km, HO, periodic BC) with ISSM the maximumn velocity decline from 80m/a to 0.1m/a compared to n=3. In COMSOL, I run the same experiment (FS and HO) and velocities increase to ₁₀⁶ m/a for n=4.

cheers,
Martin