Evaluating velocities at depth with the hybrid model

[icepack]

Evaluating velocities at depth with the hybrid model

[danshapero]

Hi Bryan! The code in that notebook to compute u_shear isn't evaluating the basal velocity as such, but rather the projection of the velocity field onto the 1st-degree Legendre polynomial, which is orthogonal to the constant functions. So it is to be expected that the basal velocity (or the surface - basal difference) is not the same as this projection.

The code that you've written certainly ought to work to compute the basal velocity. If you want something that will make a Function defined on the base mesh instead of a grid of values, you can also do this by a clever weighted depth-average. Let me know if that's what you need and I'll send some more complete code for it.

[bryanvriel]

Ah, thanks Daniel for clearing that up! The projection makes sense. Indeed, I think I could make use of a few examples of clever depth-average weighting. I was originally thinking of constructing delta functions centered at arbitrary depths, but I'm pretty sure I would've done that wrong!

[danshapero]

I made a branch with some demonstrations in the hybrid XZ notebook here. The important part is right before the conclusion section. In case you don't want to go digging through all that, here's the code to calculate the basal and surface velocities and plot them:

import sympy

ξ = sympy.symbols("ξ", real=True)
legendre_polynomials = [sympy.legendre(n, 2 * ξ - 1) for n in range(3)]
basis_functions = [p / sympy.sqrt(sympy.integrate(p**2, (ξ, 0, 1))) for p in legendre_polynomials]
basal_values = [p.subs({ξ: 0}) for p in basis_functions]
surface_values = [p.subs({ξ: 1}) for p in basis_functions]
basal_polynomial = sum(w * p for w, p in zip(basal_values, basis_functions))
surface_polynomial = sum(w * p for w, p in zip(surface_values, basis_functions))
basal_weight = sympy.lambdify(ξ, basal_polynomial.simplify())(ζ)
surface_weight = sympy.lambdify(ξ, surface_polynomial.simplify())(ζ)
u_basal = icepack.depth_average(u, weight=basal_weight)
u_surface = icepack.depth_average(u, weight=surface_weight)

fig, axes = plt.subplots()
axes.set_title("Surface & basal velocities")
firedrake.plot(u_basal, edgecolor="tab:blue", label="basal", axes=axes)
firedrake.plot(u_surface, edgecolor="tab:red", label="surface", axes=axes)
axes.legend();

Does all the sympy + Legendre polynomial hackery here make sense? You mentioned constructing delta functions at arbitrary depths. This code is doing that by instead looking at the action of a delta function on something that we know has an expansion in Legendre polynomials. So it does what a delta function would do without having to actually form the delta function as such. In any case this should be probably either be in a tutorial or in icepack itself. The attached figure shows the result.

I am a little disturbed by the fact that you can evaluate things at negative values of the vertical coordinate -- arguably those should be giving a "point not in domain" error. That's more of a Firedrake issue than an icepack issue.

Closing this for now, feel free to reopen if anything else is unclear.

[bryanvriel]

Many thanks Daniel, I think I understand now how the weights are interpreted with respect to a Legendre polynomial expansion. That's quite clever!

Yeah, I'm not too worried now about the out-of-domain evaluations. Perhaps firedrake is assuming that the user knows the vertical mesh is represented by a polynomial expansion and can thus technically be evaluated out of the domain. I'll just make sure never to do that.