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My basic idea is that PISM has two basic mass-conservingmodels which are already implemented, namely

-hydrology routing [= PISMRoutingHydrology]
-hydrology distributed [= PISMDistributedHydrology]

My suggested replacement for a true conduits (R-channels) model would be a hybrid model

-hydrology dual

which would construct one instance PRH = PISMRoutingHydrology and one instance PDH = PISMDistributedHydrology. The PRH would be the "conduits" part of this dual model and the PDH would be the distributed system of cavities.

Note that the PRH model essentially routes all water down the steepest descent path of the geometrical potential psi_o = rho_i g H + rho_w g b. That is, the water velocity is not diffusive for the "deep" reason that is true in PDH, namely that the pressure gradient term acts diffusively anywhere near steady state. In the PRH model the pressure is the overburden pressure.

In "-hydrology dual":

1. The standard melt sources (bmelt and inputtobed) would go into PDH. If water input was uniform and small, it might be that W=0 forever in the "conduits" system PRH.
2. But there would be an additional exchange term between the parts. When either the water amount W was locally large, or the flux in the distributed model was locally large, then PDH would "dump" the excess water into the PRH: PDH --> PRH.
3. The water which is dumped into the PRH model will no longer influence the effective pressure seen as basal resistance by the ice.
4. In circumstances where the fast water inputs stop the the dumping should cease after some time, and the PRH should fully drain after some additional time in those circumstances.
5. This "dumping" term would be one way only, water would never go back PRH --> PDH.

A major issue is in 2.: what should the exchange/dumping term depend on?

But I envision the dumping rate as being roughly related primarily to a measure of the excited state in the PDH. A "spring event" might start with W=0 in the PRH (conduit) model while the water builds up in the PDH (distributed) model. At some point water starts to be transferred into PRH so that the effective pressure would go back up in the PDH model. Sliding of the ice will generally cause a pressure drop in the PDH model, which goes with increasing ability of the PDH model to "hold" water. But when the PDH is full, in the sense that W exceeds the W_r scale, there should be a tendency to empty through the conduits, not just hold onto the water.

This idea takes advantage of existing code. That is, "-hydrology dual" would be implemented in a new derived class like "PISMDualHydrology". It would hold pointers to
one PDH object and one PRH object. The new class would implement the "dumping"
term.

An observational (?) or physics question about glaciers is: Do R-channels essentially always follow the steepest descent path of the geometrical potential psi_o = rho_i g H + rho_w g b? I've assumed this in suggesting this "-hydrology dual" idea.

This idea probably echos ideas in Flowers and Pimentel papers. This should be examined.

This would be a python script using matplotlib. It could go first in pism/examples/nbreen/.

At this point the model uses "bmelt" from the conservation of energy routine. This is not described in the manuscript, but it could be. But also there is heat dissipation from turbulent heating. This could be added.

This is about both the paper and the PISM implementation.

Recall that the flux formula is

q = - {k}{rho_w g} W^alpha |grad psi|^{beta-2} grad psi

Recall that

psi = P + rho_w g (b+W)

Together these imply a splitting

q = V W - D grad W

where V would depend on everything including grad W.

But we can approximate the "|grad psi|^{beta-2}" by assuming W ~~ 0:

|grad psi|^{beta-2} = |grad (P+rho_w g (b+W))|^{beta-2} ~~ |grad (P+rho_w g b)|^{beta-2}

This gives

V = - {k}{rho_w g} W^{alpha-1} |grad (P+rho_w g b)|^{beta-2} grad (P+rho_w g b)

and

D = k W^alpha |grad (P+rho_w g b)|^{beta-2}

These seem essentially as computable as what we already have.

As the title says, Ntilfunctions is missing, so the tex file won't compile.

See pism/examples/jako/hydrocent.sh in the dev branch for a minimal start.