The last blog post explained a little about how we need roughness to describe the friction between flowing water and the surface underneath it, and how measuring that roughness can give you different understandings of friction than modeling it.
It mentioned in passing that there are different ways to represent the relationship between flow and the friction it experiences from the land. These are called roughness schemes or resistance formulations and the idea behind them is shown in panel (a) in this figure.
The problem is that there are ... well quite a few ... of these schemes. Different ones for turbulent, laminar or transitional flow. Darcy weisbach, Manning, Chezy, mixed ... what's an engineer to do when trying to describe runoff? How do you pick one? And having picked it, how do you make a sensible estimate of it's parameters? As panel (b) above shows, they differ in how they describe the relationship between the depth of flow and its mean velocity, so you might think it's pretty important to pick the right scheme!
To address this Octavia Crompton again used the large dataset of USDA rainfall-runoff plots and the trusty Saint-Venant equations. She calibrated these to discharge ... and found that just about all the schemes performed the same (panel c)! This is a problem called equifinality that pops up a lot in hydrological modeling - you can't get enough information from just one kind of data (discharge only in this case) to help you work out how to describe the complicated system you're looking at.
Did it matter? If you wanted to understand velocity variation, yes. As you can see in panel (d), the velocity is more sensitive than the discharge to the scheme, and looks pretty different. So Octavia calibrated the model using BOTH velocity and discharge. She found that most commonly the transitional resistance formulation behaved best.
What she did then was see if, having chosen any of the schemes, she could predict the optimal value for the roughness coefficient. That is, if she picked Mannings, could she predict the best "n"? And she could, not too badly! This figure shows fitted versus predicted roughness coefficients for a bunch of different schemes.
What this suggests is that it's hard to pick the "best" scheme, but once you have picked one, surface proprties can help guide your choice of roughness parameters - a very cool finding.
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