September 21, 2018

Rivers V: Muddy Waters

Now that I have the speed of rivers, I can start to figure out the balance between the deposition of sediment and the erosion of the terrain.

To reiterate, the map scale makes deposition a bit of a challenge, because the amount of sediment being picked up is enormous. This also assumes that every hex has a single drainage point.

If 1 ft of height is removed from a 347 sq.mi hex, that's nearly 275 million cubic meters of sediment that's got to go somewhere. Even if we assume that not all of it is pushed along, it's still literally tons of material.

There's a lot of studies on sediment capacities and things, but less general than I'd like - since sediment capacity and transport is something we can measure, it's not usually predicted. However, we can still draw a few conclusions.

First of all, the amount of deposited sediment is however much material is eroded away $\partial h \cdot A_h$ (where $A_h = 347$ sq.mi), plus all that's coming down the river, over the time step in question $\partial t = 2.5\cdot10^5$ yr. That's pretty straightforward.
\[Q_s = {\partial h \cdot A_h \over \partial t}\]
For a river with 70 drainage points, $Q_s = 0.369$ cu.ft/s.

We can do something similar for the amount of material that the river can hold, where $A$ is the drainage area and $S$ is the slope.
\[Q_t = k_t A^{1.5} S\]
For the river in the previous example, $Q_t = 0.836$ cu.ft/s. So the capacity is greater than that which is added to the load, and nothing happens.

$Q_s$, of course, grows as we continue down the river. So at some point it will be bigger than $Q_t$, at which point it will deposit $\partial t (Q_s - Q_t)$ cu.ft of material, or add ${\partial t \over A_h} (Q_s - Q_t)$ feet to the height of the hex. $Q_s$ is set to $Q_t$ and continues on its way.

Now, this turns out to not use velocity at all! This is alright, because knowing the velocity is still useful. There is still a lot of tweaking and testing I have to do to get this to work, and the velocity will be more helpful when determining which river routes are suitable for trade.

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