Hydrology is flat, and its buckets all the way down!
For some reason, much of my recent work keeps coming back to buckets, and re-thinking the conceptualization of natural hydrologic systems as buckets. I am generally sick of talking about buckets. I'm hoping that this post is my farewell to thinking about buckets, at least for a while.
Sir Edmond Leakybucket
When I was first learning differential equations the professor told us a silly story about Sir Edmond LeakingBucket, some ol' timey English royal who had to drink his ale quickly because his ale bucket leaked. I went on to study hydrology, so I've had to think about Sir Edmond for the past fiteen years. I can't escape him. Sometimes he mixes two kinds of ales together, sometimes his ale bucket is more complicated or simpler, but he is always losing his ale. Poor guy. Sir Edmond and his bucket do two important things: 1) gives nice differential equation examples, but more importantly for hydrology 2) Leaking buckets are a primary conceptualization for hydrologic processes.
A simple differential equation for the ale level in Sir Edmond's bucket is:
Where \(h(t)\) is the ale at time \(t\), k is a proportionality constant that governs the rate of outflow. Its solution through seperation of variables is:
Where \(h_0\) is the initial ale level in the bucket at time \(t = 0\). This gives us the opportunity to track volumes of ale through this bucket, and match the fluxes from buckets with data collected on real-world hydrological systesm. This is, in a nutshell, the field of computational hydrology, we just need to dress up and add complications to this bucket, and off we go.