Can I get some help with the calculations of my mountain?

Question

Two days ago, I asked a question how to stop a waterfall. I got great replies, and had to reconsider some of my numbers. Now I came up with this calculations, and I would like to check 'm with all the geniuses here. Thanks in advance.

I have to figure out where all the water for my huge waterfall comes from, and rain seems to be the best (only) option. I limited the width of my waterfall from 1km to about 300m, kinda like the American Falls of the Niagara Falls, who have an estimated water flow of 75,000 gallons/second. This way, all the water would fit through a tunnel with a diameter of 5m, where it was 18m before (thx @hde-226868). This thing should be possible to block.

A smaller waterfall also needs fewer water, so I could go with 6,000 km² in stead of the 50,000 km² @RobWatts measured my 1km width waterfall would need. That's a surface of 75 and 80 km, but I would guess less when we have a cone like a mountain is.

6,000 km² on a mountain. What would be a realistic height here for the mountain, with an average grade of 7-10, to make it big enough to collect all the water needed for my waterfall? And how big would the ground surface be?

Small follow-up on my previous question: would it be possible for this amount of water to disappear in a cave construction beneath the water, after rerouting it like @mikey said, if there's an underground passage to the sea?

Answer 1

Your mountain will have to be a bit bigger than you realize, I think. Assuming you need 6,000 $km^2$ of catchment to produce your flow, all of this needs to be sloped toward the river rather than toward the surrounding territory. So what you need is sort of like a crater lake, although not necessarily quite as pretty.

If this is the case, you can model the catchment area as a circle about 88 km in diameter, with a fairly small lake near one side which feeds the exiting river which has the water fall. The reason the lake needs to be fairly small is to keep the river flow rate close to constant. If the lake is the same size as the catchment area, the lake will drain and flow will cease until more rain feeds it. A small lake allows the surrounding soil and vegetation to act as a buffer on incoming rainfall. The lake itself does not need to be very deep, since it does not noticeably buffer the water flow.

This means that the mountain can be almost any combination of sizes you want, consistent with the slope and the need to allow rainfall to reach the lake. If you make the height 30,000 feet, for instance, there is no way for rain clouds to make it over the rim to dump water in the lake. This also means that the mountain will not be impressively snow-capped, since a wide snow belt will mean most moisture has fallen as snow before the clouds pass over the rim to the catchment area.

Let's assume a frustrum of a cone, with top 3 km above the surrounding terrain, with a 10% average grade. Then the slope to the rim will be 30 km. This will result in a base diameter of 88 km (for the catchment area) and 60 km for the slopes, for a total base diameter of 148 km..

Actually, a better idea is probably an asymmetric cone which has a catchment rim at relatively low altitude on the side facing the prevailing winds, and much higher elevations on the downwind side. This will allow rainclouds to be driven into the catchment area at low levels, then dump their moisture on the other side as they are forced upwards. This will allow the mountain to be arbitrarily high and impressive around much of its body, while still allowing rainfall to produce the waterfall. But it can't be high except for a relatively small pass, or air will be trapped over the catchment area and backpressure will keep most rainclouds from entering the desired area.

How the tunnel was formed, how it was sealed, and how it was reopened are a sequence which you as a writer will have to deal with. Generally, such tunnels are caused by water eroding rock like limestone, which means that the channel was formed by water from the catchment area. Something must have happened to stopper it, and then something must have happened to remove the plug.

Answer 2

TL;DR: A Bigger Multnomah Falls

Yes, I think all of this is possible.

I'm trying to visualize your construct. Consider something like Multnomah Falls. At a rate of 4m3/second you can have make a tiered waterfall (see image). Unlike Multnomah Falls, your ascent should not be so easily undertaken - the cliffs should be higher and sheer. Where there is a bridge here is where your tunnel is. Where it says 4m3/sec you want significantly more.

The source of these falls is mostly a freshwater artesian spring, augmented by some snow melt; but it maintains a seasonal flow. You very well could have the same setup.

When the settlements upstream (above the top of the falls) accidentally re-route the river, it simply goes down the other side of their high plateau. This makes it much easier, since they simply have to reroute from the source of the springs, which may be closer to the other side of their 'mountain'.

The steepness and size of your mountain is no longer restricted to the waterfall's design and source but you must now consider how your mountain functions where people cannot come up or go down it. Perhaps your mountain is really a plateau like Mount Roraima? But with a natural river and springs.

EDIT: A waterfall occurs (usually) when the river has worn out the softer stone around harder, igneous stone. The harder stone remains meaning the water goes over it instead of wearing it away as well. On your plateau mountain, the tunnel could have been caused by having a patch of softer stone in the center of harder stone (like granite). It is not unusual for magma to create weird stone formations of hard stone through soft stone like limestone.

In this image, the volcanic igneous rock will remain one day long after the limestone washes or 'blows' away to processes. In your mountain, you have a volcanic tube and flow that created a wall with a hole in it, with sandstone or limestone in it millions of years ago. Over the millenia the river disolved the soft stone allowing for a ring or tunnel through which your waterfall now travels.