Any physicists here?

Disclaimer: I am not a physicist--just a guy interested in the subject.

There was once a huge ice dam near Missoula, Montana, USA. One day (literally: scientists believe it happened in a single day) that dam burst, releasing a very large lake behind it. That deluge carved out most of the Columbia River Gorge (4000 feet deep, 80 miles long). (Actually, this event may have recurred as many as 40 times.) The lake measured about 3,000 square miles and contained about 500 cubic miles of water, half the volume of Lake Michigan.

Your 5.7 mile x 5.7 mile x 2 mile cube of water is significantly smaller. But reading about the Missoula floods will give some insight as to the incredible forces that can be produced.

However: you seem to have it in a "perfect" container. At least you don't describe the materials of these boxes. Assume they're mostly smooth sided, and that when the walls "suddenly" disappear, they do so instantly (one moment the walls exist, the next moment they do not). This is so that the only calculations you need worry about are for the movement of the water itself. If your containers are made of, as the states of Washington and Oregon were during the flood mentioned above (basalt and soils and very terrified squirrels), then things get more complex.

I don't know how long it'd take in Question 1 for half the water to move from Box #4 into Box #1. Not very long. Minutes, I'd think. The dude in the northwest corner? If he's not equipped with scuba gear and lots of padding, I think he's done for. The water would first reach the intersection of the bottom of the box and its northern wall, then rush upward at a high rate.

I think it would take as long for the water in Question 2 to rush into Box #1, turn the corner and drain away in Box #2. Because of the absence of the wall between Box #1 and Box #2, however, there might be some mitigation in the up-rushing of water at the north wall. The dude may survive.

In Question 3, I assume this floaty thing is maneuverable? Like in three dimensions? I doubt there would be a high wind. Water seeks the lowest level--rather than a wall (as though made of some solid substance, like ice) falling over, the whole mass of water would just "slump down" if you will, sliding along the floor first. Any wind associated with that would be caused by turbulence at the surface (I think). If the guy's floaty thing is fixed, he would likely notice only the very loud noise made by several cubic miles of moving water. Chances are he could be fixed on a disk at center of the floor plane in Box #2, and not suffer any consequences other than a rather rapid rise in humidity.

As for the material making up these boxes, it has to be pretty strong. Not just to withstand the pressure of nearly 40 billion tons of unmoving water, but to withstand the impact of it as it rushes against the opposing wall.

If I can find my fluid mechanics book, I can probably spin off some ball park numbers.

Just thinking on this: you might be able to get a pretty good idea of what happens by building a scale model--something that would fit in your kitchen sink. Make a video of it, analyze what happens frame by frame.

ETA: I used to play around with model railroads, HO and N scale stuff. I was very anal about realism. Discovered that most model railroad trains operate at very unrealistic speeds. HO scale is 1:87 (in other words, an inch equals about 7'-3" in an HO layout--or 3.5mm = 1 foot). A scale mile is about sixty real-world feet. A train traveling at 60 MPH would take a minute to cover the distance--or it would take one second to cover a real-world distance of one foot.

In the Missoula floods, the peak flow is estimated to have been about 80 mph. Of course, this velocity didn't happen at the ice dam initially--it was the velocity somewhere downstream. Also, the water depth maxed out at around 400 meters.

If each box in your model was 12" x 12" x 4.4", the scale is 1:30,000. To accurately represent the Missoula flood, the water in your model would take about 4 minutes, 15 seconds to cross the floor of Box #1. (The Missoula flood depth at your scale of 1:30k is about 1/32".) I bet that the water in your model would move out significantly faster than that!

I would not want to be that dude of yours in the NW corner of Box #1

I'm a physicist but not a fluid mechanics expert. My expertise is in rocket and missile guidance.

For starters, there is no 'equation' for what you describe. Typically equations get developed to handle actual, physical problems. What you describe isn't an actual physical problem and because of the boundary conditions in your scenario, gravity wave propagation doesn't make sense. However, I gather that you are working on some kind of Sci-Fi/Fantasy scenario but you still want physical realism. I respect that.

You are going to have to think about your problem differently. When that barrier disappears two things are going to happen. The water at the top is going to want to fall under the influence of gravity. The water at the bottom, however, is going to be under enormous pressure. Thousands of pounds per square inch. The water near the bottom is going to be squeezed out as the water on top pushes down on top of it.

At the instant of barrier disappearance the pressure at the bottom of the tank, which I assume is 11,000 feet, is going to be around 5,000 psi, (4,783 if salt water.) It's easier to work metric so that is 3447 newtons/cm^2. Now, 1 cc of water weighs 1 gram. So when the barrier disappears, 1 cc of water at the edge will have 3447 newtons acting on it. F=ma so the acceleration of the 1cc parcel of water will be on the order of 3.5 million feet per second. The entire bottom edge is going to experience this displacement. Naturally this acceleration won't be sustained because of expansion and the rapidly changing geometry of the situation, but what you are looking at isn't what your daily experience would translate as a "classic" flow problem.

What you're going to get is an explosion. A big-ass explosion that might create a true shockwave that would propagate across the cavity well in excess of the speed of sound. Naturally the water wouldn't propagate that fast but it is going to be getting forced out by the two miles of water pushing down on top of it. I'd guess your peon on the far side might have just enough time to put his head between his legs and kiss his ass goodbye. To calculate the effects with any accuracy you'd need to recast the problem as a partial differential equation in mass and potential then solve it numerically. And since it'd probably be a good Ph.D. dissertation topic I'm not going to take the year it would require.

Yes, it would be loud, but it wouldn't be the sound of rushing water. It would sound more like a nuclear detonation.

The Missoula flood isn't really a very good analogy to this problem. In this case, even though it seems that the event is happening quickly, the stress relief, even if the flood developed over less than a minute, is going to be easily long enough to prevent the nightmare scenario described above. Such a flood, FYI, is called a jokulhaup.

And my name is Alexander Julius Centurion. At least, I wish it was...

MelancholyMan said:

... what you are looking at isn't what your daily experience would translate as a "classic" flow problem.

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Nope. LOL

The Missoula flood isn't really a very good analogy to this problem. In this case, even though it seems that the event is happening quickly, the stress relief, even if the flood developed over less than a minute, is going to be easily long enough to prevent the nightmare scenario described above. Such a flood, FYI, is called a jokulhaup.

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The Missoula flood is the only natural disaster of this type I know of. But it's obvious that there is a vast difference in scale between that event and the one proposed here. The height of the water column alone is greater by a factor of ten. And the ice dam didn't vanish in an instant.

I imagine that the mythical substance that contains the water can be considered to be frictionless, so that the 3.5 mfps velocity of the water remains virtually constant across the 5.7 miles of the empty container. (ETA: it occurs to me that if this figure is correct, bottom of the water in container #4 would reach the opposite side of container #1 in 1/10 second. Or so. At that speed, who's gonna count fractions?)

Just out of curiosity, I made a (very crude) model of the situation (taped up cheerios box, kitchen sink). Obviously, I was unable to remove the wall of the box instantaneously, but I rigged it up so that I got it out of the way in maybe 1/4 second. The water leaving the box nearly climbed the vertical wall of the sink 6 inches away. I was impressed.

ps, Navigator, I am curious as to how you came up with such a scenario, and why these immense containers of water are needed.

You do realize, I'm sure, that these are big-ass jail cells.

This additional info might have some bearing on the disaster of releasing the water. If #2 has a floor of deep canyons, do the other containers have natural-type terrain for bottoms, too? Are the walls just plunked down on, say, northern Arizona, say, and #2 happens to be over the Grand Canyon?

If that's the case, and you just blow a hole in the wall, I think you need to address the particles of the wall along with the rushing water. AND how that water might be affected by irregularities on the bottom. As I mentioned earlier, a smooth bottom will allow the water to zoom! But a more irregular one could slow the water down some (5.7 miles square is big enough to contain a small mountain range). But the force of all that water being released would likely cause severe destruction to those mountains. And yeah, anyone in box #1 is mush.

If all you need is to kill everyone off in a 32 square mile area (one of your boxes), I think you can do it just fine without all that water. Like maybe 20% as much. If your murders are really psycho, they can sit atop the walls and rejoice in the screams of the drowning victims.

XThe NavigatorX said:

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question 1.

If the border between 4 and 1 suddenly disappeared, I know once the water eventually settled, 4 would be filled w/ 5,500 feet of water and 1 would be filled with 5,500 of water. My question is, how long would it take for the massive, two-mile high wave of water to travel the 5.7 miles or so to the northern edge of 1?

I assume it would happen pretty quickly, but is there like specific terminal velocity of moving water? Is there a layman's formula that would deal with such a thing? If I have a dude standing in the top corner of 1, and he sees that wall of water suddenly rush at him, how long would he have before he would get obliterated? Would he have time to pick up his cell, call his mom and tell her not to wait up for him?

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I'll just guestimate. If he's looking in that direction he'll see it quickly (a second or two?) turn into whitewater as it accelerates into the air and starts to mix with it. That's such a lot of force I think the water at the bottom could accelerate up to near the speed of sound in a few seconds. The force needed to break the sound barrier MIGHT be enough to limit how fast the water goes, but I'm only guessing - with a block of water two miles high and an opening five miles wide, it might not.

But assuming the bottom 50 feet of water accelerates in the first few seconds up to around the speed of sound, then that's 1100 feet per second, that gives him about 30 seconds of water-free life left.

People can dive into water from large heigths and live, perhaps approaching the human "terminal velocity" in air of about 80 to 100MPH. This wall of water will be going about seven times that, and I have no doubt it will kill the character when it hits.

Meanwhile, the water would slosh back and forth between blocks 1 and 4, like a pendulum, at the rate of once every few minutes, perhaps taking hours before the surface appears reasonably flat and stable again.

XThe NavigatorX said:

Question 2...

Same scenario, but this time the wall between 4 and 1 and the wall between 1 and 2 disappears at the same time.

I know the water would eventually all drain away into the bottomless hole of #2, but approximately how long would it take? Seconds? Minutes? An hour?

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I'd "swag" five minutes for half the water to pour through the hole, and maybe after an hour the left walls of 1 and 4 would be down to 10 or 50 feet of water, and maybe another hour or two before it's down to a couple of feet deep.

XThe NavigatorX said:

I also suspect the initial violent outpouring of water would cause the dude in the top left corner of 1 to still be destroyed. Am I right? Would the amount of time it takes to smash up against that northern barrier be mitigated at all by the bottomless pit to the immediate east of the rushing water? Does this question even make sense at all?

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It may be slightly 'mitigated' but not enough to save the guy.

XThe NavigatorX said:

Question 3...

Assume the above scenario, and my dude is now standing on a small floaty thingy in square #2

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I'll presume he's floating at least a thousand feet up, whatever it takes to keep him above the water stream torrent that pours into #2.

XThe NavigatorX said:

When the water first rushes out and toward #1, would there be a big displacement of air that would cause our friend to be knocked off his floaty thingy? If so, would it be a constant thing or more like a quick shockwave?

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Hmm, I'm thinking it would START as a quick shockwave, then be a "constant" wind of maybe 50 to 200 MPH (depending on exactly where he is in 2) for the first 30 seconds or so, and it would drop off as the water drains out.

XThe NavigatorX said:

Also, am I right to assume this would be really, really, really godsdamned loud?

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Yeah, perhaps like the "freight train" sound they say accompanies a tornado.

Dunno if I dare mention this as it may or may not have any relation to your story, but this thing is a little reminiscent of the "Well world" series of novels I read long ago (when there were only five novels in the series), thought there were a lot more than nine cells on the Well World, and I think each cell was even larger (the cells covered a whole planet).

XThe NavigatorX said:

Ha. Thank you very much, Alexander Julius Centurion. Your help is much appreciated.




That's exactly the type of information I'm looking for. The time doesn't need to be exact; I'm just looking for a close approximation as to what would happen. The top-down explosion is exactly what I hadn't counted on, and I'm going to have to rethink and rewrite how this plays out.

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Two points. The explosion is bottom up. The bottom water being under the greatest pressure will be blasted out the hardest, and the pressure on it will drop very fast as it expands. This water won't propagate as a slosh but as a pressure shockwave. It would probably leave a vacuum behind it that would suck more water in. So what you're going to get is a sheet of water blasting across the bottom at successively less velocity as you move up the column. Very likely, the water at the top will just sit there and then fall straight down into the void left by the expansion at the bottom. That's like, billions of tons, falling... down. What are the walls of this cell made of? I can't imagine anything that could withstand this kind of shock. Indeed, on a planetary scale you're talking about initiating significant seismic waves, probably strong enough to induce earthquakes.

There was jokulhaup that issued from Hudson Bay at the end of the last ice age. The outflow was apparently into the North Atlantic, but it is estimated that global sealevel changed by something like an inch in less than 24 hours. It caused massive Earthquakes world wide. You might want to toss that into the mix as well if this takes place on a planet.

You might also want to consider that this event would flash millions of cubic feet of water directly into water vapor. This would manifest itself in two ways. First, you'd probably get some really thick clouds and fog that would make it hard to 'see' the wall of water coming. Secondly, since the latent heat of vaporization of water is over 300 joules per gram, the sudden conversion to water vapor would cause the temperature to drop rapidly, probably turning the advancing wall of water into a wall of ice shards travelling at hundreds, perhaps even thousands of miles per hour.

Since shock waves travel through solids faster than through the air, here is a rough pheonomenology that might be experienced by some poor schmuck on the other side of the barrier at a distance of a few miles.

The barrier would disappear and there would be a flash. If someone happened to be looking in that direction they'd see, for an instant, a towering, two-mile high, quivering wall of water. At the bottom they'd see what might be described as a rapidly advancing sand storm, but white. Then they'd feel the ground beneath them shake. Those who weren't looking would then feel the shock and look around. This would happen in the first two to five seconds. From the time they saw the advancing sheet of white to the time it hit them might be ten to fifteen seconds. It would hit as ice shards and the momentum alone would destroy anything it hit. It is unlikely they would have time to actually figure out or think about what was happening, only that it was bad. Those far enough away to avoid the ice blast would be enveloped in an ice fog blowing at hurricane force from which issued a rumbling like Zeus himself couldn't conjure, couple with huge seismic waves travelling through the ground. This could last for a couple of minutes at six miles away. Then the water would hit. Sounds bad...