Oceanography/question about shallow ocean wave height
QUESTION: Got this question today from a library customer - can you help?
If I am designing a planet the size of Earth covered by a consistently 4.5 foot depth ocean, and want to have a single small island as the only land surface, how high above sea level must my island be to remain above the wave height if a 500 mile an hour wind blows constantly?
ANSWER: Wow. What a fanciful question. I have a feeling that you meant a 4.5 MILE deep ocean. This makes much more sense, although the idea of a constant 500 mph wind is still a bit daunting to contemplate. Anyway, the bottom line is that the top of the island wouldn't have to be that high, only on the order of 100 meters (330 feet) or less.
First of all, I'll assume that the 500 mph wind generates DEEP water waves with the same 500 mph speed (waves that have a wavelength less than the water depth). Without going into technical detail, I get a wavelength of about 1000 meters (actually 786m). A simple formula for the RUNUP (= vertical distance the water reaches on land) for a 10 degree "beach" slope and a wave height of 100 meters (way overestimate) gives a value of 63 meters = 200 feet. This would be the necessary height of the island. The PENETRATION DISTANCE (how far inland the water goes) is less than a kilometer (0.6 miles)
This result depends on the beach slope (larger slope --> larger runup), but if the library customer is designing the island, 10 degrees or thereabouts would be recommended.
This situation is of course very similar to that of tsunamis. Tsunami "waves" also can travel about 500 mph and, if SHALLOW water waves are assumed, this speed matches an ocean depth of about 4.5 miles (this makes me think the customer knew a little about oceanography). A big difference is that tsunamis can have enormous wavelengths (hundreds of miles) but very modest wave heights (10 meters-ish or less). These characteristics tend to counter each other which basically means that the runup and penetration distances for tsunamis is on the same order as the wind generated waves.
The scenario of an isolated island on an Earth-sized planet with 500 mph winds is very weird and a whole lot of oceanography is being ignored (spherical Earth, Coriolus force, friction, etc.), but there you have it.
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QUESTION: I agree, a most fanciful question - it's intriguing. The questioner did emphasize to me a shallow "ocean" of 4.5 feet, however. I see from my limited research on the web that the models would be quite different for such a shallow sea. Would you be willing to run the logic/math for a hypothetical shallow body of water? (I will never look at a wave the same way again!)
ANSWER: 4.5 feet it is! I first contemplated the problem with this depth and immediately ran into the problem of where the water would go when subjected to such an enormous wind stress on a spherical earth, never mind an island. The problem is that the topology of the earth (sphere) requires a node where the wind and water velocities are zero (maybe 2 nodes, located at the poles to fix the geometry in our minds). I have no idea how this would be set up. On top of that, a spinning earth, as in the actual case, would add to the complexity since this represents a non-inertial reference frame (Coriolus force). It then occurred to me that feet should be miles and so I thought I was off the hook.
If we really do have an extremely shallow ocean (4.5 ft), then lets assume that the water velocity is equal to the wind velocity of 500 mph in some patch where we want to put the island. This is actually a very tough problem. If we ignore friction in the water (i.e., assume it is an inviscid fluid), then it can be considered a so-called potential flow problem and is described by Bernoulli's equation and, bottom line, the water will just flow around the island without piling up. Not very realistic. This approximation also ignores waves (which might be interesting to work out) and also assumes that there is still viscosity between air and water, which is sort of inconsistent with the inviscid assumption. Anyway, the no-friction case is obviously flawed.
Adding friction means having to solve a simplified (!) version of the Navier-Stokes equations with proper boundary conditions. The problem of flow past a sphere and cylinder at high Reynolds number (very fast velocities) embedded in a flow has been done but in our case, we would have to put a bump sitting 'proud' on a horizontal boundary and sticking up above the water depth, which is much more involved.
Rather than trying to do a full up calculation (very complicated), it might be possible to invoke a hand-wavy energy "conservation"+turbulent dissipation argument to get a rough estimate of the flow's interaction with the island. A little out of my depth there but I'd be willing to look into it if it is still of interest.
Can you give me any more details as to the origin of the question?
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QUESTION: It is intriguing! The question comes from a call to the university library, and I realized pretty quickly that I would not find a simple calculation or model given the unusual parameters - I needed an oceanographer to give a proper answer. I think the questioner is creating/describing a fictional world where only some of the bounds of our universe/planet apply. Perhaps the least-fixed of all the variable is the wind speed. The questioner seemed most interested in finding a maximum plausible wave height so the island could be tall enough to remain above water. I think a hand-wavy argument would be most appropriate, and it's clear to me that your hand waving would be very well informed! Hope that helps.
I think a very simple solution will work for this question, namely the conversion of kinetic energy to potential energy as the water gets pushed upward due to the presence of the island (assuming the island has a reasonably gentle slope).
For a ridiculously shallow "ocean" of 4.5 feet and an equally ridiculous high wind speed (~ a few hundred mph), I'll assume that the flow is uniform in depth and traveling at the wind speed of V. This means that all energy dissipation due to friction, including turbulent kinetic energy, is continually compensated by the body force due to the wind. The kinetic energy (per unit volume) of the flow can be equated to the potential energy it gains when it flows up onto the island
(1/2)ρV^2 = ρgH
where H = vertical deflection of the flow. Solving gives
H = (1/2)V^2/g.
For V = 100 m/s and g = 10 m/s^2
H = 500 meters = 1600 feet.
This is the maximum height the water would reach. It has been assumed that the shoreline is much longer than this height, say a couple of miles.
I think the questioner may have been thinking that the gigantic wind speed would have produced gigantic waves that would dramatically crash onto the shore, but with such a shallow ocean on a big earth and relatively tiny island, I think the situation would be more like water flowing along a channel and encountering an obstacle (bridge pylon). For a steady wind with momentum being constantly transferred to the water, the flow would be slab-like and only 4.5 feet deep on average. Any waves that are generated could not have an amplitude much higher than a fraction of the depth unless a freaky non-linear process evolved to create soliton like structures, but that is an entirely different story.