tsunami physics?

[Machine_Elf]

Wikipedia has a pretty good article explaining how tsunamis propagate across the open ocean and how they transform into a destructive wall of water upon reaching the shore.

Can someone point to more detailed information about how tsunamis move through the ocean, and how they interact with the sea floor? For starters, I understand that the wave itself moves at ~500 MPH in deep water, but how fast and how far does any given particle of water move under these circumstances? I found this reference which gives (on slide 56) a maximum horizontal current for a tsunami, in deep water, of just 3 cm/s. Which doesn’t seem like much, but if the water is 1-2 miles deep, and the entire depth of it is shifting at that velocity, then you of course can end up with a very large total volume of water moving toward shore.

Can anyone confirm at least a correct order of magnitude for that?

[jjimm]

@Machine_Elf

Which doesn’t seem like much, but if the water is 1-2 miles deep, and the entire depth of it is shifting at that velocity, then you of course can end up with a very large total volume of water moving toward shore.

Except that the velocity of inundation at landfall isn’t caused by horizontal movement of water behind it - it’s caused by the temporary raising of the sea level at the land edge caused by the extremely long wavelength of the tsunami. Just like gently dropping small pebbles into a bucket eventually causes the relatively stationary water at the edge of the bucket to cascade out horizontally with high velocity.

Don’t know the physics intimately, but it’s more a Newton’s cradle phenomenon: i.e. it’s not the horizontal movement of the water that matters, but the energy transfer radiating out from the causal event. Shallow wave animation: while the wave crest of a deep-sea tsunami is thousands times bigger than that wave crest, it no doubt mimics that action, with individual molecules returning to near their original position post-wave.

[NinetyWt]

@Machine_Elf

Can someone point to more detailed information about how tsunamis move through the ocean, and how they interact with the sea floor? For starters, I understand that the wave itself moves at ~500 MPH in deep water, but how fast and how far does any given particle of water move under these circumstances?

The thing (for lack of a better term) which moves through the ocean is the energy wave, not the water. The molecules of water don’t move through the deep ocean. So when you say “how fast and how far” do the particles of water move, the answer is, they may bob up and down but they are not traveling forward. The energy is traveling forward.

When it gets to shore that’s a different thing, as jjimm pointed out.

[Chronos]

Except that the velocity of inundation at landfall isn’t caused by horizontal movement of water behind it - it’s caused by the temporary raising of the sea level at the land edge caused by the extremely long wavelength of the tsunami.

Which must, necessarily, involve a significant horizontal movement of water.

[Machine_Elf]

@NinetyWt

The thing (for lack of a better term) which moves through the ocean is the energy wave, not the water. The molecules of water don’t move through the deep ocean. So when you say “how fast and how far” do the particles of water move, the answer is, they may bob up and down but they are not traveling forward. The energy is traveling forward.

When it gets to shore that’s a different thing, as jjimm pointed out.

The water must move laterally, even before it gets to shore. Pick any thin vertical “slice” of water in the open ocean, from the sea floor to the surface. When a wave crest passes through this slice, the slice is taller, but it’s the same density as it ever was, ergo water must have been transported laterally into the slice to make it taller. Wikipedia has an animation that shows this quite clearly. There may be a back-and-forth action to it (rather than a single unidirectional displacement), but the bottom line is that there’s definite lateral movement.

So…how fast, and how far (in deep water)?

[jjimm]

@Machine_Elf

So…how fast, and how far (in deep water)?

The first question is in your OP: 3 cm/s. The second… dunno.

Thought experiment: block of steel bolted to the floor, with a cup, bisected vertically and crazy-glued hermetically to the steel, then filled to the brim with water.

Whack the far side of the steel with a large mallet. What happens?

[jjimm]

(Or in other words, how fast were the steel molecules moving?)

[Machine_Elf]

@jjimm

(Or in other words, how fast were the steel molecules moving?)

You’re talking about a compressive wave traveling through the steel at steel’s speed of sound.

I don’t believe tsunamis are related to the compressive properties of water; the biggest evidence of this is that the speed of sound in seawater is about 3400 MPH, whereas the speed of propagation of a tsunami in deep water is only about 500 MPH.

[Interconnected_Series_of_Tubes]

Overview and equations here.

“In deep water the speed (or velocity) of a water wave depends only on its wave length. Specifically, the speed is proportional to the square root of the wavelength. Thus, the longer the wave length, the faster the wave, or vice versa.”

There’s a whole lot of energy in an entire water column being shifted 3cm/s in one direction.

IOW: It’s more like having your hypothetical body of water attached to the steel plate by a slightly flexible joint, and giving the plate a little push. The energy’s being provided by the movement of the tectonic plates themselves, not by some outside force (a hammer) transmitting energy THROUGH them.

As we’ve covered over and over on this forum, compressive waves travel through matter at the speed of sound. But this isn’t a compressive wave. It’s just a huge piece of the earth shoving a huge volume of water.

[jjimm]

@Machine_Elf

I don’t believe tsunamis are related to the compressive properties of water

They must surely be partially related - otherwise the size of a tsunami that is 1 metre in deep water would remain at 1 metre during inundation - rather than the rough average of ~7 metres from the Indian Ocean inundation.

A strike-slip earthquake displaces a massive amount of water very quickly. That volume of water slams into the water next to it (and upwards to an extent) and the pressure ripples outwards. With a maximum current in deep water of 3 cm/s per your cite.

[Interconnected_Series_of_Tubes]

@jjimm

They must surely be partially related - otherwise the size of a tsunami that is 1 metre in deep water would remain at 1 metre during inundation - rather than the rough average of ~7 metres from the Indian Ocean inundation.

What? No. Tsunami get higher at shore because of wave shoaling which is covered extensively in the wiki article.

[rbroome]

@Machine_Elf

The water must move laterally, even before it gets to shore. Pick any thin vertical “slice” of water in the open ocean, from the sea floor to the surface. When a wave crest passes through this slice, the slice is taller, but it’s the same density as it ever was, ergo water must have been transported laterally into the slice to make it taller. Wikipedia has an animation that shows this quite clearly. There may be a back-and-forth action to it (rather than a single unidirectional displacement), but the bottom line is that there’s definite lateral movement.

So…how fast, and how far (in deep water)?

and of course you are correct. But remember, a wave, even a solitary wave, has both a crest and a trough. The water particles move in uniform ellipses. First one way (I assume that is the 3 cm/sec cited above) then, as the wave passes, the particle moves back. A wave does not, can not, result in a net translation of the medium through which it travels. The problem is what happens when something changes the closed, simple conditions of the open ocean. Such as when the water moves forward a little bit and runs into land. When the wave extends from the surface to the deep bottom, all that little bit adds up. Wave physics is straightforward especially when there are no boundaries like land. And it is easy to describe what happens even on land. The trick is the generation when you have a point source, such as an earthquake. Those require a separate treatment from the usual case of winds blowing over a large area. But once the wave starts, Newton sets the rules.

[JWT_Kottekoe]

A small amplitude deep water wave (depth much larger than wavelength) is easy to analyze. A wave with a single wavelength is a perfect sine wave and the water molecules at the surface trace out a perfect circle, returning to the same place as each wave goes by, neither going forward nor backward. Beneath the surface the amplitude of the motion decreases with depth and is negligible several wavelengths down. It is easy to compute the maximum velocity of the surface water in this limit. It is equal to the phase velocity of the wave times pi*a/l, where a is the amplitude and l is the wavelength. Numbers for the Indian Ocean tsunami, given in the reference provided by the OP, are wavelength ~ 200 km, amplitude ~ 60 cm, water depth ~ 4 km.

This is clearly a shallow water wave, since 4 km << 200 km. If we applied the deep water formula, the lateral motion of the wave as it zips along at 450 mph would be a miniscule 2 mm/s, smaller by an order of magnitude than the number in the OP. In shallow water, the circular motion with zero forward progress is replaced by a more elliptical trajectory, with forward progress. The solution to this is much more complicated, but the result referenced by the OP seems quite plausible. More than 10x the deep water result, but still many orders of magnitude smaller than the phase velocity. In shallow water the phase velocity is just sqrt(g*h), where g is the acceleration due to gravity and h is the depth. For 4 km, this is about 200 m/s or 450 mph.