How close...is "close vicinity"? That is, how close to the sea floor? Or how close, in straight line, to the epicenter?
Tsunamis do not pose significant peril until they overrun a shore. That is, when they enter the breaker zone (near the surface) adjacent to a shore. At sea they are barely noticed, if at all--a surface vessel might detect one because there is view of the horizon--which the submerged sub would not have--but the vessel would simply float up with any change in sea level--no danger--no breaking waves as the tsunami moves mostly through seas below the wave levels near the surface.
If you're thinking of vertically upwelling under the ship (in the contrariwise sense of downdraft effects on ships aloft), a helmet wouldn't hurt (if there were time to don one--which is doubtful), but any up current (that which did not immediately spread laterally under the overlying weight of water) would, for the most part, simply flow around the rounded hull of the ship--possibly going unnoticed or barely noticed.
In actually, a greater hazard is likely to be posed to a sub if its caught in inland water during the passing of a hurricane--it could become landlocked on the bottom...until normal water level recovered.
So, if close means, close to the sea bottom, you'd hope not close enough so that you ran the ship into shifting sea floor--but you'd want to avoid that even if there were no earthquake.
"Tsunamis do not pose significant peril until they overrun a shore. That is, when they enter the breaker zone (near the surface) adjacent to a shore. At sea they are barely noticed, if at all--a surface vessel might detect one because there is view of the horizon--which the submerged sub would not have--but the vessel would simply float up with any change in sea level--no danger--no breaking waves as the tsunami moves mostly through seas below the wave levels near the surface."
Wait a minute, are you saying that a Tsunami, or any other great wave, far out at sea, so it does not "break", poses no danger and cannot overturn large boats and ships? You had better re-think that one.
Unless the vessel can power straight into the oncoming wave and is long enough so that its bow gets "over the top" before it gets over its own center of gravity, it will surely flip. Even without flipping, the extreme angle will do a lot of internal damage, possibly breaking or cracking the hull which may not have been designed for that kind of stress. Any vessel hit broadside by a giant wave, "breaking" or not, is likely to roll right over. In this case, a submarine, being totally watertight, would surely do much better than a surface vessel, which is normally only watertight near or below its own waterline.
I would say that a submerged submarine may ride out the giant wave just fine, depending on how near the surface it was. The nearer to the surface, the more extreme would be its vertical displacement, velocity, and acceleration, and, yes, you just might need that "bump cap".
Personal experience as a snorkeler backs that up. While floating near the surface, a large wave would tend to make me bob up and down like a cork, with some lag due to my mass and momentum, causing me to go underwater when I was not ready. If I timed my breathing, I could hold my breath as I bobbed under, or even better yet, do a surface dive, getting under the wave completely and watching it roll harmlessly overhead. My brother the Scuba diver, was very susceptible to seasickness. When we went diving/snorkeling together, he was always the first one into the water when the boat dropped anchor, because he never got seasick underwater!
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"What, me worry?" Alfred E. Neuman
The facts are that a tsunami in the open ocean even though it packs tremendous energy will cause barely a ripple on the surface. It is when the tsunami runs up on the beach in shallow water that they rise above the surface of the ocean to break on the shore at potentially extreme heights. Therefore any surface ships at sea would be largely unaffected. For example after the big tsunami in Indonesia last year there were a number of sailboats that were unaffected by the tsunami that came in ashore after the tsunami receded to help in search and resue operations.
Tsunamis at sea should not be confused with rogue waves which can be very distructive to surface ships. However rogue waves are not caused by tsunamis but are caused by interacting waves that were usually storm created or driven.
Doesn't it depend on the depth of the water? You can have "open ocean" that is not very deep (Caribbean Sea for example or Gulf of Mexico or other large areas on a continental shelf) and the tsunami will rise to a much greater height, without being in a "breaker" mode, than it will in very deep waters ("barely a ripple" as you say). So I guess it depends to some extent on your definition of "open ocean". If you cannot see any land on the horizon in any direction, wouldn't that be "open ocean"? I once travelled in a small, single engine motorboat across the Gulf of Mexico from Gulf Shores, Alabama to the Florida Keys and there was a long time that we could see no land! I have also been atop the observation tower of one of the largest cruise ships in the Caribbean, and even at that height we were out of sight of land, yet the ocean depth was no more than a few hundred feet.
And since tsunamis vary greatly in magnitude, that would also have an effect on what surface displacement happens in any particular "normal" water depth. Yes, I understand the difference between a "rogue wave" and a tsunami, but in practical terms there may not be much difference in the effect on ocean-going vessels at some point.
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"What, me worry?" Alfred E. Neuman
Not being an expert I can only repeat what I have heard or read. In Indonesia the only boats I heard that were sunk were those that were tied up along shore. I did not hear of any reported sinkings of boats in the open ocean. From the articles I read my impression was that boats in 30-40 ft of water were undisturbed. Those in 10 feet of water had problems. Being a sailor I read first hand accounts of other sailors that were there and that is what they described.
OK, this is one out of the blue. Not what I was thinking about thinking about (!) today, but puzzling and worth a couple of lines, ...maybe.
My memory is a bit hazy on this, but isn't there an element of earthquake shock waves traveling through water in order to create a tsunami. Yes, the visible and destructive element of the tsunami as where the sea disappears over the horizon line and then reappears with wave height orders of magnitude over its normal levels is what we are familiar with, but that displacement of water is the result of shock waves traveling under the ocean..
Isn't there also an element of this energy transfer associated with "P" waves and "S" waves, where Pressure waves travel in one plane and shear waves travel perpendicular to them ? And if that is so, depending on the attitude of the submarine to the epicenter of the earthquake ( broadside or pointy end on ) wouldn't those waves potentially create some damage?
From what I have read, a tsunami at sea is a very broad wave, perhaps a few feet high traveling at something like 500 miles per hour. When it reaches shore, the wave slows down and piles up upon itself. Looking at videos of the recent tsunami, it seemed like the ocean just kept rising and rushing ashore, it was not a huge crashing wave falling onto the coastline. The people were carried along in the swift flowing water along with large amounts of debris pounding into obstacle's rolling over them and making it impossible to swim or even grab onto something.
Most probably, you won't need a helmet inside a submarine, unless you feel its "cool" to wear one.
The Bernoulli's equation is at work here. Deep sea water due to its higher density vis a vis the shallow sea water, has a much faster pressure velocity. As the surge from the deep sea water moves inland into shallow sea water, it slows down, converting the kinetic head ( energy ) of the surge into elevation head i.e., high tsunami wave.
It is the same principle observed in a hydraulic jump.