If you detonate a nuclear bomb, it will not be good. Nowhere and to anyone. Therefore, it is strongly discouraged to detonate nuclear bombs without parents.
Now watch. The depth in the Challenger Abyss, the deepest point in the Mariana Trench, is 11 kilometers.
The power of the most powerful thermonuclear (hydrogen) bomb ever tested by humans is 50 million tons of TNT (50 megatons). This is three and a half thousand times more powerful than the bomb dropped by the Americans on Hiroshima in 1945.
What won't
In "YouTube" there is a video in which some guy claims that a tsunami wave after such an explosion will wash away all of Japan, half of America and half of Australia. This is, to put it mildly, nonsense.
For example, two and a half million years ago, the so-called Eltaninsky asteroid fell into the ocean between South America and Antarctica. As a result of the explosion, a tsunami wave of a kilometer (estimated) height was indeed raised, which badly battered a piece of Antarctica and the southern part of the South American continent. Although in general for the Earth, no catastrophic consequences were observed.
Promotional video:
So, the energy of the explosion of the Eltaninsky asteroid is 5,000,000 megatons (that is, 5 TERATONS), it is 100,000 times more than the energy of the Tsar Bomba explosion. The moral of this fable is as follows: no matter how deadly and destructive the action of a man-made weapon may seem to us, the scale of natural disasters is simply incomparable.
What will happen
When exploded in the air, such a bomb forms a fireball or "bubble" with a diameter of four and a half kilometers. However, water is not air. Water, unlike air, is very difficult to compress; do not forget about the monstrous pressure at a depth of 11 kilometers.
Calculations show that the diameter of the primary "bubble" will be about 1 kilometer. However, as soon as the pressure of the hot gas weakens, the surrounding water “collapses”, again compressing the “bubble”. As a result, we get a series of several shock waves, more and more weak. Finally, a stream of very hot and radioactive water will float to the surface of the ocean. However, on the surface, we will not see an explosion, or a water column, or a tsunami wave.
The main danger of such an explosion (apart from radiation) lies elsewhere. The Mariana Trench is one of the most unstable geological sites on Earth; at this point, the Pacific Plate plunges under the Philippine Plate. A powerful explosion at the bottom can provoke earthquakes and underwater landslides within a radius of several hundred, or even thousands of kilometers. In this case, a tsunami wave (or even a whole series of such waves) can no longer be avoided.
The explosion of a thermonuclear bomb by itself can generate a tsunami wave only if it occurs at a shallow depth near the coast. In this case, the water pressure will be small, and the “bubble” will reach a diameter of 4 kilometers. A very large (about 30 cubic kilometers) "funnel" is formed into which ocean waves will rush; the collision energy will generate a tsunami wave - however, by natural standards, very small. A city or village on the coast near the explosion site will certainly be destroyed - but this is likely the end of the matter. The energy of such a wave is simply not enough to cause significant damage at a great distance.
Natural tsunamis, causing severe damage at large distances from the epicenter, initially have significantly higher energy. For example, the giant European tsunami of 6100 BC arose as a result of an underwater landslide with a total volume of 3500 cubic kilometers, which is 115 times more than the volume of the "bubble" in the underwater explosion of the Tsar Bomba.