Let’s say that God did not like the idea of land and decided that planet Earth would look much better with no earth at all. Planet ehm… water would be spinning around just like Earth. Now, how would water behave on a planet with no land masses? Would the surface of this planet spin at the same speed than the crust? would it spin slower, maybe? What if there was no air on the planet, or at least no winds disturbing the water?
I don’t think there’s any way you could avoid having an atmosphere while having liquid water at the surface; unless the surface is entirely frozen then you’ll end up with an atmosphere of water vapor. As for how it would flow, that would depend on the shape of the undersea surface, and how deep it is.
Europa is not a planet, but it has an all-water surface. It’s ice of course, but may have liquid oceans under the ice surface. Its a fascinating place…I imagine it to be a giant skating rink in the heavens.
It doesn’t answer the questions, but it illustrates how an all-water world might exist.
Is the OP referring to a planet with land that’s entirely covered with water, or is it water (or ice) “all the way down”?
The oceans would turn at the same rate as the rock core/seafloor. After all, if they did not then the seafloor would be like a giant stirring paddle. Friction would impart movement to the nearby water, and so on, until the entire planet, liquid solid and air, was turning with the same angular momentum as the original system.
Of course, just like Earth, temperature differentials from sunlight and coriolis force might create flows like the gulf stream or the atmospheric jet stream. IIRC the Gulf of Mexico is tropical and shallow, so the water there heats up a lot. Coriolis force (from the earth turning) makes it flow toward the poles and counter to the rotation of the earth, since the closer to the poles, the slower the planet turns so the faster stream goes relatively - diagonally northeast to warm Iceland and Europe.
If the oceans are very deep, perhaps the difference in rotational speed of rising water from convection might be a factor. We see in Jupiter and Saturn (admittedly, gases) that between the flowing bands there are eddies and whirlpools on a massive scale.
Can we assume that a water-covered planet has no plate tectonics? It’s hard to imagine enough water gathering to cover faults thrust several miles above the rest of land. I’d imagine no volcanoes either, as those would allow islands to build up. The composition of the undersea surface makes a huge difference, therefore. Would it automatically get worn down to smoothness or would different minerals behave so differently as to form underwater mountains and valleys?
Sea level would not be constant either, because of tidal bulges at the equator. If the planet is tilted, seasons would create warmer and cooler areas of water annually and that would change currents and air frictions and densities and whatnot.
Really, lots and lots of variables come into play. I don’t know which would affect the surface the most, but without specifying any I can’t see that you can get any good answer.
First one. Flat lands down the ocean.
Even if you had a perfectly smooth and equipotential sea floor, you’d still get currents from differential heating. If it’s salt water like the Earth’s oceans, then you’d get more evaporation near the equator, hence higher salinity, hence higher density, which would cause water to sink near the equator, rise elsewhere, and flow horizontally at the top and bottom. If it’s fresh, then you’d instead get it rising at the equator and sinking elsewhere due to thermal expansion, though the effect wouldn’t be as large in that case. Either way, as the water flowed north or south, it’d be deflected by the Coriolis effect, so you’d get currents (or at least, components of currents) circulating around the planet, too. You’d also get currents in the atmosphere, for the same reasons. I’d expect that the meteorological features of both would be fairly steady: At any given location, you’d always get current in the same direction and speed, and likewise for the wind.
If you had only water and no solid rock, the pressure and temperature at the center would effectively turn it solid.
Everything would be really wet 
Actually, according to the Bible, the oceans don’t really sparkle with God and he is planning to destroy them…when he gets around to it.
For one thing, it would depend on whether your planet has moons, to gravitationally influence the movement of the fluid on the surface. As well as how fast it rotates in relation to its sun, if it rotates at all. On a planet that rotated, it would turn under the water, because even its sun would exert a gravitational pull on the water. If it did not rotate, the water would be frozen on the dark side and liquid on the light side, or if warm enough, liquid on the dark side and a vapor on the light side.
Eh, tides aren’t actually that big a deal, on the open ocean. It’s only when the water sloshes up against land masses that they get amplified into significance.
But without continents to slosh up against, there would be nothing to mitigate their multiplying effect, and there would likely be huge variations in water level. The OP asked what would happen, and I described what would happen. There wold simply be, as you point out, no observation points a fixed distance from the center of the planet, from which the effects could be observed.
On a planet completely covered with water, tides would in fact be the only big deal. unless there were volcanic sea-mounts or something, which would exist whether or not the planet is covered with water. If those existed, the surface of the planet would probably sooner or later form areas that are above sea level and continents would evolve.
Both the ocean and the atmosphere are fluids and would behave similarly as Chronos detailed above. There would be rain, there would be life, there would still be waves. Water World indeed.
I hate when I type a long entry, and I get to a point that I realize my WAGs are dead wrong and have to delete it all.
Okay, water is a fluid (assuming it’s not frozen). That much is obvious. Well, we have several planets in our own solar system that are completely covered by fluids that we can use to see how it would react. They’re called the Gas Giants. Gasses are, of course, fluids too. Because of the striation in the the gasses on these planets, particularly Jupiter, we can get a good idea of how the fluids react to having nothing to redirect them.
Now, obviously we don’t know enough about what goes on beneath the atmosphere of these planets to know how fast the surface spins in comparison to its core. But we do know how fast the surface spins. Jupiter has a day that is just under 10 hours. Saturn’s is a little more than 10 1/2 hours. Neptune’s is a bit over 16 hours. And Uranus’ is around 17 1/4 hours. Considering how big these bad boys are, they’re spinning pretty darn quick.
Now there is a lot of guess work about what is going on down deep in the gas giants, but even so there are hints at what our Waterworld would be like. The stripes on Jupiter are formed by high velocity winds, as well as convection. The light stripes (called “zones”) are cooler and represent where the atmosphere is descending downward. The dark stripes (“bands”) are warmer and are bringing stuff up. This is, of course, how our atmosphere behaves as well. And guess what, this is how our oceans behave, too. If you remove the landmasses, I suspect the ocean currents would become faster, and you’d see more regular up/down patterns emerge. It’s probably a good guess that after a few billion years, our Waterworld would get stripes. An interesting side-effect of this is that you’d probably end up with each band and zone having it’s very own special blend of flora and fauna, since the environment from one to the next would differ considerably.
Then there’s the storms. if Jupiter’s GRS (“great red spot”) is any indication, with no landmasses and high speed fluid movements, hurricanes and typhoons could become even stronger and last much longer.
At first I thought that the water wouldn’t move faster than the planet beneath it, but now I’m almost certain it would. There is more than enough evidence to show that solar radiation is enough to drive air and water currents, and that these fluids would be more susceptible to this influence than the land beneath it. Waterworld would be windy and stormy, with warm and cool bands encircling it. Sounds kind of neat.
Of course, I’m not an exoplanet physicist, so take all this for what it’s worth.
There has been some modelling of this situation, mostly to test weather and climate models in extreme situations. Here for instance is a quite technical paper on the subject
(.pdf)
Mean Climate and Variability of the Atmosphere and Ocean on an Aquaplanet
They predict floating ice caps down to 55° of latitude, but mostly the climate remains quite similar to that on our world.
There would still be Hadley Cells caused by heat transfer in the atmosphere between the equator and the poles, which would be affected by Coriolis forces just as on our world. A region of doldrums and permanent trade winds around the equator would probably present a slightly banded appearance from space, but the midlatitudes would be a region of chaotic mixing and display complex cloud formations.
The mid-latitudes of such a world would probably resemble the Southern Ocean of our world, which can get quite windy with mountainous seas. But there would be calm regions on this world as well. You can’t really use the gas giants as a model, because they have very deep atmospheres and depth mixing is very important in the formation of these bands.
Some ocean planets will probably have very deep oceans, hundreds of kilometers deep, wheras others might only have a few km of water over an Earth-like ocean floor. The deep ocean worlds will have high-pressure ice at the bottom of the water column, but presumably won’t display the sort of banded mixing you get in a gas giant because of the formation of a thermocline - though I might be wrong there.
Shallower ocean planets will probably be geologically active, and have at least a few islands here and there. Our planet might have been mostly ocean in its earliest phases, with smaller landmasses that grew into the continents we know today.
Just a quick point of fact: The Earth’s own radiation from the heat deep inside is almost always ignored, it’s not much compared to the solar energy input. The Gas Giants this is NOT the case, for example, Jupiter’s radiation from it’s internal heat is more than the energy received from the sun.
To my knowledge, we have no direct observation of these cyclones on Jupiter in three dimensions. They may be like Earth’s tropical cyclones (cyclonic motion at the surface, anticyclonic motion at the tops), but they may well be entirely anticyclonic.
Just another of my $1.34’s worth
Cool paper, eburacum45. I skipped to the end, where it has its conclusions. Even there, I barely understood what they were saying.
It didn’t occur to me that currents that are unobstructed wouldn’t deliver warmth to the polar regions, which would result in larger ice caps. I should have realized that. Still, I’m kind of proud that some of my assumptions were close to being on target. Not bad for a humanities major.
I think tides would cause the water to circulate in the similar way to earths atmospher, which is similar to atmosphere of other planets too… Given a planet that is nearly upright of course (or nearly completely upside down … one or the other ), at the equator its going to be spinning slower than the planet… the number of stripes depends on the size of the strip which is controled by the size of the Coriolis effect and the material properties… and depths… and temperature changes, and so on…