How would geothermal heating affect ocean currents on a rogue planet?

Question

On earth, the ocean's surface currents are influenced somewhat by the tides, but more by wind, and we have wind because the air at the equator heats up more than the air at the poles, causing low- and high-pressure systems. Surface currents follow the wind in a horizontal cycle, with some coriolis effect, because planetary rotation. Deep ocean currents happen because as the surface currents move warm equatorial water poleward, the water gets colder, saltier, and denser, so it sinks and is replaced with warmer water in a mostly vertical cycle. Temperature differences due to solar heating is the main factor for both surface currents and deep ocean currents. But what about a planet that doesn't have a sun?

Theoretically, a rogue terrestrial planet would still have some heating going on because of its molten core. If it has a moon, tidal flexing would help. Rogue planets can have oceans of liquid water, which may or may not be covered in thick ice, depending on what kind of atmosphere the planet has. But these oceans wouldn't have any sunlight to heat them from above; all the heat would be coming from below. If a planet's oceans were only heated by geothermal energy instead of solar energy, what sort of currents would form?

I expect that water would heat at the bottom of the sea, becoming less dense and rising and be replaced by cold, dense water from above. If we can ignore the effects of wind due to an ice cap, I'd also imagine that the temperature differences caused by warm water rising in some areas but not others would create a surface layer with horizontal currents, and that the planet's rotation would cause gyres, or chains of gyres, to form. What the gyres would look like is hard for me to picture, but perhaps they'd be centered around the upwellings of warm water. But where will the warm water actually surface? Over continental shelves where the sea is shallower? Over areas where the planet's crust is thinner? Along the boundaries of tectonic plates? Or are my speculations way off? Will the differences in heating be negligible, resulting in a stagnant sea where the water is slightly warmer down below than it is up above (the opposite of earth lakes, where if you tread water, your torso will be warm, but your toes will dip into a layer of cold water)? Or would something completely different happen?

Answer 1

Liquid water molecules are never at stand still. They are always moving (vertically, horizontally, at angles).

Gyres are created by three forces: the rotation of the Earth, wind patterns, and the landmasses of the Earth as told here.

Low temperatures at sea bed

If temperature at the sea bed is around 25°C ± 20°C, then water pressure is so high at the bottom that it will not become less dense than on the surface. Therefore convection may take place only near the bottom. Some heat may transfer to above by conduction. Surface will be frozen. There may be no gyres as there are no wind patterns and surface is covered with ice.

Medium temperatures at sea bed

If temperature at the sea bed is around 65°C ± 20°C, heat may transfer by convection and over the centuries, ice may melt at some places.

High temperatures at sea bed

If temperature at the sea bed is around 105°C ± 20°C, water may not boil because of enormous pressure. Heat transfer by convection and conduction may be fast. On the surface, water may be liquid but cold (10°C ± 5°C). There may be gyres because of the rotation of the planet depending on shape of the oceans.

Answer 2

The Earth's geothermal heat flux is something in the neighborhood of 0.087 W/m^2. Or 0.03 percent of the heat from the sun. There is some variation, being higher in areas with volcanic action for example. But as an average, it's this.

To keep oceans liquid you need something not too far off the average solar flux. That is, you would need about 3000 times the geothermal heat flux the Earth has. One epxects such a planet would have an uncomfortable number of volcanos.