How do ocean currents work on a tidally locked planet?

I am currently designing a tidally locked world with a big enough ocean to transfer heat from the day side to the night side and vice versa. Here is a map of my world:

The darker sides are supposed to be the dark side of the planet, while the brighter area is the day side, with the center being directly facing its parent star.

I'm trying to figure out how the ice caps in the dark side would grow in order to add them in my map, and for that I feel like I need some information about the planet's ocean current's. Assuming the planet rotates from left to right, what directions would the currents of my world take?

• If the planet is tidally locked, it won't 'rotate' at all relative to its sun. Jan 20 at 9:18
• What do you mean? Doesn't tidally-locked mean that the time it takes for a planet to rotate around its axis is the same as the time it takes to complete one orbit around its parent star? Or am I missing something? Jan 20 at 9:32
• I mean that there won't be any relative rotation on this world as in earth-sun, or earth-moon from the point of view of earth. The sun will always be in the same position, and the water will always bepulled in the same direction. Any circulation will be thermal or caused by libration. Jan 20 at 10:17
• Does this answer your question? How would the ocean currents of a tidally locked super-Earth work? (Also, for reference see this answer and know that currents are quite similar to wind.)
– JBH
Jan 20 at 17:10
• See why Euler's rotation theorem prevents the axis from facing the star throughout the whole orbit. @Pica I.e. it wouldn't be tidally locked as per spec. Apr 19 at 17:56

By far the greatest forces your atmosphere and water are going to be moved by are thermal.

Things like the Coriolis force driving differential circulation (and various turbulent competing cells) in the different hemispheres will be negligible, the same with trade-winds.

You would get a permanent thermal rise at the centre of the map, the point where the star is overhead. This (or these, there's no reason it won't split just like at Jupiter's North pole, but less stable) will tend to rotate and produce rotating winds, but the direction, hardly affected by the Coriolis effect will be quite unstable - both in direction and they'll wander around.

Between the centre and the twilight-zone, the prevailing wind at ground level would be towards the centre, in the higher atmosphere, it would be away from the centre. Picture a big rolling doughnut of a light-side vortex.

Around the rim, the twilight-zone, there would be turbulence and instability where the cold-air meets the high-up hot air from the light-side. There would be downward-streaming air currents in swirling vortices - and just as at the midday point, they would be unstable and mobile. On perhaps an hourly basis, the temperature might vary by many tens of degrees Celsius.

Needless to say perhaps, the moisture will be carried away from the heated areas to the cold areas. I imagine a frozen ridge of precipitated snow over packed-ice, several kilometres high ranging several hundred kilometres past the twilight-zone. There would be continuous melt-water producing lakes and rivers flowing towards the light-side - inevitably evaporating before reaching a "Sea". Desiccated salt-pans would form in depressions over much of what might once have been oceans

I probably can't help you much, but i think this or this kind of papers kinda interesting:

I. Surprisingly, pattern of heat distribution may look like a lobster, due to carioles force that in our case depends on distance of tidally-locked planet to the star (btw planet in this model isn't really hot):

Gliese 581g. The synchronous rotating period is 36.7 Earth days, radius is 1.5 times Earth radius, gravity – 13.5 m/s2, and incident stellar radiation 866 W/m2.

The two cyclones at each side of the equator are the tropical Rossby-wave modes, and the eastward long tail along the equator is the tropical Kelvin wave.

II. But the shape and size of the continent can change heat distribution dramatically:

Substellar continents reduce the efficiency of ocean heat transport. Our simulations suggest that Proxima Centauri b should transition from a lobster to an eyeball state as substellar continent size increases and inhibits ocean heat transport.

Maybe I'm wrong, but at this point I think, it may be problematic to accurately calculate this phenomenon without simulating your solar system with your continents (which are nice btw) in some fancy scientific program with an old-school interface... Well, i think we can get creative with this matter xD

So i was 'inspired' by wiki/Ocean_current and made some speculative untrustworthy draft which can easily be ignored:

I think you'll easily make better:)