Just throwing some thought on the subject, it is easy to me to understand that there is a big pump moving large amount of water from one pole to the opposite one along the year, that is during summer in north hemisphere the ice melts at Arctic and mix to the sea; at the same time water start precipitating more and more on the Antarctica, piling up as snow and ice. During the other six months the pump reverts, moving water from Antarctica to the Arctic.
In this hypothetical situation this pump will behave differently. The water will evaporate from the sea, and precipitate towards the poles. Those rain in midway to the pole would provide vegetation life, erosion, and all sort of things rain water does. That water falling in the north and south pole would have no reason, in principle, to melt and reinsert in the oceans.
If the rain started to fall mostly on high latitudes, possibly the region around equator would be more exposed and deserter, increasing the heat, evaporation, and the convection of the air at high speed. Possibly very intense winds blowing everything, moving quickly the clouds and vapor to higher latitudes, where the rain would fall. Anyway, the hotter atmosphere should increase the loss of the lighter gases to the space due to kinetic energy gain, decreasing the atmosphere pressure little by little while becoming hotter. In this loss there could be even some water vapor.
Then I think that in the long run the water would tend to move all the way to the poles draining drastically the remaining of the planet. If we think of an ocean as salty (3.5%) as ours, possibly there would be deposition of large layers of salt, most of it sodium chloride. I don't really know if there would be a "life zone" around some high latitude. Certainly tropic would have no meaning in this configuration.
That is the "big picture", but not all the story.
We have different wind flow in atmosphere and water flow in the oceans due to others factors. One of those being the tides that move water in waves half circumference of the planet. The main component being that from the moon, but we have other of smaller intensities like the solar influence. Other factors are the convection due difference temperatures and pressures, and Coriolis force making it go round in its trip across latitudes.
Another factor is that we have a non homogeneous distribution of mass on planet surface, the removal of large amount of water, moved to the poles, possibly would increase the amount of wobbling, then the precession ( https://en.wikipedia.org/wiki/Axial_precession) of the planet. Perhaps this wobbling would be responsible to a certain amount of melting at the poles, keeping the pumping working longer and, who knows, preserving some life in the planet.
A hotter atmosphere could increase the amount of clouds, if it covers all the planet we could have a very long freezing of the planet.
If we think on a dry planet with water moved to poles as ice, we could perhaps include sublimation of ice in the figure, adding the loss of water to space due to heating of the atmosphere. I don't think we would become a Mars, our gravity is about three times Mar's gravity.