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I read that thermohaline circulation is caused by cold and saline water sinking, while warm and less dense water rises. link
I understand that saline water is more dense, as there is a higher mass due to dissolved salts.
However, decreasing the temperature of water decreases the solubility, so in theory, shouldn't cold water hold less salt than warm water?
Vice-versa, shouldn't warm equatorial water increase in density as it is able to hold more salt?
Warm water does hold more salt - at saturation, but seawater isn't even close to saturation. Surface sea water gets warm, starts to evaporate (hence clouds) and therefore gets denser. So it sinks to the ocean floor, where it cools down to about 4 deg Celsius, and gets denser still. But it still is a very long way from being saturated with salt, so 'how much salt the water can hold' doesn't come into it.
Surface sea water gets warm, starts to evaporate (hence clouds) and therefore gets denser. So it sinks to the ocean floor, Does this mean in equatorial water where there is a high amount of evaporation and salinity in surface waters, the surface equatorial waters sink down to the ocean floor at the equator?
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An usual salinity in the world's major oceans is 35 g/kg. The solubility of sodium chloride in water is around 360 g/kg (Sodium Chloride, Solubility in water). Therefore, the solubility is not the limiting factor.
Sea salt consists mainly of sodium chloride (NaCl). Therefore, this approximation should be OK.
The salt in the sea water 'moves together' with the water. Thus, when warm water flows from the equator to the poles it transports salt to the poles. At the poles water cools down and sinks (together with salt).
If we had a direct relation between sea surface temperature and salinity, in some region we would need a net flux of salt in the opposite direction to the water net flux.
Global salinity gradients exist. The salinity is above the global mean salinity in some regions -- particularly in warm regions --, whereas in other regions the salinity is below this mean. These salinity gradients are caused by a net loss of oceanic water (evaporation > precipitation + fresh water inflow; enhanced salinity) in warm regions and by a net input of water by (evaporation < precipitation + fresh water inflow; reduced salinity), respectively. These gradients are not related to solubility of salt.
