This is basically a diffusion/mixing problem between two fluids of different temperature and density, where the density difference is primarily due to salinity differences.  I think you can effectively model these two fluids easily enough - a mixing time to first order would be the characteristic time it would take for a water molecule from one end of fluid 1 to reach the other end of fluid 2 (this is the crossing time).   So you would have to assume some average mixing velocity.   Temperature equilibrium should occur have one crossing time - salinity equilibrium probably takes longer as there is likely some vertical structure in the salinity distributions but it would mix out in a few crossing times.  This is enough to figure out if the timescale is 100 years or 1000 years - which is probably what your after.

There are several processes involved: the mixing in surface Black Sea layer, the mixing at the Turkish straits system, the exchange between basins, the Aegean Sea circulation and the overall Mediteranean circulation: the corresponding timescales range between 1 year to several hundred years. Could you better define which equilibrium are you mailnly interested?

Right, this question of equilibrium in systems can be complicated.  A general rule in most dynamical systems, specifically gravitational systems, is that there are two timescales:

1) a dynamical timescale - this is the characteristic time it takes for the system to know that it is a system and this is usually the "crossing time" for a particle in the system.

2) relaxation or equilibrium generally takes several dynamical timescales to occur.

once you have figured out the dynamical timescale then that sets many overall timescales to withing order of magnitude accuracy

The so and similar tasks, the Institute Water Problem RAS is engagaed durially