But this particular instance is not fog. It is liquid moisture falling rapidly, aerating and misting, cooling, condensing, and then lifting into a low cumulus type cloud.
Here's my caveman understanding of the phenomena:
There is always a mist 'cloud' from the water's impact. That mist cloud either falls to the ground if the droplets are large enough or they float around for a few tens-of-seconds and evaporate. The floating mist is composed of water droplets hundreds- or thousands-of-microns in size or larger, orders of magnitude larger than droplets in fog or clouds.
When the droplets evaporate, the liquid water absorbs heat from the surrounding air to transform to gas vapor. Cold mist needs to absorb more heat from the air to evaporate than warm mist. Now the surrounding air is both cooler and has a higher relative humidity. That always happens anytime water droplets of any size evaporate - no matter what the temperature of the droplets and surrounding air - as long as the air temperature was
above the dewpoint.
The difference here with these particular temp/dewpoint conditions is that the evaporating mist absorbs enough heat from the surrounding air to lower its temperature
below the dewpoint. That means 1) some mist droplets will no longer evaporate right away, and 2) some humid air from evaporating mist will condense back out into droplets of a few microns or so in size and produce fog (which in this case will also not immediately evaporate).
Fog and clouds are both micron- or tens-of-micron-sized droplets. Ground fog tends to
keep condensing fairly quickly and has proportionally more larger (50-100 um) droplets. The fog produced from the spillway mist does not continue to condense - the droplets remain smaller micron- or tens-of-micron-sized. They can't evaporate because the temperature in their fog/cloud bubble is still
below the dewpoint.
Cloud droplets continue to condense over time, but very slowly (hours/days) and most droplets are usually micron or tens-of-microns in size. The temperature/dewpoints in a cloud are nearly the same and often rely on some kind of sub-micron-sized particle to act as a nucleus. That is not always the case though - a cloud is droplets whether they formed spontaneously or around a nucleus.
Whether you call the spillway phenomenon fog or clouds, it is still composed of micron- or tens-of-micron-sized condensation droplets that are floating in localized air whose temperature is below the dewpoint. They will remain in liquid form until the surrounding air temperature rises above the localized dewpoint, or the dewpoint drops below the localized temperature. When either of those conditions is met, the droplets evaporate into transparent gas.