Explain, in layman's terms and with examples, Adiabatci Fog?

The simple answer is "as air rises, it cools" thus when air flows UP a slope (mountian, hill, etc) the air cools (because of pressure dropping). If that air contains enough moisture it can and will produce a fog as it goes upward. When the temperature and dew point match fog will appear in the updraft. After all fog consists of visible cloud water droplets or ice crystals suspended in the air at or near the Earth's surface. Cooling suspended water will cause water drops to become visible. Thus the adiabatic (or upslope) fog is born. (My words and an oversimplified statement.)

Here is a good picture that really explains it better https://mikealger.net/2016/12/16/frigid-weekend-an...

There is no such classification as 'adiabatic fog.' Fog formed by adiabatic cooling is called 'upslope' fog. This type of fog occurs when sloping terrain lifts air, cooling it adiabatically to its dew point and saturation.

In the U.S., the east slope of the Rocky Mountains is a prime location for this type of fog.

When air is compressed the temperature goes up, when decompressed the temperature goes down. when high humidity air is decompressed the temperature will drop below the dew point and condensation forms as fog or clouds. In weather the most common decompression usually happens in up-drafts either from warm air rising or wind against a mountain being forced up the mountain.

The condensing water actually releases quite a large amount of heat which adds some complication to the calculation of how much and how fast. This is why some clouds formed by rising warm air will flatten horizontal at a single altitude and are thin in the vertical dimension, while other clouds continue to rise and are very thick in the vertical dimension. This is because of the heat released by the condensing water in balance with the vertical temperature gradient (meters per degree) and the vertical pressure gradient (meters per pascal) of the surrounding air mass.