Freezing water on a warm night

Everybody knows that water freezes at 32 deg F. 

Then how come frost (which is just ice that has come from the water vapor in the air) can sometimes be seen in the morning on nights when the temperature never got below 32 deg F?  

 Similarly, backpackers and desert dwellers occasionally report a film of ice on a puddle or bucket of water on not-quite-freezing nights.

The answer has to do with the nature of heat transfer.  Heat can move by conduction (movement through a solid, or a fluid at rest) by convection (movement between a solid and a flowing fluid) or by radiation (direct exchange of energy via electromagnetic waves).  The temperature of a surface depends on the temperatures of the surroundings plus the effects of all three modes of heat transfer.

  

For example, let’s think about a point on the surface of the earth.  This cartoon shows a particular point on the ground that is exchanging radiation (black arrows) with its surroundings including the sun, the car, the tree, and the sky.  Simultaneously, heat is being conducted (green arrow)  into or away from the underground while convection (blue arrow) by the air is carrying heat into or away from the surface of the ground.  The direction of the heat flows depends on the temperature of the various surroundings (always from hot to cold).  

The temperature of that particular point on the ground depends on a combination of all of the surrounding temperatures, as well as the strength of the heat flows.  For example, if the wind were blowing really hard, the convection mechanism might be stronger than all of the others, and the surface of the ground would be forced very close to the air temperature.  In this case, the radiation and conduction heat flows would still be present, but would be so overwhelmed by the convection that their effect would be negligible.

At night, the same mechanisms are still active.  Now, let’s imagine a situation where the air is very dry and clear (because water vapor and especially water droplets in the form of clouds tend to interfere with radiation heat exchange) and still (thus minimizing the strength of convection).  If the air temperature in such conditions were fairly close to freezing already, say, in the thirties (Fahrenheit), then the radiation exchange with the night sky, (which can have a very cold effective temperature since it is influenced by the temperature of space)  might be sufficient to bring the temperature at the ground surface well below freezing.  Once the local surface temperature drops below freezing, water vapor in the air can condense and form frost, or the surface of a small puddle can freeze to ice even though the air temperature might remain above freezing.