A Prognostic Temperature Equation

In the physics of motion, the total derivative often appears.

where f is any variable.

From Metr 201, we remember that

and the above equation can be rewritten in the rectangular coordinate system

and in the natural coordinate system

This is a general equation that can be used to forecast the future value of a variable, f,  at a given location.

Rearranging terms and and using temperature as the variable

which is a PROGNOSTIC EQUATION that allows us to forecast the change in temperature at a given location (called the local derivative on the right) to the total changes experienced by an air parcel (the total derivative) and the temperature changes due to the motion of air with different temperatures into the local air column (termed TEMPERATURE ADVECTION).

The temperature changes due to horizontal temperature advection are

Thus the local changes in temperature are a function of temperature changes due to advection plus a correction term related to however the temperature of air parcels change as they move from place to place.

Evaluating temperature advection is very important in the process of weather foreasting.