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.