Material derivative

dt dt

In general dr can be in any direction, but in this special case it should be the displacement dr that occurs due to the motion of the fluid during the time dt. In other words, dr = v(r, t) dt where v(r, t) is the velocity of the fluid motion field evaluated at (r, t). After this crucial substitution we can write:

Stated again, this is the explicit recognition that the small increment dr which is associated with the time interval dt is identified with the motion of the fluid, namely it is the spatial increment that is advanced by the fluid itself during the time dt. We again encounter the important combination of derivatives which we termed the material derivative,

The change of the temperature field along the motion, DT/Dt, is composed of two terms: the first term, dT/dt, represents the local change of the temperature in a certain volume fixed in space (a weather station for example); the second term, v ■ VT, is due to the advection of hotter or colder air into this fixed volume by the wind.

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