Metr 430
Handout:__ Lagrangian and Eulerian
Derivatives__

The total derivative relates the relationship of the
derivative (or total infinitesmal
change) of a variable to the partial contributions or indirect infinitesmal dependencies. To put it in meteorological terms, the
total derivative measures the change of the value of a variable in a fluid
parcel and relates that change to the change measured at a fixed location and
the distribution of the variable in three-dimensional space.

Consider the total change of a variable f which can change
in time and along the cartesian coordinate axes:

(1)

Using rules of calculus, one can divide by the differential
dt.

(2)

But the definition of the wind components in the cartesian
coordinate system is:

u = dx/dt, v=dy/dt and w=dz/dt (3)

Substitute (3) into (2) and we have the equation for the
Lagrangian Derivative.

(4)

Commonly, in weather forecasting, equation (4) is solved for
the local derivative.

(5)

Any variable can be substituted for f. For example, temperature or T.

(6)

or solving for the local derivative,

(7)

Equation (7) is called the simplified Temperature Tendency
equation. It relates the
temperature observed by a thermometer at a fixed location in an air column to
the advection of temperature into air column and the changes in temperature as
the air parcels move towards the fixed location.

Equation (7) can be used to calculate the rate of change of
temperature at a fixed location. To determine the change in temperature
over a fixed time interval, equation (7) must be multiplied by the total time
period over which the changes quantified in (7) occur, as shown in equation (8) This, then, can be algebraically added
to the initial temperature at the location to obtain a forecast temperature

(8)

**Exercise: Answer on Separate Sheet**

Say that the temperature at San Francisco International
Airport (SFO) is changing at a constant rate of -0.5^{ o}C /hr.

1. What is
the net temperature change at SFO at the end of that 6 hour period?

2. What are
the symbolic forms of -0.5^{ o}C /hr and 6 hours (in differential
calculus notation)?

3. What is
the symbolic form of the expression you used to answer this question, in
differential calculus notation?

4. Now,
suppose that you know that radiative cooling is responsible for about 1^{o}C
of the cooling that occurred in that 6 hour period. What was total contribution of temperature advection to the temperature change (over the 6 hours) that occurred?