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 oC /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 oC /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 1oC 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?