Lagrangian and Eulerian Derivatives - Climate Dynamics - page 195

APPENDIX C

LAGRANGIAN AND
EULERIAN DERIVATIVES

The application of the laws of physics, such as Newton's laws of motion ap-
plied to fluid, can be visualized by considering a fluid element, for example, a
unit mass or volume of air of ocean water known as a parcel . Two perspectives
can be taken.
The Lagrangian perspective considers the parcel as a physical entity to be
followed in space, exactly as one would approach the classic physics problem
of a block on an inclined plane.
The velocity of a parcel in the x -direction (zonal direction, or east-west
direction) in local Cartesian coordinates is the Lagrangian derivative dx / dt be-
cause it tracks the changing location of the parcel with time. It is also called
the “material derivative” or the “substantial derivative.” The net balance of
forces, or the net heating, of the parcel is calculated according to the govern-
ing equations, and the resulting change in velocity or temperature with time is
expressed by the Lagrangian derivative d / dt .
Changes in any dependent variable, such as temperature, water vapor, den-
sity, or salinity, can be expressed in the same way. For example, dT / dt is the
change in temperature “following the parcel.” The Lagrangian perspective is
satisfying in a way, partly because it is so similar to the approach taken in
introductory physics texts. But it can be awkward in applications to a fluid,
because the positions of many parcels must be tracked and, in the real world,
the parcels stretch and distort and mix with the surrounding fluid over time.
More convenient, and a better match with observing systems, is the Eulerian
perspective.
The Eulerian perspective considers time rates of change within a fixed co-
ordinate system. In this case, the temperature at a given location can change
due to a net heating at that specific location but also because warmer or cooler
air or water is transported by the circulation to that location. The local time
rate of change—the Eulerian time rate of change—is denoted by the partial
derivate / t .
To translate between the Lagrangian and Eulerian derivatives, consider the
definition of the total differential, here applied to a temperature field that var-
ies in space and time, T ( x , y , z , t ):

2 T dx 2 T dy 2 T dz 2 T dt

dT / 2 +++ . (C.1)

x 2 y 2 z 2 t

Dividing through by the time interval dt we obtain