CLIMATE/EARTH 401�Geophysical Fluid Dynamics�Dynamical Meteorology
Material Derivative
Vector Momentum Equation (Conservation of Momentum)
Consider some parameter, like temperature, T
x
y
Δx
Δy
Total variations
(x0,y0)
Taylor series expansion
If we move a parcel in time Δt
Using Taylor series expansion
Higher
Order
Terms
Assume increments over Δt are small, and
ignore Higher Order Terms
Total derivative
Total differential/derivative of the temperature T,�T depends on t, x, y, z
Assume increments over Δt are small
Total Derivative
Divide by Δt
Take limit for small Δt
Total Derivative
Introduction of convention of d( )/dt ≡ D( )/Dt
By definition:
u, v, w: these are the velocities
This is done for clarity.
Definition of the Total Derivative
The total derivative is also�called material derivative. �
describes a ‘Lagrangian viewpoint’
describes an ‘Eulerian viewpoint’
Return to the Momentum Equation
Remember, we derived from force balances
This is in the Lagrangian reference frame
In the Eulerian reference frame, we have
Non-linear advection of wind velocity
This comes from the Eulerian point of view
End: Material Derivative