1. Richard B. Rood (Room 2525, SRB) University of Michigan
AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: Mass Conservation / Continuity
Richard B. Rood (Room 2525, SRB)
University of Michigan

2. Weather National Weather Service Weather Underground
Model forecasts:
Weather Underground
NCAR Research Applications Program

3. Outline
Conservation of Mass
Continuity

4. Newton’s Law of Motion
Where i represents the different types of forces.

5. Momentum Equations
We have u, v, w, ρ, p which depend on (x, y, z, t).

6. Conservation of Mass
Conservation of mass leads to another equation; the continuity equation
Continuity  Continuous
No holes in a fluid
Another fundamental property of the atmosphere
Need an equation that describes the time rate of change of mass (density)

7. Remember our particle of atmosphere, our parcel
r ≡ density =
mass per unit volume (DV)
DV = DxDyDz
m = rDxDyDz
p ≡ pressure =
force per unit area
acting on the particle of
atmosphere Dz Dy Dx

8. The Eulerian point of view our parcel is a fixed volume and the fluid flows through it.Dz Dy Dx

9. . Introduce mass flux, ρu (x, y, z)Dx Dy Dz
(x, y, z) x
ρu = mass flux at (x, y, z) in the x direction.
Flux is mass per unit time per area
Mass flux out =

10. . Introduce mass flux, ρu (x, y, z)
ρu = mass flux at (x, y, z) in the x direction.
Flux is mass per unit time per area
Mass flux in = Dz . Dy Dx x

11. What is the change of mass inside the fixed volume?
The change of mass in the box is equal to the mass that flows into the box minus the mass that flows out of the box = (flux) x (area)
Mass out right (downstream) face
Mass in left (upstream) face

12. Extend to 3-Dimensions
The change of mass in the box is equal to the mass that flows into the box minus the mass that flows out of the box = (flux) x (area)
Note: this is change in mass per unit volume.
Recognizing the definition of divergence

13. Eulerian Form of the Continuity EquationDx Dy Dz
In the Eulerian point of view, our parcel is a fixed volume and the fluid flows through it.

14. The Lagrangian point of view is that the parcel is moving.Dx Dy Dz
And it changes shape…

15. In-Class Exercise: Derive the Lagrangian Form
Remember, we can write the continuity equation
Use the chain rule (e.g., ) to go from the above equation to

16. Lagrangian Form of the Continuity Equation
The change in mass (density) following the motion is equal to the divergence
Convergence = increase in density (compression)
Divergence = decrease in density (expansion)

17. The Lagrangian point of view is that the parcel is moving.Dx Dy Dz
And it changes shape…

18. Our System of Equations
We have u, v, w, ρ, p which depend on (x, y, z, t). We need one more equation for the time rate of change of pressure…

19. Summary
The conservation of mass is one of three basic conservation laws we use in atmospheric dynamics
Momentum
Mass
Energy
The mass continuity equation connects mass, r, to the velocity field.
Also connects the thermodynamic variables to the velocity (momentum) field.