1. 1.The continuity equation 2.The vertically averaged divergence 3. The primitive equations ATOC 4720 class36

2. Equations we have learnt up to now: ()

3. Where,

4. Leading term, Change T field Even when, which is very small, Pronounced T change with time Moreover, Precipitation and latent heat release is CONTROLED by

5. 1. The continuity equation () : However, Newton’s second law of motion: provides no diagnostic information about As we shall see, Is subject to kinematic constraint based on: Conservation of mass Continuity equation Example: pancake; cloud anvil

7. The continuity equation However, there is a significant difference: Pancake: uncompressible; Air parcel: compressible: volume changes. Two types of volume changes: 1.Gradual, hydrostatic changes due to expansion and compression; [hydrostatic] In P-coordinate, this process is automatically taken into account; 2.Nonhydrostatic fluctuations--vertically propagating sound waves. [Small amplitude, high frequency, not important for large-scale atmospheric motion.

8. The continuity equation Mass: As the flow deform the shape of the parcel, its mass is conserved:

9. Mathematically, the above equation can be expanded as: Because,

10. The continuity equation: mass conservation

11. To gain a clearer understanding of the physics, let’s rewrite Equation: Let Using

12. We have, divergence convergence

13. Convergence Divergence IfZ-coordinate: