ATOC 4720 class 36 1 The continuity equation

ATOC 4720 class 36 1. The continuity equation 2. The vertically averaged divergence 3. The primitive equations

Equations we have learnt up to now: ( )

Where,

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

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

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 2. and compression; [hydrostatic] 3. In P-coordinate, this process is automatically 4. taken into account; 2. Nonhydrostatic fluctuations--vertically 3. propagating sound waves. [Small amplitude, 4. high frequency, not important for large-scale 5. atmospheric motion.

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

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

The continuity equation: mass conservation

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

We have, divergence convergence

Convergence Divergence Z-coordinate: If