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

ATOC 4720 class36

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 massContinuity equation

Example: pancake; cloud anvil

The continuity equationHowever, 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.

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

IfZ-coordinate: