Transcript Document

ATOC 4720 class32
1. Forces
2. The horizontal equation of motion
2. Forces
1. PGF
2. Gravity
3. Friction-important in PBL (planetary boundary layer)
4. Coriolis force (apparent force, due to the earth’s rotation)
z
Derivation of PGF term:
The net force in x-direction:
y
x
Similarly, we obtain PGF in y and z directions as:
Therefore, the total PGF acts on the fluid element is:
Where,
For unit mass, the PGF becomes:
PGF in P-coordinate
Let
be either x or y direction:
(see fig on next page)
Using hydrostatic equation:
We obtain:
On “p” surface from Q to R:
Since “p” is consstant pressure surface,
So,
and
Gravitational force
(includes vertical sum of gravitational attraction
And apparent force, centrifugal force
)
Frictional force
: linear drag approximation.
Drag coefficient
Coriolis force
Apply Newton’s second law of
motion
For unit mass, m=1, we obtain HORIZONTAL equation of motion
Note: gravity does not appear, because it is a vertical component
The geostrophic winds: scaling
Let’s estimate the “order” of magnitude for each term
In the above equation. Observations show that:
Velocity V: 10-30 m/s;
Time t-days:
s;
In mid latitude;
So, acceleration term:
Coriolis term:
The Coriolis term is one order of magnitude larger
Than the acceleration term.
In free atmosphere, friction is negligible. Therefore, to the
lowest order:
0