1. ATOC 4720 class32
1. Forces
2. The horizontal equation of motion

2. 2. Forces
PGF
Gravity
Friction-important in PBL (planetary boundary layer)
Coriolis force (apparent force, due to the earth’s rotation)

3. z y
Derivation of PGF term: x
The net force in x-direction:

4. Similarly, we obtain PGF in y and z directions as:
Therefore, the total PGF acts on the fluid element is:
Where,

5. For unit mass, the PGF becomes:

6. PGF in P-coordinate Let be either x or y direction:
(see fig on next page)
Using hydrostatic equation:
We obtain:

7. On “p” surface from Q to R:
Since “p” is consstant pressure surface,
So,
and

8. Gravitational force (includes vertical sum of gravitational attraction
And apparent force, centrifugal force )

10. Frictional force
: linear drag approximation.
Drag coefficient

11. Coriolis force

12. 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

13. The geostrophic winds: scaling
Let’s estimate the “order” of magnitude for each term
In the above equation. Observations show that:
Velocity V: m/s;
Time t-days: s;
In mid latitude;

14. 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: