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

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

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

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: