6.1Helmholtz Decomposition is wrong(Chapter 6-1)

6. Ageostrophic wind (1)

Here, we have to learn about ageostrophic motion.

Suppose you put an air parcel quietly on a surface which has the pressure gradient and

the air parcel has the same density as the surrounding air at that height. How does this
air parcel behave? To make things easier, I assume that the surrounding air never changes

its pressure gradient and the air parcel never mingles with the surrounding air.

The forces which act on the air parcel are the pressure gradient and Coriolis forces.

At first the Coriolis force doesn¡Çt act on it, because the speed of the air parcel is zero.
Only the pressure gradient force acts on it. So the air parcel starts to sink perpendicular

to contours of height.

The following equation of motion can be obtained:

¢ßV¡¿¢ßt¡á¡½f¡¦V¡Ýg¡¦¢àh
whereV is the air parcel¡Çs velocity after t time and f, g and ¢àh denote the Coriolis

parameter, the acceleration of gravity and the gradient of the geopotential height,

respectively.

Geostrophic wind at this moment Vg gives

¢ßVg¡¿¢ßt¡á0¡á¡Ýf¡¦Vg¡Ýg¡¦¢àh

Subtracting the latter from the former leads to

¢ß¡ÊV¡ÝVg¡Ë¡¿¢ßt¡á¡Ýf¡¦¡ÊV¡ÝVg¡Ë

This equation reduces to

¢ßA¡¿¢ßt¡á¡Ýf¡¦A
where a vector A keeps on rotating with a frequency f.

Its period T is
2¡¦¦Ð¡¿f¡á2¡¦¦Ð¡¿2¡¦¦Ø¡¦sin¦Õ¢â12¡¿sin¦Õ

where ¦Õ means latitude and ¦Ø denotes the rotating angular velocity of the earth,
which is approximately equal to 2¡¦¦Ð¡¿24 hours. So at 30¡¬N the rotation has a
24-hour period.

Fig6. 1 shows what the above equations mean.

Fig6.1 Geostrophic motion

1. ­¡ : An air parcel is at rest on the pressure field.
2. ­¢ : First, it starts to move toward low height perpendicular to height
contours in response to the pressure gradient force, but as soon as motion develops,
the Coriolis force also acts on it. So the parcel moves acceleratingly with both
forces acting on it. The pressure gradient force remains unchanged, whereas the Coriolis
force acts deflecting the parcel¡Çs motion toward the right in proportion to its speed.
So descending along contours of isobaric height, the parcel gradually accelerates
parallel to contours of height.
3. ­£ : Eventually the parcel¡Çs motion becomes parallel to height contours and
has the same direction as geostrophic wind¡Çs. Converting potential energy into kinetic
energy, the parcel moves downward along the slope of the pressure surface and its speed
becomes twice as high as geostrophic wind¡Çs. The Coriolis force also becomes twice as
strong as the one needed for geostrophic balance and acts on the parcel the way it makes

the parcel move upward perpendicular to contours of height, just as strongly as the

force acting on the initial motionless air parcel, but reversely.
4. ­¤ : The parcel¡Çs direction is gradually deflected to the right and the parcel

decelerates upward across contours of isobaric height.

5. ­¥ : Getting to its original height, the parcel is again at rest for a moment.
Then the parcel repeats the motions from ­¡ to ­¥.

Fig. 2 shows the forces acting to an air parcel while one period of ageostrophic motion.

There contours of isobaric height are drawn in straight lines.

Fig6.2 Forces acting on an air parcel at every quarter of cycle

And Fig6.3 shows that the force acting in any time of cycle has the same stlengths as the

pressure gradient force, and take a clockwise tangential direction.

Fig6.3 Forces acting on an air parcel at any time of cycle

continue