6.2Helmholtz Decomposition is wrong(Chapter6-2)

Chapter 6-2 Ageostrophic motion (2)

Fig6.4 shows one period of ageostrophic motion and the forces acting on it. There contours of isobaric height are not drawn in straight lines but in curved lines.

Fig6.4 ageostrophic wind, geostrophic wind

Here the differental vector means ageostrophic component in Chapter 5.

Fig6. 5 shows ageostrophic wind, geostrophic wind and their differential vectors.

Fig6.5 ageostrophic wind, geostrophic wind

When isobaric surfaces surround the Northern Hemisphere high in the south and low

in the north, the motion follows the trajectory in Fig6. 4.

Fig6.6 The trajectory of Ageostrophic motion

Now I would like to calculate how much kinetic energy an air parcel obtains when evolving

from its motionless state ­¡ to state ­£ where the speed reaches maximum. The force a unit
volume of air parcel undergoes and the distance it covers constitute work. The energy of
this work is converted into kinetic energy. Of the two forces the Coriolis force is
perpendicular to the direction of the motion and does not contribute to the work.
( refer to Fig6. 5 )

Fig6.7 The estimit of energy

Therefore, kinetic energy obtained by a unit volume of air parcel during this motion

is provided only by the pressure gradient force. The pressure gradient force is always
perpendicular to contours of height. So the total amount of the work during this motion
is obtained by integrating from ­¡ to ­£ ( in Fig.4 ) the inner product of the pressure
gradient force and the line segment along the path of the air parcel.

The amount of work given is

¢é ¡Ý¦Ñ¡¦g¡¦¢ßh¡¿¢ßn¡¦ds
=¦Ñ¡¦g( height ­¡¡Ýheight ­£)

where ¦Ñ is the density of air, h is the height of isobaric surface and n and s denote
the unit vector directed to the steepest slope of isobaric surface and the unit vector
directed to the path of the parcel, respectively.

This equation means that the energy produced during the parcel¡Çs transferring from ­¡

to ­£ is equal to the lost amount of potential energy. This amount of work becomes kinetic
energy, which is 1¡¿2¡¦¦Ñ¡¦v2

That means an air parcel in ageostrophic motion obtains kinetic energy by being compressed

by the pressure gradient force, and this kinetic energy is equal to the potential energy
which is lost while moving down the isobaric surface. In other words, an air parcel obtains

the velocity by moving on isobaric surface and lowering its height.

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