Helmholtz Decomposition Theorem

7.2 Helmholtz Decomposition is wrong(Chapter7-2)

Classification of the directions of ageostrophic winds

It is useful to sort out which direction does the force take, or does the ageostrophic wind

take. Because, the force will change the wind (movement).

I want to divide the directions of force foura quadrants.

Fig7.5 First quadrant and Second quadrant of the force applied in ageostrophic motion

So, Fig7.5 show us that if an air parcel has an ageostrophic wind on the left side of actual wind, the force will accelerate the speed.

Fig7.6 Third quadrant and Fourth quadrant of the force applied in ageostrophic motion

And, Fig7.6 show us that if an air parcel has an ageostrophic wind on the right side of actual wind, the force will decelerate the speed.

Let us see the example. Fig7.7 shows the distributions of the actual winds(
Black arrows) and the ageostrophic winds(red arrows) around Asia at 12Z on June 9th in 2011.

Fig7.7 The distribution of actual winds and ageostrophic winds

We can see the accelerating area around Japan sea. Acceleraing area tend to be divergent
area, and divergent area tend to cloud area.

Meanwhile, we can see the decelerating area around Yellow Sea. Decelerating area tend to be

convergent area, and tend to be fine weather area. We can see black area on the water vapor
cloud images.

Fig7.8 shows the distribution of divergence on the water vapor image.

Fig7.8 The distribution of divergence on the water vapor image

These divergent distributions have been gotten from the real(analyzed) wind, not from
divergent wind(from velocity potential).

I think that the ageostrophic wind is very useful to explane these meteorological phenomena.

I think it is no need to use divergent wind, nor velocity potential.

If you want to think the Meteorology as a science which includes dynamics,
you should not use the divergent wind driven from the velocity potential.
Because, the divergent wind leaves from an air parcel(mass), force and acceleration.
Dynamics is a science which deals in masses, forces and accelerations etc..

continue

7. 1Helmholtz Decomposion is wrong(Chapter 7-1)

7. On the forces acting on the air parcel in the actual ageostrophic motion

We have been thinking an air parcel on many kinds of hypothetical condition. I wanted you
to know what the theoretical ageostrophic motion is.

From now, I want to say about actual air parcels.
Fig.7.1 shows a sample of ageostrophic wind in the real world.

Fig7.1 We can see the ageostrophic wind in the real world

We can see the ageostrophic wind which can’t be denied. Especially, near the boundary of

the westerlies zone and the tropical zone. The air parcels which have gone up from the lower
layer are thrown out into the upper layer. They have been mingled with the air on the way of

middle layer, and have gotten a little similar velocity of upper layer air which has been

took a balance of geostrophic wind. But they have not gotten perfect geostrophic balance,
and are beginning to act as ageostrophic(non-geostrophic) wind in the westerlies zone.

Fig.7.1 shows that the actual ageostrophic component is not as large as the theoretical

ageostrophic wind, but smaller than it.

Fig 7.2 shows the real wind model.

Fig7.2 The wind can be decomposed into two components

A black arrow shows an actual wind. An actual wind can be decomposed into a geostrophic

wind component and an ageostrophic(non-geostrophic) wind component. A blue arrow shows

a geostrophic wind component, and a red arrow shows an ageostrophic wind component.

The force acting on the real air parcel

Fig 7.3 shows the velocities of an real air parcel, and forces acting to it.

Fig7.3 We can consider the Coriolis force with each component wind

The upper side of this illustration shows north. So, contours are stretched east to west.

The thinner arrows show velocity vectors, and bolder arrows show force vectors.

If the wind have a balance of force, the pressure gradient force is equal to the Coriolis

force of the wind, but they are oppositely-oriented. The balanced wind is the geostrophic
wind.

The Coriolis force is proportional to the speed at which the air is moving, and deflected

to the right. So, we can indicate how to calculate the Coriolis forces by drawing right

triangles which has velocity vectors and force vectors. These triangles are similar to each
others.

Fig7.4 The total force acying on the air

 

Fig7.4 shows the actual wind and its component, and acting forces. From this illustration,

we can understand that the actual force working the air is the Coriolis force obtained with

an ageostrophic wind component as the followings.

The solid thin blue arrowAB shows the geostrophic wind component, and the thinner black

arrowAE shows actual wind.

The real forces which act to this air parcel are the pressure gradient forceAG and the

Coriolis forceAD with actual wind velocity. So, the total force is given a composition of AG

and AD. If you draw a parallelogramADHG, it is given as a diagonalAH.

Coriolis force for the geostrophic wind is given as a broken blue arrowAC, From geostrophic

theorem, AG =AC.

Here, △ADE∽ △ACB、∠DAE=∠CAB=right angle.

Then, ∠ＢＡＥ＝∠ＣＡＤ

Here, we pay attention to △ACD. The ratio of the segmentAC to the segmentAB is the same as

the ratio of the segmentAD to the segmentAE.

So, △ACD∽ △ABE

Here, I want to call △ABE a velocity triangle, and want to call △ACD a force triangle.

If you turn this force triangle counterclockwise, it get the place △AC’D’.

Then, this segmentC’D’ is parallel to segment BE.

So, we can confirm that

the real force that is acting on an air moving as an ageostrophic wind motion is given in
 the same way as Coriolis force, but with an ageostrophic component instead analyzed wind

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