Helmholtz Decomposition Theorem

5. I want you to use another Decomposition (Chapter 5)

5. I want to recommend to use another Decomposition

For example, from Wikipedia Helmholtz Decomposition

Let F be a vector field on a bounded domain V in R3, which is twice continuously differentiable. Then F can be decomposed into a curl-free component and a divergence-free component

We sometime use weather map to do meteorological analysis.

It belongs in R2, that is an analysis in the plane. And many meteorological analyst
applys Helmholtz decomposition theorem in this weather map belonging in R2.

But, we can see many discontinuity lines in the isobalic weather map as shown in Fig5.1.

In these area, I think there are many matters to apply Helmholtz Decomposition.

Fig5.１ Discontinuity line

Drawing a conceptual rendering,

Fig5.2 at divergent point

And I have already proved that Helmholtz Decomposition is mathematically wrong in

chapter 2.

There is another way to decompose any flow without any matters. And it is very similar

to Helmholtz Decomposition in the sence that it devices into solenoidal component and
another component which has divergent component.

It is the way to decompose any wind into geostrophic wind and ageostrophic wind.

Below are qaoted from http://nsidc.org/arcticmet/glossary/geostrophic_winds.html
Geostrophic wind Theoretical wind which results from the equilibrium between horizontal components of the pressure gradient force and the Coriolis force (deviating force) above the friction layer. Only these two forces (no frictional force) are supposed to act on the moving air. It blows parallel to straight isobars or contours.
Below are qaoted from http://jp.termwiki.com/EN:ageostrophic_wind

The vector difference between the real (or observed) wind and the geostrophic wind, that is, uag = u − ug. Sometimes the magnitude of this vector difference is meant.

This decomposition has not any matter at all.
It is just applied to basic vector difference.

Fig5.3 Another Decompositon

Helmholtz Decomposition demands that the wind must be continuous. But this decomposition

does not demand it. At any point, the wind can be decided, but it must not be continuous
in the plane.

And according to a definition, geostrophic winds blow in a parallel direction with a

inversely proportional to interval of contours . The contour of geopotential are supposed

to be continuous. So geostrophic winds are supposed to be continuous, and solenoidal

winds. Contours of geopotential looks like stream function from Helmholtz Decomposition
theorem.

Actually, we can see that the contour are similar to stream function. For example,

I show the weather map at 12Z on July 31 in 2011 inFig.5.4.

Fig5.4 Similarities between geostrophic wind and the wind driven from stream function

Meanwhile, ageostrophic wind is the vector which is the rest after substructing geostrophic

wind from the actual wind. And actual wind blows nearly geostrophic motion. So, ageostrophic

wind is generaly small, but it has all divergent component.

So, ageostrophic wind is similar to divergent wind from Helmholtz Decomposition theorem.

Fig5.6 shows the similarities between ageostrophic wind and divergent wind driven from

velocity potential.

Fig5.5 similarities between ageostrophic wind and divergent wind driven from velocity potential

By this decomposition, ageostrophic wind has all of divergence component of actual(or

analysis) wind. And divergent wind is supposed to have all of divergence too. So, The
distributions of divergence from both ageostrophic wind and divergent wind are supposed
to be same.

Fig5.7 shows two distributions from two types of wind.

Fig5.6 Conparison of distributions of divergences by using two types of winds

These divergent distributions are drawn on the water vapor imagery. The plus divergence

of the upper layer are closely-linked to clouds, and minus divergence( convergence) are
closely-linked to black area.

Whichever wind you choose to calculate the distribution of divergent, you can get almost

the same consquence.

But if you choose the divergent wind from Helmholtz Decomposition, it is the end. If you choose ageostrophic wind, you can go more.

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4. Why are there both(Chapter 4)

4. Why are there both velocity potential χ and stream function A

All flows are classified to one of four kinds of flows as below.

Fig4.1 Classification of flows

So, I have said that Helmholtz Decomposition is wrong, and
neither velocity potentials nor stream functions can exist

in the flow which have both divergent component and rotational
component.

But actually, JMA publish the distribution map of the velocity

potentialχand the stream function A every day.

Why can they make them?

Although I don’t like to make the composite model of flow,

I must draw the model as follows. Any flow in the real world has

vorticities and divergences as follows.

Fig4.2 The model of decomposition

As I show in the upper illustration, you can directly calculate the distribution of

divergence and vorticity from wind F .

After calculating a divergent distribution field and a curl

distribution field, they do think that they could perfectly
separate the flow into two kinds of flow. And they do make up
a χ distribution and a A distribution.

Generally, they use following equations for the proof of Helmholtz Decomposition.
Supposing F=∇・χ＋∇×A,
taking divergence of this flow,
∇・F=∇・(∇・χ＋∇×A)= ∇・(∇・χ), ∵∇・(∇×A)＝０
And,　
Taking curl of this flow,
∇×F=∇×(∇・χ＋∇×A)= ∇×(∇×A), ∵∇× (∇・χ)＝０
So, they think that Helmholtz Decomposition is right.

But, even if the flow consist of three component as showing upper illustration, you can calculateχand A, and these equation can be right.

So, if you want to stick around Helmholtz Decomposition, you need to prove that there is no flow component which play the role for both vorticity and divergence.

When you want to prove Helmholtz Decomposition, you cann’t preliminarily use the decomposition model like Fig4.3.

Fig4.3 It’s not the proof, if you preliminarily suppose like this

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