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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