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“True” actual components of a flow

July 16, 2014

We used to use isobaric charts for the analyses of weather phenomena.

For example, 200hPa charts show the situations on the plane of about 12000m height.

We can consider the wind data on 200hPa as a flow vectors in two-dimensions.

JMA and NOAA are making “velocity potentials” and “stream functions” from these wind data sets by applying “Helmholtz Decomposition”.

But I think that Helmholtz Decomposition is wrong, so these velocity potentials and stream functions are made by their mistakes.

I want to show you that how these potentials are made.

1) On the Helmholtz Decomposition theorem

There is a theorem called Helmholtz Decomposition that says any flow can be separated into irrotational divergent flow and non-divergence rotational flow. And velocity potentials can be calculated from the irrotational flow and steam functions can be calculated from the non-divergence flow.

JMA and NOAA are publishing velocity potentials and stream functions on net.

Acording to Helmholtz Decomposition, two kinds of these potentials are independent from one another. So, if you wanted to analize the distributions of divergence in some layer, you could do it by just analyses of velocity potential map.

From Wikipedia, Helmholtz Decomposition is given as follows
Equation1 ・・・・・・・・・・・・・1)



Here, Fl means irrotational divergent flow, and Ft means non-divergence rotational flow.

I show you Fig.1 to image the Helmholtz Decomposition..

Fig.1 Illustration for Helmholtz Decomposition

2) The third component

When we think about composition of vector, that is generically considered as projections of a vector which is given at one point onto the reference axes.

But the components in the Helmholtz Decomposition are given as roles which play as a flow in the set of neighbor flows. So, you should think about the component which play both role of curl and divergence, over and above the irrotational divergence component and the non-divergent rotational component.

If there is the third component which play both roles of curl and divergence in any flow, the components of any flow should be shown as Fig.2.

Fig02 Fig.2 The components of any flow

Even if there is the third component, you can calculate nabra_dotF and ∇×nabra_productF distributions, and therefor you can get velocity potential and stream function. And furthermore, you can get Fl and Ft.

But as you can see inFig.2, the composition of these two components does not match the original flow.

So, I can say that Helmholtz Decomposition is wrong.

3) the components of a “true” actual flow

Actually, there is a fair percentage of non-divergence and irrotational component in actual flow.

So, when you divide a flow into some components, you should think about the fourth component which has neither divergence nor curl(rotation).

Fig03 Fig.3 the 4 components of a general flow


According to Equation 2), φ is calculated from the term of nabra_dotF. So, Fl should consist of the components 1+3.

 Fig.4 is calculated with the components of 1+3


Fig.4 Fl is calculated with the component of

And, vector potential A is calculated with the term of nabra_productF.

So Ft should consist of the components ②+③.
Fig05Fig.5 Ft is calculated with the components of ②+③


Therefore, Equation 1) is not correct. Therefore we should say that Helmholtz Decomposition is wrong.
Fig.6 Helmholtz Decomposition is not correct

After publishing this article, I should edit or remove my latest blog “On the components in Helmholtz Decomposition Theorem”, but I daringly keep it on the Net.

4) Another Decomposition

There is another way to decompose any wind into two components. You can decompose any wind into geostrophic wind component and ageostrophic wind component.

Geostrophic winds are theoretically given from the contours(heights of an isobaric surface). Geostrophic winds blow along contours in inverse proportion to the gap of contour lines. So, geostrophic winds are perfectly non-divergent wind.

And, because the natural winds blow as quasi-geostrophic winds, they mostly consist of geostrophic winds.

Ageostrophic winds are given as the difference calculated by subtracting geostrophic wind from the original(analyzed) winds.

So, there is no doubt in this way to divide any flow into geostrophic wind and ageostrophic wind.

I show you a illustration to image the decomposition which make a flow divided into geostrophic wind and ageostrophic wind inFig.7.
Fig.7 The Decomposition into Geostrophic wind and ageostrophic wind

And, Fig.8 is an example for analyzed(original) wind(black arrow), geostrophic wind(blue arrow) and ageostrophic wind(red arrow).
Fig.8 an example for geostrophic wind(blue), ageostrophic wind(red)

and analyzed wind(black) on 20th Jun 2011

Fig.8 shows that the composition of geostrophic wind and ageostrophic wind is nearly equal to original analyzed wind. It might be expected.

We can’t say that Geostrophic wind (blue arrow) take out 100% of the non-divergent component from natural(analyzed) wind(black arrow). But it mostly consist of them.

Ageostrophic wind component(red arrow) is approximately compounded of divergence component which is shown (1+3) in Fig.3.

Therefore, ageostrophic wind is nearly divergent wind which would be given from velocity potential.

Here I want to show the divergent wind and curl wind on the same day. The divergent winds were calculated from velocity potentials, and the curl winds were calculated from stream functions in Fig.9.
Fig.9 an example for curl wind(blue), divergent wind (red)

and analyzed wind(black) on 20th Jun 2011

After seeing Fig.9, I had been left speechless for a while. Because the composition of divergent wind and curl wind is nearly equal to the original(analyzed) wind.

Had I mistook in former article related Fig.6?

Please put it aside for a while, and confirm that the divergent winds are fairly equal to ageostrophic winds.

This is an example for that Fl in Fig.4 is nearly equal to the component of 2 in Fig.7. We can say that ageostrophic winds are nearly equal to divergent winds.

5) about stream function

Comparing Fig.9 to Fig.8, curl winds Ft calculated from stream functions nearly equal to geostrophic winds.

And we have confirmed that the composition of divergent wind Fl and curl wind Ft is nearly equal to the original(analyzed) wind. This can be a proof for that Helmholtz Decomposition is crrect.

Here, I doubt if these stream function was truly calculated by using equation 3).

Please look at Fig.5 again.

Stream function must be driven from a vector potential expressed as below.
Therefore, the components of just 2 and 3 in Fig.5 is useful to calculate nabra_productF. Because, even if the component 1 and 4 were used, they came to 0 as a consequence. So, Ft definitely not be nearly equal to geostrophic wind. Ft should be fairly small than geostrophic wind.

According to equation 2) and 3), the composition of Fl and Ft should be smaller than original analyzed wind.

There is a way to make these stream functions published from JMA or NOAA.

If you priliminaly beleaved Helmholtz Decomposition is right(Fig.1), you could get stream function from the difference calculated by subtracting divergent wind from the original(analyzed) wind.

But, there are 4 kinds of components in any actual wind.
Fig.10 Actual way to get “Stream function”

It is all right to get Fl from the equation 2). But Ft must be calculated as the differences calculated by subtracting Fl from original wind, for getting equation 1). In any another way, Fl+Ft would not be equal to original F.

To take this way is definitely distinct from Helmholtz Decomposition. This way is the same way to separate a wind into geostrophic wind and ageostrophic wind.

Here, I must confess that I don’t know exactly how to make stream function. Please ask some person who know how to calculate the stream function, if you know. And ask him to publish the way how to calculate the stream function. I think it have been top-secret among them.

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