I made inquiries about Helmholtz Decomposition to some Doctor. Here is

He is kind to answer my question as shown below,

I feel very weird in his sentence “This is referring to an abstract mathematical orthogonality of the curl and divergence operators, and not to a literal geometric orthogonality of the vectors.” .

These idea may be popular among hydrokinetic scientists, or among meteorologists.

But, I think these idea is infantile mistakes which is out of the basic mathematical vector concepts.

It’s Ok to think about an abstract mathematical orthogonality of the curl and divergence. But you have to check the orthogonality of them goes clear the orthogonality in the real world.

I think the components of vector is mistakenly thought among the hydrokinetics. “The components” are given at a point. Forexample, please see next page.

But, “the components in Helmholtz Decomposition” should be given in small space, not at a point. They should be determined in small space, or with neighbor vectors.

The basic mistake among hydrokinetics is that they think the non-divergence components and irrotational components are given at a point. They think that they can get these components at appoint by using delta functions and Fourier transformations. But, the “components” in Helmholtz Decomposition are not innate concepts such as “the vertical components and the horizontal component”. So, they should be given after calculation of “the divergence field and the curl field” in very small erea, but not at a point.

To determine the differential value at a point is the same mistake as Zeno paradox. One of Zeno paradox says that the flying arrow is stopping. You should think the arrow is flying or not, with velocity. But if you infinitely shorten the time of the arrow flying, and you can think the arrow is stopping at each second.

To be stopping or not should be considered by velocity, or “the length”/”the time”.

To shorten the time infinitely is Ok as long as time interval is not 0, for thinking the arrow is flying or not.

I don’t agree with using the ward “components” in Helmholtz Decomposition, but in this article, use it according to commonly used.

The non-divergence components and curl components are given as bellow

The divergence field and the curl field can be given by calculating from any vector field without Helmholtz Decomposition. But, these two field are not guaranteed to separately exist. If you just had mistook these field separately exist, you can get a scalar potential and a vector potential.

But if you want to prove Helmholtz Decomposition, you should prove that there is no component that contribute to both the divergence and curl composition.

If any vector has a component which contribute to both 0f the divergence component and curl component,

But, if any vector has not the component which contribute to both components,

Acording to Wikipedia, Helmholtz Decompose theorem states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field

I want to show you a simple example in two dimension cooerdinates system as below figure.

Fig 1 Calculus equation for the divergence and the curl

The divergence of vector F at a point o is calculated by using blue vectors, or V1, U2,V3 and U4 in the figure above , and the curl of vector F is calculated by using red vectors, or U1,V2,U3 and V4.

All of vectors which are decomposed from vector F for calculating the divergence and curl have been used.

So, you can say this is Helmholtz Decomposition.

Certainly, you can think about an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field with any vector field.

But if vector F has some component that contribute to both vector field, you can’t say that the vector field decomposed into an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field as showing bellow.

Fig 2 If F(r) has some component that contribute to both

If vector F is smooth, U4 bears distinct relationship to U3 in Fig 1.

Fig 3 the conditions for smooth vector field

U4 that is constitutive part of divergence is given by a equation include U3 that is constitutive part of curl.

So, you can say that Any vector field has some component which contributes to both roles(components) of divergence and curl.

Therefore, Helmholtz Decomposition is wrong.

Acording to Helmholtz Decomposition theorem, a vector can be decomposed into a curl-free component and a divergence-free component.

*A vector and its components*

What is a vector? A vector is a quantity that has both magnitude and direction.

Vectors in multiple-dimension coordinate systems can **be broken up(=decomposed)** into their *component **vectors*. In the three-dimensional case, this results in a *x- component, *a

*y-*and a

**component***z-*. The next picture is an example of a Force vector (

**component****F**) broken into its components (

**F**and

_{x}**F**) in the two-dimensional case. When breaking a vector into its components, the vector is a sum of the components.

_{y}The components are broken up from a vector at just a point.

The components generally mean the elements which are broken up (=decomposed) from a vector in Cartisian coordinate system. Each component is independent to each other.

* The components in Helmholtz Decomposition*

But the components used in Helmholtz Decomposition are not the same components of vector in Catisian coordinate system. They represent spatial aspect in a vector field.

I think that the “components” in Helmholtz Decomposition are rather the “roles” of a vector within the neighbors for divergence and curl. They should be decided within neighbor vectors.

Please, think about a vector field without Helmholtz Decomposition. Thare is just a vector field **F** at first.

So you can calculate and get the divergence field by **F**, and the curl field by **F**. Helmholtz Decomposition is not needed to calculate these calculation.

The divergence field made by**F** has an irrotational vector by (**F**), and the curl field made by ∇×**F** has a non-divergence vector by (**F**) .

An irrotational vector field has a scalar potential =∇・**F**, and a non-divergence(solenoidal) vector field has a __vector potential A__

__＝__

__∇__

__×__

__F__In Meteorology, is called velocity potentials, and **A i**s called stream functions.

Here, you have gotten the divergence vector field which is curl-free, and the curl vector field **A **which is non-divergence from a vector **F**.

** **Many articles about Helmholtz Decomposition are similer to these expositions above.

But, you have never guaranteed that the sum of these two vectors make up the original vector, or the original vector **F** can be decomposed into these two of vectors. You have just gotten an irrotational vector and non-divergence vector from F, but the sum of two may not 100% of **F**. Because, these elements, or a curl-free component and a divergence-free component are not the components in Cartisian coordinate system.

The original vector **F** may have the rest of them. Or the vector **F** might have contribute to both “components”( I want to say them “roles”).

If the vector included some components( or roles) which contributes to both of curl components and divergent components, the original vector could not perfectly decomposed into a curl-free component and a non-divergence component.

* the right way to prove Helmholtz Decomposition 1*

For all that, I can prove that Helmholtz Decomposition theorem is right.

I will show you it with diagram bellow.

I want to get the divergence and curl at a point (r), at the center of figure above. Therefor, I need F2 and F4 to get differential value of F with x axis, and F1 and F3 to get the differential value of F with y axis. F1,F2,F3 and F4 act for 100% of F(r) to calculate the differential of F at this point

And each of F1,F2,F3 and F4 can be decomposed into x components and y components in the cartesian coordinate system. They are U1 and V1 from F1, U2,V2 from F2, U3 and V3 from F3, U4 and V4 from F4.

Therefor, U1,U2,U3,U4,V1,V2,V3 and V4 act for 100% of a vector F at this point for calculation of space derivative. And half of them that is V1,V3,U2 and U4 contribute only to divergence, and another half of them that is V4,V2,U3 and U1 contribute only to curl.

So, we can say that we use all of differential element of F and use half of them to calculate the divergence**F**, and use another half of them to calculate the . So, The vector **F** can be decomposed into （**F**） and （）. This is the perfect proof for Helmholtz Decomposition.

*the right way to prove Helmholtz Decomposition 2*

I know that there is another way to prove Helmhltz Decomposition thorem. If you can find an identity that shoes some vector function equal to the sum of non-divergence vector and curl-free vector, it is the another way that can prove Helmholtz Decomposition.

A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction.

It can be expressed like

based on three perpendicular spatial axes generally designated x, y, and z. Here, **i,j,k** are the unit vectors in the coordinate system respectively.

While, the differential operator “nabra” is similarly expressed as well as a vector.

So, this can be thought as a kind of vector.

But, we need to know there is big differences between a general vector and this differential operator, “nabra”.

**A general vector is defined at a point. But, differential operator “nabra” is defined in the small area, not at a point.**

Many scientist neglect this differences. For example, there is an article about Helmhltz Decomposition in Wikipedia like bellow

In the upper quoted article, the first(upper) group of equations is OK, because they are expressed at a point. But the second(under) group of equations is not allowed.

We can easily get a sample flow which has divergences by transvers components and rotations by longitudinal components. They are not 0.

For example, let us think a flow like following Figure.

The sample shows that a flow field is squared off at 50m intervals. And we can get the velocities at each grid point. Their values are shown in the Table.

So, we can get two components at any grid. They are the longitudinal component and transverse component.

And, we can calculate the divergence with transverse components, and rotation with longitudinal components.

They both may not be 0.

The mistake in Wiki’s article is caused by the neglect of the differences between the general vector and the difference operator.

Someone might say that you could decompose a vector into two components which are the longitudinal component and transvers component at the center point by using Fourier transformation and Invers Fourier transformation.

But, I want to say that the differential is not given at a point even if using Fourier transformation and invers Fourier trans formation. It is given in the very small area, but not at a point.

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

・・・・・・・・・・・・・１）

Here, **F**l means irrotational divergent flow, and **F**t 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.

Fig.2 The components of any flow

Even if there is the third component, you can calculate **F** and ∇×**F** distributions, and therefor you can get velocity potential and stream function. And furthermore, you can get **F**l and **F**t.

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).

Fig.3 the 4 components of a general flow

According to Equation 2), φ is calculated from the term of **F**. So, **F**l should consist of the components +.

Fig.4 is calculated with the components of +

Fig.4 **F**l is calculated with the component of

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

So **F**t should consist of the components ②＋③.

Fig.5 **F**t 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 (+) 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 **F**l in Fig.4 is nearly equal to the component of 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 **F**t calculated from stream functions nearly equal to geostrophic winds.

And we have confirmed that the composition of divergent wind **F**l and curl wind **F**t 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 and in Fig.5 is useful to calculate **F**. Because, even if the component and were used, they came to 0 as a consequence. So, **F**t definitely not be nearly equal to geostrophic wind. **F**t should be fairly small than geostrophic wind.

According to equation 2) and 3), the composition of **F**l and **F**t 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 **F**l from the equation 2). But **F**t must be calculated as the differences calculated by subtracting **F**l from original wind, for getting equation 1). In any another way, **F**l＋**F**t 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.

There are two kinds of vector decompositions. One is a normal vector decomposition, and the second appears in Helmholtz Decomposition.

**1. The normal decomposition of vector**

Talking of the decomposition, you generally think about projected components of a given vector on the reference axes or rectangular coordinates.

And the vector projected on the reference axes is called the components of a given vector.

Fig.1 a normal decomposition and components

But you can also think about a decomposition into non orthogonal directions.

If a given vector is equal to the vector sum of other two vectors, these two vectors are called as components of a given vector.

Fig.2 a decomposition into non-right angle components

The basic concept of vector is that two vectors which are not perpendicular to each other have the components of each other. The dimension of it is given by the length from the foot of perpendicular to the point of intersection of two vectors.

Fig.3 a vector component of a vector on another vector direction

When the directions of two vectors are perpendicular to each other, the foot of perpendicular to other axis is on the foot of other vectors. So, in these situations, you can say that these two vectors are independent of each other.

Fig.4 the component on orthogonal axis

In other words, when two vectors are perpendicular to each other, one does not influence the other.

The normal vector decomposition is thinkable at a point.

**2. The decomposition in Helmholtz Decomposition theorem**.

Generally you can’t think about the components of vectors which can’t be decomposed.

But the components that come up by the decomposition of Helmholtz Decomposition are not simply the projections of the flow vectors. They should be called “Roles”. They are a curl-free divergence component and a divergence-free curl component. Those components are given as “Roles” of a flow at a point in the set of adjacent flows.

These components are not thinkable at just a point. It is given in a region.

**3. It is not needed to get a curl distribution and divergence**

Helmholtz Decomposition is not needed to get distributions of curl and divergence.

You can calculate “a distribution of curl” and “a distribution of divergence” by calculating F and F respectively.

It is not necessary to use Helmholtz Decomposition theorem to get them. Therefor to be able to uniquely obtain these distributions can not be the proof of Helmholtz Decomposition.

You just be able to calculate these distributions, so you should not think that any flow is decomposed into two flows.

**4. The component which plays the both roles of curl and divergence**

Because a decomposition in Helmholtz Decomposition can be thought as roles of any flow at a point in the neighborhood, you should think about a component which plays both roles of curl and divergence.

If you don’t think of it, you preliminary use the Helmholtz Decomposition theorem without the proof.

If you would prove that there is no component which plays the both role of curl and divergence in any flows, you can say that Helmholtz Decomposition is correct.

At first you should think about a component which plays the both roles of curl and divergence.

Fig.5 If any flow has the component which play the both roles

In Fig.5, **F** is any flow which has both of curl and divergence. And **F**l shows a divergence component as a curl-free component, and **F**t shows a curl component as a divergence-free component, and **F**b shows a component which plays the both roles of curl and divergence.

These **F**l and **F**t are borroeing character from the article of ”Longitudinal and transverse fields” in Wikipedia.

The substitute character “l” means as shown in Fig.6,

Fig.6 the longitudinal component

and “t” means as shown inFig.7,

Fig.7 the transversal component

I might have mistook “**k**” what he/she wrote, but I beleave that “k” should be the gradient directin of potential.

These illustrations are made by myself, so if there is any fault in these illustrations, it is my fault.

As we know, they can calculate “a distribution of curl” and “a distribution of divergence” by calculate**F** and**F** .

Here, if they had mistook a flow had been decomposed into a curl-free flow and a divergence-free flow, they could calculate a velocity potential from the distribution of divergence, and a stream function from distribution of curl.

But actually they just calculated the distributions of divergence by (**F**l+**F**b）, not by （**F**l’）, and the distributions of curl by **(****F**t+**F**b）, not by （**F**t’）.

Therefore the composition **G** of these two flows does not match the original flow **F**.

If there is no component which plays the both roles of curl and divergence , Helmholtz Decomposition theorem is correct as showing in Fig.8.

Fig.8 If any flow has not the component which play the both roles

In fact, there is an example in real world. I may say that an electromagnetic wave has an electric field as a scalar potential φ, and a magnetic field as a vector potential **A** as shown in Fig.9. But I have to say that I don’t know exactly the electromagnetic wave.

Fig.9 is drawn from Wiki.

Fig.9 The electromagnetic wave

In the article of Wikipedia, an electromagnetic wave has an electric field as a scalar potential and a magnetic field as a vector potential, and they are perpendicular to each other.

In this case, the force given in an electric field and the force in a magnetic field are perpendicular to each other.

In electromagnetic wave case, you can confirm to exist a set of a scalar potential and a vector potential. But in fluid case, we can not confirm to exist these potential.

**5. How to verify the Helmholtz Decomposition**

As I said before, if Helmholtz Decomposition is right, there is not a component which plays the both roles of curl and divergence.

Then

1) The composition of a curl component and a divergence component is equal to the original flow(vector)

2) The two of components, a curl component flow and a divergence flow are perpendicular to each other.

And a curl component flow is directed along the line of stream functions, and a divergence flow is directed toward perpendicular to iso-velocity potential lines.

Therefore at any point, the lines of stream function and the line of iso-velocity potential line are on a parallel with each other.(Please see Fig.6 and Fig.7)

Stream function is expressed by the dimension of “vector potentials” in the horizontal plane as vector potential. The direction of the stream function as a “vector potential” stands up perpendicular to the plane.(Please see Fig.7)

**6. Current status**

I will show you some current status of “the velocity potential” and “stream function”.

The natural wind in our atmosphere generally blow horizontally, because the air is generally in the condition of hydrostatic equilibrium. So we can think the natural wind as a flow in a horizontal plane.

NOAA(National Oceanic and Atmospheric Administration) and JMA(Japan Meteorological Agency) etc. beleave Helmholtz Decomposition is correct, and they publish their data of “the velocity potential” and “the stream function”

**6.1) Do the compositions of “the divergent flow” and “curl flow” match the original winds?**

JMA had been publishing their data on the net until October 2011.

By using those data, I will show you how “the velocity potentials” and “the stream functions” are going. The following examples are about on 20 Jun 2011.

Fig.10 shows the distributions of the velocity potentials and divergent winds on the water vapor imagery.

Fig.10 distribution of the velocity potentials and divergent wind

And Fig.11 shows the distribution of stream functions and curl winds.

Fig.11 distribution of stream functions and curl winds

If Helmholtz Decomposition theorem is correct, the vector sum of these divergent winds and curl winds should match the original winds.

Fig.12 shows the original winds(black) and the vector sums of two kinds of components(purple).

Fig.12 The comparison between analyzed winds(black) and composed winds

I first drew analyzed wind with black arrows, and after that I drew composed wind with purple arrows. So if the composed winds perfectly match the analyzed winds, all of arrows should be purple.

You may think that analyzed winds( original winds) match the vector sum of divergent winds and curl winds. But I can see the difference between these two kinds of winds.

Fig.13 shows the differences between original winds and the vector sum of divergent winds and curl winds.

Fig.13 the differences between analyzed winds and

vector sum of two components

There are some differences about 5m/sec in some area.

Here, what do you think with Fig.13. Permissible? or Impermissible?

I do not know how to make these potentials, but I want to applaud the efforts of them. Good jobs!

**6.2) Are “the divergent wind” and “the curl wind” perpendicular to each other?**

If “Helmholtz Decomposition” is correct, “the divergent wind” and “the curl wind” should be perpendicular to each other.

But these two kinds of winds are not perpendicular to each other as we can see in Fig.10 and Fig.11.

And if “Helmholtz Decomposition” is correct, two kinds of isolines of “the velocity potential” and “the stream function” should be parallel to each other as we have seen in Fig.6 and Fig.7.

But these two kinds of iso-lines are not parallel to each other as we can see in Fig.13.

<by NOAA data>

We can also see the same consequences in NOAA data. Fig.14 and Fig.15 show the velocity potential and stream function respectively. They were published in NOAA home page(http://www.cpc.ncep.noaa.gov/products/hurricane/).

Fig.14 The example of Velocity potential distribution

(30days mean from 16 May to 14 June in 2014)

Fig.15 The example of Stream function distribution

(30days mean from 16 May to 14 June in 2014)

Fig.14 shows 30days means of 200hPa Velocity potential and divergent wind from 16 May to 14 June in 2014(top of them).

Fig.15 shows 30days mean of 200hPa stream function as the same term as Fig.14. The bottoms of them show anomalies of velocity potential, but now it is out of our argument.

To see how to go on the two potentials, I draw stream function on velocity potential in Fig.16. In Fig.16, blue lines show stream functions, and red lines show velocity potential. And red arrows show divergent wind.

Fig.16 is made by Fig.14 and Fig.15.

Fig.16 velocity potential on stream function at 200hPa as same term as Fig.14,15

The curl wind(=divergence-free wind) blow parallel to blue line(stream function). So we can see that divergent wind and curl wind are not orthogonal to each other.

**As far as has been hitherto seen, we can say Helmholtz Decomposition is not correct**.

**I think Helmholtz Decomposition has been preventing the progress of Meteorology**.