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１． ベクトル3重積を用いたヘルムホルツの分解定理の証明
３つのベクトルをA、B、Cとすると、A×（B×C）＝B（A・C）−C(A・B)と言う恒等式があります。このA×（B×C）をベクトル3重積と言います。この恒等式のAとBを微分演算子∇で、CをWで置き換えると、
∇×（∇×W）＝∇（∇・W）−∇２W
となり、右辺第2項を左辺に、左辺の式を右辺に移項すると、
∇２Ｗ＝∇（∇・W）−∇×（∇×W） ・・・・・・１）
が得られます。
近傍の状態に影響を受ける微分演算子を1点で定義できるベクトルと同一のものとして扱うわけには行けませんが、数式の展開方法としては、微分演算子の各偏微分をベクトルの成分と同じように扱って、Wが滑らかに連続しておれば厳密に、１）式は得ることができます。
一方、ヘルムホルツの分解定理とは、任意のベクトル（関数）が、渦の無い発散のみのベクトル（関数）と、発散の無い渦のみのベクトル（関数）に分解できると言う定理で、数式で書くと
V=∇φ＋∇×Ａ ・・・・・・・・・２）
の様に表現することができます。
ここで、１）式の左辺はWの各成分をそれぞれの軸方向に2階偏微分をしただけのベクトルですから、単にベクトルVとしても問題はありません。
さらに右辺の∇・WはWのその点における発散量ですからスカラー量です。∇を一つのベクトルとして考え、ベクトルの内積はスカラーになる事からもスカラーです。これをφとすると∇（∇・W）＝＞∇φとすることができます。
また、∇×Wはその点の回転ですから一つのベクトルであり、これをAとすると∇×（∇×W）＝＞∇×Aとすることができます。
かくして、１）式から２）式が導かれました。
すなわち、ヘルムホルツの分解定理が証明できたことになります。
２． 実は2つの式は同一では無い
先に書いた２）式を書き直すと
∇２W＝V=∇φ−∇×Ａ ・・・・・・・・・２‘）
ですが、１）式から2式に書き直すには、∇・W＝φ、∇×W＝A と言う条件が付いています。
つまり１）式と全く同一の式としては
∇２Ｗ＝V=∇φ−∇×A ただし、φ＝∇・W, A=∇×W ・・・・３）
と付帯条件を付けておかなければなりません。
すなわち、φとAは互いに独立ではなく、φが決まればAも、逆にAが決まればφも必然的に決定されて居る事を忘れてはなりません。
２）式は、いわゆるヘルムホルツの分解定理そのものです。何度も書きますが、任意のベクトル関数Vがスカラー関数φのdivergenceと、ベクトル関数Aのrotationで示されることを示して居ます。
一方、３）式は、滑らかでな連続ベクトル関数Wと言うものが存在すると、そのWを共通に基の関数とするスカラー関数φ＝∇・Wとベクトル関数A=∇×Wが存在して、そのφから渦の無いベクトル∇φが、Aから発散の無いベクトル−（∇×A）が求められ、その和で基のベクトル∇２Ｗ＝Vを現すことができることを示して居ます。
一見２）式と３）式は同じように見えるかも知れませんが、２）式ではφとAが互いに独立で、いろいろの組み合わせが考えられ、φが決まっても、Aは決まっていませんし、いろいろな分布が考えられると言っています。
一方、３）式は、φが決まればAも決まっており、例えば具体的な例として、電磁波が考えられます。と言うか、私の考えでは、３）式を具体的に具現化している物理量としては、電磁波のみがこの世に存在すると思います。
すべての電磁波（ガンマ線、エックス線、光、電波など）は、一つ一つ固有のWが存在し、∇・Wがその電磁波固有の電界を、∇×Wが固有の磁界となって、瞬間瞬間にそれらの組み合わせができあがっています。
より具体的に例を挙げますと、ABCテレビの電波には、ABCテレビの電界と磁界が交互に互いに影響を与えて振動しながら伝搬していますが、この組み合わせはABCテレビ固有の電界（∇・WABC）と磁界（∇×WABC）です。
一方、CNNテレビも固有の電波を持っていて、その電波は固有の電界（∇・WCNN）と磁界（∇×WCNN）を持っており、決してABCテレビ固有の電界（∇・WABC）とCNNテレビ固有の磁界（∇×WCNN）とがペアを組むことはありません。３）式では、このことを但し書きで示して居ます。
２）では、但し書きがありませんから、任意の∇φと∇×Aが考えられると言っています。ABCテレビ固有の電界（∇・WABC）とCNNテレビ固有の磁界（∇×WCNN）とがペアを組むことも考えられ、そのような“任意のベクトル”が存在すると言っています。
すなわち、ヘルムホルツの分解定理を信じる人たちは、光の持つ電界と放送局が出す電波の磁界さえペアを組んで電磁波として存在するとも言っていることになります。
単一の光や電波などの電磁波が２）式を満足していると言う事実だけで、執拗に「ヘルムホルツの分解定理」を擁護する電磁気学の関係者が居ます。
いったい、電磁気学にこの「ヘルムホルツの分解定理」は必要なのでしょうか。彼らは、何を分解しているのでしょう。
実際に分解定理が必要なのは、気象学などの流体力学の関係者です。その気象学者たちのやっている「分解」の実態を見ると、理論的には、あきれることをやっています。
2次元の平面上（２００ｈPaや８５０ｈPa)の風を、同じ平面内にφとその面に垂直なAの分布を考えて、それらから渦なし発散風と発散無し渦成分風を、同じ面内に導き出しています。
気象学に対しては、早く「ヘルムホルツの分解定理」から脱却することを、電磁気学の世界の人には基のベクトルWが何かを研究することを願って居ます。
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Helmholtz Decomp
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ヘルムホルツの分解定理は、「任意の流れ」が「発散の無い渦だけの流れ」と「渦の無い発散だけの流れ」に分解できると言う定理ですが、この定理に関して考え方の違う二つのグループがあることが分かりました。一つ目のグループは、分かれた二つの流れが直交して居るはずとしている理論家グループであり、もう一つのグループは、直交して居なくてあたりまえとするグループです。 |
８. Helmholtz Decomposition is wrong (Chapter 8)
Chapter 8-1 Meteorological disturbance
Generally, a meteorological disturbance consists of a front area and a back area.
In the front area, cumulonimbuses are developing up, and the airs which are tossed off in
the upper layer are starting to ageostrophic motin. And in the back area, the airs have come to the end of the ageostrophic motion of the western
neighbor trough.
Fig8.2 shows a meteorological disturbance from 8th to 9th of Octorber 2010.
A developing low pressure system was passing through Japan on 9th of Octorber 2010. In the
front part of that system, we can see the flows which have considerable ageostrophic
components faceing to the left hand side of the direction of analyzed wind. They force the airs to accelerate, and tend to make divergent area in upper layer as I have said in Chapter 7. There, we can see developing clouds(white area on the water vapor image). And on the way to the area of downstream, their ageostrophic components turn around clockwise, and come to the end of ageostrophic motion. Their ageostrophic component come to
face to the right hand side of the analyzed wind, and tend to decelerate the speed, and make
As I have just described, a ageostrophic component can explain a divergent area and a convergent area. It is shown as black area on the water vapor image. convergent area with a meteorological disturbance in the westerlies zone. Violent heat in August 2010 These meteorological disturbance usually disappear on the monthly weather map. But, if that disturbance stays or moves very slowly, sometimes it appears even on monthly weather map. We had violent heat in summer in 2010. It was for the first time in decades. Fig8.3 shows the changes of monthly mean temperature in August from year to year in West Japsn.
Fig8.4 shows the mean height and the distribution of ageostrophic component of 200hPa.
We can see the statianally trough at around 115 degrees east longitude. And there was an area where the ageostrophic components was facing to the left hand sides of
analyzed winds(that is in first or forth quadrant shown in Chapter 7) on the east of the
The downstream of that flows were deflected clockwise, and at the east of Japan ageostrophictrough. So, the flows would have come to diverge. It happened around Korea Peninsula.
component came to face to the right hand sides of analyzed wind (that is in second or third
Fig8.5 shows the analyzed wind and the distribution of divergence on thequadrant).So, the flows would have come to converge.It happened near Japan.
In comparison between Fig8.4 and Fig8.5, the area where the ageostrophic component belonged
to the first or forth quadrant conformed to the divergent area around Korea Peninsula.
OLR value is small in these area. And the area where the ageostrophic component belonged to
the second or third quadrant conformed to the convergent area near Japan.
So, we can say that = the cause of violent heat in summer 2010 is the statinally trough at around 115degrees east longitude. = Meanwhile, let us see if we can find the cause of violent heat in summer 2010 by using the velocity potential and divergent wind.
Fig8.6 shows that there is a center of divergence near Philippines islands, and some divergent winds go out northward, and converge at the east of, or near Japan. So, we can find that the cause of violent heat in summer 2010 comes from the center of velocity potential near the Pilippines islands. Actually, which is right? Is the cause of violent heat stationary trough at 115degrees east
longitude, or from the center of velocity potential near Philippines islands?
We need to do more research of another summers.(to be continued |
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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.
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.
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.
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
Fig7.8 shows the distribution of divergence on the water vapor image.cloud images.
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.. |
7. On the forces acting on the air parcel in the actual ageostrophic motionWe have been thinking an air parcel on many kinds of hypothetical condition. I wanted you to know what the theoretical ageostrophic motion is. Fig.7.1 shows a sample of 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.
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.
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 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.
Coriolis forceAD with actual wind velocity. So, the total force is given a composition of AG
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
and AD. If you draw a parallelogramADHG, it is given as a diagonalAH.
Then, ∠ＢＡＥ＝∠ＣＡＤ
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.
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
continueHere, 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 |
