1403 - Coriolis Effect(2) 地轉偏向力(2)

We have examined the nature of Coriolis effect in the last article. In a nutshell, it is all about the manipulation of reference frames and inertia. In writing the last article, I have been struggled in introduction of physics concept - it is not easy to understand in short period of time, and become challenges to the readers. But without these concepts, meteorology is less beautiful.

Sometimes, to learn something more requires "to dive deep into the water, go back to the water surface and re-immerse your head again." Without this process, what we have encountered will easily lapse in the passage of time. So, don't hesitate to get back and forth in this process.

It is Coriolis force to deflect polar air masses to the right. Thus, from the perspective of people in North Hemisphere, winter monsoons comes from the North-east direction.

However, winds in the South Hemisphere are not deflected to the right, but to the left. Why would this happen? Again, we have to re-examine the nature of Coriolis effect by the roundabout concepts. In the last article, we have assumed that the roundabout rotates anti-clockwise. In fact, it can be rotating clockwise. Because Henry is sitting on the origin, from his perspective, everything is moving just as in the Inertial Reference Frame. So, the ball will move in straight line.

From the perspective of Thomas, he will see the ball bending leftward.

Therefore, the difference between rotating anti-clockwise and clockwise is just the direction of bending. If the roundabout rotates anti-clockwise, the track bends to the right; if the roundabout rotates clockwise, the track bends to the left.

So, we can replace the roundabout analogy by the reality now: Henry is sitting at the South Pole, Thomas is sitting on the Equator and the ball is substituted by many tiny air particles. 

If we look from another angle, we will know winds in South Hemisphere are blown from Southeast to Northwest.

We may combine the case in North Hemisphere and South Hemisphere into one global picture. We will know that Thomas, in both situation, is in fact sitting on the same place, no matter whether it is viewed from the North Hemisphere or from the South Hemisphere.

The only difference is the direction of air movement: In North Hemisphere, polar air masses goes south; In South Hemisphere, polar air masses goes north. Therefore, the wind direction is Northeast in North Hemisphere and Southeast in South Hemisphere.

Then, you may now know understand the underlying concepts on why our Northeast Monsoon is from Northeast. 

Up to now, we know how would Coriolis Force as a fictitious force affect polar winds with the assumption that high pressure area is constantly stayed at the poles. In reality, high pressure areas can be developed in other places as depicted below.

Think intuitively: winds from other areas should also follow the rules applied to the winds from the poles, taking a re-curving track because even in other places of the Earth, the air masses cannot escape the Coriolis effect as a result of the rotation of the Earth. Thus, the route of the winds will also be twisted. 

However, the degree of bending is different. In polar areas, Coriolis effect is much larger than that in Equatorial areas. 

To explain why this would happen, we need to know the fundamental concept in circular rotation. A circle with a longer radius has a longer circumference.When we move on a circular track, the closer our position to the centre, the shorter the distance we need to travel.

Using the above diagram, let's assume the plane rotates at 45 degrees in 1 minute. If so, that means the objects rotate near the centre will have a lower speed and objects away from the centre will have a higher speed.

In this case, suppose Henry is sitting at Point A, and would like to throw the ball to Thomas. Without any rotation, the ball moves in straight-line and reaches Thomas. Just like inertial reference frame, everything can be explained by the laws of physics properly without incorporating "fictitious force". Hence, Coriolis force is zero.

In the case when Henry is sitting at the centre of the roundabout, he does not move at all. However, when Henry is sitting at the "Point A", he will possess an initial speed when the roundabout rotates. In fact, apart from Henry himself, everything away from the centre of the roundabout will have this speed because their positions are changed after 1 minute.

This initial speed is called "momentum". Suppose Henry has a ball in hands and throws the ball out of the window of the train (Warning: this behaviour is absolutely discouraged). Suppose the train is moving at a speed of 100 km/h. Would the ball fall vertically (dotted red line track in the diagram below)?

Should not be. Because the ball is originally in Henry's hands, and Henry is sitting on the train which is travelling at 100 km/h. From a third observer's perspective, the ball is in fact travelling at 100 km/h. Even the ball leaves Henry's hands, the law of inertia preserves the speed. Therefore, the ball should have a forward momentum. Air resistance will reduce the forwarding speed gradually and ball slows down. 

In simplicity, the forces adding on the ball are (1) momentum (horizontal); (2) gravity (downward) and (3) air resistance (counter-horizontal).

Back to the roundabout case. Since Henry is affected by rotation, he is no longer situated in the inertial reference frame now. Similar to Thomas, when Henry pushes the ball to Thomas, the trajectory of the ball is not a straight line, but bending to the right at a certain degree. Moreover, because Henry is moving at certain speed, the ball on his hands have a momentum. 

There are 2 forces adding on the ball: 

(1): Momentum because of the background rotation (Like the train, it provides background velocity to Henry and the ball) and

(2): Henry pushes the ball to Thomas

Apart from the above forces, Coriolis force as a fictitious force also comes into the picture because Henry and Thomas are on rotating reference frame. Therefore, if we add the forces affecting the ball step by step, we can derive a slightly re-curving path.

From the perspective of Thomas, the major difference between the previous case (Henry is sitting at the centre) and this case (Henry is sitting at Point A) is the initial momentum. The initial momentum has, to a certain degree, offset the effect of Coriolis force because they act on the ball in opposite direction. We can subtract the initial momentum from the Coriolis force. So, the ball pushed from the centre has a larger net Coriolis Force, while the ball pushed from Point A has a smaller net Coriolis Force. 

If Henry is sitting more closer to the edge of roundabout, he and his ball will encounter even higher initial momentum, which will further offset the effect on Coriolis Force. As a result, the ball is bending only slightly.

If we visualise the above roundabout case on the Earth, we could have the following conclusion: Coriolis Force is less significant at Equatorial areas because the rotation of the Earth makes objects moving faster and producing a larger momentum to counteract with Coriolis force. 

Now, we know the following major properties of Coriolis force:

1. It is a fictitious force and does not appear in inertial reference frame.
2. It arises due to inertia acting on a rotating reference frame.
3. It is inversely proportional to the speed of the moving object and proportional to the speed of rotation.
4. It renders objects bending to the right when the rotation is in anti-clockwise direction; and bending to the left when rotation is in clockwise direction.
5. It can be offset by momentum of the object when the object is affected by background rotation.

Applying all these fundamental concept to the Meteorology, we can explain and describe the phenomenon such as the direction of wind and the track of a cyclonic system (e.g. Typhoon). However, Coriolis effect is not the only cause affecting wind direction. Back to the first picture you encountered in this channel, we have took something for granted and do not explain at all: Why wind flows from a high pressure area to a low pressure area? We will discuss next time.

1403 - 地轉偏向力(2)

上一回說到科氏力的由來。簡單來說, 是看事物角度("參考系")以及慣性互相摻合下的一種假想力, 以便解釋在地表風的物理特性。因為科氏力是物理的概念, 有可能對從未接觸過這些概念的讀者們造成困難。但是, 假如沒有背後的物理意義, 氣象學也不會如此引人入勝。

在學習的進程中, 少不免在理論和現實, 在簡單的框架和複雜的假設下不斷地進進出出, 才可以了解到一個概念的奧妙之處。為了讓讀者們對一個概念有更加深刻的了解, 請不要介意我把例子重複。

在科氏力的影響下, 在北半球高氣壓吹出的風將會向右偏移, 所以, 在這個簡化了的框架下,北半球將會在冬季受到東北風影響。

但是, 南半球的情況則完全相反, 風在南半球全部都是向左偏。到底是甚麼原因令到在南半球的情況下給出一個與北半球完全相反的結果呢? 要解釋這個問題, 我們要重新回到氹氹轉的例子。在上一章氹氹轉是被假設以逆時針方向旋轉的。這次, 氹氹轉以順時針的方式旋轉,。在Henry 的角度(慣性參考系)下, 球的路徑仍然是直線一條。

基於Thomas 是處於旋轉參考系下, 他會看到球的路徑是向左偏的。

現在, 我們不難發現, 順時針和逆時針的轉動帶來唯一的區別是: 當氹氹轉是以逆時針旋轉時, Thomas會看到球是向右偏; 當氹氹轉是以順時針旋轉時, Thomas會看到球是向左偏。

再把氹氹轉這個例子套回現實: Henry 坐在南極, Thomas 坐在赤道, 接著以空氣粒子代替球。

以另外一個角度去看, 可以看到南半球的風是由東南方吹向西北方。

假如把南半球和北半球的情況聯結起來, 便會發現不論在北半球,抑或在南半球, 他也是坐在赤道上的同一位置。(不信的話可以試試把北半球的圖倒轉來看)

在這個簡化了的情況下, 南北半球的唯一分別在於: 南半球的風是由南向北, 而北半球的風是由北向南. (囧, 冷笑話!?)

我們現在能夠明白為何香港冬季會吹東北風了。 (註: 科氏力只是其中一種因素, 現實上還有其它不同的因素影響季節風的風向。)

以上所考慮的情況, 都是以南北極作為高壓中心。現實上, 其它地方也可以產生出高壓中心的。那麼, 如果把這個局限消除,  結果會是怎樣呢? 科氏力會不會因撤銷了這個限制後消失呢?

可以想一想, 因為地球是一個旋轉的球體, 而科氏力則是旋轉的平面上出現的假想力。 所以, 不論在極地上, 抑或在其它地方的氣團, 都不能夠 "逃離科氏力的魔掌"。在極地和其它地方的分別, 是科氏力的"力度"。

在極地上, 科氏力的作用最大; 在赤道上, 科氏力的作用最少。

要解釋為何會出現上述情況, 或許需要以數學的角度去理解旋轉運動的基本概念。當一個圓形的半徑愈長, 它的圓周愈長, 其關係可以以圓周公式來表示 (對不起, 用了一條數學公式)。

上圖是描述物件在一分鐘內的旋轉運動。 在圓形中心附近的物體只是移動了一個極短的距離, 反之,在圓周的物體則走得最遠。因為"速度"是表達"物件在有限的時間內所移動的距離", 所以, 圓周附近的物件移動得比圓心附近的物件快。

我們可以再次運用"氹氹轉"的例子來簡單地想像風是從極地以外的地方吹出。假設Henry 是坐在下圖中的"A點"上, 並想把球拋向Thomas。當氹氹轉靜止的時候, 球能夠輕易地到達Thomas 的手中。在這個情況下, 所有物件的運動能夠直接地以牛頓運動定律描述, 不需要加入任何假想力的元素, 因此科氏力不存在。

Henry 坐在中心點時不受到任何背景旋轉力所影響, 一直能保持靜止。但只要他離開了中心點, 那怕只是在紅線這麼一丁點的距離, 他就會受到背景旋轉所影響, 位置在一分鐘後會改變。所以, 當氹氹轉處於旋轉狀態時,  除了中心點以外, 所有物件的位置都會改變, 並出現一種初始力。這是圓周運動的特性。

這種初始力被稱為"動量"。我們可以利用列車的例子來了解"動量"這種物理特性。 假設Henry手中有一個球, 並把球拋出窗外 (本人絕對不鼓勵這類構成自身或他人危險的動作)。假如列車現正以時速一百公里的速度行駛, 球會以下圖紅色虛線的方式垂直地向下墜嗎?

因為球在Henry 的手中, 而Henry 是坐在一列以時速一百公里開動的列車上, 可以想像, 列車把它的動能"帶"給了球, 令球具備時速百公里的初始動能。當球在Henry手中甩出, 那個球會在慣性影響下, 仍然以時速一百公里向前走。空氣阻力會使到那個球逐漸減速, 並轉為向下方(受引力影響) 移動。

總括來說, 有三種力作用在球上, 分別是: (1). 動量 (向前) ; (2) 地心吸力 (向下) ; (3) 空氣阻力 (向後)

回到氹氹轉的例子。因為Henry 不在處於圓心, 他的角度已轉為"旋轉參考系"。跟Thomas 一樣, 他看到球不再以直線運動, 而是會向右偏。更重要的是, Henry的球在氹氹轉旋轉的影響下, 具備了動量。 

基本上, 有2種力作用在球上:

(1). 因旋轉所產生的動量

(2). Henry的推力

除上述的力外, 因參考系的轉變下, 增加了科氏力(假想力)的元素。要是我們把每種力逐項加上, 最終會得出一條向右彎的曲線。

從Thomas的角度上看, 跟前一個情況(當Henry坐在氹氹轉正中的不同之處是額外的動量。因為動量受慣性影響下令球繼續向前走, 抵消科氏力導致球轉向的作用。假如我們把動量從科氏力中去除, 我們便會發現由圓心拋出的球具有更大的科氏力, 而從A點拋出的球的科氏力便會較小。

 當Henry 坐得愈接近邊緣的位置, 球所擁有的動量便會更大, 進而抵消較多的科氏力。

如果把氹氹轉的情況應用在地球表面上, 我們便會得出下面的結論: 科氏力在極地比赤道較為明顯。(這是因為赤道上的物體具有較多的動量)

現在, 我們大概知道科氏力的五大特性:

(1) 它是一個假想力, 並不會在慣性參考系上出現。

(2) 它的出現是由在旋轉參考系上的慣性驅使。

(3) 它跟物件(球)的移動速度成反比, 與平面的旋轉速度成正比。

(4) 當平面以逆時針方向旋轉, 它的作用下會令物件右偏; 當平面以順時針方向旋轉, 它則會令物件左偏。

(5) 它可被物件的初始動量所抵消。

29. 如果把這些概念應用在氣象學上, 我們可以解釋風的特性 (如氣旋 / 颱風) 。但是, 科氏力並不是影響風的唯一因素。從1401 開始, 我們就不言而喻地把氣壓的概念假設了: 為何風是由高壓往低壓走呢? 下回再說。 拜 ~

1402 - Coriolis Effect 地轉偏向力

1.  The previous article introduces that the rotation of the Earth can influence the direction of wind and affect the propagation of polar air masses in a simple setting. Air masses originating from the North Pole will first deflect to the right. To know more about what is Coriolis Force, the physical interpretation of Coriolis effect is of the fundamental importance. Here, we shall start looking at this without any formulaic expression of Coriolis effect. (which can be a headache to those who dislike numbers and symbols)

2. The concept of Coriolis effect adheres to the concept of "reference frames" in physics. Reference frame can be easily understood by the following analogy. 

3. Suppose there is a platform in the train station and suppose there is a train moving westward at 100 km/h passing through the platform. Now, consider Thomas is standing on the platform and Henry is standing on the train.

4. From the perspective of Thomas, Thomas himself is standing still and Henry is moving westward at 100 km/h, which is in the same direction and at the same speed of the train. 

5. From the perspective of Henry, Henry himself is standing still on the train. Thomas is moving eastward at 100 km/h. In fact, the platform is also moving eastward at 100 km/h.

6. Therefore, the perspective of Thomas and perspective of Henry form distinctive frames of reference. The reference frame of Thomas is to "view" everything with the reference to the platform. The reference frame of Henry is to "view" everything with the reference to the train. (i.e. the platform and the train are two separate reference frames.) It is correct to say either one of them is moving or at rest, depending on which reference frames you are referring to.

7. The reference frame which possesses the following characteristics is called Inertial Reference Frame.

• No change in velocity (i.e. no acceleration)
• Moving in a straight line

So, the analogy above demonstrates the characteristics of Initial Reference Frames. By definition, other reference frames without one of the above 2 characteristics are called Non-inertial Reference Frame.

8. The adjective "inertial" comes the word "inertia".

9. The Newton's first law has defined Inertia in this sense:

"Without exerting additional force, the object, whether it is at rest or moving forward in a straight line, is resisting to the change of its velocity."

10. This is the reason why a vehicle will not suddenly move since the mass of the object prevents any change of velocity. On the other hand, objects which is at first moving forward will not decelerate suddenly without external forces (which implies a change in velocity). At the time when Newton comes up with this idea, it is difficult to accept because at the first glance, objects on the Earth can stop spontaneously. For example, balls will stop when they traveled distant enough. But, in fact, deceleration of objects are due to friction on the surface of ground. Newton's Laws of Motion are always interpreted separately with fiction - deceleration will not occur unless an addition force has affected the movement of object.

11. In the inertial reference frame, the Newton's laws can be easily fitted without any adjustment for the purpose of describing the motion of the objects. Every motion and activity can be described by the Cartesian Coordinate System. Here are several simple examples:

From Henry's perspective,

Thomas is travelling in opposite direction. 

A boat is surfing slowly. 

Another train is standing still.

A balloon came closer to Henry.

Without additional force, all of them are moving in the same speed at the same direction forever.

12. The Earth is a Rotating Reference Frame (a type of non-inertial reference frame) as a result of the Earth's rotation at an angle of 23 1/2 degrees. 

13. When the Newton's laws of motion applies to the area of non-inertial reference frame (such as rotating reference frame), fictitious force (also known as Inertial Force) must be incorporated so that the laws will give a valid prediction and explanation on the motion of objects, creating the same prediction of motion as they are used to predicts motion of objects in inertial reference frame.

14.  Therefore, the fictitious forces are something extra for the ease of describing motions just as what is done in inertial reference frame. These forces also render the Newton's Laws of Motion to be useful in describing things in Rotating Reference Frame, without being distorted by the circular motion and background rotation in a rotating reference frame.

15. Suppose Henry pushed a ball from the origin of the circle towards the circumference on a large roundabout. Suppose Thomas is sitting on the edge of the roundabout and would like to take the ball. Suppose there is no rotation on the roundabout. Now, Henry throws the ball out from the centre of the roundabout towards Thomas. Without any additional force, the ball follows a straight line motion due to inertia. From the perspective of Henry and Thomas, they are all agreed on the straight line motion. Therefore, no modification by adding fictitious force is needed to describe the motion of the ball. 

16. Now, suppose someone has turned the roundabout anticlockwise barely. Henry will NOT move since he is sitting on the origin of the roundabout. However, Thomas who is sitting on the circumference, started to rotate. Let's assume that in 1 minute, the roundabout turned anti-clockwise by 45 degrees.

17. From the perspective of Henry, he finds that the ball moves in a straight-line. To him, the ball is just as moving in the inertial reference frame and without incorporating fictitious forces, the inertia of the ball can be described by the Newton's Law of Motion.  He also find that the ball will miss Thomas. This is because of the ball has inertia to move (without change its direction). 

18. Suppose now we take the perspective from Thomas. From his point of view, the ball that Henry throws out is subject to background rotation: at first, it is pointed towards Thomas. With the effect of inertia, the ball keeps moving towards the circumference. But due to rotation, the ball starts to take a re-curving direction and does not point towards Thomas. In the above case, we see the ball reaches the circumference on the right to the Thomas. In this case, we get the same result, but the trajectory of the ball is a curvature, from Thomas's point of view.  

19. We can also interpret the re-curving route of the ball like this: Due to inertia of the ball, the ball will continue moving in a straight line in a rotating roundabout. When we stare on the origin, the ball is moving in a straight-line motion. When we are also subject to the background rotation, we will see the ball taking re-curving trajectory. The re-curving trajectory cannot be explained by Laws of Motion. If we want this re-curving trajectory possible to be explained by the Laws of Motion in a simple way, what can we do?  What can we do to explain the motion of objects in this frame under Cartesian coordinate, just as in the Inertial Reference Frame?

20. To describe the motion of objects and render laws of physics to be usable in a rotating reference frame just as in the inertial reference frame, a fictitious force is added. To simplify the explanation why it would be in that case, we look at the answer showing the vector on Coriolis force first:

21. To dig out this fundamental concept of force, let's imagine a ball on the plane. Suppose the ball is originally moving forward in straight line. Suddenly, a force to the left is adding on it. Since there is inertia, it will not abruptly turn to the left immediately. Rather, the force will slightly and continuously modify the track. Eventually, if we wait long enough, the original forwarding inertia will become negligible and the ball will turn left.

22. With reference to the case 2b, the ball is originally pointing towards Thomas straight. However, by observation, the ball is deflected to the right, contradicting to the law of inertia (Newton's First Law). Since the law of inertia cannot explain deflection in a rotating reference frame, we add a fictitious force to rescue the law of inertia, so that it can still be used in a rotating reference frame. 

23.  According to the previous rule regarding modification of track by an additional force, the ball starts deflecting to the right because of more and more "force" has been continuously added to drive the ball turning right. We give this force a name, called Coriolis Force.

24. In the above example, we assumes the speed of rotation is 45 degrees in 1 minute. What happens if the roundabout rotates faster, e.g. 90 degrees in 1 minute?

25. Similarly, we can deduce the track when it rotates 180 degrees in 1 minute.

26. We can compare the result in case 1, 2 and 3, from the perspective of Thomas.

27. If the ball is moving really fast (think it as fast as sound or light), we can intuitively deduce that the ball is almost not subject to the deflection and hence Coriolis Force. Therefore, Coriolis Force is proportional to the speed of rotation and inversely proportional to the speed of the ball.

28. Accordingly, we can also apply this fundamental concept to understand the direction of wind. Just imagine Henry is standing on the North Pole and Thomas sitting on the Equator. We use infinitesimal small size of air particles to substitute the ball. Deflection will still occur and the track of those tiny particles (as known as air masses) will be affected by Coriolis Force as well. 

29. If we use another angle to view the Earth, we can find the reason why in Northern Hemisphere, the polar air masses come from the North-east direction but not straightly from the North direction.

30. How about in South Hemisphere? In the South Hemisphere, winds are deflected to the left instead of to the right. Why this will happen? 

 1402 - 地轉偏向力

"地轉偏向力"好像只是"中文"才有。英文我只聽過Coriolis Force (科里奧利力)。

1. 上回說到地球自轉會影響到風向和極地氣團的流動。以最簡化的方式來說: 氣團從極地吹出會首先往右偏向。假若要了解科氏力的基本成因和由來, 我們不得不利用物理的概念。(這裡我不會把令人拒氣象及物理於門外的一大堆數學公式擺出一副說教的姿態來!) 

2. 科氏力必然跟物理上"參考系"的概念有關。以下的例子(不是我發明的) 可以很簡單地描述"參考系"這個概念。

3. 現在假設有一個地鐵月台。Henry 坐在的列車現正以每小時100公里的速度向西移動。Thomas 現在正站在月台上。

4. 從Thomas 的角度而言, 他會發現自己是站著不動 (囧, 不難發現吧!), Henry 則是以每小時100公里的速度向西移動 (即使Henry 坐著列車上不動, 他在Thomas的眼中是正在以高達每小時100公里移動的傢伙!), 跟列車的速度相同。  

5. 從Henry 的角度而言, 他會發現自己是坐著不動, 而Thomas 和月台則以每小時100公里的速度向東移動。對於Henry 來說, 那個月台彷彿能夠就像列車一樣高速地逆向移動。

6. 這裹, 我特別地把Thomas 和 Henry 的角度區分出來。在物理上, 這是必要的, 因為Thomas 和 Henry 他們二人有著不相同的參考系。Thomas 是以月台作為其"參考系", 他是以"月台"的角度去看所有東西。而Henry 是以列車作為其"參考系", 所以會把列車作為中心去了解其他事物。(即使你認為月台不能移動, 現在我認為月台是可以移動的, 你不能罵我錯)。 所以你也可以說Thomas 是移動或停留, 視乎你是從那個"參考系" 去看事物。

7. 我們把有以下特性的參考系, 統一地稱為"慣怔參考系"

• 勻速 (速度沒有改變)
• 直線移動

所以, 以上的例子是描述在慣性參考系內物件的運動。慣性參考系必須同時擁有這兩種特性。假如缺少了其中一種特性, 這個參考系便稱為"非慣性參考系"。

8. 牛頓運動第一定律把慣性定義為:

" 當沒有任何外力施加的情況下, 一個物件會避免改變其自身的運動速度, 無論在該物件處於靜止或是運動的狀態下。" 

9. 所以, 一輛處於靜止狀態下的車不會無緣無故往前進。另一方面, 一輛在公路上行走的貨車也不會無緣無故突然減速。(對, 就連減速也不會, 更遑論把它停下來)。當年牛頓指出這個匪夷所思的概念曾遭到很大的衝擊: 因為地面上移動的物件會隨時間的流逝而減速, 最後會自然地停下來。在那個年代, 是很難把磨擦力和物件運動分開來看: (1) 物件減速是因為磨擦, 而不是其慣性消失。 (2) 必定要受外力影響才可使物件減速, 慣性令物件不會貿然減速。

10. 在慣性參考系下, 所有運動定律都很容易便能夠在無須任何修正下, 直接地解釋物件的運動。故此, 物理學家引用笛卡兒坐標系統來描述一切物件的運動, 如以下例子:

從Henry 的角度來看

Thomas 現在正逆向移動

有一艘船在緩慢地行駛

另一列地鐵是靜止不動

有個氣球正向Henry 飄來

這些物件假如沒有外力 (外在環境所給予的力), 它們必定會以勻速直線的方式前進, 直到永遠。

11. 地球是一個"旋轉參考系" (是"非慣性參考系"的一種)。這是因為地球圍繞傾斜23.5度的地軸旋轉。

12. 當需要把牛頓運動定律應用到非慣性參考系的時候 (如"旋轉參考系"), 我們必須假設一種假想力, 使運動定律仍然能夠適用於解釋和預測物件移動, 就正如把這些定律應用到慣性參考系的時候一樣。

13. 所以, 我們可以把假想力視為一種額外的力, 以更方便地描述非慣性系內物件的運動方式。這種力也對三個牛頓運動定律提供了補救, 使這些定律可以適用於非慣性系上。

14. 現在假設Henry 坐在一個巨型的氹氹轉的中心點 (小時候玩過吧), 向邊界推出一個球。現在假設Thomas 正坐在氹氹轉的邊界上, 打算接著Henry 推過來的球。假設現在氹氹轉本身是靜止, 非轉動的。 在無外力的情況下, 由於慣性的關係, 球必定是以勻速直線運動的。不論從Henry 或是 Thomas 的角度, 他們都會一律認同球是以直線的方式移動。這是球在慣性參考系移動下的描述, 不需加入假想力的元素去描述該球的運動。

15. 假設現在有一陌生人讓氹氹轉開始以逆時針的方式轉動 (每分鐘移動45度)。由於Henry 坐著中心點, 無論轉動速度有多大, 他都不會移動的。另一方面, 由於Thomas 坐在邊界上, 他會感到自己的位置改變了 (囧, 也不難發現吧!)。

16. 從Henry 的角度, 他會發覺球仍然以直線的方式運動, 就如在慣性參考系一樣, 所以他並不需要在解釋物體的慣性運動時加入假想力的元素。 除此以外, 他還會發覺Thomas未能把球接到手, 因為他的位置改變了。

17. 從Thomas 的角度而言, 他會發覺那個球會受到氹氹轉的旋轉影響: 起初, 球是向Thomas 的方向移動。由於慣性的關係, 該球會繼續移動直至氹氹轉的邊界。但是, 因為旋轉的關係, Thomas 會發現那個球開始轉向, 並離自己愈來愈遠。於是, 對於Thomas 而言, 那個球的軌跡是一條曲線。

18. 我們也可以把這個現象如此演繹: 由於慣性的關係, 那個球會以直線的方式在旋轉的"氹氹轉"上移動。假如我們凝視著中心點, 我們會發覺該球的確以直線的方式移動, 但是, 如果我們站在Thomas 身後, 跟他一起旋轉, 我們便會發覺那個球是以曲線移動。

19. 牛頓運動定律並不能解釋和描述曲線運動 (因為這是違反慣性的)。那麼, 究竟還有甚麼方法簡單地解釋這種現象, 以及精準地在笛卡兒坐標系統描述這種曲線運動? 

20. 為了讓運動力律能夠適用於旋轉參考系上, 我們需要添加一種假想力。(下圖中的箭咀代表力的方向)

21. 我們可以透過以下的例子去深入了解力的特性。假設有一個球體, 球體現在向前移動。在沒有外力的情況下, 該球體在慣性下將會永久地以勻速直線的方式移動。假設有一股引力令球開始朝向右方移動, 那個球絕不會突然立即改變方向, 而是慢慢減速, 逐漸地增加其向右的份量才是。當一段時間過去後, 球才會完全向右方移動。

22. 現在再運用這個概念去重新理解第2個情況中球的軌跡:

23. 我們給予這種力一個名子, 名為科氏奧利力。

24. 再假如上述例子中的氹氹轉不是每分鐘轉45度, 而是轉90度, 那麼球的軌跡又如何?

25. 同樣, 我們也可用直覺推出每分鍾轉180度時的情形。

26. 再把三種情形比較一下, 我們可以得出以下結果。

27. 假如那個球的移動速度很快, 可以想像到在旋轉的平面上, 它幾乎不受這種假想力所影響。所以, 科氏力與物件的運動速度成反比, 與背景旋轉的速度成正比。

28. 同樣地可以把這個概念應用到風向上。與之前的例子一樣, 只要假設Henry 是坐在北極的極點, 而Thomas 坐在赤道上, 並且把球幻想為一些細小的空氣粒子即可導出結論。

29. 如果用第二個角度去觀看, 我們就可以導出為何北極吹來的風都不是從正北方吹來, 而是從東北方向吹來。

30. 那麼, 南半球的情況又是怎樣? 下回再續。