Interactive Blackbodies - Henry: 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:

It is a fictitious force and does not appear in inertial reference frame.
It arises due to inertia acting on a rotating reference frame.
It is inversely proportional to the speed of the moving object and proportional to the speed of rotation.
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.
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。