Rotation and the Coriolis effect Introduction to fluid

Rotation and the Coriolis effect Introduction to fluid mechanics September 1, 2018

Reference frames and forces Remember that a reference frame is just a convenient coordinate system. It can be fixed (inertial) or moving (non-inertial). Convention: real forces are positive and imaginary forces are negative. Fixed reference frame: Only real forces Acceleration Moving reference frames: real and imaginary forces Inertia Acceleration Forwa rd velocit y ion t a r cele locity c a l eta e of ve p i r t Cen o chang time t h due tion wit c dire And these will do for most classical mechanics problems. But another “force” becomes important sometimes when the reference frame itself rotates …Coriolis force gal u f i e tr Cen tia forc iner

Why does rotation matter? You may want to: • launch a satellite into orbit • predict where cyclones will form and where they will make landfall • track garbage and debris in the ocean • calculate how much fuel your private jet will need to fly from Guangzhou to Reykjavik - Coriolis effect is important.

Coriolis effect: an apparent deflection due to the rotation of a reference frame Looking from above Standing near the pig Bodies in motion in a counter-clockwise rotating reference frame: • experience a fictitious force due to inertia that wants to push them out of the frame – centrifugal force • experience a deflection to the right in that frame because of inertia – Coriolis force Angry Birds graphics copyright: http: //www. Vecteezy. com

What about cyclones again?

To see this mathematically … On a plane tangential to the earth’s surface Rad Veloc We look at some objects moving on earth at various latitudes Cor ioli e ob Ro In the north Equato r city s -gl n o i t ta Deflection to the right lo Ve On ity Lo ve cal rti ca l ius Ou to ft he glo be • Invoke the right hand rule: city o l iu Ve s Co (fu rio In the south th lly lis no e ou glo on glob t of Deflection cto be left th e: omthe e po Earth ne nt ) Ra d

To recap •

And when does the earth’s rotation not matter? •

Things to think about … 1. A rocket is launched towards the international space station. Can you tell which way it will be deflected? 2. A hungry humpback whale off the coast of New Zealand has smelled krill about 1 Km deep and is diving straight down for lunch. Should it course correct as it dives? 3. One million gallons of 2, 3 -Disgustyl toxilate has accidentally spilled into the Yangtze and the resulting plume outfell into the South China Sea. To avoid an ecological disaster, the Chinese government has asked the Zhang Jiang, which is mapping the Kuroshio current due west of Shangai, to measure the maximum concentration of the chemical when the plume hits the current. Should the captain of the Zhang Jiang decide to bear North or South to detect the plume? 4. [Hard question] You want to sell a medallion to Mr. Jin. When you show it to him, he will first weigh it and then buy it only if it is heavier than 10 grams. Mr. Jin will be traveling by train next week and has given you two options, either meet him on the westbound from Qingdao to Pingyao, or on the eastbound two days later. Which option will you pick? 5. [Hard question] Can you explain the factor 2 and the minus sign in the Coriolis acceleration? 6. [Very hard question] Since the Coriolis force is fictitious and only occurs within the rotating reference frame, why can’t we just use a fixed frame, i. e. , from space and do all our math?

Things to think about … answers 1. Using the right hand rule with the direction of the rocket launch vertically from the equator on the index finger, and the axis of the earth’s rotation perpendicular to it on the middle finger, the resulting force on the rocket is westward on the thumb. So the rocket will be deflected to the west. 2. As we are now off the coast of New Zealand, we are in the southern hemisphere. Here, the earth’s axis is pointing downwards, and so the rotation is clockwise. As the whale dives, using the right hand rule as before, we see that it will be deflected to the east. So yes, it should course correct if it doesn’t want to miss its lunch! 3. As before, using the right hand rule, the plume of toxic chemical is going to be deflected to the south from the mouth of the Yangtze. So, the Zhang Jiang should bear due south of Shanghai to detect the plume. 4. Using the right hand rule, as the train moves west, the force on the medallion will be towards the center of the earth, and will add to the gravitational force. So, it will weigh heavier. When the train moves east, the force on the medallion will be away from the center of the earth. So, it will weigh lighter. So, you absolutely have to meet Mr. Jin on the Westbound. The force we see is the local vertical component of the Coriolis force. This is also known as the Eötvös effect. It is typically small enough to ignore but can become important in high-precision work.

Things to think about … •