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Posts Tagged ‘Coriolis effect’

Southern Storm

Sorry for yesterday’s error. Of course a storm in the southern hemisphere is like a letter “p”; both 6 and 9 are left-turning spirals. But a storm in the southern hemisphere is a right spiral, a backwards 6, with the tail upwards towards the equator.

Here is a hurricane off the coast of Brazil. The image comes from an astronaut in 2004. A hurricane in this location is unprecedented, so it has no name. It’s just The Hurricane off the coast of Brazil. You can see the letter “p” on its side, or perhaps a lower case “e”. In any case, a left spiral.

Hurricane off Brazil

Hurricane off Brazil

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Just a note, before coming to the end of Corey’s story: Some children are content to learn to speak, but others make up words very freely. Chowmega is the name of a wood that belongs to family lore; you needn’t look for it anywhere else. But as Pooh might have said, it wanted to come into the story, so I let it.

#12  Gaspard’s Apple

It was a perfect, friendly day on the asteroid. Gaspard was sitting at a low table, quietly fletching arrows made of his own chowmega shoots, beautiful red-streaked, pale golden wood, straight as ash, strong as cedar, light as bamboo. His feathers were deep yellow, occasionally streaked with red; their spines seemed to attach very readily to the arrow, as if they belonged there. Corey watched him for a while, then set himself to shooting from the equator to the northern daisies and back again while Gaspard watched.

Afterwards, they had lunch, wonderful French baguettes with bright yellow butter, and the same deliciously winy drink as before. As they ate, Gaspard told stories about his childhood. Not all were happy stories, but it was clear that the orderliness of mechanics and of the universe had comforted his soul during many of the disorders of his beloved France. This was a human view of mechanics that had never occurred to Corey.

Indeed, he got so involved in Gaspard’s stories that he wasn’t even thinking of his bow when Gaspard finished the last bit of bread and said, “So you want to see my apple tree eh?”

“Oh please,” answered Corey jumping up. “I would like it very much.”

“Well, and then you could shoot yourself a few apples, of course,” continued Gaspard.

“So then, Monsieur Coriolis,” Corey said, feeling a little strange, “you do really have an apple tree? For I have never seen it.”

Gaspard smiled, wiped his hands on his white cloth napkin, brushed away the crumbs, and stood up. “But of course!” and he led the way through the sunflowers and turned left. For a while, they walked along the gardens, past many beautiful plantings that Corey couldn’t quite remember seeing before. Everything seemed a little different from what he remembered. The roses were redder and the daisies more delicately petaled; the peppers hung long and tapered and the squash lay ribbed and streaked… At first, he thought it was a trick of his memory or of the light, but then – well, then he saw the apple tree, just its top.

“It must be growing on a hill,” he thought, “like the peaches. A real hill, not just a round-the-asteroid hill.” The tree was still very far away, with apples so dark they were almost purple, but there was no mistaking them. A moment later, the ferns came in sight in a little dip of the land, and, remembering the peaches, Corey thought perhaps the apples were at the same latitude; he did feel he had walked a similar distance from the equator. In fact, he meant to ask if Gaspard had two gardens, but even as the question formed in his mind, his bow seemed to lift into his hands on its own. He looked at the apple and aimed very carefully six inches to the left, just as with the peaches.

Something was wrong.

Gaspard came alongside him and stood perfectly silent.

He lowered his bow.

The sun was wrong. It was going down on his right instead of his left. Suddenly, he realized why. He was in the southern latitudes!

“Why! We crossed the equator,” he cried out half-accusingly. Gaspard smiled and bowed slightly. Corey raised his bow again, aimed six inches to the right of the topmost apple, and, as his arrow curved beautifully left, brought it down with a thwack and a rustle that could be heard a mile away.

“Alors! Tu as gagné ton arc et des fleches,” smiled Gaspard. “So! You have won your bow and arrows.”

“Merci, Monsieur Coriolis,” answered Corey with a bow. “Thank you! Oh, and voici! I have brought back your pine nuts – roasted!” He reached into his quiver and handed Gaspard the little bag, lifting his hand to shade his eyes from the setting sun and also to brush away sudden tears.

When he could see clearly again, the afternoon sun was sparking through his own trees. His bow hung at his side, and the quiver on his shoulder was heavy with red and golden arrows.

The End.

But if you want to start over, go to Corey’s New Bow.

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Coriolis animation

In case you don’t read comments faithfully, I want you to be sure to look at this animation found by my friend Sam, the Polish chef. Notice that it has movement from north to south in the northerh hemisphere, and from south to north in the southern hemisphere. Notice also that lovely ring of clouds over the equator. As I mentioned, they are always there, not terribly conspicuous if you don’t look for them, but perfectly definite when you do.

As always, animations only do the easy part of the Coriolis effect. What happens from south to north in the northern hemisphere is where people get confused; and what happens from north to south in the southern hemisphere is hopeless. But not for you because we talked about it in class, using a globe, and  explaining, to the surprise of many, that a man standing still on the equator is going 1000 miles an hour towards the east, and the arrow he shoots keeps that eastwards motion. So, unless he shoots along the line of the equator itself, his arrow lands many miles east of any target. At our latitude, I think we’re going eastwards at about 750 miles an hour; the arrow would land 250 miles east, assuming no friction…

All very unexpected. Named after a French engineer of the 19th century. You can research his life for your science project. I’ve never been able to find much; he was a good Catholic when that was not the way to get advancement.

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#9  Latitude Gardens

That night, Corey fell asleep wishing he could tell Gaspard what he had learned in New Zealand. He wondered if GGC-1835 had a southern hemisphere…

“Of course I have a southern hemisphere!” said Gaspard.

They were walking along with a bright patch of buttercups to their left, a most unusual planting in what Corey concluded must be the most unusual garden ever planted. It was only a few rows wide, but it stretched away to the horizon both north and south as far as he could see. It wasn’t all buttercups, of course. And it wasn’t all level, either, for soon they were walking up a steep hill towards a patch of merry forsythia in golden bloom. The forsythia was a massive bush, and more of the golden shrubbery was spaced 25 yards to the left and right, crosswise to the garden. Gaspard called it the Summer Circle, “Yes, yes of course! It goes all round the asteroid.”

Corey thought he must have seen one of these bushes from the top of the pine tree. Gaspard agreed: “it’s about 30 degrees north of the equator.”

Past the forsythia, they found anise, carrots, daisies, bush beans, and then they were looking down along some ferns and asters into a little valley with cattails and a willow at the bottom and several more herbs and flowers and some kind of fruit tree up on the farther hillside.

“Peaches,” said Gaspard, “and ripe enough, but too high to reach. I suppose you might drop me one – just knock the branch of course, so we don’t have peach soup.”

It was not hard to drop several. Corey missed the first shot, being uncertain of how much turning to expect, but in the slow motion of the flight, and in the disturbance of the leaves, he knew exactly where his arrow went – six inches to the right — and he was on target after that.

They crossed the valley and ate their peaches which had hardly bruised in the thick grass below the tree. Then Gaspard said he wanted to walk south and let Corey see how the arrows changed their flight curve with latitude.

It turned out that the plantings were Gaspard’s latitude garden, and each bed marked a change of 1º of latitude, “so I know where I am,” he said, “but the arrow’s flight doesn’t change the same amount with each latitude,” he added.

“I know,” answered Corey. “I figured it out once and then Pete had an equation. But I don’t know how much it changes.”

Gaspard smiled. “Pete,” was all he said, shaking his head.

They faced south and Corey shot his arrows one by one at the start of each planting: towards the beans, towards the daisies, towards the carrots and then the anise, even towards a spray of forsythia hanging outside the line of the garden.

By the time he ran out of arrows, they were retrieving them as they walked, and he could see how they were closer and closer to his visual mark. When a bright patch of sunflowers came in sight, Gaspard proposed one more shot and then a picnic at the equator, for indeed the sunflowers marked the center latitude on GGC-1835.

Corey agreed; he was hungry again, and he remembered Gaspard’s French bread and butter. They passed some white peonies, broad leaved comfrey, golden daylilies, deeply fringed parsley, and several other plantings before they reached the equator and sat in the shade of the sunflowers. The north equatorial section was taller with pale yellow flowers while the south equatorial planting was a crowd of shorter sunflowers with flaming red centers. A narrow path of white stones passed between them and beyond, apparently wrapping itself around the whole asteroid. Corey’s arrow to the equator was off his sights by less than half an inch.

He ate contentedly for a few minutes and then gave a quick little sigh of frustration. “Isn’t there anywhere that you just shoot where it looks right and the arrow goes there?”

“Hmmm,” said Gaspard. “Good question!” He was buttering his last crust very carefully and looking sideways at Corey, a twinkle in his eye.

“I mean, a kind of Isabela Island,” continued Corey. “It made me like to use my eyes! But here we are at the equator and I’m still off a little. Not very much, but…”

“Hmmm,” said Gaspard again, nodding his head with a slight frown betrayed by his twinking eyes.

Corey sat very still. Into his mind flew the image of Dad with the verse about turning left in the south…

Suddenly, he jumped up, grabbed his bow, and ran south along the latitude garden. As he ran, he noticed that the plantings were the almost the same as those in the northern gardens, only warmer and richer colors. He passed golden clover, fragrant cilantro, red Hibiscus, a borage bed full of bees, and more until he came to the red peonies.

When he reached these, he turned around to aim away to the north at a drooping white peony that hung to the side of its plot. Gaspard had moved well away from the garden and was smiling broadly.

“I wonder will my arrow turn left, then right?” wondered Corey. “No, that’s silly. The arrow goes straight but it might seem to be going left into the garden at first, and then right back out to my target. I wonder…” He aimed precisely where his eye told him, pulled his bow and let the arrow fly. It made a beautiful arc just slightly towards tallest sunflower, and then straightened itself back out to cut the white peony cleanly from its stem.

Gaspard went to pick up the arrow — and the peony — and turned to meet the young bowman as he came running up. Corey was so happy he was almost crying.

“I did it; I did it!” he called.

When he had caught his breath, he had to settle one question.

“So then the arrows went straight by the Wolf Volcano because I was shooting across the equator?” he asked eagerly.

“You may have been shooting across the equator,” said Gaspard. “Certainly Isabela Island sits at your equator, and Wolf Volcan is precisely upon it. But more importantly, your world is bigger than mine and your arrows move faster there. The sine anywhere near the equator is very small, and you could not have measured the curve whether you were precisely on the equator or just nearby.”

Corey gave a sigh of contentment.

“Hey,” he said shyly, rubbing his toe in the grass. “Do you have an apple tree?”

But Gaspard did not answer. When Corey looked up, he was in Wellington, and his arrows were beside him, smelling slightly of peaches. The sun was rising, and a large vase of red and white peonies stood on the windowsill. He did not remember seeing them before.

(To be continued…)

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Corey’s on his way to New Zealand. Have you figured out what he will find there? I wish his drawing were larger. I’ll work on it.

Also, I realized that Corey’s Dad and Mom needed names. So his Dad’s name is George. His Mom gets a name soon. I think it’s Marie.

Corey’s Bow #8 Windy Wellington

“New Zealand! When did we decide to go to New Zealand?” demanded Corey. He really didn’t want to go. He was afraid he would miss Gaspard somehow.

Mother smiled. “I told you right after the trip to Victoria Island,” she said, “but you might have been thinking o other things. We’re going to visit my brother Richard and your Aunt Gabrielle, and their new baby Jack.”

Pete figured out what was bothering him. “Never mind anyway. It’s no farther to GGC-1835 from there than from here.”

Corey wasn’t so sure, but anyway, he had to pack. Mother had already done the laundry, so it didn’t take long, and soon they were on the way to the airport. First they took half a day to get to California; then a second airplane took all night to cross the ocean. When they arrived, the weather was very cold and they were so tired of sitting they could hardly walk straight, but Corey remembered that this was the uncle who had sent him his bow and arrow, and thanked him as soon as they met.

“So you like it?”

“Very much,” answered Corey as everyone bundled into a warm car. It was rainy but beautiful and he liked the clouds scudding above the hills as they drove to Uncle Richard’s home.

“I’m sorry it’s so cold; it’s late winter here when you have Indian summer at home,” said Aunt Gabrielle when they reached her house. “But shh. Jack is sleeping; come in and get some breakfast.”

As the other adults settled by the fire, Aunt Gabrielle took Corey and Pete into the kitchen. They found that the light switches were up-side down – up was ‘off’ and down was ‘on’ — and that the door handles turned the wrong way to open doors.

“Is everything up-side down here?” asked Corey.

She laughed and shook her head. “I guess someone thought it would be a good joke to make handles and light switches that way. No, nothing is really upside-down. Only the Moon and the constellations, the few you can see in both hemispheres – like Orion.” She bustled about her stove.

“George says you like to shoot your bow,” she said, as the pot began to whistle. “So I made sure you can visit my Dad in his vineyard. Plenty of room to shoot there, and no grapes to bother right now. And,” she was pouring hot cocoa into two large mugs, “I don’t think your arrows will turn upside-down or go backwards.”

Pete and Corey took their hot chocolate in by the fire where they had biscuits and fruit and cheese, and then set out to visit the town. At first they wanted to sleep, but Aunt Gabrielle said that if they slept, they would never get used to the time of day in New Zealand.

“We’re not only down under,” she reminded them, “we’re around back.”

They went to visit the Parliament house, which looked like great skep and was even called the Beehive; then they climbed to the top of a hill to look out at the bay. It was windy everywhere; Corey thought he had never seen so much wind. No place for a bow and arrow!

But the following morning, they took a long ride across the hills into vineyard country, finally coming to Mr. Barker’s place. When they got out of the car, there were hugs all around and Mr. Barker wasted no time encouraging Corey to shoot his bow wherever he wished.

“‘Tis a surprise you’ll have,” he warned.

Corey was very excited and he chose the farthest thing he could see, the top of a fencepost, directly north across a slight dip in the field. Of course he was not sure of his latitude, so he didn’t exactly expect to hit his first target, and it was foolish to choose such a distant one, but imagine his astonishment when the arrow went substantially left of his aim.

“What?” he lowered his bow.

A brief disturbance in the gorse beyond the fence had told him exactly where his arrow had gone. He shook his head and stood very still. Slowly a new picture formed in his mind.

Sure enough, a surprise. Something else was upside-down in this place, and it did matter. He pictured himself standing on the bottom of the globe and shooting upwards. The globe then came round from west (left) to east (right), carrying his target along so that his arrow fell left of target.

Corey sees how an arrow shot from the South Pole will fall left of its target.

Corey sees how an arrow shot from the South Pole will fall left of its target.

“Wow!” he thought. “I suppose everything will be turned around now, — north, south, east and west, all falling left of target, instead of right.” He stood very still, going over everything he had learned in the northern hemisphere, re-drawing everything in his mind until he was satisfied that all the directions were reversed. Then he shook his head and aimed very carefully right of the fencepost. Thwack!

He ran to get his arrows and then aimed one east – to the right of east, that is, and one west – or rather, slightly northwest. Now he had a lot of walking to do, but as he set out, he saw his Dad was coming towards him, holding up his hand. He broke into a run.

“Dad,” he shouted, “did you see me shoot? Did you see?” He was so proud of himself, he couldn’t stop smiling and he was almost crying at the same time.
His dad came up and shook his shoulder.

“You sure got that figured out fast,” he said. “Let’s get those arrows.”

They set out together, and as they walked, Corey’s dad began to recite this verse:

Your southern trips arc to the left

For of straight is each journey bereft

Wherever you aim

No flight is the same

The hand and the mind must be deft!

“Dad!” grinned Corey. “You knew about it all along! But I didn’t know you could write poetry.”

“Oh, it’s not really poetry, just a rhyme,” said Dad.

But he was pleased.

(Continue at Latitude Gardens…)

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Corey shot his bow all the next day. Somehow, being able to think clearly about his arrows’ inevitable swing to the right made it okay with him. “North, south, east, or west, every direction’s bound to go to the right,” he mused. Only a few inches. It wouldn’t even show from outer space. But for a bow and arrow, and an apple or a rabbit, enough to matter totally.

As he perfected his aim, he began to wonder about battlefields. Did guns have to worry about bullets going right? Did cannons have their cannonballs going to the right?

A flock of geese honked above him. “What about migrating birds?” he wondered, lowering his bow. Did they get confused about where they were going?

“At least the birds have memory, as I do,” he mused, “and can learn from their mistakes. They probably have it all figured out.” He didn’t shoot the geese; but he watched them gradually disappear into the depths of the southern sky and thought of his first arrow on GGC-1835. What a beautiful flight that had been!

Then his mind wandered back to shooting. “Wow! I bet those cannons were a bear to operate,” he smiled to himself. “Maybe if they went fast enough, the earth didn’t turn very much. My arrow is a bit slow, of course.” He considered this for a minute, then shrugged.

He had to cross a little brook as he went to retrieve his arrows, and he began to wonder about rivers. “Rivers stick to the ground,” he mused, “but not very tightly. I wonder how many of their bends are right turns from the turning earth. Of course they can’t make all right turns; eventually ‘down’ would be to the left.” But he wondered if that was why they never went straight. “Never, ever.” He shook his head.

“Trains stay on their tracks of course,” his thoughts continued. “So they’re ok. But I wonder about airplanes. Do they have to turn left and aim southwest to go due west?”

After a while, he sat down and drew this picture:

Corey's latitudes

“Near the North Pole, the correction would be greatest, and near the equator, the least,” he mused. “I suppose pilots have must some kind of chart of how the turning changes with latitude.” He smiled at his drawing. “Pilots and geese.”

Behind him, Pete gave a satisfied sigh. Corey looked up. His brother was standing by the sidewalk with his arms folded and surveying an odd piece of handiwork. It said, in large chalk writing:

Fc = 2MΩV sin(Θ)

“What does that stuff mean?” asked Corey. “I can’t even read it.”

Pete drew himself up to his full height, (which was not very high.) “It means that the amount your arrow is going to turn depends partly on how heavy it is, partly on how fast the earth is turning, partly on how fast the arrow is going, and partly on your latitude.”

“It doesn’t mean all that,” said Corey. “It’s too short. Anyway, you didn’t say ‘equals.’”

“Don’t have to,” answered Pete. “When I say ‘depends on’ it mean that all the stuff afterwards has to be figured out and then it equals the turning.”

Corey was silent. Then he said, “Say what it means again.”

Peter squared his shoulders and said very firmly, “It means that things that fly above the ground turn right partly because of how heavy they are, partly because the earth is turning, partly depending on how fast they go, and partly depending on how far they are from the equator.”

“There!” exclaimed Corey. “Which is the part that says about how far from the equator.”

“Sine theta,” answered Pete. “It’s the pink part. Sin means sine, and theta means whatever latitude. Every latitude has a special number called its sine, and that’s what you need to know.”

And does the sine get bigger with the latitude?” asked Corey.

“Maybe. I don’t know,” answered Pete.

“Why do you like equations if you don’t know what they mean?” asked Corey, who found Pete’s mixture of understanding and indifference completely frustrating and incomprehensible.

“I just like equations,” answered Pete, arching his eyebrows. “They’re cool.”

Just then, their mother came to the door.

“Pack your bags for New Zealand,” she called.

(To be continued…)

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If you have not been following the story of Corey and his bow, you’ll want to go to Corey’s page and catch up.

Part 5: Lunch with Gaspard

“Let’s have some lunch,” Gaspard proposed. “I have my lunch basket here by the tree, and we can talk. Perhaps you have some questions, and then we will retrieve your arrow.”

Lunch was wonderful – French bread with real butter and some kind of cool drink that tasted suspiciously like a light wine. Corey ate in silence, reflecting on his morning adventure.

Of course when he shot from the north, he thought, he was not moving at all – except for turning. He frowned.

“When I shot from the North Hole,” he began, “I was not moving very fast, but I was turning. If the arrow keeps my motion, why didn’t it continue to turn left when I shot it? Then it would not have gone to the right.”

Gaspard smiled. “A wise question, and the answer is not obvious.” He stopped for a moment to empty his glass, and added, shaking his head, “Pas de tout!” (Which means “Not at all.”) Then he looked up and smiled.

“The fact is that things only keep their motions in one direction. Turning is not a direction; it’s a continual correction of a motion.”

He buttered his bread as he went on.  “For example, your turning, right now with my asteroid, is a motion straight east, continually corrected by gravity pulling you inwards so you don’t fly off into space. When you let go an arrow, it can only keep your eastwards motion, not your ongoing correction. It gets its own correction, of course, from its own gravity; otherwise it would fly off into space.” He looked up from his buttering and raised his bushy eyebrows a moment before continuing:

“Therefore, flying from the North Hole, the arrow carries no motion. But from anywhere else,” he hesitated and shrugged, “from anywhere else except the South Hole — there is an eastwards motion.”

Corey thought about this, closing his eyes and watching his arrows in flight. “That was a beautiful flight,” he murmured.

But then he had another question.

“So when I shot north,” he began, “why did the arrow curve around before it fell?”

Gaspard smiled. “I rather suspect,” (and he winked) “that if you had ridden on the back of the arrow, you would only have seen yourself flying eastwards. A satellite flying around your earth seems to me, from here, to be flying first east, then west, then east again. But it only flies right around, eh?”

“Yes, of course.” Corey munched thoughtfully. He sighed and took another drink.

“I have one more question then.”

“Yes?”

“I understand north and south turning right, but what if I shoot east and west? Will those arrows also turn right, as they do at home?”

“Let’s try it,” answered Gaspard, standing up and brushing off the crumbs.

Corey stood up too and picked up his bow. He fitted an arrow and then looked around.

“Which way is west?” he asked.

Gaspard pointed to the tower. “There’s north. West is at a right angle to north,” he replied.

Corey fitted an arrow, drew back his bow and let it fly. Once again, he watched a soaring arc – turning to the right. A distant “plink” told him that it had landed on the stones of the North Park Circle, distinctly to the northwest, and indeed to the right of his aim.

He shook his head. “What happened this time?” he asked. “If I shoot west, all the motions should be the same and the arrow should fly true.”

Gaspard nodded. “The arrow flies true enough. The problem lies with you.”

“With me?”

(To be continued)

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