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do it yourself

CORIOLIS FORCE

you didn’t explain what use Coriolis is ok … so it explains thingsbut they would happen anyway evie wells ‘ 84

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Herein are two lab-type exercises that will provide an understanding of how the perception of motion depends on the nature of the coordinate system in which it is observed and measured. Treat them as thought-experiments if you wish.

Consider this example : One’s ride on a merry-go-round can be thought of as riding on a disk that is rotating counter-clockwise around an axis anchored to the earth. Here is an alternative explanation : the merry-go-round is not moving at all and everything external to it is rotating in a clockwise direction. In oceanography, we also are confronted with, and constrained to deal with, motion of a rotational kind.

The reason for this ppt is the fact that our models of ocean circulation are used to represent motion for actual situations observed in nature with an earth-bound coordinate system. And that system is one that rotates around an axis through the Earth’s poles at a constant rate of one revolution [360 degrees] every 24 hours.

You will find that the measured path of a real moving object observed in the [ latitude, longitude ] system will be distorted by the coordinate system’s own movement. The exercises will help you demonstrate this fact to yourself.

The motion of a body moving along a path such as Mars-seen-from-Earth implies that a complex localized force is at work that varies in space and time. The planet’s trajectory does not directly suggest that the actual Newtonian gravitational force which results in Mar’s movement produces a smooth elliptical orbit about the sun. First, if you need to, check out Newton’s laws of classical mechanics.

Imagine a straight line, constant speed drift of an object on the Earth’s ocean [all the forces acting on the object are in balance, that is, sum to zero]. The object’s trajectory in lat.lon coordinates will give the impression that some unbalanced force is causing its path to continuously and smoothly deflect from a straight line.

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Gaspard-Gustave de Coriolis

Coriolis was a French mathematician (1792-1845) and a member of the Academie des Sciences (1836), who studied theoretical and applied mechanics.

Newton’s laws of motion are pertinent to movement measured against a sta-tionary Cartesian (x,y) coordinate system. In 1835, Coriolis generalized the laws to include rotating coordinate systems and so showed that an additional force was required to model the motion.

Observations of an object, moving at a constant speed in a straight line, that are made in a rotating coordinate system will show a curved trajectory. The curving deviation from what actually is a straight path gives evidence of a force that deflects the object by changing its velocity.

This force, the Coriolis force, must be included in models of ocean currents cast for a rotating planet. Without it, models will yield incorrect velocities and positions. Coriolis is a real force, but real only in our geophysical models.

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Follow some broad shoulders and pretty heavy footprints … those of

Tycho Brahe Johannes Kepler Isaac Newton

Observations made by Brahe and Kepler’s subsequent analysis were used to untie the knotted path of Mars with its mystifying, retrograde loop. A sun-centered model of the solar system did eventually explain the occurrence of loops in the paths of outer planets in terms of the earth perspective and the planet’s relative speeds along their orbits. The Earth travels more rapidly around the sun and will overtake and pass Mars. Then an earthbound observer will see Mars, over several months, slow, stop, move backward, slow, stop, and resume its expected path about the sun.

In fact, the interpretation of Mar’s loop takes on a bizarre aspect when the forces required to accomplish the reversals are considered, a sequence of decelerations and subsequent accelerations along a trajectory. So make some observations in the next few slides [be sure to keep your nose, Brahe lost his in a duel] and then try your hand at understanding Kepler and Newton’s game. First step : review Newton’s laws and prepare to use them to understand why Coriolis turns up in our models.

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NEWTON’S AXIOMS, or LAWS OF MOTIONOn May 8, 1686 at Cambridge, Trinity College, Isaac Newton finished his preface to the first edition of the Principia [The Mathematical Principles of Natural Philosophy] the whole of which, in the academic style of the day, was written in Latin. It was tran-slated into English by Motte in 1729. Here are the laws as Newton wrote them and Motte interpreted them :

Law I : Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.

Law II : The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

Law III : To every action there is always opposed an equal reaction: or the mutual attraction of two bodies upon each other are always equal, and directed to contrary parts.

For the word motion read momentum [ the product of speed and mass ] and the laws are as modern as one needs.

Law I is known as the law of inertia, Galileo called it his Principle of Inertia, and Descartes put Galileo’s statement in a form very similar to Newton’s. Although their work was prior to Newton’s, neither Galileo nor Descartes are credited for the concept of inertia, Newton is, by those not knowing enough history. When the correspondence between their works was pointed out to Newton, Isaac said : “ If I have seen further, it is only because I have stood on the shoulders of giants”; references in a bibliography, which were not forthcoming, would have been sufficient. Newton also had a dust-up with the German math major, Gottfried Leibniz, over who first invented differential and integral calculus.

6for openers, lets use part of our solar system

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The “screen” represents a portion of the sky a great distance from the Earth and Mars. It is a backdrop against which we will trace a trajectory of Mars seen from earth.

On your copy of the previous power point slide, at positions 1, take a sighting of Mars from Earth by drawing a straight line through the 1’s to a point on the middle of the upper half of the screen, put a dot there and label it with a 1. Continue with dots for sightings at positions 2, 3 and 4.

Now place the dot for positions 5 on the centerline of the screen and continue with positions 6, 7, 8 and 9 on the lower screen half. The numbered positions on the screen should be similar to those on the next slide, slide 8.

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Connect dot-to-dot and you have a rough approximation of Tycho Brahe’s observations and acquired the early astronomer’s headache. What kind of force, operating throughout Mar’s entire orbit, could produce that wiggle ? Check it out on slide 9.

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What Brahe originally saw, still weird, after all these years.

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Given the trajectory you have developed or for that matter what developed when you connected your dots : What would Newton infer from the deviations in the planet’s path using his first law of motion ?

There is a force or forces acting on Mars that can change its momentum and trajectory in a very odd way over a part of its orbit.

So : When you predict the position and movement of Mars, you would have to include this force in your model computations. Or, depending on where Mars was in its orbit, most likely get a wrong solution. Bad physicist …. you get no doughnuts.

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The next exercise will allow you to demonstrate for yourself how apparent deflections of a object’s path can arise when the coordinate system moves. In particular, when the coordinate system rotates.

The effect of rotation was first encountered by astronomers in some early observations of planetary motion, Mars in particular. This was just replicated in the first exercise.

The second case is linear motion of an object measured on a two-dimensional analogy to the surface of a rotating Earth : the rotating three-dimentional globe will be approximated by a rotating flat sheet of clear plastic. The steps taken in the exercise will show how a curved path of the same object can been seen. In general, any curvilinear motion of an object will exhibit an apparent path deviation generated by observing its change in position against a coordinate system that is itself moving.

We must recognize that in order to correctly model the observed motion of the object a force must be introduced that accounts for the apparent deviation in the object’s path; the Newtonian way to model this physical situation. Voila : the Coriolis Force that accounts for the Coriolis Effect. Thank you, Zac and Gasparde.

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This second exercise will have you create two trajectories for an object that is, in reality, moving only in a straight line at a constant speed.

One trajectory, in black, shows this movement observed against a stationary Cartesian coordinate system.

The second trajectory, in green, is found by observing the object’s positions during the same straight line motion but against a rotating coordinate system.

The first trajectory is a simple straight line; the other is curved in proportion to the angular speed of the coordinate system’s rotation. [This is our way of approximating the three-dimensional rotation of the earth].

Now, step-by-step:

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So, this is what we are after, the trajectory of the object tracked and marked in the rotating coordinate system on the plastic sheet. Mind you, the object was really moving in a straight line at a constant speed in the black coordinate system. Consider the black one a small part of the globe’s latitude and longitude system. Now, to model the object’s motion as far as Newton’s classical physics is concerned, a force must be introduced that will produce its deflection as shown above. In our ocean models such as the Ekman model and Geostrophic model, we call that force the Coriolis force. It is a real force, real in the model, that is. Thanks once again, zac and gus.

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Aurora Australia 10 March 2011 RVIB Palmer 67 deg S, 153.6 deg W image: Juan Botella

In the background, what constellation do you see ?