Transcript PowerPoint Presentation - EARLY RECORDED OCEAN …

do it yourself
CORIOLIS FORCE
you didn’t explain what use Coriolis is
ok … so it explains things
but 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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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 stationary 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.
Gaspard-Gustave de Coriolis
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 MOTION
On 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 translated 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
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first invented differential and integral calculus.
for 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 ?
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