In physics, the
Coriolis effect is an apparent
deflection of moving objects when they are viewed from a
rotating reference frame.
Newton's laws of motion
govern the motion of an object in an
inertial frame of reference.
When transforming Newton's laws to a rotating frame of reference,
the
Coriolis force appears, along with the
centrifugal
force. If the rotation speed of the frame is not constant, the
Euler force will also appear. All three
forces are proportional to the mass of the object. The Coriolis
force is proportional to the speed of rotation and the centrifugal
force is proportional to its square. The Coriolis force acts in a
direction perpendicular to the rotation axis and to the velocity of
the body in the rotating frame and is proportional to the object's
speed in the rotating frame. The centrifugal force acts outwards in
the radial direction and is proportional to the distance of the
body from the axis of the rotating frame.
These three additional forces are termed either inertial forces,
fictitious forces or
pseudo
forces. These names are used in a technical sense, to mean
simply that these forces vanish in an inertial frame of
reference.
The mathematical expression for the Coriolis force appeared in an
1835 paper by a French scientist
Gaspard-Gustave Coriolis in
connection with
hydrodynamics, and
also in the
tidal equations of
Pierre-Simon Laplace in 1778.
Early in the 20th century, the term
Coriolis force began
to be used in connection with
meteorology.
Perhaps the most commonly encountered rotating reference frame is
the
Earth.
Moving objects on the surface of the Earth
experience a Coriolis force, and appear to veer to the right in the
northern
hemisphere, and to the left in the southern. Exactly on the equator, motion east or
west, remains (precariously) along the
line of the
equator. Initial motion of a pendulum in any other direction will
lead to a motion in a
loop. Movements of air in the
atmosphere and water in the ocean are notable examples of this
behavior: rather than flowing directly from areas of high pressure
to low pressure, as they would on a non-rotating planet, winds and
currents tend to flow to the right of this direction north of the
equator, and to the left of this direction
south of the equator. This effect is responsible for the rotation
of large
cyclones (see
Coriolis effects in meteorology).
History
Gaspard-Gustave Coriolis
published a paper in 1835 on the energy yield of machines with
rotating parts, such as waterwheels. This paper considered the
supplementary forces that are detected in a rotating frame of
reference. Coriolis divided these supplementary forces into two
categories. The second category contained a force that arises from
the
cross product of the
angular velocity of a
coordinate system and the projection of a
particle's
velocity into a plane
perpendicular to the system's
axis of rotation. Coriolis referred to this
force as the "compound centrifugal force" due to its analogies with
the
centrifugal force already
considered in category one. By the early 20th century the effect
was known as the "
acceleration of
Coriolis." By 1919 it was referred by to as "Coriolis' force" and
by 1920 as "Coriolis force".
In 1856,
William Ferrel proposed the
existence of a
circulation cell in the
mid-latitudes with air being deflected by the Coriolis force to
create the
prevailing westerly
winds.
Understanding the kinematics of how exactly the rotation of the
Earth affects airflow was partial at first. Late in the 19th
century, the full extent of the large scale interaction of
pressure gradient force and
deflecting force that in the end causes air masses to move
along isobars was understood.
Formula
In non-vector terms: at a given rate of rotation of the observer,
the magnitude of the Coriolis acceleration of the object is
proportional to the velocity of the object and also to the sine of
the angle between the direction of movement of the object and the
axis of rotation.
The vector formula for the magnitude and direction of the Coriolis
acceleration is
- \boldsymbol{ a}_C = -2 \, \boldsymbol{ \Omega \times v}
where (here and below)
v is the velocity of the
particle in the rotating system, and
Ω is the
angular velocity vector which has
magnitude equal to the rotation rate ω and is directed along the
axis of rotation of the rotating reference frame, and the
× symbol represents the
cross product operator.
The equation may be multiplied by the mass of the relevant object
to produce the Coriolis force:
- \boldsymbol{ F}_C = -2 \, m \, \boldsymbol{\Omega \times
v}.
See
fictitious force for a
derivation.
The
Coriolis effect is the behavior added by the
Coriolis acceleration. The formula implies that the
Coriolis acceleration is perpendicular both to the direction of the
velocity of the moving mass and to the frame's rotation axis. So in
particular:
- if the velocity is parallel to the rotation axis, the Coriolis
acceleration is zero.
- if the velocity is straight inward to the axis, the
acceleration is in the direction of local rotation.
- if the velocity is straight outward from the axis, the
acceleration is against the direction of local rotation.
- if the velocity is in the direction of local rotation, the
acceleration is outward from the axis.
- if the velocity is against the direction of local rotation, the
acceleration is inward to the axis.
The vector cross product can be evaluated as the
determinant of a matrix:
- \boldsymbol{\Omega \times v} = \begin{vmatrix}
\boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k} \\ \Omega_x
& \Omega_y & \Omega_z \\ v_x & v_y & v_z
\end{vmatrix}\ = \begin{pmatrix} \Omega_y v_z - \Omega_z v_y \\
\Omega_z v_x - \Omega_x v_z \\ \Omega_x v_y - \Omega_y v_x
\end{pmatrix}\ ,
where the vectors
i,
j,
k are unit
vectors in the
x,
y and
z
directions.
Causes
The Coriolis effect exists only when using a rotating reference
frame. In the rotating frame it behaves exactly like a real force
(that is to say, it causes acceleration and has real effects).
However, Coriolis force is a consequence of
inertia, and is not attributable to an identifiable
originating body, as is the case for electromagnetic or nuclear
forces, for example. From an analytical viewpoint, to use
Newton's second law in a rotating
system, Coriolis force is mathematically necessary, but it
disappears in a non-accelerating, inertial frame of reference.
For a mathematical formulation see Mathematical
derivation of fictitious forces.
A denizen of a rotating frame, such as an astronaut in a rotating
space station, very probably will find the interpretation of
everyday life in terms of the Coriolis force accords more simply
with intuition and experience than a cerebral reinterpretation of
events from an inertial standpoint. For example, nausea due to an
experienced push may be more instinctively explained by Coriolis
force than by the law of inertia. See also
Coriolis effect . In
meteorology, a rotating frame (the Earth) with its Coriolis force
proves a more natural framework for explanation of air movements
than a hypothetical, non-rotating, inertial frame without Coriolis
forces. In long-range gunnery, sight corrections for the Earth's
rotation are based upon Coriolis force. These examples are
described in more detail below.
The acceleration entering the Coriolis force arises from two
sources of change in velocity that result from rotation: the first
is the change of the velocity of an object in time. The same
velocity (in an inertial frame of reference where the normal laws
of physics apply) will be seen as different velocities at different
times in a rotating frame of reference. The apparent acceleration
is proportional to the angular velocity of the reference frame (the
rate at which the coordinate axes change direction), and to the
component of velocity of the object in a plane perpendicular to the
axis of rotation. This gives a term
-\boldsymbol\Omega\times\boldsymbol v.The minus sign arises from
the traditional definition of the cross product (
right hand rule), and from the sign
convention for angular velocity vectors.
The second is the change of velocity in space. Different positions
in a rotating frame of reference have different velocities (as seen
from an inertial frame of reference). In order for an object to
move in a straight line it must therefore be accelerated so that
its velocity changes from point to point by the same amount as the
velocities of the frame of reference. The effect is proportional to
the angular velocity (which determines the relative speed of two
different points in the rotating frame of reference), and to the
component of the velocity of the object in a plane perpendicular to
the axis of rotation (which determines how quickly it moves between
those points). This also gives a term
-\boldsymbol\Omega\times\boldsymbol v.
Length scales and the Rossby number
The time, space and velocity scales are important in determining
the importance of the Coriolis effect. Whether rotation is
important in a system can be determined by its
Rossby number, which is the ratio of the
velocity, U, of a system to the product of the Coriolis
parameter,f, and the length scale, L, of the motion:
- Ro = \frac{U}{fL}.
The Rossby number is the ratio of inertial to Coriolis forces. A
small Rossby number signifies a system which is strongly affected
by Coriolis forces, and a large Rossby number signifies a system in
which inertial forces dominate. For example, in tornadoes, the
Rossby number is large, in low-pressure systems it is low and in
oceanic systems it is of the order of unity. As a result, in
tornadoes the Coriolis force is negligible, and balance is between
pressure and centrifugal forces. In low-pressure systems,
centrifugal force is negligible and balance is between Coriolis and
pressure forces. In the oceans all three forces are
comparable.
An atmospheric system moving at
U = 10 m/s occupying a
spatial distance of
L = 1000 km, has a Rossby number
of approximately 0.1.A man playing catch may throw the ball at
U = 30 m/s in a garden of length
L = 50 m. The
Rossby number in this case would be about = 6000.Needless to say,
one does not worry about which hemisphere one is in when playing
catch in the garden. However, an unguided missile obeys exactly the
same physics as a baseball, but may travel far enough and be in the
air long enough to notice the effect of Coriolis. Long-range shells
in the Northern Hemisphere landed close to, but to the right of,
where they were aimed until this was noted. (Those fired in the
southern hemisphere landed to the left.) In fact, it was this
effect that first got the attention of Coriolis himself.
Applied to Earth
Rotating sphere
Consider a location with latitude \varphi on a sphere that is
rotating around the north-south axis. A local coordinate system is
set up with the x axis horizontally due east, the y axis
horizontally due north and the z axis vertically upwards.The
rotation vector, velocity of movement and Coriolis acceleration
expressed in this local coordinate system (listing components in
the order East (
e), North (
n) and Upward
(
u)) are:
- \boldsymbol{ \Omega} = \omega \begin{pmatrix} 0 \\ \cos \varphi
\\ \sin \varphi \end{pmatrix}\ , \boldsymbol{ v} =
\begin{pmatrix} v_e \\ v_n \\ v_u \end{pmatrix}\ ,
- \boldsymbol{ a}_C =-2\boldsymbol{\Omega \times v}= 2\,\omega\,
\begin{pmatrix} v_n \sin \varphi-v_u \cos \varphi \\ -v_e \sin
\varphi \\ v_e \cos\varphi\end{pmatrix}\ .
When considering atmospheric or oceanic dynamics, the vertical
velocity is small and the vertical component of the Coriolis
acceleration is small compared to gravity. For such cases, only the
horizontal (East and North) components matter. The restriction of
the above to the horizontal plane is (setting
v_{u}=0):
- \boldsymbol{ v} = \begin{pmatrix} v_e \\ v_n\end{pmatrix}\ ,
\boldsymbol{ a}_c = \begin{pmatrix} v_n \\
-v_e\end{pmatrix}\ f\ ,
where f = 2 \omega \sin \varphi \, is called the
Coriolis parameter.
By setting
v_{n} = 0, it can be seen immediately that (for
positive \varphi and \omega\,) a movement due east results in an
acceleration due south. Similarly, setting , it is seen that a
movement due north results in an acceleration due east. In general,
observed horizontally, looking along the direction of the movement
causing the acceleration, the acceleration always is turned 90° to
the right and of the same size regardless of the horizontal
orientation. That is:
As a different case, consider equatorial motion setting φ = 0°. In
this case,
Ω is parallel to the North or
n-axis, and:
- \boldsymbol{ \Omega} = \omega \begin{pmatrix} 0 \\ 1 \\ 0
\end{pmatrix}\ , \boldsymbol{ v} = \begin{pmatrix}
v_e \\ v_n \\ v_u \end{pmatrix}\ , \boldsymbol{ a}_C
=-2\boldsymbol{\Omega \times v}= 2\,\omega\, \begin{pmatrix}-v_u
\\0 \\ v_e \end{pmatrix}\ .
Accordingly, an eastward motion (that is, in the same direction as
the rotation of the sphere) provides an upward acceleration known
as the
Eötvös effect, and an
upward motion produces an acceleration due west.
The Sun and distant stars
The motion of the Sun as seen from Earth is dominated by the
Coriolis and centrifugal forces. For ease of explanation consider
the situation of a distant star (with mass m) located over the
equator, at position \boldsymbol r, perpendicular to the rotation
vector \boldsymbol \Omega so \boldsymbol{\Omega \cdot r} = 0. It is
observed to rotate in the opposite direction as the Earth's
rotation once a day, making its velocity \boldsymbol v =
-\boldsymbol \Omega \times \boldsymbol r. The fictitious force
consisting of Coriolis and centrifugal forces is:
- \boldsymbol {F_f} = -2 \, m \, \boldsymbol{\Omega \times v} - m
\, \boldsymbol{\Omega \times { (\Omega \times r)}}
- : = +2 \, m \, \boldsymbol{\Omega \times (\Omega \times r)} - m
\, \boldsymbol{\Omega \times {(\Omega \times r)}}
- : = m \, \boldsymbol{\Omega \times (\Omega \times r)}
- : = m \, (\boldsymbol{\Omega (\Omega \cdot r) - r (\Omega \cdot
\Omega))}
- : = - m \, \Omega^2 \, \boldsymbol r
This can be recognised as the centripetal force that will keep the
star in a circular movement around the observer.
The general situation for a star, not above the equator is more
complicated. Just as for air flows on Earth's surface, on the
northern hemisphere a star's trajectory will be deflected to the
right. After rising at a certain angle, it will bend to the right,
culminate and start setting.
Meteorology
Figure 14: Schematic representation of
inertial circles of air masses in the absence of other forces,
calculated for a wind speed of approximately 50 to 70 m/s.
Note that the rotation is exactly opposite of that normally
experienced with air masses in weather systems around
depressions.
Perhaps the most important instance of the Coriolis effect is in
the large-scale dynamics of the oceans and the atmosphere. In
meteorology and ocean science, it is convenient to use a rotating
frame of reference where the Earth is stationary. The fictitious
centrifugal and Coriolis forces must then be introduced. Their
relative importance is determined by the
Rossby number.
Tornadoes have a high Rossby number, so Coriolis
forces are unimportant, and are not discussed here. As discussed
next, low-pressure areas are phenomena where Coriolis forces are
significant.
Flow around a low-pressure area
If a low-pressure area forms in the atmosphere, air will tend to
flow in towards it, but will be deflected perpendicular to its
velocity by the Coriolis acceleration. A system of equilibrium can
then establish itself creating circular movement, or a cyclonic
flow. Because the Rossby number is low, the force balance is
largely between the
pressure
gradient force acting towards the low-pressure area and the
Coriolis force acting away from the center of the low
pressure.
Instead of flowing down the gradient, large scale motions in the
atmosphere and ocean tend to occur perpendicular to the pressure
gradient. This is known as
geostrophic
flow. On a non-rotating planet fluid would flow along the
straightest possible line, quickly eliminating pressure gradients.
Note that the geostrophic balance is thus very different from the
case of "inertial motions" (see below) which explains why
mid-latitude cyclones are larger by an order of magnitude than
inertial circle flow would be.
This pattern of deflection, and the direction of movement, is
called
Buys-Ballot's law. In the
atmosphere, the pattern of flow is called a
cyclone. In the Northern Hemisphere the direction of
movement around a low-pressure area is counterclockwise. In the
Southern Hemisphere, the direction of movement is clockwise because
the rotational dynamics is a mirror image there. At high altitudes,
outward-spreading air rotates in the opposite direction. Cyclones
rarely form along the equator due to the weak Coriolis effect
present in this region.
Inertial circles
An air or water mass moving with speed v\, subject only to the
Coriolis force travels in a circular trajectory called an 'inertial
circle'. Since the force is directed at right angles to the motion
of the particle, it will move with a constant speed, and perform a
complete circle with frequency f. The magnitude of the Coriolis
force also determines the radius of this circle:
- R=v/f\,.
On the Earth, a typical mid-latitude value for f is 10
^{−4}
s
^{−1}; hence for a typical atmospheric speed of 10 m/s the
radius is 100 km, with a period of about 14 hours. In the
ocean, where a typical speed is closer to 10 cm/s, the radius
of an inertial circle is 1 km. These inertial circles are
clockwise in the northern hemisphere (where trajectories are bent
to the right) and anti-clockwise in the southern hemisphere.
If the rotating system is a parabolic turntable, then f is constant
and the trajectories are exact circles. On a rotating planet, f
varies with latitude and the paths of particles do not form exact
circles. Since the parameter f varies as the sine of the latitude,
the radius of the oscillations associated with a given speed are
smallest at the poles (latitude = ±90°), and increase toward the
equator.
Other terrestrial effects
The Coriolis effect strongly affects the large-scale oceanic and
atmospheric circulation,
leading to the formation of robust features like
jet streams and
western boundary currents. Such
features are in
geostrophic balance,
meaning that the Coriolis and
pressure gradient forces
balance each other. Coriolis acceleration is also responsible for
the propagation of many types of waves in the ocean and atmosphere,
including
Rossby waves and
Kelvin waves. It is also instrumental in the
so-called
Ekman dynamics in the ocean,
and in the establishment of the large-scale ocean flow pattern
called the
Sverdrup balance.
Eötvös effect
The practical impact of the
Coriolis effect is mostly
caused by the horizontal acceleration component produced by
horizontal motion.
There are other components of the Coriolis effect.
Eastward-traveling objects will be deflected upwards (feel
lighter), while westward-traveling objects will be deflected
downwards (feel heavier). This is known as the
Eötvös effect. This aspect of the
Coriolis effect is greatest near the equator. The force produced by
this effect is similar to the horizontal component, but the much
larger vertical forces due to gravity and pressure mean that it is
generally unimportant dynamically.
In addition, objects traveling upwards or downwards will be
deflected to the west or east respectively. This effect is also the
greatest near the equator. Since vertical movement is usually of
limited extent and duration, the size of the effect is smaller and
requires precise instruments to detect.
Draining in bathtubs and toilets
Coriolis rotation can conceivably play a role on scales as small as
a bathtub. It is a commonly held myth that the every-day rotation
of a bathtub or toilet vortex is due to whether one is in the
northern or southern hemisphere. An article in Nature, by Ascher
Shapiro, describes an experiment in which all other forces to the
system are removed by filling a 6 ft. tank with water and allowing
it to settle for 24 hrs (to remove any internal velocity), in a
room where the temperature has stabilized (temperature differences
in the room can introduce forces inside the fluid). The drain plug
is then very slowly removed, and tiny pieces of floating wood are
used to observe rotation. During the first 12 to 15 mins, no
rotation is observed. Then, a vortex appears and consistently
begins to rotate in a counter-clockwise direction (the experiment
was performed in the Northern hemisphere, in Boston, MA). This is
repeated and the results averaged to make sure the effect is real.
The Coriolis effect does indeed play a role in vortex rotation for
draining liquids that have come to rest for a long time. ["Bath-Tub
Vortex", Nature. Dec 15th, 1962. Vol 195, No. 4859, p.
1080-1081]
In reality, this experiment shows that the Coriolis effect is a few
orders of magnitude smaller than
various random influences on drain direction, such as the geometry
of the container and the direction in which water was initially
added to it. In the above experiment, if the water settles for 2
hrs or less (instead of 24), then the vortex can be seen to rotate
in either direction. Most toilets flush in only one direction,
because the toilet water flows into the bowl at an angle. If water
shot into the basin from the opposite direction, the water would
spin in the opposite direction.
The idea that toilets and bathtubs drain differently in the
Northern and Southern Hemispheres has been popularized by several
television programs, including
The
Simpsons episode "
Bart vs.
Australia" and the
The
X-Files episode "
Die Hand
Die Verletzt." Several science broadcasts and publications,
including at least one college-level physics textbook, have also
stated this. Some sources that incorrectly attribute draining
direction to the Coriolis force also get the direction wrong,
claiming that water would turn clockwise into drains in the
Northern Hemisphere.
The Rossby number can also tell us about the bathtub. If the length
scale of the tub is about
L = 1 m, and the water moves
towards the drain at about
U = 60 cm/s, then the
Rossby number is about 6 000.Thus, the bathtub is, in terms of
scales, much like a game of catch, and rotation is unlikely to be
important.
Ballistic missiles and satellites
Ballistic missiles and satellites appear to follow curved paths
when plotted on common world maps mainly because the earth is
spherical and the shortest distance between two points on the
Earth's surface (called a
great circle)
is usually not a straight line on those maps. Every two-dimensional
(flat) map necessarily distorts the Earth's curved
(three-dimensional) surface in some way. Typically (as in the
commonly used
Mercator
projection, for example), this distortion increases with
proximity to the poles. In the northern hemisphere for example, a
ballistic missile fired toward a distant target using the shortest
possible route (a great circle) will appear on such maps to follow
a path north of the straight line from target to destination, and
then curve back toward the equator. This occurs because the
latitudes, which are projected as straight horizontal lines on most
world maps, are in fact circles on the surface of a sphere, which
get smaller as they get closer to the pole. Being simply a
consequence of the sphericity of the Earth, this would be true even
if the Earth didn't rotate. The Coriolis effect is of course also
present, but its effect on the plotted path is much smaller.
The Coriolis effects became important in
external ballistics for
calculating the trajectories of very long-range
artillery shells.
The most famous historical example was the
Paris gun, used by the Germans during
World War I to bombard Paris from a range
of about .
Special cases
Cannon on turntable
Figure 6: Coriolis acceleration,
centrifugal acceleration and net acceleration vectors at three
selected points on the trajectory as seen on the turntable.
Figure 1 is an animation of the classic illustration of Coriolis
force. Another visualization of the Coriolis and centrifugal forces
is
this animation clip. Figure 3 is a graphical
version.
Here is a question: given the radius of the turntable
R,
the rate of angular rotation ω, and the speed of the cannonball
(assumed constant)
v, what is the correct angle θ to aim
so as to hit the target at the edge of the turntable?
The inertial frame of reference provides one way to handle the
question: calculate the time to interception, which is
t_{f} =
R /
v . Then, the
turntable revolves an angle ω
t_{f} in this time.
If the cannon is pointed an angle θ = ω
t_{f} = ω
R /
v, then the cannonball arrives at the
periphery at position number 3 at the same time as the
target.
No discussion of Coriolis force can arrive at this solution as
simply, so the reason to treat this problem is to demonstrate
Coriolis formalism in an easily visualized situation.
The trajectory in the inertial frame (denoted
A) is a
straight line radial path at angle θ. The position of the
cannonball in (
x,
y ) coordinates at time
t is:
- \mathbf{r}_A (t) = vt\ \left( \cos (\theta ),\right.\left.
{\color{white}..}\ \sin (\theta )\right) \ .
In the turntable frame (denoted
B), the
x-
y axes rotate at angular rate ω, so the trajectory
becomes:
- \mathbf{r}_B (t) = vt\ \left( \cos ( \theta - \omega
t),\right.\left. {\color{white}..} \sin ( \theta - \omega t)\right)
\ ,
and three examples of this result are plotted in Figure 4.
To determine the components of acceleration, a general expression
is used from the article
fictitious
force:\mathbf{a}_{B} =\mathbf{a}_A - 2 \boldsymbol\Omega
\times \mathbf{v}_{B} - \boldsymbol\Omega \times
(\boldsymbol\Omega \times \mathbf{r}_B ) - \frac{d
\boldsymbol\Omega}{dt} \times \mathbf{r}_B \ ,in which the term in
Ω × v_{B} is the Coriolis acceleration and
the term in
Ω × ( Ω × r_{B}) is the
centrifugal acceleration. The results are (let α = θ −
ω
t):
- \boldsymbol{\Omega} \mathbf{\times r_B} = \begin{vmatrix}
\boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k} \\ 0 & 0
& \omega \\ v t \cos \alpha & vt \sin \alpha & 0
\end{vmatrix}\ = \omega t v \left(-\sin\alpha, \cos\alpha\right )\
,
- \boldsymbol{\Omega \ \times} \left( \boldsymbol{\Omega}
\mathbf{\times r_B}\right) = \begin{vmatrix}
\boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k} \\ 0 & 0
& \omega \\ -\omega t v \sin\alpha & \omega t v \cos\alpha
& 0 \end{vmatrix}\ \ ,
producing a centrifugal acceleration:
- \mathbf{a_{\mathrm{Cfgl}}} = \omega^2 v t \left(\cos\alpha,
\sin\alpha\right )=\omega^2 \mathbf{r_B}(t) \ .
Also:
- \mathbf{v_B} = \frac{d\mathbf{r_B}(t)}{dt}=(v \cos \alpha +
\omega t \ v \sin \alpha, \ v \sin \alpha -\omega t \ v \cos \alpha
,\ 0)\ \ ,
- \boldsymbol{\Omega} \mathbf{\times v_B} = \begin{vmatrix}\!
\boldsymbol{i}& \! \boldsymbol{j}& \! \boldsymbol{k} \\ 0
& 0 & \omega \\v \cos \alpha\quad &v \sin \alpha\quad
&\quad \\ \; + \omega t \ v \sin \alpha & \; -\omega t \ v
\cos \alpha & 0
\end{vmatrix}\ \ , producing a Coriolis acceleration:
- \mathbf{a_{\mathrm{Cor}}} = -2\left[ -\omega v \left(
\sin\alpha - \omega t \cos\alpha\right),\right.\left.
{\color{white}...}\ \omega v \left(\cos\alpha + \omega t \sin
\alpha \right) \right]\
- :=2\omega v \left(\sin\alpha,\ - \cos\alpha \right) -2\omega^2
\mathbf{r_B}(t) \ .
Figure 5 and Figure 6 show these accelerations for a particular
example.
It is seen that the Coriolis acceleration not only cancels the
centrifugal acceleration, but together they provide a net
"centripetal", radially inward component of acceleration (that is,
directed toward the center of rotation):
- \mathbf{a_{\mathrm{Cptl}}} = -\omega^2 \mathbf{r_B}(t) \ ,
and an additional component of acceleration perpendicular to
r_{B} (t):
- \mathbf{a_{C\perp}} = 2\omega v \left(\sin\alpha,\ -\cos\alpha
\right) \ .
The "centripetal" component of acceleration resembles that for
circular motion at radius
r_{B}, while the perpendicular component is
velocity dependent, increasing with the radial velocity
v
and directed to the right of the velocity. The situation could be
described as a circular motion combined with an "apparent Coriolis
acceleration" of 2ω
v. However, this is a rough labeling: a
careful designation of the true centripetal force refers to a
local reference
frame that employs the directions normal and tangential to the
path, not coordinates referred to the axis of rotation.
These results also can be obtained directly by two time
differentiations of
r_{B} (t).
Agreement of the two approaches demonstrates that one could start
from the general expression for fictitious acceleration above and
derive the trajectories of Figure 4. However, working from the
acceleration to the trajectory is more complicated than the reverse
procedure used here, which, of course, is made possible in this
example by knowing the answer in advance.
As a result of this analysis an important point appears:
all the fictitious accelerations must be included to
obtain the correct trajectory. In particular, besides the Coriolis
acceleration, the
centrifugal
force plays an essential role. It is easy to get the impression
from verbal discussions of the cannonball problem, which are
focussed on displaying the Coriolis effect particularly, that the
Coriolis force is the only factor that must be considered;
emphatically, that is not so. A turntable for which the Coriolis
force
is the only factor is the
parabolic turntable. A somewhat more
complex situation is the idealized example of flight routes over
long distances, where the centrifugal force of the path and
aeronautical lift are countered by
gravitational attraction.
Tossed ball on a rotating carousel
Figure 7 illustrates a ball tossed from 12:00 o'clock toward the
center of a counterclockwise rotating carousel. On the left, the
ball is seen by a stationary observer above the carousel, and the
ball travels in a straight line to the center, while the
ball-thrower rotates counterclockwise with the carousel. On the
right the ball is seen by an observer rotating with the carousel,
so the ball-thrower appears to stay at 12:00 o'clock. The figure
shows how the trajectory of the ball as seen by the rotating
observer can be constructed.
On the left, two arrows locate the ball relative to the
ball-thrower. One of these arrows is from the thrower to the center
of the carousel (providing the ball-thrower's line of sight), and
the other points from the center of the carousel to the ball.(This
arrow gets shorter as the ball approaches the center.) A shifted
version of the two arrows is shown dotted.
On the right is shown this same dotted pair of arrows, but now the
pair are rigidly rotated so the arrow corresponding to the line of
sight of the ball-thrower toward the center of the carousel is
aligned with 12:00 o'clock. The other arrow of the pair locates the
ball relative to the center of the carousel, providing the position
of the ball as seen by the rotating observer. By following this
procedure for several positions, the trajectory in the rotating
frame of reference is established as shown by the curved path in
the right-hand panel.
The ball travels in the air, and there is no net force upon it. To
the stationary observer the ball follows a straight-line path, so
there is no problem squaring this trajectory with zero net force.
However, the rotating observer sees a
curved path.
Kinematics insists that a force (pushing to the
right of
the instantaneous direction of travel for a
counterclockwise rotation) must be present to cause this
curvature, so the rotating observer is forced to invoke a
combination of centrifugal and Coriolis forces to provide the net
force required to cause the curved trajectory.
Bounced ball
Figure 8 describes a more complex situation where the tossed ball
on a turntable bounces off the edge of the carousel and then
returns to the tosser, who catches the ball. The effect of Coriolis
force on its trajectory is shown again as seen by two observers: an
observer (referred to as the "camera") that rotates with the
carousel, and an inertial observer. Figure 8 shows a bird's-eye
view based upon the same ball speed on forward and return paths.
Within each circle, plotted dots show the same time points. In the
left panel, from the camera's viewpoint at the center of rotation,
the tosser (smiley face) and the rail both are at fixed locations,
and the ball makes a very considerable arc on its travel toward the
rail, and takes a more direct route on the way back. From the ball
tosser's viewpoint, the ball seems to return more quickly than it
went (because the tosser is rotating toward the ball on the return
flight).
On the carousel, instead of tossing the ball straight at a rail to
bounce back, the tosser must throw the ball toward the right of the
target and the ball then seems to the camera to bear continuously
to the left of its direction of travel to hit the rail
(
left because the carousel is turning
clockwise).
The ball appears to bear to the left from direction of travel on
both inward and return trajectories. The curved path demands this
observer to recognize a leftward net force on the ball. (This force
is "fictitious" because it disappears for a stationary observer, as
is discussed shortly.) For some angles of launch, a path has
portions where the trajectory is approximately radial, and Coriolis
force is primarily responsible for the apparent deflection of the
ball (centrifugal force is radial from the center of rotation, and
causes little deflection on these segments). When a path curves
away from radial, however, centrifugal force contributes
significantly to deflection.
The ball's path through the air is straight when viewed by
observers standing on the ground (right panel). In the right panel
(stationary observer), the ball tosser (smiley face) is at 12
o'clock and the rail the ball bounces from is at position one (1).
From the inertial viewer's standpoint, positions one (1), two (2),
three (3) are occupied in sequence. At position 2 the ball strikes
the rail, and at position 3 the ball returns to the tosser.
Straight-line paths are followed because the ball is in free
flight, so this observer requires that no net force is
applied.
A video clip of the tossed ball and other experiments are found at
youtube: coriolis effect (2-11),
University of Illinois WW2010 Project (some clips
repeat only a fraction of a full rotation), and
youtube.
Visualization of the Coriolis effect
Figure 9: A fluid assuming a parabolic
shape as it is rotating
To demonstrate the Coriolis effect, a parabolic turntable can be used. On a flat turntable, the inertia of a co-rotating object would force it off the edge. But if the surface of the turntable has the correct parabolic bowl shape (see Figure 9) and is rotated at the correct rate, the force components shown in Figure 10 are arranged so the component of gravity tangential to the bowl surface will exactly equal the centripetal force necessary to keep the object rotating at its velocity and radius of curvature (assuming no friction). (See banked turn.) This carefully contoured surface allows the Coriolis force to be displayed in isolation.
Discs cut from cylinders of
dry ice can be
used as pucks, moving around almost frictionlessly over the surface
of the parabolic turntable, allowing effects of Coriolis on dynamic
phenomena to show themselves. To get a view of the motions as seen
from the reference frame rotating with the turntable, a video
camera is attached to the turntable so as to co-rotate with the
turntable, with results as shown in Figure 11. In the left panel of
Figure 11, which is the viewpoint of a stationary observer, the
gravitational force in the inertial frame pulling the object toward
the center (bottom ) of the dish is proportional to the distance of
the object from the center. A centripetal force of this form causes
the elliptical motion. In the right panel, which shows the
viewpoint of the rotating frame, the inward gravitational force in
the rotating frame (the same force as in the inertial frame) is
balanced by the outward centrifugal force (present only in the
rotating frame). With these two forces balanced, in the rotating
frame the only unbalanced force is Coriolis (also present only in
the rotating frame), and the motion is an
inertial circle. Analysis and
observation of circular motion in the rotating frame is a
simplification compared to analysis or observation of elliptical
motion in the inertial frame.
Because this reference frame rotates several times a minute, rather
than only once a day like the Earth, the Coriolis acceleration
produced is many times larger, and so easier to observe on small
time and spatial scales, than is the Coriolis acceleration caused
by the rotation of the Earth.
In a manner of speaking, the Earth is analogous to such a
turntable. The rotation has caused the planet to settle on a
spheroid shape, such that the normal force, the gravitational force
and the centrifugal force exactly balance each other on a
"horizontal" surface. (See
equatorial
bulge.)
The Coriolis effect caused by the rotation of the Earth can be seen
indirectly through the motion of a
Foucault pendulum.
Coriolis effects in other areas
Coriolis flow meter
Figure 11: Object moving
frictionlessly over the surface of a very shallow parabolic
dish.
The object has been released in such a way that it follows an
ellipse-shaped trajectory.
Left: The inertial point of view.
Right: The co-rotating point of view.
A practical application of the Coriolis effect is the
mass flow meter, an instrument that measures
the
mass flow rate and
density of a fluid flowing through a tube. The
operating principle, introduced in 1977 by Micro Motion Inc.,
involves inducing a vibration of the tube through which the fluid
passes. The vibration, though it is not completely circular,
provides the rotating reference frame which gives rise to the
Coriolis effect. While specific methods vary according to the
design of the flow meter, sensors monitor and analyze changes in
frequency, phase shift, and amplitude of the vibrating flow tubes.
The changes observed represent the mass flow rate and density of
the fluid.
Molecular physics
In polyatomic molecules, the molecule motion can be described by a
rigid body rotation and internal vibration of atoms about their
equilibrium position. As a result of the vibrations of the atoms,
the atoms are in motion relative to the rotating coordinate system
of the molecule. Coriolis effects will therefore be present and
will cause the atoms to move in a direction perpendicular to the
original oscillations. This leads to a mixing in molecular spectra
between the rotational and vibrational
levels.
Insect flight
Flies (
Diptera) and moths (
Lepidoptera) utilize the Coriolis effect when
flying: their
halteres, or antennae in the
case of moths, oscillate rapidly and are used as vibrational
gyroscopes. See
Coriolis effect in insect stability. In
this context, the Coriolis effect has nothing to do with the
rotation of the Earth.
See also
References
- Dugas, René and J. R. Maddox (1988). A History of Mechanics. Courier Dover
Publications: pg 374. ISBN 0486656322
- Retrieved on 2009-01-01.
- MIT essays by James F. Price, Woods Hole
Oceanographic Institution (2006). See in particular §4.3 in the
Coriolis lecture
- The claim is made that in the Falklands in WW I, the British
failed to correct their sights for the southern hemisphere, and so
missed their targets. . For set up of the calculations, see
- Cloud Spirals and Outflow in Tropical Storm
Katrina from Earth Observatory (NASA)
- (Vorticity, Part 1)
- Knew? The No-Spin Zone"from Berkeley
Science Review (PDF)
- "Flush Bosh"fromsnopes.com
- "X-Files coriolis error leaves viewers
wondering" from Skeptical Inquirer
- "Bad Coriolis" from Penn State
College of Earth and Mineral Sciences
- "Bad Coriolis" from Penn State
College of Earth and Mineral Sciences
- Here the description "radially inward" means "toward the axis
of rotation". That direction is not toward the
center
of curvature of the path, however, which is the direction of
the true centripetal force. Hence, the quotation marks on
"centripetal".
- When a container of fluid is rotating on a turntable, the
surface of the fluid naturally assumes the correct parabolic shape. This fact may be
exploited to make a parabolic turntable by using a fluid that sets
after several hours, such as a synthetic resin. For a video of the Coriolis effect on such a
parabolic surface, see Geophysical fluid dynamics lab demonstration John
Marshall, Massachusetts Institute of Technology.
- For a java applet of the Coriolis effect on such a parabolic
surface, see Brian Fiedler School of Meteorology at the
University of Oklahoma.
- "Antennae as Gyroscopes", Science, Vol. 315, 9 Feb 2007, p.
771
- Halteres for the micromechanical flying insect (Wu,
W.C.; Wood, R.J.; Fearing, R.S.) Dept. of Electr. Eng. &
Comput. Sci., California Univ., Berkeley, CA; This paper appears
in: Robotics and Automation, 2002. Proceedings. ICRA '02. IEEE
International Conference on Publication Date: 2002 Volume: 1, On
page(s): 60- 65 vol.1 ISBN 0-7803-7272-7 Date Published in Issue:
2002-08-07 00:46:34.0
Further reading: physics and meteorology
- Coriolis, G.G., 1832: Mémoire sur le principe des forces
vives dans les mouvements relatifs des machines. Journal de
l'école Polytechnique, Vol 13, 268–302.
( Original article [in French], PDF-file, 1.6 MB,
scanned images of complete pages.)
- Coriolis, G.G., 1835: Mémoire sur les équations du
mouvement relatif des systèmes de corps. Journal de l'école
Polytechnique, Vol 15, 142–154
( Original article [in French] PDF-file, 400 KB,
scanned images of complete pages.)
- Gill, AE Atmosphere-Ocean dynamics, Academic Press,
1982.
- Durran, D. R.,
1993: Is the Coriolis force really responsible for the
inertial oscillation?, Bull. Amer. Meteor. Soc., 74,
2179–2184; Corrigenda. Bulletin of the American Meteorological
Society, 75, 261
- Durran, D. R., and S. K. Domonkos, 1996: An apparatus for demonstrating the inertial
oscillation, Bulletin of the American Meteorological
Society, 77, 557–559.
- Marion, Jerry B. 1970, Classical Dynamics of Particles and
Systems, Academic Press.
- Persson, A., 1998 [866] How do we Understand the Coriolis
Force? Bulletin of the American Meteorological Society 79,
1373–1385.
- Symon, Keith. 1971, Mechanics, Addison-Wesley
- Phillips, Norman A., 2000 An Explication of the Coriolis Effect,
Bulletin of the American Meteorological Society: Vol. 81, No. 2,
pp. 299–303.
- Akira Kageyama & Mamoru Hyodo: Eulerian derivation
of the Coriolis force
- James F. Price: A Coriolis tutorial Woods Hole
Oceanographic Institute (2003)
- MIT essays by James F. Price, Woods Hole
Oceanographic Institution (2006)
Further reading: historical
- Grattan-Guinness, I., Ed., 1994: Companion Encyclopedia of
the History and Philosophy of the Mathematical Sciences. Vols.
I and II. Routledge, 1840 pp.
1997: The Fontana History of the Mathematical Sciences.
Fontana, 817 pp. 710 pp.
- Khrgian, A., 1970: Meteorology — A Historical Survey.
Vol. 1. Keter Press, 387 pp.
- Kuhn, T. S., 1977: Energy conservation as an example of
simultaneous discovery. The Essential Tension, Selected Studies
in Scientific Tradition and Change, University of Chicago
Press, 66–104.
- Kutzbach, G., 1979: The Thermal Theory of Cyclones.
A History of Meteorological Thought in the Nineteenth
Century. Amer. Meteor. Soc., 254 pp.
External links