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Effects of the Earth’s Rotation
C. Chen
General Physical Oceanography
MAR 555
School for Marine Sciences and Technology
Umass-Dartmouth
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One of the most important physical processes controlling the temporal and
spatial variations of biological variables (nutrients, phytoplankton,
zooplankton, etc) is the oceanic circulation. Since the circulation exists on
the earth, it must be affected by the earth’s rotation.
Question:
How is the oceanic circulation affected by the earth’s rotation?
The Coriolis force!
Question: What is the Coriolis force? How is it defined? What is the
difference between centrifugal and Coriolis forces?
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Definition:
• The Coriolis force is an apparent force that occurs when the fluid moves
on a rotating frame.
• The centrifugal force is an apparent force when an object is on a rotation
frame.
Based on these definitions, we learn that
• The centrifugal force can occur when an object is at rest on a rotating
frame;
•The Coriolis force occurs only when an object is moving relative to the
rotating frame.
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Centrifugal Force
Consider a ball of mass m attached to a string spinning around a
circle of radius r at a constant angular velocity ω.
ω ω
r
Conditions:
1) The speed of the ball is constant, but its direction is continuously changing;
2) The string acts like a force to pull the ball toward the axis of rotation.
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Let us assume that the velocity of the ball:
V + !V " V = !V
V at t
V + !V at t + !t
! V = V!"
!V
!"
=V
, limit !t # 0,
!t
!t
!V
"!
V
V = % r, and
V
ω
"!
r
dV
d"
d"
r
=V
=V
($ )
dt
dt
dt
r
d"
= %,
dt
Therefore,
dV
= $& 2r
dt
To keep the ball on the circle track, there must exist an
additional force, which has the same magnitude as the
centripetal acceleration but in an opposite direction.
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This force is called “centrifugal force”, and is equals to
Fcf = ! 2r
Ω
On the earth, the centrifugal force is equal to
R
Fcf
Fcf = !2 R
!
where Ω is the angular velocity of the earth’s rotation and R is the position
vector from the axis of rotation to be object at a given latitude.
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The Coriolis Force
ω
t1
t2
t3
t1
t2
t3
When an objective is moving with respect to a rotating frame, an additional
apparent force appears, which tends to change the direction of the motion.
The Coriolis force!
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Important Concepts:
• Any object on a rotating frame is subject to a centrifugal force no matter
whether or not it moves.
• The Coriolis force exists only when the object moves on a rotating frame.
• The Coriolis force only changes the direction of the motion.
• The centrifugal force could accelerate the motion.
Questions:
How do we define the Coriolis force on the rotating earth?
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Ω
R
u
Assume that a fluid parcel moves eastward at a speed of u. Since
this parcel moves faster than the earth rotation, so the angular
velocity acting on this parcel should be equal to a sum of the
angular velocities of the earth and movement of the parcel as
follows:
!+u/R
Therefore, the centrifugal force exerting on this parcel is
equal to
u
Fcf = (! + ) 2 R
R
Then,
u
Fcf = (! + ) 2 R = !2 R
R
2!uR
+
+
R
Centrifugal force
u2R
R2
Too small
Coriolis force component
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( Fc ) y = #2"u sin ! , ( Fc ) z = #2"u cos !
2"u cos !
R
2! u
θ
2"u sin !
R
R
Since (Fc)z << g in the vertical, it can be ignored.
Therefore,
u
Coriolis force
on the northern hemisphere
Usually, we define that f = 2" sin ! as the Coriolis parameter.
Fc = fvi ! fuj = ! fk " v
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Properties
1.
The Coriolis force is a three-dimensional force. The vertical
component of the Coriolis force is generally ignored in the large-scale
ocean study because it is much smaller than gravity.
2.
In the northern hemisphere, the Coriolis force acts 90o degree to the
right of the current direction, while in the southern hemisphere, it is
90o degree to the left of the current direction. This is a very important
concept.
3.
The Coriolis force changes with latitude and the amplitude of the
currents. At the equator, the Coriolis force equals zero and it increases
as the latitude increases towards the poles.
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Questions: How could the Coriolis effect influence the oceanic circulation?
Example 1: Inertial (or near-inertial) motion
A fluid parcel
Movement direction
without the Coriolis
effect
The Coriolis force
t = 0,
u = 0, v = vo
Discussion:
a) u 2 + v 2 = vo2
b) Inertial period:
This is a circle!
Tf =
2!
f
t=π/2f,
u = v o, v = 0
t=3π/2f,
u =-vo, v= 0
t=π/f,
u= 0,, v = -vo
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The inertial period decreases with latitude,
1)
at equator: Tf →∞: no inertial motion because f = 0;
2)
at 30o N: Tf = 23.9 hours
3)
at 45o N, Tf = 17 hours
4)
at 90o N, Tf = 12 hours
In the real ocean, an inertial oscillation is usually caused by a sudden change
of the wind stress. If you trace a drifter, its trajectory would look like
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The Louisiana-Texas Shelf Monitoring Sites
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Cross-shelf distribution of the variance of the near-inertial currents
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Clockwise rotation of the wind direction during the cold-frontal passage
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Example 2: Defining the scale of motion
Distance: L
Advective time scale: O(L/U)
Speed: U
For the Coriolis force-induced inertial motion,
Ro =
Inertial time scale: O(1/f)
Inertial time scale
O(1/ f )
U
=
= O( )
Advective time scale O(L /U)
fL
Rossby number
The scale of motion is defined by the magnitude of the Rossby number
Ro <<1, Large - scale :
Coriolis force is dominant
R o ~ 1,
Coriolis force is important and can not be ignored
Meso - scale :
R o >>1, Small - scale
Coriolis force can be ignored
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Example 3: Geostrophic currents
FP : The pressure gradient force;
Fc : The Coriolis force
t0
Low
P0
FP (The pressure gradient force)
t1
t2
P1
P2
High
Vg
Fc
FP
Fc
Fc
Fc (The Coriolis force)
Coriolis force = Pressure gradient force
Geostrophic currents
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Example 4: Ekman Transport, Currents and Pumping
Without the Coriolis force, the water moves following
the force direction.
With the Coriolis force,
(wind stress)
!s
!s
to
t1
90o
VE Ekman transport
Fc
Fc
Ekman mass transport:
!
VE = s
f
Ekman volume transport: VE =
!s
"f
Fc
Coriolis force = Surface wind stress
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Consider the wind-driven Ekman currents in the vertical water column
!s
At the surface:
V
!s
45o
!1
45o
vE
Fc
hE
Below the surface:
!2
VE (Ekman volume transport)
!v
V
Fc
surface current
!
vE
45o
!
90o
VE
Current below the surface
Clockwise rotates with depth
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Ekman pumping
hE
hE
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Discussion:
a) Current profile:
The Ekman velocity decreases and rotates clockwise with depth:
Ekman spiral.
b) The Ekman layer thickness (depth):
hE =
directly proportional to turbulent viscosity coefficient and inversely
proportional to the Coriolis parameter.
2K m
f
b) The direction of the surface Ekman current:
v
tan ! = E = 1 , ! = 45 o
uE
The angle between the wind stress and surface Ekman current
is 45o. On the northern hemisphere, the surface Ekman current
is 45o on the right of the wind stress.
c) The total volume transport:
τs
45o
!
vE
transport
The volume transport is always 90o to the direction of the
wind stress. In the northern hemisphere, it is to the right of the
wind stress.
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Suggested reading:
Chen, C., R. O. Reid, and W. D. Nowlin, 1996. Near-inertial oscillations over the
Texas-Louisiana shelf, Journal of Geophysical Research, 101, 3509-3524.
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