Effects of the Earth’s Rotation - HOME - FVCOMfvcom.smast.umassd.edu/Courses/MAR555/Lectures_pdf/MAR555_Lec_3.pdf · Effects of the Earth’s Rotation C. Chen General Physical Oceanography

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Effects of the Earth’s Rotation

C. Chen

General Physical OceanographyMAR 555

School for Marine Sciences and TechnologyUmass-Dartmouth

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Question:

One of the most important physical processes controlling the temporal andspatial variations of biological variables (nutrients, phytoplankton,zooplankton, etc) is the oceanic circulation. Since the circulation exists onthe earth, it must be affected by the earth’s rotation.

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 thedifference between centrifugal and Coriolis forces?

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Definition:

• The Coriolis force is an apparent force that occurs when the fluid moveson a rotating frame.

• The centrifugal force is an apparent force when an object is on a rotationframe.

Based on these definitions, we learn that • The centrifugal force can occur when an object is at rest on a rotatingframe;

•The Coriolis force occurs only when an object is moving relative to therotating frame.

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Centrifugal Force

Consider a ball of mass m attached to a string spinning around acircle of radius r at a constant angular velocity ω.

ωr

ω

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 at t

V + !V at t + !t

rω!"

�

V

!"

�

V

�

!V

�

V + !V "V = !V

�

!V = V!"

�

!V

!t= V

!"

!t, limit !t # 0,

d V

dt= V

d"

dt= V

d"

dt($

r

r)

V = % r, and d"

dt= %,

Therefore,

d V

dt= $&

2r

To keep the ball on the circle track, there must exist anadditional force, which has the same magnitude as thecentripetal acceleration but in an opposite direction.

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This force is called “centrifugal force”, and is equals to

�

Fcf = !2r

On the earth, the centrifugal force is equal to

�

Fcf = !2R

where Ω is the angular velocity of the earth’s rotation and R is the positionvector from the axis of rotation to be object at a given latitude.

�

R

! �

Fcf

Ω

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The Coriolis Force

t1 t2 t3 t1

t2

t3

ω

When an objective is moving with respect to a rotating frame, an additionalapparent 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 matterwhether 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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u

Assume that a fluid parcel moves eastward at a speed of u. Sincethis parcel moves faster than the earth rotation, so the angularvelocity acting on this parcel should be equal to a sum of theangular velocities of the earth and movement of the parcel asfollows:

Ω

Ru /+!

Therefore, the centrifugal force exerting on this parcel isequal to

�

Fcf = (!+u

R)2R

R

Then,

�

Fcf = (!+u

R)2R = !

2R +

2!uR

R+

u2R

R2

Centrifugal force Too small

Coriolis force component

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�

2!uR

R

!sin2 u"

!cos2 u"

θ�

R

!! cos2)(,sin2)( uFuF zcyc "#="#=

Since (Fc)z << g in the vertical, it can be ignored.

Therefore,

u

Coriolis forceon the northern hemisphere

Usually, we define that !sin2"=f as the Coriolis parameter.

�

Fc = fvi ! fuj = ! fk " v

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Properties

1. The Coriolis force is a three-dimensional force. The verticalcomponent of the Coriolis force is generally ignored in the large-scaleocean study because it is much smaller than gravity.

2. In the northern hemisphere, the Coriolis force acts 90o degree to theright of the current direction, while in the southern hemisphere, it is90o degree to the left of the current direction. This is a very importantconcept.

3. The Coriolis force changes with latitude and the amplitude of thecurrents. At the equator, the Coriolis force equals zero and it increasesas 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

Discussion:

�

u2

+ v2

= vo

2a) This is a circle!

t=π/f, u= 0,, v = -vo

t = 0, u = 0, v = vo

t=π/2f, u = vo, v = 0

t=3π/2f, u =-vo, v= 0

b) Inertial period: f

T f!2

=

A fluid parcel

Movement directionwithout the Corioliseffect

The Coriolis force

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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

The inertial period decreases with latitude,

In the real ocean, an inertial oscillation is usually caused by a sudden changeof 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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�

Ro =Inertial time scale

Advective time scale=O(1/ f )

O(L /U)=O(

U

fL)

�

Ro

<<1, Large - scale : Coriolis force is dominant

R o ~ 1, Meso - scale : Coriolis force is important and can not be ignored

R o >>1, Small - scale Coriolis force can be ignored

The scale of motion is defined by the magnitude of the Rossby number

Example 2: Defining the scale of motion

Distance: L

Speed: U

Advective time scale: O(L/U)

For the Coriolis force-induced inertial motion, Inertial time scale: O(1/f)

Rossby number

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Example 3: Geostrophic currents

Coriolis force = Pressure gradient force

�

FP

: The pressure gradient force;

Fc

: The Coriolis force

(The pressure gradient force)

�

FP

�

Fc

�

Vg

(The Coriolis force)

t0

P2

t1

t2

�

Fc

�

Fc

�

Fc

�

FP

High

LowP0

P1

Geostrophic currents

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Example 4: Ekman Transport, Currents and Pumping

Without the Coriolis force, the water moves followingthe force direction.

With the Coriolis force,

�

!s

(wind stress)to

Fc

t1

Fc

�

!s

Fc

VE Ekman transport90o

Coriolis force = Surface wind stress

�

VE =! s

fEkman mass transport:

Ekman volume transport:

�

VE =! s

"f

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Consider the wind-driven Ekman currents in the vertical water column

At the surface:

�

!s

45o

Fc

V

�

!1

Below the surface:

�

!v

FcV

�

!2

Clockwise rotates with depth

surface current

Current below the surface

hE

�

VE

(Ekman volume transport)

�

vE

�

!s

�

!

�

vE

45o

45o

90o

�

VE

�

!

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hE

Ekman pumping

hE

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directly proportional to turbulent viscosity coefficient and inverselyproportional to the Coriolis parameter.

Discussion:

b) The Ekman layer thickness (depth):

f

Kh mE

2=

b) The direction of the surface Ekman current:

o

E

E

u

v45,1tan === !!

The angle between the wind stress and surface Ekman currentis 45o. On the northern hemisphere, the surface Ekman currentis 45o on the right of the wind stress.

c) The total volume transport:

The volume transport is always 90o to the direction of thewind stress. In the northern hemisphere, it is to the right of thewind stress.

�

! v

E

τs45o

transport

a) Current profile:The Ekman velocity decreases and rotates clockwise with depth: Ekman spiral.

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Suggested reading:

Chen, C., R. O. Reid, and W. D. Nowlin, 1996. Near-inertial oscillations over theTexas-Louisiana shelf, Journal of Geophysical Research, 101, 3509-3524.