Earth Rotation

• Earth’s rotation gives rise to a fictitious

force called the Coriolis force

• It accounts for the apparent deflection of

motions viewed in our rotating frame

• Analogies

– throwing a ball from a merry-go-round

– sending a ball to the sun

Earth Rotation

• Earth rotates about its axis wrt sun (2

rad/day)

• Earth rotates about the sun (2 rad/365.25 day)

• Relative to the “distant stars” (2 rad/86164 s)

– Sidereal day = 86164 sec (Note: 24 h = 86400 sec)

• Defines the Earth’s rotation frequency,

= 7.29 x 10-5 s-1 (radians per sec)

Earth Rotation

• Velocity of Earth surface

• Ve(Eq) = Re

Re = radius Earth (6371 km)

Ve(Eq) = 464 m/s

• As latitude, , increases,

Ve() will decrease

• Ve() = Re cos()

Ve Decreases with Latitude

-80 -60 -40 -20 0 20 40 60 800

50

100

150

200

250

300

350

400

450

500

latitude (N)

Vearth

(m/s)

Ve() = Re cos()

Earth Rotation

• Moving objects on Earth move with the rotating frame

(Ve()) & relative to it (vrel)

• The absolute velocity is vabs = vrel + Ve()

• Objects moving north from Equator will have a larger

Ve than that under them

• If “real” forces sum to 0, vabs will not change, but

the Ve() at that latitude will

Rotation, cont.

• Frictionless object moving north

vabs = const., but Ve() is decreasing

vrel must increase (pushing the object east)

• When viewed in the rotating frame, moving objects appear deflected to right (left SH)

• Coriolis force accounts for this by proving a “force” acting to the right of motion

Coriolis Forcean object with an initial east-west

velocity will maintain that velocity, even as it passes over surfaces with different velocities.

As a result, it appears to be deflected over that

surface (right in NH, left in

SH)

Coriolis Force and Deflection of Flight Path

http://www.youtube.com/watch?v=_36MiCUS1ro

http://www.youtube.com/watch?v=49JwbrXcPjc

http://www.youtube.com/watch?v=KdD3Wq2DCWQ

Earth Rotation

• Motions in a rotating frame will appear to

deflect to the right (NH)

• Deflection will be to the right in the northern

hemisphere & to left in southern hemisphere

• No apparent deflection right on the equator

• It’s a matter of frame of reference,

there is NO Coriolis force…

Wind Stress

• Wind stress, w, accounts for the input

of momentum into the ocean by the wind

• Exact processes creating w is complex

• w is a tangential force per unit area

• Units are Newton (force) pre meter squared

F = ma -> 1 Newton = 1 N = 1 kg (m s-2)

N m-2 = kg m-1 s-2

Wind Stress

• Wind stress is modeled as w =

C U2

where C ~ 2x10-3 & U is wind speed

• Values of C can vary by factor of 2

Wind Stress

• Calculations…

If U = 15 knots, what is the wind stress?

• Steps

– Convert U in knots to U in m/s

– Calculate w

Wind Stress

Facts:

1o latitude = 60 nautical miles = 111 km

15 knots = 15 nautical miles / hour

€

15knots = ...

15nmile

hour

⎛

⎝ ⎜

⎞

⎠ ⎟1hour

602 sec

⎛

⎝ ⎜

⎞

⎠ ⎟111x1000m

60nmile

⎛

⎝ ⎜

⎞

⎠ ⎟=

7.7m/s

Wind Stress

Finishing up the calculation...

w = C U2

= (2x10-3) (7.7 m/s)2

= 0.12 N/m2

We’re done!!

But what were the units of C?

What are the units of C?

•We know that w = C U2

w =[N/m2] = [kg m-1 s-2] & U2 = [(m/s)2]

C = [kg m-1 s-2] / [m 2 s-2] = [kg m-3]

-> C ~ 2x10-3 kg m-3

•Typically, C is defined as a CD

a = density air & CD = drag

coefficient

Wind Stress

• Many processes contribute to transfer of momentum from wind to the ocean

– Turbulent friction

– Generation of wind waves

– Generation of capillary waves

• Key is the recognition that the process is turbulent

Wind Stress

Vertical eddy viscosity quantifies the air-sea exchanges of horizontal momentum

Vertical Eddy Viscosity

• Vertical eddy viscosity, Az, controls the efficiency

of wind momentum inputs

• High values of Az suggest deeper penetration of

momentum into the ocean

• Values of Az are functions of – turbulence levels– wave state– stratification near the surface

Vertical Eddy Viscosity

•Similar to discussion of eddy diffusion (turbulence mixes scalars & momentum similarly)

– Values of Az (vertical) << Ah (horizontal)

– Az decreases as stratification increases

– Az is at its greatest in the mixed layer

Review

• Wind stress accounts for the input of

momentum into the ocean by the wind

• Calculated using wind speed, w = C

U2

• Processes driving wind stress &

vertical eddy viscosity are very

complex

Ekman Transport

• Ekman transport is the direct wind driven transport of seawater

• Boundary layer process

• Steady balance among the wind stress, vertical eddy viscosity & Coriolis forces

• Story starts with Fridtjof Nansen [1898]