Angular Momentum Cycle

Angular Momentum Cycle

1. Balance Equations

2. Angular Momentum in the Climatic System

3. Observations

4. Closing the Cycle of Angular Momentum

Balance EquationsThe total angular momentum of the Earth remains constant

M Angular momentum of a parcel with unit mass

acrM Frdt

Md

If the total torque vanishes 0dt

Md

n

ncrrM .)(

coscos22 urrM icu .M Mr

rMMM

Fr is the moment of force (torque).

rR

smR /464Eq.:

Angular Momentum in the Climatic System

1. Earth

s

mkgRmI ee

2332 .

1086.55

2

2a. Atmosphere (solid rotation)

s

mkgXIRmI eaa

22862 .

1001.1103

2

sphere

spherical shell

2b. Atmosphere (zonal wind, the relative angular momentum)

MM

dmuRdmM

r

r

01.0

cos Mr Mr (DJF-JJA)NH~5.31025kg m2 s-1 NH~9.41025kg m2 s-1

SH ~7.61025kg m2 s-1 SH ~-4.6 1025kg m2 s-1

me= ma~106

3. Ocean (very coarse estimate, no reliable measurements exist)

a) Zonal circulation

450

00

100Sv

NP

-0.51025kg m2 s-1

The observed changes of the angular momentum of atmosphere are 51025kg m2 s-1, Oceanic ones are< 11025kg m2 s-1

-600

600

12254

2

6

36

6

34

108.0

cos2cos2

skgmzR

ddzRM

b) Meridional shift of air and water masses

300

00

+ + +

z=2cm

Patm=2mb0.81025kg m2 s-1

Conclusion: Adjustment of the solid Earth‘s rotation to the rotation of fluid

Conclusion: Adjustment of the solid Earth‘s rotation to the rotation of fluid

Because the angular momentum is conserved

atmV

e constdvurI cos

Because MrDJF > Mr

JJA , JJA > DJF

LOD[ms ]=0.168 Mr[1025kg m2 s-1]

If Mr=5x1025kg m2 s-1 LOD=0.8ms

The relative angular momentum can be computed actronomically, as well as from the observed velocities:

Mr =5x1025kg m2 s-1 correponds to u=2m s-1.

dmuRdmM r cos

Angular Momentum in the Atmosphere

Multiply the equation of momentum

...cos

Fp

dt

dca

with r

cos

1

1

Rx

RyHave in mind:

...cos

RF

p

t

M

Pressure, Friction Torques

Tropics-sourceMid-latitudes-sink of angular momentum} Meridional

Transportt

Observations

uvCD 0

Ship reports

Source

Sink

Closing the Cycle of Angular Momentum

2

2

10 ~] [s

mvu

Wind~10ms-1

Currents~ 10-2ms-1 -10-1ms-1

2

2310~][s

mvu

410~][

][

ocean

atm

vu

vu 210~atm

ocean

m

m

Small contribution of the oceanic transport