Conservation of Angular Momentum

1
Conservation of Angular Momentum

• Jeff Gawrych
• Met. 280
• Spring 2004

2
Introduction

• The conservation of angular momentum explains
  some basic physical properties of rotating
  objects (like the earth).
• It relates to many weather and climate phenomena
  such as
• winds around tornadoes and hurricanes
• mid-latitude westerly winds and tropical
  easterlies
• ENSO

3
(No Transcript)
4
Review of Conservation Laws

• Conservation of Mass Matter cannot be created
  or destroyed
• Conservation of Energy Energy cannot be created
  or destroyed
• Conservation of Linear Momentum Linear momentum
  cannot be created or destroyed
• Conservation of Angular Momentum Angular
  Momentum cannot be created or destroyed
• Thus, mass, energy, and momentum are transferred
  back and forth within the earth and atmosphere

5
Momentum

• Momentum Mass X Velocity
• Can be linear or angular
• Linear deals with issues such as collisions.
• E.g..Billiard balls and collisions
• Angular due to rotation of planet.
• Seen jet stream winds, flow around cyclones,
  mid-latitude westerlies, tropical easterlies
• In meteorology, wind is the biggest player
• Examples of conservation of angular momentum
  include a figure skater going into spin and water
  going down a drain.

6
Linear momentum
7
Angular momentum
8
Conservation of Angular Momentum (COAM) definition

• The conservation of angular momentum (COAM) is a
  law of physics that states the total angular
  momentum of a rotating object with no outside
  force remains constant regardless of changes
  within the system.

9
On earth

• COAM holds provided there are no outside forces
  acting on planet (torques)
• Earth system consists of solid earth and
  atmosphere so AMx mtvtRt meVeRe mavaRa
  constant, where AMx is the angular momentum of
  the component
• AM balance depends on both AM of earth and AM of
  atmosphere.
• AM of earth and AM of atmosphere are inversely
  proportional.
• E.g., if westerlies winds increase ? AM atm inc.
  ? AM earth dec. ? length of day inc.

10
COAM Facts

• Atmosphere gains AM from the earth in the tropics
  where surface winds are easterly (i.e. where AM
  of atmosphere lt AM of earth)
• Atmosphere gives up AM in the mid-latitudes where
  surface winds are westerly.
• There is a net pole ward transport of AM w/in
  atmosphere, otherwise the torque owing to surface
  friction would decelerate both the easterlies and
  westerlies.
• Easterlies and westerlies must balance out to
  stay in balance, and conserve AM

11
COAM Facts

• The sum of the angular momentum (push) of the
  solid Earth plus atmosphere system must stay
  constant unless an outside force (torque) is
  applied.
• So if the atmosphere speeds up (stronger
  westerly winds) then the solid Earth must slow
  down (length-of-day increases).

12
COAM Facts

• Also, if more atmosphere moves to a lower
  latitude (further from the axis of rotation), and
  atmospheric pressure increases, it also gains
  angular momentum and the Earth would slow down as
  well.
• Other motions of the atmosphere such as larger
  mass in one hemisphere than the other can lead to
  a wobble (like a washing machine with clothes
  off-balance) and the poles move, in accordance to
  the law of the conservation of angular momentum.

13
COAM example

• Consider the following An object initially at
  rest with respect to the earth at 20N, where it
  has the same angular velocity of the earth, is
  taken to 30N.
• Question What happens to the object's
  tangential and angular velocities?
• Answer As the object moves to the North the
  distance to the earth's axis decreases. Thus,
  both the angular velocity and tangential velocity
  must increase the object moves to the east at a
  faster rate than the earth itself. Therefore,
  there is a deflection to the east. This is the
  Coriolis effect.

14
So what?

• It explains basic principles that are often taken
  for granted.
• COAM influences general circulation and jet
  streams
• Explains why the general circulation is not one a
  single Hadley-cell.
• How?
• The conservation of angular momentum prevents a
  single cell from occurring by causing air
  transported from the equator to flow eastward in
  the Northern Hemisphere.

15
A single-cell model of the general circulation
(Hadley Cell)

• At the equator, atmosphere is warmer due to more
  solar insolation --gt warm air rises
• At poles, atmosphere is cold, and therefore dense
  --gt air sinks

16
COAM in the General Circulation

• In single-cell model, upper levels winds are from
  equator to poles To conserve angular momentum,
  as r decreases, v must increase.
• This would create immense westerlies -gt
  unrealistic, and dynamically impossible
  conditions --gt not observed
• Transfer of momentum from Earth to atmosphere.
  Earths rotation toward east would stop surface
  wind motion toward west (an easterly wind) in
  time.

17
COAM in the General Circulation

• Not everywhere on Earth can there be surface
  winds from the same direction.
• Easterly winds must be balanced by westerly winds
  somewhere else.
• Result
• Subtropical jet steam occurs between Hadley and
  Ferrel cell

18
Figures a,b,c, illustrate a single-cell
circulation.
Figure d is the 3-cell circulation we experience
due to conservation of angular momentum.
19
(No Transcript)
20
Another COAM Example

• Winds around a strong cyclone (e.g. a hurricane)
  are very strong.
• The quantity VR (angular momentum) is constant
  for any given air parcel (can differ from parcel
  to parcel).
• V is the tangential wind
• R is the radial distance from the center of the
  low (e.g., eye of a hurricane)

21
Conservation of Angular Momentum in a Hurricane
Eye
V x R Constant
V
V is the tangential wind R is the
radialdistance of the air parcel from the
hurricane eye
Eye
R
Hurricane
22
Conservation of Angular Momentum in a Hurricane
Eye
V x R Constant
V
What happens asan air parcelspirals
inwardtoward the centerof the hurricane?
Eye
R
23
Conservation of Angular Momentum in a Hurricane
Eye
V1
V x R Constant
What happens asan air parcelspirals
inwardtoward the centerof the hurricane?
Eye
R1
24
Conservation of Angular Momentum in a Hurricane
Eye
V1
V x R Constant
What happens asan air parcelspirals
inwardtoward the centerof the hurricane?
Eye
R1
V2
25
Conservation of Angular Momentum in a Hurricane
Eye
V1
V x R Constant
What happens asan air parcelspirals
inwardtoward the centerof the hurricane?
Eye
R1
R2
V2
26
Conservation of Angular Momentum in a Hurricane
V x R Constantsimply means that
Eye
V1
V1 x R1 V2 x R2
Eye
R1
R2
V2
27
Conservation of Angular Momentum in a Hurricane
Eye
V1
Let V1 10 ktsR1 500 km If R2 30 km,
thenusing the equation V1 x R1 V2 x R2 we find
that V2 (V1xR1)/R2 V2 167 kts!!!
Eye
R1
R2
V2
28
Conservation of Angular Momentum in a Hurricane
Note spiral bands converging towardthe center
29
What else?

• The same mechanism is at work in tornadoes, or
  any rotating weather system.
• The flow is in the tangential direction, or in
  the direction of spin, but there also exists a
  radial inflow towards the center of the vortex,
  or a spiraling inward flow.

30
ENSO

• El Nino events cause trade wind inversion, so
  easterlies become westerlies.
• This increases the AM of the atmosphere
• AM of the solid earth must decrease to satisfy
  COAM
• Results
• earth rotation rate decreases
• Length of day increases

31
Seasonal Variations

• Seasonal variations in the AM budget are due to
  wind and pressure distributions
• For example, AM of the atmosphere reaches an
  annual maximum in winter/spring. Why?
• Larger N-S temperature gradient ? strong jet
  stream (westerlies)
• More topography in NH ? increase of mountain
  torque ? pressure and wind fluctuations

32
Conclusion

• Angular momentum is important because we live on
  a rotating planet.
• Have a huge role in atmospheric circulation and
  weather events
• Atmosphere gains AM from the earth in the
  tropics, therefore winds are easterly.
• Atmosphere gives up AM to the earth in the
  mid-latitudes, therefore winds are westerly.
• Westerly and easterly flow must balance to
  satisfy COAM.
• El Nino events or other circulation anomalies
  significantly alter angular momentum budget of
  earth.

33
Conclusion

• The angular momentum balance is highly variable
  and sensitive to additional torques such as winds
  and ocean currents
• Westerly winds atmosphere is rotating quicker
  than the earth
• Paves the way for the subtropical jet stream
• mV1R1 mV2R2 constant