Download Conservation of Angular Momentum

Conservation of Angular
Momentum
Jeff Gawrych
Met. 280
Spring 2004
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
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
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, midlatitude 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.
Linear momentum
Angular momentum
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.
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.
COAM Facts




Atmosphere gains AM from the earth in the
tropics where surface winds are easterly (i.e.
where AM of atmosphere < 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
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).
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.
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.
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.
A single-cell model of the
general circulation (Hadley Cell)
•
•
At the equator,
atmosphere is warmer
due to more solar
insolation --> warm
air rises
At poles, atmosphere
is cold, and therefore
dense --> air sinks
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 > unrealistic, and dynamically impossible
conditions --> not observed
Transfer of momentum from Earth to
atmosphere. Earth’s rotation toward east
would stop surface wind motion toward west
(an easterly wind) in time.
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
Figures a,b,c, illustrate a single-cell circulation.
Figure d is the 3-cell circulation we experience due to
conservation of angular momentum.
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)
Conservation of Angular
Momentum in a Hurricane
V x R = Constant
V is the “tangential”
wind
R is the radial
distance of the air
parcel from the
hurricane eye
V
Eye
Eye
R
Hurricane
Conservation of Angular
Momentum in a Hurricane
V x R = Constant
V
What happens as
an air parcel
spirals inward
toward the center
of the hurricane?
Eye
Eye
R
Conservation of Angular
Momentum in a Hurricane
V x R = Constant
What happens as
an air parcel
spirals inward
toward the center
of the hurricane?
V1
Eye
Eye
R1
Conservation of Angular
Momentum in a Hurricane
V x R = Constant
What happens as
an air parcel
spirals inward
toward the center
of the hurricane?
V1
Eye
Eye
R1
V2
Conservation of Angular
Momentum in a Hurricane
V x R = Constant
What happens as
an air parcel
spirals inward
toward the center
of the hurricane?
V1
Eye
Eye
R1
R2
V2
Conservation of Angular
Momentum in a Hurricane
V x R = Constant
simply means that
V1 x R1 = V2 x R2
V1
Eye
Eye
R1
R2
V2
Conservation of Angular
Momentum in a Hurricane
Let V1 = 10 kts
R1 = 500 km
If R2 = 30 km, then
using the equation
V1 x R1 = V2 x R2
we find that
V2 = (V1xR1)/R2
V2 = 167 kts!!!
V1
Eye
Eye
R1
R2
V2
Conservation of Angular
Momentum in a Hurricane
Note spiral bands
converging toward
the center
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.

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
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
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 midlatitudes, 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.
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