Conservation of Angular Conservation of Angular MomentumMomentum

Jeff GawrychJeff Gawrych

Met. 280Met. 280

Spring 2004Spring 2004

IntroductionIntroduction

The conservation of angular momentum The conservation of angular momentum explains some basic physical properties explains some basic physical properties of rotating objects (like the earth).of rotating objects (like the earth).

It relates to many weather and climate It relates to many weather and climate phenomena such asphenomena such as– winds around tornadoes and hurricaneswinds around tornadoes and hurricanes– mid-latitude westerly winds and tropical mid-latitude westerly winds and tropical

easterlieseasterlies– ENSO ENSO

Review of Conservation Review of Conservation LawsLaws

Conservation of Mass: Matter cannot be created Conservation of Mass: Matter cannot be created or destroyedor destroyed

Conservation of Energy: Energy cannot be Conservation of Energy: Energy cannot be created or destroyedcreated or destroyed

Conservation of Linear Momentum: Linear Conservation of Linear Momentum: Linear momentum cannot be created or destroyedmomentum cannot be created or destroyed

Conservation of Angular Momentum: Angular Conservation of Angular Momentum: Angular Momentum cannot be created or destroyedMomentum cannot be created or destroyed

Thus, mass, energy, and momentum are Thus, mass, energy, and momentum are transferred back and forth within the earth and transferred back and forth within the earth and atmosphereatmosphere

MomentumMomentum Momentum = Mass X VelocityMomentum = Mass X Velocity Can be linear or angularCan be linear or angular Linear: deals with issues such as collisions.Linear: deals with issues such as collisions.

– E.g..Billiard balls and collisionsE.g..Billiard balls and collisions Angular: due to rotation of planet.Angular: due to rotation of planet.

– Seen jet stream winds, flow around cyclones, Seen jet stream winds, flow around cyclones, mid-latitude westerlies, tropical easterliesmid-latitude westerlies, tropical easterlies

– In meteorology, wind is the biggest playerIn meteorology, wind is the biggest player– Examples of conservation of angular momentum Examples of conservation of angular momentum

include a figure skater going into spin and water include a figure skater going into spin and water going down a draingoing down a drain..

Linear momentum

Angular momentum

Conservation of Angular Conservation of Angular Momentum (COAM) Momentum (COAM)

definitiondefinition

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

On earthOn earth COAM holds provided there are no outside forces COAM holds provided there are no outside forces

acting on planet (torques)acting on planet (torques) Earth system consists of solid earth and Earth system consists of solid earth and

atmosphere so AMatmosphere so AMxx = m = mttvvttRRtt = m = meeVVeeRRee + m + maavvaaRRaa = = constant, where AMconstant, where AMxx is the angular momentum of is the angular momentum of the componentthe component

AM balance depends on both AM of earth and AM of AM balance depends on both AM of earth and AM of atmosphere.atmosphere.

AM of earth and AM of atmosphere are inversely AM of earth and AM of atmosphere are inversely proportional.proportional.

E.g., if westerlies winds increase E.g., if westerlies winds increase AM atm inc. AM atm inc. AM earth dec. AM earth dec. length of day inc. length of day inc.

COAM FactsCOAM Facts Atmosphere gains AM from the earth in the Atmosphere gains AM from the earth in the

tropics where surface winds are easterly (i.e. tropics where surface winds are easterly (i.e. where AM of atmosphere < AM of earth) where AM of atmosphere < AM of earth)

Atmosphere gives up AM in the mid-latitudes Atmosphere gives up AM in the mid-latitudes where surface winds are westerly.where surface winds are westerly.

There is a net pole ward transport of AM w/in There is a net pole ward transport of AM w/in atmosphere, otherwise the torque owing to atmosphere, otherwise the torque owing to surface friction would decelerate both the surface friction would decelerate both the easterlies and westerlies.easterlies and westerlies.

Easterlies and westerlies must balance out to Easterlies and westerlies must balance out to stay in balance, and conserve AMstay in balance, and conserve AM

COAM FactsCOAM Facts

The sum of the angular momentum (push) The sum of the angular momentum (push) of the solid Earth plus atmosphere system of the solid Earth plus atmosphere system must stay constant unless an outside force must stay constant unless an outside force (torque) is applied.(torque) is applied.

So if the atmosphere speeds up (stronger So if the atmosphere speeds up (stronger westerly winds) then the solid Earth must westerly winds) then the solid Earth must slow down (length-of-day increases).slow down (length-of-day increases).

COAM FactsCOAM Facts Also, if more atmosphere moves to a lower Also, if more atmosphere moves to a lower

latitude (further from the axis of rotation), latitude (further from the axis of rotation), and atmospheric pressure increases, it also and atmospheric pressure increases, it also gains angular momentum and the Earth gains angular momentum and the Earth would slow down as well. would slow down as well.

Other motions of the atmosphere such as Other motions of the atmosphere such as larger mass in one hemisphere than the larger mass in one hemisphere than the other can lead to a wobble (like a washing other can lead to a wobble (like a washing machine with clothes off-balance) and the machine with clothes off-balance) and the poles move, in accordance to the law of poles move, in accordance to the law of the conservation of angular momentum.the conservation of angular momentum.

COAM exampleCOAM example Consider the following: An object initially at rest Consider the following: An object initially at rest

with respect to the earth at 20N, where it has the with respect to the earth at 20N, where it has the same angular velocity of the earth, is taken to 30N. same angular velocity of the earth, is taken to 30N.

Question:Question: What happens to the object's What happens to the object's tangential and angular velocities? tangential and angular velocities?

Answer:Answer: As the object moves to the North the As the object moves to the North the distance to the earth's axis decreases. Thus, both distance to the earth's axis decreases. Thus, both the angular velocity and tangential velocity must the angular velocity and tangential velocity must increase; the object moves to the east at a faster increase; the object moves to the east at a faster rate than the earth itself. Therefore, there is a rate than the earth itself. Therefore, there is a deflection to the east. This is the Coriolis effect.deflection to the east. This is the Coriolis effect.

So what?So what?

It explains basic principles that are It explains basic principles that are often taken for granted.often taken for granted.

COAM influences general circulation and jet COAM influences general circulation and jet streamsstreams

Explains why the general circulation is not Explains why the general circulation is not one a single Hadley-cell.one a single Hadley-cell.

How?How?– The conservation of angular momentum The conservation of angular momentum

prevents a single cell from occurring by causing prevents a single cell from occurring by causing air transported from the equator to flow air transported from the equator to flow eastward in the Northern Hemisphere.eastward in the Northern Hemisphere.

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

• At the equator, At the equator, atmosphere is warmer atmosphere is warmer due to more solar due to more solar insolation --> warm insolation --> warm air risesair rises

• At poles, atmosphere At poles, atmosphere is cold, and therefore is cold, and therefore dense --> air sinksdense --> air sinks

COAM in the General COAM in the General CirculationCirculation

In single-cell model, upper levels winds are In single-cell model, upper levels winds are from equator to poles: To conserve angular from equator to poles: To conserve angular momentum, as r decreases, v must increase. momentum, as r decreases, v must increase.

This would create immense westerlies -This would create immense westerlies -> unrealistic, and dynamically > unrealistic, and dynamically impossible conditions --> not observedimpossible conditions --> not observed

Transfer of momentum from Earth to Transfer of momentum from Earth to atmosphere. Earth’s rotation toward east atmosphere. Earth’s rotation toward east would stop surface wind motion toward west would stop surface wind motion toward west (an easterly wind) in time.(an easterly wind) in time.

COAM in the General COAM in the General CirculationCirculation

Not everywhere on Earth can there be Not everywhere on Earth can there be surface winds from the same direction.surface winds from the same direction.

Easterly winds must be balanced by Easterly winds must be balanced by westerly winds somewhere else.westerly winds somewhere else.

Result: Result: – Subtropical jet steam occurs between Subtropical jet steam occurs between

Hadley and Ferrel cellHadley 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 ExampleAnother COAM Example

Winds around a strong cyclone (e.g. a Winds around a strong cyclone (e.g. a hurricane) are very strong. hurricane) are very strong.

The quantity VR (angular momentum) is The quantity VR (angular momentum) is constant for any given air parcel (can constant for any given air parcel (can differ from parcel to parcel).differ from parcel to parcel).– V is the tangential windV is the tangential wind– R is the radial distance from the R is the radial distance from the

center of the low (e.g., eye of a center of the low (e.g., eye of a hurricane)hurricane)

Conservation of Angular Conservation of Angular Momentum in a HurricaneMomentum in a Hurricane

EyeEye

V

R

V x R = Constant

Hurricane

V is the “tangential” wind

R is the radialdistance of the air parcel from the hurricane eye

Conservation of Angular Conservation of Angular Momentum in a HurricaneMomentum in a Hurricane

EyeEye

V

R

V x R = Constant

What happens asan air parcel

spirals inwardtoward the centerof the hurricane?

Conservation of Angular Conservation of Angular Momentum in a HurricaneMomentum in a Hurricane

EyeEye

V1

R1

V x R = Constant

What happens asan air parcel

spirals inwardtoward the centerof the hurricane?

Conservation of Angular Conservation of Angular Momentum in a HurricaneMomentum in a Hurricane

EyeEye

V1

R1

V x R = Constant

What happens asan air parcel

spirals inwardtoward the centerof the hurricane? V2

Conservation of Angular Conservation of Angular Momentum in a HurricaneMomentum in a Hurricane

EyeEye

V1

R1

V x R = Constant

What happens asan air parcel

spirals inwardtoward the centerof the hurricane? V2

R2

Conservation of Angular Conservation of Angular Momentum in a HurricaneMomentum in a Hurricane

EyeEye

V1

R1

V x R = Constantsimply means thatV1 x R1 = V2 x R2

V2

R2

Conservation of Angular Conservation of Angular Momentum in a HurricaneMomentum in a Hurricane

EyeEye

V1

R1

Let V1 = 10 ktsR1 = 500 km

If R2 = 30 km, thenusing the equationV1 x R1 = V2 x R2

we find thatV2 = (V1xR1)/R2

V2 = 167 kts!!!V2

R2

Conservation of Angular Conservation of Angular Momentum in a HurricaneMomentum in a Hurricane

Note spiral bands converging towardthe center

What else?What else?

The same mechanism is at work in The same mechanism is at work in tornadoes, or any rotating weather tornadoes, or any rotating weather system.system.

The flow is in the tangential The flow is in the tangential direction, or in the direction of direction, or in the direction of spin, but there also exists a radial spin, but there also exists a radial inflow towards the center of the inflow towards the center of the vortex, or a spiraling inward flow.vortex, or a spiraling inward flow.

ENSOENSO

El Nino events cause trade wind inversion, El Nino events cause trade wind inversion, so easterlies become westerlies.so easterlies become westerlies.

This increases the AM of the atmosphereThis increases the AM of the atmosphere AM of the solid earth must decrease to AM of the solid earth must decrease to

satisfy COAMsatisfy COAM Results: Results:

– earth rotation rate decreasesearth rotation rate decreases– Length of day increasesLength of day increases

Seasonal VariationsSeasonal Variations

Seasonal variations in the AM budget are Seasonal variations in the AM budget are due to wind and pressure distributionsdue to wind and pressure distributions

For example, AM of the atmosphere For example, AM of the atmosphere reaches an annual maximum in reaches an annual maximum in winter/spring. Why?winter/spring. Why?– Larger N-S temperature gradient Larger N-S temperature gradient strong strong

jet stream (westerlies)jet stream (westerlies)– More topography in NH More topography in NH increase of increase of

mountain torque mountain torque pressure and wind pressure and wind fluctuationsfluctuations

ConclusionConclusion Angular momentum is important because we live on Angular momentum is important because we live on

a rotating planet.a rotating planet. Have a huge role in atmospheric circulation and Have a huge role in atmospheric circulation and

weather eventsweather events Atmosphere gains AM from the earth in the tropics, Atmosphere gains AM from the earth in the tropics,

therefore winds are easterly. therefore winds are easterly. Atmosphere gives up AM to the earth in the mid-Atmosphere gives up AM to the earth in the mid-

latitudes, therefore winds are westerly.latitudes, therefore winds are westerly. Westerly and easterly flow must balance to satisfy Westerly and easterly flow must balance to satisfy

COAM.COAM. El Nino events or other circulation anomalies El Nino events or other circulation anomalies

significantly alter angular momentum budget of significantly alter angular momentum budget of earth.earth.

ConclusionConclusion

The angular momentum balance is The angular momentum balance is highly variable and sensitive to highly variable and sensitive to additional torques such as winds and additional torques such as winds and ocean currentsocean currents

Westerly winds: atmosphere is rotating Westerly winds: atmosphere is rotating quicker than the earthquicker than the earth

Paves the way for the subtropical jet Paves the way for the subtropical jet streamstream

mVmV11RR11 = mV = mV22RR22 = constant = constant