Conservation of Angular Momentum Jeff Gawrych Met. 280 Spring 2004

Introduction n The conservation of angular momentum explains some basic physical properties of rotating objects (like the earth). n 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 n Conservation of Mass: Matter cannot be created or destroyed n Conservation of Energy: Energy cannot be created or destroyed n Conservation of Linear Momentum: Linear momentum cannot be created or destroyed n Conservation of Angular Momentum: Angular Momentum cannot be created or destroyed n Thus, mass, energy, and momentum are transferred back and forth within the earth and atmosphere

Momentum n Momentum = Mass X Velocity n Can be linear or angular n Linear: deals with issues such as collisions. –E.g..Billiard balls and collisions n 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.

Linear momentum

Angular momentum

Conservation of Angular Momentum (COAM) definition n 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 n COAM holds provided there are no outside forces acting on planet (torques) n Earth system consists of solid earth and atmosphere so AM x = m t v t R t = m e V e R e + m a v a R a = constant, where AM x is the angular momentum of the component n AM balance depends on both AM of earth and AM of atmosphere. n AM of earth and AM of atmosphere are inversely proportional. n E.g., if westerlies winds increase  AM atm inc.  AM earth dec.  length of day inc.

COAM Facts n Atmosphere gains AM from the earth in the tropics where surface winds are easterly (i.e. where AM of atmosphere < AM of earth) n Atmosphere gives up AM in the mid-latitudes where surface winds are westerly. n 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. n Easterlies and westerlies must balance out to stay in balance, and conserve AM

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

COAM Facts n 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. n 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 n 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. n Question: What happens to the object's tangential and angular velocities? n 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? n It explains basic principles that are often taken for granted. n COAM influences general circulation and jet streams n Explains why the general circulation is not one a single Hadley-cell. n 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 n In single-cell model, upper levels winds are from equator to poles: To conserve angular momentum, as r decreases, v must increase. n This would create immense westerlies - > unrealistic, and dynamically impossible conditions --> not observed n 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 n Not everywhere on Earth can there be surface winds from the same direction. n Easterly winds must be balanced by westerly winds somewhere else. n 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 n Winds around a strong cyclone (e.g. a hurricane) are very strong. n 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 Eye V R V x R = Constant Hurricane V is the “tangential” wind R is the radial distance of the air parcel from the hurricane eye

Conservation of Angular Momentum in a Hurricane Eye V R V x R = Constant What happens as an air parcel spirals inward toward the center of the hurricane?

Conservation of Angular Momentum in a Hurricane Eye V1 R1 V x R = Constant What happens as an air parcel spirals inward toward the center of the hurricane?

Conservation of Angular Momentum in a Hurricane Eye V1 R1 V x R = Constant What happens as an air parcel spirals inward toward the center of the hurricane? V2

Conservation of Angular Momentum in a Hurricane Eye V1 R1 V x R = Constant What happens as an air parcel spirals inward toward the center of the hurricane? V2 R2

Conservation of Angular Momentum in a Hurricane Eye V1 R1 V x R = Constant simply means that V1 x R1 = V2 x R2 V2 R2

Conservation of Angular Momentum in a Hurricane Eye V1 R1 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!!! V2 R2

Conservation of Angular Momentum in a Hurricane Note spiral bands converging toward the center

What else? n The same mechanism is at work in tornadoes, or any rotating weather system. n 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 n El Nino events cause trade wind inversion, so easterlies become westerlies. n This increases the AM of the atmosphere n AM of the solid earth must decrease to satisfy COAM n Results: –earth rotation rate decreases –Length of day increases

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

Conclusion n The angular momentum balance is highly variable and sensitive to additional torques such as winds and ocean currents n Westerly winds: atmosphere is rotating quicker than the earth n Paves the way for the subtropical jet stream n mV 1 R 1 = mV 2 R 2 = constant