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

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 < 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 --> warm air rises • At poles, atmosphere is cold, and therefore dense --> 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 -> 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.

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.

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 V x R = Constant Eye 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 V x R = Constant V1 What happens asan air parcelspirals inwardtoward the centerof the hurricane? Eye R1

24. Conservation of Angular Momentum in a Hurricane Eye V x R = Constant V1 What happens asan air parcelspirals inwardtoward the centerof the hurricane? Eye R1 V2

25. Conservation of Angular Momentum in a Hurricane Eye V x R = Constant V1 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 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!!! V1 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