Angular Momentum About Spin Axis

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Frank Franklin
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1 Angla Momentm

2 Angla Momentm Abot Spin Axis Ω AM (pe nit mass) = velocity moment am M =( + ) = a cos φ Thin-shell appoximation: take distance fom any point in atmosphee to planet s baycente eqal to constant adis a

3 Length of Day 2 4 ΔLOD (ms) Yea ΔLOD (ms) Yea LOD vaies seasonally becase of AM consevation in solid-eath/atmosphee system and seasonal vaiations of atmospheic AM. LOD vaiations on longe timescales becase of AM exchange with moon (tides).

4 Angla Momentm in Atmosphee dm dt se deivative hee, be In axisymmetic inviscid flow, DM/Dt =. So no axisymmetic e-aangement of ai masses can geneate an angla momentm extemm. Hide s theoem is a conseqence: AM extema mst occ at bondaies. Inteio AM cannot exceed AM at eqato at the sface: M max = a 2 This implies an ppe bond on the zonal flow, the angla momentmconseving zonal + D

5 AM and Hadley cell AM-conseving zonal wind (m s -1 ) N 3 N 45 N 6 N 75 N 9 N Latitde Zonal wind (m s -1 ) Pesse (hpa) Steamfnction (Sv) 15 N 3 N 45 N 6 N 75 N 9 N Latitde

6 Pimitive Momentm Eqations The sal pimitive momentm eqations follow fom angla momentm consevation and hydostatic balance by expanding ot definitions of angla momentm and making the thin-shell appoximation. Fo t + M + M The Coiolis toqe nd side, aises the st with tem M epesents a 2 cos 2 thefom local ate o M a@ a momentm com v2 sin a cos fv paamete 2 sin. Pimitive Momentm + D. d dt fv v tan a 1 + D dv dt + f D

7 Aveaged AM Eqation Reynolds aveaging spin AM eqation in flx t ( ) + (M + M ) gives in a statistically + D oe convenient fo aveaging than the advective fom Ū M Ū M M D, tmosphee appoximation, o, in thin-shell appoximation, f V Ū M M D mosphee

8 o, in thin-atmosphee M f V U appoximation, M U M Fee Atmosphee U M (2.2b) M D, f V U M D, o, f V in thin-atmosphee U M appoximation, M D, M Dominant Balances low Rossby Feenmbe Atmosphee (2.2b) (2.2b) FeeFee Atmosphee U M M (2.3a) Atmosphee lowatmosphee: Rossby nmbe Fee Whee Rossby nmbe is negligible, f V small, U M is Mhence low Rossby nmbe low Rossby nmbe U M M (2.3a) o, in thin-atmosphee appoximation, U U M M M M Fee Atmosphee (2.3a) (2.3a) o, in thin-atmosphee2.3.1 appoximation, fappoximation, V M ino,thin-atmosphee appoximation, in thin-atmosphee o, in o, thin-shell appoximation, (2.3b) low Rossby nmbe M f V eddy meidional ow balances (2.3b) Coiolis toqe on mean angla momentm f V f V M MU (2.3b) (2.3b) M M x divegence whee dag is weak. Coiolis toqe on mean meidional ow balances eddy angla momentm Coiolis toqe on in mean meidional ow balances eddyangla anglamomentm momentm Coiolis toqe on o, mean meidional ow balances eddy thin-atmosphee appoximation, Coiolis toqe on mean meidional flow balances eddy angla momentm flx x divegence whee dag is weak. divegence whee dag isfeel weak.cells). x x divegence whee dag is weak. divegence (ppe banches Hadley, Nea Sface f V M Nea Sface Nea Sface low Rossby nmbe Nea Sface Nea sface: Eddy flxes geneally weak. When Rossby nmbe is small, dominant balance is Ekman balance Coiolis ow balances edd low Rossby nmbe lowlow Rossby nmbe U toqe M on mean D meidional, (2.4a) Rossby nmbe x divegence whee dag is weak. U M D, U U M D, M D, o, in thin-atmosphee appoximation, o,o, in in thin-shell appoximation in thin-atmosphee appoximation, appoximation, o,thin-atmosphee ino,thin-atmosphee appoximation, (2.4a) (2.4a) (2.4a) Nea Sface f V D, (2.4b) f V D, (2.4b) f V DD,, (2.4b) f V low Rossby nmbe Coiolis toqe on mean meidional flow balances mean zonal dag. Hence, (2.4b) eqatowad flow whee thee ae eastelies, polewad flow U whee M westelies. D,

9 Eath (Annal Mean).2 Se> Se< σ Latitde Se >, D Flow towad poles (towad spin axis) Se<, D Flow towad eqato (away fom spin axis)

10 AM Flxes (Tansient) and Zonal Wind

11 Seasonal Cycle RG31 RG31 Schneide et al.: WATER VAPOR AND CLIMATE CHANGE RG31 Schneide et al.: WATER VAPOR AND CLIMATE CHANGE RG31 Fige 5. Eath s Hadley ciclation ove the cose of the seasonal cycle. Black contos show the 5. Eath s Hadleywith ciclation the cose of the seasonal cycle. Black contos the massfige flx steam fnction, dashedove (negative) contos indicating clockwise motionshow and solid 9 1 mass contos flx steam fnction, with dashed (negative) indicating clockwise and solid s ). Colos indi(positive) indicating conteclockwise motion contos (conto inteval is kgmotion 1 kg s ). Colos indiindicating conteclockwise 25 1 denoting cos!), inteval with theis oveba the seasonal cate (positive) hoizontalcontos eddy momentm flx divegence motion div( v(conto 6 cos!), with the oveba denoting the seasonal cate hoizontal momentm divegence div( v(conto m s 2 2, with ed tones and zonal mean andeddy pimes denotingflx deviations theefom inteval m s, Ro with ed tones and zonal mean pimes theefom (conto inteval 8 1in which fo positive and bleand tones fo denoting negativedeviations vales). Gay shading indicates egions >.5. The fo positive and ble tones fo negative vales). Gay shading indicates egions in which Ro >.5. The vetical coodinate s = p/ps is pesse p nomalized by sface pesse ps. Compted fom eanalysis vetical coodinate s = p/p is pesse p nomalized by sface pesse ps. Compted fom eanalysis data fo the yeas spovided by the Eopean Cente fo Medim Range Weathe Foecasts data fo the yeas povided by the Eopean Cente fo Medim Range Weathe Foecasts [Kållbeg et al., 24; Uppala et al., 25]. [Kållbeg et al., 24; Uppala et al., 25]. climate changes onlyonly via via changes in the eddy momentm 1% incease inceaseininthe thestength stengthofof mean climate changes changes in the eddy momentm the thehadley Hadley cells, cells, aa 1% thethe mean (Schneide et al. 21) flx divegence S, and possibly via changes in the width of meidional mass flx eqies a 1% incease in S, an

12 Eddy AM Flx Conveges in Midlatitde Jets.2 σ Pesse (hpa) Altitde (km) 9 S 45 S 45 N 9 N Latitde

13 AM Balance at Sface Ekman balance f v = D Zonal dag balances Coiolis foce on meidional flow Z!"#$%&'( )&*+(

14 Vetically Integated AM Flxes

Chapter 12: Kinematics of a Particle 12.8 CURVILINEAR MOTION: CYLINDRICAL COMPONENTS. u of the polar coordinate system are also shown in

Chapter 12: Kinematics of a Particle 12.8 CURVILINEAR MOTION: CYLINDRICAL COMPONENTS. u of the polar coordinate system are also shown in ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle Chapte 1 Kinematics of a Paticle A. Bazone 1.8 CURVILINEAR MOTION: CYLINDRICAL COMPONENTS Pola Coodinates Pola coodinates ae paticlaly sitable fo solving