Atmospheric Dynamics 11:670:324. Class Time: Tuesdays and Fridays 9:15-10:35

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1 Amospheric Dnamics 11:67:324 Class ime: esdas and Fridas 9:15-1:35 Insrcors: Dr. Anhon J. Broccoli (ENR Dr. Benjamin Linner (ENR As: Websie: ebook: Jenn Kaka Ma Ninik Amospheric Dnamics Sp13 on Sakai (sakai.rgers.ed Marin Mid-Laide Amospheric Dnamics Wile. Grading Qi (mahemaical mehods inclding ecor analsis: 5% Homework problems: 15% GEMPAK eercises: 15% Firs horl eam: 2% Second horl eam: 2% Final eam: 25% 1

2 Wha Are Dnamics? Deiniion: he sd o amospheric and oceanic moions wih emphasis on he phsical laws ha goern sch moions. Corse Objecies o la a mahemaical and heoreical ondaion o be sed in laer applicaions. o appl he laws goerning lid moion (laws o hdrodnamics and hermodnamics o he amosphere in order o ndersand and predic is behaior. o share wih o he eciemen o how mahemaics can be sed o describe wha occrs in he real world. 2

3 Learning Goals Deelop a concepal ndersanding o amospheric dnamical processes. Maser he ondaional mahemaical and phsical principles o amospheric dnamics. Appl he concepal ndersanding and mahemaical and phsical principles o sole problems. Use specialied soware o anale real-ime and hisorical meeorological daa. Basic Laws Conseraion o mass (conini eqaion Conseraion o energ (1 s law o hermodnamics Conseraion o energ (1 s law o hermodnamics Newon s 1 s Law (no reslan orce no change in moion Newon s 2 nd Law (rae o change o moion o a bod is proporional o reslan orce acing on i Conseraion o anglar momenm Newon s Law o Graiaion Ideal Gas Law (eqaion o sae 3

4 Coordinae Ssems o describe he locaion in space o a poin in a lid a coordinae ssem is sed. A commonl sed coordinae ssem is he recanglar or ssem (also known as Caresian. ( Recanglar coordinaes are oen sed o describe moions o he amosphere or ocean een hogh he earh is a sphere. In so doing one assmes ha he - plane is angen o he srace o he spherical earh. General conenion or se o recanglar coordinaes: is a measre o disance rom some origin and increases oward he eas. is a measre o disance rom some origin and increases oward he norh. is ero a srace o earh and increases pward. 4

5 Fndamenal Mahemaical Conceps and Operaions Fndamenal sae ariables sch as wind speed emperare and pressre are ncions o (i.e. depend pon he independen ariables (. For eample amospheric pressre can be epressed as a ncion o space and ime: P P ( Assme Δ he qoien Deriaies represens a small disance in he direcion. Δ Δ he deriaie o a ncion Δ represens he slope. d d lim Δ ( is deined as ( Δ Δ ( In he limi (as goes o his becomes he slope a a poin and his is he deriaie ( or he gradien or rae o change. d d 5

6 6 Parial Deriaies Wih sandard deriaies or ncion aried in one dimension. Howeer some ariables sch as emperare ar no onl in ime ( b also in space: he parial deriaie o wih respec o will ell s how as changes as we moe in he direcion and is deined as ollows: ( ( Δ Δ Δ ( lim Similarl ( Δ Δ Δ ( lim Chain Rle O Diereniaion Assme: hen: ( ( (

7 More Ideniies ( Order o parial diereniaion does no maer. 2 2 ( ln 1 Epansion o oal Deriaie I ( hen d B d d d d w d d wes-eas componen o lid eloci soh-norh componen o lid eloci w erical componen o lid eloci 7

8 w d d d d Eler s relaion (epansion o oal deriaie: d A d B w w C D E erm A: Local rae o change o a a ied locaion erm B: oal rae o change o ollowing he lid moion erm C: Adecion o in direcion b he -componen low erm D: Adecion o in direcion b he -componen low erm E: Adecion o in direcion b he -componen low oal Deriaie s. Local Deriaie oal deriaie is he emporal rae o change ollowing he lid moion. Eample: A hermomeer measring changes as a balloon loas hrogh he amosphere. d Local deriaie is he emporal rae o change a a ied poin. Eample: An obserer measres changes in emperare a a weaher saion. 8

9 Adecion erms Assme ha hin lines are conors o a scalar qani and hick arrows indicae he lid moion. We wish o ealae he adecion erm low A B C high A poin A: A poin B: A poin C: > > < > < > > ranspor rom low o high: negaie adecion o neral adecion o ranspor rom high o low: posiie adecion o alor Series A ncion ( can be comped b alor epansion gien he ales o he ncion and is deriaies a a poin : ( ( ( ( ( ( n 1 ( n ( ( n n! 2! p ( ( 2 ( ( 3! 3... A rncaed alor series can be sed o approimae (. 9

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