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

Transcription

1 Dnamics o the Atmosphere 11:67:34 Class Time: Tesdas and Fridas 9:15-1:35 Instrctors: Dr. Anthon J. Broccoli (ENR 9) broccoli@ensci.rtgers.ed Dr. Benjamin Lintner (ENR 5) lintner@ensci.rtgers.ed TA: Anthon DeAngelis (ENR 1) deangelis@ensci.rtgers.ed Website: Tetbook: Martin, Mid-Latitde Atmospheric Dnamics, Wile. Grading Qiz (mathematical methods, inclding ector analsis): 5% Homework problems: 15% GEMPAK eercises: 15% First horl eam: % Second horl eam: % Final eam: 5% 1

2 What Are Dnamics? Deinition: The std o atmospheric and oceanic motions, with emphasis on the phsical laws that goern sch motions. Corse Objecties To la a mathematical and theoretical ondation to be sed in later applications. To appl the laws goerning lid motion (laws o hdrodnamics and thermodnamics) to the atmosphere in order to nderstand and predict its behaior. To share with o the ecitement o how mathematics can be sed to describe what occrs in the real world.

3 Basic Laws Conseration o mass (continit eqation) Conseration o energ (1 st law o thermodnamics) Newton s 1 st Law (no resltant orce no change in motion) Newton s nd Law (rate o change o motion o a bod is proportional to resltant orce acting on it) Conseration o anglar momentm Newton s Law o Graitation Ideal Gas Law (eqation o state) Coordinate Sstems To describe the location in space o a point in a lid, a coordinate sstem is sed. A commonl sed coordinate sstem is the rectanglar, or,,z sstem (also known as Cartesian). z (,,z ) z 3

4 Rectanglar coordinates are oten sed to describe motions o the atmosphere or ocean, een thogh the earth is a sphere. In so doing, one assmes that the - plane is tangent to the srace o the spherical earth. General conention or se o rectanglar coordinates: is a measre o distance rom some origin and increases toward the east. is a measre o distance rom some origin and increases toward the north. z is zero at srace o earth and increases pward. Fndamental Mathematical Concepts and Operations Fndamental state ariables sch as wind speed, temperatre and pressre are nctions o (i.e., depend pon) the independent ariables (,, t). For eample, atmospheric pressre can be epressed as a nction o space and time: P P (,, t ) 4

5 Assme Δ The qotient Deriaties represents a small distance in the direction. Δ Δ The deriatie o a nction Δ represents the slope. d d lim Δ ( ) is deined as ( Δ) Δ ( ) In the limit (as goes to ), this becomes the slope at a point and this is the deriatie ( ), or the gradient or rate o change. d d Partial Deriaties With standard deriaties, or nction aried in one dimension. Howeer, some ariables sch as temperatre ar not onl in time, bt also in space: T (,, t ) The partial deriatie o T with respect to will tell s how ast T changes as we moe in the direction and is deined as ollows: T lim Δ T ( Δ,, t) T (,, t) Δ Similarl, T T lim Δ (, Δ, t) T (,, t) Δ 5

6 6 Chain Rle O Dierentiation Assme: Then: ), ( ), ( (, ) More Identities ( ) Order o partial dierentiation does not matter. ( ) ln 1

7 Epansion o Total Deriatie I (,,t) then d t d d z dz Bt d, d, w dz west-east component o lid elocit soth-north component o lid elocit w ertical component o lid elocit w d t d d z dz Eler s relation (epansion o total deriatie): d t A w t z d w z B C D E Term A: Local rate o change o at a ied location Term B: Total rate o change o ollowing the lid motion Term C: Adection o in direction b the -component low Term D: Adection o in direction b the -component low Term E: Adection o in z direction b the z-component low 7

8 Total Deriatie s. Local Deriatie Total deriatie is the temporal rate o change ollowing the lid motion. Eample: A thermometer measring changes as a balloon loats throgh the atmosphere. dt Local deriatie is the temporal rate o change at a ied point. Eample: An obserer measres changes in temperatre at a weather station. T t Adection Terms Assme that thin lines are contors o a scalar qantit and thick arrows indicate the lid motion. We wish to ealate the adection term low A B C high At point A: At point B: At point C: >, > <, > <, > > Transport rom low to high: negatie adection o netral adection o Transport rom high to low: positie adection o 8

9 9 Talor Series A nction () can be compted b Talor epansion gien the ales o the nction and its deriaties at a point : ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n n!...!! p A trncated Talor series can be sed to approimate ().