Synoptic Meteorology I. Some Thermodynamic Concepts

Transcription

1 Synotic Meteoroloy I Some hermodynamic Concets

2 Geootential Heiht Geootential Heiht (h): the otential enery of a nit mass lifted from srface to. Φ d 0 -Since constant in the trooshere, we can write Φ Δ m m m 2 2 s s 2 J k We can now define eootential heiht!

3 Geootential Heiht h 1 Φ d (no nits) eootential meters (m) m 2 /s 2 m 2 s -2 denotes a otential enery Ultimately, h is emloyed to ensre NO v in horiontal flow.

4 Geootential Heiht Here s how it works: i. Earth is an oblate sheroid R R e R e > R

5 Geootential Heiht ii. As sch, increases as yo travel S -> N (yo re decreasin distance from the center of mass) iii. Frther, srfaces at a constant heiht above the earth (sea level) will also be misshaen. 10,000m 000m MS * So, ravity chanes, y on those srfaces!

6 Geootential Heiht lower over ole eootential srface srface A v hiher over eqator v On a srface, is not always a (arallel allel to local kˆ ) If we assme it is, then we se a v that does not oint to the center of Earth s mass. Doin so, artificially introdces an eqatorward acceleration, A v (Moral: Use h).

7 Geootential Heiht On a constant heiht srface (above MS) N 0 o 10,000m ole > eqator * is not always to (a ball on this srface tends to the eqator) On a eootential srface N 0 o Φ constant

8 Geootential Heiht On a eootential srface N 0oΦ constant ball is at rest eootential srface NO comonent of v actin alone the srface v does not affect horiontal flow On constant srfaces, we se lines of constant h (m), e.., 5640 m at 500 mb. A constant srface does not eqal a eootential srface, BU v is to each heiht line.

9 Geootential Heiht m ,021m 10,000m 9.78 o 9.84 o o Δ N h o m 9970m s ,000m m 10,000m 45 o Δ h ,000m s m 10,021m 0o Δ h ,000m s o o 9.79

10 Geostrohic Winds y f ρ 1 x f v ρ 1 - coordinates f Φ 1 v y f y f Φ 1 - coordinates x f x f v

11 Geostrohic Winds Qick Conversion Rle (or h) 1. relace 1/ρ with 2. exchane with (eometric heiht) or h (eootential heiht) *We se eootential heihts to et Δh more accrately from RAOBS. Δh R 9.8 v ln U eliminated variable, Δ R v ln U

12 Hysometric Eqation Derivin the hysometric eqation or thickness eqation assme a hydrostatic atmoshere ρ and introdce the eqation of state: ρ 1 R 2 Rd

13 Hysometric Eqation Now, sbstitte 2 into 1 R d R l d 1 Now, assme d R l d 1 d R l d d R ln

14 Hysometric Eqation [ ] ) ( ln ln R R d ) ( ln ln R R d ) ( ln R d R ) ( ln d R ) ( ln and that brins s to...

15 Hysometric Eqation R d Δ ln he Hysometric Eqation! Ex 1: 287 Δ J KK 9.81 km ks m 2 s 2 2 (269 K ) ln m 2 s m mb mb msl 297 (75 o F) (- 24 o F) 269.5

16 Hysometric Eqation Ex. 2: 287 Δ Δ J KK R d ln (285 K ) 1000 ln m s mb mb K K K 5779 km ks m s m known 500mb (1000mb) known

17 Hysometric Eqation Ex. 3: Δ R d Δ ln Δ R d R d e Δ 450 m,, ln e ( 450 ) ( 303 ) R Δ ha e ha d 954 ha known Δknown MS

18 Hysometric Eqation So Hysometric Eqation is Vital for: 1. Redcin Station ressre to sea level. 2. Determinin heihts of er-level ressre srfaces.

19 Hysometric Eqation *Crcial Qestion* Δ R d constant constant R d ln If we fix and, then Δ is deendent only on. Δ α

20 Hysometric Eqation cold 5160 thickness attern warm thickness stron radient of stron radient of r lare Δ

21 hickness and hermal Wind hickness and thermal wind: 1. hels exlain how winds chane in vertical 2. assmes eostrohic atmoshere tells s somethin abot vertical strctre of 3. tells s somethin abot vertical strctre of atmoshere

22 hickness and hermal Wind v v v V V - V V r V r V shear of V r eostrohic cold wind V r 1 kˆ V v R f d ln kˆ n- 3 n- 2 n-1 n warm thickness thermal wind blows ll to thickness contors

23 hickness and hermal Wind chanes in the wind in the vertical (vertical wind shear) α the mean thermal radient over the layer V r r V r V V r V r V r V V r

24 hickness and hermal Wind With mltile ressre levels: backin conterclockwise (cold air always to left) V v V300 cold V r V r 850 warm * backin with heiht mean layer V is casin cold advection

25 hickness and hermal Wind Veerin- clockwise chane of wind warm V v 850 r V cold V r 300 * veerin with heiht mean layer V is casin warm advection

26 otential emeratre Consider the First aw of hermo dh C d- α d 1 First aw states that, for a closed system, no heat may be added to/removed from a system (arcel). dh 0 C d α d 2 Now, eqation of state ives s ρr or ρ /R or α R/ 3

27 otential emeratre Sbstitte 3 into 2 to et C d R d Next, divide by to et C d R d d Now divide throh by C to et Rd J/k K C 1005 J/k K d Rd C d

28 otential emeratre We then et θ d Rd C 1000 d d θ tem at 1000mb Τ tem at some ressre level ln θ -ln Rd ln 1000 ln C ln θ Rd C ln 1000

29 otential emeratre ake exonent of both sides θ Rd C 1000 let θ 1000 Rd C θ κ 1000 θ otential tem. tem. a arcel will have if broht down to 1000mb from some level, havin some tem.,.

30 otential emeratre Examle θ 1000 κ Given, θ 28 o C301K, what will be at 850 mb? 301K K K 14.3oC