1. THE MOMENTUM EQUATIONS FOR SYNOPTIC-SCALE FLOW IN THE ROTATING COORDINATE SYSTEM

Transcription

1 NOTES FO THE THEOY OF WKD 35. THE MOMENTUM EQUATIONS FO SYNOPTIC-SCALE FLOW IN THE OTATING COODINATE SYSTEM Scalin o the momentm eqations or snotic scale circlation (>000km dimension) reslted in the elimination o relatiel small terms to ield the olloin eqations: i - direction : d sin (.) j - direction : d sin (.) k - direction : (hdrostatic aroimation) (.3) These terms reresent orces on a nit mass (m=) o the atmoshere: d ) Acceleration orces:, ) Coriolis orces:, d here sin 3) Pressre radient orces:,, ) Grait orce: The horiontal eqations (.) and (.) ma also be ritten in ector ormat: d i d j i j sin i sin j i j i j d (i j) ( i j) ( i j)

2 Let the horiontal ind be (i j). Thereore, or the last terms on the riht ( i j) k i 0 j 0 k 0 and let i j The momentm eqation or horiontal lo ma then be ritten as: d k (.)

3 . NATUAL COODINATES The natral coordinate sstem is a coordinate sstem ith orthoonal (indeendent or erendiclar to each other) nit ectors, n and k that makes it easier to describe horiontal cred lo in the atmoshere. The nit ectors are orientated in sch a a that oints toards the direction o lo (here horiontal lo), n oints 90 o toards the let o the direction o lo ( ) and k ards (90 o to the horiontal lane o lo). The coordinates ill ollo the cratre lo as illstrated in ire. O B D C Q s n A P Fire In natral coordinates the horiontal elocit ector hich oints toards the direction o lo ( -direction) ma be ritten as (.) 3

4 here is the horiontal ind seed. The horiontal ind seed, hich is deined to be ositie, ma be eressed as ds (.) here s is a increment o the cre olloed b a arcel o air moin in the horiontal lane. It is imortant to note that is rearded as ositie ( > 0) i n oints toards the centre o the circle ormed b the cratre lo, and is neatie ( < 0) i n oints aa rom the centre o the circle ormed b the cratre lo. 3. THE MOMENTUM EQUATIONS IN NATUAL COODINATES For alication in natral coordinates the horiontal momentm eqation (eqation (.)) needs to be conerted to natral coordinates. Acceleration term o eqation (.): d B alin the rodct rle o dierentiation, eqation (.), hich reresents the horiontal acceleration (orce er nit mass) that ollos the motion, becomes: d d d (3.) The second art o the second term on the riht hand side o eqation (3.) ma be ritten in a more aroriate a. We start b

5 d d ds ds (3.) Let s irst look at the irst art on the riht hand side o eqation (3.). I e consider a small increment dislacement PQ in ire, e ma assme that, hich in realit orms a cre, is close to the lenth o CD, and that oints in the same direction o nit ector n. In eneral the arch lenth o a comlete circle (circmerence) is eqal to meanin that the arc lenth o PQ and CD is s and, resectiel. Note that becase is a nit ector it ollos that. It thereore ollos that s d ds lim s0 s d ds n (3.3) Secondl, accordin to eqation (.) the second art on the riht hand side o eqation ds (3.) ma be ritten as Eqation (3.) ma no be ritten as d n and b relacin this back into eqation (3.) the acceleration in natral coordinates becomes d d n (3.) 5

6 Pressre radient term o eqation (.): In the horiontal lane ressre radient orce can thereore be ritten as: n in the natral coordinate sstem and the s n n (3.5) s n Coriolis term o eqation (.): k Accordin to eqation (.) and thereor k. The cross rodct o nit ectors k and ields the nit ector in the direction o n (riht hand rle). The Coriolis term then becomes k 0 n 0 0 k 0 n (3.6) From eqations (3.), (3.5) and (3.6), the horiontal momentm eqation (.) ma no be ritten in terms o natral coordinates d n n n (3.7) s n Comonents o eqation (3.7) in the and n directions are - direction : d (3.8) s 6

7 n - direction : (3.9) n These to eqations orm the basis rom here e ill inestiate the roerties o dierent tes o snotic-scale circlation in the atmoshere.. SUMMAY OF CONDITIONS FO NATUAL COODINATES Condition d 0 in eqation (3.8). s The eqation alies or lo in the direction o moement ( ). We are considerin lo alon isobars, meanin that ressre ill not chane oer the ininitesimal dislacement s. The seed is constant olloin the motion. Condition The soltion or in eqation (3.9) is alas ositie. Condition 3 is rearded as ositie (>0) i n oints toards the centre o the circle ormed b the cratre lo, and is neatie (<0) i n oints aa rom the centre o the circle ormed b the cratre lo. Condition The nit ector n oints 90 o toards the let o the direction o lo ( ) 7

8 5. GEOSTOPHIC FLOW For eostrohic lo ( ) condition in section alies. Note that the eostrohic ind is deined as a ind here the ressre radient orce is in balance ith (or eqal to) the Coriolis orce, and thereore here the centrietal orce is inored ( 0 ) in eqation (3.9). n is not an eact soltion o eqation (3.9) since the centrietal orce is inored, and condition does thereore not al ( miht be ositie or neatie). The eostrohic lo describes constant motion (no acceleration) in a straiht line since the to orces actin on the motion are eqal and thereore. The Coriolis arameter ( sin ) is alas neatie in the Sothern Hemishere. It is thereore onl the ressre radient alon the n -direction that can chane. I the ressre radient is ositie, mst be ositie (riht belo), and i the ressre radient is neatie, mst be neatie (let belo). H = hih ressre; L = lo ressre. L n 0 n H H n 0 n L 8

9 6. INETIAL FLOW For inertial lo the ressre ield is niorm (isobaric srace), and thereore here the ressre radient orce is inored ( 0 ) in eqation (3.9). n mst be ositie (condition ), and is alas neatie in the Sothern Hemishere. The onl otion is that mst be ositie to ield a ositie. Accordin to condition 3 is ositie (>0) i n oints toards the centre o the circle ormed b the cratre lo. Inertial lo is alas anti-cclonic or conter-clockise in the Sothern Hemishere. This te o lo does not oten occr in the atmoshere. n 9

10 7. CYCLOSTOPHIC FLOW For cclostrohic lo the Coriolis orce is deined as small and can be inored. This miht haen i the horiontal scale o the circlation is small (tornado) or in eqatorial reions here the Coriolis orce is ero, and thereore here the Coriolis orce is inored ( 0 ) in eqation 3.9. n mst be real and non-neatie and thereore mst be ositie. This ill onl n be tre i >0 and the ressre radient neatie, or i <0 and the ressre radient ositie. is rearded as ositie and constant. > 0 and 0 n < 0 and 0 n L n L n For cclostrohic lo circlation can either be clockise or conter-clockise arond a lo-ressre sstem in the Sothern Hemishere. 0

11 8. GADIENT FLOW For radient lo e sole b sin all three terms in eqation (3.9). The radient ind is thereore a three-a balance beteen the Coriolis orce, the ressre radient orce and the centrial orce in the horiontal lane and thereore n Since the eqation or radient ind takes accont o the centrial orce, the radient ind is a better aroimation o the actal ind than is the eostrohic assmtion. The radient ind eqation (3.9) ma be rearraned to orm the olloin qadratic eqation: n 0 ith soltions (or roots) (8.) n mst be ositie and real (eression nder the root mst be ositie). Not all the mathematical ossible roots in eqation (8.) ill corresond to hsicall ossible soltions. While is neatie in the Sothern Hemishere and is constant, and the ressre radient in the n -direction miht be neatie or ositie. We hae the olloin eiht otions - (a) to (h):

12 > 0 < 0 0 n 0 n (a) : hsicall ossible (c) : hsicall ossible (b) : hsicall ossible (d) : hsicall imossible (e) : hsicall ossible () : hsicall imossible () : hsicall imossible (h) : hsicall imossible Otions (a) and (b) Ste : Dra the circlation sstem The olloin to conditions reslt in conter-clockise lo arond a H n hih (anti-cclone) 0 ( n oints toards centre) 0 n (Hih ressre in centre) Ste : Ealate the term otside the root o eqation (8.) > 0 since > 0 and < 0. I e add this ositie term to the ositie root one ill alas obtain a ositie ale or. Otion (a) is thereore hsicall ossible. Otion (b) needs some rther inestiation.

13 Ste 3: Ealate the terms nder the root o eqation (8.) > 0 hile n > 0 and thereore take the sqare root n bt e are interested in the neatie root n add n both sides to obtain in (8.) n bt < 0 and > 0 > 0 is thereore hsicall ossible Otions (c) and (d) Ste : Dra the circlation sstem The olloin to conditions reslt in clockise lo arond a lo (cclone) 0 ( n oints aa rom centre) 0 n (Lo ressre in centre) L n Ste : Ealate the term otside the root o eqation (8.) < 0 since < 0 and < 0. I e add this neatie term to the neatie root one ill alas obtain a neatie ale or. Otion (d) is thereore hsicall imossible. Otion (c) needs some rther inestiation. 3

14 Ste 3: Ealate the terms nder the root o eqation (8.) > 0 hile n < 0 and thereore n take the sqare root n add both sides to obtain in (8.) n bt < 0 and < 0 > 0 is thereore hsicall ossible

15 Otions (e) and () Ste : Dra the circlation sstem The olloin to conditions reslt in conter-clockise lo arond a lo (cclone) L n 0 ( n oints toards centre) 0 n (Lo ressre in centre) Ste : Ealate the term otside the root o eqation (8.) > 0 since > 0 and < 0. I e add this ositie term to the ositie root one ill alas obtain a ositie ale or. Otion (e) is thereore hsicall ossible. Otion () needs some rther inestiation. Ste 3: Ealate the terms nder the root o eqation (8.) > 0 hile n < 0 and thereore take the sqare root n bt e are interested in the neatie root n add n both sides to obtain in (8.) n bt < 0 and > 0 < 0 is thereore hsicall imossible 5

16 Otions () and (h) Ste : Dra the circlation sstem The olloin to conditions reslt in clockise lo arond a hih (anti-cclone) H n 0 ( n oints aa rom centre) 0 n (Hih ressre in centre) Ste : Ealate the term otside the root o eqation (8.) < 0 since < 0 and < 0. I e add this neatie term to the neatie root one ill alas obtain a neatie ale or. Otion (h) is thereore hsicall imossible. Otion () needs some rther inestiation. Ste 3: Ealate the terms nder the root o eqation (8.) > 0 hile n > 0 and thereore take the sqare root n add n both sides to obtain in (8.) n bt < 0 and < 0 < 0 is thereore hsicall imossible 6

17 9. FINITE DIFFEENCE METHODS TO CALCULATE WINDS Consider the eostrohic ind eqations, as discssed in section 5, bt in this case or snotic lo in the OTATING COODINATE SYSTEM (see eqations (.) and (.)): i - direction : j - direction : sin sin (9.) (9.) From eqations (9.), the eostrohic ind comonent in the i direction ( ais: est to east) and j direction ( ais: soth to north) becomes: (9.3) sin Geostrohic ind comonent in the -direction : sin (9.) The term sin can be easil soled since and are constants and the latitde () is knon. The to deriaties, hoeer, are not that eas to sole. Meteoroloical data is oten roided in the orm o rid ields, here rid oints are arraned alon latitde () and lonitde () lines ith interal distances o and., i,, j, On sch a rid ield, artial deriaties in the and directions ma be nmericall estimated b sin the olloin inite dierence method: 7

18 : West East. The ble circles reresent 3 rid oints (,, 3 ) in the -direction alon latitde and lonitdes,, 3. The ressre ales at the 3 rid oints are,, 3. The distance beteen the rid oints is. We assme that lim 0 And thereore 3 (9.5) Where, on the sherical Earth: r cos ( ) r cos ( 3 ) : North Soth 8

19 The ble circles reresent 3 rid oints (,, 3 ) in the -direction alon lonitde and latetdes,, 3. The ressre ales at the 3 rid oints are,, 3. The distance beteen the rid oints is. We assme that lim 0 And thereore 3 (9.6) Where, on the sherical Earth: r( ) r( 3 ) EXECISE 9. Consider the olloin ressre ale rid ield on the srace o the Earth ith Pressre: 990 hpa Pressre: 989 hpa Pressre: 988 hpa 7 o Pressre: 989 hpa Pressre: 988 hpa Pressre: 987 hpa 8 o Pressre: 988 hpa Pressre: 987 hpa Pressre: 986 hpa 0 o E o o E 9 o S Calclate and at the black circle. 9

20 EXECISE A O Soth Pelican 00 Harbor hpa Paradise Island O hpa 998 hpa.00 O.50 O.00 O O Consider an Island (Paradise Island) somehere in the Paciic Ocean. Shis in the icinit o the Island hae measred the ressre at each one o the rid oint at a ien time. The ressre at Pelican Island Harbor is also knon as 000 hpa. A shi, not ar rom the Island, asked or ermission to enter the harbor. It ma be daneros or shis to enter the harbor i the reailin inds are too stron ( > ms - ) Yo are in chare o Pelican harbor, and the onl inormation aailable are the ressre ales at the ien rid oints. Yo also hae the olloin constants aailable: adis o the earth: a = m Earth s anlar seed o rotation: = rad.s - Standard sea-leel densit o air: O =.5 k. m -3 PLEASE ANSWE THE FOLLOWING QUESTIONS:. The shi is crrentl at rid oint A. Determine the aroimated distance beteen the shi and Pelican Harbor (in kilometers).. Dra the 00hPa, 000 hpa and 999hPa isobars on the ma (dra the aroimated direction o ind lo beteen these isobars b assmin the ind is eostrohic). 3. Write don an eression or the eostrohic ind in the - and - directions. 0

21 . Use the eqations in 3 to calclate the eostrohic ind seed and direction at Pelican Harbor and dra the ind ector on the ma. Wold it be sae or the shi to enter the harbor? BACKGOUND Beore e calclate the distance beteen the harbor and shi, e irst need to eress the - and - comonents o distances on the sherical srace o the Earth in terms o lonitde () and latitde (). emember that e are not dealin ith a lat srace! To do that e consider the circmerence o a circle namel r circmerence = anle radis = r and thereore = r Accordin to this e ma rite and on the sherical srace o the Earth as: = r = = r cos() here r is the radis o the Earth s. ANSWE FO QUESTION D A shi = r harbor C = r cos() B We kno that: r = = = 0.5 o

22 These to anle increments mst be in radiance. To conert rom derees to radiance, e se the olloin: = 80 o rad = rad = 0.5 = First calclate and : r cos (line CB) = ( ) cos(-35)( ) = ( )(0.855)( ) = meters = 5.5 km r (line AB) = ( )( ) = meters = 55.6 km (AC) = (CB) + (AB) = (5535.6) + ( ) = ( ) + ( ) = and thereore AC = meters = 7.9 km ANSWE FO QUESTION In the Sothern Hemishere, lare-scale ind circlation is clockise arond loressre sstems and conter-clockise arond hih-ressre sstems. The eostrohic lo is arallel to the isobars (lines o eqal ressre measred in hpa thick lines in the diaram). The ind ector shold thereore oint toards the northeast (as indicated b black arros in the diaram). This eercise miht be a a to eri that or inal calclated eostrohic ind oints in the aroriate direction.

23 Hiher ressres 000 A O Soth 00 Harbor hpa Paradise Island Loer hpa 998 hpa ressres.00 O West Pelican.50 O West.00 O West O Soth O Soth ANSWE FO QUESTION 3 Geostrohic ind in the -direction (ointin rom est to east): sin sin Geostrohic ind in the -direction (ointin rom soth to north): sin sin ANSWE FO QUESTION We irst calclate the constant term sin hich occrs in both the - and - direction eostrohic ind eqations at latitde = 35.5 o S. sin (.5)()(7.90 From QUESTION e kno that )(sin 35.5)

24 ()(5535.6) 907.m ()( ) 77.m I e consider the ressre dierences (radients) across these distances in the - and - directions, e can calclate the deriaties in the eostrohic ind eqations o QUESTION 3 (remember that ressre on the ma is ien in the nit hpa, and mst be conerted to Pa or calclations: hpa=00pa): We no relace all the reslts aboe in the eostrohic ind eqations o QUESTION 3 Geostrohic ind comonent in the -direction: sin.0370 ( ) 7.35m. s Geostrohic ind comonent in the -direction: sin.0370 ( ).7m. s The manitde o the total eostrohic ind seed is thereore: (7.35)(7.35) (.7)(.7) 7.37ms 98.5kmh Note that the ind direction is alas ien as an anle that oriinates rom the North ais, and is ositie clockise rom the North ais. The ector that is ormed b this anle indicates the direction FOM hich the ind blos. For or roblem, this is illstrated as ollos here is the anle o the ind ector rom the North ais:

25 North () East () sin o rom here This anle allo or the ind to blo in the estimated directions indicated in QUESTION here e hae considered lo arond hih and lo ressre sstems. Hoeer, ind direction is alas measred rom here the ind blos, in other ords: The ind direction is ( 80) ( ) o 9.3 A ind ith direction 9.3 o bloin at a seed o 7.37 m.s - old be too stron and it is thereore not sae or the shi to enter the harbor. 5

26 0. THEMAL WIND The thermal ind describes chane (ertical ind shear) o the eostrohic ind ith altitde in the resence o a horiontal temeratre radient. The thermal ind thereore occrs in areas here one inds stee temeratre radient on horiontal leels in the ertical - here on constant ressre leels. A tical eamle is a olar ront here arm air rom the troics meets ith cold olar air. Isobar - line o constant ressre (or eamle the 500 hpa line) Area o temeratre radient (olar ront) Isoterm - line o constant temeratre A B ertical heiht Cold air Warm air Soth North Consider a ertical cross section in the atmoshere ith a cold air mass on the let and a armer air mass on the riht. The isotherms, and thereore also isobars, ill lit to a hiher altitde oer the arm air mass that ill reslt in a isotherm, and isobar, sloe in the area o temeratre radient. A horiontal temeratre and ressre radient ill thereore aear oer the horiontal area AB. This temeratre and ressre radient ill reslt in airlo (or inds). The horiontal inds ma be aroimated as eostroic inds, meanin that the Coriolis and Pressre radient orces are in balance. The horiontal lo ill thereore be in a direction that is aroimatel arallel to the isobars in the area o temeratre radient. Instead o considerin inds oer the horiontal area AB e ill rather analse inds on CONSTANT PESSUE leels - remember that a horiontal temeratre radient ill still occr oer a constant ressre leel - or eamle the 500 hpa leel. We ill 6

27 thereore derie an eression or the chane (ertical ind shear) eostrohic ind on dierent constant ressre leels rom the srace o the Earth ards into the atmoshere as a reslt o a horiontal temeratre radient oer these leels. Scalin o the horiontal momentm eqations or snotic scale circlation (>000km dimension) reslted in the elimination o relatiel small terms to ield eqations (.) and (.). When makin the eostrohic assmtion (Coriolis Force = Pressre Gradient Force; d d 0 ), it ollos that: i - orce comonents : (0.) j - orce comonents: here sin (0.) Since the thermal ind has to do ith a ertical ind shear e also need to incororate the ertical atmosheric eqation or hdrostatic eqation (.3): k - direction : (0.3) In eqations (0.), (0.) and (0.3) ressre is a ariable that chanes oer,, and sraces. Since e ant to ork on constant ressre leels, e need to relace the crrent deriaties that elain chane oer constant altitdes () ith deriaties that elain chane oer constant ressre () leels. In a sstem here (,,,t) are indeendent coordinates one ma rite (the ootnotes denote constant and sraces): 7

28 8 t t t t Since (,,,t) are indeendent coordinates in the -sstem and since ressre does not chane in the and directions in the -sstem, these eqations become 0 and thereore 0 and thereore relace eqation (0.3) hich are eressions here the deriaties on constant altitdes () are reritten in terms o deriaties on constant ressres (). elace these reslts into eqations (0.) and (0.) (0.) (0.5)

29 9 hich ield eqations here the deriaties are ealated ith ressre held constant (on constant ressre leels in the ertical). Since e are interested in the ertical ind shear o the eostrohic ind on dierent constant ressre leels in the ertical, e dierentiate eqations (0.) and (0.5) ith resect to : since is not a nction o and since is continos. From eqation (9.3) ollos that: and i is continos and T e obtain T Thereore T T T ln T T T ln since is constant and since is constant in the and directions as deined (ootnote denote constant ressre). In ector orm the aboe ma be ritten as j T i T j) i ( ln i ) j i ( and j T i T T T T 0 0 k j i T k

30 e end ith the THEMAL WIND EQUATION ln k T (0.6) Hoeer, eqation (0.6) is actall a relationshi or the ertical eostroic ind shear or the chane o the eostrohic ind ith resect to ln in the ertical. Strictl seakin the THEMAL WIND reers to the ector dierence beteen the eostrohic inds at to constant ressre leels. This becomes obios i e interate eqation (0.6) beteen ressre leels 0 and. () (0) 0 k () (0) T ln k T (ln ln 0) here T is the aerae temeratre beteen 0 and This ields an eqation or the THEMAL WIND T () (0) ln k T (0.7) 0 Interretation o eqation (0.7) I can be shon that the thickness o the laer beteen ressres 0 and is roortional to the aerae temeratre T beteen these to laers. The armer the atmoshere, the thicker the laer ill become. The thermal ind eqation is an etremel sel dianostic tool or hiher latitdes. It ma be sed to check data o the obsered ind and temeratre or consistenc, or it ma be also be sed to estimate the mean horiontal temeratre adection in a laer. 30

31 Since the temeratre radient in ire 0. is alas erendiclar to the isotherms, the cross rodct in eqation 0.7 indicates that the thermal ind ector ill alas arallel to the isotherms (line o constant temeratre) ointin let in the Sothern Hemishere here the cold air is soth and the armer air is north. As illstrated in Fire 0., a eostrohic ind that trns conter clockise ith heiht (backs) is associated ith cold air adection (cold air lo in the direction o the srace eostrohic ind). Clockise trnin (eerin) o the eostrohic ind ith heiht imlies arm air adection in the laer. It is thereore ossible to estimate the horiontal temeratre adection at a ien location solel rom data on the ertical roile o the inds. The eostrohic ind at an leel in the atmoshere can also be calclated i the srace eostrohic ind and the mean horiontal temeratre radient are knon. (a) Warm T o +T T T 0 T o Cold T o -T (b) Warm T o +T 0 T T o T Fire 0. T o -T Cold elationshi beteen trnin o the eostrohic ind and temeratre adection (a) conter clockise ind direction shit ith heiht (backs) and (b) clockise ind direction shit ith heiht (eerin) 3

32 3. CICULATION AND OTICITY Additional notes on the hsical meanin o orticit ill be roided (not or eam roses) Circlation and orticit are to rimar measres o rotation in a lid. Circlation, hich is a scalar interal qantit, is a macroscoic measre o rotation or a inite area o the lid. orticit, hoeer, is a ector ield that ies a microscoic measre o the rotation at an oint in the lid. In this section e ill ocs on orticit, since it is a concet commonl sed in meteorolo to determine the deree o rotation in the atmoshere. The relatie orticit is deined as the crl o the relatie elocit, in other ords elocit as eerienced in a rotatin coordinate sstem on the Earth. The ector o relatie orticit ma thereore be ritten as: k j i U For snotic scale circlation e onl consider the ertical ( k ) comonent, hich describes chanes on the horiontal leel. elatie orticit thereore becomes: U k (.) For snotic scale motion, the orticit eqation o the atmoshere can be deried sin the aroimated horiontal momentm eqations (.) and (.). The deriaties in eqations (.) and (.) ma be ritten in terms o artial deriaties to ield: t (.) t here sin (.3) Dierentiate eqation (.) ith resect to and eqation (.3) ith resect to t (.) t (.5) a c b d e

33 33 The olloin ste is to sbtract eqation (.) rom eqation (.5). We ill do that searatel or each set o terms (terms a to ). Terms a: t t t t t relace eqation (.) t (.6) Terms b: relace eqation (.) (.7) Terms c: Similar to Terms b ollos (.8)

34 3 Terms d: relace eqation (.) (.9) Terms e: ) ( ) (.. (.0) Terms : (Note: The Coriolis arameter is a nction o onl - soth-north direction) () ()

35 35 bt 0 (.) B ttin terms a to in eqations (.6) to (.) toether e et t and ith some minor simliications ) ( t (.) I e assme that Coriolis () is onl a nction o it ollos that t 0 0 t 0 ) ( d hich is eqal to the let hand side o eqation (0.). Eqation (.) thereore becomes ) ( ) ( d (.3) to orm the orticit eqation. Eqation (.3) states that the rate o chane o the absolte orticit (orticit as ieed rom sace - addition o that contains the anlar seed) olloin the motion is ien b the sm o three terms on the riht, called the dierence term, the tiltin or tistin term and the solenoidal term, resectiel.