CSU ATS601 Fall Other reading: Vallis 2.1, 2.2; Marshall and Plumb Ch. 6; Holton Ch. 2; Schubert Ch r or v i = v r + r (3.

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1 3 Eath s Rotaton 3.1 Rotatng Famewok Othe eadng: Valls 2.1, 2.2; Mashall and Plumb Ch. 6; Holton Ch. 2; Schubet Ch. 3 Consde the poston vecto (the same as C n the fgue above) otatng at angula velocty. An nfntesmal change n due to ths otaton can be wtten as sn(90 - )m (3.1) whee m s the unt vecto n the decton of change of and (90 - ) s the angle between and. Theefoe, d d d sn m sn m ) (3.2) whee we have used the defnton of a coss-poduct. Ths s the velocty assocated wth the otaton of the coodnate system. Imagne now that thee s addtonal ate of change n due to some othe pocesses. In ths case, d whee and denote the netal and otatng fames espectvely. d + o v v + (3.3) In geneal, a vecto B that changes n the netal fame s seen n the otatng fame as ( B) ( B) + B (3.4) E. A. Banes 32 updated 21:15 on Tuesday 29 th Septembe, 2015

2 Applyng ths elaton between the netal and otatng fames to the velocty v n place of B leads to dv dv + v (3.5) d (v + ) + (v + ) (3.6) dv d + + v + ( ) (3.7) dv + v + v + ( ) (3.8) dv + 2 v + ( ) (3.9) (3.10) o dv dv - 2 v - ( ) (3.11) whee we have assumed that d / 0,.e. the ate and decton of otaton s constant. Thus, we see that n the otatng fame thee ae two addtonal acceleaton tems (appaent foces). Cools tem -2 v : deals wth motons wthn the otatng efeence fame Centfugal tem - ( ): deals only wth otaton of the coodnate system These addtonal acceleaton tems ae fundamentally elated to the fact that otaton epesents an acceleated moton (even at constant angula velocty, due to the contnual change n decton). The Cools and centfugal contbutons only appea due to the specfc choce of an acceleatng coodnate system. They ae theefoe often called appaent o fcttous foces (fo example, they do not do any wok and theefoe dsappea n the enegy budget). Ths should not be taken to mply that these appaent foces ae unmpotant o that they don t do anythng (fo example, takng a tun too fast on you bke o wth you ca makes the centfugal foce qute eal fom the pont of vew of the otatng fame). Let s take a moment and get a feelng fo the scales of these tems fo lage-scale atmosphec flows, whee /s. Cools tem: 2 U /s 10m/s m/s 2 Centfugal tem: 2 a ( /s) m m/s 2 Thus, the centfugal tem s appoxmately an ode of magntude lage than the Cools tem. It sn t untl you have flow speeds of about 200 m/s that the two tems become compaable. E. A. Banes 33 updated 21:15 on Tuesday 29 th Septembe, 2015

3 Why don t we eve talk about the centfugal tem due to Eath s otaton when we talk about lage scale weathe systems? Cools effect On the sphee, we have (0, cos, sn ) (3.12) whee s lattude and the 3-component dectons ae n Catesan coodnates (local x-y-z). Thus, the components of the Cools tem n Catesan coodnates ae: Thus, we see that -2 v (-2 w cos + 2 v sn )î +-2 u sn ĵ + 2 u cos ˆk (3.13) polewad moton ) eastwad acceleaton equatowad moton ) westwad acceleaton upwad moton ) westwad acceleaton downwad moton ) eastwad acceleaton Thus, the Cools effect appeas to deflect objects to the ght. Note that all of these acceleatons/deflectons completely conseve angula momentum consevaton - t s just that ou ntuton s poo n otatng efeence fames, and so, thee seem to be exta foces. Also, emembe that the Cools foce s eally an appaent foce, that s, t does no wok, namely Wok: v F v F cos whee s the angle between the two vectos (3.14) Thus, fo the Cools foce, Wok: v ( v) 0 (3.15) Snce fo lage-scale dynamcs vetcal veloctes ae much smalle than hozontal veloctes, the Cools contbuton nvolvng w s much smalle than the contbutons nvolvng hozontal veloctes and s usually neglected; the Cools tem n the vetcal momentum equaton s much smalle than othe tems (e.g. gavty) and s also usually neglected (see shallow flud appoxmaton late): -2 v ((((((( -2 wcos + 2 vsn )î +-2 usn ĵ + ((((( 2 ucos ˆk (3.16) 2 v sn î +-2 usn ĵ f(v, -u) f(vî - uĵ) (3.17) E. A. Banes 34 updated 21:15 on Tuesday 29 th Septembe, 2015

4 whee f 2 sn ˆk (3.18) and f s the Cools paamete. Note that wth these appoxmatons, the Cools tem can also be wtten as: -2 v -f v -(-fvî + fuĵ + 0ˆk) fvî - fuĵ (3.19) The momentum equatons n a otatng catesan coodnate system (attached locally to the sphee) wth constant otaton ae: + 2 v -1 p - (3.20) and wth the appoxmatons to the Cools foce descbed above, these become: Du g (3.21) Note that we ae dopped the subscpt. Also, g now holds new meanng and efes to the effectve gavty (.e. ncludes a centfugal contbuton) to be descbed next Centfugal effect Fgue: In ths mage s the same as ou?, and g denotes the effectve gavty g eff. Taken fom Fgue 6.18 of Mashall and Plumb. The centfugal foce can be wtten as F ce - ( ) - (? ) 2? (3.22) E. A. Banes 35 updated 21:15 on Tuesday 29 th Septembe, 2015

5 whee? stands fo pependcula to denote the pojecton of onto the plane pependcula to. (Usng the ght-hand-ule, you can see that the fnal esultng vecto s n the same decton as? ). Ths can be wtten as the gadent of a potental: ce -( 2 2? )/2 ) F ce - ce (3.23) Thus, usng some tg denttes, usng the fact that? cos, and efeng to the fgue above, one can show that the components of the centfugal tem ae: - ( ) (0, - 2 cos sn, 2 cos 2 ) (3.24) go though sketch The vetcal component of the centfugal foce leads to a steady upwad acceleaton that opposes gavty. Howeve, ths upwad acceleaton s much smalle than gavty (by thee odes of magntude; see above) and s conventonally combned wth gavty to fom effectve gavty. In sphecal coodnates we wll defne whee now g g eff g gav + 2? - (3.25) s the effectve potental, ncludng gavty and the centfugal tem: The medonal component of the centfugal tem leads to a steady equatowad acceleaton whch ove geologcal tmes defomed the eath nto an ellpsod wth an equatoal bulge (even though ths defomaton s vey small compaed to the sze of the planet). The shape of ths ellpsod causes tue gavty to exhbt a polewad medonal component whch n (geologcal) equlbum exactly opposes the equatowad centfugal component. In ode to etan the moe smple sphecal coodnates we make the assumpton that these two components cancel each othe and fo sake of consstency exclude the medonal centfugal component. In dong so, we assume that the local vetcal s n the exact opposte decton of the geopotental/effectve gavty (that s, ẑ s n the opposte decton of g eff. Thus, we have ou momentum equatons, wth the vetcal centfugal effect hdden nsde. + 2 v -1 p - (3.26) How does the Cools effect opeate fo noth/south movement? How about east/west? E. A. Banes 36 updated 21:15 on Tuesday 29 th Septembe, 2015

6 3.2 f-plane appoxmaton Moton on small enough hozontal scales (e.g. mesoscale phenomena) do not feel the sphecty of the Eath (e.g. moton of hozontal scale that s at least one ode of magntude smalle than the adus of the Eath). In ths case, we may appoxmate ou coodnate system as a local Catesan system attached at one lattude and longtude of nteest (denoted by efeence values 0 and 0 ): (x, y, z) (a( - 0 ) cos, a( - 0 ), z) (3.27) that s, we descbe moton on a tangent plane. In ths case, the Cools paamete s a constant: f 0 2 sn 0. Hence the name f-plane appoxmaton. In ths case, the momentum equatons become (neglectng the Cools tems elated to the vetcal): Du - f 0v - xp + f 0u - yp Dw zp - g (3.28) 3.3 -plane appoxmaton Movng to a slghtly moe complex stuaton - thee ae cetan lage-scale motons (e.g. planetay waves) that feel the sphecty of the eath (and thus changes n f). One way to smplfy thngs s to allow the Cools paamete to change wth y (that s, lattude) but lnealy. In ths nstance, you stll have you nce tangent plane geomety. To do so, we expand the Cools paamete nto ts Taylo sees to fst ode about a lattude 0. Recallng that a Taylo sees expanson about x 0 looks lke: we obtan f(x) f(x 0 )+(x-x 0 )f 0 (x 0 )+ (x - x 2 0) f 00 (x 0 )+... (3.29) 2! f 2 sn 2 sn ( - 0 ) cos 0 f 0 + y 0 2 a cos 0 and y a( - 0 ) (3.31) In the above equaton, we have elated y to the lattude usng the fact that dy ad. Ths appoxmaton, when f f 0 + y s known as the beta-plane appoxmaton. Note, unlke the f-plane, the beta-plane allows fo vaatons n the Cools paamete wth lattude - t s just a lnea vaaton w..t. lattude athe than the full sn vaaton w..t. lattude. E. A. Banes 37 updated 21:15 on Tuesday 29 th Septembe, 2015

7 3.4 Contnuty and themodynamc equatons wth otaton Snce we have ntoduced the effects due to ou change of coodnate systems fom an netal one to a otatng one n the momentum equatons, we need to check whethe thee ae any changes n the emanng equatons of moton. Both, the contnuty and the themodynamc equatons nvolve the mateal ate of change of a scala popety of the flud (densty and potental tempeatue). Recall that a scala flud popety does not depend on the coodnate system they ae measued n - the value s what t s at evey pont, no matte what you call that pont! Futhe, the mateal ate of change of scala flud popetes s coodnate ndependent snce we ae followng flud pacels. That s, and D D (3.32) In the case of the themodynamc equaton, we ae done snce D Q (3.33) T c p and so thee ae no othe tems to woy about. Ths s also tue fo the evoluton of any othe tace. Howeve, fo the contnuty equaton, we stll have the dvegence tem to deal wth snce we only know the equaton s tue n the netal fame: Usng a smla set of steps as we dd fo the momentum equatons, - ( v ) (3.34) v (v + ) v (3.35) snce s paallel to but s pependcula to, the dvegence s zeo. Theefoe, - ( v ) (3.36) E. A. Banes 38 updated 21:15 on Tuesday 29 th Septembe, 2015