The Coriolis force. Atmospheric Motion. Newton s second law. dx 2. Dynamics. Newton s second law. Mass Acceleration = Force.

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1 Atmspheic Mtin Dynamics Newtn s secnd law m F Mass Acceleatin = Fce x Themdynamics dx m dt = F TMD Lecte Cncened with changes in the intenal enegy and state f mist ai. Newtn s secnd law The Cilis fce Newtn s secnd law x m F dx m dt = F Mass Acceleatin = Fce applies in an inetial fame f efeence. A line that tates with the ndabt Ω A line at est in an inetial system Bt we like t make measements elatie t the Eath, which is tating! T d this we mst add cectin tems in the eqatin, the centifgal and Cilis acceleatins. Appaent tajecty f the ball in a tating cdinate system

2 The centipetal acceleatin/centifgal fce Effectie Gaity g is eeywhee nmal t the eath s sface Ω twad fce =Ω R g* g Ω R g* R g Ω R Inwad acceleatin = Ω g= g* +Ω R effectie gaity g n a spheical eath effectie gaity n an eath with a slight eqatial blge Effectie Gaity If the eath wee a pefect sphee and nt tating, the nly gaitatinal cmpnent g* wld be adial. Becase the eath has a blge and is tating, the effectie gaitatinal fce g is the ect sm f the nmal gaity t the mass distibtin g*, tgethe with a centifgal fce Ω R, and this has n tangential cmpnent at the eath s sface. g = g * +Ω R When fictinal fces can be neglected, F is the pesse gadient fce ttal pesse F = p T d Ω = F + g dt d = p + T g Ω dt fce pe nit lme pe nit mass This is Ele s eqatin f mtin in a tating efeence fame.

3 Ω The Cilis fce des n wk the Cilis fce acts nmal t the tatin ect and nmal t the elcity. Ω is diectly pptinal t the magnitde f and Ω. Nte: the Cilis fce des n wk becase ( Ω ) 0 Petbatin pesse, byancy fce Define p = p ( T z ) + 0 p whee p 0 (z) and 0 (z) ae efeence pesse and density fields p is the petbatin pesse Ele s eqatin becmes dp0 dz = g 0 D 0 + Ω = p + g Dt g = (0, 0, g) the byancy fce Imptant: the petbatin pesse gadient p 0 and byancy fce g ae nt niqely defined. 0 Bt the ttal fce p + g is niqely defined. Indeed p 0 + g = T + p g Mathematical fmlatin f the cntinity eqatin f an incmpessible flid δz w + δw δy δx w + δ + δ

4 The mass cntinity eqatin Rigid bdy dynamics Incmpessible flid = 0 Mass m Fce F Cmpessible flid + ( ) = 0 t x Anelastic appximatin ( (z) ) = 0 Newtn s eqatin f mtin is: dx m dt = F Pblem is t calclate x(t) gien the fce F Flid dynamics pblems Flid dynamics pblems Ø The fce field is detemined by the eall cnstaints pided by the eqiement f cntinity the bnday cnditins Ø In paticla, the pesse field at any instant is detemined by the flw cnfigatin Ø The aim f any flid dynamics calclatin is t calclate the flw field U(x,y,z,t) in a gien egin sbject t apppiate bnday cnditins and the cnstaint f cntinity. Ø The calclatin f the fce field (i.e. the pesse field) may nt be necessay, depending n the sltin methd. I will nw illstate this with an example! Let s fget abt density diffeences and tatin f this example

5 U LO isbas A mathematical demnstatin D = p ' Dt = 0 Mmentm eqatin Cntinity eqatin steamlines LO U pmp The diegence f the mmentm eqatin gies: p' = ( ) This is a diagnstic eqatin! Assmptins: iniscid, itatinal, incmpessible flw Bt what abt the effects f tatin? Newtn s nd law etical cmpnent byancy fm mass acceleatin = fce Dw p = T g Dt z Pt Then p = p (z) + p T = (z) + Dw p = + b Dt z whee whee dp g dz = b g =

6 byancy fce is NOT niqe Byancy fce in a hicane b g = () z it depends n chice f efeence density (z) bt pt p' g = + b z z is niqe () z z Initiatin f a thndestm tppase negatie byancy tflw θ = cnstant iginal heated ai θ = cnstant psitie byancy LFC LCL Τ+ΔΤ T inflw U(z) negatie byancy

7 Sme qestins psitie byancy tflw p' Ø Hw des the flw ele afte the iginal themal has eached the ppe tpsphee? Ø What dies the pdaght at lw leels? Obseatin in seee thndestms: the pdaght at cld base is negatiely byant! iginal heated ai LFC LO Answe: - the petbatin pesse gadient LCL inflw negatie byancy F fictinless mtin (D = 0) the mmentm eqatin is Let R 0 The gestphic appximatin D + Ω = p Dt Ω = p This is called the gestphic eqatin petbatin pesse We expect this eqatin t hld appximately in synptic scale mtins in the atmsphee and ceans, except pssibly nea the eqat. Chse ectangla cdinates: k = (0,0,) z k x Ω = Ωk elcity cmpnents = (,,w), = h + wk h = (,,0) is the hizntal flw elcity h y

8 Take k Ω = p (k )k = (0, 0 w) The gestphic wind Ωk ( k ) = Ω[ ( k ) k ] = k p h = k hp Ω h and = k hp Ω 0 = p z h This is the sltin f gestphic flw. s h p = ( p/ x, p/ y, 0) Ø The gestphic wind blws paallel t the lines ( me stictly sfaces) f cnstant pesse - the isbas, with lw pesse t the left. Ø Well knwn t the layman wh ties t intepet the newspape "weathe map", which is a chat shwing isbaic lines at mean sea leel. Ø In the sthen hemisphee, lw pesse is t the ight. Chice f cdinates Gestphic flw Ø F simplicity, let s ientate the cdinates s that x pints in the diectin f the gestphic wind. Ø Then = 0, implying that p/ x = 0. p = Ω y Ø Nte that f fixed Ω, the winds ae stnge when the isbas ae clse tgethe and, f a gien isba sepaatin, they ae stnge f smalle Ω. pesse gadient fce isba isba Cilis fce (Nthen hemisphee case: > 0) lw p high p

9 A mean sea leel isbaic chat e Astalia Nte als that the sltin h = k hp Ω and 0 = p z L tells s nthing abt the etical elcity w. Ø F an incmpessible flid, = 0. Ø Als, f gestphic flw, h h = 0. H H H Ø then w/ z = 0 implying that w is independent f z. If w = 0 at sme paticla z, say z = 0, which might be the gnd, then w 0. The gestphic eqatin is degeneate! Vtex flws: the gadient wind eqatin Ø The gestphic eqatin is degeneate, i.e. time deiaties hae been eliminated in the appximatin. Ø We cannt se the eqatin t pedict hw the flw will ele. Ø Sch eqatins ae called diagnstic eqatins. Ø In the case f the gestphic eqatin, f example, a knwledge f the isba spacing at a gien time allws s t calclate, 'diagnse', the gestphic wind. Ø Stict gestphic mtin eqies that the isbas be staight,, eqialently, that the flw be ni-diectinal. Ø T inestigate balanced flws with ced isbas, inclding tical flws, it is cnenient t expess Ele's eqatin in cylindical cdinates. Ø T d this we need an expessin f the ttal hizntal acceleatin D h /Dt in cylindical cdinates. Ø We cannt se the eqatin t fecast hw the wind elcity will change with time.

10 The adial and tangential cmpnents f Ele's eqatin may be witten w p f = t θ z The case f pe cicla mtin with = 0 and / θ 0. p + f = t w f = θ z p θ Ø This is called the gadient wind eqatin. The axial cmpnent is w w w w p w = t θ z z Ø It is a genealizatin f the gestphic eqatin which takes int accnt centifgal as well as Cilis fces. Ø This is necessay when the cate f the isbas is lage, as in an exta-tpical depessin in a tpical cyclne. The gadient wind eqatin Fce balances in lw and high pesse systems Wite p 0 = + + f Cyclne Anticyclne tems intepeted as fces V V Ø The eqatin expesses a balance f the centifgal fce ( /) and Cilis fce (f) with the adial pesse gadient. Ø This intepetatin is apppiate in the cdinate system defined by and θ, which tates with angla elcity /. LO PG CE CO LO CO PG CE

11 The eqatin p 0 = + + f p = f+ f + 4 is a diagnstic eqatin f the tangential elcity in tems f the pesse gadient: Ø In a lw pesse system, p/ > 0 and thee is n theetical limit t the tangential elcity. p = f+ f + 4 Ø In a high pesse system, p/ < 0 and the lcal ale f the pesse gadient cannt be less than f /4 in a balanced state. Chse the psitie sign s that gestphic balance is eceed as (f finite, the centifgal fce tends t ze as ). Ø Theefe the tangential wind speed cannt lcally exceed f/ in magnitde. Ø This accds with bseatins in that wind speeds in anticyclnes ae geneally light, wheeas wind speeds in cyclnes may be qite high. Limited wind speed in anticyclnes In the anticyclne, the Cilis fce inceases nly in pptin t : => this explains the ppe limit n pedicted by the gadient wind eqatin. V CO CO = f CE = PG CE End f L