Dynamics of the Atmosphere
Dynamics of the Atmosphere ATOC 5060
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Geostrophic adjustment enpo tmtial height 16 crass genial I an isobaric su rface Speed perpendimlar Geostrophic adjustment 1 39iI39ime initial Heights dirs u39lz an EU Winds disiuman m 2500 km TheI COMET Program Geopotsntial height in ems sectim I an isobaric svu rface Speed mrmmdiwlar Geostrophic adjustment 2 Time Water Heig hts dis J39barIcE 02139 CI Um s Winds 4 25am km The COMET Frugram GEDP t nti l heigiht 1 train sanctum l on isuba ric 5m rface l Speed perpendicular Geostrophic adjustment 3 39l n ne firmal Heights ditsiLI39bLmcc L EI IIE39I Unit3 Winds diEKLIEtIEI I ca BEHHBI39 time 25M km The rCUMET Frugram Inertia gravity waves disturbance is heating with condensationlatent heating mva pss 5m mb CDNTRDL run 2 ETA model test Geostrophic adjustment Large scale state tends to wards balance geostrophic ow Flow must satisfy conservation of PV the entire time thus defining a path to the steady state Specifically PV conservation offers a constraint on the way a steady state is reached Balance reached by emission of interia gravity waves A third of potential energy liberated converted to potential energy rest radiated away by waves Flow adjusts to disturbance larger than Rossby radius otherwise disturbance adjusts to ow Fluid away from the disturbance X gtgt RR does not feel the disturbance Important problemissue for numerical simulations prediction initialization generation of waves Quasigeostrophic system or why we love elliptic equations for QGPV Charney s QG the motion of largescale atmospheric disturbances is governed by Laws of conservation of potential temperature and potential vorticity and by the conditions that the horizontal velocity be quasigeostrophic and the pressure quasihydrostatic Section 2 On the scale of atmospheric motions 1948 Baroclinic cyclogenesis Hg ns thm nnl mm H n nu Cyclone development 500 mb heights and MSLP for OUZIONOV1975 Cyclone development 500 mb heights and MSLP for OUZHNOV19 Cyclone development 500 mb heights and MSLP for OUZIJNOV19 Cyclone development 500 mb heights and MSLP f 00215NOV1975 Geostrophic vorticity Define quasi geostrophic vorticity 4 g 8x 8y Geostrophic wind equations give 1 6CD 1 acD Mg g f 6x f0 6y 2 2 gigaaaam fn 3X 0y fn Since ow is nondivergent we may de ne a streamfunction 2 1 2 4g v 11 7v c1gt Knowing stream lnction geopotential can invert for vorticity QG vorticity equation Much like derivation of vorticity equation take BBx of vg equation minus 66 of ug equation 0 7 vg vgg ffo 0p cf primitive vorticity equation scaled vorticity equation 7 E It 7 v vf map fV Vk dprw Recall vorticity equation lead us to the Rossby potential vorticity Expect to nd a QG potential vorticity in the not to distant future Steps in QG derivation Horizontal ow is geostrophic horizontal advection by geostrophic ow Coriolis effects beyond that needed for geostrophic balance due to l Ageostrophic ow 2 Variation in Coriolis parameter with latitude Vertical motion due to ageostrophic divergence Atmosphere is hydrostatic Atmosphere air is an ideal gas Conservation of mass Charney s QG the motion of largescale atmospheric disturbances is governed by Laws of conservation of potential temperature and potential vorticity and by the conditions that the horizontal velocity be quasigeostrophic and the pressure quasihydrostatic Section 2 On the scale of atmospheric motions 1948 Filtered equations Primitive equations QuasiGeostrophic equations V q a T givenJ Vg Va q a givenJ W W Er7fkgtltV7VltIgt dtg7f0kaa7 ykag 01 R T 01 7 7 R T 0 p 0p p Ou 6v 6a 0M 0v Ba 0 a a 0 6x By 617 0x 6y 0p EJrVVJTnS39Fai 3VgVI7gj7aw i 51 CF at 017 CF 7 5 independent quantities All variables related to q or q QG system Advantages changes in Z T and V are linked relations between variables are simple expediting the math and the interpretation energetically consistent captures the gross features observed in middle latitudes Some properties T or e is proportional to 6 6p advected velocities are geostrophic advecting velocities are geostrophic Vertical advection contribution in thermodynamic equation associated with adiabatic work expansioncompression dedt OJS Where S is a horizontal mean static stability vorticity equation has 3 parts 7 00 c ange 7 horizontal advection of geostrophic absolute vorticity both relative and planetary 7 fD divergence term QG prediction A barotopic situation would dictate the advection ofvorticity can be used to show estimate changes in vorticity an s geopotential height e thermal wind balance describes balance between horizontal ow at different evels So change in the vertical wind shear will induce vertical motion stretching and ageostrophic ow to maintain is required the thermal e These changes can be larger than the advective role However it is the vertical motion induced by different vorticity advection at two different layer that is of central interest Similarly thermal advection will be associated with vertical motion as the vorticity at one height will change relative to another To close our system we must remove the vertical velocity from the equa 39ons it appears in BOTH vorticity and thermodynamic equation It is from here that the QG potential vorticity emerges PV 1nvers1on 1 PV related to geopotential by elliptic operator since g TVZCD Consider PV eld 4 44f D q rgb 0 rgtb l Recall Laplacian tends to smooth as in weather maps and Photoshop CD obtained from PV tends to spread beyond PV features This gives a type 0 action at a distance Example PV inversion with ozone Ozone concentrated in stratosphere PV concentrated in stratosphere stronger stability PV conservation means PV behaves like a passive tracer much like ozone So could use measurements of ozone to guess PV eld then invert this derived PV to obtain wind eld Then use thermal wind equation to deduce temperature eld Would be useful near tropopause and lower stratosphere where wind measurements are dif cult Would this work Ozone d 39 PV calculated measure from weather from TOMS nym 1 Iuwgmmlumle t Mmumnu sznd pm mmv ll39VymIrgmzd in mu m l m mi mm wimullml m I m rvu m mm Comm mm M the Ian 4 m mum nghv u n 1 Wu mm mm w mm D dzrhzd m umgnpm Kw Islam 3 m Aummm 1m unwork chhmao um looman I2 quot51 3391 ZOOhPa wind from PV OZOIIC looman I2 quot51 3391 Atmospheric dynamics ATOC 5060 Class aims to Develop understanding of what drives atmospheric motions Behavior of dynamic structures of the atmosphere on short and long time scales Understand whyhow the atmosphere changes Be able to predict howwhen changes will occur Explain changes in terms of transfer of energy from one form to another kinetic potential chemical Explain role of time mean ow in contrast to transient ow Do this as simply as possible ie use some good approximations Today s weather ltquot i 3 quotquot gt NW 3 Sea level pressure and o th 39 M b We i 7 I 1000 500 hPa tluckness x 39 v r t Geopotential height and k relative vorticity at 500 hPa Wm Global weather today 500 hPa geopotential height and vorticity Qquot i E stronger less wavy Northern Hemisphere Southern Hemisphere To quantify is to understand How do we quantify the atmospheric state State variables e g temperature humidity density pressure wind in 3d chemical concentration Others Vorticity divergence potential temperature potential vorticity angular momentum kinetic energy potential energy Things that are homogenous and do nothing are boring H 0w do we describe things happening Rules for changes in state variables Changes can be in time Change can be in space Change can be relative to other state variables example Rules can contains interactions between different state variables Sound like differential equations Xanosphereconserves Energy mechanical heat chemical Mass Momentum and angular momentum of air and other gases To describe motion we can make use of 0 Newton39s 2nd law 2F ma 0 First Law of Thermodynamics 0 We need to account for fact that earth is rotating spherical and there is gravity Sy abus Dynamics of the middle atmosphere 201 00 km Tropical dynamics The General Circulation of the Atmosphere on Earth mostly Instability of large scale waves Waves in the atmosphere Ways to simplify the full equations to be more useful Governing laws of atmospheric motion Polar stratospheric vortex Low pressure center over very cold winter pole 01 Sep 1996 Colors show potential vorticity at 10 hPa N 30 km Energy lost by winter night cooling but balanced by deposition of kinetic energy by atmospheric waves Also exists on Mars and Jupiter See dynamics from chemicals Aug 10 1997 Satellite measurements of nitrous oxide N20 Source in low latitudes can not penetrate vortex wall for dynamic reasons N20 destroyed inside vortex by chemical reactions Similar to ozone EPITOMS Total Ozone for Jun 1 2002 Dobson Units EENnSam Damey lt 100 Red gt 500 DU Tropical variability e g MJO Outgoing longwave radiation OLR Time months 2 39 Central America L ongitude Tropical Paci c Ocean Indonesia l Westerly Wind and temperature Temperature and wind eld related thermal wind balance ie alBZN if0y 39Knowing temp erature gradient estimate jet 39Knowing jet estimate gradient We want to understand What balances give rise to this robust structure Wind and temperatures related to overturning circulation WA Annual average U vr pnrssunz am Stationary ow HA7 h A7 n 7N r NCEP Reanalysis Z500 January mean QON GrADS image 5050 5 50 5250 5350 5450 5550 5650 5750 5850 0 Growth of waves in the atmosphere 0 baroclinic spinup How do waves grow Temperature increases in tropics increasing potential energy Temperature decreases at poles decreasing potential energy Eventually this becomes unstable and waves grow heating 1 I cooling mixing by waves Unstable Stable Common themes in topics Conservation laws hold Atmosphere can sustain wave motions on many scale There are mechanism that lead to instability and wave growth Waves transport momentum and energy which can alter behavior of the mean state The atmosphere is always moving Stable states represent a balance between two or more opposing in uences Atmosphere conserves Energy mechanical heat 0 Mass of air and other gases Momentum and angular momentum To describe motion we can make use of 0 Newton39s 2nd law 2F ma 0 First Law of Thermodynamics 0 We need to account for fact that earth is rotating spherical and there is gravity ATOC 5060 Focus on being able to use theory for atmospheric problems Tuesdays lectures covering foundation material review last homework exercises Thursdays application of theory to problems set up new problemsexercises Weekly homework exercises many from the text to be reviewed in the following class by a random person Projects or a research nature Strongly encouraged to read ahead and read from other sources Many of the details needed for homework assignments will require you to study the text I expect you to spend at least as much time in class as out of class Doing so will get you an average passing grade Homework Three questions 1 Approximating the role of earth s rotation To go over in class 2 Review of partial derivatives potential temperature and coordinates Especially review advection 3 Review of scale analysis a theme in the first few weeks ie read chapter 1 2 and 3 0f the text book Vonicity equation Geopotential height at 1000 hPa Flow parallel to contours outside tropics geostrophic Flow peipendicular to contou near equator ie divergent Walker circulation Rotating table apparatus R0 ATLNG ANNULUS COLD DEMONSTRATION L3 ROTATION my 6100 GMT I ubxrrvc J u z u mlnpuu d hm x I m y t and he um I u a km imcs whom b t r urlmz a von Neumann is a coauthor prediction With computers Worlds rst numerical 1950 O D m B D 39lt 9 E Gravity waves external and internal Lenticular lg 7l0 Slrcamllium in slr udy llmv over an iulinil um upward displacemcm Altar Dumm 991 Trapped waves narrow ridges periodic waves broad ridges 39 N of sinuwidal ridge Ior he mum ndgv xlnrl brmul villgc mm h The dashed IIIIC in b shows the phzm of lumimum Long Term Mean uwnd ms we Surface drag Long Term Mean ufl NmAZ NOMrCREsCh39ma e Dwaqnbs cs Gamer amp f v 39 e 50w GrADS image 705 704 703 73912 70A 0 0A 02 03 Gravity wave stress Long Term Mean ugwd NmAE am a Vertical velocity simulation 3km 7 i 30 KM httpmoemetfsuedurhartmtnwavehtml x km Figum 1413 39 39 5 mm s cxcncd by uniform ow 0ch u wwdimemiunul ridge um km wide Arm Quency 194s Homework exercise for Thursday Show that acoustic waves have a phase speed proportional to RT What is the speed of sound near the surface What is the speed of sound neat the tropopause Gravity waves and adjustment Trapped waves narrow ridgesquot periodic waves broad ridgesquot ig mu Strenmlmuxm mm mm mm mnnm wuth muwum mlgc Iquot nu nurqu Imgu we I and mm xulgc one In The mm lmr m my mum nu phuw m maxuuum upward dlxpluucmclll mum Ummu I99 4m y a0nu yHyuunuum bl c 1 25 25 20 3 2 N 5 15 I0 0 5 5 1 I l I Alllllllllllll lkllllIIIIII IIIIIIA1 III q o 2 3 4 5 G I 0 5 20 25 30 35 45 x km Wm mix 10quot lmquot Figure 1414 ammumn cl Rm 10139 Iwn mum 39 nun v 1 m v I gtH H x 391 A y puun umer u l m In J Am mulmn pussushllhl u IIII llumwquotl39 2jz um 3923 A q 7 UL 1ruuud mmk mnlnlm Incxcnmnh u umu moving mmy me my m lhc gmle where Wm WM M V9quot m39m quot I39m W quot L ckxm39 c mupumnl 1 Much rm slum hurl uxllid n39 n 39x 4 Iunmw ImI m 1 V V r e39 th u 39nuuuk I m Iml mufth mm InlulIL Hummus lhc quotupmphcru Mu 1mm mmmmm an mncln 2 l n hm 2 lungur lmnuul y r Y tn 1 h ml mmcngm prupngmm inln um Mmlmplwlc mm c m39uumcnng u Hung Icch Salby Intro to Arm Physics I no v I v v I 40 39 39 39 39 39 39 39 39 a b C 35 35 so 30 A 25 25 an g m 2c N 2 6 2 ms m x10 m x km Figure bus muuumn Fiuurw ms As in Fig 14 cpl m cumlikiuni mpmumnuvc uhunmmruml fur cum pnmnlIThcvuvcisDopplershiftedmmmimr39 plmxc c r 7 Lnnhccrincnl lchwhcrc m1 r 39 Mn quotr my v I mluccd rm clamy h39unIimm m w hiulmr lmmk 39I hc ruuinn of small In2 in In upper Impusphcrc coincidus Salby Intro to Arm Physics Diagnosis of vertical motion Geopotential tendency QG vorticity equation 66 VE Veg f4r f0 if QG thermodynamic equation a jy 6cm R J V V 7 iaw at E K 6p j cp p As a consequence of conservation 0fQG PV V2 f vg Vivzq3fif zvg 6p 0 6p at f 6p 0 6p Notice left hand side is an elliptic operator for the tendency which can be solved given boundary conditions Terms on the right are all in terms of geopotential 00 mb heights and MSLP 500 mb heights and abs vort 500 mb heights and height tend for 00213NOV1975 Vertical motion 0 In deriving geopotential equation we canceled omega 0 Could combine to cancel geopotential tendency to get omega as a function of geopotential QG vertical motion pa was eliminated in prediction equation but is important for for instance stretching Specifically if we want the ageostrophic divergence need to be able to extract it from horizontal geostrophic ow 7 recall divergence appear in vorticity equation Consider momentum equation Continuity Thermodynamic Each OJ as a res1dual from a difference in large quantities 6 aiam0 2V V 0a i 6x 6y 6p at g 6p 0p p Expect most accurate method to be based on state of geopotential rather than change in geopotential as eg in vortiCity equation Omega equation Merge QG vorticity and thermodynamic equations To derive the geopotential tendency combined to eliminate 03 Now eliminate tendency terms and retain n vhf Ozijm f 3 viv2ltigtfj ivzvgv7 vv 0 p 0 6p2 Fap g f0 6 op vorticity temperature diabatic advection advection heating For adiabatic conditions scale analysis gives diagnostic equation 2 Z v2 f 00 2 mi gv ivzquf 039 0p 039 0p f0 Depends only on geopotential Not divergence Not vorticity tendency Not temperature tendency Recall thermal wind WOC V2 f026m N 6Vg VLV2 f 6 6p 6 617 f0 ie since Laplacian give opposite sign 6V R VT g kxvr1nfampi 6 f K 17 J So vertical motion associated with advection of absolute vorticity by the thermal wind Thermal wind parallel to isotherms eg 5001000 hPa thickness This has immediate use in understanding weather Properties of omega Obtain accurate 0 diagnostically recall no acceleration in hydrostatic equation Ascent 0 lt0 wgt0 Descent 03gt0 wlt0 Vertical velocity exists to ensure thermal wind balance in the presence of different vorticity advection at different levels All good 500 1000 m geopotential E V 0 temperature gt 500 N w 4000 Vaful l vfivicml K cap 6 6p m I 4500 D 1000 2000 3000 433m 5000 soon 7000 a w QC VT V Offset isobars and isotherms due to westward tilt Cold temperature advection implicates thermal wind So vertical velocity must exist for balance 03 qualitatively by looking at vorticity advection ALONG isotherms Vorticity advection at ma imum above surface low thus ascent mb heights and temp for 00213NOV197 Q Today 57R 39r 1 r39 3 1 ngm k 1 39 r 4 GFS Anu ys ws 22 Tue 12 FEB 2003 500mb Genpoten a Heigth dam Vumcity W ssec QG wrap up What are the key assumptions What is the role of divergence Why is QG PV useful what are its special properties and where does it come from What are the relationships between the various variables of interest mathematical sure but what is the physical meaning Why was is QG useful for weather forecasting QG can allow new weather features develop To understand this in more detail it is useful to consider how atmospheric waves can amplify MRS 5 3 39 crs Anu ysis 122 Tue 12 FEB 2003 500mb Genpoten a Heights dam Vumcuy W ssec camms 7 r 5 as mm 22 Tue 12 FEB 2003 was EULAch5 crs Anu ys ws 22 w M FEB 2003 500mb Genpoten a Heights dam Vumcuy e isec was in GFS Anal sis lZZ Thu M FEB 2008 l 554 SLP mHUUU lOUD7SDUmbThickness dam Prediction for tomorrow Cyclone will not change too much might weaken a bit 7 cold advection upper leve s A Atmosphere has become somewhat barotropic of yesterday quite baroclinic 7 but will continue to move southeast following upper level flow ie 39 Temperature contours mostly parallel to SLP so not much advection of vorticity so not as much vertical velocity snow will slow and stop Notice too develoment of a high means subsidence so probably clear sky 7 so sunny Probably few clouds as the new system moves off to the east and causes oca mixing PS 7 check humidity field need water for snow and atmosphere has dried out over last few ays quot Mn 5 a gam 739 ki39 g 55 2 b 5 quot r m 1 SLP mwuou wouoisoum Thickness dam by czs 24m NAM ssueu wzzwiazuoa win 122 Fri 15 FEB 2003 700mb Vev ca Vemcny muw 12m Pvecip mm Atmospheric waves Inn w 039 w Individual Obs hgt m 5151 GTAD S image 1 4850 4950 5050 5150 5250 53 550 5650 5750 5850 39d Mer H 1 from a MS I mini w Individual Obs E GrADS image 7350 i300 7250 7200 71507100 750 100 150 200 250 300 350 mt pmtej mm 7 m mm m a Juh Monthly Longterm Mean 196871996 hgt m 56 mow NL EP GrADS Image J 5050 5 50 5250 5550 5650 5750 5850 7225 7175 4 25 r75 r25 25 75 25 175 Geopotential tendency and vertical motion 0 mb heights nd 330 K pott vortt fo 00213NOV1975 Recall PV inversion Knowing the PV we can estimate everything else Temperature wind geopotential In QG since the flow is geostrophic we can obtain the wind field exactly Then with thermal wind balance obtain the temperature exactly More generally ie not QG need to assume a number of balance conditions to do the PV inversion Example PV inversion with ozone Ozone concentrated in stratosphere PV concentrated in stratosphere stronger stability PV conservation means PV behaves like a passive tracer much like ozone So could use measurements of ozone to guess PV eld then invert this derived PV to obtain wind eld Then use thermal wind equation to deduce temperature eld Would be useful near tropopause and lower stratosphere where wind measurements are dif cult Would this work gt PV calculated from weather measured 3 J observations from TOMS u m End mm mmrhy mvu quotWm burn on my mm m mm H w mm Cnnumv mum an m Irquot 41 an m n u 1 Wu nuh Wm Mm 1mm amyupum mm D 55w 1 ll mum 1nKI 1NI3UI4 HItNu n mm 3 am In 10000TRM I2 quot51 3391 ZOOhPa wind from PVOzone mm mama Mum mm a um um untamed mm xnvcmn m mm mm mnqu wv Mulls smg e smaullvwg m rm hm Wm ummumm looman I2 quot51 3391 Example more PV anomalies PV near tropopause PV near surface PV anomaly shaded Potential temperature and Wind anomalies contoured Hoskms 22 a1 1985 QJ39RMS 111 877946 Example Vertical coupling forced at 250 hPa Amount of vertical inversely proportional to length scale as Laplacian selects for smaller scale namlilud mam mudnay Consider scale via m4 1 sin m 6w iimcosmz 07 Again an artifact of the smoothing associated with the eleptic Example Impact of aspect ratio Recall vertical and horizontal scales are related to f and 6 Specifically km N mmmpun WW I 39 quot 2 quot39 r m Con51der scale Via k1 39 k L m 11 sin m N sin 10 a W N k cos lo 62 2 a 7k sin kx Z N 43quot So more vertical propagation for longer a waves k small k391 large 5m vm wquot In lture we will see that this meansfor long waves upper troposphere ow can mess with lower troposphere Geopotential tendency a synoptic meteorology View of QG 139e PV is nice and all but geopotential makes F more sense to me Prognostic geopotential equation QG vorticity equation 66 Vg Veg fdr f0 if QG thermodynamic equation a K 6cm R J V V 7 iaw at E K 6p cp p As a consequence of conservation 0fQG PV V2 f vg Vivzq3fif zvg 6p 0 6p at f 6p 0 6p Notice left hand side is an elliptic operator for the tendency which can be solved given boundary conditions Terms on the right are all in terms of geopotential Geopotential tendency equation VZZ ZJJfOV V 1 quyf3fozv g f 8 0p 0 01 0t 0 0 0p vorticity temperature advection adVeCtion Thermal advection dominates development stronger in lower troposphere Through thermal wind balance ensures that low level development is associated with upper level change surface cyclogenesis causes upper level trough etc Notice this suggests a transfer of energy from potential energy related to thicknessgoepotential height to kinetic energy more vorticity and stronger wind speeds This conversion is fundamental to the general circulation QG prediction 6 9quot 6w QG vorticit 5 V V y at g a f f0 6p i 2 VOITiCity vorticity g 7 f0 V q advection stretching ageostrophic Given d can compute ug and vg What about vertical velocity Eliminate it Use thermodynamic equation V2 fOV VELVqudrrjii iv 8 f0 0 8 0p 0 0p at 0p vorticity Vertical difference advection in temperature advection Recall Laplacian gives dd lt 0 when dQ gt 0 etc Inverting Laplacian tends to smooth And give a minus sign Also elliptic equations have solutions Geopotential tendency zifii i Lz ifiE V 6Pa at r ngg VfDVlIf apa Vg VBP 0 CD falls with positive vorticity advection cyclonic 0 CD falls with when warm air advection increases with height or when cold air advection decreases with height Example 0 Warm air advection at the surface causes increased thickness 0 Increased thickness causes high pressure at layer top 0 High pressure creates ageostrophic horizontal motion 0 Unbalanced ageostrophic drives divergence mass lost from layer 0 Divergence lowers heights and creates low pressure at surface still high pressure at layer top 0 Surface low causes convergent ageostrophic motion for balance 0 S0 knowing geopotentiul can estimate quotomegaquot with QG 500 mb heights and MSLP for 0021 NOV197 l j V 500 mb heights and abs vort for 002 3NOV1 Q75 QG vert1ca1 mot1on quot 39 39 quot 39 equation Fnr instance stretching Speci cally ifwe mm the ageostrophic divergence need to be able to extract it from hon39zontal geostrophic ow so w practical issue c iderrnornenturn equation Continuity Thennodynamic Each was a residual from a di erence in large quantities Bu 3H v E map i at g 8p op p a 6v 6m 6x 6y 0 611 Change in geopotential as in vorticity equation Luau Turbulence and boundary layers HW graph Eddy heat flux VT jan 2007 lev500 hPa heat flux K mls planetary wave number Total stationary eddy heat transport 2 85 K ms Total transient eddy heat transport 2 95 K ms Weather and turbulence Big whorls have little whorls which feed on the velocity and little whorls have lesser whorls and so on to viscosity Lewis Fry Richardson This class Start to consider role of interaction with the surface Momentum transfer at the surface also heat and gas exchange Role of turbulence in mixing momentum Reynolds averaging Richardson number Momentum equations fv ia pFrx dt pax dv lap F dt f pdy y l7 Frlmanav Divergence Away from the surface we can ignore friction Led to geostrophic assumption By introducing friction drag change the way we can balance the equations of motion Importance of Turbulent Eddies Turbulent eddies are important in the atmospheric boundary layer because they can transport momentum heat and moisture As a result the dynamical equations that we have discussed during this semester must be modi ed for use in the atmospheric boundary layer 0 We will now introduce a strategy for including the effects of turbulence in the dynamical equations To do this we will attempt to separate the turbulent variations in atmospheric properties from the largescale variations Vertical structure in the ocean K m emuquot Data urmssa u new rule s l MDW K A l l I l NoAAPMELTAU Praieci Umee tum Character of turbulence The effects of turbulence can be ignored in the free atmosphere but can not for ow near the surface eg drag ensures ow is ageostrophic Viscosity ensures Wind speed is zero very close to the surface Turbulent transfer much more efficient than molecular effects or thermal condition Turbulent eddies exist at all time and space scales between the limits of the boundary layer depth and the scale at Which molecular diffusion take over millimeter Sonic anemometer Measures very small scale variation in 3d Wind field An Example of Turbulence The effects of turbulence are evident in this record of surface wind speed measured by an anemometer The gusts and lulls in the wind which typically last for less than a minute are indicative of the passage of turbulent eddies During this gust the wind speed increases by 50 Wind Speed ms Turbulence 39 Eddies small but still much larger than Viscous scale Energy transferred to smaller scales Where ultimately dissipated by molecular diffusion 39 Small scale eddies generated by Wind shear dIVIdX and by buoyancy ie convection Equations for turbulence We Wish to evaluate the F r terms in the momentum equation and equivalent terms in the heat and moisture equations Make use of random nature of turbulent eddies in a statistical representation Turbulent eddies small compared to synoptic scale motions can ignore Coriolis acceleration Also can make some further simplifications of primitive equations for near surface conditions Boussinesq assumption Boussinesq equations Horizontal momentum Thermodynamic dl i Zip Fr a po a X rw dt dz dv 1 3p ditf piaiyFry Continuity 0 mean density does not change Vertical momentum 31 37V 31 Z 0 non hydrostatic Bx By Bz dw 1 317 A19 7 iigiFrz dt pO dz 60 Note 19 0A6 Where 19 is appropriate given 0 Reynolds averaging Define all quantities to be composed of a time mean and a deviation 1quot 2 u Iquot At a given point this deviation v V v39 from the time mean gives a w W W measure of turbulent eddies 9 9 Compare this with spatial deviations we examined with waves Stationary topographically forced waves H l i u 3 quot 39t NCEPlkmn ynsZSOO January mean l M Long Term Mean hgl m 90 GrADS hnage 5250 5350 5450 5550 5550 5750 5850 NCEP Reanalysis ZS 00 m 1 Vi mm r a 5 a a la H 9 am 39 H000 M Dev1ations from zonal mean January mean Separate eddy variations from background a g 3 E E E39 39 5 i5 J E E a EE u a g ann i mantra quot E 5quot 5539 55555 siiiifsss si lmifigiEE umem A simple way to separate the turbulent variations from the largescale variations is to average our wind measurements over a period of 3060 minutes Turbulent variations Instantaneous velocities can be decomposed into mean and 7 turbulent components M M M u u L7 turbulent f mean velocity velocity instantaneous vel city Turbulent velocities are the positive and negative deviations of the instantaneous velocities about the mean Rules of averaging quotquot 4 L ft11rbulenceinL 39 quot l we need to have some asic rules for dealing with the mathematics of averagingr mputing the time average of a variable Axyzt that is a function of space and time 7 1 quot1 AXyz Zer gt511 discrete function 10 1 P Axyz77 IAxyztdt continuous PH function Averagin g Let A and B be two variables that vary over time and let c represent a constant We will show that A B X E Computing the time average of a variable Axyzt that is a function of space and time Ni ABlzt3 HEW NH i0 1 p p 72Aza IAE39H de N a i0 1 1 p 1 p WZA WZB LAW 34 AB discrem continuous More averaging ough similar mathematical manipulation we can derive the following averaging rules Next we will apply these rules to variables that are split into mean and turbulent componentsr Reynolds Averaging Let AXa and B b What is the average ofA K Xa m X In order for the above to be true then 1 0 Notice important outcome that the mean of the deviations is zero sum of positive deviations must equal the sum of the negative deviations Averaging products E K a if b K a Kb a b KFaFKHEm What is the average of the product ofA and B However a F a 7 m 0 Thom Kb Xb K00 AB a39b39 Note that the second term on the right nonlinear product or cross product is not necessarily zer N71 fb 23 Note that Reynolds averaging governing equations Substitute into momentum equations ML V Wiii a 39f nB my 3911 E V Lii a az39fm my Noticing advection can be wrinen in the form Mia 3 am 3W 3 B a J y Bz LagaltmaltwgtMMMM dt at ax By az ax By Bz Use fact that mean of mean times deviation is zero Turbulent terms Expanding and taking the time mean of all terms a 13 at 39 MW 8 8v Blw a So defining F7 in 1 37 74 777 1 p a V We have an expression of the frictional term In general the z component is largest and of most inmest Turbulent governing equations r riir l 4 pm Ex 3 By Bz At pm By Bx By az la az Note a A Example kinetic energy kguuvuwz per unit mass Substituting for mean and deviations 7 l 2 7 2 7 2 EEZEuuVVww Collecting wrms and taking time av rage n e mean ofmean times deviation is zero g 17 2 2 Mean kinetic energy E Eb H W Turbulent kinetic energy EEWW Diurnal TKE development Hnlnhl km Time in Fig 51 Modeled lime and 59m va zlxm m a mmulence mam energy w ms m s i Iorwzmava From Yams a and new 975 TKE equation Evolution of TKE E 72 Use eddy form of governing equations to derive TKE equation dz 73 73v g BM 13w 7 7 u 9 7 7 d Waz VWBzW 0 Bz z A B C D E F A local change and advection by the mean ow B mechanical production due to wind shear C production by buoyancy convection D transport by eddies E redistribution by pressure gravity waves Fdissipationmnerionof L 39 to heat Flux Richardson number Ratio of buoyant and shear eddy generation buo anc oduction an av Rf mecil aniCZ ryroduction W 9 9 w W a 7 V W 37 A measure of the stability or degree to which the atmosphere wants to resist developing turbulenc Rfgt1 stability dominates and ow is laminar turbulence wnds to Rflt1 ow unstable and turbulence develops eddies generated by strong wind shear Can be used to define top of the boundary layer by looking for altitude where Rf 1 CHIMU FDLH m uuly AUUv ERA40 PBLH m January 2000 0 400 R00 1900 1800 9000 9400 9900 Equations for the boundary layer Bu au 8 BE iap at a vaywaz ax 8 MW av av av 3V ii atua vaywazf pay azv Simplify for balanced ow making use of geostrophic definition fv Vg BEAM W fi ig zv w States that turbulent transfer is responsible for a geostrophic ow 0 Frictional mvargenca Vorticity equation 2 Why did Charney call it PV 500 mb relative vorticity s 1 for OOZO1JAN2000 in DODHMe08e 6054e052905 0 22705 705 65705 32705 term 500 mb absolute vorticity 165 squot for OOZONANZOOO 315 K potential vorticity Meg m 25 1kg 1 for OOZO1JAN2000 m The Vorticity Equation Want to understand the processes that produce changes in vorticity So derive an expression that includes the time derivative of vorticity d 7 d 6v 7 Bu dt dt 6x 6y Sum of forces in X direction Recall that the momentum equatlons dt 7 Sum of forces in y direction Thus we will begin our derivation by taking d d d y momentum equation7 X momentum equation dt dx dy Equivalently we could use the vectorform and compute the curl of the vector momentum equation d W k Vx dt dt Vort101ty equatlon We will work in Cartesian coordinates The addition termsfar spherical earth came out more naturally in vectarfarm ualv wa ial u v w LaP at 8x 8y 82 pay d d y momentum equation7dix momentum equation y gm 6 a3um w l6pfu 6y 0 6x 6 az ax ax ay 62 ay 3 alwalwal Lalfv 6y 6x By 62 p6x Vorticity equation continued 00v 82v avau 82v avav 82v avaw f ni 2 La al axe 8x2 Bxax 0ng gzax axaz azax Bx 1 ax Ban 8214 014014 8214 auav 8214 auaw fav 0 Lapap 2 ax Bx awe W M at limeP1P ua w Eau avj FfE F v r v 7 2 azaxayazaxay axay aypayaxaxay axaz ayaz Thus the vorticity equation 1 a a a au L 222 dt4 fi 4f8x8y 8x 82 8y 82pz8y 8x 8x 8y Physical intuition 0v Owav Owau l J L LFigax My 7 apap apapw 0y KOxOZ 3202 d Bu E fe fg Terms in vorticity equation Minimal 1 0104091101 ay KBxaz ayaZfFKayax ax y A B c D d Bu Elt f fg A Rate of change of absolute vorticity following the uid motion B Effect of horizontal velocity divergence on vorticity C Transfer of vorticity between horizontal and vertical components twisting term or tilting term D Effects of baroclinicity solenoidal term For pressure coordinates solenoidal term disappears d5 7 55 av E77VV f7wgifVVkEXVQ In practice solenoidal quite term is small in height coordinates too shaded across the pres Hum gradicnl force The Iatlcr sxerl It torque on the malaria clement l d W ide a sulenuid de ned by two intersecting isobar solid lines and isothurc dashed lines Absolute vorticity term A Expanding the rate of change of absolute vorticity is written d f emf1 M0 f 0 f V0 f dt at 0x V 0y 39 02 local tendency horizontal vertical of absolute advection of advection of vorticity absolute absolute vorticity vorticity f ZQ sinq Being independent of x and z we can write d f0 u0 va fw0 dt 01 0x 0y Absolute vorticity change due to 3d advection of relative vorticity and meridional advection of planetary vorticity Stretchingdivergence term B Effect of horizontal velocity divergence on vorticity 7 c fla l K 5x 5y J m 0 Drvergence 5 gt 0 Vorticity will decrease if absolute vorticity is positive Vorticity will increase if absolute vorticity is negative 64 0 Convergence E lt0 Vorticity will increase if absolute vorticity is positive 0 Vorticity will decrease if absolute vorticity is negative Consider vorticity of a region being squished into a smaller area This mechanism is quite important for largescale midlatitude systems Tilting term C Transfer of vorticity between horizontal and vertical components 6w 6v BW 61 6x62 1632 Vertical shear in ofv Wind is gives shear vorticity about an eastWest axis consider paddle rotating around the solid vorticity vector 7 here Q is zero EastWest variations in the vertical velocity tilw the vector to be more vertical dashed vorticity vector This new vector has a vertical component and thus Q is non zero 6v BW 62 6x Solenoidal term term D Vorticity generation due to baroclinic structure density function of pressure and temperature gangrene pzk y 0x 0x 0y or a uniform pressure gradient orizontal variations density means there is a nonuniform acceleration due to the pressure gradient force F h 1 dpy PGW 7 pm dy Variations in acceleration produce vorticity dCdt N dPGFydx This can occur in a baroclinic atmosphere but is absent in a barotropic atmosphere 9 gt9 gtp gtp a 3 2 1 or 1sentroplc ow PagtP3gtP2gtP1 amiabmw 6y 0x alt Thought experiments For dry adiabatic ow 1 p2 al l lip ay 0x 0x 6y E 0x 62 ay 62 l The solenoid term disappears in pressure coordinates Why Pressure variations along pressure suifaces is zero The solidnoidal AND tilting term disappears in isentopic coordinates Here the vertical velocity is now d0 W dt Adiabatic condition means no vertical motion 7 so certainly no sheer in it 639 Recall vorticity advection Advection of relative vorticity kUk2l2 A coskx cosly Advection of planetary vorticity bk A coskx cosly The advection of relative vorticity dominates for short waves Advection of planetary vorticity dominates for small waves thus con rming what was seen in the project small waves move slower when there is only advection Given a westward mean ow expect short waves individual lows etc tend to move eastward while longer waves tend to move westward in proactive long wave linked to topography and land se contrast to tend not to move very much at all This can be all seen with the barotopz39c model Baroclinic instability Baroclinic cyclogenesis Hum mm heal llux K ms Heat flux by waves Eddy heat ux vT39 39 n 2007 lev500 hPa Stationarle os sby waves Baroclinic waves 4 10 planetary wave number Total stationary eddy heat transport 85 K ms Total transient eddy heat transport 95 K ms Normal mode analysis Understand method way in which instabilities occur and develop P Nt w Come up with asome wave equations Linearize very small initial disturbance Assume solution II39 Aexpikx ct ie construct wave equation knowing we want to do this Examine conditions fo defining k real and positive Develop dispersion relationship 0 cku f 39 39 r growth 00 94 4 04 Growth r39aLe 02 1 f f r39 r v on u 1 u 2 o Zonal waverumber Fig 523 Growth rate versus wavenumber for the twolevel model for variol values of I calculated from Eq 559 Growth rates are scaled by KRAU an wavenurn ers y Kg Basic parameters are for a plane centred 5 wil N2 I 2 d a vertical shear of 40m 5quot roughly corresponding to norther hemisphere winter Solid curve 1 o Dashed curve i 81 x IOquot rnquots Is I Dotted curve E 216 x IO l 111 James 1994 Unstable 8 lt 0 Stable 8 gt0 O 05 LO 22k2 gt Fig 83 Neutral stability curve for the two level baroelinic model Baroelinic growth rate alpha January July Baroclinic growth predictions 1 Thermal wind balance geostrophichydrostatic continually maintaine Vertical motion causes stretching and vorticity generation at lower eve s Short waves always stable k2 gt ZkRZ Beta effect stabilizes flow for long waves leads to a minimum UT or AT for instability For given stability increasing U will give a wave number that becomes unstable first the most unstable wave Depends on S stability F rotation rate and UT temperature gradientdifference Baroclinic waves in midlatitude typically on scale of most unstable wave kkR gt L 4000km planetary wave 67 Baroclinic cycle 39 Warming in the tropics cooling at poles UT increases For given stability moist adiabatic UT reaches critical value and most unstable wave grows Potential energy is converted to eddy kinetic energy Geopotential gradient is reduced thus UT is reduced below to critical threshold Net effect is to transport heat poleward This avoiding geostrophic paradox in zonal mean Analogies with rotating annulus experiment What is B What is f What is stability 6 What is Rossby radius LR or 7 1 What is the critical UT for instability AT 9 G 2 What is the most unstable wave numberwave length for a given rotating rate and temperature difference First need to justify the analogy between the incompressible Boussineq case with the stratified atmosphere case from Chapter 8 Hints in chapter 10 Review of basic eqlationships Key concepts for today De ne coordinate systems Thermodynamic equation Lapse rate de ne De nition of potential temperature Scale analysis 7 homework revision and more next week Geostrophic and thermal wind balance Keep your eye on the quantity RT which we will see again and again Notice quantity gcp which we will see again and again NCEP Reanalysis data Surface pressure mb for 00215NOV1975 u hPa 540 580 630 670 720 780 810 850 900 940 990 1030 Christopher Godfrey39s NCEP Reanalysis Page m 0L A NCEP Reanalysis data 500 mb wind field and isotochs ms for OUZISNOV1975 1 no so 35 4c 45 50 Christopher Godfrey39s NCEP Reanalysis Page m 0L A NCEP Reanalysis data V50ms VT 301510deg 500 mb wind field and temp for OOZISNOV1975 722 719 45 713 rm 77 Chnstopher Godfrey39s NCEP Reanalysls Page m 0L A Zonal mean temperature httpirid11de0columbiaedu Zonal mean potential temperature Pressure mb enn n htt irid11de0c01umbiaedu Homework Three questions 1 2 3 Approximating the role of earth s rotation To go over in class Review of partial derivatives potential temperature and coordinates Especially review advection Review of scale analysis a theme in the first few weeks ie read chapter 1 2 and 3 0f the text book Homework assignment Baseline road in Boulder is exactly 40 N Perform a Taylor expansion of the Coriolis parameter f Retain the rst linear term to estimate the value of the Coriolis parameter at 45N based on a reference latitude of 40 N Determine the size of the error in the approximation as a percentage Approximate forms off 2 00E704 1 80EVO4 1 60E704 1 40EVO4 1 NEW 1 00E704 8 00E705 6 00E705 4 00E705 2 00E705 0 00E00 0 f ZQsin d fplane plan latitude Zeroth order approx f plane is OK but not great First order approx Is not bad Works over reasonably large range of latitudes say 10 degrees Second order getting quite close but seems like a lot of work for just the midlatitudes We will see the first term beta is often enough Homework exercise for next class 1 Show that the horizontal pressure gradient force in height coordinates can be rewritten in isentropic coordlnates as 0 6x 2 6x 3 2 Do a scale analysis to show that an approximate form of the thermodynamic equation tells us that temperature changes are mostly due to imbalance between temperature advection and work done by expansion or compression 3 Read Hadley s paper from 1735 and gure out what physical principle he is relying on PDF le of assignment with details on class web site Atmosphere conserves Energy mechanical heat 0 Mass of air and other gases Momentum and angular momentum To describe motion we can make use of 0 Newton39s 2nd law 2F ma 0 First Law of Thermodynamics 0 We need to account for fact that earth is rotating spherical and there is gravity Primitive equations 7i0p dt 0 0x bas1c building blocks 7 ii an Momentum horizontal dt 0 ay 0 p pg Hydrostatic vertical 02 lip J 01 g Continuity p dt Kax 0y OZ dT dp Thermodynamic p E aE J p pRT and equation of state ideal gas Boundary layers add an extra force for momentum equations and contributes to J in thermodynamic equation Otherwise a closed set ofPDEs See Holton Ch2 for full derivation NCEP Reanalysis data 500 mb wind field and heights for OOZI5NOV1975 5100 5200 5300 5400 5500 5500 5700 5300 Christopher Godfrey39s NCEP Reanalysls Page A Some assumptions 0 Atmosphere is thin so distance from center is about the same as the earths radius 0 Gravity is constant 0 Given these also neglect horizontal component of rotation ensures momentum equations conserve momentum 0 These are called the traditional approximation Westerly Wind and temperature Temperature and Wind eld related thermal Wind balance ie amap N aTay Knowing temperature gradient estimate jet Knowing jet estimate gradient Gravity waves and adjustment Trapped waves narrow ridgesquot periodic waves broad ridgesquot ig mu Strcmnlmuxm and mm mm mnnm wuth muwum mlgc Iquot nu nurqu Imgu we I and mm xulgc one In The mm lmr m my mum nu phuw m maxuuum upward dlxpluucmclll mum Ummu I99 Inertia gravity waves disturbance is heating with condensationlatent heating mva vss 5m ms CDNTRDL run i2 ETA model test Momentum Atmosphere conserves Energy mechanical heat 0 Mass of air and other gases Momentum and angular momentum To describe motion we can make use of Newton39s 2ndl law 2F ma 0 First Law of Thermodynamics 0 We need to account for fact that earth is rotating spherical and there is gravity Some assumptions Atmosphere is thin so distance from center is about the same as the earths radius Gravity is constant Given these also neglect horizontal component of rotation ensures momentum equations conserve momentum These are called the traditional approximation Primitive equations basic building blocks Momentum horizontal Hydrostatic vertical Continuity Thermodynamic iil dt pox if iil dti pay 0p 027 pg i a pdt 0x By 02 at d aPJ F alt alt p pRT and equation of state ideal gas Boundary layers add an extra force for momentum equations and contributes to J in thermodynamic equation Otherwise a closed set ofPDEs See Holton Ch2 for full derivation J NCEP Reanalysis data 500 mb wind field and heights for OOZI5NOV1975 5100 5200 5300 5400 5500 5500 5700 5300 Christopher Godfrey39s NCEP Reanalysls Page A Westerly Wind and temperature Temperature and wind field related thermal wind balance ie Ouap N 6T6y Knowing temperature gradient estimate jet Knowing jet estimate gradient Zonal mean Circulation Global energy budget Incoming Omgoing S lar Longwave Radiation Hadiaiionc 342 W m L 7 40 Emuted b Annosphenc Atmosphere 165 Window Greenhouse wase 324 350 Back Radiaiion F 39 I 168 A 78 ul f Absurbed by Surface Thennais Evap0 Radiation 324 transpiration Absorbed by Suriac e Houghton IPCC 2001 Diabatie heating PRESSURE kPa n u n v i i i i u i i u n x I I 10 A Rn quot A s sas aos 0 JUN Amqu A V 60N 90N i 17 Latitude pressure cross sections of the lime and zonal mean healing IQ n six years of F MWF d l Conluur inlcn39ul 02Kdny positive vuluca h 1 James shgfitcd a DJF h JJA Zonal Wind and potential M temperature l DJF quot 2 Ig 39 3 I 6 V 56 mn i i I l l l 39 OS 60 i305 0 JON SON BUN 2 Mia e Hg 42 animus of i7 and 7 to a DJF and lil JJA 1mm nn six ycnls nl ECMWquizL milnur inlcnul I39m ii us in Fig 4 39onlour inlcrvul I39m39 1 IS IU K James 1995 Mass flux stream function 200 9amp5 91 Fig 107 Streamfuncuon units 102 kgmils l for the observed Eulcrian mean meridional circu Imion for Nonhcm Hemisphere winler based on lhc dam of Schubert cl 1 I990 Hadley 1735 Dove 1837 Ferrel 1856 Mass ux stream function sm cq j a 905 SOS i 4i 4 44 90H 107 Streamfunction units 0 kg mquot1gtquot I for the observed Euleriun mcun meridional circu lation for Nonhcm Hemisphere winter bawd on the dam of Schubert c1 31 1990 Mass ux streamfunction and Zonal Wind James 1995 PRESSURE kFa 71477 litin JJA pnassunz kPq as I J J i 39 303 o JON sbN N wmm Fig 44 The zonal mean wind it and vcclors or he meridional wind for l Dccembcrilmiuary February DJ F b JuncJulyAugusl JJA n AA 05 Eddy heat transport V T James 1995 lPu nassua pa FRESSU n a c Ann 505 305 0 LATINDE 9 ON DJF JJA Fig 57 Latitudwpressurc scclions showing the poleward lmnsicnl eddy temper a me ux W for a DJF and bi JJA Contour interval 2 t values shaded littscd on six years of ECMW 39 ata Km 539 39 negalivi Eddy momentum transport u v James 1995 massma Kan JON PRESSURE up 0 lAllTUDE Fig 55 Latituderprcssure cross sections ofthe poleward transient eddy momentum uxes observed in a the DJF and b the JJA Negative values shaded Based on six years of ECMWF data N JJA seasons Contour interval 5 n12 5 1 Zonal mean meridional circulation PRESSURE db 0 ION 20 30 40 50 60 70N 10 mama Zonalmumcro licmispllcric And I clium nl39lllc quotum slruum l uncllon m m quot kg 5 Wm Innual DJF Ind JJA mczm comlilimm ch39ncuI prolilcs 01 llc mum vulucx are shown on hc gm Zonal mean Circulation Was Hadley right all along Hadley 1735 Mass ux stream function SON 90 Fig 07 Streamfunction units 102 kgm I s for the observed Euleriun mean meridional circu lation for Nonhem Hemisphere winter bmed on the data of Schubert cl 3 I990 Zonal Wind and potential X temperature W DJF 5 Qh m v i E quot JJA um t v 4 u t quot V t 39 1 J os 7605 305 0 30M SON 90M LAUTUOE Ifig 42 f onlmus or n and 7 to a DJF and h JJA based m ycnls n LCMWI dam Contour int w in Fig 4 39onlour inlcrvu cmll rm n 4 James 1995 Isentropic mass circulation 400 l l Tropopause frequency l l llv iilIIH 39N llllll39l Hsozzzle l l fquot L O C e 39I 444 M 4 H r r r H8 lit 1 Ill4 IQ lllIlllllA IllllIlIIIIII 1 zzrr KL 39 L22d N 00 C i Mi surface 3 frequency Potential Temperature K M J O 294 billion kgsec 132 billion kgsec I 90 60 30 0 30 60 90 Latitude Noone2008 MUGCM R21 L9 5 years Mass transport of mass by mean and eddies Perpetual July simulation 360 day average ll 1 quoti quotil r HMLAL H illllHH 7 HHH rH H 360 m l 5s l l l l l l i ll r r l rl quot ZZ CC Potential Temperature K or 142 billion kgsec 294 billion kgsec 90 60 3O 0 30 60 30 0 30 60 9O Latitude Latitude 1 Eddies remove air mass from subtropics 2 Equatonvard return flow below mean height of surface 3 Summer circulation very different Noone 2008 MUGCM R21 L9 5 years 10 Residual and isentropic diabatic mass streamfunction a DJF TEM b JJA TEM Tanaka et a1 JAS 2004 Isentropic diabatic mass streamfunction a DJF st 1 m s 50 33 Zonal no mean 300 500 700 505 c DJFu l m 10 ED 0 5 Eddy Tanaka et a1 JAS 2004 908 306 30 JON 60N 90N Fig 109 Residual mean meridional stream unclion unils 02 kg mquot squot for Nurlhcrn Hemiwhcm winlcr based on the data of Schuberl cl ul I900 Fig 108 EliusxenJ nlm ux dn39ei39genrc divided by lhc slumlm39d density m for Nnrlhern Hemisphere wmlcr based on hc dam ul39Schuben el al 990 llnils I39ll squot day 1 Holton Hadley 1735 was mostly right just didn t mention eddies explicitly 400 L G 0 Potential Temperature K W N O 9o 60 g R39gi 9o Latitude 12