The Angular Momentum Budget of the Transformed Eulerian Mean Equations

Transcription

1 OCTOBER 008 E G G E R A N D H O I N K A 3305 Te Angular Momentum Budget of te Transformed Eulerian Mean Equations JOSEPH EGGER Meteorologisces Institut der Universität Müncen, Munic, Germany KLAUS-PETER HOINKA Institut für Pysik der Atmospäre, Oberpfaffenofen, Germany (Manuscript received 6 December 007, in final form 4 April 008) ABSTRACT Te axial angular momentum (AAM) budget of onal atmosperic annuli extending from te surface to a given eigt and over meridional belts is discussed witin te framework of conventional and transformed Eulerian mean (TEM) teory. Conventionally, it is only fluxes of AAM troug te boundaries and/or torques at te surface tat are able to cange te AAM of an annulus. TEM teory introduces new torques in te budget related to te vertically integrated Eliassen Palm flux divergence and also new AAM fluxes of te residual difference circulation. Some of tese torques are displayed for various annuli. In particular, te application of TEM teory generates a large positive torque at troposperic upper boundaries in te global case. Tis torque is muc larger tan te global mountain and friction torques but is cancelled exactly by te new vertical AAM fluxes troug te upper boundary. It is concluded tat te TEM approac complicates te analysis of AAM budgets but does not provide additional insigt. Isentropic pressure torques are believed to be similar to te TEM torques at te upper boundary of an annulus. Te isentropic pressure torques are evaluated from data and found to differ in several respects from te TEM torques. 1. Introduction Corresponding autor address: Josep Egger, Meteorologisces Institut der Universität Müncen, Teresienstr. 37, Munic, Germany. j.egger@lr.uni-muencen.de Te transformed Eulerian mean (TEM) equations attracted enormous interest and sparked intense researc activities immediately after tey were introduced by Andrews and McIntyre (1976). Tey offered new ways to look at te interaction of waves and te onal mean flow (see Andrews et al for a concise outline of TEM teory). One of te main tecnical points of tis approac is te emergence of te so called Eliassen Palm flux divergence (EPD) in te prognostic equation for onal mean momentum. Tis divergence (convergence) represents te source (sink) of wave activity (see Andrews et al. 1987). Hence, it appeared to be a breaktroug tat tis term can be sown to be part of te onal momentum equation. Edmon et al. (1980) expressed it succinctly, stating te particular combinations of eddy fluxes wic are represented on an Eliassen Palm flux cross section are fundamental for te interaction between eddies and mean flow more so tan te eddy eat and momentum fluxes considered separately. Since ten, climatologies of wave-driving ave been presented (e.g., Edmon et al. 1980; Mecoso et al. 1985) and detailed correlation analyses of various terms of te TEM equations ave been performed (Pfeffer 1987, 199). Stratosperic warming events ave been interpreted in terms of te TEM teory (Dunkerton et al. 1981; Palmer 1981). Randel and Stanford (1985) applied tese concepts to observed baroclinic life cycles; Plumb (1986) extended te TEM approac to tree dimensions. TEM teory is also discussed and applied in te oceanograpic community (see, e.g., Eden et al. 007 and references terein) and as found its way into textbooks (Pedlosky 1987; Holton 199; Vallis 006). Tere ave been also critical voices. Pfeffer (1987) found tat te transient EP flux and its divergence provide muc more direct information on te sources, sinks, and propagation caracteristics of synoptic-scale waves in te atmospere tan tey do about te response of te mean onal current to wave action. Moreover, Pfeffer (199) compared observed canges of te onal mean wind wit te vertical component of te Eliassen Palm flux but did not find a correlation. DOI: /008JAS American Meteorological Society

2 3306 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 65 Holton (199) remarks in is text book tat if we are primarily concerned wit te angular momentum balance for a onal ring of air extending from te surface to te top of te atmospere... it proves simpler to use te conventional Eulerian mean formalism. It is te purpose of tis article to complement Pfeffer s (1987, 199) approac and furter explore te comments of Holton (199) by concentrating on an aspect of tis problem tat as received little attention so far. Altoug many autors ave discussed te application of TEM teory to te axial angular momentum (AAM; e.g., Edmon et al. 1980; Pfeffer 1987), te conservation form of te angular momentum equation as not been exploited. In particular, te calculations tat led Holton (199) to make is remarks on te utility of te TEM formalism in angular momentum budgets ave not been publised. Let us consider a onal annulus of widt W and dept D. Te AAM conservation equation in coordinates states tat te AAM of tis annulus can be canged only by AAM fluxes troug its lateral and vertical boundaries (e.g., Egger and Hoinka 005) and by torques at te lower boundary if te annulus intersects te topograpy. TEM teory reformulates te onal momentum equation. It is of obvious interest ow tis transformation affects te structure of te AAM conservation equation. Wic types of fluxes and torques are introduced tis way? In particular, observations must be used to calculate tese fluxes and torques. Are tese terms large wen compared to tose found in standard AAM investigations? Te basic equations are given in section. An application to te atmosperic time mean state is presented in section 3. Te discussion in section 4 includes remarks on AAM budgets in isentropic coordinates.. Budget equations First, a brief derivation of te AAM budget equations will be given, including topograpy at te lower boundary. Next, te additional terms due to TEM teory will be incorporated. Readers wo are esitant to go troug all tese budget equations may first ave a look at te simple example [(4.1) and (4.)] presented in te discussion. A main message of tis paper is contained in tis example. Te angular momentum equation is m vm p t,.1 were m u a cosa cos. is te specific axial angular momentum. Te notation is conventional, wit density, velocity v, pressure p, eart s radius a, and as an angular momentum stress. It is convenient for te comparison wit TEM equations to separate in (.) te specific relative angular momentum term m w ua cos.3 from m m a cos.4 and to introduce for m w a specific prognostic equation: t m w vm w fa cos p,.5 wic follows from (.1) after invoking te equation of continuity. Budgets for not only te AAM but also te wind term m w will be derived in te following because te TEM approac as been introduced to better understand te onal mean flow predicted by (.5). Vertical integration of (.1) over te dept of a layer is straigtforward wen bot and are constant and is above te eart s topograpy. Te result is t m d v m d wm pd,.6 were v (u, ) is te oriontal velocity. It is, owever, also attractive for budget calculations to coose te topograpy as lower boundary. It follows tat t m d v m d wm pd p s,.7 were te lower boundary condition w v.8 at as been taken into account and p s is te surface pressure. Zonal averaging as to face te dependence of on longitude. Tis rules out te application of te more elegant barycentric onal averages (e.g., Juckes et al. 1994). Instead, we introduce te integral b a cos bd.9 0

3 OCTOBER 008 E G G E R A N D H O I N K A 3307 and te onal average b 1 bd 0.10 for a variable b. Averaged vertical integrals are written bc d b cd e bc,.11 d were all deviation terms are lumped togeter in te eddy term (symbol e) and te first term on te rigt is called te mean flow term. Note tat b sb were s acos. After onal integration, (.7) becomes smd t p s. cos m d w m cosm de e swm.1 Integration of (.1) over te widt W a( 1 )of a onal belt completes te derivation of te budget equation for an annulus extending from te surface to te eigt and from latitude 1 to. It is seen from (.1) tat te angular momentum of tis annulus canges indeed only troug mean flow and eddy fluxes at te latitudinal and upper boundaries, te mountain torque at te lower boundary, and te friction torques at bot te upper and lower boundary. It is customary to omit te small upper friction torque. Te surface stress is denoted by f. Te flux terms in (.1) drop out if we integrate over te globe and if. Tis yields te standard budget d dt M T o T f,.13 were M is te global axial angular momentum, T o is te global mountain torque, and T f is te global surface friction torque. Wit lower boundary we ave to replace wit in te integrals in (.1), remove te mountain torque term, and add a term (w m) 1, on te leftand side and a term (swm) e on te rigt. Te barycentric average would be suitable in tis case. Te relative momentum equation, by analogy wit (.1), is t sm w d cos m w d w m w sfa cos e cosm w d d sfa cos swm w e p s f. e d.14 Te Coriolis term due to te Eulerian mean meridional flow is te main new feature on te left-and side. Tis term as an eddy companion. Te global version of (.14) wit also contains Coriolis torques tat vanis for climatic mean conditions. Of course, (.1) and (.14) are closely related. For example, te mean flow Coriolis term in (.14) is simply idden in te second term on te left-and side of (.1). It is a key point of TEM teory (Andrews and McIntyre 1976) to introduce te residual circulation * * and.15 w* w w*,.16 were te residual difference velocity as te components * 1 and w* 1 a cos cos Te primes denote deviations from (.10). As stated by Andrews and McIntyre (1976), tere are many possibilities to introduce a residual circulation. We coose ere te simplest version. We ave to replace (, w) in (.1) and (.14) by ( * *, w w* w *) and to distribute te new transport terms to bot sides of (.1) and (.14). Te result for (.1) is

4 3308 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 65 t sm d cos *m d w *m cos e m d *m d swme w *m p s f. Integration of (.19) over te belt widt W a( 1 ) gives te TEM formulation of te AAM budget for tis annulus: t w *m ad 1 sm da d a cos *m d 1 a cosm e 1 ps 1 fa d. 1 cos*m d 1 Te first four terms on te rigt-and side result from te vertically and meridionally integrated EPD. Te first two terms stem from te integration of te meridional component of te EPD; te last two result from te vertical component. Tus, TEM teory states tat te AAM budget is affected by AAM fluxes troug te boundaries because of te residual circulation by Eliassen Palm fluxes troug te boundaries and by mountain and friction torques. Our terminology wit respect to (.0) calls all terms on te rigt-and side swm e w *m ad torques and tose on te left fluxes (except for te tendency). Following common practice, Coriolis terms are also called torques. By setting * w* 0, we recover te AAM budget of te annulus in standard form. Te TEM formulation is obviously more complicated but is compatible wit standard global angular momentum budgets. Te global case wit leads to te correct budget Eq. (.13). We may switc as before from to as a lower boundary to obtain, instead of (.0), t 1 sm da d a cos *m d 1 a cosm e *m d 1 1 w *m swm e w *m 1 ad..1 Te TEM budget for te relative angular momentum is analogous to (.0): t sm w d ad a w *m w ad sfa cos * dad 1 sf cosa 1 e d swmw e w *mw 1 cos *m w 1 d1 ad a cos ad sf cosa 1 1 m w d e cos*m w d1 ad ps 1 1 fa d..

5 OCTOBER 008 E G G E R A N D H O I N K A 3309 As wit (.0), te first four terms on te rigt-and side represent te vertically integrated form of te EPD. Note in particular tat te Coriolis term on te rigt-and side of (.) as been integrated vertically [see (.17)] to yield torques at te upper and lower boundary. Tere is also a Coriolis torque due to te residual meridional wind at te left-and side of (.). Te global version of (.) is wit : d dt M w sfa cos * da d sfa cos d e * d ad T o T f..3 Tis time, te global Coriolis torque due to te residual circulation on te left-and side does not vanis even in te time mean, nor does te Coriolis torque due to te residual difference circulation on te rigt-and side. 3. Results Data ave been used in te past to study (.14) on a term by term basis at least in approximate forms (Peixoto and Oort 199; Egger and Hoinka 005). Tere is no need to repeat tese calculations altoug te formulation of te budget Eq. (.14) is more accurate tan usual because of te proper incorporation of te lower boundary conditions. Neverteless, tis improvement is not expected to lead to a substantial revision of te results obtained so far. Wat as to be done, owever, is to evaluate and discuss te new TEM terms on te rigtand side of (.0) and (.). We reduce te complexity of tese terms by noting tat m a cos is an excellent approximation. Wit tat and (.17) and (.18), te TEM contribution to te EPD in (.19) is cos *m d w *m fa cos a cos Te vertical integration in (3.1) as been carried out using (.17); ence, TEM teory introduces a new torque, T fa 3 cos d, 3. 1 at te upper boundary of te annulus. Tere is also a new torque at te lower boundary, but let us first concentrate on T. Tis torque does not vanis if we integrate over te globe so tat te lower atmosperic layer of dept ( ) excanges angular momentum wit te atmospere above. For an estimate of its order of magnitude we assume a simple profile A sin, 3.3 were A 5 15mKs 1 (e.g., Peixoto and Oort 199; Juckes 001) close to te surface and near te tropopause wile A 5 for a midtroposperic value of. Wit / (km 1 ), te new torque is T 50A Hadley (1 Hadley J) for te global case. Tis torque is positive and implies a perpetual gain of angular momentum in te annulus. Te global friction and mountain torques amount to a few Hadley (e.g., Peixoto and Oort 199), so tat te new torque dwarfs tese torques. Te stratospere appears to lose a substantial amount of angular momentum according to TEM teory. Of course, te stratospere does not contain a source of AAM and te mean fluxes of AAM troug te tropopause ave to vanis (Egger and Hoinka 007). Te second term on te rigt-and side of (3.1) gives te torque T a cos 3 1, 3.4 wic contributes little to te global budget in te longterm mean but may be quite important in midlatitude belts were T 0(0) in te Nortern (Soutern) Hemispere. Moreover, te seasonal variation of T is important (see Fig. ). Note tat T 0 for. Tus, TEM teory yields te smallest additional torques for deep atmosperes. Te mean torque T for Nortern Hemispere win-

6 3310 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 65 FIG. 1. TEM torque T (3.) in Hadley for Nortern Hemispere (a) summer (JJA) and (b) winter (DJF) for belts wit D and as a function of te eigt of te upper annulus boundary. All figures are based on 40-yr European Centre for Medium-Range Weater Forecasts Re-Analysis (ERA-40) data for te years ter [December February (DJF)] and summer [June August (JJA)] is displayed in Fig. 1 for belts wit D 4.5 and for various annulus depts. Te evaluations in Fig. 1 ave been made at levels being 1000 m apart. Results ave been interpolated. Torques are mostly positive and can be as large as 50 Hadley close to te ground. Of course, te torque reflects mainly te eddy eat transport. Te summer torques are almost completely restricted to te Soutern Hemispere, wereas te distribution is more symmetric in winter. Te global mean of T is displayed in Fig. wit a pronounced maximum near te ground. Global torques are 500 Hadley close to te ground and 100 Hadley in te midtropospere, in reasonable agreement wit (3.3). Te seasonal variation of te global torques is small. Of course, similar results ave been found also by oters (Juckes 001; Tanaka et al. 004). If is cosen as a lower boundary, (3.1) is to be replaced by cos *md w *m T T 1, 3.5 were T 1 is defined by analogy to T. Obviously, TEM introduces te difference of large torques. 4. Discussion Altoug it is a great attraction of TEM teory tat EPD represents te eddy forcing in terms of te potential vorticity flux, wic is dynamically more fundamental tan eiter te momentum or eat-fluxes separately (Pedlosky 1987), we learn ere tat te wave forcing described by EPD is not suitable for studying te AAM budget. Tere is no dynamical mecanism in te atmospere tat induces te torques T and T or T 1. Moreover, tese torques do not ave any effect on

7 OCTOBER 008 E G G E R A N D H O I N K A 3311 Obviously, te terms on te rigt-and side correspond wit T T 1, and we know for sure tat tey do not correlate wit te tendency on te left-and side. Andrews et al. (1987) point out tat tese terms would represent a form drag if te analysis were carried out on material surfaces [see also (4.9) and te related discussion]. However, (4.) is written in coordinates were suc a form drag does not exist. Note also tat te results of a numerical integration of te twodimensional model wit (4.1) as a onal mean flow equation would not be affected at all by a switc to te TEM formulation. Tus, (4.) does not provide new insigts but is just more complicated tan (4.1) in integrated form. It as been pointed out by Juckes et al. (1994) tat isentropic analysis offers a principal advantage over te TEM equations, namely, te clean treatment of te lower boundary and a more direct portray of te diabatic eating. Moreover, te formulation of te angular momentum equations in isentropic coordinates offers a particularly clear picture of te interaction of eddies and te mean flow. In particular, Juckes et al. (1994) argued tat te pressure torque [see (4.8)], wic is an important part of te isentropic angular momentum balance, is closely related to T. Tey argued tat 1 gp M f g cosp, 4.3 FIG.. Globally integrated value of T in JJA and DJF in Hadley as a function of eigt. te AAM of te annulus. It is ard to see wat we learn from introducing suc torques. Tey are balanced, of course, by te corresponding fluxes associated wit te residual difference circulation on te left-and sides of (.0) (.). A similar but less general result follows from considering a two-dimensional f-plane model for sallow Boussinesq flow wit flat lower boundary. Te onal mean flow equation is in tat case t u f o uw 0, 4.1 wit f o constant. Te switc to TEM and vertical integration yields t ud f o * d uw f o. 4. were te averages on te rigt ave to be carried out on isobaric surfaces (see also Tanaka et al. 004 for a somewat different deviation of (4.3) were g is te geostropic wind). Tis approximation is, of course, of ig interest because we ave argued tat T cancels exactly and as no effect on te AAM budget in coordinates, wereas (4.3) suggests tat T is an important part of isentropic angular momentum budgets. To resolve tis issue, we briefly discuss te related isentropic budget equations. Te isentropic AAM equation is t m v m m M, 4.4 were g 1 p/ represents te density, relates to te diabatic eating, and M is now te Montgomery potential. Budget equations for annuli are derived by first integrating (4.4) vertically from te surface wit s (subscript s for surface values) to an isentropic surface wit wic does not intersect te ground. Te result is

8 331 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 65 FIG. 3. Isentropic pressure torque T p (4.9) in Hadley for (a) JJA and (b) DJF for belts wit D as a function of potential temperature. t md v m d m s s M d f, s were te surface potential temperature equation t s v s s as been invoked. Tere are momentum transports at te top of te layer coupled to diabatic eating. Te first term on te rigt-and side of (4.5) essentially represents te pressure torque. Its evaluation is easy to perform if we return to coordinates, so tat M s d p d pd p, 4.7 were is te geometric eigt of te upper isentrope. Zonal integration yields smd t s s a cos md m p f, 4.8 were te integration as to be carried out along te upper and lower surface; tat is, we obtain tis way a correct pressure torque term in isentropic coordinates. Te mountain torque acts at te lower boundary. A direct effect of te eating is found at te upper isentrope. We arrive at (.13) for te global case and. Of course, (4.8) is not new. Jonson (1989, ereafter J89) presented, for example, a detailed observational analysis of te onally averaged angular momentum budget at isentropes. However, a vertical integration was not carried out by Jonson (nor by Juckes et al.

9 OCTOBER 008 E G G E R A N D H O I N K A 3313 FIG. 4. Globally integrated value of T p in Hadley in JJA and DJF as a function of potential temperature. 1994), so tat te role of te various terms in J89 s budget differs necessarily from tat in (4.8). Te pressure torque T p p d a is displayed in Fig. 3 for comparison wit Fig. 1. Te approximation (4.3), if reliable, would lead one to expect a close similarity of bot figures. It is seen tat tese internal torques peak at midtroposperic eigts at midlatitudes to decrease iger up. Maxima are 30 Hadley. Weak negative torques are found in te tropics. Tis means tat atmosperic eddy motion removes angular momentum from te upper tropospere at midlatitudes and brings it down to te lower tropospere. As in Fig. 1, tere is a pronounced asymmetry of winter and summer cases. Te patterns in Fig. 3 agree quite well wit te result of J89, altoug we ave to keep in mind tat Jonson displays essentially te derivative of T p wit respect to, so tat a ero-line in J89 is found in te midtropospere. Moreover, te calculations of J89 are based on different data. Te global mean of T p is presented in Fig. 4 wit a maximum of 300 Hadley in winter for 90 K. A comparison of Figs. 1 and wit Figs. 3 and 4 sows tat (4.3) provides some guidance but is not fully satisfactory. In particular, T as its maximum at te ground wereas T p peaks in te tropospere. We ave to keep in mind tat T p represents a torque tat acts on te angular momentum wereas T is uncorrelated wit te angular momentum tendency. Altoug we calculated te torque T only for climatic mean conditions, it is clear tat te conclusions would be te same if we applied te TEM formalism to, say, daily AAM budgets. Te torque T would, of course, also vary from day to day but would ave no effect on te onal mean. Finally, it sould be pointed out tat te relevance of te Eliassen Palm flux as a diagnostic tool is not restricted to its role in te onal angular momentum budget. Te relation of te flux to quasigeostropic potential vorticity transports and wave activities is well establised, and corresponding results are not at all affected by te negative outcome of our analysis, nor are any nonacceleration teorems. Neverteless, TEM teory does not offer any advantage wen it comes to analying AAM budgets for te atmospere as observed. Acknowledgments. We are grateful to te referees for constructive criticism. REFERENCES Andrews, D., and M. McIntyre, 1976: Planetary waves in oriontal and vertical sear: Te generalied Eliassen Palm relation and te mean onal acceleration. J. Atmos. Sci., 33, , J. Holton, and C. Leovy, 1987: Middle Atmospere Dynamics. Academic Press, 489 pp. Dunkerton, T., C.-P. Hsu, and M. McIntyre, 1981: Some Eulerian and Lagranigan diagnostics for a model stratosperic warming. J. Atmos. Sci., 38, Eden, C., R. Greatbatc, and D. Olbers, 007: Interpreting eddy fluxes. J. Pys. Oceanogr., 37, Edmon, H., B. Hoskins, and M. McIntyre, 1980: Eliassen Palm cross sections for te tropospere. J. Atmos. Sci., 37, Egger, J., and K.-P. Hoinka, 005: Torques and te related meridional and vertical fluxes of axial angular momentum. Mon. Wea. Rev., 133, , and, 007: Stratospere tropospere excange: A

10 3314 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 65 onal mean perspective of angular momentum. Dyn. Atmos. Oceans, 43, Holton, J., 199: An Introduction to Dynamic Meteorology. Academic Press, 511 pp. Jonson, D., 1989: Te forcing and maintenance of global monsoonal circulations: An isentropic analysis. Advances in Geopysics, Vol. 31, Academic Press, Juckes, M., 001: A generaliation of te transformed Eulerianmean meridional circulation. Quart. J. Roy. Meteor. Soc., 17, , I. James, and M. Blackburn, 1994: Te influence of Antarctica on te momentum budget of te soutern extratropics. Quart. J. Roy. Meteor. Soc., 10, Mecoso, C., D. Hartmann, and J. Farrara, 1985: Climatology and interannual variability of wave, mean-flow interactions in te Soutern Hemispere. J. Atmos. Sci., 4, Palmer, T., 1981: A diagnostic study of a wavenumber- stratosperic sudden warming in a transformed Eulerian-mean formalism. J. Atmos. Sci., 38, Pedlosky, J., 1987: Geopysical Fluid Dynamics. Springer, 710 pp. Peixoto, J., and A. Oort, 199: Pysics of Climate. Springer, 50 pp. Pfeffer, R., 1987: Comparison of conventional and transformed Eulerian diagnostic in te tropospere. Quart. J. Roy. Meteor. Soc., 113, , 199: A study of eddy-induced fluctuations of te onalmean wind using conventional and transformed Eulerian diagnostics. J. Atmos. Sci., 49, Plumb, R. A., 1986: Tree-dimensional propagation of transient quasi-geostropic eddies and its relationsip wit te eddy forcing of te time mean flow. J. Atmos. Sci., 43, Randel, W., and J. Stanford, 1985: Te observed life cycle of a baroclinic instability. J. Atmos. Sci., 4, Tanaka, D., T. Iwasaki, S. Uno, M. Ujiie, and K. Miyaaki, 004: Eliassen Palm flux diagnosis based on isentropic representation. J. Atmos. Sci., 61, Vallis, G., 006: Atmosperic and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press, 745 pp.