1074 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 43, No. 1 IThe Integrated Enstrophy Budget of the Winter Stratosphere Diagnosed from LMS Data*. MARK R. SCHOEBEIUAtmospheric Chemistry and Dynamics Branch, NASA/Goddad Space Flight Center, Greenbelt, MD 20771 ANNE K. SMITHCooperative Institute for Research in Environmwld Sciences, University of Colorado. Boulder, CO 80309 (Manuscript mived 6 March 1985, in final form 25 November 1985)ABSTaACFThe Northern Hemisphere, qpasi-geostrophic, integrated enstrophy budget for the 1978/79 winter has beenanalyzed from 10-0.1 mb using LIMS data. The stratospheric integrated enstrophy builds up duriag early winteras a result of the diabatic forcing of the polar vortex. Starting in January, an irregular and generally irreversibletransfer of enstrophy from the zonal mean reservoir to planetary waves begins to reduce the total. This transfer.of enstrophy to the waves produces significant imbalances in the integrated enstrophy budget at 10 mb. Theimbalances appear to result from the transf" of enstrophy to smaller scales not resolved by the LRvB instrumentWe believe that these imbalances are a signature of -by wave breaking, as the imbalance episodes correspondwell to the appearance of Ertel vorticity fdaments recently shown by McIntyte and Palmer.In the mesosphere the total enstrophy shows We seasonal trend. Our analysis indicates that the mesospheremay be a region of continuous wave breaking during winter.1. IntroductionExtensive satellite observations of two dramaticstratospheric sudden warmings took place during thewinter of 1978/79. A synoptic description of theseevents was reported by Labitzke (198 1) and Quiroz(1 979). About 2 1 January 1979 a minor warming occurred, sssociated with a strong upward pulse of zonalharmonic (wave) 1. This "minor" event was Bctuallymore disruptive to the upper stratosphere than the major warming which OccUfcBd on 23 February, associatedwith an amplification of wave 2 below 10 mb. The neteffect of the January warming was to move the polarnight jet poleward. At midlatitudes, the zonal meanpotential vorticity gradient was reduced, a process attributed by McIntyre and Palmer (1984) to irreversiblemixing of the potential vorticity (pv) field. The poleward movement of the jet is, in this case, a direct effectof the southward pv transport by waves.In the periods before and after the January event,the amplitudes of waves 1 and 2 vary rapidly, withwave 2 decreasing when wave 1 increases and viceversa. Palmer and Hsu (1983) and Smith (I 983) havepartially explained the interaction of wave 1 and 2 overthis period as a &It of the coordinate representation.The vacillation of waves 1 and 2 results from choosinga coordinate system not aligned with the displaced* Contribution No. 25 of the Stratospheric General Circulationwith Chemistry Project, NASA/GSFC.Q 1986 American Meteorological Society(wave zero plus wave 1) polar vortex. This type ofwave-wave interaction appears mostly at high latitudesthroughout the winter (Smith et al., 1984). It is notgenerally associated with irreversible cascade of enstrophy to smaller scales.At lower latitudes, McIntyre and Palmer ( 1984) havesuggested that actual down-scale transfer of enstrophyby nonlinear interaction occurs in zones surrounding .critical lines. There, smaller scale disturbances grow atthe expense of the larger scale waves propagating upwards and equatomrd. McIntyre and Palmer termthis process "wave breaking"; it generally connotes irreversible mixing of both pv and constituents. McIntyreand Palmer call the region of wave breaking the surfzone. Such zones may be uarrow when the wave amplitudes are small, but their breadth increases with increasing wave amplitude. The surf zone may be expected always to be present at low latitudes surroundingthe critical line for stationary planetary waves, sincethe stationary waves are always present at some nonzero amplitude.In addition to the continual transfer of wave eqstrophy to smaller scales at low latitudes, the NorthernHemisphere winter is punctuated by larger wavebreaking events associated with stratospheric warmings.During the warmings surges in the wave amplitudebring the critical line poleward and greatly expand thesurf zone into high latitudes. During this process largeamounts of potential vorticity are transferred from thezonal mean flow to the eddies. It is the sudden warmings which produce the largest perturbation in the poI JUNE 1986 MARK R. SCHOEBERLtential vorticity distribution of the midlatitude stratosphere.McIntyre and Palmer (1983, 1984) argue that thedown-scale transfer of enstrophy and irreversible mixing can be viewed synoptically as the wrapping up ofErtel vorticity isopleths near the critical lines. Indeed,such processes appear to be evident from satellite data.Ozone mixing ratio contours also show similar structures (Leovy, et al., 1985a) although the structures arenot expected to be identical since there is no reason tosuppose that chemical tracers are perfectly correlatedwith pv.A quantification of the vortex erosion and potentialvorticity mixing was suggested by McIntyre and Palmer(1983), hereafter MP and implemented by Butchartand Remsberg (1986), hereafter BR using LIMS(Limb Infrared Monitor of the Stratosphere) data. Theanalysis stems from the idea that the polar vortex isseparated by a region of tight potential vorticity contours from a low latitude mixed region, where the gradients are small. Nonlinear mixing is continually takingplace at the vortex edge. During the large wave breakingevents there is a rapid decrease of the vortex area aslarge amounts of high potential vorticity air are drawnout of the vortex and subsequently mixed. As a measureof this process, BR computed the area enclosed by individual contours of Ertel's potential vorticity. Theyfound that in the middle stratosphere the area withinmany of the contours shrank during the course of thewinter as predicted by the wave breaking hypothesis.One interpretation of the analysis of BR is that wavebreaking is continuously eroding the area of the polarvortex as thin tongues of pv are stripped from the edgeand brought into the surf zone. During the two dramatic warming events, when MP clearly show largefilaments of pv separating from the vortex and beingadvected toward low latitudes, BRs analysis shows veryrapid shrinking of the area within some of the pv contours. However, BR found it difficult, in practice, tofind the boundary between the vortex and the mixedregion in their analysis, thus adding uncertainty to theoverall interpretation.The analysis we present in this paper is similar inbasic intent to that of BR; we also are concerned witha measure of the overall changes in the potential vorticity distribution on a surface. However, the focus ofour work is complementary to theirs. Rather than concern ourselves with a differentiation between the vortexand surf zone, as in BR we focus on.the interaction ofRossby waves with the mean flow.Our approach to understanding the evolution of thepotential vorticity during warming events is to examinethe horizontally integrated quasi-geostrophic enstrophy(IE). This quantity has an unusually simple mathematical description since the integration removes theadvective terms. For conservative flow which conformsto quasi-geostrophic scaling, the potential vorticity andenstrophy are conserved along pressure surfaces andAND ANNE K. SMITH 1075are therefore constrained by a number of theorems(Schoeberl, 1982).In the absence of external forcing (e.g., radiative dissipation), the total integrated enstrophy (eddy pluszonal mean) will remain constant. However, when dynamic events are highly nonlinear over a period thatis short compared with the dissipation time scale, potential vorticity is simultaneously transferred. to smallerand larger scales (Fjortoft, 1953). Provided all wavenumbers can be resolved and no waves are dissipated,the total integrated enstrophy remains constant. If thehigh wave numbers cannot be resolved, and there issignificant transfer to those wavenumbers then therewill be an apparent decrease in the total enstrophy.This decrease should therefore be a characteristic signature of enstrophy cascade in a coarsely resolved system such as that constructed from satellite observations.When the decrease is permanent, it is a signature ofirreversible enstrophy cascade. This decrease is exactlythe kind of process which should take place duringstratospheric "wave breaking."The analysis of large-scale dynamics in this study isbased on the conservation of Matsuno's (1 970) quasigeostrophic potential vorticity, and concerns its distribution between the zonal mean and waves. There areboth advantages and disadvantages to this approach.The applications of quasi-geostrophic potential vorticity are more limited than for Ertel's potential vorticitybecause the former is only meaningful when the flowconforms to quasi-geostrophic scaling. Nevertheless, itis straightforward to determine geostrophic horizontalwinds, used in computation of the quasi-geostrophicpotential vorticity, from satellite retrievals of temperature and geopotential. In addition, the quasi-geostrophic system of equations can be formulated entirelyin terms of the geopotential and the diabatic (or othernonconservative) forcing, and therefore it is entirelyconsistent with the available data. Although it may bepossible to derive ageostrophic horizontal winds fromsatellite observations, it is usually quite complex (Elson,1986).Another advantage of using a quasi-geostrophicanalysis is that, when there are no sources or sinks, thepotential vorticity is approximately conserved on a twodimensional pressure surface, i.e., there is no verticaladvection term. Thus maps and analyses of quasi-geostrophic potential vorticity on pressure surfaces havethe same advantages as maps and analyses of Ertel'spotential vorticity on potential temperature surfaces(e.g., see Hoskins et al., 1985). Conservative motioncan rearrange the pv distribution on a pressure surface,but cannot change'the values.As was demonstrated so clearly by McIntyre andPalmer (1 983), it can be quite illuminating to look attwo-dimensional maps rather than at the results of amathematical analysis such as Fourier decomposition.This is particularly true when wave amplitudes are solarge that it is not meaningful to linearize around the1076 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 43, No. 1 Izonal mean flow. However, as long as the assumptionis not made that waves are linear, a wavenumber analysis includes all the available information and can bea useful tool even for disturbed conditions.Furthermore, the identification of the zonal meanflow is important since it is the pv field associated withthat flow which is diabatically forced by stratosphericsolar absorption while the nonzonal disturbances areonly radiatively damped. Finally, we are concerned, inpart, with the nature of dynamical events which develop over the course of a winter. During that time thestratosphere progresses from an almost symmetric statein early winter to a highly disturbed state during thelater part, `and then back to near symmetry by spring.A wavenumber analysis provides a simple means ofquantifying these changes, and aids in the interpretation of events during the winter.The purpose of this paper is to present an analysisof the integrated enstrophy budget derived from theLIMS data. In section 2 we review the constraints onIE and present simple models of IE evolution to helpanalyze our results. Since only a small fraction of thezonal mean enstrophy can be utilized by the waves wealso introduce the concept of available IE. In section3, the observations are discussed. In section 4 we prese'nt results for the 1978/79 winter season. In additionto focussing on periods of wave activity, we briefly discuss the seasonal trends in IE. Our conclusions aresummarized in section 5.2. TheoryWe use Matsuno's (1 970) quasi-geostrophic potentialvorticity (4) to compute the enstrophy (q2) and IE. Adiscussion of this quantity and its relation to the Ertelvorticity is given in the Appendix. The continuityequation for q is" dq dtwherez = In(po/p), J is the net heating by radiative transfer,and M is the momentum input from external sources(i.e., breaking gravity waves) which we assume atthis point to be zero. The total derivative in (1) isgiven byda a adt at ax "- - +u--+vwhere u, and vg are the geostrophic winds. The quantityq is defined asq=f+v,,-- f (`"s" - ug), + f ; (& fl' P) 'cos4where y = uC$, C$ = latitude, x = ah cos4 where X = longitude, f = 20 sinA fl is the potential temperature,8, = d(@/dz where the brackets, ( ), indicate a normalized meridional integration, 8' = fl - (d), and p isthe density. The enstrophy equation isId- - q2 = qs.2 dtIntegrating (2) over a pressure surface we obtain theintegrated enstrophy equation- - ((2) + (2)) = (49 + (41s').la2 dtHere the overbar represents the zonal mean and theprime represents the eddy components in all zonalwavenumbers. Note that it was not necessary to assumethe waves are linear in order to separate the wave andmean terms in (3).A region of positive horizontal potential vorticitygradient must be present in order to support Rossbywave propagation. As a wave begins to grow in thepresence of such a gradient, a down-gradient eddy fluxof potential vorticity is required by conservation. Inother words, the potential vorticity for the wave comesfrom the basic state whose gradient is weakened in theprocess of forming the wave. The transfer of potentialvorticity from the basic state to the wave can continue(and the wave can grow in amplitude) until the basicstate vorticity gradient is destroyed.Although the Rossby wave growth is limited by thepresence of a horizontal, but not necessarily zonal meanmeridional, potential vorticity gradient, the zonal meangradient is still relevant. Consider the following hypothetical case where pv is conserved as illustrated inFig. 1. Initially a symmetric state with a region of highpv over the pole and a strong zonal mean pv gradientexists. Next, a large wave 1 manifests itself. The pvmaximum shifts substantially off the pole producing avery weak zonal mean pv gradient; however, the gradient around the distorted maximum is still strong.Merely shifting the location of the pv maximum inlongitude cannot change hemispheric average enstrophy due to wave 1 ((z)). Note further that there isan absolute maximum to the wave amplitude; it isconstrained by the difference between the highest andlowest pv values initially found on the pressure surface.Although there is no way for the integrated wave Ienstrophy to grow beyond this predetermined limit,higher wavenumbers could also exist and propagatearound the distorted vortex. To do this they wouldstretch or break apart the single pool of high pv, andin the process decrease the IE due to wave 1. The totaleddy IE still does not increase (but may decrease slightlyif part of the stretched pool crosses the pole). This hypothetical case represents a qualitative two-dimensionalpicture of the Rossby wave amplitude saturation limitworked out by Lindzen and Schoeberl ( 1982) andSchoeberl (1982) for a &plane and, in fact, roughly1 JUNE 1986 MARK R. SCHOEBERLSYMMETRIC VORTEX?-"7IPOTENTIAL VORTICITY ZONAL MEAN/*I ASYMMETRIC VORTEXAL POLEII / \?\ IFIG. 1. The change in the zonal mean vorticity due to the displacement of the vortex off the pole. Panel (a) shows potential vorticitydistribution due to a symmetric vortex; Panel (b) indicates the presenceof a strong wave 1 plus higher wavenumbers at low latitudes. Thechange in the zonal mean potential vorticity is shown on the left.describes the situation in the lower stratosphere in latewinter.To the extent that the stratosphere starts from a zonally symmetric state and the quasi-geostrophic potentialvorticity is conserved, the original zonal mean pv gradient gives an indication of how large waves can eventually grow. However, an even more quantitative measure of the ability of the hemispheric mean pv gradientto produce wave can be defined. The IE in the zonalmean flow that can be utilized by a Rossby wave is thedifference between the actual zonal mean IE and thezonal mean IE computed by conservatively rearrangingthe pv so that the zonal mean gradient is flat. A quantitative measure of the potential for pv rearrangementby waves in terms of IE is D, defined as' D = (b)- (Q2 which is zero when there is no mean gradientin pv. If Q is 4 - (ij) then D = (Q2). By analogy tothe available potential energy defined by Lorenz (1959,D can be thought of as the zonal mean available potential enstrophy.Forming an equation for D from the usual linearizedform of (I), the zonal mean IE equation isAND ANNE K. SMITH 1077The eddy enstrophy equation is- (4") + 2(QQy) = 2(s'g').d (4b)The term (u'q'Q,) in (4a) and (4b) is the potential enstrophy conversion term.Equations (4a, b) can be interpreted as componentsof an integrated enstrophy cycle. Figure 2 shows thiscycle during wave growth and decay includingfhe effects dissipation. Even though in this figure (4's') appears as a sink and (@) as a source, the usual situation,it is possible for the sign of either term to be reversed.In general, waves are able to utilize only a smallfraction of the zonal mean enstrophy. For this reasonthe available potential enstrophy, with its emphasis onthe pv gradient, may be a better predictor of the potential for wave activity than the zonal mean enstrophyitself. For example, for a @ channel at 60"N, an earthradius in width with ij = J; it is easy to show that the .available enstrophy is only 8.4% of the zonal meanenstrophy. This further illustrates the point that themean pv gradient is far more critical to wave formationthan the absolute magnitude of the pv.When wave amplitudes reach their saturation limit,the flow will be so distorted that very little of the pvstructure will be in the zonal mean, and nonlinearitywill be a dominant influence on the dynamics. Overlarge areas the vorticity gradient will be weak or reversed. This is just the situation during the large wavebreaking events discussed by MP. In general, we expectthat when wave amplitudes are appreciable and wavepv is close to its saturation limit that wave breakingwill occur over a large portion of the hemisphere.23. DataatThe observations used in this study are the same asdiscussed by Smith et al. (1984). They were derivedfrom the LIMS instrument which was flown on theNimbus-7 satellite. The instrument operated from 25October 1978 through 28 May 1979, and measuredinfrared radiance which was inverted to give profilesof temperature. Geopotential height was determinedby integrating hydrostatically from a standard level inthe lower stratosphere. The data are mapped using aKalman filter, applied independently at each latitudeand pressure level, which simultaneously interpolatesin time and in the zonal direction, and spectrally decomposes the data.An extensive effort has been made to compare theLIMS data with data from other sources in order todetermine the resolution and to evaluate errors. Gille' Because nonlinearity is always present, in some sense waves arealways breaking. However, we have used the term wavebreaking, asdefined by MP, to indicate a rapid and irreversible deformation ofmaterial contours. This process is not always occurring.JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 43, No. 1 I1078ENSTROPHY CYCLECGROWING WAVE DECAYING WAVED = <q > - <q>2-2 'SE = <ST>EP = <qI2> sz = <Si>FIG. 2. The integrated potential enstrophy cycle; the zonal mean available IE is (3) - (3);the eddy IE is (p); agthe conversion term, C, is (u'q'aQ/a?). The two source/sink termsare SZ, (3) and SE, (4's'). The cycle for a growing wave is shown in the lower left; enstrophyis transferred from the wave to the mean flow. The mean flow forcing term SZ tries to maintainthe mean flow while the wave forcing, SE acts as dissipation. In the lower right, the conversionterm is reversed for a decaying wave.and Russell (1984) give an overview of the LIMS experiment and the validation studies. Details of temperature comparisons are given in Gille et al. (1984);Leovy et al. (1985b) discuss the mapped temperatureand height fields. The validation studies indicate thatthe LIMS temperatures, \and presumably also heightand derived horizontal winds, have a resolution of3-4 km in the vertical. This good resolution, combinedwith the nearly global coverage typical of satellite measurements, makes the LIMS data very useful for determining such features as the structure of planetarywaves. The data have'been smoothed in the meridionaldirection with a least squares cubic spline and in timewjth a '/4-'/2-!/4 running mean filter. The diabatic calculation is made using an updated version, of the radiative transfer model of Ramanathan ( 1976) and Ramanathan and Dickinson (1979). The meridional integration, ), used to determine IE is defined as1-84In the results presented in section 4, we have notintegrated over the entire hemisphere due to the difficulty of computing pv near the equator. Our domainfor integration extends from 20" to 84'N. The highlatitude limit is a result of data cutoff due to the LIMSorbital geometry; the neglected area is so small that ithas no impact on the results. The enstrophy changedue to wave boundary fluxes, given by1 JUNE 1986 MARK R. SCHOEBERLwas computed at both the equatorward and polewardedges of the domain. The boundary flux term was sosmall as to be completely negligible at all times andlevels, and has not been included in the results.4. Resultsa. The integrated enstrophy budgetThe basic findings of our study are presented on timeseries plots of the integrated enstrophy (IE) for theNorthern Hemisphere during winter. The .zonal ormean IE refers to that due to the zonal average pv((3)>,and the eddy or wave IE to that due to the eddypv ((q'*)). Available IE (D) is defined in section 2. InFig. 3 the variation of the total IE, eddy and zonalmean IE as well as the available IE are shown for fourpressure levels, 10,5, 1 and 0.1 mb, which correspondroughly to altitudes 31, 36, 48 and 65 km.There are similarities in the trends at the 10, 5 and1 mb levels. During November when the polar vortexis being established by radiative forcing, the zonal meanIE increases steadily, while the eddy IE (zonal wavenumbers 1-6 summed) remains roughly constant. Inearly December a small dip in the zonal mean IE withan associated increase in the eddy IE occurs, coincidentwith the appearance of a Canadian warming. This is avery barotropic warming so changes in IE appear simultaneously throughout the stratosphere.In late December and early January, regular pulsations in the enstrophy at 1 mb appear with a 13-1 5day period. These pulsations are only weakly evidentat the other levels. In mid-January a large jump in theeddy IE occurs at 5 and 10 mb; at 1 mb this jumpappears as part of the steadily increasing amplitude ofthe pulsations which begin with the Canadian warming.The pulsations in total IE at 1 mb may be due to sharpfeatures in the pv which are marginally resolvable inthe satellite analyses and thus come in and out of focus.The signal-to-noise ratio of the data is less at, and above,1 mb, which also contributes to the irregularity.With the increase in the eddy IE at the 10-1 mblevels in mid-January there is a corresponding decreasein the available IE and the zonal mean IE. The totalIE shows a small decrease at 10 mb and an increase at5 and 1 mb as the wave builds. After the eddy IE reachesits peak amplitude and begins to decrease, the totalenstrophy decreases at all levels but the zonal meanremains relatively constant except at the highest level.At 0.1 mb both the mean and eddy IE vary irregularly with time and appear to be for the most part uncorrelated. Dunkerton and Delisi (1985) found evidence that the mesosphere, especially in high latitudes,is a region of intense lateral mixing by planetary waves.This is consistent with the results in Fig. 3d which showsthat the available IE is small and the region is permanently saturated with Rossby waves. In the mesosphere the large mean forcing by radiative transfer andAND ANNE K. SMITH 1079gravity wave breaking, in addition to the mixing associated with breaking Rossby waves, will contributeto the rapid decay of planetary wave disturbances.The changes in the total IE at 1 mb and below duringthe January minor warming are noteworthy. At 5 and10 mb the changes in the total enstrophy occur afterthe wave amplitude reaches a peak. Following the January warming the decrease in the total IE generallycontinues at 10 and 5 mb. There is a second peak inthe eddy IE in early February associated with a minorwarming which is more visible with increasing altitude.In late February a small increase in the eddy IE occursagain, followed by another rapid drop in total IE thisis the February major warming. It is not very prominentin the midstratosphere but, nevertheless, is discernableat all levels. Wave activity in March is weak.The overall picture of winter IE budget in the stratosphere is that of a decrease in the zonal mean IE andavailable IE from the beginning of December at 1 and5 mb and from mid-January at 10 mb. The decreaseapparently results from the overall transfer back to themean when the waves decay which is not compensatedby a transfer from the waves back to the mean. Theapparent loss of enstrophy because of the incompletetransfer from the wave to the mean is greatest followingthe sudden warnings. This result is consistent with thefindings of BR which show a slow persistent erosionof the polar vortex over late winter with acceleratedlosses of high pv area during the sudden wannings.In the mesosphere (Fig. 3d) there are strong fluctuations in the eddy and zonal mean IE throughout thewinter. The late January, early February, and late February increases which are prominent in the eddy IE at10 mb are also apparent at 0.1 mb, but they do notappear to be well correlated with changes in thetotal IE.Figure 4 shows the total IE tendency, the diabaticforcing and (minus) the residual in the enstrophybudget [Eq. (3)] at 10 mb. This figure is derived fromthe results shown in Fig. 3a and smoothed with a 5day running mean filter to reduce the high frequencynoise. The 5-day running mean filter, used only on thedata shown in Fig. 4, produced no significant changein the overall IE budget during the warming periods.Note that the budget makes no distinction between theintegrated enstrophy inJhe waves and that in the mean.In the early winter (q2)It and (@) are roughly balanced, which indicates that the increase in '(2) overthat period is due to radiative buildup of the polar vortex. The zonal mean radiative forcing continues to actas a source of enstrophy throughout the winter. However, once the waves attain appreciable amplitude inDecember, the eddy radiative damping acts to decreasethe total enstrophy. This explains why the sign of thediabatic forcing varies with time.Four periods of large imbalance in the IE budgetshown in Fig. 4 appear after 1 Jan. The first occursINTEGRATED ENSTROPHY 10.0 mba.30.20.NI060-lI0X.10.0.4Total EnstrophyZonal Mean Enstrophy - Available Enstroohy . - . Eddy Enstrophy - - - b.INTEGRATED ENSTROPHY 5.0 MB30.20.cvI0v)Q)0-lI0Xc10.0.C' -\Zonal Mean Enstrophy - Available Enstrophy . - . Eddy Enstrophy ""NOV DEC . ' JAN FEB MARFIG. 3. The seasonal 5uctuations of the zonal mean integrated enstrophy, the total IE, theavailable IE and the eddy IE at (a) 10, (b) 5, (c) 1 and (d) 0.1 mb for the winter 1978/79 computedfrom LIMS data. Points A and B on the ordinate indicate the IE and available IE due to PlaIEtarYrotation.1 JUNE 1986C.20.NI0cz 10.mI0FX0.MARK R. SCHOEBERL AND ANNE K. SMITHd.20.NI0a,cnI 10.0XF0.INTEGRATED ENSTROPHY 1.0 MI3I I I I Total Enstroohv1NOV DEC JAN FEB MARINTEGRATED ENSTROPHY 0.1 MBI I I I108 1108210.mI0rn P)0.zX- 10.JOURNAL',OF THE ATMOSPHERIC SCIENCES VOL, 43, No. I1ENSTROPHYBUDGET 10.0 mbI I !Diabatic Forcing lord Tendency - - I I I INOV DEC JAN FEB ' MARFIG. 4. The various terns in the IE budget equations at 10 mbthe total tendency, a(T)fat, the total radiative dissipation, (4s) andthe residual. Imbalances in the IE budget occur when the total IEtendency and the radiative dissipation do not match.about mid-January during a very minor warming, better seen at 1 mb in Fig. 3c; the second occurs after theJanuary minor warming; the third during a minorwarming in early February; and the last occurs duringthe major warming in late February. The second andfourth IE imbalance periods, which are the larger,clearly result from a decrease in eddy IE without acorresponding increase in the zonal mean IE (seeFig. 3).' As mentioned earlier, a possible cause for the IEimbalance in the post-warming periods is irreversibledown-scale cascade of enstrophy into unresolvablezonal harmonics or meridional structures, most ofwhich would then be dissipated. To see this moreclearly we rewrite (3) asN2 atm= I(1 1)where N is the number of resolvable components ofthe zonal spectrum from LIMS data. The growth ofpe modes greater than N would then decrease the totalon the left-hand side of (1 1) as well as produce a decrease in the eddy enstrophy. The LIMS data set canresolve up to wave number 6, but the data are lessreliable beyond wave number 3.From the middle of January in Fig. 3a there is rapiddecrease in zonal mean IE with an associated increasein eddy IE. At the end of January,.the eddy IE tendencychanges sign rapidly. We'have calculated the conversion term from mean IE to wave IE [see sec. 2, Eq.(4a-b)] and it also reverses sign as the wave decays.This indicates that there is some transfer from the waveback to the zonal mean. This transfer, however, is sosmall that the mean IE, as well as that due to the wave,decreases during this period. The interpretation suggested by Eq. (1 1) is that significant irreversible downscale enstrophy transfer takes place during these twoimbalance periods which is suggestive of wave breaking.In fact, the two large IE imbalance periods coincidewith McIntyre and Palmer's (1983) Figs. 2 and 4, whichshow large-scale wave breaking poleward of 20"N on27 January and at the end of February. The imbalanceassociated with the early February minor warming coincides with the development of another tongue of highpv air as discussed by Dunkerton and Delisi (1986).An IE imbalance can also result from the manipulation of the data prior to computation of the enstrophy. In section 3 it'was noted that the,data have beensmoothed with a least-squares cubic spline in the meridional direction. The latitudinal smoothing couldproduce an imbalance as follows: the growth or decayof the large scale wave produces a change in the me.ridional structure of the zonal mean flow with abouttwice the meridional wavenumber of the wave. Thesubsequent meridional smoothing of the flow will reduce the magnitude of changes in the zonal mean flowmore than it reduces the wave itself.. As a result thezonal mean enstrophy will appear to respond sluggishlyto the growth or decay of the wave and an imbalancewill appear in the IE. This latter mechanism is probablypartially responsible for the periodic imbalances associated with traveling waves at 1 mb and the smallerimbalances in the IE budget prior to the January andFebruary warmings. We do not believe that this mechanism is responsible for the large imbalances at 10 mb.There is obvious evidence of the annual cycle in thestratospheric radiative forcing in the zonal mean IE.For example, Figs. 3a, b, c show that the peak in thezonal mean IE appears near winter solstice when thepolar night jet usually attains its greatest velocity. Asthe polar night region decreases-in area toward springthere is also a general decrease in the zonal mean IE.However this decrease appears as a series of short jumpsassociated with enhanced wave activity rather than asmooth decline. It appears that the waves play an important role in accelerating the decrease in the zonalmean IE; that is, the wave activity tends to push thezonal mean toward the summer circulation faster thanradiative processes alone. This picture is also consistentwith the late winter final warming circulation reversalwhich sets up the summer circulation usually beforeequinox.While MP and BR both emphasized nonlinearity a.sthe principal mechanism behind the erosion of the polar vortex, it appears from Figs. 3 and 4 that radiative1 JUNE 1986 MARK R. SCHOEBERLprocesses contribute significantly to the enstrophybudget. Once the vortex is shifted off the pole it is subject to radiative decay which can be substantially enhanced if the system has a baroclinic structure. It isclear from Fig. 3 that while the radiative processes cannot account for the large enstrophy imbalances in latewinter, they could very well account for much of theQUASI-GEOSTROPHIC POTENTIAL VORTICITY0.1 mb 121 1 178QUASI-GEOSTROPHIC POTENTIAL VORTICITY0.1 mb 1/16/79RG. 5. The potential vorticity for (a) I December 1978 and (b)I5 January 1979 at 0.1 mb. Note the absence of an organized polarvortex. Units are a"; the symbol GM is located at 0" long.AND ANNE K. SMITH 1083slow areal decrease in the polar vortex noted by BR.Note that the diabatic forcing and the total tendencymove together except during late February.b. The available potential enstrophyFigure 3 shows the available IE, D, as well as the IEand available IE due to planetary rotation ((f`) and(f') - (f)'). Both at 10 and 5 mb the actual zonalmean IE is a factor of 2 or more larger than (f') whichindicates that the zonal flow configuration (polar nightjet) contributes significantly to the potential vorticity.Note, however, that the available IE is still only a smallfraction O( 10%) of the total, just as (f ') - (f)' is asmall fraction of (f').A rough quantitative measure of the total saturationof the basicstate by Rossby waves is the ratio of theeddy IE, (4") to the available IE. Total saturation,when the waves have used up all the available IE, wouldbe indicated by a ratio of infinity and no waves by arat0 of zero. A mean ratio of greater than one, i.e.(4") on the average larger than D, would indicate ahigh degree of saturation. At 1 mb and below D isusually larger than.the eddy enstrophy in early winter;that is, the basic state is unsaturated. However as winterproceeds the wave IE increases and the basic state becomes more saturated with waves. This starts in midDecember at 1 mb and in mid-January at 5 and 10mb when the first warming occurs. Even though thewave IE decreases after late January, the stratosphereremains nearly saturated with waves until the end ofFebruary. In the mesosphere (Fig. 3d) D is always lessthan the eddy IE. This suggests that Rossby waves inthe lower mesosphere have used nearly all the availableIE even in early November, and that this entire regionmay be a permanent zone of Rossby wave breaking(Dunkerton and Delisi; 1985). As an illustration, Fig.5 shows the pv computed for the 0.1 mb surface for 1Dec 1978 and 15 Jan 1979. There is little evidence ofan organized polar vortex at this level. If the lowermesosphere is nearly saturated with Rossby wavespropagating upward from the troposphere then thesewaves would tend to decelerate the flow as they dissipate (or break). The momentum and heat transport bythe breaking of these waves could very well augmentthe stress on the flow believed to be produced by breaking gravity waves.c. Limits of the wave amplitudeLindzen and Schoeberl (1982) used the potential.vorticity constraints like those above to argue that thewave height amplitude might not be greater than about700 gpm; however, Labitzke (198 1) and others haveshown that the height of wavenumber 1 at 1 mb exceeded 2400 gpm at 60"N during the January warming.In deriving a numerical value for the wave amplitude1084 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 43, No. 11maximum, Lindzen and Schoeberl presumed that theplanetary vorticity was the major contributor to thepotential vorticity. Clearly, from Figs. 3 and 5, thisassumption was incorrect since the planetary vorticityis greatly enhanced by the basic zonal mean flow inthe winter stratosphere. When the observed zonal meanpv structure is taken into account, the maximum amplitude for the 1978/79 winter computed using theLindzen and Schoeberl method give "2700 gpm whichcorresponds better to the peak wave 1 amplitude seenin late January.The limit to wave amplitude is also consistent withthe qualitative picture of the vortex area discussed byMcIntyre and Palmer (1983; 1984) and diagnosedquantitatively by Butchart and Remsberg ( 1986). Whenthe vortex is completely distorted and moved off thepole, so that all of the vorticity gradient is in the eddiesand none is in zonal mean, then the basic state is saturated. If, as we have assumed, there is no rapid sourceof potential vorticity, then the location of the vortexcan change but its area and the maximum and mini' mum values of pv cannot change. Therefore the amplitude of the wave is limited by the initial (zonal mean)vorticity distribution, quantified as the available IE.When, because of irreversible mixing during the courseof the winter, the range of pv values is reduced, themaximum amplitude attainable is also reduced. Thussaturation during the January 1979 warming was characterized by an exceptionally large wave amplitude,but in the February 1979 warming the saturation amplitude was much smaller.5. Summary and discussionThe integrated enstrophy (IE) budget in the stratosphere and mesosphere (1 0 to 0.1 mb) has been analyzed using LIMS data. An important assumption inour analysis is that the flow is quasi-geostrophic so thata budget of potential vorticity is meaningful. This assumption is certainly not true in the tropics and Elson(1986) has shown that there may be other difficultiesat high latitudes. Nevertheless, the theoretical expectations of the integrated enstrophy budget equationsare largely borne out in our analysis. The meridionalintegration tends to reduce the noise normally generated in computing a highly differentiated quantity suchas the potential vorticity from data, and undoubtedlyhelps to isolate and clarify the observational signal.Another key factor of our study is the division ofthe pv field into zonal mean and eddy parts. Since weare concerned with the development in the stratosphereover the time scale of a winter, this turns out to be auseful separation. The transfer of enstrophy betweenthe mean flow and the waves occurs irregularly, andour results indicate that the transfer is related to theobserved enstrophy loss. In addition, the wave-meanflow separation provides insight into how the radiatively forced pv structure, which is initially nearly symmetric, is related to Rossby waves which appear later.The results show that increases in the eddy IE.tendto coincide with decreases in the zonal mean IE, andthe total is approximately conserved. In the lowerstratosphere, radiative effects have a time scale substantially longer than the rapid exchanges between eddyand zonal mean IE. There are, however, significantperiods where the total 1E is not conserved even whenradiative effects are considered. At 10 and 5 mb themost significant periods of nonconservation appear after large amounts of potential vorticity are transferredto eddies from the zonal mean, which occurs after thetwo observed sudden warmings. An imbalance in theIE budget then develops, probably because a significantamount of eddy IE moves to smaller scales not resolvedby LIMS. This idea is consistent with the wave breakingpicture of McIntyre and Palmer (1 984) and the arealanalysis of Butchart and Remsberg (1 986). The timingof the periods of largest imbalance corresponds to theirobserved periods of strongest wave breaking and rapidshrinking of the high pv area of the polar vortex. Ouranalysis provides evidence that potential vorticitytransfer to smaller scales during wave breaking eventsis both real and irreversible.Smaller 'imbalances in the IE budget also appearduring periods of rapid eddy growth and decay at 10,5 and 1 mb. Some of these imbalances probably resultfrom data smoothing which tends to reduce the effectsof waves on the zonal mean flow. Our largest imbalances are obtained at 1 mb possibly because of thepoor signal-to-noise at this level, Finally, at 0.1 mbconservation of 1E appears to be very poor. At thislevel external forcing by breaking gravity waves probably plays an important role in maintaining the basicstate.We have given theoretical evidence that the eddiescan tap only a small fraction of the zonal mean IE andthis fact is borne out in our analysis. The time scalefor radiative forcing in the stratosphere is slow relativeto that for dynamical changes during warming events.The presence of waves distorts the pv field but cannotchange the values, so the waves are limited by theavailable distribution of pv values. For a symmetric,state, the range of pv values is quantified as the availableIE; for a distorted state, it is given by the sum of eddyand available IE. When the waves are saturated, themean pv gradient essentially disappears and the integrated eddy enstrophy is large. This occurs in thestratosphere during the two sudden warmings. For theremainder of the winter an appreciable amount of thebasic state pv gradient is still contained in the zonalmean, or in other words the available IE is significant..The available IE is a hemispheric integrated quantity,and as such does not determine to what amplitude aparticular wavenumber will grow. It does, however, givean indication of how much additional Rossby wave1 JUNE 1986 MARK R. SCHOEBERLactivity as a whole can be supported at a particulartime. The sum of the available and eddy IE is equivalentto the mean square derivation of pv from its areal average. This quantity decreases in an irregular mannerfrom January through March at 10-1 mb. Thus, theability of the stratosphere to support more Rossby waveactivity decreases as the winter progresses.The situation is different in the mesosphere; the eddyIE exceeds the available IE throughout the winter. Because of the rapid time scales for forcing by radiativetransfer and gravity wave breaking, pv in the mesosphere is conserved only for a relatively short time.Therefore, the idea of available IE, which stems fromthe notion that Rossby waves rearrange pv on a surfacebut that it cannot change, is not directly applicable. Acomparison of eddy and available IE, which comprisethe deviation of pv from its zonal and meridional mean,respectively, suggests that pv varies as much or morein longitude as in latitude in the mesosphere. This region appears to be permanently saturated with Rossbywaves. Some of these waves may result from nonzonalgravity wave forcing (Schoeberl and Strobel, 1984;Holton, 1984), but the largest perturbations in the eddyIE in the mesosphere do appear to propagate from below and can be traced to the lower stratosphere.The overall IE budgets give an interesting picture ofthe stratosphere and mesosphere climatology duringthe 1978/79 winter. The lower stratosphere showsstrong brief periods of wave saturation and associatedmixing of the zonal mean potential vorticity gradient.The mixing that coincides with sudden warmings appears to be largely irreversible and reduces the availableIE and total IE during January and February. At higheraltitudes, wave amplitudes increase and the weak wavefluctuations at 10 and 5 mb become large at 1 mb. Thedecrease in the available IE begins in December at thatlevel. Even higher, at 0.1 mb, the basic state appearsto be dominated by the eddies from early winter. Thezonal mean pv gradient is continually mixed, as indicated by smaller values of the available IE than eddyIE (Fig. 3d).APPENDIXErtel Vorticity, Matsuno's Vorticity and EnergeticsThe Ertel vorticity is conserved on potential temperature surfaces, and the conservation of quasi-geostrophic potential vorticity on 'pressure surfaces is avalid approximation to the vertical component of theErtel vorticity under most stratospheric conditions(Hartmann, 1977; Hoskins et al., 1985). Matsuno(1970, 197 1) derived an energetically consistent approximation of the potential vorticity equations forlarge-scale perturbations in the stratosphere. However,the system derived by Matsuno does not locally conserve Matsuno's potential vorticity (as will be discussedbelow) although it globally conserves Matsuno's vorAND ANNE K. SMITH 1085ticity divided by the sine of the latitude. On the otherhand, a system which .conserves Matsuno's potentialvorticity does not conserve the global integrals of energyin the sense of Lorenz (1967). In both situations thenonconservation is weak; it is on the order of otherterms removed by scale analysis in the derivation. Wediscuss the conservation properties of the various potential vorticity approximations herein as this is an important concern for the budget computations performed in this paper.The quasi-geostrophic vorticity on a sphere is givenbywhere y = a4 and x = aX cost$; X is longitude and [ isthe local vorticity. Matsuno (1970) developed a modified vorticity defined asMatsuno included isallobaric advection of planetaryvorticity in the perturbation equation which is Roll2smaller than the geostrophic advection. Matsuno ( 1970,197 1) then formulated the following energetically consistent linearized for planetary scale motion:where the overbar indicates zonal mean and the primeindicates deviation from the zonal mean. Aside fromglobal energy conservation in the sense of Lorenz, theseequations conserve the global integral of ijm(sin4)-',but they do not locally conserve qm or conserve theglobal integral equivalent to the enstrophy, qm2(sint$)-2.By local nonconservation we mean that (A3a, b) cannotbe derived by linearization of a conservation equation.Using qm in (A3) to enforce local conservation weobtain an alternate system:a- la-qm+-- (cos4vah) = 0, (A4a)at COS+ ayThese equations globally conserve qm as well as theenstrophy qm2. They do not globally conserve energyin the sense of Lorenz (1967) as there are spuriousenergy conversion terms. However, these terms areO(RO'/~) so the nonconservation is weak-as is the localnonconservation of qm in (A3a, b).1086 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 43, No. 11For most diagnostic studies it seems reasonable toprefer (A4a, b) over (A3a, b) as- &HVq, =V*FIta cos4where F is the EP flux (Palmer, 1982) and local conservation of qm is enforced. The differences betweenthe two systems is small inside the range where thescale analysis is valid (e.g., poleward of 20").As an aside, it is possible to construct a totally conservative system by requiring the local conservation off+ cm/sin4. Eq. (A3b) becomes- q$ + U - q& + vk sin4 - (,&in4 + f) = 0.a a aat ax aY(A5)The combination of (A3a) and (A5) now globally conserve f + tm/sin4 as well as the equivalent enstrophy.In addition, (A3a) and (A5) globally conserve energy.Unfortunately, the division by sine may cause b/sin4to become very large near the equator. As a result, theintegrated enstrophy computed with f + ,&in4 willbe much less sensitive to mid- and high latitude fluctuations; qm therefore appears preferable for this study.Acknowledgments. Observational analyses of LIMSdata were done at the National Center for AtmosphericResearch (NCAR) while A.K.S. was a visitor to theAdvanced Study Program. NCAR is sponsored by theNational Science Foundation.REFERENCESButchart, N., and E. 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