JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 86, NO. CI0, PAGES 9707-9714, OCTOBER 20, 1981 Turbulenceand StressOwing to Gravity Wave and Tidal Breakdown R. S. LINDZEN Naval ResearchLaboratory, Washington,D.C. 20390 Centerfor Earth and PlanetaryPhysics, Harvard University,Cambridge, Massachusetts 02139 It has been suggested(Lindzen, 1967, 1968a,b; Lindzen and Blake, 1971; Hodges, 1969) that turbu- lencein the upper mesospherearises from the unstablebreakdown of tides and gravity waves.Crudely speaking,it was expectedthat sufficientturbulence would be generatedto preventthe growth of wave amplitudewith height(roughly as (basicpressure)-•/2). This workhas been extended to allowfor the generationof turbulenceby smalleramplitude waves, the effectsof mean winds on the waves,and the effectsof the waveson the mean momentumbudget. The effectsof mean winds, while of relatively small importancefor tides,are crucialfor internal gravity wavesoriginating in the troposphere.Winds in the troposphereand stratospheresharply limit the phase speedsof wavescapable of reachingthe upper mesosphere.In addition,the existenceof critical levelsin the mesospheresignificantly limits the ability of gravitywaves to generateturbulence, while the breakdownof gravitywaves contributes to the devel- opmentof criticallevels. The resultsof the presentstudy suggest that at middle latitudesin winter, eddy coefficientsmay peak at relativelylow altitudes(50 kin) and at higher altitudesin summerand during suddenwarmings (70-80 kin), and decreasewith heightrather sharplyabove these levels. Rocket obser- vationsare used to estimatemomentum deposition by gravity waves.Accelerations of about 100 m/s/ day are suggested.Such accelerations are entirely capableof producingthe warm winter and cold sum- mer mesopauses. 1. INTRODUCTION discussedin connectionwith the diurnal tide. Again, the tur- The possibilityof breakingwaves generating turbulence in bulencewas assumedto persistup to someturbopause height. the mesospherewas noted someyears ago by Lindzen [1967; These studiestended to ignore the fact that the relevant fre- 1968a]and Hodges[1969] for tidesand gravity waves,respec- quencieswere the Doppler shifted frequenciesand that in- tively. The idea, in thesepapers, was simply that vertically trinsic frequenciesmight be very different. Lindzen [1971] presenteda crude model of turbulence in the propagatinggr.avity waves that, in the absenceof damping, growin amplitudeas po -•/• (wherePo is the basicunperturbed mesospherethat summarized the above mechanismsand, in pressure)could at someheight reach amplitudes for whichthe addition, argued that 'turbulent' diffusion could result from wave fields themselveswould be strongly unstable (i.e., the nonbreakingwaves (NB tidal modesare simply specialcases combinationof mean and wave fieldswould have a negative of internal gravity waves [viz. Lindzen, 1970]. The specificar- staticstability or at leastits Richardsonnumber would drop gument was based on wave •transiency,but other arguments below 0.25). Above such a height it was suggestedthat the have been advanced by Weinstock[1976]. This matter is dis- cussedfurther in a separate note by R. S. Lindzen and J. waveswould generatesufficient turbulence, on the average,to preventfurther wave growthwith height.Theoretical results Forbes(manuscript in preparation,1981) wherein we consider implied that amongtidal modes,only the first propagating the cascadeof energy from stable waves to waves of suffi- ciently small vertical wavelengthsto permit unstable break- diurnal mode would prove important in this respect.Lindzen and Blake [1971]estimated that this modewould break down down. Returning to Lindzen [1971], the model of eddy diffu- between80 and 90 kin, generatingturbulence up to a height sion presentedconsisted in a diffusioncoefficient 'that of about 108 km above which molecular viscosityand con- increasedas (po)-•/2 up to a levelat whichwave breaking oc- ductivity are sufficientto inhibit further wave growth. The curred, increasedmarkedly at this level (50-90 kin), and re- turbulencegenerated by this tidal modeis restrictedto within mained constantup to someturbopause level. Both the mag- nitude and the distribution of the diffusion coefficient were about 30ø latitude of the equator as is the mode itself. The cessationof turbulence at 108 km correspondsclosely to the reasonablycompatible with what was called for by measure- ments of composition[Hunten, 1975;von Zahn et al., 1980] but heightusually designated for the turbopauseand in a loose fashionsupports the notionof tidal generationfor turbulence. the range of uncertainty was large. The notion of ex- The situationwith respectto internal gravity wavesis less ponentially increasing (with height) eddy coefficientshas, certain.Structures with moderatelyshort vertical wavelengths however,gained fairly general acceptance. Recently,Holton and Wehrbein[1980] have concentrated (0(12 kin)) are commonlyobserved in rocketsoundings, pole- ward of 30ø (thus excludingdiurnal tides, at least in some on the momentum depositionby breaking waves, modeling measure),especially in winter. Examplesof such soundings this by Rayleigh friction (acting to bring zonal flow to zero). are shownin Figure 1. On the basisof dispersionrelations To be sure, wave breaking does lead to deposition of wave givenby Hines[1960], one commonly--butarbitrarilymiden- momentum flux, quite apart from the generationof diffusive turbulence; the two effects are, however, distinct as will be titled the gravity wavesas oscillationsof relativelyshort pe- shown in section 2 of this paper. Moreover, in the case of riod (0(3 hours)). Observationssuggested that such gravity wavesalso broke down, and Hodges[1969] estimatedthe re- breaking tides, the depositedmomentum is not attempting to sultinggeneration of turbulencealong similar lines to those bring the mean flow to zero [Fels and Lindzen, 1974]. Never- theless, as was noted by Leovy [1964] and Lindzen [1968b, This paper is not subjectto U.S. copyright.Published in 1981 by 1973],if one anticipatesthat friction is responsiblefor the re- the American GeophysicalUnion. versal of the latitudinal temperature gradient at the meso- Papernumber 1 CO 154. 9707 9708 LINDZEN: TURBULENCE AND STRESS (a) Winter 65 (b) Winter 64 (c) Winter 62163 lOO 8o ..... .-'• _--,.... • 60 ß? 40 ' -' •"'-' 40 --"""'"='•'•-,40J-' _ 2O ..... febOõlOGMT20 _._ 5febO320GMT'-'"J.... 28feb63 2211 GMT ----8 feb 2253 GMT .... 13feb 0430GMT J'..... 7dec. 631311 GMT -oo' 21 ' jan.2224 ' GMT of --- 2D, jan 0411 'GMT oJ- Ternperatu re (" K) Tempera t u re ("K) Ternperat u re (d) Summer 65 (e) Summer 64 100- 100- L•.• .... _• 6O ß 40 4O ..... 23jul, 170õ (•MT -- 7 aug. 0100 GMT --- 12 aug 0149 GJVlT 20 I ------I aug. 20030340 (•MTGMT 20 .... 16 aug. 0315 GMT • 8 aug. 1015 GMT ---- 18 aug. 0125 GMT ß • . i . i . , O ' ' ' ' ' ' ' ' ' ' ' i . , , C20'' 160' 200' 240 280 120 160 2O0 24O 28O Temperatu re Temperature (" K) Fig. 1. Winter and summertemperature sounding at Wallops Island (38øN). pause,then one really doesneed a mechanismthat actsto re- wintersand summers.For example,in Figure 3 we showwin- duce zonal velocitiesrather than gradients. ter and summerwind profiles for the latitude band 30ø-45øN The purposeof the presentpaper is to reassesscarefully, but basedon rocket data for the period 1960-1964 [U. S. Standard simply,the role of wavebreakingin the mesosphere,empha- AtmosphereSupplement, 1966]. Also shown are geostrophic sizingthe role of the mean flow in determiningthe behaviorof winds based on observedtemperatures. There is qualitative gravity waves.Assuming that mosthigh latitude gravity waves similarityto Figure 2, but the quantitativedifferences are ob- originate in the troposphere,it is shown in section 3 that vious.In particular,Figure 3 showswind magnitudespeaking changesin the mean zonal flow accompanyingchanging sea- at the levelsof wavebreakingdeduced from Figure 1. For our sons,sudden warmings, etc., dramatically alter the nature of calculationswe will use the resultsin Figure 3. For reference gravitywaves in the mesosphere.Such changes lead to simple purposes,we showin Figure 4 standardmodels for the height explanationsof variousmesospheric observations. distributionof temperaturein winter and summer (CIRA, 1972). The information in Figure 4 will be needed in sub- 2. GRAVITY WAVES IN THE MESOSPHERE--SIMPLE WKB sequentcalculations. However, it should be noted that in the ANALYSIS absenceof wave perturbations,mean lapserates are only 0 (+_ In this section,gravity waves typical of middle to high lati- 2.4ø/km) which are (for many purposes)small when com- tudeswill be emphasized,though theoretical results are easily paredwith (g/%,)= 9.8ø/kin. extendedto tidal gravity waves. It will be assumedthat the We now proceedto use the mathematicaltheory of internal reader is familiar with the mathematicaltheory of internal gravity waves to exploit the above observationallyderived gravity waves. Only selectedresults will be presentedhere. features. The crudeness of these features leads us to restrict The following referencesare but a few of the referenceswhere ourselvesto only the simplest, approximate, theoretical re- more completetreatments are presented:Hines [1960], Lind- sults.Our equation for the vertical velocity perturbation, w, zen [1970], Gossardand Hooke [1975],and Holton [1979]. associatedwith internal gravity waves is We will begin by presentingsome data relevantto our sub- N2(1+ 12/k') 1 sequentdiscussion. Figure 1 showsvarious temperature pro- ß =o (i) files