On the Solution of the Homogeneous Vertical Structure Problem for Long-Period Oscillations

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Abstract

The problem of determining the long-period normal modes of a barotropic atmosphere with realistic temperature, distribution and in uniform motion is examined over the entire range of equivalent depth. It is demonstrated that there are only two solutions which satisfy a radiation/finite energy condition and approximately satisfy the homogeneous surface boundary condition. Outside of a particular interval of equivalent depth, it is shown that there exist no solutions. The problem is solved numerically over the restricted interval where two sharp dips in the surface error are found.

The first of these corresponds to a 9.6 km equivalent depth and a Lamb structure. This mode, which is the counterpart of the thin film solutions on a sphere, is due to the hydrostatic nature of the basic state and exists despite the temperature variation. The second dip, corresponding to a 5.8 km equivalent depth here, is a result of buoyancy ducting and is a consequence of the temperature variation. The energy density of this structure has maxima at the surface and at a level appropriate to the stratopause. Because of its more fundamental nature, the Lamb mode satisfies the homogeneous problem to a greater degree and has the greatest likelihood of being realized in the atmosphere. The second structure indicates the potential for wave ducting in the stratosphere and mesosphere. Because of its dependence on the particular nature of the temperature profile, however, its realization in the atmosphere would be more variable.

Local energy residence times are calculated for each mode. The Lamb mode has its greatest value near the surface where the dissipationless estimate exceeds several hundred wave periods in a layer appropriate to the troposphere. The second structure resides longest in a layer appropriate to the stratosphere and mesosphere where energy may remain on the order of 10 periods in the absence of dissipation.

Abstract

The problem of determining the long-period normal modes of a barotropic atmosphere with realistic temperature, distribution and in uniform motion is examined over the entire range of equivalent depth. It is demonstrated that there are only two solutions which satisfy a radiation/finite energy condition and approximately satisfy the homogeneous surface boundary condition. Outside of a particular interval of equivalent depth, it is shown that there exist no solutions. The problem is solved numerically over the restricted interval where two sharp dips in the surface error are found.

The first of these corresponds to a 9.6 km equivalent depth and a Lamb structure. This mode, which is the counterpart of the thin film solutions on a sphere, is due to the hydrostatic nature of the basic state and exists despite the temperature variation. The second dip, corresponding to a 5.8 km equivalent depth here, is a result of buoyancy ducting and is a consequence of the temperature variation. The energy density of this structure has maxima at the surface and at a level appropriate to the stratopause. Because of its more fundamental nature, the Lamb mode satisfies the homogeneous problem to a greater degree and has the greatest likelihood of being realized in the atmosphere. The second structure indicates the potential for wave ducting in the stratosphere and mesosphere. Because of its dependence on the particular nature of the temperature profile, however, its realization in the atmosphere would be more variable.

Local energy residence times are calculated for each mode. The Lamb mode has its greatest value near the surface where the dissipationless estimate exceeds several hundred wave periods in a layer appropriate to the troposphere. The second structure resides longest in a layer appropriate to the stratosphere and mesosphere where energy may remain on the order of 10 periods in the absence of dissipation.

2350 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME36On the Solution of the Homogeneous Vertical Structure Problem for Long-Period Osci~Bations MURRY L. SALBYNational Center for Atmospheric Research~, Boulder, CO 80307(Manuscript received 2 March 1979, in final form 31 May 1979) ABSTRACT The problem of determiningthe long-period normal modes of a barotropic atmosphere with realistictemperature distribution and in uniform motion is examined over the entire range of equivalent depth.It is demonstrated that there are only two solutions which satisfy a radiation/finite energy conditionand approximately satisfy the homogeneous surface boundary condition. Outside of a particularinterval of equivalent depth, it is shown that there exist no solutions. The problem is solved numericallyover the restricted interval where two sharp dips in the surface error are found. The first of these corresponds to a 9.6 km equivalent depth and a Lamb structure. This mode, which isthe counterpart of the thin film solutions on a sphere, is due to the hydrostatic nature of the basic stateand exists despite the temperature variation. The second dip, corresponding to a 5.8 km equivalent depthhere, is a result of buoyancy ducting and is a consequence of the temperature variation. The energydensity of this structure has maxima at the surface and at a level appropriate to the stratopause. Becauseof its more fundamental nature, the Lamb mode satisfies the homogeneous problem to a greater degree andhas the greatest likelihood of being realized in the atmosphere. The second structure indicates the potential for wave ducting in the stratosphere and mesosphere, Because of its dependence on the particular natureof the temperature profile, however, its realization in the atmosphere would be more variable. Local energy residence times are calculated for each mode. The Lamb mode has its greatest value nearthe surface where the dissipationless e stimate exceeds several hundred wave periods in a layer appropriateto the troposphere. The second structure resides longest in a layer appropriate to the stratosphere andmesosphere where energy may remain on the order of 10 periods in the absence of dissipation.1. Introduction Numerous investigations have considered thenormal modes of oscillation of a hydrostatic atmosphere. Preliminary work was begun by Laplace (seeLamb, 1932) in his ocean tidal theory and extendedby Hough (1897, 1898), Haurwitz (1940) and otherswho investigated two-dimensional solutions on asphere. A comprehensive numerical treatment ofthe two-dimensional problem and the correspondingHough solutions was presented by LonguetHiggins (1968). The quest for obtaining three-dimensional solutions in the atmosphere was also started by Laplace,who conjectured the existence of oscillationsanalogous to those of an ocean of uniform densityand depth. These were examined by Taylor (1936)who suggested that there would be an analog to theocean depth in Laplace's tidal equation for theatmosphere. This spawned the term equivalent depthwhich has persisted to date. This terminology isperhaps unfortunate, since not all of the solutionsof the three-dimensional problem necessarily cor ~ The National Center for Atmospheric Research is sponsoredby the National Science Foundation.0022-4928/79/122350-10506.50c 1980 American Meteorological Societyrespond directly to the two-dimensional solutions ona sphere. The introduction of this additional dimension creates the possibility for oscillations dueprimarily to the vertical nature and not possible ina two-dimensional film. An issue of considerable debate in the ensuinginvestigations of this problem has been the natureof the upper boundary condition. Although manystudies have employed more traditional conditions(e.g., free surface, rigid lid), Lindzen et al., (1968)showed that this necessarily introduces a set ofspurious modes due to reflection at the upperboundary. The linearized equations for a motionless basicstate are separable, the separation parameter beingrelated to the equivalent depth. The equationgoverning the horizontal behavior is essentiallyLaplace's tidal equation, while that governing thevertical has been labeled the vertical structureequation. For an isothermal basic state there existsonly one solution to the problem with zero surfacevertical velocity and either finite energy or no energyradiating in from infinity. This has the vertical structure of a Lamb wave (Wilkes, 1949) and correspondsto an equivalent depth of 10 km.DECEMBER 1979 MURRY L. SALBY 2351 Siebert (1961) examined the atmosphere's responseto tidal forcing for simplified temperature profiles.This study, which was conducted over a restrictedrange of equivalent depth appropriate to resonancetida. l theory, also predicted a characteristic valufi of10 km. One of the more comprehensive treatmentsof the problem was made by Dikii (1965) Who considered a broad range of motions. In that study thetemperature was assumed to grow unbounded in theuppermost layers, which forced the upper boundarycondition to be one of finiteness. Unfortunately,this condition, appropriate at infinity, was necessarily applied at a finite height. This effectively introduced an artificial upper boundary which produced a spurious set of vertical modes due toreflection. Imbedded in this set is a solution with acorresponding equivalent depth of 10 km whichexists for a broad spectrum of frequencies. Kasahara (1976) examined the problem in a multi-levelmodel, also with a rigid lid. Aside from a solutioncorresponding to the 10 km equivalent depth andoscillatory modes due to reflection, a mode withenergy constant in height and corresponding to aninfinite equivalent depth also resulted. The latter isalso due to the rigid lid since it necessarily resultsfor the completeness of the set of orthogonal functions satisfying the two-point boundary valueproblem. We will examine the normal mode problem foroscillations on the order of a day or longer in ahomogeneou's, barotropic atmosphere with realistictemperature distribution and in uniform motion.Only solutions resulting from the lower boundarycondition and/or from the character of the basic statewill be considered. In particular, the results arefree of modes arising from the nature of the upperboundary condition. The entire range of the separation parameter (equivalent depth) is examined. Solutions are retained which have either finite disturbanceenergy or no energy radiating in from infinity, andapproximately satisfy a homogeneous surface condition. Attention will focus on the Rossby solutions; however, the same analysis applies to thegravity modes.2. Development of the boundary value problem Linear, adiabatic disturbances will be consideredin a hydrostatic, barotropic atmosphere, in uniformmotion. Consistent with the traditional formulationof this problem, the basic state is assumed to be ofuniform composition, and variations in gravity areneglected. In this case, the basic-state variablesdenoted by zero subscripts are a function of heightonly and may be written in terms of the backgroundpressure Po(~,4,) = po(O) e-e(-, (1.1)where I2 ~' --In( po 1~:(0 = fi-~) \ po(O) J '0.2)fi(O - RTo/g _ po/pog (i.3)is the normalized scale height, fi being a mean value,R is the specific gas constant and /; = z/_,~ (1.4)is the normalized geometric height. The following dimensionless disturbance quantities (in standard notation) are appropriate for oscillations on the order of a day or longer (cf. Salby,1980):(2)where e = fi/a is the atmosphere's shallowness andis of the order 10-a, a is the earth's radius, g thegravitational constant, fl = fl + A is the total angular velocity of the basic state (the angular velocityof the surface, fl, plus that of the atmosphere, A)and is assumed constant, and fi is an arbitrary disturbance scale. Solutions are sought which may exist in the limiting sense of no forcing. Thus the problem is mathematically homogeneous. The linearized equationsare separable in time and longitude, and hence, asteady disturbance may be assumed of the formei(mx-~t) with integer zonal wavenumber m (for cycliccontinuity) and frequency tr. The normalized in-'trinsic frequency is defined as & = to/2fl = (tr -Arn)/2~.To first order in small quantities (Salby, 1980) theequations are-i&fi - sin4,~ = - --/5, (3.1)-i&~ + sin4,fi = -fir0/5 , (3.2)O4, 1 0/5b=~/5- o~ '(3.3)2352 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME36 ..- .~__1 (O~ ) im--ttmgp -- j92[0~ + 1 - + H- cos0/~ 1 0 1 0-+ (cosO~) + B cos~ aT ~9 og ~i '&----99/3 +' --+~c ~ y 0gHere y-1-- - O, (3.4)=-i&~gJgb. (3.5)y'is the ratio of specific heats and ,/ = (21~)2a2/gI~is Lamb's parameter and is of order 10. Theboundary conditions are ~,=0 (~=0) radiation/finite energy condition (~--> m)where one of the latter is chosen where appropriate. These equations may be reduced by making thespherical transformation/2 = sin(O), and the velocities obtained in terms of the pressure as a = ~U[.b], (4.1) 5 = -iBV[13], (4.2) comg [ , op ] v- =i OH ' O~ -'el3 , (4.3) o[where /22)1/2 U[,]= ( - /2 - +--. ,.(5.1) co~ _ /22 0/x ' 1 - /22 (1-/2~)"2[c') m~] v[.] - ~; ---7 co0/2 ' + --! -/22 ' - (5.2)Eqs. (4) imply that the zonal and meridional velocities are in quadrature. The velocities and densitymay be eliminated leaving the following partial differential equation in pressure: L. j[/3] = M[15l, (6)whereL[.I = = - (7.1) H 1 ~-~ [~e] (7.2) ~3 0bt co2-_ ~2 0/2 . 1_ (_~ CO2 "~ /.12 ,__ m2.]r/~co2 /22) co2 /22 1 _--~2 ' (7.3)J[/3] is effectively the velocity divergence. Eq. (6)bears some resemblance to the eigenvalue problemfor the normal modes of a contained, rotating fluid.The latter, termed Poincar6's problem, is mathematically ill-posed in that it is hyperbolic, yetboundary conditions are necessarily imposed aroundthe domain (Greenspan, 1968). The equation hereis hyperbolic inside the latitude band l/2[ < [col.Nonetheless, Eq. (6) is separable, and by assuming15 = Z(OY(lx) the following two boundary valueproblems result with separation parameter ~: L'J[Z] + aZ = 0 ] DZ'- ~ =0 (~=0) J , (8) radiation/finite energy condition (~ --~ o~) M[Y] + aY = 0 /. (9) Y bounded at/2 = _+ 1. J Eq. (9) is equivalent to Laplace's tidal equationif ~/a is replaced by the equivalent depth.3. Solution of the vertical structure problem Unlike Dikii's study, the motions that are beingconsidered here lie in the hydrostatic regime; therefore, the two boundary value problems uncouple.The value(s) of a, i.e.~ the equivalent depth(s), maybe obtained from the vertical problem and then substituted into the horizontal problem which may thenbe solved for the frequency co. As posed, the problem is homogeneous. A nonzero solution can occur exactly only if no disturbanceenergy escapes from th~ system. For this semiinfinite volume this means that either the energypropagates solely in the horizontal, or it encountersa barrier of infinite extent in the vertical and is completely reflected. Any other possibility will allowenergy to eventually radiate to infinity. It will be ~seenthat there exist an infinite number of solutions to theequations with finite energy and no energy flux and,hence, completely contain the disturbance energy.It is not immediately obvious that these cannot beconsidered as normal modes of the system. For the isothermal case, the only solution to theproblem is the Lamb structure corresponding to , Z(O = e~and c~ = 1 - ~ ~0.714.This solution has zero vertical velocity throughoutand therefore no vertical energy flux. As Dikii noted, in general for a non-isothermalatmosphere, there are no exact solutions to thisproblem. We shall relax its homogeneous nature byrequiring that at a characteristic value of a, thevertical velocity achieves a small value at the surfacerather than vanishing exactly.DECEMBER 1979 M U R R Y L. S A L B Y 2353 Eq. (8) may be written in normal form (cf.,Rektorys, i969) by letting Z(0 = e~2fir-~[/~' + Mu2v(0, (10)in which casev" + k2(~; oOv = 0~v'+ [~-(~'+g) l (11)+2(~;7~)v=- (~=o)where k2(~; a) is a complicated expression in ~ anda, and plays the role of a refractive index or verticalwavenumber squared. Regions where k2 > 0 correspond to oscillatory vertical behavior, whereas thesolutions are primarily evanescent and considerablereflection occurs where k~ < 0. Actually, k:.(~; a)defines a family of refractive index profiles with aas a parameter. Solution of the problem may beviewed as varying a, i.e., k~(~; a), so that the upwardand downward propagating waves approximatelycancel the vertical velocity at the surface. Although the expression for k~(~; a) is complicated, its sole dependence on a is through the terma[([t' + t0//4]. The second factor is proportionalto the square of the Brunt-Vfiisfiilfi frequency.Since most standard temperature profiles are positively stable. (Gossard and Hooke, 1975), it followsthat k~(~; a) depends monotonically on a for all ~.Therefore, a can be chosen sufficiently large/smallso that k: >> O/k~ ~ O.v=k-Ua[Aexp(-iI~kd,) + Bexp(+i I2kd~)] - (i3)For the Rossby solutions, the frequencies are negative. As is traditionally the case for buoyancy waves(the vertical motion here is that of a gravity wave),the energy propagates opposite to the phase in thevertical; hence, the radiation condition requiresthat A -- 0. It is shown in the Appendix that in this range thereexist no values of a in which the remaining solution approximately satisfies the surface boundarycondition. This follows from the physical consequence of the short-wavelength approximationwhich implies that the transmission coefficient isunity (Morse and Feshback, 1953). Thus, the energycontainment mechanism is absent. For a sufficiently large negative, (~ = -k~ ~ 0)the general solution is given bywith k arbitrarily taken positive. For a realistic temperature distribution, the finite energy conditionrequires that A = 0. Any one of this family of remaining solutions has finite energy and zero energyloss. However, as is shown in the Appendix thereexist no values of a in this range in which the remaining solutions approximately satisfy the surfaceboundary condition of zero vertical velocity.a. Solution for large [c~ { The solution of (11) may be approached with theWKB approximation. This procedure is valid ff thefunction 5 0 1 02 --(td)~ 1 02) 4(k2)~ 0~for all ~ (Nayfeh, 1973). One way that this inequalitymay be satisfied is if k~ ~ dk/d~ (Morse and Feshbach, 1953), i.e., the refractive index varies slowlyover a wavelength. In the light of previous remarksthen, it should not be surprising that For a sufficiently large positive (k~ 0), thegeneral solution of (11) isb. Solution for intermediate ~ The search for characteristic values of a hasnow been restricted to a finite interval over whichthe profiles of refractive index may exhibit severalbarriers to vertical propagation. In the perfectlyhomogeneous problem there is no excitation andhence no flux of energy. The relaxation to "approximate'' surface homogeneity, however, canamount to the introduction of energy through surface work. Since the system is conservative, theresultant energy must eventually tunnel through anybarriers along its route to infinity. These barriers,however, may cause energy to build up in certainregions due to .repeated reflection. The height profile used in the numerical part of thisstudy corresponds to the 1976 U.S. StandardAtmosphere temperature (Fig. 1). The mean scaleheight was taken to be 7.3 km appropriate to a 250 Kmean temperature. Profiles of F(~; o0 were evaluated for different a, and it was determined that themaximum value fell within 0.10 outside the a interval.(-3., 15.). Thus, outside this interval, the WKB2354 JOUR,NAL OF THE ATMOSPHERIC SCIENCES' VOLUME36 60 55A 50~0~ 45{..9rn40'I,,,35.~~<3o03~2tal20~_~-I.~<~ 10NORMALIZED HEIGHT PROFILE0 ,,,I,, 0 .5 I~0 1.5 :~.0 2.5 3.0 3.5 4.0 FIG. '1. Normalized scale height from the 1976 U.S. standard Atmosphere,temperature. 6O 55~ .ISI i IREFRACTIVE INDEXci;.764/o=I.P5.~35o ~-_.30~w25~~ 2o5.'~ 10~ 0 ~ n5 -.4 '~3 -.2 nl 0 .I .~ ka FiG. 3. Ve~ical refractive index as a function of a..3formulation previously discussed will be assumedvalid. As is commonly the case for reference atmospheres, the temperature and hence /-/ approachesa constant at sufficiently great heights. Eq. (11)may then be solved analytically in this region andthe appropriate upper boundary condition imposed.Since the solution is unique only up to a complexmultiplicative constant, the remainder may be obtained by marching downward. This was done fordifferent values of a in the restricted interval, andfor each o~ the normalized surface vertical velocity(error) was evaluated. The spectrum of I-/~l isshown in Fig. 2. The two distinct dips correspondto a values of 0.764 and 1.25, i.e., equivalent depthsof 9.6 and 5.8 km. The associated refractive index SURFACE VELOCITY SPECTRUMI ~ I ~ I ~ I ~ I ~ I ~ I '~ IOt ioo~1~= I0-1. I0-~ I0' -4-2. 0 2 4 6 8 tO 12 14FIG. 2. Normalized surface error as a function of a.profiles are shown in Fig. 3. It can be seen that eachhas a region of positive k2 bounded by slabs Of negative refractive index, and for each, k2 approaches aconstant positive value at sufficiently great heights.Also, it can be seen that the surface error has noappreciable drop at the value o~ = 0 correspondingto the infinite equivalent depth. The normalized velocity and temperature magnitudes corresponding to the first dip a~e shown inFig. 4a. Also shown for ~eference is the velocitymagnitude of the Lamb structure adjusted for thisnonisothermal basic state by replacing e~ by e~e.The close similarity in both character and equivalentdepth indicates that this is the Lamb structuremodified by the variation of temperature. Bretherton (1969) has' determined the Lamb wave, i.e., theacoustic mode, in the presence of small variationsin background temperature and velocity. As noted inthat study, in the limit of the temperature variation going to zero, this solution would tend to theLamb structure. This mode is a consequence of thehydrostatic distribution of mass and exists in spiteof the temperature variation. Lindzen and Blake(1972) also analyzed-the Lamb wave with realistictemperature distribution. In that study which included dissipation for shorter periods than are beingconsidered here, a peak in the surface response wasfound near the 10 km equivalent depth. The phase profile shown in Fig. 4b has a verysmall variation over the first 13 scale heights followed by a linear trend above, indicating free transmission. This suggests that the solution may bethought of as a purely evanescent wave with a~much smaller amplitude propagating componentsuperimposed upon 'it to carry off energy from theDECEMBER 1979 M U R R Y L. S A L B Y 2355surface. The actual solution is represented quite'well by the adjusted Lamb mode up to levels wherethe evanescent energy falls below that of the propagating component (see Fig. 4c). As previously noted, the horizontal problem isessentially that of Laplace's tidal equation with theLamb's parameter replaced by the effective Lamb'sparameter aT, or equivalently, with the depth replaced by the equivalent depth ///a. Because thedifference in aT for the two modes is small compared with the solutions' dependence on its valuethe horizontal characters of the two modes are veryclose. The Hough modes are approximately thosecorresponding to a Lamb's parameter value of 10(see Longuet-Higgins, 1968). It follows from theresults of Longuet-Higgins thatII (U2 + V2)dS = 2~'a2vSS(15)where v is the ratio of kinetic to total energy of aparticular Hough mode. With these we may form global averages of kineticand potential energy densities. These are given by(16)(Gossard and Hooke, 1975) where N is the BruntVfiis/iil/~ frequency, and the angle braces and theoverbar denote time and global averages, respectively. These are shown in Fig. 4c along with thetotal energy for the first Rossby-gravity mode(zonal and meridional indices equal to 1). Theenergy densities are reminiscent of a classical edgewave of which the Lamb mode is a particular case. The second, smaller dip in the surface velocityhas been largely ignored in the literature. However,apparently it corresponds to the equivalent depth of6.8 km determined in Dikii's investigation. Inanother resonance tidal study, Jacchia and Kopal(1952) determined an equivalent depth of 7.9 kmbesides the 10 km value. This was accomplished byvarying the temperature profile to yield the greatesttidal amplification at the surface. The final profilewas not greatly removed from the reference profiles examined. It was also noted that the responsewas sensitive to the stratopause temperature. Thisfeature may be understood by examining the refraczo1816141210~ 6w- 4tlJT 2'J 0 20tD~- 18~.~~ 1614 VELOCITY AND TEMPERATURE MAGNITUDE ~ ~ ~ ~ ~ ~ ~ ~1 ~ ~ ~,"l-r~ .,' ~' ~,'~' ./-' . ~~ Ma~ (U) / ~ --~ M~G (T) / ....... ; I I ] lllll[ t [ I II1[~[ [ I [ { ~ [ [[I0o I0~ I0~ I0~ VELOGITY PH~SE , ,,(~,, ,,i, , ,, I,, ,, I,,,, I ''" t "'' I ''~ '~ /12l086"; 2 ~,,,I,,t,lt,[~ll~,l,,~,l -200 -150 -tO0 -50 0 50 ARG (U) (DEGREES) ENERGY DENSITIESilit,,I,il~00 15014,I,I1111~1,1,1,1,1~1~1'200...... <~>---~ TOTAL 0 i ~ .I ,2 .3 ~ .5 .6 .7 .8 .9 1.0 I.I 1.2 c~ --.764 Fro. 4. Velocity and temperature magnitudes, velocity of theadjusted Lamb mode shown for reference (a); phase (b); andmean energy densities for the first Rossby-gravity mode,normalized by the maximum kinetic energy (c). a = 0.764.2356 JOUR'NAL OF THE ATMOSPHERIC SCIENCES VOLUME362O181614IO 6tsJ42O' i0-~VELOCITY AND TEMERATURE MAGNITUDE i r iiilll[ i i !iiHiI i iIIllllI I IIIIHII i ll,llllll/ - i~llll~_ / I'/ . (a) / (,"- f,?_ - //// /;' t~I /.,y . ~// '},//'/ MAG(U)' ..4/7 -----MAG~T, . ~-~/;" - .... ~SO~.ERMaL ~ ~' ~ ~11~' ~1 illlnl! H illUll I i i~11~1 i i~1~1 i llllln i0o i0~ i0~ i0~ i0~WJI0 VELOCITY PHASE 2OI: '' I ' ' ''1 ' ' r '1 ~ I r I J~18, (b) '11' r"l~''r ~'1 '3 'll '' r '1~]~ ~. / - 16f ~ ' ~ 14 . ~2~ -~ 4I- -'rc2_ 2~ ~ O~,,~r)*~l~l~, ,~ -200 -150 ,-IOO -50 O 50 IOO I~~ ARG (U) (DEGREES)~ ENERGY DENSITIES~ 20 , [ ~ i., i ~.1 , i ~ i , i ~ i [ i ~ i ~ i , g~.~8~ (c) ~ ...... ~> ~ ---<~> . I TOTAL I ,, ; 6 I ........ ~ S , ,- ,. 0 .~ .2 .~ A .5 .~ .7 .8 .~ 1.0 I.I I.~0=1.25200FIO. 5. As in Fig. 4 except for a = 1.25. Magnitude of a freelypropagati~ng internal~ wave is also shown for reference.tive index profile corresponding to the second dip.There exists a layer of positive k2 whose maximumis near the stratopause. Unlike the previous case,here the minimum wavelength 2*r/k,,ax is of the sameorder as the thickness. Fig. 5c shows that the energyhas indeed built up in this region indicating significantwave trapping between the bounding layers. Thiscavity effect 'is also suggested by the velocitymagnitude and phase shown in Fig. 5a and 5b. Avelocity node is nearly formed at the base and thephase jumps 180-. Aside from this jump, the phasevaries only slightly up to 13 scale heights after whichit indicates free propagation, as was the case for theLamb structure. Unlike the Lamb mode, this solutionexists because of the .variation in temperature.4. Application to the atmosphere The occurrence of the second structure wasexamined by Giwa (19,68) for several model profiles.As Jacchia and Kopa! also noted, he found itsexistence to depend upon the nature of the temperature profile, and its equivalent depth to vary between5.3 and 8.0 km. In this light the particular equivalentdepth determined in this study should not be takentoo seriously. The height profile used here might beviewed as, a global mean, but more realistically,horizontal temperature variations would result in aperturbed annulus of positive refractive index. Thismode also appears among the ducted acousticgravity waves discussed in Francis (1973).' Since the homogeneous surface boundary condi-'tion is satisfied to a lesser degree by some twoorders of magnitude than for the Lamb mode, itmay be more meaningful to view this second structure as a ducted mode. The greatest likelihood ofits observation would be in the upper stratospherewhere its energy density might be more comparableto that of the Lamb structure and it would offer amechanism for containing disturbance energyoriginating there.' Since these modes leak energy, a characteristicrelaxation time may be defined as the ratio of thetotal integrated energy to total leakage rate, i.e., v (17) Swhere (I,} is the time-averaged vertical energyflux. Discussion of a transient time scale for thisconservative system would only be realistic in anatmosphere where the dissipation occurred aboveany reflection, i.e., 13 scale heights. Since this'isunlikely in the real atmosphere, the relaxation timesDECEMBER 1979 M U R R Y L. S A L B Y 2357TABLE l, Relaxation time (wave periods).rn 0 I 2 3 4 ot = 0.764I 0.2471E + 040.7215E + 03 0.1047E + 04 0.1545E + 04 0.2186E 4 042 0.2581E + 040.7215E + 03 0.1047E + 04 0.1647E + 04 0.2369E + 04 ot = 1.25I 0.2418E + 020.1442E + 02 0.1556E + 02 0.1759E + 02 0.1961E + 022 0.2479E + 020.1470E + 02 0.1487E + 02 0.1830E + 02 0.2070E + 02defined here are really academic. These are shownfor several Rossby modes in Table 1 where rn andn refer to zonal and meridional indices, respectively. As is to be expected, there is considerabledifference between the two vertical structures.Table 1 may be compared with the value 923 whichLindzen and Blake determined for a planar Lambwave. The differences are due to horizontalstructure. Perhaps a more meaningful quantity that can bedefined is the signal time or time that would be required for a disturbance to traverse a given regionin the vertical. Rather than the steady forcing portrayed here, it is more likely that normal structuresin the atmosphere are excited by transient behaviorof the basic-state and boundary conditions. Therefore, energy would be injected in pulsations andeventually either dissipate or propagate out of aparticular region. The results obtained here can beused to estimate the latter effect. Although this problem is steady, one might envision the constant energy flux as being composedof successive slugs of energy moving upward at avelocity determined by the local refractive index.The movement of a single pulse of energy would notdiffer radically from those in this steady stream,particulary in this linear context. The vertical energyflux and energy density are related by (lz) = [(E) + (P)]c.,,where c, is the group velocity. The signal time between the surface and a level z is then = I~ (E) + (P) dz, T(z,/x) J0and the global average becomes ~(z)- 4vra2 ~ (I~) VGlobal-averaged signal times corresponding to thefirst Rossby-gravity mode (period slightly greaterthan 1 day) are shown in Fig. 6. The residence timefor a particular layer may be obtained by differencingthe signal times at the top and bottom. For the Lambmode, the residence time is greatest near the surfacewhere it exceeds several hundred days in a layerappropriate to the ~roposphere. This is indicative ofthe fundamental nature of this structure, i.e.,horizontal propagation. Realistic residence timesfor this structure would be dictated by dissipation.One conclusion that may be drawn from this, however, is that apparently, this structure should be.sensitive to dissipation only locally, at least withregards to vertical transport. In particular, if thereis an energy sink restricted to some level, e.g.,critical surface absorbtion, the time scale for transporting energy vertically to the sink should be muchgreater than other dissipative time scales, except in.the immediate vicinity. The time scale for horizontalenergy transport, however, should be comparable to dissipative time scales. Energy deposited inthe second structure resides longest in a layerSIGNAL TIME1614 (~= .76412I0864 0 200 400 600 800 I000 1200 1400 1600 1800 02 4 6 8 I0 i2 14 16 18 20 TIME (DAYS)FIG. 6. Mean signal time between surface and a given height for the first Rossby-gravity mode.2358 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME36appropriate to the stratosphere and mesospherewhere it may remain on'the order of 10 days in theabsence of dissipation. Unlike in the previous case,this ~is at least comparable to dissipative time /scales. However, even at these levels, the residencetime predicted for the Lamb mode in the absence ofdissipation greatly exceeds that of the secondstructure. 5. Summary The problem of determining the long-period nor mal modes of a barotropic at.mosphere with realistic temperature distribution and in uniform motion has been examined over the entire range of separation parameter. It was demonstrated that there are only two modes which satisfy the radiation/finite energy condition and approximately satisfy the homogeneous surface boundary condition. There exist no approxi mate solutions to the homogeneous problem outside a given interval of equivalent depth. The problem was solved numerically inside this interval, where two distinct dips in the error of the surface boundary condition were found. The first, corresponding to a 9.6 km equivalent depth and a Lamb structure, is a- consequence of the hydrostatic distribution of mass and exists desp!te the temperature variation. The second, having a 5.8 km equivalent.depth here, is caused by buoyancy ducting and is a consequence of the variation in temperature. Because of the fundamental nature of the first-structure,' it satisfies the homogeneous boundarycondition to a greater degree than the second, andhas the greatest likelihood of being realized in theatmosphere. The associated solutions are the threedimensional counterparts of the thin film solutionson a sphere, since they correspond to perturbationsof zero vertical motion in the isothermal case. Theducted mode has energy densities largest near thesurface and at a level appropriate to the stratopause.If this ducting is not precluded by other effects,the greatest likelihood of its 6bservation would be inthe upper atmosphere, where the energy densitymight be comparable to that of 'the Lamb mode.Mean residence times were calculated for bothstructures. The Lamb mode has its greatest residencetime near the surface where the dissipationlessestimate exceeds several hundred wave periods for alayer appropriate to the troposphere. The ducted- mode remains longest in a layer appropriate to the stratosphere and mesosphere where it resides on the order ofl0 periods in the absence of dissipation. Acknowledgrnents. I would like to express my gratitude to Drs. Alan Pierce, Ri6hard Lindzen and Akira Kasahara for their helpful comments and criticisms. I also wish to thank Walter Jones -for pointing out the reference to Francis and to Ms. Ursula Rosner and Ms. Verlene Leeburg for typing the manuscript. APPENDIXThe Surface Condition for the WKB Approximation1. tz large positive (k2 ~ 0) Substitution of (13) into the surface boundary condition results in the following equation in the surface values of k:k 2~2 d~ 1 ! . ~' - _ (~0' + K) +' (A1) 210 H 200' + ~)Validity of the WKB approximation implies that thefirst term in parentheses is much smaller in magnitude than the second. Hence, (h' + 'By assumption, however, k is real and much greaterthan zero; hence (A2) cannot be satisfied..2. a large negative (-~2 = k~ ~ 0) Substitution of(14) into the surface boundary condition results in the equation 1 + 1 1 - -- (/~' + K) + (A3) 2/- ~0 200' +As in the previous case, this reduces to 1 1 /~ 2/~ H (~' + K) + 2(10' + ~) (A4)and ~ is by assumption much greater than zero. Fora realistic temperature distribution near the surface,the right-hand side is small; hence, the problem hasno solution in this regime either. REFERENCEsBretherton, F. P., 1969: Lamb waves in a nearly isothermal at-mosphere. 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London, A189, 201-257.--, 1898: On the application of harmonic analysis to the dy namical theory of the tides II. Phil. Trans. Roy. Soc. London, A191, 139-185.Jacchia, L. G., and Z. Kopal, 1952: Atmospheric oscillations and the temperature profile of the upper atmosphere. J. Meteor., 9, 13-23.Kasahara, Akira, 1976: Normal modes of ultra long waves in the atmosphere. Mon. Wea: Rev., 104, 669-689.Lamb, H., 1932: Hydrodynamics. Dover, 730 pp.Lindzen, R. ~;., and D. Blake, 1972: Lamb waves in the presence of realistic, distributions of temperature and dissipation. J. Geophys. Res., 77, 2166-2176.--, E. S. Batten and J.-W. Kim, 1968: Oscillations in atmos pheres with tops. Mon. Wea. Rev., 96, 133-140.Longuet-Higgins, M. S., 1968: The eigenfunctions of Laplace's tidal equation over a sphere. Phil. Trans. Roy. Soc. London, A262, 511.Morse, P. M., and H. Feshback, 1953: Methods of Theoretical Physics, Vols. I and II. McGraw-Hill, 1978.Nayfeh, A. H., 1973: Perturbation Methods. Wiley, 425 pp.Rektorys, K., 1969: Survey of Applicable Mathematics. The MIT Press, 1369 pp.Salby, M. L., 1980: Global-scale disturbances and dynamic similarity. J. Atmos. Sci., 37 (in press).Siebert, M., 1961: Atmospheric tides. Advances in Geophysics, Vol. 7. Academic Press, 105-187 pp.Taylor, G. I., 1936: The oscillations of the atmosphere. Proc. Roy. Soc. London, A156, 318-326.Wilkes, M. V., 1949: Oscillations of the Earth's Atmosphere. Cambridge University Press, 76 pp.

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