Gravity wave propagation in a nonisothermal atmosphere with height varying background wind

Transcription

1 GEOPHYSICAL RESEARCH LETTERS, VOL. 34, L2383, oi:.29/27gl36, 27 Gravity wave propaation in a nonisothermal atmosphere with heiht varyin backroun win Qihou Zhou an Yu T. Morton Receive 2 July 27; revise 3 Auust 27; accepte 5 October 27; publishe 4 December 27. [] We erive a ravity wave propaation euation or a compressible an non-isothermal atmosphere with a variable backroun win proile. Impact o all the raient terms on the vertical wavenumber epens only on the intrinsic horizontal phase velocity an the backroun atmosphere. For the backroun win variation, any one o the linear irst orer erivative, secon orer erivative an the suare o the irst orer erivative terms can be the ominant term uner ierent conitions. For temperature variation, only the linear irst orer erivative is important or waves havin a slow intrinsic horizontal phase velocity. Our euation inicates that the eect o win shear on the vertical wavenumber is opposite to that preicte by the Taylor-Golstein euation, which assumes an incompressible lui. We also erive an expression or the amplitue o the vertical win perturbation. Citation: Zhou, Q., an Y. T. Morton (27), Gravity wave propaation in a nonisothermal atmosphere with heiht varyin backroun win, Geophys. Res. Lett., 34, L2383, oi:.29/27gl36.. Introuction [2] Stuy o ravity waves is an important topic in atmospheric sciences. All ravity wave theories are base on the unamental principles o momentum, enery an mass conservation. However, ue the complexity o the real atmosphere, linear approximations are neee to obtain analytical solutions. Taylor [93] an Golstein [932] ormulate the theory or an incompressible lui havin a vertical variation in the backroun horizontal velocity. The Taylor-Golstein euation is wiely applie to the lower an mile atmosphere. The theory presente by Hines [96] is or a compressible, isothermal an winless atmosphere. The case o a non-isothermal atmosphere with a constant win is iscusse by Einaui an Hines [97] as well as by Beer [974]. Avances on ravity wave theories an comparison with observations are summarize in a review paper by Fritts an Alexaner [23]. Despite numerous publications on ravity waves, a linear theory consierin a compressible atmosphere with heiht varyin temperature an backroun win is not oun in the literature. In the ollowin, we present analytical results o ravity wave propaation in a compressible atmosphere havin a vertical variation in both backroun win an temperature. 2. Wave Propaation Euation [3] The atmosphere we assume is invisci an irrotational but compressible with the backroun temperature an win havin a vertical variation. The euations o motion, enery an mass conservation are [e.., Beer, 974; Fritts an þ U ru ¼ r þ U rp þ U þr ð Þ ¼ In the above euations, U, p, r are win velocity vector, piiiiiiiii pressure an ensity, respectively. Parameters c = H an are the spee o soun an ravitational constant, respectively. is the aiabatic inex, an H = kt/(m) isthe scale heiht, where k is the Boltzmann s constant, T is the temperature an m is the mean molecular weiht. Usin subscript an or backroun an perturbe parameters, respectively, an urther lettin U o =(U x, U y, ) be the backroun win an u = (u, v, w) be the perturbe velocity in the Cartesian coorinate with upwar bein positive, the linearize euations or small þ þ U ru þ þ U rv þ U rw r ð2þ þ u rp þ U rp ¼ c þ þ U U þ r u Þ ¼ We will seek wave solutions o the ollowin orm: ðþ Electrical an Computer Enineerin Department, Miami University, Oxor, Ohio, USA. Copyriht 27 by the American Geophysical Union /7/27GL36 p ; r ; u; v; w / WðÞe z iðwt kxx kyyþ ð3þ p r where k x an k y are the two horizontal wavenumbers, which are assume to be constants. L2383 o5

2 L2383 ZHOU AND MORTON: GRAVITY WAVE PROPAGATION L2383 [4] With the assumption o euation (3), reuction o variables rom the linear euations will lea to the ollowin couple euations between p an the vertical perturbe velocity w: i W 2 H þ k U W H w þ p w þ i W2 k 2 c 2 WH þ c 2 p ¼ p ¼ p ð4þ where W = w U. k is the Doppler shite anular reuency, k = k x^x + k y^y with ^ inicatin unit vector. From euation (4), we obtain the ollowin ierential euation overnin the vertical win motion: 2 2 þ ðþ@w þ rz ðþw¼ c2 þ k2 ð5þ ¼ 2Wk U H þ H W 2 ð6aþ k 2 c 2 " r ¼ W2 c 2 þ w 2 k2 W 2 k þ W 2 U 2 H þ 2k2 k W 2 k 2 c 2 W : U þ 2 k U 2 # c 2 W 2 k 2 c 2 W 2 k4 k 2 c 2 W 2 H k 2 c 2 þ W 2 k k 2 c 2 W U ð6bþ H In the above euations, w ( ) /2 /c is the Brunt- Vaisalla reuency, an k = jkj. Euation (5) is reuce to the Taylor-Golstein euation [e.., Nappo, 22, p. 29] i there is no temperature variation an =. The eneral expression or w is reuce to that or a temperature varyin atmosphere iven by Beer [974, p. 65] i U oes not vary with heiht. [5] Euation (5) can be rewritten into a stanar waveeuation o the orm with R w ¼ ~w ðþe z 2 ðþ 2 2 þ 2 ~w ¼ 2 ¼ r Deinin the intrinsic horizontal phase velocity in the rame o the backroun win as v W k, an v 2 c 2 ð7þ ð8þ v 2, we have the ispersion relation or an atmosphere with varyin temperature an backroun win as 2 ¼ w2 v 2 4H 2 þ k 2 v 2 c 2 þ c2 ^k v v 2 2 U 2 þ 2 c 2 ^k H v v 2 U 3c2 ^k v 4 U 2 þ c2 2v 2 3c4 4v 4 2 H H 2 þ 2c 4 2H 2 v 2 v 2 v 2 2 þ 3 v2 c 2 H v 2 v v 2 ^k U H [6] Quantity in the above can be larely interprete as the vertical wavenumber. Althouh the ispersion relation, euation (9), is a cumbersome euation, it is easily simpliie to other known ispersion relations uner more restricte assumptions. By lettin U = an =,the2 expression is the ispersion relation erive by Hines [96] or a winless isothermal atmosphere. It is reuce to the Taylor-Golstein euation i there is no temperature variation an = [e.., Nappo, 22]. We urther note that the sins o (^k. U )2 an ( )2 terms are neative. This is consistent with the act that ravity waves cannot propaate reely at iscontinuous bounaries. Otherwise, when (^k. U )2 or ( )2 is very lare, as at a iscontinuous bounary, 2 coul be positive, siniyin a reely propaatin wave. [7] In orer or the linear theory to be applicable, the atmosphere nees to be stable. In the mile an lower atmosphere, stable conitions reuire j U j <2w.4/s an T < K/km. K/m, which leas to j j <.3. In orer to estimate the secon orer erivatives, we assume that the backroun win an temperature are sinusoial unctions o altitue with a vertical wavelenth larer than a couple o scale heihts. The ispersion relation or waves havin a slow horizontal intrinsic phase velocity (v <.5c), applicable to most o the airlow observations in the mesosphere [Hecht, 24], is simpliie to 2 þ k 2 ¼ w2 v 2 4H 2 þ ^k 2 U 2 2 þ H 3 c 2 ^k U þ 2 v v 2 H ^k U v ð9þ ðaþ [8] Since v = ^k. U, we can rewrite euation (a) in terms o v as 2 þ k 2 ¼ w2 v 2 4H 2 2 v v v 3 c 2 2 v H 2 v 2 þ H v 2 ðbþ [9] At even smaller intrinsic horizontal phase velocity (v <.2c), the suare term can be omitte. Note that 2 + k 2 is the total wavenumber, an it epens on the instrinsic horizontal phase velocity an the state o the backroun 2o5

3 L2383 ZHOU AND MORTON: GRAVITY WAVE PROPAGATION L2383 temperature an backroun win is relatively small in a vertical wavelenth, the so-calle WKB solution o euation (7) is ~w ðþ z w R p ii e i z z ðþ where w o is a constant an z is a reerence heiht. In the ollowin iscussion, we will assume the above euation is applicable an is larely inepenent o z. 3. Discussion [] It is o interest to point out that unction, as expresse in euation (6a), etermines how the wave amplitue chanes uner the conition that 2 is larer than an not a stron unction o z. With the einition o v 2 iven in the above, the expression or can be expresse as, ¼ H v 2 v 2 ð2þ The amplitue o the vertical win perturbation as a unction o z becomes wz ðþ/ p ii e R z 2 z H þ ln v2 ¼ w ðz o Þ v ðþ z piii e 2 R z z H ð3þ Fiure. (a) Eect o backroun win chane on the vertical wavenumber. Horizontal axis is the intrinsic horizontal phase velocity relative to the spee o the soun. Vertical axis is the vertical wavenumber suare. Soli an otte lines are or isothermal an constant backroun win. Lines with symbols are backroun win moiications to the suare o the vertical wavenumber by assumin U =.2/s an 2 U =4.2 6 m s. (b) Eect o temperature or scale 2 heiht chane on the vertical wavenumber. Soli line is the same as in Fiure a. Lines with symbols are temperature moiications to the suare o the vertical wavenumber by assumin =.5, 2 H =3. 5 m, an U 2 =.2/s. atmosphere. The backroun win aects the wave propaation characteristics throuh its eect on phase velocity. For brevity, we will use phase velocity to mean intrinsic horizontal phase velocity, v rom here on. [] Plane wave solution reuires that be inepenent o altitue z. Strict inepenence is iicult uner the assumption that backroun win an temperature are both varyin with z. Nevertheless, i the atmospheric variation in [2] It is well R z unerstoo that the exponential increase in amplitue (e 2 z H ) is to balance the atmospheric ensity chane. Our result shows that the vertical win amplitue has a relatively simple expression or heiht varyin backroun win an temperature as well. I the total eect ro iiiiiiiiiiiiiiiiiiiiiiiii the temperature an backroun win variation makes c 2 v 2 = a weak unction o altitue, the vertical win amplitue, as in an isothermal an winless atmosphere, rows exponentially. Gravity waves cannot have a vertical velocity component at a (horizontal) phase velocity o the spee o the soun. This is the same conclusion that one can raw rom the ravity wave polarization relations an it is a characteristic o the Lamb waves. [3] To see how vertical wavenumber is aecte by the heiht variation o the backroun win an temperature, let us assume U =.5, =.2/s, which are approximately hal o the values beore instability occurs in the mesosphere. (Since it is only the projection o the backroun win in the irection o wave propaation that matters, we will assume that backroun win is aline with the propaation irection in the ensuin iscussion.) Let us urther assume that the backroun win an temperature are sinusoial unctions o heiht with a vertical wavelenth o 3 km. This puts values o 2 H an 2 U 2 at 3. 5 m an m s, respectively. Usin 2 these values an assumin H = 6 km an = 9.5 ms 2, we plot how each term in euation (9) varies with the phase velocity in Fiures a an b. In orer or a ravity wave to reely propaate in the vertical irection, 2 nees to be larer than zero. The soli black line is the irst term in euation (9), which is the suare o the total 3o5

4 L2383 ZHOU AND MORTON: GRAVITY WAVE PROPAGATION L2383 wavenumber in the absence o temperature or win raient. The otte line is the secon term in euation (9) by assumin a horizontal wavelenth o 4 km an its sin is always neative. The terms associate with ( U o )2 an ( )2 are always neative as well. The linear st an 2n erivative terms can be either positive or neative. [4] For the conitions iven above, we see, rom Fiure a, that the 2 U term ominates the U 2 an ( U )2 terms or small phase velocities. In orer or the (win) shear term to be about the same as the curvature term, the euivalent vertical wavelenth o the backroun win nees to be loner than 2pH/(2 ), which is about 5H or =.4. The ( U )2 term ominates when the phase velocity excees.6c. The shear term ominates only when U is small an the backroun wavelenth is very lare. With U <.2/s, a ravity wave havin a horizontal wavelenth larer than 2 km (corresponin to k 2 < 7 m 2 ) will likely be able to propaate with a real vertical number in the reion where the phase velocity is smaller than.2c. For the iven conitions, a ravity wave havin a phase velocity larer than.4c is likely evanescent since the ( U )2 term, which is always neative, excees w 2 v 2 4H. 2 [5] In Fiure b, we see that the linear variation o the scale heiht (or euivalently temperature) ominates other temperature variation relate terms when the phase velocity is below.5c. In a stable atmosphere an with v <.5c, ominance o the term over ( )2 an ( U ) terms is always true. Uner the same conition, term will also ominate the 2 H term as lon as the euivalent vertical wavelenth o the 2 backroun temperature is larer than about one scale heiht. [6] In the above, we iscusse how the heiht raients o temperature an backroun win aect 2. Plane wave solution, euation (), epens on the conition that the manitue o the ollowin secon orer resiue, R 2,is much smaller than unity [Einaui an Hines, 97]: In terms o 2, this conition is R 2 ¼ ð4þ jr 2 j ¼ ð5þ One can take the irst an secon orer erivatives o euation (9) an substitute them into euation (5) to see how the temperature an backroun win variations aect the above conition. Such an exercise or a eneral case involvin both temperature an win variations will prouce complicate euations. Einaui an Hines [97] iscusse the case o temperature variation in etail. Here we consier the isothermal atmosphere with j v H v jan the secon orer erivative neliible. For this simpliie case, the conition in euation (5) becomes v v 6 5 w 2 2 v 2 v 2 w 2 ð6þ [7] Finally, we note that Taylor-Golstein euation is oten use to iscuss ravity wave propaation an uctin. Since it is erive uner the assumption o an incompressible atmosphere ( = ), it is o interest to compare it with our result here or a compressible atmosphere. Taylor- Golstein euation [e.., Nappo, 22] is 2 ¼ k 2 þ w2 v 2 4H 2 þ 2 U 2 U v H v ð7þ With = an v 2 c 2, our euation or a compressible atmosphere is 2 ¼ k 2 þ w2 v 2 4H 2 þ 2 U 2 þ 2 v U H Usin =.4, our euation becomes, 2 ¼ k 2 þ w2 v 2 4H 2 þ 2 U 2 þ :43 U v H v v ð8þ ð9þ We see that the Taylor-Golstein euation is only vali at slow phase velocities or a compressible atmosphere. For v 2 c 2, in aition to the act that the w einition epens on, the shear terms are opposite in sin an the manitue o the shear term in the Taylor-Golstein euation is about twice o our result. I the shear term ever plays a siniicant role in moiyin the propaatin characteristics o a ravity wave, past uantitative conclusions rawn rom Taylor-Golstein euation are likely incorrect. 4. Conclusions [8] We present the ispersion relation o ravity waves or a compressible atmosphere with temperature an backroun win variation usin linearize euations. In its eneral orm, there are seven terms associate with the temperature an win variations: linear irst/secon orer erivatives an uaratic o irst orer erivative or each o the two variables, an the prouct o the two irst orer erivatives. The coeicients o the uaratic terms are neative, implyin that a ravity wave cannot propaate reely throuh a iscontinuous bounary. The ispersion relation contains previously known ispersion relations erive uner more restricte assumptions. Horizontal phase velocity in the neutral win rame plays a key role in eterminin how the raient terms aect the vertical wavenumber. The hiher the intrinsic horizontal phase velocity, the more iicult it is or a ravity wave to propaate in the vertical irection. For moerate win an temperature variations, ravity waves havin an intrinsic horizontal phase velocity larer than.5c are likely to become evanescent. For ravity waves havin a relatively slow intrinsic horizontal phase velocity (v <.5c), 2 U term ominates U 2 an (U )2 terms i the euivalent vertical wavelenth o the backroun win is less than 5H. For temperature variation, the term in eneral ominates all other temperature variation terms. We also show that ravity verticalr iiiiiiiiiiiiiiiiiiiiiiiii win amplitue rowth is also proportional to c 2 v 2 = in aitional to the exponential term or an isothermal an winless atmosphere. 4o5

5 L2383 ZHOU AND MORTON: GRAVITY WAVE PROPAGATION L2383 [9] Acknowlements. The work presente in this paper is supporte by NSF rants ATM an ATM We thank the reviewers or their careul reains o the manuscript an constructive comments. Reerences Beer, T. (974), Atmospheric Waves, John Wiley, New York. Einaui, F., an C. O. Hines (97), WKB approximation in application to acoustic-ravity waves, Can. J. Phys., 48, Fritts, D. C., an M. J. Alexaner (23), Gravity wave ynamics an eects in the mile atmosphere, Rev. Geophys., 4(), 3, oi:.29/2rg6. Golstein, S. (932), On the instability o superpose streams o luis o ierent ensities, Proc. R. Soc. Lonon, Ser. A, 32, Hecht, J. H. (24), Instability layers an airlow imain, Rev. Geophys., 42, RG, oi:.29/23rg3. Hines, C. O. (96), Internal atmospheric ravity waves at ionospheric heihts, Can. J. Phys., 38, Nappo, C. (22), An Introuction to Atmospheric Gravity Waves, Acaemic, New York. Taylor, G. I. (93), Eect o variation in ensity on the stability o superpose streams o lui, Proc. R. Soc. Lonon, Ser. A, 32, Y. T. Morton an Q. Zhou, Electrical an Computer Enineerin Department, Miami University, Oxor, OH 4556, USA. (zhou@muohio. eu) 5o5