Generation Mechanisms of Convectively Forced Internal ...

1960 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

q 2003 American Meteorological Society

Generation Mechanisms of Convectively Forced Internal Gravity Waves and TheirPropagation to the Stratosphere

IN-SUN SONG AND HYE-YEONG CHUN

Department of Atmospheric Sciences, Yonsei University, Seoul, South Korea

TODD P. LANE

National Center for Atmospheric Research, Boulder, Colorado

(Manuscript received 13 November 2002, in final form 18 March 2003)

ABSTRACT

Characteristics of gravity waves induced by mesoscale convective storms and the gravity wave sources areinvestigated using a two-dimensional cloud-resolving numerical model. In a nonlinear moist (control) sim-ulation, the convective system reaches a quasi-steady state after 4 h in which convective cells are periodicallyregenerated from a gust front updraft. In the convective storms, there are two types of wave forcing: nonlinearforcing in the form of the divergences of momentum and heat flux, and diabatic forcing. The magnitude ofthe nonlinear source is 2 to 3 times larger than the diabatic source, especially in the upper troposphere. Threequasi-linear dry simulations forced by the wave sources obtained from the control (CTL) simulation areperformed to investigate characteristics of gravity waves induced by the various wave source mechanisms.In the three dry simulations, the magnitudes of the perturbations produced in the stratosphere are comparable,yet much larger than those in the CTL simulation. However, the sum of the quasi-linear perturbations generatedby the nonlinear and diabatic sources compare well with the mesoscale circulations and gravity waves in theCTL simulation.

Through the spectral analysis, it is found that the stratospheric gravity waves in all simulations have similarvertical wavelengths (6.6–9.9 km), horizontal wavelengths (10–100 km), and periods (8–80 min). Also, themagnitudes of gravity waves in the quasi-linear dry simulations are comparable with each other in spite ofthe differences in the magnitude of the nonlinear and diabatic sources. This is because the vertical propagationcondition of linear internal gravity waves, in both the troposphere and stratosphere, restricts wave sourcesin the horizontal wavenumber (k) and frequency (v) domain, and therefore all of the forcing cannot generategravity waves that can propagate up to the stratosphere. Compared with the diabatic sources, the nonlinearsources are inefficient in generating linear gravity waves that can propagate vertically into the stratosphere.These results suggest that wave generation mechanisms cannot be accurately understood without examiningthe vertical propagation condition of the gravity waves. Also, the ‘‘effective’’ wave sources are of comparablemagnitude, yet mostly out of phase. Therefore, although the wave amplitudes produced by simulations withnonlinear forcing and diabatic forcing are about 2 to 3 times too large, their sum compares well to the controlsimulation.

1. Introduction

It has been recognized that vertically propagatinginternal gravity waves have profound effects on thelarge-scale circulation of the middle atmosphere (e.g.,Lindzen 1981; Holton 1982; Matsuno 1982). The pa-rameterization of orographically generated gravitywave drag can successfully alleviate excessive west-erly bias appearing in the upper troposphere and lowerstratosphere of the Northern Hemisphere wintertime inlarge-scale numerical models (e.g., Palmer et al. 1986;

Corresponding author address: Prof. Hye-Yeong Chun, Dept. ofAtmospheric Sciences, Yonsei University, Shinchon-dong, Seodae-mun-ku, Seoul 120-749, South Korea.E-mail: [email protected]

McFarlane 1987). In the atmosphere, there are variousgravity wave sources generating nonstationary gravitywaves relative to the ground other than orography gen-erating (stationary) waves. Fritts (1984) noted that alarge portion of gravity waves observed in the middleatmosphere consist of high frequencies. Bergman andSalby (1994) showed that the upward propagation ofenergy and momentum, in the tropical stratosphere, ismainly associated with gravity waves with frequencieshigher than 0.5 cycles day21 . Recently, cumulus con-vection has received a great deal of attention as a pos-sible source of the nonstationary gravity waves, es-pecially in the Tropics where persistent cumulus cloudsexist.

There have been several numerical modeling studiesof convectively generated gravity waves and their ef-

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15 AUGUST 2003 1961S O N G E T A L .

fects on the large-scale circulation of the middle at-mosphere. Alexander et al. (1995) investigated thespectral characteristics of internal gravity waves in-duced by squall lines. Their spectral analysis indicatedthat gravity waves with vertical wavelengths of 6–10km, horizontal wavelengths of 10–100 km, and intrin-sic periods of 10–60 min are dominant in the strato-sphere. Alexander and Holton (1997) showed that con-vectively generated gravity waves can provide aboutone-fourth of the momentum forcing required for driv-ing the quasi-biennial oscillation (QBO) in the tropicalstratosphere. Piani et al. (2000) examined an interac-tion between the typical zonal wind of the QBO andthe gravity waves induced by three-dimensional deepconvection. They found that the gravity wave momen-tum flux can accelerate the zonal-mean zonal wind byas much as 1.0 m s21 day21 and 20.3 m s21 day21 inthe QBO westerly and easterly phases, respectively,and induces the downward propagation of shear layersin both QBO phases.

Compared with orographic gravity waves, it is moredifficult to understand how gravity waves are generatedby convective sources. There are currently three pro-posed generation mechanisms (thermal forcing, obsta-cle, and mechanical oscillator mechanisms) for con-vective gravity waves. In the first mechanism, con-vective clouds are regarded as thermal forcings thatgenerate gravity waves (nonstationary or stationarywaves relative to the thermal forcing) in a stably strat-ified environment. Some analytic studies on the re-sponse of a stably stratified atmosphere to a prescribedthermal forcing (e.g., Lin and Smith 1986; Bretherton1988; Chun and Baik 1998; Baik et al. 1999) can beincluded in this category. Based on this mechanism,Chun and Baik (1998) proposed a parameterization ofgravity waves induced by cumulus convection(GWDC) for stationary waves relative to forcing re-gion. Chun et al. (2001b) investigated the effects ofGWDC on large-scale circulation using the YonseiUniversity atmospheric general circulation model withthe GWDC parameterization proposed by Chun andBaik (1998). Recently, Chun and Baik (2002) updatedChun and Baik (1998)’s parameterization using morerealistic environmental conditions, including basic-state wind shear and stability difference between theconvective region and above. In the obstacle mecha-nism, convective clouds are considered obstacles to theflow, and gravity waves are generated when the flowis subsequently blocked. This mechanism was pro-posed by Clark et al. (1986), supported by observationsabove the convective boundary layer by Kuettner etal. (1987), and examined in numerical simulations byHauf and Clark (1989). Pfister et al. (1993) appliedthis idea to time-dependent clouds, which were con-ceptually called moving mountains. Recently, Beres etal. (2002) used this obstacle mechanism, in their squallline simulations with tropospheric shear, to explain theenhancement of the momentum fluxes of gravity waves

propagating in the opposite direction of the tropo-spheric wind shear. In their simulations, the thermalforcing mechanism, as well as the obstacle mechanism,is also responsible for the generation of gravity wavesbecause nonsteady diabatic wave forcings related totransient convective cells exist in their simulated squallline. Finally, the mechanical oscillator mechanism wasfirst proposed by Pierce and Coroniti (1966). Fovell etal. (1992) explained the characteristics of gravitywaves in the stratosphere induced by the midlatitudesquall lines through simple numerical simulationsbased on mechanical oscillator arguments. The basisof this mechanism is that strong convective updraftsin convective storms stimulate the stable stratosphereand generate gravity waves above clouds. Convectivecells acting as mechanical oscillators, as well as a non-steady diabatic forcing, can generate gravity waveseven under zero background wind relative to the con-vective cells. Alexander et al. (1995) showed that thedominant period of gravity waves obtained from theirspectral analysis corresponds to the regeneration periodof convective cells, and accounted for their gravitywave spectra using this mechanical oscillator mecha-nism. Lane et al. (2001) simulated a three-dimensionalmulticellular convective system in the Tropics and an-alyzed the sources of linear forced wave equation inorder to examine the generation mechanism for con-vectively generated gravity waves. They showed thatwave sources associated with nonlinear advection rath-er than latent heat release are dominant in their sim-ulated convective system and emphasized the mechan-ical oscillator mechanism because the nonlinear wavesources are strong near the level of neutral buoyancy,through which strong convective updrafts overshoot.

In this study, we examine characteristics of internalgravity waves in the stratosphere induced by mesoscaleconvective storms and their generation mechanisms us-ing a two-dimensional cloud-resolving model. Twotypes of numerical simulations are performed: a non-linear moist simulation and three quasi-linear dry sim-ulations forced by wave sources associated with thenonlinear and diabatic processes in the nonlinear moistsimulation. In section 2, the numerical model used inthis study is described. In section 3, results from thenonlinear moist simulation are presented. In section 4,the sources of linear forced internal gravity waves inthe nonlinear moist simulation are analyzed. In section5, the three quasi-linear dry simulations forced by thewave sources are described, and results from those drysimulations are presented. In section 6, the character-istics of the gravity waves in the stratosphere simulatedin the nonlinear moist and three quasi-linear dry sim-ulations are analyzed using spectral analysis and thelinear internal gravity wave theory. The summary andconclusions are presented in the last section.

2. The numerical modelThe numerical model used in this study is the Ad-

vanced Regional Prediction System (ARPS; Xue et al.



FIG. 1. The basic-state (a) Brunt–Vaisala frequency and (b) zonal wind in astorm-relative frame.

1995), which is a nonhydrostatic, compressible modelwith explicit liquid–ice phase cloud microphysical pro-cesses. In this study, two-dimensional (x–z) simulationsare performed, with the ice phase processes and therotational effects of the earth ignored. In the ARPS,horizontal wind and thermodynamic state variables aredefined as the sum of horizontally homogeneous andtime-invariant basic-state variables and perturbationvariables. The basic states for vertical wind, cloud water,and rainwater are assumed to be zero. The two-dimen-sional prognostic equations are

]ru ]u ]u ]p95 2 ru 1 rw9 2 1 D , (1)u1 2]t ]x ]z ]x

]rw9 ]w9 ]w9 ]p95 2 ru 1 rw9 2 1 rB 1 D , (2)w1 2]t ]x ]z ]z

]ru9 ]u9 ]u9 du5 2 ru 1 rw9 2 rw91 2]t ]x ]z dz

1 Q 1 D , (3)u

]p9 ]p9 ]p9 ]u9 ]w925 2 u 1 w9 1 rgw9 2 rc 1 ,s1 2 1 2]t ]x ]z ]x ]z

(4)

]rq ]q ]q ]5 2 ru 1 rw9 1 (rV q) 1 S 1 D , (5)q q q1 2]t ]x ]z ]z

where u 5 U(z) 1 u9 is the zonal wind; U(z) and u9 are

the basic-state and perturbation zonal wind; w9, u9, andp9 are the perturbations of vertical wind, potential tem-perature, and pressure, respectively; and and are ther ubasic-state density and potential temperature, respective-ly. In (2) the buoyancy term B is given as 2gr9/ , whererg is the gravitational acceleration; and r9 is the densityperturbation. In (4), cs(5 ) is the speed of sound,ÏgRTwhere g, R, and are the ratio of the specific heat ofTair at constant pressure and volume, the gas constant fordry air, and the basic-state temperature, respectively. In(5), q represents the mixing ratio of water vapor, cloudwater, or rainwater. The sedimentation term, Vq (the ter-minal velocity of rainwater), is nonzero only when q isa rainwater mixing ratio. Here Dc represents turbulentand background mixing for any prognostic variable c.For the subgrid-scale turbulence parameterization, thefirst-order closure by Lilly (1962) and Smagorinsky(1963) is used. For the background mixing used to sup-press numerical noise, the second- and fourth-orderschemes are used in the vertical and horizontal, respec-tively. The vertical and horizontal background mixingcoefficients are 0.002 and 0.004 s21, respectively. Notethat Q and Sq represent the diabatic forcing source orsink for water vapor, cloud water, or rainwater due tomicrophysical processes.

The model domain is 1200 km wide and 51 km deep.Horizontal and vertical grid sizes are 1 and 0.3 km,respectively. A sponge layer with a depth of 15 km isincluded in the uppermost part of the model domain. Atthe left boundary, a radiation condition based on Or-


15 AUGUST 2003 1963S O N G E T A L .

FIG. 2. The w9 5 2 m s21 contour (thick) superimposed on total potential temperaturefield at (a) t 5 3 h and (b) t 5 5 h in the CTL simulation. The contour interval fortotal potential temperature is 7 K.

lanski (1976; with a vertically averaged outgoing phasespeed) is used. At the right boundary, a sponge layerwith a width of 100 km is included to prevent wavereflection detected in long-term simulations (Chun et al.2001a). The ARPS uses the mode-splitting time inte-

gration presented by Klemp and Wilhelmson (1978) toeffectively calculate acoustically active terms in thegoverning equations of the compressible model. In allsimulations, the large and small time steps are 3 and 1s, respectively.



FIG. 3. (a) Time series of domain maximum vertical velocity inthe CTL simulation and (b) power spectral density calculated using(a) after t 5 6 h. Significant peaks are indicated in numeric valuesin min.

3. Nonlinear moist simulation

In the nonlinear moist (control) simulation, convec-tion is initiated by an ellipsoidal warm bubble with amaximum potential temperature perturbation of 2 K cen-tered at x 5 800 km and z 5 1.5 km. The time inte-gration for the control (CTL) simulation is carried outfor 12 h. The analytic sounding of Weisman and Klemp(1982) is used for the thermodynamic basic state. Thissounding has a convective available potential energy(CAPE) of 2436 J kg21. The Brunt–Vaisala frequencycalculated from this sounding is about 0.01 s21 in thetroposphere and 0.021 s21 in the stratosphere (Fig. 1a).The basic-state wind is assumed to have a constant shearbelow z 5 6 km and be uniform above it (Fig. 1b). Thisbasic-state wind is relative to a gust front propagatingeastward with a speed of 18 m s21. Thus, the resultsshown hereafter are in the reference frame moving withthe gust front, and therefore the convective system isconfined to the model domain for the long time inte-gration.

Figure 2 shows the contours of vertical velocity of 2m s21 and total potential temperature fields at t 5 3 hand t 5 5 h. In the CTL simulation, the convectivesystem reaches a quasi-steady state after t 5 4 h witha periodic cell regeneration. Convective cells are tilteddownshear before t 5 3 h, become upright at t 5 3 h,and are tilted upshear after t 5 4 h. This can be explainedby vorticity balance between the vertical shear of thebasic-state wind and the cold pool induced by the evap-oration of rainwater (Rotunno et al. 1988). Before t 53 h, the positive vorticity induced by the basic-statewind with a positive shear dominates the negative vor-ticity induced by the cold pool, because rainwater is notsufficient to produce a strong enough cold pool. In thequasi-steady state after t 5 4 h, the evaporative coolinggradually strengthens due to the increase of rainwatermixing ratio and dry inflow from the rear side of theconvective system. As a result, the vorticity generatedby the cold pool begins to overcome the positive vor-ticity of the basic-state wind shear, and convective cellsare tilted upshear. As the simulated convective systemreaches the quasi-steady state, the structure of convec-tively generated internal gravity waves in the strato-sphere also begins to change. At t 5 3 h, upright con-vective cells induce gravity waves propagating west-ward and eastward with comparable magnitude. On theother hand, at t 5 5 h, the westward-propagating gravitywaves are dominant in the stratosphere due to the con-vective cells propagating rearward (westward in the pre-sent case) relative to the gust front located near x 5770 km (Fig. 2b). This result is consistent with Fovellet al. (1992).

Figure 3 shows the time series of domain maximumvertical velocity and its power spectral density (PSD)calculated after t 5 6 h. In the quasi-steady state, theaverage value of domain maximum vertical velocity isabout 12 m s21, and its fluctuation is quite regular after

t 5 7 h. Before calculating the PSD, a trend with time-scale larger than the maximum period resolvable in thetime series is removed by fitting a second-order poly-nomial. In addition, a Welch window (Press et al. 1992)is applied to the time series to prevent energy leakageat each frequency to neighboring frequencies in thePSD. Using the time series modified by the conditioningprocesses, the PSD is calculated as follows:

2Dt2PSD( f ) 5 |FFT(w9v )( f . 0)| , (6)j jvss

where Dt (560 s) is the time interval in the time series;FFT is the fast Fourier transform, v j is the Welch win-dow function given by 1 2 [(j 2 N/2)/N/2]2, j 5 0,. . . , N 2 1, where N is the number of data; and vss 5

. In the PSD (Fig. 3b), there is a primary peakN21 2S vj50 j

at 18.1 min and a secondary peak at 45.1 min. From t5 4 to 12 h, most of the domain maximum verticalvelocities coincide with the maximum updrafts of con-vective cells being separated from the gust front, andtherefore the primary periodicity of 18.1 min is asso-


15 AUGUST 2003 1965S O N G E T A L .

FIG. 4. Forcing terms (a) Fu, (b) Fw, (c) Fu, and (d) Q at t 5 6 h. Contour interval for Fu and Fw (Fu and Q) is 0.005 kg m22 s22 (0.005kg K m23 s21). Negative values are plotted with dashed lines.

ciated with the convective cell regeneration. However,in the period from t 5 6 h to 6 h, 45 min, the cellregeneration cannot be seen in the domain maximumvertical velocity, although periodic cell regenerationsstill exist (not shown). This is because the periodicallygenerated convective updrafts are weaker than the gustfront updraft during this period.

In the next section, the wave sources that generatethe stratospheric internal gravity waves in the CTL sim-ulation, shown in Fig. 2, are analyzed. As mentionedby Raymond (1975) and Lane et al. (2001), there aregenerally two types of wave sources in a convectivesystem. These sources are diabatic forcing due to latentheating and cooling and nonlinear forcing associatedwith strong perturbations in the convective system. Be-cause the mechanisms for generating the convectivelygenerated gravity waves are closely related to thosewave sources, it is necessary to understand their char-acteristics before analyzing the gravity waves in thestratosphere.

4. Gravity wave sourcesFor the calculation of the gravity wave forcings, it is

assumed that the compressible flow simulated in theARPS can be approximated by the anelastic flow (e.g.,Ogura and Phillips 1962; Wilhelmson and Ogura 1972).The validity of such an approximation can be examinedthrough a scale analysis of the prognostic equation forp9 given in (4). For the scale analysis, (4) can be re-written as

1 ]p9 ]p9 p9 ]u9 ]w91 U 2 1

2 21 2 1 2c ]t ]x c ]x ]zs s

rg ]u9 ]w92 w9 1 r 1

2 1 2c ]x ]zs| | | |

| |A B

1 ] ]5 2 (p9u9) 1 (p9w9) . (7)

2 [ ]c ]x ]zs



FIG. 5. PSDs of the forcing terms Fu, Fw, Fu, and Q as functionsof (a) vertical wavelength, (b) horizontal wavelength, and (c) period.The PSDs of Fu, Fw, Fu, and Q are plotted with solid, dashed, dotted,and dashed–dotted lines, respectively. Tick labels for Fu and Fw (Fu

and Q) is plotted at left- (right) hand sides of each panel, respectively.All PSDs are plotted in area-preserving forms.

When the terms in (7) are calculated in the CTL sim-ulation, the terms A and B are from 10 to 100 timeslarger than the other terms. Note that the term A canbe rewritten as ( g/ )w9 5 2[(d /dz) 1 (d ln /dz)]w9,2r c r r us

where the magnitude of d ln /dz is about 10% andr u40% of that of g/ in the troposphere and stratosphere,2r cs

respectively. Therefore, d ln /dz cannot be ignored inr uthe strongly stratified stratosphere. Durran (1989) crit-icized the anelastic approximation based on the adia-batic basic-state stratification and developed an im-proved approximation to the compressible flow calleda pseudo-incompressible system. However, providedmost of gravity wave sources are in the tropospherewhere the basic-state stratification is weak comparedwith the stratosphere, the anelastic system can be usedin the calculation of gravity wave forcing. As a resultof the scale analysis based on the anelastic approxi-mation, the prognostic equation for p9 becomes a well-known anelastic continuity equation:

] ](ru9) 1 (rw9) 5 0. (8)

]x ]z

Under the anelastic approximation and by separatinglinear and nonlinear terms, (1)–(3) in the dry and in-viscid atmosphere can be rewritten as

Du9 dU ]f9r 1 rw9 1 r 5 F , (9)uDt dz ]x

Dw9 u9 ]f9r 2 gr 1 r 5 F , (10)wDt u ]z

Du9 dur 1 rw9 5 F 1 Q, (11)uDt dz

where f9 5 p9/ and /Dt 5 ]/]t 1 U]/]x. In (9)–r D(11), the forcing terms Fu, Fw, Fu, and Q are expressedas

] ]F 5 2 (ru9u9) 1 (ru9w9) , (12)u [ ]]x ]z

] ]F 5 2 (rw9u9) 1 (rw9w9) , (13)w [ ]]x ]z

] ]F 5 2 (ru9u9) 1 (ru9w9) , (14)u [ ]]x ]z

Q 5 r[H 1 C], (15)

where H and C are diabatic heating and cooling ratesdue to microphysical processes, respectively.

In (12)–(15), Fu and Fw are the divergences of per-turbation Reynolds stress, Fu is the divergence of per-turbation heat flux, and Q is the diabatic forcing. Theseforcing terms are generally confined to being importantin small regions of strong convective activity and actas forcing for the linear momentum and thermodynamicenergy equations. The forcing terms Fu, Fw, and Fu are

calculated using perturbation variables saved at a 1-mininterval in the CTL simulation, and the diabatic forcingterm Q is calculated using diabatic heating and coolingrates obtained at a 1-min interval from the saturationadjustment and rain evaporation process in the CTLsimulation. These forcing terms at t 5 6 h are shownin Fig. 4. The magnitude of the forcing terms above z5 15 km is much smaller than below z 5 15 km. Belowz 5 15 km, strong forcing is concentrated in the con-vectively active region (720 km # x # 770 km), behindthe gust front updraft located at x 5 770 km. In thestratiform cloud region (x # 720 km), the forcing ismuch weaker than in the convectively active region andappears to be related to weak convective cells propa-gating rearward in the upper troposphere. For a quan-titative analysis of the spatial and temporal structuresof the forcing terms, their PSDs as functions of vertical


15 AUGUST 2003 1967S O N G E T A L .

and horizontal wavelengths and period are calculated(Fig. 5). The domain used for the spectral analysis is1100 km wide from x 5 0 to 1100 km and 15 km deepfrom z 5 0 to 15 km. The forcing terms calculated every1 min from t 5 4 to 12 h are used. The PSDs as functionsof vertical and horizontal wavelengths and period areaveraged over x–t, z–t, and x–z surfaces, respectively.The PSDs are plotted in area-preserving forms. All forc-ing terms have spectral peaks at periods of 9.25–9.62and 16.58–18.5 min. However, the dominant spatialscales of the forcings are different from each other: Fu,Fw, and Q have peaks at the vertical wavelength of 7.5km, whereas Fu has a peak at 5.0 km. In the horizontalwavelength spectra, Fw is dominant only in the regionnear 12 km; Fu, Fu, and Q have peaks at the horizontal

wavelengths larger than 25 km as well as near 12 km.Particularly, the peak of Q at a horizontal wavelengthof 47.8 km is much larger than that near 12 km. In thisstudy, the forcing terms above z 5 15 km are ignoredbecause the objective of this study is to investigate thegravity waves generated by the tropospheric convection,which is mostly confined below z 5 15 km. Even thoughthe potential temperature (Fig. 2) showed that theboundary between the troposphere and stratosphere isnear z 5 12 km, the tropospheric wave forcings arecalculated below z 5 15 km since convective cells pen-etrate above z 5 12 km frequently during the integra-tion.

Following Lane et al. (2001), (8)–(11) can be writtenas a single equation for w9:

2 2 2 2D ] w9 ] 1 ] d U D ]w9 dU D 1 dr ]w9 ] w921 (rw9) 2 1 1 N

2 2 2 25 6 1 2 1 2[ ]Dt ]x ]z r ]z dz Dt ]x dz Dt r dz ]x ]x

2 2 2 2 2] D F ] D F g ] F g ] Q dU ] Fw u u u5 2 1 1 1 ,2 2 2 21 2 1 2 1 2 1 2 1 2[ ] [ ]]x Dt r ]x]z Dt r u ]x r u ]x r dz ]x r

(16)| | | | | || | |A B C

where N is the Brunt–Vaisala frequency. Equation (16)implies that the generation of linear gravity waves isrelated to the nonlinear and diabatic processes withinconvection. These nonlinear terms will be large withinthe cloud, and negligible some distance away from thecloud, and therefore form a compact source of gravitywaves. This wave generation is analogous to Lighthill’s(1952) radiation tensor. Lane et al. (2001) grouped thosewave sources into three terms—A, B, and C—accordingto the related physical processes (nonlinear advection,diabatic heating and cooling, and the effects of envi-ronmental wind shear), respectively. Consistent withLane et al., the term involved in shear generation C isabout an order of magnitude smaller than the terms in-volved in A and B in the region of nonzero verticalshear below z 5 6 km. Nevertheless, the low-level shearis important in maintaining the convective system andaffecting the propagation of lower-tropospheric waves.The effects of the wind shear on the vertically propa-gating gravity waves will be examined in section 6.

5. Quasi-linear dry simulations

In order to investigate characteristics of linear internalgravity waves induced by the wave sources given in(16), three quasi-linear dry simulations (DRYMH,DRYM, and DRYQ) are performed. The methodologyof this quasi-linear dry simulation is almost the sameas that in Pandya and Alexander (1999). The magnitudes

of Fu, Fw, Fu, and Q, calculated from the CTL simu-lation, are reduced by factor of 10 000. These reducedforcing terms are applied to the governing equations toelicit a quasi-linear response in the nonlinear numericalmodel, and they will induce small-amplitude perturba-tions that will behave similarly to those in a forced linearmodel. In these quasi-linear dry simulations, the modelnumerics, configurations related to physical processes,and basic-state wind and thermodynamic structures arethe same as those in the CTL simulation except thatcloud microphysical processes are ignored. Thus, thecomparison between the nonlinear simulation and theforced quasi-linear simulations is probably more appro-priate than the same comparisons with a forced linearmodel. In the quasi-linear dry simulations, those forc-ings are added in the right-hand sides of the prognosticequations given in (1)–(3). The DRYMH simulation isforced by Fu, Fw, and Fu; the DRYM simulation forcedby Fu and Fw; and the DRYQ simulation forced by Q.In those quasi-linear dry simulations, the forcings belowz 5 15 km are imposed, and the analysis of gravitywaves is carried out above z 5 15 km. All quasi-lineardry simulations are carried out for 12 h. For time in-tegration, the nonlinear and diabatic forcings at eachlarge time step (3 s) are linearly interpolated from theforcings calculated at a 1-min interval.

Figure 6 shows the wave sources in the DRYMH,DRYM, and DRYQ simulations at t 5 6 h. The wavesources in Figs. 6a–c are



FIG. 6. Wave sources in the (a) DRYMH, (b) DRYM, and (c) DRYQ simulations at t 5 6 h. Contour intervalis 0.1 3 1029 m21 s23 for each panel. Negative values are plotted with dotted lines.

2 2 2] D F ] D F dU ] Fw u u2 12 21 2 1 2 1 2[ ] [ ]]x Dt r ]x]z Dt r dz ]x r

2g ] Fu1 , (17)21 2u ]x r

2 2 2] D F ] D F dU ] Fw u u2 1 , (18)2 21 2 1 2 1 2[ ] [ ]]x Dt r ]x]z Dt r dz ]x r

and

2g ] Q, (19)

21 2u ]x r

respectively. Strong wave sources are concentrated inthe convectively active region (720 km # x # 770 km).The horizontal and vertical scales of each source areless than 10 and 6 km, respectively. The horizontalscales of these sources are considerably reduced com-pared with that of the forcings shown in Fig. 5b. Thisis due to the second- or higher-order horizontal deriv-


15 AUGUST 2003 1969S O N G E T A L .

FIG. 7. Perturbation vertical velocity fields in the (a) CTL, (b) DRYMH, (c) DRYM, and (d) DRYQ simulations at t 5 6 h. Contourinterval for the CTL simulation is 1 m s21 and for the other simulations is 2.0 m s21, with negative values shaded. The perturbation verticalvelocities in the DRYMH, DRYM, and DRYQ simulations are multiplied by factor of 10 000 for comparison with the CTL simulation.

atives of these forcings. Overall, the sources in the con-vectively active region in the quasi-steady state are tiltedupshear because they are strong in the ascending flowtoward the rear side of the convective system from thegust front updraft. The diabatic source is tilted upshearin the convectively active region (Fig. 6c), while thenonlinear sources are locally tilted downshear, espe-cially in the mid- to upper troposphere (Figs. 6a and6b). The spatial structure of the diabatic forcing is quitesimilar to that of convective cells, but the magnitude ofdiabatic source is 2 to 3 times smaller than the nonlinearsources. In the stratiform cloud region, nonlinear anddiabatic wave sources with horizontal scales less than10 km exist. However, the diabatic source in this regionis about 10 times smaller than the nonlinear sources.These differences in the magnitude and spatial structureof the nonlinear and diabatic wave sources persist duringthe quasi-steady state after t 5 4 h. On the other hand,in an initial stage before t 5 3 h, the magnitude of thenonlinear sources is comparable to the diabatic source,and their dominant spatial scales are almost the same(not shown). Considering that the periodic generationof strong convective updrafts is the result of the non-

linear flow dynamics (e.g., Baik and Chun 1996; Pandyaand Durran 1996), and the wave sources associated withFu and Fw are much stronger than those associated withQ, one can expect that the mechanical oscillator mech-anism might be important in generating convectivelyforced internal gravity waves. In fact, Lane et al. (2001)made the same conclusion based on the relative mag-nitude between the nonlinear and diabatic source terms.However, as will be shown later, stratospheric gravitywaves induced by each wave source in quasi-linear drysimulations are all similar in their spectral character-istics as well as magnitude.

Figure 7 shows the perturbation vertical velocity fieldin the CTL, DRYMH, DRYM, and DRYQ simulationsat t 5 6 h. The perturbation vertical velocities in thequasi-linear dry simulations are multiplied by factor of10 000 for comparison with the CTL simulation. Asalready mentioned in some previous studies (e.g., Pan-dya and Alexander 1999; Lane and Reeder 2001), themagnitude of perturbations in the quasi-linear dry sim-ulations is about 2 to 3 times larger than that in theCTL simulation. Also, as shown in Fig. 7, the spatialstructures of perturbation vertical velocities in the three



FIG. 8. Perturbation vertical velocity fields in (a) the CTL simu-lation and (b) DRYMH 1 DRYQ at t 5 6 h. (c) The differencebetween perturbation vertical velocities shown in (a) and (b). Contourinterval is 1.0 m s21 for each panel. Negative values are shaded.

quasi-linear simulations are also different from those inthe CTL simulation. In the troposphere, none of thequasi-linear simulations reproduce the perturbation ver-tical velocity field associated with the mesoscale cir-culation simulated in the CTL simulation. In the con-vectively active region, the perturbation vertical veloc-ities in the DRYQ and CTL simulations are mostly inphase with each other, while the perturbation verticalvelocities in the DRYMH and DRYM simulations areroughly out of phase with those in the CTL simulation.Particularly, in the DRYMH and DRYM simulations,there is strong downdraft near x 5 765 km, that is, wherethe gust front updraft is located in the CTL and DRYQsimulations. Unlike in the troposphere, the gravitywaves in the stratosphere in the DRYMH simulation arein phase with those in the CTL simulation, while thewaves in the DRYQ simulation are almost out of phasewith those in the CTL simulation. In the stratosphere,it is very interesting that the magnitudes of the gravitywaves in all of the quasi-linear dry simulations are al-most identical, although the magnitudes of the nonlinearwave sources are 2 to 3 times larger than that of thediabatic source (as shown in Fig. 6). In order to explainthis result, factors that control the spatial and temporalstructures and magnitude of quasi-linear perturbationsin the troposphere as well as in the stratosphere will beexamined in section 6.

Figure 8 shows the sum of the perturbation verticalvelocities in the DRYMH and DRYQ simulations (here-after, DRYMH 1 DRYQ). The perturbations inDRYMH 1 DRYQ are believed to be directly com-parable to perturbations in a single quasi-linear simu-lation forced by a combination of the nonlinear anddiabatic forcings used in the DRYMH and DRYQ sim-ulations, because the response of the atmosphere tothose forcings is quasi-linear. As can be seen in Fig. 8,the perturbations in the DRYMH 1 DRYQ compareswell to the stratospheric gravity waves, as well as thedetailed mesoscale and convective cell-scale circula-tions in the CTL simulation. The differences betweenvertical velocities shown in Fig. 8c can be regarded asinsignificant, considering that the wave forcings are de-rived under the anelastic assumption, and that the timevariation of the forcing terms on a subminute scale isassumed to be linear. There are, however, relatively largedifferences near z 5 30 km in the region of 650 km ,x , 700 km. In this region, strong wave perturbationsoccur after t 5 4 h in the CTL simulation, and thesemay give rise to nonlinear wave–wave interactions inthe stratosphere. Thus, the differences near z 5 30 km,shown in Fig. 8c, are probably caused by nonlinear pro-cesses such as wave–wave interaction that do not occurin the quasi-linear simulations. Pandya and Durran(1996) discussed the differences between linear and fi-nite amplitude perturbations forced by the diabatic forc-ing. They concluded that linear models fail to accuratelyrepresent the mesoscale circulations in the lower tro-posphere because they cannot support nonlinear pro-

cesses that modify the basic-state stability associatedwith the development of a cold pool. In the presentstudy, the perturbations in the nonlinear moist simula-tion are reasonably reproduced using a quasi-linear sim-ulation forced by both the time-varying nonlinear anddiabatic forcings in the troposphere. Thus, including thenonlinear wave forcings given in (12), (13), and (14),in addition to the diabatic forcing, is a crucial factor forsuccessful quasi-linear simulations of the circulationsand gravity waves.

It is interesting to note that the magnitude of the


15 AUGUST 2003 1971S O N G E T A L .

gravity waves in the DRYMH 1 DRYQ is close to thatin the CTL simulation, which is 2 to 3 times smallerthan that in either the DRYMH or DRYQ simulation.One can speculate that it is due to some cancellationbetween the forcing terms in the DRYMH and DRYQsimulations. However, considering that the nonlinearforcing terms are much stronger than the diabatic forc-ing (Fig. 6), the magnitude of the combined forcing isclose to the forcing in the DRYMH simulation, evenwith a complete cancellation of the diabatic forcing bysome part of nonlinear forcing. This result, along withFig. 7, implies that the magnitude of the linear gravitywaves in the stratosphere cannot be explained simplyby the magnitude of the imposed tropospheric forcingshown in Fig. 6. In section 6, we will show why thegravity waves in the DRYMH 1 DRYQ resemble thosein the CTL simulation.

6. Characteristics of gravity waves in thestratosphere

a. Spectral analysis of gravity waves

A spectral analysis of the stratospheric gravity wavessimulated in the CTL, DRYMH, DRYM, and DRYQsimulations is carried out to characterize convectivelygenerated gravity waves and understand the influenceof the different generation mechanisms. The quasi-steady convective system in the CTL simulation inducesthe westward- and eastward-propagating gravity wavesrelative to the (almost stationary) gust front updraft,located near x 5 770 km. In all simulations, the verticalvelocity fields saved at a 1-min interval from t 5 6 to12 h are used for the spectral analysis. The domain usedfor the spectral analysis is 900 km wide from x 5 170to 1070 km and 19.8 km deep above z 5 15 km. Asshown in Fig. 2, the westward-propagating waves aremuch stronger than the eastward-propagating waves inthe CTL simulation. As pointed out by Lane et al.(2001), it is, however, possible that the energy of theeastward-propagating waves is underestimated when thevertical velocities are used in the gravity wave analysis.This is because the intrinsic frequencies of the eastward-propagating waves are smaller than the westward-prop-agating waves on account of the Doppler shifting by thebasic-state wind (2 m s21 in the stratosphere). Therefore,the PSDs of the stratospheric gravity waves are calcu-lated separately in the western and eastern regions of x5 770 km. In the eastern region of x 5 770 km, west-ward-propagating waves are negligibly weak becausethere is no significant convective activity in the easternregion of gust front updraft in the quasi-steady stateafter t 5 4 h. However, in the western region of x 5770 km, there may be some eastward-propagating wavesgenerated by westward-moving convective cells in theconvective system. In order to examine the magnitudeof eastward-propagating waves in the western region ofx 5 770 km, phase speed spectra are calculated using

the perturbation vertical velocities of the stratosphericgravity waves in the region of 170 km , x , 770 km(not shown). The eastward-propagating waves in thewestern region of x 5 770 km are found to be negligiblyweak compared with the westward-propagating waves.Thus, the gravity waves in the western and eastern regionof x 5 770 km can be regarded as the westward- andeastward-propagating gravity waves in this study.

For the PSD of the gravity waves in the stratosphere,the preprocesses applied to the PSD in Fig. 3b are used,except a linear trend is removed from the the verticalvelocity field. In the quasi-linear dry simulations, thevertical velocities in the stratosphere are multiplied bya factor of 10 000 before the PSD calculation. The PSDsof the westward- and eastward-propagating waves, asfunctions of vertical and horizontal wavelengths andperiod, are averaged for x 2 t, z 2 t, and x 2 z surfaces,respectively. Because exponentially decreases withrheight, and the magnitude of upward-propagating grav-ity waves increases with height in proportion to 21/2,rthe vertical velocity perturbations are multiplied by

(z0 5 15 km) before calculating the PSDsÏr(z)/r(z )0

in order to prevent the bias of the PSDs toward thespectral wave characteristics at higher altitudes.

Figure 9 shows PSDs of the gravity waves as func-tions of vertical wavelength (lz); horizontal wavelength(lx); and period in the CTL, DRYMH, DRYM, andDRYQ simulations and in DRYMH 1 DRYQ. Resolv-able vertical wavelengths and periods range from 0.6 to19.8 km and 2 min to 6 h, respectively. Resolvablehorizontal wavelengths range from 2 to 600 km (300km) for the westward- (eastward) propagating waves.The dominant vertical wavelengths range roughly from6.6 to 9.9 km in all simulations regardless of the hor-izontal propagation direction. For the westward- (east-ward) propagating waves, the dominant horizontalwavelengths and periods are 20–100 (30–80) km and8–80 (15–50) min, respectively. In all quasi-linear drysimulations, the peak at 17.1 min associated with theconvective cell regeneration is dominant. The PSDs inthe DRYMH 1 DRYQ are also almost identical to thosein the CTL simulation, even though the gravity wavesin all quasi-linear dry simulations are generally muchstronger than in the CTL simulation. This will be ex-plained in section 6b.

Salby and Garcia (1987) showed that the verticalwavelength of large-scale waves induced by thermalforcing depends on the depth of thermal forcing ratherthan a detailed vertical structure of thermal forcing. Al-exander et al. (1995) related the dominant peaks at lz

5 6–10 km to the vertical scale of diabatic forcing insimulated convective storms. Recently, however, Holtonet al. (2002) showed that the horizontal wavenumberspectrum of wave forcing with a given frequency iscrucial for determining dominant vertical wavelengthsof gravity waves, and noted that the relation of the dom-inant vertical wavelengths of waves to the depth of thewave forcings should be regarded as a ‘‘rule of thumb.’’



FIG. 9. PSDs of the gravity wave perturbation (w9) as a function of (a) vertical wavelength, (b) horizontalwavelength, and (c) period in CTL (thick solid), DRYMH (thin solid), DRYM (dotted), DRYQ (dashed), andDRYMH 1 DRYQ (thick gray solid).

The dependence of the vertical wavelengths of gravitywaves on their horizontal wavenumbers and frequencieswill be examined in section 6c.

Figure 10 shows the storm-relative horizontal phasespeed (c) spectra in all simulations and in DRYMH 1DRYQ. To obtain the storm-relative phase speed spec-trum, a two-dimensional spectrum is calculated using

the vertical velocity field over the horizontal area of 900km from x 5 170 to 1070 km and the period of 6 hfrom t 5 6 to 12 h in all simulations. The two-dimen-sional spectrum is calculated as follows:

2DxDt2PSD(k, v) 5 |FFT2D[w9(x, t)](k . 0)| . (20)

N Nx t


15 AUGUST 2003 1973S O N G E T A L .

FIG. 10. Storm-relative phase speed spectra of the (a) westward-and (b) eastward-propagating gravity wave perturbations (w9) in thestratosphere in CTL (thick solid), DRYMH (thin solid), DRYM (dot-ted), DRYQ (dashed), and DRYMH 1 DRYQ (thick gray solid).

Here, Dx is the horizontal grid size; FFT2D is the two-dimensional FFT; and Nx and Nt are the number of datapoints in the horizontal direction and time, respectively.The vertical velocity is multiplied by as inÏr(z)/r(z )0

the case of the one-dimensional PSD. The phase speedspectrum is calculated using 2 m s21 phase speed bins.The spectral power at each bin c0 represents the inte-gration of the two-dimensional PSD with respect to kand v over the area where c0 2 1.0 , v/k , c0 1 1.0.The phase speed spectrum is then averaged for 19.8-kmdepth above z 5 15 km. In Fig. 10, PSDs of westward-and eastward-propagating waves are separately plottedbecause the magnitude of vertical velocity perturbationsof the eastward-propagating waves is about 50 timessmaller than that of the westward-propagating waves.Westward-propagating waves in all simulations havephase speeds between 252 and 26 m s21 relative to

the convective system. The strong spectral peaks existat 220 and 224 m s21 in the CTL simulation. The phasespeed spectra in the quasi-linear dry simulations arequite different from the CTL simulation in their re-spective shapes and the location of strong peaks.

As can be seen in Fig. 10a, the waves generated bynonlinear forcing in the DRYM simulation tend topropagate slowly westward from the gust front com-pared with the waves induced by the nonlinear forcing(including the heat flux) in the DRYMH simulation.This implies that the effect of the heat flux forcing onthe characteristics of gravity waves can be significant,although the heat flux forcing is not likely to greatlymodify the overall structure and magnitude of nonlin-ear wave sources when Figs. 6a and 6b are compared.Despite the large differences between the phase speedspectra in the CTL and quasi-linear dry simulations,the spectrum in DRYMH 1 DRYQ is quite similar tothat in the CTL simulation except in the phase speedregion between 224 and 220 m s21 , in which themagnitude of gravity waves is about twice as large asthan those in CTL. As mentioned in Fig. 8, gravitywaves in the CTL simulation can be affected by thenonlinear wave–wave interaction caused by stronggravity wave perturbations. Thus, in the present study,it can be also seen that the nonlinear processes in thegravity wave field in the CTL simulation reduce themagnitude of gravity waves with phase speeds of 224to 220 m s21 . In the phase speed spectrum for theeastward-propagating waves in the CTL simulation, thegravity waves with storm-relative phase speeds of 20–44 m s21 are dominant. The magnitude of phase speedspectra in the quasi-linear dry simulations is too large,and their dominant phase speeds are shifted by about25 m s21 compared with the CTL simulation. In theeastward-propagating waves, the PSD in DRYMH 1DRYQ compares well to the CTL simulation for mostphase speeds.

b. Wave propagation

In this section, a more detailed analysis of the indi-vidual wave sources are performed using a combinationof spectral analysis of the wave source, and the verticalpropagation condition for linear gravity waves. Thisanalysis attempts to explain why the amplitudes of thegravity waves in all of the quasi-linear simulations aresimilar, even though the magnitudes of the respectivesources are very different. Also, the reason whyDRYMH 1 DRYQ compares well to CTL, even thoughboth DRYMH and DRYQ produce gravity waves whoseamplitudes are too large, is examined.

Taking Fourier transform [(x, t) → (k, v)] of (16)gives

2] w2 2 2(U 2 c) 1 (U 2 c) m w 5 S, (21)

2]z

where



FIG. 11. Base-10 logarithm of vertically averaged (z 5 0–15 km) two-dimensional PSDs of the wave sources in the (a) DRYMH, (b)DRYM, and (c) DRYQ simulations as a function of zonal wavenumber (k) and (storm relative) frequency (v). Logarithmic contour intervalis 1.0. The line corresponding to the phase speed of 226.7 m s21 is plotted with thick white line in each panel.

2 2N 1 d U 1/H dU2m 5 2 2

2 2(U 2 c) (U 2 c) dz (U 2 c) dz

122 2 k , and (22)

24H

ˆ ˆ ˆF ] F g Fw u uS 5 ik(U 2 c) 2 (U 2 c) 15 [ ]r ]z r u r

ˆ ˆg Q dU Fu 2z /(2H )1 1 e . (23)6u r dz r

In (21)–(23), k is the horizontal wavenumber, m is thevertical wavenumber; c (5v/k, where v is a storm-relative frequency) is the storm-relative horizontal phasespeed; and w 5 w exp[z/(2H)], where w is the Fouriertransform of w9; H is the scale height of basic-statedensity, which is 9.6 km in the troposphere and 6.3 kmin the stratosphere. As already shown in several theo-retical studies (e.g., Smith and Lin 1982; Chun and Lee1994; Chun and Baik 1998), the magnitude of w isproportional to the magnitude of S in the troposphere.

Figure 11 shows the two-dimensional (k–v) PSDs ofthe wave source S for each DRYMH, DRYM, andDRYQ. (The shown spectra are constructed by aver-aging the PSD of S from the surface to 15 km.) Thisfigure shows a number of differences between the char-acter of the respective sources. For example, the sourcein DRYQ (Fig. 11c) has negligible power at small wave-numbers (k , 0.25 3 1024 cycles m21), whereas thesources in DRYMH (Fig. 11a) and DRYM (Fig. 11b)possess significant power at those wavenumbers for allfrequencies. Also, the source in DRYQ does not possesssignificant power at k , 1.0 3 1024 cycles m21 and2v . 2.0 3 1023 cycles s21, whereas the sources inDRYM and DRYMH do. One similarity, however, isthat the sources in all of the quasi-linear simulationspossess strong power at v ; 0, for a wide range ofhorizontal wavenumbers. These quasi-stationary (rela-tive to the gust front) sources are mainly confined belowz 5 4.5 km (not shown).

Thus, based on the forcing spectra in Fig. 11, it isreasonable to expect that stratospheric gravity waveswith c , 226.7 m s21 [i.e., along the line connecting


15 AUGUST 2003 1975S O N G E T A L .

FIG. 12. Base-10 logarithm of two-dimensional PSDs of the gravity wave perturbations (w9) at z 5 18 km in the (a) DRYMH, (b) DRYM, and(c) DRYQ simulations as a function of zonal wavenumber (k) and (storm relative) frequency (v). Logarithmic contour interval is 1.0.

(k 5 0, 2v 5 0) and (k 5 1.5 3 1024, 2v 5 4.0 31023)] should be much stronger in DRYMH and DRYMthan in the DRYQ. Also, the quasi-stationary wavesshould be important in the stratosphere in all quasi-linear simulations. However, the two-dimensional (k–v) PSDs of gravity waves at z 5 18 km in the quasi-linear simulations (Fig. 12) are different from their re-spective source spectra (Fig. 11). The spectra of gravitywaves in the stratosphere occupies a small region in thek–v domain, and their PSDs are all qualitatively similarregardless of the imposed forcing. Compared with Fig.11, it is clear that the nonlinear wave sources with k ,1.0 3 1024 cycles m21 and 2v . 2.0 3 1023 cycless21 are not contributing to the gravity waves that reachthe stratosphere. Also, the quasi-stationary gravitywaves are negligibly weak at z 5 18 km because of theabsorption of those gravity waves near the critical levellocated at z 5 5.4 km (see Fig. 1b).

A linear gravity wave generated in the tropospherecan propagate vertically to the stratosphere, for eachvalue of k 2 v, if m2 in (21) is larger than 0 in theforcing region and aloft (provided U ± c). Therefore,in the absence of critical levels, a linear gravity wave

that reaches the stratosphere satisfies the condition thatm2 . 0 at every level between the wave source and thestratosphere. Thus, the vertical propagation conditiondefines a region in k–v space, within which verticallypropagating gravity waves can be generated. Using thispropagation condition, we define the effective waveforcing as the forcing able to generate waves that prop-agate vertically.

Figure 13 shows horizontal wavenumbers and fre-quencies at which the vertical propagation condition atevery level from z 5 12, 6, 3, and 0 km to z 5 15 kmis satisfied. For convenience, those horizontal wave-numbers and frequencies included in the shaded regionare denoted as kp and vp, respectively. From the verticalpropagation condition, it can be seen that for wavesources in each layer of z 5 12–15, 6–15, 3–15, and0–15 km, there is some region in k–v space that cansupport linear gravity waves that can propagate abovez 5 15 km.

The area defined by kp–vp is determined by a com-bination of stability and the basic-state wind throughthe dispersion relation. In comparison to Fig. 13a, thereduced area in Fig. 13b is due to the lower stability in



FIG. 13. Two-dimensional (k–v) spectral region where the vertical propagation conditions at every level from (a) z5 12, (b) 6, (c) 3, and (d) 0 km to z 5 15 km are satisfied.

the troposphere. With its lower stability, the tropospherefilters the higher-frequency waves, reducing the effec-tiveness of high-frequency wave sources (in DRYMHand DRYM). The area difference between Figs. 13b and13c is mostly due to changes in the basic-state wind.The wind shear can significantly restrict vertical prop-agation, and hence the effectiveness of wave sourcesbetween z 5 3 and 6 km. In particular, the short wave-length part of the wave sources can be mostly filteredby the wind shear in the troposphere. This can be clearlyseen by comparing the PSDs of the wave forcing (Fig.5b) with those of the linear gravity waves in the strato-sphere (Fig. 9b). This suggests that the basic-state windshear is important not only for determining the meso-scale convective structure (Weisman and Klemp 1982;Fovell and Dailey 1995) but also for controlling thewave propagation condition, especially in the majorconvective region. Recently, Beres et al. (2002) inves-tigated the effect of tropospheric wind shear on con-vectively forced gravity waves and showed that windshear increases (reduces) momentum flux of wavespropagating in the opposite (same) direction of basic-state wind. Because the wave sources in the troposphere(Fig. 6) are concentrated between z 5 3 and 10 km(except near the gust front), the wave sources withinthe area defined by kp–vp in Fig. 13c, may be responsiblefor most gravity waves that arrive in the stratosphere.The difference between Figs. 13c and 13d is mainly dueto the magnitude of the basic-state wind between thesurface and z 5 3 km, because the vertical wind sheardoes not change with height below z 5 6 km. The stronglow-level wind tends to prevent the vertical propagationof gravity waves. However, the difference between Figs.13c and 13d may not be important for source restriction,in the present dry simulations, because the wave sources

in the troposphere are weak below z 5 3 km (exceptnear the gust front).

We now attempt to define the effective wave sourcefield. The effective source at a vertical level (z0) is obtainedfrom the inverse transform of the Fourier coefficients ofthe source given in (23). Coefficients with k and v outsideof the area defined by kp and vp (within which the verticalpropagation condition at every level between z 5 z0 and15 km are satisfied) are removed. The remaining coeffi-cients are multiplied by 2 exp[z0/(2H)] so that the ef-2kp

fective sources in Figs. 14a and 14b can be expressedas (17) and (19). Figure 14 shows the effective wavesources in the DRYMH and DRYQ simulations and inDRYMH 1 DRYQ at t 5 6 h. The shading in Fig. 14represents regions where the source in DRYMH is outof phase with the source in DRYQ. Compared with Fig.6, the magnitude of the nonlinear and diabatic sourcesshown in Fig. 14 is about 10 to 20 times smaller thantheir original value. A significant part of the originalsource fields are removed because they do not projectto a part of the spectrum where the propagation con-dition is satisfied. Also, the reduction in magnitude ofthe nonlinear sources is more than the reduction of thediabatic source. The result is that the magnitudes of theeffective nonlinear sources are comparable to the ef-fective diabatic source. This explains why the magni-tude of the gravity waves in all of the quasi-linear sim-ulations are similar, even though the diabatic source ismuch weaker than the nonlinear sources.

Local maxima and minima of the effective nonlinearand diabatic sources are mostly out of phase. Therefore,the net effective source in DRYMH 1 DRYQ is smallerthan the individual effective sources in DRYMH andDRYQ, respectively. This interference of the two sourc-es explains why the stratospheric gravity waves in


15 AUGUST 2003 1977S O N G E T A L .

FIG. 14. Wave sources in the DRYMH and DRYQ simulations, and in DRYMH 1 DRYQ at t 5 6 h that cangenerate the linear internal gravity waves propagating to the stratosphere above z 5 15 km. Contour interval is0.2 3 10210 m21 s23. The shading represents the region where the nonlinear source in the DRYMH simulationis out of phase with the diabatic source in the DRYQ simulation.

DRYMH 1 DRYQ resemble those in CTL, even thoughthe waves in DRYMH and DRYQ are 2 to 3 times larger.

Two important points arise from Fig. 14. First, be-cause of the restrictions placed on the wave source bythe vertical propagation condition, the nonlinear anddiabatic sources derived from the convective system areinefficient in generating gravity waves. The effectivenonlinear and diabatic sources are significantly different

from and smaller in magnitude than the original sources.Therefore, the propagation condition must be consideredwhen analyzing gravity wave sources. Second, there issignificant cancellation between the nonlinear and thediabatic sources, resulting in a reduced net gravity wavesource. Therefore, both nonlinear and diabatic sourcesare important in generating the gravity waves that reachthe stratosphere.



FIG. 15. Contour plot of the (a) vertical wavelength (km) and (b) zonal and (c) vertical components of group velocities (m s 21) of gravitywaves superimposed on base-10 logarithm of the two-dimensional power spectral density of gravity wave perturbations (w9) at z 5 18 kmin the CTL simulation.

c. Dominant vertical wavelength and group velocity

Figure 15 shows the contours of vertical wavelengthand zonal and vertical components of group velocitiessuperimposed on the two-dimensional (k–v) spectrumof gravity waves at z 5 18 km in the CTL simulation.The vertical wavelengths and group velocities are cal-culated from (22) using the basic-state wind and stabilityin the stratosphere. For the westward-propagating waves(Fig. 15a), the vertical wavelengths of 4–20 km aredominant. Gravity waves with vertical wavelengths of6–14 km have horizontal wavenumbers less than 1.253 1024 cycles m21 (lx 5 8 km), while gravity waveswith vertical wavelengths of 4 and 20 km have hori-zontal wavenumbers less than 1.0 3 1024 and 0.5 31024 cycles m21, respectively. For the eastward-prop-agating waves, the vertical wavelengths of 6–14 km aredominant, and those waves have horizontal wavenum-bers less than 0.5 3 1024 cycles m21. The shape of thistwo-dimensional spectrum in the CTL simulation is al-most the same as those in the quasi-linear dry simula-tions. Figure 15 clearly shows that the dominant verticalwavelengths depend on the horizontal wavenumbers and

frequencies via the dispersion relation. The verticalwavelengths of gravity waves are determined by thebasic-state wind and stability, horizontal wavenumber,and frequency, not only by the depth of diabatic forcing,as suggested by Salby and Garcia (1987) and Alexanderet al. (1995).

The westward- and eastward-propagating gravitywaves have zonal group velocities (cgx) of 250 to 210and 20–50 m s21 and vertical group velocities (cgz) of2–20 and 4–10 m s21, respectively. The fastest west-ward- and eastward-propagating waves have cgz of 10–14 and 6–10 m s21, respectively. A ray composed ofthe fastest westward- (eastward) propagating waves canpropagate as far as 85 (142) km in the horizontal, and17 km from z 5 18 km to the model top in 28.3 (47.2)min. Given that the gust front updraft is almost sta-tionary near x 5 770 km, and that the model domain(used in phase speed analysis) is 900 km wide from 170to 1070 km, the vertically averaged phase spectra shownin Fig. 10 are not biased toward high intrinsic frequen-cies. This is because the fastest horizontally propagatingwaves generated in the CTL simulation reach the phys-


15 AUGUST 2003 1979S O N G E T A L .

ical model top before propagating out the lateral bound-aries of the model domain, and there are no significantwave sources from x 5 170 to 255 km and x 5 928 to1070 km.

7. Summary and conclusions

The characteristics of internal gravity waves in thestratosphere generated by mesoscale convective storms,and the generation mechanisms for those waves wereinvestigated using a two-dimensional version of ARPS.In the nonlinear moist (control) simulation, a typicalmulticellular-type convective system was developed un-der a basic-state wind and thermodynamic condition ap-propriate for midlatitude squall lines. The simulatedconvective system reaches a quasi-steady state after t5 4 h, in which a gust front is quasi stationary in themodel domain and convective cells are periodically re-generated from the gust front at an interval of about18.1 min. The gravity waves in the stratosphere gen-erated by convective storms are almost isotropic beforethe quasi-steady state, but the westward propagation rel-ative to the gust front becomes dominant in the quasi-steady state.

In a convective system, there are the two types ofwave forcings for linear internal gravity waves: nonlin-ear forcing in the form of momentum and heat fluxdivergences and diabatic forcing. In the control (CTL)simulation, the nonlinear forcing is much stronger thanthe diabatic forcing, especially in the upper troposphere.In order to examine differences in the responses of astably stratified atmosphere to the different forcings,three quasi-linear dry simulations forced by the mo-mentum flux forcings (DRYM), momentum and heatflux forcings (DRYMH), and diabatic forcings (DRYQ)are conducted. The magnitude of the stratospheric grav-ity waves in these quasi-linear dry simulations are com-parable with each other but are about 2 times larger thanthose in the CTL simulation. However, the stratosphericgravity waves in the stratosphere in the CTL simulationare reproduced reasonably well by DRYMH 1 DRYQ.

For a quantitative understanding of the gravity waves,spectral analysis in the vertical, horizontal, and timedomains were carried out. In all simulations, gravitywaves with vertical wavelengths of 6.6–9.9 km, hori-zontal wavelengths of 10–100 km, periods of 8–80 minare dominant for both westward- and eastward-propa-gating waves. In the CTL simulation, westward- andeastward-propagating gravity waves with storm-relativephase speeds of 225 to 220 and 20–44 m s21 aredominant, respectively. The shape and the location ofpeaks of phase speed spectra in the quasi-linear drysimulations are different from each other, and also quitedifferent from those in the CTL simulation. However,again, the phase spectrum in the CTL simulation is re-produced by DRYMH 1 DRYQ.

The results raise two interesting questions. 1) Whyis the magnitude of the gravity waves in all quasi-linear

dry simulations comparable, even though the magnitudeof the nonlinear forcing is 2 to 3 times larger than thatof the diabatic forcing? 2) Why do the linear gravitywaves in DRYMH 1 DRYQ compare well, in both themagnitude and spectral characteristics, to waves in theCTL simulation? In order to answer these questions,two-dimensional (k–v) spectra of the wave sources andinduced gravity waves in the stratosphere were exam-ined. From these spectra, it was found that a large por-tion of the nonlinear wave sources cannot generate grav-ity waves that can reach the stratosphere, due to thevertical propagation condition of the linear internalgravity waves. Using the vertical propagation condition,the following can be found.

1) The effective nonlinear and diabatic sources that cangenerate gravity waves that reach the stratosphereare strongly confined by the vertical propagationcondition. This condition is determined by the basic-state wind, vertical wind shear, and the stability inthe troposphere and stratosphere.

2) A large part of the nonlinear wave source cannotgenerate linear gravity waves that reach the strato-sphere.

3) The effective nonlinear and diabatic forcings havecomparable magnitudes, and they are largely out ofphase in major source regions.

4) Consequently, the magnitude of gravity waves in theDRYMH 1 DRYQ is comparable to that in the CTLsimulation, which is 2 to 3 times less than either theDRYMH or DRYQ simulation, due to cancellationof the effective sources. This implies that both thenonlinear and diabatic forcings are important forconvective wave generation.

The above points have interesting implications forgravity wave source parameterizations. First, 1) and 2)suggest that the wave propagation condition and theresultant filtering play a crucial role in shaping the spec-trum of stratospheric waves. Therefore, in a parameter-ization, realistic filtering by the background flow, whichdetermines the effectiveness of wave forcing and thevertical propagation of gravity waves above the forcing,should be considered. Second, 3) and 4) imply that asource parameterization based on diabatic forcing alonewill not include the cancellation due to nonlinear forc-ing. Therefore, such a parameterization may give a poorrepresentation of wave characteristics in parts of thespectrum. In summary, the results of this study suggestthe convective systems, like the one modeled here, gen-erate gravity waves through a complicated relationshipbetween diabatic and nonlinear forcing and the wavepropagation condition. This relationship should be a top-ic of continuing research.

Acknowledgments. The authors thank two anonymousreviewers for their careful reading the manuscript andhelpful comments. This research was supported by theClimate Environment System Research Center spon-



sored by the SRC Program of the Korea Science andEngineering Foundation.

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