Momentum Flux Spectrum of Convectively Forced Internal Gravity Waves and Its Application to Gravity Wave Drag Parameterization.

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1 JANUARY 005 S O N G A N D C H U N 107 Momentum Flux Spectrum of Convectively Forced Internal Gravity Waves and Its Application to Gravity Wave Drag Parameterization. Part I: Theory IN-SUN SONG AND HYE-YEONG CHUN Department of Atmospheric Sciences, Yonsei University, Seoul, South Korea (Manuscript received 7 January 004, in final form 8 June 004) ABSTRACT The phase-speed spectrum of momentum flux by convectively forced internal gravity waves is analytically formulated in two- and three-dimensional frameworks. For this, a three-layer atmosphere that has a constant vertical wind shear in the lowest layer, a uniform wind above, and piecewise constant buoyancy frequency in a forcing region and above is considered. The wave momentum flux at cloud top is determined by the spectral combination of a wave-filtering and resonance factor and diabatic forcing. The wave-filtering and resonance factor that is determined by the basic-state wind and stability and the vertical configuration of forcing restricts the effectiveness of the forcing, and thus only a part of the forcing spectrum can be used for generating gravity waves that propagate above cumulus clouds. The spectral distribution of the wave momentum flux is largely determined by the wave-filtering and resonance factor, but the magnitude of the momentum flux varies significantly according to spatial and time scales and moving speed of the forcing. The wave momentum flux formulation in the two-dimensional framework is extended to the threedimensional framework. The three-dimensional momentum flux formulation is similar to the twodimensional one except that the wave propagation in various horizontal directions and the threedimensionality of forcing are allowed. The wave momentum flux spectrum formulated in this study is validated using mesoscale numerical model results and can reproduce the overall spectral structure and magnitude of the wave momentum flux spectra induced by numerically simulated mesoscale convective systems reasonably well. 1. Introduction Gravity waves propagating upward from tropospheric source regions can significantly affect momentum budgets by depositing their momentum to the large-scale flow in the rarefied upper middle atmosphere as they are broken, filtered at their critical levels, or dissipated by eddy viscosity during their vertical propagation (Lindzen 1981; Matsuno 198). The gravity wave momentum deposition is balanced by the Coriolis torque, and adiabatic warming or cooling accompanied by the meridional circulation induced by the Coriolis torque can result in substantial departures from the radiative equilibrium temperature distribution, especially in polar regions (Holton 198). According to the downward control theory proposed by Haynes et al. (1991), the meridional circulation at a vertical level is determined by vertically integrated eddy momentum forcing due to vertically propagating planetary and gravity waves above that level. In fact, gravity wave Corresponding author address: Hye-Yeong Chun, Department of Atmospheric Sciences, Yonsei University, Shinchon-dong, Seodaemun-ku, Seoul , South Korea. chy@atmos.yonsei.ac.kr momentum deposition in the polar mesosphere can significantly affect temperature distribution and the strength of the polar night jet in the stratosphere (Garcia and Boville 1994). This implies that impacts of gravity waves on large-scale circulation are not confined to the upper stratosphere or mesosphere where strong gravity wave drags appear but can also be significant in the lower stratosphere. In the tropical stratosphere, it is known that gravity waves with a wide spectrum of phase speed, as well as planetary waves such as Kelvin and Rossby gravity waves, can significantly contribute to momentum forcing required for driving the quasi-biennial oscillation (QBO) of the zonal-mean zonal wind (Lindzen and Holton 1968; Holton and Lindzen 197). Dunkerton (1997) showed that the dynamics of the QBO cannot be understood without considering a broad tropical wave spectrum, ranging from high-frequency gravity waves to low-frequency planetary waves. Also, some observational and numerical modeling studies (e.g., Holton and Tan 1980, 198; Baldwin and Dunkerton 1998; Hamilton 1998) have shown that the QBO can be coupled to variability of large-scale zonal-mean zonal wind and temperature in extratropical regions. Therefore, understanding the interaction between the large-scale flow 005 American Meteorological Society

2 108 J 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 VOLUME 6 and gravity waves in the Tropics is also important in accounting for variability of the global circulation. It is cumulus convection that has been accepted as the most important gravity wave source in the Tropics where persistent convective clouds exist and in the SH high latitudes where mountains are rare as compared with the NH. In the Tropics, significant portions of waves generated by cumulus clouds have high frequencies [0.5 cycles per day (cpd)] rather than the lowfrequency Kelvin or Rossby gravity wave (Bergman and Salby 1994). Momentum forcing estimated from mesoscale gravity waves observed above clouds can be comparable to that due to Kelvin waves (Pfister et al. 1993). Also, several observational and numerical modeling studies (e.g., Sato 1993; Alexander and Pfister 1995; Piani et al. 000; Vincent and Alexander 000; Beres et al. 00) have shown that gravity waves induced by active cumulus convection have substantial momentum flux ( Nm ) comparable to that from orographic gravity waves. Therefore, these convectively forced gravity waves can supply strong eddy momentum forcing to the large-scale flow in the middle atmosphere when they are either breaking or dissipating. The importance of the convectively forced gravity waves in large-scale circulation has led to a demand for the parameterization of their effects in large-scale numerical models (e.g., Rind et al. 1988; Kershaw 1995; Chun and Baik 1998, 00; Beres et al. 004). Chun and Baik (1998, CB98 hereinafter) first analytically formulated momentum flux of convectively forced internal gravity waves under a vertically uniform background wind and stability condition and proposed a gravity wave drag (GWD) parameterization induced by cumulus convection (GWDC). The parameterization of CB98 can significantly alleviate zonal wind and temperature biases in the SH wintertime in the Yonsei University AGCM (Chun et al. 001) and in the Tropics in the National Center for Atmospheric Research Community Climate Model, version 3 (Chun et al. 004). Chun and Baik (00, CB0 hereinafter) updated the parameterization of CB98 by including effects of vertical wind shear in the forcing region and using different static stabilities between the forcing region and above. However, the parameterizations of CB98 and CB0 only take account of gravity waves that are stationary relative to moving convective clouds, and thus they cannot consider high-frequency gravity waves above cumulus clouds that have been observed and numerically simulated (Pfister et al. 1993; Bergman and Salby 1994; Alexander and Pfister 1995; Alexander et al. 1995; Vincent and Alexander 000; Beres et al. 00; Song et al. 003). Therefore, the parameterizations of CB98 and CB0 need to be extended to include nonstationary as well as stationary gravity wave spectra generated by convection. In this study, we analytically formulate the phase speed spectrum of the momentum flux of internal gravity waves induced by diabatic forcing, as a natural extension of CB0. Beres et al. (004, BAH04 hereinafter) analytically formulated the wave momentum flux spectrum induced by diabatic forcing in a simple background flow with vertically uniform wind and stability. Using the linear formulation, they suggested a method of specifying the gravity wave spectrum above convection in large-scale models and validated their method using diabatic forcing and gravity wave spectra obtained from mesoscale convection simulations. The main purpose of the current study, to formulate the momentum flux spectrum at cloud top for GWDC parameterization in climate models, happens to be similar to BAH04. There are, however, several differences between this study and BAH04, and they will be discussed in detail in section 6. The remainder of this paper is organized as follows. In section, two-dimensional (x z) analytic solutions for convectively generated internal gravity waves are obtained in a spectral domain, and the wave momentum flux at cloud-top height is formulated as a function of phase speed. In section 3, dependency of the wave momentum flux on background conditions and the structure and magnitude of diabatic forcing is examined. In section 4, the wave momentum flux formulation in a two-dimensional framework is extended to a three-dimensional (x y z) framework. In section 5, the wave momentum flux spectrum in this study is validated using mesoscale numerical model results. The summary and conclusions are presented in the last section.. Theory a. Governing equations and solutions Governing equations for a two-dimensional, smallamplitude perturbation induced by diabatic forcing in a hydrostatic, nonrotating, inviscid, and Boussinesq airflow can be expressed as u t U u x du dz w 0, x 1 b, z b b U t x N w gq, and 3 c p T 0 u x w 0, z 4 where u and w are the horizontal and vertical perturbation wind velocities, respectively; (p/ 0, where p is the perturbation pressure and 0 is the basic-state density) is the perturbation kinematic pressure; b (g/ 0, where g is the gravitational acceleration, is the perturbation potential temperature, and 0 is the reference potential temperature) is the buoyancy per-

3 JANUARY 005 S O N G A N D CHUN 109 turbation; U is the basic-state wind; N is the Brunt Väisälä frequency; c p is the specific heat of air at constant pressure; T 0 is the reference temperature; and Q [q(x, t) q (z), where q(x,t) is the horizontal and temporal structure of Q and q (z) is the vertical structure of Q] is the diabatic forcing representing the latent heat released by a cumulus cloud. In this study, U and N are assumed to depend on the altitude z only. Equations (1) (4) can be combined into a single equation for w: D Dt w d U D z dz Dt w N x w x g q q c p T 0 x, 5 where D/Dt /t U/x. Taking double Fourier transform in x ( k) and t ( ) of (5) yields the Taylor Goldstein equation for convectively forced linear internal gravity waves: ŵ z N U c d Udz gqˆ q U cŵ c p T 0 U c, where ŵ and qˆ are the Fourier transforms of w and q, respectively, and c (/k, where is the ground-based frequency and k is the horizontal wavenumber) is the ground-based horizontal phase speed. Here, the double Fourier transform of variable A is defined as Âk, z, 1 6 Ax, z, t e ikxt dx dt, 7 where k 0. In this study, a three-layer atmosphere is assumed. A schematic diagram of the three-layer atmosphere and diabatic forcing is shown in Fig. 1. In this three-layer atmosphere, the basic-state wind has constant shear from the ground to a vertical level z s within the forcing region and is uniform above. The stability is piecewise constant in the forcing region and above. The vertical structure ( q ) of the forcing is specified as follows: q 1 z z mz d for z b z z t, 8 0 elsewhere, where z m (z t z b )/ and z d (z t z b )/. General solutions of (6) for the uniform wind case (U 0 U t ) can be written as ŵ 1 A 1 e i 1z B 1 e i 1z ŵ A e i 1zz b B e i 1zz b for 0 z z b, 9 gqˆ c p T 0 N q u for z b z z t, and 10 1 ŵ 3 A 3 e i zz t B 3 e i zz t for z z t, 11 where 1 N 1 / U t c, N / U t c, and u /( 1 z d ). FIG. 1. Schematic diagram of the basic-state wind and stability used in this study. The symbols z b, z t, and z s represent the bottom and top heights of the diabatic forcing and the top height of shear layer, respectively. Here U 0 and U t are the magnitudes of the basic-state wind at the surface and forcing top, respectively. Also, N 1 and N are the buoyancy frequencies below the forcing top and above, respectively. For the shear case (U 0 U t ), the basic-state wind U is given as U 0 z for 0 z z s and U t for z z s, respectively. Here, is the vertical wind shear below z z s given by (U t U 0 )/z s. General solutions of (6) for the shear case are written as ŵ 1 C 1 z z 1i D 1 z z 1i for 0 z z b, ŵ C z z 1i D z z 1i 1 gqˆ c p T 0 N q s for z b z z s, 13 1 ŵ 3 C 3 e i 1zz s D 3 e i 1zz s gqˆ c p T 0 N q u for z s z z t, and 14 1 ŵ 4 C 4 e i zz t D 4 e i zz t for z z t, 15 where z (c U 0 )/, Ri 1/4, where Ri is the Richardson number (N 1/ )] is a real number because only dynamically stable basic states (i.e., Ri 1/4) are considered in this study, and s [/(Ri )](z z ) /z d. All unknown coefficients in the general solutions for each case (U 0 U t or U 0 U t ) are determined by applying the flat-bottom boundary condition (ŵ 0atz 0), the upper radiation condition (Booker and Bretherton 1967) that allows for upward propagation of gravity wave energy in the uppermost layer (z z t ), and the interface conditions for continuity of vertical displacement and pressure across z z b, z s, and z t to (9) (11) and (1) (15) (see appendix A). Thus, ŵ is obtained for both uniform wind and shear cases.

4 110 J 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 VOLUME 6 b. Momentum flux The space- and time-averaged momentum flux M is calculated from û and ŵ using Parseval s relation (Arfken and Weber 1995): M 1 L x L t L x L t 0 0 uw dx dt 0 Reûŵ* dk d, 16 where L x and L t are appropriate spatial and time scales, respectively, used for averaging. Note that the momentum flux calculated from (16) is not always due to gravity waves; in fact, the momentum flux below z z t is mainly due to the diabatic forcing itself. In real convective storms, momentum flux inside storms is mainly due to turbulence associated with convective activities rather than gravity waves. For this reason, the momentum flux due to gravity waves is calculated from the cloud top. Therefore, in the GWDC parameterization, the reference level for the parameterization is the cloud top, unlike the mountain wave drag parameterization for which the reference level is the mountain surface. For the calculation of wave momentum flux at the diabatic forcing top, û at z z t is calculated using ŵ in the uppermost layer and the continuity equation [(4)]: û sgnu t c kŵ, and thus Re (ûŵ*) at z z t becomes N Reûŵ* sgnu t c k U t c Here X is given as g c p T 0 N 1 qˆ X XuYu1 Xu1XuYu1 XuYu1 X X u1 X u X X u1 X u X Xu1XuYu1 X s4 Y s3 X s X s3 X s0 X s4 Y s3 X s3 X s X s0 X s5 X X s4 X s X s3 X s0 X X s4 X s3 X s X s0 for U 0 U t, for U 0 U t, 19 where X s5 4(X a X a X s0 ). Here X a (z s z ) 1/i {Re(Y s ) ( 1 i)[ s(z s ) u (z s )]} i(z s z )Y s1. Terms consisting of X in (19) are presented in appendix A. Also, X is a function of the ground-based phase speed c of gravity waves, the vertical configuration ( q, z b, and z t ) of diabatic forcing, the basic-state wind at the ground and at z z b and z t (U 0, U b, and U t ), the vertical shear of the basic-state wind below z z s, and the stabilities below and above z z t (N 1 and N ). When Ri 1/4, X for the shear case asymptotically approaches X for the uniform wind case in phasespeed regions of c U 0 U t U 0. To represent the wave momentum flux as a function of phase speed, (16) is rewritten in a coordinate system of nondimensionalized spectral radius r (k ) and phase speed c (c s c ). Here, k k/k s and / s, where the tilde denotes dimensionless variables, and k s, s, and c s ( s /k s ) are dimensional constants for k,, and c, respectively. Note that the nondimensionalization is required to define the spectral radius r, but the dimensional constants disappear in the final formulation of the momentum flux [() and (4)]. After some algebraic manipulations, the integration of Re(ûŵ*) becomes 0 Reûŵ* dk d g N sgnu t cc p T 0 N 1 U t c X c dc, 0 where k s c qˆr, c 1 cc s 0 dr. 1 Details of the derivation of (0) and (1) are presented in appendix B. From (16) and (0), the phase-speed spectrum of wave momentum flux is obtained: g Mc sgnu t c 0 L x L t c p T 0 N 1 N U t c X c. Here (c) is the diabatic source function. In this study, the diabatic forcing is assumed to be a Gaussian-shaped function in x and t to mimic a strong convective cell localized in space and time: qx, t q 0 exp x x 0 c q t t 0 x exp t t 0 t, 3 where q 0 is the maximum magnitude of the diabatic forcing; x 0 and t 0 are the location and time, respectively, when the forcing has its maximum magnitude; c q is the moving speed of the forcing; and x and t are spatial and time scales of the forcing, respectively. The calcu-

5 JANUARY 005 S O N G A N D CHUN 111 lation of (1) using the diabatic forcing given in (3) yields c q 0 x x 16 t, 4 1 c c q c 0 where c 0 x / t. Details of the derivation of (4) are given in appendix C. 3. The wave momentum flux The structure of the wave momentum flux in the phase-speed domain is determined by three terms (c), X, and N / U t c, as shown in (). The (c) is exclusively dependent on the magnitude, horizontal and temporal scales, and horizontal moving speed of diabatic forcing. The X depends on basic-state wind and stability and on the vertical configuration (structure, height, and depth) of the diabatic forcing. It will be discussed in detail in this section. The N / U t c corresponds to N / U t in CB0 [see (1) in CB0] for the stationary case (c 0). Figure shows X for six different basic-state wind profiles: positive shear with U 0 0ms 1 and U t 0ms 1 and with U 0 0ms 1 and U t 0ms 1 ;no shear with U 0 U t 0ms 1 and with U 0 U t 0 ms 1 ; and negative shear with U 0 0ms 1 and U t 0ms 1 and with U 0 40ms 1 and U t 0ms 1. Here z b.5 km, z s 6 km, z t 11 km, N rad s 1, and N 0.0 rad s 1 are used. The X is plotted in the phase-speed grid with a grid size c of1ms 1 by numerically integrating X over c, and thus X at c c p represents X averaged over the interval from c p c/ to c p c/. For the numerical integration, the adaptive numerical integration routine (Piessens et al. 1983) is used. In the case of uniform wind (Fig. b), X is symmetric with respect to c U t. However, in the shear case (Figs. a,c), such a symmetry is not found; rather X shows a mirror image with respect to c U t for positive and negative shears with the same magnitude. When the cloud-top wind speed increases from0to0ms 1 (bottom panels of Figs. a c), the shape of X is identical to that for U t 0ms 1 but moves 0 m s 1 to the right for both the uniform and shear cases. This result demonstrates the Galilean invariance of X. The X has singularities at three phase speeds (c U 0, U b, and U t ; see appendix A). Because the numerical integration for intervals including such singular points does not converge or blows up, X in such an interval is not calculated and thus is not plotted in Fig.. In physical terms, it means that the waves with singular phase speeds are filtered out below the cloud top and thus cannot propagate above. In the shear case, the FIG..The X values for six different basic-state wind profiles with (a) positive shear, (b) no shear, and (c) negative shear below z z s (6 km) when z b.5 km, z t 11 km, N rad s 1, and N 0.0 rad s 1. Phase speeds at which X has its peaks are numbered over the peaks. In the phase-speed interval that includes U 0, U b,oru t at which the X has its singularity, X is not plotted.

6 11 J 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 VOLUME 6 magnitude of X is very small at phase speeds between U b and U t. This condition implies that waves with the same phase speeds as the basic-state wind between the cloud bottom (z b ) and the top of the shear layer (z s ) can encounter critical levels when they are propagating to the cloud top. Therefore, only a small amount of momentum flux by these filtered waves can be seen at the cloud top. Song et al. (003; SCL hereinafter) showed that gravity waves at the cloud top consist of waves that are generated by forcing at every level within storms and can propagate through the flow above each forcing level up to the cloud top. Note that the vertical propagation of the waves is allowed only when the vertical propagation condition [m 0 and c U(z), where m is the vertical wavenumber defined by () of SCL] is satisfied. In the present case of hydrostatic and Boussinesq system under constant basic-state wind shear, m is always positive, and thus gravity waves are filtered out only when they reach critical levels. Therefore, X can be regarded as a factor representing gravity wave filtering by the vertical propagation condition introduced by SCL. However, X has a structure with its peaks in phasespeed regions satisfying the vertical propagation condition. This structure implies that X is not characterized by the wave filtering alone. To get some insights into such a structure of X, we examine the vertical configuration of diabatic forcing, on which X also depends. For this, we consider a uniform wind case with N 1 N and a fictitious cooling between z z t and z b, together with an actual heating between z z b and z t, to ensure ŵ 0atz 0. Here, both the heating and cooling have identical magnitude (q 0 ) and vertical structure. In this case, q in (6) is changed into q (z)u s (z b, z t ) q (z)u s (z t, z b ), where u s is a function that becomes unity (zero) when z is between (outside of) its two arguments. Taking a Fourier transform of (6) with respect to z along with this modified q yields W g 1qˆ c p T 0 N 1 i m f cosm f z t cosm f z b m f z d sinm f z t sinm f z b, m f z d m f To ensure the existence of this Fourier transform, small vertical damping is considered, but factors due to the damping are ignored in (5). Nevertheless, we can obtain finite ŵ by appropriately choosing the path of integration for inverse Fourier transform of W in the complex plane. where W is the Fourier transform 1 of ŵ in z and m f is the vertical wavenumber associated with the vertical configuration of forcing. As m f approaches 1, wave resonance occurs, and the magnitude of W increases indefinitely. When the resonance occurs, W has a factor of {cos( 1 z t ) cos( 1 z b )] 1 z d sin( 1 z t ) sin( 1 z b )]}/( 1z d). This factor is exactly the same as X for the uniform wind case with N 1 N, which can be derived using (19). This result means that vertically propagating waves generated by a forcing are already in resonance for a given vertical configuration of the forcing that determines the vertical wavenumber spectrum of the forcing. Based on this reasoning, a resonance relationship will also exist in a shear flow, albeit it is not straightforward to derive. Thus, X should be regarded as a wave-filtering and resonance factor that represents resonance between vertical harmonics consisting of diabatic forcing and natural wave modes with vertical wavenumbers given by the dispersion relation, as well as wave filtering by the vertical propagation condition. The wave resonance relationship implies that the vertical structure of diabatic forcing ( q ) has direct impact on X and, as a result, on the spectral structure of gravity waves induced by the forcing. To examine the effects of vertical structure of forcing on X, we redefine q (z) as1 [(z z m )/z d ] nq, where n q is a positive integer. Figure 3 shows q for n q 1,, 4, 6, and ; and shows X corresponding to each q.asn q increases, forcing gradually becomes flat in the vertical direction, and thus the total amount of heating increases. As a result, the overall magnitude of X increases as n q increases. Figure 3 also shows that, as forcing becomes flat in the vertical direction, the forcing includes largem f harmonics, and at that time X has strong peaks near c U t. This result has fundamental implications for the analytic representation of the gravity wave spectrum beyond simply demonstrating such a dependency of X. The linear formulation, as in the current spectral momentum flux formulation, may be invalid at cs near U t because the nonlinearity {gq 0 x /[c p T 0 N(U t c) ]} of thermally induced internal gravity waves (Lin and Chun 1991) becomes significantly larger than unity near c U t. In hydrostatic assumption, waves with large vertical wavenumber (m) must have small c U. Therefore, if a forcing includes large-m f harmonics, the forcing generates large-m (small c U ) waves that are likely to violate the linear assumption. Thus, the momentum flux of such large-m waves may not be properly represented by (). Rather, it is better to consider them as saturated waves in large-m tails of their m spectrum, which should be distinguished from the sourcedependent m region. Therefore, to obtain a linear formulation that can be valid in the whole phase-speed region, in this study the vertical profile of the diabatic forcing was specified as the second-order polynomial in which only few large-m f harmonics are included. BAH04 compared their momentum flux formulation for stationary waves with that of CB98 and reported significant differences between the two in the case of small U t (i.e., large m; see Fig. 5 of BAH04). BAH04 attributed the difference to different spatial distribu-

7 JANUARY 005 S O N G A N D CHUN 113 FIG. 3. Vertical structure ( q ) of diabatic forcings with four different n q values (1,, 4, and 6) and X corresponding to each q for z b 3 km, z t 9 km, U 0 U t 0ms 1, and N 1 N 0.01 rad s 1. For comparison, q and X when n q approaches are plotted with gray lines. tions of the diabatic forcing and, based on numerical model results, asserted that their formulation is more realistic. The difference between BAH04 and CB98 is likely a result of the fact that CB98 used a box-shaped forcing with many large-m f harmonics whereas BAH04 used a sine-shaped forcing with only a single m f harmonic. Although the sine-shaped heating profile or second-order polynomial form used here with its maximum near midtroposphere is more realistic than the box-shaped profile, the momentum flux spectrum induced by such a smooth heating profile is not always more realistic. Using a high-resolution mesoscale model with a spatial grid size of 50 m, Lane et al. (003) recently demonstrated that there exist convective sources with extremely small spatial scales and that those sources can actually induce strong large-m gravity waves. Given that smooth heating profiles can exclude most large-m waves that might be generated by real convection, the reality of such smooth heating profiles should not be overestimated. The wave resonance relationship implies that the depth and height of forcing can also be important factors in determining X and as a result, shaping wave spectrum above the forcing. To examine the dependency of X on the depth and height of diabatic forcing, diabatic forcings with two different depths (z t z b ) of 4 and 7 km are considered. The dependency of X on the center height [(z t z b )/] of those forcings in the vertically uniform wind and stability case is shown in Fig. 4. Because the hydrostatic assumption is used in this study, the vertical wavelength * z corresponding to peak phase speed c* of gravity waves above the diabatic forcing is given by * z U t c* /N. In Fig. 4, * z s are plotted in parentheses over peaks of X. When z b 0 km (top panels in Fig. 4), X s have two dominant peaks that appear at c* 11 (0)ms 1 for z t z b 4 km (7 km). Salby and Garcia (1987) showed in their analytic study that vertically propagating waves forced by convection have vertical wavelength spectra with a broad peak centered on the vertical wavelength that is times the heating depth. When z b 0 km, the * z s are a little smaller than times the diabatic heating depth. Nonetheless, the relationship between * z and heating depth seems approximately valid considering that the variation of X in the range of m s 1 near c c* (corresponding to z * z 1.3 km) is negligible.

8 114 J 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 VOLUME 6 FIG. 4.The X values for diabatic forcings with two different depths (z t z b ) of (a) 4 and (b) 7 km and their dependency on the center height [(z t z b )/] of forcing under the vertically uniform basic-state wind (U 0 U t 0ms 1 ) and stability (N 1 N 0.01 rad s 1 ) condition. Peak phase speeds are numbered over peaks, and numbers in parentheses are vertical wavelengths corresponding to the peak phase speeds. However, when forcing is lifted from the ground (i.e., when z b 3 or 6 km), * z s are not determined by the heating depth alone and there appear several new peaks with * z s much larger or smaller than times the diabatic heating depth. This result is because, as z b increases, the vertical wavenumber spectrum of the forcing has peaks at new m f s other than the vertical wavenumber corresponding to times the forcing depth, and waves generated by the forcing have * z s that correspond to such new m f s according to the wave resonance relationship. This has already been observed by Beres et al. (00) and Alexander and Holton (004) in their numerical studies, and their results can be explained theoretically in the current study through the wave resonance concept. Some theoretical and numerical modeling studies on convectively forced waves (e.g., Salby and Garcia 1987; Fovell et al. 199; Alexander and Holton 004) have shown that waves above convection appear as a superposition of waves whose phase lines are inclined at different angles with respect to the vertical direction. This arrangement implies that waves above convection can have several dominant m s. Such a wave structure can be closely related to the resonance between elevated forcing and gravity waves, as shown in the middle and bottom panels of Fig. 4. The current momentum flux

9 JANUARY 005 S O N G A N D CHUN 115 formulation explicitly takes into account elevated forcing and thus can represent such a wave structure. In the momentum flux formulation by BAH04, the bottom of heating is assumed to be located at the ground. As shown in the top panels of Fig. 4, forcing with z b 0 induces gravity waves with only two dominant phase speeds. Therefore, the detailed wave structure above convection may not be explicitly represented by BAH04 s formulation. Although the spectral structure of wave momentum flux is affected by (c), the structure might not be modified significantly by (c) as long as (c) is smooth or does not have several strong peaks. Effects of (c) will be discussed later in this section. Figure 5 shows the dependency of X on N 1 (Fig. 5a) and N (Fig. 5b) for the uniform (U 0 U t 10 m s 1 ) and the sheared (U 0 10ms 1 and U t 10ms 1 ) basic-state wind profiles when z b.5 km, z s 6 km, and z t 11 km. The shape of X in the phasespeed domain strongly depends on N 1.AsN 1 increases, peaks of X in the uniform wind case (in the shear case) move away from c U t (U 0 and U t ), and phase speed widths ( c ) of the peaks increase in both the uniform and sheared wind cases. In the uniform wind case, this dependency is readily explained using the resonance relationship. The resonance relationship indicates that m* f where the vertical wavenumber spectrum of forcing has a peak is equal to * 1 (N 1 / U t c*, where c* is peak phase speed). Therefore, as N 1 increases, U t c* also increases. That is, peaks in the phase-speed domain should move away from U t. Also as N 1 increases, the numerator of the phase (e.g., N 1 z t / U t c ) of cosine and sine terms in (5) increases and, FIG. 5. Dependence of the X on (a) N 1 and (b) N for the (left) uniform and (right) sheared basic-state wind when z b.5 km, z s 6 km, and z t 11 km. Stippled areas represent phase-speed intervals including singularities. Contours of 0.5, 1, 1.5, and are plotted, and values larger (smaller) than 1 (0.1) are darkly (lightly) shaded.

10 116 J 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 VOLUME 6 as a result, the phase change of the same amount occurs over a wide range of U t c. That is, c increases as N 1 increases. In contrast to N 1, the dependency of X on N is not significant. As N increases, only the magnitude of X decreases slightly without a change in its shape. However, the magnitude of momentum flux spectrum, which has a factor of N / U t c [see ()], can be significantly changed by N, especially when strong convection penetrates the tropopause, near which the stability varies rapidly in the vertical direction. To account for the way in which wave momentum flux is determined, the phase-speed spectra of the wave momentum flux generated by three different diabatic forcings are examined (Fig. 6). For this, a basic state with N 1 N 0.01 rad s 1, U 0 18ms 1, and U t ms 1 is considered, and diabatic forcing with n q 1, z b 3 km, and z t 9 km is assumed. The basic-state wind profile used in Fig. 6 is almost identical to that used in numerical simulation by SCL. All three diabatic forcings have the same x (5 km) and t (0 min) but different moving speeds (c q 0, 10,0ms 1 ). The x, t, and c q are chosen based on the characteristics of dominant convective cells in the convective system numerically simulated by SCL. The maximum magnitude FIG. 6. (a) The N X / U t c for n q 1 (thin solid) and (c) with x 5 km, t 0 min, and q 0 Jkg 1 s 1 (thick gray) and (b) the wave momentum flux. Parameters considered are N 1 N 0.01 rad s 1, U 0 18ms 1, U t ms 1, z b 3 km, z s 6 km, and z t 9 km. The function (c) and wave momentum flux induced by the forcing function are plotted with lines of the same pattern (dashed for c q 0ms 1, solid for c q 10ms 1, and dotted for c q 0ms 1 ). q 0 of diabatic forcing is assumed to be Jkg 1 s 1 for all three cases to produce a magnitude of the wave momentum flux comparable to that presented in previous numerical modeling studies (e.g., Alexander and Holton 1997; Beres et al. 00). The wave momentum flux is calculated in a range from c 100 to 100 m s 1 and is assumed to be zero in the phase-speed interval in which X is singular. In (), L t is set equal to t (0 min) and L x is set to 40 km taking into account that energy of waves with the largest c (100ms 1 ) horizontally propagates as much as 10 km during the period of L t. In (), 0 and T 0 are set to the air density and temperature, respectively, at z z t in the atmosphere with constant stability N 1. The three values with different c q s result in wave momentum flux spectra that are somewhat different from each other. This result is because the three (c) s are projected onto different portions of N X / U t c according to c q. Overall, the wave momentum flux spectra, however, have strong peaks at phase speeds at which (c) is enhanced by peaks of X, which approximately coincide with peaks of N X / U t c. This condition implies that the basic-state wind and stability and the vertical spectrum of diabatic forcing, which determine the X, strongly constrain the structure of the wave momentum flux spectrum. However, it does not mean that the forcing spectrum is not important. The forcing spectrum (c) can significantly change the magnitude of the peaks in the wave momentum flux spectra. The magnitude of the momentum flux at the cloud top, which is one of the important factors for the parameterization, varies significantly at some phase speeds (e.g., near 0 and 0 10ms 1 ) according to values of c q s. When c q 0, 10,and0ms 1, the net wave momentum fluxes are , , and Nm, respectively. Because the momentum flux is positive (negative) for c U t (c U t ) (where U t ms 1 in this case) and N X / U t c is relatively small between c 10ms 1 and U t, the diabatic forcing with c q 0ms 1 tends to produce a more positive momentum flux while that with c q 0 ms 1 produces a more negative momentum flux for given X and N / U t c. The wave momentum flux spectrum also depends on spatial and temporal scales ( x and t ) of diabatic forcing because (c) can have a broad or narrow distribution according to x and t. However, the dependency of the wave momentum flux spectrum on (c) s with various x and t is not examined in this study because x and t are highly dependent on the type of convective system, and thus it is not easy to choose various scales of convective cells. 4. Extension to three-dimensional wave propagation The formulation () of the cloud-top wave momentum flux is obtained in the two-dimensional framework. Even though most of the GWD parameterization

11 JANUARY 005 S O N G A N D CHUN 117 schemes have been developed within a two-dimensional framework (e.g., Holton 198; Kershaw 1995; CB98; CB0), and the two-dimensional formulation given in () is still useful for the GWD parameterization, the main limitation of the two-dimensional formulation is that wave propagation allows for only one horizontal direction (positive and negative), which is usually determined to be parallel to the horizontal wind velocity at a certain reference (launch) level. To get a more reliable formulation of cloud-top momentum flux for use in GCMs, the two-dimensional theory in previous sections is extended to three dimensions. Similar to (5), a single equation for w is derived from three-dimensional governing equations (not shown) for linear perturbations in an inviscid, hydrostatic, and Boussinesq airflow system: D Dt w d u D z dz Dt w x d D dz Dt y w N w w g q x y c p T 0 q q x y, 6 where D/Dt /t u/x /y. Here, u and are zonal and meridional background winds, respectively, and N is the static stability. Also, u,, and N are assumed to be functions of the altitude z only. Taking the three-dimensional Fourier transform (x k, y l, t ) of (6) gives the Taylor Goldstein equation for the linear internal gravity waves propagating in a given horizontal direction ( ) induced by a diabatic forcing: ŵ z N U, zz U, z c U, z c ŵ gqˆ q c p T 0 U, z c, 7 where ŵ and qˆ, the three-dimensional Fourier transforms of w and q, are functions of magnitude of horizontal wavenumber vector k h ( k h k l, where k and l are the zonal and meridional wavenumbers, respectively), wave propagation direction, groundrelative frequency, and the altitude z. The propagation direction is the angle measured counterclockwise from the east to k h, and thus k and l are represented as k h cos and k h sin, respectively. In (7), the basicstate wind U(, z), defined as u cos sin, isthe background wind in the direction of, and c (/ k h ) is the phase speed of gravity waves. In the threedimensional theory, gravity waves with c 0(c 0) propagate in the direction of ( ). The three-dimensional Taylor Goldstein equation (7) is almost of the same form as (6) in the twodimensional case except that basic-state wind U additionally depends on. After some mathematical manipulations, the wave momentum flux spectrum in a three-dimensional framework can be written as 3 g Mc, sgnu, z c 0 L x L y L tc p T 0 N 1 N U, z c X c,. 8 It is of interest that the term X in this three-dimensional case is exactly the same as that in the two-dimensional case except that the basic-state wind and phase speed are defined as being parallel to wave propagation direction. However, the definition of function in the three-dimensional case is somewhat different: k s c, qˆr, c, 1 cc s 0 r dr. 9 To obtain (c, ) using (9), the diabatic forcing function should be defined as a function of y as well as x and t. Similar to (3), the three-dimensional Gaussianshaped diabatic forcing function is given as qx, y, t q 0 exp x x 0 c qx t t 0 h exp y y 0 c qy t t 0 h exp t t 0 t, 30 where c qx and c qy are the zonal and meridional components of the moving speed of diabatic forcing, respectively. This Gaussian diabatic forcing q is isotropic with respect to x and y, and thus qˆ(r, c, ) calculated from (9) does not depend on. By substituting (30) into (9), the diabatic wave source function is calculated as follows: c, q 0 h t c c qh c, 31 0 where c 0 h / t and c qh (c qx cos c qy sin) isthe speed of the convective cell moving in the direction of wave propagation. 5. Validation of the wave momentum flux spectrum In this section, the wave momentum flux spectra formulated in this study are validated using mesoscale numerical model results to prove the reliability of the formulation. The two-dimensional formulation of the wave momentum flux spectrum () is compared with momentum flux spectra obtained from three twodimensional convection simulations (Fig. 7). One of the three simulations is a midlatitude squall-line case in SCL, and the other two are tropical-storm cases in

12 118 J 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 VOLUME 6 FIG. 7. (top) The basic-state wind profiles and (bottom) the two-dimensional analytic wave momentum flux spectra and numerically calculated momentum flux spectra for (a) a midlatitude squall line simulated by Song et al. (003) and for two tropical storms under (b) no upper-level shear and (c) strong upper-level shear simulated by Beres et al. (00). The wave momentum flux spectra and wind profiles in numerical modeling are plotted with dotted lines, and the analytic wave momentum flux spectra and wind profiles used to obtain the analytic spectra are plotted with solid lines. Shading in the top panels denotes the vertical region of diabatic forcing used to obtain the analytic spectra. Beres et al. (00, BAH0 hereinafter). BAH04 validated their wave momentum flux spectrum using the frequency spectrum of diabatic forcing obtained from the simulations by BAH0 as well as a red-noise spectrum assumed for use in large-scale models. However, in this study, validation is performed using the Gaussian-shaped forcing with n q 1 used in the our analytic calculation to focus on a direct application of the current formulation to the GWDC parameterization in GCMs. It is not straightforward to estimate z b and z t from a numerically simulated mesoscale convective system because there exist convective cells with various z b and z t during the evolution of the convective system. To avoid such uncertainties, we determine z b and z t using the level of free convection (LFC) and the level of neutral buoyancy (LNB), respectively, of the thermodynamic soundings used in simulations. Given the broad depth of low-level inflow to convection, z b and z t are determined by averaging the LFC and LNB for air parcels lifted between the ground and z 3 km and thus are obtained as.75 and 10.5 km (.75 and 8.75 km), respectively, for the SCL (BAH0). The basic-state wind used in our study is appropriately matched with the wind profile used in each simulation. The match is achieved by determining z s between z b and z t for U 0, U b, and U t given by the wind profiles used in simulations. When the match cannot be made (e.g., z s z b or z s z t ), z s is set to the closest one of z b and z t, and then U b is modified. Because z t is located far below the tropopause in both the SCL and BAH0 cases, N 1 and N are set equal to 0.01 (0.01) rad s 1 for the SCL (BAH0), respectively. For the calculation of (c), x and t are assumed to be 5 km and 0 min, respectively. In the BAH0 cases, q 0 is assumed to be 3.8 J kg 1 s 1. The CAPE, averaged in the same way as in z b and z t,in SCL is about.4 times that in BAH0. However, q 0 in the SCL case is set to 1. times the q 0 used in the BAH0 cases to consider the cancellation between waves induced by nonlinear and diabatic wave sources in the SCL case. The moving speed of the diabatic forcing c q in the BAH0 cases is set to 10 m s 1 as presented by BAH04. In the SCL case, c q is set to 0 m s 1, the mean value of the moving speeds of the convective cell (1ms 1 ) and nonlinear forcing (7 m s 1 ), to include the effects of nonlinear forcing.

13 JANUARY 005 S O N G A N D CHUN 119 As in the case of Fig. 6, L x and L t are set to 40 km and 0 min, respectively. Numerically simulated momentum flux spectra for the SCL and BAH0 cases are calculated at z 18 and 15 km, respectively. In Fig. 7c, for comparison with model results at z 15 km, the analytic momentum flux between c 1and0ms 1 is removed, assuming critical-level filtering between z z t (8.75 km) and 15 km. Overall features of the numerically simulated momentum flux spectra are represented reasonably well by the current formulation. However, in any cases, there are discrepancies between the analytic and simulated spectra. In the SCL case, the main difference is in the region of large negative phase speeds (c 30 m s 1 ) and small positive phase speeds. The magnitude of the formulated fast westward- (slowly eastward-) moving waves is smaller (larger) than that of the simulated. The discrepancy for the westward-moving waves occurs mainly because the effects of fast-moving nonlinear forcing are not well approximated by diabatic forcing. The discrepancy for the eastward-moving waves likely occurs because (c) derived from Gaussian-shaped forcing does not accurately represent (c) obtained from the convective system in the SCL case (not shown). The cases in BAH0 cannot easily be directly compared with the current formulation because the wind profiles used in this study are very different from those used in the simulations. Despite these discrepancies due to the simplified structure of forcing and the basic-state wind profile, the momentum flux spectra formulated in the two-dimensional framework can represent spectra obtained from mesoscale convection simulations reasonably well. The three-dimensional wave momentum flux formulation is also validated using the momentum flux spectra over the convective system simulated by Piani et al. (000, PDAH hereinafter; Fig. 8). For the calculation of the analytic spectrum, z b is set to 3.5 km based on the horizontally and temporally averaged (i.e., large scale) heating rate shown by PDAH in their Fig. 10, and z t is set to 10.5 km given that the maximum of the largescale heating rate (about 3Kday 1 ) is located at about z 7 km. The basic-state wind profile is found through the matching process, and N 1 and N are set to 0.01 rad s 1. As in the two-dimensional case, h and t are set to 5 km and 0 min, respectively. In this three-dimensional case, q 0 is set to 6.1 J kg 1 s 1, which is much larger than that in the two-dimensional case. This large q 0 is required because the three-dimensional forcing is highly localized in the horizontal in comparison with the two-dimensional forcing, which is assumed to be cyclic in the direction perpendicular to the twodimensional domain. In (8), L x L y is set to (10 km), L t is set to 0 min, is set to zero, and c qh is assumed to be 10 m s 1 relative to the convective system. The momentum flux spectrum formulated in the threedimensional framework can approximately reproduce the magnitude and phase-speed region of dominant peaks in the zonal wave momentum flux spectrum in PDAH. However, there are also some discrepancies between the formulated and numerically simulated spectra, as in the two-dimensional case, and the discrepancies are attributed to assumed forcing and wind structure. In this three-dimensional case, the discrepancies are also caused by considering only a single propagation direction. Despite such discrepancies, the momentum flux spectrum formulated in the three-dimensional framework can also represent a spectrum obtained from mesoscale convection simulations reasonably well. 6. Summary and conclusions In this study, the momentum flux of convectively forced internal gravity waves in an inviscid, hydrostatic, and Boussinesq airflow system was analytically formulated as a function of ground-based phase speeds in both two- and three-dimensional frameworks. For the analytic formulation, we considered a three-layer atmosphere with piecewise constant stability in the forcing region and above, a constant vertical wind shear in the lowest layer, and a uniform flow above. Because the formulation of the cloud-top momentum flux in the two- and three-dimensional frameworks are similar (even though the three-dimensional framework allows for various wave propagation directions and threedimensionality of forcing), detailed analysis of cloudtop momentum flux and its dependency on the basicstate wind and stability were only performed for the two-dimensional case. Space- and time-averaged wave momentum flux was formulated as a function of phase speed for the given basic-state wind and stability conditions and Gaussianshaped diabatic forcing. It was found that the wave momentum flux spectrum in the phase-speed domain is determined by the spectral combination of N / U t c, X, and diabatic source function. The X depends on the basic-state wind and stability and the vertical configuration (structure, height, and depth) of diabatic forcing. This X includes two physical meanings. First, X acts as a wave-filtering factor that controls the effectiveness of the diabatic source (c). Considering that N / U t c also implies the critical-level filtering, the combination of N / U t c, X, and (c) represents the effective source mechanism that was found in SCL through explicit gravity wave calculation using a mesoscale numerical model. Second, X acts as a factor that represents resonance between the vertical harmonics consisting of forcing and natural wave modes with the vertical wavenumbers given by the dispersion relation of internal gravity waves. This resonance relationship explains why the structure of X with peaks strongly depends upon the vertical wavenumber spectrum of forcing. As a result, X can be regarded as a wave-filtering and resonance factor. Thus, the spectral

14 10 J 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 VOLUME 6 FIG. 8. (a) The basic-state zonal wind profiles, (b) three-dimensional analytic zonal wave momentum flux spectrum, and (c) zonal momentum flux spectrum above a three-dimensional tropical convective system simulated by Piani et al. (000). The wind profile in numerical modeling is plotted with dotted lines, and the analytic wave momentum flux spectrum and wind profile used to obtain the analytic spectrum are plotted with solid lines. Shading in (a) denotes the vertical region of diabatic forcing used to obtain the analytic spectrum. The momentum flux spectrum shown in (c) is taken from Fig. 16 in Piani et al. (000). distribution of the wave momentum flux is determined by a combination of the wave-filtering and resonance factor and the diabatic forcing spectrum according to the degree of overlapping between the two. To develop a more realistic wave drag parameterization, the phase-speed spectrum of the wave momentum flux was analytically formulated in the three-dimensional framework. The formulation of the threedimensional momentum flux is similar to the two-dimensional case. However, unlike the two-dimensional case where wave propagation is in only one direction (with positive and negative) in a horizontal plane, the three-dimensional framework allows wave propagation in any direction in a horizontal plane and thus makes it possible to more realistically parameterize effects of convectively forced internal gravity waves in large-scale numerical models. The wave momentum flux spectrum formulated in this study is validated using the momentum flux spectra obtained from mesoscale convection simulations in two- and three-dimensional frameworks. The analytic spectrum formulated in this study can reproduce explicitly modeled wave momentum flux spectra reasonably well. However, there exist several discrepancies between the analytically formulated and numerically simulated spectra. One of the main reasons for the difference, besides the degree of simplification in analytical models as compared with numerical modeling including nonlinearity and cloud microphysical processes, is the neglect of the nonlinear forcing of the mesoscale convective system. Convective gravity waves are generated by two forcings: nonlinear forcing, representing the mechanical oscillator mechanism; and diabatic forcing, representing the thermal forcing mechanism; and