Effects of Nonlinearity on Convectively Forced Internal ...

Effects of Nonlinearity on Convectively Forced Internal Gravity Waves: Application toa Gravity Wave Drag Parameterization

HYE-YEONG CHUN, HYUN-JOO CHOI, AND IN-SUN SONG*

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

(Manuscript received 21 July 2006, in final form 14 May 2007)

ABSTRACT

In the present study, the authors propose a way to include a nonlinear forcing effect on the momentumflux spectrum of convectively forced internal gravity waves using a nondimensional numerical model(NDM) in a two-dimensional framework. In NDM, the nonlinear forcing is represented by nonlinearadvection terms multiplied by the nonlinearity factor (NF) of the thermally induced internal gravity wavesfor a given specified diabatic forcing. It was found that the magnitudes of the waves and resultant momen-tum flux above the specified forcing decrease with increasing NF due to cancellation between the twoforcing mechanisms. Using the momentum flux spectrum obtained by the NDM simulations with variousNFs, a scale factor for the momentum flux, normalized by the momentum flux induced by diabatic forcingalone, is formulated as a function of NF. Inclusion of the nonlinear forcing effect into current convectivegravity wave drag (GWD) parameterizations, which consider diabatic forcing alone by multiplying thecloud-top momentum flux spectrum by the scale factor, is proposed. An updated convective GWD param-eterization using the scale factor is implemented into the NCAR Whole Atmosphere Community ClimateModel (WACCM). The 10-yr simulation results, compared with those by the original convective GWDparameterization considering diabatic forcing alone, showed that the magnitude of the zonal-mean cloud-top momentum flux is reduced for wide range of phase speed spectrum by about 10%, except in the middlelatitude storm-track regions where the cloud-top momentum flux is amplified. The zonal drag forcing isdetermined largely by the wave propagation condition under the reduced magnitude of the cloud-topmomentum flux, and its magnitude decreases in many regions, but there are several areas of increasing dragforcing, especially in the tropical upper mesosphere and lower thermosphere.

1. Introduction

Vertically propagating gravity waves generated byvarious sources in the troposphere have profound ef-fects on the large-scale circulation in the middle atmo-sphere where waves are broken, filtered at their criticallevel, or dissipated by eddy viscosity during their propa-gation (Lindzen 1981; Matsuno 1982). Among the vari-ous sources, cumulus convection is one of the majorsources of nonstationary gravity waves with a widephase speed spectrum. In the tropics, convectivelyforced gravity waves can contribute to momentum forc-

ing required to drive the quasi-biennial oscillation andsemiannual oscillation (Alexander and Holton 1997;Sassi and Garcia 1997).

There are several numerical modeling studies of con-vectively forced gravity waves and their generationmechanisms. Pandya and Alexander (1999) showedthat the spectral characteristics of convective gravitywaves in a quasi-linear simulation forced by diabaticforcing alone is similar to that in a fully nonlinear simu-lation, although the amplitude of waves is much largerthan that in the fully nonlinear simulation. Lane et al.(2001) showed that convective gravity waves are gen-erated by two forcings, diabatic and nonlinear forcing,and demonstrated the importance of the nonlinear forc-ing mechanism in the generation of convective gravitywaves by comparing the magnitude of the nonlinearand diabatic forcing of a simulated storm. Song et al.(2003, hereafter SCL) performed quasi-linear simula-tions of gravity waves generated by the diabatic forcingand nonlinear forcing separately and found that the

* Current affiliation: Global Modeling and Assimilation Office,NASA GSFC, Greenbelt, Maryland.

Corresponding author address: Prof. Hye-Yeong Chun, Depart-ment of Atmospheric Sciences, Yonsei University, Shinchon-dong, Seodaemun-ku, Seoul 120-749, South Korea.E-mail: [email protected]

FEBRUARY 2008 C H U N E T A L . 557

DOI: 10.1175/2007JAS2255.1

© 2008 American Meteorological Society

JAS4133

characteristics of the gravity waves induced by either ofthe forcings are determined by the effective forcing thatis filtered by the vertical propagation condition of grav-ity waves in the spectral domain. The effective diabaticforcing and effective nonlinear forcing are comparableand largely out of phase with each other, suggesting theimportance of both forcing mechanisms in generationof convective gravity waves.

To understand the implications of SCL on convectivegravity parameterization, Chun et al. (2005) calculatedthe momentum flux spectrum of convective gravitywaves in the stratosphere using SCL’s numerical simu-lations. They showed that the momentum flux inducedby either forcing differs significantly from each other aswell as from the momentum flux obtained from thecontrol simulation including both forcings. This is be-cause cancellation of momentum flux by cross-corre-lation terms between the two forcings cannot be repre-sented in the momentum flux by a single forcing. Basedon this result, Chun et al. suggested the inclusion ofnonlinear forcing in recently developing convectivegravity wave drag (GWD) parameterizations with adiabatic forcing alone, such as Beres (2004) and Songand Chun (2005, hereafter SC05), although those pa-rameterizations are still much more realistic than thespectral gravity wave drag parameterization used incurrent large-scale models that have a prescribedsource spectrum.

A most straightforward method that includes thenonlinear forcing effect on the momentum flux spec-trum is likely to obtain an analytical solution of gravitywaves induced by both forcings. However, one of theproblems, aside from the additional effort required forits mathematical treatment, is that compared with thediabatic forcing we do not know much about the struc-ture of nonlinear forcing and its relationship with thediabatic forcing, which might be strongly coupled witheach other. Given this situation, one feasible way toinclude the nonlinear forcing effect on the momentumflux of convective gravity waves is to solve weakly non-linear perturbations forced by diabatic forcing, as wasdone by Chun and Baik (1994) in a uniform back-ground wind and stability condition and by Chun (1997)that included background wind shear. In weakly non-linear systems the zeroth-order solution representswaves induced by diabatic forcing alone, while the first-order solution represents waves induced by nonlinearforcing (nonlinear advections of the zeroth-order mo-mentum and heat fluxes).

There are, however, several difficulties in this ap-proach. First, obtaining an analytical formulation ofmomentum flux [the sum of at least three terms, u0w0,�u0w1, and �u1w0, where u0, w0 (u1, w1) are the zeroth

(first)-order perturbation zonal and vertical velocitiesand � is the nonlinearity factor (NF) of the thermallyinduced gravity waves, respectively] is very tediousmathematically, especially under a relatively realisticbackground condition such as the three-layer structureconsidered in SC05. For some terms in the first-ordersolution, numerical calculation might be involved(Chun 1997). Second, and perhaps more importantly,the weakly nonlinear solution may not apply directly tothe highly nonlinear flow regime such as mesoscale con-vective storms considered in the present study.

To overcome these analytical difficulties, while stillobtaining some insight into the influence of nonlinearforcing on the momentum flux of convective gravitywaves in a simple dynamic system, in the present studywe use a nondimensional numerical model (NDM) in atwo-dimensional framework, which is similar to thatused in Baik and Chun (1996, hereafter BC96). TheNDM simulates gravity waves forced by a specified dia-batic forcing according to a nonlinearity factor of thethermally induced gravity waves. In NDM, nonlinearadvection can be considered as the nonlinear forcing, ofwhich its magnitude depends largely on the nonlinear-ity factor.

This paper is organized as follows: section 2 describesthe model hierarchy. In section 3, perturbations in-duced by a specified diabatic forcing under various non-linearity factors are presented. In section 4, the cloud-top momentum flux spectrum is calculated and com-pared with that derived from diabatic forcing alone.Based on this result, we determine a scale factor for themomentum flux normalized by the momentum flux in-duced by the diabatic forcing alone. The scale factor isthen included in the cloud-top momentum flux formu-lation by SC05, and impact of the new convective GWDparameterization in a GCM is examined in section 5.Summary and discussion are given in the final section.

2. Nondimensional numerical model

The equations governing perturbations in a two-dimensional, hydrostatic, nonrotating, Boussinesq air-flow system forced by diabatic forcing can be written as

�u

�t� U

�u

�x� w

dU

dz� u

�u

�x� w

�u

�z� �

��

�x, �1�

�b

�t� U

�b

�x� N2w � u

�b

�x� w

�b

�z�

gQ

cpT0, �2�

�u

�x�

�w

�z� 0, �3�

��

�z� b. �4�

558 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 65

Here u and w are the perturbation zonal and verticalwind velocities, � is the perturbation kinematic pres-sure, b is the perturbation buoyancy, U is the basic-statewind, N is the Brunt–Väisälä frequency, cp is the spe-cific heat of air at constant pressure, T0 is a constantreference temperature, g is the gravitational accelera-tion, and Q is the diabatic forcing. For direct compari-son with SC05’s linear analytical calculation of the mo-mentum flux spectrum, we consider the basic-statewind and stability condition in a three-layer structureand diabatic forcing as in SC05 (see Fig. 1 of SC05):

U�z� � �U0 � �z for 0 � z � zs,

Ut for z � zs,�5�

N�z� � �N1 for 0 � z � zt

N2 for z � zt,�6�

Q�x, z, t� � q�x, t��z�, �7�

q�x, z� � Q0 exp��x � x0 � cq�t � t0�2

��x �2�2 � exp ��

�t � t0�2

��t �2�2 �, �8�

��z� � �1 � ��z � zm��zd2 for zb � z � zt

0 elsewhere.�9�

Here U0 is the basic-state wind at the surface, zb and zt

are the bottom and top heights of the specified diabaticforcing respectively, zs is the top of the shear layer, Ut

is the background wind above zs, N1 and N2 are thebuoyancy frequencies below and above zt respectively,� is the wind shear [�(Ut � U0)/zs], Q0 is the magnitudeof the diabatic heating, x0 and t0 are the center of thediabatic forcing in the zonal direction and time for themaximum forcing respectively, cq is the moving speed,�x and �t are the horizontal and temporal scales of thediabatic forcing, respectively, zm � (zt � zb)/2, and zd �(zt � zb)/2.

As suggested by BC96, the following nondimensionalvariables are introduced:

t �L

Uct, x � Lx, z �

Uc

Ncz, U � UcU,

N � NcN, � � Nc�,

u �gQ0L

cpT0NcUcu, w �

gQ0

cpT0Nc2w, � �

gQ0L

cpT0Nc�,

b �gQ0L

cpT0Ucb, Q � Q0Q. �10�

Here L, Uc, and Nc are the characteristic scales of thehorizontal length, basic-state wind, and stability, re-spectively, and the caret quantities are dimensionless.The other variables in (1)–(9) related to height (zb, zt,zs, zm, and zd), horizontal (�x and x0) and time (�t andt0) scales, wind (U0, Ut, and cq), and stability (N1 andN2) are nondimensionalized by using the correspondingcharacteristic scales in (10). Substituting (10) into (1)–(9) yields (with all the carets dropped hereafter) thefollowing nondimensional governing equations:

�u

�t� U

�u

�x� w

dU

dz� �u

�u

�x� w

�u

�z� � ��

�x, �11�

�b

�t� U

�b

�x� N2w � �u

�b

�x� w

�b

�z� � Q, �12�

�u

�x�

�w

�z� 0, �13�

��

�z� b. �14�

Here � is the nonlinearity factor of the thermally in-duced gravity waves defined by Lin and Chun (1991) as

�gQ0L

cpT0NcUc2 . �15�

The nonlinearity factor can be interpreted as a scaleratio of the perturbation horizontal wind to the basic-state wind. The nonlinearity factor is proportional tothe heating rate and the horizontal scale of heating,while it is inversely proportional to the square of thebasic-state wind speed and the Brunt–Väisälä fre-quency. For cases with stronger inflow in the convectiveforcing region, convective cells may blow downstreamrather quickly before they generate gravity waves, andthe amplitude of perturbations will be smaller. Whenthe stability is smaller in the region where diabatic forc-ing exists, stronger perturbations are expected, as usu-ally seen from the numerical simulations of mesoscalecirculations associated with convection. The depen-dency of the basic-state wind and stability as well asdiabatic heating on thermally induced internal gravitywaves has been demonstrated clearly in previous ana-lytical studies (e.g., Smith and Lin 1982; Lin and Smith1986; Lin and Chun 1991). The nonlinearity factor isalso a key parameter in the convective GWD param-eterization scheme proposed by Chun and Baik (1998).

Note that calculating the nonlinearity factor for realor numerically simulated storms is not straightforward,especially under vertically various basic-state wind andstability conditions, since there are multiple convectivecells with different horizontal scales. For the SCL case,one can estimate � 2 with L � 50 km (the dominant


horizontal wavelength of diabatic forcing as shown inFig. 5b of SCL), Q0 � 3.63 J kg�1 s�1 (diabatic heatingaveraged over 50 km in the major convection region),Uc � 17.8 m s�1, Nc � 0.01 s�1, and T0 � 273 K. HereUc is estimated by Um � cm, where Um (2 m s�1) is thebasic-state wind in the middle region of convection (z �6 km) and cm (�15.8 m s�1) is the mean speed of con-vective cells. Even if we consider a single convectivecell with a smaller horizontal scale (say L � 5 km), thenonlinearity factor will not be changed significantly,given that the heating rate averaged over a smallerhorizontal domain will be larger. The nonlinearity fac-tor is sensitive to the magnitude of the basic-state windand stability for a given diabatic heating structure. Onecan calculate the nonlinearity factor using the basic-state variables at the cloud top, as in the convectiveGWD parameterization by Chun and Baik (1998).However, for the SCL case, the basic-state wind is con-stant with height above z � 6 km and the basic-statestability is almost constant in the troposphere. Conse-quently, NF evaluated at the cloud top is the same asthat in the middle region of convection.

The basic-state wind, stability, and diabatic forcing in(5)–(9) are nondimensionalized with exactly the sameforms. Note that (11)–(14) are identical to those of theNDM used in BC96 except that the Rayleigh frictionand Newtonian cooling terms are excluded in (11) and(12), respectively. Because an inviscid flow is consid-ered in the current NDM, the linear part of the gov-erning equations (11)–(14) is identical to the governingequation set of SC05. Therefore, the current NDM re-sults, compared with SC05’s linear solution, can isolatenonlinearity effects of the thermally induced internalgravity waves. Equations (11)–(14) are solved numeri-cally using a finite-difference method. The time deriva-tive is calculated using a leapfrog scheme, and a fourth-order compact implicit scheme (Navon and Riphagen1979) and a centered difference scheme are used for thefirst derivative terms with respect to x and z, respec-tively. A flat bottom boundary condition (w � 0 at z �0) and the upper radiation condition as proposed byKlemp and Durran (1983) are imposed. At the lateralboundary, the radiation condition proposed by Betzand Mittra (1992) is imposed, which allows waves topropagate at the lateral boundary more efficiently, atleast in the present simulations, compared with the ra-diation condition of Orlanski (1976) used in BC96. TheAsselin (1972) time filter and the space smoothing ofthe fourth-order type (Perkey 1976) also are applied.

The model domain is 200 wide and 20 deep with ahorizontal and vertical grid size of 0.1. The model isintegrated up to 25.92 with a time interval of 0.0018. Fordirect comparison with SC05’s analytically calculated

momentum flux spectrum we choose L� 5 km, Uc �18 m s�1, and Nc � 0.01 s�1. For the given characteristicscales, the dimensional model domain is 1000 km 36 km with horizontal and vertical grid sizes of 500 and180 m, respectively. In all the numerical simulations, weset zb � 1.53, zt � 5.83, zs � 3.33, U0 � �1, Ut � 0.11,�x � 1, �t � 4.32, cq � �1.1, x0 � 100, t0 � 12.96, andN1 � N2 � 1. Note that a vertically uniform Brunt–Väisälä frequency was assumed as in SC05 since the topof clouds is far below the tropopause in SCL. For thegiven diabatic forcing structure, the nonlinearity factoris varied through the range NF � 0�5 between thedifferent experiments, and the influence of the nonlin-earity on the momentum flux spectrum above the dia-batic forcing is investigated.

In the present NDM simulations, nonlinear advec-tion terms are retained from the surface to model top.Therefore, two factors associated with the nonlinearityaffect the momentum flux above the forcing region:nonlinear forcing in the troposphere and nonlinearityabove the forcing region. Note that in the present studywe call the nonlinear advection below and above thetop of the specified diabatic forcing as “nonlinear forc-ing” and “nonlinearity,” respectively, because the non-linear advection in the region of the specified diabaticforcing is largely determined by the forcing itself, whilenonlinear advection above the forcing representspurely a nonlinearity of wave disturbance. The latterfactor is not likely to be significant in the momentumflux spectrum above the forcing region, as shown inSCL (their Fig. 8c), except for extremely high values ofNF. (This is examined through additional simulations,discussed in detail in section 4, in which nonlinear forc-ing is limited to the height of the specified diabaticheating.) Therefore, nonlinear forcing and nonlinearityare used interchangeably in the present paper.

3. Nondimensional model results

Figure 1 shows the vertical velocity field from theoriginal NDM simulation (CTL hereafter) with respectto NF at t � 12.96, the time for the maximum forcing.The region for the specified diabatic forcing defined in(7)–(9) is overlaid with thick gray line. It is clear that asNF increases, the magnitude of the vertical velocityabove the forcing region decreases, while the maximumupdraft in the forcing region increases slightly. At thistime, the maximum updraft exists near the center (x0 �100) of the maximum forcing but at different heightswith respect to NF, slightly below the maximum heatingheight (zm � 3.68) for NF � 0 (z � 3.4) and above it forother NFs ranging from z � 3.7 to 4.5. As NF increases,an eastward-tilted updraft below the forcing top ex-


tends upward with a relatively smaller horizontal sizeand larger magnitude. On the other hand, the waveamplitude above the forcing deceases as NF increases.

The dependency of the magnitude of waves on thenonlinearity factor can be explained by Fig. 2, whichshows the difference of the vertical velocity betweeneach nonlinear case and the linear (NF � 0) case. Thedifference field represents the impact of nonlinearity onthe vertical velocity for a given diabatic heating. Shad-ing in Fig. 2 represents the area where the vertical ve-locity induced by diabatic forcing alone (wNF�0) is outof phase with that induced by nonlinear forcing alone(wNF � wNF�0). When the nonlinearity is included,downdraft on the upstream and updraft on the down-

stream of the specified heating are induced in the forc-ing region, and their magnitudes increase as NF in-creases (Figs. 2b–f). Note that the upstream is east(west) of the heating center below (above) z � 3, giventhat the basic-state wind blows from east to west belowz � 3 and changes its direction above. In most regions,in particular where nonlinearity-induced perturbationsare strong, vertical velocity perturbations induced bythe nonlinear forcing are out of phase with those in-duced by diabatic forcing alone. The only exceptionoccurs in the region where the two centers of the maxi-mum positive difference connect (from the surface toabout z � 6, tilted eastward). As NF increases, the areaof this positive phase shrinks, while the magnitude of

FIG. 1. The perturbation vertical velocity fields for the nonlinearity factors of (a) 0, (b) 1, (c) 2, (d) 3, (e) 4, and(f) 5 at a time step t � 12.96. The contour interval is 0.05. The thick solid box denotes the area of diabatic forcingspecified by (7)–(9).


the positive difference between the two fields increases.Because of this nonlinearity-induced pattern of the ver-tical velocity in the specified heating region, the loca-tion of the maximum updraft shifts upward and east-ward as NF increases (Fig. 1). Above the forcing region,the magnitude of the nonlinearity-induced upwardpropagating waves also increases as NF increases. How-ever, since those nonlinearity-induced waves (Figs. 2b–f) are largely out of phase with the waves induced bydiabatic forcing alone (Fig. 2a), the magnitude of wavesabove the forcing top is smaller than that induced bydiabatic forcing alone, as shown in Fig. 1.

Note that the present result for vertical velocity withrespect to NF is significantly different from that re-

ported in BC96 based on a similar NDM. In BC96, themagnitude of the vertical velocity is about 5 timeslarger than in the current simulation, and the domain-maximum vertical velocity shows a periodic oscillationfor NF � 3, with multiple moving updraft and down-draft cells that propagate downstream of the specifiedheating. The differences between the present resultsand BC96 likely are due to the different backgroundcondition and forcing structure imposed, along with aminor contribution from use of different numericalschemes. BC96 imposed a uniform basic-state wind andstability condition along with vertically uniform (fromthe surface to z � 2) and stationary heating. It remainsfor future research to determine the response of a sta-

FIG. 2. (a) The perturbation vertical velocity field for the case NF � 0 and the difference in the perturbationvertical velocity between each nonlinear case with NF � (b) 1, (c) 2, (d) 3, (e) 4, and (f) 5 and the case NF � 0.The shading represents the region where the nonlinearity-induced vertical velocity is out of phase with the verticalvelocity induced by diabatic forcing alone. The contour interval is 0.05.


bly stratified nonlinear flow to specified heating underdifferent basic-state conditions and imposed forcing.

To understand the structure of the nonlinear forcingwith respect to the diabatic forcing, we calculate thenonlinear and diabatic sources of internal gravitywaves, defined per SCL [see Eq. (16) of SCL], in non-dimensional form:

��2

�x�z �DFu

Dt � ��2Fb

�x2 �dU

dz

�2Fu

�x2 , �16�

�2Q

�x2 , �17�

where,

Fu � ��

�x�uu� �

�

�z�uw��, �18�

Fb � ��

�x�bu� �

�

�z�bw��. �19�

Here D/Dt � �/�t � U�/�x. The nonlinear source in (16)is the simplified form under hydrostatic approximation,as considered in the present NDM.

Figure 3 shows the nonlinear source of (16) superim-posed on the diabatic source of (17) for different NFs att � 12.96. Note that in the present NDM system, thediabatic source is fixed while the nonlinear source canbe changed according to NF, directly by the NF valueitself and indirectly through changed magnitude of per-turbations resulting from different NF values. The dia-batic source (Fig. 3a) has its minimum value near thecenter of the specified forcing and is positive on theupstream and downstream edges of the forcing, due tothe second-order derivative of the Gaussian-type heat-ing structure given by (8). As NF increases, a positivenonlinear source is induced at the center of the negativediabatic source, while a negative nonlinear source isinduced near the boundaries of positive and negativediabatic sources. As a result, cancellation between thediabatic and nonlinear sources occurs in the center ofthe specified heating, while the diabatic source locatedat the bottom right and top left of the negative region isenhanced by the negative nonlinear source there. It isnoteworthy that not all sources shown in Fig. 3 can beused to generate gravity waves that propagate aboveforcing region, but the effective forcing that satisfies thevertical propagation condition of internal gravity wavesin the spectral domain (k � �) for a given basic-statewind and stability condition (Song et al. 2003) can. Fur-ther discussions about this issue will be given in the nextsection.

The nonlinear source has a wavelike structure, asexpected, since it originates from the nonlinear advec-

tion of the waves induced by diabatic forcing, and it hassmaller horizontal and vertical scales than those of thediabatic source. The structure of the nonlinear sourceshown in Fig. 3 is very complicated, even for a singleconvective cell with a Gaussian-type spatial and tem-poral structure, as considered in the present study. Formore realistic situations with multiple convective cellswith different temporal and spatial scales, nonlinearforcing is likely to be much more complicated than thepresent result. This is one of the factors making anyanalytical approach to formulate a momentum fluxspectrum by including nonlinear forcing technically dif-ficult.

4. Momentum flux spectrum

Figure 4 shows the gravity wave momentum fluxabove forcing (at z � 10) with respect to zonal phasespeed (cp) for different NFs. It is clear that the magni-tude of the momentum flux above the convective forc-ing decreases gradually as NF increases. This is particu-larly pronounced at phase speeds from �0.4 to �1.6,where the maximum magnitude of momentum flux ex-ists for each NF case. The maximum magnitude of mo-mentum flux for NF � 5 is one order of magnitudesmaller than that induced by diabatic forcing alone(NF � 0). Besides the magnitude, the phase speed atwhich the maximum magnitude of momentum fluxoccurs shifts slightly toward the right (a slightly lessnegative value) with an increase in NF. For westwardpropagating components with larger phase speeds(cp � �1.6) and for eastward propagating components,the magnitude of the momentum flux decreases withNF but without significant difference. This implies thatthe magnitude change of the momentum flux by includ-ing the nonlinear forcing is phase speed dependent.

To get some insight into how each forcing mechanismcontributes to the momentum flux spectrum above theconvective forcing, as shown in Fig. 4, additional nu-merical simulation is conducted with nonlinear forcingalone, which is similar to the dry simulation forced bynonlinear forcing (DRYMH) in SCL. The nonlinearforcing by (18) and (19), obtained in the CTL simula-tion from the surface to the top of diabatic forcing, issaved at intervals of 0.036 (10 s in the time dimension)and is then included on the right side of the linearizedversion of the NDM equations without specified dia-batic forcing Q. Note that the CTL simulation withNF � 0 is equivalent to the linear simulation with dia-batic forcing alone, similar to the dry simulation forcedby diabatic forcing (DRYQ) in SCL. Figure 5 shows themomentum fluxes induced by the CTL simulation andlinear simulations by diabatic forcing alone (u0w0) and


nonlinear forcing (u1w1) alone, cross-correlation mo-mentum fluxes between the two forcings (u0w1 andu1w0), and momentum flux induced by the sum of thediabatic and nonlinear forcings (u0 � u1)(w0 � w1) fordifferent NFs. As NF increases, the momentum flux bynonlinear forcing increases with the same sign as the

momentum flux by diabatic forcing alone. This is some-what expected, given that the magnitude of momentumflux by any single source is proportional to the squareof vertical velocity that must be proportional to forcingmagnitude, in general, as shown analytically by Chun etal. (2005). Nonetheless, the momentum fluxes of CTL

FIG. 3. The diabatic (solid and dashed lines) and nonlinear (shading) sources of internal gravity waves for different NFs of (a) 0, (b)1, (c) 2, (d) 3, (e) 4, and (f) 5 at a time step t � 12.96, calculated by (17) and (16), respectively. Solid (dashed) lines denote positive(negative) values in the diabatic forcing. Contour intervals of diabatic and nonlinear forcing are 1 and 2, respectively.


Fig 3 live 4/C

and (u0 � u1)(w0 � w1) decrease as NF increases. Thisis because cross-correlation momentum fluxes with asign opposite to the momentum fluxes by the singlesources increase more rapidly with NF and, conse-quently, the resultant magnitude of total momentumflux becomes smaller as NF increases. The presentNDM simulation, which isolates specified diabatic forc-ing and nonlinear forcing associated in a simple dy-namical framework, clearly demonstrates that the can-cellation of the momentum flux by the cross-correlationterms is due to the fundamental characteristics of thediabatic forcing and induced nonlinear forcing. This im-plies that the magnitude of the momentum flux inducedby both the diabatic forcing and nonlinear forcingshould be smaller in any case than that induced by dia-batic forcing alone, and this effect should be included inconvective GWD parameterizations. Note that the dif-ference in momentum flux between CTL and (u0 �u1)(w0 � w1) represents a nonlinearity of wave distur-bance above the forcing region, and it is not negligiblefor highly nonlinear flow regimes (NF � 5).

This result can be explained also by relative magni-tude of forcing terms and their phase difference. Al-though the magnitude of the nonlinear forcing is muchlarger than that of the diabatic forcing, especially forlarger NF cases as shown in Fig. 3, the magnitude of theeffective nonlinear forcing is actually smaller than thatof the effective diabatic forcing for all NFs considered,and it increases as NF increases (not shown). This isbecause a large part of the nonlinear forcing, especiallyhigh frequency and short horizontal wavelength com-ponents, cannot satisfy the vertical propagation condi-

tion of internal gravity waves. The magnitude of theeffective nonlinear forcing relative to that of the effec-tive diabatic forcing and its relationship with NF areevidenced by the momentum flux spectrum in Fig. 5. Inaddition, the phase-difference spectrum (not shown)between the effective nonlinear forcing and effectivediabatic forcing indicates that the two effective forcingterms are largely out of phase with each other. Detailsin calculation of effective forcing and phase differencecan be found from Song et al. (2003) and Chun et al.(2005), respectively. Because of the above two facts [(i)the magnitude of effective nonlinear forcing is smallerthan that of effective diabatic forcing for all NFs con-sidered in the present study and it increases as NF in-creases and (ii) effective diabatic and effective nonlin-ear forcings are out of phase with each other], the mag-nitude of total effective forcing decreases as themagnitude of NF increases. This is why the momentumflux induced by two forcing terms decreases as NF in-creases.

Figures 4 and 5 showed that the magnitude change ofthe momentum flux by including the nonlinear forcingis prominent at phase speeds where the magnitude ofmomentum flux is large, although it is generally phasespeed dependent. Therefore, in the present study, ascale factor for the total momentum flux, normalized bythe momentum flux induced by diabatic forcing alone,is obtained as a function of nonlinearity factor onlyusing the maximum magnitude of momentum flux foreach NF in Fig. 5. Although it is possible to make ascale factor as a function of phase speed and a nonlin-earity factor using the result of Fig. 4, we do not thinkit is critically necessary at this point for use in convec-tive GWD parameterizations. One of the reasons, be-sides simplicity in formulation and computational effi-ciency, is that the detailed spectral shape of the mo-mentum flux obtained in the present NDM result maynot be directly applicable to the cloud-top momentumflux spectrum in SC05’s parameterization, mainly dueto the cloud-moving speed. The magnitude of the mo-mentum flux is prominent near the phase speeds similarto the moving speed of diabatic forcing (cq), which isfixed in the present NDM simulation, while it ischanged spatially and temporally based on the large-scale wind averaged over a cloud layer in each GCMgrid (Song and Chun 2006). Therefore, obtaining thescale factor as a function of phase speed and NF usingthe present NDM result may not be necessary, andmight even include additional uncertainties in convec-tive GWD parameterizations.

Figure 6 shows the maximum magnitude of momen-tum flux as a function of NF normalized by the maxi-mum magnitude of momentum flux by diabatic forcing

FIG. 4. The momentum flux spectrum above (z � 10) the speci-fied diabatic forcing with respect to the zonal phase speed fromthe CTL simulations for different NFs.


alone, obtained using Fig. 4 (dots), and its best fittingcurve (solid line) that represents the scale factor for themomentum flux by including nonlinearity effect. Thescale factor is obtained as

F �� 1��1 � ab�, �20�

where a � 0.487 92 and b � 1.648 96. According to (20),the magnitude of cloud-top momentum flux reduces tobe less than 0.4 when the nonlinearity becomes larger

than 2. It is rather straightforward to include the non-linear forcing effect in current convective GWD param-eterizations that consider diabatic forcing alone by sim-ply multiplying the cloud-top momentum flux spectrumby the scale factor. For this, the nonlinearity factor (15)can be obtained at each grid using the model-produceddiabatic forcing rate (from cumulus parameterization)and wind and stability with a specified horizontal size ofcloud (5 km is used in the present study, as in SC05).

FIG. 5. The momentum flux spectrum at z�10 induced by CTL simulation, diabatic forcing alone (u�0w�0),nonlinear forcing alone (u�1w�1), cross-correlation terms between the two forcings (u�0w�1 and u�1w�0), and sum ofdiabatic and nonlinear forcings [(u�0 � u�1)(w�0 � w�1)] for different NFs.


Figure 7 compares the momentum flux spectrum ob-tained from diabatic forcing alone (CTL with NF � 0),the CTL simulation with nonzero NF, and diabatic forc-ing alone multiplied by the scale factor for NFs of 1, 2,3, 4, and 5. The updated momentum flux spectrum isreasonably good and indeed much better than the mo-mentum flux induced by diabatic forcing alone as com-pared with that of CTL simulation. Discrepancies in themomentum flux between the updated one and the CTLsimulation exist, as expected, mainly due to using ascale factor that is phase speed independent: (i) thephase speed at which the maximum magnitude of mo-mentum occurs shifts to the west, (ii) the magnitudes ofmomentum flux with relatively high negative and posi-tive phase speeds are smaller than those by CTL simu-lation, and (iii) the momentum flux at c � 0 for theupdated one has a sign opposite to that for the CTLsimulation with NFs larger than 3. However, these dis-crepancies occur mostly at phase speeds where themagnitude of the momentum flux is insignificant. Over-all, the updated momentum flux is much better thanthat induced by diabatic forcing alone, and the scalefactor obtained in the present study is considered to bereasonably good to use in convective GWD parameter-izations.

To examine the generality of the fitting-curve formu-lation, NFs calculated by (15) and that estimated by(20)–(21) using the maximum magnitude of the mo-mentum flux in a nonlinear simulation (MCTL) and thatin a quasi-linear simulation forced by diabatic forcingalone (MDRYQ) are compared for all cases simulated in

Choi et al. (2007) and shown in Fig. 8. Choi et al. per-formed ensemble numerical simulations of convectivegravity waves under various ideal and real convectioncases. For ideal cases, the basic-state wind profiles areassumed to linearly increase with height from theground to z � 6 km with different surface winds (Us ��27, �24, �21, �18, �15, �12, and �9 m s�1) and tobe uniform (2 m s�1) above z � 6 km. The Us � �18m s�1 case is identical to the SCL case. In addition, anonshear case with zero basic-state wind (Us � 0 m s�1)is conducted. For real storm cases, a case observed dur-ing the Tropical Ocean and Global AtmosphereCoupled Ocean–Atmosphere Response Experiment(TOGA COARE) on 22 February 1993 (Trier et al.1996) and a case observed in Koto Tabang, Indonesia,on 11 April 2004 (Dhaka et al. 2005) have been con-sidered.

The variables for calculating NFs for each storm caseare shown in Table 1. For calculating (15), the horizon-tal scale (L) of convection is assumed to be 50 km (thedominant horizontal wavelength of diabatic forcing formost cases) and Q0 is the heating rate averaged over 50km in the major convective region for 1 h from t � 5 to6 h for ideal cases and from t � 2 to 3 h for real cases.The Uc is estimated by Um � cm, where Um is the basic-state wind in the middle region of convection (z � 6km) and cm is the mean speed of convective cells in thedirection of the maximum momentum flux (eastwarddirection for positive momentum flux and westward di-rection for negative momentum flux). The referencetemperature T0 is fixed to be 273 K and the momentumflux is evaluated at z � 18 km. Figure 8 demonstratesthat NFs calculated by (15) and (20)–(21) are reason-ably well matched with each other for most cases, ex-cept for the TOGA COARE case. Also, a scatterplot ofNF versus MCLT/MDRYQ (not shown) for all the casesconsidered in Choi et al. (2007) shows that the resultlocates mostly near the fitting curve, except for theTOGA COARE case, where NFs calculated by (15)and (20)–(21) are not matched well, as shown in Fig. 8.Note that NF by (15) is very sensitive to the magnitudeof Uc, and that Uc is sensitive to the moving speed ofconvective cells (cm), which is not straightforward todetermine when there are multiple convective cells withdifferent speeds. Also, the height at which the basic-state wind and stability are evaluated is important todetermine NF by (15). The same calculation can beconducted using the variables at the cloud top. Forideal simulations, NFs evaluated at the cloud top arethe same as those in the middle region of convectionsince the basic-state wind is constant with height abovez � 6 km. For real cases, NF evaluated at the cloud topis larger (smaller) for the TOGA COARE case (Indo-

FIG. 6. Maximum magnitude of the momentum flux with respectto NF normalized by the maximum magnitude of the momentumflux induced by diabatic forcing alone (NF � 0) at z � 10 calcu-lated using the result in Fig. 4 (solid dots) and its best fitting curve(solid line) as a function of nonlinearity factor.


nesia case). Overall, NFs calculated by (15) and (20)–(21) are reasonably well matched with each other formost simulated convection cases. This implies that thescale factor obtained in the present study can be ap-plied to various types of convective storms.

5. Impact of the updated convective GWDparameterization on a GCM

The cloud-top momentum flux formulation by SC05is updated by including the nonlinear forcing effect as

Mct�, c, � � Mcto �, c�F ��, �21�

where Moct(�, c) is the cloud-top momentum flux origi-

nally proposed by SC05, � is the wave propagation di-rection, and c is the phase speed in the wave propaga-tion direction. The detailed formulation of Mo

ct(�, c)can be found in SC05. Song and Chun (2006) developedconvective GWD parameterizations based on thecloud-top momentum flux spectrum by SC05 with twodifferent wave saturation methods and performed off-

FIG. 7. Momentum flux spectrum by diabatic forcing alone (NF � 0) (dotted curves), CTL simulation (solid blackcurves), and diabatic forcing alone but updated with the scale factor obtained in (20) (solid gray curves) fordifferent NFs.


line tests of the schemes using global reanalysis data.Then, Song et al. (2007) implemented the scheme, usingthe Lindzen-type wave saturation method, into theWhole Atmosphere Community Climate Model ver-sion 1b (WACCM1b) developed at the National Centerfor Atmospheric Research (Sassi et al. 2002) and inves-tigated the effects of convective GWD parameteriza-tion on large-scale circulations. In this study, we per-form the numerical simulation identical to Song et al.(2007) [spectral GWDC (SGWDC) simulation hereaf-ter] but with an updated cloud-top momentum fluxspectrum by (21).

WACCM1b is a global spectral model with T63 hori-zontal resolution at 66 vertical levels from the sur-face to about 140 km. The model description andphysical processes included can be found in Song et al.(2007) and references therein. As in Song et al., in thecurrent simulation with the updated parameterization

(SGWDC_NL simulation hereafter), WACCM1b runsfor 12 years starting from an initial condition (1 July1978) using the climatological ozone and sea surfacetemperature. The result shown is the last 10-yr average.In this study, we only present a cloud-top momentumflux and zonal drag forcing to show how the resultsfrom the updated parameterization, including a nonlin-earity effect on the cloud-top momentum flux spec-trum, differs from the results by the SGWDC simula-tion that was analyzed in detail in Song et al. (2007).

Figure 9 shows the 10-yr averaged nonlinearity factorby (15) at cloud top, zonal-mean zonal momentum fluxspectrum at the cloud top by the SGWDC simulationand its difference between SGWDC_NL and SGWDCsimulations in January and July. Shading in Fig. 9c de-notes area where the magnitude of the cloud-top mo-mentum flux in the SGWDC simulation is reduced inthe SGWDC_NL simulations. NF is calculated using

TABLE 1. The variables used to calculate NFs by (15) and (20)–(21) and resultant NFs. The variables are obtained from thenumerical simulations for eight ideal and two real storm cases considered in Choi et al. (2007). Details are given in the text.

CASEQ0

(J kg�1 s�1)Um

(m s�1)cm

(m s�1)Uc � Um �cm (m s�1)

N(s�1)

MCTL

(10�2 N m�2)MDRYQ

(10�2 N m�2)F(�) �

MCTL/MDRYQ

NF by(15)

NF by(20)

Us � �27 2.84 2.0 18.6 �16.6 0.01 0.277 0.734 0.378 1.85 2.09Us � �24 2.68 2.0 14.2 �12.2 0.01 0.188 0.674 0.279 3.23 2.75Us � �21 4.14 2.0 �10.9 12.9 0.01 �0.132 �0.936 0.141 4.47 4.62Us � �18 3.62 2.0 �15.8 17.8 0.01 �0.232 �0.651 0.356 2.05 2.21Us � �15 3.56 2.0 �14.1 16.1 0.01 �0.338 �0.993 0.340 2.47 2.31Us � �12 3.41 2.0 �10.8 12.8 0.01 �0.318 �1.32 0.241 3.74 3.10Us � �9 3.22 2.0 �9.3 11.3 0.01 �0.337 �2.37 0.142 4.53 4.60Us � 0 1.73 0.0 �6.7 �6.7 0.01 �0.106 �1.23 0.086 6.92 6.47

TOGA 1.57 �6.3 �26.2 19.9 0.012 �0.110 �0.43 0.256 0.59 2.95Indo 0.65 �4.0 3.0 �7.0 0.013 0.160 0.41 0.390 1.83 2.03

FIG. 8. Comparison of NFs calculated by (15) and (20)–(21) for ideal and real storm casessimulated in Choi et al. (2007). Here Us denotes the basic-state wind at the surface, and Us �0 is the case with no wind shear. TOGA and Indo denote the cases observed during TOGACOARE and in Indonesia. Details are given in the text.


the basic-state wind and stability at the cloud top witha maximum heating rate from convective heating pro-file at each model grid. Relatively large values of NFexist in the tropical region, where convective cloudsfrequently occur, and in the Southern Hemisphere(SH) at high latitudes and the Northern Hemisphere(NH) middle latitudes in July. However, even in thetropical region, the 10-yr mean of NF is mostly between0.2 and 1.0 and is larger than 2.0 at only a few localpoints where horizontal wind speed at the cloud top is

very small. Given that the scale factor F(�) is 0.67 forNF � 1 and 0.97 for NF � 0.2, the magnitude change ofthe momentum flux due to the nonlinearity factor can-not be significant. Therefore, the difference in the zonalmomentum flux spectrum at cloud top between theSGWDC_NL and SGWDC simulations (Fig. 9c) isone order of magnitude smaller than the cloud-top mo-mentum flux obtained from the SGWDC simulation.The cloud-top momentum flux spectrum from theSGWDC_NL simulation (not shown) is basically simi-

FIG. 9. The WACCM simulation results in 10-yr average of (a) global distribution of nonlinearity factor of convectively forcedinternal gravity waves calculated by (15) and (b) zonal-mean zonal cloud-top momentum flux spectrum by the SGWDC simulation andits difference from the SGWDC_NL simulation (SGWDC_NL � SGWDC) in (left) January and (right) July. Contour intervals in (b)and (c) are 5.0 10�7 N m�2 (m s�1)�1 and 5.0 10�8 N m�2 (m s�1)�1, respectively. Areas where the magnitude of cloud-topmomentum flux in the SGWDC simulation is reduced in the SGWDC_NL simulation are shaded.


lar to that of the SGWDC simulation (Fig. 9b), but itsmagnitude is reduced in most latitudes and phase speedranges, except in the NH (SH) middle latitudes of stormtrack regions over 30°–60° in January (July), where thepositive momentum flux with phase speed ranging fromabout 20 to 50 m s�1 and the negative momentum fluxwith phase speed ranging from about �10 to �10 m s�1

are amplified.Figure 10 shows the zonally averaged zonal drag forc-

ing due only to the convective GWD process in theSGWDC simulation and its difference between theSGWDC_NL and SGWDC simulations for Januaryand July. Shading in the difference field (Fig. 10b) rep-resents areas where including the updated parameter-ization reduces the magnitude of drag by SGWDCsimulation. The drag forcing is concentrated in thetropical mesosphere with its maximum positive (nega-

tive) of 24.6 m s�1 day�1 (�9.5 m s�1 day�1) in Januaryand of 28.7 m s�1 day�1 (�50.2 m s�1 day�1) in July.The difference is concentrated where the drag forcingwas large, with maximum (minimum) values of 1.5m s�1 day�1 (�1.8 m s�1 day�1) in January and of 3.7m s�1 day�1 (�5.4 m s�1 day�1) in July, which is oneorder of magnitude smaller than the drag value in theSGWDC simulation. This is somewhat to be expectedsince the shape of the cloud-top momentum flux in theSGWDC_NL simulation is not significantly differentfrom that in the SGWDC simulation. The nonlinearforcing effect reduces the magnitude of drag forcing inmany regions, but there are several areas (the tropicalupper mesosphere and lower thermosphere betweenabout 20°S–0° in January and 10°S–10°N in July, andthe lower mesosphere in the SH middle latitudes) ofincreasing drag forcing.

FIG. 10. The WACCM simulation results in 10-yr average of zonal-mean zonal drag forcing inducedexclusively by convective GWD parameterization (a) in the SGWDC simulation and (b) its difference fromthe SGWDC_NL simulation in (top) January and (bottom) July. The contour intervals of (a) and (b) are 3and 0.3 m s�1 day�1, respectively; negative values are dashed. Areas where the magnitude of drag forcingin the SGWDC simulation is reduced in the SGWDC_NL simulation are shaded.


In the stratosphere, the magnitude of drag forcing ismuch smaller than that in the mesosphere, thus it is notwell represented using the same contour intervals asFig. 10. Figure 11 shows the same fields as Fig. 10 butwith z � 45 km. The difference occurs mainly in theNH (SH) upper stratosphere in January (July) withmaximum (minimum) values of 0.081 m s�1 day�1

(�0.023 m s�1 day�1) in January and of 0.076 m s�1

day�1 (�0.066 m s�1 day�1) in July. Although the dif-ference is much less than that in the mesosphere, it isgreater than 20% of the maximum magnitude of dragforcing in the SGWDC simulation. This implies that thenonlinear forcing effect on the zonal drag forcing iseven more significant in the upper stratosphere than inthe upper mesosphere. In addition, the difference in thedrag forcing occurs mostly in the shaded area, indicat-ing that including the nonlinear forcing effect generallyreduces the magnitude of drag forcing in the strato-sphere. The only exception exists in the winter hemi-sphere middle latitudes of storm track regions, where

the magnitude of the cloud-top momentum flux in-creases by including the nonlinear forcing effect, asshown in Fig. 9c.

It is noteworthy that the reduced magnitude of thecloud-top momentum flux resulting from the nonlinearforcing effect can either increase or decrease the mag-nitude of the drag forcing in convective GWD param-eterizations, depending on wave dissipation processes.When the waves are dissipated through the critical-level filtering process, drag forcing can be decreased byreduced magnitude of the cloud-top momentum fluxbecause the vertical gradient of the momentum fluxdecreases by reduced magnitude of cloud-top momen-tum flux. On the other hand, for vertically propagatingwaves, reduced magnitude of the momentum flux canallow the waves to reach relatively higher altitudes be-fore they are saturated and break, and consequentlycan produce a larger magnitude of drag forcing due tolower density, if the vertical gradient of the momentumflux is assumed to be the same. In the present WACCM

FIG. 11. As in Fig. 10 except below z � 45 km. The contour intervals are (a) 0.02 and (b) 0.005 m s�1

day�1.


simulation with Lindzen-type linear wave saturationprocess (Song et al. 2007), wave breaking occurs mostlyin the upper mesosphere, so the reduced magnitude ofcloud-top momentum flux decreases drag forcing in thestratosphere but increases in the upper mesosphere.

Figure 12 shows time series of the zonal-mean zonalmomentum forcing by convective GWD parameteriza-tion in the tropical region averaged over 15°S–15°N at10 and 30 hPa. It shows that zonal momentum forcing inthe SGWDC_NL simulation either increases or de-creases compared with that in the SGWDC simulation,although the overall magnitude of momentum forcingin SGWDC_NL decreases somewhat, especially at 30hPa. According to the relationship between drag forc-ing and wave dissipation processes, the momentumforcing by convective GWD, shown in Fig. 12, shoulddecrease under the reduced magnitude of cloud-topmomentum flux, given that the critical-level filteringprocess is the major wave dissipation process in the

stratosphere. The argument of wave dissipation, how-ever, is valid only for a single wave at an instant timewhen the other conditions (source spectrum, wind, sta-bility, etc.) are exactly the same. What is likely to hap-pen in GCMs is that, once reduced magnitude of thecloud-top momentum flux changes drag forcing, thewave source spectrum as well as wave propagation con-dition can be changed through the change of dynamicand thermodynamic variables at each model grid.Therefore, the difference in drag forcing between thetwo simulations results from multiple positive andnegative feedback processes.

6. Summary and discussion

The nonlinear-forcing effect on the momentum fluxspectrum of convectively forced internal gravity wavewas investigated using a nondimensional numericalmodel (NDM) in a two-dimensional framework. To iso-late nonlinearity effects, the governing equations of the

FIG. 12. Time series of zonal-mean zonal drag forcing in the tropical stratosphere averagedover 15°S–15°N at (upper) 10 and (lower) 30 hPa by the SGWDC (gray curves) andSGWDC_NL (black curves) simulations.


NDM, basic-state wind and stability conditions, and thediabatic forcing structure used for all simulations aremade identical to those used in the linear analyticalstudy of Song and Chun (2005), which considered dia-batic forcing alone. The NDM simulations with variousNFs revealed that the magnitude of wave perturbationabove the specified diabatic forcing decreases as NFincreases because the waves induced by nonlinear forc-ing are largely out of phase with those induced by dia-batic forcing.

The magnitude of the momentum flux above theforcing decreases as NF increases. This is because themomentum flux induced by nonlinear forcing alone in-creases with NF with the same sign of the momentumflux induced by diabatic forcing alone, but the cross-correlation momentum fluxes with an opposite sign in-crease more rapidly with NF. This represents a reducedmagnitude of total effective forcing as NF increases,due to the cancellation of the effective diabatic forcingby the effective nonlinear forcing that increases as NFincreases. Using the maximum magnitude of the mo-mentum flux obtained by the NDM simulations foreach NF case, we obtained an analytical formulation ofa scale factor for the momentum flux, normalized bythe momentum flux induced by diabatic forcing alone,as a function of NF. The updated momentum flux in-duced by diabatic forcing alone and using the scale fac-tor is reasonably good as compared with the CTL simu-lation including both forcings.

The updated convective GWD parameterization ofSC05 using the updated cloud-top momentum fluxspectrum was implemented into a GCM (WACCM1b),and 10-yr simulation (SGWDC_NL) results were com-pared with those (SGWDC) by the original SC05 pa-rameterization. It was shown that the magnitude of thecloud-top momentum flux by the SGWDC_NL simula-tion is generally smaller than that by the SGWDC simu-lation in most latitudes and phase speed ranges, exceptin the middle latitudes (30°–60°) of storm track regionsin both January and July, where the positive (negative)momentum flux with phase speed ranging from about20 to 50 m s�1 (�10 to �10 m s�1) is amplified. Thedifference in the cloud-top momentum flux betweenthe SGWDC_NL and SGWDC simulations was foundto be one order of magnitude smaller than the cloud-top momentum flux by the SGWDC simulation. Thisrather small difference results mainly because the non-linearity factor of the gravity waves, calculated usingthe GCM-produced variables (wind, temperature, dia-batic heating rate), is mostly less than 1.0 even in thetropical region and, consequently, the scale factor isclose to 1 in most regions.

The impact of the new parameterization on zonal-

mean zonal drag forcing is to reduce its magnitude inmany regions, for both positive and negative drags, butthere are several regions with increased magnitude ofdrag forcing, especially in the tropical upper meso-sphere and the SH middle latitude lower mesosphere.Under the reduced magnitude of cloud-top momentumflux in the updated parameterization, drag forcing canbe either increased or decreased depending on wavedissipation processes: It generally increases in the up-per mesosphere where wave saturation is the major dis-sipation process, while it decreases in the stratospherewhere critical-level filtering is the major wave dissipa-tion process. However, in GCM, drag forcing is deter-mined not only by wave dissipation process but also bythe complicated relationship between the wave sourcesand wave propagation condition through the multiplepositive and negative feedback processes.

In this study, we presented a way to include a non-linear forcing effect on the cloud-top momentum flux ofconvective gravity waves by using a scale factor derivedfrom NDM results with respect to NF. Although thisapproach is rather straightforward, there are severalfactors that could influence the results, mainly the wayto derive the scale factor. First, the scale factor is ob-tained using the NDM for a single convection case witha simplified heating structure and basic-state wind andstability condition. Although we considered such anidealized situation to directly apply the nonlinear forc-ing effect to SC05’s convective GWD parameterization,one may ask whether the current result can be appliedto various convective storms. For the SCL case, themaximum value of the cloud-top momentum flux in theCTL simulation (MCTL) is �0.232 10�2 N m�2, whilethat in the DRYQ simulation (MDRYQ) is �0.651 10�2 N m�2. Therefore, MCTL/MDRYQ � 0.357. Giventhat MCTL/MDRYQ � F(�) in (21), 0.357 corresponds to� � 2.2 according to the curve fitting formulation of(20). This value of NF matches well with that previouslyestimated (� 2) using (15). To examine the generalityof the fitting-curve formulation, we used the result ofensemble numerical simulations of convective gravitywaves by Choi et al. (2007), who considered eight idealcases with different low-level wind shears and two realcases observed in the tropics. We found that the fitting-curve formulation by (20) represents reasonably wellthe result of most cases considered in Choi et al. (2007).Sensitivity of the fitting-curve formulation to factorsother than the low-level shear, such as different con-vective available potential energy (CAPE), three-di-mensionality, directional shear, and upper-level shear,remains to be done in a future study.

Second, GCM-produced diabatic heating rate with ahorizontal grid area of 40 000 km2 can be smaller than


the diabatic heating by mesoscale convective stormsconsidered in the present study and, consequently, thenonlinearity factor obtained at the GCM grid can beunderestimated somewhat. Although this might be aninevitable limitation to parameterization of any meso-scale processes to GCM, given that the nonlinearity ofthermally induced internal gravity waves cannot be di-rectly calculated in the current GCM, some correctioncan be included to represent the mesoscale impactproperly. A reasonable way to include mesoscale im-pact on the nonlinearity factor needs to be found.

Acknowledgments. The authors thank two anony-mous reviewers for their helpful comments. This workwas supported by the Korea Science and EngineeringFoundation (KOSEF) through the National ResearchLab. The program was funded by the Ministry of Sci-ence and Technology (M10500000114-06J0000-11410)and by the Korea Research Foundation Grant fundedby the Korean government (MOEHRD) (R02-2004-000-10027-02).

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