a toy model of the instability in the equatorially trapped convectively coupled waves...

A Toy Model of the Instability in the Equatorially Trapped Convectively CoupledWaves on the Equatorial Beta Plane

JOSEPH ALLAN ANDERSEN

Department of Physics, Harvard University, Cambridge, Massachusetts

ZHIMING KUANG

Department of Earth and Planetary Science, and School of Engineering and Applied Sciences, Harvard University,Cambridge, Massachusetts

(Manuscript received 29 February 2008, in final form 12 June 2008)

ABSTRACT

The equatorial atmospheric variability shows a spectrum of significant peaks in the wavenumber–frequency domain. These peaks have been identified with the equatorially trapped wave modes of rotatingshallow water wave theory. This paper addresses the observation that the various wave types (e.g., Kelvin,Rossby, etc.) and wavenumbers show differing signal strength relative to a red background. It is hypoth-esized that this may be due to variations in the linear stability of the atmosphere in response to the variouswave types depending on both the specific wave type and the wavenumber. A simple model of the con-vectively coupled waves on the equatorial beta plane is constructed to identify processes that contribute tothis dependence. The linear instability spectrum of the resulting coupled system is evaluated by eigenvalueanalysis. This analysis shows unstable waves with phase speeds, growth rates, and structures (vertical andhorizontal) that are broadly consistent with the results from observations. The linear system, with anidealized single intertropical convergence zone (ITCZ) as a mean state, shows peak unstable Kelvin wavesaround zonal wavenumber 7 with peak growth rates of �0.08 day�1 (e-folding time of �13 days). Thesystem also shows unstable mixed Rossby–gravity (MRG) and inertio-gravity waves with significant growthin the zonal wavenumber range from �15 (negative indicates westward phase speed) to �10 (positiveindicates eastward phase speed). The peak MRG n � 0 eastward inertio-gravity wave (EIG) growth rate isaround one-third that of the Kelvin wave and occurs at zonal wavenumber 3. The Rossby waves in thissystem are stable, and the Madden–Julian oscillation is not observed. Within this model, it is shown that inaddition to the effect of the ITCZ configuration, the differing instabilities of the different wave modes arealso related to their different efficiency in converting input energy into divergent flow. This energy con-version efficiency difference is suggested as an additional factor that helps to shape the observed wavespectrum.

1. Introduction

If the satellite record of outgoing longwave radiation(OLR; Liebmann and Smith 1996)—a good proxy fordeep tropical convection (see, e.g., Arkin and Andanuy1989)—within 15° of the equator is analyzed in zonalwavenumber–frequency space (Wheeler and Kiladis1999, hereafter WK99) it shows a number of statisti-cally significant peaks (Fig. 1). These propagating dis-turbances form a large part of the tropical synoptic-scale variation, organizing individual convective ele-

ments (typically 100 km across, persisting for a fewhours) on large spatial (thousands of kilometers) andtemporal (days) scales (e.g., Chang 1970; Nakazawa1988). The wave activity peaks have been identifiedwith the equatorially trapped waves of shallow watertheory (see, e.g., WK99; Yang et al. 2007). Many of theproperties of these waves are well described by rotatingshallow water wave theories (Matsuno 1966). Here weattempt to address the observation that the variouswave types (e.g., Kelvin, Rossby, etc.) and wavenum-bers show differing amplitudes. It is hypothesized thatthis is due in part to the (linear) stability of the atmo-sphere to perturbations of the various wave types de-pending on the specific wave type and the wavenumberin question.

Corresponding author address: Joseph Allan Andersen, Depart-ment of Physics, Harvard University, Cambridge, MA 02138.E-mail: [email protected]

3736 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

DOI: 10.1175/2008JAS2776.1

© 2008 American Meteorological Society

JAS2776

We explore this using the simple model of the con-vectively coupled waves using the first two baroclinicvertical modes, midtropospheric moisture, and the sub-cloud layer developed in Kuang (2008b, hereafterK08b) extended to the equatorial beta plane. Thismodel is part of a continuing effort in this field to con-struct models of minimal complexity to identify the ba-sic instability mechanisms of these waves. There is along history of such simple models, going back at leastas far as 1969 (e.g., Yamasaki 1969; Hayashi 1970;Lindzen 1974; Neelin et al. 1987, Wang 1988; Mapes2000; Majda and Shefter 2001; Khouider and Majda2006a; Raymond and Fuchs 2007). The early modelsemphasized a first baroclinic structure that is of onesign over the full depth of the troposphere (e.g., Wang1988; Neelin et al. 1987; Emanuel 1987). The impor-tance of the second baroclinic component was recog-nized and included in the model of Mapes (2000), andan instability mechanism involving the second baro-clinic temperature anomaly and deep convective andstratiform heating was identified (Mapes 2000). Morerecently, Khouider and Majda (2006a) included the in-teraction between free tropospheric moisture and deepconvective and congestus heating in their model andfound that moisture plays a major role in destabilizingthe system. This is an important finding. However, the

actual instability mechanism was not clearly identified.Khouider and Majda (2006a) described the process of adry episode causing more congestus heating, which moist-ens and preconditions the free troposphere, leading todeep convection, which then returns the free troposphereto a dry condition. This description, while correct, doesnot reveal how the initial perturbation gets amplifiedand thus falls short of an instability mechanism. Thismotivated the study of K08b, which continued the ef-fort by Khouider and Majda (2006a) but sought to de-velop conceptually simple treatments of convection andto identify the basic instability mechanisms.

In K08b, the convective parameterization is based onthe quasi-equilibrium (QE) concept—namely, that con-vection responds quickly to the changes in the large-scale flow and so can be considered to be in statisticalequilibrium with the flow (Arakawa and Schubert 1974;Emanuel et al. 1994) by keeping the convective avail-able potential energy (CAPE) constant for parcels ris-ing from the boundary layer to the midtroposphere.The tropospheric moisture is included as an input to theconvection calculation to represent the effect of en-trainment of the drier tropospheric air into a rising con-vective cloud. This acts as a control on the depth ofconvection because with a moister midtroposphere, up-drafts lose less buoyancy from the entrainment of en-

FIG. 1. OLR power divided by background spectrum for signals that are (a) antisymmetric and (b) symmetricabout the equator. The power spectrum is averaged over the region 15°S to 15°N and is constructed from NationalOceanic and Atmospheric Administration (NOAA) daily OLR data running from June 1974 to September 2007.Contours begin at 1.1 and are drawn at intervals of 0.1, only for statistically significant values. Shallow waterdispersion curves drawn are for equivalent depths of 8, 12, 25, 50, and 90 m, in order of increasing frequency. (AfterWK99).

DECEMBER 2008 A N D E R S E N A N D K U A N G 3737

vironmental air and can reach higher to form deep con-vection and subsequent stratiform rain; thus, for a givenamount of lower-tropospheric convective heating, theupper-tropospheric heating will be greater.

Analysis in K08b identified a moisture–stratiform in-stability, in addition to the stratiform instability ofMapes (2000). To see it in its most basic form (as illus-trated in Fig. 11 of K08b), consider the horizontalpropagation of a sinusoidal wave with a second-modetemperature structure in the vertical. The propagationof the wave modulates deep (first baroclinic) convec-tive heating by perturbing the statistical equilibrium be-tween the lower troposphere and the subcloud layer.More specifically, deep convection is enhanced behinda warm lower-tropospheric temperature anomaly (aslarge-scale ascent cools the lower troposphere) and isreduced behind a cold lower-tropospheric temperatureanomaly. The modulation on deep convection peaks aquarter-cycle behind temperature anomaly peaks. Thecombined effect of an enhanced deep convective heat-ing and its induced vertical advection of moisture is tomoisten the midtroposphere, leading to deeper (i.e.,more stratiform) convection in a region where the ini-tial temperature anomalies are positive in the uppertroposphere and negative in the lower troposphere,amplifying the initial perturbation. It was also foundthat in addition to the moist convective damping ef-fect (MCD; e.g., Emanuel 1993; Neelin and Yu 1994)whereby the finite response time of convection dampshigh-frequency waves, the tendency of the second baro-clinic mode convective heating to reduce existingmidtropospheric moisture perturbations tends to re-duce the instability at low wavenumbers. These twoeffects were suggested as contributors to the wavenum-ber selection in convectively coupled waves.

Extension of the model of K08b to the equatorialbeta plane allows investigation of the equatorial Rossby(ER), mixed Rossby–gravity (MRG) and inertio-grav-ity (IG) wave types. These waves have significantly dif-ferent convection/temperature/wind structures and ob-served amplitudes from the Kelvin-like waves studiedin most previous work that are confined to a line alongthe equator. Our emphasis will be on identifying thebasic factors that contribute to the differing instabilityof the different wave types and wavenumbers.

The linear instability spectrum of the resultingcoupled system is found by eigenvalue analysis. We userealistic model parameters estimated from Cloud Sys-tem Resolving Model (CSRM) studies of the convec-tively coupled waves. The instability analysis producesunstable waves with phase speeds, growth rates, andstructures (vertical and horizontal) that compare rea-sonably well with observations.

We will show that the instability’s dependence uponwave type is related to the projection of mean-stateintertropical convergence zone (ITCZ) heating ontothe mode (as has been suggested in the past; e.g.,Takayabu 1994; WK99) and to an energy feedback ef-ficiency effect. This latter effect stems from the differ-ing roles of divergent and rotational flows in the con-vective parameterization and the differing efficiencieswith which the waves convert wave energy input fromconvective heating into divergent wind. We investigatethis by analytical investigation of a simplified case toexplore the details of the mechanisms responsible forwavenumber and wave type selection in the instabilityspectrum. The simplified case is chosen to remove thealready understood mechanisms that exist within themodel. Considering the instability in terms of the en-ergy flow between the geopotential and wind fieldsshows that the instability strength is strongly influencedby the differing efficiencies of conversion from the en-ergy input due to convective heating to divergentwinds.

The paper is organized as follows: Section 2 providesa brief description of the model used in this paper.Section 3 describes the results of an eigenvalue analysisconducted to determine the spectrum of unstable wavespresent in our model, as well as tests of the sensitivity ofthe system to ITCZ configuration and to the param-eters used. Section 4 describes a simplified limiting caseof our model and an energy flow–based feedbackmechanism that explains the differing instabilities ofthe different wave modes. This is followed by a sum-mary and discussion (section 5).

2. Model description

a. Model equations

Our basic equations are a simple extension of themodel of K08b to the equatorial beta plane. The rela-tion to previous models was discussed in K08b and inthe introduction. The model equations can be derivedfrom the anelastic primitive equations for an atmo-sphere linearized about a resting mean state:

�tu� � �y�� 1

��xp�, �1�

�t�� yu� � �1

��yp�, �2�

�x��u�� y�� z��w�� 0, �3�

�tT� � w��zT �g

cp� � J� � �T�, and �4�

�zp� �g�

TT�, �5�


where � and T represent the density and temperature;u, �, and w represent the velocities in the zonal, merid-ional and vertical directions, respectively; and J repre-sents the convective heating. Overbars represent meanquantities and primes denote deviation from this meanstate (assumed to be small). A linear damping on tem-perature, denoted by , represents radiative cooling/heating toward the mean temperature profile. Thebuoyancy frequency N is given by

N2 �g

T��zT �

g

cp�, �6�

and y is the linear expansion of the Coriolis parameterf for small displacements away from the equator.

Then, with the additional assumptions of a rigid lidand constant buoyancy frequency, we can expand Eqs.(1)–(5) in terms of the baroclinic modes numbered j,thus:

g�

T�T�, J�, w��zT �

g

cP�� j�T, J, w� j�x, y, t�

�

2sin�j�z

HT� and �7�

��u�, ��, p�� j�u, �, p� j�x, y, t�HT

2jcos�j�z

HT�, �8�

where HT is the height of the lid at the top of thetroposphere. We note that despite the above deriva-tion, the use of these modes is based on empirical evi-dence rather than on their being normal modes of thedry atmosphere because the atmosphere does not pos-sess a rigid lid at the tropopause—a fact emphasized inthe context of convectively coupled waves by, for ex-ample, Lindzen (1974, 2003) and Raymond and Fuchs(2007).

Substituting the expansion forms [Eqs. (7) and (8)]into our perturbation velocity equations [Eqs. (1)–(3)]yields

�tuj � ��yk � uj � �HTj and �9�

wj � �cj2�H � uj, �10�

where T is the temperature perturbation scaled to actsimilarly to a geopotential perturbation and �H denotesthe horizontal gradient vector operator �H � (�x, �y).The wave speeds cj are given by

cj �NHT

j�. �11�

We can also make a modal expansion of the thermo-dynamic Eq. (4) by combining it with Eq. (10), gener-ating

�tTj � cj2�H � uj � Jj � �Tj �12�

For the rest of this paper, we restrict ourselves to j �1, 2—the first two baroclinic modes, which are suffi-cient to explain most of the convective activity (Mapesand Houze 1995). The heating anomalies associatedwith these two modes, J1 and J2, represent the convec-tive heating broken into deep convective and conges-tus/stratiform components, respectively. The modal de-

composition of convection heating is based on cloudphysics of convection. We note that the work by Ray-mond and Fuchs (2007) showed that one can produce aboomerang temperature structure with a first baroclinic–only convective heating profile and a radiating upperboundary condition, so the question of whether thetwo-mode structure observed in convectively coupledwaves arises from cloud physics of convection or large-scale dynamics remains a matter of debate.

For the purposes of the convective parameterization,the heating is rewritten in terms of lower- (L) and up-per- (U) tropospheric heating anomalies:

L �12

�J1 � J2� and �13�

U �12

�J1 � J2�. �14�

The total upper-tropospheric heating is considered tobe a fraction of the lower-tropospheric heating:

U0 � U

L0 � L� r0 � rqq, �15�

where U0 and L0 are the mean-state heating and r0 isthe mean-state ratio between L0 and U0; q is themidtropospheric moisture anomaly and is evolved ac-cording to

�tq � �a1c12�H � u1 � a2c2

2�H � u2 � d1J1 � d2J2, �16�

where aj and dj give the midtropospheric moisture ten-dencies due to vertical advection and convection, re-spectively. We have neglected the effect of the horizon-tal advection of moisture in our formulation for sim-plicity and recognize that its effect should be evaluatedin the future.


In a region of the midtroposphere that is anoma-lously moist with positive q, the entrained air is moisterthan average and the associated evaporative cooling issmaller, leading to relatively increased buoyancy ofparcels in this region. This in turn leads to parcels risingto greater altitude and heating the upper tropospheremore. As parcels rise to form deep convection, the sub-sequent stratiform heating further enhances heating inthe upper troposphere relative to the lower tropo-sphere. The opposite holds for a region with an anoma-lously dry midtroposphere.

The lower-tropospheric heating is considered to berelaxing quickly (over a time scale L, typically a fewhours) toward the quasi-equilibrium situation given bythe constancy of lower-troposphere nonentrainingCAPE. In our context, quasi-equilibrium will be en-forced by keeping the difference between the boundarylayer moist static energy (MSE) hb and the verticallyaveraged lower-tropospheric saturation moist static en-ergy �h*� approximately constant. Note that hb and �h*�both vary under the action of convection—a positiveconvection anomaly dries and cools the boundary layerthrough the action of downdrafts and turbulent trans-port and modifies the lower-tropospheric saturationMSE primarily (in our model) by heating the atmo-sphere—to reach a consistent equilibrium state. Thus,for equilibrium to be reached instantaneously,

�thb � ��th*�LT, �17�

where LT is the region of the lower troposphere inwhich this QE formulation is assumed to hold, extend-ing vertically up to the level where the effect of entrain-ment begins to become important. We shall neglect thesurface flux anomalies in this paper and write

�thb � �b1J1 � b2J2. �18�

The term bj describes the reduction of boundary layermoist static energy by the convection Jj.

Meanwhile, we also assume that the average satura-tion moist saturation energy in the lower tropospherecan be well approximated by a linear combination ofthe two baroclinic mode temperature anomalies:

��th*�LT � �t {F �T1 � �1 � �T2� }. �19�

Here, F is a proportionality constant relating thechange in lower-troposphere temperature to the changein moist static energy in the same region, and � de-scribes the relative influence of the two modes on thetemperature of the lower troposphere. It is, looselyspeaking, equivalent to the fractional depth of the tro-posphere through which this QE formulation can beassumed to hold before entrainment becomes an im-portant effect.

Substituting these two relations [(18) and (19)] into

Eq. (17) allows us to quantify the quasi-equilibrium interms of our model:

�b1J1 � b2J2 � F ��tT1 � �1 � ��tT2�. �20�

By expanding this equation using Eqs. (12)–(15), wecan find a lower-tropospheric heating Leq that satisfiesthe quasi-equilibrium assumption for the large-scalevariables (q, u, and T):

Leq �AL0rq

Bq �

Fc12

B ��H � u1 ��

c12 T1�

�F �1 � �c2

2

B ��H � u2 ��

c22 T2�, �21�

where

B � F � �b1 � b2� � A�rqq � r0� and �22�

A � �b2 � b1� � F �1 � 2�. �23�

We can improve the realism of the model by assum-ing that the convection does not respond to changes inthe large-scale environment instantly. Rather, we allowL to relax toward Leq over a short time scale L:

�tL �1L

�Leq � L�. �24�

By introducing the meridional velocity fields �j andthe Coriolis force terms yk � uj, this extension pre-vents the simple reduction of the dry part of the prog-nostic equations to temperature fields that occurs inK08b, leading to the presence of velocity fields in theequations for Leq (21) and q (16) [cf. his Eqs. (23) and(11)].

b. Parameter choices

As described in K08b, our parameter values, such asaj, bj, dj, F, �, r0, rq, , and L, are estimated from theCSRM simulations of Kuang (2008a) which showedspontaneously developing convectively coupled waves.1

The estimation is conducted by projecting the CSRMresults onto the vertical structure of our modes andinterpreting regression results in the framework of ourconvective parameterization. We consider this ap-proach to be somewhat more rigorous than tuning ourmodel to match observations, as well as more efficientgiven the pre-existence of the CSRM results.

Parameter values are given in Table 1 and are essen-tially identical to those used by K08b. The units of the

1 We would like to note that, unknown to Kuang (2008a), asimilar modeling strategy was used by Brown and Bretherton(1995), although their method is more applicable to a convectiveparameterization than to CSRM.


parameters are based on measuring energies in theequivalent temperature change (in K) for 1 kg of dry airgaining that amount of energy. Moisture content is like-wise measured by the temperature change that wouldresult if all the stored latent heat in a volume of air wereconverted to thermal energy in an identical, but dry, airmass.

3. Beta-plane linear instability analysis

In WK99, the observations are of waves of finite am-plitude. If the waves are indeed unstable, in the obser-vations the linear growth of the unstable waves is bal-anced by nonlinear terms and an equilibrium isreached. In this case, the equilibrium is of a statisticalnature, with waves growing and dying as they movearound and enter regions of differing forcing (due tothe changing background state and wave–wave interac-tions), and is also time varying on various scales. In thispaper we investigate the linear regime of infinitesimalwaves with a zonally and temporally fixed backgroundstate. We see this as a first step toward an understand-ing of the spectrum. Further investigation of the non-linear effects is planned. Also planned is an investiga-tion of other interpretations of the observations, suchas stable waves continually excited by stochastic or ex-tratropical forcing (e.g., Zhang and Webster 1992;Hoskins and Yang 2000) in the context of this model.

a. Linearization

Calculation of the instability spectrum assumes thatour system is growing from infinitesimal anomalies, sowe can remove any nonlinear terms from the equationset. In our case, linearization is simply achieved by ig-noring the first term in the second parentheses from thedefinition of B [Eq. (22)],

B � F � �b1 � b2� � Ar0, �25�

and linearizing Eq. (15) into

U � L0rqq � r0L. �26�

Our linearized model consists of Eqs. (9), (12), (13),(14), (16), (21), (23), (24), (25), and (26).

Our basic state, represented by the parameter L0 (y),is an idealization of the zonal mean state. As shown inFig. 2, the control profile consists of a Gaussian peak(of amplitude 0.7 K day�1) in mean-state convectioncentered on the equator, with a width of 5°, above asmall, nonzero uniform offset that extends to the edgesof the domain.

The linear system is then Fourier transformed tozonal wavenumber–frequency space and the resultingeigenvalue problem is solved on a meridionally discretedomain extending approximately 6000 km to the northand south, with 181 regularly spaced points. This largedomain is utilized to reduce the influence of any edge-related errors and can be implemented without signif-icant computational cost. The resulting complex fre-quency eigenvalues can be separated into the real part�, which gives the modal frequency, and the imaginarypart �, which gives the modal growth rate. The corre-sponding eigenvectors give the modal structure in theprognostic fields (u1, �1, etc).

b. Results

The results from the linear analysis are shown in Figs.3 and 4. Unstable waves are indicated in the frequency–wavenumber diagram, with the marker radius propor-tional to the growth rate. Unstable modes are assignedto an equivalent dry class (dispersion curve rangesshown dashed) based on their position in Fig. 3, relativeto the dry dispersion curves indicated. Investigation ofthe modal structure (discussed below) shows that theseidentifications are warranted. The system contains un-stable modes corresponding to Kelvin waves, n � 1 and

TABLE 1. Parameter values as used in the control cases discussed in section 3.

Symbol Normative values Description

b1, b2 1.0, 2.0 Tendency for reduction in boundary layer moist static energy per unit heating J1 and J2

a1, a2 1.4 K, 0.0 K Increase in q tendency per unit vertical velocity (�xu � �y�) by advectiond1, d2 1.1, �1.0 Decrease in q tendency per unit heating J1 and J2

r0 1.0 Background mean U/L ratiorq 1.0 K�1 Linear dependence of U/L ratio on moisture deficitF 4 Ratio between moist static energy and temperature in the lower-tropospheric “non-entraining”

convection region� 0.5 Relative contribution of the first mode temperatures to the lower-tropospheric temperature

anomaly L 2.0 h Adjustment time to approach QE over the lower tropospherec1, c2 (50.0, 25.0) m s�1 Dry gravity wave speeds for the first and second modes 0.15 day�1 Temperature anomaly damping coefficient�L 5.0° Meridional width of L0 profile, representing the ITCZ structure


FIG. 2. Background state lower-tropospheric convective heating used in the control insta-bility experiment. The background state is an idealization of the zonal mean state, with anITCZ-like Gaussian peak over the equator and a small mean convection over the rest of thedomain.

FIG. 3. Eigenfrequencies for the linear system with an idealized zonal mean state. Data pointdiameter corresponds linearly to growth rate. Stable modes are omitted. Lines show theoret-ical dispersion curves for dry waves with equivalent depths of 40 and 60 m (in order ofincreasing frequency).


n � 2 inertio-gravity waves, and mixed Rossby–gravitywaves.

Kelvin waves are the most unstable wave type.Wavelengths between about 13 000 and 2000 km (plan-etary wavenumbers 3 and 20) are unstable, with a maxi-mum growth rate of �0.10 day�1 occurring at wave-lengths of around 5700 km (wavenumber 7). Thesewaves have phase speeds between 20 and 25 m s�1,slightly slower than the second baroclinic wave, withthe speed deficit becoming stronger as the wavenumberdecreases. The growth rate curve rises sharply betweenwavenumbers 3 and 7 and then decays more slowly asthe wavenumber is further increased to 17.

Figure 5 shows the structure of this mode, along withvertical structure of temperature. The first baroclinicmode is approximately in quadrature with the second,leading to a tilted vertical temperature structure, simi-lar to observations. Zonal winds blow in the standarddry wave directions. The meridional wind has a smallnonzero component. This is not unexpected becausethe moist physics that couples our system mixes themodes together so that the convectively coupled“Kelvin” wave observed, although dominated by aKelvin wave profile, will still contain a small amount ofother waves in superposition with it. The wind fields areplotted for the lower troposphere at z � 0.

Figure 3 shows unstable MRG waves, approximatelybetween wavenumbers �15 and 10, with a peak growthrate of �0.03 day�1 occurring at wavenumber 3(�13 000 km wavelength). The phase speed of theMRG waves also appears to drop below the dry speedas the wavenumber decreases, although this becomeless visible at large negative wavenumbers as the dis-persion curves converge. The modal structure of the(eastward) planetary wavenumber-7 mixed Rossby–gravity wave is shown in Fig. 6.

The system also shows instability in the n � 1 (east-ward and westward) IG waves, with a bias toward west-ward IG (WIG) waves.

The n � 2 and IG modes are only marginally un-stable in this model. We consider growth rates below afew percent of the peak values to be only marginallyunstable and far too dependent on assumptions aboutparameters for us to draw solid conclusions about them.The sensitivity of our model to parameter variation isexplored in section 3d below.

The reconstruction of the planetary wavenumber-7eastward IG (EIG) wave is shown in Fig. 7. In Fig. 8,the k � �7 WIG wave structure is also shown for com-parison. These modes also show horizontal and verticalstructures that compare well with observations.

All the reconstructed modes show first- and second-

FIG. 4. Growth rates for the linear systems with an idealized zonal mean state. Data pointsize encoding is included for ease of crossreferencing with Fig. 3. The smooth curves visible tothe eye correspond to the unstable wave types found in the system—Kelvin, n � 1 IG, MRG,and n � 2 IG.


FIG. 5. Structure of the coupled Kelvin wave at planetary wavenumber 7. (top), (middle)Temperature (contours) and wind anomalies (vectors) for the (top) first and (middle) secondbaroclinic modes; (bottom) reconstructed vertical temperature anomaly above the equator.

FIG. 6. Structure of the coupled MRG n � 0 EIG wave at planetary wavenumber 7. (top),(middle) Temperature (contours) and wind anomalies (vectors) for the (top) first and(middle) second baroclinic modes; (bottom) reconstructed vertical structure at y � 1000 km(north of the equator), approximately the peak of the MRG profile.


mode winds with similar amplitudes, broadly consistentwith observations (see, e.g., Haertel and Kiladis 2004), aswell as a vertically tilted structure, again generally consis-tent with observations (see, e.g., Straub and Kiladis 2002).

Most noticeably absent from the instability spec-trum are significant equatorial Rossby waves and theMadden–Julian oscillation (MJO). The complete ab-sence of the MJO signal indicates, not unexpectedly,

FIG. 8. As in Fig. 5, but for the coupled WIG wave at planetary wavenumber 7 (k � �7).

FIG. 7. As in Fig. 5, but for the coupled n � 1 EIG wave at planetary wavenumber 7.


that the current simple model is missing the physicsresponsible for the MJO. The Rossby waves exist in oursystem but are barely unstable within a range of param-eter choices around our nominal values (discussed be-low in section 3d). Because both these missing wavetypes are at similar, low frequencies, the model mayhave either a too strong damping at low frequencies ormay be missing some other physics that destabilizeswaves at these frequencies.

Comparison of Figs. 3 and 4 with Fig. 1 shows that

(i) peak Kelvin wave growth occurs at a wavelengthcomparable to the most active waves in the obser-vations;

(ii) the ranges of wavenumber and wave type showinginstability in our model match the ranges of wavesthat are observed in the OLR signals;

(iii) the Kelvin wave dispersion curves shift to higherequivalent depth at higher wavenumbers, oppositeto the trend observed (the reason for this discrep-ancy is unclear);

(iv) the model possesses a preference for unstablewestward n � 1, 2 IG modes (as opposed to east-ward n � 1, 2 IG), similar to observations; and

(v) The waves possess a tilted vertical structure that isgenerally consistent with the observations, with

low-level temperature anomalies preceding a briefperiod of deep anomaly followed by a period of“stratiform” temperature anomaly.

c. Varying ITCZ configurations

To address the question of whether the dominance ofKelvin waves is due to the amount of overlap betweenthe mean-state heating and the Kelvin wave profile be-ing greater than that for other waves, we have con-ducted several additional experiments with differentITCZ configurations in our mean state.

Figure 9 shows a double-ITCZ mean state, with thetwo peaks approximately a second baroclinic deforma-tion radius north and south of the equator, collocatedapproximately with the MRG temperature anomalypeaks determined from Fig. 6. Unstable modes for thiscase are shown in Fig. 10. Although moving the peakheating away from the equator does reduce the overallgrowth rate of the unstable Kelvin waves and increasesthat of the unstable MRG waves, the growth rates ofthe Kelvin waves are still strongest. The MRG growthrates surpass the n � 1 IG wave in the small wavenum-ber region of the spectrum. The ER waves are slightlymore unstable, but still not significant. The recon-structed mode structures (not shown) are not signifi-cantly different from the single ITCZ case.

FIG. 9. Background state lower-tropospheric convection used in double ITCZ experiment.The background state is an idealization with a pair of ITCZ-like Gaussian peaks centered 1000km north and south of the equator and a small mean convection over the rest of the domain.


Two further experiments (not shown) placed (i) adouble ITCZ at the off-equatorial peaks of the n � 1IG temperature anomaly field and (ii) a single ITCZ atthe northern MRG peak temperature anomaly posi-tion. Although the wave that had the greatest overlapwith the convective envelope showed some enhance-ment of growth rate, neither of these experiments al-tered the general dominance of the Kelvin waves.

Another aspect of the observed activity profile thatstands out as possibly being related to ITCZ configu-ration is the asymmetry between eastward and west-ward inertia-gravity waves—the observed activity ofWIG waves is much stronger than that of EIG waves. Ithas been suggested that this is primarily because thelatitudinal structure of the EIG waves is significantlymore complex and a greater proportion of the EIGtemperature anomaly is located away from the equatorand the ITCZ (see, e.g., Takayabu 1994; WK99). Toexplore this possibility, a further experiment is shownin which the ITCZ is replaced with a meridionally con-stant mean-state heating of 0.75 K day�1.

The results from the linear analysis are shown in Fig.11. Kelvin waves are still the most unstable wave type.Wavelengths between about 13 000 and 2300 km (plan-etary wavenumbers 3 and 17) are unstable, with a maxi-mum growth rate of �0.11 day�1 occurring at wave-length of around 5000 km (wavenumber 8). Unstable

MRG waves (also referred to as n � 0 EIG waves forthe k � 0 part of the spectrum) appear in the regionapproximately between wavenumbers �5 and 15, witha peak growth rate of �0.05 day�1 occurring at wave-number 7 (�5700 km wavelength). The system alsoshows instability in the (eastward and westward) IGwaves, with a significant instability bias toward WIGwaves. Because of the extreme poleward extent of themean-state convection in this case, there is significantinstability in high meridional wavenumber (IG) modes.

In this situation, the projection of the ITCZ heatingprofile onto the EIG and WIG structures should beessentially the same. As can be seen, some but not all ofthe east–west asymmetry has been removed. It will bedemonstrated below that some of this asymmetry is dueto the differing efficiencies with which the eastward andwestward IG waves convert convective heating intomore convective heating.

d. Parameter sensitivity

As mentioned above, the model does display somesensitivity to parameter variations. Figure 12 shows thegrowth rate curves for sensitivity experiments con-ducted with the damping and the convective equilib-rium relaxation time L increased and decreased by10%. The instability for the Kelvin, equatorial Rossby,mixed Rossby–gravity and n � 1 inertio-gravity waves

FIG. 10. As in Fig. 3, but for a background state with a double ITCZ (Fig. 9), located nearthe MRG meridional structure peaks (see Fig. 6).


are seen to be generally robust, with the location of thepeak instabilities in these cases remaining approxi-mately the same. The peak growth rates do changeslightly, as expected, when the damping is varied. How-ever, the n � 2 and higher IG modes can be observedto vary greatly in their instability character in this rangeof parameters. As would be expected from the moistconvective damping theory, decreasing the convectiverelaxation time decreases the moist convective dampingeffect, which then decreases the damping applied tohigh-wavenumber instabilities, leading to an increase ininstability for the n � 2 and n � 3 IG modes. For thesame reason, the Kelvin wave peak instability can beseen to shift very slightly to higher frequencies when is decreased.

A similar sensitivity test was applied to all of thesemiempirical parameters contained within our model.Generally, for a 10% change in parameter value, varia-tions in the spectrum were similar in character andamount to those discussed for damping above (notshown). The parameters r0 and c2 showed interestingsensitivity:

(i) When r0 is varied, the position of the peak insta-bility shifts to higher wavenumber for higher r0.The reverse holds for reduced r0.

(ii) A variation in the c2 parameter leads to a similar

variation in the speed of the unstable “Kelvin”modes. Any unstable mode speed variation causedby modification of c1 is negligible. This indicatesthat the speed of the unstable modes is largely de-termined by the second mode.

The equatorial Rossby waves are still not observed un-der any of these parameter variations.

These results are similar to the more detailed sensi-tivity study conducted for the 1D version of this modelincluded in K08b.

e. Midtropospheric moisture versus freetropospheric moisture

As in K08b, we have used midtropospheric moistureinstead of the vertically averaged free troposphericmoisture �q� in Eq. (16). This choice was based on thenotion that �q� is not necessarily the most relevantquantity as a control on convection, which may be tiedmore strongly to certain weighted averages of free tro-pospheric moisture. We have used midtroposphericmoisture although other and potentially better choicesare certainly possible. The use of the column moiststatic energy conservation constraint was not necessaryin our case because the effects of convection on themidtropospheric moisture and on the boundary layer

FIG. 11. As in Fig. 3, but for a background state with a meridionally constantlower-atmospheric heating.


moist static energy were estimated separately from theCSRM simulations. It is also not an essential elementfor indentifying the basic instability mechanisms.

However, for other issues such as understanding thecolumn MSE budget of the system, it is desirable to use

�q� so that the column MSE conservation constraint isautomatically satisfied, as done in Khouider and Majda(2006). Let us (in this subsection only) replace qmid with�q� in Eq. (15), so that the column MSE budget equa-tion can be written as

�t ��q� � T1�MT � hbMb� � MT ��1 � �a1��c12�H � u1 � �a2�c2

2�H � u2� � ��1 � �d1��MT � b1Mb�J1

� ��d2�MT � b2Mb�J2, �27�

where MT and Mb are the mass in the troposphere andthe boundary layer, respectively, and the other param-eters retain their meanings defined above, except witha � � to denote that they are associated with �q�. ColumnMSE conservation requires that

�1 � �d1��MT � b1Mb and �28�

��d2�MT � b2Mb �29�

so that one can therefore infer values of �d1� and �d2�from b1 and b2. We estimate that MT /Mb is �10, whichgives �d1� � 0.9 and �d2� � �0.2. The CSRM results ofKuang (2008a) can also be viewed in this frameworkfollowing the procedure described in the appendix ofK08b. Regression of the column MSE tendenciesagainst vertical velocities of the two vertical modes givea value for 1 � �a1� that is effectively zero and a valueof �a2� � 0.4. This indicates, allowing for uncertainties

FIG. 12. Damping sensitivity studies. As in Fig. 2, but for modification of the damping parameters as follows: (a) → 1.1 � , (b) → 0.9 � , (c) L → 1.1 � L, and (d) L → 0.9 � L. Peak growth rates for each case are (a)10 � 10�2 day�1, (b) 11 � 10�2 day�1, (c) 9 � 10�2 day�1, and (d) 11 � 10�2 day�1.


of the estimates, a close to neutral gross moist stabilityto the first baroclinic mode vertical motion, and thecolumn MSE variations are almost entirely associatedwith second baroclinic mode vertical motion. When �q�is used, we have also estimated that L0 � r�q� is about0.55. The resulting instabilities (not shown) are verysimilar to those resulting from the instability analysis ofmodel using midtroposphere moisture.

4. The role of energy flow efficiency

This model has previously been analyzed for Kelvin-like waves on the equator (K08b), revealing some ofthe mechanisms determining the instability spectrum.Specifically, the MCD effect (e.g., Emanuel 1993; Nee-lin and Yu 1994) damps high-frequency waves, and thetendency of the second baroclinic mode convectiveheating to reduce existing midtropospheric moistureperturbations tends to reduce the instability at lowwavenumbers. Expansion of the model to the betaplane in this paper has revealed the existence of at leastone more mechanism within the model, responsible forthe differences in growth rates for waves with similarfrequencies. We have already shown that the ITCZconfiguration and projection of the mean-state heatingon to the modes’ temperature anomaly profiles exertsome control on the shape of the growth rate curves.However, this control does not seem to explain all theaspects. We hypothesize that the growth rate curves forthe various wave types are further shaped by the rela-tive efficiency with which the various waves convertinput energy into the divergent winds, which in turngenerate more energy input. In this section, we willshow that the growth rates can be directly linked to thegeneration of divergent winds in the modes. To explorethe mechanisms that control the shape of the instabilityspectrum, in this section of the paper we consider asimplified system that has been modified to remove themechanisms already identified in K08b and any ITCZconvective projection amplitude effects (discussed insection 3, above) through the choice of specific param-eters. Some more detailed exploration of limiting caseswithin the model (in 1D) is conducted in K08b.

a. Physical description of simplified system

Our simplified system is based on the second limitingcase of K08b, in which the convective mass flux is domi-nated by entraining parcels and the boundary layer is inquasi-equilibrium with the second mode temperature(� � 0). The first temperature mode is essentially un-coupled from our system and can be ignored. The sys-tem is further simplified by setting b2 � 0 to remove the

influence of the second mode heating on the boundarylayer moist static energy and � 0 to remove the ra-diative damping; d2 is also set to zero to remove theeffect of second mode heating upon the midtropo-spheric moisture q. These changes remove the scaleselection of the moisture–stratiform instability.

Let us further tighten quasi-equilibrium to “strictquasi-equilibrium” by enforcing instantaneous relax-ation to the equilibrium values of L (by setting L � 0).This change removes the MCD effect.

A uniform L0 profile is used, removing the ITCZprojection effects.

Importantly, the parameter rq, which controls thecoupling between moisture and heating, is set to a verysmall value so that the heating can be approximated asan infinitesimal perturbation on the dry system.

This modified case is evaluated numerically (Fig. 13)in the same manner as the control case. The simplifiedsystem shows many unstable modes, but for this sectionwe will concentrate on the Kelvin, MRG, n � 1 ER, andn � 1 IG modes, so the modes with n � 1 are deliber-ately omitted in Fig. 13. The Kelvin wave has a constantgrowth rate for all wavenumbers greater than zero. TheMRG wave growth rate increases monotonically fromthe very small for large negative wavenumbers to rela-tively large at high positive wavenumbers, with rapidincrease in the region between k � �5 and k � �5. Inthe domain evaluated, the MRG growth rate neverreaches that of the Kelvin wave. The system also showsunstable ER and IG waves, with ER growth rates in-creasing from a low value for large (negative) wave-numbers to close to the Kelvin value for wavenumberzero, and n � 1 IG waves have a large growth rate forall wavenumbers, with growth increasing as |k| in-creases.

Within the spectrum of unstable modes are severalmodes located at k � 0, � � 0. These modes indicate aninstability in our mean state that would lead to a mean-state drift away from the values we use here on a timescale comparable to the growth of the waves we areinterested in. This is not considered a serious shortcom-ing here because we are looking at a very specific lim-iting case. In our more realistic cases treated earlier,such “climate drift” modes are not observed—all un-stable k � 0 modes posses nonzero frequency and thusconstitute oscillations about the mean state that, withinthe linear wave regime, do not affect other waves.

b. Energy flow from heating into divergent windsand heating feedback

To demonstrate our hypothesis, let us consider howthe energy that flows into the wave due to our convec-tive heating goes into increasing the divergence of the


flow. As we will show, the amplitude of the convectiveheating is proportional to the amplitude of the diver-gence, so this component of the energy drives the posi-tive feedback, whereas energy that flows into the rota-tional component or the potential energy of the wave isessentially trapped there and cannot contribute to thefurther growth of the wave.

For this section, let us consider a wave that is essen-tially dry in character and that is forced by a diagnosticheating that approximates the simplified case discussedabove. We will further consider how this wave changeswith time over a short interval while the heating acts.

The heating can be approximately written as a diag-nostic function of the dynamical variables as follows:

In our limiting case, Eqs. (16) and (20) become

�tq � �a1 � d1�J1 and �30�

�b1J1 � F�tT2, �31�

where we have further assumed that the vertical veloc-ity field is well approximated by the heating field (agood assumption for the first mode heating, as con-firmed by the modal eigenstructures for the controlmodel discussed above). Equation (30) states the com-bined effect of deep convection and the associated ver-tical moisture advection is to moisten the midtropo-

sphere. Equation (31) is a statement of deep convectionmaintaining strict quasi-equilibrium between theboundary layer and the lower troposphere.

Furthermore, we can also write [from Eqs. (13), (14),and (15)]

J2 � �rqL0q �32�

(letting r0 � 1).Combining the above equations, we have

�t J2 � �rqL0�tq

� �rqL0�a1 � d1�J1

�rqL0�a1 � d1�F

b1�tT2. �33�

We can see that J2 keeps the same phase as T2. Fur-ther, because J2 is a very small perturbation on the drywave, we can also write

�tT2 � c22�2, �34�

where the divergence �2 is defined by

�2 � �H � u2; �35�

thus,

FIG. 13. As in Fig. 2, but for the simplified system described in section 4 and with thedispersion curves for heq � 80 m also plotted. All modes with meridional index n � 1 aredeliberately not plotted, despite being unstable within this system.


�t J2 �rq�a1 � d1�Fc2

2

b1�2. �36�

So we now have

|J2| �|�2|

. �37�

Combining our above relations for the heating field,we define

J2 � �B|�2|T2

|T2| , �38�

where B is a constant of proportionality, considered tobe very small, so that the heating is only a small per-turbation to the waves:

B � �rq�a1 � d1�Fc2

2

b1. �39�

In this limiting case, our equation set reduces to:

�tu2 � �y�2 � �xT2, �40�

�t�2 � ��yu2 � �yT2, and �41�

�tT2 � c22�2 � J2. �42�

From this point, the first mode is mathematically su-perfluous, so we can drop the subscripts without ambi-guity.

c. Kelvin wave heating–divergence feedback

We start with a neutral Kelvin wave, with the stan-dard form (assuming the dry dispersion relationship tobe sufficiently accurate given the arbitrarily small heat-ing) for the complex wave:

u � u0e��

�y2

2c�ei�kx� t�, �43�

� � 0, and �44�

T � �c2k

u0e

��y2

2c�ei�kx� t �

� cu0e��

�y2

2c�ei�kx� t �, �45�

where the amplitude of T is determined using the stan-dard wave relations (in wavenumber–frequency space)and the dry dispersion equation (see, e.g., Gill 1982).

We can then calculate the total energy (per unitlength in the x direction) stored in the wave (H is theequivalent depth of the second mode):

E �12

u2H �12

T2

g�

H

2 �u02 �

�cu0�2

c2 ��e��

�y2

c�

cos2�kx � t��

� Hu02

2�

k

2� �0

2�

�0

2�

k cos2�kx � t� dx dt ��

�

e��y2

c�

dy �H

2�c�

�u0

2. �46�

The overbar denotes the time and zonal mean of a quantity, integrated over its meridional extent.The energy input in time �t is given by

�E � JT�t ��B|�|T2

|T | �t �B|�| cu0

c2u02e

��y2

c�

cos�kx � t��t �B|�|cu0�t

2k � c�

��47�

(|T | � �cu0, |�| is defined similarly).With the increased energy in the wave after this in-

terval, the amplitude of the wave (given by u0) will haveincreased:

�u0 � ��u0E��1�E

�Bc|�|�t

2Hk. �48�

Now, the divergence in the wave can also be deter-mined:

� � �xu � �ku0e��y2

2c�

sin�kx � t�. �49�

We can calculate how the divergence increases in thistime interval:

�� k�u0e��y2

2c�

sin�kx � t�. �50�

Because �� is exactly in phase with �, we can also saythat

�|�| � k�u0

� kBc|�|�t

2Hk,

�|�|�t

�Bc

2H|�|. �51�


As shown, the amplitude of the Kelvin wave diver-gence grows exponentially with time at a rate that isindependent of wavenumber, as observed in the nu-merical analysis above.

d. Mixed Rossby–gravity wave heating–divergencefeedback

The basic dry MRG wave is well described by itsy-velocity field

� � �0e��y2

2c�ei�kx� t� �52�

and a dispersion relationship

� �kc

2 �1 ��1 �4�

k2c�. �53�

Then, using the standard forms,

T � �i c2

2 � c2k2 ��y �k

�y��

�ic�0�ye��y2

2c�

ck � ei�kx� t�, and �54�

u ��y� � ikT

�i

� �i�o�ye��y2

2c�

ck � ei�kx� t�. �55�

Similarly to the Kelvin wave example, we can find theenergy per unit length stored in this wave:

E �H

4�0

2�c�ck2 � �� 2ck � 2�

� � ck�2 ��c

�. �56�

Then, the increase in the wave amplitude due to anincrease in energy �E is given by

�� 2�E� � ck�2

H�0�c�ck2 � �� 2ck � 2��

�c, �57�

and the increase in energy over �t is

�E � JT �Bc4�0|�|�t

4� � � ck�� c

��58�

(|T| � c�0/(ck � �), |�| is defined similarly).Leading to an increase in the wave amplitude of

�� Bc3|�|�t� � ck�

2H �c�ck2 � �� 2ck � 2�, �59�

the divergence of the MRG wave is

� ��0 �

c2k � c ye

��y2

2c�

cos�kx � t� �60�

and the magnitude of the divergence is

|�| ��0�

c� � ck��61�

(note the change of sign so that the magnitude of thedivergence is positive—this simplifies interpretation).Thus,

�|�|�t

�Bc

2H|�|� c�

c�ck2 � �� 2ck � 2� �62�

�Bc

2H|� |

�

2� �k2c

2 ��1 ��1 �4�

k2c� , �63�

where the upper sign applies for positive wavenumbersand the lower sign applies for negative wavenumbers.

e. Equatorial Rossby and inertio-gravity waveheating–divergence feedback

A similar analysis for the n � 1 Rossby and inertio-gravity waves yields

�|�|�t

�Bc

2H|�|� c�3c2k2 � 2ck n�1 � 2 n�1

2 �

c4k4 � 3c3k2� � 3c� n�12 � n�1

4 � 2c2k n�1�� k n�1��, �64�

where

n�1 � ��k

k2 � 3�

c

, �k2c2 � 3�c� �65�

for the Rossby and IG waves respectively (see, e.g., Gill1982), obtained from the dispersion relationship

�

c �2

� k2 � �k

� 3

�

c�66�

by ignoring the first and last terms on the left-hand side,respectively.

f. Energy feedback results

Figure 14 shows the growth rate of the MRG, n � 1ER, and n � 1 IG waves in this construction, normal-


ized by the constant growth rate of the Kelvin waves.The growth rate of the MRG increases as the divergentcomponent of the wave increases from a small value atnegative wavenumbers (where the wave is very muchlike the low-divergence ER waves) to a large value atpositive wavenumbers, where the wave most resemblesthe high-divergence Kelvin wave. Just as observed inthe numerical study (Fig. 13), the growth rate increasesrapidly between k � �5 and k � �5.

The growth rate of the n � 1 ER wave increases as kincreases from �� to zero, as observed in the numericalresults. The growth rate of the n � 1 IG wave increasesas |k| increases from a small nonzero value at k � 0, asobserved.

It is interesting to note that the Rossby waves achievegrowth rates very close to those of the Kelvin wavesbecause the Rossby waves are typically considered tobe dominated by vorticity. However, this can be ex-plained by visualizing a very long-wavelength Rossbywave (k � 0; refer to the left column of Fig. 15). Thiswave will consist of nearly uniform motion away fromand toward the equator, with zonally uniform meridio-nal velocity and near-zero zonal velocity. In this case,the wave clearly has a large divergence and very smallvorticity. As the wavenumber increases, the contribu-tion of the zonal divergence increases, offsetting the

meridional divergence (because of the phase differ-ence). At large wavenumbers, the two components can-cel, leading to reduced divergence and a wave that hassignificantly larger rotational flow (right column of Fig.15).

The inertio-gravity wave spectrum is explained bysimilar logic, although the details are different becauseof the different structures of the waves, particularly thenonzero vorticity at zero wavelength and the increasingdivergence with wavelength.

The numerical values of the corresponding growthrates (also normalized to the numerical Kelvin wavegrowth rate) are also plotted on this figure for compari-son. The MRG and ER growth rates are essentiallyindistinguishable from the analytic solutions. The dis-crepancy for the inertio-gravity waves stems from theapproximation used for the wave frequency in Eq. (65),which is only accurate to within 13% in this case (see,e.g., Gill 1982), whereas the error in the ER frequencyis less than 2%.

The physical content of this argument is containedmore in the method than the final expressions. Theconstruction of instability mechanism here emphasizesthe differing efficiencies with which the modes can con-vert input energy into divergent flow, driving more en-ergy input. As the growth rates derived from this con-

FIG. 14. Analytic growth rate for the waves in the simplified limiting case, relative to theKelvin wave growth rate (lines) and numerical growth rates (points), as described in thelegend.


struction of the model resemble the numerical resultsclosely, we feel that this vindicates our hypothesis, illu-minating a further mechanism that influences the linearinstability spectrum of the equatorial convectivelycoupled waves.

The specifics of our derivation assume the basic formof the moisture–stratiform instability where low-leveldivergence leads to second baroclinic convective heat-ing a quarter-cycle later as discussed in the introduc-tion. The detailed result is thus dependent on the exis-tence of the moisture–stratiform (or similar) instabilitymechanism.

5. Summary and discussion

In this paper, we have demonstrated that our exten-sion of the model of K08b to the equatorial beta planeallows us to produce a linear system with a spectrum ofunstable modes that bear a good resemblance to themodes visible in the global OLR data, both in terms ofexcited wavenumber range and also in terms of the

wave types present. The unstable Kelvin, MRG, and IGwaves visible in the results of WK99 are reproducedwell using this model.

Our model is closely related to that of Khouider andMajda (see, e.g., Khouider and Majda 2006a,b, 2008),especially in the inclusion of midtropospheric moistureas a control of convection depth. However, by makingour model conceptually simpler than their model, thevarious instability and spectrum-shaping mechanismsare made clearer. The further extension of our model tothe beta plane also allows the investigation of instabilityshaping due to the mean-state profile and the meridi-onal winds of the various waves observed on the equa-torial beta plane.

The model does fail to produce any significantly un-stable ER or any MJO waves. As stated above, this maybe due to our model possessing too great a low-fre-quency damping or lacking an effect that serves to de-stabilize the low-frequency waves. However, the obser-vations of Yang et al. (2007) indicate that the Rossbywave signals detected in the OLR spectrum might have

FIG. 15. (top) Analytic divergence and (bottom) vorticity fields for dry equatorial Rossby waves with � � (left)25 � 103 km and (right) 1.1 � 103 km. It is important to note that the x axes of the two columns have very differentscales, as demanded by the very different wavelengths depicted. Grayscale is in arbitrary units.


a different instability character. For example, they maybe unstable only to finite disturbances, requiring a largeextratropical Rossby or near-resonant MRG wave toexcite them. Or possibly the waves are simply stable,with their decay in balance to near-constant stochasticforcing from the extratropics and the MRG activity.The reason for the lack of MJO activity is at presentunknown.

Previous interpretations of the spectrum, like thoseof Takayabu (1994) and WK99, use the differing pro-jections of the mean-state heating onto the wave struc-ture to explain the east–west asymmetry. In the contextof our model, it was also observed that the position andsize of the ITCZ affects the growth rates of the differ-ent waves: modes that possess a large spatial overlapwith the ITCZ profile are enhanced, whereas thosewith a small overlap are weaker. However, even whenthe ITCZ configuration provides large overlap for theMRG or IG waves and small overlaps for the Kelvinwaves, the Kelvin wave growth rates remain large com-pared to those of the MRG and IG waves, indicatingthat additional factors affect the growth rates of differ-ent wave types. We suggest that the efficiency withwhich waves convert input energy into divergent windsis an important factor. We note that Lindzen (1974)also argued that the east–west asymmetry in the n � 1,2 IG waves may be related to the differing amounts ofconvergence in the different propagation directions.Our mechanism shares some characteristics with his ar-gument in that the growth of modes is related to thestrength of the divergent winds of that mode, exceptthat our mechanism is based on the moisture–stratiforminstability instead of the classical wave–conditional in-stability of the second kind (CISK) as in Lindzen(1974). We return with more discussions of wave-CISKat the end of the paper.

With the addition of our energy flow divergencefeedback mechanism, the basic shape of the wave spec-trum is starting to become clearer. The linear part ofthe spectrum is shaped by the combined actions ofmoist convective damping, which damps high-fre-quency waves; the damping effect of the second modeconvective heating on midtropospheric moistureanomalies, which damps more strongly the lowest-fre-quency waves; the ITCZ projection effect; and the en-ergy flow divergence feedback, which selects the wavetypes with larger divergent components.

Analytical extension of the energy flow feedback toour full set of equations is expected to be nontrivial;furthermore, our discussion is based on a simple linearmodel of convectively coupled waves. However, itseems likely that the principle of “wasted energy” in therotational component of the velocity fields of the equa-

torial waves will also apply in the real atmosphere,helping to shape the observed spectrum of convectivelycoupled equatorial waves.

The present model does not make use of the columnMSE budget so that it does not make a statement of thegross moist stability. One could use the vertically aver-aged free tropospheric moisture in place of the midtro-pospheric moisture and take advantage of column MSEconservation, as done in Khouider and Majda (2006a)and discussed in section 3f. The gross moist stability tofirst baroclinic motions in the CSRM simulations ofKuang (2008a) is found to be close to neutral. Becausewe also neglect radiative and surface flux feedbacks,the moisture mode found in, for example, Fuchs andRaymond (2002), Raymond and Fuchs (2007), andSugiyama (2008a,b, both manuscripts submitted to J.Atmos. Sci.) is absent here. We plan to include theradiative and surface feedbacks to examine the mois-ture modes in the future.

Last, given the long history of wave-CISK (Yamasaki1969; Hayashi 1970; Lindzen 1974), we would like tobriefly discuss the relation between wave-CISK, the di-rect stratiform instability in Mapes (2000), and themoisture–stratiform instability in K08b and the presentstudy. The direct stratiform instability may be viewedas a two-mode discretization of the classical wave-CISKenvisioned in Lindzen (1974),2 in which waves modu-late convection, which then further generates waves,and for waves of certain structures and phase speeds,the two can reinforce each other. Indeed, as shown inK08b, when the lag time between stratiform heatingand deep convective heating is neglected, the growthrate of the direct stratiform instability increases linearlywith wavenumber, similar to that observed in classicalwave-CISK. Our moisture–stratiform instabilitymechanism relies on the effect of midtroposphericmoisture anomalies on convection, and hence differsfrom the classical wave-CISK (unlike the classicalwave-CISK, it has bounded growth rates at short wave-lengths even without the MCD effect). The moisture–stratiform instability could be also viewed as wave-CISK in a very broad sense, if one defines wave-CISKas a self-exciting cooperative instability between waveand convection that does not require surface flux orradiative feedbacks (Bretherton 2003). However, it isunclear that this association is particularly meaningfulbecause the definition of wave-CISK in this case wouldbe so broad that it would not contain the actual physicalprocesses that are at work.

2 This was pointed out to us by Chris Bretherton.


Acknowledgments. This work was partly supportedby the Modeling, Analysis, and Prediction (MAP) Pro-gram in the NASA Earth Science Division and by NSFGrant ATM-0754332. The authors thank William Boos,David Romps, and Christopher Walker for helpfulcomments and suggestions during the drafting of thispaper. Comments by Matthew Wheeler, David Ray-mond, and an anonymous third reviewer also improvedthe paper. The discussions on the moist static energybudget and wave-CISK were prompted by discussionswith Adam Sobel, Chris Bretherton, and David Ray-mond. Comments by Chris Bretherton considerablysharpened our view on wave-CISK.

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a toy model of the instability in the equatorially trapped convectively coupled waves...

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