The Interaction of Waves and Convection in the...

15 DECEMBER 2003 3009L I N D Z E N

q 2003 American Meteorological Society

The Interaction of Waves and Convection in the Tropics

RICHARD S. LINDZEN

Program in Atmospheres, Oceans and Climate, Massachusetts Institute of Technology, Cambridge, Massachusetts

(Manuscript received 9 June 2002, in final form 9 July 2003)

ABSTRACT

Interest in tropical waves and their interaction with convection has been rekindled in recent years by thediscovery, using satellite infrared data to track high clouds, that such waves closely display the dispersiveproperties of linear, inviscid wave theory for an atmosphere with a resting basic state and equivalent depthsbetween 12 and 60 m. While several current approaches focus on internal modes in the atmosphere, this isinconsistent with the absence of internal modes in the atmosphere, which is characterized by a single isolatedeigenmode and a continuous spectrum. It will be shown, using an extremely simple approach to convection,that the observed properties of waves are consistent with a continuous spectrum. The approach assumes thatthe total convection is determined by mean evaporation, but that the convection is patterned by zero-averagedperturbations to triggering energy following the recent approach of Mapes. This is, perhaps, the simplest hy-pothesis that can be applied. The observed convection associated with the migrating semidiurnal tide is usedto calibrate the time scale for the convective response to patterning, which is the only adjustable parameter inthis formulation. It is shown that this time scale leads to not only the observed phase of the semidiurnal heatingbut also the observed phase lead of low-level convergence in tropical waves vis-a-vis the convective heating.Finally, it is shown that this phase is sensitive to the equivalent depth, which it is suggested is the basis for theselection of equivalent depth. Reasonable simulations of observed waves are readily obtained.

1. Introduction

An intriguing feature of tropical waves is the fact thata wide spectrum of such waves seems to be character-ized by equivalent depths in the range of about 12–60m. This is illustrated in Fig. 1 taken from Wheeler andKiladis (1999), and was earlier noted by Takayabu(1994). A notable exception is the Madden–Julian os-cillation (MJO; in that its dispersive properties do notcorrespond to a particular equivalent depth), but thereare other exceptions as well, including tides. We willreturn to the exceptions later in this paper; for the mo-ment we will concentrate on the 12–60-m waves. Thenotion of ‘‘equivalent depth’’ arises in tidal theory andis discussed in a more general context in Lindzen (1967)and in Lindzen and Matsuno (1968)—all of whom focuson equatorial waves of the sort investigated by Wheelerand Kiladis (1999). A pedagogical treatment may befound in Lindzen (1990). In section 2, we review thetheory of linearized waves on a static basic state—fromwhich the concepts and terminology for tropical wavesarises. Equivalent depth is a measure of vertical wave-length (or of exponential scale, depending on the signand magnitude of the equivalent depth). An equivalentdepth of 40 m corresponds approximately to a vertical

Corresponding author address: Richard S. Lindzen, MIT, Building54, Room 1720, 77 Massachusetts Avenue, Cambridge, MA 02139.E-mail: [email protected]

wavelength of 8 km. The possible importance of suchequivalent depths was noted by Stevens and Lindzen(1978, hereafter SL) who pointed out that a quarterwavelength, which was about the depth of the tropicalconvective boundary layer, might maximize conver-gence in such a layer. More recently, Wheeler and Ki-ladis (1999) showed with satellite data that equivalentdepths in this neighborhood actually dominated thespectrum. Early attempts to explain the prevalence ofparticular equivalent depths are reviewed in section 3.It remains tempting to associate these equivalent depthswith internal normal modes, but as noted in section 2,the atmosphere does not, in general, possess such nor-mal modes. Rather, from the beginning it was recog-nized that the existence of such preferred equivalentdepths was likely to arise from the interaction of thewaves with cumulus convection and the ‘‘effective’’ at-mospheric heating associated with the latent heat releasein these clouds. However, in one fashion or another, theattempts described in section 3 proved unsatisfactory.These attempts mostly sought explanations based oninstabilities arising from the wave–convection interac-tion.

In section 4 of the present paper, another approach istaken based on a simplification of the ‘‘triggering’’ ap-proach to convection presented by Mapes (2000). Here,convection, in the gross, is in statistical equilibrium withevaporation, but low-level convergence can act so as to

3010 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

FIG. 1. Normalized power in OLR as a function of period and wavenumber for westward and eastward waves. Shaded regions showwhere signals are distinguishable from the background at the 95% confidence level. Superposed are dispersion curves for equivalentdepths h of 12, 25, and 50 m. (a) Even-numbered meridional modes; (b) odd-numbered modes. From Wheeler and Kiladis (1999).

pattern this convection, with convection, itself, actingto guarantee that the low convergence is consistent withthe redistribution of convection. It should be stressed atthe outset that this is simply an hypothesis; however, itturns out to be an hypothesis whose implications appearto be consistent with various observations.

In this approach, we usefully distinguish patterningprovided by disturbances that are primarily forced bymechanisms other than convection from those that areforced by the patterned convection itself. Among theformer is the solar semidiurnal tide, which we will useto determine the characteristic time scale for convectivepatterning. For the latter, the low-level convergence pro-vided by the excited disturbance must be consistent withthe heating needed to force the disturbance. Assumingthat there is a characteristic time for the patterning tooccur, the convergence must lead the heating in phaseby a specified amount given from the tidal study.

This is confirmed by a recent data analysis by Strauband Kiladis (2003). In section 5, classical atmosphericwave theory (for an unbounded atmosphere without in-ternal normal modes) is used to examine the conditionsfor self-consistent patterning of convection by wavesforced by the patterned convection. Consistency leadsto the selection of the observed equivalent depths[though for shorter period waves, the small heating as-sociated with congestus clouds described in Mapes(2000) improves agreement]. Moreover, the verticalstructure associated with these equivalent depths agreeswith the observed structure in that it follows the shapeof the heating in the troposphere while taking the formof upward-propagating internal waves in the lower

stratosphere. Finally, in section 6, we discuss the im-plications of the present results, as well as some re-maining difficulties in the theory.

2. Classical atmospheric wave theory

The applicability of classical atmospheric wave the-ory to tropical waves is by no means self-evident. How-ever, the fact that observed tropical waves display thedispersive properties obtained from classical atmospher-ic wave theory (see Fig. 1) suggests that it is a reason-able place to start. This theory dates back to Laplace(1825), and its description in its present form can al-ready be found in Lamb (1932). The version of thetheory where the spherical earth is approximated by anequatorial beta plane is given in Lindzen and Matsuno(1968). All this material is covered in current textbooks(Andrews et al. 1987; Lindzen, 1990), and there is littlepoint in repeating this material here. However, for con-venience we will sketch the theory, emphasizing pointsof special relevance to the present study.

Beginning with the linearized equations of motion fora shallow perfect gas on a rotating sphere, we can reducethe equations to a single equation for a particular var-iable; for example, vertical velocity w. Given that thecoefficients of the differential equation are independentof time t and longitude f, and that the equation is sep-arable in its colatitude u and altitude z dependence, wecan obtain solutions of the form

x /2 i(vt1sf )w(u, f, z, t) 5 e e y (x)Q (u), (1)O n n

where x 5 (dz/H ), and H 5 RT/g is the local scalez# 0


FIG. 2. Equivalent depth vs vertical wavelength.

height, where R is the gas constant for air, T is thebasic unperturbed temperature (taken to depend onlyon z because we are taking the basic state to be static),and g is the acceleration of gravity. The height in scaleheights is x. The equations for yn(x) and Qn(u) aregiven by the following, where L and M are operatorswhere the subscript indicates the appropriate variableand the superscripts show the parametric dependences(on the frequency, zonal wavenumber, rotation rate V,earth’s radius a, static stability G, etc.). The separationconstant (known as the equivalent depth) is hn , andF v,s is the forcing associated with frequency v andwavenumber s:

h ,v,s,V,anL {Q } 5 0 (Laplace’s tidal equation);u n

(2)h ,H,G v,snM {y } 5 F (x) (vertical structure equation),x n n

(3)

where (x) is the projection of F v,s(x, u) on Qn(u).v,sF n

In the present paper, we will primarily use the verticalstructure equation, which is characteristically of theform

2d yn 21 l y 5 F , (4)n n2dx

where

1 1 dT g 102l 5 1 2 . (4a)21 2h T dz c 4Hn 0 p

Essentially, the equivalent depth (which is the separationconstant) is a measure of the vertical wavenumber (or,obviously, the vertical exponential scale when l2 is neg-ative), and depending on the relation of hn to zonalwavenumber, frequency, etc., the wave will essentiallybe an internal gravity or Rossby wave or some com-bination of the two. The relation between equivalentdepth and vertical wavelength for T0 characteristic ofthe troposphere is shown in Fig. 2.

It will be important to distinguish between the use ofclassical wave theory for forced and free waves.

Free versus forced waves

For forced waves, we are given v and s. Laplace’stidal equation is solved for hn and Qn, where hn is aneigenvalue and Qn is an eigenfunction. The forcing is


expanded in these eigenfunctions (known as Houghfunctions), and the vertical structure equation is solvedfor the response to each component of the forcing, thecomponents being the projections of the forcing on theHough functions. If there exists a complete set of ver-tical eigenfunctions, then one could just as well haveexpanded the forcing in terms of these, but, as we shallnote, this is not the case for the present problem.

For free waves, the vertical structure equation issolved in the absence of forcing. The eigenvalues arethe equivalent depths of the fluid system. For shallowwater, the only eigenvalue is the depth of the fluid. Fora stably stratified liquid with a lid, the equivalent depthscorrespond to an infinite set of vertical modes. For theunbounded atmosphere, there is generally only a singleeigenvalue, corresponding to a Lamb mode with anequivalent depth of about 10 km. There is also a con-tinuous spectrum. For each equivalent depth and zonalwavenumber, Laplace’s tidal equation is solved for theeigenfrequencies vn and the associated Hough func-tions. For each Hough function, one obtains a relationbetween frequency and wavenumber.

The name ‘‘equivalent depth’’ was chosen by analogywith the shallow water case where the equivalent depthwas the actual depth of the fluid, and there was no needfor a vertical structure equation.

Despite the fact that the atmosphere generally is foundto have only a single equivalent depth (;10 km), manytropical waves are observed to behave as though theyhad a relatively unique equivalent depth of around 12–60 m as seen in Fig. 1 from Kiladis and Wheeler (1999).This new equivalent depth is generally attributed to theinteraction of waves with tropical convection. Note aswell that the MJO is not associated with a particularequivalent depth but rather with a specific period. Whilethe MJO is not the focus of the present paper, it shouldbe mentioned that there have been many suggestions asto how this period arises including the observation thatthis is the expected period for planetary-scale baroclinicinstability, which might, in turn, pattern tropical con-vection (Straus and Lindzen 2000). There is also a longrecord of attempts to account for the MJO with varia-tions of wave–conditional instability of the second kind(CISK) mechanisms (see section 3; Lindzen 1974b;Chang and Lim 1988; Wu 2003).

It should also be noted that recent attempts to interpretthe Wheeler–Kiladis results in terms of internal verticalnormal modes (Mapes 2000; Majda and Schefter 2001;Emanuel et al. 1994) are inconsistent with the spectralproperties of the vertical structure equation for realisticatmospheres without lids. Traditional arguments that thetropopause acts as a lid are inconsistent with the lowreflectivity of the tropopause, as well as the fact thatthe tropopause is not absolutely horizontal. Similarly,vertical normal modes such as those calculated by Ful-ton and Schubert (1985) are also based on the assump-tion of a perfectly reflecting upper boundary, and asnoted by Lindzen et al. (1968), this leads to spurious

resonances. To be sure, the set of vertical modes thusobtained constitute a complete set, which can be usedto expand solutions, but no physical meaning attachesto such modes since the essential lid does not exist forthe real atmosphere, and the failure of such modes tosatisfy an appropriate upper boundary condition leadsto Gibbs-type phenomena.

3. Early approaches to the interaction of wavesand cumulus convection

The initial approach to such waves was the so-calledwave-CISK theories (Lindzen 1974b; Yamasaki 1969;Hayashi 1970). These followed the approach of Charneyand Eliassen (1964) in assuming that, if a large-scaledynamic system could lift air to the lifting condensationlevel, then a cooperative interaction between convectionand the large-scale disturbance could lead to the am-plification of the disturbance. In the Charney–Eliassenversion of CISK, the lifting was due to Ekman pumping,while in wave-CISK, the lifting comes from the wavefield itself. Maximum lifting at the lifting condensationlevel implied an equivalent depth of 10 m (with a quarterwavelength corresponding to about 500 m). This led tothe suggestion that a spectrum like that displayed inKiladis and Wheeler (1999) should exist—though witha smaller equivalent depth (Lindzen 1974a).

In these early approaches, the cumulus mass flux wastaken to be proportional to convergence at 500 m. How-ever, the constant of proportionality was generally un-known. To remedy this, Cho and Ogura (1974) soughtto determine with observations the relation between cu-mulus mass flux and vertical velocity at the lifting con-densation level. They found that the cumulus mass fluxwas approximately 4 times the ambient vertical massflux at the lifting condensation level.

Although more recent discussions tend to ignore this,there was a substantial reassessment of wave-CISK overthe following 5 yr. However, even in the 1960s, A. Elias-sen (1975, personal communication) noted a basic prob-lem with the very concept of CISK that was associatedwith his name: namely, that the lowest 2 km of thetropical atmosphere formed a turbulent trade windboundary layer in which air was constantly being liftedabove the lifting condensation level—even in the ab-sence of any larger-scale system. In general, the break-out of deep convection is limited by the presence of atrade inversion (or more generally, the convective in-hibition energy; Mapes 2000).

As noted by SL and Lindzen (1988), the ratio foundby Cho and Ogura (1974) became unity if one consid-ered ambient vertical mass flux at 2 km instead of 500m. This led to an approach to cumulus parameterizationwherein local cumulus mass flux was taken to be de-termined by evaporation and large-scale convergencewithin the trade wind boundary layer (Lindzen 1988;Geleyn et al. 1982). The resulting parameterization,modified for use with the European Centre for Medium-


FIG. 3. Schematic of cumulus, trade layer, and mixed layer withand without triggering.

FIG. 4. Schematic of hierarchical organization of convection.

Range Weather Forecasts (ECMWF) model, has cometo be known as the Tiedke parameterization, though theparameterizations used by the ECMWF have evolvedsince. The geometry involved is schematically illus-trated in Fig. 3. It was noted by SL that tropical waveswere more nearly characterized by an equivalent depthof 30 m, which corresponded approximately to a verticalwavelength of 8 km, with a quarter wavelength (whereone would expect a maximum in convergence) corre-sponding to the depth of the convective boundary layer(note that different papers associate the same verticalwavelengths with somewhat different equivalent depthsbecause of the use of different basic-state s). Unfor-Ttunately, SL found that their interaction was unable toproduce instability except for gravity waves correspond-ing to squall systems.

In Stevens et al. (1977), emphasis shifted from wave-CISK to a view of equilibrated waves whose convergencefield below 2 km served to simply reorganize convectionthat would occur anyway. As noted by Reed and Recker(1971) and many since, the amplitude in precipitation oftropical waves tends to equal the mean precipitation(where mean refers to a mean over the wave considered),suggesting a reorganization of existing precipitation rath-er than the production of additional precipitation. This isillustrated in Fig. 3. Consistent with this observation,Stevens et al. (1977) assumed that the amplitude of thewave forcing was approximately equal to the mean latentheating. Equivalently, if the average cumulus mass fluxis c, then equilibration occurs when the zero-averageMwave contribution to the cumulus mass flux has anM9camplitude equal to c. Thus, convection patterned onMone scale can be repatterned on smaller scales, much asschematically illustrated in Fig. 4. This situation will bediscussed further in the next section.

A question has long remained as to whether thereactually is a causal relation between large-scale con-vergence and cumulus mass flux or whether it is simplya necessary balance (Arakawa and Schubert 1974;

Emanuel 2000) in an equilibrated system. The work ofSL already noted that wave-CISK really no longerworked with the mass budget parameterization. The pur-pose of the present paper is to examine the behavior oftropical waves from a perspective more nearly in linewith equilibrium views.

4. Wave patterning of convection

Our present approach is a very substantially simpli-fied version of the convective triggering approach de-scribed by Mapes (2000). In general,

EM 5 1 = · rV, (5)c q

where E is evaporation and q is specific humidity in themixed layer, and the divergence is evaluated below thetop of the convective boundary layer.

In the absence of large-scale convergence, we expectconvection to be randomly occurring with a spatiallyuniform probability distribution. In the mass budget ap-proach, Mc responds to directly determined convergencewithin the convective boundary layer. However, in thepatterning approach, convection automatically providesself-consistent low-level convergence, but perturbationsto convergence determine the pattern of convection(Mapes 2000, it should be noted, emphasized other per-turbations). Of course, if the perturbation provides moreconvergence, the convection will not have to provideas much. This situation is somewhat analogous to thesituation in Benard convection where small irregulari-ties in the bottom plate can determine the plan form ofthe convection (Koschmieder 1993).

Note that in both the mass budget and the patterningapproaches, waves do not change the total amount ofconvective activity. In the patterning approach, the low-level fields of the wave perturbation biases the randombreakdowns of convective inhibition energy (CIE) pro-duced by boundary layer turbulence so as to pattern theconvection that would occur anyway. Thus, the meanamount of convection is essentially determined by themean evaporation. While there is evidence that squall


systems play an important role in the convection itself,other systems ranging from gravity waves to easterlyand Kelvin waves to the Hadley and Walker circulationsserve primarily to pattern the convection that wouldotherwise exist. Moreover, all the sources of patterningcan simultaneously coexist as schematically illustratedin Fig. 4.

Recall that the amplitude in precipitation of tropicalwaves tends to be about equal to the mean precipitationaveraged over the wave, suggesting a reorganization ofexisting precipitation rather than the production of ad-ditional precipitation. Note that precipitation rate andMc are closely related since the moisture rising in thecumulus tower condenses due to adiabatic cooling. It istempting to assume that the latent heat thus releasedserves to heat the atmosphere. However, as shown byArakawa and Schubert (1974) and Ooyama (1971), cu-mulus convection does not directly heat the ambientatmosphere. Rather, that portion of the mean verticalvelocity that is carried in cumulus towers also does notcontribute to adiabatic cooling of the ambient atmo-sphere. Thus, we must subtract this part from the adi-abatic cooling; that is, the adiabatic cooling term be-comes

]u(rw 2 M ) . (6)c ]z

The term, Mc(]u/]z), constitutes an effective cumulusheating term. The patterning of the convection givesrise to a contribution to the effective cumulus heatingin the form of the pattern. The contribution of the pat-terning to the mean is, however, zero. This effectiveheating forms an essential link in the interaction oflarge-scale dynamics with convection since it serves asa forcing for motions. However, as previously men-tioned, it proves useful to distinguish two differentmodes of interaction. In the first, we are dealing withmotion systems that have their origin in processes sep-arate from effective cumulus heating. These motion sys-tems, however, provide low-level convergence with theresulting effective cumulus heating modifying the mo-tion system.

The second mode involves self-excitation where, forexample, motion systems forced by effective cumulusheating provide low-level convergence that, in turn, trig-gers the convective pattern that forces the wave. Self-excitation requires that the phase of the patterning besuch as to produce the wave required for the patterning.We will discuss this further in the next section whereit will be seen that this leads to the selection of a par-ticular equivalent depth (or depths).

An important and well-studied example of the firstmode of interaction is the solar semidiurnal migratingtide (Chapman and Lindzen 1970). Forcing of this tideis primarily due to insolation absorption by ozone (But-ler and Small 1963) and water vapor (Siebert 1961).Such forcing leads to approximately the observed am-

plitude of the surface pressure oscillation, but phase isabout 1 h off (maxima at 0900, 2100 UTC instead of1000, 2200 UTC). Lindzen (1978) and Hamilton (1981)showed that the observed semidiurnal component ofrainfall provided additional forcing that would correctthe discrepancy. While it is easy to imagine local factorscausing daily variations in rainfall, there is reason toexpect that the semidiurnal component is primarily dueto patterning by the preexisting global migrating tide.In both Lindzen (1978) and Hamilton (1981), when dai-ly variations in precipitation were Fourier decomposedat individual stations, it was found that the diurnal (24h) component varied in local phase according to thenature of the station (land, atoll, island, etc.) as onemight expect for local factors. On the other hand, thesemidiurnal (12 h) component was found to have ap-proximately the same phase in local time everywhereas would be characteristic of forcing primarily by thepreexisting migrating tide. However, the calculated con-vergence due to the global migrating tide was one orderof magnitude less than needed to account for the ob-served precipitation. It was already noted by Lindzen(1978) that this implies that patterning rather than directforcing of the convective pattern is involved. However,in contrast to tropical waves, the tidal component ofrainfall is only a fifth of mean rainfall. Assuming thatpatterning is involved, this would imply that the timeneeded for the convective response to the patterningperturbation is long compared to the tidal time scale (12h/2p). Indeed, one can use the amplitude of the semi-diurnal tide in rainfall to estimate the characteristic re-sponse time. The simple calculations used to make thisestimate also have two additional implications that canbe checked in order to test the patterning hypothesis.First, the ratio of wave time scale (period/2p) to con-vective response time also determines the phase lag be-tween the effective heating and the low-level conver-gence responsible for the patterning. Thus, one can im-mediately check if this phase lag is such as to correctthe discrepancy in the observed semidiurnal tide. Sec-ond, the finite convective response time also impliesthat there must be a specific phase lead for low-levelconvergence relative to effective heating for tropicalwaves of the sort described by Wheeler and Kiladis(1999), if these waves do, in fact, involve the hypoth-esized patterning mechanism. The recent analysis ofStraub and Kiladis (2003) allows us to check this atleast for equatorial Kelvin waves. As we will see, thereis quantitative agreement in both cases supporting thepresent form of the patterning hypothesis. Note, thatconsistent with our attempt to consider the simplest pos-sibility, we have taken the convective response time tobe independent of the horizontal scale and magnitudeof the perturbation. Presumably, shortcomings in thisassumption should manifest themselves in obvious dis-crepancies in our results.

To crudely analyze this situation, we will take thevertical velocity within the convective boundary layer


FIG. 5. Perturbation cumulus mass flux as a fraction of mean cu-mulus mass flux vs time scaled by wave period. Also shown is theFourier projection of the solution on frequency v: (a) a/v 5 0.16;(b) a/v 5 5.

FIG. 6. Phase lead vs period and approximate result.

to be given by w sin(vt). Let be the cumulus massM9cflux responding to wave patterning induced by pertur-bations in low-level convergence, while c is the meanMmass flux. Let a21 be the characteristic response timeof Mc to the patterning provided by the perturbation inw. Patterning is taken to concentrate convection in re-gions where w is positive and suppress convection wherew is negative. The following equation roughly describeshow we expect to behave:M9c

1 d1 1 M9 ø M sgn(sinvt), (7)c c1 2a dt

where sgn(x) 5 1 for x . 0, and sgn(x) 5 21 for x, 0.

For convenience, we will let vt 5 x, so that the above

equation becomes

v d1 1 M9 ø M sgn(sinx). (7a)c c1 2a dx

Although will, of course, be distorted from a sineM9cwave, its impact on the wave will be associated withits projection on the sinusoidal v component.

Figures 5a and 5b show the behavior of / c forM9 Mc

a/v 5 5 and 0.16. In general, as a/v becomes large,/ c approaches one, and the phase lag goes to zero.M9 Mc

This is already evident in Fig. 5a. On the other hand,as a/v becomes small, / c decreases, and the phaseM9 Mc

lag for convection approaches 908. For the solar semi-diurnal tide, Lindzen (1978) finds that / c ø 0.2.M9 Mc

This, as can be shown, corresponds to a/v ø 0.16, ora21 ø 11.94 h. The phase lag is about 81.88. The ef-fective heating associated with this phase lag is, indeed,what is needed to correct the phase of the semidiurnaltide forced by ozone and water vapor heating alone.This offers some confidence that the value of a deter-mined by means of the semidiurnal tide is reasonable.It should be added that this value is also compatiblewith / c being on the order of unity for tropicalM9 Mc

waves with periods on the order of 5 days or longer.Note that for | a/v | $ 1, the phase lag becomes essen-tially | v/a | 3 908. This is illustrated in Fig. 6.

The waves studied by Straub and Kiladis (2003) haveperiods in the neighborhood of 3–5 days. From Fig. 6,we see that there must be a low-level phase lead inconvergence [relative to outgoing longwave radiation(OLR)] of about 408 if patterning is to be appropriate.This is, in fact, what they find, though the time reso-lution of their data is only just adequate to determinesuch a phase lead albeit with some uncertainty.

5. Explicit calculation of equivalent depth for self-consistent patterning of convection

Our final task in this paper is to show that the pat-terning mechanism acts to select the observed range of


FIG. 7. Mean temperature profile for our standard basic state.

equivalent depths as well as to replicate the observedstructure of the tropical waves—within the context ofclassical atmospheric wave theory. We will solve Eq.(4) using a forcing distribution corresponding to theeffective cumulus heating for a continuous range of hin order to see if a particular choice of h leads to aconsistent phase for the low-level convergence to pat-tern the convection needed to produce the patterningitself. If the phase is inconsistent, it is essentially anal-ogous to pushing a swing at a frequency different fromthe swing’s natural frequency, and the result will be tocancel the oscillation. As can be deduced from Fig. 6,the appropriate phase depends on the period of the os-cillation. For long periods, the low-level patterning con-vergence should be approximately in phase with theeffective convective heating; however, for shorter pe-riods (about 5 days), there should be a discernible phaselead for the convergence.

We will basically follow the analysis of SL. We willconsider the linearized equation for vertical structure ofw[the vertical velocity in logp coordinates, w*, weightedby exp(2x/2)], where x 5 ln( p/ps), p is pressure, andps is surface pressure:

2d w R x21 l w 5 exp 2 Q(x), (8)

2 1 2dx gh 2

where

S 12l 5 2 and

h 4

R dTS 5 1 kT .1 2g dx

The equivalent depth is h, is the basic-state temper-Tature, g is the acceleration of gravity, k 5 R/cp, R 5the gas constant for air, cp is the heat capacity of air atconstant pressure, and Q(x) is the vertical distributionof effective convective heating. Stevens and Lindzen(1978) took S to be constant (620 m), which plausiblyreplicates the gross thermal structure of the tropical tro-posphere. We will also use a more detailed specificationof the thermal structure given by

x 2 x1S(x) 5 S 1 (S 2 S ) 1 1 tanh1 2 1 1 2[ ]d1

x 2 x21 (S 2 S ) 1 1 tanh3 2 1 2[ ]d2

x 2 x31 (S 2 S ) 1 1 tanh , (9)4 3 1 2[ ]d3

which allows us to represent the reduced stability in themixed layer and the trade wind boundary layer, as wellas the increased stability in the stratosphere. Figure 7shows the distribution of basic temperature with heightfor the following choice of parameters: S1 5 5 m, S2 5

200 m, S3 5 500 m, S4 5 2400 m, x1 5 0.0625, x2 50.25, x3 5 2.2, d1 5 0.01, d2 5 0.03, and d3 5 0.25.For the heating distribution we take

x 2 xcbxQ 5 e sin p , (10)1 2x 2 xT c

which closely follows the form observed by Reed andRecker (1971) and Yanai et al. (1973) for b 5 20.33,zo 5 0.15, and zT 5 2.01. For the small values of h(equivalent depth) that we will consider, there is neg-ligible difference between w in z or log p coordinates(Lindzen 1990). Hence we can take w 5 0 for our lowerboundary condition. For the upper boundary conditionwe take the radiation condition applied at x 5 10. Ourvertical resolution is 0.005; that is, we have a total of2000 levels. Note that for the present analysis, the mag-nitude of Q is irrelevant; only its shape matters. Thus,we ignore the multiplicative constant that should be inexpression (10).

Our procedure will be to solve Eq. (2) for values ofh ranging from 1 to 100 m [using Gaussian eliminationas described in Lindzen (1990); if the user wishes touse the algorithm in Lindzen (1990), he should obtainan errata sheet from the author]. In order for the waveto be consistent with the effective convective heating ittriggers, the phase of w in the boundary layer will haveto be somewhat greater than zero (i.e., it must lead heat-ing) in order to allow for the response time inferred insection 2. For periods greater than a few days, this isgenerally small. Figure 8 shows the variation of phasewith h. We see that approximately appropriate valuesare found for h around 15 m and between about 35 and45 m. While the behavior shown in Fig. 8 does notchange much for reasonable changes in basic state orheating profile, such changes can lead to appropriatevalues occurring throughout the region 12 m , h , 60m. For example, in Fig. 9, we show results using thebasic state and heating from SL. This is a very different


FIG. 8. Phase of perturbation w in subcloud layer as a function ofequivalent depth for standard conditions.

FIG. 9. Same as Fig. 8, but for basic state and effective heatingdistribution in SL.

FIG. 10. Same as Fig. 8, with congestus heating. Dashed horizontallines indicate range of phases that are sufficiently close to beingconsistent so as to be associated with minimal suppression.

basic state from what we refer to as our standard basicstate. Here S 5 620 m and there is no attempt to de-lineate either the boundary layer or the stratosphere. Theheating differs from that adopted in this paper in thatzc 5 0.07 (instead of 0.15). There are obvious changesto the solution, but none alter our earlier conclusionseither qualitatively or even quantitatively. It is importantto recognize that one does not, in general, need precisephase consistency for low-level convergence. Whilewhat one takes as ‘‘close enough’’ is, to some extent,subjective, the underlying principle is clear. For ex-ample, if the low-level convergence is off by 308, thepatterned heating will still reinforce the wave for aboutthree cycles, after which it will begin cancelling thewave. Thus, for a packet with three cycles, 308 may beclose enough, though not for longer wave trains.

While the preceding analysis yields results consistentwith Wheeler and Kiladis (1999) for relatively long pe-riods, for relatively short periods (about 5 days), con-sistency requires a phase lead on the order of 308, andthis would lead to equivalent depths somewhat largerthan suggested by the data. The results presented byMapes (2000) show that we may have misrepresentedthe effective cumulus heating. Mapes stresses that thebreak out of cumulus towers is accompanied and some-what preceded by the development of congestus cloudswhose precipitation is smaller than that of the tallertowers (see Fig. 10 of Mapes). The inclusion of effectiveheating of this sort turns out to be able to bring theabove results into good agreement with observations.As long as the congestus heating is shallower and weak-er than, and somewhat in advance of that due to thetowers, the details are relatively unimportant.

To show this, we will add the following to our ex-pression for Q:

x 2 xciwQ 5 Ke sin , (11)congestus 1 2c 2 xtc c

where xtc 5 0.75.

For K, we will take 0.2. For f, we will try 0 and p/6.The latter choice of phase lead is appropriate for rela-tively short periods, while zero is more nearly appro-priate for longer periods. Note that Mapes (2000) findsa lead time, and a given lead time corresponds to asmaller phase lead as the period gets longer. The resultsfor both the SL basic state, and what we have referredto as the standard basic state, are shown in Figs. 10 and11. We see that for both basic states, consistent andnearly consistent phases are found in the range of hbetween 15 and 60 m. It should be understood that thepresent approach to the interaction of waves with con-vection consists in finding consistent phases since in-consistent phases will lead to the self-destruction ofwaves with the corresponding equivalent depths. Suchself-destruction will, however, be slow for nearly con-sistent phases. It is evident that the presence of a rel-


FIG. 11. Same as Fig. 9, with congestus heating. Dashed horizontallines indicate range of phases that are sufficiently close to beingconsistent so as to be associated with minimal suppression.

FIG. 12. Vertical structure of w for h 5 18 m: (a) amplitude;(b) phase.

atively small amount of congestus heating, even in theabsence of a phase lead for such heating, significantlychanges the phase lead of the low-level convergence.The reason is mainly that the shallowness of the con-gestus heating allows more wave leakage into theboundary layer since leakage depends on the ratio ofthe vertical wavelength of the wave (in the absence offorcing) to the thickness of the forcing (Lindzen 1966).

Finally, Figs. 12 and 13 show the vertical structureof the vertical velocity field for waves with h 5 18 mand h 5 52 m. We see that within the troposphere, thewaves mostly follow the distribution of the effectiveheating, but above this region they behave like verticallypropagating disturbances. This is exactly what is foundby Straub and Kiladis (2003, see their Fig. 3). The math-ematical reason for this is simply that for thick forcingand large l2 the solution to (8) consists approximatelyof a balance between the second term on the left-handside of the equation and the forcing (i.e., the particularsolution), while above the forcing the balance must bebetween the two terms on the left-hand side (the ho-mogeneous solution). Note as well, that there is morewave leakage for h 5 52 m than for h 5 18 m; hereagain, this is because the ratio of the vertical wavelengthto the thickness of the forcing is greater. It should finallybe noted that the vertical structure of the temperaturefield will differ from that of the vertical velocity. Thisis because the wave temperature field is proportional tothe difference between the second term on the left-handside of Eq. (8) and the forcing, and hence, gives moreemphasis to the homogeneous solution. Thus, the tem-perature field displayed in Straub and Kiladis (2003) is,not surprisingly, more structured than the vertical ve-locity field. This property of the temperature field isalso evident in Stevens et al. (1977).

6. Discussion and summary

In this paper, we have examined the hypothesis thatlow-level convergence can sequentially pattern convec-tion that would occur anyway due to evaporation. (Bysequential, we simply mean that the patterning by aperturbation of a given scale will pattern the convectionalready patterned by larger scales.) It was further as-sumed that patterning is associated with a time scalethat is largely independent of wave scale or amplitude,this being the simplest situation to consider. In supportof the proposed mechanism, we found that it leads tothe correct phase of the semidiurnal component of theobserved daily variations of precipitation and to the ob-served equivalent depths and vertical structures of trop-ical waves as analyzed by Wheeler and Kiladis (1999)and Straub and Kiladis (2003). (It should be added thatthe time scale found for patterning, about 12 h, is longerthan the time it typically takes for the decay and re-


FIG. 13. Same as Fig. 12, but for h 5 52 m.

generation of an individual cumulus tower. It is morenearly the characteristic time for mesoscale cloud com-plexes. This suggests that patterning is organizing thesecomplexes rather than individual clouds, and is doingso in the shortest time consistent with the formation ofthese complexes regardless of the amplitude or scale ofthe perturbation.) There were several important steps toour argument.

1) Using the solar semidiurnal tide as an example ofpatterning by low-level convergence, we obtained anestimate for the time scale for such patterning tooccur. This, of course, will be longer than the timescales for squalls, which are likely characteristic ofthe convection itself. In addition, the patterning hy-pothesis led to the prediction of the observed phasefor the semidiurnal component of convective pre-cipitation. The assumption that the time scale arrivedat from considering the zonal wavenumber 2 tidewould also be characteristic of other scales tested inthe second step.

2) In this step we check whether the phase lead requiredby step 1 for self-excited equatorial waves corre-sponds to observations. As it turns out, the phaselead observed by Straub and Kiladis (2003) is es-sentially what is called for by the patterning timederived from the semidiurnal tide. This provides im-portant support for both the notions that convectionis patterned and that waves are forced by the re-sulting ‘‘effective cumulus heating.’’

3) Finally, patterning requires only a consistent (ornearly consistent) phase in the trade wind boundarylayer, and the paper simply shows that phase variessubstantially with equivalent depth, hovering aroundacceptable phases for the observed equivalentdepths. This serves to preferentially select theseequivalent depths over other equivalent depths thatdo not present consistent phases.

Note that the fact that the congestus clouds play arole in the cumulonimbus response to dynamic pattern-ing (at least for relatively short periods), and that theinclusion of congestus heating is important in order toachieve consistency in phase between patterning andcumulus heating at the observed equivalent depth whenshort periods are considered suggests that the interactionof waves and convection may be more subtle than an-ticipated.

Despite our success in accounting for many observedfeatures of tropical waves, as noted in Stevens et al.(1977), solutions of simple inviscid linear theory fortropical waves suffer from one significant drawback: forobserved values of rainfall (i.e., effective cumulus heat-ing), amplitudes of temperature and horizontal velocityoscillations are too large. Models commonly replicatetropical easterly waves, but with reduced wave com-ponents of rainfall. Stevens et al. (1977) showed thatthese problems could readily be eliminated by the in-clusion of a simple model for cumulus momentum trans-port (Schneider and Lindzen 1980). Since then, therehas been much interest and controversy over the formor even the existence of so-called cumulus friction. Sar-deshmukh and Hoskins (1987), for example, argued thatthere was no evidence for any such phenomenon. Tungand Yanai (2002a,b), however, have recently presentedevidence to the contrary. Until these issues are resolved,the present approach must be considered somewhat ten-tative. Nevertheless, as noted earlier, the dispersion re-lation observed by Wheeler and Kiladis (1999) suggeststhat such physics remains relevant, and provides a plau-sible starting point for the study of such waves.

Acknowledgments. The preparation of this paper wassupported by Grant DE-FG02-93ER61673 from the De-partment of Energy. Conversations with K. Emanuel,M. Yanai, and G. Kiladis are gratefully acknowledged,as are the helpful comments of the reviewers of a pre-vious version of this paper.


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