## 1. Introduction

Tropical convection forces a full spectrum of waves that propagate into the middle atmosphere. Tropical mean flow oscillations such as the quasi-biennial oscillation (QBO) and the semiannual oscillations (SAOs) in the stratosphere and mesosphere are driven by the deposition of the momentum flux carried by these waves. It is still unknown precisely which waves are primarily responsible for driving the mean flow oscillations, but it is currently believed that the full tropical wave spectrum, from planetary scale to mesoscale, plays a role. Horinouchi and Yoden (1998) and Kawatani et al. (2010) have successfully performed simulations of the QBO and SAO forced by waves that are resolved in a model with T40 resolution. Scaife et al. (2000) and Giorgetta et al. (2006) needed to supplement waves resolved by the model with gravity wave momentum flux parameterizations in order to reproduce a QBO-like oscillation. The resolved waves in these models are forced by convective heating parameterizations. Ricciardulli and Garcia (2000, hereafter RG00) found that various convective parameterizations did not produce heating variability consistent with observations. The goal of this study is to calculate the tropical wave spectrum forced in a model by heating derived from observations of rainfall rates by the Tropical Rainfall Measuring Mission (TRMM), with the idea that the wave spectrum so produced may be more realistic than the wave spectrum produced by convective parameterizations. There are still uncertain assumptions that must be made, however, so the effects of these assumptions will be examined in detail.

One of the first studies of the full tropical wave spectrum was presented by Bergman and Salby (1994). They calculated the waves generated by convection using high-resolution global cloud imagery. Latent heating was deduced from the cloud imagery and used to force the primitive equations linearized about a uniform background flow. Solutions to these equations were obtained using the Hough mode theory developed in Salby and Garcia (1987).

Our study incorporates a suite of models, beginning with further development of the linear Hough model. The studies of Salby and Garcia (1987), Bergman and Salby (1994), and RG00 all made the assumption that heating was uniform in altitude below a fixed cloud height of 10 km. However, observations suggest that convective heating is better represented with a half-sine shape profile (see, e.g., Houze 1982; Schumacher et al. 2004; Shige et al. 2004). Furthermore, global IR cloud data can provide estimates of cloud-top height. Hence, we shall examine the effect of altering the former assumptions about heating profiles by comparing results using these heating profile shapes to results obtained with the half-sine shape and variable cloud height. The paper by Ryu et al. (2011) examines how including stratiform-type heating profile shapes affect the wave spectrum.

Our experiments with a time-dependent primitive equation model use realistic background zonal mean zonal winds and temperatures from the National Centers for Environmental Prediction (NCEP) reanalysis data provided by the National Oceanic and Atmospheric Administration’s Office of Oceanic and Atmospheric Research, Earth System Research Laboratory, Physical Sciences Division (NOAA/OAR/ESRL PSD, Boulder, Colorado) from their Web site (http://www.esrl.noaa.gov/psd; Kalnay et al. 1996). We find that wave flux entering the lower stratosphere is sensitive to the structure of the buoyancy frequency profile *N*(*z*) in the tropical troposphere. The sensitivity of the flux spectrum to both heating profile shape and *N*(*z*) is demonstrated here and further explained by Ortland and Alexander (2011).

A nonlinear time-dependent model is used to examine the wave flux propagating through realistic zonal mean zonal winds and in the presence of nonlinear interaction among the waves. Mean winds are found to significantly limit the propagation of Rossby waves into the stratosphere. Returning to the Hough model once again, this is explained with an ad hoc argument relating to the structure of the projection of the heating spectrum onto the Hough functions. Nonlinearity is also found to limit wave amplitude in the troposphere (e.g., Manzini and Hamilton 1993).

The nonlinear model allows us to calculate the wave driving of the mean flow through the divergence of vertical momentum flux. Bergman and Salby (1994) examined the wave spectrum in detail and found that the bulk of wave flux relevant for forcing the QBO (with phase speeds from 10 to 20 m s^{−1}) is provided by gravity waves with zonal wavenumber greater than 20. We present a similar budget analysis by calculating the contribution made to mean flow acceleration by various portions of the wave spectrum, and we find that ⅔ of the westward mean flow forcing and ½ of the eastward mean flow forcing comes from waves with zonal wavenumber greater than 20. This is in rough agreement with the results obtained by Kawatani et al. (2010).

Several basic states are used for the model simulations to be discussed in this paper. The tropical averages of the relevant profiles are shown in Fig. 1. We shall compare results from two opposite phases of the QBO. The left panel shows the mean zonal wind at the equator for May 2005 and 2006 from NCEP. During 2005 the easterly QBO jet peaks near 30 km, whereas in 2006 the westerly QBO jet peaks at 28 km. Several simulations will also employ mean zonal winds set to zero. To illustrate the effects of background temperature, these simulations employ two different mean temperature profiles independent of latitude. The corresponding buoyancy frequency profiles are shown in the right panel of Fig. 1. These profiles are derived from May 2006 NCEP temperatures by either a tropical average between 15°S and 15°N or an average between 45°S and 45°N. The latter average has warmer temperatures in the upper troposphere and hence larger buoyancy frequency than the tropical average.

The paper is organized as follows. Section 2 presents an outline of the solution to the linear equations via Hough modes. This section also presents a description of the heating spectrum and its projection onto Hough functions. Section 3 presents solutions to the vertical structure equation for various *N*(*z*) and heating profiles. These solutions are used to derive the flux response to heating as a function of equivalent depth. The Hough projections of heating and the flux response functions are combined in section 4 to obtain the full spectrum of vertical Eliassen–Palm (EP) flux. This section also describes the sensitivity of the flux spectrum to the structure of the heating profiles and to the structure of *N*(*z*) in the troposphere. Section 5 describes mean flow and nonlinear effects on the flux spectrum. Mean flow forcing in the stratosphere for two opposite phases of the QBO is analyzed in section 6. The final section presents a summary and discussion.

## 2. Linear wave response to convective heating

### a. Solution method

In this section we discuss the analytic solution to the forced undamped primitive equations linearized about a zero zonal mean zonal flow. The solution method is outlined by Andrews et al. (1987) and is described in considerable detail by Salby and Garcia (1987). The notation and conventions of Andrews et al. are adapted here.

*λ*, sine-latitude

*μ*= sinφ, and log-pressure height

*H*= 7 km is a constant scale height. The horizontal and vertical structure functions of each mode are indexed by a nonnegative integer

*n*, integer zonal wavenumber

*s*, and a nonnegative frequency

*ω*. The index

*n*, which may be considered a proxy for meridional wavenumber, is a shorthand notation for a pair of indices as in Salby and Garcia (1987), one of which determines the wave manifold containing Kelvin, Rossby, or gravity waves. Negative zonal wavenumbers correspond to westward traveling waves. Discrete frequency samples

For each spectral index, the primitive equations reduce to a pair of coupled ordinary differential equations. The Hough functions

*f*(

*μ*) may be expressed as a serieswhere the coefficients of the expansion are obtained via the horizontal projectionThe continuity and thermal energy equations determine the relationsbetween the two vertical structure functions, where

*N*is the buoyancy frequency and

*a*is the planetary radius. The heating field

*J*is decomposed as in Eq. (1) with horizontal structure given by the Hough functions and

*W*to

*V*is made to avoid carrying around inconvenient factors in what follows. Solutions are also required to satisfy the bottom boundary conditionand the radiation or exponential decay condition at the top. A different vertical structure equation is satisfied by

*f*is the Coriolis parameter. This definition of

*F*differs from the standard definition by factor of

_{z}*a*so that it has units of momentum flux. Solving for the wave components of the wind fields in terms of geopotential, one obtains the wave component of flux:The average EP flux serves as a measure of wave activity and its divergence will be used to estimate the forcing of the tropical winds. If the latitude average extends from pole to pole then the orthogonality of the exponential and Hough functions enable the average EP flux above the heating to be written as a sum of contributions from each separate Hough mode. Each such contribution is given byThe second expression is obtained from the first by using Eqs. (4) and (5). The full-latitude average of the EP flux [Eq. (9)] will only be used in the following section. For an equatorially trapped mode, the flux is confined to the tropics over an effective width that depends on the mode. When the latitude average is confined to a range relevant to the tropics, orthogonality will essentially hold only among those modes for which the Hough function amplitude is confined within the selected range. Hence, in general, a decomposition [Eq. (9)] of the flux average for a given wave component over a restricted range in terms of individual modes is only approximate. The exact result is obtained by performing the latitude average of Eq. (8).

### b. Hough projections

*N*= 0.02 s

^{−1}were calculated. The range of equivalent depths for the tropical waves discussed in this study runs from 0.3 to 2500 m. This range roughly corresponds to waves with vertical wavelengths from 0.5 to 50 km. The values of the indices include zonal wavenumber values of

*s*= −120, … , 120, and 224 discrete values of

*ω*ranging from 0 to 4 cpd with an interval of

*n*is zero for the single Kelvin and mixed Rossby–gravity wave (RGW) modes but runs through the positive integers for the inertia–gravity wave (IGW) and Rossby wave (RW) modes.

*R*is the TRMM 3B42 rainfall rate,

*ρ*is the mass density of water,

_{W}*L*is latent heat of condensation,

_{c}*c*is specific heat at constant pressure, and

_{p}*M*is the integral of air density up to the cloud top. The cloud top is obtained from global-merged IR brightness temperatures matched to NCEP temperatures to determine a height. We use 28 days of data from May 2006. In this paper we assume that all of the precipitation is convective and do not include ice processes or evaporation. A comparison of zonal means of the monthly-mean heating with these assumptions to the TRMM monthly convective/stratiform heating product appears in Ryu et al. (2011).

These heating rates will be converted into heating profiles using two different profile models. The simplest model is to assume that the top of the heating profile is fixed at 10 km and that the heating rate at each altitude below the cloud top is given by Eq. (11). A second model is the half-sine profile *z*_{top} is the cloud-top height. This model will incorporate either a cloud-top height fixed at 10 km or determined from the TRMM merged IR brightness temperature.

Two figures illustrate the spatial and temporal structure of the heating ^{−1}. The top panel of Fig. 3 shows the heating power spectral density averaged from 15°S to 15°N as a function of zonal wavenumber and frequency. To make this figure, the heating was binned on a 0.75° latitude grid so that the results could be directly compared to the power spectra shown in RG00. Most of the amplitude is confined to small zonal wavenumber and frequency, except for noticeable peaks at the diurnal and semidiurnal frequencies. It is shown below that the low-frequency heating is inefficient in producing a response that propagates into the stratosphere.

The solid curve in the bottom panel of Fig. 3 shows the heating power spectrum from the top panel summed over zonal wavenumber. This spectrum has the same structure and magnitude as the heating spectrum derived from GCI by RG00 (their Fig. 8b). However, those authors examined heating over a 128-day range, about 4.5 times longer than used here. Since the power spectrum for statistically steady heating is proportional to the time period, the TRMM heating variance is roughly a factor of 4 larger than the variance derived by RG00. The dashed curve shows the power spectrum derived from heating retained in the original 0.25° latitude bins, and the dotted–dashed curve shows the power spectrum for full latitude and longitude resolution up to zonal wavenumber 720. The deviation of these spectra from the spectrum derived from lower resolution shows that there is significant power in the small spatial scales. The waves excited by this heating variance at small spatial scales and at frequencies smaller than 4 cpd have relatively small vertical wavelengths and group velocities. Hence they do not readily propagate very high into the stratosphere.

The heating spectrum is projected onto all of the Hough functions for each frequency and zonal wavenumber to obtain the complex amplitude profile ^{−1}, which lies roughly along lines through the origin out to the corner of the plot at (*s*, CPD) = (120, 4). The amplitude of projection onto the IGW is larger for westward moving waves. The amplitude plots are overlaid with curves of constant equivalent depth. Since the vertical wavenumber is related to the equivalent depth by Eq. (10), these curves represent a dispersion relation between vertical wavenumber and mode indices. As the mode index is increased, the dispersion curves for each type of mode (IGW, RW, etc.) shift slightly in the zonal wavenumber–frequency plane but otherwise maintain similar structure. The Hough projections are computed for a fixed range of equivalent depths that correspond to vertical wavelengths between 0.5 and 50 km when *N* = 0.02 s^{−1}. The black regions in the plots correspond to waves outside this range.

### c. Mean wind effects

The method used above to solve the linear primitive equations can be generalized to the case of nonzero background flow only if the zonal mean zonal wind *ω* with the Doppler-shifted frequency *U*_{0} the difference between Ω and *ω* with opposite sign to the value associated to the same mode when the mean winds are zero.

The wind in the source region changes the pattern of the zonal wavenumber–frequency spectrum entering the stratosphere by altering the horizontal projection of heating. In Fig. 5, the value of *U*_{0} = −5 m s^{−1} was chosen since it is close to the value of the zonal mean zonal winds from the May 2006 NCEP data in the tropical troposphere. Figure 5 is obtained by projecting the horizontal dependence of the heating spectrum for each point in the (*s*, *ω*) plane onto the set of Hough functions *n*. The result for the Kelvin, RGW, RW, and first IGW modes are shown. The new dispersion curves plotted in Fig. 5, when compared to the same curves in Fig. 4, should help visualize how the mapping takes place. Notice that there are now new westward Kelvin and IGW wave projections, and that westward RW and RGW have frequencies much larger than permitted when *U*_{0} = 0.

This picture of the change in horizontal projection as a function of the zonal mean winds helps to explain why most Rossby waves do not propagate into the stratosphere, especially if we consider the multiple-scale picture of propagation described by Ortland (2005). Consider the Rossby modes shown in Fig. 5 as they propagate upward. The waves with large zonal wavenumber originate with phase speed close to −5 m s^{−1}. If the mean winds become more easterly then these waves will reach a critical level and be filtered out. If the winds become more westerly then the equivalent depth for a particular mode at a given zonal wavenumber and frequency can become arbitrarily large, pass through infinity (i.e., Lamb’s parameter goes to zero), and then become negative. Modes with negative equivalent depth (which are not shown in the figures) are evanescent with altitude and decay quickly. Rossby waves can only propagate upward if the mean winds remain close to their value at the source or if the zonal wavenumber is small enough to provide a wide range of phase speeds at which they can propagate.

## 3. The vertical structure equation

Solutions to the vertical structure Eqs. (5) and (6) determine how the vertical EP flux entering the stratosphere depends on the structure of *N*(*z*) and *J*(*z*) within the troposphere. We shall examine some numerically determined solutions that illustrate the dependence on these profiles. Analytic solutions that provide some insight into the physical mechanisms that control the solution structure are given by Ortland and Alexander (2011).

*h*is proportional to the invariant wave flux of the vertical structure Eq. (5):We view

*h*as well as

*z*since

*h*appears as a parameter in the vertical structure equation. Both

*z*above the heating. Once the heating and buoyancy frequency profiles have been fixed, the equivalent depth serves as a parameter in the vertical structure equation. Thus, for a fixed altitude

*z*, so we suppress that variable and view the wave flux as a function of equivalent depth alone. Note that

*I*and

*F*, together with the structure of the heating spectrum in the

_{z}*N*(

*z*) profiles. The heating profile will be either a half-sine shaped profile

The *N*(*z*) profile has value equal to the other two profiles at 5 km in the center of the heating profile. The constant profile was studied by Salby and Garcia (1987) and Bergman and Salby (1994), while a two-step profile was used by Horinouchi and Yoden (1996). We shall examine how using more realistic profile shapes affects the results. The dependence of the wave flux response on *N*(*z*) may be relevant for investigating the effects of climate change, since it is conceivable that trends in the tropical temperatures may produce significant trends in tropical *N*(*z*).

Figure 6 shows the response of *I*(*h*) obtained from numerically integrating the vertical structure equation for the heating and buoyancy profiles described above. The top panel shows the response to constant-step heating as a function of equivalent depth for the three *N*(*z*) profiles, while the bottom panel shows the wave flux response to half-sine heating. As shown by Salby and Garcia (1987), the response function for constant *N* (blue curves) has a primary peak near *m*_{0} corresponds to an equivalent depth of 172 m.

The response function also has a series of secondary peaks. The secondary peaks for constant step heating drop off gradually, but for the half-sine profile the secondary peak amplitudes drop so rapidly that they are not visible on this scale. The same holds true for the responses relative to the other *N*(*z*) profiles. The primary peaks are similar for the two heating profiles, but the secondary peaks are much stronger for the step profile. This sensitivity to the heating profile may be important because it occurs at small equivalent depths where, as we saw above, the Hough projections of heating onto IGW have large amplitude. The secondary peaks in the EP flux response with equivalent depths between about 10 and 50 m correspond to waves observed in the stratosphere that are relevant for QBO forcing. In this range of equivalent depths, the maximum at the first secondary peak in the EP flux response is about a factor of 4 smaller for the half-sine profile than for the constant-step profile. Hence, since the half-sine profile is a more realistic model for the heating profile, wave forcing calculated using the constant-step profiles is likely to be an overestimate. The sensitivity to the *N*(*z*) profiles shows up as both an alteration of the primary peak structure and as an enhancement of all peaks relative to the response for constant *N*. The variation in the magnitudes of the secondary peak due to the different *N*(*z*) profiles examined here is a roughly a factor of 2.

The primary peak is near 290 m for the NCEP tropical *N*(*z*) profile and 360 m for the warm tropopause *N*(*z*) profile. Because relationship (10) between local vertical number and equivalent depth involves *N*(*z*), for general profiles it is not possible to assign a definite vertical wavenumber to the vertical depths at the peak of the response functions. For a heating depth of 10 km, and for values of 290 and 360 m for the equivalent depths at the peak response for the NCEP and warm tropopause profiles, a vertical wavelength of twice the heating depth gives values of 0.017 and 0.019 s^{−1} for *N*, respectively. Although these values do occur somewhere in these profiles, there does not appear to be any reason to single them out. The vertical wavelengths in the stratosphere that corresponds to these values of equivalent depth are roughly 15 and 17 km. Therefore, it is not clear how Salby and Garcia’s rule of thumb is applicable for general *N*(*z*) profiles. Ortland and Alexander (2011) show that the peak response for the NCEP *N*(*z*) profile is enhanced relative to the response for constant *N* = 0.0134 s^{−1} because of both internal wave reflection and the reduction of *N* within the heating region. In fact, as we shall see, small changes in the buoyancy frequency can produce a large change in the total magnitude of EP flux.

## 4. Linear EP flux spectra

The wave flux response as a function of equivalent depth that was studied above will now be mapped onto the *I*(*h*) from Eq. (12) for *h* of the mode, and then multiplying by the additional factor

*Q*(

*z*). By linearity, the wave flux response of

*Q*(

*z*). The complex profile

In the previous section it was shown that, for a given *N*(*z*), the wave flux response to *Q*(*z*) is a function of the equivalent depth alone and hence is constant along the dispersion curves shown in Figs. 4 and 5. The wave flux response to *Q*(*z*) for the first IGW mode with half-sine heating profile and cloud-top height fixed at 10 km is shown as a function of zonal wavenumber and frequency in the top panel of Fig. 7. If the cloud height changes from profile to profile, then the vertical projections for modes along a dispersion curve will not be constant. The wave flux response to each

The total EP flux spectrum integrated over latitude, as a function of zonal wavenumber and frequency, may now be obtained by multiplying the wave-flux response spectra (as shown in Fig. 7) and the horizontal projection spectra (as shown in Fig. 4) for each mode and then summing over the mode index (which also includes summing over the different wave types). It ends up that after summing over all the modes, the spectra for either the variable or fixed cloud height models are nearly indistinguishable (not shown).

To understand the dynamics of the tropical mean flow, it is more relevant to consider the tropical average of the vertical EP flux rather than the global average. We now turn to a time-dependent linear primitive model to calculate the tropical wave flux average. Both the linear and nonlinear models to be used in the remainder of this study are described at the beginning of section 5. The linear model was run with zonal mean zonal winds set to zero and a zonal mean temperature profile that is independent of latitude and equal to the zonal mean NCEP temperature profiles averaged either between 15°S and 15°N or between 45°S and 45°N as shown in Fig. 1. No wave damping is used except in the top layers to prevent wave reflection from the model lid. The global average EP flux spectra at 15 km derived from these experiments are nearly identical to the spectra derived from the Hough mode model.

The EP flux spectrum for two simulations using the half-sine or constant step heating profiles that incorporate the tropical NCEP background temperature profile are shown in the top-left and middle-left panels of Fig. 8, respectively. All other panels in this figure use the half-sine heating profile. The bottom-left panel of Fig. 8 shows the spectrum obtained using the warm tropopause profile shown in Fig. 1. The structure of these spectra follows from what has already been shown for the Hough projection and the wave flux response function. First, the Hough projections have larger amplitude for small zonal wavenumber and frequency. The Hough projection amplitude is also larger for modes with small *n* and broad latitude structure. Second, the wave flux response spectra have local minima and maxima along the curves of constant equivalent depth. The curves that correspond to the same equivalent depth for modes with small index occur close together in the

## 5. Wave response in the nonlinear model

The linear Hough mode model is useful for a physical understanding of wave generation by convective heating, but it is not suitable for a detailed calculation of the wave response when there are zonal mean winds with vertical shear. For this purpose we employ a time-dependent primitive equation model to calculate the wave flux through numerical integration. An alternate approach would be to use a numerical steady-state wave model as in Garcia and Salby (1987). The horizontal structure of the fields in this model is represented by a spherical harmonic expansion with triangular truncation T80. The vertical log-pressure altitude grid extends from the surface to 60 km with 0.5 km spacing. The model is initialized with various configurations of zonal mean winds and temperatures. The spectral representation allows zonal mean fields to be retained throughout the runs by means of a linear relaxation that applies only to the zonal mean component. For the experiments described here, the model is initialized with three configurations: NCEP zonal mean zonal winds and temperatures for May 2005 or 2006, representing two opposite phases of the QBO, or with zero zonal mean winds and a zonal mean temperature profile independent of latitude. Rayleigh friction above 40 km acts on the wave components to prevent wave reflection from the upper boundary. Radiative damping is parameterized by Newtonian cooling with a rate of 0.05 day^{−1} above the tropopause. The model is forced by heating obtained from the convective heating spectrum, derived from 28 days of the TRMM 3B42 data product sampled every 3 h as described in section 2, by summing the associated Fourier series evaluated on the model time grid every 7.5 min. The large zonal mean component of the heating is set to zero to prevent the model zonal mean from evolving away from the initial state. To achieve steady wave amplitudes, the time integration is performed for 84 days, with the 28 days of heating repeated 3 times.

Results from a linear version of the time-dependent model have already been described in the previous section. The linear version of the model is identical to the nonlinear version except that the advection terms in the primitive equations are replaced with products that only involve the zonal mean flow and a wave component, and the zonal mean components are fixed with time. The nonlinear simulations further employ ∇^{8} horizontal hyperdiffusion that damps the smallest scales at a rate of 1 day^{−1} to prevent wave energy accumulation at the smallest scales. The linear model serves as a bridge between the analytic Hough model and the nonlinear primitive equation model and was primarily used to check that the time-dependent model is consistent with the Hough model.

The vertical EP flux is calculated at every model level from a Fourier analysis of the fields output every 3 h over this time period. The vertical EP flux component is averaged from 15°S to 15°N in order to estimate the relevant wave driving of the tropical stratospheric mean flow. Henceforth this average of the vertical EP flux is referred to simply as the EP flux. We focus only on the vertical component of EP flux because we have found that the horizontal divergence of the horizontal component of EP flux is an insignificant component of the total divergence above the tropopause.

Several experiments were performed to examine the effects of radiative damping and of resolution on the wave spectrum produced by the time-dependent models. Comparing runs in both the linear and nonlinear models with the Newtonian cooling either set to zero or to the values described above, we found that there was no noticeable effect of radiative damping on the EP flux divergence calculations. The time resolution of the spectra is limited to frequencies smaller than 4 cpd because of the 3-h interval of data provided by TRMM. Staying within this frequency range, it was found that including zonal wavenumbers higher than 120 produces no significant increase in the total EP flux. Furthermore, there is only a small difference in the total flux between the T120 and T80 resolutions. Hence all experiments are performed at T80 resolution. There is no advantage to decreasing the vertical spacing to less than 0.5 km. A larger vertical 0.75-km grid interval is found to produce a small decrease in flux relative to the runs with 0.5-km spacing. The reduced flux occurs for low-frequency waves with small vertical wavelength that are relevant for wave driving of the mean flow in the lower stratosphere. Waves whose wavelengths are shorter than 5 km make only a small contribution to the total EP flux in the upper stratosphere because they are susceptible either to critical level filtering because of their small horizontal phase speed or to radiative damping at lower altitudes because of their small vertical group velocity.

The EP flux spectra from two nonlinear model simulations are shown in the right panels of Fig. 8. Comparison of the top panels shows the effect of switching from a linear to nonlinear model while keeping the mean state and the heating the same. A significant reduction in the spectral amplitude for the nonlinear run is evident. Note in particular that most of the Rossby waves at wavenumbers larger than 10 have been eliminated from the spectrum. In the linear model, one finds that potential temperature profiles are overturned within the troposphere at the location of large heating events. This does not produce instability in the linear simulation because it merely represents the superposition of all the linear wave components that evolve in the model independently of each other. In the nonlinear model, processes that limit the wave amplitude prevent an unstable superposition from occurring at the location of large heating events. Manzini and Hamilton (1993) also found that slow large-scale wave amplitudes in a nonlinear model were limited by Richardson number-dependent vertical mixing and dry convective adjustment.

The middle-right and bottom-right panels in Fig. 8 show the EP flux spectrum from a nonlinear simulation initialized from May 2005 conditions. The spectrum at 20 km (middle-right panel) shows only minor differences from the spectrum for the zero wind case (top-right panel). The tropical mean zonal wind speed below 20 km is less than 5 m s^{−1} (see Fig. 1). Waves in the spectrum whose phase speed are equal to the maximum wind speed occurring below that level are shown with a red diagonal line. The bottom-right panel shows the wave spectrum at 30 km. At this altitude, easterly mean zonal wind speed has increased to 37 m s^{−1}. Waves with westward phase speed smaller than this have been effectively removed from the spectrum. Between 20 and 30 km there is a slight reduction of the EP flux magnitude for waves that have not encountered a critical level due to both Newtonian cooling and nonlinear interaction among the waves in the spectrum.

EP flux spectral densities at 20 km as a function of frequency are shown in Fig. 9. This figure provides a quantitative comparison of the spectral amplitudes. The top panel shows the spectra obtained from the linear experiments, comparing the two heating profile shapes and two buoyancy frequency profiles. The middle panel shows the same comparison for parallel nonlinear experiments. The change in the heating profile shape (cf. black and blue curves) from half-sine to constant step redistributes spectral amplitude from lower to higher frequency. Increasing the buoyancy frequency in the upper half of the troposphere decreases the amplitude for waves with frequency less than 2 cpd by more than a factor of 2. However, these differences in the spectra for the linear model are substantially reduced in the nonlinear model, although the differences are still significant for large-frequency waves. Nonlinearity within the forcing region appears to limit the amplitude of waves with frequencies smaller than 2.5 cpd for reasons mentioned above, and the difference between the nonlinear spectra for frequencies less than 1 cpd becomes negligible. An increase in heating amplitude may produce only a limited increase in the wave flux amplitude at the lower frequencies because of nonlinear effects described above.

The bottom panel of Fig. 9 shows how the flux spectrum responds to changes in the mean zonal wind below 20 km. The mean zonal winds in the tropical troposphere for May 2005 and 2006 are easterly near 5 m s^{−1}. As shown in Fig. 4, this wind speed in the source region tilts the RW response in the spectral domain along a line representing waves with phase speed −5 m s^{−1}. In 2005 these waves encounter westerly winds above 15 km and as a result most of the RW are trapped in troposphere. In 2006 the easterly wind speed increases to around −10 m s^{−1} at 20 km and hence most of the RWs encounter a critical level. In both cases the large RW EP flux amplitude in the case of zero mean winds is substantially reduced. On the other hand, easterly winds in the source region are amenable to Kelvin waves and we see that the EP flux amplitude for Kelvin waves with very slow phase speeds is somewhat increased for both 2005 and 2006 wind configurations.

## 6. Wave driving of the mean flow

As waves propagate upward into the stratosphere they will be subject to radiative damping, dissipation due to interactions among themselves, saturation and breaking, or filtering by critical levels. All of these processes cause the waves to impart momentum to the zonal mean flow. EP flux divergence produces a mean flow response that is divided between zonal wind acceleration and driving of a mean meridional circulation, but EP flux divergence at the equator primarily drives mean flow acceleration (Plumb and Eluszkiewicz 1999). The EP flux divergence has been calculated over the 28-day interval that begins 28 days into three nonlinear simulations that are initialized with either zero mean zonal winds or with mean zonal wind from May 2005 and 2006.

The EP flux divergence from these simulations is averaged from 15°S to 15°N and plotted as a function of phase speed and altitude in Fig. 10. The top panel shows the result for the mean zonal wind set to zero. EP flux divergence is positive (negative) in the source region below 15 km for westward (eastward) waves. Near the tropopause the RW, RGW, and Kelvin waves with slow phase speed are subject to radiative damping and lose most of their momentum to the mean flow. Waves with increasing phase speed lose momentum to the mean flow via the various mechanisms described above at increasing altitude. Westward (eastward) waves produce westward (eastward) mean flow acceleration as they propagate upward. The net mean flow acceleration is close to zero except near the tropopause. The middle and bottom panels of Fig. 10 show the EP flux divergence for the simulations with mean winds from May 2005 and 2006. The zonal mean flow at the equator on day 28 of the simulation is superimposed on the plots. The RW flux divergence is much weaker than in the previous simulation because of the mean wind filtering mechanisms already discussed above. Westward (eastward) acceleration occurs below westward (eastward) mean flow jets. Most, but not all, of the acceleration comes from waves that encounter a critical level where their phase speed matches the mean flow speed. The vertical group velocity of waves that do not encounter a critical level will slow as they propagate upward into a region where the mean flow speed approaches their phase speed and hence is subject to increased radiative damping. Some waves may also reach unstable amplitudes and dissipate. Thus, waves with phase speeds much larger than the maximum jet speeds are seen to contribute to the mean flow acceleration near the altitudes of the jet speed maxima. Note that the mean flows shown in this figure have evolved somewhat from the initial profiles shown in Fig. 1.

The net mean flow acceleration profiles for the May simulations are shown in Fig. 11. The mean flow acceleration due to only westward or eastward phase speed waves is also shown. The main question we wish to address is whether there is enough EP flux divergence to drive a QBO. An estimate is that the acceleration required must be 0.3–0.4 m s^{−1} day^{−1} (Dunkerton 1997). This estimate includes the 0.2 m s^{−1} day^{−1} acceleration required to counteract the vertical advection due the upward vertical velocity of the Brewer–Dobson mean meridional circulation. Figure 11 shows that the eastward acceleration at 23 km in May 2006 is well within the range required, but westward acceleration in May 2005 falls short since the peak acceleration around 22 km is only 0.2 m s^{−1} day^{−1}. Note the large accelerations near 35 km in both cases that appear to be initiating the next opposite QBO phase in the upper stratosphere.

The evolution of the mean winds during the 84-day simulation at 28-day intervals is shown in Fig. 12. The eastward jet initially at 28 km in 2006 descends at the correct rate of about 1 km month^{−1} and the peak jet speed is maintained. However, there is no significant zonal mean vertical velocity in this simulation at these altitudes to oppose the descent. Hence a more realistic simulation might produce a jet descent rate that is too slow. At 35 km the weak westward jet in the initial zonal wind profile strengthens and descends rapidly at a rate of about 2 km month^{−1} after the first 28 days. Some of this initial acceleration may be due to wave transience since it is during this period that these waves arrive in the upper stratosphere and develop steady amplitude. Nevertheless, there appears to be sufficient wave flux to initiate the next QBO phase aloft. The westward jet, initially at 32 km in 2005, descends at a rate of 1 km month^{−1} as well, but the peak jet speed is not maintained during the descent. The jet speed reduction indicates that there are not enough westward waves in the model to drive the westward phase of the QBO. Eastward winds develop quickly in 2005 at 38 km within the first 28 days and then descend rapidly at a rate of 2 km month^{−1}. There appears to be sufficient eastward wave flux to initiate the eastward QBO phase in May 2005.

## 7. Summary and discussion

The question of the degree to which small-scale gravity waves are required to drive a QBO is persistent, still currently unresolved, and the primary motivation behind the study presented here. Horinouchi and Yoden (1998) showed that it is possible to produce enough flux in resolved waves at T40 resolution to drive a QBO-like oscillation, but it is uncertain if the parameterized convection produced in such models is realistic. Other GCM simulations by Scaife et al. (2000) and Giorgetta et al. (2006) produce QBO-like oscillations from both resolved and parameterized gravity waves. Kawatani et al. (2010) produce a QBO in a high-resolution T213 model and find that small-scale waves with horizontal wavelength less than 1000 km (zonal wavenumber *s* > 40) contribute at least 50% of the wave driving of the QBO in their model.

All of these studies rely on various cumulus convection parameterizations. The goal of our study has been to constrain the wave driving with observations. Even with this constraint there are various uncertainties and limitations to our approach. The main uncertainties include the conversion from TRMM rainfall rates to heating rates and the structure of the heating profile. We have attempted to describe the sensitivity of the wave flux to assumptions about the heating profile structure by contrasting the results for two simple profile models and by showing how these profile structures affect the solution to the vertical structure equation. Evaporation, which we have neglected, could affect both the strength and shape of the heating profile. We have also found that the wave flux is very sensitive to the tropical temperature profile, which may have implications for future study of seasonal and interannual variability. Alexander and Ortland (2010) found a correlation between Kelvin wave potential energy derived from High Resolution Dynamics Limb Sounder (HIRDLS) temperature measurements and the buoyancy frequency in the tropical tropopause layer. It remains to be determined if the observed Kelvin wave seasonal variability can be reproduced in our model when forced with seasonally varying TRMM heating and background flow.

The model experiments with nonzero zonal mean flow show that the zonal winds have a filtering effect on equatorial Rossby waves, confining them to the lower stratosphere. It is possible that Rossby waves are confined by other mechanisms, such as thermal damping acting on waves with slow vertical group velocity or the waves becoming evanescent through changes in the buoyancy frequency. However, these last two mechanisms do not apply in our simulations. Substantial Rossby wave flux was noted when the background flow was set to zero, and there was no effect on the Rossby wave flux when the thermal damping was set to zero.

Horinouchi (2002) pointed out that it is necessary for both the mean and variance of rainfall rates derived from satellite observations to be correct in order to have confidence in the validity of the wave spectrum produced by the derived heating. It is beyond the scope of this paper to present such a validation study of the TRMM 3B42 rainfall rates. However, we have done some validation of portions of the wave spectrum produced by our model when forced with heating derived from TRMM rainfall. Ryu et al. (2011) find that mean heating in the TRMM monthly convective/stratiform heating product compares well with the mean heating derived from the TRMM 3B42 rain rate as used here. They also find that the model Kelvin wave excited by this heating compares well with Kelvin wave amplitudes seen in HIRDLS temperatures. The Rossby wave amplitudes in the troposphere and lower stratosphere of our model also agree with amplitudes found in reanalysis data products. This will be reported in future work. Thus, our model response to heating derived from TRMM rainfall does not appear to overestimate low-frequency wave amplitudes, providing some indirect validation of the low-frequency part of the TRMM rainfall rate spectrum.

Our attempt to quantify how broad a spectrum of tropical waves is required to force the QBO should be regarded as a preliminary study with the TRMM dataset. Heating power and vertical EP flux spectra have been examined for a full year of simulations using TRMM heating. The spectra shown here for May 2006 are close to the annual average, with month-to-month variability about the average no more than 25%. A comparison of the variability in Kelvin wave amplitudes between model and HIRDLS-*Aura* data (e.g., Alexander and Ortland 2010) and an investigation of the implications for QBO driving will be presented elsewhere.

Another limitation to our study is the temporal resolution of the TRMM data. The 3-h sample interval limits our frequency resolution to less than 4 cpd. At this temporal resolution we find that increasing the horizontal resolution beyond T80 does not produce a significant increase in wave flux. Hence we cannot explore the full region of spectral space that is examined by Kawatani et al. (2010). It is likely that a significant amount of wave flux resides in the spectrum for frequencies higher than 4 cpd, which may account for why our simulated wave acceleration of the mean flow is somewhat below the level required to drive a QBO. Furthermore, examination of Fig. 8 shows that the amplitude in the branch of the spectrum along the primary peak of the flux response decreases rather slowly as it approaches the top edge of the plot. A method for extrapolating the EP flux to higher frequencies would be valuable for assessing the heating spectrum obtained in the various convective parameterizations used in GCMs or for determining an appropriate source flux to be used in parameterizations of gravity waves that cannot be resolved by the model.

Figure 13 shows the EP flux divergence spectrum at 25 km. The left panel is for westward waves during the westward QBO phase in 2005 and the right panel is for eastward waves during the eastward QBO phase in 2006. The plot has been divided into spectral regions and labeled with the percentage of the total wave acceleration provided by waves within that region. Kelvin waves with zonal wavenumber less than 20 and frequency less than 1 cpd provide 37% of the wave driving in the model for the eastward QBO phase. Given the percentages near the upper boundary of the plot, it is conceivable that wave acceleration might increase significantly if the spectral region is extended to 8 cpd and the *s* increased to 160. The wave acceleration contribution from westward waves during the westward phase of the QBO is spread more evenly through the spectral domain resolved by the model. Waves with frequencies greater than 2 cpd and with zonal wavenumbers between 20 and 60 provide 34% of the total wave acceleration. The percentage of mean flow acceleration provided by the westward waves is seen to drop off only gradually as the boundary of our spectral domain is approached. Hence it is conceivable that westward wave driving will also increase significantly, by an even larger percentage than for eastward waves, if the spectral domain is extended. These crude estimates of where the contribution to QBO forcing comes from within the spectral domain are consistent with what has been provided by the high-resolution simulations of Kawatani et al. (2010).

## Acknowledgments

This work was supported by the NASA Atmospheric Composition: *Aura* Science Team Program under Contract NNH08CD37C, by the NASA Geospace Science Program under contract NNH08CD34C, and by NSF Grant 0121564.

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