## 1. Introduction

Atmospheric gravity waves occur because of vertical displacements within a stable environment, which can be caused by a variety of processes including flow over topography, jet streams, frontal systems, and moist convection. Gravity waves generated by these tropospheric sources may propagate vertically into the stratosphere and mesosphere, contributing to the momentum budget of the middle atmosphere (see Fritts and Alexander 2003 for a review). Within the troposphere, these waves can also modulate and initiate clouds (e.g., Balaji and Clark 1988), influence mesoscale weather systems (e.g., Zhang et al. 2001), and, therefore, modify regional precipitation processes. Furthermore, tropospheric gravity waves generated by deep convection have been shown to promote convective development in the vicinity of their source, leading to cloud clustering in environments with low convective inhibition, such as the tropics (e.g., Mapes 1993; Liu and Moncrieff 2004).

Considerable progress has recently been made in our understanding of the generation of gravity waves by deep convection in numerous numerical modeling studies (e.g., Fovell et al. 1992; Holton and Alexander 1999; Piani et al. 2000; Lane et al. 2001; Beres et al. 2002; Song et al. 2003). These studies have increased our insight into the dynamics of the wave source and have provided estimates of the influence of waves generated by these systems on the middle atmosphere. Single cloud systems, such as squall lines, have been shown to generate a spectrum of gravity waves with strongest power at horizontal wavelengths between 5 and 50 km and periods from 10 to 60 min. These temporal and spatial scales correspond to the scale and life cycle of individual updrafts through to the scale and life cycle of the system itself. Larger-scale, more organized systems like hurricanes have been shown to generate gravity waves with longer horizontal scales and longer periods (e.g., Kim et al. 2005).

To date, most previous studies of gravity wave generation by deep convection have focused on single cloud systems, such as squall lines, in relative isolation from the larger-scale environment. This single-system approach presumably provides robust estimates of the spectrum of waves with scales less than or equal to the system scale, but it cannot represent those waves generated by clusters of clouds and multiscale organized systems. For example, much of the tropics feature multiscale convection that forms into organized systems and clusters of clouds over the ocean. Although this oceanic convection may be relatively weak in comparison to continental convection, clusters of clouds are long lived and occur over large regions exceeding hundreds of kilometers in diameter. Such long-lived, large-scale clusters have the potential to generate gravity waves with long periods and long horizontal scales that could be responsible for a substantial vertical flux of horizontal momentum into the tropical stratosphere. Moreover, squall lines represent a special category of convective organization, and in the tropics they are far less frequent than the widespread, less intense convection that occurs over the tropical oceans.

Long-wavelength gravity waves may also play a role in organizing tropical convection. For example, Mapes (1993) showed that mesoscale convective systems (MCSs) generate borelike disturbances that propagate horizontally away from an existing MCS, thereby communicating the effect of the diabatic heating to the surrounding environment. These bores were shown to possess a modal structure in the vertical, related to the depth of the convection (and/or tropopause), with horizontal phase speeds characterized by hydrostatic gravity waves. The deepest propagating mode caused compensating subsidence as a response to the deep diabatic heating [or, equivalently, as a response to localized ascent; e.g., Bretherton and Smolarkiewicz (1989)] and was shown to initially stabilize the region surrounding the MCS. However, the influence of this deep mode is limited because of its fast phase speed (∼50 m s^{−1}). Higher-order vertical modes associated with low-level diabatic cooling were shown to propagate more slowly (∼15–25 m s^{−1}) away from the cloud, producing low-level lifting and destabilizing the environment. This destabilization defines a region surrounding an existing convective system that is prone to new convective development, encouraging cloud clusters to form in low-convective-inhibition environments. Lane and Reeder (2001) illustrated that such a process can also occur in the neighborhood of individual convective cells, and a preexisting MCS is not necessary to promote new convective development. The radius of influence of these (high order) destabilizing modes is affected by the rotation of the earth. Larger clusters are more likely to form in the tropics than in the midlatitudes because of the latitudinal variation in the Coriolis parameter (Liu and Moncrieff 2004).

Mapes’s (1993) arguments explaining the gregarious nature of deep convection have also been used to understand the propagating systems that are identified in satellite observations in the tropics and that form spontaneously in multiscale cloud-system-resolving simulations of tropical convection (e.g., Tulich et al. 2007; Grabowski and Moncrieff 2001; Liu and Moncrieff 2004). These cloud-system-resolving simulations readily form horizontally propagating convective structures associated with storm outflow (gust fronts) and high-order vertical wave modes (gust front modes) with phase speeds of approximately 5–10 m s^{−1}. Larger-scale (>100 km) propagating clusters of clouds also form, and the envelope of the cluster moves with slightly larger phase speeds (10–20 m s^{−1}). It is generally appreciated that these faster, larger-scale systems are a response to deep propagating gravity wave modes, but the details of the interaction are still an active area of research (e.g., Tulich and Mapes 2008). These multiscale structures in tropical convection are formed, at least in part, by the interaction between convective systems and the gravity waves they generate. Not only do these gravity waves contribute to the organization of cloud populations, they also influence the width, lifetime, and spatial separation of individual convective clouds (e.g., Balaji et al. 1993), which may feed back to the gravity wave generation. Multiday cloud-system-resolving simulations allow such feedbacks to occur, placing no external constraints on the scales of convection; these scales are freely selected by the interactions among the population of clouds, the waves they generate, and the evolving environment.

The aim of this study is to conduct simulations of multiscale tropical convection to characterize the gravity waves generated by freely evolving cloud populations. Such an approach generates a broader spectrum of waves: from the cloud-system scale to the cluster scale. The resultant spectrum of waves will be examined in detail to quantify links between the gravity wave characteristics and the properties of the tropospheric convection.

This paper is organized as follows: section 2a describes the numerical model and presents the multiscale cloud simulation, which is analyzed in terms of spectral gravity wave components in section 2b. Section 3 presents additional simulations that explore the sensitivity of the results to aspects of the model simulation, including domain size and model resolution. Section 4 summarizes the results and describes possible implications of this study.

## 2. Multiscale cloud simulation

This section describes the results of one multiscale cloud simulation, along with spectral analyses of the accompanying stratospheric gravity waves.

### a. Simulation description

The numerical model used here was originally described by Clark (1977), with subsequent updates and improvements that are summarized by Clark et al. (1996). The model is a finite-difference approximation to the nonhydrostatic anelastic equations of motion and utilizes second-order accurate spatial and temporal differencing to solve the model equations. In its current configuration, the model is two-dimensional slab symmetric (*x–z*), and features explicit treatments of cloud processes via a combination of the Kessler (1969) warm-rain and the Koenig and Murray (1976) ice parameterizations.

The two-dimensional simulation presented in this section uses a domain that is 4000 km wide and 40 km deep, with 1-km horizontal grid spacing and vertical grid spacing that varies from 50 m near the surface to 200 m above 1 km. The time step is 5 s. The uppermost 10 km feature a Rayleigh friction absorber, which acts to damp vertically propagating disturbances before they reach the model’s upper boundary. The initial wind is zero everywhere, and the lateral boundaries are periodic. The initial thermodynamic profile was observed at 1330 UTC 26 November 1995 over the Tiwi Islands, Australia during the Maritime Continent Thunderstorm Experiment (MCTEX). This sounding (or an approximation to it) has been used in a number of studies of convection and gravity waves (e.g., Crook 2001; Lane et al. 2001; Lane and Reeder 2001). It exhibits a high surface water vapor mixing ratio (19 g kg^{−1}) and the mean tropospheric Brunt–Väisälä frequency equals 0.0095 s^{−1}. The tropopause is located 16.5 km above the surface. Convection is initiated and sustained during the simulation using a combination of surface fluxes and imposed tropospheric cooling, which mimics net radiative cooling. A sensible heat flux of 10 W m^{−2} and a latent heat flux of 100 W m^{−2} are imposed at the surface, along with small-amplitude random perturbations to these fields. (These surface fluxes are representative of those observed over tropical oceans.) Below 9.5 km the troposphere is cooled by 2 K day^{−1}, and above 9.5 km the rate of cooling decreases linearly to be zero at 15.5 km. The simulation is run for 120 h (5 days).

Figure 1 shows the horizontal space–time distribution of the simulated rain rate for the entire simulation. Relatively early into the simulation (∼10 h), convection develops and shows evidence of the formation of eastward- and westward-propagating cloud systems, exhibiting familiar v-shaped propagation patterns. These v-shaped patterns represent the initiation of *secondary* convective cells resulting from propagating disturbances (density currents and/or gravity waves) that emerge from *primary* convective systems and propagate in the positive and negative horizontal directions at approximately 5 m s^{−1} (marked “A” on Fig. 1). At later times, larger convective systems emerge, which are associated with clusters of clouds that propagate in either horizontal direction at speeds between 5 and 15 m s^{−1}. These propagating systems eventually manifest themselves as larger clusters, with regions of enhanced convective activity hundreds of kilometers wide (e.g., *t* = 100 h, 800 < *x* < 1500 km).

At 120 h, the modeled clouds (Fig. 2a) possess a variety of depths, widths, and horizontal separations and are at different stages of their individual life cycles. The clouds occur at irregular locations in the horizontal, and similarly the depth of the cloud base has substantial spatial variance. The mature clouds extend to about 10–11 km in depth, with occasional clouds deepening to 13–15 km (not shown). Therefore, the majority of the convection falls many kilometers short of the tropopause. Figure 2 also highlights the small-scale perturbations in the potential temperature and vertical velocity field throughout the troposphere and stratosphere; these are characteristic signatures of gravity waves. In particular, the vertical velocity perturbations show complex interference patterns as the gravity waves from adjacent clouds or cloud systems interact as they propagate vertically and horizontally.

Even though the clouds form into propagating clusters and many are deeper than 10 km, the clouds are of modest intensity. A time series of the domain-maximum vertical velocity *w* (Fig. 3) shows the convection to be strongest in the first 20 h of the simulation (with updraft speeds of about 8–10 m s^{−1}), and it evolves to a statistically quasi-steady state at later times, with maximum vertical velocities of about 5 m s^{−1}. The strength and depth of the simulated convection resemble observed oceanic tropical convection. Also, the modeled convection is *multiscalar*; that is, it features an ensemble of clouds with spatial and temporal variation in scale, intensity, and organization. To the extent that the results of this simulation are insensitive to domain size and model resolution (discussed in section 3), there is no a priori control over either the scale of the clouds or their degree of convective organization. The simulated clouds spontaneously select a preferred horizontal scale and a preferred separation from other clouds. When the local interactions are constructive, organized systems and cloud clusters form. To the extent that the modeled convective organization is neither a direct result of external forcing nor caused by the imposed initial conditions, the simulation provides synthetic datasets that enable a comprehensive investigation of the influence of multiscale convection (i.e., clouds, cloud ensembles, and their life cycles) on the spectrum of convectively generated gravity waves.

Figure 4 illustrates the spatial variability of the convection and the horizontal velocity at *t* = 120 h at three selected locations. The average horizontal velocity is approximately zero across the domain. The positive and negative flows are a result of convective momentum transport, which is generally indiscernible from gravity wave effects. These flows are associated with the vertical tilt of the cloud systems with height. In Fig. 4a, the tilt is in the negative direction, consistent with the local vertical wind shear in that region. In the lower troposphere, the horizontal flow is positive, shifting to negative near the cloud top (i.e., negative shear). Figure 4a also shows a relationship between the low-level flow and the depth of the cloud base: there are high bases above regions of deep negative flow and lower bases over shallower negative flow. Figure 4b illustrates another morphology consisting of negative horizontal flow between 3 and 8 km and positive flow aloft. This positive shear generates systems with a weak positive tilt. Figure 4b also shows a shallow surface layer (∼1–2 km deep), and in this layer flow is in the positive direction. Finally, Fig. 4c features suppressed convective activity and a distinct wavelike pattern with a vertical wavelength of 4–6 km. In summary, these three examples show that the simulated multiscale convection is associated with distinctive patterns of horizontal flow on a scale that greatly exceeds that of individual clouds. These patterns, which transport horizontal momentum and are ultimately convectively generated, have distinct morphology (e.g., tilt, flow organization and propagation) and cause feedbacks to larger scales. At locations remote from the active convection (e.g., Fig. 4c), the layered flow is wavelike, featuring short vertical wavelengths and long horizontal scales. These ubiquitous layered flows are clearest in the convectively suppressed regions and possess more complex vertical structures in regions of active convection.

In the lower stratosphere, a rich spectrum of propagating gravity waves is excited by the underlying convection. For example, a horizontal space–time section of horizontal velocity over a subset of the spatial and temporal domain at 20-km altitude (Fig. 5) illustrates propagating wave patterns of a variety of horizontal scales and propagation speeds. These waves propagate in both horizontal directions, interact locally, and produce complicated interference patterns. In this figure, waves with horizontal wavelengths of approximately 20 km through to larger scales (∼150 km) are evident, with propagation speeds between approximately 5 and 10 m s^{−1}. The spectral characteristics of these waves are analyzed in the following subsection.

### b. Spectral decomposition of the gravity waves

To examine the spectral components of the gravity waves, the horizontal velocity *u* and the vertical velocity *w* are recorded at every horizontal grid point every 2 min throughout the simulation at 20 km above the surface. These data allow the gravity wave spectrum to be examined in terms of frequency, horizontal wavenumber, and phase speed. It is assumed that perturbations in velocity are signatures of gravity waves that have propagated vertically from the troposphere.

#### 1) One-dimensional velocity spectra

To examine the spatial structure of the velocity perturbations, a discrete Fourier transform is applied separately to the two velocity components in the horizontal domain at each time level between 20 and 120 h. (Restricting the analysis to after 20 h minimizes artifacts resulting from the initial convective development.) Each amplitude spectrum when multiplied by its complex conjugate forms the power spectrum. The power spectra from the different time levels are averaged and then scaled appropriately to form the (one sided) power spectral density (PSD). The PSD curves versus horizontal wavenumber for the vertical and horizontal velocities are shown in Figs. 6a and 6c, respectively. Similarly, the signals are analyzed in the time–frequency domain by applying the Fourier decomposition every 2 min from 20 to 120 h to the time series defined at each individual grid point. The frequency spectra from each grid point are averaged and the PSD versus frequency for the vertical and horizontal velocities are shown in Figs. 6b and 6d, respectively.

The horizontal spectrum of vertical velocity (Fig. 6a) shows that the vertical velocity is dominated by wavenumbers between approximately 10^{−4} and 6 × 10^{−4} rad m^{−1} (horizontal wavelengths between 10 and 60 km), with a rapid reduction in power at higher wavenumbers (mostly resulting from numerical diffusion). Similarly, the temporal spectrum of vertical velocity shows a broad maximum at frequencies between approximately 10^{−3} and 7 × 10^{−3} rad s^{−1} (periods between 15 and 100 min, Fig. 6b). The rapid reduction in power at higher frequencies is due to the fact that the maximum frequency of waves generated in the troposphere is the local Brunt–Väisälä frequency (∼0.01 s^{−1}). The reduction in PSD toward lower frequencies is relatively shallow, and for frequencies less than approximately 2 × 10^{−4} rad s^{−1} (periods greater than 9 h) the PSD is almost constant. Both the horizontal and temporal spectral decompositions of the vertical velocity show that the signatures of vertical velocity resulting from gravity waves are dominated by the scales associated with individual cloud systems (viz., their horizontal scale and their individual life cycles from initiation to decay).

The wavenumber spectrum of the horizontal velocity (Fig. 6c) is different in character from the vertical velocity, with a consistent increase in power toward smaller wavenumbers (longer wavelengths). Like the vertical velocity, a strong reduction in power occurs at the highest wavenumbers that are influenced by numerical diffusion (<10 km). Similarly, the frequency spectrum of horizontal velocity (Fig. 6d) is dominated by the longest time scales (lowest frequencies). This temporal behavior is also linked to the evolution of the long-lived organized convection in the troposphere, which implies larger spatial scales and slower manifold variability.

A comparison between the spectra of the vertical and horizontal velocities clearly shows that the horizontal velocity derives its strongest signatures from low-frequency, long-wavelength waves, and the vertical velocity is most strongly influenced by shorter-wavelength, high-frequency waves. This result is consistent with simple wave theory, which relates the respective velocity signatures to the wavenumber–frequency controls on the phase tilt of propagating gravity waves. The stronger signatures in horizontal velocity at long temporal and large spatial scales imply that the horizontal velocity is strongly influenced by the longer-wavelength, longer-period waves generated by large-scale variability within the cloud population. This behavior is a key component of the *upscale effects of tropospheric deep convection,* which allows coherent structures (convective organization) that are embedded within fields of stochastic cumulus convection to evolve. The interaction among convection, convective downdraft outflows (density currents), and convectively generated gravity waves is vital to the upscale process (e.g., Mapes 1993; Liu and Moncrieff 2004).

#### 2) Momentum flux spectra

To characterize the modeled gravity waves in more detail, we perform two-dimensional spectral analysis: the two-dimensional (*x–t*) fields of horizontal and vertical velocity are decomposed using two-dimensional Fourier analysis. The result is a two-dimensional amplitude spectrum for each velocity component in terms of wavenumber and frequency. The gravity wave momentum flux *ρuw*, where *ρ* is the density, is the appropriate quantity to measure the influence of gravity waves on the mean horizontal flow in the middle-to-upper atmosphere. The spectrum of momentum flux is calculated from the cospectrum of the vertical and horizontal velocities. For example, in constructing a two-dimensional spectrum of momentum flux, the two-dimensional amplitude spectrum of horizontal velocity is multiplied by the complex conjugate of the two-dimensional amplitude spectrum of the vertical velocity, and the real part of the result is the cospectrum. The horizontal phase speed *c* = *ω/k*, and in a two-dimensional system it can be shown that gravity waves with upward energy propagation (downward phase propagation) possess momentum flux of the same sign as their horizontal phase speed (see Beres et al. 2002 for details).

One-dimensional horizontal wavenumber spectra are constructed by integrating the two-dimensional momentum flux spectrum across all frequencies. This integration is conducted separately for waves with positive and negative phase velocities, and the magnitude of these momentum flux spectra (after smoothing over five adjacent wavenumber bins) is shown in Fig. 7a. The momentum flux spectra are scaled in such a way that the integral in horizontal wavenumber space is equal to the magnitude of the momentum flux propagating in the given direction. Similarly, the spectra of momentum flux versus frequency are constructed by integrating across all positive horizontal wavenumbers (for waves with positive phase speed) and all negative wavenumbers (for waves with negative phase speed). The integral of these spectra in frequency space is equal to the total momentum flux in the respective direction. The spectra of momentum flux versus frequency are shown in Fig. 7b. Both the horizontal wavenumber and the frequency spectra show very little asymmetry (i.e., there is almost equal momentum flux propagating in the positive and negative directions). In particular, the total positive momentum flux is 7.07 × 10^{−3} N m^{−2}, and the total negative momentum flux is −6.95 × 10^{−3} N m^{−2}.

The horizontal wavenumber spectrum of momentum flux (Fig. 7a) is remarkably flat, being approximately constant across almost 2 decades of scale. The momentum flux decreases at wavenumbers greater than approximately 4 × 10^{−4} rad m^{−1} (wavelengths less than 15 km), that is, at scales less than the cloud-system scale. (This reduction is probably due to a combination of this physical scale and the influence of model resolution.) The momentum flux also suffers a slight decrease in amplitude at wavenumbers less than or equal to 3.1 × 10^{−6} rad m^{−1} (scales greater than 2000 km). These low-wavenumber signals are likely influenced by the size of the model domain, in which case they should be disregarded. The flat momentum flux spectrum implies an equal contribution to the total momentum flux from each spectral component (in the horizontal), highlighting the importance of gravity waves with both long and short horizontal scales. This spectrum is similar to white noise (except that white noise does not feature the reduction in amplitude at high wavenumbers). The frequency spectrum of momentum flux (Fig. 7b) is not flat; it maximizes at approximately 2 × 10^{−3} rad s^{−1} (a period of approximately 1 h). The frequency spectrum also exhibits relatively strong momentum flux components from low frequencies, with the slope of the spectrum toward lower frequencies being rather shallow.

The two-dimensional momentum flux spectrum is smoothed using a simple average of the nine adjacent frequency–wavenumber bins, and its normalized magnitude is shown in Fig. 8. (Note that the smoothing applied to this spectrum is kept to a minimum to retain important features.) Figure 8 shows only positive frequencies, and therefore the horizontal phase speed is the same sign as the horizontal wavenumber. Similar to Fig. 7, Fig. 8 clearly shows that the normalized momentum flux is almost exactly symmetric about *k* = 0, implying that the flux is apportioned equally between those waves propagating in a positive direction and those waves propagating in a negative direction, and the total momentum flux is approximately zero. Throughout the simulation, the domain-averaged horizontal wind remains close to zero, and there is no systematic direction of propagation of the convective systems; therefore, a symmetric wave spectrum is to be expected.

*m*is the vertical wavenumber,

*N*is the average upper-tropospheric Brunt–Väisälä frequency (0.0098 rad s

^{−1}), and

*H*is the density-scale height (6500 m). Vertical wavelengths of 33, 16.5, and 11 km are shown; these wavelengths correspond to twice the depth of the troposphere, the tropospheric depth, and 2/3 the depth of the troposphere. Tropospheric waves with these wavelengths are typically referred to as the

_{S}*n*= 1, 2, and 3 modes, respectively. (This modal nomenclature is derived from the corresponding number of antinodes of tropospheric standing waves that possess nodes at the surface and the tropopause.)

Figure 8 shows that the momentum flux spectrum is generally strongest within the region of frequency–wavenumber space that corresponds to vertical wavelengths smaller than 11 km, in a relatively small phase speed (*ω*/*k*) range. The strongest individual spectral components are at small wavenumbers and frequencies. The momentum flux spectrum also evinces contributions from the *n* = 1 and *n* = 2 modes, with regions of local maxima clearly following the corresponding dispersion curves. Although these contributions are relatively weak, the contribution from the *n* = 2 mode increases in amplitude as it extends to low frequencies. There is also evidence of an *n* = 3 signal; however, this signal is on the edge of the contiguous region of relatively large momentum flux and is difficult to unambiguously separate from those signals. It is important to note that the vertical wavelength corresponding to the *n* = 3 mode is also (perhaps coincidentally) the approximate cloud-top height. Therefore, Fig. 8 shows that the strongest momentum flux signal is contained in tropospheric vertical wavelengths less than or equal to the cloud-top height. The responses from the *n* = 1, *n* = 2, and *n* = 3 modes are harmonics of the depth of the troposphere, and occur because of a natural vertical-scale selection associated with the location of the tropopause. The occurrence of these modes is probably unrelated to the specific details of the convection, and waves with these deep vertical scales are only efficiently generated if the horizontal and temporal scales of the wave source project onto the appropriate vertical scale via the dispersion relation (see Holton et al. 2002 for details). Furthermore, the gravity waves with vertical wavelengths smaller than 11 km (*n* > 3) do not show evidence of discrete vertical modes, and the spectrum is controlled by the broad response in frequency and horizontal wavenumber space.

To analyze the gravity waves in terms of their phase speed, the two-dimensional momentum flux spectrum is converted into a phase speed spectrum using the technique of Beres et al. (2002). Each spectral component of the (unsmoothed) momentum flux spectrum is added to 1 m s^{−1} wide bins, with the total momentum flux in each bin representing the contribution from that phase speed. The integral of the phase speed spectrum is equal to the total momentum flux. The momentum flux versus phase speed is shown in Fig. 9, and because the spectrum is approximately symmetric only the positive phase speeds are shown.

Figure 9 shows that the strongest contribution to the momentum flux spectrum is at a horizontal phase speed of 5 m s^{−1}, with relatively small contributions extending to about 17 m s^{−1}. In addition, there is a weaker isolated peak corresponding to about 22 m s^{−1}.

Also shown in Fig. 9 are the phase speed spectra constructed by examining the early stages of the simulation (from 20 to 70 h) and the later stages of the simulation (from 70 to 120 h) separately. These curves show that the momentum flux amplitude is stronger earlier in the simulation and weaker later in the simulation, in agreement with the change in convective intensity (see Fig. 3). However, despite the change in momentum flux amplitude, the phase speed spectrum preserves its shape throughout the simulation. Thus, even though there is a preference for increased cloud clustering as time progresses, the momentum flux spectrum is relatively insensitive, displaying only subtle changes in the shape of the spectrum. This lack of sensitivity is also exhibited in the one-dimensional wavenumber and frequency spectra (not shown).

*k*

^{2}≪

*m*

^{2}) in a flow with zero background horizontal velocity is

*λ*= 2

_{z}*π*/

*m*is directly proportional to the phase speed, and

*λ*= 2

_{z}*π*|

*c*|/

*N*. To the extent that the gravity waves can be considered hydrostatic, the phase speed spectrum can be converted to a (pseudo) vertical wavelength spectrum using a simple linear transformation. Such an approach is arguably more reliable than a spectral decomposition in the vertical, because of the limited vertical extent of the model domain and the changes in the vertical wavelength over the depth of the model resulting from changes in stability. The phase speed spectrum (Fig. 9) is converted into a (tropospheric) vertical wavelength spectrum using the upper-tropospheric Brunt–Väisälä frequency, and shown in Fig. 10. This tropospheric vertical wavelength is approximately twice that in the stratosphere.

The majority of the momentum flux can be attributed to tropospheric vertical wavelengths less than 11 km (Fig. 10), that is, vertical modes with *n* > 3. The absolute maximum corresponds to a vertical wavelength between 3 and 4 km. The isolated peak at *c* = 22 m s^{−1} in Fig. 9 is representative of a vertical wavelength of approximately 15 km, that is, slightly shorter than the *n* = 2 mode. (Note that it is likely that this peak is the *n* = 2 signal, and the slight disagreement in scale could be a result of the fact that a single average value of *N* is used to convert from phase speed to vertical wavelength.)

It is important to note that the strongest gravity wave response has a vertical wavelength that is much shorter than the depth of the convection and the corresponding vertical scale of the diabatic heating and cooling. (Note that the cooling resulting from evaporation extends from the cloud base to the surface). The strongest wave response is at tropospheric vertical wavelengths of about 3–4 km, and, as illustrated by Fig. 4, the strongest convective response is at about twice this scale. This result suggests that it is inappropriate to relate the vertical structure of the diabatic forcing to the vertical wavelength of the gravity waves, which is in agreement with Holton et al. (2002). Holton et al. showed that it is more appropriate to consider the wave generation as a response to the horizontal and temporal scales of the forcing, with the vertical scale of the gravity waves defined by the dispersion relation.

To determine to what extent the gravity waves can be categorized as truly hydrostatic, a *hydrostatic vertical wavelength spectrum* is calculated and also shown in Fig. 10. The hydrostatic spectrum is constructed by calculating the vertical wavenumber for each spectral component from the nonhydrostatic dispersion relation, and only those components that satisfy *m*^{2} > 10 *k*^{2} are included in the spectrum. This (somewhat arbitrary) separation results in about 70% of the momentum flux being sourced from hydrostatic waves, with the shape of the hydrostatic spectrum consistent with the total spectrum. The largest differences, that is, the strongest nonhydrostatic waves, occur either at short vertical wavelengths or equivalently slow horizontal phase speeds. (Note that this result implies that it is necessary to include nonhydrostatic effects in parameterizations of convectively generated gravity waves.)

The phase speed and vertical wavelength spectra presented in Figs. 9 and 10 clearly illustrate that the strongest contributors to the stratospheric momentum flux are slow-moving signals with relatively short tropospheric vertical wavelengths. These vertical wavelengths are less than the depth of the convection. The one exception is a weak signal of the faster-moving *n* = 2 tropospheric mode, which appears as an isolated peak in the spectra. The shorter vertical wavelengths could be interpreted as the *gust front modes* identified by Tulich and Mapes (2008), that is, high-order vertical modes that are a response to the precipitation-driven storm outflow at the surface. These wavelengths also correspond to the range of vertical structures seen in the horizontal velocity (Fig. 4). These modes represent patterns of inflow and outflow throughout the cloud depth and the shallow region of inflow or outflow near the surface (i.e., the gust fronts).

## 3. Additional simulations

In this section, additional simulations are conducted to examine some sensitivities of the previously analyzed simulations to some aspects of the model setup and the environment, including the model domain size and resolution and the depth of the convection relative to the tropopause.

### a. Sensitivity to domain size

To examine the sensitivity of the momentum flux spectrum to domain size, the simulation was rerun twice using a 1000- and 2000-km-wide domain with all other aspects of the simulation unchanged. The momentum flux versus phase speed spectrum for the 1000-, 2000-, and 4000-km-wide domains are extremely similar in shape and amplitude, with only minor differences in detail. The difference between the momentum flux spectrum from the 2000- and 4000-km simulations are shown in Fig. 11a, along with the difference between the momentum flux spectrum from the 1000- and 4000-km simulations. In both cases, the differences between the spectra amount to only 5%–10% of the maximum momentum flux. Also, all three spectra (4000, 2000, and 1000 km) possess maximum amplitude at a phase speed of 5 m s^{−1}, and the one- and two-dimensional spectra of momentum flux from the two additional simulations are also extremely similar to the original 4000-km simulation (not shown). Therefore, the momentum flux spectrum is relatively insensitive to domain size. This lack of sensitivity may seem inconsistent with the flat spectrum of momentum flux in wavenumber space, with larger domains including more spectral components at small wavenumbers than smaller domains. However, the total momentum flux in the spectrum is equal to the integral in wavenumber space of the momentum flux spectrum (Fig. 7a), and the contribution of small wavenumbers to that integral is minor in comparison to the contribution from larger wavenumbers.

The peak in the momentum flux at *c* = 5 m s^{−1} is a robust feature of the spectrum, which is not sensitive to domain size. To examine the physical structures that comprise this peak, the contributions to the *c* = 5 m s^{−1} momentum flux bin are separated into their original horizontal wavenumber components and shown in Fig. 11b. (Note that because *c* = 5 m s^{−1}, this wavenumber spectrum can be interpreted as a frequency spectrum by multiplying the wavenumber by 5.) Figure 11b identifies a broad spectrum of waves that contribute to the *c* = 5 m s^{−1} signal, with similar amplitude signals from the shortest and longest wavenumbers. The contribution is strongest from wavenumbers 3.5 × 10^{−4} < *k* < 5.5 × 10^{−4} rad m^{−1} (horizontal wavelengths between 11 and 18 km), which corresponds to frequencies 1.75 × 10^{−3} < *ω* < 2.75 × 10^{−3} rad s^{−1} (periods between 38 and 60 min). These scales represent a physical property of the individual clouds or cloud systems, their horizontal scale, and the time scale of system development; therefore, this momentum flux signal derives from individual cloud systems. Although this result does not provide insight into which mechanism is actually generating the gravity waves, it does highlight the importance of the cloud life cycle in the wave generation by defining preferred temporal scales of the gravity waves.

Despite the robustness of the contribution from the cloud-system-scale gravity waves, the larger-scale wave signals remain important. Not only do these large-scale signals contribute to the total momentum flux (recall that approximately 70% of all momentum flux is derived from hydrostatic waves), but they may also play a (somewhat minor) role in filtering the stratospheric wave spectrum through nonlinear wave–wave interactions. Linear theory of critical levels (e.g., Booker and Bretherton 1967) explains that gravity waves cannot propagate vertically if their phase speed is equal to the background wind speed. In the scenario here, the large-scale waves define the background flow through which the cloud-scale waves propagate. For example, the large-scale waves produce signatures in horizontal velocity with wavelike vertical structure (e.g., Figs. 4c and 5). This vertical structure comprises vertical shear zones with horizontal extents that are much greater than the horizontal wavelength of the cloud-scale waves. These large-scale perturbations in horizontal velocity may act to dissipate the slowest (cloud scale) waves through critical-level interactions. To investigate this hypothesis, the standard deviation of the horizontal velocity, calculated at 20 km from 20 to 120 h, is equal to 1.0 m s^{−1}. (Note that the amplitude of a monochromatic wave is approximately √2 times the standard deviation.) This suggests that nonlinear wave filtering may act to filter the slowest gravity waves (*c* < 2 m s^{−1}). Further, the maximum and minimum horizontal velocities at 20 km are approximately ±5 m s^{−1}, suggesting that critical-level interactions may occur in some locations for slightly faster waves. Further research is required (e.g., using ray-tracing techniques) to investigate the importance of this filtering mechanism.

Finally, although the momentum flux spectrum shows little sensitivity to domain size, it is possible that a larger domain and longer simulation times might facilitate the formation of larger-scale convectively coupled waves (e.g., Tulich and Mapes 2008), which might also contribute to the momentum flux spectrum.

### b. Sensitivity to horizontal resolution

Given that the modeled gravity waves are insensitive to domain size, a 2000-km-wide domain is adequate to explore the sensitivity of the solutions to model resolution. (The shorter domain readily allows smaller horizontal grid spacing to be used.) Two additional simulations are conducted: one with 2-km horizontal grid spacing and one with 500-m horizontal grid spacing. The phase speed spectra from these simulations are shown in Fig. 12, along with the 1-km grid spacing simulation. Two aspects of the phase speed spectra are sensitive to horizontal grid spacing. First, the amplitude of the momentum flux gets progressively smaller as the grid spacing is reduced. This reduction in amplitude is anticorrelated with the intensity of the convection, which shows slightly larger maximum vertical velocities for the higher-resolution simulations (not shown). The second aspect of the sensitivity is that the spectrum shifts toward slightly slower phase speeds as the resolution increases. The higher-resolution simulations produce narrower convective updrafts and narrower clouds and hence waves with shorter horizontal wavelengths. Provided that the frequency of the waves remains constant, a reduction in the wavelength results in a reduction in the wave phase speed. Also, these shorter-scale gravity waves tend to have a weaker momentum flux signature. These trends with resolution are consistent with the findings of Lane and Knievel (2005), who demonstrated similar sensitivities and trends for squall lines.

### c. Sensitivity to the depth of the convection relative to the tropopause

The two-dimensional momentum flux spectrum (presented in Fig. 8) and the phase speed and vertical wavelength spectra (Figs. 9 and 10) highlighted a clear signal from the tropospheric *n* = 1 and *n* = 2 modes (vertical wavelengths of 33 and 16.5 km, respectively). However the contribution of these modes to the momentum flux were weak, with the amplitude of the *n* = 2 mode about 10% of the maximum momentum flux. The majority of momentum flux was sourced from higher-order vertical modes (*n* > 3), which in this case corresponds to waves with vertical wavelengths smaller than the depth of the convection (∼11 km). The response from the *n* = 1 and *n* = 2 modes is seemingly ubiquitous in convectively active regions, and occurs regardless of the depth of the convection (e.g., Bretherton and Smolarkiewicz 1989) because the tropopause acts as a preferential location of a node in the vertical (i.e., a region of suppressed vertical displacement). This ubiquity suggests that the small momentum flux corresponding to the fast response at *c* = 22 m s^{−1} (Fig. 9) is not necessarily related to any property of the convection but is rather a result of the depth of the troposphere. We hypothesize that if the depth of the troposphere were reduced, the response from the *n* = 2 mode (and corresponding stratospheric wave response) would shift to slower phase speeds (shorter vertical wavelengths). If the tropopause height corresponds to the cloud top, then the response from the *n* = 2 mode should be enhanced.

To test these hypotheses, the 2000-km-wide (1-km horizontal grid spacing) simulation is rerun with a modified thermodynamic sounding with the tropopause at 11 km. Above 11 km, the Brunt–Väisälä frequency is constant and equal to 0.0225 s^{−1}, and below the tropopause the thermodynamic sounding is the same as in the previous simulations. The two-dimensional momentum flux spectrum for this low tropopause simulation is shown in Fig. 13, and the corresponding phase speed spectrum is shown in Fig. 14, along with that from the original simulation. (In Fig. 14 the two spectra are normalized by the respective value of the total positive momentum flux to aid comparisons.)

Figure 13 shows that the two-dimensional momentum flux spectrum for the 11-km tropopause resembles the result from the simulation with the 16.5-km tropopause (Fig. 8), with spectral power occurring along lines of constant vertical wavelength that correspond to the *n* = 1 and *n* = 2 modes. In this case, however, the *n* = 1 and *n* = 2 modes are relative to the new tropopause height, and correspond to vertical wavelengths equal to 22 and 11 km, respectively. The phase speed spectra in Fig. 14 show that there is very little difference between the spectrum from the original simulation and that from the simulation with the lower tropopause. The two simulations show minor differences at intermediate phase speeds (*c* = 10–15 m s^{−1}), which may be caused either by minor changes in aspects of the convective evolution or by a change in the high-order vertical modal response. The notable difference between the two spectra is that the momentum flux signal at 22 m s^{−1} is not present in the low-tropopause case, in line with the above-hypothesized process. In contrast to the hypothesis, however, the 15 m s^{−1} phase speed, which corresponds to the depth of the convection (and now the *n* = 2 mode), does not show an enhanced signal in momentum flux.

Many previous studies (e.g., Lane et al. 2001) have shown that overshooting updrafts, especially those that are directly influenced by the tropopause, play an important role in gravity wave generation. Therefore, this simulation, which has the tropopause close to the cloud top, should show an enhanced momentum flux signal and possibly generate stronger high-frequency waves. There is some evidence of this enhancement. The phase speed spectrum possesses greater momentum flux at faster phase speeds between 27 and 38 m s^{−1}, approximately double that of the original simulation. The two-dimensional spectrum shows that these faster waves arise from a slight enhancement of the momentum flux at higher frequencies. However, these changes do not translate to a strongly enhanced total momentum flux. Nevertheless, the simulated convection is relatively weak, and therefore the overshooting updrafts are unlikely to penetrate very far above their level of neutral buoyancy. Stronger convection would probably show larger differences between the two simulations (e.g., convective updrafts are usually more intense over land than over the ocean).

## 4. Discussion and summary

A two-dimensional model was used to examine the generation of gravity waves by tropical multiscale convection. The modeled clouds evolved from an initially thermodynamically uniform and motionless base state and self-organized into systems of preferred width, depth, intensity, and horizontal separation. The cloud population evolved via a combination of stochastic convection and the subsequent formation of weakly organized propagating cloud clusters. The depth and intensity of the convection resembled oceanic tropical convection, and formed a rich spectrum of gravity waves that propagated from the troposphere to the stratosphere.

Spectral analysis of the gravity waves showed that, as expected, the gravity wave signatures in vertical velocity were a response to the shorter-scale, higher-frequency gravity waves. Likewise, the signatures in horizontal velocity were a response to the longer-scale, lower-frequency gravity waves. The spectrum of momentum flux was remarkably flat in horizontal wavenumber space on scales of *O*(1000 km) down to the cloud-system scale *O*(10 km). The flatness of the spectrum implies an equal contribution from each individual wavenumber component to the total momentum flux spectrum and reflects the stochastic nature of the cloud population. Throughout the simulation, a variety of scales evolved, but no single spatial scale emerged as the dominant signal.

Further analysis of the gravity wave momentum flux spectrum showed that it contained important contributions from at least three categories of wave modes. The first is the relatively slow cloud-scale waves that emerge from individual convective clouds and are a robust feature of the wave spectrum. The second is the relatively long horizontal-scale and short vertical-scale modes that propagate horizontally and modify inflow and outflow patterns on spatial scales greatly exceeding the scale of individual convective clouds. These modes also propagate vertically, and may filter the slowest of the cloud-scale waves through nonlinear wave–wave interactions. The third category is the deep tropospheric modes that form with vertical scales that are harmonics of the tropospheric depth. The occurrence of these modes is ubiquitous in regions of active convection. Previous research has shown that they modulate and organize the convection on scales exceeding the individual cloud-system scale.

These three wave categories suggest controls over the character of the waves, of which some were briefly examined herein. For example, the horizontal scale of individual clouds modifies the horizontal scale of the cloud-scale waves; such a modification can occur either through (unphysical) changes in model resolution or through changes in the physical characteristics of the clouds themselves. Also, changes in the thermodynamic profile and microphysical characteristics will influence the cold pool depth and rate of propagation, influencing the generation of the long-horizontal-wavelength, short-vertical-wavelength waves (e.g., gust front modes). Finally, the depth of the convection relative to the tropopause affects the strength and/or occurrence of the response from the deep tropospheric modes.

The occurrence of such a broad wave spectrum of gravity waves has important implications for the parameterization of convectively generated waves in general circulation models. Multiscale convection generates waves that occur on scales exceeding the cloud scale and even exceeding the mesoscale convective system scale (e.g., squall lines). The longer-wavelength waves contribute to the total momentum flux, help organize the source, and are also responsible for nonlinear wave–wave interaction. This implies that efforts at parameterization may need to include information about the statistics of the cloud population in addition to details of the individual clouds or systems. Current parameterizations lack this additional detail and only assume a single wave source (i.e., a single convective system) whose intensity is related to some aspect of the convective parameterization scheme (such as the grid-averaged heating).

The flatness of the momentum flux spectrum in the horizontal wavenumber domain is a property that may also be exploited to quantify the total momentum flux from cloud populations. Observational techniques usually suffer some limitations, meaning that each platform has an *observational filter* (Alexander 1998); that is, it can only observe part of the wave spectrum. For example, satellite observations of gravity waves (e.g., Ern et al. 2004) usually cannot observe short-horizontal-wavelength gravity waves. However, by assuming a simplified flat momentum flux spectrum, observations can potentially be extrapolated to other (unobservable) scales; that is, knowledge of the amplitude of the momentum flux in one part of the spectrum can be used to infer the amplitude in other parts of the spectrum, allowing estimates of the total momentum flux. Such estimates are necessary to constrain gravity wave source parameterizations.

The momentum flux spectrum for the multiscale (oceanic) convection considered herein differs in important ways from the single-system squall line simulations that have been presented to date. The most notable difference is that the strongest waves occur at substantially slower phase speeds than are characteristic of the more intense systems. This difference may occur for a number of reasons. More intense systems are more likely to overshoot deeper into the stratosphere, producing higher-frequency (faster) gravity waves. Also, more intense systems are likely to possess stronger regions of outflow and cloud-induced wind shear. The enhanced shear will act to filter the spectrum, virtually eliminating the slowest wave modes. Finally, the multiscale approach models the clouds from their initiation to decay, allowing lower-frequency (slower) waves to develop. In summary, it is anticipated that significant differences between convectively generated gravity waves over land and ocean would occur.

Despite the differences in the above aspects of the phase speed spectrum, the maximum amplitude of the momentum flux presented here resembles that from previously reported squall line simulations (e.g., Choi et al. 2007). Squall lines might be expected to produce much stronger momentum fluxes than the relatively benign oceanic convection that is represented here. However, the multiscale convection simulation contains many more clouds, each contributing to the total momentum flux. There are numerous (somewhat weak) sources in the simulations here, which produce a similar response to a single intense source (over a similar area). These results suggest that oceanic convection, with its more widespread occurrence, may be equally as important as more intense isolated systems, and future research efforts should reflect this.

The simulations presented herein are idealized, with the following two notable simplifications: (i) two-dimensionality and (ii) quiescent flow. First, the two-dimensional system is used here to simplify the spatial interactions and to reduce computational costs. The addition of the third dimension results in a reduction in the amplitude of gravity waves as they propagate horizontally away from their source. This might reduce the influence of deep modes on the surrounding atmosphere, reducing their effect on the organization of convection. Nevertheless, recent three-dimensional studies (e.g., Shutts 2006) have shown multiscale interactions in three dimensions that appear similar to those occurring in two dimensions. Second, the lack of background horizontal flow results in a symmetric spectrum of gravity waves and a population of clouds that, on average, transports no momentum and remains relatively disorganized.

The inclusion of environmental wind shear is anticipated to generate more highly organized and long-lived systems, which would change the spectrum of gravity waves and their vertical propagation. Preliminary simulations in the multiscale framework show that (in line with previous squall-line studies) tropospheric shear removes the symmetry in the gravity wave spectrum and enhances the momentum flux signature at faster phase speeds. The results of this continuing study will be reported in a future paper.

## Acknowledgments

Todd Lane is currently supported by the Australian Research Council Discovery Projects Scheme (DP0770381). We thank two anonymous reviewers for their helpful comments.

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* The National Center for Atmospheric Research is sponsored by the National Science Foundation.