## 1. Introduction

Using a two-dimensional numerical model, Bretherton and Smolarkiewicz (1989) showed that gravity waves propagated radially from convective clouds, and that the passage of these waves produced compensating subsidence and adiabatic warming in the stable environment surrounding the cloud. In this way, the net effect of the latent heat release and adiabatic cooling within the cloud updraft is communicated to the cloud environment. Such waves play a central role in modifying the cloud environment, and are the focus of the present study.

Bretherton and Smolarkiewicz (1989) also showed that the gravity wave response led to a *buoyancy adjustment time,* which was much shorter than the *mixing time* of the cloud. That is, the fast nature of gravity waves caused the cloud environment to adjust much more quickly than any effects caused by advective motions. Shutts and Gray (1994) examined the role of gravity waves in adjusting the environment of isolated clouds in highly rotating reference frames. However, the primary focus of their study was on the final balanced flow, rather than the gravity waves themselves.

The ideas of Bretherton and Smolarkiewicz (1989) were applied to a larger scale by Mapes (1993), who considered the role of gravity waves in the upscale growth of mesoscale convective systems (MCSs). Mapes showed that a *prescribed* heat source with a vertical profile like that of observed MCSs generates horizontally propagating (bore) waves. Such waves were shown to produce upward displacements at low levels in a region surrounding the heating. These displacements destabilized the atmosphere, and consequently conditions near an MCS but beyond the area of low-level outflow became more favorable for further convection. Mapes argued that this mechanism promotes cloud clustering, and he describes MCSs as “gregarious.” Similar calculations were reported by Nicholls et al. (1991) and Pandya et al. (1993). Furthermore, McAnelly et al. (1997) used a combination of radar observations and linear modeling to describe the development of MCSs. In agreement with Mapes their results showed that gravity waves are an important mechanism contributing to MCS upscale growth.

The studies by Bretherton and Smolarkiewicz (1989), Nicholls et al. (1991), Mapes (1993), and Pandya et al. (1993) all assumed that the waves responsible for adjusting the cloud environment were generated by convective heating. In order to make the problem tractable, these studies represented the convection using an externally imposed heat source which was switched on at an initial time and subsequently maintained. However, the net heating is strongly controlled by the cloud's own circulation and it is questionable whether it can be imposed externally. Moreover, Lane et al. (2001) have shown that gravity wave emission is inextricably tied to the internal dynamics of the cloud, such as the (nonlinear) advection of buoyancy by the cloud's circulation. Like Bretherton and Smolarkiewicz, the study presented here models the convective cloud and the waves it generates using a cloud-resolving model that explicitly treats moisture and thermodynamic processes. One aim of this study, however, is to determine how well convective heating, along with linear theory, describes the way convective clouds generate gravity waves.

A detailed understanding of how simple isolated convective clouds affect their environment is an essential stepping-stone to understanding how highly organized cloud ensembles, such as MCSs, promote and regulate further convection in their vicinity. With this in mind, the central aim of this study is to determine whether the ideas of Mapes (1993) and McAnelly et al. (1997), which concern relatively large-scale organized MCSs, can be applied to smaller-scale isolated convection. Like the study by Bretherton and Smolarkiewicz (1989), this is a fundamental study designed to determine the properties of the gravity waves generated by an isolated cloud, and to examine the effect of these waves on the local environment of the cloud. To do this, the numerical calculations will be relatively idealized, limited to two dimensions, and will ignore any background flow. Such a configuration simplifies the interpretation of the gravity waves. Extending the work of Bretherton and Smolarkiewicz, the effect of the gravity waves generated by the isolated cloud will be quantified in terms of the convective available potential energy (CAPE) and the convective inhibition (CIN). Such a quantification allows us to estimate the importance of these waves in adjusting the cloud environment. The spectrum of the gravity waves will also be examined to determine the character of the gravity waves generated by the cloud.

The paper is organized as follows. Section 2 describes the numerical model, the initial conditions, and the evolution of the modeled convective clouds. Section 3 quantifies the adjustment of the troposphere in terms CAPE and CIN, and describes the changes in CAPE and CIN produced by radiating gravity waves. The power spectrum of the gravity wave field is calculated in section 4, and the applicability of linear theory and prescribed diabatic heating for deep convective clouds is briefly discussed. Our conclusions are set out in section 5.

## 2. Numerical model

This section outlines briefly the numerical model, the initial conditions, and how the modeled convection evolves.

### a. Model outline

The numerical model used in this study has been described in detail by Clark (1977) and Clark and Farley (1984). The model is nonhydrostatic, anelastic, and the finite difference approximations are second order in both space and time. The present study is idealized; the model calculations are two-dimensional, and neglect both the Earth's rotation and topography. On the time and spatial scales considered here, the effects of rotation on the convection and gravity waves are not considered important. In order to help expose the fundamental dynamics, surface processes such as friction, sensible heat flux and moisture flux are neglected also. The model includes a parameterization of warm rain processes (Kessler 1969), ice microphysics (Koenig and Murray 1976), and subgrid-scale turbulence (Lilly 1962; Smagorinsky 1963). However, for most of this study the effects of ice processes are not included. Furthermore, the effects of internal absorption and scattering of radiation are neglected.

The model domain is 240 km wide and 35 km high with 250-m grid length in both the horizontal and vertical. A numerical time step of 3 s is used, ensuring that the fast gravity wave modes do not violate the CFL stability criterion. A rectangular cartesian coordinate system is used with *x* measuring the horizontal distance and *z* the height. A Rayleigh-friction sponge is incorporated in the uppermost 10 km of the domain to absorb vertically propagating disturbances without reflection. Radiation conditions are imposed at the lateral boundaries, and are described in detail in Clark (1979). The boundary conditions have been tested in a number of numerical experiments with different domain sizes. In general, the gross features of the cloud and gravity-wave fields are insensitive to domain size, suggesting that the boundary conditions are satisfactory. However, the lateral boundary partially reflects some relatively slow-moving waves with short horizontal wavelengths. For this reason, care must be taken to ensure that the lateral boundary is sufficiently far from the cloud so that reflections do not affect the model solution in the region of interest. It will be shown, that it is important to handle these waves accurately as they have the largest effect on the environmental CIN.

### b. Initial conditions

The background potential temperature profile is characterized by constant Brunt–Väisälä frequencies (*N*) in both the troposphere and stratosphere. Specifically, *N* = 0.0115 s^{−1} in the troposphere and *N* = 0.025 s^{−1} in the stratosphere (which correspond to buoyancy periods of 9.1 and 4.2 min, respectively). The surface potential temperature is 299 K and the height of the tropopause is 16 km. The background water vapor mixing ratio *q* is defined at the initial time by *q* = 19.2 exp(−[*z*/3100]^{1.5}) g kg^{−1}, which is an approximation to the sounding used by Lane et al. (2001). [This moisture sounding was taken during the Maritime Continent Thunderstorm Experiment (Keenan et al. 2000) over the Tiwi Islands, Australia.] The background profiles of temperature and dewpoint temperature are plotted on a skew *T*–log*p* diagram (Fig. 1).

At the initial instant, the atmosphere is assumed to be at rest. This initial state is chosen for three main reasons. First, wind shear affects the propagation, absorption and dissipation of gravity waves (e.g., see Bretherton 1966). These effects complicate the analysis of the gravity waves and therefore the analysis of the source. Second, tropospheric wind shear causes convective structures to tilt with height (e.g., squall lines). This tilting of the convection complicates the interpretation of the wave generation mechanism (e.g., see Fovell et al. 1992). Finally, low-level wind shear may interact with gust fronts or cause ducting of gravity waves (e.g., Lindzen and Tung 1976; Yang and Houze 1995). These two mechanisms affect the environment of the cloud, and it would be difficult, if not impossible, to distinguish them from each other. Thus, although the modeled convection will be idealized, the effects of the gravity waves on the environment should be easily identified.

A localized potential temperature perturbation *θ*′ is added to the background thermodynamic profile. This perturbation, or “warm bubble,” is defined by the expression *θ*′(*x,* *z*) = Θ exp[ −(*x* − *x*_{0})^{2}/*σ*^{2}_{x}*z* − *z*_{0})^{2}/*σ*^{2}_{z}*z* ≤ 2.5 km and |*x* − *x*_{0}| ≤ 4 km with *θ*′(*x,* *z*) = 0 outside the region. The constants are Θ = 4 K, *x*_{0} = 120 km, *z*_{0} = 1 km, *σ*_{x} = 4 km, and *σ*_{z} = 1 km. In the region of this bubble, the modeled atmosphere is convectively unstable and develops a deep convective system within approximately 30 min.

### c. Evolution of the convective cloud

The modeled convection is similar in a number of respects to that considered by Bretherton and Smolarkiewicz (1989) and Shutts and Gray (1994). The convection is short-lived, with moderate depth (penetrating to a depth of approximately 11 km) and moderate intensity (with maximum updrafts exceeding 20 m s^{−1}). Figure 2 depicts the vertical velocity and the cloudy air at 40-min intervals. A broad updraft with a horizontal scale similar to that of the warm bubble is established after about 20 min of integration (Fig. 2a). The maximum vertical velocity is approximately 20 m s^{−1} and is located at *z* ≈ 4 km. Outside the cloud, most of the tropospheric air is subsiding, with the maximum downward motion adjacent to the maximum updraft.

The vertical velocity and the perturbation pressure are averaged horizontally on the interval 116 ≤ *x* ≤ 124 km, and the evolution of these averaged quantities are shown in Fig. 3. The convection rapidly deepens, and at about 25 min into the integration the cloud top overshoots the level of neutral buoyancy (LNB), which is located approximately at a height of 9 km (Fig. 3a). At the same time, a strong precipitation-driven downdraft develops and lasts for about 30 min (until time, *t* = 55 min). A mesohigh forms at low levels in conjunction with the downdraft (Fig. 3b).

The 60-min vertical velocity field shows prominent gravity waves in the cloud environment (Fig. 2b). At this time, most of the cloudy air is above *z* = 5 km, and the cloud updrafts have substantially weakened. Forty minutes later, the convective updrafts have decayed, and the cloudy air, which is centered on the LNB, has spread horizontally (Fig. 2c). Apart from minor asymmetries later in the evolution due to round-off error, the cloud remains symmetric about the *x* axis throughout its evolution because the background flow is at rest. Hence, all subsequent figures will show only the half plane *x* ≥ 120 km.

## 3. Adjustment in the cloud environment

In this section we examine how the cloud affects its environment, and in particular, we investigate the role of propagating gravity waves. First, these waves are identified in the potential temperature field. Second, the effect of these waves are quantified in terms of the CAPE and CIN.

### a. Gravity waves

*ω*is the frequency of the wave, and

*k*and

*m*are the horizontal and vertical wavenumbers respectively. Assume that

*ω*> 0,

*k*> 0, and

*m*< 0, which corresponds to a wave propagating in the positive

*x*and positive

*z*directions. From (1), the horizontal and vertical phase speed for the waves are

*k*≪

*m*; hence

*C*

_{z}≪

*C*

_{x}and

*C*

_{gz}≪

*C*

_{gx}. The key points to note are that (i) the horizontal phase and group propagation are equal and depend only on the vertical wavenumber (and Brunt–Väisälä frequency); (ii) the larger the vertical wavelength, the faster the horizontal propagation; (iii) upward group propagation implies downward phase propagation; and (iv) the rate at which the wave propagates vertically is much smaller than the rate at which is propagates horizontally.

*Z*

_{T}is the height of upper boundary and

*n*is an integer. Note that the lower-order modes travel faster than higher-order modes. To the extent that the tropopause behaves like a rigid surface, the gravity waves generated in the model will be governed by Eq. (3). In the calculations presented below, the height of the tropopause

*Z*

_{T}is 16 km. As we shall see, it proves useful to discuss the gravity wave field in terms of vertical modes, where the

*n*th mode has a vertical wavelength

*λ*

_{z}= 2

*Z*

_{T}/

*n.*The horizontal phase speeds for the first three internal modes are approximately 58, 29, and 19 m s

^{−1}, respectively.

Figure 4 shows four cross-sections of the perturbation potential temperature at 20-min intervals. Large wavelike perturbations are evident in the cloud environment. In so far as the motion is adiabatic, the vertical displacements in the flow are proportional to the potential temperature perturbations. A positive perturbation is associated with air that has been displaced downwards, and a negative perturbation is produced by air that has been displaced upward.

While a variety of waves of different wavenumber contribute to the perturbation potential temperature field, the first three wave modes are particularly prominent. For example, 40 min into the run the perturbation potential temperature has the vertical structure of the *n* = 1 mode at *x* ≈ 170 km. Similarly, the *n* = 2 and *n* = 3 modes make major contributions to the vertical structure at *x* ≈ 150 km and *x* ≈ 130 km, respectively. Both the *n* = 1 and *n* = 2 modes produce positive potential temperature perturbations near the surface, whereas the *n* = 3 mode has a negative perturbation associated with low-level lifting.

Figure 4 also shows the effect of dispersion in separating out waves of different wavenumber. For example, consider the evolution of the perturbation potential temperature at *x* = 180 km. The *n* = 1 mode travels fastest, and arrives at *x* = 180 km first (*t* = 40 min). This is followed by the *n* = 2 mode (*t* = 60 min). Twenty minutes later, the *n* = 2 mode has partially propagated out of the plot and, at the plot's right-hand boundary, its phase has advanced through roughly 180°. The *n* = 3 mode arrives at *x* = 180 km in the final panel (*t* = 100 min). Notwithstanding dispersion, higher-order modes seem to be generated later in the cloud's lifetime than the lower-order modes. Close to the cloud, the perturbation potential temperature field is dominated by slower moving higher-order modes.

### b. CAPE and CIN

We examine now how the cloud environment is affected by the waves identified in the perturbation potential temperature, and quantify these effects in terms of CAPE and CIN.

CIN is the amount of work that the atmosphere must do to lift an air parcel from a given height to its level of free convection (LFC). CAPE is the amount of energy that can be released by lifting a parcel from a specified height to its (uppermost) level of neutral buoyancy. Of course, the magnitude of the CIN and the CAPE depend on the level from which the parcel is lifted, among other things. In this study, the air parcel is assumed to be lifted from the surface. The CAPE also depends on whether the condensed water remains with the air parcel or whether it precipitates out. In the calculations that follow, the condensed water is retained by the parcel, and consequently the thermodynamic processes are reversible (see, e.g., Emanuel 1994). In any case, the calculations below focus on the changes in the CIN and the CAPE from their initial values, and these changes are far less sensitive to the details of the calculations than the values of CIN and CAPE themselves.

The CAPE of the background atmosphere, calculated by lifting a parcel from the surface, is 1649 J kg^{−1}. The corresponding CIN is 18.4 J kg^{−1}, and the level of LFC is 1.1 km. In the absence of moisture sources, changes in CIN and CAPE are directly related to vertical parcel displacements, and hence the perturbation potential temperature. Departures of the CIN and the CAPE from their initial values are calculated at every time step for each column in the model, and the evolution of these perturbations are shown in Fig. 5. As expected, local mixing rapidly stabilizes the atmosphere close to the cloud, and this is reflected in decreasing CAPE within the cloud and increasing CIN below cloud base. These perturbations expand horizontally as the cloud grows.

The first three wave modes describe the changes in CAPE and CIN seen in Fig. 5 very well, and in particular, the speed at which the changes radiate outward. For the first 30–50 min, the CAPE decreases at a fairly uniform rate in the cloud environment. These falls spread outward from the center of the domain at approximately 55 m s^{−1} and are associated with the *n* = 1 mode. The wave mode acts to stabilize the troposphere, with downward displacements of parcels warming and drying the atmosphere. This reduces the CAPE and increases the CIN. However, the change in CIN by the *n* = 1 mode is small because the maximum displacements occur in the middle troposphere.

Following the *n* = 1 mode, the CAPE increases briefly in the environment, before developing a wavelike pattern with a period of around 30 min. The CIN evolves in a markedly different way to the CAPE. Initially, the CIN increases strongly near the center of the domain, and this effect propagates outward at approximately 30 m s^{−1}.

The *n* = 2 mode produces negative displacements at low levels and positive displacements at upper levels. This rearrangement cools and moistens the upper part of the troposphere and hence increases the CAPE. At low levels the atmosphere is warmer and drier, and consequently the CIN increases also. The change in CIN is more pronounced for the *n* = 2 mode than the *n* = 1 mode because the maximum parcel displacement occurs at approximately *Z*_{T}/4 rather than *Z*_{T}/2.

After about 30 min, the CIN decreases below the cloud, and this decrease spreads outward at approximately 15 m s^{−1}. At about 100 min, a second strong perturbation develops near the edge of the cloud and propagates into the environment at less than 15 m s^{−1}.

The *n* = 3 mode produces positive displacements at low levels, thereby reducing the CIN. This reduction is pronounced because the maximum upward displacement occurs at *Z*_{T}/6, which strongly affects the subcloud layer. However, the *n* = 3 has little effect on the CAPE.

Finescale structure in the CAPE develops in the environment after about 100 min of integration, although the perturbations remain mostly positive. In the environment far from the cloud, the perturbations in the CIN are generally positive, until the passage of the *n* = 3 mode when a negative perturbation originates near the center of the cloud at *t* = 30 min and propagates outward.

Taken together, the results show that the environmental CAPE is reduced by approximately 240 J kg^{−1} (15% of the initial CAPE) early in the life of the cloud by a fast-moving (*n* = 1) disturbance. (Recall the horizontal phase speed of the *n* = 1 mode is approximately 55 m s^{−1}.) In contrast, the CIN suffers large fractional decreases of approximately 6 J kg^{−1} (33% of the initial CIN) later in the cloud lifetime. These decreases are produced by a slower moving (*n* = 3) disturbance. (Recall, the horizontal phase speed of the *n* = 3 mode is approximately 15 m s^{−1}.) By definition, the CIN depends only on the thermodynamic structure in the subcloud layer. Therefore, only the higher modes affect the CIN as it is these modes that have maxima in the subcloud layer. Conversely, the CAPE is affected significantly only by the first two modes. This is because the CAPE is an integral quantity calculated over a deep layer, and consequently the effects of higher modes are averaged out.

There are localized effects in addition to the changes in CAPE and CIN induced by horizontally propagating gravity waves. In particular, there is a distinct region directly adjacent to the cloudy air with reduced (increased) values of CAPE (CIN). Presumably this region is caused by local subsidence surrounding the cloud rather than by gravity wave induced motions.

With the passage of the third wave mode, the CIN is reduced in a region surrounding the cloud (but beyond the area of its outflow). This region is favorable for the development of new convection. In his study of MCSs, Mapes (1993) predicted such a region surrounding the cloud would result from the passage of the second wave mode rather than the third mode. When the depth of the subcloud layer is comparable to the height of maximum parcel displacement of the second mode (*Z*_{T}/4) it may play a significant role in reducing the CIN. However, in the case presented here, the subcloud layer is relatively shallow, and is only affected by higher order wave modes. The convection modeled here has a maximum depth of approximately 11 km, whereas Mapes' (prescribed) convective heating spanned the depth of the troposphere. Therefore, it may not be relevant to compare in detail the effect of individual modes on the environment in Mapes' study to ours because deeper convection probably generates stronger low-order modes. Furthermore, the generation of the *n* = 3 mode coincides with the onset of precipitation at about 30 min. A similar result has been found by previous authors (e.g., Mapes 1993; McAnelly et al. 1997) who attributed the source of the mode causing these upward parcel displacements (*n* = 2 in their case) to evaporative cooling in the lower part of the convective system.

The region surrounding the cloud that is favorable to new convection is relatively short lived. Thus, any influence of the gravity waves on the generation, organization, or propagation of the convective system would only occur shortly (∼hours) after the development of the cloud. Nevertheless, the existence of this region suggests that these gravity waves may play an important role in the development of convective systems, on a smaller scale than considered previously.

### c. Sensitivity tests

The results above demonstrate that gravity waves radiate from the modeled clouds and affect the environment. Therefore, it is necessary to establish that the waves are not an artifact of initializing the model with a warm bubble. The sensitivity of the numerical solution to the background profile and to the external forcing was investigated in a series of numerical experiments. For example, in one experiment an island was placed in the center of the model domain instead of the warm bubble. As the island was heated, a sea-breeze circulation developed, which eventually initiated convection. All of the numerical experiments produced convective systems with similar depths and intensities, although the horizontal structures and the growth rates differed from run to run. Nonetheless, each numerical experiment generated a similar spectrum of tropospheric gravity waves, and very similar changes in CIN and CAPE.

It was stated earlier that a number of tests on domain size were carried out to determine how effectively the lateral boundaries transmit gravity waves. One such test was to run the same numerical experiment with a domain half as wide. The cloud field, gravity wave field, and temporal evolution of CAPE were nearly identical to the original experiment. However, the temporal evolution of the CIN showed marked differences in the far-field, and this is shown in Fig. 6. (Note that this figure has a different coordinate axis to Fig. 5b and the center of the cloud is at *x* = 60 km.) The effect on the CIN of the third mode extends approximately 30 km from the center of the cloud, whereas using a larger domain the effect of this mode on the CIN (Fig. 5b) continues horizontally for the entire modeled domain. With a reduced domain, the third internal mode does not cause lifting in the subcloud layer past *x* = 90 km, and therefore does not reduce the CIN. Although the gross features of the wave field in the two cases are very similar (not shown), a small change in the lifting of the subcloud layer creates a large change in the CIN. For example, the maximum difference in CIN between the two cases is approximately 10 J kg^{−1}, and in some regions far from the cloud the perturbations are of opposite sign. Such a difference between the two cases has important implications when considering numerical models of cloud clusters, as the organization and initiation of convective cells may be sensitive to domain size.

Among these tests on domain size, a number of cases were run with the vertical boundary of the model at different heights, with differing absorber depths. These tests showed that the vertical boundary condition works very well in transmitting vertically propagating waves. Furthermore, these tests also confirmed that the vertical modal structure of the tropospheric gravity waves are due to the change in stratification at the tropopause, not the model lid.

## 4. The role of diabatic heating in generating the waves

In general, gravity waves will be generated whenever isentropes are displaced vertically in a frame of reference moving with the fluid. Depending on the frequency of the forcing, the waves will either decay with height or will propagate vertically. Isentropic displacements may be the result of adiabatic motion within the fluid (e.g., vertical advection), forced ascent or descent by imposed boundary conditions (e.g., mountain waves), or diabatic processes such as latent heating. Lane et al. (2001) have emphasized the role played by the motion within the cloud in generating high frequency, upward propagating gravity waves of the type observed in the stratosphere. On the other hand, many theoretical studies (e.g., Bretherton and Smolarkiewicz 1989; Nicholls et al. 1991; Mapes 1993; Pandya et al. 1993) assume that clouds generate gravity waves mainly though the effects of diabatic heating. These studies parameterize the convection by an external heat source that is switched on at the initial time and subsequently maintained. This section determines the extent to which diabatic heating controls the gravity wave spectrum, and examines the accuracy of neglecting nonlinear effects.

### a. Spectral analysis

In this section spectral analysis is used to distinguish between the low-frequency (hydrostatic) waves that play an important role in adjusting the cloud environment, and the high-frequency (nonhydrostatic) waves that propagate nearly vertically. Such an analysis is necessary to understand the source of the waves, and to determine what part of the spectrum modifies the cloud environment.

*ω*−

*k*spectrum. The spectrum is calculated at a height of 16 km because this is well above the cloud top, and consequently the spectrum will not be unduly affected by the convective updrafts within the cloud. The spectrum is one-sided in

*ω*and two-sided in

*k.*As there is no background flow the spectrum is symmetric about

*k*= 0, and therefore only

*k*> 0 will be shown. The power spectrum is averaged over 5 adjacent frequency/wavenumber bins and is shown in Fig. 7. The vertical wavenumber

*m*is calculated from the (nonhydrostatic) dispersion relation,

*π*/

*m*) at harmonics of the tropospheric depth are overlayed on the power spectrum (Fig. 7). Also shown is a one-dimensional frequency spectrum, which is calculated by summing the two-dimensional spectrum over the horizontal wavenumber domain (Fig. 8).

The two-dimensional spectrum shows a distinct maximum corresponding to waves with frequencies between 3.5 × 10^{−3} s^{−1} and 5.2 × 10^{−3} s^{−1} (periods between 30 and 20 min) and horizontal wavenumbers between 1 × 10^{−4} m^{−1} and 2 × 10^{−4} m^{−1} (horizontal wavelengths between 60 and 30 km). The vertical wavelengths of these modes range from 10 to 32 km, which encompass the *n* = 1, 2, and 3 modes discussed earlier. In addition to this maximum, there exists a broad maximum for all horizontal wavenumbers with frequencies between 7 × 10^{−3} s^{−1} and 1 × 10^{−2} s^{−1} (periods between 15 and 10 min). There is relatively little spectral power at frequencies greater than 1 × 10^{−2} s^{−1} (periods less than 10 min), as these modes are evanescent in the troposphere. Modes with vertical wavelengths less than 1 km (*n* > 32) are numerically unresolved and therefore do not contribute significantly to the spectrum.

When the gravity waves are approximately hydrostatic (*m* ≫ *k*), isopleths of vertical wavenumber are linear in *ω* − *k* space. This is approximately the case in Fig. 7 in the rectangular region bounded by (*k,* *ω*) = (0, 0) and (*k,* *ω*) = (3 × 10^{−4} m^{−1}, 6 × 10^{−3} s^{−1}). The waves that contribute to the maximum in the power spectrum are in this region and, therefore, are approximately hydrostatic. Furthermore, the vertical group velocity of these waves is approximately zero [see Eqs. (1)–(3)]. Outside this region in *ω* − *k,* the waves are nonhydrostatic and may propagate vertically. This is an important and necessary distinction because the hydrostatic waves will remain mainly in the troposphere and affect the cloud environment. On the other hand, the nonhydrostatic waves may propagate vertically and can strongly affect the momentum budget of the middle atmosphere (see Hamilton 1997 for details).

The maximum in the two-dimensional spectrum represents the waves, which have been the focus of this paper, and are most important in adjusting the cloud environment in the troposphere. The vertical wavelength corresponds to twice the depth of the convection and the period corresponds to the approximate lifetime of the convective system. Moreover, the horizontal wavelength is approximately twice the width of the convective cloud. It appears that the depth and lifetime of the convection determines the position of this maximum. In fact, if the same experiment is rerun including ice microphysics (not shown), an increase in the depth and lifetime of the system occurs, and an identical increase in the period and vertical wavelength of the maximum in the spectrum is observed. This suggests that these low-frequency (hydrostatic) gravity waves are a direct response to the gross features of the convection, possibly the diabatic heating.

In contrast, the broad maximum for all wavenumbers with frequencies between 7 × 10^{−3} s^{−1} and 1 × 10^{−2} s^{−1} (periods between 15 and 10 min) appears unrelated to either the temporal or spatial distribution of the gross features of the convection. These higher-frequency waves are similar to those investigated in Lane et al. (2001), and are due to an oscillation of the convective updrafts about the level of neutral buoyancy. The horizontal wavenumber of these high-frequency waves ranges from 6 × 10^{−4} to 2.5 × 10^{−3} m^{−1} (horizontal wavelengths from 12 to 2 km), encompassing scales less than the width of the cloud down to the (resolved) turbulent eddies within the cloud. It is difficult (if not impossible) to distinguish between the propagating gravity waves and the turbulent eddies with frequencies close to *N.* Nevertheless, it can be seen from Fig. 7 that there is a difference in spectral power of approximately two orders of magnitude between *k* = 6 × 10^{−4} m^{−1} and *k* = 2.5 × 10^{−3} m^{−1} at *ω* = 7 × 10^{−3} s^{−1}. In fact, later in the cloud's lifetime the waves with *k* = 6 × 10^{−4} m^{−1} (a horizontal wavelength of 10.5 km) appear to dominate the wave field at the tropopause (Fig. 4d).

This convective system was initialized with a warm bubble, and consequently it is possible that higher-frequency waves are contaminated by spurious modes due solely to the initialization. However, this broad maximum at high frequencies in the power spectrum is characteristic also of other numerical experiments where the convective systems were initialized by the sea breeze convergence that developed over an isolated island (not shown).

Lane et al. (2001) focused on the higher-frequency waves that dominated the stratospheric wave field in their study, whereas in this case it has been shown that the maximum in the two-dimensional power spectrum represents lower-frequency waves. However, if we look at the frequency spectrum alone (Fig. 8) it is clear that the higher-frequency waves dominate. In fact, the peak in the one-dimensional spectrum corresponds to a frequency of approximately 6 × 10^{−3} s^{−1} (period of approximately 17 min). These higher-frequency waves tend to propagate nearly vertically, while the lower-frequency waves tend to propagate more horizontally. In a three dimensional atmosphere, the amplitude of gravity waves decreases by the inverse of the radial distance from the source and consequently the amplitude of lower-frequency horizontally propagating waves will be reduced in comparison with the higher-frequency vertically propagating waves. Therefore, we expect that in a real atmosphere the contribution to the power spectrum from the lower-frequency waves would be significantly less than that from the higher-frequency waves. Thus, the principal effect of the gravity waves generated by such a cloud may be on the middle atmosphere rather than the troposphere. There is also the possibility that the hydrostatic gravity waves may trigger new clouds which are themselves new sources of (both hydrostatic and nonhydrostatic) gravity waves.

### b. Representation of the wave source

Consider how a localized heat source affects a two-dimensional stably stratified fluid at rest. This problem has been investigated in detail by Bretherton and Smolarkiewicz (1989) as a first step in understanding the interaction between convective clouds and their environment. Assume that the geometry is the same as that for the numerical model described in section 2. For simplicity, let the fluid be inviscid, hydrostatic, nonrotating and confined between two rigid plates separated by a distance *Z*_{T}. Suppose that the heat source is confined to an infinitely thin sheet along the *z* axis, that its vertical structure is given by sin(*πz*/*Z*_{T}), and the heat source is switched on at the initial time and maintained thereafter. As shown by Bretherton and Smolarkiewicz, the linear response comprises two bore waves that are generated at the initial instant, and propagate away from the heat source with phase speeds *C*_{x} = ±*NZ*_{T}/*π.* These bores can be thought of as a sum of gravity waves, although the component waves do not disperse as they all have the same vertical structure [see Eq. (3)]. Bretherton and Smolarkiewicz show that there is subsidence in the environment only in an infinitely narrow region at the leading edge of bore wave. As the bore waves pass, the environment is adiabatically warmed as isentropes are displaced downward. The flow is strictly horizontal following the passage of the bore; the flow is directed toward the leading edge of the bore at upper levels but toward the heat source at low levels. There is ascent only along the *z* axis. As the fluid rises along the *z* axis it cools adiabatically at a rate that exactly balances the rate of heating by the source. If the heat source is switched off, two further bores waves are generated. Importantly, these bore waves have amplitudes which are equal and opposite to those generated at the initial time. Hence, as the bore waves radiate outward, they reverse the effects of the original waves, leaving no net change in the environment.

The results of Bretherton and Smolarkiewicz (1989) have been extended by Nicholls et al. (1991) and Pandya et al. (1993), who examined the effects of a localized heat source on an infinitely deep fluid. Removing the upper rigid boundary has little effect on results. This is because the waves generated by the heat source have horizontal wavelengths that are long compared to their vertical wavelengths, and consequently they propagate at a shallow angle to the horizontal [see Eq. (2)]. A similar results was found in section 3. Mapes (1993) applied a slightly modified form of Bretherton and Smolarkiewicz's analysis to MCSs. The key difference between Mapes' work and that of Bretherton and Smolarkiewicz is that the vertical heating profile used by Mapes is proportional to sin(*πz*/*Z*_{T}) − sin(2*πz*/*Z*_{T})/2. Consequently, the linear response comprises two bore waves in each direction with phase speeds *C*_{x} = ±*NZ*_{T}/*π* and *C*_{x} = ±*NZ*_{T}/2*π.* Mapes argues that the heating profile he used is more representative of that produced by MCSs.

Section 4a showed that the vertical wavelength and period of the low-frequency waves that adjust the cloud environment in the troposphere are controlled by the depth and lifetime of the convection. However, the depth and lifetime of the convection are inextricably coupled to the diabatic heating, which in turn, is dependent on the cloud circulation. This raises the question: How well do the idealized heating profiles used in theoretical studies represent that in the cloud model? We shall see that these differences have an important effect on the gravity wave generation by the cloud and the subsequent changes to the cloud environment.

*u*is the horizontal velocity,

*w*is the vertical velocity, and

*Q*is the diabatic heating rate. The (dry) buoyancy is defined as

*b*=

*g*[

*θ*−

*θ*

*z*)]/

*θ*

*z*), where

*g*is the acceleration due to gravity,

*θ*is the potential temperature, and

*θ*

*Q*

_{prescribed}is an externally imposed heat source. In contrast, the right-hand side of Eq. (4) is determined by the cloud circulation among other things.

Figures 9a and 9b show the evolution of *Q* and *wN*^{2} from the cloud model averaged over the central 10 km of the domain (i.e., over the convection). Physically, *wN*^{2} represents the adiabatic warming or cooling associated with the vertical advection of the background potential temperature. As the cloud grows, the height of the diabatic heating maximum rises. The heating lasts for about 30 min and is followed by evaporative cooling, predominantly in the middle troposphere. While both the diabatic heating and the adiabatic cooling terms have similar vertical and temporal distributions, the adiabatic cooling term is approximately half that of the diabatic heating. It is also less compact than the diabatic heating. These figures imply that the terms from Eq. (4), which are neglected in Eq. (5) are almost as large as those retained.

One quantity neglected in Eq. (5) is the vertical advection of buoyancy, *w*∂*b*/∂*z.* The evolution of this term, averaged horizontally over the central 10 km of the computational domain, is shown in Fig. 9c. The vertical advection of buoyancy has approximately the same amplitude as the diabatic heating, and has a similar compact shape. The main difference is a negative perturbation marking the top of the convective cloud, and absence of any significant cooling as the cloud decays. The key point to note is that *wN*^{2} = *Q* does not accurately describe the local response to the diabatic heating within the model cloud.

Figure 10 shows the averaged diabatic heating, the averaged nonlinear advection of buoyancy, *u*∂*b*/∂*x* + *w*∂*b*/∂*z,* and the averaged linear rate of change of buoyancy ∂*b*/∂*t* + *wN*^{2}. (Note that for ease of comparison, the diabatic heating shown in Fig. 9a is replotted in Fig. 10a.) Although the linear rate of change of buoyancy (Fig. 10b) has a similar shape to the diabatic heating (Fig. 10a), its amplitude is about one and a half times larger than the heating. This is because the contribution made by the nonlinear advective term to Eq. (4) (Fig. 10c) is at least as large (in magnitude) as the heating. Note also that the timescale of the nonlinear advective term is about half that of the other terms; the nonlinear terms have a period of approximately 10–15 min, in comparison to the other terms whose *half-period* is 10–15 min. Near the top (bottom) of the convective updraft, the nonlinear terms increase (decrease) the net effect of diabatic heating on the linear buoyancy. These nonlinear terms cannot be neglected and, therefore, the assumptions underlying the linear approximation are violated within the convective cloud.

As emphasized by Lane et al. (2001), the nonlinear advections within the cloud play a central role in generating gravity waves. The assumption that diabatic heating alone approximates the local rate of change of buoyancy is clearly violated within an isolated deep convective cloud. The buoyancy-driven circulation within the cloud acts to balance or oppose the effects of latent heating, and therefore, contributes significantly to the gravity wave generation. It follows that parameterization of the wave source should include the effects of these nonlinear motions. The relatively short timescale of the nonlinear terms suggests that they will be responsible mainly for the higher-frequency waves. The gravity waves with periods closer to the lifetime of the convection are probably associated with time-variations in the diabatic heating rate.

Pandya and Alexander (1999) completed a study in which diabatic heating derived from a nonlinear simulation was used to force a linear model. Although their spectra from the linear model had similar shapes to those from the nonlinear model, the amplitude was grossly overestimated. One possible reason for this is that the adiabatic cooling associated with the vertical advection of buoyancy was neglected in their linear model.

For simplicity, the discussion above has focused on the buoyancy equation. However, similar arguments applied to the momentum equations show that they too cannot be linearized. Finally, in this case the heating is transient, whereas in other studies (such as Bretherton and Smolarkiewicz 1989; Nicholls et al. 1991; Mapes 1993; Pandya et al. 1993) the heating is maintained and spans the depth of the troposphere; these differences will presumably affect the resulting gravity wave spectra.

## 5. Conclusions

This study has examined how convectively generated gravity waves modify the local environment of an isolated cloud, using an idealized two-dimensional, numerical model. The modeled convection generated a spectrum of gravity waves whose vertical wavelengths were harmonics of the depth of the troposphere. It was shown that the first three wave modes account for the changes in the environmental CAPE and CIN.

The fastest wave had the vertical structure of the first internal mode of the troposphere. As it radiated away from the convection, it displaced air parcels downward, which had the effect of warming and drying the atmosphere. Consequently, this mode reduced the environmental CAPE. The second internal mode increased both the CAPE and the CIN. On the other hand, large decreases in the CIN occurred later in the cloud lifetime and were produced by a slower moving disturbance with the vertical structure of the third internal mode.

The study concluded that the CAPE was affected significantly by only the first two modes. This is because the CAPE is an integral quantity calculated over a deep layer, and consequently the effect of higher modes is averaged out in the integration. Conversely, only the higher mode significantly affected the CIN as it is the only mode that produced large parcel displacements in the subcloud layer.

The maximum far-field perturbations in CAPE and CIN were approximately 15% and 33% of the initial background values respectively. These results agreed with previous studies of more organized convection, which predicted a region surrounding the convective system that favors the development of new convection. However, this region was transient, and the results suggested that any influence of gravity waves on the cloud environment occurs shortly after the development of the original cell. This influence may be particularly important in tropical regions where the CIN is relatively small. These results suggest that gravity waves generated by isolated clouds may influence the organization and generation of new convection. Previously, this possibility has only been considered for systems that already showed some level of existing organization. Further work is required to determine the net effect of the environmental modification on convective systems.

It was also shown, that within a deep convective cloud the linear approximation is violated. Nonlinear terms were shown to enhance the effects of diabatic heating near the top of the cloud, and reduce the effects of diabatic heating near the bottom of the cloud. These results showed that diabatic heating may not be the only mechanism required to accurately characterize the source of linear gravity waves.

To simplify the analysis, the results presented here were based on a two-dimensional calculation. However, in three dimensions, the waves would attenuate much more rapidly as they propagated away from an isolated cloud. Therefore, the waves would presumably have a correspondingly smaller affect on the cloud environment. Furthermore, the effects of wind shear would probably have a profound effect on the results presented here. This is a topic for future research.

This paper has focused on the relatively simple problem of how an isolated convective cloud generates gravity waves and the role played by these waves in modifying the cloud environment. This problem is an essential first step in understanding how populations of convective clouds, including MCSs, affect their environment, and whether these populations promote further convection in their neighborhood.

Finally, surface processes such as friction, sensible heat flux, and moisture flux were neglected in order to clearly focus on the role played by the gravity waves in modifying the cloud environment. However, in the atmosphere, it is likely that boundary layer fluxes are the most important mechanism modifying the CIN.

## Acknowledgments

We are grateful to Terry Clark for allowing us to use his numerical model, and we would like to thank Roger Smith for his CAPE diagnostics program.

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