## Abstract

Tropical convection is inherently multiscalar, involving complex fields of clouds and various regimes of convective organization ranging from small disorganized cumulus up to large organized convective clusters. In addition to being a crucial component of the atmospheric water cycle and the global heat budget, tropical convection induces vertical fluxes of horizontal momentum. There are two main contributions to the momentum transport. The first resides entirely in the troposphere and is due to ascent, descent, and organized circulations associated with precipitating convective systems. The second resides in the troposphere, stratosphere, and farther aloft and is caused by vertically propagating gravity waves. Both the convective momentum transport and the gravity wave momentum flux must be parameterized in general circulation models; yet in existing parameterizations, these two processes are treated independently. This paper examines the relationship between the convective momentum transport and convectively generated gravity wave momentum flux by utilizing idealized simulations of multiscale tropical convection in different wind shear conditions. The simulations produce convective systems with a variety of regimes of convective organization and therefore different convective momentum transport properties and gravity wave spectra. A number of important connections are identified, including a consistency in the sign of the momentum transports in the lower troposphere and stratosphere that is linked to the generation of gravity waves by tilted convective structures. These results elucidate important relationships between the convective momentum transport and the gravity wave momentum flux that will be useful for interlinking their parameterization in the future.

## 1. Introduction

Convective clouds in the tropics possess a variety of scales and form a crucial component of the atmospheric water cycle, heat budget, and radiative balance. In addition to these influences, organized convective clouds and the gravity waves they generate can induce a vertical flux of horizontal momentum that can have important influences on the larger-scale flow. There are two main contributions to this momentum transport. The first contribution resides entirely in the troposphere and is due to the organized circulations associated with convective systems. This *organized convective momentum transport* (OCMT) is strongly linked to tilted convective structures, can vary significantly throughout the system lifetime, and is sensitive to the regime of convective organization. The second component of this momentum transport occurs in the troposphere, stratosphere, and farther aloft and is caused by vertically propagating gravity waves generated by convection. This *gravity wave momentum transport* (GWMT) has important influences on the momentum budget of the middle atmosphere. Both of these contributions to the momentum transport are unresolved in current general circulation models and their effects should be parameterized. The focus of this study is to characterize these momentum transports associated with idealized organized convection and explore the links between the two processes.

The OCMT associated with convective systems forms an integral part of their influences on the surrounding atmosphere, the system’s longevity, and the evolution of the convective regime. Furthermore, OCMT can play a key role in the upscale growth of those systems, contributing to the circulations that maintain them and allowing them to encompass larger areas (e.g., Moncrieff and Liu 2006). Parameterized OCMT can induce accelerations of the resolved scale flow of general circulation models exceeding a few meters per second per day in magnitude, which can improve the representation of simulated winds in the tropical troposphere and correct some of the biases in the distribution of precipitation (e.g., Richter and Rasch 2008). Thus, OCMT can influence the Hadley circulation, both directly by modifying the tropospheric winds and indirectly through feedbacks associated with convective heating (e.g., Song et al. 2008; Wu et al. 2007). There exist a number of parameterizations of OCMT (e.g., Gregory et al. 1997; Wu and Yanai 1994) that can be coupled to a parent cumulus parameterization scheme. Mindful of thermodynamic and dynamical consistency, OCMT should be parameterized in all models that apply cumulus parameterization. For example, Moncrieff and Liu (2006, their Fig. 10) showed that a regional-scale simulation with parameterized cumulus convection (but no parameterization of OCMT) transported momentum in the opposite direction to the transport by explicit (convection-permitting or cloud-system-resolving model) organized convection. This fundamental dichotomy between convective momentum transport by organized and disorganized convection was also shown to occur in a global numerical weather prediction model (Moncrieff and Klinker 1997, their Fig. 8).

The accelerations associated with GWMT significantly affect the momentum budget of the stratosphere and mesosphere by influencing the meridional circulation, temperature structure, and the underlying dynamics. GWMT from tropical convection also contributes to the stratospheric quasi-biennial oscillation (Dunkerton 1997). For these reasons convectively generated GWMT should be parameterized to achieve more realistic circulation patterns [see Fritts and Alexander (2003) and Kim et al. (2003) for reviews]. Improvements in the understanding of the generation of gravity waves by convective clouds have resulted in parameterizations of GWMT generated by deep convection (e.g., Kershaw 1995; Beres et al. 2005; Chun et al. 2004). Specifically, the Beres et al. and Chun et al. parameterizations each represent convection as an idealized heat source, with the GWMT derived from the analytic solution of a linearized system. While uncertainties in this representation remain, including the effects of nonlinearity on the representation of the wave source (e.g., Lane et al. 2001; Chun et al. 2008) and the lack of sufficient observations to constrain the parameterized spectrum, these parameterizations are a vast improvement over previous methods that used an ad hoc background spectrum to represent the gravity waves.

A major challenge in the design and implementation of a parameterization of GWMT by convection is how to link the parameterization to the convective source, which itself is usually parameterized in global models. Ideally, such a parameterization should be consistent with the underlying physics of convection and convectively generated gravity waves, which are assumed to be interlinked by dynamical processes (e.g., Yang and Houze 1995). However, our present lack of understanding of convection–wave interaction and, especially, the involvement of organized convection has resulted in independent parameterizations for OCMT and GWMT. It is the purpose of this study to explore relationships between OCMT and GWMT that may be suitable for interlinked parameterizations of these processes in the future.

As demonstrated by Moncrieff (1992, hereafter M92; see his Fig. 10), OCMT is a product of vertically tilted coherent convective circulations. When such circulations have positive tilt, the horizontal and vertical velocity perturbations are positively correlated and the mean vertical flux of horizontal momentum is positive. Similarly, updrafts or downdrafts with negative vertical tilt have horizontal and vertical velocity perturbations that are anticorrelated and therefore induce a negative momentum transport. The horizontal pressure gradient is intimately associated with the vertical transport of horizontal momentum. For steady convective overturning in a vertically bounded domain the vertical integral of the divergence of momentum transport is zero. Therefore, horizontal momentum can be redistributed but the total change in horizontal momentum is balanced by the horizontal pressure gradient. It follows that both *downgradient* and *countergradient momentum* transport can occur in the convective layer, independent of the vertical shear profile [M92, his Eq. (15)]. It is the tilt of the coherent circulation and the involvement of the horizontal pressure gradient in maintaining this tilt that is ultimately responsible for the sign of the momentum transport by long-lasting convection [e.g., mesoscale convective systems (MCSs)]. The morphology and regime of long-lived convective organization is strongly affected by vertical shear (Moncrieff 1981). For example, in its early stage of development the tilt of convective updrafts in the lower troposphere involves local interaction between the low-level wind shear and precipitation-downdraft driven cold pools (e.g., Weisman and Rotunno 2004). On longer time scales, the low-level shear will evolve as a response to far-field convectively induced circulations on spatial scales larger than individual convective systems (e.g., due to acceleration of the mean flow associated with vertical momentum transport), thus providing upscale influence on the mature cloud population. Ultimately, cloud populations exhibit significant complexity and a variety of regimes can exist simultaneously (e.g., M92, his Fig. 11). These different regimes have been described at length in the literature (e.g., Houze 2004) and may be tilted upshear or downshear; this relationship may change throughout the convective life cycle (e.g., Weisman and Rotunno 2004).

An excellent example of organized precipitating convection is the squall line or MCS complete with a prominent trailing stratiform region, (e.g., Houze 2004). This observationally based conceptual model is approximated by the two-dimensional slantwise layer overturning model of M92. The MCS propagates in the direction of the low-level vertical shear vector and is associated with a backward (i.e., upshear) tilted organization. Therefore, the sign of the mean OCMT is opposite to the direction of system propagation. On the other hand, an upward propagating gravity wave has a vertical flux of horizontal momentum that is of the same sign as the intrinsic phase speed of that wave (e.g., Lane and Moncrieff 2008, hereafter LM08). It is important to be mindful that while OCMT and GWMT are inherently linked, they are the result of two dynamically distinct processes. Nonetheless, previous studies of gravity wave generation by squall lines (e.g., Fovell et al. 1992; Beres et al. 2002) have shown that the stratospheric gravity waves usually show a dominance of those waves that propagate backward relative to the traveling convective systems. Therefore, a typical squall line might be expected to induce GWMT of the opposite sign to the propagation speed of the convective system but of the same sign as the OCMT; such relationships will be examined in detail in this study.

To explore the processes underlying the links between the OCMT and GWMT we use idealized multiscale cloud-system-resolving model simulations. In contrast to studies of single systems, this multiscale framework allows fields of convective clouds to freely evolve and the cloud populations to self-organize into preferred scales and regimes, forming propagating systems and involving convectively coupled waves (e.g., Grabowski and Moncrieff 2001; Tulich and Mapes 2008). The idealized cloud-system-resolving simulations of LM08 examined the generation of gravity waves by multiscale convective systems in an unsheared environment. In those simulations the convection was weakly organized and generated a spectrum of gravity waves that was very broad in horizontal wavenumber, encompassing scales from the cloud scale to the cluster scale. The gravity wave spectrum was dominated by relatively slow horizontal phase speeds that corresponded to the approximate life cycle and spatial scale of the individual cloud systems.

This paper extends the LM08 results with attention to the effects of changes in cloud organization and propagation induced by tropospheric shear on the spectrum of GWMT and its relationship to the OCMT. To achieve this, the LM08 model configurations are extended to incorporate low-level vertical wind shear, which modifies the morphology and distribution of the convective organization and, in particular, produces tilted mesoscale convective systems associated with nonzero OCMT (e.g., M92). In all cases, the vertical shear is confined to the lower troposphere in order to limit any explicit influence of background shear on the wave generation, as was considered by Beres et al. (2002). In other words, the wind shear will only have an *implicit* influence on the wave generation through altering the regime of convective organization, notably the tilted nature of the simulated convective systems.

As in LM08 the modeled convection presented here is in a simplified two-dimensional framework. This simplification will, of course, have some influence on the gravity wave generation and the associated momentum transport signals, and presumably the transports in two dimensions exceed those that would be found in three dimensions. Lane and Sharman (2008) showed this using high-resolution cloud simulation, demonstrating a moderate increase in stratospheric gravity wave amplitude in two dimensions compared to three dimensions but strong similarities in wave character. Furthermore, Piani et al. (2000) noted that stratospheric gravity wave drag in their three-dimensional experiments was comparable to similar two-dimensional estimates.

Kingsmill and Houze (1999) examined the momentum fields in all the mesoscale convective systems observed by airborne Doppler radar in TOGA COARE. These systems contained the fundamentals of the M92 two-dimensional model. Kingsmill and Houze also explored the three-dimensional aspects of the MCSs and examined how the two-dimensional archetypal model agrees with the more complex three-dimensional natural MCSs. Kingsmill and Houze show that even though natural mesoscale systems are three-dimensional, the basic morphology of the two-dimensional approximation is upheld. Furthermore, LeMone and Moncrieff (1994) investigated the effects on the environment of quasi-two-dimensional convective bands by comparing mass and momentum fluxes and horizontal pressure changes derived from field experiment case studies to those predicted by the M92 archetypal model. This simple model predicts the vertical mass flux, the vertical divergence of the vertical flux of line-normal momentum, and the pressure change across the system, as well as the vertical transport of line-parallel momentum. M92 successfully replicates the shapes of the vertical mass flux and line-normal momentum flux profiles and correctly predicts both the magnitude and shape of the profiles for the cases in near-neutral environments (low buoyancy or high shear) and with width-to-depth aspect ratios close to the dynamically based value of 4:1. The model is less successful for systems in more unstable environments, likely because of the neglect of the generation of horizontal momentum by buoyancy in the archetypal model. The prediction of the band-parallel momentum, which assumed a periodic distribution of cumulonimbus along the bands, was fair.

Finally, many of the regimes of convective organization have been shown to form readily in both two-dimensional (e.g., Grabowski and Moncrieff 2001; Tulich and Mapes 2008) and three-dimensional (e.g., Shutts 2006) simulations. Grabowski et al. (1998) performed a detailed investigation of the effects of the third spatial dimension on three regimes of convective organization that occurred over a week-long period. They found that the *mean* thermodynamic fields, precipitation, mass flux, and the overall regime of convective organization were remarkably similar in two and three dimensions. The biggest change concerned the *evolution* of the regimes. While a vector quantity such as momentum transport will be affected by the two-dimensional approximation, it should be noted that mesoscale convective systems characteristically have a preferred linear orientation mainly as the result of the organizing effect of the continual excitation of new convection at downdraft outflow (cold pool) boundaries. In other words, despite some uncertainty in the strength of the fluxes, in our opinion the processes that link OCMT and GWMT will be represented by two-dimensional geometry well enough to capture the essence of how these processes interact. The fully three-dimensional problem will be examined in due course using the two-dimensional framework for insight.

This study has two primary objectives. The first objective is to characterize changes in OCMT and GWMT associated with the regime of convective organization associated with the lower-tropospheric wind shear. The second objective is to use these characterizations to explore the links between key features in the stratospheric gravity wave spectra and the organized convective momentum transport in the troposphere. The remainder of the paper is organized as follows: section 2 describes the numerical model, its cloud-system-resolving configurations, and the features of the modeled convection. Section 3 characterizes the gravity wave momentum transport and the organized convective momentum transport using Fourier analysis. The links between the OCMT and GWMT are discussed in section 4, and our conclusions and their future implications are summarized in section 5.

## 2. Cloud-system-resolving model simulations

### a. Model configuration

The numerical model was originally described by Clark (1977), with subsequent updates and improvements (Clark et al. 1996). It is a finite-difference approximation to the anelastic equations of motion of the form described by Lipps and Hemler (1982). The model utilizes a first-order Smagorinsky–Lilly (Lilly 1962; Smagorinsky 1963) subgrid-scale closure and an explicit treatment of moist processes via a combination of the Kessler (1969) warm rain and Koenig and Murray (1976) ice parameterizations.

The model configurations reported here are based on those described by LM08. All simulations use a two-dimensional model domain that is 2000 km long 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 farther aloft. The time step is 5 s. The uppermost 10 km of the model domain features a gravity wave absorbing layer and the lateral boundaries are periodic. The initial conditions are constructed from a single tropical thermodynamic profile derived from the 1330 UTC 26 November 1995 Tiwi Islands, Australia, sounding (see LM08). Convection is maintained by constant surface fluxes of latent heat (100 W m^{−2}) and sensible heat (10 W m^{−2}) and imposed cooling of 2 K day^{−1} below 9.5 km that decreases linearly in magnitude to be 0 K day^{−1} at 15.5 km. All the simulations are run for 180 h.

The sensitivity of the modeled convection to domain size and grid spacing are discussed in LM08, and for this reason these same tests are not conducted herein and all cases use the same domain geometry. Specifically, LM08 found that the gravity wave spectra were insensitive to domain size but there was some sensitivity to model grid spacing. Finer-resolution simulations were qualitatively similar but showed a decrease in the horizontal phase speed of the modeled gravity waves, consistent with a reduction in horizontal wavelength associated with smaller-scale convective structures. Nonetheless, it should be noted that resolution convergence will likely not be achieved until the grid spacing approaches very fine scales *O*(100 m) (e.g., Lane and Knievel 2005).

The difference between the calculations reported here and those reported by LM08 is the presence of lower-tropospheric wind shear. We employ a wind profile to promote changes in convective mode such as squall lines (e.g., Weisman and Rotunno 2004). In these calculations a constant wind shear near the surface is imposed at initial time such that

where *U*(*z*) is the initial wind profile, *U*_{0} is the initial surface wind speed, *z* is height above the surface, and *h* is the depth of the initial shear layer. This profile with nonzero surface wind and zero wind aloft was designed to simplify the analysis of gravity waves (the absence of surface friction and the constant surface fluxes means that this system is Galilean invariant). Furthermore, recalling the introduction, the absence of shear above 2.5 km is designed to remove any direct role the shear may have on the wave generation or dissipation. The set of wind configurations described in this study is listed in Table 1, with all cases possessing positive low-level shear. After initiation the domain-averaged wind is permitted to evolve (i.e., it is not relaxed toward this initial wind profile), and there are minor changes in the mean wind profile throughout the simulations (discussed in more detail later).

### b. Regimes of convective organization

The time evolution of the overall convective activity behaves similarly in all simulations and resembles that described by LM08 for unsheared mean flow. Small and shallow cumulus clouds develop shortly after model initialization. The cloud field gradually deepens and after approximately 10 h contains clouds with tops above the middle troposphere. However, throughout the simulations the convection remains relatively weak, which is characteristic of convection over the tropical oceans, with maximum updraft strengths between 5 and 10 m s^{−1} and many cloud tops around 10–12-km altitude. For the most part, the strongest convective updrafts occur after about 20 h and become progressively weaker until a quasi-steady state is attained after about 100 h. As expected, the principal difference between the simulations with different wind shear occurs in terms of the dynamical structure and propagation of the regimes of convective organization.

Figure 1 shows the time evolution of the total-column cloud field (cloud water plus ice) for four simulations: *h* = 2.5 km and *U*_{0} = 0, −5, −10, and −15 m s^{−1}, referred to as the U00, U05, U10, and U15 simulations, respectively (see Table 1). Figure 1a (U00, one of the simulations described by LM08) shows that in the absence of background shear the convection eventually self-organizes into cloud clusters that become longer lived as time progresses. While many of these cloud systems propagate, their propagation speed is relatively slow (∼5 m s^{−1}) with no preferential direction of propagation, so they can be considered as weakly organized.

For those simulations involving low-level shear (Figs. 1b–d; U05, U10, U15), distinct regimes of convective organization gradually emerge. Each simulation contains short-lived clouds that travel close to the surface wind speed as well as larger long-lived cloud systems that remain approximately stationary and eventually merge into large cloud clusters. Because the clouds in these three simulations are qualitatively similar, for the sake of brevity our discussion is focused on the U10 simulation (Fig. 1c).

Figure 1c shows that the U10 simulation undergoes at least three distinct stages of evolution. In the first stage the cloud field is dominated by the small, short-lived clouds organized into coherent lines in *x*–*t* space and traveling in the negative horizontal direction. The first of these lines begins at approximately *x* = 2000 km, *t* = 10 h and travels across the domain in approximately 45 h (i.e., 12 m s^{−1} in the negative *x* direction). As time progresses this propagation speed increases by a few meters per second. We refer to these clouds as the *upshear propagating regime* because they travel upshear (negative *x* direction) and the inflow in the frame of reference traveling with the system is unidirectional (positive *x* direction). This regime of shallow convective organization was modeled by Moncrieff and Liu (1999; see our Fig. 1b).

The upshear propagating regime dominates the cloud field until approximately 70 h when the second stage of cloud evolution begins with the emergence of a longer-lived *quasi-stationary regime* of convective organization (i.e., quasi-stationary in the frame of reference of the initial wind profile). These quasi-stationary regimes are well-defined contiguous cloud patterns that last for tens of hours, broaden to encompass almost 100 km in the horizontal, and remain approximately stationary in this frame of reference. During this second stage, which lasts until approximately 120 h, the quasi-stationary regimes and the upshear propagating regimes coexist. After 120 h the final stage of the cloud evolution begins, with the formation of a large cloud cluster wider than 500 km. The envelope of this cluster moves in the negative direction at a speed slower than the earlier upshear propagating regime, and therefore slower than the surface wind speed. The cluster is a merger of adjacent quasi-stationary regimes. Around the time of its formation the upshear propagating regime is suppressed and by the end of the simulation is no longer evident. The other simulations with shear (U05 and U15) evolve similarly. The primary difference is the speed of the upshear propagating regimes (which move faster in U15 and slower in U05) and the timing of the emergence of the quasi-stationary regime and the larger-scale clusters (which occur earlier in the U15 simulation).

Figure 2 demonstrates the variety of convective structures present in the U10 simulation at any given time. Figure 2a shows a region of the simulation that is comprised of shallow clouds (heights less than 4 km) with a distinct positive (downshear) tilt with height (i.e., in the same direction as the shear vector). These small clouds are characteristic of the upshear propagating regime described above. Also in Fig. 2a is an approximately upright deep cloud (at *x* = 815 km) except a weak anvil-like structure extends in the positive direction. Figure 2b shows a region demarked by deeper convection. This regime of organization is distinguished by its negative tilt (opposite to the low-level shear vector) and the marked anvil-like structure extending toward the negative direction. Furthermore, a new cloud is deepening at approximately *x* = 1825 km. These properties of this deep cloud are consistent with the upshear tilted, trailing stratiform convective archetype (e.g., M92) that models the most common squall lines (e.g., Parker and Johnson 2000; Houze 2004). This deep cloud is characteristic of the quasi-stationary regime described earlier. In summary, although the cloud population and the convective organization within that population are complex, their morphology is described by convective archetypes discussed in the literature, many of which coexist.

### c. Regimes of momentum transport

To evaluate the collective effect of the cloud population on the large-scale flow, the vertical flux of horizontal momentum is calculated for the U10 simulation. These fluxes are calculated from hourly model data using a simple horizontal mean of the product of the horizontal and vertical velocity perturbations (multiplied by the background density), that is, *ρu*′*w*′; the background wind field used to define these perturbations is calculated every hour. The mean momentum transport for three 50-h time periods (20–70, 70–120, and 120–170 h) is shown in Fig. 3a. These three time periods correspond to the three stages of cloud system evolution described earlier. The momentum transport shown in Fig. 3a comprises a combination of OCMT and GWMT. For the purposes of discussion, it is reasonable to assume that within the convectively active regions (*z* < 10 km) the momentum transport is primarily associated with OCMT whereas in the stratosphere (*z* > 16 km) it is almost entirely GWMT. (In the next section, spectral analysis will make this separation more rigorous). The single positive peak in momentum transport near the surface in Fig. 3a is related to resolved momentum fluxes in the surface layer that are mostly balanced by the parameterized fluxes associated with turbulent mixing and surface fluxes of sensible heat. These fluxes in the surface layer will not be considered further.

Figure 3a highlights a systematic change in the vertical structure of the momentum transport above the surface layer through the three stages of cloud evolution described above. At early time (20–70 h) the momentum transport is positive throughout the troposphere and stratosphere. The momentum transport generally increases from the surface until it maximizes above the shear layer at approximately 3-km altitude, then decreases with altitude until approximately 6 km, and is approximately constant farther aloft (with the exception of a slight reduction at the tropopause). In the second stage of evolution (70–120 h) the momentum transport is weakly negative. In the final stage of evolution (120–170 h) the near-surface transport is strongly negative, decreases in magnitude until approximately 5 km, contains a shallow layer of positive transport, and then maintains a negative sign above 8 km.

The positive momentum transport that is characteristic of the first stage of evolution represents an upward transport of positive momentum, and within the shear layer the transport is *countergradient*; that is, the environmental shear is enhanced as a result of momentum transport due to the downshear (positive) tilt of the organized convection (Moncrieff and Green 1972). Above the convection, the positive momentum transport is characteristic of a gravity wave spectrum that is dominated by upward propagating gravity waves with positive intrinsic phase speed. The negative momentum transport during the final stage of evolution is *downgradient* (i.e., the opposite sign from the shear). Such transport can arise from a combination of simple mixing within the shear layer, organized convection tilted backward into the shear as in squall lines (M92), and/or a gravity wave spectrum dominated by upward propagating gravity waves with negative intrinsic phase speed.

Under the assumption that the momentum transport in the lower troposphere is predominantly associated with OCMT and in the stratosphere is associated with GWMT, it is notable that the GWMT has the same sign as the OCMT in the lower troposphere. This result is consistent across all of the simulations with shear (U05, U10, U15) and will be discussed in more detail in the next section.

The organization of convection during the three stages of evolution is consistent with the profiles of domain-averaged momentum transport. During the first stage, the dominant shallow upshear propagating regime features a positive (downshear) tilt consistent with the positive convective momentum transport. The positive tilt originates within the shear layer but continues to transport momentum above that shear layer where the transport attains its maximum value. In the two later stages, which are dominated by the deeper quasi-stationary organized regime and the large cluster, the negative (upshear) tilt is consistent with the negative momentum transport. There is an intermediate period when the upshear propagating and quasi-stationary regimes coexist, contributing positive and negative momentum transports, respectively. (A similar scenario is depicted by M92’s Fig. 11, where shallow downshear and deep upshear convective systems coexist). An approximately equal contribution from these two regimes explains the near-zero momentum transport for the second period (70–120 h). Furthermore, the magnitude of the momentum transport during 120–170 h is smaller than during 20–70 h, likely because the mean convective activity is weaker at these later times.

### d. Role of convectively generated large-scale circulations and shear

Large-scale circulations and the associated shear profiles are an important part of mesoscale convective dynamics, especially the evolution of the vertical tilts that are vital to the OCMT distribution. To examine the evolution of these larger-scale circulations a perturbation streamfunction *ψ* is calculated from the standard equation ∂*ψ*/∂*z* = *ρu*′, where *ρ* is the density and *u*′ is the perturbation horizontal velocity. The time-averaged streamfunction for each of the three periods of convective evolution in U10 is shown in Fig. 4. In the first stage of evolution (Fig. 4a) these circulations show little coherence and are dominated by perturbations on scales less than ∼100 km. During this period the vertical tilt of the cloud systems and the associated momentum transports are controlled primarily by the mean shear rather than the larger-scale circulations. Between 70 and 120 h (Fig. 4b), the tropospheric circulations increase in scale and show further evidence of multiple regimes of convective organization. For example, at approximately 700 km the small oscillations of the streamlines near the surface are consistent with shallow clouds, and at approximately 900 and 1300 km the upright streamlines near the surface are characteristic of upright convection (cf. Fig. 2a). Tilted circulations also emerge, with ascent at approximately 1700 km being directed in the negative horizontal direction, contributing to negative OCMT (cf. Fig. 2b). A coherent circulation pattern emerges during the third stage of evolution (Fig. 4c), which is dominated by the large organized cluster between 1000 and 1500 km. The ascending branch of this circulation between 1000 and 1500 km and below 5-km altitude is mostly tilted in the negative direction (i.e., negative OCMT). Moreover, the descent around 500 km that acts to suppress convection away from the convectively active region, is directed in the positive direction and therefore also contributes to the negative OCMT below about 5 km. Above 5 km and below 8 km the circulations have the opposite tilt (e.g., at 1400 km), causing the shallow layer of positive OCMT in Fig. 3a. Upright structures present at this time, for example at 1000 and 1800 km, make a zero net contribution to the OCMT. These meso-*α*-scale circulations are the result of evolving feedback causing local variations in the shear, which promotes organization and further upscale growth of the cloud population. It is clear from Fig. 4c that this circulation is limited by the computational domain, suggesting the need for a larger domain.

Finally, it is useful to consider the net effect of the momentum transport on the mean horizontal wind. Figure 3b shows the initial wind profile and the domain average wind profile at 170 h. Consider the momentum transport from 20 to 70 h associated with the upshear propagating regime (Fig. 3a). Within the shear layer (*z* < 2.5 km) the momentum transport generally increases with height, leading to a negative effect on the mean flow; above the shear layer the momentum transport decreases with height until about 6 km altitude, leading to a positive mean flow effect at these heights. Thus, within the shear layer the momentum transport accelerates the mean flow, which is consistent with the increase in surface wind speed (Fig. 3b). (Note that this increase in surface wind is responsible for the slight progressive increase in propagation speed of the upshear propagating regime). Above the shear layer the momentum transport acts to accelerate the wind in the positive direction, resulting in the slight increase in wind speed at around 4 km. During the final period of evolution (120–170 h) the effects become more complicated. Nevertheless, the effect of the negative momentum transport in the shear layer is the same sign as in the early stages [because the nonzero near-surface (∼500 m) negative momentum transport allows a positive vertical gradient], causing a further increase in the magnitude of the surface winds.

## 3. Characterization of momentum transport

### a. Spectral analysis

To investigate the signals that contribute to the mean momentum transport presented in the previous section, spectral analysis is conducted on the simulated velocity fields. In each case, the momentum transport is examined at two altitudes. The first altitude is 2.5 km (i.e., the top of the initial shear layer in those simulations with shear), which should be associated with the OCMT. The second altitude is 20 km in the stratosphere where the momentum transport is almost entirely due to GWMT. At each altitude the momentum transport is determined from the cospectrum of the horizontal and vertical velocity perturbations. The cospectrum is a scale-dependent measure of the covariance of the in-phase component of two variables (see Jenkins and Watts 1968). To obtain the cospectrum, the amplitude spectrum of each of the two variables is determined using Fourier analysis and the cross-spectrum is determined from the product of one of these amplitude spectra by the complex conjugate of the other; the cospectrum is the real component of this cross-spectrum. Such an analysis is common for gravity waves (e.g., Beres et al. 2002) but to the best of our knowledge has not previously been applied to OCMT.

Two-dimensional cospectra of the simulated horizontal and vertical velocity perturbations are constructed using 2-min data for the entire horizontal domain. Multiplied by the density, this procedure provides two-dimensional, frequency *ω*–horizontal wavenumber *k* spectra of *ρu*′*w*′. These spectra are simplified by allocating each spectral component into 1 m s^{−1} bins determined by the value of *ω*/*k*, the horizontal phase speed *c*. See LM08 for an additional description of this procedure.

#### 1) Unsheared flow

First consider the momentum transport spectrum for the simulation without mean shear, U00 (Fig. 5). The spectrum at 2.5 km (Fig. 5a) shows the net momentum transport as well as the positive and negative contributions to that net transport at each value of *ω*/*k*. At this altitude the net momentum transport spectrum is almost exactly antisymmetric around *ω*/*k* = 0. By definition, the integral of the *ω*/*k* spectrum is equal to the horizontal mean of the momentum transport and therefore the total momentum transport at 2.5 km is approximately zero. At each value of *ω*/*k* there are negative and positive contributions to the net momentum transport. For example, the net transport for stationary signals (*ω*/*k* = 0) is zero and consists of a balance between positive and negative momentum transports. Such a balance is consistent with stationary and upright convective systems that have zero net momentum transport: the patterns of inflow (and outflow) of such systems are horizontally symmetric about the convective core and contain momentum transports of opposite sign. The strongest net transport (magnitudes) are found at *ω*/*k* = ±2 m s^{−1}. This is consistent with slowly traveling convective systems, such as those present in Fig. 1a, which possess a tilt with height that is opposite to their propagation direction, analogous to the M92 trailing stratiform organized convective archetype. We refer to these slow and stationary momentum transport signals as the *quasi-steady response*.

At 20 km the gravity wave spectrum (Fig. 5b) is also antisymmetric about *ω*/*k* = 0, with a net transport equal to zero. This spectrum is discussed in detail by LM08, but it is important to note that this spectrum peaks at *c* = *ω*/*k* = ±5–6 m s^{−1} (i.e., at speeds faster than the peaks in OCMT). In this spectrum only the net transport at each phase speed is shown because there are no contributions from signals of opposite sign to the phase speed (i.e., the entire signal at 20 km is consistent with upward propagating waves).

The fact that upward propagating gravity wave packets have a momentum flux of the same sign as the (intrinsic) horizontal phase speed allows further interpretation of the momentum transport signal at 2.5 km. The momentum transport signal at each phase speed is separated into the positive and negative contributions to the net momentum transport at 2.5 km (Fig. 5a); these contributions are assigned a line style to depict whether they are consistent or inconsistent with upward propagating gravity waves. This separation demonstrates that there is a contribution to the momentum transport signal that is consistent with an upward propagating gravity wave (thin solid line in Fig. 5a) and the magnitude of this transport has local maxima at *c* = *ω*/*k* = ±4 m s^{−1}. However, the sign of the net momentum transport is inconsistent with an upward propagating gravity wave at all values of *ω*/*k*. While this is consistent with a downward propagating gravity wave, we argue that the majority of the quasi-steady momentum transport at 2.5 km arises from unstable convective motions.

To further examine the vertical structure of the contributions to the momentum transport, the spectra at heights of 5, 7.5, and 10 km are shown in Fig. 6. (The spectra are almost exactly symmetric about *ω*/*k* = 0 and therefore only half of the spectra are shown). These three spectra and Fig. 5 illustrate the transition between the lower troposphere, where the net momentum transports are dominated by signals that are inconsistent with upward propagating gravity waves, and the upper troposphere and stratosphere where upward propagating gravity waves dominate. For *ω*/*k* > 0, as height increases the relative amplitude of the negative contribution decreases relative to the positive contribution (and vice versa for *ω*/*k* < 0), with the strongest relative reductions at larger values of |*ω*/*k*|. By 10-km altitude (Fig. 6c) the signal is almost entirely consistent with upward propagating gravity wave packets, except for positive and negative signals at *ω*/*k* = 0 that balance. The result is that the net momentum transport signal at each value of *ω*/*k* eventually changes sign between the lower troposphere and altitudes above the convectively active region. As mentioned above, within the convectively active regions most of the fluxes are likely associated with unstable convective motions and not gravity waves (cf. Kershaw 1995). Yet, Figs. 5 and 6 demonstrate that the convective momentum transport signal contains a component that strongly resembles the stratospheric gravity wave spectrum farther aloft, adding further credence to the idea that these processes are intimately linked.

#### 2) Sheared flow

To explore the changes in the characteristics of the OCMT and the GWMT associated with low-level shear and changes in the convective regime, the U10 simulation results are analyzed in the same way as the U00 simulation above. The 2.5- and 20-km spectra are shown in Figs. 7 and 8, respectively, and to highlight the changes associated with convective regime changes during the three stages of evolution discussed earlier they are analyzed separately over three 50-h periods. Also shown in each panel is the integral of the momentum transport for each time period. (Note that the value of these integrals are consistent with the profiles shown in Fig. 3 but not identical because the phase speed spectra are calculated from 2-min data and the domain average profiles are calculated from hourly data).

The changes to the 2.5-km spectrum associated with the introduction of shear are subtle, yet important. The net momentum flux is still dominated by a quasi-steady signal, and the sign of the total momentum flux is, in part, related to the relative amplitudes of the two peaks in the quasi-steady signal at the positive and negative values of *ω*/*k*. An additional feature is present when *ω*/*k* is less than approximately −7 m s^{−1}, extending to approximately *ω*/*k* = −15 m s^{−1} in this case. We call this spectral tail the *transient response*, which is not present in the unsheared simulations.

During the first period of the cloud evolution (Fig. 7a) the net momentum transport at 2.5 km is positive, associated with an enhancement of the positive component of the quasi-steady signal at *ω*/*k* = −2 m s^{−1}, representing either stronger or more prevalent slow convective structures with positive tilt. A weakly positive signal also emerges as part of the transient response at *ω*/*k* = −14 m s^{−1}. This positive momentum transport signal is consistent with the upshear propagating regime that propagates in the negative direction at this approximate speed and possesses a positive tilt. The propagating regime becomes less prevalent as time progresses (Fig. 7b); the amplitude of the positive quasi-steady signal (at *ω*/*k* = −2 m s^{−1}) reduces to be almost in balance with its negative counterpart (at *ω*/*k* = 2 m s^{−1}). By the final stage of convective evolution (Fig. 7c) the net momentum transport is, however, negative. This negative dominance arises from two changes in the spectrum. The first change is an enhancement of the negative momentum transport for stationary signals (*ω*/*k* = 0), consistent with the dominance of the negative tilt of the quasi-stationary regime. The second change in the momentum transport is associated with the transient response at (*ω*/*k* < −7 m s^{−1}): the positive contributions at these speeds weaken considerably and the entire spectral tail has a net negative momentum transport by the third stage of convective evolution. We postulate that these negative transient signals are associated with updrafts with negative tilt that usually propagate rearwards within the quasi-steady organized circulation. As can be seen in Fig. 7, however, the transient response always contains a positive contribution (associated with positive tilted structures). Yet, as the quasi-stationary regime becomes more prevalent, the signals associated with negative transport/tilt dominate the transient response. At higher altitudes (e.g., 5 km), the transient signal remains a significant feature of the spectrum (Fig. 9a). At this altitude the positive contribution to the transient signal is almost negligible, presumably because the positively tilted, upshear propagating regime is shallow and does not penetrate significantly above the shear layer. By 10-km altitude (Fig. 9b), as in the U00 case, those signals inconsistent with upward propagating gravity waves have almost disappeared. Positive and negative steady signals remain; however, at this time they do not exactly balance because of the domain-scale circulation that has evolved.

There are important differences between the gravity wave spectrum for the U10 simulation (Fig. 8) compared to the spectrum for the U00 simulation (Fig. 5b), and the U10 spectrum also evolves as the convective regime changes. First, the shape of the spectrum is no longer symmetric: the spectrum of gravity waves with positive phase speed is of a similar shape to U00, but the spectrum is much broader at negative phase speeds with an enhancement in magnitude at phase speeds between −10 and −20 m s^{−1}. While the respective shapes of the positive and negative components are consistent for each of the three stages, the amplitudes change. During the first stage of evolution (Fig. 8a) the maximum momentum transport at positive phase speeds is approximately double (in magnitude) the maximum at negative phase speeds. (Recall that during this time the convection is dominated by the upshear propagating regime with positive momentum transport and tilt.) As time progresses the peak of the positive momentum transport weakens to become similar to the negative signal, yet because the negative part of the spectrum is broader the total momentum transport is negative. (Note that there are also changes to the amplitude of the spectrum associated with changes in convective intensity.) Thus, for the mature cloud population that is dominated by the quasi-stationary regime, the gravity wave spectrum has broadened toward faster negative phase speeds. These faster phase speeds that are responsible for the asymmetry in the gravity wave spectrum are also those phase speeds that characterize the transient response in the OCMT transport field in the lower troposphere. This is a key result. It suggests a direct link between the tilted structures that are responsible for the transient OCMT signal and the contributions to the gravity wave spectrum that define its asymmetry.

This picture developed from the U10 simulation is consistent across the other simulations with shear shown in Fig. 1. Figure 10 shows the net momentum transport at 2.5 and 20 km for the U05 and U15 simulations between 120 and 170 h. These simulations show that at 2.5 km the mature momentum transport spectra (Fig. 10a) are similar to those in Fig. 7c for the U10 simulation. Notably, both spectra evince a transient response with a negative momentum transport. The width of the transient response in *ω*/*k* space is related to the strength of the low-level shear/wind speed [i.e., the stronger shear simulation has transient signals that are faster (in magnitude)]. The time evolution of the 2.5-km spectrum is slightly different between the cases, with the U05 simulation maintaining a positive transient response until 120 h and the U15 simulation shows the negative transient signal to be dominant at an earlier time (not shown). These differences in evolution are explained by the timing of dominance of the quasi-stationary regime being earlier in the simulations with larger low-level shear. Similarly, the gravity wave spectra at 20 km (Fig. 10b) demonstrate a broadening toward the faster negative phase speeds as the low-level shear/wind speed is increased (in magnitude). As in the U10 simulation, this broadening in the gravity wave spectrum coincides with the phase speeds of the transient response farther below, providing further evidence that the two are linked.

Importantly, the negative momentum transport associated with the transient response at 2.5 km is consistent with upward propagating gravity waves. It is possible that the asymmetry in the gravity wave spectrum is simply the result of these signals propagating from 2.5 km to the stratosphere. At 2.5-km altitude, however, upward propagating gravity waves could arise only if (i) there were wave sources below 2.5 km or (ii) they were the result of downward propagating waves reflecting off the lower boundary. First, it is unlikely that a wave source below 2.5 km would have a strong influence on the stratospheric gravity wave field, especially because these waves would need to propagate upward through the convectively active troposphere. Second, signals with comparable amplitude corresponding to downward propagating waves (positive momentum transport) are not present at these phase speeds, and if they were the net momentum transport would be close to zero. In conclusion, it is likely that these transient signals in OCMT are associated with unstable convective updrafts with negative tilt, as discussed further in the next section.

## 4. Synthesis and discussion

The spectral analyses of the multiscale cloud simulations have revealed important information about the characteristics of the OCMT in the troposphere and its variability associated with convective regime. In particular, in the absence of background shear the convection does not show any preferred direction of propagation and is weakly organized. The OCMT in the weakly organized case consists of a quasi-steady signal containing positive and negative momentum transports, which when integrated across all propagation speeds provides zero net vertical momentum transport.

When low-level wind shear is included in the simulations the convective regime changes as a result and the characteristics of the OCMT change in unison. In sheared conditions the quasi-steady response is still present, but the integral effect of this response is not necessarily zero. For example, early in the U10 simulation the positive momentum transport within the quasi-steady response was dominant, consistent with the dominance of propagating structures with positive tilt (Moncrieff and Liu 1999; see their Fig. 1b). Later in the U10 simulation, the positive momentum transport within the quasi-steady response weakened and a stationary signal with negative momentum transport developed, consistent with the presence of stationary structures with negative tilt. In addition to the quasi-steady response, the addition of low-level shear introduced a transient signal in the momentum transport. Early in the simulation, the transient response propagated and possessed a weakly positive momentum transport. Later in the simulations, the transient response was entirely negative and encompassed a relatively broad range of negative phase speeds. The net result is that the momentum transport (OCMT) is positive in the early stages of evolution of the population and mostly negative in the later stages of evolution.

These changes in the sign and characteristics of the OCMT are consistent with the modeled convection and the organized convective systems described by many previous studies. In the early stages of evolution, when the OCMT is positive, the weak and positive transient response stems from the shallow upshear propagating regime. In addition, the positive bias of the quasi-steady signal is consistent with downshear tilted, slow-moving deep clouds. Such downshear tilted clouds are consistent with Weisman and Rotunno’s (2004) description of the early stage of squall-line evolution (e.g., see their Fig. 2a) when precipitation-driven cold pools are poorly developed and the tilt of the system is predominately downshear (e.g., Fig. 11a; Moncrieff and Liu 1999) and associated with positive OCMT and countergradient momentum transport (Moncrieff and Green 1972).

The long-lived upshear tilted systems propagate into the low-level wind and in the U10 wind profile are quasi-stationary. This is consistent with the strengthening of the stationary negative momentum transport in the later stages of the simulation (Fig. 7c). The dynamics of the upshear tilting systems are explained by the M92 steady-state model. In this model the work done by the horizontal pressure gradient is a key quantity which directly affects the OCMT.

As described by many authors (e.g., Lafore and Moncrieff 1989; Fovell and Tan 1998; Lin et al. 1998; Yang and Houze 1995), long-lived upshear tilted squall lines and MCSs commonly consist of several convective updrafts initiated at the leading edge of the cold pool and traveling rearward relative to the squall system. These rearward-moving updrafts are embedded within the larger-scale quasi-steady circulation and their tilt is usually defined by that circulation (i.e., the schematic diagram in Fig. 12). It is likely that the transient response with negative phase speed and negative momentum transport, which are present late in all simulations in shear flow (e.g., Figs. 7c and 10a), are related to these rearward traveling updrafts embedded within the quasi-stationary system. In other words, the quasi-stationary regime of organization possesses multiple scales of motion, all of which contribute to OCMT of the same sign, namely regions of mesoscale ascent and descent, precipitation-driven downdrafts, and transient updrafts.

As the environment and the cloud population evolve during the simulation, several regimes of convective organization coexist. In the early stages the clouds are less organized and are relatively short lived, consistent with weak cold pools. In the later stages, the cloud systems organize into long-lived mature squall lines or MCSs. In the intermediate stages of development, the net effect of the clouds on the momentum transport is close to zero, consistent with upright convective clouds (e.g., Fig. 11b). This is likely due to the fact that upshear and downshear tilted clouds coexist within the population (e.g., Fig. 2) and their respective contributions to the OCMT approximately cancel at these times.

From the examination of the momentum transport spectra it is also clear that there are links between the properties of the OCMT spectrum and the asymmetries that emerge in the GWMT spectrum aloft. In the absence of shear, the gravity wave spectrum is symmetric, and the features of this symmetric spectrum occur in all simulations with shear. In the sheared simulations, however, the most prominent feature of the gravity wave spectra is a spectral broadening toward negative phase speeds, coinciding with the phase speeds of the transient OCMT that we associate with tilted updrafts embedded within the convective system. We argue that the gravity waves generated by these transient features generate the asymmetry in the wave spectrum, defining an important link between the OCMT and GWMT. It is likely that these gravity waves are generated via the mechanism described by Fovell et al. (1992). In addition, Yang and Houze (1995) highlighted the direct connection between rearward propagating updrafts and the gravity waves farther aloft. Yang and Houze argued that the updrafts themselves were trapped gravity waves. However, our spectra (at 2.5 km; Figs. 7 and 10a) are inconsistent with vertically trapped gravity waves because the net momentum transport associated with the transient response is nonzero. We argue it is more appropriate to ascribe these transient signals to unstable convective structures. Finally, an additional asymmetry in the gravity wave spectra is also present earlier in the simulations, when the positive GWMT is enhanced in amplitude relative to the negative GWMT (e.g., Fig. 8a). This asymmetry occurs during the stage of convective evolution that has positive OCMT and is therefore dominated by convective systems with positive tilt.

For the most part, the analysis of OCMT and GWMT shows that the integral of these two transports have the same sign, suggesting that convective systems preferentially generate gravity waves propagating in the same direction as the tilt of the convective system. The dynamics underlying this process was illustrated by Fovell et al. (1992), who showed that a tilted momentum source akin to a convective updraft would generate a spectrum of waves propagating in both directions, but those waves propagating in the direction of the tilt were stronger. (This directional spectral bias arises because the tilted source projects more efficiently onto the waves propagating in the direction of the tilt.). Fovell et al.’s result is consistent with our finding that the sign of the mean OCMT is dominated by gravity waves that propagate in the same direction as the tilt of the convection.

To illustrate this point, the organized cloud with negative tilt shown in Fig. 2b is further considered in Fig. 13. As this cloud develops (at 66 h; Fig. 13b) the streamlines demonstrate the profound negative tilt of the convective core, which is marked by the upward pointing arrow aligned at 65° from the horizontal. An hour later, once a number of periods of gravity waves have been generated (Fig. 13a), the vertical velocity identifies a dominance of gravity waves with negative tilted phase lines (marked on figure) directly above the developing storm. Consistent with Fovell et al.’s discussion, positive tilted gravity waves are noticeably weaker. Moreover, the temporal evolution of the vertical velocity in the lower stratosphere above the storm (Fig. 14) demonstrates numerous coherent waveforms propagating at *c* = −11 m s^{−1}, and at least one wave propagating at *c* = 7 m s^{−1}. Incidentally, these phase speeds are close to the peaks in the domainwide gravity wave spectrum at 20 km (Fig. 8), demonstrating that important features in the spectrum of the cloud population can be explained using arguments based on individual systems.

Finally, despite the complexity of the simulated cloud population, the consistency of sign between the OCMT and GWMT is a robust property of our multiscale simulations. The relationship among the convective regime, gravity wave generation, and the mean profile of momentum transport is simply schematized by Fig. 11. This schematic is consistent with the time evolution of gravity wave generation in previous studies of isolated squall lines (e.g., Fovell et al. 1992; Beres et al. 2002). This schematic also represents the net effects of the multiscale cloud population throughout the three stages of evolution simulated herein. It is important to note that the consistency in sign of the OCMT and GWMT may not be generally applicable in environments with middle or upper tropospheric wind shear, which could suffer significant influences from the effects of that shear on wave propagation/dissipation.

## 5. Conclusions and future outlook

Idealized two-dimensional multiscale cloud-system-resolving model simulations were used to examine the characteristics of organized convective momentum transport (OCMT) and gravity wave momentum transport (GWMT) and to determine linkages between these two fundamental dynamical processes. Characteristics of the momentum transports were determined using spectral analysis, and the sensitivity of those spectra to variations in convective regime was explored by analyzing simulations with different low-level wind shear.

All simulations (with shear) showed variability in the regime of convective organization as the cloud population evolved. The net effect of the population is consistent with downshear tilted clouds at early times and upshear tilted quasi-stationary systems at later times. Accordingly, the wave spectrum was dominated by gravity waves with positive phase speed at early times and negative phase speeds at later times. These variations illustrate the importance of the evolution of the convective regime on the sign of the momentum transports: the sign of the net GWMT and the net OCMT are mostly consistent. Furthermore, the OCMT separates into a quasi-steady response and a transient response, the latter being consistent with rearward-traveling updrafts embedded within upshear tilted organized convective systems. These aspects explain the asymmetry in the spectrum of GWMT aloft and define a clear link between OCMT and GWMT.

The strong dynamical linkages between OCMT and GWMT imply that progress can be made by deriving consistent parameterizations of OCMT and GWMT in the future and also evaluating the internal consistency of such parameterizations in global models. Our simulations were idealized by design and were based on two-dimensional geometry to simplify the physical interpretation of the results. We have taken a first step toward quantifying important linkages between OCMT and GWMT, laying a path toward consistent parameterizations of organized precipitating convection and convectively generated gravity waves and defining a methodology and theoretical basis for future parameterization developments. Based on this progress, work is underway to explore the organized convection and gravity wave problem in the more realistic framework of three-dimensional geometry.

## Acknowledgments

TPL acknowledges support from the Australian Research Council Future Fellowships (FT0990892) and Discovery Projects (DP0770381) Schemes, and Rit Carbone and NCAR/TIIMES for funding his travel to NCAR in 2008 to conduct this research. We also thank Terry Clark for supplying his numerical model and Tiffany Shaw for useful discussions.

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## Footnotes

*Corresponding author address:* Todd Lane, School of Earth Sciences, University of Melbourne, Melbourne, VIC 3010, Australia. Email: tplane@unimelb.edu.au

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