## 1. Introduction

During the past few decades, a remarkable amount of attention has been given to mesoscale convective systems (MCSs), a key category of organized precipitating convection in both midlatitude and tropical regions. This has involved field campaigns, satellite measurements, high-resolution numerical simulations, and theoretical models of propagating mesoscale circulations. Review papers by Houze (2004) and Moncrieff (2010) describe some of the excellent progress that has been accomplished in our understanding and observational validation of the dynamics, transport properties, and scale interactions associated with MCSs.

The lifting to saturation of the planetary boundary layer air by cold pools (density currents) is a well-known physical–dynamical mechanism for initiating and maintaining long-lived propagating squall lines and MCSs. In that context, the outflow from evaporation-cooled downdrafts that transport air with low equivalent potential temperature downward to Earth’s surface is a primary, albeit not the only, category of density current. The effectiveness of the evaporative cooling depends critically upon the degree of subsaturation of the boundary layer, although convective downdrafts do contribute to cold pool formation even when the boundary layer is near saturation. This two-part paper will show that a near-saturated boundary layer can significantly change not only the density current kinematics but also the morphology of the entire mesoscale system in which the density current is embedded.

Indeed, there have been significant advances in our knowledge of long-lived mesoscale convective systems, squall lines in particular, in terms of the interactions between the background low-level shear and strong surface-based cold pools. Such interactions can give rise to a range of convective archetypes (e.g., Parker and Johnson 2000, 2004), with the front-fed trailing-stratiform squall-line archetype being the quintessential example (Houze 2004). Rotunno et al. (1988) devised a theory for such systems, now known as the Rotunno–Klemp–Weisman (RKW) theory, which explains the dynamics underlying their longevity [see also Weisman and Rotunno (2004), hereafter WR04]. A key feature of this theory is the preferential ascent on the downshear edge of cold pools that helps maintain the systems. The vast majority of attention has been focused on downshear-propagating systems explained by RKW; there are only a few studies that document observed upshear-propagating systems [e.g., Knupp and Cotton (1987); Corfidi (2003); though these particular cases are influenced by additional terrain and/or synoptic forcing]. It is also noteworthy that the majority of attention on mesoscale convective organization has tended to focus on systems featuring strong mesoscale downdrafts that feed the surface disturbance.

Studies of organized convective systems in tropical and/or low convective inhibition environments (e.g., LeMone et al. 1998; Johnson et al. 2005; Robe and Emanuel 2001; Anber et al. 2014) show a broad range of organizational behavior, some of which is consistent with RKW, including strong dependence on wind shear. However, near-saturated tropical environments have the potential to form weaker downdrafts and density currents, which could affect their ability to organize convection via typical processes. Arguably there has been far less attention placed on understanding organized mesoscale convective systems in low–convective inhibition environments with weaker downdrafts and density currents, in comparison to the cold pool–driven canonical systems. These environments could lead to atypical regimes of organization that do not conform with the common models of long-lived convection. Examples include tropical systems that propagate faster than their cold pools would explain (e.g., Aves and Johnson 2008; Webster et al. 2002) and systems with orientations and propagation directions poorly explained by the direction of the low-level shear (e.g., Johnson et al. 2005). Atypical systems have also been considered theoretically, such as the Moncrieff (1981) steady dynamical models of distinct regimes of organized convection. The mathematical theory that underpins these steady models of organized convection and density currents has a general validity. An objective of this two-part study is to utilize these steady models that explain the longevity and morphology of the numerical simulated convective systems.

Low–convective inhibition environments also allow the potential for ascent by gravity waves to play a role in convective initiation and maintenance. Deep tropospheric gravity waves have been shown to destabilize the environment of mesoscale convective systems and promote clustering (e.g., Mapes 1993; Numaguti and Hayashi 2000; Lane and Reeder 2001; Stechmann and Majda 2009). Moreover, in numerous modeling studies of tropical and extratropical environments, gravity waves have been linked to convective initiation (e.g., Balaji and Clark 1988; Lac et al. 2002; Morcrette et al. 2006; Tulich and Mapes 2008) and can become coupled to the convective cloud population at mesoscales (Lane and Zhang 2011). Further, gravity waves propagating ahead of convective systems have been shown to be responsible for discrete propagation episodes (Fovell et al. 2006; Shige and Satomura 2001), where convection develops ahead of the surface-based cold pool.

In this study, we examine the kinematics of long-lived quasi-steady propagating mesoscale convective systems that emerge spontaneously from long-running convection-permitting idealized simulations of tropical convection in low–convective inhibition conditions. These systems form through self-organization, are robust features of the simulations, and differ from the standard mesoscale convective systems in that they have distinctive density currents and are maintained, in part, by interactions with gravity waves. Our study has two parts. This paper (Part I) examines upshear-propagating convective systems with particular attention to an upshear-propagating density current. Moncrieff and Lane (2015, hereafter Part II) examines downshear-propagating convective systems and presents a new analytic dynamical model of the mesoscale airflow.

The remainder of this paper is organized as follows. The numerical model, its configuration, and an overview of the results of the main simulation are presented in section 2. The properties of the upshear-propagating convective systems are presented in section 3. Section 4 presents results from an additional simulation aimed at describing the dynamical structure of the convective systems, their distinctive propagation characteristics, and their robustness. In section 5, a nonlinear steady model of the upshear-propagating density current and the linear theory of ducted gravity waves explain the dynamical processes that maintain the convective systems. Also presented in section 5 are trajectory calculations to further expose the kinematics of the upshear-propagating regime. Finally, the main conclusions and results are summarized in section 6.

## 2. Model configuration and simulation overview

### a. Numerical model

This study uses Cloud Model 1, revision 15 [CM1; see Bryan and Fritsch (2002) for details]. In its current configuration, the model is three-dimensional, nonhydrostatic, and compressible. Temporal differencing uses an explicit Runge–Kutta time-split technique (Wicker and Skamarock 2002) with fifth-order advection in the horizontal and vertical. Subgrid-scale mixing is via a 1.5-order turbulence kinetic energy closure; microphysical processes are parameterized using the Lin et al. (1983) scheme, which contains separate bulk categories of cloud water, rain, snow, graupel, and cloud ice. Surface friction and Coriolis terms are neglected.

Each simulation uses the same model geometry, which builds upon the previous work in two dimensions by Lane and Moncrieff (2008), Lane and Moncrieff (2010), Lane and Zhang (2011), and Shaw and Lane (2013). The domain is 2040 km long (*x* direction), 120 km wide (*y* direction), and 30 km deep (*z* direction). The horizontal grid spacing is 1 km, and the lateral boundary conditions are periodic. There are 150 vertical levels: 5 levels are in the lowest 500 m, and above that height the vertical grid spacing is 200 m. The uppermost 8 km of the domain features a Rayleigh friction damping layer to minimize reflections off the upper boundary. The model time step is 3 s, with acoustic terms calculated using a small time step of 0.375 s. Each simulation is integrated for 240 h.

The initial thermodynamic sounding, shown in Figs. 1a and 1b, used in Lane and Moncrieff (2008), was observed near Darwin, Australia, during the Maritime Continent Thunderstorm Experiment (MCTEX) on a day with intense island thunderstorms. Random perturbations, with maximum amplitude of 0.1 K, are applied to the lowest two model levels at the initial time. The initial wind profile is idealized and features a surface wind of −20 m s^{−1}, which increases linearly to zero at *z* = 5 km; above 5 km, the background wind is zero (Fig. 1c). Although a deep shear layer has mostly been used in previous simulations of midlatitude squall lines (e.g., Thorpe et al. 1982; WR04), similar shear is observed in the tropics and during the Australian monsoon (e.g., Wapler et al. 2010; Lin and Johnson 1996). This profile is also part of the parameter space explored by Anber et al. (2014), although admittedly the low-level shear is at the strong end of the expected range for tropical environments. The mean wind profile is allowed to evolve in time (i.e., it is not nudged to the initial state), allowing the effects of convection to modify the wind profile and produce dynamically consistent self-organization and notable atypical new regimes of mesoscale convective organization. It will be shown later in this paper that the self-organization acts to decrease the wind shear, and that likely factors into our successful set of numerical simulations.

Convection occurs and is sustained by a combination of surface fluxes and imposed tropospheric cooling. The cooling rate is −2 K day^{−1} below 9.5 km and decreases linearly in magnitude to zero at 15.5 km and further aloft. This cooling and the surface fluxes significantly modify the thermodynamic environment compared to its initial state. For example, Fig. 1b shows that the lower troposphere gets close to saturation as the simulation proceeds, significantly weakening the evaporative cooling and the strength and structure of the convective downdrafts.

Two categories of experiment are presented in Part I: 1) constant fluxes (EX-CF) uses constant surface sensible heat flux of 10 W m^{−2} and constant surface latent heat flux of 100 W m^{−2}; and 2) interactive fluxes–perturbation wind (EX-IF-PW) assumes that the sea surface temperature is 300 K and surface (sensible and latent heat) fluxes are determined using a wind speed–dependent formula for the coefficient of enthalpy *C*_{E}, defined using Deacon’s formula [see Rotunno and Emanuel (1987), their Eq. (36)]. For this experiment, the horizontal velocity perturbations from the initial state are used to determine *C*_{E}. Note that, because of the absence of surface friction and the surface flux formulations used in Part I, the modeled environment is Galilean invariant. The nonzero surface wind was chosen for convenience to make the upper-level winds close to zero and reduce the expected domain-relative propagation speeds of the organized systems. The results are equivalent to those that would be expected if 20 m s^{−1} were added to the initial wind profile (making the initial surface wind zero).

### b. Simulation results

The temporal evolution of the total cloud water path (i.e., the vertical integral of liquid water plus ice) for EX-CF is shown in Fig. 2. Convection initiates after approximately 10 h and within a few more hours becomes dominated by systems that move in the negative *x* direction. The temporal coherence of these convective systems is exemplified by the *y* average of the cloud water path (Fig. 2b), with the cross section along *y* = 0 (Fig. 2a) showing intermittent cloud coverage. The differences between Figs. 2a and 2b suggest three-dimensional variability in cloud occurrence. The cloud field implies a dominant convective regime that is maintained until approximately 100 h. Figure 1 shows the environment at 100 h: the lower troposphere has cooled, the middle troposphere has moistened considerably, the surface winds have weakened, and the shear has deepened to about *z* = 10 km. Around this time, the convective available potential energy (CAPE) for parcels originating at the lower boundary is about 1500 J kg^{−1}, and the convective inhibition (CIN) is negligible, only about 2 J kg^{−1}. After about 100 h, slower-moving systems dominate the remainder of the simulation. These slower systems exhibit coherence in both panels of Fig. 2, suggesting the systems are more two-dimensional (i.e., contiguous in the *y* direction).

Figure 2b shows two lines that represent the travel speed in the *x* direction of the two aforementioned convective regimes: line A corresponds to −22 m s^{−1}, and line B corresponds to −5 m s^{−1}. The domain-averaged wind in the *x* direction *u* at 100 h (Fig. 1c) shows the mean surface wind at this time is −15 m s^{−1}. Therefore, those systems with speeds depicted by line A are moving at approximately −7 m s^{−1} relative to the mean surface wind. Moreover, because the wind profile has positive wind shear, these systems have no mean-flow steering level; that is, they propagate (wavelike) in the upshear direction. We denote these systems as the upshear-propagating regime (UPR). Those quasi-two-dimensional systems depicted by line B are moving in the positive direction relative to the surface wind, which is the downshear direction. We denote these systems as the downshear-propagating regime (DPR). Although the UPR dominates the first 100 h of the simulation and the DPR dominates thereafter, the UPR is evident throughout most of the simulation and appears to interact with the DPR. The UPR is the focus of this paper, and the DPR is the focus of Part II.

## 3. Simulated upshear-propagating systems

### a. Dynamical structure

Two snapshots of the spatial patterns of cloud water path for limited portions of the model domain are shown in Fig. 3, depicting the typical structures associated with the UPR. As described in the previous section, the cloud water path has significant three-dimensional structure and is characterized by a few strong isolated convective cells adjacent to a population of numerous weaker cells. The strongest cells have broader cloudy regions that extend downshear. Although the cells are isolated and relatively short lived (not shown), they are embedded in long-lived organized mesoscale convective systems that last tens of hours or more and therefore far exceed the lifetime of the individual cells. Indeed, the two snapshots shown in Fig. 3, which are 10 h and more than 700 km apart, are embedded in the same propagating convective system (cf. Fig. 2). Moreover, Fig. 3 imply the (small and large) convective cells are arranged along a quasi-linear disturbance that is aligned approximately parallel to the *y* axis (i.e., perpendicular to the mean shear).

The longevity of the UPR in the simulations makes it appropriate to analyze its quasi-steady structure in the frame of reference moving with the system. A two-dimensional composite is created by constructing a moving average following one upshear-propagating system (in the *x* direction) and then averaging in the *y* direction (perpendicular to the propagation direction). Figure 4 shows a composite created for 10 h of simulation beginning at 65 h (i.e., the system shown in Fig. 3), depicting the quasi-steady structure of the UPR. Although only one system is used to create this composite, different composites created from other upshear-propagating systems in this simulation are in close agreement.

Figure 4 reveals the key kinematic structure of the UPR. First, the cloudy air tilts toward the positive direction with height, extending approximately 200 km in the *x* direction from its leading edge near the surface. Thus, the system is upshear propagating but downshear tilted and is thereby distinct from the standard model of long-lived (leading line, trailing stratiform) squall lines and MCS (e.g., Houze 2004; WR04). Figure 4a shows the wind relative to the moving system (i.e., the *x* component of the velocity minus the system-propagating speed), demonstrating that the wind passes through the system from left to right at all levels above the surface. In other words, this system propagates in a wavelike manner. There is a strong negative wind perturbation immediately below the system (Fig. 4b; *x* ≈ 970 km), and its strength reduces the system-relative velocity at the surface to approximately 2 m s^{−1} (Fig. 4a) (i.e., a near-stagnant region exists in the system-relative reference frame). The system-relative streamlines (Fig. 4c), defined in the normal way as *ρ* is the density, *ψ* is the system-relative streamfunction, and *u* is the *y* average of the system-relative wind), show little vertical deviation of the system-relative flow. In this view, the streamlines appear almost flat, with the exception of an up–down draft (cf. Parker and Johnson 2004) confined to the lowest few kilometers below the system and collocated with the near-stagnant region. The flow is moist (Fig. 4d), with the subcloud layer in the near-stagnant region close to saturation (relative humidity > 90%), and the relative humidity below the stratiform part of the cloud system (980 < *x* < 1100 km) is greater than 80%. This high relative humidity inhibits evaporation leading to relatively weak cold pools (discussed later).

The streamlines (Fig. 4c) show no significant mean (mesoscale) ascent in the convective region. This behavior is explained by the localized convective cells (cf. Fig. 3) and the considerable three-dimensional structure of the system. Convective updrafts and compensating subsidence between the cells essentially average out, as demonstrated later by trajectories in section 5.

As mentioned above, a key feature of the perturbation wind (Fig. 4b) is the strong negative near-surface perturbation underneath the cloud in the same direction as the system propagation. In addition, above the surface, the perturbation wind fluctuates between a localized positive perturbation at *z* ≈ 2 km and a negative perturbation that is maximized at *z* ≈ 8 km. That is, the perturbations are wavelike in the vertical, representing layers of organized inflow and outflow.

### b. Surface disturbance

A close-up view of the composite system structure, including negative buoyancy and rainwater mixing ratio, is shown in Fig. 5. It identifies a well-defined surface cold pool (region of negative buoyancy) immediately underneath the system and coinciding with the region of rainfall. The cold pool resembles a density current propagating in the negative direction, with its leading edge at *x* ≈ 925 km, its “head” at *x* ≈ 970 km, and a long shallow “tail” extending more than 100 km in the positive direction. The strongest negative surface buoyancy coincides with the near-stagnant region identified in Fig. 4. The coincidence of rain and negative buoyancy, combined with the slight reduction in rainfall mixing ratio in the cold pool suggest that evaporation makes some contribution to the cold pool formation. However, as shown above, the relative humidity is high in the subcloud layer, which inhibits evaporation. Horizontal cross sections of equivalent potential temperature *θ*_{e} (not shown) demonstrate that the negative buoyancy in the cold pool coincides with negative *θ*_{e} perturbations with strengths exceeding −4 K. These *θ*_{e} reductions show that convective downdrafts play a role in the cold pool formation.

The horizontal structure of the cold pool at the surface shown in Fig. 6 demonstrates that the cold pool extends across the entire domain in the *y* direction, albeit with variations in intensity. The regions of largest negative buoyancy correspond to the regions with the most intense convection (cf. Fig. 3). The system-relative surface wind is also shown and depicts that the surface flow ahead and behind the system is positive, but with large regions below the convective cells having negative system-relative flow. The boundary between the negative and positive flow defines a stagnation point in this reference frame; stagnation occurs along large portions of the leading edge (left side) of the cold pool. Stagnation along the edge of the cold pool implies that the cold pool determines the system propagation. The cold pool is also clearly responsible for the initiation of convection, with most of the convective cells in Fig. 3 forming along the leading edge of the cold pools shown in Fig. 6.

The strength of the cold pool is determined using the formula *C* is the cold pool strength, *B* is the buoyancy, and *D* is the cold pool depth (Weisman 1992). The value of *C*, calculated from the composite buoyancy (Fig. 5) at *x* = 970 km and assuming *D* = 1 km, equals 6.2 m s^{−1}. This strength is close to the speed of convective system-relative to mean surface wind determined earlier (≈7 m s^{−1}). Of course, this composite estimate is lower than instantaneous values because the *y* average results in smaller buoyancies as a result of the spatial variation in intensity and the complicated shape of the cold pool’s leading edge (cf. Fig. 6). Moreover, although the composite wind (Fig. 4a) does not identify the actual position of stagnation because of this averaging, Fig. 6 clearly shows stagnation along parts of the leading edge of the cold pool.

To place this system in the context of previous work on long-lived convective systems and squall lines (e.g., WR04), we estimate the shear strength Δ*u* to be about 10 m s^{−1} (viz., the vertical change in wind over the lowest 3 km at this time determined from Fig. 1c). Thus, *C* < Δ*u* and the downshear tilt of the UPR is consistent with the shear dominating any opposing baroclinic vorticity generation associated with the cold pool. However, the development and maintenance of the UPR is inconsistent with the theories underlying WK04. In particular, as shown above, it is the upshear edge of the cold pool that determines the propagation and development of the UPR. In contrast, WK04 describe how the downshear edge of a density current is preferential for sustained vertical ascent that underlies long-lived systems and show that the upshear edge is not favorable. Moreover, the baroclinic vorticity tendency at the upshear edge of the cold pool would be of the same sense as the mean shear, also consistent with the downshear tilt of the system. Clearly, the UPR is atypical in being a long-lived, downshear-tilted, upshear-propagating convective system maintained at the upshear edge of the density current.

### c. Discussion

The shear profile considered here is strong and deep and, coupled with the weak cold pools, takes conditions away from the RKW optimum state, allowing our atypical regime to emerge. This shear also supports ducted gravity waves, which are discussed further in section 5. The UPR only dominates in the earlier stages of this simulation, with the transition to the DPR occurring at about 100 h. At this time, there is a reduction of the CIN (reduction from about 2 to less than 1 J kg^{−1}) and CAPE (reduction from about 1500 to about 1300 J kg^{−1}), and the low-level shear continues to weaken slowly. These changes in the environment should explain the regime transition, which will be discussed again briefly in Part II.

Previous three-dimensional radiative–convective equilibrium (RCE) simulations (e.g., Robe and Emanuel 2001; Anber et al. 2014) have examined the sensitivity of convection to wind shear. These studies, along with observational studies (e.g., LeMone et al. 1998; Johnson et al. 2005) have demonstrated that wind shear has significant influence on organizational morphology, consistent with distinct dynamical regimes of organized mesoscale convection (Moncrieff 1981). Many of the RCE-simulated systems were consistent with RKW-like arguments and therefore different from the UPR discussed above. Both Robe and Emanuel (2001) and Anber et al. (2014) note changes in organizational behaviors at stronger shear. Notably for Anber et al.’s (2014) stronger- and deeper-shear cases, with similar shear profiles to that considered here, there was a transition in organizational behavior and a tendency for the systems to aggregate instead of forming quasi-linear MCSs. Neither of these studies, however, identifies a UPR.

Aspects of the simulations presented here that are different from those cited above might explain the emergence of the UPR. First, the simulation domain in EX-CF is considerably larger than the RCE experiments, which show a tendency for producing only one aggregated mesoscale convective system at a time. The larger domain allows less constrained self-organization, including the coexistence of multiple organized systems and multiple regimes. In EX-CF (and other simulations in this study), the mean wind profile is allowed to evolve, and there is no relaxation to the initial shear as is normally used for RCE simulations. It is likely that the deepening and weakening of the shear with time is essential for the selection of both the UPR and the DPR regimes of convective organization. Finally, at least in EX-CF, the UPR is a transient regime that is dominant in the early stages of the simulations. Later in the simulation, as EX-CF approaches equilibrium, it is difficult to identify the UPR, and it might easily be overlooked. In other words, the focus on equilibrium states in the RCE approach may lead to an undesirable omission of interesting transitional regimes.

In the following section, we show results from an additional simulation (EX-IF-PW) aimed at further elucidating the structure and dynamics of the UPR and their generality. This simulation uses interactive surface fluxes instead of the constant surface fluxes reported earlier.

## 4. Interactive surface flux simulation

Figure 7 depicts the space–time evolution of the cloud water path for EX-IF-PW. This simulation takes about 40 h for convection to develop, and then propagating convective systems emerge almost immediately. The propagating structures show close resemblance (in both the *y* average and *y* cross section) to the UPR in EX-CF until about 150 h. After 150 h, systems continue to propagate at approximately the same speed as earlier in the simulation, except the cloud cover has increased and, at any one time, one large system dominates the domain. As in EX-CF, the *y* cross section (Fig. 7a) shows less temporal coherence, even after 150 h. This indicates significant variability in cloud cover in the *y* direction throughout the systems’ evolution.

The systems shown in Fig. 7 propagate at a similar speed to the UPR in Fig. 2. As before, the structure of the systems is diagnosed by creating composites based on the system-relative averages. Two composites are created for this simulation: one at 130 h (Figs. 8a,b) and the other at 220 h (Figs. 8c,d). In these figures, the winds relative to the moving systems and their perturbations from the domain average are shown. Figures 8a and 8b show that this convective system at 130 h in EX-IF-PW is remarkably similar to the UPR in EX-CF (cf. Fig. 4), except that the system depth is slightly smaller (note that the domain shown in Fig. 8 is larger than Fig. 4, but the horizontal cloud extent is similar in both cases).

The kinematic structure of this convective system at 130 h is essentially the same as in EX-CF. Moreover, when the system is more intense (e.g., at 220 h; Figs. 8c,d), the kinematic structure of the convective system is simply an enhancement of its structure at earlier times. (Other fields presented in Fig. 4 are not shown in Fig. 8 because they closely resemble those in EX-CF.) Thus, the UPR is also a robust and dominant feature of EX-IF-PW. With interactive surface fluxes, the UPR evolves into a large mesoscale convective system about 300 km in horizontal extent but nevertheless retains the three-dimensional structure that characterized the (weaker) systems in EX-CF. For example, Fig. 9 shows the horizontal distribution of the cloud water path at 220 h. The cloud field that composes the UPR extends in the positive direction and features three or four large convective cells. Although not shown here, similar plots for an earlier time (e.g., 130 h) from EX-IF-PW closely resemble Fig. 3 from EX-CF.

In the following section, we show that the UPR morphology is consistent with an alternative dynamical theory of upshear propagation, and linear wave theory will explain how the UPR is maintained by coupling to a ducted gravity wave.

## 5. Dynamical models of upshear propagation

The previous sections have documented the existence and persistence of the UPR within the idealized three-dimensional simulations. As discussed previously, the propagation of the convective systems and their longevity are firmly linked to the surface-based disturbances manifested as quasi-two-dimensional density currents that initiate vertical development on their upshear side. Recall that the sustained vertical development of deep convection on the upshear edge of density currents is counter to the WR04 theory of squall lines, where the downshear side is the preferred region for vertical development and the initiation of convection cells. In this section, we use theory to help explain the structure and maintenance of this atypical archetype.

### a. Density current dynamics

The classical theory of density currents developed by Benjamin (1968) is based on mass and energy conservation together with an integral constraint on the vertical transport of horizontal momentum. Moncrieff and So (1989, hereafter MS89) generalized these principles by adding vorticity conservation, enabling the effects of shear on density currents to be modeled analytically. Moncrieff and Liu (1999, hereafter ML99) used this extended dynamical theory to model upshear- and downshear-propagating density currents. The schematic airflow in the density current part of the circulation in Fig. 10a originates from ML99 (their Fig. 1b), which is a special case of the MS89 results. We now show that the ML99 model approximates the simulated upshear-propagating density current for the negatively buoyant regions of Fig. 5.

*h*; and

*θ*is the potential temperature. Defining the Froude number

*θ*is nonnegative,

*z*=

*h*.

*S*

_{L}and

*S*

_{R}are the mean shears to the left and right of the current, respectively (see Fig. 10a). Because

For comparison of the simulation results with the analytic model and the ML99 schematic (Fig. 10a), consider instantaneous sections through a region of surface stagnation in EX-CF at 65 h (cf. Figure 6a). Because these sections shown in Figs. 10b and 10c are constructed from an average between *y* = 80 and 90 km, the rear inflow to the density current *y* direction, these streamlines are only approximate.) The sections from the simulation approximate the key features of the schematic (Fig. 10). Moreover, estimates of the parameters from the model sections (Figs. 10b,c) are: *h* = 500 m, *U*_{L} = 5 m s^{−1}, *U*_{R} = 4 m s^{−1}, *S*_{L} = 4 m s^{−1} km^{−1}, *S*_{R} = 14 m s^{−1} km^{−1}, and the mean buoyancy in the density current is calculated as −0.02 m s^{−2}. These model-derived parameters approximately satisfy Eq. (1). Using these values, *z* = *h* is

The above analysis has shown, using energetic and continuity arguments, that the mean ascent on the upshear side of propagating density currents acts to maintain the upshear-propagating convective systems that characterize the UPR, provided the CIN is sufficiently small. Notice that, on the downshear side of a density current (see ML99, their Fig. 1a), the formula for the horizontally averaged ascent is different from the formula for the upshear side [cf. our Eq. (2) with ML99’s Eq. (4)]. Specifically, on the downshear side, the two shear-effect terms have the same sign and work together to decrease the mean ascent.

In the next section, we also consider the role of gravity waves in contributing to that maintenance and selecting the preferred direction of propagation.

### b. Linear wave theory

*k*is the horizontal wavenumber,

*c*is the phase speed of the disturbance, and

*w*

_{k}(

*z*) are components of the vertical velocity and vertical amplitude function, respectively, for a given wavenumber

*k*] is governed by the Taylor–Goldstein equation:where

*H*constant, and

*l*

^{2}is the Scorer parameter, defined aswhere

*U*is the background wind (parallel to the direction of wave propagation), and

*N*is the Brunt–Väisälä frequency. Examination of the Scorer parameter can help determine the vertical propagation characteristics of gravity waves because, for simple flows,

*l*

^{2}−

*k*

^{2}is equal to the square of the vertical wavenumber, and vertical variations are related to changes in the vertical group velocity. In situations where

*l*

^{2}>

*k*

^{2}, vertically propagating solutions exist, and when

*l*

^{2}<

*k*

^{2}, solutions are evanescent. An elevated region with

*l*

^{2}<

*k*

^{2}above a surface-based layer with

*l*

^{2}>

*k*

^{2}is known to form a duct, which permits (ducted) waves to propagate significant distances horizontally. More complicated vertical variations in

*l*

^{2}result in partial (or total) internal wave reflections and interference between upward- and downward-propagating waves that lead to nontrivial vertical amplitude functions satisfying Eq. (3).

Figure 11 shows the vertical structure of the Brunt–Väisälä frequency and Scorer parameter for mean profiles calculated over the 300-km-long area ahead of the upshear-propagating system (from 600 to 900 km in the coordinate system shown in Fig. 4).^{1} The phase speed *c* = −22 m s^{−1} (i.e., the propagation speed of the system) is used to calculate *l*^{2}, and solid lines show profiles using the dry definition of *N* and are therefore representative of the environment in the absence of moist convection. With the exception of the reduced stability near the surface, the tropospheric stabilities are close to the typical value of 0.01 s^{−1}. As is common in the tropics, there is a slightly reduced stability in the upper troposphere associated with convective outflows. The wind profile at this time is similar to that shown in Fig. 1c (so it is not repeated here). The Scorer parameter is maximized about 1 km above the surface, decreases rapidly to possess a local minimum at *z* ≈ 4.5 km, and continues to decrease to its minimum in the upper troposphere at *z* ≈ 13 km. The rapid decrease in the Scorer parameter above the low-level maximum could result in wave ducting; for example, *l*^{2} < 0.5 × 10^{−6} m^{−2} for *z* > 5 km would prohibit gravity waves with horizontal wavelengths less than approximately 9 km (and *c* = −22 m s^{−1}) from propagating above 5 km, while they are permitted below 5 km.

To investigate the vertical structure of the wave modes supported by the environment ahead of the convective system, Eq. (3) is solved for the eigenfunctions, *w*_{k}(*z*), and corresponding eigenvalues, *λ* = 2*π*/*k* (under the assumption that *c* = −22 m s^{−1}) using the same method as Lane and Clark (2002).^{2} [The boundary conditions are *w*_{k}(0) = 0 and that the downward-propagating component is zero at *z* = 20 km.] Two solutions that satisfy these boundary conditions are found: the first [*w*_{1}(*z*)] has the eigenvalue *λ*_{1} = 7.9 km, and the second [*w*_{2}(*z*)] has the eigenvalue *λ*_{2} =13.1 km; these (normalized) solutions are shown in Figs. 12a and 12b. The first eigenfunction (Fig. 12a) is indicative of a ducted wave mode with an antinode at *z* ≈ 2 km. The second eigenfunction (Fig. 12b) shows a partially ducted mode with nodes at *z* ≈ 3 km and near the tropopause (*z* ≈ 16 km) and antinodes near the surface (*z* ≈ 1.5 km) and in the middle troposphere (*z* ≈ 7 km). Of relevance to the maintenance of the UPR, the locations of the antinodes near *z* ≈ 1.5–2 km in *w*_{1}(*z*) and *w*_{2}(*z*) imply that the environment ahead of the convective system is supportive of ducted gravity wave modes that cause localized ascent or descent concentrated in the subcloud layer.

Using the two-dimensional anelastic continuity equation, it can be shown that the vertical structure of the horizontal velocity perturbations that correspond to the eigenfunctions (Fig. 12) are defined as *z* = 2 km (*u*_{1}) and *z* = 1.5 km (*u*_{2}). Unlike the ducted mode (*u*_{1}), the partially ducted mode (*u*_{2}) shows relatively large horizontal velocity perturbations aloft that are maximized for 3 < *z* < 5 km.

To assess the relevance of these linear solutions, the profile of the actual horizontal velocity perturbations from the model composite in the near-stagnant region is calculated and shown in Fig. 14. The model solution shows strongest perturbations at the surface (within the density current), a flow reversal at about *z* = 750 m, and a localized maximum near *z* = 2 km that decays to zero aloft. This model profile shows some resemblance to the wave modes from the linear solutions (Figs. 13a,b). In particular, both linear solutions show a perturbed flow that is maximized near the surface, with the flow reversal aloft. However, the reversal is about 1 km and 750 m too high in *u*_{1}(*z*) and *u*_{2}(*z*), respectively. The ducted solution [*u*_{1}(*z*); Fig. 13a] shows the best resemblance to the model above the reversal because the magnitude of the perturbations is less than at the surface, but the perturbed flow does not decay to zero aloft like the model.

The above linear solutions do not, however, take into account the effect of the convective system itself on the stability and wave ducting properties, which may be important because the strongest velocity perturbations occur directly below saturated air. As shown in Fig. 5, the cloud base is at about *z* = 2 km. To crudely represent this effect, the Brunt–Väisälä frequency used for the linear calculations is modified to use the regular dry definition below 2 km and the saturated definition [Eq. (5) of Durran and Klemp (1982)] above 2 km. The modified stability profile and the corresponding profile of *l*^{2} are shown in Fig. 11 (dashed lines); the inclusion of moist effects reduces *l*^{2} considerably for 2 < *z* < 13 km (for *c* = −22 m s^{−1}), which creates a more effective wave duct near the surface. Equation (3) is solved again using the moist profiles, and in this case, only one solution is found and has *λ*_{1} = 11.7 km; the corresponding vertical structure functions are shown in Figs. 12c and 13c. This new solution is a ducted wave mode, with a peak in *w*_{1}(*z*) at about *z* = 1.5 km. The profile of *u*_{1}(*z*) shows better resemblance to the actual model profile: specifically, there is a local maximum at *z* = 2 km that decays to zero aloft. The flow reversal is still about 750 m too high in the linear solution. Nonetheless, taking into consideration the approximations made to create this linear solution, the agreement between the vertical structure of the linear solution (Fig. 13c) and the model perturbations in the near-stagnant region (Fig. 14) is remarkable.

These linear solutions show that the model environment is supportive of ducted and partially ducted wave modes that propagate at speeds that match the speed of systems that compose the UPR. Importantly, the wind perturbations from the (nonlinear) convection-permitting model in the region of the surface disturbance show good agreement with the vertical structure of these linear modes, suggesting that the surface disturbance (density current) is somehow coupled to these ducted wave modes. The horizontal wavelengths of the ducted modes (viz., the eigenvalues *λ*) are shorter than the scale of the perturbations revealed by the composite (Fig. 4b). Nevertheless, the averaging broadens the horizontal scale of the perturbations significantly; the actual instantaneous horizontal scale of convergence associated with the density current is approximately 5 km (not shown) and the scale of the forward surface flow is approximately 10 km (e.g., Fig. 6a)—both consistent with the eigenvalues. Our preferred explanation is that the ducted wave modes are coupled to the region of localized ascent and forward flow near the leading edge of the density current, thus helping to maintain the convective development on the upshear edge of the cold pool and the continued upshear propagation. The broader-scale horizontal velocity perturbations, which define the organized mesoscale inflow and outflow, are then a product of the convective system itself.

It is also worth mentioning that the speed of the UPR is approximately −18 m s^{−1}, relative to the approximately −4 m s^{−1} mean-tropospheric flow. This places the propagation speed of this system in the range of values expected for deep hydrostatic tropospheric gravity waves (with *n* = 2 or *n* = 3, where *n* is the number of antinodes in the vertical) that have been shown to affect convection (e.g., Mapes 1993; Tulich and Mapes 2008; Lane and Zhang 2011). Such waves may play a role in the UPR longevity and propagation, and the vertical structure of the wind perturbations in Figs. 4 and 8 seems consistent with this idea. However, these deep modes do not explain the stagnation at the upshear edge of the density current, which ultimately defines the propagation of the system. It is the shallow nonhydrostatic ducted waves revealed by linear theory that appear to be most relevant for coupling to the density current.

### c. Parcel trajectories

The kinematics of the UPR is further illustrated by a set of trajectories released to illustrate parcel motion. The trajectories are each treated as massless and frictionless parcels advected by the resolved-scale velocity; the temporal integration is conducted “online,” which means the calculation occurs every model time step. All trajectories are released at 220 h in EX-IF-PW^{3} at *x* = 700 km (i.e., ahead of the system shown in Fig. 9) and integrated for 5 h. For this example, a trajectory is initiated at every grid point in the *y* direction at heights of *z* = 2 and 7.5 km; for ease of interpretation, the location of the trajectories in the *x* direction are adjusted to place them in a frame of reference moving with the convective system.

Figure 15a shows the locations in the *x*–*z* plane of all of the trajectories. Many trajectories released at *z* = 2 km remain near that height because the convective cells only cover a relatively small portion of the domain. However, many parcels do ascend a significant distance when they are embedded within the convective updrafts, with parcels eventually being detrained at almost every level in the range 2 < *z* <13 km. A smaller number of these parcels approach the surface when they become embedded within convective downdrafts. The trajectories released at *z* = 7.5 km are dominated by descent, with many parcels descending more than 1 km; this descent is associated with “compensating subsidence” of the unsaturated air that attempts to balance the increase in mass at upper levels. Moreover, a number of other trajectories initiated at *z* = 7.5 km ascend up to *z* ≈ 10 km as parcels are entrained within the convective updrafts. Comparison of these trajectories with the composites presented earlier (e.g., Fig. 4c) that show limited mean vertical ascent, suggests that the ascent within convective updrafts is mostly balanced by descent (convective downdrafts and compensating subsidence); such balance is facilitated by the three-dimensional structure of the convective cells.

To examine the horizontal movement of the trajectories, specifically those that ascend from the lower troposphere and descend from the middle troposphere, they are separated into those initialized at *z* = 7.5 km that descend more than 500 m (Fig. 15b) and those initialized at *z* = 2 km that ascend more than 500 m (Fig. 15c). The ascending air is relatively localized in the *y* direction occurring at *y* locations that roughly correspond to the largest convective cells in Fig. 9. The descending air (from 7.5 km) occurs for the most part between the convective cells. Moreover, there are a few occurrences of descending trajectories that diverge in the *y* direction but then eventually cross, with the crossing occurring in line with ascending air [e.g., (*x*, *y*) = (950, 95) km and (*x*, *y*) = (930, 40) km]. Other analysis not presented here shows that in such cases the midtropospheric air is diverted laterally around convective cells and behind those cells the trajectories descend, converge, and cross.

In addition to those trajectories shown in Fig. 15, trajectories originating from the first model level were also computed (not shown here). Many of these trajectories were entrained into convective updrafts; about half reached the upper troposphere, and the other half were detrained in the middle troposphere. The trajectories not entrained into updrafts remained in the lower troposphere after being displaced vertically over the leading edge of the upshear-propagating density current.

These trajectory calculations and other analysis are used to inform the development of a schematic of the UPR, which is shown in Fig. 16. Key attributes of this regime are shown, including the localized convection, the mean up–down near-surface drafts, the ascending air within the convective updrafts that originate near the surface, and the air that descends between and around the convective cells. The purely propagating character illustrated in Fig. 16, (i.e., inflow entirely from the leading side of the system) resembles the morphology of propagating tropical cumulonimbus and squall lines identified by Moncrieff and Miller (1976, their Fig. 1) and Moncrieff (1981, his Fig. 1).

## 6. Summary and conclusions

This study has examined a class of long-lived convective system that occurred spontaneously in idealized convection-permitting simulations. The simulations were configured in a three-dimensional doubly periodic channel, initialized with an observed tropical thermodynamic sounding and an idealized wind profile. Convection was initiated and maintained by surface fluxes and imposed tropospheric cooling. Lasting more than 200 h, these simulations enabled self-organization to occur and two classes of long-lived regimes to evolve: the upshear-propagating regime (UPR) and the downshear-propagating regime (DPR). The UPR was the focus of this article and the DPR is the focus of Part II. The propagation and maintenance mechanisms for both parts differ substantially from the standard model of long-lived mesoscale convective systems.

The two simulations in Part I differ with respect to their treatment of surface fluxes. In the simulation with constant surface fluxes, the UPR was dominant in the first half of the simulation, and in the simulation with interactive surface fluxes, the UPR was always dominant. The kinematics of the UPR was similar in the two simulations, even when the convective systems grew upscale and intensified considerably. In each case, the UPR manifested as a group of isolated convective cells, which were organized into a line that was approximately perpendicular to the mean low-level wind shear. The envelope of these cells, the convective system, propagated approximately 7 m s^{−1} relative to the mean surface wind in a direction opposed to the mean low-level shear vector. The individual convective elements that characterized the systems are plumelike and tilted downshear.

The propagation of the UPR was associated with the generation of new convective cells at the upshear edge of a surface-based density current. The matching propagation speeds of the density current and the convective system were exemplified by the stagnation of the surface flow in the system-relative frame of reference. Composites of the across-line quasi-steady structure identified a downshear-tilted mesoscale system with a notable up–down draft in the vicinity of the density current. Trajectory calculations elucidated the approximate balance between ascent in the convective updrafts and descent in the surrounding air that acted to maintain the approximate mean horizontal mesoscale flow.

The morphology of the UPR differs substantially from the morphology of the standard model of long-lived mesoscale convective systems and squall lines (e.g., WR04; Parker and Johnson 2004; Houze 2004). In the standard model, new convection is initiated by the ascent of boundary layer air over the *downshear* edge of the cold pool, and the mesoscale system propagates downshear. The UPR differs in two ways. First, the UPR has new convective development at the *upshear* edge consistent with the ML99 density current model. Second, the UPR propagates relative to the environmental winds at all levels in a wavelike manner. Because of their up–down nature, the drafts through the cold pool contribute little to the overall vertical transport of mass, and the compensating descent occurs mainly between the convective systems. It follows that the mesoscale systems are truly three-dimensional. Moreover, their morphology is a generalization of the highly idealized three-dimensional Moncrieff and Miller (1976) propagating-tropical-squall-line model.

The linear gravity wave theory suggests that the maintenance of this convective system and selection of the upshear propagation direction is related to coupling between the surface density current and ducted gravity waves. Specifically, by solving the Taylor–Goldstein equation for the vertical structure of disturbances within the simulated environment, we identify ducted and partially ducted gravity wave modes that match the propagation speed of the convective systems and surface density current. The ducting of upshear-propagating disturbances was maintained in the (dry) environment because of a combination of the background shear and stability profiles that led to a notable reduction in the Scorer parameter with height in the troposphere. The vertical structure of these theoretical wave modes closely resembled the perturbed horizontal flow within the density current from the convection-permitting model, implying that the density current and the ducted waves could become coupled. Further, the best agreement between the theoretical ducted mode and the convection-permitting model was obtained when the influence of the reduction in stability above the density current, caused by the convection, was taken into account.

The occurrence and robustness of the simulated UPR was reliant on (i) the relatively weak surface cold pool/density current; (ii) the low convective inhibition (CIN); (iii) the near-saturated subcloud layer; and (iv) the low-level shear profile. The near-saturated subcloud layer inhibited the formation of a stronger cold pool, which, along with the low CIN, allowed the gravity waves to play an important role, whereas a stronger cold pool would likely have dominated the wave response. The shear profile contributes substantially to the duct formation and defines a preferred direction of propagation of these wave modes. It is likely that such ducting and the occurrence of the UPR is reliant on the relatively strong shear considered here, though this should be investigated as part of future research. The up–down drafts and the role of convectively generated gravity waves featured prominently in two-dimensional numerical simulations by Crook and Moncrieff (1988). The dynamical structure of the mesoscale airflow organization in their simulation, although with significant differences in certain respects, broadly resembles the downshear-propagating system in Part II.

It should be noted that none of the above thermodynamic and cold pool properties [viz., (i)–(iii) in the previous paragraph] are unusual over the tropical oceans. The shear profile is consistent with observed profiles albeit at the stronger end of the observed distribution. The near-saturated and low–convective inhibition environments over the tropical oceans should maximize the influence of gravity waves on the organization and maintenance of deep convection. Moreover, the occurrence of weaker cold pools in these near-saturated environments implies that cold pool–based arguments for convective longevity may be less important, especially for weaker systems like those herein. Indeed, as mentioned in the introduction, there are examples of observed systems that do not conform to the standard squall-line archetypes in that their structure and propagation cannot be explained by advection or cold pools alone. The UPR may help explain some of these observed systems, and more detailed examination of observed cases is planned in the future.

In conclusion, our study has described a convective regime that is part of a broader set of nonconforming long-lived mesoscale convective systems. In Part II, another nonconforming regime is described; namely, a long-lived, downshear-propagating regime whose longevity and propagation is also incompletely explained by the standard theories.

TPL acknowledges support from the Australian Research Council’s Future Fellowships (FT0990892) and Centre of Excellence (CE110001028) programs. We thank George Bryan (NCAR) for providing his numerical model and useful discussions. Other discussions with Rich Rotunno were particularly helpful. We also thank Morris Weisman and the two anonymous reviewers for their useful and constructive comments.

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^{1}

A similar analysis was performed with the averaging conducted over the entire 700-km-long domain that was used to create the composites shown in Fig. 4; the results were very similar to those reported here that average ahead of the system. We choose the averaging ahead of the system because it better characterizes the environment into which the system propagates.

^{2}

Lane and Clark (2002) used a simpler version of the Scorer parameter.

^{3}

Note that we chose to calculate the trajectories for EX-IF-PW instead of EX-CF because in EX-IF-PW the UPR is the only long-lived regime.