## 1. Introduction

High-resolution numerical simulations and dynamical models have exposed fundamental properties of the mesoscale organization, longevity, and propagation of precipitating convection. The standard model of the mesoscale convective system is characterized by a rearward-tilted slantwise circulation consisting of a trailing stratiform region, an overturning updraft, and a mesoscale downdraft (Houze 2004). Density currents generated by evaporation-driven downdrafts are key features of the standard model [see Weisman and Rotunno (2004, hereafter WR04) and the discussion in Lane and Moncrieff (2015, hereafter Part I)]. However, the standard model does not represent all categories of mesoscale convective organization, especially when the planetary boundary layer and/or subcloud layer is close to saturation and the density currents are weak. In such situations, as in Part I, other processes, such as gravity waves, can play a significant role in the propagation and longevity of convective systems.

Our two-part study examines the kinematics of long-lived, quasi-steady propagating convective systems in a near-saturated, low–convective inhibition environment that emerge spontaneously through self-organization and differ in important respects from the standard model. Both papers present atypical categories of mesoscale convective organization. In Part I, we examined strictly upshear-propagating systems that involve interactions with gravity waves and dynamical lifting at the upshear boundary of the density current instead of at the downshear boundary as in the standard model. Here, in Part II, we examine downshear-propagating convective systems that travel with the environmental flow at a steering level. The atypical character is featured as lower-tropospheric jumplike ascent beneath an elevated downshear-tilted overturning updraft circulation that replaces the mesoscale downdraft and density current associated with the standard mesoscale system.

Long-lived downshear-tilted, downshear-propagating convective systems have been considered previously, mostly in the context of midlatitude squall lines. Of particular relevance are the front-fed leading stratiform systems considered by Parker and Johnson (2004) and Storm et al. (2007), among others. Other downshear-tilted systems have been explored in the context of low-level shear–cold pool interactions (e.g., WR04), where the downshear tilt is explained by the dominance of environmental shear over the baroclinically generated vorticity at the edge of the cold pool.

In the downshear-propagating mesoscale system examined herein, unlike the aforementioned studies, the cold pool is of secondary importance to the propagation and maintenance of the system. Instead, the system has borelike characteristics that resemble some previous idealized simulations (e.g., Crook and Moncrieff 1988; Parker 2008) and involve gravity wave mechanisms, as in Part I. Specifically, a three-dimensional dynamical model of the Lagrangian-mean circulation approximates the simulated mesoscale system.

The remainder of this paper is organized as follows. The numerical model, its configuration, and an overview of the simulation results for constant surface fluxes are presented in section 2. The properties of the simulated downshear-propagating systems are presented in section 3 along with results for an experiment with interactive surface fluxes. The dynamical model of the downshear-propagating system is developed in section 4, and the paper concludes in section 5.

## 2. Model configuration and simulation overview

### a. Numerical model

This part of the study uses the same Cloud Model 1, revision 15 (CM1; Bryan and Fritsch 2002), numerical model configurations as Part I, and the reader is referred there for details. In brief, the numerical experiments use a domain that is 2040 km long (*x* direction), 120 km wide (*y* direction), and 30 km deep (*z* direction). The horizontal grid spacing is 1 km, the lateral boundary conditions are periodic, and the model has 150 vertical levels. The initial thermodynamic sounding in Figs. 1a and 1b is a tropical sounding observed near Darwin, Australia. The initial wind profile shown in Fig. 1c is allowed to evolve with time in order for the model to achieve dynamically consistent self-organization. Convection is sustained by a combination of surface fluxes and imposed tropospheric cooling.

We present two experiments: the constant-fluxes experiment (EX-CF), which was examined in Part I; and the interactive fluxes–full wind experiment (EX-IF-FW), where the interactive surface fluxes are calculated as in the interactive fluxes–perturbation wind experiment (EX-IF-PW) in Part I, except that the total horizontal velocity is used to determine the wind speed–dependent formula for the coefficient of enthalpy *C*_{E}. Although the latter configuration produces unrealistically large surface fluxes because of the strong background wind speed of 20 m s^{−1}, this experiment is dominated by the downshear-propagating regime and is thereby relevant here.

As described in Part I, because of the absence of surface friction and the surface flux formulations used in EX-CF, 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. By design, the additional experiment EX-IF-FW is not Galilean invariant, because the full surface wind speed is used to determine the surface fluxes in order to create the unrealistically large surface fluxes.

### b. Simulation overview

The evolution of the convection and clouds for EX-CF in Fig. 2 was described in Part I. The constant-fluxes experiment features two regimes that emerge through self-organization. The first, which dominates the first 100 h of the simulation and was the focus of Part I, marked as line A in Fig. 2b, propagates upshear at a speed of −22 m s^{−1}. After 100 h, slower-moving systems dominate the simulation. Line B denotes a speed of approximately −5 m s^{−1}, which characterizes these slower systems. As shown in Fig. 1c, at 200 h into the simulation, the surface wind is approximately −12 m s^{−1}, so the systems that emerge later in the simulation travel at approximately 7 m s^{−1} relative to the surface wind in the downshear direction. Unlike the strongly three-dimensional upshear-propagating systems in Part I, these downshear-propagating systems are weakly three-dimensional, if not quasi-two-dimensional, since their propagation and temporal coherence are captured by both the *y* average and the *y* cross section of cloud water path (Fig. 2). This downshear-propagating regime (DPR) is the focus of Part II.

In the latter half of the simulation, the environment has moistened considerably, with the relative humidity close to 90% in the lowest 4 km of the atmosphere (Fig. 1b). The shear has deepened and weakened, extending over a 10-km depth (Fig. 1c). Around this time, the convective available potential energy (CAPE) for surface parcels is approximately 1300 J kg^{−1}, with very small convective inhibition (CIN) of 0.1 J kg^{−1}. Hence, CAPE and CIN have both reduced compared to earlier times that are dominated by the UPR (see Part I). The CAPE is smaller for parcels originating from above the surface and vanishes for parcels originating above about 1.2 km (see Table 1 for specific values, calculated from composite model fields described later in the paper).

## 3. Simulated downshear-propagating systems

The horizontal structure of the DPR is shown in Fig. 3 at two times during EX-CF. The cloud water path demonstrates the quasi-two-dimensional structure of the convective systems, suggested by Fig. 2. Although the cumulonimbus clouds that compose these systems are not linearly aligned per se, they nevertheless evolve into a complex squall-line-like structure that spans the domain in the *y* direction. Unlike the UPR, which occurred as a line of strongly three-dimensional plumes, the DPR features a weakly three-dimensional system with more intense convective cells embedded within it.

As with the UPR, the near-constant propagation speed of the DPR lends itself to a composite analysis of its quasi-steady mesoscale structure. A Lagrangian average, following the system shown in Fig. 3 and averaging in the *y* direction, is constructed (Fig. 4). The composite demonstrates that the flow relative to the traveling system (Fig. 4a) is right to left in the lowest 4–5 km and left to right aloft. The relative surface flow is between −5 and −7 m s^{−1}, and no surface stagnation occurs in this reference frame. The convective system has a steering level between 4 and 5 km.

The perturbation wind (Fig. 4b) and system-relative streamlines (Fig. 4c), defined in the usual way as *ρ* is the density, *ψ* is the system-relative streamfunction, and *u* is the *y* average of the system-relative wind, identify two quasi-steady branches:

- A deep
*overturning updraft*, where air moving from the right of the system between 3- and 5-km altitude ascends within the cloud, reverses direction, and then flows back toward the right of the system between 5 and 8 km - A
*jump updraft*, where air below 3 km moving from the right of the system decelerates, ascends, and then maintains that vertical displacement as it continues to move to the left of the system. The ascent within the jump updraft resembles a bore or hydraulic jump, which is more clearly evident in Fig. 5.

*x*= 780 km,

*z*= 5 km), the vertical motion inferred by the streamlines is all upward. The cloud outline of the DPR tilts toward the right with height (i.e., downshear), as does the upper part of the overturning updraft. The subcloud layer is near saturated, with relative humidity exceeding 90% (Fig. 4d). The perturbation pressure (Fig. 4e) has a positive surface-based perturbation that extends to the left from

*x*= 870 km for the entire shown domain. This pressure perturbation implies a quasi-permanent change in the near-surface flow, consistent with a bore. The pressure perturbation extends forward of the cool surface anomaly (Fig. 5), similar to that in Adams-Selin and Johnson (2013).

The spatial structure of the upper-level streamlines and perturbation flow are what might be expected from a downshear-tilted quasi-steady heat source, as was demonstrated by Pandya and Durran (1996) for upshear-tilted systems. The forward tilt of the downshear-propagating, regime and the presence of a prominent jump updraft resemble the observed front-fed convective-line leading-precipitation archetype described by Parker and Johnson (2004). The main difference is that, for our system, the near-surface motion involves a jump updraft instead of the up–down draft in Parker and Johnson (2004). The quasi-steady flow in this regime closely resembles the numerically simulated convective system in Fig. 16 of Crook and Moncrieff (1988), which was produced through the artificial suppression of the subcloud-layer evaporation.

### a. Surface cold pool

A close-up view of the composite convective system (Fig. 5) elucidates the vertical structure of the near-surface flow. The streamlines show that the jump updraft is characterized by weak ascent near the surface, with the largest vertical displacement associated with air originating between 1- and 2.5-km altitudes. Below 1 km, the vertical displacement is small, and air above 2.5 km feeds into the overturning updraft. Therefore, the quasi-steady mesoscale overturning circulation has the appearance of an elevated convective system. The precipitation falls into the ascending air and should reduce the vertical acceleration by hydrometeor loading and evaporation (Parker and Johnson 2004), although the high relative humidity below cloud base will presumably reduce the evaporation. The surface cold pool is weak, with a peak composite negative buoyancy anomaly of approximately 0.03 m s^{−2}, which is slightly weaker than the cold pool associated with the upshear-propagating regime (e.g., Fig. 5 of Part I).

As in Part I, 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* = 820 km and assuming *D* = 700 m, is 5.0 m s^{−1}. This estimate of the cold pool strength (i.e., propagation speed) is about 2 m s^{−1} less than the actual speed of the system relative to the surface flow. This two-dimensional composite calculation likely underestimates the instantaneous strength of the actual three-dimensional cold pool. As with the estimates in Part I, *C*/Δ*u* is small, which is consistent with the downshear tilt of the system.

The streamlines in Fig. 5 show no evidence of surface stagnation in the frame of reference relative to the convective system, implying mesoscale flow through the front of the surface cold pool, rather than over it, as in density currents. Horizontal cross sections of the surface buoyancy and system-relative wind (Fig. 6) confirm this structure, illustrating that there are no regions of stagnation along the edge of the cold pool, nor are there any large regions of forward (positive) flow within the cold pool. The *x* velocity of the system-relative surface flow through the rightmost boundary of the cold pool is approximately −4 to −6 m s^{−1}. Given that the system is traveling at approximately −5 m s^{−1} and the surface wind speed is approximately −12 m s^{−1} (Fig. 1), the flow depicted in Fig. 6 implies very little deceleration of the surface wind (consistent with Fig. 4b). Moreover, Figs. 5 and 6 suggest that the cold pool does not behave like a downshear-propagating density current^{1} and does not play a direct role in determining the propagation speed of the downshear-propagating mesoscale system. The cold pool may still initiate convective-scale updrafts within the mesoscale system, but the net effect of these small scales may not contribute to the mesoscale propagation. Indeed, the horizontal displacement of the convective activity (Fig. 3) from the surface cold pool (Fig. 6) implies that the cold pool is simply left behind the propagating system. Thus, with reference to the propagation of the system, the near-surface flow speed is subcritical (e.g., Raymond and Rotunno 1989). Other subcritical examples in the literature for idealized mesoscale convective systems include Parker (2008) and the strongest shear cases of WR04.

The flow through the leading edge of the surface cold pool is consistent with numerical experiments described by Parker (2008) and Raymond and Rotunno (1989), among others. In such subcritical cases, the propagation speed of the system is defined by a (linear or nonlinear) gravity wave speed instead of the speed of a propagating density current. Indeed, the borelike appearance of the jump updraft and its properties noted earlier are consistent with this explanation. As in Parker (2008), we estimate the gravity wave speed *c* using the linear relation *c* = *N*/*m*, where *m* is the vertical wavenumber and *N* is the Brunt–Väisälä frequency. The low-level inflow has *N* = 0.012 s^{−1}, and the vertical wavelength of the horizontal velocity near the surface below the storm is between approximately 3 and 4.5 km (cf. Fig. 4b), giving speeds between 5.7 and 8.6 m s^{−1}. This range is consistent with the propagation speed of the system relative to the low-level flow but is a crude estimate because of the inherent nonlinearity of the system. Moreover, a critical level at about 4.5 km (Fig. 4a) is important for bore and gravity wave maintenance since it traps energy between this level and the surface and could give rise to disturbances with a vertical wavelength equal to this depth. Crook and Moncrieff (1988) explored the role of the critical level for systems with a notable similarity to those considered here.

Given the three-dimensionality of the convective activity (cf. Fig. 3) it is worthwhile to consider the local flow in addition to the *y* average (along-line average). Figure 7 shows the same composites as Fig. 4, except the flow is averaged between *y* = 40 and *y* = 50 km, which encompasses a region of convective activity (Fig. 3) and local maxima in negative buoyancy (Fig. 6). Here, the system-relative streamlines (Fig. 7a) are approximate because ∂*υ*/∂*y* (Fig. 7c) is nonzero due to the limited averaging in the *y* direction. Even locally, the system maintains the same overall structure as the along-line average in Fig. 4. The overturning updraft is a key feature of the vertical circulation, and the perturbation wind maintains its qualitative structure of inflow or outflow, though with smaller-scale structure in the vertical. The low-level flow does not reproduce the borelike jump updraft but instead resembles an up–down draft. However, there is significant horizontal divergence at the low levels beneath the cloud (implied by Fig. 7c); therefore, these streamlines are not a reliable representation of the surface flow. Moreover, ∂*υ*/∂*y* (Fig. 7c) exhibits a horizontal dipole structure below cloud (at around *z* = 2 km) and in the upper parts of the cloud (centered on *z* = 9 km). These dipoles identify horizontal trajectories that diverge and then converge as they flow from right to left (*z* = 2 km) and left to right (*z* = 9 km) around the regions of convective activity. This weakly three-dimensional structure will be further explored in the next section using trajectories.

### b. Parcel trajectories

Here, trajectory calculations will be used to further quantify the vertical and horizontal structure of the downshear-propagating regime. As in Part I, a series of trajectories is launched during the simulation. Three sets of trajectories are released at 170 h at (*x*, *z*) values of (900 km, 500 m), (900 km, 2 km), and (800 km, 9 km), respectively. For each set, a line of trajectories are initialized at each grid point in the *y* direction and are integrated for 10 h. The lower-tropospheric trajectories initialized at *x* = 900 km are designed to expose the near-surface (right to left) flow that feeds the convective system, and the upper-tropospheric trajectories expose the left-to-right flow through the upper part of the convective system (cf. Figs. 3 and 4). For ease of interpretation, the horizontal locations of the parcels are adjusted to place them in a frame of reference moving with the system (Figs. 8 and 9).

To examine the near-surface flow, consider the near-surface trajectories (initially at *z* = 500 m; Fig. 8). These trajectories have been separated into three groups based on their final altitude to identify the key dynamical properties of the mesoscale system. Many trajectories remain near their original altitude (Fig. 8a), highlighting that the surface-based convection has a three-dimensional cellular structure. The jump updraft is reflected in the trajectories (Fig. 8b), with many parcels ascending to 2 km, as implied by the streamlines in Fig. 5. Some surface parcels ascend through the depth of the convection and feature in the overturning updraft path (Fig. 8c). An approximately equal number of parcels follow the jump and overturning updrafts. The remaining parcels (not shown) ascend within the cloud and are detrained in the middle troposphere.

It is worth remarking that some of the surface trajectories ascend to the upper troposphere in Fig. 8c, even though the composite streamlines in Figs. 4c and 5 do not show such ascent. Instead, these streamlines show that air originally above about 2.5 km feeds the overturning updraft, implying an elevated convective system. These differences occur because the streamlines represent the mesoscale circulation, whereas the trajectories can capture the convective-scale features. The streamlines represent the Lagrangian-mean mesoscale circulation. Contributions of convective-scale updrafts and downdrafts average out and are therefore not represented in the mean mesoscale structure. On the other hand, the trajectories capture this convective-scale motion. In other words, there is no inconsistency, because the streamlines and the trajectories characterize different scales of motion.

The above remarks are consistent with the changes in the profile of equivalent potential temperature *θ*_{e} between the right and left of the system (Fig. 10b). Specifically, the midtropospheric *θ*_{e} increases to the left of the system, presumably because of the midtropospheric detrainment of air parcels of surface origin, and the surface *θ*_{e} decreases to the left of the system, presumably related to convective downdrafts.

The trajectories released at 2 and 9 km, designed to sample the ∂*υ*/∂*y* dipoles identified in Fig. 7c, are shown in Fig. 9. Here, only those trajectories that deviate vertically less than 2 km from their original altitude are shown in order to isolate the horizontal trajectories. As suggested by Fig. 7c, both sets of trajectories show parcel paths that deviate around the convective system. The lower trajectories (Fig. 9a) show lateral deviations that are generally less than 5–10 km, leaving holes in the trajectory paths about 10 km wide [e.g., at (*x*, *y*) = (840, 45) km]. These holes indicate deviation of the flow around individual convective updrafts that have small horizontal scales, in part because of their proximity to the surface. In contrast, at upper levels (Fig. 9b), the lateral displacements of the trajectories exceed 20–30 km in some cases, passing around much of the upper-level cloud.

### c. Effects of interactive surface fluxes

An additional simulation with enhanced interactive surface fluxes (EX-IF-FW) is now described. The temporal evolution of the total cloud water path for EX-IF-FW is shown in Fig. 11. As in EX-CF in the early stages of the simulation, the cloud population is dominated by fast-moving systems (the UPR), but, after about 80 h, slower-moving systems dominate. These systems have a similar propagation speed to the DPR in Fig. 2 for EX-CF, but instead of being persistent, these systems undergo intensification and decay on about a 30-h time scale.

The composite vertical structure of a system from EX-IF-FW (Fig. 12) shows very similar structure to Fig. 4. The main differences are that, in this case, there is very little surface deceleration and the near-surface flow undergoes an elongated up–down path instead of a pure jump. The overturning circulation and the pressure perturbations are much stronger. Nevertheless, the systems that dominate the later stages of the EX-IF-FW simulation are characteristic of the DPR in EX-CF.

### d. Discussion

As described in Part I, the emergence of the DPR follows a reduction of CIN and CAPE, and there is also a notable increase in the relative humidity below 5 km (Fig. 1b). This suggests that the thermodynamic environment might play a key role in determining the regime transition. The results from EX-IF-FW also support this hypothesis, with the larger surface fluxes leading to an earlier transition to the DPR. Though there is also a gradual weakening and deepening of the shear throughout EX-CF, which may be important as well. Investigating these regime transitions is a subject of future investigation.

Other modeling studies that simulate convection in low-CIN and/or moist environments (e.g., Robe and Emanuel 2001; Anber et al. 2014) demonstrate the sensitivity of organized regimes of convection to wind shear. Many cases are well explained by WR04-type arguments, though for stronger shears there is a change in organizational behavior. Herein, only one initial shear profile is considered, but other two-dimensional (2D) simulations reported in Lane and Moncrieff (2010, hereafter LM10) expose the sensitivities of convective organization in this thermodynamic environment to wind shear (e.g., LM10, their Fig. 1). For weaker shears, the LM10 organized systems were broadly consistent with WR04-type arguments, likely because they were closer to the optimum ratio of cold pool strength to shear. LM10’s case with 15 m s^{−1} of shear over the lowest 2.5 km gives some suggestion of the formation of a DPR, and an additional 2D simulation (not shown) with 20 m s^{−1} of shear over the lowest 5 km (i.e., the same wind profile used here) clearly identified the emergence of the DPR. This latter simulation served as the original motivation for this present study. Thus, these 2D simulations suggest that the strength/depth of the shear is of paramount importance in determining the existence of the DPR. Of interest, LM10 also identify an upshear-propagating regime of weaker clouds, which is evident for most sheared cases and has relevance for Part I.

As discussed above, other simulations using the same initial thermodynamic and wind environments to those used here (Fig. 1) spontaneously produce the DPR. Specifically, the 2D LM10 simulations using the Clark–Hall model (Clark et al. 1996) produced the DPR, as did equivalent 2D simulations using CM1 (Bryan and Fritsch 2002). The vertical structure of the DPR in these other simulations was almost identical to that in Fig. 4. Moreover, they all exhibited almost constant propagation speeds, as in the DPR here, regardless of the instantaneous intensity of the system. These results suggest that, for this shear profile, the emergence of the DPR is insensitive to the choice of the numerical model and also the two- or three-dimensionality of the simulated system.

In the following section, we develop an archetypal dynamical model for the DPR to approximate the quasi-steady system-relative composite streamlines in Figs. 4 and 5 where the small-scale convective updrafts and downdrafts are averaged out. Nevertheless, noting the comments in section 3b, the small-scale convection and the parcel trajectories do affect the equivalent potential temperature of the mean state. An archetype is, by definition, the simplest possible (minimalist) idealization; for instance, the middle inlaid diagram in Fig. 13, adapted from Moncrieff (1992, hereafter M92), is the minimalist model of the squall line simulated by Thorpe et al. (1982).

## 4. Archetypal model for the downshear-propagating regime

Based on the general nonlinear conservation properties of mass, momentum, total energy, thermodynamics, and vorticity in the system-relative (Lagrangian) frame of reference (Moncrieff 1981), the archetypal model exactly relates outflow and inflow and provides the associated transport. An important point is that, despite the brutal simplification of setting CAPE = 0, the archetypal models reliably approximate propagating organized convection (Moncrieff 2010). The quotient of the work done by the horizontal pressure gradient across the system and the kinetic energy of the system-relative flow,

The inlaid diagrams in Fig. 13 identify three forms of the M92 archetype model for the Bernoulli number in the range −8 ≤ *E* ≤ 8/9. The inlay on the far right of Fig. 13 for *E* = −8 shows the relative inflow is entirely right to left, indicating a strictly propagating two-dimensional counterpart of the three-dimensional strictly propagating system of Fig. 16 in Part I. The inlay for *E* = 0 has three branches: an overturning updraft, an overturning downdraft, and a jump updraft, representing the archetypal standard mesoscale convective system. The inlay for *E* = 8/9 has two branches, an elevated overturning updraft and a full-depth jump updraft, but no downdraft, reminiscent of the following three-branch DPR model.

Figure 14 shows the idealized three-branch downshear-propagating system traveling with the environmental flow at height *z* = *h*_{*}, the steering level. In branch 1 of Fig. 14a, the outflow is higher and slower than the inflow, as is characteristic of a hydraulic jump and transition from supercritical to subcritical flow. Figure 10a shows that the inflow shear and the outflow shear for the jumplike branch 1 are about equal, consistent with the zero baroclinic generation of vorticity. In contrast to branch 1, the outflow speed from branch 2 exceeds the inflow speed. Figure 9b shows the three-dimensional upper-level flow separating around the cloud system. To aid mathematical tractability, we assume the inflow to branch 2 has a constant speed *U*_{2} equal to the vertically averaged inflow in the simulation. The flow separation of the upper-tropospheric streamlines is modeled by *W*, the ratio of branch-2 outflow and inflow widths. Table 1 shows that CAPE for the elevated branch 3 is negligible, so CAPE = 0 approximately. Therefore, the DPR model is automatically archetypal.

*E*

_{1}= 0.6 and

*U*

_{2}= 0.25, approximately. No real roots of the quadratic equation exist for

*W =*1 which eliminates a purely two-dimensional DPR archetype. Real roots of the quadratic exist for

*W*significantly larger that unity. For example, for

*W*= 2, the sole physically consistent solution is

*h*

_{*}= 0.4. The values of the other variables in Table 2 are calculated using the equations in the appendix.

Comparison of archetypal model parameters (*W* = 1, 2) with those derived from the simulation EX-CF. Dashes for *W* = 1 indicates that no real solutions exist for the quadratic equation.

The values of the remaining variables in Table 2 are estimated from Figs. 1, 4, 9, and 10. Table 2 shows that the archetypal model variables verify satisfactorily with the simulation results. The less satisfactory agreement between the archetypal and simulated values of *h*_{1} is likely because of the two-dimensional assumption used for branch 1. Comparison of the trajectories in Figs. 9a and 9b shows that branch 1 is three-dimensional, albeit much weaker than branch 2.

## 5. Summary and conclusions

The forward-tilted mesoscale system in Part II differs substantially from the rearward-tilted standard mesoscale convective system and the M92 archetype in particular. A rearward tilt has the constructive physical property that the updraft, where precipitation forms, overlies the mesoscale downdraft. Therefore, the evaporating precipitation can readily maintain the mesoscale downdraft via a combination of hydrometeor loading and evaporative cooling. In contrast, a subsaturated planetary boundary layer is detrimental to the forward-tilted system: hydrometeor loading and evaporation will disrupt the low-level inflow. A near-saturated boundary layer alleviates that issue by replacing the mesoscale downdraft with a jump updraft and establishing an elevated overturning branch.

Convective momentum transport is another distinction. The rearward-tilted standard mesoscale convective system is associated with negative momentum transport (M92, his Fig. 10a) distinct from the positive momentum transport of the downshear-propagating system (M92, his Fig. 10b). The positive momentum transport is consistent with the positive *u* perturbation in the overturning branch outflow of the EX-IF-FW simulation (Fig. 12b) and in the EX-CF simulation (Fig. 4b).

Part I and Part II of our investigation of long-lived mesoscale systems in a low–convective inhibition environment revealed atypical regimes of mesoscale organization. In Part I, we demonstrated that for upshear-propagating, rearward-tilted mesoscale systems, new convection is generated at the upshear edge of the density current rather than at the downshear edge, as for the standard model of mesoscale convective systems. The reasons were explained by the Moncrieff and So (1989) and Moncrieff and Liu (1999) dynamical analysis summarized in Part I. The upshear-propagating system is strongly three-dimensional in terms of the dry jumplike descent between the convective systems and the moist spiral-like overturning of the mesoscale updraft and downdraft (see Fig. 16 in Part I). This particular category of convective organization is reminiscent of the highly idealized Moncrieff and Miller (1976) tropical squall-line model.

The downshear-propagating, forward-tilted convective system examined in Part II is characterized by borelike jump updrafts, an elevated overturning updraft, and the absence of a mesoscale downdraft, which contrasts with the key role of density currents in the standard mesoscale convective system. The downshear-propagating system is weakly but necessarily three-dimensional, as shown by the separation of the upper-tropospheric flow around the cloud system.

We noted some resemblance between our results and the Parker and Johnson (2004) mesoscale system observations. It would be interesting to examine if these results have other observational applications.

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

# APPENDIX

## Mathematical Formulation

*L*

_{1}≤

*x*≤

*L*

_{2}, 0 ≤

*z*≤ 1 of Fig. 14 is given byFor region 1,Since

*u*(

*L*

_{1},

*z*) = −

*U*

_{1}+

*Az*for 0 ≤

*z*≤

*h*

_{1}and

*u*(

*L*

_{2},

*z*) = −1 +

*Az*for 0 ≤

*z*≤

*h*

_{0}, it follows thatwhere

*W*≥ 1 is the width of the outflow region in the transverse (

*y*) direction. The transverse width of the other two regions is unity.

*h*

_{0}≤

*z*≤

*h*

_{2}, it follows thatCombining Eqs. (A2) and (A4) giveswhere

*z*= 0) and the upper boundary (

*z*= 1) is given by the sum of the kinetic energy and the work done by the horizontal pressure gradient energy: the Bernoulli equation,

*p*is the pressure. Applied along the horizontal lower and upper boundaries, the Bernoulli equation givesandThere are no sources of vorticity, so in region 1 the outflow shear

*A*is equal to the inflow shear, giving

*A*=

*U*

_{1}/

*h*

_{1}= 1/

*h*

_{*},andIntegrating the momentum equation ∂/∂

*x*(

*ρu*

^{2}+

*p*) + (∂/∂

*z*)(

*ρuw*) = 0, over the domain

*h*

_{*}< 1 are physically acceptable.

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