Convective momentum transport and multiscale organization in simulated shear parallel mesoscale convective systems

The way in which moist convection interacts with large scale flows is a major contemporary research issue. Organized mesoscale systems are, in particular, important for the interactions between convection and the ambient shear. Here we present numerical simulations of mesoscale systems evolving in a background shear using the Research and Weather Forecasting model. We are particularly interested in the long time integration, allowing the systems to repeatedly develop and die and effectively interact with the background shear. Starting with a typical African and equatorial jet-shear, the simulated solution goes through various phases or stages. First, a transient state, consisting of scattered squall-like systems that are aligned perpendicular to the background shear, develops and then evolves into a regime of multiscale mesoscale systems with large stratiform anvils. During the latter period the background wind changes substantially through the effect of both up scale and down scale convective momentum transport. At this stage, the systems become aligned parallel to the wind shear, with elongated stratiform anvils in which meso-beta scale convective cells evolve and propagate in the shear direction, relative to the stratiform anvils. These results are reminiscent of the development of shear parallel mesoscale convective systems observed for instance in the Eastern Pacific ITCZ and corroborate recent theoretical results obtained with a simple multi-cloud model. As such they have important implications for the parameterization of CMT in climate models.


Introduction
A major contemporary research issue in organized tropical convection is a good understanding of the way in which moist convection interacts with the large scale flow and waves in particular, at various spatial and temporal scales (Houze 1982;Chen et al. 1996;Stevens 2005;Moncrieff et al. 2007;Wu and Moncrieff 1996;Moncrieff 2010;Tompkins and Semie 2017). Organized mesoscale systems constitute a conveyer belts through which small scale convective cells interact with the ambient shear (Dudhia and Moncrieff 1987;Tao and Moncrieff 2009;Houze 2004;Parker and Johnson 2004;Weisman and Rotunno 2004;Lane and Moncrieff 2010). They are also important to study in their own right as they induce a great deal of variability in winds and precipitation, especially in the monsoon regions and in the Eastern Pacific inter-tropical convergence zone (ITCZ) and their impact on the global circulation and precipitation budget is poorly understood (Khouider and Moncrieff 2015;Liu and Moncrieff 2017).
Moreover, mesoscale convective systems are believed to strongly interact with convectively coupled waves and the Madden Julian Oscillation (MJO), see for example Madden and Julian (1972), Zhang (2005), Moncrieff and Klinker (1997), Nakazawa (1988), Fovell and Tung (2016), Majda (2007) and Majda and Xing (2010). To address the two-way interactions between mesoscale systems and the large scale flow, several studies have been devoted to the evolution of mesoscale convective systems in a sheared environment, combining the analysis of observational records, theory and numerical modelling (Dudhia and Moncrieff 1987;Chen et al. 1996;Houze 1982Houze , 2004Liu and Moncrieff 2017;Majda and Xing 2010;Parker and Johnson 2004;Khouider and Moncrieff 2015;Majda and Stechmann 2008).
Using numerical simulations, Dudhia and Moncrieff (1987) pointed out the importance of the low-level shear in the alignment of organized rain bands along the shear direction, that are abundant in the GATE observational records. They used a small domain of 50 × 25 km 2 and imposed a large-scale ascent to maintain organized deep convection. The smallness of the domain prevented the development of self-sustained organized convection.
In a similar vein, here, we present the results of new numerical simulations of mesoscale systems, using the Research and Weather Forecasting (WRF) model, on a large enough domain, with a focus on the long time integrations allowing the systems to repeatedly develop and die and interact with the background shear. Starting with the sametypical African and equatorial jet-shear used by Dudhia and Moncrieff (1987), as a background, the simulation goes through various phases or stages. First, a transient state, consisting of scattered squall-like systems that are aligned perpendicular to the background shear which then evolve onto multiscale mesoscale systems with large stratiform anvils. During this period the background wind goes through a major transformation through the effect of both up scale and down scale convective momentum transport (CMT), resulting in a first baroclinic flow structure resembling the MJO zonal wind. At this stage, the mesoscale systems turn around and become aligned parallel to the shear and are characterized by elongated stratiform anvils in which mesobeta scale convective systems evolve and propagate parallel to the shear direction, with much slower speeds compared to the main stratiform envelope, which moves at a much faster speed consistent with an upper level steering level wind. This is reminiscent of recent theoretical results obtained by Khouider and Moncrieff (2015), using a multi-baroclinicmode-multi-cloud model that exhibit instability of mesoalpha and meso-beta scales modes of similar nature. Here, the three-way interactions between the meso-beta convective cells, the meoscale stratiform systems and the background wind shear, in the WRF simulation, will be discussed and particular attention will be given to the transport of momentum and moisture across scales, which are found to be the key dynamical drivers of these disturbances.
It is worth noting that the long period run is necessary as it allows not only for the systems-due respectively to the meso-alpha and meso-beta instabilities-to develop but to interact with each other and with the environment. This would among other aspect provide sufficient data to cary statistical analysis of the morphological and dynamical features of the systems as well as their impact on the background shear by drawing for instance time averages turbulent transports. As already seen in Khouider and Moncrieff (2015), the spontaneous development of shear parallel systems-due to the double multicloud instabilitybegins with the development of scatted convections which first aggregates into shear-perpendicular squall-like systems before they develop into shear parallel systems composed of slowly moving meso-beta rain bands embedded within fast moving meso-alpha stratiform systems. Also, an important issue of particular interested is whether the mesoscale systems are able to create a significant change in the background shear to provoke their own demise as it occurs in the idealized experiment of Majda and Stechmann (2008) or the systems will instead persist, maybe under different forms. As we will see below, although the systems are capable to induce enough momentum transport both upscale and downscale (friction) and significantly alter the initial shear, the changes in the shear morphology are made in such a way that the systems persist for a long time-at least throughout the simulation period of 30 days. Instead of weakening abruptly, the changes of the environment are gradual. Once developed the coupled convective and stratiform (or slow and fast as referred to below) seem to feed into one another and somehow self sustain for at least the duration of the simulation.
The paper is organized as follows. The WRF simulation environment is described in Sect. 2. Section 3 discusses the results starting in Sects. 3 and 3.1 with the description of the multiscale organized rain bands through which the mesoscale organization of convection first manifest itself in the form of scattered squall lines right after the convection onset. It is followed by the illustration of the two-way interactions between the convection organized systems and the mean flow in Sects. 3-3.2. A more in depth investigation of the dynamical and physical features of the convective and stratiform wave modes is given in Sects. 3-3.3. A concluding discussion is giving in Sect. 4.

The simulation set up
We use the Weather Research and Forecasting model version 3.2.1 on a 450 × 450 km 2 square domain with a 1 km resolution in the horizontal and 50 vertical levels. We note that the 1km resolution is typical for cloud permitting models and allows to resolve fairly well mesoscale cloud systems although the clouds themselves are not fully resolved (Janiga and Zhang 2018;Liu et al. 2017;Guichard and Couvreux 2017). The time step is 3 s. For microphysics, the WRF single-moment (WSM) five-class scheme of Hong et al. (2004) is used. Further, the sophisticated rapid radiative transfer model (RRTM) scheme is used for longwave radiation (Mlawer et al. 1997) and the Goddard scheme (Chou and Suarez 1994) is adopted for shortwave radiation. The parameterization of boundary layer processes is handled by the Yonsei University (YSU) scheme (Hong et al. 2006).
The surface is defined to be an ocean with a fixed uniform temperature of 302 • K. The initial conditions consist of a double jet mimicking the African Easterlies at 600 hPa and the Tropical Easterlies at 200 hPa (Dudhia and Moncrieff 1987), which is also used in both Dudhia and Moncrieff (1987) and Khouider and Moncrieff (2015) and a stably stratified potential temperature profile, typical for the equatorial atmosphere. The zonal wind and potential temperature initial profiles are illustrated on the top two panels in Fig. 1. All other variables, including moisture, are initially set to zero, and particularly unlike Dudhia and Moncrieff (1987), a large scale ascent is not imposed here as the comparatively large domain allows the development of self-sustained mesoscale convective systems. One could alternatively set the moisture to its observed climatological profile over the ITCZ for instance and such setting would have perhaps the merit for leading to an earlier onset on convection in the simulation but since we know that the main dynamical instabilities are not very sensitive to changes in the moisture background parameter, based on the previous theoretical work Fig. 1 Initial Dudhia-Moncrieff shear (A) and potential temperature (B) profiles. Growth rates diagram (C) and physical structure (D, heating (solid red) and cooling (dashed-blue) contours and velocity arrows) of meso-beta and meso-alpha mode in the multicloud model of Khouider and Moncrieff (2015) of Khouider and Moncrieff (2015), we do not expect the present results to change, qualitatively at least, if a non zero moisture profile was used instead.
The equations of motion are solved as an initial value problem with uniform sea surface temperature, upper atmospheric gravity wave damping, boundary conditions. Horizontally, the boundary conditions are periodic in the initial shear direction, x, and symmetric boundary conditions are used in the transversal direction, y. Although, the effect of Earth's rotation is negligible at such scales, we will sometimes refer to the direction of initial wind shear as the zonal direction or longitude while the transversal direction is sometimes referred to as latitude. For the vertical coordinate we interchangeably use the pressure, p, or the elevation from the sea surface, z, and t refers to time. We are particularly interested in the longtime behavior to allow two-way interactions between meso-scale organized convective systems and the large scale-domain mean flow.

Multiscale organized rain bands
The simulation is run for 30 complete days, between the fictitious dates of December 1, 2000 to December 30, 2000 and the six (6) hourly accumulated precipitation during the period of convective initiation, corresponding to the first 2 days of the simulation, and during a mature period between midday December 13 and midday December 15, are reported in Figs. 2 and 3, respectively. The simulated atmosphere remains dry during the first 24 h or so ( Fig. 2a) but as soon as convective precipitation develops, it quickly intensifies and gets organized into rain bands oriented at a roughly 45 • angle with respect to the imposed shear direction (Fig. 2b, c). Shortly after, the rain bands flatten and become mostly parallel to the imposed wind direction and appear to evolve in horizontal channels or corridors Fig. 2d-f), which suggests that conditions favourble for the underlying dynamics are persisting. Similar formation of such a convection favorable corridor has been observed in the numerical simulation of Khouider and Moncrieff (2015) using an idealized primitive equations model with a crude vertical resolution reduced to the first four baroclinic modes and a multicloud convective parametrization (Khouider and Majda 2006). However, as can be seen from Fig. 3, here, the transversal location of the rain-favoring corridors changes rather rapidly from one frame to another.
This can be evidenced for example by noticing that the rain bands in the bottom panels of Fig. 3 for example move from being mostly focussed along the line of zero relative latitude a time 00:00DEC15 to extending mostly towards negative latitudes at time 06:00DEC15 to refocussing around the line of relative latitude 0.5. In Khouider and Moncrieff (2015) the convection-favoring corridor is associated with a Hadley-like mean circulation characterized with transversal convergence at low-level and divergence in upper troposphere allowing the development and persistence of stratiform clouds along the flanks of the convection corridor. Such local traversal flow seems to be lacking in the present WRF simulation as demonstrated below in the next subsection showing a rather weak and oscillating meridional wind.
In Fig. 4A and B, we show the colour-shading of the vertical velocity overlaid by the contours (black) of cloud water, at time 12:00 Dec. 15, averaged over lower and upper troposphere, respectively. We note that unlike the corresponding 6 hourly accumulated precipitation plot on the last panel in Fig. 3, the convection activity is confined to a narrow region of the domain. The snapshot exhibits narrow cells of strong updrafts expanding over the whole troposphere-a signature of deep convective cells that are flanked by weaker updrafts and downdrafts. In the lower troposphere the downdrafts form an almost contiguous region south of the mostly redupdraft dominated patch. Comparing to the upper tropospheric plot on the right, these lower tropospheric downdrafts are topped by weak updrafts suggesting a region dominated by stratiform clouds. On the other hand, there are upper tropospheric downdrafts narrowly surrounding the strongest upper tropospheric updrafts that are themselves on top of weak updrafts in the lower troposphere suggesting the prevalence of congestus clouds. The regions of congestus activity are less organized, compared to the stratiform wake, but they consistently flank the deep convection cells. The maximum updraft values of about 1 m s − 1 are relatively low compared to observations or large eddy simulation results. This is likely due to the fact that our grid resolution of 1 km does not fully resolve the updrafts and these values should instead be interpreted as averages over 1 km wide horizontal areas of updrafts.
In Fig. 5, we present the Hovmöller diagrams of precipitation and 850 hPa zonal wind along the initial wind direction, averaged in the meridional direction. Two distinct periods are shown. Panels a and b represent the evolution of zonal wind and precipitation, separately, during the early stages of the simulation, namely the first 4 days, while the panel c shows the zonal wind colour shading beneath the precipitation contours for the period of 3 days, between 14 and 16 December. Despite the complex vertical structure of the initially imposed wind, convective disturbances appear as eastward moving streaks in both the precipitation and zonal wind fields at the start of the precipitation onset at Day 2.
Wave streaks, more visible on Fig. 5c, in the total zonal winds become evident only at later times, when the mean wind weakens considerably, as shown below. Of particular interest is the fact that the wind streaks seem to move at a fast speed of about 20 m s −1 as indicated on that panel together with envelopes of active and break phases, in terms of precipitation activity. The individual precipitation events on the other hand appear to move at a much slower speed of about 6 m s −1 .
The individual convective events last only a short period of a few hours, as indicated by the small and elongated closed contours, while the zonal wind bearing streaks run thought the length of the domain and often circle around according to the periodic boundary conditions. This superposition of two wave modes one carrying the individual rain events and moving slowly (here at roughly 5 m s −1 ) and the other representing a modulating mesoscale wave envelope and moving at a much faster speed (here at roughly 20 m s −1 ) are consistent with the multiscale theory for shear parallel convective bands put forward in Khouider and Moncrieff (2015) based on a modified version of the multicloud model of Khouider and Majda (2006).
According to Khouider and Moncrieff (2015), in the presence of the same mean wind shear as in Fig. 1A, the modified multicloud model (MCM) presents a double instability of mesoscale-like convective systems one at the mesoscale alpha and one at the mesoscale beta wavelengths, in addition to the intensively documented instability at synoptic scales ( Fig. 1C), representing convectively coupled equatorial waves, which characterized the earlier versions of the Fig. 2 Successive snapshots of horizontal distribution of 6 hourly accumulated precipitation precipitation during the first 2 days. Precipitation starts during Day 2 and quickly intensifies and forms organized rain bands aligned at an angle of roughly −45 • with respect to the shear direction at 12:00. The rain bands then become elongated and align parallel to the shear direction from time as seen on the bottom panels multicloud model. As demonstrated by nonlinear simulations, the meso-beta wave carries upright convective lifting while the meso-alpha instability is associated with mesoscale stratiform systems that provide the convective envelopes within which the convective cells-events evolve (Khouider and Moncrieff 2015).
The structure of the associated two linear modes and instability diagram from the Khouider and Moncrieff (2015) study are illustrated on the bottom two panels (d and e) of Fig. 1 suggesting two unstable wave-modes moving in the same direction though with considerably different speeds. Nonetheless, in Khouider and Moncrieff (2015)'s simulation, the stratiform waves move to the left (eastward) while the individual convection cells move to the right (westward) whereas in the present case both waves move to the left. As will be apparent below, the main reason behind this switching around in the propagation direction of the meso-beta convective events is related to the fact that the near surface wind remains westerly through the Khouider-Moncrieff MCM simulation despite a few important other changes in the mean flow while in the present simulation the near surface wind completely changes direction from westerly to easterly by Day 5 as will be seen in the subsection below. This may be due to small scale turbulent eddies that are not captured by the four baroclinic modes in Khouider and Moncrieff (2015)'s simulation.
Before we dig further into the role played by the background wind in the development of the multiscale waves, we present in Fig. 6 the Hovmöller diagrams of the vertical velocity (a-c) and water vapour mixing ratio (d-f), both taken at three different vertical layers, namely near the surface (a, d), in the mid-troposphere (b, e), and in the upper troposphere (c, f), for the time period between Day 14 and Day 17. We note that the vertical velocity in a-c is taken at the three fixed levels 850 hPa, 600 hPa, and 300 hPa, respectively while the water vapour in Panels d-f is the average between 950 and 850 hPa, between 850 and 500 hPa, and 300 and 100 hPa, respectively. The surface precipitation contours are overlaid on each panel. From the vertical velocity panels (a-c), we see that the updrafts are in phase with the slow propagating precipitation events concurring with the convective lifting while the subsidence streaks, especially in the upper troposphere, are carried by the fast moving stratiform waves. Notice however that the slow moving convective events often terminate with traces of significant subsidence (the blue coloured patches at the tips of the precipitation events), more visible at 600 hPa, which mark the detrainment of the deep convective clouds. For the water vapour on the other hand (d-f), we can distinguish three scales of variability. In addition to the slow and fast waves, the first corresponding to positive moisture anomalies running parallel to the precipitation events, that are visible at all three levels, we can see super slowly-propagating positive disturbances, especially on Panel d (950-850 hPa), moving roughly at 3.5 m s −1 (slower than the precipitation events), concurring with the near surface mean zonal wind speed hPa, c 300 hPa) and water vapor mixing ratio averaged over the indicated atmospheric layers (d 950-850 hPa, e 850-500 hPa, f 1000-300 hPa) with surface precipitation contours overlaid. Lifting is carried by slowly moving rain bands while the bulk of the subsidence follows the fast-moving stratiform wave. There is also sign of divergent gravity waves moving in the opposite direction visible in the 300 hPa vertical velocity contours. The dashed lines on the top right panel indicate wave disturbances moving at three separate speeds at the same time of the simulation, as we will see below. These (moisture) disturbances presumably correspond to moistening events associated with non precipitating shallow cumulus and/or cumulus congestus clouds that detrain in the lower troposphere as they are advected by the low-level background wind.

Two-way interactions between the convection organized systems and the mean flow
In Fig. 7a-c, we plot the time evolution of the zonally averaged meridional velocity (v) in the time-latitude space during roughly the first 17 days of the simulation. During this period, the evolution of v undergoes two main phases, in addition to the quiet-transient period preceding the convection onset, between Day 1 and Day 2. There is a first short episode of roughly 3 days, spanning from Day 2 to Day 5, characterized by meridionally propagating disturbances corresponding essentially to the squall line-like convective bands seen on the first few panels of Fig. 2. This first phase is followed by a second period characterized by oscillations of horizontally elongated strips of wind disturbances. It is clear from these plots that a mean Hadley-like circulation doesn't take place, unlike the Khouider and Moncrieff (2015) idealized simulation using a multicloud parameterization. Also, the oscillation between negative and positive mean wind values suggest upscale and downscale momentum transport in the transversal direction by the prevailing mesoscale convective systems; The oscillation period of roughly 12 h is right between the 18 h-time it takes for slow moving convective-precipitation events and the 6 h-time it takes for slow moving-stratiform waves to cross the periodic-computational domain.
As shown in Fig. 7d, similar oscillations occur in the mean zonal wind as well. This suggests that the upscale momentum transport which drives the mean wind oscillation seen in Fig. 7 results from a cooperation between the two waves so that depending of the wave alignment (trough to trough or crest to crest) may result in acceleration (deceleration) of near surfaces northerlies (southerlies) and vice versa. This however remains purely hypothetical which merits further investigation and quantification and will be conducted and reported elsewhere by the authors.
In Fig. 8A and C, we show the time evolution of the mean zonal and meridional wind vertical profiles. Compared to its initial state, the zonal wind undergoes important transformation resulting mainly in severe reduction of the shear with a tendency to an overtime homogenization. This is in fact confirmed with the instantaneous plots in Fig. 8B which show the mean zonal wind profile at various times during the simulation. Interestingly, while the strength of the wind and wind shear decreases overall, there are two particular heights at which the mean wind experiences some radical changes. The first such change occurs near the surface where the mean wind goes from being westerly with roughly 3 m s −1 speed to becoming easterly by Day 13 with a −3 m s −1 speed and remains so throughout the rest of the simulation while it continues to evolve at higher levels. The other location is between 500 hPa and 400 hPa. There, the mean wind goes from being neutral to acquiring an easterly wind of nearly −9 m s −1 which then slowly decays to about −3 m s −1 at Day 30. The intermediate wind profiles at Day 5 and Day 13 suggest a vertical mixing due to turbulent eddies of some sort. However, the systematic development of easterly wind near the surface, out of initial westerlies, warrants against a purely turbulent dissipation mechanism. Upscale and downscale mesoscale convective momentum transport is most certainly at play.
The evolution of the meridional wind profile plot in Fig. 8C confirms that the oscillatory behaviour of the meridional wind is achieved throughout the troposphere and is most significant between Days 12 and 17, consistent with Fig. 7c. We note in particular the nearly third baroclinic structure of the meridional mean velocity consistent with mesoscale convective momentum transport due to tilted mesoscale systems (Majda and Biello 2004;Majda and Stechmann 2008;Khouider et al. 2012;Khouider and Moncrieff 2015). Also, the oscillatory nature of the meridional mean wind profile is carried through the tropospheric height results in nearly neutral mean meridional wind so there is no energy transfer from the initial mean zonal flow to the meridional component, instead these meridional oscillations result from upscale and downscale momentum transport as the two convective mesoscale systems, moving at separate speeds, interact with each other and force the transversal large scale wind.

Dynamical and physical features of the convective and stratiform wave modes
In Figs. 9 and 10 we plot the composited vertical structure of the fast moving-stratiform wave-mode and the slow-moving convective disturbance, respectively, including zonal and vertical velocity components, temperature and water vapour anomalies. The structure in Fig. 9 shows a clear wave form resembling stratiform dominated mesoscale systems with a significant backward tilt within the troposphere, below 250 hPa, in all the shown fields (Moncrieff 2004;Khouider and Moncrieff 2015;Liu and Moncrieff 2017;Houze 2004;Parker and Johnson 2004;Lafore and Moncrieff 1989). This backward tilt of the wave dynamical fields, with respect to the direction of propagation, is common to almost all propagating convectively coupled disturbances, from meso-to synoptic to planetary scale disturbances  (Mapes et al. 2006;Kiladis et al. 2009;). It is termed the self-similarity feature of tropical convective systems and is believed to play an important dynamical and thermodynamical role in both the propagation and the maintenance of the wave (Mapes 2000;Majda and Shefter 2001;Khouider and Majda 2006).
Notice the boomerang shape of the zonal wind and temperature fields with the forward tilt above 250 hPa, reminiscent of convectively coupled Kelvin and inertiagravity waves as seen both in observations and numerical simulations (Wheeler and Kiladis 1999;Straub and Kiladis 2002;Haertel and Kiladis 2004;Khouider and Han 2013), although the wavelength is much smaller to that of Kelvin waves and the wave propagates in a direction is opposite to that of Kelvin waves. The forward stratospheric tilt is believed to be a signature of vertically propagating stratospheric gravity waves due to a moving heat source in the troposphere, associated with the wave itself (Wheeler and Kiladis 1999;).
The enhanced positive moisture disturbance in the midtroposphere, between 850 and 450 hPa on the bottom right panel of Fig. 9, is probably associated with the evaporation of stratiform rain in the wake of the wave which is followed by strong subsidence, evident from the blue colour in the w contours on the bottom left panel.
The slow convection wave shown in Fig. 10, on the other hand, doesn't seem to display the same coherent tilted structure. Instead, we see mostly a strong vertical velocity disturbance in the upper troposphere, from the bottom left panel, topped by a strong divergence as can surmised from the zonal wind field on the top-left panel, roughly between 250 and 150 hPa. This strong lifting in the upper troposphere coincides with warm temperature anomalies (top-right), suggesting a region of strong deep convection. In the lower troposphere, the vertical velocity is very weak and appears to be very scattered. This is more likely due to cancelations between updrafts and surrounding downdrafts as can be inferred from Fig. 4. Notice the warm temperature anomalies near the surface (to the right, roughly between grid point 100 and 150) that lead the deep convection phase which is characterized by cold temperatures at the surface (cold pool) and warm temperatures aloft. This suggests a build up of a convective instability prior to the onset of deep convection which in turn consumes the instability. There is also a significant moisture anomaly ahead of the convection centre which is followed by strong drying due to convective moisture transport. However, the most striking feature resides in the strong positive moisture anomaly in the mid-troposphere slightly leading the convection centre by maybe 50 grid points (50 km). Clearly, this is a result of detrainment of congestus clouds, which are noticed in Fig. 4, that lead deep convection and which moisten and precondition the environment prior to deep convection (Khouider and Majda 2008;Waite and Khouider 2010).
The convective mode represented in Fig. 10 is arguably the engine that drives the stratiform dominated mesoscale system of Fig. 10 while the later is the conveyer belt that supplies the fuel. They are believed to be the manifestation of the meso-beta and meso-alpha instabilities reported by Khouider and Moncrieff (2015) and illustrated on the bottom panels of Fig. 1.
In Fig. 11, we plot the composite of the total moisture tendency, q∕ t (a, b), the contribution to this tendency due to phase change (c, d), namely, the evaporation minus condensation, including sublimation and deposition, and the total liquid and solid water (e, f), cloud condensate (ice + liquid water) and rain, overlaid by the temperature contours. The corresponding u, w flow arrows are overlaid on top in each panel. The two waves are shown simultaneously. The faststratiform mesoscale system is shown on the right panels (a, c, e) and the slow-convective disturbance is shown the left panels (b, d, f). For the stratiform wave, we see significant drying, q∕ t < 0 , in the mid-to-upper troposphere followed by substantial moistening at the same elevation. A quick look at Fig. 9 shows that the moistening phase coincides with the region of positive vertical velocity, at those heights, suggesting active convection.
The corresponding phase change contribution (condensation minus evaporation) in the middle left panel (c) shows evaporation dominating near the surface, likely due to the evaporation of rain, which is substantially enhanced between grid points 250 and 400 and negative tendencies (condensation) taking place almost everywhere else and more importantly and more significantly in the mid-to upper troposphere between 500 and 450 hPa, where the convective lifting takes place. However, this is at odds with the total moisture tendencies on the top panel. A close look at the wave's moisture profile in Fig. 9 suggests that the drying and moistening near the dominant mid-troposphere is due essentially to the advection of the moisture anomalies by the anomalous wind which also seem to transport moisture near the surface upward and backward, following the wave tilt, directly to where the strong condensation/deposition takes place. Notice the maximum moisture anomaly on the bottom right panel of Fig. 9 which coincides with the maximum upward and rearward winds between zonal coordinates 250 and 400 km and pressure levels 750hPa and 350 hPa. This yields a negative moisture tendency (drying) below and to the right of the moisture maximum, where the gradient of  , d, e) waves. Notice that the condensation and evaporation go only to 300 hPa moisture is positive, and positive moisture tendency above and to the left of it, where the gradient of moisture is negative. The drying phase in front is simply the opposite phase of what has been described, with now the descending winds coinciding with the minimum moisture anomaly in Fig. 9. The cloud condensate plus rain colour plot on the bottom panel, clearly shows a large stratiform anvil in the upper troposphere extended downward with supposedly rain and snow falling to the ground while some of it eventually evaporates near the surface. The near surface moisture is instantly transported upward and backward leading to the moisture profile in Fig. 9 and the substantial moistening in the mid-to-upper troposphere. Thus the mesoscale stratiform wave is primarily driven by the dynamical moisture transport by the wave wind anomalies, fuelled by surface evaporation and maintained by recirculation of rain evaporation near the surface.
For the convective wave on the other hand, a close inspection of Fig. 10 and the panels (b, d, f) on the right of Fig. 11, in the light of the above discussion, suggests that to the contrary this wave-mode is essentially driven by grid scale updrafts in the lower troposphere forming congestus clouds that detrain and moisten the mid-troposphere. The organization of convection takes place only in the upper troposphere where the upward wave winds dominate and coincide with the negative moisture gradients above 550 hPa and which can explain only the upper tropospheric moistening while the drying in the lower troposphere is arguably due to grid scale updrafts including congestus cloud formation. Notice the strong near surface evaporation (right-middle panel of Fig. 11) that coincides with the cold pool visible from the temperature contours on the lower panel, which also shows significant cloud condensate in the mid-to upper troposphere which extends into a stratiform-like wake to the right.
Similarly, to Fig. 11, we plot on the panels a and b of Fig. 12 the running average momentum transport, (u � w � ) z , where u ′ , w ′ are the anomalous zonal and vertical wind components, respectively, associated with the fast and slow waves. The averaging is carried in the frame of reference moving with the wave between times 00Z15DEC and 12Z15DEC. This quantity is often referred to as convective momentum transport (CMT) due to mesoscale convective organization and in essence it is the amount of forcing the waves exert on the mean flow. As can be seen from the left and right panels, both the stratiform and the upright convective wave-modes provide a positive CMT in the middle troposphere, and negative in the upper and lower troposphere. This translates into the dampening of the already negative mean wind (Fig. 8a@15DEC) in the middle troposphere and a weak acceleration in the lower and upper troposphere, which thus provides a downscale and upscale transport of energy from the mesoscale convective system into the mean flow.
On the two panels c and d of Fig. 12, we plot the projected CMT onto the first three baroclinic modes, namely, cos(z), cos(2z), cos(3z) . We note that the trigonometric profiles are not the actual baroclinic mode profile in the presence of non-uniform stratification but they represent a good approximation (Tulich et al. 2007). As it can surmised from the CMT projected plots and mean shear profiles the first baroclinic mode is damped by both the slow and fast waves while the second and third baroclinic modes appear to be accelerating.
The acceleration of the third baroclinic mode is consistent with the simulations of Khouider and Han (2013) and the theoretical work of Majda and Stechmann (2008) and Khouider et al. (2012). However, instead of seeing a sustained acceleration of the wind towards a third barocliniclike vertical structure, the successive profiles in Fig. 8b suggest rather a uniform wind limit. The homogenization and mixing is presumably due to cumulus friction and turbulent eddies due to purely shear instability. However, the complex path followed by the evolution, through time, of the mean wind, as suggested by Fig. 8a, b, hints to non trivial impact of the mesoscale CMT on the mean wind apparent from the upper tropospheric maximum exhibited by the yellow profile corresponding to the mean wind at 13 Dec 2000 (20001213). Moreover, the third baroclinic mode structure is clear in the meridional wind in Fig. 8c although we only computed the CMT in the x-direction.
Also worth noting, theoretical studies suggest that only tilted systems would result in non zero CMT (Majda and Stechmann 2008;Khouider et al. 2012;Moncrieff 2004) but as can be seen from Figs. 9 and 10, only the stratiform wave is tilted. The theory is based on purely linear sine waves while here the wave are highly nonlinear and particularly the convective mode is a collection of grid scale updrafts.

Summary and concluding discussion
The results of a numerical simulation of multiscale convective systems evolving in a background vertical shear are presented here. We run the WRF model for a period of 30 days on a 450 × 450 km 2 domain, with the 1 km horizontal resolution. We used a uniform SST and the initial conditions consist of a stably stratified temperature profile and a background zonal wind with a vertical shear mimicking a combination of the tropical and African jets, in the upper and middle troposphere, respectively. After an adjustment period of roughly 2 days, scattered convective events begin to appear randomly within the domain. A few hours later, the precipitation is organized into squall-like rain bands that run at angle with respect to the background shear and by Day 3, mature systems that are aligned parallel to the shear direction, exhibiting a clear multiscale nature as can be surmised from Fig. 2, develop and evolve.
These systems are reminiscent of the shear parallel mesoscale systems simulated by Dudhia and Moncrieff (1987) and reproduced by a multicloud model in Khouider and Moncrieff (2015), that are abundant over the Tropical Atlantic and the Eastern Pacific intertropical convergence zone (Dudhia and Moncrieff 1987;Khouider and Moncrieff 2015). In fact, the Hovmöller diagrams in Figs. 5 and 6 suggest that the solution consists mainly of two propagating mesoscale disturbances one moving at a slower speed of 6 m s −1 and forms and envelope carrying the individual grid scale storms while the other moves at a faster speed of 20 m s −1 and is associated with the larger scale stratiform anvils, as predicted by linear theory (Khouider and Moncrieff 2015). This is indeed confirmed by the vertical structure composites shown in Figs. 9 and 10 for the fast-stratiform wave and the slow-convective envelope, respectively. The two waves coincide respectively with the meso-alpha and mesobeta instabilities seen in the multicloud model linear solution reported in Fig. 1 (Khouider and Moncrieff 2015).
A close inspection of the moisture budget for the two waves, in Fig. 11, suggests that the fast-large scale-stratiform wave is self sustained through a return flow that carries moist air from the near surface layer, dominated by the evaporation of rain, to the upper troposphere where it condenses and forms stratiform clouds. Whereas the slow-convective disturbance is maintained by grid scale convective events among which congestus clouds detrain in and moisten the mid-troposphere. The deposited moisture is then transported upward by an organized convection loop in the upper troposphere as can seen on the right panels of Fig. 11. In the lower troposphere grid scale updrafts and surrounded by strong downdraft and cancels each other on average.
Even though only the fast-stratiform wave exhibits a strong background tilt, reminiscent of tropical convective systems, both disturbances have a strong impact on the background wind shear, namely, through CMT. As can be seen in Fig. 12, both waves induce an acceleration of the mean wind in the third baroclinic mode as predicted by theory and modelling results (Moncrieff 2004;Majda and Stechmann 2008;Khouider et al. 2012;Khouider and Han 2013). Nonetheless, the mean wind slowly homogenizes and decays as suggested by the plots in Fig. 8. While this systematic homogenization and decay is presumably due to cumulus friction and other shear instability driven turbulent eddies, the impact of CMT from the two mesoscale waves on the mean wind is evident from the apparent regular oscillations in the zonally averaged meridional and zonal flows in Fig. 7. The period of the oscillations is roughly 8-12 h. This time is roughly equivalent to the time period it takes the two waves move relative to each other; this is equivalent to the period of a wave moving at the relative speed of 20 − 6 = 14 m s −1 and a wavelength of 400 km (the domain width). Indeed, 14 ms −1 × 8h ≈ 403 km.
The simulation results presented here suggest an intriguing multi-scale structure of shear parallel mesoscale systems, unlike the commonly known shear perpendicular systems and squall lines. They provide substantiated evidence for the theoretical work of Khouider and Moncrieff (2015) that shear parallel MCS's are effectively controlled by the combined mesoscale alpha and mesoscale beta dynamical instability mechanism exhibited through the multicloud model. Moreover, the nontrivial synoptic scale oscillation, of 8-12 hin the mean wind attributed to CMT due to these waves warrant against the dismissal of such systems in the dynamical budget of the large scale atmospheric circulation. It speaks in favours of including the parameterization CMT due such systems in global circulation models.
It is worth recalling that convective momentum transport due to titled mesoscale systems and in particular shear parallel systems of the type simulated here is quite distinct from the classical CMT due to upright convection which is often included as part of a mass flux parameterization for instance (Zhang and Cho 1991). Because convective parameterizations typically used in GCMs are column based, they cannot represented titled mesoscale systems and the associated stratiform 2nd baroclinic mode that characterize them but instead they are designed for upright convection due to deep convection only and as such the resulting CMT is only downscale. As previously noted the tilt is important in order to produce upscale CMT (Majda and Biello 2004;Majda and Stechmann 2008;Khouider et al. 2012;Khouider and Moncrieff 2015). A CMT parameterization for shear parallel systems based for example on the Khouider and Moncrieff (2015) model or a simplified/modified version of it-especially in the light or more recent developments Grant et al. (2020) and in the lines of Khouider et al. (2012) for example-can be used in tandem with a classical convective parameterization as done for instance in Moncrieff et al. (2017) albeit the mesoscale model used there is much simpler and shear parallel systems were not particularly targeted.