The simulations presented in this study are conducted with the cloud-resolving model CM1 version 19.8 . The default horizontal grid size is 120 km by 120 km, with a default simulation time of 2 hours (9600dt). The vertical extent of the domain is 20 km. A sponge layer exists in the upper 5 km, which damps upward-propagating gravity waves. Output is stored per 5 min interval. The simulations are run in large-eddy-simulation (LES) mode at dx,dy=200 m and dz=100 m by default. In addition, extra simulations are run with additional grids where dx,dy=100 m, dx,dy=500 m and dx,dy=1 km. In the latter two, the vertical grids are adjusted to 250 and 500 m intervals. In one last simulation with an adjusted resolution, dz is set to 200 m.

In LES mode, a turbulent kinetic energy (TKE) scheme after handles the subgrid turbulence. For the microphysics, the default CM1 scheme is used (the two-moment scheme of ). Boundary conditions of the simulations are of the non-periodic, open type. Hence, derivatives of any quantity are set to 0 at the boundaries in every direction: an infinite reservoir of inflow air is theoretically available to the convective systems. As a consequence, reflection of wave signals can also occur at the boundaries. For more details on the model settings (dynamics and physics), we also refer to .

The initial thermodynamics are prescribed using the profile of (Fig.  in Appendix ), a standard in CM1. Two basic local potential temperature perturbations have been set at t=0 to trigger convective cells with various kinds of organization. Furthermore, the initial wind profiles are varied at t=0 to realize systems that manifest with a certain organizational mode of convection, numbered 1–3 (see the left side of Fig.  in Appendix ; ; and Bryan's CM1 code). Each of the four combinations of local temperature perturbations and wind profiles is introduced below. These prescribed profiles establish the four modes of convection. An overview of all the scenarios introduced in this section is shown in Fig. .

Diagnostics that represent latent heating by precipitation, upper divergent outflow, and convective momentum transport are evaluated over two separate time intervals. The first time interval ends after 75 min for the squall lines and after 90 min for the regular multicells and supercells. Diagnostics are also evaluated over the second time interval, running from the end of the first interval until the end of the simulation (120 min). This approach with two time intervals creates temporal subsamples. In the first interval's subsamples, effects close to the selected box boundary are relatively unimportant. On the other hand, such effects have a comparatively stronger impact on the diagnostics during the second time interval. Comparison between the two intervals helps to determine the relevance of, for instance, the propagation of gravity waves influencing the larger-scale environment.

Furthermore, a restricted rectangular horizontal area within the whole domain is selected, over which diagnostics are averaged spatially, further limiting boundary effects. The exact extent of the boxes is depicted in Sect.  that follows.

Figure  depicts the temporal evolution of the four convective systems in the control simulation together with their corresponding simulated patterns of radar reflectivity.

Figure  shows the vertical velocity at a level near the tropopause (within 0.5 km). The boxes over which further diagnostic quantities are integrated are also outlined accordingly as black rectangles. The simulations are split into the following two time intervals, as mentioned in Sect. : the first interval before the snapshot in Fig.  and the second interval, covering the part of the simulation afterwards. The box is chosen such that the flow effect of the convection through rippling in the wave signal at the tropopause level is still limited to the region (mostly) within that horizontal extent. By the time of the snapshot in the figure, only a fraction of the longwave gravity wave signal left the region of the box.

Comparing the four modes of convection, Fig.  shows clear contrasts. The supercell simulation (panel a) has a large region over which shortwave signals occur near the tropopause in comparison to the regular multicell. The box size over which outflow is diagnosed has to be sufficiently large that it covers the adjustment region of shortwave gravity wave activity. On the other hand, shortwave variability only occurs in an oval that is restricted to a comparatively smaller region in the regular-multicell simulation (panel b). For a fair comparison, the supercell and regular-multicell integration boxes are set over the same spatial extent relative to the initial warm bubble. A region where near-tropopause boundary reflection effects (as a result of the open boundary conditions) are suspected to occur can be identified in the infinite-length squall line simulation; specifically, regions away from the centre (y=0) have a wider zonal extent (about x>±35 km) over which shortwave w variability is strong (namely at y<-20 and y>40) compared to the central region. That pattern in w is not present in the display of the finite-length squall line as a consequence of the arch shape, which only covers a restricted part of the domain.

The patterns of vertical motion as a consequence of gravity waves and convection occurring in the middle troposphere are discussed in further detail in Sect. S2 of the Supplement. This is the level where the wave amplitude of the fastest mode of gravity waves maximizes.

Figure  shows the time evolution (x axis) of the vertical divergence profiles (y axis) for each of the four basic types of convective organization (starting from t=5 min). Initially, the convection has not developed. It requires about 30 min before intense convection develops, as can be visualized with the selected threshold and corresponding colour scale in the figure. Note that the colour scales of the isosurfaces and the isolines use different values to allow for a distinction between the pattern of individual ensemble members and the ensemble mean value of strong divergence. In the top and bottom rows, strong convergent low-level (z<3 km) signals start to exceed the isoline threshold after about 45 min (±20 min). With a slight delay (about 15–20 min), strong signals of mass divergence set in and stick to a layer between 8 and 13 km, around and just below the tropopause.The mass divergence (convergence) signals in the regular multicell both set in after 1 h in the upper (lower) troposphere. Afterwards, they expand their vertical extent with time. In this set of simulations, the upper-tropospheric divergence also sticks to the 8–13 km layer. The low-level convergence expands quite a lot after about 1.5 h so as to nearly cover the full lower half of the troposphere.

The ensemble spread is narrow for Fig. a and b, as suggested by the spaghetti lines. Among the infinite-length-squall-line ensemble members there is more spread: the lower panel (panel e) suggests a typical spread up to about 1–2 km in the vertical direction for each contour of mass divergence (convergence) during the second hour (maximum and minimum level of isoline). This is confirmed by numerical analysis; the time average of the layer depth over which μens±σens overlaps with zero divergence, from t=45 min until t=120 minutes and over the troposphere (2–12 km altitude), is 300–350 m for the regular multicell and the supercell. Conversely, for the infinite-length squall line, the value is 975 m. provide a detailed analysis of the ensemble evolution in this set of simulations.

Of particular interest is the mid-tropospheric contour of neutral convergence in Fig.  (white isolines), as this marks the boundary between the upper-tropospheric divergent outflow and the entrainment or inflow region of convection in each simulation. It settles at about 5–6 km altitude after rather noisy behaviour in the first 30 min due to (mostly) undeveloped convection. It rises to 7–8 km altitude for each mode of convection eventually (after about 90 min of simulation) but not before shortly dropping to about 4 km altitude in the infinite-length-squall-line simulations (at about 60 min). The strength of the upper-tropospheric divergence signal gradually increases towards the end of each simulation.

Moving to panels b and d, with regard to the finite-length squall lines, a substantial amount of ensemble spread is identified (the earlier-discussed ensemble standard deviations of divergence overlap with zero over typical depths of about 700 m, which is smaller than for the infinite-length squall lines). The outflow divergence settles to levels of about 7.5 to about 13 km quite soon and remains at those levels along the outer parts (at the edge of the finite-length squall line). The convergence zone at low levels seems to slowly lift with time in this ensemble, reaching an upper bound of about 7 km after 100–120 min. The divergence signal seems to be much stronger in the edge region of the finite-length squall line than at its centre as well. Even though the time evolution of divergence (convergence) in the finite-length-squall-line-centre simulations shares many similarities with that of the infinite-length squall line, the first hour has a contrasting evolution. Signals are nonetheless rather weak during that hour.

The lower boundary of the integration mask for the upper-tropospheric divergence is best set at 7 km, as this boundary is most suitable for differentiating the regions of convergence and divergence. Therefore, results using this altitude threshold are used for the analysis of the next section (Sect. ). The upper boundary is quite stagnant at 13–14 km altitude, and therefore, the upper boundary is set to 14 km, which is about 2 km above the tropopause (see also Fig.  in Appendix ).

Note that the figures illustrate the ensemble spread. That means they exclude simulations with physics perturbations. Especially for these perturbed simulations (notably the -40 % latent-heating constant), the vertical profiles may differ due to lower tops of the convective clouds. Corresponding extra panels for these simulations are available in the Supplement (Fig. S3).

Figure  presents the integrated strength of divergent outflows as a function of net latent heating by precipitation. Purely focusing on the three main scenarios (see leftmost three of the four green boxes in Fig. ), the separation between the infinite-length squall lines on the one hand and both the supercells and regular multicells on the other hand at given net latent heating is obvious. The latter two have increased divergence compared to all the squall line simulations. Nonetheless, this contrast is reduced at higher latent-heating rates (i.e. over roughly 3000 W m-2).

More specifically, the initial ratio between mass divergence and column latent heating is much higher for the regular multicells and supercells than for the squall lines. For increased precipitation rates, the ratio apparently decreases compared to that at low latent-heating rates. On the contrary, the same ratio for squall lines even increases with precipitation intensity, although this is not so clear. The robustness of the increment in the ratio between the mass divergence and precipitation rate of squall lines is questionable. The typically lower ratios for the supercell at high precipitation rates reduce the gap between the two regimes at higher latent-heating rates; on the one hand, there is a regime that the supercell and regular multicell seem to follow, and on the other hand, there is another regime that the (infinite-length) squall line seems to follow.

Interestingly, the physics perturbations do not substantially affect the suggested regimes, and resolution also has no noticeable effect in the plane of Fig.  (it should be noted that simulations with physics perturbations and a different resolution than the control runs are included in this figure). The isolated convection regime (i.e. nearly 3D outflow, applicable to regular multicells and supercells) is surprisingly linear at precipitation intensities up to about 2000 W m-2. Moreover, the contrast in the typical ratio of mass divergence and latent-heating rates (normalized mass divergence) between the two regimes that are identified is suggested to exceed a factor 2: the steep sloped line that would describe the two isolated modes of convection reaches an upper-tropospheric divergence of almost 4×10-5 kg m-3 s-1 at 2000 W m-2, whereas only one data point for the infinite-length squall line reaches about 2×10-5 kg m-3 s-1 at a precipitation rate of about 2000 W m-2 (≈3 mm h-1).

When one now focuses on the finite-length-squall-line simulations (pink and red symbols in Fig. ), its end region data points (i.e. pink crosses) are in line with the mass divergence of supercell simulations. At low precipitation intensities, the mass divergence increases as rapidly as for the supercell. In addition, both supercells and end regions of finite-length squall lines have a ratio between the mass divergence and precipitation rate that reduces at higher precipitation rates in very similar ways. That decrease of the ratio between the mass divergence and precipitation rate with higher precipitation rates also occurs for central regions of the finite-length squall line (i.e. pink squares), but the mass divergence is systematically reduced compared to the end regions of the finite-length squall lines. Hence, the behaviour of the finite squall line centres aligns better with the infinite-length squall line simulations. In spite of that, the mass divergence is systematically reduced even more strongly in the infinite-length squall lines compared to the finite squall line centres.

In general, the full domain integration of the finite-length squall line leads to intermediate behaviour, with the amount of divergence in between the centre and the line end. That is a consequence of averaging over the centre and end regions. Reduced normalized mass divergence at a high precipitation intensity also occurs for the full integration over the finite-length squall lines.

Infinite-length squall lines represent nearly 2D convection (e.g. as in ). On the other hand, initially isolated convective cells that are circular in an environment without convection follow the 3D regime (supercell and multicell). With increasing precipitation or latent-heating rates, outflows from deep convective cells are more likely to collide. They mimic idealized point or line sources less well. This effectively creates an intermediate dimensionality between 2D and 3D. The shift from or variability between purely 3D, intermediate, and purely 2D convection largely explain the variability in divergent outflows as detected. It could be seen as a non-linear effect on divergent outflow from deep convection by convective organization and aggregation. On the other hand, the absence of the effects of upscaling convective organization could have lead to a better resemblance of idealized 2D or 3D regimes.

The initial ratios between precipitation intensity and mass divergence suggest that the low precipitation intensity can obey the following two limit regimes:

a 2D outflow regime with reduced mass divergence

The existence of two outflow regimes (2D and 3D divergence) has been

suggested in the previous section

Figure a and b show that, during the initial time interval, the meridional component of the divergence is negligible throughout most of the domain. The pattern in the v term of the divergence is consistent with this part of the finite-length squall line resembling the 3D outflow regime. Similar patterns have been identified for the infinite-length squall line (not shown) but without an enhanced meridional component of divergence at the squall line end. Practically, all divergence occurs initially in the u component, which is consistent with the manifestation of a 2D-like outflow regime in the infinite-length squall line and the centre region of the finite-length squall line (Fig. ).

During the second time interval, the v components develop particularly strongly on the northern- and southern-arching regions of the squall line. Both components are of the same order of magnitude, but that is to be expected for a strongly curved squall line. The pattern of squall line curvature is clear from the accumulated precipitation (and, similarly, the simulated radar imagery in Fig. l). The possibility that outflow collisions occur and that the effective flow resembles a regime governed by a mixture of 2D and 3D regimes as a result of interference between outflows from individual convective cells is strongly supported by both Figs.  and . Hence, Fig.  provides further evidence for the explanation of Sect. .

The (mean) divergence in Fig. a is about 1 order of magnitude larger than in Fig. b in the squall line centre. This is also consistent with the infinite-length squall line, where the difference is about 1 order of magnitude too. In the bottom panels of the figure, that contrast has strongly reduced as a result of the arching squall line, which is in accordance with expectations of bending mixed 2D–3D convection. Therefore, the leading order divergence variability in Fig.  can confidently be attributed to the dimensionality of outflows (2D and 3D) and, correspondingly, the presence of a pulse or line source of latent heating.

Upper-tropospheric outflow from deep convection has been quantified for each simulation by integrating the mean mass divergence in 3D over a region surrounding the convective cells (horizontally) and over 7–14 km altitude. Figure  and divergence profiles (not shown) revealed that the lower boundary is the most critical of the two. Therefore, the integration has been repeated for the 6–14 km layer. The resulting patterns in a diagram of latent heating versus divergence comparable to Fig.  are not sensitive to the lower boundary.

The procedure has been executed for two time intervals separately: a first time interval where the fastest gravity waves escape the box of integration and a second time interval where a large proportion of the gravity wave signal has escaped that box. Even though some potentially relevant flow effects with consequences for upper-tropospheric (UT) divergence could have escaped this integration box with the gravity waves, the results suggest that this is not the case. That such an escape did not have consequences for diagnosed UT divergence can be justified with the following arguments:

The mechanism of gravity waves is to restore density anomalies with fluctuations, which are averaged out when integrated over longer distances in the quasi-horizontal plane.

Figure  suggests that the equivalent of about 1500 W m-2 of precipitation is needed for 1×10-5 kg m-3 s-1 of divergence at low precipitation rates in an infinite-length squall line, whereas this is only about 500 W m-2 for the regular-multicell and supercell regimes. The proportionality between these two regimes is well over a factor 2 and likely very close to 3 and π.

The idea of the finite-length-squall-line simulations is that the outer part at both of the squall line ends mimics a regime where convective cells can freely induce their outflow in a 3D space when both ends are combined (as if the centre is removed), as in the case of a multicell and supercell. The centre, however, is geometrically somewhat restricted, as in that of an infinite-length squall line. Infinite-length squall lines only allow for outflow in one horizontal direction. The idea is supported by the magnitudes of two components of the mass divergence in finite- and infinite-length squall line simulations: outflow in the zonal direction exceeds that in the meridional direction by 1 order of magnitude, which is consistent with findings pointed out by . Similarly, in the finite-length squall line, divergent outflow is much larger in the direction normal to the finite-length squall line than in that parallel to the squall line initially, with dynamics induced by small cells (short wavelengths) compensating for each other in the parallel direction (Fig. ). That provides further support for the idea of two outflow regimes.

Outflow simulations with expressions based on the numerical model of and suggest that the ratio of convective outflows between a line source and point source is a factor of 2π in the limit case. That stems (in their calculations) from a conversion of the delta function from radial geometry to an x–y plane in their derivation . A mechanism that could explain the deviation of about a factor of 2 between their linear gravity wave models and large-eddy cloud simulations as performed here has not been found yet. Theoretical support for different regimes of updraught and pressure perturbations between 2D (a line source) and 3D (a point source) in a weak-shear environment is also provided by . The updraughts and pressure perturbations as studied by drive the outflows as studied here; outflows relate to updraughts through continuity. derived a deviation of a factor of 2 theoretically and subsequently compared the findings to cloud simulations .

The robustness of the results, together with the arguments above, give high confidence in the impact of outflow dimensionality on the magnitude of the divergent winds. Furthermore, the intermediate ratio of mass divergence to precipitation rate at high precipitation intensities compared to the initial 2D (low ratio) and 3D (high ratio) regimes suggests that convective aggregation likely affects the dimensionality of convective outflow in the upper troposphere. Outflow likely adapts to a mixture of 2D and 3D regimes due to the convective organization and interference between outflows of individual cells. When aggregates of convective cells collide with upper-tropospheric outflows of other convective cells, the effective dimensionality would be something intermediate between 2D and 3D: the outflows first collide along the line through the updraughts and become nearly 2D along the line, but on the outer regions, the outflow can still move as if the convective cell was isolated. That corresponds to a nearly 3D outflow regime, and any mixture creates ovals of outflow similar to the finite squall line in our conceptual framework (even if the supercells also reveal such behaviour and collisions of outflow after some time).

The mass divergence found in the dataset (Fig. ) is, in terms of magnitude, in good agreement with linear gravity wave adjustment models where heating is imposed as proxy for a convective system .

The initial 2D and 3D regime behaviour with reduced divergence at higher precipitation intensities due to convective aggregation found in this study contrasts with the modelling and observation studies by and . In , it is suggested that a stratiform contribution by the vertical half wavelength due to a stratiform fraction of a mesoscale convection system actually increases the divergence at any given heating rate or at a given precipitation intensity. On the one hand, this could imply that the convective systems simulated in this study do not extend sufficiently and so do not generate any sizeable stratiform precipitation system. Indeed, the stratiform contribution to the simulated squall line and supercell clouds is not substantial, especially when looking at the precipitation that reaches the surface (turned into net latent heating); see Figs.  and S1 in the Supplement (see also ). The formation of a stratiform precipitation regime usually coincides with lifting of the level of neutral divergence, on average . In Fig. , such a continuous gradual rising of the level of neutral divergence for supercells and squall lines during the second hour cannot be detected. However, at least a fraction of an MCS is usually stratiform within the second hour. This suggests that the simulations are only representative for purely convective systems or for those with minor stratiform fractions, which is a reasonable approximation to convection in certain regions . In spite of this, these purely convective systems are important to study for a better understanding of the role of deep convection in the climate system and to improve deep-convective parameterizations.

On the contrary, using Fig. , one could argue that a stronger increase of the mass divergence with latent-heating rate occurs for the infinite-length-squall-line simulations at rather high latent-heating rates rather than at low latent-heating rates (as the highest latent-heating rates occur in the second hour). Furthermore, stratiform precipitation systems may form out of the squall line anvils. That would be in agreement with arguments by and , assuming that the stratiform fraction of the system increases with time. However, an infinite-length squall line is, in practice, not very representative for most deep convection in the real world. In addition, the level of neutral divergence does not seem to rise at all for the infinite-length squall line in Fig. , which would not support possible compatibility of our results with the arguments of and . Furthermore, the more realistic finite-length squall line does not behave in agreement with these studies. In the finite-length squall lines, the ratio between the mass divergence and net latent heating decreases at higher precipitation rates during the second (later) time interval (Fig. ).

Theoretical 2D squall line models have been extensively studied by Moncrieff and co-authors (e.g. ). In , it was argued that such 2D models could be very beneficial for parameterizing convective momentum transport. pointed out that one should be careful and that processes such as convective momentum transport in actual squall line convection can have characteristics of a mixture of 2D and 3D convection. Certain sections have more characteristics of a 2D convection regime, and others have more characteristics of a 3D convection regime. The finite squall lines suggest that the same is true regarding divergence profiles. That means that a comparison of traditional and theoretical models of idealized convection (e.g. 2D models) to cloud-resolving and large-eddy simulations is needed. By stimulating such model intercomparisons, the applicability of traditional practices and findings to the more complex simulation techniques can be scrutinized (as done here and in ).

Theoretically, convectively induced divergence profiles are suggested to mimic 2D or 3D regimes in some cases. On the contrary, in practice, intermediate behaviour is suggested to be more likely, especially for intensive systems. That can be a worthwhile consideration for the development of convective parameterizations that would take convective organization and aggregation into account (e.g. ).

In and , it was found that numerical weather prediction errors are initially established predominantly in regions of enhanced and mostly convective precipitation. were able to attribute initial error growth (<12 h into the simulation) in their stochastically perturbed simulations to non-conservative processes and predominantly to the deep-convection parameterization. That parameterization represents the collective effect of organized convective systems and isolated convective cells. At later times, the induced ensemble variability corresponds predominantly to variability in upper-tropospheric divergent winds. inferred that this upper-tropospheric variability is likely associated with latent-heat release below and corresponding deep convection as precursors.

The quantitative understanding of upper-tropospheric outflow and the uncertainty quantification achieved in this work could support an extension of the potential vorticity diagnostics of towards smaller scales. Consequently, it may lead to better insights into the role of individual convective systems in certain forecast errors. Furthermore, it may reveal biases between certain modelling approaches, namely large-eddy simulation, cloud-resolving simulations with explicit deep convection, and global simulations with parameterized deep convection (see for example ). Intercomparison of divergent outflows among the simulations investigated here and various simulation set-ups could provide structural insights into (potential) biases between each of them. Such insights into structural biases depending on the treatment of deep convection may be beneficial to the understanding of weather and climate simulations. In this work, a set-up for the intercomparison of divergent outflows is established, focusing on large-eddy simulations with advanced small-scale-process representation. The effects of differential convective organization or differential diabatic heating could potentially be followed to synoptic-scale uncertainty days ahead using a thorough understanding of upper-tropospheric divergence variability (e.g. ). This study builds the insights into which main factors control the upper-tropospheric divergence, which is required to trace mesoscale uncertainty induced by diabatic heating, while its uncertainty can evolve upscale to synoptic-scale uncertainty.