Following the classic work of , we consider the convective ensemble consisting of a spectrum of plumes with varying entrainment rates ϵ′. The mass flux of each plume m(ϵ′) increases with height z as it entrains and is governed by 11m∂m∂z=ϵ′ from the cloud base zb up to its LNB, at which point it is assumed to detrain into the environment. We further assume that each plume is saturated above cloud base, and that the moist static energy of each plume hp is governed by, 2∂hp∂z=-ϵ′(hp-he), where hp=cp(T-T0)+gz+lv0q and he is the moist static energy of the environment, defined analogously. Here T is the temperature, q is the specific humidity, cp is the isobaric specific heat capacity of the dry air–vapour mixture, g is the gravitational acceleration constant, and lv0 is the latent heat of vaporisation at a reference temperature T0=273.15 K. For simplicity, we neglect any contribution of freezing to the moist static energy, and we assume no condensate (liquid or ice) is present in the parcel.

Since each plume within the spectrum has a different entrainment rate, it will become neutrally buoyant and detrain at a different level. showed that one could solve for h(z), the moist static energy of the plume that detrains at level z. Note that h(z) does not represent the vertical profile of any individual plume. Specifically, h(z) is approximately governed by the equation 3∂h(z)∂z-λh(z)-h(zb)=-ϵ(z)h(z)-he(z). Here ϵ(z) is the entrainment rate of the plume that detrains at level z and 4λ=11+0.75ϵ(z)(z-zb)dln⁡ϵdz, as defined in .

Since h(z) refers to a plume at its LNB, we can relate its properties to those of the environment. Neglecting the effect of moisture on density, h(z) is equal to the saturation moist static energy of the environment he∗(z), and we may therefore write Eq. () entirely in terms of environmental quantities: 5∂he∗(z)∂z-λhe∗(z)-he∗(zb)=-ϵ(z)lv0qe∗(z)-qe(z), where qe and qe∗ are the specific humidity and saturation specific humidity of the environment, respectively.

Given the tropospheric relative humidity RH and the entrainment rate of the plume detraining at each level ϵ(z), either Eq. () or Eq. () may be used to infer the saturation moist static energy of the environment, which we take to be that of the large-scale mean atmosphere. From this, one may derive the mean temperature and humidity profiles assuming hydrostatic balance.

assumed power-law profiles of ϵ(z), finding that their solutions compared well to simulations of radiative–convective equilibrium. Here, we instead relate ϵ(z) to the mass flux required to produce a given areal-mean precipitation rate, allowing us to explore how the thermodynamic structure of the atmosphere varies with precipitation.

At each level z, the total mass flux M(z) is given by a sum over all plumes that are positively buoyant: 6M(z)=∫0ϵ(z)m(ϵ′)dϵ′. Using Eq. (), this may be written in terms of the cloud-base mass flux distribution mb(ϵ′): 7M(z)=∫0ϵ(z)mb(ϵ′)eϵ′(z-zb)dϵ′. The distribution mb(ϵ′) may be thought of as representing the spectrum of thermals in the boundary layer, a fraction of which are positively buoyant and may contribute to the free-tropospheric convective mass flux. As a simple starting point, we assume this distribution to be uniform: mb(ϵ′)=M0/ϵ0, where M0 is the cloud-base mass flux required under the conditions of radiative–convective equilibrium, and ϵ0 is a characteristic entrainment rate. Since the maximum size of boundary-layer thermals scales with the depth of the boundary layer and we expect larger thermals to lead to plumes with smaller entrainment rates , we set the characteristic entrainment rate ϵ0=E/zb, for a non-dimensional parameter E. Figure  shows mb(ϵ′) for different cloud-base heights and cloud-base mass fluxes. Buoyant plumes exist up to a maximum entrainment rate ϵb=ϵ(zb). A higher cloud base results in a set of buoyant plumes limited to relatively small entrainment rates, while a larger cloud-base mass flux implies a wider entrainment distribution.

With uniform mb(ϵ′), the integral in Eq. () may be evaluated analytically: 8M(z)=M0ϵ0eϵ(z)(z-zb)-1z-zb. This equation may be rearranged for the entrainment rate ϵ(z), 9ϵ(z)=ln⁡MbM0η(z)Ezb(z-zb)+1z-zb, where we have expressed the mass flux M(z)=Mbη(z) as a product of the cloud-base mass flux and a vertical structure function η(z). Finally, we relate ϵ(z) to the areal-mean precipitation rate. To do so, we assume that the precipitation efficiency is invariant, so that the precipitation rate may be taken to be proportional to the mass flux of humidity through cloud base: P∝Mbq∗(zb), where q∗(zb) is the saturation specific humidity at cloud base. This allows us to write 10ϵ(z)=ln⁡PP0η(z)Ezb(z-zb)+1z-zb where P0≈0.125 mm h⁻¹ is the precipitation rate in radiative–convective equilibrium e.g.,. While the precipitation efficiency is likely to be sensitive to properties such as the environmental relative humidity, we assume it is constant here for simplicity and postpone incorporating a varying precipitation efficiency to future work.

Equation () gives ϵ(z) as a function of the precipitation rate, the cloud-base height and the structure function η(z). For the present discussion, we adopt a simple profile of η(z) as shown in Fig. a; it remains constant up to 10 km and then decreases smoothly to zero at 15 km following 11η(z)=1z≤10km,12+12cos⁡π(z-10km)5km10km<z≤15km,0z>15km. Based on this mass-flux profile, Fig. b shows profiles of ϵ(z) for the control case of E=0.15, zb=0.7 km and P=0.5 mm h⁻¹, and for cases with a larger cloud-base height (zb=1.4 km) and smaller precipitation rate (P=0.25 mm h⁻¹). As expected, the entrainment rate decreases with cloud-base height and increases with the precipitation rate. In each case, ϵ(z) decreases with height, partially to counteract the growth of plumes through entrainment, and additionally, at heights above 10 km, due to the decrease in total mass flux following η(z).

While the form of η(z) has important effects on the entrainment and moist static energy at upper levels, the relationships between convection and the large-scale thermodynamic profiles described in the next section are insensitive to, for example, the inclusion of a mid-tropospheric maximum in η(z). We note, however, that a valid solution of the model only exists if ϵ(z) is monotonically decreasing with height, and this puts some limits on the mass-flux profile.

Armed with the profile of ϵ(z) and a given value of the tropospheric relative humidity RH, we may solve the SPM for the thermodynamic profiles of the plume spectrum and the environment following . We initialise the plume spectrum and environment with a surface temperature of 300 K and a surface humidity calculated iteratively such that zb is at the lifted condensation level (LCL). An exact formula for the LCL was derived by , but it depends implicitly on the surface humidity, necessitating an iterative solution. In the sub-cloud layer, we assume the plume conserves its specific humidity, while the environmental relative humidity is linearly interpolated from the surface value to that of the free-troposphere. Both plume and environment are assumed to have constant virtual potential temperature below zb.

Above cloud base, we solve Eq. () for the moist static energy profile h(z) and the moist static energy of the environment he(z) using a simple first-order Euler method. We do not consider the effect of liquid or solid condensate, but, unlike in the approximate equation Eq. (), we consider the effect of moisture on the density of air. At each step, the moist static energy of the detraining plume h(z) is inverted assuming saturation to find the plume virtual temperature, which is equal to the virtual temperature of the environment. Using the free tropospheric relative humidity RH, the environmental temperature, specific humidity, and moist static energy he may be calculated, allowing the integration to continue upwards until the level at which M(z)=0 at z=15 km. An example solution is shown in Fig. c.

Once the environmental properties have been found, the moist static energy of individual plumes may be calculated via Eq. (). Examples are shown in Fig. c for entrainment rates equal to (0,0.1,…,0.9)ϵb, where ϵb is the maximum entrainment rate of buoyant plumes at cloud base. Each of these plumes therefore represents a decile of the mass-flux distribution at cloud base.

For each plume, we may calculate the integrated buoyancy 12Bint=∫zbztgTvp-TveTvedz, where Tvp and Tve are the virtual temperatures of the plume and environment, respectively, and zt is the height at which the plume becomes neutrally buoyant. We consider two measures of the convective intensity. Firstly, we define CAPE as the value of Bint for a plume with zero entrainment. Secondly, we define the extreme convective updraft velocity as 13wext=2Bint(0.1ϵb), that is, a simple measure of the maximum vertical velocity associated with the top decile of the updraft distribution.

We visualise the solutions of the SPM using the “stability–humidity plane” introduced by to quantify the effect of entrainment on the tropospheric lapse rate . This phase plane plots the vertical difference of the saturation moist static energy Δhe∗ on the ordinate against the mean saturation deficit lv0(qe∗-qe)‾ (expressed in energy units) on the abscissa, each defined between the 500 and 850 hPa levels in the lower troposphere. Since h∗ is a function of temperature and pressure only, the difference Δhe∗ is a function of the mean lapse rate of the lower troposphere and represents a measure of stability, with less negative values representing a more stable lower troposphere. The mean saturation deficit lv0(qe∗-qe)‾ represents a measure of humidity, with larger values corresponding to a drier atmosphere.

The variables Δhe∗ and lv0(qe∗-qe)‾ are chosen because their ratio is closely connected to the entrainment rate ϵ(z) of the SPM. To see this, consider Eq. () and vertically integrate both sides between the 850 and 500 hPa levels. Neglecting the variation of ϵ(z) with height, so that λ≈0, this gives 14Δhe∗=-ϵ‾Δzlv0(qe∗-qe)‾, where Δz is the vertical distance between 850 and 500 hPa, and ϵ‾ represents an average entrainment rate that we refer to as the effective entrainment rate. For a fixed Δz, constant values of the effective entrainment rate are represented by diagonal lines on the stability–humidity plane (Fig. ).

showed that, during precipitation, the atmosphere tends to lie in an elongated region centred on the line ϵ‾=0.25 km⁻¹ and interpreted this as evidence for the effect of entrainment on the lapse rate. Here, we examine how convection varies in different regions of the stability–humidity plane according to the SPM. To do this, we calculate solutions of the SPM for zb=1.4 km and for a range of tropospheric relative humidities RH between 2 % and 98 % and areal-mean precipitation rates P between 0 and 1.25 mm h⁻¹. For each solution, we calculate the parameters Δhe∗ and lv0(qe∗-qe)‾, providing its location on the stability–humidity plane. The solutions spread out in two dimensions to fill most of the plane, except for the lower left corner. Interpolating the solutions to a regular grid allows us to contour various convection-related quantities in the stability–humidity plane in Fig. .

Variations in the maximum entrainment rate at cloud base ϵb closely follow those of the effective entrainment rate ϵ‾, albeit with slightly higher magnitudes (Fig. a). This supports our use of the stability–humidity plane and indicates it is a useful method to constrain convective entrainment using only large-scale fields .

According to the SPM, cloud-base entrainment is a strong function of the areal-mean precipitation rate (cf. Eq. ), and as a result, precipitation increases with both instability and humidity, peaking along the lower left portion of the phase plane (Fig. b). While the strong relationship between precipitation and environmental humidity is well known , the sensitivity of precipitation to both environmental humidity and lapse rate in the SPM is consistent with recent work casting the precipitation–humidity relationship in terms of the buoyancy of an entraining plume e.g.,.

Figure c and d show the distribution of SPM-derived convective intensity measures in the stability–humidity plane. Because CAPE is calculated for an undiluted parcel, it is largely a function of instability, with little humidity dependence, increasing downwards in the stability–humidity plane. On the other hand, the extreme updraft wext increases primarily with instability but also exhibits a weak dependence on humidity owing to entrainment, with larger values occurring in a drier atmosphere. This dependence is weak across much of the phase space but becomes more pronounced for high entrainment rates (ϵ‾>0.25 km⁻¹), where strongly entraining plumes can maintain large vertical velocities only if the entrained air is relatively moist, which limits the dilution of updraft buoyancy. The top decile of the updraft distribution is therefore largest when the atmosphere is far from saturation, in stark contrast to the largest area-mean precipitation rates, which are highest near saturation. The explanation for this is that, in a dry atmosphere, the convective mass flux tends to be smaller, and only plumes with relatively small entrainment rates remain buoyant, resulting in higher average buoyancy within clouds. Thus given percentiles of the mass-flux distribution increase in intensity as the atmosphere becomes drier, even though the maximum possible updraft, associated with an undilute plume, does not. This is consistent with the argument of that only larger and more intense convective cells are able to penetrate into the free troposphere when it is dry. We note, however, that the SPM does not directly consider convective inhibition, which found to be larger in a drier atmosphere and could also prevent weaker convective cells from developing.

Finally, we consider the effect of changing cloud-base height zb on the SPM solutions. Plotting results in the stability–humidity plane reveals that, when the cloud-base height is varied, the SPM predicts similar dependencies between the convection and its environment, but with different absolute magnitudes of the convective variables (not shown). To visualise these differences, Fig.  plots the effective entrainment rate as a function of the areal-mean precipitation rate for two different values of zb. Solutions for all values of RH are shown, but they almost collapse to a line. Given our definition of the cloud-base mass-flux distribution, we expect entrainment to decrease with zb. Indeed, for low-to-moderate precipitation rates, the effective entrainment rate is lower when the cloud base is increased to 1.4 km, although the increase is relatively modest, and in fact reverses at very heavy precipitation rates (Fig. ). This reversal is associated with a mismatch between the cloud-base entrainment rate ϵb and the effective entrainment rate measured over the region between 500 and 850 hPa because of a sharp decrease in saturation moist static energy below 850 hPa. For very high areal-average precipitation rates, entrainment may be better estimated from the variation in he∗ nearer to cloud base.

In summary, the SPM solutions show large area-mean precipitation rates to be concentrated in the lower-left of the stability–humidity plane, while the strongest local convection, which we expect to be associated with the strongest local precipitation rates, is concentrated either in the lower or the lower-right portion of the phase plane, depending on whether it approaches the undilute limit. Increases in cloud base height can shift the results to a lower effective entrainment rate with the same overall dependencies. We now seek to test these predictions using radar observations and reanalysis data. A caveat to note in these comparisons is that the SPM does not predict the relative frequency in the phase plane; it does not indicate how rarely or commonly a given position in the stability–humidity plane is occupied in the tropics. We return to this point in the discussion.

We use radar data provided by the Australian Unified Radar Archive (AURA), which is maintained by the Australian Bureau of Meteorology . We focus on the operational C-band radar located at Berrimah, Darwin (12.456° S, 130.927° E) for the 20-year period from December 2001 to 2021. We use the Level 2 rain rate and the convective–stratiform partitioning products in AURA. The rain rate fields are derived using the reflectivity fields from the lowest available scan above ground level, with reflectivity and rain rate related by an empirically derived relationship of the form Z=260.6R1.21, where Z is the reflectivity and R is the rain rate in mm h⁻¹. This is a standard approach for obtaining instantaneous rain rate from single-polarisation radar data. The convective–stratiform classification is based on the algorithm developed by and applied to gridded reflectivity at 2.5 km above radar level. The radar data have been quality controlled to remove non-meteorological echoes and artefacts. Details of data calibration and the convective–stratiform classification can be found in and citations therein.

The Level 2 AURA data provide precipitation estimates at a spatial resolution of 1 km within a radius of 150 km from the radar site. The temporal resolution is 10 min before October 2018 and typically 5–6 min thereafter. Following , we restrict the analysis to pixels within a 100 km radius to minimise range-related artefacts. No minimum radius exclusion is applied, although manual inspection indicates that some near-radar pixels affected by the cone of silence are already masked in the radar product. Rainfall events are identified using a threshold applied to the time series of domain-mean rain rate, calculated as the total rainfall within the 100 km radius divided by the full area of that domain (i.e., including non-precipitating pixels). An event begins when the domain-mean rain rate first exceeds 0.1 mm h⁻¹ and ends when it falls below this threshold. Only events with a duration of at least 1 h are retained to exclude non-meteorological echoes. These identified rainfall events contain a total of 175 610 individual radar scenes. Only these radar scenes are used in the analysis.

For each radar scene, the domain-mean rain rate is used to characterise large-scale rainfall, with higher values indicating precipitation distributed over a larger fraction of the analysis domain. To quantify local intensity, individual pixel values are not used directly because they may be contaminated by ground clutter and random noise. Instead, we identify convective cells as connected regions of convective pixels using eight-neighbour connectivity. We then compute the cell-mean rain rate for each identified convective cell, and we define the maximum cell-mean rain rate among all convective cells within the domain as the measure of local intensity. A high maximum cell-mean rain rate indicates that intense precipitation is concentrated within a small spatial footprint. To reduce the influence of small, short-lived echoes and residual non-meteorological artifacts, only convective cells with an area greater than 5 km² are considered. As a sensitivity test, we alternatively use the 90th-percentile pixel-wise rain rate within each cell as a measure of local convective intensity. This produced results similar to those obtained using the cell-mean rain rate (not shown); therefore, we only present results based on the maximum cell-mean rain rate. To minimise edge effects, cells intersecting the 100 km radius boundary are retained in their entirety, rather than restricting the analysis to only the portion within the domain. Very large systems (e.g. long squall lines) may still be truncated at 150 km, but would still be identified as large systems.

The convective and stratiform contributions to the domain-mean precipitation are calculated using the AURA convective–stratiform partitioning product. For each radar scene, convective rainfall contribution is defined as the sum of rain rate over pixels classified as convective, divided by the sum over all precipitating pixels of the 100 km-radius analysis domain; stratiform rainfall is calculated analogously using pixels classified as stratiform.

For spatial characteristics of convection, we focus on the identified convective pixels and exclude pixels classified as stratiform. With this convective–stratiform separation, only moderate to deep convection is considered, and weakly or non-precipitating shallow clouds are excluded. Following , we calculate the number of convective cells (N) and the area (A) of each cell for each radar scene. This gives the convective area fraction (CAF) of each radar scene as: 15CAF=1D∑i=1NAi, where D is the total area of the radar domain, including non-raining pixels (∼ 31 400 km²) and Ai is the area of each individual convective cell. The morphology of convective cells is represented by the number of cells (N) and the mean cell area: 16A‾=1N∑i=1NAi.

Large-scale environmental conditions are obtained from the fifth generation of the European Centre for Medium-Range Weather Forecasts (ECMWF) atmospheric reanalysis ERA5;. The ERA5 data are on a 0.25° × 0.25° grid on 37 pressure levels with hourly temporal resolution. Following , the environmental conditions are linearly interpolated to the temporal resolution of the radar data. To characterise the large-scale environment, we use ERA5 data from a 3° by 3° area centred on the radar (11–13.75° S, 129.5–132.25° E; 12 by 12 grid points).

Our primary environmental variables are the stability Δhe∗ and the saturation deficit lv0(qe∗-qe)‾ defined in the previous section. They are calculated using ERA5 temperature and specific humidity at pressure levels between 850 and 500 hPa, and averaged over the radar domain. We also use CAPE, convective inhibition (CIN), and boundary-layer height (BLH) obtained from the ERA5 single-level dataset. CAPE is calculated as the integrated positive buoyancy of an air parcel lifted pseudo-adiabatically, including freezing and a mixed-phase range. Parcels are lifted from all model levels below the 350 hPa level, and the “most unstable parcel”, leading to the maximum CAPE, is retained . Note that, for computational efficiency, the calculation procedure of CAPE and CIN in ERA5 uses an approximation that neglects the effect of water vapour on density. This may lead to an underestimation of CAPE and overestimation of CIN . Nevertheless, for simplicity, we use the CAPE and CIN provided by ERA5 and discuss these limitations in Sect. .

CIN quantifies the energetic barrier that must be overcome for a parcel to reach its level of free convection (LFC). It is calculated as the integrated negative buoyancy below the LFC of the most unstable parcel defined above . This gives CIN positive values, with larger values indicating stronger inhibition. While CIN suppresses weak ascent, sufficiently strong lifting can overcome it and subsequently permit vigorous deep convection. The spatial maximum CAPE and maximum CIN within the analysis domain are used to represent the most unstable conditions and the strongest inhibition, respectively. The BLH in ERA5 is diagnosed using the bulk Richardson number, following . As a secondary measure of the depth of the sub-cloud layer, we calculate the LCL from ERA5 surface pressure and 2 m temperature and dewpoint temperature using the MetPy package , based on .

To compare the predictions of the SPM to observations, the radar domain mean rainfall and the maximum convective cell intensity of each radar scene are plotted in the stability–humidity plane in Fig. b and c, respectively. The value of each 2D histogram bin is the mean of all scenes falling within that bin. Only bins with more than 30 scenes are included in the analysis. Consistent with the SPM predictions (Fig. ), the areal-mean precipitation increases with both humidity and instability, peaking in the lower-left region of the phase plane. That is, the mean rain rate over the radar domain increases with the effective entrainment rate ϵ‾.

It may seem counter-intuitive that heavier large-scale precipitation is associated with higher effective entrainment rates ϵ‾. But in the framework of the SPM, ϵ‾ does not represent the dilution experienced by individual clouds. Rather, it characterises how the large-scale environment adjusts to restore buoyancy neutrality under convective activity. When precipitation is widespread, a large number of plumes must be buoyant, requiring an unstable and moist environment. In this context, a higher ϵ‾ is consistent with more extensive convective activity.

In the radar observations, the rainfall may be separated into stratiform and convective types; stratiform rainfall contributes an increasing fraction of the domain-mean rainfall as the atmosphere becomes moister, and as ϵ‾ becomes larger (Fig. d–e). This agrees with previous studies demonstrating that the strong observed increase in precipitation with moisture is largely associated with enhanced stratiform rainfall through expansion of the rain area, rather than intensified convection . The SPM, however, does not have an explicit representation of stratiform rainfall, and it can only produce large area-averaged precipitation through widespread convection.

Unlike the domain-mean rain rate, the maximum cell-mean convective rainfall intensity is largest in a drier and more unstable atmosphere, peaking in the bottom right of the stability–humidity plane (Fig. c), consistent with the behaviour of the extreme updraft velocity wext in the SPM (Fig. 3d). This is in agreement with , who found that the intense and localised “showers” develop in drier environments with larger instability. As the cell intensity depends on both total rainfall within a convective cell and the cell area, we tested the sensitivity of this relationship to the definition of intensity. The pattern shown in Fig. c remains evident when intensity is represented by the 99th percentile of pixel-wise rain rate calculated from rainy pixels only (not shown), indicating that the relationship is not primarily controlled by the area of the most intense convective cell. Additionally, this relationship is not limited to the most intense convective cells: when intensity is instead defined as the mean convective rain intensity across all cells, calculated as total convective rainfall divided by total convective area, the same pattern persists (Fig. f).

The higher convective intensity also coincides with the largest values of CAPE and CIN when binned in the stability–humidity plane (Fig. a–b), and shows little systematic relationship with ϵ‾. However, unlike in the SPM, reanalysis CAPE is not simply a function of instability Δhe∗, but also increases with the saturation deficit. In the observational analysis, CAPE and convective intensity are well aligned across a number of different intensity measures, but this is not true for the strongest updrafts wext in the SPM. This difference may be because CAPE integrates parcel buoyancy over the depth between the LFC and the LNB, whereas our stability measure characterises only the 850–500 hPa layer. The 850–500 hPa lapse rate (Fig. c) does not exhibit the same pattern as CAPE (Fig. a), indicating that lower-tropospheric stability alone does not determine CAPE in ERA5. While the idealised SPM links CAPE solely to stability, CAPE is also influenced by the properties of the source parcel and the thermodynamic structure outside the 850–500 hPa layer. Although reanalysis-derived CAPE is known to exhibit biases relative to sounding-derived estimates, ERA5 has been shown to be one of the most reliable reanalyses for convective parameters . A detailed interpretation of this behaviour of CAPE in the stability–humidity plane is left for future work.

In summary, the observational analysis confirms important aspects of the behaviour of the SPM. In both cases, the areal-mean rain rate increases with the effective entrainment rate, while the intensity of convection peaks when the atmosphere is drier and more unstable.

The correspondence between the SPM and observational analysis above suggests that the effective entrainment rate ϵ‾, diagnosed purely from large-scale fields, provides information about the level of mixing between the convective ensemble and its environment. We therefore use the stability–humidity plane to investigate factors that influence ϵ‾ and, by extension, convective mixing.

Previous studies have suggested that larger convective cells may be associated with lower entrainment because of their lower surface-area to volume ratio e.g.,. However, the observed convective cell morphology does not support a strong influence of cell size on ϵ‾. Instead, as found by , drier and more unstable environments are associated with a smaller number of convective cells (Fig. d) and a larger mean convective cell area (Fig. e). The largest cells occur under similar environments to the most intense convection, with a weak dependence on ϵ‾. Restricting the environmental conditions to raining pixels or considering environmental conditions 3 h prior to each radar scene does not substantially change this pattern (not shown).

A possible confounding variable in the above analysis is that of the convective area fraction (CAF). This increases strongly with the areal-mean precipitation and is therefore highest for large values of ϵ‾ (Fig. f). But a large CAF may also favour larger cells , thereby obscuring the effect of cell size on entrainment. To address this, we compare scenes with similar overall convective extent by partitioning radar scenes into four groups according to CAF quartiles. Characteristics of each quartile in the stability–humidity phase plane are shown in Fig. , where Q1 denotes the lowest quartile of CAF. Because partitioning the data into quartiles reduces the sample size within each group, we lower the minimum number of radar scenes required per 2D histogram bin from 30 to 20.

For a given CAF quartile, the relationships identified in Fig.  persist: drier and more unstable environments are associated with fewer but larger convective cells, and the relationship to ϵ‾ is weak (Fig. a–h). Because CAF is constrained within each quartile, this shift toward larger cells reflects increased spatial clustering rather than greater total convective coverage, indicating enhanced convective organisation. Previous studies have found contrasting effects of convective organisation on entrainment, with overall dilution tending to decrease because of the moist near-cloud environment, but an increase in mixing through increases in turbulent kinetic energy .

Note that the above relationships of maximum cell intensity to the mean area of cells are specific to convective cells. When all raining areas are included (stratiform and convective), the mean rain cell area follows areal-mean precipitation more closely (not shown), because high areal-mean rainfall often reflects an expansion of raining area associated with stratiform precipitation .

As for the overall statistics, when stratified by CAF, CAPE and maximum cell intensity exhibit patterns similar to those of mean convective cell area (Fig. i–p). Higher CAPE in drier and more unstable environments coincides with stronger maximum cell intensity. This pattern is weaker in the lowest CAF quartile, for which maximum cell intensity and mean cell area depend more strongly on humidity than on stability. At low CAF, the limited convective coverage favours smaller cells, which are expected to experience greater entrainment and therefore be more sensitive to environmental humidity. Nevertheless, these results give a consistent picture of the thermodynamic conditions that favour more spatially organised and locally intense convection. Such storms form in environments in which stronger instability supports deeper, more coherent updrafts, while reduced background humidity limits the number of simultaneously active cells.

Another hypothesis is that the depth of the boundary layer may influence convective entrainment through its effect on the size of convective updrafts . Indeed, this hypothesis informed our parameterisation of the entrainment distribution in the SPM. While our preceding results suggest that size, at least when it comes to convective rain cells, is not a strong predictor of entrainment, we nevertheless examine the behaviour of the boundary-layer height (BLH) and the LCL, as diagnosed in ERA5, binned in the stability–humidity plane. We present both the spatial maximum BLH (Fig. a) and the spatially averaged BLH (Fig. b) within the radar domain, where the maximum captures local variations, while the mean reflects the bulk boundary layer state. Only the spatially averaged LCL is shown (Fig. c) as the maximum LCL exhibits a similar pattern (not shown).

The results show that the mean BLH increases with both saturation deficit and stability, with contours that roughly follow those of ϵ‾. This is consistent with the argument that a deeper boundary layer reduces convective entrainment. The LCL and the maximum BLH have slightly different dependencies, with a stronger dependence on saturation deficit and weaker dependence on stability. Nevertheless, when plotted as a function of the areal-mean precipitation rate, ϵ‾ generally decreases with all measures of boundary layer depth: mean and maximum BLH and LCL (Fig. ). Comparing Fig.  with Fig. , we see that the effective entrainment diagnosed from ERA5 has a similar dependence on entrainment to the SPM. This supports our use of the cloud-base height as a key parameter within the model's entrainment distribution.

In addition to a lower effective entrainment rate, higher values of the LCL and maximum BLH correspond to greater maximum cell intensity in the radar observations (cf. Figs. c and a, c). This is consistent with previous theoretical and observational studies relating boundary layer depth to the dilution experienced by convective updrafts. For the SPM, however, a fixed CAPE has a similar value of wext even when the cloud-base height is reduced from zb=1400 m to zb=700 m (not shown). This is because the SPM considers the effect of entrainment on the environment as well as the convective ensemble, and wext therefore increases with a wider entrainment distribution rather than a lower mean value of entrainment. However, this result is likely sensitive to our assumption of a uniform distribution of mass flux with entrainment; further work is needed to investigate different mass flux distributions based on cloud-resolving simulations.

Finally, we consider the possible role played by spatial variability in humidity in the diagnosis of ϵ‾. The SPM assumes that the environment for each plume within the plume spectrum is the same and equal to that of the domain mean. However, small-scale humidity variability may result in a near-cloud environment substantially different to that of the mean, thereby affecting dilution produced by entrainment mixing. To test this hypothesis, we calculated the spatial standard deviation of the saturation fraction (σSF), using ERA5 data. Here the saturation fraction is defined as the ratio of the column-integrated water vapour divided by the same quantity assuming saturation, with the vertical integrals taken from 1000 to 100 hPa following . Note that we are limited by the 0.25° resolution of the ERA5 data to consider only relatively large-scale humidity variability. Figure d shows that σSF increases with the mean saturation deficit, and is highest in environments in which the most intense convection occurs. This supports the notion that intense convection develops within moist, localised regions embedded in a drier large-scale environment e.g.,. The enhanced moisture variability may also partly reflect the effects of convection itself as represented in ERA5, for example through data assimilation, rather than a pre-existing environmental control. Although we do not infer causality from this relationship, this finding points to the possible importance of small-scale variability in the environment, unaccounted for in the SPM when characterising convective mixing.

In summary, the results of this subsection provide support for a role of the depth of the boundary layer in setting the bulk entraining properties of convection. On the other hand, we found that the average size of convective rain cells was not related to the effective entrainment rate, suggesting either that the size of updrafts relevant for their mixing properties is not well captured by measures of their rainfall area, or that boundary-layer depth is related to the effective entrainment rate for reasons other than its effect on updraft size. We also noted that small-scale variability in the convective environment may cause variations in mixing that are not captured in the effective entrainment rate. Nevertheless, the overall agreement between the SPM and the observational analysis is promising, and suggests the stability–humidity phase plane is a useful tool for examining the mixing and other properties of convection, particularly for understanding and evaluating storm-resolving models.