The PPE is based on the composite stratocumulus-to-cumulus transition case in . computed thousands of forward and backward air parcel trajectories from areas of extensive cloud cover in the NE Pacific between May and October for 2002 to 2007. Boundary layer properties were retrieved over a six day period of advection from satellite data and meteorological reanalysis. found that the climatological, or averaged, trajectory represented the key characteristics of the transition well. developed this into a reference case for numerical simulation that represents a typical trajectory in the NE Pacific for June to August in 2006 and 2007 from a subset of trajectories for the three days in which the majority of the transition occurred. The meteorological state in this reference case is a good starting point for simulating a typical transition in the NE Pacific, from which we perturbed a range of cloud-controlling factors to explore variations in cloud behaviour.

The ensemble was simulated using the UK Met Office and National Environmental Research Council (NERC) LES model, called the MONC (Met Office/NERC Cloud) model . The model solves a set of Boussinesq-type equations, using an anelastic approximation here, which is based on a reference potential temperature profile that depends only on height. The subgrid turbulence parameterization is an extension of the Smagorinsky-Lilly model and is based on that described in . Version 0.9.0 of the Leeds-MONC Github repository was adapted for this study and released as version 0.9.1 . Here, MONC was coupled to the two-moment Cloud AeroSol Interaction Microphysics scheme CASIM, version 6341: and the Suite of Radiation Transfer Codes based on Edwards and Slingo SOCRATES, version 1012:.

CASIM is a two-moment bulk microphysics scheme that represents hydrometeors using gamma distributions for mass and number . Only warm-cloud processes (cloud liquid and rain) were used since ice processes are not part of the stratocumulus-to-cumulus transition in the NE Pacific. Simulations were initiated with soluble aerosol represented by prognostic mass and number concentrations in the Aitken and accumulation modes. The Aitken mode distribution has a standard deviation of 1.25 and a mean radius of 25 nm. The accumulation mode distribution has a standard deviation of 1.5 and a mean radius of 100 nm. The density of all aerosol particles was assumed to be 1500 kg m⁻³. All aerosol size modes were represented by a lognormal distribution. At saturation, the number of aerosol particles activated into cloud droplets was calculated using the scheme of , and these activated aerosol were represented using a separate in-cloud aerosol prognostic. Aerosol material contained within droplets can grow through droplet collision and coalescence with the assumption that one aerosol particle was present in each droplet, and is returned to the appropriate aerosol size mode on evaporation of the cloud droplets (including the coarse mode). Accretion and autoconversion are represented by the parameterization. Rain can evaporate in the subsaturated grid boxes, but aerosol is not returned to the size modes through this process.

Stratocumulus-to-cumulus transitions are often simulated in a Lagrangian style in which the domain moves with the advected cloudy air . As in other studies, we simulated the advection towards the equator by forcing the SSTs to increase over the course of the simulation. Wind profiles were retained to ensure appropriate ocean surface evaporation, but the model has periodic boundary conditions so the domain was always focused on the same cloud cell. The temperature and specific humidity profiles were allowed to evolve freely and the large-scale divergence was set to a constant value of 1.86 × 10⁻⁶ s⁻¹. The large-scale subsidence is calculated in the model as -Divergence × vertical height above sea level. Simulations were run for 3–4 d with a spin-up period of around an hour being discarded. The SST was increased by nearly 1.5 K per day, from 293.75 to 300.93 K, following , and . The domain was 12.8 by 12.8 by 3.1 km³. The horizontal resolution was 50 m, and the vertical resolution varied from 20 m near the surface, to 5 m around the temperature inversion, and gradually increased above that. It is worth noting that the domain size affects precipitation formation, with precipitation onset occurring earlier in larger domains where mesoscale organisation can be simulated. showed sensitivity tests for different domain sizes, and found that a large domain size encouraged earlier precipitation and onset of the stratocumulus-to-cumulus transition. The LES setup is idealised because realistic profiles would be specific to an individual transition case rather than being representative of a typical case. Although this may limit the realistic nature of the simulations, it simplifies the perturbation method for a study such as this where perturbations are made from a reference case to learn broadly about the transition behaviour across parameter space. This idealised setup also enabled comparison with previous studies that used the same approach e.g.,.

PPEs are a valuable tool for understanding the joint effects of parameters on model output. Perturbing parameters simultaneously in a space-filling way maximizes information from the model about how parameters jointly affect the outputs of interest. Five cloud-controlling factors were perturbed plus a sixth factor that alters the dependence of the autoconversion rate on Nd. Table  shows the individual ranges for each parameter, which form the boundaries of the 6-dimensional hypercube that the ensemble covers.

The parameter ranges were chosen to span the breadth of studies on stratocumulus-to-cumulus transitions in the subtropics. Often case studies are designed for LES simulation from observations of particularly fast or slow transitions, so a broad range of behaviours was included in the parameter space by spanning these reported cases . Although SST varies along the airmass trajectory, we chose not to include perturbations to SSTs or SST gradients among the parameters we investigated. To be useful, such a study focusing on cloud feedback would need to consider realistic covariations of SSTs with the cloud-controlling factors under investigation. Since many LES studies have not focused on the aerosol effect, the range for the accumulation mode concentrations was informed by the Cloud System Evolution in the Trades (CSET) and Marine ARM GPCI Investigation of Clouds (MAGIC) campaigns, which took place in the NE Pacific . Note that we have not included extremely polluted cases, such as the biomass burning region off the western coast of Africa. There are many studies of the aerosol semi-direct effect on the stratocumulus-to-cumulus transition in the Atlantic ocean, with some contradicting results . Further understanding of transition mechanisms will help to untangle these joint effects.

The transition properties analysed here are the transition time and the mean rain water path (R). The transition time is the time taken to transition from a stratocumulus regime (beginning at T1) to a cumulus regime (beginning at T2). Figure  shows two examples of how this was calculated from the cloud fraction (fc) for all the ensemble members based on fc>0.9 for stratocumulus and fc<0.55 for cumulus. The value of 0.55 for cumulus was chosen as a reasonable value for a cloud transition that maximised the number of transitioning ensemble members available for emulation. The sensitivity test in Appendix  shows that the key conclusions are statistically significant down to a threshold fc of 0.47, after which not enough simulations transition within the simulation time to give statistically significant information about the effects of different factors on the transition time. Figure a shows the base simulation, which has stratocumulus from the start of the simulation (T0) so T1 is set equal to T0, although realistically T1 could be earlier. The fc decreases below the cumulus threshold just after 50 h, but it recovers until the final time step when it reaches the threshold again, T2, giving a transition time of about 68 h. It is possible the cloud could recover again if a longer simulation were conducted, which creates some noise in the calculation of transition time. Figure b shows a simulation that takes about 12 h to build up stratocumulus, hence subtracting T2 from T1 gives a transition time of about 32 h.

Gaussian process emulation is a Bayesian machine learning method to learn the relationship between a set of input parameters and an output of interest . It uses a prior specification of the relationship consisting of a mean function (e.g., constant or linear) and a covariance structure. Here we use a constant mean function for transition time, a linear mean function for mean R, and the Matérn 5/2 covariance structure. The prior is updated using a set of training data, which is the set of perturbed inputs and corresponding outputs from the PPE, to create a posterior specification. Once validated, the emulator can be used to predict values for new sets of input values, with quantified accuracy.

The emulators of transition time and mean R were validated using the leave-one-out method. Here, an emulator is created from all but one of the training points and then used to predict a value for that left-out point. This is repeated for each point in the training set and the differences between the predicted values and the actual values are used to gauge how reliably the emulator can reproduce model output. Figure  shows that the training points were predicted within the 95 % confidence intervals for all of the points for transition time and mean R. However, the confidence intervals are quite large, especially in the transition time where some points are up to 10 h out in the predictions. The mean R emulates slightly better, which may be because it is easier to quantify than the transition time. There is some noise in the transition time calculation due to the simulation sometimes ending before it is obvious that the cloud has fully transitioned. The noise incurred in the transition time calculation is discussed in Sect. . We additionally validated the emulators by calculating the ratio of the standard deviation of the mean values at the training data (a measure of variation in emulated output) to the mean of the standard deviation of those points (the uncertainty in emulated values). For both emulators, this ratio is larger than 1, which tells us the function changes more than the underlying emulator uncertainty. If the ratio was less than 1, the emulator uncertainty would be too large compared to changes in the function, so it would not be a useful approximation of the relationship. This validation shows that the emulators predict model output with sufficient accuracy for us to gain important insights into the processes that drive transitions.

Following , we ran some initial condition ensembles to gauge the internal variability of the model, so that a “nugget” term could be added to the emulators. The nugget term allows the posterior mean function to have a buffer around each training point, rather than interpolating them exactly. This is useful when the data are noisy or, as in this case, for incorporating internal variability. At four training points we ran four extra simulations and varied the random seed in the model that initiates turbulence, which allowed us to calculate the approximate variance due to internal variability in the transition time and mean R. In three of these initial-condition ensembles, the members all transitioned within a few hours of each other, but in one ensemble the cloud recovered and did not fully transition until early the next day (approx. 10 h later). Adding this variance into the emulators accounts for some of the noise created in the transition time calculation (Sect. ). Details of this calculation can be found in and in the code repository .

We used a Python package to calculate the Sobol indices, which obtain the contributions of variance in each parameter to the variance in the outputs that we are emulating . We discuss the “main effect”, which is how much of the variance in the output is due to the variance in the individual parameter, and the interactions which are the portion of the variance that cannot be explained by linear combinations of the individual parameters, and is attributed to the interactions between parameters.

The stratocumulus-to-cumulus transition in the base simulation is similar to that of previous LES studies based on the composite case see also,. Figure a–c all show a distinct diurnal cycle in fc, liquid water path (L) and R. The fc is defined as the fraction of cloudy columns with a cloud liquid mass-mixing ratio greater than 0.01 g kg⁻¹. The stratocumulus initially has a high fc and is not drizzling. Through the first night the cloud begins to drizzle and through the second day the fc and L are lower than the first day. The cloud drizzles more through the second night, further depleting L and Na (Fig. d). During the third day it breaks up more into cumulus-like clouds and only recovers slightly through the night. Figure d shows that the domain-mean accumulation mode aerosol decreases gradually throughout the simulation and the Aitken remains fairly constant. Figure e–j shows three snapshots from 09:00 p.m. local time for each day of the simulation. At 09:00 p.m. local time on the first day (e–f) there is a uniform stratocumulus cloud with fc=0.99. The inversion height, and cloud top, are around 1000 m with a cloud layer thickness of about 300 m. At the same time on day 2 (g–h) there is a slightly more broken cloud but still a high fc of 0.94. The cross section shows that the boundary layer deepened and cloud top rose by a couple of hundred metres during the intervening day. The lowest cloud base is around 800 m, but now the base marks the bottom of cumulus-like plumes that feed into the higher stratocumulus cloud base, around 100 m above. Since the first day, L has decreased towards the edges of the cloud and thickened towards the middle of the cell. as the stratocumulus layer thinned. At the end of the third day (i–j) there is a much more broken cloud, with fc=0.53, and cumulus plumes in a layer below. At this stage the boundary layer is around another 100 m deeper, and the cloud top has risen with it.

Compared to other studies that simulated this composite case, the boundary layer did not deepen to the same degree and there was less drizzle. Other LES models simulated a boundary layer depth between 1.5 to 2.5 km, whereas our simulation has a maximum depth of 1.4 km . This could be due to the different numerical methods, radiation schemes and mixing processes in the models, or to the stretching of the vertical layers in the top of the domain. is the only study using aerosol processing that we compared our base simulation to. In our simulation, Fig. c shows that R peaks at about 25 g m⁻² at the beginning of the third day, which aligns roughly with the sensitivity tests in , which also used the parameterisation in a similar domain size. However, it is much less than the peak of 150 g m⁻² for the same domain size using their bin-emulating bulk microphysics scheme. The transitions in our simulations may be slower than those in the previous studies because the shallower boundary layer may limit the boundary layer decoupling and the lower R may limit the potential for a drizzle-depletion mechanism.

The whole PPE is summarised in Fig.  by splitting it into three categories: members that did not transition, members that transitioned with low mean R, and members that transitioned with high mean R. The simulations with high mean R generally started with a higher temperature inversion, i.e., a deeper boundary layer, and on average the boundary layer deepened less throughout the simulation than those that had lower mean R or did not transition. The lifting condensation level lowers throughout the simulations for all members, but slightly more for the high mean R set. The decoupling factor was calculated as the relative decoupling index from , zCB-zLCLzLCL, which is based on , where zCB is the cloud base and zLCL is the lifting condensation level. Cloud base was calculated as the domain-mean cloud base where cloud is present in the column. The high mean R set also decouples faster than the other sets, with the non-transitioning set being slowest to decouple. The high mean R set has significantly more surface precipitation than the other sets, with the non-transitioning set having the least. The non-transitioning set maintains a high fc until the very end of the simulation time, while the transitioning sets show fc decreasing significantly from day 2 (high mean R) and day 3 (low mean R). There is not much distinction in L between the sets, except the non-transitioning set increases more during the second night. The high mean R set has a much higher mean R in the first night and second day, but the other sets increase steadily from the second night onward. On average, the low mean R set has a lower median initial BLNa than the other sets, but the median BLNa decreases at the same rate as the non-transitioning set. The high mean R shows a faster initial decrease through the first night and second day, when it has the highest R.

Figure a shows that the range of initial fc produced across the PPE is large, as expected from perturbing many initial conditions over a large range of environmental conditions. Those that form stratocumulus (67 simulations, Fig. b) and those that form cumulus (37 simulations, Fig. c) make up the ensemble subset that transition. The subset mean in Fig. c has a similar shape to the base simulation, but the subset mean transitions a few hours earlier. However, the PPE members show a wide range of behaviours. On average, fc stays near one through the first day and night, before dipping in the second day to fc ≈ 0.75 and on the third day it crosses the cumulus threshold and stays below. A diurnal cycle can be seen in many of the members, with some members dipping to fc ≈ 0.4 and still recovering in the second night. Additionally, some members keep fc ≈ 1 until the third day and then transition rapidly.

While many of the simulations that transitioned formed stratocumulus within the first day, there were three simulations that only formed stratocumulus beyond the end of the second day when the SST had increased by at least 1 K and these transitioned very quickly. The transitioning simulations are “epoch aligned” in Fig. d by aligning T1 for each member, and the high SST members are highlighted. These fast transitions occur despite being in areas of parameter space where you might not expect it, for example in a very shallow boundary layer with a low autoconversion rate. This subset of simulations shows that warmer initial SSTs may act to considerably speed up the transition, above meteorological conditions, which has implications for the future warmer climate. However, the PPE does not have enough simulations with warm SST to draw a definitive conclusion. The warm SST simulations have been removed from this analysis (leaving 34 simulations) since the difference in SST at initial stratocumulus is akin to perturbing a seventh parameter, but one that was not initially accounted for in our experimental design.

We analysed mean R to determine whether the drivers of the transition might have acted through a drizzle-depletion mechanism. The PPE mean R is summarised in Fig. , with the domain-averaged timeseries for each member shown in Fig. a. The PPE is split into “low” (red) and “high” (blue) R by a temporal mean threshold of 7 g m⁻² (approximately half of the highest member). The BLNa (panel b) and the fc for the transitioning simulations (aligned by T0s in Fig. c and epoch aligned by T1s in Fig. d) have also been coloured low and high for R with corresponding subset means. The histograms in Fig. e and f show the number of points being averaged over at a given time in each subset, which varies because of the different stratocumulus formation times (Sect.  and Fig. ).

We find that the set of simulations with higher mean R transitioned approximately 24 h ahead of those with lower mean R (Fig. d). Figure a shows that those with higher mean R mostly produced drizzle in the first two days, whereas for those with lower mean R the drizzle gradually builds through the simulation. Figure b shows that in most simulations the BLNa decreases. It also shows that the higher mean R subset has a median concentration that is initially higher than the lower mean R subset, but it decreases more sharply over the first 20 or so hours from T1. After 20 h, the gradient of the higher mean R subset levels out to be similar to the lower mean R subset. In Fig. c, the timeseries are lined up with the diurnal cycle and it shows that the high R subset mean recovers more than the low R mean during the nights. This could be similar to the behaviour shown in , where the drizzling stratocumulus case recovers to higher L values through the night compared with the suppressed precipitation case, which is driven more by entrainment than longwave cooling. However, Fig. c shows that some of the high R cases follow a more extreme diurnal cycle than the low R cases.

Figure  shows that by splitting the ensemble into the high and low mean R subsets, some of the marginal correlations become stronger. Figure a shows Δθ has a stronger correlation with transition time when only considering the low mean R cases, and the correlation is otherwise insignificant. Conversely, BLNa has a stronger correlation with transition time when only considering the high mean R cases. Figure c shows that although the fastest transitions do have a higher mean R, drizzle is clearly not the only important factor determining the transition time. Rather, other factors affect the characteristics of the transition, such as the degree of decoupling and the ability to recover through the night. It should be noted that with the inclusion of the high SST ensemble members, any correlation of mean R with transition time vanishes. This suggests that this correlation may not be significant if a wider array of deepening-decoupling mechanisms were represented.