In order to evaluate the model simulations of individual ship tracks, rather than a larger-scale temporal or spatially aggregated assessment, we define the following criteria.

Accurate representation of changes in CRE. The cloud radiative effect (CRE) in overcast scenes is largely dependent on the inside ship track Nd and LWP values , which encapsulate both the Twomey brightening and subsequent LWP adjustments. We want to ensure our model is producing simulations with the correct numerical values inside the ship track. Due to difficulties in obtaining perfect background simulations, our first criteria considers solely the absolute values where the ship perturbation has been applied, since this is ultimately where it is key to get the behaviour correct.

We apply the above criteria by considering temporal evolutions (treating the length of a ship track as a time since aerosol injection axis) of inside track Nd and LWP, and the associated enhancements from the background (the calculation of which is explained in Sect. ). We discuss the “correctness” with respect to each criterion in terms of the similarities and differences between the observed responses compared to the simulated responses.

Our case study is simulated using the Met Office Unified Model (UM, version 13.0; ), in the Regional Atmosphere and Land (RAL) configuration 3.1 . Our domain covers the Eastern and Central Pacific, off the coast of California (Fig. ). The 2625 km by 1125 km domain is centred on (40° N, 135.25° W) with a horizontal resolution of 1.5 km. There are 90 levels up to 40 km, with 16 levels below 1 km. The model time step is 75 s, and the lateral boundary conditions to the nested regional model are provided hourly by the UM global model at N216 resolution (≈ 60 km) in Global Atmosphere (GA) 6.1 science configuration . Our simulations are initialised from global model analysis at 00:00Z on 11 July 2018 and run for 48 h.

To properly simulate the cloud adjustments to an aerosol perturbation in a cloud resolving model, a double moment cloud microphysics scheme is recommended . Double-moment microphysics schemes prognose both the mass and number concentration of each hydrometeor species (e.g. ). A double-moment microphysics scheme allows Nd to be recalculated at each time step, and therefore can subsequently modify process rates (such as autoconversion), impacting the cloud evolution over time . In single moment schemes, the number concentration of each species (from which the process rates are derived) are either constant or diagnosed from other meteorological parameters using empirical relationships. These schemes, whilst computationally cheaper, can fail to accurately represent the indirect effect from aerosol-cloud interactions .

For cloud microphysics, we employ the Cloud AeroSol Interacting Microphysics (CASIM) scheme . CASIM is a double moment scheme with five hydrometeor species (cloud liquid, rain, ice, snow, and graupel) in which both mass and number concentration mixing ratios are prognosed. Each cloud species' particle size distribution (PSD) is described using a generalised gamma function with constant shape parameters. The autoconversion of cloud droplets into rain droplets and accretion of cloud water by rain are parametrised in CASIM following . CASIM also contains other warm cloud processes, such as aggregation (rain collecting rain), and we refer the reader to for details on these parametrised processes.

We couple CASIM to the double moment modal Global Model of Aerosol Processes aerosol microphysics scheme (GLOMAP-mode; ), within United Kingdom Chemistry and Aerosols (UKCA) sub-model, where aerosols are represented by five log-normal modes. These are the nucleation, Aitken, accumulation, coarse, and “Aitken insoluble” modes. Both the number and mass of each of the four chemical components (sulphate, sea salt, black carbon, and organic carbon) are prognosed in this double moment scheme. Black carbon cannot enter the nucleation mode, and sea salt cannot enter either the nucleation or Aitken modes. Dust is represented separately within the CLASSIC binned sectional scheme . GLOMAP-mode simulates aerosol microphysical processes such as the nucleation of new particles, coagulation of particles, condensation, and cloud processing . Particles can grow through condensational growth or coagulation between particles, and is represented by an increase in particle diameter, which can move particles into a greater size mode. Particles in soluble modes can also absorb atmospheric water and undergo hygroscopic growth .

Coupling between GLOMAP-mode and CASIM allows for aerosol mass and number concentrations of the Aitken, accumulation, and coarse aerosol modes to be passed into the CASIM activation scheme for the production cloud droplets , which is parametrised by ARG. Autoconversion and accretion rates from CASIM are summed and passed back into GLOMAP-mode to calculate the removal of aerosol (that are inside droplets) by rain. Rain rates also determine the removal of aerosol through impaction scavenging. GLOMAP-mode also uses the liquid water content from CASIM to calculate the rate of conversion of sulphur dioxide into sulphate inside cloud droplets .

Our simulations are coupled to the standard radiative transfer scheme in the UM (using the RADAER module; ) in order to calculate the aerosol radiative effects, however the direct radiative effect of aerosols from shipping is not the main focus of this study.

Within our simulations, we only consider aerosol in the soluble accumulation mode and initialise our simulation with a constant sulphate aerosol number concentration of 200 cm⁻³, characteristic mass 3 × 10-18 kg, and aerosol diameter of ≈ 0.1 µm, which is typical of sulphate aerosols . Aerosol is allowed to evolve freely after the model simulation begins, with the boundaries determined by the global driving model. As described in , aerosol mass and number concentrations can vary as determined by the microphysical processes in GLOMAP-mode, however CASIM does not have the capability to track aerosol tracers within hydrometeors and therefore cloud microphysics does not alter aerosol mass/size/composition except through irreversible removal via wet-scavenging.

Whilst we only consider our initial aerosol field to be in the accumulation mode, there are still background emissions into other modes. The different types of non-ship aerosol sources are described in , with emissions from marine phytoplankton, SO₂ from volcanoes, fires, and industrial sources. In the simulations of our study, we use anthropogenic emissions from the Coupled Model Intercomparison Project (CMIP6) inventory . The emissions are at a low resolution of ≈ 135 km . Natural emissions of sea salt, primary marine organic aerosol, and dust are parameterised as described in and . The emissions of sea spray, sulphate and carbonaceous aerosols are allocated to modes according to their size distribution , whereas biofuel, fossil fuel, and biomass burning emissions are emitted into the Aitken insoluble mode and then undergo “ageing” into the Aitken mode in certain conditions. Secondary organic aerosol (SOA) are produced from the oxidation of monoterpenes . Further details on other trace gas and aerosol emissions within this regional model set up are described in . These emissions can enter different aerosol size modes, not just the accumulation mode that the simulation is initialised from, however it is the accumulation mode aerosol mass/number concentrations that will dominate.

This assumption whereby we initialise our aerosol fields only in the soluble accumulation mode enables us to consider differences between control and ships runs as solely a function of increasing aerosol number concentration at the ship location, and the effects of aerosol chemistry are neglected. Differences in hydroscopicities in other aerosol-modes are assumed to be a less important factor in droplet activation than the mass/diameter of the mode . A similar initialisation from a uniform aerosol field of accumulation mode aerosol has been conducted before in , and was found to produce Nd that were in approximate agreement with observations. However, this assumption is evidently a significant simplification to our aerosol configuration, as the accumulation mode will dominate the aerosol mass and number concentrations, with the concentrations of other modes only impacted by the natural and anthropogenic emissions described above. We discuss the potential consequences of this in Sect. .

As a result of this simplified aerosol configuration, our only discussion of aerosol-microphysics concerns how the order of magnitude of the injected ship aerosol influences the activation scheme, and relies mainly on offline calculations. Subsequent investigations therefore consider only the model's representation of cloud adjustments to a droplet number perturbation, without evaluating aerosol microphysics or aerosol realism. Future work that increases the level of complexity and considers aerosol properties through to cloud adjustments within more realistic aerosol configurations would be valuable. Nevertheless, this study is intended as a first step toward understanding model representation of cloud adjustments within ship tracks in this context.

In order to simulate the shipping impacts on clouds, we add five ships to our simulation in our ships run. We use container ship locations identified in AIS data and matched to observed ship tracks from 11–12 July 2018, to add realistic moving sources of aerosol . Details of ship start/end positions and velocities can be found in Table . Container ships are selected because of their large size and high emissions, meaning that the resulting ship tracks are the most distinct in the observations. Our five selected ships also cover both directions of the shipping route within the domain. Aerosol is added in a model level at 10 m above sea level and immediately dispersed throughout the 1.5 km by 1.5 km grid box using the following equation: 1Nx,y,z,t+1=Nx,y,z,t+dNdt×ΔtV where dNdt is the rate of production number (the rate of ship aerosol emissions) in s⁻¹, Δt is the model time step (75 s), and V is the grid box volume. This describes the aerosol number concentration added to the entire grid box per time step. At this model resolution, ship tracks form at a minimum width of 1.5 km. Typical ship tracks have a width of on the order of 10 km , therefore this model resolution is suitable to capture the horizontal extent. Aerosol is almost immediately transported up to cloud base, and there is little deposition of aerosol at the bottom model level, therefore the 10 m injection height (whilst an underestimation) is approximately correct.

Realistic emissions of condensation nuclei (CN) from ships fall in the range 1016 to 1018 kg s⁻¹ . find that the ARG activation parameterisation exhibits a sharp “drop off” in activated fraction at increasing aerosol number concentrations (albeit, for NaCl), therefore we investigate the use of a high ship aerosol case (ship production number 1016) and a lower ship aerosol case (ship production number 5 × 1014) in this study (Sect. ). We add the aerosol into the accumulation soluble mode with a mass 3 × 10-18 kg, the same as background aerosol, meaning that both our background and ship aerosols are only in the accumulation mode with the same mass. This is done to simplify simulations and isolate causal cloud adjustments in our analysis, since differences between the ships and control run can be attributed to the increased aerosol number concentration at the ship locations. Since these are not the actual emissions from the real ships, there is a considerable source of uncertainty introduced by this “best guess” of the emissions. However, since in this study we are not investigating the model representation of aerosol microphysics and instead are concerned with the model representation of the cloud adjustment to changes in Nd, this estimation of the emissions is deemed reasonable.

To evaluate our model simulations, we use observations from NASA's Aqua and Terra satellites to compare to our model output. We use Level-2 Collection 6.1 data from the Moderate Resolution Imaging Spectroradiometer (MODIS; ) to obtain LWP and derive Nd following .

We collocate Aqua and Terra overpasses the occur during our simulation with our model domain, and only consider images the contain any part of a ship track, obtaining six snapshots of our domain, as shown in Fig. . Model data is output hourly, therefore we compare satellite data to model output from the nearest time step. When model data is compared to satellite data, the model output is masked only to consider the region contained within the associated satellite snapshot.

Simulated precipitation in the domain is evaluated against ERA5 surface precipitation , as well as Global Precipitation Measurement (GPM-IMERG; ) and overpasses from the Cloud Profiling Radar (CPR) onboard CloudSat (both the 2C-Precip-Column and 2C-Rain-Profile products; ). We investigate the probability of precipitation across Nd – LWP space using observations from the CCCM (CERES–CloudSat–CALIPSO–MODIS) combined product . We use the CloudSat CPR precipitation flag (from 2C-Precip-Column; ) to define a probability of precipitation as counts of liquid precipitation or drizzle divided by all counts, per Nd-LWP bin (from MODIS Aqua). Since in 2018 CloudSat was not part of the A-train (and therefore not collocated with MODIS), we use observations from 2007–2011, as in .

We follow a methodology similar to that of , , and to obtain our ship track locations. Using ship positions (from AIS data) we infer the locations of clouds which experienced an aerosol perturbation (neglecting the time taken for aerosol to reach cloud base). These positions are advected in wind fields over time to predict ship tracks which consist of not only location, but the associated time since that location experienced the aerosol perturbation. This allows the length of a ship track to be treated as a time axis, and discern time evolution after an aerosol perturbation even in snapshot imagery .

In our observations of ship tracks in this case study, the hourly ship locations are advected in ERA5 reanalysis winds at 0.25° resolution and 3 h intervals between the surface and the boundary layer top, with vertical motion within the ship plume following (see for further details). In model output, the same ship locations are advected instead in the model winds following the same methodology.

Since this study is comparing only five ship tracks between model output and satellite data, it is essential that the exact ship track location is logged as to properly quantify the aerosol effects. We apply small corrections to our model and satellite output track locations from the above methodology to ensure that our track location falls exactly on each visible ship track, for both model output and observations. This is done by hand logging the nearest point of the visible track, perpendicular to the predicted track location. This is not possible in large composite studies which contain thousands of ship tracks, however due to the small number of ships considered in this study it is feasible, and ensures the most accurate measurement of the cloud responses.

Following , we regrid our datasets to 2D space for each ship track, at each model time step/MODIS snapshot. This 2D space consists of the “time along track” (binned to 1 hourly windows), and the perpendicular “distance away from track” (binned to 1 km). This allows us to investigate exactly how the cloud properties vary with time since aerosol perturbation and distance from the centre of the perturbation.

A key challenge in this study is determining the most appropriate method for quantifying the impact of ship emissions on clouds across our two datasets: model simulations and satellite observations.

For the UM-CASIM simulations, the impact of ship-emitted aerosols can be directly quantified by comparing the enhancement in cloud properties between the ships run and the control run. To mitigate stochastic noise between model runs, we apply Gaussian smoothing to the model fields (smoothing with a Gaussian kernel with standard deviation of 0.75 km) before calculating the percentage difference to obtain enhancements in Nd and LWP (ϵN and ϵL). This approach allows for an estimation of aerosol-induced changes while minimising the influence of small-scale variability in between model runs.

In contrast, defining a control region in observational data is less straightforward . Traditional observational studies of ship tracks typically identify an “unpolluted” reference region located approximately 30 km perpendicular to the ship track . This region serves as a control from which the percentage enhancement in cloud properties is calculated. However, this methodology presents potential biases. Specifically, non-linear background gradients can introduce systematic errors in the estimated aerosol effect . Additionally, in scenes containing multiple ship tracks, this approach becomes increasingly uncertain. For instance, in our case study, many “outside track” regions that are used as controls inadvertently overlap with other ship tracks. Consequently, the percentage enhancement derived from such comparisons likely underestimates the true aerosol impact . Despite these limitations, this method remains the most viable approach given the constraints of observational data.

In Fig. j, l we compare the use of (a) the control simulation and (b) the “outside” region to define the unperturbed cloud for the model simulation (solid orange and dashed orange lines, respectively). For the model, we find that both methods for calculating model enhancements produce the same results (to within 15 % and 10 % for Nd and LWP enhancements in the first 15 h of ship tracks, respectively). Our model background has little background variability, and any variability is much smaller than the large perturbation inside the ship track, therefore both methods provide similar results. This would not necessarily be the case in a model simulation with greater background variability.

Using Eq. (1) from , we can calculate expected enhancements in cloud droplet number concentration (ϵN) based on the background Nd as a function of aerosol mass emission rates 2ϵN=MEγα+βNcln+1 where ME is the mass emission rate (in kg s⁻¹), Ncln is the unperturbed background Nd, and the constants α, β, and γ are 0.1041, 0.0038 and 0.36 as obtained from the fit in . This equation provides a useful method of evaluating whether or not our model is producing enhancements of the right order of magnitude, based on its background conditions.

Using Eq. (1), and a realistic ship production number of 1016 s⁻¹, we obtain aerosol number concentrations within the ship track of roughly 20 000 cm⁻³. This is realistic given in-situ observations of ship tracks . However, with these values, our simulated ship tracks exhibit bizarre “split” behaviour, with no cloud droplets activated inside the centre of the track where the aerosol number concentrations are highest (see Fig. ).

This unphysical split-ship track effect is due to non-monotonic behaviour of the Abdul-Razzak and Ghan (ARG) activation parameterisation scheme with increasing aerosol number concentrations, beyond some critical aerosol number concentration (dependent on temperature, pressure and updraft velocity) . To demonstrate this, we utilise pyrcel , a python package for implementing a simple adiabatic cloud parcel model, to isolate the impact of the activation parameterisation.

We plot both the ARG and NS parameterisations as a function of aerosol number concentration in Fig. a, using values of temperature, pressure and updraft velocity that are representative of our domain. We find that with using our realistic ship production numbers (1016 s⁻¹) the aerosol number concentrations (20 000 cm⁻³) are beyond the critical concentration of ARG, producing the unrealistic split ship tracks. NS, whilst not containing this non-monotonic behaviour, would contain a large over estimation of cloud droplets at the realistic ship production number and therefore would also be unsuitable with these emissions. This detail of the ARG activation parameterisation has been documented in , and emphasises that commonly used aerosol activation parameterisations are not suitable for MCB applications.

Reduction of our ship production number to 5 × 1014 s⁻¹, produces only a 200 cm⁻³ aerosol perturbation, but recovers Nd that match much closer to observations. This artificial reduction in ship production number is necessary to obtain more realistic ship tracks (without the splitting effect), but we must verify whether or not the resultant enhancements in these ship tracks is as expected based on observations. Using Eq. (2), we calculate expected enhancements (ϵN) based on the background Nd as a function of aerosol mass emission rates. Using our ship production of numbers of 1016 and 5 × 1014 s⁻¹, and a mean aerosol mass of 3×10-18 kg, we obtain mass emissions rates of 1.5×10-1 and 1.5×10-3 kg s⁻¹, respectively.

We find that the model enhancement obtained from our reduced emissions are consistent with what we would expect from the real emissions (Fig. b), despite the emissions being unrealistically low. Since the enhancements obtained are as expected based on observations, we assume that this reduced ship production number is sufficient to drive this realistic test case, and suitable for evaluation of the model's process representation. For the remaining results of this work, we only simulate ships with ship production number of 5 × 1014 s⁻¹, however acknowledge that this undermines any ability to make claims relating to the injected aerosol number concentration. As such, we investigate the model representation of the cloud adjustments (specifically LWP adjustments) subsequent to the simulated Nd perturbation, which is of the correct magnitude as established in this section.

Using the model configuration detailed in Sect. , our simulation is run for 48 h from 00:00Z 11 July 2018 to 00:00Z 13 July 2018. In our control run, no ship emissions are added to the model, and we use this simulation to characterise our “unpolluted” cloud. Typically, in ship track observations, we have no pure “control” cloud, and therefore we must treat a region outside of a ship track as being representative of what the cloud would have looked like if there was no aerosol perturbation present. In the model case, however, we can simply turn ships on and off to isolate the aerosol impact on the cloud.

The aerosol number concentration (Na), cloud droplet number concentration (Nd), liquid water path (LWP), surface rain rate (RR), and top of atmosphere outgoing short wave radiation (SW) for the control run 44 h after initialisation can be found in the left-hand column of Fig. . Figure a demonstrates how the coupled GLOMAP-mode aerosol scheme allows depletion of aerosol in locations where there is precipitation (in the Central Pacific region of the domain), and the land sources of aerosol are the most significant over the course of the simulation. The coupled UM-CASIM configuration produces open/closed cellular convection in this marine stratocumulus (Fig. c in the south-west of the domain), with the presence of drizzle (surface rain rates on average of roughly 0.5 mm h⁻¹; see Fig. g).

We evaluate our simulated precipitation in Fig. . Surface rain rates are shown for UM-CASIM control simulation (regridded to the same 0.25° resolution as ERA5) at 12 hourly intervals, and compared against ERA5 and GPM-IMERG large scale surface rain rates at the same time. Evidently, there is a significant disagreement between ERA5 and GPM on the presence of precipitation in the domain, with GPM being unable to capture drizzle rates smaller than 0.2 mm h⁻¹ . UM-CASIM is in rough agreement with ERA5 on the presence, location, and magnitude of the precipitation, however ERA5 contains its own uncertainties in the realism of precipitation . The UM-CASIM precipitation rates lie between those from GPM-IMERG and ERA5.

There are 3 overpasses of this domain by the CloudSat Cloud Profiling Radar (CPR; ) during our simulation run. Figure a shows the location and presence of surface precipitation (in the 2C-Precip-Column product; ) in these overpasses. There is significant disagreement in between the 2C-Precip-Column and 2C-Rain-Profile products from the CPR, with UM-CASIM underestimating precipitation along this overpass when compared to 2C-Precip-Column, and having a small mean bias when compared to 2C-Rain-Profile.

Evidently, this evaluation demonstrates that there is some uncertainty in the simulated precipitation within our domain, as well as within the observed precipitation. The UM-CASIM precipitation falls between what is predicted from ERA5 and GPM, with different CloudSat products giving largely different precipitation rates. More generally, double moment CASIM is shown to have increased skill at representing precipitation in a frontal system over the UK, and a tropical storm in Darwin , as compared to other model microphysics schemes (evaluated against GPM and radar), however there are still limitations in its representation of moderate precipitation, in the range of 4–16 mm h⁻¹ .

With our initialised background aerosol field of 200 cm⁻³, we produce a relatively clean cloud droplet background of roughly 50 cm⁻³. This is largely in line with that seen in satellite observations (Fig. a, c), however we note that two large sources of cloud droplets are missing from our model simulation. This is likely due to our initialisation from a constant aerosol field, and therefore we are missing these sources of aerosol in our simulation. Future work in initialising from a more realistic aerosol field would be beneficial, and the limitations of our simplified aerosol configuration are discussed in Sect. . Due to simplifications in the composition of aerosol field, we do not evaluate the model representation of model microphysics and consider mainly the LWP adjustment to the ship Nd perturbation.

As outlined in Sect. , we evaluate our simulations against three criteria in order to determine the realism of our simulations, and the ability of our model to accurately simulate ship tracks/MCB. In the following sections, we present our findings with respect to each of these criteria.

We confirm that the commonly used activation parameterisation by is inadequate to simulate aerosol-cloud interactions at very high aerosol number concentrations. We demonstrate the potential result of the non-monotonicity of the ARG activation when simulating ship tracks – ship tracks appear split (Fig. ). In practical terms, this necessitates an artificial reduction in ship aerosol emissions to obtain ship track structures that resemble those seen in satellite imagery. While this workaround may yield visually plausible results in non-precipitating conditions, it undermines the physical realism of the simulation's temporal evolution. An alternative activation parameterisation from would also be unsuitable at these high aerosol number concentrations, as activated cloud droplet concentration would be overestimated. Further work is necessary in determining whether other parameterisations (such as that of ) would be suitable, or if entirely new schemes for high aerosol number concentrations are needed.

We find that model-observation discrepancies become more pronounced in broken cumulus clouds, where there is greater precipitation. Under such conditions, it is possible that the autoconversion parameterisation of KK00, which governs the conversion of cloud water to rain, is too sensitive to increases in Nd. This scheme tends to suppress production of rain droplets too readily when the cloud droplet number concentration (Nd) is high – as demonstrated by the complete shut off of precipitation inside these ship tracks that form in precipitating scenes (see Fig. ).

In Fig. , the PoP is almost binary in our simulation, with either 100 % chance of precipitation above some threshold Nd-LWP line, and almost 0 % chance below it. In observations, there exists a much wider space in which precipitation can occur. This means that for any conditions close to this threshold line in Nd-LWP in our simulation, a small increase of Nd (such as through the injection of ship emissions), will move us beyond this threshold and decreases the PoP too extremely.

Consequently, the model predicts unrealistic suppression of precipitation, resulting in overestimation of the ship track lifetime and, by extension, the radiative cooling associated with the perturbation. This is not surprising, since the KK00 autoconversion scheme was designed for mid-latitude and extratropical stratocumulus layers under typical meteorological conditions, with their evaluated “polluted” case only having an Nd of 175 cm⁻³. This is aligned with the findings of with respect to GCM parameterisations, where precipitation inhibition can lead to a wet scavenging feedback which can increase aerosol loading somewhat non-physically. KK00 is possibly not well-suited to simulate very high Nd perturbations, as it tends to easily suppress precipitation – even in conditions where precipitation could still occur.

As discussed in Sect. , there is some uncertainty in the realism of the simulated precipitation in the domain, since it is difficult to obtain conclusive precipitation measurements across different satellite and reanalysis products. In order for the enhancement of LWP in a ship track to occur, precipitation need only be suppressed at cloud base, and therefore it is also possible that no impact in surface precipitation would occur. This highlights the difficulty in evaluating not only precipitation in models, but specifically the precipitation suppression effect.

In their designed configurations, these parameterisations limit the ability of models to provide meaningful insight into high Nd perturbations and subsequent adjustments over longer timescales or under precipitating conditions. This has important implications with respect to modelling across many scales, not just regional simulations. The development of global cloud resolving models would also benefit from improved cloud microphysics parameterisations .

Additionally, for simulations aimed at informing MCB strategies, it is particularly critical that parameterisations are suitable for high-Nd scenarios. Without modifications, models will likely overstate the cooling impacts, leading to inaccurate assessments of MCB efficacy. Improved autoconversion schemes that remain valid at high Nd (such as a scheme where autoconversion rate does not tend to zero as quickly with increases in Nd for a given LWP, allowing for a more diffuse PoP within Nd-LWP space), along with more robust activation parameterisations that avoid the need for artificial aerosol reductions, and the inclusion of cloud-processing such that precipitation suppression feedbacks are avoided, are essential for advancing the realism of such simulations.