Aviation's total radiative forcing arises not only from CO2 emissions but also significantly from non-CO2 effects, including nitrogen oxide (NOx) emissions and the formation of persistent contrail cirrus clouds . The estimation of the contrail cirrus' radiative impact requires a realistic representation of contrails in large-scale models. Therefore, the general circulation model (GCM) ECHAM has been expanded by the contrail module CCMod . Due to the coarse resolution of GCMs, small-scale processes in the early stages of contrails can not be explicitly resolved and therefore have to be parameterized. The early ice crystal number is the key quantity of young contrails, which strongly influences the later contrail cirrus properties, life cycle and radiative forcing . For kerosene combustion, parameterizations have been developed for both the initial number of ice crystals formed and the loss of ice crystals due to wake vortex interaction . These parameterizations have been successfully implemented in CCMod . To assess the radiative impacts of contrails from alternative aviation technologies such as hydrogen combustion, these parameterizations need to be revised. Recently, extended the wake vortex loss parameterization to hydrogen combustion scenarios. However, a parameterization for the initial number of ice crystals formed is still lacking for hydrogen combustion cases.

Contrail formation theories/models range from binary decisions on whether contrails form or not , to time-squeezed microphysical models , to fully time-resolved microphysical models. In the latter models, the dynamical representation ranges from 3D large-eddy simulations (LES) e.g., and 3D Reynolds-Averaged Navier-Stokes (RANS) models e.g.,, to 0D box models e.g.,. A 0D box model approach uses an offline approach, meaning that the microphysics is calculated without feedback on the dynamics. Instead, the dilution, which describes plume expansion, cooling, and humidity evolution, is externally prescribed. This can be either through analytical formulations e.g., or based on output from prior performed LES or RANS simulations. The big advantage of these 0D box models is their low computational cost due to their simplified dynamic representation, which allows exploring a large parameter space of emission, aerosol and meteorological parameters with a detailed microphysical contrail formation model. At their minimal computational cost, when relying on an average trajectory, 0D models neglect any plume heterogeneity. The plume heterogeneity is accounted for when an ensemble of trajectories so-called multi-0D approach as described in is used, but as long as the box model is run for each trajectory separately, diffusive processes between nearby trajectories are neglected. Therefore, introduced the 1D framework RadMod as a compromise between computational expensive 3D models with detailed dynamical core and low-cost 0D box models with strongly simplified dynamics. So far, only the dynamical component of this 1D framework has been described.

Previous studies on contrail formation from hydrogen combustion showed that microphysical processes strongly control the number of contrail ice crystals formed . For a comprehensive understanding of contrail formation, however, it is also essential to consider how engine characteristics influence the microphysical processes. This includes a realistic representation of how various aspects of commercial turbofan engines affect contrail formation, as well as how these influences may change with advancements in modern and future engine technologies.

Turbofan engines generate thrust through two streams: a core flow, where combustion takes place, and a bypass flow ducted around the core . The overall efficiency of propulsion represents the fraction of the fuel’s combustion heat converted into the work rate performed by the thrust . The overall efficiency of propulsion depends on the flight state and can be expressed as the product of the thermal efficiency and the propulsive efficiency. Thermal efficiency measures the proportion of combustion heat converted into power delivered by the engine core. According to the second law of thermodynamics, this value is always less than one. Thermal efficiency has increased over the years due to higher overall pressure ratios and higher turbine inlet temperatures . Current values are around ∼ 0.45, with a practical upper limit near ∼ 0.55 due to physical limitations (e.g., material properties) and constraints on NOx emissions . Propulsive efficiency measures how effectively core delivered power is converted into thrust. Some energy inevitably remains as kinetic energy in the exhaust, so this efficiency also remains below one. The values of propulsive efficiency have increased over the years, with current values around ∼ 0.8, and are expected to increase to approximately ∼ 0.85 . This increase over the years is primarily due to the adoption of higher bypass ratios, which is the ratio of the bypass to core mass flow. A higher bypass ratio means that a larger mass of air is accelerated at a lower velocity reducing the kinetic energy in the exhaust. Bypass ratios have increased from approximately ∼ 2 in the 1960s, to ∼ 4–6 in the 1980s, to ∼ 8–10 in the 2000s, and up to around ∼ 12 in the 2010s with the advent of geared turbofans . Future ultra-high bypass ratio engines may reach values up to ∼ 20 . However, as the bypass ratio increases, so does the fan diameter, which leads to increased aerodynamic drag and installation challenges , imposing a practical upper limit. To overcome this technical limit, a potential future technology could be the use of open rotor fans with bypass ratios ≳ 30 , and theoretical propulsive efficiencies of up to ∼ 0.95 .

In a comprehensive study, investigated contrail formation for kerosene combustion using a LES and an average 0D box model approach. As part of this study, Lewellen investigated the impact of engine size on dilution speed and developed a corresponding scaling law for the number of ice crystals formed on ambient aerosols. To the best of the authors’ knowledge, the influence of engine size has not yet been studied for hydrogen combustion. This aspect could, however, be of particular relevance, as hydrogen-powered aviation is expected to be introduced first in regional, short- and medium-haul aircraft with smaller engines, and potentially later extended to long-haul aircraft equipped with larger engines .

By comparing a wide range of simulations, also found that a 0D box model approach generally overestimates ice crystal numbers relative to LES results when simulating contrail formation on emitted particles with high numbers. In contrast, box model results closely match the LES outcomes for low particle numbers or contrail formation on entrained ambient aerosols (which is the case in our setups). Hence, a 0D box model approach is deemed appropriate for our study, which deals with contrail formation on entrained ambient aerosols for hydrogen combustion.

Here we literally repeat the according subsection “Scope and outline of the study” of Part 1 (see Sect. 1.2 in ): Our overarching goal is to develop a parameterization for the final number of ice crystals formed for aircraft with hydrogen combustion, suitable for implementation in GCMs and other large-scale contrail models. Specifically, we seek a functional relationship of the form 1Nice,f=Nice,f(aatmosphere,aaerosol,aaircraft), where aatmosphere denotes a set of variables characterizing the background atmosphere (e.g., ambient temperature), aaerosol represents properties of ambient aerosols (number concentration, size distribution and solubility) and aaircraft includes aircraft-related parameters (e.g., engine size).

Our objective is to identify a functional relationship that balances simplicity with physical fidelity. In other words, we aim to capture the impact of the key processes that govern ice crystal formation in a form that is simple enough to be easily applied in other models (level of complexity, computing time, ease of implementation). In Parts 1 and 2 of a trilogy of papers, we systematically explore a broad parameter space and address aspects that have not been explored before to gain a deep insight into the physical processes influencing the ice crystal formation. This allows us to select the set of variables that constitute the inputs to the parameterization of Nice,f.

While a previous Nice,f parameterization of contrail formation used a first-principles-based concept, the current work across Parts 1 to 3 employs a hybrid approach. It combines a data-driven advanced regression method to fit a multidimensional database of contrail formation simulations with analytical scaling relations. To keep the number of dimensions in the simulation database as small as possible, we identify conditions under which the sensitivity to specific variables is negligible, and also employ analytical scaling relations derived from sensitivity simulations.

In Part 1, we focus on the roles of atmospheric variables aatmosphere and aerosol properties aaerosol. Aircraft-related parameters aaircraft are addressed in Part 2. The final parameterization, synthesizing insights from Parts 1 and 2, will be presented in Part 3.

In the current study, we incorporate the leading effects of engine-related aspects into the simulations. In Sect.  we provide the theoretical background about the extensions made to our used box model (described in Sect. ). We then investigate the impact of these aspects on the number of ice crystals formed (Sect. ) before we discuss the findings in the light of existing literature and propose possibilities for future research (Sect. ). Finally, we conclude with a discussion of how our findings can be used in the development of a generalized parameterization of ice crystal number (Sect. ).

We review and present the equations necessary for understanding the considerations in the subsequent sections. Detailed derivations and explanations can be found in .

Fuel is combusted in an aircraft engine to produce thrust, propelling the aircraft forward in accordance with Newton’s third law of motion. In this context, the overall efficiency of propulsion 2η=FU∞m˙FQ is defined as the ratio of work rate performed by the thrust F at aircraft speed U∞ and the chemical energy supplied by the fuel, which is determined by the fuel flow rate m˙F and the mass-specific heat of combustion Q (also known as lower calorific value). The remaining part of the chemical energy m˙FQ(1-η), which does not contribute to generating the thrust, is contained in the exhaust air in the form of excess total enthalpy above ambient levels. The total enthalpy is the sum of static enthalpy and kinetic energy. Additionally, the combustion of fuel adds water vapor to the core exhaust, quantified by the fuel-dependent water vapor (mass) emission index EI_v.

We refer to the core exhaust as plume and to individual parts of it as plume parcels. Each plume parcel mixes with cold ambient air downstream of the engine. Classical thermodynamic contrail formation theory assumes that the core and bypass air of a turbofan engine are fully mixed by the time contrail formation begins (or, equivalently, that the emitted enthalpy is entirely contained within the core flow at the engine exit). Additionally, it assumes a stagnant exhaust plume, such that the entire kinetic energy is presumed to have been dissipated into heat at the onset of contrail formation. Furthermore, the theory assumes the same entrainment rate ω for both enthalpy and mass, implying a Lewis number of one. This assumption is generally reasonable, as turbulent diffusion is expected to dominate over molecular diffusion under contrail formation conditions within a strong jet. The entrainment rate is defined as 3ω(t)=-dln⁡Ddt, where D is the dilution factor. The value of D is one at the engine exit and gives the mass fraction of core air in a plume parcel downstream of the engine (hence, it is a monotonically decreasing function with time/downstream position).

In addition to the stated assumptions, an isobaric mixing process is assumed. Furthermore, when relating static enthalpy and temperature, the slight temperature dependence of the specific heat capacity at constant pressure cp is neglected . The temperature evolution of a plume parcel is then governed by 4dTdt=-ω(t)T(t)-Ta+SLH(t). In Eq. (), Ta is the ambient temperature and SLH denotes a source/sink term accounting for the microphysical latent heat release/consumption during phase transitions. Analogously, the equation for the plume parcel's water vapor mass fraction χv is expressed as 5dχvdt=-ω(t)χv(t)-χv,a+SPH(t), where χv,a is the ambient level and SPH is a source/sink term due to phase changes. The plume parcel's water vapor mass fraction χv and partial water vapor pressure ev are related by 6ev=paϵχv with the ambient pressure pa and the ratio of molar masses of dry air and water vapor ϵ=0.622.

Equations () to () lead for each plume parcel in the absence of microphysical processes (SLH=SPH=0) to a universal mixing line in an ev-T diagram. The slope of this mixing line is determined by 7G=EIvcppaϵQ(1-η). The entrainment rate ω just tells how fast the thermodynamic quantities describing the plume parcel's state evolve along this line.

At engine exit, the exhaust is already a mixture of ambient air and fuel mass, which is described by the air-to-fuel ratio 8CE=Q(1-η)cp(TE-Ta), with TE denoting the exit temperature. The time-dependent overall dilution of a plume parcel is then defined as 9C(t)=CE+1D(t)≈CED(t), where the fuel mass is neglected in the approximation due to the typically large air-to-fuel ratios. For kerosene combustion, air-to-fuel ratios are on the order of 50–70 . These ratios may be even higher for hydrogen combustion, as the mass-specific heat of combustion is larger by a factor of 2.79 and therefore less fuel is needed for the same amount of energy.

The fuel mass flow rate can be expressed as 10m˙F=paA^EU∞RdTECE with the specific gas constant Rd and effective core exit area 11A^E=AEUtot,EU∞. The effective core exit area A^E combines the true physical exit area AE with the ratio between total exit jet speed Utot,E and aircraft speed U∞. The total jet speed Utot,E is the sum of excess jet speed Ujet,E with respect to the ambient air and the aircraft speed U∞. The effective area accounts for the axial divergence of the jet and is representative of a plume parcel's volume (see Sect. 3.2.2 in for details). Using A^E, the number of entrained ambient aerosols per meter of flightpath is given by 12Naer(t)=naerTaA^ETED-1(t)-1, where naer is the ambient aerosol number concentration.

We aim to study the impact of varying supersaturations that arise from the influence of η on the slope G of the mixing line (Eq. ). Clearly, an increase in η reflects technological advancements in engine configuration (Sect. ). Here, we do not explicitly address the underlying causes of varying efficiencies among different engines (e.g., differences in engine design such as bypass ratios or jet velocities), and we therefore neglect their potential influence on exhaust plume dynamics and the associated dilution. Such a dilution time scaling is proposed in Sect.  for other purposes. Instead, our focus is to investigate the thermodynamic influence of η on the formation of ice crystals. To this end, we present a physically consistent model setup as follows.

We assume the same ambient conditions and the same flight Mach number but an engine that is more/less efficient, i.e., a change of overall efficiency from a reference value η* to another value η. According to Eq. () this implies that the fuel flow has to be adjusted to 13m˙F=η*ηm˙F* to maintain the thrust. We now assume that the core mass flow is preserved, as then the plume's mass and number of entrained ambient particles (Eq. ) stay the same for a given dilution state downstream of the engine. From preserving the core mass flow, a change in the air-to-fuel ratio 14CE=ηη*CE* follows. In combination with Eq. () this implies a change in core engine exit temperature 15TE=η*(1-η)η(1-η*)TE*-Ta+Ta, where Ta is the ambient temperature. Equations () to () together with Eq. () imply a change in the core exit effective area 16A^E=TETE*A^E*.

The length and velocity scale of a free turbulent round jet are the nozzle exit radius rE and exit velocity Ujet,E, respectively . We write down the Reynolds-averaged mean momentum equation in polar-coordinates , which reads in non-dimensional form 17ρ^U^∂U^∂x^+ρ^V^∂U^∂r^=D^Tr^∂∂r^ρ^r^∂U^∂r^ with the scaled axial velocity U^=U/Ujet,E, radial velocity V^=V/Ujet,E, axial coordinate x^=x/rE, radial coordinate r^=r/rE, density ρ^=ρ/ρ0, and turbulent viscosity D^T=DT/(rEUjet,E).

Since D^T≈const. , solving Eq. () together with the continuity equation, temperature equation, and equation of state gives a universal solution for the scaled variables. Looking at a time increment 18dt=dxU=dx^U^rEUjet,E and setting dx^=dx^*⋅1 and U^=U^*⋅1 (due to the universal solution for the scaled variables), it follows the relation 19dt=Ujet,E*Ujet,E⋅rErE*⋅dt*=sshear⋅sE⋅dt*=sdil⋅dt*, where asterisks symbols denote reference values. In Eq. () we use the definitions 20asshear:=Ujet,E*/Ujet,E20bsE:=rE/rE*20csdil:=sshear⋅sE.

This dilution time scaling (Eq. ) was first proposed by . By using Ujet,E as velocity scale, we implicitly assume a strong jet speed regime , where the jet dynamics are similar to a setup without relative movement between jet source (here aircraft engine) and environment. Far downstream of the aircraft engine, however, when the jet has been decelerated, the dilution is influenced by the coflow velocity U∞ . Moreover, D^T≈const. might be only a valid assumption when a self-similar flow has been established . In addition, the dynamical influence of a bypass flow is not explicitly treated in the above derivation. Nevertheless, as suggested by , Eq. () may contain the leading factors of a dilution time scaling. In the analysis of our simulation results, the scaling properties presented so far will help to reveal underlying and possibly hidden relations between seemingly different scenarios.

As the combustion takes place in the core of a turbofan engine, excess water vapor at the engine exit is only contained in the core flow. However, part of the chemical energy released during combustion is used to drive the fans. Due to the work performed on the fluid by the blades, the bypass air contains excess static enthalpy and kinetic energy above ambient levels. Consequently, water vapor and total enthalpy are distributed differently at the engine exit of a turbofan engine.

investigated contrail formation criterion for various propulsion systems. Assuming that the core and bypass of a turbofan are already well-mixed at engine exit, this mixed exhaust is already supersaturated in his investigated hydrogen combustion case (see his Fig. 4b). As this well-mixed state is one of the assumptions of the classical thermodynamic contrail formation theory (Sect. ), this approach becomes insufficient when simulating the time-resolved contrail formation process. Therefore, a more sophisticated modeling approach is needed accounting for the inital separation of emitted total enthalpy into the core and bypass flow.

The dilution is derived from a LES conducted for a single uninstalled CFM56 engine representative of an A340-300 aircraft. From this LES, generated an ensemble of 25 000 trajectories, each representing a fraction of the plume’s mass/volume. then used a suitably merged subset of 1000 trajectories for their box model simulations. In our simulations, we utilize both this 1000-trajectory ensemble and a single average trajectory, which represents the mass-weighted average of the ensemble see Supplement in. The dilution factor is determined from the data by treating the temperature as a passive tracer. In the trajectory ensemble approach, box model simulations are performed for each member and results are presented as sum over all trajectories for extensive variables (e.g., ice crystal number) and as mass-weighted average for intensive variables (e.g., temperature). For the average trajectory approach, a single box model simulation is performed.

We use the average dilution factor obtained from a temporal LES simulation for an isolated engine representative of a B737 aircraft . The dilution factor is determined by an exhaust-weighted average (see Sect. 2d in for details). These dilution data have previously been used alternatively to the FLUDILES data in .

We use the average dilution factor obtained from the RANS data we obtained from Airbus. In addition to the average dilution factor D‾ we also incorporated the extracted β, γ‾ and E‾kin. Details have been described in Sect. . The engine is designed for a Short-Medium Range (SMR) scenario (a A320 like aircraft).

We use the analytical formula presented in , which reads 35D(t>τm)=τmtB and D=1 for t≤τm with τm≈0.01s. The exponent B determines the dilution speed with a default value set to B=0.9 after . Setting B=1.15 leads to a dilution evolution more comparable to the other data sources (Fig. a).

The time series of D differ substantially between the various CFD data sources (Fig. a). Consistent with our theoretical understanding, the dilution data show a much smaller spread after the dilution time scaling based on the engine size and jet speed (Eq. ) is applied (Fig. b).

In the result section (Sect. ) the box model initializations differ depending on the purpose of each section. Therefore, we briefly summarize the settings at the beginning of each subsection. When not stated differently, we use the FLUDILES trajectory ensemble as dilution data and the baseline values listed in Table  as model setup. The prescribed baseline properties of ambient aerosol particles correspond to highly-soluble Aitken-mode particles. Ambient aerosol particle sizes, solubility, and number concentrations show large variability in the atmosphere . In Part 1 , we systematically examined how these properties influence droplet activation and the subsequent homogeneous freezing of droplets into ice crystals. Although the primary focus of the current study is on engine-related parameters, we nonetheless examine a variety of ambient aerosol scenarios, covering a wide range of number concentrations (Sect. ).

In Sect. , we presented scaling relations describing how engine exit conditions and dilution speed vary with ambient temperature and pressure changes. The linear scaling of density and fuel flow with ambient pressure (Table  and Eq. ) has already been consistently applied in previous box model simulations by . In these studies, however, they held the engine exit temperature TE constant and did not adjust the dilution speed when they changed the ambient temperature. Here, we investigate whether neglecting these changes significantly affect Nice,f.

The FLUDILES data were produced with an ambient temperature Ta*=220K and an exit temperature TE*=580K (Table ). If the ambient temperature Ta=210K is specified, the scaling suggested in Sect.  results in TE≈554K and sdil≈1.02. For Ta=232K these two values are TE≈612K and sdil≈0.97. However, the difference in the number of ice crystals formed is negligible between simulation runs with the scalings applied and simulation runs with fixed TE=TE*=580K and without dilution time scaling (Fig. ). Therefore, we can safely neglect these scaling relations in subsequent analyses and instead prescribe the same exit temperature and the same unscaled dilution regardless of the actual ambient temperature.

This subsection analyses how Nice,f changes with variations of the overall efficiency of propulsion η and how Nice,f(η) can be expressed in terms of Nice,f(η*) for a fixed reference value η*. In a first step, we perform sensitivity studies with the FLUDILES trajectory ensemble, where a variation of η is reflected by corresponding adaptations of the engine exit conditions (as outlined in Sect. ). From theoretical considerations, a higher η value implies a steeper mixing line and higher and longer-lasting supersaturation. Consistent with this, Fig.  shows increasing ice crystal numbers Nice,f with increasing η. Clearly, Nice,f itself and the change of it with η depend on the meteorological background conditions. In situations with high plume supersaturations and large Nice,f values (i.e., lower ambient temperature and higher ambient pressure), a change in η has less pronounced effect on Nice,f. For an higher ambient temperature (Ta=232K), however, an increase in η leads to a noticeable increase in Nice,f.

The slope G of the mixing line depends on both, the overall efficiency of propulsion and the ambient pressure (Eq. ). The value pair η and pa corresponds to the the same slope as the value pair η* and p̃a with the definition 36p̃a=pa1-η*1-η. Therefore, we try to mimic the effect of a variation of the overall efficiency by fixing its value to η*=0.4 but adjusting the ambient pressure after Eq. (). In the box model setup of this simulation series with η*=0.4, we fix the exit temperature to TE*=580K, neglecting the impact of the ambient temperature on the exit condition (Sect. ) while maintaining consistency with the scaling relations presented in Sect. . Consequently, the exit conditions for the tuple (η,pa) are different from the tuple (η*,p̃a), meaning the two approaches start from different points on the mixing line. For an ambient temperature Ta=225K, the engine exit temperature is TE=665K for η=0.35, and TE=514K for η=0.45 using Eq. ().

For the aerosol number concentration naer=100cm-3, the approximation with η* and p̃a works almost perfectly (Fig. a). For naer=1000cm-3, the approximation slightly overestimates the ice crystal number for η values above and slightly underestimates it for values below η*=0.4 (Fig. b). The deviations come from the fact that the ambient pressure goes into other calculations within the box model (e.g., mean free path of air molecules, diffusion coefficient of water vapor) and that the starting points on the mixing line are different. However, as the deviations are only marginal, this suggests a low sensitivity of Nice,f on the exact engine exit conditions.

In summary, changes in overall efficiency of propulsion can be well approximated by adjusting the ambient pressure. Therefore, when developing a parameterization for the number of ice crystals formed, it will be sufficient to represent the pressure dependence accurately.

In the following, we investigate the effect of the engine size rE on contrail formation and propose scaling relations between engine size and the number of ice crystals formed Nice,f. To do so, we perform box model simulations based on the FLUDILES trajectory ensemble. While keeping all other parameters fixed, the core exit radius rE is scaled by the factor sE (see definition in Eq. ). According to Sect. , this involves also a change of the dilution time scale by a factor sdil (Eq. ). While the scaling factors sE and sdil refer to different aspects of the problem, in our particular case, where only rE is varied and sshear=1, sdil takes the value of sE.

The (effective) plume area at engine exit scales as A^E∝sE2. If the engine size variation had only this trivial geometric effect and the plume dilution time scale had no effect on contrail formation, then a first-order estimate would give Nice,f∝A^E∝sE2. This first-order estimate holds, if the normalized ice crystal numbers Nice,f/sE2 are independent of sE and lie on a universal curve for different sE values. Figure a reveals that this is approximately the case for low aerosol number concentrations naer at ambient temperatures Ta≲230K. However, the curves corresponding to the different sE values start to deviate at higher aerosol number concentrations. A slower dilution leads to lower (scaled) Nice,f values because aerosol particles that are entrained and activated into ice crystals at an early stage have more time to consume the water vapor. While these early-activated ice crystals grow, they reduce the supersaturation, preventing later-entrained aerosols from being activated into ice crystals.

derived a scaling relation by considering the time interval during which a plume parcel can potentially activate aerosols and the time interval required for ice crystals to grow large enough to deplete the available water vapor significantly. In our application, this scaling is expressed as (see Appendix ) 37Nice,fsE2⋅sdila∼f(sdil3/2⋅naer) with a=-3/2 and f being a function that depends on ambient conditions and aerosol properties but not explicitly on the engine size/dilution speed. This means that the curves for different sE values should collapse when the scaling is applied. Indeed, this scaling approach works reasonably well for temperatures Ta≲230K (Fig. b). However, for Ta=233K, the curves for the different sE values do not collapse, hinting that another timescale might become important at high ambient temperatures (which is not relevant in kerosene combustion scenarios, for which this scaling was originally developed).

We will extend the existing scaling approach by considering the limiting factor of droplet freezing. Aerosols that are activated into water droplets have to grow large enough before the freezing process is triggered . The duration between droplet activation and freezing into ice crystals depends on the time point at which an aerosol particle is entrained into the plume, specifically on the prevailing plume temperature, plume supersaturation, and their subsequent evolution. As a first-order estimate, we use the time difference between the first droplet activations and the first ice crystal formations in our simulations as a proxy for the freezing timescale δtfrz (Table ). We compare this timescale δtfrz to the maximum possible duration of water supersaturation δtsupersat. In the absence of microphysical processes, the latter depends on dilution speed and scales linearly with sdil. In Table , we present the average of δtsupersat over all trajectories for a prescribed value of sdil=1.

At low ambient temperatures, supersaturations are high and long-lasting. With increasing Ta, however, δtsupersat decreases. In contrast, δtfrz is short at low ambient temperatures and increases with increasing ambient temperature. Hence, the relative importance of δtfrz grows with increasing temperature. This means that, at high ambient temperatures, a substantial portion of aerosols that are entrained and activated into droplets at a later stage are unable to freeze before the relative humidity over water falls below 100%. These droplets evaporate subsequently. Clearly, this is more pronounced for faster dilution, i.e., shorter δtsupersat, which explains the relative order of the three curves for Ta=233K in Fig. b.

However, these three curves are shifted by an approximately constant factor. This allows the freezing time scale to be accounted for by adapting the exponent a in Eq. (). We propose maintaining a=-3/2 up to Ta=230K, followed by a linear increase with ambient temperature, specifically, 38a=-32forTa≤230K16Ta/K-239for230K<Ta≤235K. This adapted scaling approach performs reasonably for temperatures Ta≲233K (Fig. c). For Ta=234K, however, the symmetry breaks between the decelerated dilution (sdil=sE=2) and the accelerated dilution (sdil=sE=0.5). Whereas for sdil=2 the scaling still works reasonably well, the (scaled) Nice,f values are lower for sdil=0.5. This means that the function f introduced in Eq. () becomes dependent on the dilution speed, thus limiting the applicability of the scaling approach above Ta∼233K.

In Sect. , we described several data sets to prescribe the dilution in the box model. In the following, we investigate the influence of the dilution data on our model results.

In a first step, we compare results for the FLUDILES trajectory ensemble and the average trajectory (as already done by for contrail formation on soot particles for kerosene combustion). Both approaches yield a comparable number of ice crystals Nice,f when the ambient temperature Ta is below ∼230K (Fig. ). At higher temperatures, however, Nice,f values for the ensemble approach tend to be higher than for the average trajectory approach. Furthermore, scenarios exist where no ice crystals form in the average trajectory, whereas Nice,f is larger than zero in the ensemble approach. How large the discrepancies between the two approaches are depends on the peak plume supersaturation (e.g., affected by ambient pressure, Fig. a–b) and on the aerosol properties (Fig. a, c, d). The key factor in understanding these differences is the interplay between the timescale of supersaturation δtsupersat and freezing δtfrz (Sect. ). In the ensemble approach, the 1000 members dilute at different speeds. Some trajectories dilute more slowly, allowing more droplets to grow sufficiently to freeze into ice crystals. In contrast, the freezing is more of a binary process in the single average trajectory approach, resulting in lower or zero ice crystal number at the same temperature.

The different dilution data were produced for different engine characteristics and background conditions (Table ). However, we only extract the information on dilution (in terms of D and ω) and the information on engine size and jet speed in the subsequent analysis. Particularly, we still rely on classical contrail formation theory and do not consider the thermodynamic influence of the bypass flow and kinetic energy dissipation. This will be investigated in Sect. . In each simulation setup we use the same exit temperature TE=580K and overall efficiency η=0.4, obviously neglecting differences among the engines. However, it disentangles the impact of those values from the influence of the dilution on Nice,f. Moreover, we intend to use the theoretically derived dilution time scaling (Sect. ) with associated scaling of Nice,f (Eq. ) neglecting any further details in the configuration of the engines.

Since the number of entrained particles (Eq. ) and thus the number of ice crystals formed is proportional to the effective core exit area A^E (Eq. ), we use the geometric scale s^E2=A^E/A^E* to make the results for the different-sized engines comparable. We use the FLUDILES engine as reference, thus A^E* refers to its effective core area. Relating the effective core areas instead of the physical core areas (as done with the scale sE2 in Sect. ) additionally accounts for the different ratios of total jet and aircraft velocities in the CFD data. The dilution time scale sdil, however, is still determined with Eq. () using sE and sshear. As the FLUDILES engine is treated as reference, the associated values are s^E=sdil=1 for this engine. Moreover, we set s^E=sdil=1 for the Kärcher dilution as the analytical formula does not contain an explicit dependence on engine size or jet speed. Instead, the dilution speed is governed by the exponent B in Eq. ().

Simulations using the default Kärcher dilution (B=0.9 in Eq. ), yield significantly different ice crystals numbers compared to those using the FLUDILES dilution (Fig. a). This difference arises from the markedly slower dilution speed (Fig. ). However, when a faster dilution with B=1.15 is used (Fig. b), the resulting ice crystal numbers are similar to those from the FLUDILES simulations. This suggests that the exact shape of the dilution curve over time (Fig. ) is of lesser importance for the final number of ice crystals formed.

The faster Lewellen dilution (Fig. ) caused by the smaller core radius (Table ), leads to higher (scaled) Nice,f values at high aerosol number concentrations (Fig. c). When the scaling after Eq. () is applied, the resulting curves for a given ambient temperature align reasonably well (Fig. d), demonstrating the consistency of the scaling approach across different dilution scenarios.

The scaling performs somewhat less ideally for the Turbofan dilution data (Fig. f). Especially for Ta=215K, the scaled curves have slightly different shapes. This may be partially attributed to the dynamic influence of the bypass flow on the dilution speed. Additionally, droplets do not freeze in the Airbus Turbofan case for naer≳1000cm-3 at Ta=232K in contrast to the FLUDILES simulations. Nevertheless, the scaling after Eq. () based on core exit conditions still accounts for the leading impact of the dilution speed on Nice,f. Moreover, Nice,f is only weakly sensitive on the exact choice of the plume boundary in Eq. () (Fig. e–f).

Using the Airbus Turbofan dilution data, we investigate the impact on the number of ice crystals formed for the two approaches outlined in Sect. . The first approach explicitly considers the thermodynamics of the bypass flow, including kinetic energy effects. The second, simplified approach assumes that the whole emitted combustion heat is contained as static enthalpy in the core flow at the engine exit. In the former approach we prescribe the estimated exit conditions for the Airbus Turbofan data (Table  and β=0.62) and use the extracted time evolutions of D‾, E‾kin and γ‾ (Sect. ). In the latter approach we make use of only D‾ and prescribe the adjusted exit temperature T̃E after Eq. (). Using the same TE=564K for all ambient temperatures leads to T̃E=809K for Ta=215K and T̃E=798K for Ta=232K.

The former approach leads to initially higher plume supersaturation (Fig. a) and thus to a slightly earlier onset of droplet activation (Fig. b). However, this has little influence on the final number of ice crystals formed (Fig. b–d). This insignificant impact is due to the continuous entrainment of ambient aerosols into the core plume over time. Since we assume that aerosols sucked into the bypass duct are not destroyed there, both approaches yield the same time evolution of entrained aerosols (Fig. b), governed by the core air dilution factor D (Eq. ). Due to the continuous entrainment process, activated aerosols become abundant enough to significantly deplete the water vapor (difference between opaque and transparent curves in Fig. a) only at a stage when the relative humidity profiles of both approaches have already converged. This convergence occurs once the bypass and core flows have mixed and most of the kinetic energy has been converted into heat. Consequently, aerosols entrained after this point experience similar thermodynamic conditions in both modeling approaches, leading to similar ice crystal numbers.