In a box model approach, fluid dynamics is not resolved and changes in thermodynamic properties inside the box are externally prescribed. We use the general plume dilution equations, which are described in Sect. 2.3.1 of , to calculate the cooling and expansion of the plume as well as the evolution of the humidity. Based on 3D large-eddy simulations (LESs) using the FLUDILES solver, sampled an aircraft plume with 25 000 trajectories behind the engine of an A340-300 aircraft. Thereby, the temperature evolution T3D,k(t) was tracked for each trajectory indexed by k. As in , we use these data to infer the plume dilution factor by assuming that temperature is a passive tracer: 2Dk(t)=T3D,k(t)-T3D,aT3D,E-T3D,a, where T3D,E=580 K and T3D,a=220 K are the plume exit and ambient temperatures of the FLUDILES simulation. In the remainder of the text, the subscripts “E” and “a” denote conditions at the engine exit plane and in the atmospheric background, respectively.

In the following, we describe some modifications of the FLUDILES trajectory data set compared to the original one from :

As in , we introduce a lower limit T3D,k(t)+ϵ, and all T3D,k(t) values below this lower limit are set to this value. We choose ϵ=0.2 K such that the implied dilution and plume area are consistent with the area enclosed by the trajectories.

The emitted air at the engine exit plane is a mixture of ambient air (going through the engine) and combustion products. The initial plume dilution CE (i.e., the air-to-fuel ratio) is linked to the exit temperature TE: 4CE=Q(1-η)c‾p(TE-Ta). This equation follows from the conservation of fuel mass and thermal energy flow. Moreover, c‾p=1020 J (kg K)-1 is assumed to be an average heat capacity for dry air over the temperature range of interest and the effects of the jet kinetic energy are neglected. The latter assumption is justified when (most) kinetic energy is converted into thermal energy well before the plume has cooled down to temperatures where supersaturation is reached and formation of droplets commences. The overall dilution C(t) generally increases with plume age due to continuous entrainment of ambient air and is related to the dilution factor D(t) via 5C(t)=CE/D(t), implying DE=1. Whereas C denotes an air-to-fuel ratio, the dilution factor D is defined as a fuel-to-air ratio (due to legacy reasons) and decreases with plume age. The word “overall” in the name of C(t) refers to the fact that this quantity considers the combined effect of dilution occurring in the engine and the free atmosphere. Compared to Eq. () we drop the index k from the dilution factor Dk. With this we imply that derivations in this subsection consider the total plume and not individual plume parts represented by a single trajectory.

The flow of the (total) plume mass is defined as 6m˙plume(t):=ρ(t)A(t)Utot(t)=m˙FC(t).

Here, ρ(t) is the air density, A(t) is the plume cross-sectional area (perpendicular to the axial direction) and the total exhaust velocity Utot(t)=U∞+Ujet(t). The latter quantity Utot(t) is the sum of the absolute value of the aircraft speed, U∞, and jet velocity, Ujet(t), relative to the ambient air. The fuel flow rate (units: kg s-1) is denoted by m˙F.

Applying Eq. () at the engine exit plane and at any other time for a fixed fuel flow rate, plugging in Eq. (), using the ideal gas law, and assuming isobaricity, we obtain the following relation: 7A(t)=AET(t)TED(t)Utot,EUtot(t). Note that Eq. (16) in shows a similar relation; however it misses the ratio of the velocities. The same holds for their Eq. (15). This term Utot,E/Utot(t) accounts for the fact that the plume is compressed/stretched due to the axial divergence of the jet. This clearly affects how the cross-sectional plume area changes over time. However, this divergence effect does not change the volume of the considered air parcel; it only distributes its mass over a segment with a changing extent along the axial direction. The volume of the plume changes only due to the entrainment of ambient air (reflected by the dilution factor D) and through associated density changes (reflected by the temperature ratio). Introducing an effective area 8A^(t)=A(t)Utot(t)U∞, Eq. () can be written as 9A^(t)=A^ET(t)TED(t). The effective area A^(t) is the cross-sectional area that is reached when the exhaust plume is (theoretically) decelerated from Utot(t) to U∞ in a volume-preserving manner. Since we provide all quantities in terms of “per meter of flight path”, the effective area is the appropriate quantity that is representative of the volume of the plume at a certain age. Note that Eq. () has the same form as Eq. (16) in . The only difference is that it uses the effective plume cross-sectional areas instead of the real physical ones. In our case, U∞=250 ms-1 and Ujet,E=230 ms-1. Hence, the ratio γU=Utot,E/U∞ is 1.92.

The fuel consumption per meter of flight path mC can be computed by 10mC=m˙F/U∞=paRdTEAECEUtot,EU∞. mC is the amount of fuel that is burned per unit length along the flight direction. The amount of a corresponding combustion product per flight distance is then, e.g., mPT=EIPT×mC (where EIPT is the emission index of an assumed passive tracer). However, we have to keep in mind that mPT only specifies the amount of the tracer in a segment where the jet velocity approached 0. At the engine exit plane, the total tracer amount would be lower by a factor of γU.

In the preceding paper of , formulas were expressed in terms of fuel consumption and never in terms of the fuel flow rate. There, the symbol mF was used for the fuel consumption. In the present paper, this is replaced by mC in order to avoid confusion with the fuel flow rate m˙F.

Even though we made the comment that the derivations here consider the total plume, they are also valid for individual plume air parcels represented by a trajectory. The FLUDILES simulation data feature an initial plume temperature profile with a smooth radial transition between the plume and the environment instead of a step at the plume edge. Hence, trajectories in the outer plume regions start with TE,k(t0)<TE,Utot,E,k<Utot,E and Dk(t)<1.0. In principle, one could assign a trajectory-specific value of γU to each trajectory. The fraction of trajectories having a γU value much smaller than the default is rather small. Hence, we apply a bulk-correction factor of γU=1.92 to ensemble mean values (instead of applying individual correction factors to each trajectory and then performing a suitable mass-weighted averaging/summation to trajectory ensemble data). This simplification leads only to minor quantitative deviations.

As written above, missed including the term γU in the computations of plume area and fuel consumption. However, the implications of this are not overly serious. The time evolution of intensive quantities in the box model is not affected by this oversight. Only in the final step of translating ice crystal number concentrations into a total number of ice crystals Nice per flight distance does one have to include the term γU. However, for quantities like the apparent ice emission index (AEI) or the activation fraction, as shown in the figures of , the term γU is canceled out and the displayed figures are all unaffected by our oversight. For the sake of clarity, we want to note that the simulations of kerosene contrails discussed in Sect.  of the present study are computed with the correct formulas.

We prescribe background particles as an ensemble that is characterized by a log-normal size distribution with a geometric mean dry radius (r‾d), a geometric width (σaer) and a number concentration (naer). Moreover, we specify the hygroscopicity parameter . The parameters are varied mostly independently of each other, representing different types of ambient aerosol particles and accounting for their natural variability.

According to the classical microphysical pathway of contrail formation with the liquid transition phase e.g.,, we consider only CCN or partially soluble mixed particles (i.e., an insoluble core and a hydrophilic coating), where we assign the latter simply to “weakly soluble particles”. We exclude heterogeneous ice nucleation on insoluble particles, since measured and modeled number concentrations of ice nuclei (IN) in the UT are typically several orders of magnitudes lower than those of CCN e.g.,. Even though IN may have important effects on natural cirrus cloud properties e.g.,, we expect a negligible contribution to the overall contrail ice crystal formation. Insoluble but still wettable particles like uncoated soot and oil droplets could also be treated by adsorption activation theory. This is based on standard Köhler theory but uses a specific description of the water activity that accounts for adsorption processes. In particular, the Frenkel, Halsey and Hill (FHH) adsorption approach is able to treat multilayer adsorption of water vapor onto insoluble particles and should be considered in future studies.

For simplicity, we assume that the entrained ambient particles are initially completely dry, i.e., without any previous environmental hygroscopic water uptake. We allow for condensational growth of the entrained aerosol particles not only in water-supersaturated conditions but also at plume relative humidities lying between the deliquescence relative humidity (DRH) and water saturation. Thereby, DRH is the minimum threshold relative humidity that allows for hygroscopic water uptake by a given substance.

Aerosol particles from the environment are continuously entrained into the plume. The background aerosol abundance is specified in terms of an aerosol background number concentration naer.

As a next step, we quantify the number of aerosol particles Naer(t) (here per flight distance) that are present in the expanding plume. 11Naer(t)=naerρ(t)ρaA^(t)-ρEρaA^E=naerTaA^(t)T(t)-A^ETE=naerTaA^ETED(t)-1-1:=α(t)naer The ratio of densities in the first line accounts for the fact that during an adiabatic entrainment process, the mixing ratio is conserved and not the concentration. Moreover, we assume that the initial plume is void of any aerosol particles, which implies that any aerosol particle sucked into the aircraft engine is destroyed. Hence, the plume area at the engine exit is subtracted (second term in the first line). The further re-formulations assume isobaric conditions and use the ideal gas law and the definition of the effective area (Eq. ).

If we were to also prescribe aerosol particles in the initial plume (with identical properties to those in the environment), results would not change too much as their contribution to the total aerosol particle number would become smaller and smaller while the plume expands (this case is easily treated by removing “-1” from the term “D(t)-1-1”).

The number of particles being entrained into the plume during one time step Δt is then given by 12ΔNaer(t)=(α(t)-α(t-Δt))naer.

In every time step, a new SIP ensemble representing these newly entrained particles is created. Only in the initial stages of the simulation when RHwat is still below DRH do no new SIPs have to be created as aerosol particles inside the plume are still dry. In this case, it is sufficient to only increase the SIP weight (i.e., the number denoting how many real particles are represented by a SIP).

Clearly, the continuous creation of new SIPs would cause huge values of NSIP and lead to computationally expensive or even unfeasible simulations. Hence, we apply a SIP merging algorithm if the overall SIP number NSIP gets too large (see Appendix ).

Moreover, we found that T3D happens to increase in certain (short) segments along several plume trajectories. This implies that the cross-sectional area represented by the trajectory shrinks (i.e., C(t)≤C(t-Δt)) and a negative value of ΔNaer(t) follows. To inhibit such an unwanted detrainment of particles and hydrometeors, we have smoothed our trajectory data such that the dilution C(t) is a monotonically increasing function with time (see Sect. ).

For spherical droplets, the single-droplet mass growth equation is given by as 13dmwdt=4πr(ev-eK,wat)RvTDvβm-1+eK,watLc2RvKT2βt-1, where r is the wet aerosol or droplet radius ev is the partial vapor pressure. Lc denotes the specific latent heat for condensation and evaporation, Dv the binary diffusion coefficient of air and water vapor, K the conductivity of air, Rv the specific gas constant of vapor, and T the temperature. The transitional correction factors βm and βt are calculated according to Eqs. (A4) and (A5) of based on . Note that there is a transcription error in the previous study, and the denominators of both Eq. (A4) and Eq. (A5) miss the term “+1”. With this correction, βm and βt tend towards 1 for small Knudsen numbers, as intended. The quantity eK,wat is the product of the saturation vapor pressure over a flat water surface esat,wat and the equilibrium saturation ratio over a solution droplet surface SK. As in , we calculate SK using the κ-Köhler equation : 14SK=r3-rd3r3-rd3(1-κ)⋅exp⁡2σ^MwatRTρwatr, where the first term is the activity of water (awat) and the exponential expression is the Kelvin term. rd is the particle dry radius, κ the hygroscopicity parameter, σ^ the surface tension of the solution droplet, ρwat the mass density of water, Mwat the molar mass of water and R the universal gas constant. The surface tension typically increases with decreasing awat for salt solutions due to negative adsorption and increases with decreasing awat for acidic solutions due to positive adsorption. Since we do not prescribe specific aerosol particle species but only the hygroscopicity parameter in the present study, we approximate σ^ with the surface tension of pure water droplets and use the polynomial expression by as in . Hence, a slight error is caused in the Kelvin term for more concentrated solution droplets.

The entrained ambient aerosol grows by condensation if RHwat is larger than the deliquescence relative humidity (DRH). As long as the entrained particle is completely dry, SK is not applicable, since r=rd. Therefore, we set SK to a value that is slightly lower than RHwat for the first time step with condensation. This causes the particle to grow hygroscopically so that r>rd in the next time step, and then SK is calculated according to Eq. (). During the subsequent plume evolution, the wetted particle/droplet grows further due to condensation if RHwat>SK or shrinks due to evaporation if RHwat<SK according to Eq. (). In , the soot particles were considered to be activated into water droplets if the wet radius exceeded the critical radius, which is the radius at the maximum of SK. In the present study, we consider aerosol particles to be activated into water droplets if the activity of water has exceeded a critical value awat,c=0.90 to ensure sufficient water uptake for freezing. Once an aerosol particle has been activated into a droplet, it can freeze into an ice crystal if the plume temperature drops below the homogeneous freezing temperature of that solution droplet (Tfrz). For the calculation of Tfrz, follow the approach of and assuming pure water droplets. We extend this approach by including a simple correction term, based on the parameterization of , to account for the decrease in Tfrz due to the solution effect e.g.,. Further details are described in Appendix .

The depositional growth of the ice crystals formed is calculated according to Eq. (7) of , which is based on . Note that there is a transcription error in that equation, where the correction term βv-1 occurs twice in the mass diffusion term and should be removed in the denominator.

Figure  highlights the strong impact of ambient temperature Ta on contrail ice crystal formation. As shown in Fig. a, the plume temperature at a given plume age is certainly lower in a colder environment and finally approaches the corresponding Ta value. The peak mean relative humidity over water (displayed in Fig. b) increases with decreasing Ta (reaching values of around 350 % and 550 % for Ta values of 220 and 215 K, respectively). The large increase for low Ta values is due to the non-linearity between saturation vapor pressure and temperature. The relative humidity over ice behaves accordingly (not shown).

The slight change in the evolution of Naer is a consequence of our model setup with a fixed aerosol number concentration and the varying plume air density with temperature (at fixed ambient pressure). In general, the droplet formation is basically controlled by the time period where RHwat is above DRH such that water can condense on the entrained aerosol particles. (Note that the evolution in RHwat and, therefore, this time period vary with each trajectory, and here we only display the ensemble mean quantity.) This mean time period for possible droplet and ice crystal formation substantially increases with decreasing ambient temperature (e.g., from 0.15 s up to 2.8 s for Ta= 215 K). This means that ice crystal formation is initiated earlier and comes to a halt later (see solid red and blue lines in Fig. c) compared to the baseline case. Moreover, nearly all formed droplets freeze very quickly into ice crystals so that Ndrp approaches zero. For these reasons, the final ice crystal number strongly increases with decreasing Ta over the whole temperature range (see also Fig. ). This is different to what we find for conventional soot contrails as all soot particles turn into ice crystals if Ta is several kelvins below the SA threshold temperature e.g.,. Yet, any further reduction in Ta does not lead to more ice crystals in the conventional case. Moreover, the peak plume RHwat values are substantially higher for H2 than for kerosene combustion in the same ambient conditions. Therefore, droplet and ice crystal formation on ambient particles is controlled more strongly by the time period in which the plume is water-supersaturated than by the maximum water supersaturation.

For higher ambient temperatures (Ta≥ 230 K), many droplets cannot freeze into ice crystals and evaporate thereafter (which is indicated by declining Ndrp at nearly constant Nice). This is because the homogeneous freezing temperature of the smaller and/or more concentrated solution droplets is below the plume/ambient temperature. Hence, the final ice crystal numbers are decreased further in addition to the fact that the time period for possible droplet formation is lower. The decrease in Nice with increasing Ta becomes stronger for Ta≥ 232 K (see also Fig. ), and for Ta= 233 K, only a few large droplets can form ice crystals. For higher ambient temperatures no ice crystal formation occurs anymore. This means that for H2 combustion, the freezing temperature is typically smaller than the SA threshold temperature and becomes a more limiting criterion for contrail formation. Yet, the SA threshold temperature is still relevant, as its difference from the ambient temperature determines the peak and time period of water supersaturation in the plume. The ice water mass (shown in Fig. d) in general increases with decreasing ambient temperature. The strong increase between Ta values of 230 and 233 K is mainly due to the increase in Nice.

In the following sections, we will focus our analysis on the final number of ice crystals formed (Nice,f), since the young-contrail ice number mostly impacts the further contrail (cirrus) properties and radiative forcing e.g.,. In contrast, the initial ice water mass and mean ice crystal radius were shown to have a low impact on the contrail life cycle in the dispersion phase e.g.,, but the size distribution of the contrail ice crystals formed can strongly impact the sublimation loss of ice crystals during the vortex phase e.g.,. The latter will be investigated in future studies.

Figure  displays Nice,f versus ambient temperature Ta for (a) three different pressure pa values and (b) three different values of ambient relative humidity over ice RHice,a. Note that our parameter settings are simplified in the sense that some combinations of parameter values are not realistic for the atmosphere (e.g., the lowest Ta value at the highest pa value). Compared to the previous subsection, we now include a further case with Ta= 210 K (that requires a longer simulation time, since the period when the plume is water-supersaturated is longer than 3 s). This case emphasizes the enhanced increase in Nice,f with decreasing Ta for very cold conditions and is consistent with the findings by . Nice,f is increased for a higher ambient pressure because the slope of the mixing line G, defined by Eq. (), is larger for a higher pressure. Moreover, Fig. b shows an incline of Nice,f with increasing RHice,a. Both the increased G and the higher RHice,a lead to higher peak plume relative humidities and enlarge the time period for possible droplet and subsequent ice crystal formation. Interestingly, this increase is quite strongly pronounced for the highly ice-supersaturated case (red line) at ambient temperatures between 230 and 234 K (and for 234 K ice crystals can only form at all for this case). This is because the droplet freezing is mainly limited by the droplet size in that Ta range, and for the high-RHice,a case, more larger droplets can form that turn into ice crystals.

Finally, ambient temperature is the parameter that most influences the number of contrail ice crystals formed, while the impact of ambient pressure and relative humidity is clearly smaller. This behavior is similar to the conventional case with kerosene combustion despite the very different temporal evolution in the exhaust particle number concentration.

Figure  shows the variation in Nice,f with different aerosol particle properties. The sensitivities are always shown for three ambient temperatures Ta (differentiated by the color). In general, we see an increase in Nice,f with decreasing Ta for any particle property combination, being consistent with the findings in the previous section. Figure a shows that Nice,f increases with increasing aerosol number concentration naer. This increase becomes weaker for higher naer (⪆ 200 cm-3) values. This is due to the enhanced competition for plume water vapor between the growing droplets/ice crystals for increased aerosol number concentrations.

Next, we analyze the importance of the mean dry radius of the aerosol size distribution r‾d and the hygroscopicity parameter κ. Typically, the ice crystal number strongly increases with increasing mean dry size for r‾d⪅ 10 nm and then stays nearly constant (see Fig. b). The increase is mainly due to the Kelvin effect; i.e., larger aerosol particles are easier to activate into water droplets, since they require lower plume water supersaturations and grow more quickly into water droplets e.g.,.

Figure c shows the dependence of Nice,f on κ. For Ta= 230 K, Nice,f slightly increases for a larger κ for all three r‾d values as indicated in the legend. For the lower-Ta cases, the variation in Nice,f with r‾d and κ is more complex. For the small-sized particles (dash-dotted lines), there are two counteracting effects: Nice,f increases with a rising hygroscopicity parameter for κ<0.1, since more soluble particles can more easily form water droplets. On the other hand, Nice,f subsequently decreases. This is because some of the droplets cannot freeze into ice crystals, since their water activity is lower due to the enhanced solution effect for higher κ, and, therefore, the homogeneous freezing temperature is significantly decreased (see Fig.  in the Appendix). For the other r‾d cases, Nice,f hardly changes with the solubility and mean aerosol particle size.

We also analyze in Fig. d the impact of the geometric width of the aerosol size distribution for four different r‾d–κ combinations as displayed in the legend. Our results imply a very low sensitivity of Nice,f to the geometric width.

In conclusion, the sensitivity of the ice crystal number to r‾d and κ is low for aerosol particles with a large mean dry size. For the small-sized particles, we find a quite complex variation in Nice,f with κ, mainly for low ambient temperatures, due to various counteracting effects.

So far, the aerosol particles have been prescribed with a single log-normal size distribution and a fixed hygroscopicity value. In the present section, we prescribe two co-existing ambient aerosol particle ensembles and analyze contrail ice crystal formation for two ambient temperatures. The given naer value is the total number concentration of both aerosol ensembles. We restrict our analysis to cases where each ensemble has a number concentration of 0.5⋅naer. We consider nucleation-mode (r‾d= 3 nm), Aitken-mode (r‾d= 15 nm as in our baseline case) and accumulation-mode (r‾d= 50 nm) particles. The given mean dry sizes of the single modes are prescribed consistently with typical observed UT geometric mean diameters over the Atlantic and Pacific oceans within the ATom campaign (see Fig. 12 of ). Moreover, we consider well-soluble (κ= 0.5) particles like inorganic salts and weakly soluble ambient particles (κ= 0.05) like organic species or aviation soot. We restrict our analysis to a scenario where the two co-existing particle ensembles always differ in either their mode (in our case r‾d) or the solubility κ. Moreover, we compare the total Nice of the two aerosol particle ensembles with a reference case, which uses a single aerosol population with the same value of naer and the average r‾d and κ values of the two particle ensembles.

The first two rows of Fig.  show the temporal evolution of Nice with a fixed naer of 600 cm-3. First, we analyze the contrail ice number evolution for bi-modal aerosol size distributions with the same κ value (first row). Panel (a) shows that Nice for the Aitken-mode particle is at the end around 50 % larger than that for the nucleation-mode particle for both temperatures. This is because, for a given plume relative humidity, the larger particles can better activate into water droplets (and freeze thereafter) than the very small nucleation-mode particles due to the Kelvin effect. Moreover, the total ice crystal number is at the end around 10  % lower than the reference Nice. Considering the Aitken and accumulation mode in panel (b), Nice between the two particle ensembles is quite similar and for Ta=230 K nearly identical. This is consistent with the findings for a single particle ensemble where the variation in the ice crystal number for r‾d⪆ 10 nm is low (see Fig. b). For the same reason, the total and reference Nice values are close to each other.

Now, we investigate the ice crystal formation for particle ensembles with two different solubility characteristics but the same r‾d (second row). For Ta= 230 K, Nice for the weakly soluble particles is lower than for the well-soluble particles. This is because the less hygroscopic particles are harder to activate and/or cannot grow to sufficiently large droplet sizes in order to freeze into ice crystals. Hence, the total ice crystal numbers are slightly reduced (by around 5 %–10 %) relative to the reference ice numbers. For Ta= 225 K, Nice is at the end lower for κ=0.5 than for κ=0.05 and the total ice number is lower than the reference ice number, contrary to the high-Ta cases. This is due to the decrease in Tfrz for droplets with a higher solution effect (lower awat), as explained in Sect.  and shown in Fig. c.

The last two rows show the final ice crystal numbers (Nice,f) for the same aerosol particle ensembles as in panels (a)–(d) but for different aerosol number concentrations. Basically, we see a similar trend for the ice crystal numbers of the single ensembles and the reference case to that in the panels above throughout the whole naer range. In general, the increase in Nice,f with increasing naer becomes weaker for higher naer, in particular for the low-Ta cases. This is consistent with the findings for a single-aerosol ensemble (as already shown in Fig. a). The flattening in Nice,f is pronounced most strongly for the nucleation mode and the weakly hygroscopic particles. Again, the total Nice,f of the two particle ensembles is nearly always reduced compared to the reference Nice,f of the single average particle ensemble except for the low-Ta cases in panel (g) and (h).

Finally, we find the largest differences between the total ice crystal number of the co-existing particle ensembles and the associated average single particle ensemble for the cases where nucleation-mode particles are involved. This is mainly due to the non-linearity between the (final) contrail ice crystal number and mean dry size for those very small aerosol particles.

Figure  compares ice crystal numbers of conventional and H2 contrails. Panel (a) displays the ice crystal number Nice,f behind a conventional aircraft as a function of ambient temperature and for three typical soot number emission levels. For our calculated fuel consumption (that accounts for the different Q values of H2 and kerosene), the displayed soot particle numbers (Ns= 1.54 × 1012, 3.07 × 1012 and 6.14 × 1012 m-1) represent soot number emission indices of around 5 × 1014, 1 × 1015 and 2 × 1015 kg-1, respectively. Prescribing our baseline ambient pressure and ambient relative humidity, the SA threshold temperature ΘG for kerosene is around 227 K (≈ 10 K lower than that for H2). Very close to ΘG, only a few soot and the entrained ambient aerosol particles can form ice crystals due to very low plume water supersaturations. For ambient temperatures (Ta⪅ΘG – 0.5 K), ice crystals mainly form on soot particles, since Ns is around 2 orders of magnitude higher than Naer during the contrail formation time (not shown). Consistently with previous studies e.g.,, Nice,f strongly increases with decreasing Ta and then approaches the respective Ns values for sufficiently low Ta. For higher Ns, the number of ice crystals formed rises more steeply and approaches Ns at lower Ta.

Now we investigate the ratio of the ice crystal numbers between H2 and kerosene contrails (Nice,f,rel) shown in panels (b)–(d). We constrain our analysis to that temperature range in which kerosene contrails are able to form according to the SA criterion. Mitigation is achieved for Nice,f,rel<1, where a lower value is connected with a higher mitigation potential. Panel (b) shows Nice,f,rel versus naer for the three Ns values (see line style) and for three Ta cases (different colors). In general, we see a clear decrease in Nice,f,rel with increasing Ns and decreasing naer. Thereby, Nice,f,rel is below 0.1 for naer≤ 600 cm-3 and below 0.15 for higher naer. Interestingly, we see for the higher Ns and naer cases a larger difference in Nice,f,rel between 220 and 222 K than between 220 and 225 K.

Therefore, we investigate in more detail the temperature dependency of Nice,f,rel in the second row of the figure. Panel (c) shows the relative change in the ice crystal number with respect to the three soot cases for our baseline naer. In connection with the minimum at Ta≈ 224 K, there are two different dominating effects: below 224 K, Nice,f,rel increases with decreasing Ta, in particular for the high-soot case. This is because Nice,f for the kerosene contrails approaches Ns with decreasing Ta (well below the SA threshold), while Nice,f for the H2 contrails increases further for lower Ta. The latter is because the ambient aerosol is continuously entrained into the plume and the time period for droplet and ice crystal formation increases for colder ambient conditions due to a longer-lasting water supersaturation (see Sect. ). The strong increase in Nice,f,rel above around 225 K results from the strong decrease in Nice,f for the conventional contrail. This is because ice crystal formation on the weakly soluble soot particles becomes more and more limited the closer the ambient temperature approaches the SA threshold temperature. Thereby, Nice,f,rel is around 0.3 for Ta= 226.5 K and around 20 for Ta= 227 K (the latter is not visible in the figure). Finally, we show the change in ice crystal number relative to our baseline soot case but for three different naer values. The sensitivity of Nice,f,rel to naer is quite similar to that to Ns in panel (c). The most obvious difference is that an increase in naer by a factor of 2 (brown line in d) has a much lower impact on Nice,f,rel than a decrease in Ns by the same factor (dashed line in panel c). This is due to a weaker increase in the H2 contrail ice crystal number with increasing naer for high number concentrations (shown in Fig. a).

We can conclude that a switch to H2 combustion indicates a high mitigation potential if the ambient temperature is more than 0.5 K lower than the SA threshold temperature for kerosene. This is mainly because Naer during contrail formation is around 2 orders of magnitude lower than Ns for typical naer values. Another aspect is that the fuel consumption of a cryoplane is chosen to be around a factor of 2.8 lower than that of a corresponding conventional aircraft (only based on the difference in the combustion heat of the fuels). If air traffic with H2 combustion occurs at ambient temperatures between the kerosene SA threshold temperature and the droplet freezing temperature, clearly additional contrails are produced, which would be absent in the case of kerosene combustion.

We investigate the young-contrail optical thickness (τ) both for kerosene and for H2 combustion. For the quantities analyzed and presented so far, we used a reduced trajectory data set after merging trajectories with similar radial coordinates (see Sect. ). However, the column-wise computation of τ requires spatial information of the trajectories, namely the lateral and vertical Cartesian coordinates x and z. Hence, the results presented next were obtained using the full trajectory data set from . Figure a–d show the contrail width (indicated by the x coordinate) plume age (t) distribution of τ: the first row displays the kerosene and the second row our H2 baseline case, for Ta of both 220 K and 225 K, respectively. Observations suggest that the threshold τ for the visibility of contrails is around 0.05 e.g.,. Panel (a) indicates that the kerosene contrail at Ta= 220 K becomes visible after around 0.25 s of plume age. Afterwards, the plume quickly spreads and τ tends to increase due to further formation and growth of ice crystals. Peak values of around 2 and slightly higher are reached for t between 1 and 1.5 s. For Ta= 225 K, ice crystals form later and the contrail is visible after around 0.5 s. Maximum τ is around 50 % lower than for the low-Ta case, since that contrail forms near the formation threshold and the nucleated ice crystal number is significantly reduced (see Fig. a, black line). For the H2 case, the optical thickness is substantially decreased compared to kerosene contrails, being consistent with the findings by . Moreover, these contrails become visible later than those for kerosene at the same Ta.

Finally, panel (e) shows the temporal evolution of the 90th-percentile optical thickness over the contrail width τ90. We juxtapose the kerosene contrail with three H2 contrails with different naer values. Consistent with the spatio-temporal distributions, τ90 of the H2 contrails is significantly lower than for the kerosene contrails (in particular for Ta= 220 K). The optical thickness decreases for lower naer values. We expect that the H2 contrail for naer= 100 cm-3 would be hardly visible. A lower temperature causes a slight increase in τ90 for the higher-naer cases and for t < 1.5 s. This is mainly due to the higher ice water content for lower Ta (not shown). Instead, the much stronger increase in τ for the kerosene case is due to the increased ice crystal number already at the beginning of contrail formation.

Several recent studies have considered chemi-ions that are mainly composed of sulfur species e.g.,. While sulfur is likely not produced during H2 combustion, NOx is still emitted. The reaction of NOx with H2O and OH species potentially leads to the formation of nitrogen compounds like nitric acid that have been observed in conventional aircraft plumes e.g.,. Moreover, have shown within cloud chamber experiments that nitric acid and ammonia can nucleate directly to form volatile ammonium nitrate particles at temperatures below 258 K. Finally, nitrogen species might be a potential source of the nucleation of ultrafine volatile particles in both conventional and H2 combustion plumes, but the formation process behind this is not yet sufficiently understood.

We expect that engine lubrication systems will continue to be used for H2 engines, causing emissions of oil vapors. Even though an air–oil separator recovers around 99 % of the oil emissions, the residual may contaminate the engine plume. have shown in laboratory experiments that “jet oil vapors reach gas-phase supersaturation in cooling emission plumes leading to rapid nucleation and formation of ultrafine volatile particles in the range of ∼ 10–20 nm.” These diameter ranges appear to be consistent with the ground-measured ambient UFPs downwind of Frankfurt Airport, in which organic engine oil constituents have been identified .

The UFPs formed can contribute to droplet and ice crystal formation in addition to the background aerosol. Assuming a typical oil consumption of around 1 L h-1 with 1 % volume fraction (residual) that enters the plume, our estimates suggest that the UFP number per flight distance could be even larger than that of soot particles. Since plume water supersaturations are much higher for H2 than for kerosene in the same ambient conditions, we expect that droplets and ice crystals would mainly form on those UFPs rather than on the entrained ambient particles. This means that the number of ice crystals could be similar or even increased compared to conventional contrails. One should still keep in mind that the experiments of refer to conventional kerosene combustion and that the properties of the oil particles (size and chemical composition) could significantly change for H2. Moreover, associate measurements in cruise altitude conditions are necessary to confirm the occurrence of those UFPs.

Finally, a hermetic and clean sealing of the engines from the oil system aimed at a complete jet oil recovery could be a technical means to achieve a valuable mitigation effort for contrail formation if future model studies and flight campaigns give indications of abundant droplet formation on oil particles.