In this study, we consider soot to be the only exhaust particle since it plays a major role for contrail formation behind conventional passenger aircraft e.g.. Background particles will become relevant for aircraft with soot-poor or even soot-free emissions as for liquid hydrogen propulsion . Even though ultrafine volatile particles become more substantial for contrail ice crystal formation in a 3D LES set-up , they are supposed to have a tiny impact on contrail formation for current soot-rich emissions in a box model framework . Hence, we neglect volatile particles in the present study, but we will incorporate this particle type in the next model version.

The size spectrum of soot particles ranges from a few to hundreds of nanometres with geometric-mean values of around 15 nm . Soot particles are composed of several nanometre-sized primary spherules combined into aggregates. For simplicity, we assume a spherical shape. Recent studies indicate a fractal irregular shape of the larger (more than several 10 nm) soot particles e.g.. Taking this into account would modify their effective surface area and, therefore, the activation into water droplets and condensational growth rates. But it would require additional complexity that may not be relevant in such an intermediate detailed microphysical approach.

We prescribe log-normally size distributed soot particles by an ensemble of NSIP simulation particles (SIPs) using a technique described in Appendix A of . In a priori tests, we performed simulations with different values of NSIP ranging from 50 to 200. We find converged results of the analysed contrail properties for NSIP above around 100. Since this threshold value slightly varies with the geometric-mean dry core radius and particle size distribution width, we choose a default value of around NSIP=130.

In general, the moist exhaust plume cools due to mixing with ambient air. Contrails can form if ambient temperature is below the so-called Schmidt–Appleman (SA) threshold temperature (ΘG), and the plume becomes water-supersaturated in a certain temperature range. The threshold temperature varies with atmospheric parameters (ambient relative humidity over water, air pressure) and engine/fuel properties (specific combustion heat, propulsion efficiency, water vapour emission index). The calculation of ΘG is described in Appendix 2 of .

We use the Kappa–Köhler theory (Sect. ) to calculate which soot particles are able to activate into water droplets. Thereby, we prescribe the hygroscopicity parameter of soot particles as a measure for their solubility. This is a more convenient approach than defining a fixed fuel sulfur content since other polar species may lead to active sites on soot particles as well. Smaller soot particles require higher water supersaturation to overcome the Köhler barrier than larger ones (Fig. ). Therefore, the latter particles can activate earlier and may suppress the droplet formation on smaller particles due to depletion of water vapour. The super-cooled droplets subsequently grow by condensation as long as the exhaust air remains water-supersaturated. They freeze into contrail ice crystals when the homogeneous freezing temperature (Sect. ) is reached. Larger droplets can freeze earlier (i. e., at higher plume temperature) and grow further by deposition before smaller droplets manage to freeze into ice crystals.

To calculate the equilibrium saturation ratio over a droplet or ice crystal surface (SK), we use the Kappa–Köhler equation : 1SK=aw⋅exp⁡Ker=r3-rd3r3-rd3(1-κ)⋅exp⁡2σMwRTρwr, consisting of the solution term, described by the activity of water aw, and the exponential Kelvin term. rd is the particle dry core radius, r the droplet radius and κ the hygroscopicity parameter characterizing the solubility of the aerosol particle; σ is the surface tension of the solution droplet, ρw the mass density of water and T temperature. Mw and R denote the molar mass of water and the universal gas constant, respectively. For the activation of exhaust particles into water droplets, the plume saturation ratio needs to overcome the maximum of SK, denominated by the critical saturation ratio (Sc). This maximum increases with decreasing rd due to the Kelvin effect (Fig. a) and decreases with increasing κ due to the solution effect (Fig. b). The calculation of Sc is described in Appendix .

The equilibrium saturation vapour pressures over the droplet (ice crystal) surface, used in Eqs. () and (), are the product of the saturation vapour pressure over a flat water (ice) surface and SK.

The droplet growth of activated soot particles is calculated according to and . In the steady state (small changes in the vapour flux) and for spherical droplets, the single droplet mass growth equation is given by 2dmwdt=4πr(ev-eK,w)RvTDvβm-1+eK,wLc2RvK^T2βt-1=4πrev-eK,wFMβm-1+FHβt-1, where ev is the partial vapour pressure and eK,w equilibrium saturation vapour pressure over the droplet surface. Lc denotes the specific latent heat for condensation/evaporation, Dv the binary diffusion coefficient of air and water vapour, K^ the conductivity of air and Rv the specific gas constant of vapour. The terms FM and FH represent the mass and heat diffusion term, with βm and βt denoting the transitional correction factors calculated according to .

We calculate the homogeneous freezing temperature (Tfrz) of a super-cooled droplet with radius r according to the method of . This method is suitable for a strong cooling situation typically occurring in a jet plume. We introduce the nucleation rate as the number of droplets freezing per unit time 3j=J⋅LWV, where J is the nucleation rate coefficient, and LWV=43π(r-rd3) is the liquid volume available for freezing. The temperature-dependent nucleation rate coefficient J is parameterized according to : 4J/(m-3s-1)=106⋅exp⁡(a1T+a2), where a1 = 3.574 K-1 and a2 = 858.72 are empirical constants. Assuming that the whole droplet freezes immediately if ice nucleates within its volume, the pulse-like nature of j may be expressed by a characteristic freezing timescale 5τfrz-1=∂ln⁡j∂t≃∂ln⁡J∂TT˙, with T˙ denoting the plume cooling rate. Using Eq. () we obtain τfrz≃(a1T˙)-1.

The homogeneous freezing requirement will be fulfilled if the product of j=LWV⋅J and τfrz is unity. Using this requirement, inserting the relations from above and rearranging to T:=Tfrz yields 6Tfrz(r,T˙)=1a1ln⁡3×10-6a1T˙4π(r3-rd3)⋅m3s-a2.

The homogeneous freezing temperature in contrails declines with decreasing droplet radius and increasing cooling rate. As shown in Fig. 8 of , Tfrz ranges between around 230 and 232 K for typical droplet sizes (100–500 nm) at cruise altitude conditions.

The mass change of a single ice crystal is given by 7dmidt=4πCrDvβv-1(ev-eK,i)Dvβv-1LdeK,iK^βk-1βv-1TLdRvT-1+RvT, where C is the shape factor, r the equivalent volume radius and Ld the specific latent heat for deposition/sublimation. In our study, we set C to unity assuming spherical ice crystals as they are typically observed for young contrails e.g.. Since sedimentation is of low relevance during contrail formation, we neglect ventilation effects here. The calculation of the transitional correction factors βv and βk is described in Appendix .

The temporal evolution of the mass mixing ratio of a species χ(t) is given by 8χ˙(t)=-ωχ(χ(t)-χa)+ξ(t), where the first term describes the mixing of the hot exhaust with ambient air, and the second term (ξ) represents sink and source terms. The entrainment rate ωχ characterizes the dilution, and χa is the ambient mass mixing ratio of the species. Applying the equation above to water vapour and setting up an analogous equation for the evolution of the plume temperature (T) yields 9xv˙(t)=-ωv(xv(t)-xv,a)+q˙PC(t),10T˙(t)=-ωT(T(t)-Ta)+T˙LH(t), where Ta and xv,a are ambient temperature and vapour mass mixing ratio, respectively. The subscript LH relates to latent heating, and PC stands for phase changes from water vapour to liquid water or ice and vice versa.

From the evolution of a passive tracer χp(t) (defined by Eq.  with ξ=0), we can derive the entrainment rate ωp, 11ωp(t)=-χp˙(t)χp(t)-χp,a, and set ωv=ωp since water vapour is supposed to be diluted like a passive tracer. The entrainment rate characterizing the temperature change, ωT, differs from ωv by a factor called the Lewis number, Le=ωv/ωT, which is the ratio of the thermal and mass diffusivity. We set Le to 1, assuming that heat and mass diffusion proceed equally fast since turbulent exchange processes are dominating over molecular exchange processes. Due to ωv=ωT, we skip the index from now on and simply use ω. The entrainment rate ω describes the instantaneous mixing strength. Integrating it over time, we obtain the plume dilution factor 12D(t)=exp⁡-∫t0tω(t′)dt′=χp(t)-χp,aχp,0-χp,a, where the subscript “0” refers to conditions at the engine exit at time t0:=0 s.

At the engine exit, the emitted air is already a mixture of ambient air and combustion products. The initial plume dilution N^0 can be derived from the temperature excess T0-Ta: 13N^0=Q(1-η)c‾p(T0-Ta), where Q is the specific combustion heat, η the propulsion efficiency and c‾p = 1020 J(kgK)-1 an average heat capacity for dry air over a temperature range between around 200 and 600 K see derivation in.

The overall dilution N^ generally increases with plume age due to the continuous entrainment of ambient air and is related to the dilution factor (this quantity decreases with plume age) via 14N^(t)=N^0/D(t), implying D0=1.

The initial plume area A0 results from mass conservation : 15A0=N^0mFρ0, where ρ0=paRdT0 is the air density, with pa denoting air pressure, Rd the specific gas constant of dry air and mF the fuel consumption (in kgm-1). The evolution of the plume area is given by 16A(t)=A0⋅ρ0ρ(t)D(t)=A0⋅T(t)T0D(t).

In the numerical implementation, the ordinary differential equations (ODE) like Eqs. () and () are discretized by a simple Euler forward scheme. Due to the linearity of the ODEs, the implementation of an implicit Euler backward or a second-order trapezoidal scheme is straight-forward, but we find no improvement over the forward scheme. The source terms in the ODEs are provided by the LCM. In the LCM, the computation involves summations over all SIPs with phase changes or water vapour uptake/release during the time step. All model components (plume evolution ODE and LCM microphysics) use a time step of 0.001 s. This time step is sufficiently small to account for the fast diffusional growth at high plume supersaturations.

used the LES solver FLUDILES to simulate the plume evolution in a 3D domain over 10 s starting behind the CFM56 engine nozzle of an A340-300 aircraft without treatment of the bypass. They sampled the plume with 25 000 trajectories that were randomly positioned inside the initial plume area A0=πr02 with radius r0 = 0.5 m. We assume that all trajectories represent an equal share of the plume, e.g. in terms of air mass and number of soot particles. For each trajectory, the temperature evolution T3D,k(t) has been tracked. We use these data to infer the dilution factor (or equivalently the entrainment rate) along each trajectory k by assuming that temperature is a passive tracer: 17Dk(t)=T3D,k(t)-T3D,aT3D,0-T3D,a.

T3D,0=580 K and T3D,a=220 K are the plume exit and ambient temperature of the FLUDILES simulation. These equations are only meaningful if T3D,k>T3D,a is fulfilled. However, the LES temperature occasionally falls below T3D,a due to a pressure drop resulting from the interaction of the plume with the wake vortex. In the computation of ωk(t) or Dk(t), we replace all values of T3D,k that are below T3D,a+ϵ with this lower limit. The threshold ϵ is chosen such that the maximum dilution is still reasonable for a few seconds of plume age (Dmin≈2.5×10-3).

In the FLUDILES simulation, the initial plume temperature profile is based on the Crocco–Busemann relation and introduces a smooth radial transition between the plume and the environment. This ensures numerical stability as gradients in prognostic variables that are too strong are numerically intricate. This means that T3D,k(t0) only equals T3D,0 directly behind the nozzle exit centre and decreases down to around 400 K towards the plume edge (see black line in Fig. a). These trajectories start with Dk,0<1 and an accordingly higher initial dilution N^k,0. For simplicity, we mostly neglect this fact in the following description, but the initialization in this transition region is consistently treated in our model.

The process of converting jet kinetic energy into thermal energy is considered within the 3D simulations. This means that the FLUDILES temperature is higher than a “passive tracer temperature”. Therefore, we underestimate dilution, causing a later onset of contrail formation. This effect is partially compensated for by the inhomogeneous initial temperature profile, implying that exhaust air parcels are already diluted near the plume edge.

Basically, we only infer the dilution from the trajectory data according to Eq. () and plug the corresponding entrainment rate values into Eqs. () and () for each trajectory. For the radial plots presented later on, the coordinate data of these trajectories are used, but the actual computations are independent of this type of information.

Note that the ambient temperature Ta and plume temperature at engine exit T0 in Eqs. () and () are allowed to differ from T3D,a and T3D,0 used in Eq. (). Similarly, the ambient and initial plume water vapour can be chosen independently of the 3D data. In such a case, we basically only use the information of the dilution and its variability.

Moreover, a single mean state trajectory can be derived from the trajectory ensemble. We determine the corresponding mean dilution DAT by averaging the preprocessed temperature data over all 25 000 trajectories and plugging the average temperature evolution T‾3D into Eq. ().

There are other ways of computing the mean properties of the trajectory ensemble. For instance, the mean temperature can be a weighted average, where the cross-sections or masses of the trajectories are used as weights. The internal energy is conserved when a mass-conserving temperature average is used. The Supplement shows additional results for a mass-conserving temperature average and contrasts them with our default procedure.

Table  summarizes initial conditions of our baseline scenario, which are described in the following. The plume exit temperature (near the plume centre) is set to 580 K. We prescribe an ambient temperature of either 220 K (baseline case) or 225 K (near-threshold case). The baseline SA threshold temperature (ΘG) is 226.2 K for the given atmospheric conditions and engine/fuel properties. The prescribed ambient relative humidity over ice, RHice,a, determines the ambient water vapour mixing ratio xv,a being proportional to RHice,a⋅es,ice(Ta). We use RHice,a = 120 % for the baseline case.

Engine and fuel parameters such as specific combustion heat, propulsion efficiency and water vapour mass emission index are prescribed consistently with the trajectory data set-up and have values according to an A340-300 aircraft at cruise speed (Table ). These parameters are kept constant for all sensitivity studies, even though they can slightly change with varying ambient conditions. For the fuel consumption, which is calculated by rearranging Eq. (), we obtain lower values than typical for this aircraft type since the initial plume radius given by FLUDILES is only 0.5 m (Sect. ). For given T0, specific combustion heat and propulsion efficiency, we obtain N^0≈75 according to Eq. (). Note that the initial plume water vapour mixing ratio is controlled by the water vapour emission index EIv: 18xv,k(t0)=EIvN^0,k+N^0,k-1N^0,kxv,a, where the second term is the contribution of ambient humidity entering and exiting the engine. This term is less than 1 % of the total plume water vapour and, therefore, neglected in our initialization.

Based on laboratory measurements, we prescribe log-normally distributed soot particles with a geometric-mean dry core radius (r‾d) of 15 nm and geometric width of 1.6 and set the engine soot number emission index (EIs) to 1015 kg-1 as a typical value for commercial aircraft e.g.. Furthermore, we assume the same hygroscopicity parameter value (κs=0.005) as in .

The overall number of emitted engine soot particles per flight distance is given by 19Ns,tot=EIs⋅mF.

Table  summarizes the parameter variations (PVs) that investigate the sensitivity of contrail formation to soot properties (PV1–PV4) and ambient conditions (PV5/PV6). In the first two PVs, we either vary EIs or r‾d. Flight measurements have shown that a 50:50 mixing of conventional kerosene with a biofuel causes a decrease in EIs by around 50 % . Moreover, they find a decrease of the average soot particle size. This is because the switch to alternative fuels leads to a reduced number of primary soot spherules. The subsequent coagulation between those spherules decreases, and the final soot aggregates are smaller on average. Therefore, in PV3, we vary both EIs and r‾d and combine a 50 % reduction in EIs with a reduction in r‾d to 12.5 nm. In PV4, we change the hygroscopicity parameter of soot particles in a reasonable range between 0.001 and 0.01 (Markus Petters, personal communication, 2019). Finally, we study the variation of contrail properties with ambient temperature Ta (PV5) and relative humidity over ice RHice,a (PV6).

Here, we introduce diagnostics that are used to evaluate the simulations.

The fraction of soot particles activated into water droplets (ϕact) or frozen to ice crystals (ϕfrz) is given by 20ϕact/frz=∑i=1NSIPζi,act/frzNs,iNs,tot, where ζi characterizes whether soot particles (ζi,act=ζi,frz=0) represented by a SIP i have turned into water droplets (ζi,act=1 and ζi,frz=0) or ice crystals (ζi,act=ζi,frz=1). Note that we classify ice crystals as activated; hence we get 0≤ϕfrz≤ϕact≤1.

We introduce the apparent ice number emission index, AEIice, as a measure of the number of formed ice crystals per fuel mass burnt. Since we consider only soot as exhaust particles in our study, AEIice is simply given by the product of ϕfrz and EIs.

Furthermore, we define the effective radius of the soot–droplet–ice number distribution as 21reff=∑i=1NSIPri3⋅Ns,i∑i=1NSIPri2⋅Ns,i.

Typically, reff is a quantity that is relevant in radiation problems. Here, we prefer reff over the ordinary mean radius since the time evolution is smoother, and there is no need to specify the total particle number.

For ensemble mean values, intensive and extensive physical quantities are mass-weighted averages and sums of trajectory ensemble values, respectively. For quantities that are defined as ratios like the effective radius or relative humidity, first the averages of the quantities that appear in the denominator and nominator are computed, and then the ratio over those averaged quantities is taken. Note that the mass weights cancel out in the definitions of ϕact/frz and reff.

We employ the box model in two different frameworks as outlined in Sect. . In the average traj framework, a single box model run with an average dilution is performed. In the ensemble mean framework, box model runs are carried out for the complete set of 25 000 trajectories.

Figure  compares the plume relative humidity (top row) and the activation–freezing fraction between both frameworks. This is done for baseline conditions (Fig. a and c, Ta = 220 K) and a near-threshold case (Fig. b and d, Ta = 225 K with |Ta-ΘG|≈ 1.5 K). First, we analyse the baseline case. For the average traj, RHliq surpasses water saturation after around 0.4 s and leads immediately to a rapid droplet formation on soot particles. Thereby, ϕact increases quasi pulse-like from zero to its final value. Subsequently, RHliq decreases and approaches 100 %. After 0.7 s the droplets freeze rapidly into ice crystals, and the depositional ice crystal growth causes a steeper decrease in RHliq until ice saturation is reached (see blue line in Fig. a). Regarding the ensemble mean evolution, the fraction of soot particles forming droplets/ice crystals increases smoothly over a certain time window. This is explained as follows: the spatial variability in RHliq is quite large (Fig. d) due to a systematic radial gradient and superposed turbulent deviations. Therefore, droplet and ice crystal formation occurs for different trajectories at a different plume age, mostly depending on the radial distance to the plume centre (Fig. e and f). Although the ensemble mean features a smaller RH peak value than the average traj evolution, a larger fraction of soot particles forms ice crystals (96 % vs. 88 %).

In the near-threshold case (right column), Tsat≈ 236 K is lower and closer to Ta. Hence, water saturation is reached at a higher plume age when the cooling rates are already lower. Additionally, the peak RH values are clearly reduced. According to the evolution of the plume relative humidity, droplet and ice crystal formation on soot particles is initiated a few tenths of a second later compared to Ta = 220 K ((Fig. d). The time between the activation of the first droplets and the freezing of the last droplets is also longer (1.6 versus 1.2 s). The final ϕact and ϕfrz values are clearly reduced relative to the baseline case since only the larger soot particles can activate into water droplets due to the decreased peak RHliq. Again, the ensemble mean evolution displays slightly higher final ϕfrz values than the average traj.

A closer look at the average traj evolution of the near-threshold case reveals that ϕact is not a monotonically increasing function and that the final ϕfrz value is lower than the ϕact value at some intermediate time (in this example at t=0.9 s). This is explained as follows: the largest soot particles are activated first, and over time smaller soot particles grow large enough to be considered activated as well. However soon after, RHliq drops below SK (for the smaller soot particles) as condensational growth removes water vapour. Those particles shrink, such that their radius drops below the activation radius. Once the largest water droplets freeze, depositional growth leads to a fast depletion of water vapour. Again, the smallest of the currently activated droplets face water-subsaturated conditions and also become deactivated (at around t=1.25 s). We will encounter this “deactivation phenomenon” later again. Even though it is not visible in the ensemble mean evolution, this deactivation phenomenon also occurs for most trajectories of the ensemble runs.

In this section, we investigate the sensitivity of contrail formation to various soot properties. Figure  presents results of the sensitivity studies PV1 to PV4. As in the previous section, the baseline and near-threshold temperature cases are considered. We first analyse the baseline temperature case (Ta=220 K; Fig. a, c, e and g). The ensemble mean results reveal that the temporal evolution of the activation fraction ϕact and freezing fraction ϕfrz hardly changes with any variation of the soot properties. In the average traj framework, the sensitivity to soot properties is slightly higher. There, the activation and freezing fraction moderately increases with decreasing EIs (Fig. a) and increasing r‾d (Fig. b). A higher number of soot particles leads to a faster depletion of plume relative humidity, prohibiting the activation of smaller particles. Likewise, a decrease of r‾d reduces the number of soot particles that manage to be activated. Changing the two parameters, EIs and r‾d, simultaneously (Fig. c), the effects on ϕact and ϕfrz more or less cancel out, resulting in an almost negligible effect. The variation of ϕfrz with the hygroscopicity of soot particles (Fig. d) displays the lowest impact within the considered parameter range. Note that the average traj simulations with higher prescribed EIs (dotted cyan lines) exhibit the aforementioned deactivation phenomenon.

In the near-threshold scenario, the ϕact and ϕfrz values are clearly lower due to decreased plume water supersaturation (Sect. ). While the variation with EIs (Fig. b) is still low, both average traj and the ensemble mean evolution are more sensitive to the variation of r‾d and κ than for Ta=220K. This is because of two basic reasons: first, the critical saturation ratio for the activation of soot particles varies non-linearly with the soot core size (due to the non-linear Kelvin term), displaying a strong change for rd<10 m (Fig. a). This means that far away from the formation threshold, larger changes of r‾d and κ are necessary to affect the activation of the several nanometre-sized soot particles. Second, these small soot particles are at the tail of the size-distribution, and, therefore, their activation only makes a low contribution to the overall ϕact. In the near-threshold case, only the larger soot particles (rd≳20 nm) can form water droplets. This activation threshold is close to r‾d, which simultaneously defines the maximum of the log-normal size distribution. Hence, changing r‾d or κ has a significantly higher impact on ϕact for Ta=225 K than for lower ambient temperature.

Again, we find a stronger sensitivity to rd (Fig. d) than for κ (Fig. h) Interestingly, the ensemble mean simulations show larger rd- and κ-induced variations in ϕfrz in the end than the average traj simulations, which is opposite to what we saw in Fig. c and g. This is mainly due to the deactivation phenomenon. In the combined variation of EIs and r‾d (Fig. f), the impact of r‾d on ϕact and ϕfrz dominates over that of EIs. Halving the EIs value and decreasing r‾d to 12.5 nm leads to a decrease in ϕfrz from 0.41 to 0.31 and translates into a decrease of AEIice even by 62 % (instead of around 50 % if only the EIs value is halved).

Furthermore, we have investigated the sensitivity of contrail formation to the geometric width of the soot particle size distribution σ (not shown): in contrast to the variation with r‾d and κ, there is a negligible impact near the contrail formation threshold (Ta = 225 K). This is because the activation of soot particles with dry core radii below r‾d is already inhibited due to the Kelvin effect. For our baseline case, a higher fraction of soot particles forms ice crystals if the geometric width is reduced. This is due to the fact that the smallest soot particles within the size distribution become larger and hence can be activated more easily. Decreasing (increasing) geometric width from 1.6 to 0.8 (2.4) causes a decrease (increase) in ϕfrz from 0.88 to 0.83 (0.98) for the average traj and from 0.96 to 0.93 (0.99) for the ensemble mean.

In this subsection, we study the dependence of contrail formation on ambient conditions. We vary the ambient temperature Ta and background relative humidity over ice RHice,a (see PV5 and PV6 in Table ).

Figure a and b display the variation of ambient temperature Ta at default RHice,a=120 %. As already explained above, a change of Ta affects the contrail formation time and the number of soot particles turning into ice crystals. The lower the Ta, the earlier that water saturation is first reached in the plume and the higher the attained peak water supersaturation is. Hence, reducing Ta to 215 K, droplet and ice crystal formation is initiated earlier, and more ice crystals form (Fig. a). The results for Ta=220 and 225 K have already been described in Sect. . Thereby, the difference in ϕfrz is significantly larger between 225 (dotted) and 220 K (solid) than that between 220 and 215 K (dashed).

The evolution of the effective radius reff in Fig. b shows a steep increase from the moment when the first droplets are activated. The increase slows down in most cases and is accelerated again once the droplets freeze. The curves level off as soon as ice saturation is reached in the plume. This happens after around 1 s for the lower Ta and after around 2 s for Ta=225 K. The final reff values hardly change between Ta=215 and 220 K and between the average traj and the ensemble mean. Only for Ta=225 K are higher values attained. This is because the absolute amount of ambient water vapour is higher for higher Ta and gets distributed among fewer ice crystals.

Figure c and d show the variation of RHice,a. For Ta=220 K (Fig. c), contrail ice nucleation is not at all affected by the variation of RHice,a. In contrast, the sensitivity of ϕfrz to RHice,a is very high for Ta=225 K. At ice saturation (dotted line), only few ice crystals form (ϕfrz<10 %). This is because a decrease in RHice,a causes a decrease in ΘG from 226.2 to 225.5 K so that those contrails form only 0.5 K below the SA threshold temperature, and maximum plume water supersaturation is only a few percent. Analogously, more ice crystals form for RHice,a=140 % as ΘG, and hence |Ta-ΘG|≈3 K is higher.

Our results highlight that the sensitivity of contrail ice nucleation to atmospheric parameters is large near the contrail formation threshold, which is consistent with previous studies e.g.. Therefore, it is meaningful to display the variation of the nucleated contrail ice crystal number with ΔT:=Ta-ΘG as well (see next section).