Contrail–cirrus is probably the largest contribution from aviation to climate change in terms of radiative forcing . However, its quantification is associated with large uncertainties and the confidence of those estimates is still rated low . Contrail radiative forcing is estimated by global circulation models (GCMs), whose parametrisations of contrails have been improved in the recent past. In particular, the analysis methods switched from diagnostic approaches for young (line-shaped) contrails towards a process-based treatment of contrail cirrus evolution .

Contrail microphysical, geometric and optical properties depend on a multitude of meteorological and aircraft parameters as investigated by high-resolution simulations . In the first few minutes of a contrail's lifetime, the contrail evolution is strongly affected by the downward-propagating wake vortex pair. On the one hand, this can lead to a substantial increase in contrail depth. On the other hand, adiabatic heating may result in strong crystal loss. Both processes have an impact on the evolution of the contrail–cirrus while it spreads by vertical wind shear and produces a fall streak by sedimentation.

The present study develops and provides a parametrisation for the depth and crystal number of young contrails, which accounts for the important physics and considers the dominant parameters, namely relative humidity, temperature, thermal stratification and aircraft parameters. The parametrisation is based on the evaluation of a large data set of recent large-eddy simulations (LESs). Due to its simple analytic form, it is suited to be incorporated into large-scale models where it can replace present-day contrail initialisation with ad hoc assumptions, where vortex phase processes are only very roughly captured. This approach exhibits a way to condense information gained from LESs such that it finds its way into global-scale models.

The presented parametrisation covers a much larger parameter space and is more universal than the parametrisation given by and extended by . Clearly, the present work updates the earlier versions.

In the recent past, the effect of biofuels on contrail properties aroused scientific interest. In particular, the likely reduction in the initial ice crystal number can have a significant effect, even though the initial differences in ice crystal number are reduced during the vortex phase . The effect of biofuels may be overestimated, if crystal loss is neglected. The presented parametrisation accounts for the crystal loss; hence, it avoids this problem.

Moreover, the present approach complements a parametrisation describing the long-term contrail–cirrus evolution in where the ice crystal number is an important input parameter.

Since the design of the proposed parametrisations is motivated by the physical processes involved in contrail evolution, this section will give a short overview of various aspects of the contrail evolution. In the remainder of the text, we assume that the environment is sufficiently cold and moist, such that contrails form and persist (i.e. that the air is supersaturated). In general, contrail development is divided into three temporal phases based upon the governing physical processes: the jet, vortex and dispersion phase .

The jet phase denotes the first several seconds, in which the lift generating airflow over the wings transforms into a counter-rotating vortex pair. The hot exhaust jets mix with the cold ambient air, expand and get entrained into the forming vortices . Formation of ice crystals is completed within 1 s behind the aircraft and their number Nform depends inter alia on the number of emitted soot particles Nsoot . Both quantities are usually given in units, “number per metre (of flight path)”.

In the vortex phase, which lasts several minutes, the counter-rotating vortices propagate downward up to several hundreds of metres. Whereas most of the exhaust (including the ice crystals) is carried down in the vortex system (called primary wake), some of it gets detrained and forms a curtain between the cruise altitude and the actual vortex position (called secondary wake). After vortex break-up, buoyancy may cause a major fraction of the exhaust to rise back to the original emission altitude e.g.. Ice crystals sublimate and get lost due to adiabatic heating in the primary wake . The number of ice crystals surviving the vortex phase, N, is highly variable and depends on a multitude of environmental conditions and aircraft properties . Ice crystals in the secondary wake face the fresh supersaturated air and grow, such that they often contain most of the contrail ice mass at the end of the vortex phase.

The dispersion phase treats the contrail-to-cirrus transition and starts once the vortices have lost their coherent structure and vorticity has reached ambient levels. Not only the environmental conditions during this transition are important, but also the early contrail properties matter for the later contrail–cirrus properties . More specifically, investigated the impact of aircraft type on contrail–cirrus evolution; they found that contrails differed a lot in terms of vertical extent and ice crystal number after the vortex phase and simulation tests showed that both the contrail depth and ice crystal number after the vortex phase are relevant for the later properties of the contrail–cirrus.

Compared to the timescales of natural processes like vertical turbulent mixing, the initial expansion during the vortex phase can be viewed as a sudden event. The extent of the shear-induced horizontal spreading scales linearly with the contrail depth. Later, the depth of contrail–cirrus will increase by sedimentation and the formation of a fall streak given that the supersaturated layer is sufficiently deep. Moreover, radiative lifting may increase the contrail vertical extent . The effectiveness of sedimentation depends on the ice crystal size distribution. In a simplified picture (where we think of a monodispersed size distribution of the ice crystals), the size is determined by contrail ice mass and ice crystal number. As the ice mass of the emerging contrail–cirrus is mainly controlled by the supply of ambient water vapour, contrail cirrus properties are more susceptible to changes of number than mass of the young contrail . Thus, our study focuses on ice crystal number N rather than ice crystal mass I.

The number of ice crystals after the vortex phase N is given by Nsurv=Nform×fNs, where fNs is the fraction of ice crystals surviving the vortex phase (survival fraction). Nform is the number of generated ice crystals in the beginning and is given by Nform=m˙fUEIiceno, where EIiceno is the “emission” index of ice crystals. In soot-rich regimes, homogeneous freezing of water-activated soot is the primary ice formation mechanism and EIiceno may be given by fA×EIsoot, where fA is the fraction of activated soot particles. A schematic overview is given in Fig. . Evaluation of fA and the corresponding soot emission index EIsoot is left to others . In this paper, we simply vary EIiceno and parametrise fNs.

Section 2 describes the data set of simulations and explains the vortex phase processes in more detail. Section 3 presents analytical parametrisations of the survival fraction, the depth and the width of 5 min old contrails. Section 4 discusses the relevance of the input parameters and the effects of a soot reduction. Section 5 discusses the robustness of the presented parametrisations and conclusions are drawn in Sect. 6. The Appendix A includes instructions on how to implement the parametrisation and the Supplement contains a FORTRAN source code file of the parametrisation. Throughout the paper, parametrisations and fit functions are identified by a “^” symbol, e.g. f^Ns or N^surv.

We introduce three length scales that help to understand and explain the observed processes in a young contrail. A similar attempt has already been undertaken in , where three timescales were used to better explain the observed processes in the downward sinking contrail. The three length scales are

zdesc, which measures the final vertical displacement of the wake vortex;

We will later see that the approximations of fNs and H take simple forms, if expressed in terms of zΔ, which is a linear combination of the three length scales: zΔ=αatm×zatm+αemit×zemit-αdesc×z^desc with positive weights αi. The equation describes a balance between the WV surplus (raising RHi) and wake vortex-induced adiabatic heating (lowering RHi). Note the minus sign in front of the z^desc term. In the following, we will define the three length scales. Whereas zatm and zemit are analytically defined, zdesc is determined from simulations and an analytical approximation z^desc enters the balance Eq. ().

The latter two length scales were introduced assuming an ice crystal-free parcel, yet they are also meaningful for an ice crystal laden plume. In this case, the excess moisture (WV emission + ambient supersaturation) quickly deposits on the ice crystals such that relative humidity reaches saturation and the ice mass is the sum of both contributions. Ongoing heating of the plume causes a slight subsaturation and sublimation of ice mass. After a downward displacement of zatm, the ice crystals have lost as much mass as was contributed by the ambience (neglecting a small time delay); i.e. the remaining ice mass equals that of the WV emission.

The left panel of Fig.  depicts the simulated fNs values as a function of zΔ for several subsets of simulations. The choice of the weights αi for the computation of zΔ will be discussed later. We approximate fNs by an arc tangent function (grey curve), which depends solely on zΔ. The approximated survival fraction f^Ns is defined by f^Ns=a^(zΔ), where a^(x)=β0+β1πarctan⁡α0+(x/100m). Values of a^ below 0 and above 1 are clipped. Again, values of the fit parameters β0,β1 and α0 are provided later.

The first row of Fig.  shows fNs for a basic RHi-TCA variation for one AC type. The selected simulations are listed in block 1 of Table  and all have the same z^desc. zatm and zemit vary over large ranges, as they mainly depend on RHi and TCA. Thus, this sub-panel demonstrates that the RHi and TCA sensitivity of fNs is well captured by our a^(zΔ) approach.

In a next step, the AC type is varied (block 2/row 2 of the table/figure). The aircraft type strongly affects the wake vortex properties, i.e. the descent speed, the time of vortex break-up and the final vertical displacement. Thus, this simulation subset focuses on the variation of zdesc (and zatm). fNs is smaller for larger aircraft. This trend is also captured by the approximation, as z^desc is larger and, correspondingly, zΔ is smaller.

Row 3 of the figure extends the simulation set by cases with weaker stratification where zdesc is larger and fewer ice crystals survive. The approximation is able to predict the behaviour, yet the distances between the data points and the approximation are larger than in row 1.

So far, the prediction of ice crystal number loss is based on a balance equation of the ice crystal mass. This is suitable and works for the simulations depicted in rows 1–3, as they all use the reference ice crystal emission index EIiceno, ref=2.8×1014 kg-1. In these cases a certain ice mass change can be linked to a certain ice crystal loss. As discussed in Sect. , this does not hold any longer, when EIiceno is varied. To cover the effects of a EIiceno variation, the parametrisation must be extended. Note that, so far, EIiceno did not enter the computations for the length scales, zΔ nor for a^.

It is clear that zdesc is unaffected by a EIiceno variation and our strategy is that the terms αatm×zatm and αemit×zemit in the balance equation are modified. We keep the definitions of zatm and zemit and make the weights αatm and αemit dependent of EIiceno.

We define the normalised emission index EIiceno* as EIiceno/EIiceno, ref and use a correction term of type EIiceno*-γ for the weights. Subsequently, the weights αatm and αemit are smaller for higher EIiceno*. This reduces zΔ and, correspondingly, f^Ns becomes smaller, as desired.

Rows 4 and 5 shows the simulated survival fractions for a small EIiceno variation for six AC types and a large EIiceno variation for the default AC type. These sub-panels demonstrate that the effect of an EIiceno variation can be well captured by the corrected approximation. Note that without this correction the various symbols of a specific colour would all lie on one vertical line (i.e. identical zΔ).

So far, the fuel flow changed only with AC type. As a last test, we vary the fuel flow (i.e. the WV emission V) for a fixed AC type. The sensitivity simulations are listed in block 5 in Table  (originally discussed in Fig. 9 of U2014). Figure  nicely reveals that the sensitivity to V is well represented in the parametrisation; for RHi=110 % the agreement is excellent, for RHi=120 % it is reasonable.

In the left panels of Fig. 5 the vertical distance of the symbol (fNs) to the grey curve (f^) shows the absolute error fabs=f^Ns-fNs. In the right panels, scatter plots of f^Ns vs. fNs are depicted and the errors can be more conveniently assessed. Furthermore, the root mean square of the absolute errors are given for each subset of simulations. The absolute errors are mostly below 0.1 for each subset, which proves the suitability of the proposed parametrisation.

Finally, we provide the values of the fit parameters that were used for the computation of zΔ and a^: β0=0.4,β1=1.19,α0=-1.35,αatm=1.7×EIiceno*-γatm,αemit=1.15×EIiceno*-γemit,

αdesc=0.6,γatm=0.18,γemit=0.18.

We determined these values by minimising the bias and standard deviation of the absolute and relative errors. Note that we did not apply a formal optimisation algorithm to find an optimal solution. We rather made a subjective trade-off between minimising the four error parameters.

Splitting the analysis into two separate length scales zatm and zemit, implicitly assumed linearity in es, which is not the case. Using the combined length scale zbuffer defined by 1+siesTCARvTCA+ρemit=esTCA+ΓdzbufferRvTCA+Γdzbuffer may be more physically plausible and, indeed, zbuffer is up to 15 % smaller than the sum of zatm and zemit. However, the main purpose is to design a parametrisation that approximates the simulated values the best. We found that the ansatz with two separate length scales allowed us to design a better parametrisation, as the weights αatm and αemit are individually adjustable.

The definitions of zatm and zemit (or zbuffer) rely on equating water vapour concentrations. One may argue that mixing ratios (and not concentrations) are conserved during adiabatic processes. From this follows the conservation of es(T)/Tκ with κ=3.5 instead of es(T)/T.

As most of the variability in zatm and zemit comes from es(T), using modified length scale definitions (with κ) would not improve the parametrisation.

The determination of the contrail depth was exemplified in Sect. . Table  lists H values for all simulations. Clearly, H depends on zdesc. The farther the vortices descend, the deeper the contrails can be (Fig.  right). On the other hand, the contrails may not reach the full vertical extent. In particular for slight supersaturations, the primary wake runs dry and all ice crystals in it are lost, as shown in Fig.  left. Thus, the parametrisation for H takes into account the combined effects of wake vortex descent and crystal loss.

Figure  left shows the relationship between H/z^desc and f̃Ns. Note that H is the contrail depth determined from the model simulations, whereas z^desc and f̃Ns are parametrisations (f̃Ns is similar to f^Ns and will be defined later). Using parametrised rather than simulated values for zdesc and fNs allows us to derive a parametrisation for H that is based only on available data. We suggest a piecewise linear function b^(x) as an approximation for H/z^desc as indicated by the grey curve.

Then the parametrised contrail height H^ is given by H^=z^desc×b^(f̃Ns), where b^(x)=η1xx∈[0,xs]η2x+η1-η2xsx∈xs,1 with η1=6, η2=0.15 and xs=0.2.

The piecewise definition reflects the fact that contrail depth changes only slightly with large survival fractions and is much stronger once a major fraction of the ice crystals is lost. Thus, the definition is split into two parts (above and below xs) and the slope of the approximation is much higher in the x<xs part; i.e. η1 is much larger than η2.

Note that b^ can achieve values greater than unity implying that H^ can be greater than z^desc. This is reasonable as the contrail extends above the formation altitude on the one hand. Moreover, zdesc was determined by finding the centre of the primary wake whereas the contrail bottom is defined by the lower end of the plume.

Finally, we define f̃Ns, which is used in Eq. () and is in one aspect different from f^Ns. Section  discussed that a variation of EIiceno has a small impact on the contrail mass and depth. The impact on crystal loss is however large and the values of f^Ns are distributed over a large range, as can be seen in bottom right panel of Fig. . In the definition of f̃Ns we exclude the EIiceno effect by simple means. We take the original definition for f^Ns and simply set γatm=γemit=0. Subsequently, f̃Ns does not change with EIiceno. In rows 4 and 5 of Fig.  the various symbols of a specific colour all lie on one vertical line. From Eq. () follows that H^ does not depend on EIiceno.

Analogous to Fig. , the right panel of Fig.  shows a scatter plot, now contrasting H vs. H^. Again, root mean squares of the absolute errors are supplied in the figure. They are around 50 m or below for each subset, which again proves the suitability of the proposed parametrisation.

Contrail width is a parameter whose choice is not as critical for the later evolution as that of contrail depth and ice crystal number. The horizontal spreading rate of a contrail is roughly given by W˙=H×s. For typical values H=400 m and vertical wind shear s=0.005 s-1 (considering only the component normal to contrail length axis), the width increases by 1 km every 8 min. Thus, an uncertainty in the initial width translates into a small offset in the assumed contrail age, which becomes unimportant when looking at contrail-to-cirrus evolution over several hours.

The determination of the contrail width is based on the evaluation of transverse profiles of ice crystal mass after the vortex phase, which are depicted in Fig. . The distributions resemble roughly Gaussian distributions (at least in the absence of a sheared cross-wind and when averaged along the flight direction).

For a B777 aircraft (left panel), most of the ice mass is confined to a 150m broad band centred around the aircraft body. Towards the end of their lifetime, the vortex tubes meander and some ice is laterally even farther dislocated. This effect is not apparent, when all ice of the primary wake has been lost before this stage (RHi≤110 %). The middle panel shows a modest dependence of contrail width on aircraft type. The smaller the aircraft wing span is, the smaller is the distance between the two vortex centres and exhaust plumes and the narrower is the final contrail. In the case with weaker stratification (right panel), the formation of vortex rings is pronounced, the vortex tubes oscillate more strongly and the contrail is broader.

Thus, 3-D dynamical phenomena like the well-known Crow instability of the trailing vortices lead to substantial variations along the flight direction, even if the environmental conditions are homogeneous in this direction, as exemplified by and for conserved exhaust species, and by and U2014 for contrails.

This also implies that the contrail width varies along flight direction and attains its maximum values only in certain segments along the flight direction. Figure  depicts cross sections of number concentrations from two slices, which are only 60m apart. Slice B contains a factor of 2.5 more ice crystals than slice A. Moreover, the contrail is much broader, in particular in the lower part.

Global-scale models, in which the current parametrisation might be employed in the future, cannot resolve such subtleties as contrail intrinsic heterogeneities. Thus, we propose the simple estimate W^=150m independent of the parameter settings. A more sophisticated approximation may include dependencies on wing span and Brunt–Väisälä frequency.

In global-scale models, where the mean age of the initialised contrails is typically half the time step (which is at present times larger than the contrail age used here) one may correct for this offset in time.

If one is interested in deriving number concentrations from the parametrisations of N,H and W, a more suitable width parametrisation is presented in Sect. .

The number of generated ice crystals Nform depends on the number of emitted soot particles Nsoot. The higher the corresponding emission indices are, the more crystals are lost during the vortex phase. We apply our derived parametrisation to determine an average survival fraction for EIiceno=1015, 1014 and 1013 kg-1, respectively. The average is taken over a 4-D cube varying relative humidity (100–140 %), temperature (210–226 K), Brunt–Väisälä frequency (0.006–0.014 s-1) and wing span (20–84 m). Using the empirical relationships given by Eqs. (), () and (), the wing span determines all relevant aircraft properties. For each parameter, the survival fraction is evaluated for nine values equally distributed over the given range (totalling 94 combinations). About 29, 55 and 75 % of the ice crystals survive for EIiceno=1015, 1014 and 1013 kg-1. Thus, a EIiceno reduction from 1015 down to 1014 kg-1 (or 1014 down to 1013 kg-1) implies only a factor of 5.3 (or 7.4) reduction of the ice crystal number in the end. An initial factor 100 reduction gives 40 times fewer ice crystals in the end. So far, all 94 data points (variation over RHi,TCA,NBV and b) are equally weighted, which could certainly be refined in the future.

Now more detailed sensitivity tests follow, where we analyse the behaviour of an EIiceno variation for certain subsets of the parameter space. For this, we take the average over three out of the four dimensions of the 4-D cube and show the EIiceno dependence for different values of the fourth parameter. Figure  shows the EIiceno dependence of the crystal loss for different values of RHi,TCA,NBV and b (from left to right). EIiceno runs from 1012 to 1016 kg-1. The solid lines in the top row depict the number of surviving ice crystals. The average over a 3-D cube N¯surv=∑i,j,k=08N^surv(i,j,k)/93 is shown, where N^surv=Nform×f^Ns and i,j and k are generic indices for three selected parameters. Similarly, the dashed lines show the initial ice crystal number N¯form=∑i,j,k=08Nform(i,j,k)/93. The bottom row shows the survival fraction f¯Ns=∑i,j,k=08f^Ns(i,j,k)/93. As expected, N¯surv and f¯Ns increase with increasing RHi, decreasing TCA and increasing NBV. The dependence on wing span is more complicated. Even though f¯Ns decreases with increasing b, N¯surv does increase, as this is overcompensated by an increase in Nform (following Eq. ). Now we turn the attention to the EIiceno dependence. Clearly, f¯Ns decreases with increasing EIiceno. The dependence on EIiceno is qualitatively similar for all values of TCA, NBV and b. However, for different RHi values, the function f¯Ns(EIiceno) looks qualitatively different. We can divide the analysed EIiceno range in a low EIiceno regime and a high EIiceno regime. In the low EIiceno regime, we find a strong dependence of f¯Ns on EIiceno for low supersaturation and a weak dependence for high supersaturation. In the high EIiceno-regime, it is the other way round. The slope of f¯Ns(EIiceno) is steeper, the higher the supersaturation is.

Biofuels cause lower soot emissions with a likely impact on Nform. As the initial differences are reduced during the vortex phase, mitigation studies neglecting those effects may overestimate the effect of biofuels.

In this section, we analyse the sensitivity of N and H to the various input parameters of the parametrisation. We evaluate N^surv=Nform×f^Ns and H^ for the 4-D cube as defined above and take the average over three out of the four dimensions. Figure  shows the ice crystal number and the contrail depth after the vortex phase as a function of the various input parameters. We separate between scenarios with EIiceno=1015 and 1014 kg-1. The variation in N is due to variations in Nform and f^Ns. Note that Nform depends only on wing span b and EIiceno following Eq. (), whereas f^Ns depends on all five parameters. In combination, we find that N depends most strongly on EIiceno and RHi. b,TCA and NBV (in this order) have a smaller impact.

The contrail depth depends most strongly on RHi and b and to a lesser extent on TCA and NBV. According to the design of our parametrisation, the contrail depth does not depend on EIiceno.

The sensitivity analyses reveal the most significant parameters for the determination of ice crystal number and contrail depth. It gives an estimate on how uncertainty in the input parameters translates into uncertainties in the output parameters.

Concerning the crystal loss determination, EIiceno and RHi should be well characterised in some future application of the parametrisation, whereas for NBV and TCA it may be sufficient to have a rough estimate or to assume some default values.

From the parametrisations of N,H and W one may derive the mean ice crystal number concentration n^mean=N^surv/(H^W^) of a specific contrail. Hereby, we assume that the ice crystals are spread over a rectangular cross section with width W^ and height H^. Figure  contrasts n^mean with the simulated values of nmean. nmean is given by Nsurv/A, where A is the actual cross sectional area of a contrail (averaged along flight direction). We find that n^mean underestimates the simulated values by roughly a factor of 5 (left panel). The main reason for this bias is that the parametrised area A^=W^H^ overestimates the real area A by around the same factor. In Fig. , the black boxes show the area-equivalent rectangle for this specific contrail with width Wrect=A/H. Wrect is much smaller than the actual width of the contrail. The parametrised number concentrations will become more realistic, if we prescribe an area-equivalent width Wrect=0.63b and use A^=WrectH^ instead. The constant 0.63 is chosen such that the expectation value of Δnmean=n^mean-nmean is zero. The right panel  shows a much better agreement of n^mean with nmean. The standard deviation of the relative error 2Δnmean/(n^mean+nmean) is around 50 %. The prediction of nmean is more uncertain than that of N or H, as the uncertainties add together. The parametrised number concentrations are similarly realistic when we use Wrect=36m (not shown; again the expectation value is zero and the standard deviation is 55 %).

Depending on the purpose the parametrisation is applied for, we recommend using either the W^ definition from Sect.  or the Wrect definition from this section.

As the contrails get diluted, mean concentrations decrease over time. Evaluating the simulated nmean at t=3min instead of at t=5min, we obtain 1.65 times higher values (with a standard deviation of 0.23).

Analogous to the analyses in Sect. , Fig.  (middle) shows the sensitivity of n^mean (after 5 min) to the various input parameters of the parametrisation. We find a dominant impact of EIiceno and RHi. b, TCA and NBV appear to be less important. Using Wrect=36m instead of Wrect=0.63b, various opposing trends do not cancel out and the sensitivity to b is larger (not shown).

Deriving analogous relations for ice water content or optical depth would be desirable. As we do not parametrise the contrail ice mass, this cannot be achieved with the present parametrisation.

Several further parameters may affect the early contrail evolution. Their importance, which has been partly investigated by recent 3-D simulation studies, will be discussed in the following.

Stronger vertical wind shear and higher ambient turbulence potentially speed up the vortex decay. However, for stratification values typical of the upper troposphere, variations of vertical wind shear s (assuming a linear wind profile) and turbulence are second-order effects and wake vortex descent, contrail height and crystal loss are fairly unaffected by such variations . For strongly curved wind profiles (i.e. non-zero s˙) vortex decay may be accelerated and the situation becomes more intricate.

Cruise altitude mainly determines the ambient temperature, pressure and air density (linked by the ideal gas law). Note that the definition of zatm and zemit is based on equating water vapour concentrations, which do not depend on ambient pressure/density (unlike to mixing ratios). Thus, the contrail ice mass balance and consequently crystal loss and contrail depth are in theory not affected by ambient pressure. Ambient pressure affects, for example, the diffusivity of water vapour and thus ice crystal growth . But the sensitivity is too weak to be important in this case as confirmed by 2-D contrail simulations . Note that the variation of other microphysical constants (not dependent on pressure) like the deposition coefficient induce larger uncertainties .

The initial circulation Γ0 of the wake vortices is inversely proportional to the air density (see Eq. ) and changes with flight altitude. Compared to the variation of aircraft mass and wing span, the rather small changes in ρair have a second-order effect on Γ0 and the wake vortex displacement zdesc. Thus, the dependence of contrail evolution on flight altitude is mainly a temperature effect.

In the simulations presented in Sect.  ambient relative humidity was assumed to be uniform over the entire domain. Hence, we do not cover scenarios with a thin supersaturated layer where the contrail leaves the supersaturated layer at some point during the wake vortex descent. A thin moist layer may limit the contrail depth and inhibit contrail growth in its early stage. However, such scenarios have not yet been analysed in any LES study of young contrails. So far, the impact of the depth of the supersaturated layer was only investigated for contrail–cirrus ; i.e. the supersaturated layer was deep enough to contain the young contrail, yet limited the later formation of the fall streak.

All simulations discussed in Sect.  use the standard configuration, as we call it. Further uncertainties arise from the choice of this configuration and may have an impact on the simulated crystal loss and contrail depth. Clearly, these parameters are not reflected in our parametrisations. This includes variations of the spatial initialisation and the initial width of the ice crystal size distribution as well as the disregard of the Kelvin effect in the ice growth equation. Their effects have been discussed in detail in U2014 and the effect on contrail depth was found to be negligible and is not further discussed here. Crystal loss, however, is affected and in the following we discuss whether this downgrades the universality of our parametrisations.

Figure  depicts fNs of these sensitivity simulations (coloured symbols) and contrasts them with the simulations with the standard configuration (grey symbols, all those simulations that were originally depicted in Fig. ).

In the default configuration, UG2014 and U2014 use a simple spatial initialisation; i.e. the crystal number concentrations are constant inside the two initial plumes. Following the approach of and , Gaussian-like distributions of ice crystal number concentrations were alternatively prescribed (see blue symbols). The effect on the crystal loss extent was often low (originally discussed in Fig. 7 in UG2014). In one particular case with high temperature and relative humidity, the crystal loss was stronger, once a Gaussian distribution was used (originally discussed in Fig. 13 of U2014; blue square in our Fig. ). If a Gaussian distribution is more realistic, the survival rates may be overestimated for high supersaturations.

Based on the assumption of spherical ice crystals, the standard configuration includes a correction of the local relative humidity over a curved surface (Kelvin effect) in the ice crystal growth equation. It is not clear how physically plausible this is, as ice crystals are never perfect spheres. When we switch off the Kelvin correction, fewer ice crystals are lost (brown symbols), as expected.

In U2014, the initial width of the ice crystal size distribution was varied, as it cannot be measured or numerically determined accurately enough see discussion in. The narrower the SD chosen is, the fewer ice crystals lost (green symbols). Note that reports a weaker sensitivity to this parameter which may be connected to the fact that the spatial initialisation of the ice crystals and the assumed initial wake age differ between their study and ours.

Ideally, the initialisation of the vortex phase simulations would use input from 3-D jet phase simulations that model the contrail formation with a detailed ice activation scheme and which could provide 3-D fields of the ice crystal size distribution, water vapour, temperature and velocity (only to name the most important ones). Unfortunately, such simulations are not available. Presently, 3-D jet phase simulations employ simplified ice activation schemes , whereas detailed plume microphysics are embedded in 0-D models with prescribed plume dilution .

Overall, we can state that the new data points (of the sensitivity simulations) lie within the spread of data points with the standard configuration. This implies that the uncertainties/biases due to the numerical configuration are smaller than the deviations of the simulation results from the fit function. This suggests that the uncertainties are acceptable given the accuracy of the parametrisation.

The early contrail evolution has been studied by several groups with different LES codes in the past . Qualitatively, the sensitivity trends are similar for most studies (except for one outlier model discussed below) and confirms the reliability of the model results. Nevertheless, quantitative comparisons between the various models were always hampered by the fact that in each study parameter variations started from different base states. This means that differences in the simulated contrail properties could always be attributed to slightly different input parameters and as a consequence possible systematic differences may be overlooked. Our approach offers an ideal framework to make a more in-depth intercomparison between the models and further check whether the results from other LES models support our proposed parametrisation. Block 7 of Table  lists input parameters as well as the values for fNs and H from several simulations of the above-mentioned studies. Analogous to the evaluation in Sect. , we compute zΔ and z^desc for the given input parameters and derive the approximated values for the survival fraction and contrail depth. Those are then compared to the simulated values (see Fig. ). First we discuss the survival rates depicted in the top row. All EULAG-LCM results from UG2014 and U2014 are plotted as light grey symbols; results from other groups are plotted with coloured symbols. The data set includes a combined variation of TCA and RHi (red crosses) as well as a wide range EIiceno variation (red plus symbols) by . We find that their simulated values for the survival fraction nicely follows our parametrisation; i.e. the red symbols are not farther away from the 1-to-1 line (in the right hand side panel) than the grey symbols. Similarly, the few data points by agree reasonably well with our parametrisation. This demonstrates the consistency between those models and also the robustness of the fNs parametrisation. Now we turn our attention to the survival rates simulated by (brown diamonds). Section 3.5 of U2014 already pointed out that the observed trends in several sensitivity studies of disagreed with all other models and appeared implausible. The present analysis confirms this and reveals an inconsistency of their data with our fNs-parametrisation. Obviously, their survival rates scatter strongly and there is no correlation between their fNs values and the predicted values f^Ns.

As a next step, we compare the contrail depth values of the various models (bottom row of Fig. ). Again, the data points from the other groups are added in colour. From (green squares) and (red crosses), H values are available for RHi=130 % and RHi=110 % simulations. For the other studies, only results of RHi=130 % simulations are available. For all RHi=130 % cases, the contrails reach their full vertical extent and accordingly the ratio H/z^desc is around 1.2–1.4 (left panel) as in our EULAG-LCM simulations. For the RHi=110 % simulations, contrail depth H is smaller as already observed in our EULAG-LCM simulations. The left panel reveals that the new H values support the piecewise linear approximation. The right panel shows non-normalised values of H and focuses on the variations in zdesc. In particular, the H values of three different aircraft types in (brown diamonds) are useful for comparison, as the wake vortex descent and with it the contrail depth vary over a wide range. Apparently, all H values are close to the 1-to-1 line, which leads to the conclusion that the wake vortex descent is strikingly similar in all models.

Young contrails have been measured in situ or with lidars in the past. Unfortunately, total ice crystal number cannot reliably be determined with those methods. Lidars do not measure ice crystal numbers and in situ measurements do not sample the complete contrail, which would be necessary to derive such an integral property. Thus, we limit ourselves to compare the contrail depth with lidar measurements and ice crystal number concentrations with in situ measurements. We have seen that contrail properties depend on a multitude of parameters and usually not all of them are determined with sufficient accuracy to evaluate our proposed parametrisation. Moreover, contrails are heterogeneous and in situ measured properties may depend on how and where the contrail is sampled. As an example for the contrail heterogeneity, Fig.  shows the occurrence frequencies of ice crystal number concentrations inside specific 3 and 5 min old contrails. Having in mind all the above arguments, it is clear that the following exercise can only serve as general sanity check.

used a ground-based lidar and finds the heights of a few young contrails to range between 150 and 300 m. We hypothesise that the contrails were produced from small to medium sized aircraft, in modestly supersaturated or stratified environments as their values do not reach our maximum values. In situ measurements of number concentrations show mean values of around 100–200cm-3 consistent with our data.