Large-eddy simulations (LES) with Lagrangian ice microphysics were used to study the early contrail evolution during the vortex phase. Microphysical and geometrical properties of a contrail produced by a large-sized aircraft (type B777/A340) were investigated systematically for a large parameter range. Crystal loss due to adiabatic heating in the downward moving vortices was found to depend strongly on relative humidity and temperature, qualitatively similar to previous 2-D simulation results. Contrail depth is as large as 450 m for the investigated parameter range and was found to be underestimated in a previous 2-D study. Further sensitivity studies show a nonnegligible effect of the initial ice crystal size distribution and the initial ice crystal number on the crystal loss, whereas the contrail structure and ice mass evolution is only barely affected by these variations. Variation of fuel flow has the smallest effect on crystal loss. At high supersaturations, our choice of contrail spatial initialization may underestimate the ice crystal loss. The set of presented sensitivity studies is a first step toward a quantitative description of young contrails in terms of vertical extent and crystal loss. Concluding contrail-to-cirrus simulations demonstrate the relevance of vortex phase processes and its three-dimensional modeling on the later contrail-cirrus properties.
Contrail-cirrus contributes significantly to the aviation climate forcing and probably exerts a stronger radiative forcing than the accumulated aviation CO2 emissions [Burkhardt and Kärcher, 2011]. Aged contrails lose their line shape and start to resemble natural cirrus, such that identifying and observing them in reality is a difficult task. Especially, natural cirrus may, depending on the synoptic evolution, form in the vicinity of contrail-cirrus which complicates the observation of the latter. In regions with heavy air traffic, contrail-cirrus persistence can modify or even suppress natural cirrus formation. Numerical approaches such as cloud resolving models (CRM) or global circulation models (GCM) can, by construction, discriminate between various cloud types and can yield valuable information on the evolution of contrail-cirrus and its competition with natural cirrus. With CRM, contrail-cirrus evolution can be studied in detail and important processes and parameters can be identified with this approach [Jensen et al., 1998; Unterstrasser and Gierens, 2010a]. GCMs, on the other hand, are a suitable tool to obtain global estimates of contrail-cirrus cover and radiative forcing [Burkhardt and Kärcher, 2009].
CRM simulations have shown that wake vortex related processes during the first minutes have the potential to affect the lifetime optical properties of the evolving contrail-cirrus even hours later [Unterstrasser and Gierens, 2010b]. This emphasizes the need for realistic vortex phase simulations.
In this study, the contrail evolution during the first 5 min is studied by means of 3-D large-eddy simulations (LES) and the complex interplay of vortex dynamics and contrail microphysics is examined. The early contrail properties and evolution are strongly affected by meteorological as well as aircraft parameters.
Most of the recent contrail vortex phase studies rely on 3-D models [Lewellen and Lewellen, 2001; Huebsch and Lewellen, 2006; Paugam et al., 2010; Naiman et al., 2011] with a proper representation of vortex decay features (vortex descent, Crow instability, and shortwave instability). One major drawback of the 3-D approach is the high computational requirements which so far admitted only a sparse sampling of the large parameter range. On the other hand, efficient 2-D models with less detailed vortex dynamics were used to explore a larger number of atmospheric conditions [Unterstrasser et al., 2008; Unterstrasser and Sölch, 2010].
The present study combines the advantages of both approaches. Results of nearly 50 high-fidelity 3-D simulations are presented. This is a considerable advancement to previous 3-D studies which tested only a few parameter combinations.
The early contrail evolution depends on meteorological and aircraft parameters which directly affect contrail microphysics and/or the wake vortex decay. In this study, we limit ourselves to parameter variations which have a direct impact on contrail ice microphysics, namely, relative humidity, temperature, and several parameters affecting the initial ice crystal properties.
Parameters that control wake vortex properties like aircraft type, thermal stratification, and vertical wind shear are not varied in this study. In an upcoming study focusing on the impact of aircraft type, variations of these parameters are discussed.
The paper is structured as follows: We begin with a brief description of the basic model in section 2. The various sensitivity studies and results are presented in section 3. The effect of inertia on ice crystal transport is quantified in section 4. Section 5 discusses implications of the early contrail evolution on the contrail-to-cirrus transition. Conclusions are drawn in section 6. A list of symbols is provided in Table 1.
Table 1. List of Symbols
eddy dissipation rate
initial circulation of vortex
8·106 m2 s−1
kinematic viscosity of air
density of air at the lower boundary
mean diameter of initial ice crystals
ice crystal “emission” index
normalized ice crystal number/mass
fraction of surviving ice crystals
width, length, and height of simulation domain
fuel flow rate
number of simulation particles per grid box and in total
number of time steps in simulation
pressure at the lower boundary
core radius of vortex
radius of initial discs with ice crystals
width of lognormal size distribution; see definition in equation (2)
ambient relative humidity with respect to ice
vertical wind shear
temperature at cruise altitude
maximum RMS of additional turbulent fluctuations
(initial) ice crystal mass per meter flight path
(initial) ice crystal number per meter flight path
longitudinal profile of ice crystal number/mass; see equation (6b)
number of surviving ice crystals per meter flight path
transverse profile of ice crystal number/mass; see equation (6c)
vertical profile of ice crystal number/mass; see equation (6a)
2 Model Description and Setup
The simulations have been carried out with the LES code EULAG-LCM. The base model EULAG [Smolarkiewicz and Margolin, 1997; Prusa et al., 2008] solves the anelastic formulation of the momentum and energy equations and employs the positive definite advection algorithm Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) [Smolarkiewicz and Margolin, 1998]. A microphysics module with Lagrangian ice particle tracking has been coupled to EULAG and forms the model version EULAG-LCM [Sölch and Kärcher, 2010]. The new module comprises explicit aerosol and ice microphysics, homogeneous freezing of liquid supercooled aerosol particles, heterogeneous ice nucleation, deposition growth of ice crystals, sedimentation, aggregation, latent heat release, and radiative impact on particle growth. With this complete description of ice microphysical processes accurate simulations of natural cirrus have been performed [Sölch and Kärcher, 2011]. In the present study, deposition and latent heat release are the only microphysical processes switched on. The ice crystals are assumed to have hexagonal shape. By default, LCM considers a Kelvin correction term in the diffusional growth equation. Its significance on crystal loss was shown by Lewellen et al.  and is demonstrated in section 3.4.
The initialization of the flow field is analogous to a recent wake vortex study [Unterstrasser et al., 2014]. In the latter study, wake vortex simulations with dispersing passive tracers were performed with EULAG-LCM.
There, turbulent fields are generated in a priori runs for prescribed stratification and eddy dissipation rate. These are superimposed with a counterrotating vortex pair.
For wake vortex simulations, it has been found useful to employ a variant of a nonlinear predictor scheme within EULAG (for details see Unterstrasser et al. ). This allows to run the model with Courant-Friedrichs-Lewy (CFL) numbers close to 1 which substantially reduces the computational requirements and is one reason why a large number of sensitivity experiments have been performed with a 3-D approach.
The microphysical setup is similar to our previous 2-D contrail simulations with EULAG-LCM [Unterstrasser and Sölch, 2010]. The similarity of the microphysical setup enables us to conclusively compare the 2-D and 3-D approaches. From now on, the preceding 2-D study of Unterstrasser and Sölch  will be abbreviated as “US2010.”
The first 3-D EULAG-LCM contrail simulations of the vortex phase have been presented in Jeßberger et al. . There, observations and simulations of contrails produced by three different aircraft are compared in a case study.
2.1 Simulation Setup
The model domain has dimensions Lx=384 m, Ly=400 m, and Lz=600 m. The transverse direction is along x, the flight direction (longitudinal) along y, and the vertical direction along z. The grid resolution is 1 m along the vertical and transverse direction and 2 m along flight direction. Periodic boundary conditions are applied in the horizontal and rigid boundaries in the vertical. The latter means that all three velocity components are 0 at the top and bottom layers.
The time step Δt is initially 0.03 s and increases to 0.10 s at later stages of the simulation. The simulations were run until tsim≈300 s, and the number of time steps is of the order 104.
A Lamb-Oseen vortex [Lamb, 1879] with core radius rc=4 m and initial circulation Γ0=520 m2 s−1 is prescribed initially at z = 500 m. The vortex separation distance is b0=47.3 m and corresponds to an aircraft with wing span bspan=60 m (type: A340/B777). The domain length Ly is chosen such that the formation of the most unstable Crow mode (∼8.6b0) can be triggered [Crow, 1970]. A further sensitivity simulation with Lz=800 m shows that the default domain height and the distance of the vortices from the top and bottom boundaries is large enough to avoid spurious boundary effects.
The Brunt-Väisälä frequency NBV is 1.15·10−2 s−1, and the eddy dissipation rate of the ambient turbulence field is ε = 10−7 m2 s−3. This represents a very calm atmosphere. For the given strong thermal stratification, however, vortex decay does not depend strongly on the turbulence intensity [e.g., Unterstrasser et al., 2014] and the validity of the simulation results is not confined to these very calm cases. At the lower boundary, the pressure is p0=250 hPa.
The microphysical setup is similar to that of US2010. In short, the humidity field in the domain (also in the plume area) is uniform with constant (apart from small turbulent deviations around ) and T* is the temperature at cruise altitude.
We assume that the initial ice crystals contain exactly the amount of the water vapor emitted due to fuel burning. Sensitivity simulations with increased water vapor mixing ratio and accordingly smaller initial ice mass show that this assumption is uncritical as deposition growth is fast in this regime. The initial ice mass and crystal number per meter of flight path are and . This corresponds to an “emission” index EIiceno=2.8·1014 per kg of burned fuel. The crystals are spatially uniformly distributed inside two circles around the vortex cores with radius Rinit=20 m, which is in line with plume areas obtained by jet/vortex interaction simulations [Paoli et al., 2013].
The initialization properties correspond roughly to a state a few seconds behind the aircraft.
The contrail is located in an otherwise cloud-free environment (i.e., no ice crystals of natural cirrus are present). Analytical estimates indicate that the evolution of young contrails is not affected by surrounding cirrus [Gierens, 2012]. Thus, the presented results also hold for contrails embedded in natural cirrus with in-cloud supersaturation.
Parameters of the reference setup are summarized in Table 2. All baseline simulations are finally characterized by prescribing the temperature T* at cruise altitude and the relative humidity (supersaturation ). The baseline simulation with and T*=217K is referred to as “reference simulation.”
Table 2. Numerical, Atmospheric, Aircraft, and Ice Crystal Parameters of the Baseline and Sensitivity Simulationsa
The meanings of the symbols are listed in Table 1.
Number of simulation particles (SIPs) per grid box; seesection 1
mono or 3
mono or 3
2.2 Ice Crystal Initialization
In a Lagrangian microphysics model, the ice phase is resolved by a large number NSIP of simulations particles (SIPs). Each SIP contains a certain constant number NIC of real ice crystals. In the preceding 2-D study, computer memory issues did not limit the number of SIPs and the prescribed lognormal size distribution was resolved by Np=1000 SIPs per grid box. This fairly large number guaranteed an excellent resolution in spectral space over time and all contrail areas. In the present 3-D study, this large number cannot be attained and we use Np=53 SIPs per grid box which results in 26·106 SIPs across the total simulation domain. Up to recently, all previous 3-D studies [Naiman et al., 2011; Paugam et al., 2010; Huebsch and Lewellen, 2006; Lewellen and Lewellen, 2001] started with a monodisperse size distribution. In particular for Lagrangian microphysics models, this facilitates the initialization procedure since all SIPs can be prescribed with the same NIC and radius/mass. However, starting with a monodisperse size distribution is unrealistic [Kärcher and Yu, 2009] and underestimates the crystal loss during the vortex phase (see sections 3.3 and 4 in US2010 for a detailed discussion). Recently, Lewellen et al.  also varied the initial size distribution and found their simulation results to be independent of it. Their simulations start at an earlier wake though (see section 3.4) indicating that initial size differences wash out quickly.
For the initial ice crystal mass distribution, we prescribe a lognormal distribution with given geometrical width σm and geometric mean mass m0:
The total ice crystal number concentration N is the zeroth moment of the mass distribution. Analogously, the ice mass concentration (IWC) is the first moment. For consistency with our earlier studies we introduce an alternative width definition:
The relationship of the geometric mean mass m0 and the mean mass is then simply
The sensitivity study addressing the impact of the width will be presented in terms of rSD. The technical aspects of the SIP initialization procedure are described in Appendix A of Unterstrasser and Sölch . Appendix A (in this study) examines the sensitivity to Np and proves that the default value for Np is chosen sufficiently large for this type of simulation.
3 LES of Contrail Vortex Phase
First, we shortly discuss the reference simulation and compare it with our earlier 2-D approach. Then, the impact of relative humidity and temperature on the contrail structure and crystal loss is explained in detail. Further sensitivity studies highlight the importance of parameters affecting the microphysical initialization (like the width of the size distribution, the fuel burn, or the ice crystal emission index EIiceno).
Following US2010, we first define a few quantities. The total ice crystal number (per meter of flight path and averaged along flight direction) is
The normalized ice crystal number is given by
The term “fraction of surviving ice crystals” (denoted as ) refers to evaluated at the end of the simulation.
The ice crystal number profiles are computed and scaled as follows:
i.e., the ice crystal distribution is integrated along the transverse and vertical direction and averaged along the flight direction. Corresponding quantities for ice mass like , , or are defined analogously.
3.1 Reference Simulation
We first explain a few basic features of the contrail evolution during the vortex phase by discussing a reference simulation in more detail. We keep this section short since several studies in the past already described the important phenomena [Lewellen and Lewellen, 1996, 2001; Huebsch and Lewellen, 2006]. The parameter setting of the illustrated simulation is summarized in Table 2. The temperature at cruise altitude is T*=217 K, and the ambient relative humidity is . Figure 1 shows ice crystal concentrations within the cross-sectional plane normal to the flight direction. The concentrations are averaged along flight direction. At t = 1 min, basically all ice crystals are still captured in the primary wake which traveled downward around 100m. After 2 min the highest ice crystals concentrations are still found in the core of the primary wake. This clearly depends on the ambient relative humidity and is not necessarily true in drier conditions. Ice crystals keep being detrained from the primary wake due to baroclinic instabilities. This leads to the formation of a curtain that eventually reaches the initial formation altitude of the contrail. At that stage, the contrail is already more than 300m deep. The primary wake broadens due to an oscillation of the vortex tubes (the well-known Crow instability [Crow, 1970]) and the formation of vortex rings [e.g., Misaka et al., 2012]. After 5 min the contrail is more than 450m deep and nearly 300m broad, at least in this perspective of an average along flight direction.
To emphasize the importance of microphysics and its strong dependence on dynamics, Figure 1 (fifth panel) shows the contrail structure after 5 min from a simulation with . Despite evolving in a supersaturated environment, adiabatic heating in the downward sinking vortex pair leads to a substantial loss of ice crystals. Eventually all ice crystals are completely sublimated in the primary wake. Only detrained ice crystals in the curtain survive and the resulting contrail is narrower and shallower than that of the case.
Figure 2 shows vertical profiles of ice crystal number for the (3-D) reference simulation and the corresponding simulation with the former 2-D approach of US2010. Both simulations use T*=217 K and . The 2-D approach employed a module that was based on 3-D vortex phase simulation results and monitored and interactively adapted the vortex decay and descent in order to obtain a more realistic behavior [Unterstrasser et al., 2008]. Since the 2-D simulation of the vortex phase stopped after tbreakup=140 s, the 5min profile stems from a subsequent contrail-to-cirrus simulation whose setup is similar to the one in Unterstrasser and Gierens [2010a]. In the 2-D model, the vortex lifetime tbreakup was prescribed based on a parameterization by Holzäpfel . Recently, an improved vortex tracking algorithm [Hennemann and Holzäpfel, 2011] revealed that the former study underestimated the vortex lifetime. Consistent with these new findings, our 3-D simulations also show coherent vortex structures over a longer period. The 3-D simulation stops after around 5 min, at a time the effect of aircraft-induced dynamics became small.
After 1 min the two vertical profiles in Figure 2 look similar. The vertical displacement of the primary wake agrees well between the two models. After 5 min qualitative differences are obvious. In the 3-D approach, the vortices traveled farther down and after their breakup much ice is eventually transported upward again. Whereas in the 2-D model the whole plume rises and remains compact, in the 3-D approach the detrainment along flight direction is variable and linked to the well-known 3-D dynamical phenomena. In some segments, the air parcels bounce back to cruise altitude; in other segments the ice crystals linger around the altitude of vortex breakup. Eventually, the “3-D model” contrail is vertically more extended than the “2-D model” contrail. In section 5, we examine whether these model-induced geometrical differences are significant for the later contrail evolution. In particular, the effects of shear-driven contrail spreading may differ.
3.2 Impact of Relative Humidity and Temperature
This section examines the importance of relative humidity and temperature on contrail structure and microphysics. The simulations cover scenarios at saturation with short-living contrails and high supersaturation just below the homogeneous freezing threshold . The lowest temperature T*=209 K is encountered at the highest flight levels. The warmest case (T*=225 K) corresponds to the highest possible temperatures which still allow for contrail formation following the Schmidt-Appleman criterion [Schumann, 1996]. In our basic parametric study, we neglect the fact that fewer ice crystals form in scenarios close to the threshold temperature TSA [Kärcher et al., 1998] and initialize all baseline simulations with the same . Scenarios with fewer initial ice crystals, which are treated within the EIiceno sensitivity study in section 3.3 (though at T*=217 K and not T*=225 K), give an impression on what to expect close to TSA.
3.2.1 Contrail Microphysics
Figure 3 summarizes the temporal evolution of the fractions of ice mass and crystal number for various values and fixed temperature T*=217 K. The description of the and evolution is short as it is qualitatively similar to the results of the US2010 study (see US2010, section 3.2). During the first seconds the ice mass increases until the excess water vapor fully deposited on the ice crystals. The initial ice mass add-on scales linearly with supersaturation . Subsequently, the ice mass in the primary wake starts to decrease as adiabatic heating causes a sustained slight local subsaturation [see, e.g., Naiman et al., 2011, Figure 9]. Detrained ice crystals forming the secondary wake, on the other hand, face fresh supersaturated air and grow. This ice mass increase partly balances and later largely overcompensates the ice mass loss in the primary wake, such that the total ice mass strongly increases (as depicted).
Once the ice mass in the primary wake decreases, the “primary” ice crystals shrink. After some temporal offset, the first ice crystals are small enough (ice crystal size Lcrit≈0.1μm) to be regarded as completely sublimated. The larger the initial ice mass add-on is, the longer it takes until ice crystal loss sets in. After around 3 to 4 min the vortex system breaks down and its descent stops. Then, the ice crystal loss basically ceases and reaches a quasi steady state. Eventually, between 3% and 93% of the initial ice crystals survive the vortex phase. At later stages (not covered by these simulations), contrail ice crystals in supersaturated environments can sublimate due to buoyant sloshing [Huebsch and Lewellen, 2006] or spectral ripening [Lewellen, 2012], however usually at a lower rate than observed during vortex descent.
Figure 4 shows the surviving ice crystal number fraction as a function of relative humidity and temperature (as denoted in the legend). Each symbol depicts for a specific combination of and T*. The diamonds show 3-D simulation results and the plus signs previous 2-D simulation results of US2010. Overall, this plot contains results of 18 3-D and 28 2-D simulations which gives consideration to the strong impact of the meteorological background.
The and T* dependence is qualitatively similar in the 3-D and the previous 2-D results. Again, we see a strong increase of with and higher survival fractions in colder environments. For the moistest cases considered, more than 90% of the ice crystals survive. In a warm (T≥217 K) and saturated atmosphere, barely any crystals survive. The sensitivity to temperature appears to be weaker in the 3-D model. This may be attributed to the complex interplay of dynamics and microphysics and is not further elaborated here.
3.2.2 Contrail Structure
Next, we discuss the contrail structure and dimensions at the end of the vortex phase. Aircraft parameters (wing span, aircraft mass, and cruise speed) and meteorological parameters (thermal stratification, turbulence intensity, and vertical wind shear) affect the vortex initialization and evolution and thus they control, inter alia, the (final) vertical displacement of the primary wake, and the vertical distribution of emitted (passive) tracers in an aircraft plume [Unterstrasser et al., 2014]. Previous contrail 3-D studies used setups with various vortex initialization and evolutions and examined their impact on contrail (vertical) structure. Huebsch and Lewellen  and Naiman et al. [2011, Figures 10–12] vary the thermal stratification, the aircraft type, and the vertical wind shear. Variations of the two latter parameters were already examined in Lewellen and Lewellen . Additionally, the turbulence intensity was varied in Paugam et al.  and Huebsch and Lewellen . The trends in contrail depth for each parameter variation are consistent over all mentioned studies. So, the basic dynamic mechanisms are well understood.
Additional to dynamical effects investigated in the latter studies, microphysical processes additionally affect the contrail structure, especially driven by adiabatic heating in the downward sinking vortex system. The example in Figure 1 already highlighted the importance of ambient relative humidity on the crystal sublimation which can result in a crystal-free primary wake. The sensitivity of contrail structure to relative humidity will be analyzed in more detail. For this, we discuss profiles of ice mass and crystal number along each coordinate axis for relative humidity ranging from 100% to 140% and fixed temperature T*=217 K (Figures 5-7). The displayed quantities are defined in equations (6a)–(6c).
Before starting the actual discussion, we remark that very few ice crystals are lost in the case (see Figures 5, right, and 3, right) and ice crystal number concentration can then be regarded as a passive tracer concentration. Thus, the displayed ice crystal number profiles of the case highlight the contrail structure solely affected by dynamic processes. The microphysical evolution (via latent heat release) has a negligible feedback on the dynamics, and the wake vortex evolution is thus practically identical for all displayed cases. Thus, observed differences between the case and the drier cases can then be attributed to microphysical processes (i.e., complete sublimation of ice crystals).
First, the vertical profiles shown in Figure 5 are discussed. In a saturated environment very few ice crystals survive and the contrail is around 150m deep with much fewer ice crystals present than in the beginning. Once the relative humidity exceeds 120%, the contrail reaches its full vertical extent of around 450m. Whereas the ice crystal number around cruise altitude is unaffected by a variation (between 120% and 140%), the more ice crystals survive in the primary wake, the moister it is. The ice mass scales more or less linearly with supersaturation . This linear scaling demonstrates that the evolution of ice mass has a short memory. The ice mass which was lost during the adiabatic heating stage redeposits on the ice crystals once the air parcels rise again. Also, fresh supersaturated air is entrained once the vortices are weaker and the plume dilutes.
In a weakly supersaturated environment , crystal loss is strong such that barely any ice crystals survive in the primary wake. Accordingly, the ice mass has only one peak at cruise altitude unlike to cases with a second peak farther down. For the cases, the crystal loss has irreversible effects on the ice mass distribution. No linear scaling of ice mass is valid any longer in this low-supersaturation regime.
Although relative humidity was varied in several of the studies mentioned above, only data presented in Huebsch and Lewellen  allow to reconstruct the contrail depth for similar aircraft type and thermal stratification and different values. The green and brown bars in Figure 5 (middle) show their contrail depth values for and 130%, respectively. Similarly to our model, the contrail depth depends strongly on .
A more conclusive model comparison enhancing the confidence in the wake vortex modeling capabilities was performed between EULAG and NTMIX in Unterstrasser et al.  and gave very good qualitative agreement in terms of vertical plume extent [see Unterstrasser et al., 2014, Figure 15).
Figure 5 (right) shows the number of ice crystals that sublimate at a given height. To avoid misunderstandings, we clarify that this is not equivalent to the position of the remaining soot cores as these are further advected after being released from the sublimated ice crystals. nicely illustrates where/when sublimation starts. Initially, the ice crystals grow by taking up the excess water vapor from the environment. Once the supersaturation is consumed after a few tens of seconds, adiabatic heating leads to a shrinking of the crystals. The moister it is and the larger the ice crystals initially grow, the longer it takes until the ice crystals are completely sublimated. For a specific ambient humidity, the layer where ice crystals get lost is rather large. This reflects the fact that the initial size distribution is not monodisperse.
Along the transverse direction (see Figure 6), the ice mass and crystal number distribution resemble Gaussian distributions (at least when averaged along the flight direction and in the absence of a sheared cross wind). Most ice crystals are contained in a 150m broad area. For , ice crystals survive in the meandering vortex tubes (which are present at the final stages of the vortex phase). Then, the contrail is up to 300m broad.
Figure 7 displays the ice mass and crystal number distribution along the flight direction. Although the atmospheric and microphysical initialization (see black lines) is homogeneous along the flight direction, turbulence and the Crow instability lead to ice crystal accumulation in certain segments. These variations can be attributed to dynamical processes [e.g., Crow, 1970; Misaka et al., 2012; Unterstrasser et al., 2014]. Here this is especially apparent for cases with , when ice crystals survive in the primary wake. In drier cases, mostly detrained ice crystals survive which form a curtain above the vortices. In this case, the distribution is smoother along flight direction. This is also consistent with findings of Paugam et al. [2010, Figure 11].
3.3 Sensitivity to Microphysical Initialization
In this section, we vary parameters which affect the properties of the initial ice crystal size distribution (SD). These are the width rSD of the SD (see definition of rSD in equation (2)), the fuel flow rate , and the ice crystal emission index EIiceno.
As part of the microphysical initialization we assume that the ice crystals contain exactly the amount of water vapor emitted from the engines and their number scales with EIiceno:
with water vapor emission index EIH20=1.25 kg/kg and aircraft speed U.
Each parameter affects the initial SD in a different way, which is schematically depicted in Figure 8 and summarized in Table 3. If the fuel flow rate is varied, the initial ice mass and crystal number change, while the mean and modal mass stay constant. If the ice crystal emission index EIiceno is varied, the initial ice mass stays constant and ice crystal number and consequently the mean mass change. If the width rSD is varied, the ice mass as well as ice crystal number stay constant (implying an unchanged mean mass). Here only the modal mass m0 changes (among the discussed quantities).
Table 3. Effect of Various Parameter Variations on the Moments of the Initial Size Distributiona
“Y” means that the quantity is affected by the specific parameter variation. The moment of kth order is denoted as μk. The width parameter r is defined in equation (2) and depends on μ2.
Ice crystal number
0th moment μ0
1st moment μ1
Mean mass ,
Modal mass m0,
median mass mmed
All three panels of Figure 9 show the temporal evolution of ice crystal number fraction . For all three parameter variations, the impact on the normalized ice mass evolution is small and thus not shown.
We first discuss the variation of width rSD which is a repetition of a sensitivity study performed within US2010. Figure 9 shows for the 3-D and 2-D approaches. Both models show a similar behavior, and a nonnegligible impact of the width parameter is confirmed. Nevertheless, the contrail structure is only slightly affected by a rSD variation. The ice mass profiles look similar (Figure 10, left). In any case the contrail is around 450m deep. However, fewer ice crystals are present, especially in the lower contrail parts (Figure 10, middle). Figure 10 (right) nicely shows how the ice crystal loss is affected. A broad initial SD contains smaller crystals than the monodisperse SD. Consequently, ice crystals are lost earlier, i.e., farther up, the broader the SD is. Moreover, the layer where ice crystals are lost is deeper for a broader initial SD. The lower end of the sublimation layer corresponds to maximum vertical displacement of the vortex system and is similar for all cases.
The early size distribution (a few seconds behind the aircraft) can only hardly be measured, not to say, that it is impossible (see discussion on this topic in US2010). As a consequence for the time being, the lack of precise initial data evokes an irreducible uncertainty in estimating the extent of crystal loss. Note that Lewellen et al.  reports on an insensitivity to the shape of ice crystal size distribution rendering this problem not as severe. The type of engine or operational mode of the engines affect their efficiency and thus the fuel burn. Here we vary the fuel flow rate moderately by ±20%. The corresponding values of and are listed in Table 2. Not surprisingly, a larger fraction of ice crystals survives, if more water vapor is emitted initially. For RHi=120%, the difference in the sublimation extent is almost negligible. The final (unnormalized) ice crystal number is more strongly controlled by the initial value than by the extent of the crystal loss during the vortex phase. For a lower relative humidity the initial ice mass uptake from the ambient water vapor is smaller and the amount of emitted water vapor becomes more important. Consequently, changes in have a more substantial effect on . The contrail vertical structure is qualitatively unaffected for both RHi values (not shown).
Finally, the emission index EIiceno is varied. EIiceno is difficult to determine [Schumann et al., 2013] as it depends on the uncertain soot emission index and the fraction of activated soot cores. Likely, EIiceno is more variable in reality than, e.g., the fuel flow rate and is thus varied over a larger range. takes values between 6.8·1011 m−1 and 1.7·1013 m−1 (variations by a factor of 2 and 5 from the default value). The higher is, the smaller is the mean mass of the ice crystals. Thus, a larger fraction of them sublimates during the vortex phase agreeing with the qualitative description of this sensitivity variation in Huebsch and Lewellen . As already found in US2010, the effect of a EIiceno variation can be described by a power law relationship. The (unnormalized) number of surviving ice crystal can be approximated by
The exponent α depends on the background humidity and ranges from 0.67 (RHi=110%) to 0.79 (RHi=130%) (see Figure 11). The ratio describes the variability in the ice crystal number prior to the vortex phase, whereas describes the variability in the ice crystal number after the vortex phase. For fixed ambient conditions, the vortex phase processes reduce the variability in the ice number generated by different fuel properties and various jet phase processes (different nucleation, fuel sulfur content, emission indices) as already stated in US2010.
3.4 Sensitivity to Initial Spatial Ice Distribution
The initial placement of the ice crystals is rather simple in our model. We prescribe uniform ice crystal concentrations in two Rinit=20 m circles, each centered around the vortex core. At the start of our simulation, the entrainment of jet exhaust into the fully rolled-up vortices is thus considered to be completed. Unterstrasser et al.  examined the importance of the spatial distribution by varying the radius Rinit. Besides these simple configurations, a more realistic initialization based on simulation results of Paoli et al.  was used. Paoli et al.  examined the entrainment of the jets by the trailing vortices and its consequences on contrail formation and early evolution out to 20s. The sensitivity analysis in Unterstrasser et al.  showed that the detrainment of a passive tracer during the vortex phase and its vertical distribution after vortex breakup is only moderately dependent on the spatial initialization. Moreover, the simulations by Paoli et al.  showed that our assumption of Rinit=20 m is sound (though their distribution is less homogeneous than ours) [see also Unterstrasser et al., 2014, Figure 11].
The approach in Huebsch and Lewellen , Naiman et al. , Lewellen et al. , and Lewellen  is different to the one here and in Unterstrasser et al. . They also assume a fully rolled-up vortex pair. However, their simulations start at an earlier wake age, where the exhaust jet has not yet been entrained into the vortex system. The exhaust is still more concentrated and not centered around vortex cores (see their Figure 1, respectively).
For comparison, we reproduce the Naiman et al.  initialization of their “Large 4-Engine” aircraft. We keep our vortex parameters b0,Γ0, and rc unchanged. However, the position of the four jets and the Gaussian-like ice crystal number concentration distributions are adopted from values rj, Δxj, and Δyj taken from their Table 4. The number of SIPs per grid box is variable and higher closer to the peak concentration. Overall, 116·106 SIPs are used.
Figure 12 shows averaged ice crystal number concentrations in the plane perpendicular to flight direction. Thirty seconds after the simulations start, effects of the different initializations are still apparent. The baseline simulation (labeled “Uniform”) shows a rather uniform concentration field with some fluctuations. These seem to be a numerical artifact that appears when ice crystals are initialized in the very first time step. In a test simulation with the ice crystals initialized after a few time steps, an initial spurious SIP clustering is suppressed. Then the concentration field remains extremely uniform and slightly fewer ice crystals get lost (the deviation from the baseline simulation is, however, much smaller than that of the GAUSSIAN simulation shown in Figure 13). In the sensitivity run (“Gaussian”), the entrainment of the exhaust into the vortex oval is just completed. Most notably, fewer ice crystals are present close to the vortex core as mixing into this region is inhibited. A closer look at the entrainment process of this simulation is given in Unterstrasser et al. . After 1.5 min the detrainment of the ice crystals and plume shape appears similar. Higher concentrations around the vortex cores are still present in the Uniform simulation. Figure 13 shows that crystal loss starts earlier in the Gaussian simulations (dotted lines). Eventually, 40% of the ice crystals survive compared to ∼50% for the Uniform simulation for RHi=120% (black lines). We speculate that the earlier onset of crystal loss is due to a stronger competition for available moisture in the areas with high number concentrations. Quantitatively, the differences are even larger for the moister case (RHi=140%, red lines). For the Uniform simulation, barely any ice crystals get lost . For the Gaussian case, the crystal loss is more substantial, as only 70% of the ice crystals survive. Furthermore, this figure demonstrates that the Kelvin curvature effect substantially enhances crystal loss. The Kelvin effect leads to a spectral ripening in ice clouds [Lewellen, 2012], and Lewellen et al.  first quantified its effect on crystal loss in young contrails and contrail-cirrus. The size distribution (SD) broadens, as small ice crystals grow slower/sublimate faster than the larger ones (around saturation, larger ice crystal grow at the expense of sublimating smaller ones). Once this Kelvin correction is switched off, the left tail of the SD shapes up more slowly and the onset of crystal loss is retarded. In this specific case, the extent of ice crystal loss is then only two thirds of the original case ( versus ∼55%). The enhancement is similar for the two kinds of initial spatial distributions.
3.5 Comparison With Recent LES Studies
In this section, we focus on comparing the extent of crystal loss simulated with various models. We consider results of Huebsch and Lewellen [2006, hereinafter H06] and Naiman et al. [2011, hereinafter N11]. We omit a comparison of contrail structure which was already presented in subsection 3.2.2. Qualitatively, the trends are similar for most cases. H06 and N11 similarly get a reduced crystal loss for smaller aircraft, a moderate sensitivity to thermal stratification, and a weak sensitivity to vertical wind shear. This study and H06 find a strong sensitivity to RHi and a moderate sensitivity to the initial placement of the ice crystals. Moreover, the impact of an EIiceno variation seems similar, although no quantitative data are provided in H06.
N11 disagrees with all other previous studies in terms of a low to zero RHi sensitivity and a high EIiceno sensitivity. There, the temporal ice crystal number evolutions are barely affected by ambient relative humidity. On the other hand, a decrease in EIiceno by 1 order of magnitude basically inhibits crystal loss. Unfortunately, the authors neglect to explain these unprecedented results. Having in mind the RHi-dependent onset of crystal loss (see Figures 5 and 3) and theoretical considerations [see Unterstrasser et al., 2008, section 5], the low RHi sensitivity in N11 appears implausible. N11 speculate that they used too few Lagrangian particles which may lead to the observed discrepancies. Our NSIP variation experiments (see Appendix A) indicate numerical convergence for total SIP numbers as low as those used in N11. Thus, a plausible interpretation of the N11 results is still pending.
Quantitative comparisons between the models in general remain inconclusive since parameter variations start from different base states (different aircraft and temperature among many others parameters).
In situ measurements and EULAG-LCM simulations of ∼100s old contrails are compared in Jeßberger et al. , especially focusing on the aircraft impact on contrail properties.
4 Effect of Inertia on Crystal Transport
So far, possible inertial effects of the ice crystals have not been considered in our model. If we assume that the particles follow the flow field uf, then their particle position xp is governed by
This is how advection is currently treated within the EULAG-LCM model. Since the fluid stream lines are strongly curved around the vortex core, inertial effects (i.e., centrifugal forces) may affect the trajectories of the ice crystals. This effect may be responsible for the formation of hollow contrails which are sometimes observed (see Figure 14, left). The scientific community still lacks an explanation on which physical processes cause this ice crystal-free inner contrail region. Inertial effects are one possible explanation.
Considering inertia, the particle velocity up is governed by
where uf is the fluid velocity and D is the drag force. The mass, cross section, and diameter of the particles are denoted as mp,Ap an Dp, respectively. The drag force can be approximated by
see Gerthsen et al. . The drag coefficient cd is a function of the particle Reynolds number with kinematic viscosity of air ν [Seinfeld and Pandis, 1998]. The particle position xp is governed by
Equation (10) is a nonlinear differential equation which cannot be solved analytically.
Since its numerical integration requires tiny time steps, an inclusion into the LES model is not straightforward and intended here. Thus, we test the importance of inertial effects in a simple “offline” study.
On a 2-D plane, the fluid velocity uf is prescribed by an analytical Lamb-Oseen profile with time-constant circulation Γ and core radius rc. Two-dimensional versions of equations (10) and (11) are solved numerically by employing a Euler forward scheme with a sufficiently small time step 10−5 s. We assume spherical ice crystals with various diameters Dp and constant density ρice=920 kgm−3 which are initialized at various initial distances d0 from the vortex center. Initially, the particle velocity is set to the fluid velocity. The computation is carried out until t = 100 s. Figure 14 shows the distance d of the ice crystal from the vortex center as a function of time. The outward drift of the ice crystals is larger, the heavier they are and the smaller d0 is. Ice crystals close to the vortex core (d0≤4 m) are finally more than 5m away from the vortex center. Large ice crystals (Dp≥8μm) accumulate at distances larger than 10m. The solid curves labeled with Dp=0μm consider massless particles where the original equation (9) without inertial effects is integrated in time. These particles do not spiral out and prove the appropriateness of the numerical scheme.
Thus, inertia may cause a slight crystal separation by size in the radial direction. However, it must be noted that for common atmospheric conditions and aircraft the ice crystals usually remain smaller than 8μm inside the primary wake [Unterstrasser and Sölch, 2010] and considerable inertial effects seem unlikely. Especially, the contrail displayed in Figure 14 (left) is faint and evolves in slightly subsaturated air where the bulk of ice crystals is even smaller.
To our knowledge, Naiman et al.  were the first ones to study inertial effects in contrails. They found similar results. Nevertheless, their approach has flaws. They directly include the integration of the modified advection equation (equations (10) and (11)) into the LES system. By doing so, they keep the drag coefficient constant over the whole LES time step. This is not reasonable and clearly overestimates the pushing back of the particle on the fluid streamline. Moreover, a higher-order multistage Runge-Kutta (RK) integration scheme, as they state to employ for the standard advection equation (equation (9)), cannot be applied to the modified equations. RK schemes use evaluations of the right-hand side at intermediate stages in time and space. Contrary to the fluid velocity uf=uf(x,t), up is defined for the particle and not in space. Thus, RK schemes are ill defined for such a problem. Resorting to a first-order single-stage scheme (Euler forward), on the other hand, requires the use of tiny time steps, clearly smaller than typical LES time steps.
5 Contrail-to-Cirrus Transition
This section presents contrail-cirrus simulations which aim at illustrating the importance of the vortex phase processes. The design of the presented 2-D contrail-cirrus simulations with EULAG-LCM is similar to that in Unterstrasser and Gierens [2010a]. Compared to the vortex phase simulations, the 2-D contrail-cirrus model domain is larger. In a first subsimulation over 2000s, the width Lx is ∼6km, the height Lz is 1km, and mesh sizes are 5m in both directions. In a subsequent subsimulation up to 4 h, the model domain is again enlarged.
For illustration, Figure 15 (left) shows the model domain of the first subsimulation. The data of the vortex phase simulation are incorporated inside the smaller black box. We prescribe (over a 1 km thick layer), T*=217 K, and vertical wind shear s = 2·10−3s−1. The vortex phase simulation with and T*=217 K is used for initialization. The 3-D fields (e.g., perturbations of qv and θ) of the vortex phase simulation are averaged along flight direction, interpolated on the coarser grid, and embedded into the enlarged 2-D model domain. The Lagrangian ice crystal data are unchanged except for a shift in position (the coordinate y along flight direction is neglected). Similarly, we can use the data of 2-D vortex phase simulations (in this case the averaging along flight direction is not necessary).
For this specific combination the differences between the 2-D and 3-D simulation of the vortex phase were discussed in section 3.1. The qualitatively different vertical distribution of ice crystal number can be revisited in Figure 2. The crystal loss, however, is similar. Sixty-five percent (3-D) and 50% (2-D) of the initial ice crystals respectively survive. Figure 15 (right) shows the total extinction of the evolving contrail-cirrus over 4 h. The total extinction measures the disturbance of the shortwave flux and basically represents the product of a characteristic optical thickness and width (for a definition, see Unterstrasser and Gierens [2010a, equation 12]). The two blue curves show the cases for with 2-D initialization (dotted line) and 3-D initialization (solid line). The “3-D-init” contrail is deeper in the beginning. Thus, the horizontal spreading due to vertical wind shear is larger, the cross-sectional area increases faster, more water vapor is depleted, and the ice mass and total extinction are larger than for the “2-D-init” contrail. It is well known that in situations with no vertical updraft sedimentation losses can at some point no longer be balanced by entrainment of fresh supersaturated air and deposition of the excess water vapor. Thus, total extinction starts to decline, in this case after 3 h. This reversal occurs earlier in the 3-D-init case, since the crystals are larger on average.
Whereas the above example outlined the importance of the numerical design of the vortex phase modeling, we now will illustrate the significance of the meteorological conditions during the vortex phase. For this, we use vortex phase simulations for three values of , namely, 100%, 110%, and 120%. The vertical distribution of ice crystal mass and number was shown in Figure 5. The vertical extent of the 5 min old contrails is roughly 100m, 300m, and 450m, respectively. About 3%, 30%, and 65% of the initial ice crystals survive the vortex phase. Again, we prescribe , T*=217 K, and vertical wind shear s = 2·10−3s−1 in the large model domain and embed the data of the vortex phase simulation. Since water vapor perturbations are added to background state, it is by construction guaranteed that the total water (water vapor plus ice mass) is equal over all simulations. So the differences in the evolving contrail-cirrus can be attributed to the initial differences in cross-sectional area and ice crystal number as there is no water vapor deficit inside the black box (see Figure 15, left). Figure 15 (right) again shows the total extinction of the evolving contrail-cirrus over 4 h. Again, we find differences in the contrail-cirrus for the same reasons as above. The larger the contrail is and the more ice crystals survive the vortex phase, the stronger is the spreading of the contrail-cirrus. The meteorology-induced differences are even larger than those induced by the numerical design.
This experiment is simple for several reasons. During contrail-cirrus evolution the ambient conditions do not vary over time (no vertical updraft is considered) and only one meteorological configuration is tested. Moreover, for the initializations with the relative humidity is instantaneously raised to the new value . In reality, the relative humidity increase would take some time. Nevertheless, this experiment shows that the meteorological conditions at the time of contrail formation can leave a long-lasting mark.
Three-dimensional large-eddy simulations (LES) were used to study the contrail evolution during the vortex phase. The employed flow solver EULAG [Smolarkiewicz and Margolin, 1997] is suited to study wake vortex dynamics [Unterstrasser et al., 2014]. The Lagrangian ice microphysics module LCM [Sölch and Kärcher, 2010] supports size-resolved microphysics and plausibly captures the sublimation effect in the downward moving vortices [Unterstrasser and Sölch, 2010]. The purpose of the present study was to reduce uncertainties in the quantitative description of young contrail properties inherent to the former two-dimensional approach. So far, only computationally less expensive 2-D simulation approaches have been employed to systematically study some portions of the large parameter space, however at the expense of a proper wake vortex treatment. Here a systematic exploration of the parameter space was started to be performed with a 3-D model. Overall, nearly 50 simulations are discussed which is a considerable quantitative advancement over previous contrail 3-D studies. The fundamental environmental parameters relative humidity and temperature were varied over a large range (RHi=100%–140% and T = 209 K to 225K). The strong dependence of crystal loss on relative humidity and temperature found with the 2-D approach was confirmed qualitatively. The sensitivity to temperature appears to be weaker in the 3-D model.
Further parameter studies varied the initial ice crystal properties. The narrower the width of the initial ice crystal size distribution is assumed, the more ice crystals survive. Thus, previous simulation studies starting with monodisperse SDs may have underestimated the crystal loss. The higher the ice crystal “emission” index EIiceno is, the smaller the initial ice crystals and the more susceptible to sublimation they are. A power law function fα describes the relationship between the initial and surviving number of ice crystals. The exponent α < 1 implies that for fixed ambient conditions the variability in ice crystal number (due to variations of EIsoot or fuel sulfur content) is reduced during the vortex phase. Moreover, a variation of the fuel flow was found to have the smallest effect on crystal loss. At high supersaturations, our choice of contrail spatial initialization may underestimate the ice crystal loss.
Concluding contrail-to-cirrus simulations demonstrate the relevance of vortex phase processes and its three-dimensional modeling on the later contrail-cirrus properties.
In the near future, the model setup will be used to investigate the variation of aircraft type and thermal stratification which both have a strong impact on the wake vortex evolution and indirectly affect contrail microphysics. This is a logical step toward a more complete quantitative description of young contrails properties.
Appendix A: Sensitivity to Simulation Particle Number
As a default the simulations start with 53 SIPs per grid box. Additional runs with 147 and 450 SIPs per grid box are performed for the default width rSD=3. The sensitivity runs gave practically identical results and confirm the convergence of the Lagrangian approach for this model application (not shown).
The SIP initialization with a monodisperse SD is straightforward and was applied in previous 3-D model studies. Without the need of initially resolving a distribution of ice crystal sizes, the number of SIPs per gridbox is usually chosen smaller. However, it is not clear whether the emergence of microphysical variability is inhibited as the SD broadens in the course of a simulation. Thus, we perform a further simulation series where the monodisperse SD is resolved by 147, 53, or 14 SIPs per grid box, respectively. The lowest value is only reasonable when starting with a monodisperse SD and gives a total SIP number of 8 million which matches the value used by Naiman et al. . Again, we find practically identical results (not shown). These findings are consistent with Unterstrasser and Sölch  where sensitivity analyses showed that deposition (and also sedimentation) is well resolved with rather few SIPs.
The author is partly funded by the DFG (German Science Foundation, contract UN286/1-1). I thank K. Gierens and N. Görsch for comments, I. Sölch for assistance with EULAG-LCM, and T. Jurkat for discussions. Computational resources were made available by the German Climate Computing Center (DKRZ) through support from the German Federal Ministry of Education and Research (BMBF). This work contributes to the DLR projects CATS and WeCare. The constructive review by D. Lewellen is highly appreciated.