a. Dynamical time scale

Figure 2 provides temperature histories of air parcels flowing over an idealized (Joukowski) wing of a wide-body aircraft in flight for various distances Δy above the wing surface at the leading edge, as calculated in Gierens et al. (2009). For the selected trajectory closest to the wing (Δy = 3.5 m), T decreases by ΔT ≃ 15 K below the ambient value T₀ = 220 K. The subscript 0 denotes ambient values in the surroundings of the aircraft. This roughly corresponds to the relative temperature change ΔT/T₀ = (γ − 1)/γ(Δp/p₀) estimated with the adiabatic index γ = 7/5 and a pressure difference Δp ≃ −50 mb balanced by the weight of an aircraft cruising at p₀ = 250 mb (i.e., ΔT = −2T₀/35 ≃ −13 K). Boundary layer effects are confined to a very narrow layer above the wing (Gierens et al. 2009) and are not considered here.

The ratio of wing depth (Δx ≃ 10 m) to speed (U₀ ≃ 250 m s⁻¹) of an aircraft provides a dynamical time scale (τ_d = Δx/U₀ ≃ 40 ms) associated with the formation phase of aerodynamic contrails. According to Fig. 1, these appear very close to the wing, so all microphysical processes responsible for their formation must evolve within τ_d (dynamical constraint). Typical cooling rates are on the order of ΔT/τ_d ≃ −0.5 K m s⁻¹, similar to those found in jet contrails (Kärcher and Fabian 1994). Contrary to the steady decrease of T in jet contrails caused by isobaric mixing, a warming phase follows after cooling when the accelerated airflow slows its speed and approaches U₀ upon passing the trailing edge of the wing.

In Fig. 2, we also show the temporal development of the passive (without condensation or freezing) ice saturation ratio S_i = nkT₀/e_i = e_i(T₀)/e_i(T). Here, n denotes the number concentration of H₂O molecules, k = 1.38 × 10⁻¹⁶ erg (g K)⁻¹ is the Boltzmann constant, and e_i is the saturation vapor pressure over ice, e_i = 3.445 × 10¹⁰ exp(−θ/T) mb, where T is in K and with θ = 6132.9 K (Marti and Mauersberger 1993). The resulting high supersaturation triggers subsequent condensation and ice formation. For streamlines closer to the wing surface (but still above the turbulent boundary layer), saturation ratios reach values of ∼40, as shown later in this study.

From classical nucleation theory and a more detailed molecular model, the range of S_i values at which ice nucleates homogeneously from the vapor phase is 50–90 (Pruppacher and Klett 1997). The maximum S_i values are still below this range, underscoring the key role of aerosols in generating the ice phase in aerodynamic contrails.

b. Microphysical time scales

An analysis of microphysical time scales relevant to aerodynamic contrail formation allows their basic properties to be estimated. We assume p₀ = 250 mb unless otherwise noted, corresponding to a typical cruise altitude. For convenience, we introduce a scaling factor κ = p₀/p to track the dependences of our results on the ambient air pressure p. Cruising aircraft fly in a relatively narrow pressure regime (within 150–300 mb).

1) Uptake of H₂O in liquid aerosol particles

Particles containing supercooled aqueous solutions of sulfuric acid (H₂SO₄) and other components are suspended in ambient air flowing over aircraft wings. The mass fraction ω of water in an aerosol particle containing N H₂O molecules with mass m is ω = mN(mN + M)⁻¹, where M is the total mass of water-soluble substances. Over the very short time scale τ_d, only water condensation occurs (M = const.) because H₂O molecules are many orders of magnitude more abundant than any other condensable solute. The rate of change of ω in a liquid particle of radius r is then given by

View Expanded

where υ ≃ 3 × 10⁻²³ cm³ is the volume of an H₂O molecule in solution. The H₂O uptake time scale τ_u follows from τ_u⁻¹ = d ln(ω)/dt, the characteristic time for changes in particle composition due to water uptake (dilution). It can also be interpreted as the time needed to approach local H₂O equilibrium between the gas phase and the liquid solution particle.

The number of H₂O molecules taken up in a cooling event per unit time in a liquid particle is determined by the diffusional flux (Pruppacher and Klett 1997):

View Expanded

where e is the water saturation vapor pressure over the solution droplet, β ≤ 1 is a factor correcting the flux for gas kinetic effects, u ∝ T is the mean thermal speed, D ∝ κT ² is the gas diffusion coefficient of H₂O molecules, and α ≃ 1 is their accommodation coefficient at the particle surface (Clement et al. 1996). The scaling of D with T is approximate. The characteristic radius r_c separates kinetic and diffusive uptake regimes. Values of r_c ∝ κT ^3/2 range between 0.55–0.75 μm. Underlying Eq. (2) is the assumption of a steady-state gas diffusion profile. This assumption is safe because steady-state conditions establish on a time scale of ∼ r ²/D, which is very much faster than any of the other time scales discussed in this work.

Recalling the discussion of Fig. 2 and assuming ambient ice saturation, we set e(T) ≪ nkT ≃ e_i(T₀) in Eq. (2). We approximate the water mass fraction at ice saturation by ω ≈ 0.65, a value typical for hygroscopic aerosols such as aqueous H₂SO₄ in the T range considered here. Combining Eqs. (1) and (2) yields

View Expanded

giving τ_u in units of seconds for T₀ in K and r₀ in μm. For κ = 1 and r₀ = 0.1 μm, this leads to τ_u = 0.13, 0.011, and 0.002 s at T₀ = 205, 220, and 235 K, respectively. For particles with r₀ ≪ r_c (kinetic regime), we obtain τ_u ∝ r₀, and for r₀ ≫ r_c (diffusion regime), τ_u ∝ r₀². The values of τ_u increase almost exponentially with decreasing T₀ because the equilibrium gas concentrations of H₂O decrease in proportion to e_i. Smaller particles are capable of responding faster than larger ones to changes in ambient relative humidity by taking up water; they simply need to condense fewer H₂O molecules per unit time to equilibrate.

2) Homogeneous freezing of liquid aerosol particles

Homogeneous nucleation of ice in a supercooled aqueous particle occurs after sufficient H₂O molecules have been taken up, that is, when the water activity a in the particle becomes sufficiently large. Here, a ≤ 1 is defined as the ratio of e and the saturation vapor pressure over a pure water particle. Homogeneous freezing is a threshold process, commencing at rather high ice saturation ratios (S_* > 1.5). The thresholds S_* depend on temperature and freezing particle size but are independent of the chemical nature of the solute (Koop et al. 2000).

Starting at or near S_i = 1, freezing is expected after significant uptake of H₂O, that is, after τ_u. The trajectory in Fig. 2 with Δy = 11.2 m barely reaches S_*, implying that homogeneous freezing takes place within a wing depth above the wing surface (Δy < Δx). The increase in peak S_i with decreasing Δy implies that freezing rates are larger closer to the surface.

In the background atmosphere unaffected by the aircraft’s flow field, liquid aerosol particles are in statistical equilibrium with the H₂O gas phase, except transiently in regions with large absolute cooling rates (> 30 K h⁻¹, in convective updrafts or lee waves). This is explained by τ_u values much shorter (< 1 s for T₀ > 200 K) than time scales of most atmospheric processes creating nonequilibrium conditions in aerosols. Given the much higher cooling rates and the limited time τ_d available in the flow over airfoils, only liquid particles for which τ_u + τ_f ≤ τ_d holds can participate in aerodynamic contrail formation, where τ_f is the homogeneous freezing time scale.

We derive τ_f in the appendix [Eq. (A1)] and find that τ_f/τ_d ≃ (4–8) × 10⁻⁴ ≪ 1, and that τ_f is also much smaller than τ_u, except for extremely small r₀ of a few nm. Hence, τ_f is not a limiting factor in our analysis. The dynamical constraint then simplifies to τ_u ≤ τ_d, or, using Eq. (3),

View Expanded

giving r̂₀ in μm for T₀ in K and τ_d in ms. Values of r̂₀ versus temperature are shown in Fig. 3 (upper solid curve) for κ = 1 and τ_d = 40 ms. In full water equilibrium, the homogeneous freezing rate coefficient J is similar for all particles, and the rate JV rises with the liquid particle volume V. During aerodynamic contrail formation, particles larger than r̂₀ have negligible J because they lag behind equilibrium and stay at low a, so for them JV is small despite larger V. If particles become too small, water uptake is limited by the Kelvin barrier; an increased value of the saturation vapor pressure e prevents particles with r₀ < r_K ≃ 4 nm from freezing homogeneously in conditions favorable for contrail formation (Kärcher et al. 1995).

Figure 3 depicts the range of possible initial ice particle radii r_i0 in aerodynamic contrails between the two solid curves, that is, r_K ≤ r_i0 ≤ r̂₀. In the hatched areas, freezing of liquid particles is not possible because either water uptake is too slow or the Kelvin barrier inhibits sufficient water uptake. Because freezing occurs almost instantaneously, τ_f is neglected and individual ice particles undergo an initial growth stage within their available time ∼ (τ_d − τ_u) according to the evolution of supersaturation depicted in Fig. 2.

If the homogeneous freezing temperature of activated aerosol particles (near 235 K) is not reached during adiabatic cooling (despite wing-induced peak cooling on the order of 25 K), the aerodynamic contrail generated by a large aircraft would consist of (almost) pure water droplets and would only be stable in a water-saturated environment. For this to happen, T₀ must exceed 260 K, a condition not relevant for jet aircraft typically cruising above the 300-mb pressure level. At intermediate temperatures, aerodynamic contrails would be composed of layers of ice close to the wing followed by layers of liquid droplets higher above, but the liquid layer would rapidly dissolve if the air were between water and ice saturation.

3) Depositional growth of ice particles

The volume of an ice particle with radius r_i increases during deposition of H₂O molecules from the gas phase according to υdN/dt, which leads to the radial growth rate

View Expanded

where we introduce the ice supersaturation s_i = (S_i − 1) and use α_i = 1, to compute r_c from Eq. (2). The flux in Eq. (5) approaches the maximum attainable gas kinetic limit dN/dt = α_iuπr_i²[n − e_i/(kT)] because the freshly nucleated ice particles are smaller than the mean free path of the H₂O molecules in air. The spherical approximation, neglect of the Kelvin effect, and the use of an upper limit value for molecular sticking is a good approximation for high supersaturation, in which case the role of migration of H₂O molecules to appropriate growth sites at the ice particle surface diminishes relative to vapor diffusion from the gas phase toward the ice particle. The vigorous growth under extremely high supersaturation is likely to produce amorphous ice. It would be interesting to learn more about the evolution of the mass, shape, and structure of small ice particles created in this way once the supersaturation decayed to low ambient values.

We define the growth time scale by τ_g⁻¹ = d ln(r_i)/dt, the characteristic time for a change in particle size due to water uptake. With s_i > 0, this time scale follows from Eq. (5):

View Expanded

giving τ_g in seconds with r_i0 in μm and T₀ in K. Setting s_i = 1 and identifying r_i0 with the freezing aerosol particle radius r₀ for a rough estimate, we infer that τ_g = τ_u3(1 − ω)/ω ≃ 1.6τ_u. Hence, uptake and initial growth evolve over similar time scales.

The question arises whether the nascent ice particles with an initial radius r_K ≤ r_i0 ≤ r̂₀ and total concentration n_i affect the evolution of the H₂O number density n (or S_i) by vapor deposition. The rate of change is given by dn/dt = −n_idN/dt. The time scale for relaxation of n toward equilibrium is given by τ_r⁻¹ = −d ln(n)/dt = 4πDβ_ir_in_i [combining Eqs. (2) and (5)], where n_i denotes the total ice particle number density and r_i the average ice particle radius. Inserting r_i = r_i⁰ = r̂₀ (the largest possible initial radius) yields lower limit values for τ_r > 2500 cm⁻³/n_i in seconds for T < 235 K. Because concentrations of upper tropospheric particles available for ice formation rarely exceed 2500 cm⁻³ (Schröder et al. 2002), τ_r > 1 s. Consequently, in the airflow over the wing, the evolution of S_i as shown in Fig. 2 is hardly affected by freezing and growth processes.

Integrating Eq. (5) with T = T₀ over the time Δt_f after freezing leads to

View Expanded

where ŝ_i denotes the ice supersaturation averaged over Δt_f .

(i) Initial growth stage

We next consider the early growth stage in which air flows over the airfoil and the ice particles experience (most of) the supersaturation (Fig. 2). We approximate Δt_f ≈ τ_d − τ_u(r_i0) and ŝ_i ≈ 5.5 in Eq. (7), guided by a trajectory with Δy = 3.5 m. The largest ice particles for which Δt_f = 0 form late and have no time left for growth, so r_i(Δt_f) = r_i0 = r̂₀. For smaller particles, Δt_f and r_i(Δt_f) increase beyond r_i0; their growth time is longer. At fixed r_i0, r_i(Δt_f) values strongly increase with T₀, reflecting the exponential increase of the growth rate.

We recall that a short warming phase occurs after the supersaturation pulse around x = 10 m (Fig. 2), giving rise to subsaturations (S_i ≃ 0.6) over a time span τ_d/2. This warming causes some of the ice particles to evaporate. Its effect is visible in Fig. 1, where the aerodynamic contrail appears to fade out immediately behind the innermost parts of the wings and then becomes better visible again because of further growth in ambient supersaturated air.

Small ice particles evaporate on a time scale τ_e = τ_g(|s_i|) after their initial growth phase. The criterion for particles to survive evaporation during the transient warming is τ_e ≥ τ_d/2 or, with Eq. (6),

View Expanded

giving r̂_i in μm for T₀ in K and τ_d in ms. The values r̂_i for κ = 1, |s_i| = 0.4, and τ_d = 40 ms are shown as the dashed curve in Fig. 3. Ice particles formed below this limit that have not grown past r̂_i evaporate. Hence, the stability region between the upper solid and the dashed curves, max {r_K, r̂_i} ≤ r_i0 ≤ r̂₀, includes ice particles that can undergo a second, atmospheric growth stage, provided the ambient air is supersaturated.

Particles interact with visible light of wavelength λ ≃ 0.4 − 0.8 μm in the Mie scattering regime when the optical size parameter 2πr_i/λ ≃ 1, that is, for r_i = r_υ = λ/(2π) ≃ 100 nm. Figure 3 demonstrates that only ice particles that either nucleated at a size above r_υ or grew past r_υ are large enough to be optically active in terms of Mie scattering.

We now address the question of how many ice particles form in aerodynamic contrails. Because τ_f ≪ τ_g, and given the high supersaturations within τ_d, all available liquid particles in the region r_K ≤ r_i0 ≤ r̂₀ (Fig. 3) freeze. This minimizes the role of heterogeneous ice nuclei in aerodynamic contrail formation because those constitute only a small subset of the total aerosol concentration (DeMott et al. 2003), justifying our approach that builds on homogeneous freezing. While growing into the diffusion regime (r_i > r_c), any initial ice particle size distribution becomes very narrow in the absence of turbulence, because according to Eq. (5) smaller particles grow faster. This is consistent with the appearance of colors in Fig. 1.

Concentrations of aerosol particles are highly variable in the upper troposphere, showing geographical and seasonal variations (Hofmann 1993) along with features caused by aerosol nucleation and cloud scavenging processes (Schröder et al. 2002). Total concentrations n_a usually exceed 500 cm⁻³. Number mean radii r_a fall into the accumulation (∼100 nm) or Aitken (∼10 nm) mode size range. Although an accumulation mode is always present, the Aitken mode is only populated near regions of active or recent nucleation.

We estimate the actual number density n_f ≤ n_a of freezing particles as a function of T₀ (via r̂₀) from Eq. (4). To this end, we assume the aerosol size distribution to be unimodal and lognormal, with parameters n_a, r_a, and the geometric standard deviation σ_a (typical range 1.5–2). Concentrations of particles near the Kelvin radius r_K are usually small; as shown in the appendix in Eq. (A2), those with r₀ < r̂₀ are then approximated by

View Expanded

This estimate, together with Fig. 3, suggests that n_f falls well below n_a if T₀ becomes sufficiently small. In such cases, the exact total number n_i of ice particles in aerodynamic contrails entering the stability region (r_i < r̂_i) is more difficult to quantify; robust prediction of n_f depends on an accurate knowledge of the evolving freezing aerosol size distribution.

To address the question of visibility, we compute the initial temporal evolution of optical depth τ_υ of a monodisperse ice particle population according to

View Expanded

where r_i follows from Eq. (7) within t = 0 and t = τ_d − τ_u(r_i0), and n_f ≤ n_i is given by Eq. (9). The Mie extinction efficiency at a given wavelength λ is approximated by (van de Hulst 1957)

View Expanded

This formula is valid for refractive indices 1 > μ > 2 (we take μ = 1.31 for ice and λ = 0.55 μm) and describes all salient features of the Mie extinction curve, being sufficiently accurate for our purpose. According to the discussion of S_i in Fig. 2, the geometrical aerodynamic contrail depth Δy is set equal to 5 m, about half of the peak wing depth of the airliner shown in Fig. 1. We define an approximate threshold optical depth of 0.01 for visibility, depending on human perception (Middleton 1952).

The temporal evolution of τ_υ in the initial growth stage is depicted in Fig. 4 as a function of the scaled time available for growth after freezing (i.e., 0 ≤ t/Δt_f ≤ 1) for τ_d = 40 ms, κ = 1, r_i0 = r_a = 50 nm, σ_a = 2, and four ambient temperatures T₀ including 241 K (see legend) relevant to the observation (section 2). To avoid maximizing τ_υ in this order-of-magnitude analysis, we use a background particle number concentration n_a = 500 cm⁻³ at the low end of measurements and a low average supersaturation, ŝ_i = 2, both accounting for transient subsaturations and spatial gradients in the flow field over the airfoil. These plausible choices are justified a posteriori by the numerical simulations shown in Fig. 5.

Only at the warmest T₀ do the ice particles grow rapidly enough to become visible above the wing (the curve crosses the approximate visibility threshold, the dotted line in Fig. 4). Significantly below ∼230 K ice particles form but stay near their initial size most of the time (not shown). In addition, in the coldest case n_i is already limited by small r̂₀ values of the order r_a [i.e., n_f/n_a = 0.82 at 210 K from Eq. (9)]. This finding already hints at a basic explanation of the rare occurrence of visible aerodynamic contrails.

(ii) Atmospheric growth stage

After passing the airfoil, the ice particles continue to grow provided the ambient air is supersaturated, defining a second growth phase required for contrail persistence. Before this atmospheric growth stage, the transient warming phase may reduce the ice particle concentration from Eq. (9). At low T₀, the small ice particle sizes facilitate evaporation and render aerodynamic contrails less stable. In such cold conditions, they would not form if (small) Aitken mode particles were absent.

The r_i values are largely controlled by T₀ rather than ambient S_i for times up to 1 s or ∼250-m distance behind the wing. Thereafter, r_i(t) will be governed by a more complex evolution of T and S_i because the aerodynamic contrail mixes with the jet plumes and the trailing vortices (see Fig. 1). We may expect that r_i values will then start to differ from each other significantly depending on wake dynamics and ambient supersaturation, but ice particles stay well below 1 μm for T₀ < 210 K.

In sum, particles smaller than r̂₀ from Eq. (4) but larger than ∼4 nm are expected to freeze instantaneously after τ_u from Eq. (3) because they equilibrate most rapidly and freezing occurs in a burst. The freezing process turns the available aerosol droplets into almost monodisperse, small ice particles that grow to sizes comparable to wavelengths of visible light and become visible above ∼230 K. Further growth depends on levels of ambient supersaturation and details of plume and wake dynamics. Differences between jet exhaust and aerodynamic contrails are compiled in Table 1.

a. Case study

The observation of aerodynamic contrail formation discussed above forms the basis of our case study. In a first step, we performed aerodynamic simulations to calculate the flow and thermodynamic field above an idealized wing surface near the airframe. The wing depth was 11.7 m and the effective angle of attack was 1° (Gierens et al. 2009). From this laminar flow field, we constructed a set of 20 streamlines stacked vertically above the wing, excluding the turbulent boundary layer (the width of which does not exceed ∼45 cm, as estimated in Gierens et al. 2009). Those were used as trajectories along which the APSCm was run to calculate aerosol and ice microphysics and the water budget. Finally, the ice crystal size distributions were used to calculate the vertically integrated optical depth and served as input to a radiative transfer calculation to simulate the observed colors along the aging contrail.

According to the weather data, the ambient temperature and pressure were 241 K and 300 mb during the observation. In the absence of aerosol measurements, we used a total of 1000 cm⁻³ aqueous H₂SO₄ particles, distributed lognormally with a dry mean mass radius of 50 nm and a geometric width of 2. The corresponding wet aerosol spectrum is depicted in Fig. 6 (dashed curve). Ice crystals were treated as spheres. Particle growth rates due to water uptake were computed using mass accommodation coefficients of 1 and 0.5 for aerosol droplets and ice particles, respectively. All particles were transported along the streamlines, but nucleated aerosol particles were removed from the air parcels. To enable an atmospheric growth stage, the ambient air was assumed to be mildly supersaturated (S_i = 1.1). In fact, we have chosen this value to achieve a good agreement with the observation of colors in Fig. 1, given the other constraints noted above.

Figure 5 summarizes the results of the gas dynamical and microphysical calculations, showing (top) S_i, (bottom left) n_i, and (bottom right) the effective ice particle radius r_eff, defined as the ratio of the third and second moment of the simulated ice particle size distribution. The ice saturation ratio resembles the results shown in Gierens et al. (2009), in which water vapor was treated as a passive tracer in the flow field. This confirms that only a negligible amount of H₂O is taken up during aerosol freezing in the near-field area surrounding the wing (before the atmospheric growth stage). The highest ice-supersaturation ratios of ∼40 are achieved where adiabatic expansion is largest, as indicated by enhanced streamline curvatures within the region {x = 0–5 m, y = 0–4 m}. Successively lower S_i values propagate upward as the wing-induced perturbations decay. Before and after the wing, some subsaturation is induced (darkest blue and black areas). The white contour curves separate subsaturated from supersaturated regions (S_i = 1), and regions in which ice particles can form (S_i > 1.6; see section 3) or grow by vapor deposition (S_i > 1).

Freezing commences coincident with the sharp rise of S_i at the front of the airfoil. As predicted by our time scale analysis, in a region only few meters wide above the wing surface, most of the available aerosol particles freeze. The maximum ice particle concentration appears ∼10 m (or ∼40 ms) downstream of the zone where S_i takes its maximum because aerosol particles need time to take up water in order to freeze (depending on particle size, this time τ_u may be on the order of τ_d). About 10% of these crystals sublimate in the subsaturated zone after passing the wing tail. The ice crystal concentrations fall off rapidly higher aloft, generating a sharp contrail lid after about half a wing depth. This behavior is consistent with the scaling of pressure, temperature, and relative humidity discussed in Gierens et al. (2009). The effective ice particle sizes are largest near x = 10 m in a region 2–4 m above the wing, consistent with the fact that the maximum available condensable water maximizes there, as also shown in Gierens et al. (2009). Below and above this region, r_eff values are smaller because the ice particle number density is higher and supersaturation is lower, respectively.

Figure 6 presents the ice particle size distributions (solid curves) at selected times along the y = 3.2 m trajectory (where the largest ice particles occur) and x locations past the leading edge of the wing (x = 0). As expected, the first freezing particles are relatively small (∼70 nm), although much larger droplets are available. Depositional growth becomes significant after a large portion of the aerosol particles have been transformed into ice, emphasizing the burstlike character of aerodynamic contrail formation. As estimated in the last section, the size distribution becomes rapidly monodisperse, enabling the formation of colors. Together with the high concentrations, r_eff values on the order of 0.5 μm explain the immediate visibility of the contrail, in agreement with Fig. 1. The short warming after x ≃ 10 m forces r_eff to decrease in the subsaturated region, but the ice particles grow again in the atmospheric growth stage because we have prescribed ambient supersaturation.

To simulate the colors for the resulting size distributions, the spectral scattering coefficient and scattering phase function were calculated using the Mie code. Scattering is most efficient when the ice particle radius equals the wavelength of the radiation (r_i = λ). The fact that the scattering properties depend on the ratio r_i/λ introduces a strong wavelength dependence (445, 555, and 600 nm are representative of blue, green, and red) even though the refractive index of ice is nearly independent of λ in the visible spectral region. For particles with r_i < 0.5 μm, blue is scattered more than red, whereas this is reversed for larger sizes. For particles much larger than the wavelength, geometrical optics holds and the scattering coefficient no longer depends on wavelength. This illustrates the change in color as the ice particles grow.

Figure 7 shows the optical depth τ_υ of the simulated aerodynamic contrail as a function of time, calculated by integrating the scattering coefficient over all vertical model layers. The optical depth increases rapidly to a first maximum of 0.012 once the ice crystals have finished the initial growth stage after t ≃ 40 ms. Our simple estimate from Fig. 4 yields τ_υ = 0.036 as the first peak value at the same time for the case study. The difference between the analytical estimate and the simulated value arises mainly from the assumption of vertically uniform supersaturation and monodisperse ice particle concentration in the simple model. A transient decrease of τ_υ forced by adiabatic warming and partial evaporation follows, after which τ_υ steadily rises again to 0.03 after ∼0.5 s in the atmospheric growth stage. This evolution of τ_υ is consistent with Fig. 1 and our brief description provided in section 2, suggesting that the assumed visibility threshold of 0.01 is meaningful. After 0.7 s, the ice crystal effective radius increases beyond ∼1 μm, where Mie oscillations become important and modulate τ_υ (not shown). Even slight deviations from spherical ice particle habits or surface roughness might modify τ_υ from Fig. 4, but no simple means are available to assert such effects.

To calculate the colors displayed in Fig. 7, a radiative transfer calculation was performed to simulate radiances in the visible spectral range. As a first approximation, the color of the scattered radiation was calculated with a simple single scattering approximation; the spectral scattering coefficient was multiplied with the spectral extraterrestrial irradiance. Hence, we neglect the wavelength dependence of the scattering phase function, as well as the contribution of Rayleigh scattering. The consequences of this simplified treatment are discussed below. The approximate spectral radiances were converted into colors following the procedure by of the Commission Internationale de l’Eclairage (CIE 1986). In brief, spectral radiance (calculated with a step width of 5 nm within λ = 380–780 nm) is multiplied by the three color matching functions and integrated to obtain the tristimulus values X, Y, and Z, which are in turn converted to color (R,G,B) values using the CIE system matrix. Brightness is considered by multiplying (R,G,B) by the luminosity Y.

The resulting colors are shown in Fig. 7 as a function of time. They can be divided into three main regions, again consistent with Fig. 1: up to 0.3 s (75 m), colors are mostly blue; in the range 0.3–0.7 s (75–175 m), yellow and orange; and after 0.7 s (175 m) mostly pink, which is roughly comparable to the colors on the photograph except for the yellow region that appears to start with green in Fig. 1, which is not simulated.

As mentioned above, this is only a rough approximation. An exact calculation would require detailed knowledge about the geometry of the scene, in particular the viewing direction of the camera. In the following, we present a refined calculation including Rayleigh scattering and molecular absorption and discuss the consequences for the interpretation. To simulate radiances, we use the libRadtran radiative transfer model (Mayer and Kylling 2005) with the discrete ordinate solver DISORT (Stamnes et al. 1988). LibRadtran has been shown to produce accurate sky radiances, in particular for cloudy conditions (Mayer et al. 2004). The conversion from spectral radiance to colors is performed as described above. Two simplifications were introduced. First, because the effective ice particle radii r_eff vary only little with height above the wing, we used r_eff of the layer with the largest ice water content (IWC) and applied it for all heights. Second, the vertically integrated optical depth τ_υ was set to 0.03 for this calculation.

We need to consider that in Fig. 1 not only the age of the contrail changes, but also the viewing angle. From the observation data (vertical distance 1200 m, length of the contrail 175 m), we estimate that the contrail spans about 10° in the image, which needs to be considered in the calculation. As an example, Fig. 8 shows the radiance distribution along two directions: (a) along the principal plane of the sun (relative azimuth 0°; this plane is defined by the direction of incident sunlight and the vertical direction) and (b) perpendicular to the principal plane of the sun (relative azimuth 90°), both as a function of time. The horizontal axis is the time t after the aircraft; the vertical axis, the viewing zenith angle θ. The images show the distribution of radiance over the full viewing angle, as if the sky were completely covered by the simulated contrail. (In our one-dimensional radiative transfer simulation, the contrail fills the whole sky.) Absolute brightness is not considered for the plots; rather, the colors were normalized because the result is much brighter in the vicinity of the sun and the other regions would not be visible at all. The solar zenith angle was −40°, marked by the white line in Fig. 8a. If the aircraft in the photograph flew along the principal plane, the trajectory would resemble the dotted black line in Fig. 8a, starting at θ = −35° (close to the sun) and heading directly away from the sun. In reality, the aircraft trajectory could be any straight line starting at a t = 0 and ending at θ = 10° higher or lower in the vertical. The real trajectory was probably neither in the principal plane nor perpendicular, but somewhere in between. Figure 8c finally shows the colors along the trajectory from Fig. 8a. The colors in this example are close to the observation, but they depend on the viewing geometry and the orientation of the flight path. In particular, the sequence of colors is very similar to our simple approximation in Fig. 7.

It is immediately obvious that the colors are slightly affected by the viewing geometry because the exact colors are determined not only by the extinction coefficient but also by the details of the scattering phase function. Because the scattering phase function has a strong forward peak, the most interesting effects appear in the vicinity of the sun, where the scattering by the ice particles dominate over the Rayleigh scattering. Nevertheless, the sequence of colors is always similar, irrespective of where we place the aircraft path in Fig. 8a. However, the contrast is strongly affected by the viewing geometry. The contrail is clearly visible close to the sun, but the further it is away from the sun, the less can the contrail be distinguished from the blue sky because the sideward scattering by the contrail particles is much smaller than in the forward direction and a much larger optical depth would be required for the contrail to become visible against the blue Rayleigh background. In the perpendicular plane, little can be seen because the viewing directions in the perpendicular plane are always more than 40° away from the sun.

From these studies we conclude that a contrail with τ_υ = 0.01–0.03 should be clearly visible from the flight altitude of the lower aircraft. The colors, however, appear only if the contrail is observed close to the sun. If observed from the ground, the Rayleigh scattering between surface and aircraft further reduce the contrast and a larger optical depth is probably required to clearly distinguish the contrail.

We do not overemphasize the discussion of colors at this point for two further reasons. First, slight alterations of plume turbulence or background meteorological parameters also affect the colors, as suggested by Fig. 1 of Gierens et al. (2009). Second, photographs and their reproductions are not fully conservative with respect to colors.

b. Sensitivity study

We performed a sensitivity study by changing T₀ for otherwise unchanged parameters. Figure 9 shows the simulated temporal evolution of r_eff for the case study (241 K) and three lower temperatures. The transient warming phase is most clearly observed for the highest T₀ where growth and evaporation rates are fastest. Although r_eff values grow past 0.1 μm at the two warmest temperatures, this is hardly the case for 220 K and not at all for 210 K. The labels in Fig. 9 indicate the simulated ice water contents after 1 s of contrail age. Only above 230 K are these values comparable to those in jet exhaust contrails, which are always visible at the time of formation; at lower temperatures aerodynamic contrails become extremely tenuous. The simulations also predict that the number of nucleated ice crystals decreases when T₀ decreases because slower water uptake rates prevent particles from freezing. These numerical results are entirely consistent with the estimates from the analytical study in section 3.

We briefly explore the consequences of these results for optical depth at λ = 0.5 μm and a contrail age of 0.7 s. At solar wavelengths, τ_υ scales in proportion to the ratio of ice water content and r_eff for constant extinction efficiency (Fu and Liou 1993). For the case study at T₀ = 241 K, we have τ_υ ≃ 0.03, r_eff ≃ 1μm and IWC ≃ 1 mg m⁻³, yielding the approximate relationship τ = 0.03IWC/r_eff by setting Q_ext = 2 despite r_eff values on the order of λ. For T₀ = 230, 220, and 210 K, we find r_eff = 0.43, 0.2, and 0.07 μm read off Fig. 9 and IWC = 0.12, 0.009, and 0.0002 mg m⁻³ from our simulations, respectively. Taken together, τ_υ decreases to 0.008, 0.001, and 9 × 10⁻⁵ with decreasing temperature, extending our earlier results from Fig. 4 into the atmospheric growth stage to a distance of ∼175 m (0.7 s) behind the wing. We further estimate that nascent aerodynamic contrails become invisible once the temperature falls below ∼232 K, explaining why they are difficult to observe.

Reductions or enhancements of ambient ice supersaturation would slow or enhance the atmospheric growth stage but leave the formation process unaffected. If fewer liquid-containing aerosol particles were available than assumed here, aerodynamic contrails would initially contain somewhat larger ice crystals.

a. Aerodynamic contrail cirrus

To judge the atmospheric relevance of aerodynamic contrails, we discuss temperatures measured along the flight paths of commercial airliners based on five years of Measurement of Ozone and Water Vapor by Airbus In-Service Aircraft (MOZAIC) data. Figure 10 displays the mean temperature values in different pressure layers along with the average overall layers in ice-supersaturated regions allowing contrails to persist. In the extratropics (>30°N), mean tropospheric temperatures cover the range 211.5–221 K, averaging to ∼218 K at ∼230 mb. In the tropics (0°–30°N), temperatures are significantly higher: 218.5–234 K, averaging to ∼228 K at ∼235 mb. Therefore, the occurrence frequency of aerodynamic contrails is expected to be much higher in the tropics than at middle and high latitudes.

The region enclosed by the dotted lines in Fig. 10 marks the range of threshold temperatures below which jet exhaust contrails can form, assuming ambient ice saturation. Although temperatures are very often low enough to enable jet contrail formation at midlatitudes, they would not form in the majority of cases in lower latitudes. (We recall that the contrail from Fig. 1 has been observed near 32°N at a rather high temperature too warm for the generation of jet contrails.) This demonstrates that large regions exist in which aerodynamic contrails can provide a complementary source of contrail cirrus.

The impact of aerodynamic contrails on the radiation balance depends on the evolution of their optical depth and coverage. After the decay of the wake vortices, the ice water path and the effective ice particle size of contrail cirrus are governed by atmospheric temperatures and factors including moisture supply and wind shear, among others. Optical properties of contrail cirrus are different from those of natural cirrus because of different formation conditions.

Because aerodynamic contrails preferably form at warm temperatures in low latitudes and the ice water content is a strong function of temperature (Schumann 2002; see his Fig. 11.5 showing a compilation of field data), they should result in optically thicker contrail cirrus than those existing in extratropical latitudes. In tropical and subtropical areas, mean vertical shear of the horizontal wind vector ∂_zυ in the upper troposphere calculated from daily climate model data appears to be larger than at middle latitudes due to large instantaneous directional wind shear [Burkhardt and Becker (2006), especially their Figs. 1c and 1d displaying dissipative heating rates ∝(∂_zυ)²]. This presumably increases aerodynamic contrail cirrus coverage and decreases ice water path faster in the tropics. Warmer temperatures and larger ice water content tend to generate larger ice crystals, reducing the effect of aerodynamic contrail cirrus by limiting their lifetimes due to enhanced ice crystal sedimentation.

Current subsonic air traffic is low in the tropical areas, but in the future air traffic is expected to increase strongly in these regions. Between 30°N and 30°S at 300 mb, the frequency of ice supersaturation is ∼10% with maxima located over Africa and the Maritime Continent, but jet contrail formation conditions are seldom met (Burkhardt et al. 2008). Hence, the fraction of persistent aerodynamic contrail cirrus is likely to increase significantly. The resulting global radiative forcing by aerodynamic contrails is expected to be different from current estimates of radiative forcing by jet contrails. Without prognostic models that enable the simulation of interactions among contrail cirrus, water vapor, and natural cirrus cloud fields, and because of the complex microphysical mechanisms affecting the properties and coverage of contrail cirrus, these differences in radiative forcing will be difficult to predict and quantify.

b. Aerodynamically induced preactivation

Another aspect concerns the microphysical processing of aerosol particles flowing over airfoils. The insoluble fraction in atmospheric particle populations is forced to form submicrometer ice particles in all conditions in which ice is generated aerodynamically (i.e., according to our discussion, essentially across all subsonic flight levels). This processing may be viewed as a specific type of preactivation: ice may form on these preactivated nuclei at lower supersaturations than without processing. Preactivation of exhaust soot particles may also occur after dissolution of short-lived jet contrails (Kärcher et al. 2007). Virtually nothing is known in general about preactivation in cirrus conditions, but it is possible that aerodynamic ice formation provides a low temperature source of effective ice nuclei. We attempt to provide a very rough estimate of the number concentration n_pre of preactivated ice nuclei. This value should be compared to recent estimates of the background concentration of upper tropospheric heterogeneous ice nuclei from in situ measurements in the Northern Hemisphere of some 0.01 cm⁻³ (DeMott et al. 2003; Haag et al. 2003).

The value of n_pre is given by the product of the number concentration of available insoluble particles at cruise levels n_ins capable of serving as preactivation nuclei, which may be some 10 cm⁻³ according to aircraft measurements (Minikin et al. 2003), and the ratio between the air volume processed by aerodynamic ice formation V_pre and the upper tropospheric volume V_trop in which air traffic is mainly concentrated. The latter is approximated by the zonal band within 30° and 60°N between 9–12-km altitude: V_trop ≃ 2.8 × 10⁸ km³. We assume rapid mixing of aerodynamically processed air within V_trop.

The processed air volume may be computed as the product of the initial aerodynamic contrail cross section A ≃ 100 m² and the aircraft flight path L. The current fleet travels a total distance of ∼2.75 × 10⁹ km globally per month; from this number, fractions of 0.9 and 0.75 are flown in the Northern Hemisphere and upper troposphere, respectively. Using the monthly flown distance accounts for the fact that the mean residence time of upper tropospheric air is on the order of 4 weeks. This yields L ≃ 1.9 × 10⁹ km and V_pre ≃ 1.9 × 10⁵ km³, combining to n_pre = n_insV_pre/V_trop ≃ 0.007 cm⁻³. This value is uncertain because it is not known how many insoluble particles can actually be preactivated or how often the same air mass is processed several times and also because of spatial inhomogeneities of the flight routes neglected here. Nevertheless, it is possible that aerodynamically induced preactivation contributes with some significance to the apparent background concentration of effective heterogeneous ice nuclei, possibly already affecting natural cirrus formation. This contribution will likely increase in the future given projected increases in air traffic growth rates.