a. The basic dynamic model

The basic dynamical framework employed is a modified and upgraded version of a large-eddy simulation (LES) model, which has been described in Chlond (1992). The model explicitly calculates the large (resolved) scale motions, while the small (subgrid) scale motions have to be parameterized. The equations governing the resolved-scale motions are the three-dimensional, spatially averaged, incompressible Boussinesq equations conserving mass and momentum. The Boussinesq approximation has been applied, which is appropriate since significant convective motions within contrails will be limited to a shallow layer (∼1 km) within the elevated computational domain.

In Cartesian coordinates the equations are

where overbars signify the Reynolds average over the grid volume Δx·Δy·Δz, and the horizontal average is denoted by the angular brackets. Here, u, υ, and w are the velocity components; p is the perturbation pressure;f is the Coriolis parameter; g/Θ₀₀ is the buoyancy parameter; and ρ₀₀ is the reference density, which is assumed to be independent of height. The components of the geostrophic wind u_g and υ_g are fixed in time but can depend on altitude, and τ_ij is the subgrid stress tensor given by

where δ_ij is the Kronecker delta.

In order to describe the thermodynamic properties of the cloudy atmosphere, the thermodynamics is represented by the conservation equation for the ice–liquid water potential temperature Θ_i and the total water content q = q_υ + q_i, where q_υ is the specific humidity and q_i the ice content. These equations are given by

where (∂Θ_i/∂t)_RAD is a cooling term simulating radiative cooling; (∂Θ_i/∂t)_SED and (∂q/∂t)_SED describe the effect of sedimentation of ice particles on the variables Θ_i and q, respectively. These terms are derived from parameterizations discussed in sections 2b and 2c, respectively;H_i is the turbulent heat flux and Q_i represents the turbulent total moisture flux given by

The ice–liquid water potential temperature Θ_i, suggested by Tripoli and Cotton (1981), is defined by

where Θ is the potential temperature, T is the absolute temperature, L is the latent heat coefficient of sublimation, and c_p represents the specific heat of dry air. We further assume that

where p₀₀ denotes the hydrostatic reference pressure.

The spatially averaged equations discussed above can be solved once the subgrid-scale (SGS) fluxes are determined in terms of the resolved-scale fields. Our subgrid flux model is based on a transport equation for the SGS kinetic energy,

and is similar to that of Deardorff (1980). We write the SGS equation as

where the first two terms on the right-hand side are the production terms due to local shear and buoyancy, respectively. The third term is a transport term and the last term is the viscous dissipation; Λ is a length scale and c_ε is a stability-dependent coefficient; both are specified algebraically below. The subgrid fluxes were parameterized according to first-order closure by

where K_M is a subgrid-scale eddy coefficient for momentum and K_H is a subgrid-scale eddy coefficient for scalar quantities, which are given by

where

Here R_d is the gas constant for dry air; T_i is the ice–liquid water temperature

and q_s is the saturation specific humidity, which is obtained from the relation

q_sT, pe_sTpe_sT

The saturation vapor pressure with respect to a plane surface of pure ice is determined according to

e_sT

where e_s is in 10² Pa (Laube and Höller 1986). The expressions (16a)–(16b) are simplified versions of those derived in Sommeria and Deardorff (1977), where we have neglected q_s in comparison with unity and we used the reference temperature Θ₀₀ in place of Θ_i. Here r denotes the subgrid-scale cloud fraction, which is either zero (q_i = 0) or unity (q_i > 0).

The eddy coefficients are proportional to the product of a length scale and the subgrid turbulent kinetic energy. Accordingly we have

where, according to Deardorff (1980), c_M and c_H are given by

The turbulent Richardson number Ri is defined as

where the length scale Λ is given by

z,e^1/2N⁻¹(20)

Here Δ is the average grid scale Δ = (Δx + Δy + Δz)/3, where Δx, Δy, and Δz are the respective grid intervals and N is the Brunt–Väisälä frequency, defined by

Hence, Λ is proportional to z near the surface but is limited to the average mesh spacing. The third limit is applied only in stable region where N² > 0. FollowingDeardorff (1980), the constant in the dissipation term is calculated as

b. The cloud microphysical model

To describe cloud physical processes a model with detailed treatment of microphysical processes is used. As a first approximation the ice particles are treated as spherical frozen drops. In this case, the ice particle spectrum could be represented by a one-dimensional number density function f(r, x, t). Following the density function notation of Berry (1967), this function is defined such that f(r, x, t) dr is the number of ice particles per unit mass between the radius r and r + dr at the location x = (x, y, z) and at time t. The analytical form of the equation governing the time rate of change of the number density function is written as

The second and the third term on the left-hand side of Eq. (23) represent advection of the ice particles with the air’s velocity and sedimentation of particles with their terminal velocity w_T. Values for w_T are calculated following Beard (1976). The fourth term on the left-hand side of Eq. (23) represents the spectral spreading and shifting effect on f due to vapor deposition/sublimation where dr/dt is the Lagrangian diffusional growth rate of a particle of radius r. The last term represents turbulent particle mixing. Equation (23) reveals that collision–aggregation processes and nucleation processes are not included in the model. The reason for the first assumption is that aggregation is of minor importance at temperatures below −20°C, presumably because ice crystals do not stick together when they collide at these temperatures (Hobbs et al. 1974). Admittedly, we must acknowledge that, in light of more recent results and observations, aggregation may be more significant than previously thought (Kajikawa and Heymsfield 1989; Mitchell 1988, 1991, 1994). However, we believe that ignoring this process in the present work is likely not of great consequence. The justification for the latter assumption appears more serious because it seems unlikely that no new cloud particles are allowed to form in a supersaturated atmosphere, especially in the presence of dynamical disturbances such as contrails. However, we think that due to the fact that the aircraft emits countless CCN in the air along with water vapor to produce a contrail containing a high concentration ofice particles, the most important cloud physical processes in contrails are diffusional growth and sublimation of ice particles; hence nucleation of new ice particles should be of secondary importance.

The rate of growth of ice particles is determined by considering the simultaneous transport of mass and energy from a falling ice particle. The equation governing the rate of growth by condensation is similar to that of Young (1974) except that the terms for the change in particle temperature with time and the effects of accretion of water drops on an ice particle are neglected. The governing equation for diffusional growth rate of a spherical ice particle of radius r that includes radiative heat exchange between the ice particle and its environment is

where

Here R_υ is the gas constant for water vapor and ρ_i is the density of ice. The quantity λ′ denotes the thermal conductivity of moist air and D′ the diffusivity of water vapor, where both terms are modified to take into account gas kinetic and ventilation effects on the diffusional growth following Hall and Pruppacher (1976). The first part of the right-hand side of Eq. (24) describes the usual particle growth equation without the radiation term. The quantity S is the ratio of the actual specific humidity of the ambient air q_υ and its saturation value q_s(T, p):

The second part of Eq. (24) describes the impact of radiative heat transfer on the particle growth. Here ΔF is the net longwave radiational energy gain at the particle surface. Following Bott et al. (1990), this quantity could be written as

where B, F⁺, and F⁻ denote the Planck function and the upward and downward broadband longwave radiative fluxes, respectively; Q_abs(r) is the efficiency factor for absorption, assumed to be unity for simplicity.

The ice content q_i is obtained by integrating the product of the number density function f(r, x, t) and the mass of a single ice particle m_i(s) over the entire particle spectrum:

The terms (∂q/∂z)_SED and (∂Θ_i/∂t)_SED in Eqs. (6) and (7), which describe the effect of sedimentation on the variables q and Θ_i, can be obtained as follows:

c. Radiative transfer model

The radiative transfer model is used to calculate radiative cooling rates for use in the thermodynamic equation and in the ice particle condensational growth term. We utilize a simple radiative transfer parameterization scheme for application to the LES cloud model, which only takes into account infrared absorption and emission by ice particles. Radiative cooling due to gaseous absorbers and solar radiative heating are excluded. The longwave scheme is based on the parameterization of cloud “effective emissivity” (Cox 1976). This quantity allows the calculation of broadband infrared fluxes and cooling rates within the cloud. For the flux computations at a given level z, the atmosphere is supposed to be isothermal above and below z. Vertical columns are considered independent and any lateral effect is neglected. Upward and downward radiative fluxes F⁺ and F⁻ are written

where B(T(z)) is the blackbody emission at the temperature of the level z; F⁺(z_B) and F⁻(z_T) are the impinging fluxes at the bottom and at the top of the model domain, respectively, which have to be prescribed. Here ε⁺(z, z_B) and ε⁻(z, z_T) are the effective emissivities of ice between the bottom of the model domain (z = z_B), and between level z and the top of the model domain (z = z_T), respectively. They are given by

where IWP(z_B, z) and IWP(z, z_T) are the amounts of ice contained in the columns below and above level z. From Eqs. (29a) and (29b), a₀ defines a mass absorption coefficient (m² kg⁻¹), which is given by Starr and Cox (1985) as

a⁺₀a⁻₀

The radiation computation is made at each time step for each column of air. The source term in Eq. (6) can now be written as

where ΔF is the difference between upward and downward fluxes at two consecutive levels around the level where Θ_i is defined.

d. Computational grid, boundary conditions, and numerical techniques

Since finite difference schemes suffer from numerical errors, it is particularly important to study which grid resolution and domain size are adequate to give a reliable representation of the phenomenon under consideration. Therefore, we carried out two-dimensional test simulations in a domain of size 640 × 640 m² with grid spacings of Δx = Δz = 20 m, Δx = Δz = 13.3 m, Δx = Δz = 10 m, and Δx = Δz = 5 m. It was found that the solution converges in terms of the vertical velocity variance, integrated ice content, and diffusion parameters, and that the resolution Δx = Δz = 10 m is sufficient. Moreover, it turns out that the circulations induced by the contrail often span at the end of the simulations a great portion of the model domain, so the width of the domain may be having an effect on the results. To demonstrate that this has little effect on contrail evolution we also performed a simulation with a domain 1280 m wide instead of 640 m while maintaining the grid resolution. It appears that the restricted domain size does not affect significantly the evolution of the contrail plume in the control case. The results in the extended domain are almost identical to those of the restricted domain.

Therefore, the spatial domain is specified as an elevated box that extends horizontally and vertically over a finite domain of size 640 × 640 × 640 m³. The domain is divided into 64 × 64 × 64 grid boxes (i.e., Δx = Δy = Δz = 10 m). At the lateral boundaries, periodicity is assumed. At the bottom and top, a zero slope condition on all variables except w, q, and Θ_i, is imposed. Vertical velocity w is zero at the bottom and the top and the gradients of q and Θ_i are fixed at the initial gradients Γ_q and Γ_Θi. Although ∂f/∂z = 0 is formally maintained at the lower boundary, cloud ice is permitted to exist across the lower boundary by virtue of its relative fall speed. These boundary conditions may be unrealistic if the region near the lower or upper boundaries becomes disturbed. This is prevented by on-time termination of the simulation. Furthermore, at the lower and upper right levels of the model there are sponge layers that remove fluctuations of the velocity, temperature, and humidity fields in order to damp disturbances there.

The numerical integration scheme is based on an equidistant staggered grid and finite difference approximations. All scalars are defined at the cell centers, u is evaluated at the middle of the upstream and downstream faces, υ at the middle of the two cross-stream faces, and w at the middle of the bottom and top faces.

The advection terms of the momentum equations are treated by a scheme proposed by Piacsek and Williams(1970). It is a second-order method in space, which conserves the integral of linear and quadratic quantities up to very small errors. The diffusion terms are solved with a second-order, spatial central difference scheme. Time integration of the momentum equation is performed using the Adams–Bashforth scheme. The Poisson equation for the pressure is solved using a fast Fourier (in the horizontal directions) and a finite-difference technique (in the vertical direction).

The integration method used to solve Eqs. (6)–(7) and (13) is based on a fully explicit, time-splitting method. Terms representing the divergence of the vertical flux due to advection and sedimentation are treated by the monotone, locally modified version of Bott’s advection scheme (Chlond 1994). All other terms are advanced with the Euler method in time and are resolved with the second-order central-difference scheme in space.

The numerical treatment of the ice particle continuity equation (23) is similar to that of Eqs. (6)–(7) and (13). However, special care has to be taken in the handling of the diffusional growth of ice particles. As pointed out in section 2b, ice particles are represented in terms of a one-dimensional distribution function with the radius of the particle as the independent variable.

For numerical purposes the ice particle size distribution is subdivided into 12 classes with equidistant grid spacing in logarithmic scales, resulting in a radius grid that spans from r_min = 1 μm to r_max = 64 μm. As can be seen from the fourth term of Eq. (23), the diffusional growth may be formulated as a flux divergence in radius space due to condensational growth and sublimation of ice. The numerical procedure to treat this process is a one-dimensional application of the locally modified version of Bott’s advection scheme. However, since the particle growth velocity depends on the supersaturation, it is a function of time. Arnason and Brown (1971) have shown that due to this time dependency the use of an explicit method to integrate this equation is not suitable since it requires extremely small time steps for numerical stability. In order to allow larger integration time steps, the condensational growth process is implicitly solved similar to the method of Hall (1980) and Bott et al. (1990). At time step n the diffusional growth of the particles is calculated with a constant value of the saturation value S = (Sⁿ + Sⁿ⁺¹₀)/2, where Sⁿ is the actual saturation value and Sⁿ⁺¹₀ is a first guess (e.g., Sⁿ⁺¹₀ = 1). Utilizing Newton–Raphson method, the final value Sⁿ⁺¹_m is iteratively determined by requiring |Sⁿ⁺¹_m − Sⁿ⁺¹_m−1| < 10⁻⁶.

A time step of 1.5 s was used for all runs. Each 30-min simulation required about 12 hours of CPU time on one processor of a CRAY C916 computer.

a. Control run

The flow fields provide an important indication of the nature of the resolved-scale eddies. Figures 1 and 5–8 illustrate typical realizations of the secondary flow pattern in vertical plan views. The figures show isopleths of the various fields, which have been averaged along the airplane flight direction (i.e., along the y direction). Except for Θ_υ, horizontal averages have not been removed. Figure 1 shows contours of the ice content in vertical x–z cross sections at times (a) t = 2 min, (b) t = 4 min, (c) t = 8 min, (d) t = 14 min, (e) t = 22 min, and (f) t = 30 min. Isolines are drawn for 1, 10, 20, 25, and 30 mg kg⁻¹. Superimposed are velocity vectors of the resolved-scale velocity field (maximum indicated). Figures 5–8 represent vertical cross sections for the variables N_TOT (particle concentration), r_eff(effective radius), w, and Θ_υ for the same times as Fig. 1.

At time zero, the model starts from the equilibrium mean fields already described, except for small random disturbances applied to the temperature field within the contrail. The amplitude of the temperature perturbations is found to increase by instability and, consequently, the contrail grows as well in the horizontal as in the vertical direction. After 2 min the two cloud lines interconnect to form a single cloud band. The flow field is composed of two counterrotating vortices originating at the edges of the contrail. These vortices entrain moist environmental air into the contrail, which is subsequently processed into ice within the cloud. After 30 minutes of simulation time the contrail nearly fills the entire horizontal domain. The maxima of the ice content are found in the central parts of the contrail and amount to about 20 mg kg⁻¹. Throughout the contrail broadening process, the flow field exhibits an increasingly complex structure that is composed of several vortices. The maximum amplitude of the vertical velocity perturbation w is of the order of 0.1 m s⁻¹ and is found at the upper edges of the contrail.

To get an impression of the three-dimensional structure of the evolving contrail, isosurface plots of the ice content (12 mg kg⁻¹ surface) are displayed in Fig. 2 at (a) t = 4 min, (b) t = 14 min, (c) t = 22 min, and (d) t = 30 min. Interestingly, the ice field is almost homogeneous along the y direction at t = 4 min and t = 14 min. However, at later times it is striking from the three-dimensional fields that no regular two-dimensional motions are generated by the model. The fact is also supported in Fig. 3, where we have plotted contours of the ice content (left panel) and of the vertical velocity (right panel) in horizontal cross sections (at z = 0 m) at times (a) t = 4 min, (b) t = 14 min, and (c) t = 30 min. At t = 30 min, the ice field as well as the verticalvelocity field exhibit considerable variations along the y axis is on scales of about 100 m. Therefore, the assumption of homogeneity along the streamwise direction of two-dimensional studies appears at least questionable on the basis of our three-dimensional results, especially in the mature state of the dispersion regime. The applicability of the two-dimensional approximation is also highlighted in Fig. 4, which shows time series of resolved-scale variances for a three- and a two-dimensional simulation of the control case. Depicted are mean integral variances (per meter along the y direction) of the along-wind ρ₀〈υ"²〉, cross-wind ρ₀〈u"²〉, and vertical velocity component ρ₀〈w"²〉. It is evident that the two-dimensional simulation successfully predicts the evolution of the u′ and w′ variance in the initial and the mature phase of the dispersion regime. Both three- dimensional and two-dimensional simulations produce a variance ratio 〈u"²〉/〈w"²〉 of about 2 for t > 1200 s, which is a characteristic feature of semistable-layer turbulence. In contrast, the evolution of the along-wind variance 〈υ"²〉 is not properly reproduced by the two-dimensional model. It appears that for t > 1200 s three-dimensional effects become quite important. At t = 1800 s, the three-dimensional model predicts a resolved-scale variance ratio 〈u"²〉/〈υ"²〉 of about 6, while the two-dimensional model predicts a ratio of 〈u"²〉 to 〈υ"²〉 of almost infinity (the ratio (〈u"²〉/〈υ"²〉)_2D is not identical to infinity since Coriolis forces have been included in the two-dimensional simulation). This supports the suggestion that three-dimensional effects reduce the resolved-scale variance ratio 〈u"²〉/〈υ"²〉 because, as a result of thepressure-velocity-gradient correlation 〈p"∂υ"/∂y〉, coupling of υ" momentum to u" and w" momentum is stronger than two-dimensional models suggest. Therefore, we conclude that even two-dimensional-like phenomena should be modeled with a three-dimensional model to get a proper turbulent kinetic energy evolution. Nevertheless, it turns out that the two-dimensional modeling approach has its justification as the two-dimensional model predicts many aspects of the contrail evolution quite successfully (e.g, vertical velocity variance, ice production rate, and diffusion parameters of the contrail derived from the two-dimensional solution resemble quite closely those of the three-dimensional runs).

Figures 5 and 6 show the ice particle concentration and the effective radius fields from the control run. These microphysical parameters are inferred from the calculated one-dimensional number density function f(r, x, t) and are obtained with the following relations:

in which N(x, t) is the particle concentration and r_eff is the effective radius, respectively, and r₁ and r₂ are the upper and lower boundaries of the radius interval over which the particles are calculated (1 μm ⩽ r ⩽ 64 μm). From the calculations we find that in the central part of the contrail the total concentration of ice particles is largest (N ≈ 10⁸ kg⁻¹). Hence, at z = 0 m, which corresponds to the formation level, the ice particle concentration has its maximum and decreases rapidly below z = −150 m and above z = 150 m, as shown in Fig. 5. The effective radius field exhibits a clear stratification, as shown in Fig. 6. The smallest ice particles are found in the central region of the contrail, correspondingto the same area where the largest ice content and the maximum ice particle concentration appear. In this zone the particles grow from about r_eff = 2.5 μm at t = 0 min to about r_eff = 6.5 μm at t = 30 min. At lower levels, the mean value of the effective radius increases up to about r_eff = 30 μm. This fact could be explained as follows.

The small ice particles in the central part of the contrail are formed in the early aircraft exhaust plume by homogeneous or heterogeneous nucleation. As the contrail grows by entraining supersaturated air these particles spread over a large area and grow due to vapor deposition. As a result, in the central part of the contrailthe particle size increases with time and the total particle concentration decreases with time due to dilution. A small fraction of the particles attains a size at which gravitational settling becomes important. Since the largest particles have the largest terminal velocity, the proportion of bigger particles is larger in the lower part of the contrail than in the upper part of the ice cloud. Obviously, the small particles contribute more to the ice content than to the sedimentation rate, resulting in a decrease of the ice content and the total ice particle concentration at lower levels.

To illustrate the dynamical and thermodynamical features of the contrail evolution we show vertical cross sections of the vertical velocity and virtual potential temperature anomaly fields at different times (Figs. 7 and 8). During the first stage of contrail development, which occurs during the first 8 min of simulation time, the updraft velocity pattern shows a w-shaped pattern within the contrail, as a result of the developing counterrotating vortices. These vortices produce updrafts with maxima of about 0.1 m s⁻¹ within the contrail. These maxima are due to the local extrema of the horizontal gradient of buoyancy caused by the vapor deposition process creating a source of horizontal vorticity. The second stage of contrail evolution comprises the time interval from 14–30 min. During this period the vertical velocity field shows different distinct maxima within the cloud. These maxima are located near the outer edge of the ice cloud with values of about 0.1 m s⁻¹. These maxima are coupled with downdraft velocities near the same levels inside and outside the cloud. This velocity configuration is due to the developing complex multiple vortex structure as a result of the continual production of vorticity at the cloud boundary.

The fields of virtual potential temperature anomaly shown in Fig. 8 are seen to be well correlated with the field of vertical velocity. At every time the field of vertical velocity is, for the most part, in phase with the temperature perturbation field, that is, maximum temperature perturbations coincide with strongest upward motions and vice versa, indicating an upward buoyancy flux. Maximum temperature perturbations are found to be in the range between 0.04 and 0.06 K and are determined largely by the vapor deposition process due to latent heat release. Thus, we anticipate that the turbulent kinetic energy of the contrail is produced by buoyancy, which is reinforced by condensational heating. In principle, differential radiative cooling could also provide a substantial source, via buoyancy production, for turbulent kinetic energy of the contrail. However, our numerical simulations of contrail development have revealed a surprising insensitivity to longwave radiative cooling/heating (see also next subsection). This is due to the small cooling rates that we found in the contrail compared to the much higher values present, for instance, in stratocumulus clouds. Inspection of the longwave radiation cooling field reveals (not shown here) that the cooling is almost homogeneously distributed within the contrail with maximum cooling rates of about −1 K/day (by comparison: maximum infrared cooling rates at cloud top of stratocumulus clouds are about −100 K/day). Therefore, we conclude that in contrast to the stratocumulus-topped planetary boundary layer, which is mainly driven by cloud-top radiation cooling, cloud–radiation interaction insignificantly influence the dynamics of contrails and that latent heating due to vapor deposition provides the important energy source for turbulent kinetic energy of contrails. However, despite our findings about the insensitivity of contrail dynamics on radiation, one should keep in mind that this statement is only valid on a timescale of about 30 min. On a longer timescale, on the other hand, one should observe thin cirrus/contrail lifting/sinking even due to small heating/cooling effect of importance for contrail evolution in its very mature stage (Jensen et al. 1996). Moreover, there is no doubt that thin cirrus clouds can be a significant factor in affecting the radiative energy balance of the earth and in turn may have the potential to lead to greenhouse warming (e.g., Prabhakara et al. 1993).

b. Sensitivity runs

In order to examine the influence of external parameters, initial conditions, and radiation on the development of contrails, sensitivity tests have been performed. The control run was rerun (see also Table 1) with

1. different ambient temperatures T₀₀ (runs 2 and 4);
2. different static stabilities ∂Θ/∂z (runs 1 and 5);
3. different relative humidity r_w (r_i) (runs 8 and 9);
4. mean vertical wind shear (runs 6 and 7);
5. no radiation term in the droplet growth equation and no longwave radiative effects at all (runs 10 and 11);and
6. different initial ice particle sizes (runs 12 and 13).

In Fig. 9 we have plotted integral quantities such as domain-averaged turbulent kinetic energy (a), integrated ice content per meter of flight (b), horizontal spreading σ²_x (c), and vertical spreading σ²_z (d) of the contrail as function of time for the various model experiments. These quantities are all inferred from the simulations and are obtained with the following relations

where u", υ", and w" denote the deviations of the velocity components from their horizontal means and the integration is performed over the total model domain. In order to quantify the horizontal and vertical spreading rates of a contrail it is necessary to define effective contrail cross sections A by

where C(y, t) denotes the area enclosed by the contour N^*_min = 100 kg⁻¹ of those particles that are insignificantly influenced by gravitational settling, that is,

The horizontal and vertical half-widths of the contrail are then obtained by taking the half of the projections of the effective contrail cross section on the x and z axis, respectively, that is,

As seen in Fig. 9, the evolution of contrails is mainly controlled by relative humidity, temperature, and static stability of the ambient air. The greatest reductions in turbulent kinetic energy occur in runs 2 and 8, where the TKE has dropped from 0.8 × 10⁻³ to 0.17 × 10⁻³ and 0.8 × 10⁻⁴ m² s⁻². The physical reason for this effect arises from the microphysical processes governing the diffusional growth of ice particles. Inspection of Eq. (24) reveals that the mass growth rate of a single ice particle is directly proportional to the ambient relative humidity and that for a given relative humidity the growth rate is a monotonic increasing (stronger than linear) function of temperature. As a result, reducing the relative humidity of the ambient air by 12% (run 8) or lowering the temperature of the environmental air by 10 K (run 2) produces less vigorous eddies. In both cases the conversion rate from water vapor to ice, and hence the latent heat release, is much less than in the reference run owing to the initial decrease of total water content. On the other hand, increasing the environmental humidity by 12% (run 9) or raising the environmental temperature by 10 K (run 4) results in enhanced energy levels. In both cases the conditionally available potential energy that could by converted into kinetic energy is larger than in the reference case. Therefore, we conclude that vapor deposition provides the significant energy source driving the secondary flow eddies within contrails. The influence of thermal stratification is also very pronounced. In the simulation that has been performed in a neutral stratified atmosphere (run 1) enhanced energy levels and greater ice production rates are found. The physical reason for this effect is due the fact that in this case the eddies have to perform no work against the stable stratification. On the other hand, for a more stably stratified (relative to the reference run) environment (run 5), the stratification counteracts vertical motions so that mixing is much slower in the vertical than in the horizontal directions. The ratio of horizontal to vertical variance as a measure of anisotropy amounts to about 2:1 in the control case and about 4:1 in run 5. These values for the anisotropy ratio are typical for turbulence in a stably stratified medium where rather large and almost horizontally oriented eddies cause the mixing (Gerz and Schumann 1991; Schumann and Gerz 1995). Recently, Schumann et al. (1995) analyzed atmospheric turbulence data from a measurement campaign in the North Atlantic flight corridor near the tropopause. They also found strong anisotropic air motions with almost vanishing turbulent dissipation rates, suggesting that the spectral turbulent energy transfer from larger to smaller scales was almost suppressed, or even may go upscale to form larger eddies from smaller eddies, as hypothesized for quasi-two-dimensional stable layer turbulence (Gage 1979; Lilly 1983, 1988). Figure 9 depicts the ice production rate as only slightly dependent on the initial ice particle concentration. The comparison reveals only a small change of the integrated ice content IC at t = 1800 s (3.6 × 10⁻¹ kg m⁻¹ (run 12), and 4.7 × 10⁻¹ kg m⁻¹ (run 13) versus 4.4 × 10⁻¹ kg m⁻¹ in the reference run), although the initial ice particle concentration has been varied by a factor of 1/8 (run 12) and 2.8 (run 13), respectively. The insensitivity may be explained by an overseeding of numerous ice crystals that will grow more slowly than a single iceparticle owing to the competition of many ice crystalsfor the available water vapor. The results for the runs 6, 7, 10, and 11 are nearly indistinguishable from the control run, that is, the mean vertical wind shear of the horizontal velocity components and radiative processes are of minor importance in the evolution of contrails at least during the 30-min simulation period. With respect to runs 6 and 7 we have to admit that these experiments may have only restricted applicability to the questionof the potential importance of vertical shear since the assumed shear cases contained only very weak wind shear [5 × 10⁻³ s⁻¹ (along-wind shear); 1 × 10⁻³ s⁻¹ (cross-wind shear)]. Estimates of diffusion parameters of aircraft exhaust plumes near the tropopause from nitric oxide and turbulence measurements indicate that shear dominates the dispersion after a time of about 1 hour, even for the cases with only a weak wind shearof 2 × 10⁻³ s⁻¹ (Schumann et al. 1995). The effect of radiative transfer on the particle condensational growth is almost negligible since the ice particles are too small on the average. Effects of radiation on ice crystal growth are expected to be important only for particles having sizes of more than a few tens of microns (Gierens 1994). Moreover, radiative cooling leads to insignificant changes in the structure of the modeled contrail. This is due to (as already discussed in the previous subsection) the small cooling rates that we found in contrails. The impact of radiative processes on deposition growth and thermodynamic properties may be important when cloud radiative cooling is significant as in boundary layer clouds (Ackerman et al. 1995).

Except for in situ measurements and remote sensing observations by Knollenberg (1992), Baumann et al. (1993), Schumann (1994), Freudenthaler et al. (1995) and Gayet et al. (1996), little is known about the spatial structure of microphysical and dynamical parameters of contrails. Gayet et al. (1996) analyzed natural and contrail-induced cirrus from in situ and remote sensing measurements during the International Cirrus Experiment (ICE) over the North Sea during fall 1989. These observations were made in a situation of moderate wind shear in a supersaturated (with respect to ice) layer at an altitude of about 8000 m and at temperatures of about −37°C. Although the situation during the observations is comparable to those considered here, none of our results completely fits the situation during the measurements of this case study. Nevertheless, we use these data, and an attempt has been made to perform a rough comparison of our results to these observations. For this purpose we have summarized in Table 3 mean [defined in terms of the area defined in Eq. (37)] and maximum values of bulk parameters characterizing the microphysical properties of the simulated contrails for the various model experiments.

Simulated mean and maximum ice contents are in excellent agreement with the observations of Gayet et al. (1996), who reported values in the range between 2 and 18 mg m⁻³ (note that the values in Table 3 have to be multiplied by the density of air ρ₀₀ = 0.396 kg m⁻³ to obtain ice contents and ice particle concentrations per volume), whereas the computed values lie between 2.6 and 14.3 mg m⁻³. In contrast, there seems to be an underprediction of mean effective radii. However, keeping in mind the uncertainties in experimental estimates of microphysical parameters, our results appear also quite realistic with respect to the particle size, that is, values of 〈r_eff〉 in the range between 4.6 and 15.4 μm in the simulations versus 15 and 18 μm in the observations. The most glaring discrepancy between the model and the data is the high particle number density in the simulations. The ICE data suggest ice particle concentrations up to 8.2 × 10⁵ m⁻³ within contrails, whereas the computed particle number densities are in the range between 7 × 10⁵ and 2.5 × 10⁸ m⁻³. Several possible explanations may be given for the discrepancies between this type of simulation and the observations. These include missing simulated microphysical processes, unawareness about the initial and environmental conditions, and uncertainties about experimental estimates of microphysical parameters.

With respect to the first point a few comments on the limitations of our model are appropriate at this point. A possible limitation of the model results from the assumption of spherical ice particles, and more seriously, from neglect of the aggregation process. As already pointed out in section 2b, it appears not very likely that aggregation constitutes a significant process in ice clouds at low temperatures. However, more recent results and observations suggest that aggregation may be of more importance than previously thought (Kajikawa and Heymsfield 1989; Mitchell 1988, 1991, 1994). In fact, aggregation could probably establish an important sink for small particles and would increase the number density of large crystals in our model. Hence, we conclude that the potential importance of the aggregation process is an obvious uncertainty in our simulations, but we think that the omission of this process can explain the discrepancy between simulated and measured particle concentrations only in part. However, further observational evidence is required for a truly definite evaluation.

The second possibility is that errors in the assumed initial and environmental conditions significantly altered the simulation. It should be realized that the observations of contrails during ICE possibly cannot be generalized for all cases of this type of clouds, especially for young contrails. In fact, the intercomparison of simulations and observations is hampered by the lack of information, first, about the initial particle size distribution, second, about the amount of water vapor in excess of ice saturation, and, third, about the age and, therefore, about the state of evolution of the observedcontrail. Finally, we should note that the apparent discrepancy may be partly traced back to the fact that the measured particle concentrations include contributions from ice crystals with dimensions larger than the PMS probe detection limit (3 μm) while the model results include contributions from particles in the range of 1–64 μm. Moreover, the PMS probe measurements of total ice crystal number densities are probably not very reliable due to the large uncertainty in the smallest size bin. The PMS probe probably underestimates the number of small particles, which would reduce the discrepancy between the predicted number density and the actual contrail properties.