Abstract Large-eddy simulations with size-resolved microphysics are used to model persistent aircraft contrails and contrail-induced cirrus from a few wing spans behind the aircraft until their demise after many hours. Schemes for dynamic local ice binning and updating coupled radiation dynamically as needed in individual columns were developed for numerical efficiency, along with a scheme for maintaining realistic ambient turbulence over long times. These capabilities are used to study some of the critical dynamics involved in contrail evolution and to explore the simulation features required for adequate treatment of different components. A ''quasi 3D'' approach is identified as a useful approximation of the full dynamics, reducing the computation to allow a larger parameter space to be studied. Ice crystal number loss involving competition between different crystal sizes is found to be significant for both young contrails and aging contrail cirrus. As a consequence, the sensitivity to the initial number of ice crystals in the contrail above a threshold is found to decrease significantly over time, and uncertainties in the ice deposition coefficient and Kelvin effect for ice crystals assume an increased importance. Atmospheric turbulence is found to strongly influence contrail properties and lifetime in some regimes. Water from fuel consumption is found to significantly reduce aircraft-wake-induced ice crystal loss in colder contrails. Ice crystal shape effects, coupled radiation, and precipitation dynamics are also considered. An extensive set of simulations exploring a large parameter space with this model are analyzed in a companion paper. Introduction It is not uncommon for portions of the upper troposphere to be apparently cloud free (or containing only thin cirrus) yet highly supersaturated with respect to ice (e.g., Gierens et al. 2012 ). For such conditions ice clouds seeded by the passage of aircraft (contrails) can persist and grow into significant cloud cover that at late times could easily be mistaken for natural cirrus (Minnis et al. 1998) . Given the projected increases in air traffic in the coming decades and potential impact of increased cloud cover on climate, understanding the formation, properties, and effects of persistent contrails is of clear importance (Penner et al. 1999) . Despite recent progress in evaluating global impact of contrails (Ponater et al. 2002; Rap et al. 2010; Frömming et al. 2011; Chen and Gettelman 2013) , contrails remain the most uncertain component of aviation impact on climate (Lee et al. 2010) . Introduction Contrail evolution is a complex process dependent on many properties of the aircraft and of the ambient atmosphere. Exploring these sensitivities involves dynamics on length scales ranging from nanometers for the ice microphysics to kilometers for the late-time atmospheric dispersion. This makes numerical simulation of contrails challenging. A large number of simulations are required to fully represent the phenomena given the size of the parameter space, but with an a priori unknown level of complexity required to make these representations suitably faithful. Introduction Persistent contrail evolution involves the growth of ice within an expanding wake-plume volume in an icesupersaturated environment. In the young contrail the volume expansion is driven by aircraft-induced dynamics: engine jets, trailing vortex interactions, and residual Brunt-Väisälä oscillations set up by the wake-vortex descent. Effects with longer time scales take over later on: atmospheric dispersion, ice sedimentation, and buoyant forcing due to radiative or latent heating. Previous simulation studies (e.g., Gierens 1996; Chlond 1998; Gierens and Jensen 1998; Jensen et al. 1998; Khvorostyanov and Sassen 1998; Sussmann and Gierens 1999; Lewellen and Lewellen 2001a; Chen and Lin 2001; Paoli et al. 2004; Paugam et al. 2010; Wong and MiakeLye 2010; Naiman et al. 2011; Unterstrasser and Sölch 2010) have generally been limited to specific age regimes of contrail evolution, typically with greatly simplified ice microphysics and/or fluid dynamics and sampling only small regions of the relevant parameter space. In a more extensive study (Unterstrasser et al. 2008; Unterstrasser and Gierens 2010a,b) contrails have been simulated from ;20 s to several hours over a wider sampling of parameter space, though with bulk ice microphysics and 2D, prescribed (rather than simulated) wake dynamics, and ambient atmosphere. In another approach, Schumann (2012) describes a contrail-cirrus prediction model (CoCIP), designed to model the full contrail life cycle but at a dramatically simplified level to allow the modeling of large populations of contrails. Introduction Here we describe a more comprehensive approach, simulating contrails starting at 1-s age from realistic aircraft conditions and following the evolution with 3D fluid mechanics, size-resolved (''binned'') microphysics, and coupled radiation through the wake and ambientturbulence-dominated regimes out to the demise of the resulting contrail cirrus many hours later. Novel strategies are used to improve numerical efficiency. These capabilities are used to explore some of the critical dynamics involved in contrail evolution and to assess what is required for adequate simulation of different facetsfor example, whether detailed initial conditions, 3D fluid mechanics, binned microphysics, and/or realistic ambient atmospheric motions are required to reproduce contrail-averaged quantities. A ''quasi 3D'' simulation mode is established as a useful approximation, which makes practical the coverage of large regions of parameter space with the simulations. Introduction This paper describes the model and highlights some general features of the simulation results that have not appeared elsewhere. A companion paper (Lewellen 2014, hereafter Part II) analyzes a large set of full-lifetime simulations for a single aircraft including significant shear and coupled radiation, emphasizing late-time and lifetime-integrated behavior. An extensive parametric study focused on young contrail behavior will be presented elsewhere, including different aircraft and physically based parameterization of the results. A significant and surprising result from the present work is the importance in both young and old contrails of ice crystal loss involving competition between different crystal sizes in the ice spectrum. An analytic treatment of this effect along with a more general analysis of different regimes of ice-size spectra development has been presented in Lewellen (2012) . Introduction The model and simulation procedures are described in section 2, with further details given in the appendix. Sample simulation results are given in section 3. Some key physical results are discussed at more length in section 4, followed by concluding remarks in section 5. Contrail model a. Fluid mechanics and earlier model The starting point for the present work was the largeeddy-simulation (LES) model used for studies of wake dynamics, wake-plume chemistry, and contrails described in Lewellen (1996, 2001a,b) ; only a brief summary is given here. A 3D finite-difference implementation of the incompressible Navier-Stokes equations is employed, second-order accurate in space and time, with a direct solver for the pressure and a variable time step. The piecewise parabolic method (PPM) is used for advection of the liquid-ice potential temperature Q li , total water mixing ratio q t , ice number densities, and passive species concentrations. The subgrid model uses a quasi-equilibrium second-order turbulence closure scheme with a prognostic equation for the subgrid turbulence kinetic energy (TKE) and diagnostic equation for the subgrid turbulence length scale L. The physical damping of turbulence by stable temperature or rotation gradients is included by appropriate reductions in L. The latter helps prevent unphysically large subgrid diffusion in vortex cores if they are only modestly resolved and was developed and used extensively for tornado simulations (Lewellen et al. 2000; Lewellen and Lewellen 2007) . Contrail model a. Fluid mechanics and earlier model The aircraft wake-contrail is initialized as an instantaneous line source in the downstream (y) direction. There are competing needs of relatively large domains in the cross-stream (x) and vertical (z) directions to handle sheared and precipitating contrails and fine grid resolution to treat both the wake-vortex cores and species dispersal from the exhaust jets. To meet these efficiently we employ 1) stretched grids in x and z to concentrate fine grid spacing where it is needed, 2) vertical and cross-stream grid tracking to keep the falling vortices or drifting contrail within the fine grid region, and 3) different grids and domains for different stages of the evolution as the needs change. The x and y boundary conditions are chosen periodic. The Boussinesq approximation and periodic boundary conditions in z are used for early simulation segments on smaller domains (with imposed shifts to account for mean temperature or velocity gradients); the anelastic approximation and a closed domain in z are used for later segments on deep domains. Contrail model a. Fluid mechanics and earlier model The main limitations of the contrail studies of Lewellen and Lewellen (2001a) were the use of simple bulk ice microphysics and omission of coupled radiation or any means of maintaining ambient atmospheric turbulence. We describe the model improvements to surmount these limitations below, with further details given in the appendix. b. Ice microphysics with dynamic local binning Binned microphysics schemes are more numerically costly than their bulk counterparts. For a given application, however, there is often no definitive way to test the adequacy of a bulk scheme other than to compare its results with those of a size-resolved scheme. This motivated the development of the numerically more efficient bin-resolved scheme described here. The main LES code handles the advection, diffusion, and sedimentation of ice species. For ice crystal diffusional growth and sublimation we adapted subroutines from the National Aeronautics and Space Administration (NASA) Ames Community Aerosol and Radiation Model for Atmospheres (CARMA; e.g., Jensen et al. 1994; Ackerman et al. 1995) . For later reference, the basic equation employed for the growth of a crystal of mass m is (see, e.g., Pruppacher and Klett 1997) b. Ice microphysics with dynamic local binning where dS i is the supersaturation, e i is the vapor pressure with respect to ice, L s is the latent heat of sublimation, R y and D* are the gas constant and diffusivity for water vapor (the latter modified by kinetic theory effects for small crystals), K is the thermal diffusivity, and f y is a ventilation coefficient (51 for small spheres). The C and d are the crystal capacitance and characteristic length, both taken equal to the radius r for spherical crystals, and a k 5 2s i-y / (r i R y T) is taken in analogy with the Kelvin equation for droplets but using the surface tension for an ice-water vapor interface s i-y , and ice density r i . The growth rate varies for different sized crystals, via the C factor and Kelvin term, and the size spectrum naturally develops over time, in contrast with a bulk model where the development of only one or more moments of the spectrum are followed. For example, in the simple bulk model used in Lewellen and Lewellen (2001a) , the expression analogous to (1) is for the time evolution of the average mass in the local spectrum, m, a k is set to 1, and Cf y is replaced by a parameterization of the form am b chosen to try to incorporate some of the spectral and shape effects. b. Ice microphysics with dynamic local binning In contrail evolution the range of crystal sizes that is significantly populated varies greatly with location and time (e.g., a spectrum of small crystals in the cores of young contrails versus a spectrum of larger crystals in precipitation streamers). The number of ice bins required locally is typically a modest fraction of the global requirements for the full contrail evolution. Accordingly a dynamic local ice binning, as shown in Fig. 1 , was employed to reduce the computational costs. For each grid cell only the local ice spectrum and thermodynamic variables are passed to the CARMA routines at each time step to update the ice spectrum. In each grid cell the local bin space is dynamically shifted up or down within the global space during the simulation as needed for best following the evolving crystal spectra. Details of the implementation are given in the appendix. b. Ice microphysics with dynamic local binning By default we assume spherical ice crystals, but some effects of crystal shapes have been explored as will be discussed later. Pressure perturbations due to fluid motions are accounted for in computing local saturation conditions. These have a small effect within the trailing vortex cores but are negligible elsewhere. Buoyancy due to latent heat release is included; fragmentation and aggregation of ice crystals are not (as the latter is expected to be very small at contrail temperatures; see, e.g., Connolly et al. 2012) . We have developed a new and numerically efficient scheme for including spectrally resolved homogeneous freezing nucleation but defer its description elsewhere; this process is not generally encountered during contrail evolution unless a persistent updraft is imposed, and so is not included in the simulations considered here. The crystals in this work are treated as pure ice; that is, the aerosol core options in CARMA are not utilized here. b. Ice microphysics with dynamic local binning FIG. 1. Dynamic local binning schematic. A global space of n g ice mass bins defined with fixed mass ratio between neighboring bins (m j11 /m j 5 R m ) such that the full range of ice crystal mass expected during a contrail's evolution is encompassed. In the simulation, only a subset of this bin space with n l , n g bins is employed with the choice, specified by the starting index within the global space, allowed to vary with space and time [i s 5 i s (x, t)] so as to best represent the local evolving ice-size spectrum. The overall CPU and memory needs are effectively reduced by the ratio n l /n g with no loss in simulation accuracy. c. Coupled radiation Radiation processes can play an important role in icecloud evolution (e.g., Dobbie and Jonas 2001) . The effects of radiative heating or cooling of the contrail have been included in the LES by adapting the two-stream radiation routine from CARMA, which is spectrally resolved, using 26 solar and 18 infrared wavelength bins. The work of Gounou and Hogan (2007) comparing a fully 3D radiative transfer model or the independent column approximation (ICA) applied to idealized contrails suggests that the ICA is not an unreasonable choice for contrail modeling, where the concern is the effect of radiation on the contrail dynamics itself (though for computing the radiative forcing of the contrail on Earth, the uncertainty is likely greater; Forster et al. 2012) . Taking advantage of the ICA, we choose the frequency for updating the radiative heating rates in each vertical column independently, dynamically adjusting each as is needed during the evolution. In implementing this scheme, the modification of crystalvapor exchange rates due to direct radiative heating of crystals are not included (since the microphysics updates occur more frequently); however, these contributions are minor corrections for crystal sizes of a few tens of microns or less (Gierens 1994 ) and so should not significantly impact the contrail dynamics. Further details are discussed in the appendix. Given that the contrail is generally optically thin, localized spatially, and that the radiative heating rates typically change more slowly than other dynamics in the simulation, this scheme dramatically reduces the CPU requirements without compromising the accuracy of the computation. Inclusion of both coupled longwave and shortwave radiation now generally increases the simulation time over the same case with no radiation by 20% at most, compared with an overall order of magnitude increase if the radiation update is computed everywhere at each time step. d. Late-time turbulence maintenance Our treatment of ambient turbulence for young contrails was discussed in Lewellen and Lewellen (1996) and Lewellen et al. (1998) . Beyond a few Brunt-Väisälä periods aircraft wake-induced dynamics die away leaving ambient turbulence, wind shear, and large-scale uplift/ subsidence as primary drivers of contrail evolution. The most relevant conditions are stably stratified with a mixture of low-amplitude gravity waves and turbulence since aircraft avoid regions of strong turbulence for safety. The turbulence levels are expected to be lower than generally found in natural cirrus since the latter usually requires an active source of uplift for its nucleation whereas contrails do not. The generation, maintenance, and properties of turbulence in the upper troposphere and lower stratosphere are complex and not well understood (see, e.g., Quante and Starr 2002) . Accurately simulating the interaction of gravity waves producing intermittent patches of turbulence with LES would require both very large domains (for the spectrum of traveling waves) and relatively fine grids (for resolving turbulence generation in breaking waves); however, for useful sensitivity studies we assume that it is sufficient to simulate quasi-steady ambient turbulence with realistic, statistically reproducible behavior across the range of conditions likely to be encountered at cruise flight level, regardless of how the motions are initially generated or maintained. d. Late-time turbulence maintenance Unstratified homogeneous isotropic turbulence is completely categorized statistically given the turbulent dissipation rate. This is not true for the stratified case where the mix of waves and turbulence, along with the energy transfer rate, can differ significantly on different scales and in time. Moreover, in this case there can be significant energy loss through wave transport out of the domain or temperature mixing within. The parameter space is multidimensional, including at least stratification, mean shear, gravity wave spectrum, turbulence strength on different length scales, and level of intermittency. To provide representative solutions we perform a separate LES of decaying turbulence to produce velocity perturbation fields that are then added periodically to the contrail LES. Varying the amplitude of the initial temperature perturbations T amp , the age of the added field f age , and time between additions t add along with the mean shear and stratification allow different quasi-steady turbulence states to be achieved. The fields produced have a realistic patchy character as expected since in between regenerative additions they represent LES Navier-Stokes solutions for the desired stratification and shear. This is in contrast to the scheme employed in Paugam et al. (2010) , where velocities from a simulation with an unstable Richardson number are imposed on the desired stable conditions. Further details and sample results are given in the appendix. Including turbulence regeneration is less important if either coupled radiation or mean wind shear drive more significant levels of turbulent diffusion in and around the contrail. e. Quasi-3D simulation For exploring some sensitivities, a full 3D LES should not be required. A caricature of the most important fluid motions-the strong downward motion of the wake vortices and the expansion of the plume through mixing with the ambient environment-should suffice. For this purpose we conduct some simulations with only a minimal number of downstream grid points (typically six independent points plus two additional for imposing periodic boundary conditions). On larger scales the simulation is effectively 2D, but 3D eddies are crudely resolved in a narrow spatial range of scales determined by the downstream resolution (and hence the choice of downstream domain size). This 3D eddy scale was roughly chosen in the range expected to be most important-on the order of a fraction of the vortex spacing for the wake-dominated contrail and of the contrail depth (before precipitation) for older contrails-with the results proving to be rather insensitive to the precise choice. The dramatic reduction in memory and CPU time requirements of this quasi-3D approximation (Q3D) allows a much larger parameter space to be explored as well as the ability to test bin or grid resolutions that might be prohibitive in 3D. Because of the small downstream extent, there is a larger spread in results between different turbulent realizations than arises with full 3D simulations (cf. Fig. 9 ), so averaging over a small ensemble of Q3D simulations is sometimes desirable; this is not an issue for studying sensitivities that do not strongly affect the fluid dynamics (e.g., changing ice microphysics). e. Quasi-3D simulation As will be shown below, the Q3D approximation proves to be a much better physical approximation to the full dynamics than one would expect a priori. It is sometimes true that only a limited subset of degrees of freedom need be included to represent the physics that determines some averaged quantities. A discussion of why that might be the case here is given in section 4a. The success of such a reduced treatment is not entirely unexpected-we have seen similar in reproducing some mean aspects of atmospheric boundary layer behavior using LES with greatly restricted degrees of freedom (e.g., Lewellen and Lewellen 1998 )-but ultimately the utility in any given case can be confirmed only by comparison with fully 3D simulations. and assumed propulsion efficiency of 30% are given for illustration. Table 1 lists the domains and grid resolutions used for different time periods. The simulations are initialized from 2D Boeing wake rollup calculations (Czech et al. 2005) in two steps: an initial high-resolution short-domain run without ice for the early turbulent spread of the exhaust jet followed by the introduction of ice crystals distributed spatially like an exhaust tracer species. Only cases satisfying the Appelman criterion for contrail formation (see, e.g., Schumann 1996) are considered. It is expected that essentially all of the contrail ice crystals are produced during the initial rapid mixing and cooling of the exhaust jets. Accordingly, the total crystal number is set consistent with the fuel flow rate and an assumed ice crystal number emission index EI iceno . This quantity is effectively an aircraft variable in our treatment. We consider variations in the simulations around a nominal value of 10 15 particles per kilogram of fuel, consistent with in situ aerosol measurements (Schröder et al. 2000; Anderson et al. 1998 ) and jetplume ice nucleation modeling (Kärcher and Yu 2009) . Delayed nucleation tied to the contrail is expected to be of secondary richardsonvictoria@example.com. In the wake-dominated regime vertical displacements are almost entirely downward (relative to the ambient atmosphere at rest). At late times radiative heating of the contrail can produce sustained updrafts, but supersaturations are depleted by the existing contrail itself, limiting the possibilities for new ice nucleation [a result supported by the simulation tests of Unterstrasser and Gierens (2010b) ]. Further details on simulation initialization and some sensitivity tests are given in the appendix. Contrail evolution from seconds to hours Contrails exhibit a wide range of behavior, as even casual observation of the sky under different conditions will attest. The primary sets of 3D simulations considered here are summarized in Table 2 . Temperature T and relative humidity with respect to ice (RH i ) are the two most important ambient variables governing contrail evolution. Persistent contrails are found within sizable ranges of both (Iwabuchi et al. 2012) . To best illustrate different dynamical elements that arise in different regimes we highlight in particular ''warm'' cases with T 5 225 K (toward the upper end of the observed range) and ''cold'' cases with T 5 205 K (toward the low end of the observed range) along with a more typical temperature for cruise altitude of 218 K. Note that for fixed RH i warm and cold also correspond respectively to ''relatively moist'' versus Contrail evolution from seconds to hours ''relatively dry.'' Both higher (130%) and lower (110%) values of RH i within the observed range were considered. In the naming convention used for the simulations the first number indicates RH i , the second number indicates T, T and the trailing letter indicates the ambient-turbulence level. In some cases multiple turbulent realizations of a given simulation were performed by choosing different random number seeds governing the perturbations for wake and ambient-turbulence initializations; this can give some indication of the statistical uncertainty in the results. The qualitative behavior from these sample cases is representative of that we have seen in much larger sets of simulations (e.g., in Part II). For each simulation the results constitute a fivedimensional space: time-evolving ice-size spectra in three spatial dimensions. To assess the effects of different physical parameters or modeling choices it is convenient to distill this to a few simple metrics. For this purpose we choose the time-evolving total (i.e., integrated over the x and z directions, averaged over y) ice crystal number N(t), mass M(t), and surface area S(t) per meter of flight path (Figs. 2 and 3) and the contrailaveraged ice crystal size spectrum at selected times. In the absence of crystal growth/sublimation, pure fluid dynamic motions would leave these measures invariant. It is natural to characterize the radiative effects of cloud layers in terms of an optical depth, but this is an ambiguous measure for contrails given their limited horizontal extent and nonuniform cross section. The quantity S(t) provides an approximate measure of optical depth integrated for the contrail as a whole, unchanged by these ambiguities. It is approximately twice the total extinction E defined in Unterstrasser and Gierens (2010a,b) . For a given orientation and width definition one can recover a mean optical depth from S in the limit of approximating the extinction efficiency as independent of crystal size, Q ext ' 2; for example, Contrail evolution from seconds to hours for optical depth t on a vertical path, contrail width W, and crystal number densities N j for crystal radius r j . We consider only properties of the contrail itself; to determine the radiative impact of the contrail on Earth's surface, one would need to fold in information about the incoming radiation and environments above and below the contrail as well. Contrail evolution from seconds to hours Figures 4-6 illustrate some of the contrail spatial evolution. The main events of the wake-dominated regime (cf. Fig. 5 )-exhaust entrainment-detrainment into the trailing vortex system, vortex fall, linking, ring dynamics and decay, and Brunt-Väisälä oscillation of the wake--affect the contrail appearance in the same ways as was described when simulating with bulk microphysics (Lewellen and Lewellen 2001a) or, for many facets, just a passive exhaust tracer (Lewellen and Lewellen 1996) . We emphasize here structure evolution at later times and changes that depend on spectrally resolved microphysics. Contrail evolution from seconds to hours For much of the evolution most of the contrail volume is driven close to ice saturation because crystal number densities are typically large in young contrails and time scales long in older contrails. As a consequence, for conditions with constant ambient RH i , saturation mixing ratio q s , and air density r, M(t) scales essentially with the growth of the plume cross-sectional area, A(t): Contrail evolution from seconds to hours At very low temperatures the water from the aircraft exhausts becomes competitive in the young contrail and should be added to (3). Until precipitation becomes important, the atmospheric pressure p enters the dynamics significantly only through q s and r but drops out in the combination in (3); the insensitivity to p that this predicts for young contrail dynamics was confirmed in simulation tests. The value of M(t) increases with plume growth dominantly driven by exhaust jet mixing (until ;10 s depending on aircraft), vortex fall (;300 s depending on , and no mean uplift, subsidence, or wind shear (though the ambient-turbulence fields do introduce such motions locally). To illustrate precipitation dynamics, we consider deep supersaturated layers with initial profiles with constant RH i above flight level down to 1 km below, linearly dropping to RH i 5 richardsonvictoria@example.com km below flight level and below. aircraft and stratification), Brunt-Väisälä oscillation (;20 min depending on stratification), ambient turbulence, and mean wind shear (which is not included in the simulations here but considered at length in Part II). Eventually crystals around the plume edges become large enough that sedimentation velocities exceed turbulent mixing velocities and precipitation becomes the dominant means of bringing ice crystals into contact with supersaturated air. The resultant growth in ice crystals and increase in fall velocities provides a positive feedback-essentially a ''precipitation instability.'' A rapid increase in vertical extent of the contrail ensues [(e.g., from ;(40 to 90) min in Figs. 4 and 6] until the full depth of the supersaturated layer is reached. Eventually the total ice mass falls as ice is lost to subsaturated regions below. Often a rough equilibrium is established between new ice growth on the contrail edges (driven by turbulent mixing) and precipitative ice loss within the older parts in the center. The precipitation growth of the contrail occurs much sooner in the warm cases than the cold. Consideration of a large set of simulations (Part II) shows that the time scale for this initial growth of the contrail precipitation plume downward is essentially that required for a single crystal growing in the given ambient conditions to fall through the supersaturated-layer depth, depending strongly on T and RH i but much less on EI iceno , turbulence level, shear, or coupled radiation. At late-enough times (not reached in all of the simulations), the entire remaining plume precipitates downward and N(t) and S(t) drop off rapidly (cf. Figs. 3 and 6 ). Ambient-turbulence levels or coupled radiation can affect the contrail evolution significantly over time scales longer than ;30 min (Figs. 2 and 3) . Increased turbulence increases plume dispersion and with it M(t) and S(t). In the absence of precipitation the growth seen in the simulations is roughly a power law, M(t) ; t 0.8 or t 1 for the ''low'' and ''moderate'' turbulence, respectively. This is consistent with the rough ;t 0.8 dilution rate for exhaust species out to a few hours found from in situ measurements by Schumann et al. (1998) . Including coupled radiation can either increase or decrease the Contrail evolution from seconds to hours ice crystal number, ice surface area, and contrail lifetime (e.g., Fig. 3 ), depending on the regime. Different regimes of coupled radiation behavior are explored in Part II. Early in the evolution the wavelength of the most prominent downstream periodicity in the ice distribution (e.g., Fig. 5 ) is set by the Crow instability (Lewellen and Lewellen 1996) . In the simulations there is a natural clumping that occurs so that the dominant wavelength tends to increase in time. If conditions are such as to foster more rapid precipitation, these structures will be directly reflected in the precipitation patterns as well. Since the lowest precipitation streamers originate from FIG. 4 . Mean vertical profiles (z 5 0, indicating flight level) of (a) engine tracer concentration, (b) ice crystal number, (c) ice mass, and (d) ice surface area, per meter of flight path from simulation S10T225m. Times shown are 10, 90, 300, 900, and 1800 s; and 1, 1.5, 3, 6, 9, and 12 h (colors progressing through the spectrum from red to blue). Contrail evolution from seconds to hours FIG. 5. Drift plot of cross-stream-integrated ice surface area showing young contrail development for cases (top) S10T205m and (bottom) S10T218m. The horizontal axis sweeps over both time and downstream distance (drift velocity 5 12 m s 21 ) to illustrate both temporal and spatial evolution. The plot format as described in Lewellen et al. (1998) is motivated by scanning lidar measurements. the younger contrail at flight level, and subsequent streamers precipitate from two sides of the parent population, there is a natural tendency for the precipitation streamers to exhibit a shorter periodicity than the ice mass distribution at flight level for the mature contrail. This has been reported in some contrail observations with large ice mass (Atlas et al. 2006) . Over time ambient turbulence, waves, and shear can destroy all remnants of the periodic structure inherited from the young contrail. Contrail evolution from seconds to hours In the absence of ice nucleation, N(t) monotonically decreases. The most dramatic losses are tied to adiabatic heating in the falling trailing vortex system producing subsaturated conditions (Lewellen and Lewellen 2001a; Sussmann and Gierens 1999) , but there are also significant number losses during this period and throughout the later evolution (discussed more below) that do not depend on the local q t falling below the ice saturation mixing ratio q s as well as losses at later times through precipitation. Initially the spatial evolution of ice crystal number parallels that of a passive exhaust tracer (e.g., Fig. 4 )-driven by advection and diffusion. The differences in these distributions grow in time because of sedimentation and evaporative loss of ice. The spatial distribution of crystal number is also seen to be far less uniform during the contrail evolution than that of the ice mass, with the uniformity of the ice surface area distribution lying in between. In particular the number densities in the precipitation streamers are typically orders of magnitude less than those near flight level for most of the evolution. In the contrail-averaged size spectra (Fig. 7) the precipitation plume contribution adds only a small shoulder to the number distributions; nonetheless this sometimes dominates the ice mass (cf. Fig. 7c) . Contrail evolution from seconds to hours The ice crystal size spectrum shifts to larger sizes in time as crystals are lost. There is often a long period in which the spectral shape (displayed logarithmically) is nearly constant, just changing in magnitude and mean crystal size (cf . Fig. 7a) ; the dynamics responsible are discussed more below. The local spectra differ significantly in different spatial regions: around the original flight level small crystals and high number densities dominate, at the plume edges larger crystals arise, and in the precipitation plume the crystal size progressively increases, and number densities decrease, with distance from flight level. These differences between core and FIG. 6. Drift plot of (top) vertically integrated and (bottom) cross-stream-integrated ice surface area showing long-time contrail development for case S10T225m with coupled longwave radiation with T sfc 5 298 K. The drift format is as in Fig. 5 , but over much longer times with a slower drift velocity (2 m s 21 ). Contrail evolution from seconds to hours precipitation regions are well known already from in situ measurements (e.g., Heymsfield et al. 1998; Lawson et al. 1998; Atlas et al. 2006 ) and 2D simulations Unterstrasser and Gierens 2010b) ; the latter have demonstrated as well that the vapor pressure is driven near ice saturation in the contrail core as is found here. The evolution of the spectral widths in Fig. 7 are in at least rough agreement with the measurements within contrails at different times of Schröder et al. (2000) , though differences in (and incomplete knowledge of) the ambient conditions in the measurements precludes a more quantitative comparison. Idealized simulations with some physics turned off help to isolate the different physical mechanisms influencing each contrail metric. Figure 8 provides an example. Among the four simulations only two distinct behaviors of N(t) arise, depending on whether the Kelvin term in the microphysics is included or not, and only two behaviors of M(t), depending on whether or not precipitation is allowed. The crystal size spectra are paired depending on the former effect for smaller size crystals but on the latter for larger. Of the two effects precipitation dominates the influence on the large crystal spectra and M(t) evolution, while the Kelvin effect strongly influences the small crystal spectrum and N(t) evolution. The surprising importance of the Kelvin effect even at late times is discussed below. small-scale downstream clumping, the Crow instability produces characteristic periodic downstream puffs with spacing of a few wingspans, these in turn influence subsequent patterns of precipitation streamers, and ambient wind shear and wave fields modulate different patterns in older contrails and contrail cirrus. Which if any of these 3D dynamics significantly affect contrailaveraged metrics? Figure 9 compares the cold and warm B-767 simulations from an ensemble of turbulent realizations of Q3D simulations for the same conditions. The agreement is quite respectable and is representative of results we have found when comparing Q3D simulation with 3D across the range of conditions we have considered. In contrast, a 2D simulation run within the LES (not shown) compares poorly. The Q3D simulations differ from the 2D in effectively allowing one scale of 3D motions to be crudely resolved, chosen on the order of a fraction of the vortex spacing for the wakedominated contrail and of the contrail depth (before precipitation) for older contrails. The Q3D simulation excludes 3D eddies on larger scales. The wake dynamics and exhaust dispersion for the first tens of seconds do not critically involve such modes, nor does aging contrail cirrus (where shear typically dominates the large-scale dispersion). The 3D component omitted from Q3D simulations that plausibly is of greatest quantitative importance is the wake-vortex dynamics at intermediate times. The depth reached by the wake-vortex system affects the fraction of ice crystals surviving the wake regime and the later contrail spread (because subsequent vertical growth is inhibited by stratification). A priori the Crow instability dynamics would seem critically involved in setting the average depth, but the success of the Q3D approximation over a large parameter space suggests otherwise. The results are consistent with the average vertical extent being determined by the balance between the downward wake momentum and the positive buoyancy acquired in falling through the ambient stratification, together with smaller-scale entrainment (detrainment) into (out of) the vortex system. Both Q3D and 3D simulations represent these elements in approximately the same way. Fully 3D simulations permit reconnection and reorientation of vortex segments, which affect the contrail appearance greatly (e.g., Fig. 5 ) but its average properties to a much lesser extent. Mixing scales smaller than the vortex spacing are important, however. Different 2D treatments of the descent of a wake-vortex in a stably stratified atmosphere can give widely varying predictions for the evolution [see, e.g., Table 1 of Widnall (1975) ] depending largely on the entrainment-detrainment assumptions made. In our 2D simulations (not presented here) the mixing into and out of the falling vortex system is underestimated (the subgrid model being designed for 3D rather than 2D applications). This leads to a slowly decreasing spacing of the vortex pair, a downward acceleration, and a greatly overestimated total vertical extent. The Q3D simulations develop some resolved smaller-scale mixing leading to an essentially similar trajectory of descent of the vortex system as that found in the 3D simulations. Figure 8 shows that there are crystal losses dependent on the Kelvin effect that persist even to ages of several hours. This component, which we dub ''in situ loss,'' can occur even without conditions becoming subsaturated and can be the dominant loss mechanism at late times until precipitation into subsaturated layers wins out. Figure 10 shows that in situ loss is critical in the wakedominated regime as well. Crystal-loss fractions are significant even for large RH i and greatly exceed the losses found with the Kelvin effect turned off or with the bulk microphysics of Lewellen and Lewellen (2001a) . This irreversible loss leads to significant differences in contrail properties that persist in time. b. Ice crystal number loss The in situ loss rates and the shape of the ice crystal spectrum that results are derived analytically from the governing equations for diffusional growth of spherical crystals in Lewellen (2012); here we limit ourselves b. Ice crystal number loss to summarizing the basic physical mechanism. For small spherical crystals, surface energy (Kelvin) effects shift the critical vapor supersaturation level that separates sublimation from growth away from zero to a value depending on crystal radius, 1 2 e a k /r [cf. b. Ice crystal number loss (1)]. Given that a k ; 2 3 10 23 mm is much less than the mean crystal sizes encountered in the contrail, the magnitude of the Kelvin-dependent effect seen in the figures is at first surprising. Its origin is in the local competition between different sized crystals together with dynamics forcing the system close to saturation. In the young descending contrail adiabatic heating forces the water vapor mixing ratio q y below q s but the high number densities ensure that conditions never get far from equilibrium (the simulations rarely show vapor saturation levels in these regions below 97% and generally above 99%). Even the small differences due to the Kelvin effect for modest crystal sizes are relevant in comparison to the small levels of subsaturation maintained in most regions experiencing crystal loss. This broadens the size spectrum leading to more rapid crystal loss. At late times conditions are naturally driven toward equilibrium (q y / q s ) so they appear slightly supersaturated to the larger crystals but, because of the Kelvin effect, slightly subsaturated to the smallest. Large crystals scavenge moisture from small ones leading to a slow but persistent crystal loss. This represents an example of the more general phenomenon known as Ostwald ripening [as pointed out to us by J. Ross (2009), personal communication] or ''spectral ripening,'' which has also been considered for liquid water clouds (Çelik and Marwitz 1999; Wood et al. 2002) . This effect is amplified in contrail evolution by long integration times allowing a small effect to accumulate and low turbulence and vertical velocities preventing other dynamics from taking over (in contrast to most natural-cirrus evolution). It is the combination of the growth in mean crystal size and the spectral broadening due to the moisture scavenging that permits the nearly constant spectral shape (displayed logarithmically) noted earlier in Fig. 7a to arise. b. Ice crystal number loss The mean in situ loss rate is unmodified by modest vertical oscillations but can be eliminated by sufficiently large plume mixing or ascent or low-enough number concentrations (Lewellen 2012) . While the in situ loss rates can be quantitatively examined in the simulations, the rates for actual contrails are more uncertain. The Kelvin effect is quantitatively more poorly understood for ice crystals than for water drops and, as noted below, uncertainties in the ice deposition coefficient factor in as well. While a shift in the critical vapor saturation level is certainly present for small crystals, and the basic effect of large crystals growing at the expense of smaller ones is likely robust, there may be significant quantitative changes owing to crystal growth and shape effects. There are also smaller residual crystal losses that can remain even in the absence of the Kelvin effect (barely evident in Fig. 8 ) depending on the combined effects of gravity wave magnitude and parcel mixing. c. Sensitivity to EI iceno The changes in crystal-loss mechanisms seen with spectrally resolved microphysics alter the sensitivities to EI iceno . Figure 11 shows sample Q3D results varying EI iceno . Because of the in situ crystal losses the fraction of the total crystals lost in the wakedominated regime rises with EI iceno faster than the expectations from bulk treatments (e.g., Lewellen and Lewellen 2001a; Unterstrasser et al. 2008) . Moreover, this sensitivity remains for older contrails, contrary to the expectations based on bulk treatments that the fraction of crystals lost would be approximately independent of EI iceno (e.g., Unterstrasser and Gierens 2010a). One consequence is that the sensitivity to EI iceno . ;10 14 kg 21 diminishes significantly over time; in Fig. 11 an initial factor of 1000 difference in crystal number is reduced to ;3 within a few hours. This insensitivity is one sided, however: for smallenough EI iceno the crystal number densities are reduced sufficiently so that in situ losses are absent over FIG. 10 . Total ice crystal number evolution in young contrails simulated with bulk microphysics (long dashed lines), binned microphysics (solid lines), or binned without the Kelvin effect (short dashed lines). Results from Q3D simulations with ambient T 5 218 K and RH i 5 110% (thin lines) or 130% (heavy lines). The number losses around 100 s are forced by wake-vortex descent and those around 700 s are forced by descent in a Brunt-Väisälä oscillation of the wake plume. c. Sensitivity to EI iceno the time scales available. The dramatic changes in contrail evolution suggested in Fig. 11 from reductions in EI iceno below this threshold are considered more in Part II. It is interesting that the EI iceno suggested by both jet-plume ice nucleation modeling for current aircraft [the soot-dominated regime of Kärcher and Yu (2009) ] and in situ aerosol measurements (Schröder et al. 2000; Anderson et al. 1998 ) is of the same order of magnitude (;10 15 kg 21 ) as the lower range over which in situ losses are found to be large in the wakedominated regime here. Even if the mechanisms limiting the crystal numbers in the initialization stage considered by Kärcher and Yu (2009) were absent, the ice crystal numbers in contrails would tend to be forced to the order of magnitude consistent with in situ measurements anyway by the crystal losses during the wake-dominated regime. The quantity M(t), governed dominantly by plume volume and ambient conditions, is relatively insensitive to EI iceno unless it drops below about 10 14 kg 21 (Fig. 11b) ; the time scale for reaching the equilibrium state in (3) exceeds plume expansion time scales if ice number densities are small enough. d. Sensitivity to water from fuel consumption The ice crystal number losses in the young contrail are greatly reduced at colder temperatures (e.g., Figs. 2 and 5). Reduced crystal loss for dryer conditions and thus smaller crystals seems counter intuitive, but it proves to be an effect of water from fuel combustion. The water from the engines raises q t /q s locally in the contrail (particularly in those regions with highest crystal number densities) to a much greater extent the smaller q s , and hence the smaller T. For example in S10T205 the engine water raises q t /q s in the young (;30 s) contrail core from the ambient 110% to over 200% while in S30T225 the corresponding increase is only from 130% to ;138%-explaining the enhanced crystal survival in the former case despite the decreased ambient RH i . As discussed more in Part II, this is one factor in raising the potential significance of cold contrails despite their relatively low ice water content. That very low temperature conditions lead to greatly reduced ice number losses was noted in Unterstrasser et al. (2008) but attributed to changes in growth-sublimation rates with temperature. e. Further microphysics uncertainties 1) ICE CRYSTAL SHAPE EFFECTS For the present simulations we have assumed spherical ice crystals for simplicity. In situ sampling of contrails suggest that the shapes of small (;20-mm diameter or less) crystals that are found in young contrails or contrail cores are quasi spherical (e.g., Gayet et al. 1998; Schröder et al. 2000; Febvre et al. 2009 ) while the shapes of the larger ones found in the contrail periphery or in precipitation streamers are more complex (e.g., bullet rosettes, extended columns, or irregular) (e.g., Lawson et al. 1998; Heymsfield et al. 1998; Gayet et al. 1998) . Iwabuchi et al. (2012) found results consistent with this picture from a large survey of the optical properties of persistent contrails. Crystal shapes can affect growth rates, sedimentation velocities, and optical properties. To assess their potential importance for contrail evolution Meza (2010) added to the current LES model the option of using nonspherical shapes: hexagonal columns, plates, bullet rosettes, or a mass-dependent combination of shapes (near spherical for small sizes, hexagonal columns 1) ICE CRYSTAL SHAPE EFFECTS of varying aspect ratio for midrange, and bullet rosettes for the largest). The shape effects were incorporated through modifications of C, f y , and d in the ice growth [(1)] and of sedimentation velocity as a function of crystal mass; the effects on coupled radiation were only partially included. The precise implementation and comparison of results for different shapes are given in Meza (2010), but we summarize the main findings here. For realistic shape assumptions and in the absence of coupled radiation none of the differences in N(t) and M(t) for different conditions due to shape effects were found to be large in comparison to the Q3D realization spread. Some more significant differences were found in S(t) but were consistent with being primarily geometrical rather than dynamical, representing the change in surface area for fixed number density and mass given by shape change alone. This suggests that while it may be important to consider shape differences when assessing the radiative impact of a contrail, it might be sufficient to perform simulations with spherical crystals to determine N(t) and M(t) and then assess shape effects on the radiative impact afterward. The exception would be cases where the radiative heating-cooling of the contrail itself strongly couples into its dynamics, together with a heating-cooling rate that is strongly shape dependent. Preliminary sets of simulations with coupled radiation in Meza (2010) did not identify any dramatic crystal shape-dependent effects, but more complete treatment of the radiation for nonspherical shapes is required before drawing final conclusions. 1) ICE CRYSTAL SHAPE EFFECTS The relative insensitivity of the contrail evolution to crystal shape that was found is perhaps not unexpected: small ice crystals in the contrail do not differ much from spheres, while large crystals have significant fall velocities shortening lifetimes and so limiting their contribution to contrail-integrated metrics. 1) ICE CRYSTAL SHAPE EFFECTS 1 For any compact shape, such as a hexagonal column with aspect ratio around one, the shape-dependent factors that might directly affect contrail evolution (e.g., the capacitance factor or sedimentation velocity) do not vary much from the spherical results (even though the optical properties might be quite distinguishable). Further, within the contrail core, the growth of ice mass is generally
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