Persistent Contrails and Contrail Cirrus. Part I: Large

Persistent Contrails and Contrail Cirrus. Part I: Large-Eddy Simulations fromInception to Demise

D. C. LEWELLEN, O. MEZA,* AND W. W. HUEBSCH

West Virginia University, Morgantown, West Virginia

(Manuscript received 2 October 2013, in final form 10 March 2014)

ABSTRACT

Large-eddy simulations with size-resolved microphysics are used to model persistent aircraft contrails and

contrail-induced cirrus from a fewwing spans behind the aircraft until their demise aftermany 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 evo-

lution 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 signifi-

cantly over time, and uncertainties in the ice deposition coefficient andKelvin 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.

1. Introduction

It is not uncommon for portions of the upper tropo-

sphere 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, prop-

erties, and effects of persistent contrails is of clear im-

portance (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).

Contrail evolution is a complex process dependent on

many properties of the aircraft and of the ambient at-

mosphere. Exploring these sensitivities involves dy-

namics on length scales ranging from nanometers for the

ice microphysics to kilometers for the late-time atmo-

spheric 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 representa-

tions suitably faithful.

Persistent contrail evolution involves the growth of ice

within an expanding wake-plume volume in an ice-

supersaturated environment. In the young contrail the

volume expansion is driven by aircraft-induced dynam-

ics: engine jets, trailing vortex interactions, and residual

Brunt–Väisälä oscillations set up by the wake-vortex

*Current affiliation: Inter American University of Puerto Rico,

Bayamon, Puerto Rico.

Corresponding author address: David C. Lewellen, Department

of Mechanical and Aerospace Engineering, P.O. Box 6106, West

Virginia University, Morgantown, WV 26506-6106.

E-mail: [email protected]

DECEMBER 2014 LEWELLEN ET AL . 4399

DOI: 10.1175/JAS-D-13-0316.1

� 2014 American Meteorological Society

mailto:[email protected]

descent. Effects with longer time scales take over lateron: atmospheric dispersion, ice sedimentation, andbuoyant forcing due to radiative or latent heating. Pre-vious 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 Miake-

Lye 2010; Naiman et al. 2011; Unterstrasser and Sölch2010) 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 mi-

crophysics 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.

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 ambient-

turbulence-dominated regimes out to the demise of the

resulting contrail cirrus many hours later. Novel strate-

gies are used to improve numerical efficiency. These

capabilities are used to explore some of the critical dy-

namics involved in contrail evolution and to assess what

is required for adequate simulation of different facets—

for example, whether detailed initial conditions, 3D fluid

mechanics, binned microphysics, and/or realistic ambi-

ent 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 pa-

rameter space with the simulations.

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 para-

metric study focused on young contrail behavior will be

presented elsewhere, including different aircraft and

physically based parameterization of the results. A sig-

nificant 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 dif-

ferent regimes of ice-size spectra development has been

presented in Lewellen (2012).

Themodel 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.

2. Contrail model

a. Fluid mechanics and earlier model

The starting point for the present work was the large-

eddy-simulation (LES) model used for studies of wake

dynamics, wake-plume chemistry, and contrails de-

scribed in Lewellen and 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 Qli, total water mixing ratio qt, ice number

densities, and passive species concentrations. The sub-

grid model uses a quasi-equilibrium second-order tur-

bulence closure scheme with a prognostic equation for

the subgrid turbulence kinetic energy (TKE) and di-

agnostic equation for the subgrid turbulence length scale

L. The physical damping of turbulence by stable tem-

perature 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 exten-

sively for tornado simulations (Lewellen et al. 2000;

Lewellen and Lewellen 2007).

The aircraft wake–contrail is initialized as an in-

stantaneous 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 spe-

cies dispersal from the exhaust jets. To meet these effi-

ciently we employ 1) stretched grids in x and z to

concentrate fine grid spacing where it is needed, 2) ver-

tical 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. TheBoussinesq

approximation and periodic boundary conditions

4400 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 71

in z are used for early simulation segments on smaller

domains (with imposed shifts to account for mean

temperature or velocity gradients); the anelastic ap-

proximation and a closed domain in z are used for later

segments on deep domains.

Themain 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 appli-

cation, 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 moti-

vated the development of the numerically more efficient

bin-resolved scheme described here. The main LES

code handles the advection, diffusion, and sedimenta-

tion 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 Atmo-

spheres (CARMA; e.g., Jensen et al. 1994; Ackerman

et al. 1995). For later reference, the basic equation em-

ployed for the growth of a crystal of massm is (see, e.g.,

Pruppacher and Klett 1997)

dm

dt5

4pCfy(11 dSi 2 eak/d)"L2s

KRyT2 1

RyT

ei(T)D*

# , (1)

where dSi is the supersaturation, ei is the vapor pressure

with respect to ice,Ls is the latent heat of sublimation,Ry

and D* are the gas constant and diffusivity for water va-

por (the latter modified by kinetic theory effects for small

crystals),K is the thermal diffusivity, and fy 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 ak 5 2si–y/

(riRyT) is taken in analogy with the Kelvin equation for

droplets but using the surface tension for an ice–water

vapor interface si–y, and ice density ri. 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 de-

velopment 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, ak is set to 1, and Cfy is

replaced by a parameterization of the form amb chosen to

try to incorporate some of the spectral and shape effects.

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 re-

quired 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.

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 mo-

tions 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 en-

countered during contrail evolution unless a persistent

updraft is imposed, and so is not included in the simu-

lations considered here. The crystals in this work are

treated as pure ice; that is, the aerosol core options in

CARMA are not utilized here.

FIG. 1. Dynamic local binning schematic. A global space of ng ice

mass bins defined with fixed mass ratio between neighboring bins

(mj11/mj5Rm) 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 nl, ng bins is employedwith the

choice, specified by the starting index within the global space, al-

lowed to vary with space and time [is 5 is(x, t)] so as to best rep-

resent the local evolving ice-size spectrum. The overall CPU and

memory needs are effectively reduced by the ratio nl/ngwith no loss

in simulation accuracy.


c. Coupled radiation

Radiation processes can play an important role in ice-

cloud evolution (e.g., Dobbie and Jonas 2001). The ef-

fects 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 con-

trails 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 con-

trail 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 crystal-

vapor exchange rates due to direct radiative heating of

crystals are not included (since themicrophysics 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 sig-

nificantly 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. In-

clusion of both coupled longwave and shortwave radi-

ation now generally increases the simulation time over

the same case with no radiation by 20% at most, com-

pared 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 con-

trails was discussed in Lewellen and Lewellen (1996) and

Lewellen et al. (1998). Beyond a few Brunt–Väisälä pe-riods aircraft wake-induced dynamics die away leavingambient turbulence, wind shear, and large-scale uplift/subsidence as primary drivers of contrail evolution. Themost relevant conditions are stably stratified with a mix-ture of low-amplitude gravity waves and turbulence sinceaircraft avoid regions of strong turbulence for safety. Theturbulence levels are expected to be lower than generallyfound in natural cirrus since the latter usually requires anactive source of uplift for its nucleation whereas contrails

do not. The generation, maintenance, and properties ofturbulence in the upper troposphere and lower strato-sphere are complex and not well understood (see, e.g.,Quante and Starr 2002). Accurately simulating the in-

teraction of gravity waves producing intermittent patches

of turbulence with LES would require both very large

domains (for the spectrum of traveling waves) and rela-

tively 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 re-

producible behavior across the range of conditions likely

to be encountered at cruise flight level, regardless of how

the motions are initially generated or maintained.

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 stratifica-

tion, mean shear, gravity wave spectrum, turbulence

strength on different length scales, and level of in-

termittency. To provide representative solutions we

perform a separate LES of decaying turbulence to pro-

duce velocity perturbation fields that are then added

periodically to the contrail LES. Varying the amplitude

of the initial temperature perturbations Tamp, the age of

the added field fage, and time between additions taddalong with the mean shear and stratification allow dif-

ferent 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 cou-

pled 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 themost 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 mini-

mal 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 ex-

plored 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).

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 com-

parison with fully 3D simulations.

f. Initialization and modeling choices

The initialization, modeling, and grid choices are

partly aircraft specific. Those used here for a B-767 with

wingspan of 47.25m, flight speed 237m s21 at altitude of

10.7 km, total circulation in the wake of 391m2 s21, fuel

flow 5.78 kg km21, specific combustion heat 43MJkg21,

and assumed propulsion efficiency of 30% are given for

illustration. Table 1 lists the domains and grid resolu-

tions 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 theAppelman criterion for

contrail formation (see, e.g., Schumann 1996) are con-

sidered. 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

EIiceno. This quantity is effectively an aircraft variable in

our treatment.We consider variations in the simulations

around a nominal value of 1015 particles per kilogram

of fuel, consistent with in situ aerosol measurements

(Schröder et al. 2000; Anderson et al. 1998) and jet-

plume ice nucleation modeling (Kärcher and Yu 2009).Delayed nucleation tied to the contrail is expected to be

of secondary importance at most. In the wake-dominated

regime vertical displacements are almost entirely down-

ward (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.

3. 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 rel-

ative humidity with respect to ice (RHi) are the two most

important ambient variables governing contrail evolu-

tion. 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 225K

(toward the upper end of the observed range) and ‘‘cold’’

cases withT5 205K (toward the low end of the observed

range) along with a more typical temperature for cruise

altitude of 218K. Note that for fixed RHi warm and cold

also correspond respectively to ‘‘relatively moist’’ versus

TABLE 1. Grid and ice binning parameters for 3D simulations.

Grid

Time

period

Domain

(x 3 y 3 z km3)

Fine grid spacing

(Dx 3 Dy 3 Dz m3)

1 0–2 s 0.84 3 0.052 3 1 0.4 3 0.4 3 0.4

2 2–90 s 0.84 3 0.26 3 1 0.6 3 2. 30.6

3 90–300 s 0.84 3 0.26 3 1 1.25 3 1.6 3 1.25

4 5–15min 1.52 3 1.04 3 1.6 5 3 4.8 3 5.

5 15–90min 6 3 2.08 3 4 16 3 16 3 16

6 1.5–12 h 12 3 4.16 3 4 25 3 28 3 25

Ice bins 48 global, 20 local

50-nm–2.6-mm radius; mass ratio 2


‘‘relatively dry.’’ Both higher (130%) and lower (110%)

values of RHiwithin the observed rangewere considered.

In the naming convention used for the simulations the

first number indicates RHi, the second number indicates

T,T and the trailing letter indicates the ambient-turbulence

level. In some cases multiple turbulent realizations of

a given simulationwere 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 ca-

ses 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 five-

dimensional space: time-evolving ice-size spectra in

three spatial dimensions. To assess the effects of dif-

ferent 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., in-

tegrated over the x and z directions, averaged over y) ice

crystal numberN(t), massM(t), and surface area S(t) per

meter of flight path (Figs. 2 and 3) and the contrail-

averaged 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 am-

biguous measure for contrails given their limited hori-

zontal extent and nonuniform cross section. The

quantity S(t) provides an approximate measure of op-

tical 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, Qext ’ 2; for example,

t51

WDy

ðDy0

dy

ðdx t

51

WDy

ðDy0

dy

ðdx

ðdz�

jpr 2j NjQ

jext’ S/2W (2)

for optical depth t on a vertical path, contrail width W,

and crystal number densities Nj for crystal radius rj. We

consider only properties of the contrail itself; to de-

termine 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.

Figures 4–6 illustrate some of the contrail spatial

evolution. The main events of the wake-dominated re-

gime (cf. Fig. 5)—exhaust entrainment–detrainment

into the trailing vortex system, vortex fall, linking, ring

dynamics and decay, and Brunt–Väisälä oscillation ofthe wake–—affect the contrail appearance in the same

ways as was described when simulating with bulk mi-

crophysics (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.

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 RHi, saturation mix-

ing ratio qs, and air density r,M(t) scales essentially with

the growth of the plume cross-sectional area, A(t):

M(t)’ (RHi2 1)qsrA(t) . (3)

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 dy-

namics significantly only through qs 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 dom-

inantly driven by exhaust jet mixing (until ;10 s de-

pending on aircraft), vortex fall (;300 s depending on

TABLE 2. Ambient conditions at flight level for principal 3D

simulations. Three further variations of the simulations with as-

terisks were performed with coupled longwave radiation, varying

the upwelling blackbody radiation temperature for each caseTsfc5258, 278, and 298K. The low and moderate ambient-turbulence

levels are quantified in Fig. A1. Unless noted otherwise, we assume

an air pressure of 238.5 hPa, EIiceno 5 1015 kg21, potential tem-

perature gradient of 2.5Kkm21, and nomean 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 con-

stant RHi above flight level down to 1 km below, linearly dropping

to RHi 5 50% at 1.5 km below flight level and below.

Simulation RHi (%) T (K) Turbulence

S30T225m 130 225 Moderate

S30T225l 130 225 Low

S10T225m* 110 225 Moderate


S10T218m* 110 218 Moderate



S10T205l 110 205 Low


aircraft and stratification), Brunt–Väisälä oscillation(;20min depending on stratification), ambient turbu-

lence, 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 tur-

bulent 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 RHi but much less on

EIiceno, 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 ;30min (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); t0.8 or t1

for the ‘‘low’’ and ‘‘moderate’’ turbulence, respectively.

This is consistent with the rough ;t0.8 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

FIG. 2. Time histories of total (a) ice crystal number, (b) icemass,

and (c) ice surface area per meter of flight path from sample 3D

simulations. Line colors indicate different combinations of RHi

and T, and dashed lines indicate different ambient-turbulence

levels as follows: S30T225m (black solid lines, three different tur-

bulent realizations included), S30T225l (black dashed), S10T225m

(green), S30T205m (blue), S10T205m (red solid), and S30T205l

(red dashed).

FIG. 3. Time histories of total (a) ice crystal number and (b) ice

surface area per meter of flight path from cases S10T225m with

varying coupled radiation: none (dashed line), coupled longwave

with Tsfc 5 258, 278, 298K (thin to thick solid lines, respectively).


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 distribu-

tion (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 (z5 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).

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 12m s21) 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 pop-

ulation, 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 am-

bient turbulence, waves, and shear can destroy all rem-

nants of the periodic structure inherited from the young

contrail.

In the absence of ice nucleation, N(t) monotonically

decreases. Themost 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 signifi-

cant number losses during this period and throughout

the later evolution (discussed more below) that do not

depend on the local qt falling below the ice saturation

mixing ratio qs 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 differ-

ences 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 distri-

bution lying in between. In particular the number den-

sities 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; nonethe-

less this sometimes dominates the ice mass (cf. Fig. 7c).

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 signifi-

cantly 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 de-

velopment for case S10T225m with coupled longwave radiation with Tsfc 5 298K. The drift format is as in Fig. 5, but over much longer

times with a slower drift velocity (2m s21).


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 (Jensen

et al. 1998; 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 influ-

encing each contrail metric. Figure 8 provides an ex-

ample. Among the four simulations only two distinct

behaviors ofN(t) arise, depending onwhether theKelvin

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.

4. Further discussion

a. Quasi-3D simulation

Contrails exhibit significant 3D structures at different

ages that are readily apparent in the simulations (e.g.,

Figs. 5 and 6). Turbulence in the exhaust jets leads to

FIG. 7. Evolving spectra of ice crystal numbers vs crystal radius

from (a) S10T205l and (b) S30T225m. (c) Spectra of total icemass for

S30T225m. The times and line colors are as in Fig. 4.

FIG. 8. Sample results from Q3D simulations for RHi 5 130%,

T 5 218K, and low ambient turbulence with precipitation present

(solid lines) or turned off (dashed lines), and Kelvin effect in the

microphysics present (thin red lines) or turned off (thick black

lines). (a) Ice crystal number, (b) ice mass, and (c) ice-size distri-

bution at contrail age 5 h.


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 contrail-

averagedmetrics? Figure 9 compares the cold and warm

B-767 simulations from an ensemble of turbulent re-

alizations of Q3D simulations for the same conditions.

The agreement is quite respectable and is representative

of results we have found when comparing Q3D simu-

lation 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 simula-

tions 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 wake-

dominated 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 suchmodes, 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 sub-

sequent vertical growth is inhibited by stratification). A

priori the Crow instability dynamics would seem criti-

cally involved in setting the average depth, but the

success of the Q3D approximation over a large param-

eter 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 en-

trainment (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 seg-

ments, 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 im-

portant, however. Different 2D treatments of the de-

scent 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.

b. Ice crystal number loss

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 wake-

dominated regime as well. Crystal-loss fractions are

significant even for large RHi 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.

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 spheri-

cal crystals in Lewellen (2012); here we limit ourselves

FIG. 9. Comparison for cases S10T205l and S30T225m of 3D

(dashed lines, threedifferent turbulent realizations for S30T225mcase)

and quasi-3D (solid lines, four different turbulent realizations for each

case) simulation results. Time histories of total (a) ice crystal number

and (b) ice mass.


to summarizing the basic physical mechanism. For

small spherical crystals, surface energy (Kelvin) effects

shift the critical vapor supersaturation level that sepa-

rates sublimation from growth away from zero to

a value depending on crystal radius, 12 eak/r [cf. (1)].

Given that ak; 23 1023mm is much less than themean

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 qy below qs but the high number

densities ensure that conditions never get far from

equilibrium (the simulations rarely show vapor satu-

ration 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 main-

tained 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 (qy / qs) so they appear slightly super-

saturated 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 rep-

resents 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 accu-

mulate and low turbulence and vertical velocities pre-

venting 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 logarith-

mically) noted earlier in Fig. 7a to arise.

The mean in situ loss rate is unmodified by modest

vertical oscillations but can be eliminated by suffi-

ciently 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 re-

sidual 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 EIiceno

The changes in crystal-loss mechanisms seen with

spectrally resolved microphysics alter the sensitiv-

ities to EIiceno. Figure 11 shows sample Q3D results

varying EIiceno. Because of the in situ crystal losses the

fraction of the total crystals lost in the wake-

dominated regime rises with EIiceno 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 in-

dependent of EIiceno (e.g., Unterstrasser and Gierens

2010a). One consequence is that the sensitivity to

EIiceno . ;1014 kg21 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 small-

enough EIiceno the crystal number densities are re-

duced 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 mi-

crophysics (solid lines), or binned without the Kelvin effect (short

dashed lines). Results from Q3D simulations with ambient T 5218K and RHi 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 oscil-lation of the wake plume.


the time scales available. The dramatic changes in

contrail evolution suggested in Fig. 11 from reductions

in EIiceno below this threshold are considered more in

Part II. It is interesting that the EIiceno suggested by

both jet-plume ice nucleation modeling for current

aircraft [the soot-dominated regime of Kärcher andYu (2009)] and in situ aerosol measurements (Schröderet al. 2000; Anderson et al. 1998) is of the same order of

magnitude (;1015 kg21) as the lower range over which

in situ losses are found to be large in the wake-

dominated regime here. Even if the mechanisms lim-

iting 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 EIiceno unless it drops below

about 1014 kg21 (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 qt/qs locally in the contrail

(particularly in those regions with highest crystal num-

ber densities) to a much greater extent the smaller qs,

and hence the smaller T. For example in S10T205 the

engine water raises qt/qs 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 RHi. As

discussed more in Part II, this is one factor in raising the

potential significance of cold contrails despite their rel-

atively 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; Schröderet al. 2000; Febvre et al. 2009) while the shapes of the

larger ones found in the contrail periphery or in pre-

cipitation streamers are more complex (e.g., bullet ro-

settes, 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, sedi-

mentation 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

FIG. 11. Effects of varying EIiceno on total (a) ice crystal number,

(b) ice mass, and (c) ice surface area, for Q3D simulations with

RHi 5 110%, T 5 218K, and low ambient turbulence [dashed

patterns from top to bottom in (a) indicate 1017, 1016, 1015, 1014,

1013, 1012, and 1011 per kilogram of fuel]. The twoEIiceno5 1011 per

kilogram of fuel cases shown differ by the choice of initial radius of

the ice crystals (0.2mm for double crosses and 8.0mm for double

asterisks).


of varying aspect ratio for midrange, and bullet rosettes

for the largest). The shape effects were incorporated

through modifications of C, fy, 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. Somemore 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 itmay 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 radi-

ative 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 inMeza (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.

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 veloci-

ties shortening lifetimes and so limiting their contribu-

tion to contrail-integrated metrics.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 gov-

erned by mixing rates of ambient air into the contrail or

vertical displacement of it, rather than growth rates for

individual crystals that can be affected by shape. Meza

(2010) found large differences in contrail evolution due

to shape only when the smallest crystals were assumed

(unphysically) to be bullet rosettes: the reduced d for

a given crystal mass that results amplifies the Kelvin-

dependent in situ crystal loss, thereby reducing N(t)

significantly relative to the result for spheres.

2) ICE DEPOSITION COEFFICIENT

The ice deposition (or accommodation) coefficient bi

is poorly determined experimentally, with results in the

literature ranging over more than two orders of magni-

tude (e.g., Pruppacher and Klett 1997, their Table 5.5).

The default choices made by different cirrus modelers

covers nearly as large a range and have a significant

impact on the crystal number densities produced via

homogeneous nucleation in natural-cirrus simulations

(Lin et al. 2002). The quantity bi appears in the ice

growth [(1)] effectively through modifying the water va-

por diffusion coefficient D (Pruppacher and Klett 1997):

D*5D/(11 lb/r) with lb ’D

bi

�2p

RyT

�1/2

. (4)

There is negligible effect as bi approaches 1 or as crystals

grow large compared to the mean free path of water

vapor (a fraction of a micron for contrail conditions). As

already noted, contrail dynamics are such that ‘‘small

crystal’’ effects can have an extended range of influence.

Figure 12 shows results from simulations varyingbi away

from the CARMA assumed value of 0.93. There is in-

creasing sensitivity seen in crystal losses as bi drops

below 0.1 and profound differences in M(t) as well as

N(t) when the values become small enough. The sensi-

tivity is dominantly through the effect of bi on in situ

crystal loss (Lewellen 2012), although significant sensi-

tivity remains even with Kelvin effects excluded since

bi � 1 reduces overall ice growth rates. Recent labo-

ratory measurements designed specifically for cirrus

conditions (Skrotzki et al. 2013) recommend the value of

bi 5 0.7 and may, if confirmed, significantly reduce this

uncertainty, though it remains unexplained why some

measurements give much lower values and remains

a possibility that bi may vary with crystal size, habit,

supersaturation, and/or temperature.

5. Concluding remarks

In this work, contrails have been simulated from a few

wing spans behind the aircraft through the wake-vortex

evolution, transition to contrail cirrus, and eventual

demise through precipitation of the constituent ice

crystals into subsaturated air. The use of dynamic local

ice binning, column by column dynamic updating of

radiative heating, a scheme for providing resolved am-

bient atmospheric turbulence, and the surprising success

of a quasi-3D simulation mode make practical the

1 Further discussion on the potential effects of nonspherical

crystal shapes integrated over the contrail’s lifetime are given in

Part II.


exploration of a large parameter space of detailed con-

trail simulations out to many hours of simulated time.

While the simulations are more numerically costly than

the approach of Unterstrasser and Gierens (2010a,b),

they have the advantage of permitting a direct assess-

ment of the effects on contrail dynamics of explicit

binned ice microphysics, 3D fluid dynamics, realistic

resolved aircraft wake dynamics, and resolved ambient

atmospheric turbulence/waves.

Some of these effects prove more critical than others.

While 3D dynamics can dramatically affect contrail

appearance, comparing 3D and Q3D results suggests

that including a very limited range of scales of 3D tur-

bulence may be sufficient for mean contrail properties.

Inclusion of resolved ambient turbulence/waves dem-

onstrates their importance to older contrail extent and

lifetime; it is not appropriate to consider only a single

turbulence level as if it were universal, particularly in the

absence of other turbulence production (e.g., frommean

shear or coupled radiation). More importantly, the in-

clusion of binned microphysics significantly increases

rates of ice crystal number loss in both young and old

contrails because of local competition between large and

small crystals for available moisture and how this is

modified by the Kelvin effect. This adds to previous

contrail work suggesting the need for improved micro-

physics (Huebsch and Lewellen 2006; Unterstrasser and

Sölch 2010). It might be possible to approximately in-

corporate these effects through suitable modifications of

2D fluid dynamics and bulk microphysics models a pos-

teriori, but they will not naturally be present.

For lower ambient temperatures the water from the

aircraft engines becomes important in boosting qt/qs in

the young contrail and thereby significantly reducing

crystal number losses in the wake-dominated regime.

The fractional rate of ice crystal number loss is also

strongly dependent on the crystal number densities

themselves (in contrast to simple expectations from bulk

models). As a consequence, the sensitivity to changes in

effective ice crystal number emission index are dra-

matically reduced over time for EIiceno . ;1014 kg21.

Given the sensitivity of the simulations to the Kelvin

effect and the ice deposition coefficient bi through their

impact on crystal-loss rates, improving the observational

constraints on these would be useful.

Quantitative differences aside, most of our conclu-

sions on contrails of more than a few minutes’ age are

consistent with other studies (e.g., Jensen et al. 1998;

Lewellen and Lewellen 2001a; Unterstrasser andGierens

2010a,b), including the critical importance of ambient

RHi, the primary role of temperature in determining

crystal size, the very small fraction of crystal number but

significant fraction of ice mass found in precipitation

streamers and their efficiency in removing moisture

from large volumes, and the critical importance of

coupled radiation in some regimes but not others. Sed-

imentation plays a critical role in the vertical spread of

the contrail and its ultimate lifetime. The time scale for

the initial onset of significant precipitation growth is

dominantly governed by RHi and T. The contrail/

contrail-cirrus lifetime is generally shortened by fac-

tors leading to larger crystal sizes, including higher RHi

orT, lower ice numbers, increased turbulent mixing with

supersaturated air, and increased wake-induced or in

situ crystal loss. An assessment of the effects of different

physical variations on overall contrail significance

therefore requires covering a sizable parameter space

with simulations over the full lifetime. The model de-

scribed here is efficient enough for such studies, taken

up in a companion paper (Part II).

FIG. 12. Effects of varying ice deposition coefficient: bi5 1 (solid

line), 0.5 (short dashed), 0.3 (long dashed), 0.1 (long–short dashed),

0.01 (starred–dashed). Results from Q3D simulations with RHi 5110%, T 5 218K, and low ambient turbulence.


The simulations here show a clear potential for long-

lived, large-volume contrails. Even if the ice cloud inmost

of this volume is not optically apparent it could still have

significant radiative impact. At the same time the aging

contrail can sometimes transform a very large volume

that was highly supersaturated into one that is near zero

supersaturation, extending from just above flight level

down through the depth of the supersaturated layer in the

vertical and over several kilometers in the horizontal

even in the absence of strong mean wind shear. This

could possibly provide subsequent aircraft with a corridor

in which to fly without producing significant contrails

themselves. While this would be a drifting corridor,

conditions for which it might be toughest to track (e.g.,

large cross-stream shear) would tend to give wider targets

of subsaturated horizontal extent. By removing the su-

persaturation in this region the corridor could also lead to

a reduction in natural cirrus. Because of the extensive

interaction of a single contrail over its lifetime with both

the horizontal (particularly with wind shear) and the

vertical extent of the ice-supersaturated region, these

effects may need to be accounted for explicitly in regional

and global simulations of contrail effects. Because of the

potential depletion of ice supersaturation, this may be

particularly important in regions of heavy air traffic or for

future projections of increases in air traffic.

Acknowledgments. This research has been supported

by the Boeing Company. We thank Steve Baughcum

and Mikhail Danilin for important contributions

throughout the course of this work, Eric Jensen and

Andy Ackerman for providing access to CARMA,

Michael Czech and Jeffrey Crouch for providing 2D

wake results for initialization, and Tim Rahmes and

John Ross for useful input and discussions. We thank

three anonymous reviewers for comments leading to

improvements in the presentation of this work.

APPENDIX

Additional Model and Simulation Details

a. Ice microphysics

At each time step in each grid cell where the ice

numbers are not numerically insignificant, the local ice

spectrum (cf. Fig. 1) is passed to the CARMA routine

for updating the diffusional growth, performed in a bin

space extended from the local space by one bin on either

end. Advection and diffusion updates in each spatial

direction are conducted on the subspace of the global

bin space that is populated along the given spatial line.

At the end of each time step the ice spectrum at each

spatial point x is checked to see whether the spectrum

would be better represented in local bin space if is(x, t)

were shifted up or down by 1. The spectrum is then

‘‘folded back’’ from the expanded bin space into the

appropriately shifted local space. The crystals outside

the new local space are simply added to the nearest size

bin that is included. Both total ice number and qt are

conserved in the folding process; ice mass is not. Ideally

the number of local bins should be chosen large enough

so that the number of crystals folded back is always

negligible. The latter is monitored by the code so that

a simulation may be terminated if an increase in number

of local bins is required. The simple shifting criterion of

minimizing the number of crystals at the local bin ends is

found to work well. When crystals evaporate out of the

smallest global bin they are removed completely from

the simulation. Physically crystals evaporating out of the

smallest local (but not global) bin should sometimes be

removed as well, in regimes where large crystals are

growing at the expense of small ones so that down-

shifting of the local space is not appropriate. This is di-

agnosed by crystals piling up in the lowest local bin after

the spectrum is folded back; accordingly, crystals evap-

orating out of the lowest local bin are removed if its bin

population exceeds that of the next larger bin.

Based on several tests we choose for the simulations

here Rm 5 2, ng 5 48, nl 5 20, and smallest global bin

corresponding to crystal radius of 50 nm. The local

binning scheme was critically tested by running it

concurrently with a conventional global binning within

the same simulation for different contrail conditions.

Contrail simulations were also performed varying the

ice bin resolution. In addition, in Lewellen (2012) dif-

ferent ice binning choices in parcel model tests were

compared against exact analytic solutions of ice spec-

trum development for numerically challenging regimes

with in situ crystal loss. The tests suggest that for con-

trail simulation a bin mass ratio of 3.0 is probably

marginally adequate, while a ratio of 2.0 generally re-

produces results of a much finer binning well (though

somewhat overestimating in situ crystal loss; Lewellen

2012). Parcelmodel tests showed that, for a binmass ratio

of 2.0, even as few as 10 local bins faithfully represents the

evolution in different growth regimes. Within contrail

simulations the concurrent local–global tests showed al-

most indistinguishable results (much less than typical

Q3D realization differences) using 15 local bins. An ad-

vantage of the dynamic local binning is that the largest

and smallest crystal sizes represented in the global space

may be selected conservatively without significant in-

creases in numerical cost. Sensitivity tests showed no

apparent change in results when the choice of 50-nm

minimum radius was shifted up or down by a few bins.


Early in the contrail dynamics the restriction on the

time step set by the wake fluid mechanics is generally

more stringent than that set by the ice physics. At late

times the microphysics is substepped on a shorter in-

terval than the fluid mechanics if necessary (an existing

feature of CARMA). When populations of crystals with

large fall velocities become significant, the LES time

step is reduced if necessary so that the fall velocity sat-

isfies the Courant condition. If the population is physi-

cally insignificant, then its fall velocity is simply reduced

to maintain stability with the current time step.

b. Coupled radiation with dynamically adjustedindependent column updates

For each grid column in the domain, vertical profiles

ofT, qy, and ice particle concentrations are passed to the

CARMA radiative transfer routine to compute a radia-

tive heating rate for each point in the column. The

contributions from all columns are stored and used to

update local conditions at each time step. For a single

column denote the vertical heating rate profile HR(z, t)

and time between updates dtR. Then dTE [ dtR 3maxz[HR(z, t1 dtR)2 HR(z, t)] provides an estimate of

the maximum error in temperature due to the delay in

updating the heating rates. Since the coupled radiation

mainly affects the contrail dynamics through buoyancy-

induced changes in parcel heights, a natural temperature

difference scale to consider in the simulation is dT0 [gDz, where g is the potential temperature gradient and

Dz is the finest vertical grid spacing. After each update if

dTE. fRdT0 then dtR is halved; if dTE, fRdT0/3 then it is

doubled. The tolerance fR is an input for the scheme. The

dtR’s are allowed to vary for each column independently,

with the exception of requiring that the values for

neighboring columns differ by no more than a factor of 2.

This avoids potential problems associated with horizontal

advection of sharp-edged clouds. In addition to these

updates, the heating rates in all columns are computed for

the first time step and for eachLESoutput cycle (typically

every 300 s at late times). A value of fR 5 0.01 has been

chosen for the simulations here. A set of contrail test

cases showed that changing to fR5 0.001 or 0.1 produced

little or no differences in mean contrail development.

Other components of the radiation scheme are taken

essentially as is fromCARMA. For our sensitivity studies

here and in Part II we take the upwelling blackbody ra-

diation temperature Tsfc as our primary variable. Condi-

tions above the LES domain are specified via a blackbody

temperature of downwelling infrared radiation at domain

top and an assumed integrated water column above the

domain (taken as 130K and 0, respectively, for the sim-

ulations here and in Part II). Between the LES domain

and the surface, 10 evenly spaced layers (in pressure

coordinates) are added, with the pressure and tempera-

ture interpolated between the surface values and those at

the bottom of the LES domain, and a constant relative

humidity with respect to water assumed (taken as 30%

for the simulations here). A constant CO2 concentration

of 350ppmv is assumed throughout the column and the

ozone mixing ratio taken from the U.S. standard atmo-

sphere midlatitude profile; no other trace gasses are in-

cluded in the radiation computations. When incoming

solar radiation is included (as for some runs in Part II),

latitude, date, surface albedo, and either time-varying or

constant solar zenith angle can be specified.

c. Late-time turbulence maintenance

For each of the domains in Table 1 except grid 1

(where ambient atmospheric turbulence plays no sig-

nificant role) a preliminary LES to produce ambient-

turbulence fields was performed with periodic boundary

conditions, uniform grid spacing in each direction and

the desired ambient stratification, mean shear, and do-

main size. These were initialized with temperature

perturbations �1�2Tamp with �1 and �2 randomly chosen in

each grid cell and distributed uniformly between21 and

1. Fields from these decaying turbulence simulations

were taken at a chosen age fage and these added to the

contrail LES every tadd seconds to maintain some am-

bient turbulence in the main simulations over long

times. Before adding the aged turbulence fields to the

contrail LES the horizontal mean averages were re-

moved and the domain shifted by a random amount in

x, y, and z so that the sequential additions were effec-

tively independent of each other even though for each

domain only a single field from the corresponding de-

caying turbulence LES was utilized. The accompanying

subgrid TKE field was also added but the temperature

perturbations were not, since this would interact un-

physically with the ice dynamics. The initially imposed

mean vertical shear in the contrail LES persists on its

own over the course of a simulation (without any

nudging or imposed pressure gradients required) since

only perturbations away from the horizontal average are

added, periodic horizontal boundary conditions are

used, and the shear and stratification levels are chosen

such that the mean shear does not directly drive a shear

instability.

After repeated field additions the turbulence statistics

of the simulation, averaged over periods long compared

to tadd, become quasi steady. The state is determined by

the choice of stratification, shear, Tamp, fage, and tadd as

well as the domain size and grid resolution. The turbu-

lence and gravity wave amplitudes can be increased by

increasing Tamp and/or decreasing tadd; the relative im-

portance of gravity waves compared to turbulence is


increased by increasing fage and/or tadd; and the level of

intermittency increases with increased tadd. The nu-

merical overhead of this scheme is minimal.

Given the use of different domains and grids for dif-

ferent simulation segments, a practical consideration is

the extent to which the character of the fields can be

matched across different domains or grids. This was

found empirically to be achieved approximately if the

same Tamp, fage, and tadd are chosen and in addition the

same resolution is used for the preliminary dying tur-

bulence simulation in each case; Fig. A1 provides an

example from the current simulations. For the simula-

tions of Table 2 we chose fage and tadd both equal to

10min (roughly the Brunt–Väisälä period for the givenstratification) and choseTamp5 0.2K and 2K for the low

and moderate turbulence cases, respectively. The hori-

zontal and vertical resolutions for the decaying turbu-

lence simulations were chosen to be 50 and 33m,

respectively. The mean dissipation levels produced (Fig.

A1) fall within the broad range observed in the upper

troposphere or lower stratosphere [e.g., as given in

Table 17.3 of Quante and Starr (2002)].

d. Simulation initialization

The 2D Boeing results are utilized five spans behind

the aircraft (;1-s age) and include velocities, down-

stream vorticity, a passive exhaust tracer, and amodeled

turbulent eddy viscosity. From the latter a turbulent

kinetic energy is estimated, a fraction of which is used to

initialize the subgrid TKE field for the LES and a small

fraction is used for resolved velocity perturbations. The

velocity field from the 2D calculation is combined with

these random perturbations (and a mean ambient shear

profile if desired) to form the LES velocity field. The

spreading exhaust jets rapidly develop their own re-

solved turbulence and the LES subgrid TKE adjusts on

a fast time scale as well so that the simulation results are

not very sensitive to the magnitudes of perturbations or

subgrid TKE chosen. The velocity field is extended from

the smaller domain of the 2D simulation by integrating

the vorticity field using Biot–Savart’s law. The passive

tracer from the 2D model is used directly to initialize

a passive tracer in the LES and is scaled (given a fuel

flow rate, propulsion efficiency, and specific combustion

heat) and added to a desired ambient gradient to form

the LES potential temperature field.

The fine resolution in grid 1 of Table 1 is required to

resolve the small vortex cores provided by the 2D

Boeing simulation as well as being beneficial in rapidly

developing realistic jet turbulence. Comparison be-

tween the 0.4-m-resolution LES and a 0.25-m-resolution

test run out to 15 s supported the adequacy of the coarser

resolution. It is numerically costly to run the full binned

microphysics simulation with sufficient spatial resolu-

tion to maintain the initial tight vortex cores throughout

the vortex-dominated stages of the LES. Our previous

studies initialized with more diffuse cores (Lewellen and

Lewellen 1996, 2001a) suggest that this is not required in

order to simulate the dispersion of the exhaust species

and basic vortex dynamics well. Accordingly, we run the

2-s simulation on grid 1 with the subgrid rotational

damping turned off to modestly diffuse the vortex cores

so that they can be well resolved on grid 2.

Results from the initial LES are used to initialize the

main contrail simulation on grid 2 with ice introduced.

Periodic copies were used to extend from the smaller to

larger downstream domain and an ambient-turbulence

field of desired character added as discussed earlier. The

initialization times represent a compromise. The LES

must be initialized late enough that the approximations of

taking the wake to be uniform downstream, and the hot

exhaust jets to be incompressible, are reasonable. It must

be early enough to generate the large concentration

fluctuations in the exhaust jets that are a product of 3D

FIG. A1. (a) Mean dissipation rate (m2 s23) and (b) vertical

velocity variance (m2 s22) vs time for sample simulated ambient-

turbulence fields such as those employed in the contrail simula-

tions. Thick solid lines are from low-turbulence simulation on 12343 4 km3 domain; other lines frommedium-turbulence simulation

on 123 43 4 km3 domain (solid line), 0.843 0.263 1 km3 domain

(long dashed), and 123 0.323 4 km3 Q3D domain (short dashed).

The small-scale dissipation rate and vertical velocity variance are

chosen to roughly quantify the ambient-turbulence state, as the

most important impact on contrails is through mixing rates and

vertical motions.


turbulence in the early stages. This prompted the ;1-s

initialization age and high-resolution LES. Ice initializa-

tion is delayed in part to ensure that the exhaust jets have

cooled so as to become supersaturated with respect to ice.

Amuch longer delay is undesirable because it would slow

the initial development of the Crow instability, which

necessitates the longer downstream domain.

Ice is introduced in a single size bin (generally 0.2-mm

radius). Simulation tests showed results to be insensitive

to the initial ice spectral distribution: in the young wake

plume the number densities are sufficiently high that the

crystals rapidly (in ;1 s) adjust their mean size to

achieve local equilibrium conditions regardless of their

starting point, and the mixing in the exhaust jets is rapid

enough that it quickly governs the shape of the local size

spectrum, washing out the memory of the initial bin

distributions. Figure 11 provides an example showing

that even if the initial crystal numbers are much less and

the initial bin size significantly raised, the effects wash

out after the wake-dominated regime.

For simplicity the ice has been introduced spatially

following a passive exhaust tracer. Given the complex-

ities of ice nucleation in the turbulent cooling jets this

need not be the case physically. Simulations with the ice

number density initially distributed spatially as some

power of the local tracer concentration C with the total

numbers held constant, were compared to test the sen-

sitivity. Varying over the range from C1/3 to C3, the re-

sulting changes in contrail properties still proved small.

This is consistent with changes in results due to engine

placement, which, while more significant, are generally

not dramatic. Note that in these sensitivity tests only

persistent contrails were considered; significant changes

to the evolution of short-lived contrails in subsaturated

conditions have not been ruled out.

Contrails exhibit an approximate periodicity down-

stream governed by the Crow instability. Ideally we

would perform simulations in a long enough domain to

include several Crow periods in order to produce aver-

age statistics that do not vary between different turbu-

lent realizations of the simulation. In previous work

comparing simulations on different length domains

(both without ice and with bulk ice microphysics) we

have found, however, that a single Crow period develops

in essentially similar fashion whether it interacts with

independent realizations downstream or only periodic

copies of itself. This makes simulation of a single Crow

period a physically reasonable and numerically efficient

choice. Running different realizations then gives some

measure of the quality of the averages (cf. Fig. 2).

When a new run segment is initialized on a different

grid during the course of a simulation (cf. Table 1), the

3D field variables are simply linearly interpolated onto

the new grid. The fields are sufficiently well resolved that

any nonconservation of species that this interpolation

could in principle produce are in practice confirmed to

be negligible. The new downstream domain is always

chosen to be an integer multiple of the old one, with the

fields periodically continued into the expanded domain.

A fresh ambient-turbulence field generated for the new

expanded domain is added at the start to break the

symmetry between the periodic copies of the contrail

downstream and to provide local variations in the new

regions added in the cross-stream directions. Given the

presence of the contrail, the domain expansion in the

cross-stream directions is not done as a periodic copy

and so is not required to be an integer multiple of the

old domain. Because the cross-stream boundaries

are always located far from the expanding contrail and

the fresh turbulence addition dominates the decayed

turbulence remaining on the old grid, any discon-

tinuities in the ambient velocity fields produced by this

procedure have negligible effect on the subsequent

contrail evolution.

Given the vertical grid tracking utilized in the code,

it is necessary throughout the simulation segments and

regridding steps to keep track of where the original

flight altitude lies within the grid at each time. Hy-

drostatic balance within the domain is then assumed to

obtain the mean pressure at each level in the grid from

the pressure assumed at flight level. This same capa-

bility is employed if a uniform uplift or subsidence is to

be included in the simulation (as is done for some cases

in Part II).

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Persistent Contrails and Contrail Cirrus. Part I: Large