Global-mean Vertical Tracer Mixing in Planetary Atmospheres. I.Theory and Fast-rotating Planets

Xi Zhang1 and Adam P. Showman21 Department of Earth and Planetary Sciences, University of California Santa Cruz, CA 95064, USA; xiz@ucsc.edu2 Department of Planetary Sciences and Lunar and Planetary Laboratory, University of Arizona, AZ 85721, USA

Received 2018 March 24; revised 2018 August 13; accepted 2018 August 13; published 2018 October 5

Abstract

Most chemistry and cloud formation models for planetary atmospheres adopt a one-dimensional (1D) diffusionapproach to approximate the global-mean vertical tracer transport. The physical underpinning of the key parameterin this framework, eddy diffusivity Kzz, is usually obscure. Here we analytically and numerically investigatevertical tracer transport in a 3D stratified atmosphere and predict Kzz as a function of the large-scale circulationstrength, horizontal mixing due to eddies and waves and local tracer sources and sinks. We find that Kzz increaseswith tracer chemical lifetime and circulation strength but decreases with horizontal eddy mixing efficiency. Wedemarcated three Kzz regimes in planetary atmospheres. In the first regime where the tracer lifetime is shortcompared with the transport timescale and horizontal tracer distribution under chemical equilibrium (c0) isuniformly distributed across the globe, global-mean vertical tracer mixing behaves diffusively. But the traditionalassumption in current 1D models that all chemical species are transported via the same eddy diffusivity generallybreaks down. We show that different chemical species in a single atmosphere should in principle have differenteddy diffusion profiles. In the second regime where the tracer is short-lived but c0 is non-uniformly distributed, asignificant non-diffusive component might lead to a negative Kzz under the diffusive assumption. In the thirdregime where the tracer is long-lived, global-mean vertical tracer transport is also largely influenced by non-diffusive effects. Numerical simulations of 2D tracer transport on fast-rotating zonally symmetric planets validateour analytical Kzz theory over a wide parameter space.

Key words: astrochemistry – hydrodynamics – methods: analytical – methods: numerical – planets and satellites:atmospheres

1. Introduction

The spatial patterns of atmospheric tracers such as chemicalspecies, haze, and cloud particles are significantly influencedby atmospheric transport. For most planets in the solar system,the horizontal variations of tracers are usually much smallerthan their vertical variations. For extra-solar planets, thehorizontal distributions of tracers are not well resolved byobservations. The global-mean vertical tracer distribution is theprimary focus in most chemistry and cloud models. Verticaltransport by large-scale atmospheric circulation and wavebreaking leads to a tracer profile that deviates from its chemicalor microphysical equilibrium. One example is the “quenching”phenomenon in the atmospheres of giant planets and close-inexoplanets where the gas tracer concentrations (e.g., CO onJupiter, Prinn & Barshay 1977) in the upper atmosphere are notin their local chemical equilibrium, but instead are dominatedby strong vertical tracer mixing from the deeper atmosphere.As a result, the observed tracer abundance in the upperatmosphere is “quenched” at an abundance corresponding tothe chemical-equilibrium abundance at the location where thetracer chemical timescale is equal to the vertical mixingtimescale. This disequilibrium “quenching effect” has asignificant influence on the interpretation of observed spectraof planetary atmospheres (e.g., Prinn & Owen 1976; Prinn &Barshay 1977; Smith 1998; Cooper & Showman 2006;Visscher & Moses 2011).

Investigation of global-mean vertical tracer distributioncould be simplified to a one-dimensional (1D) problem.However, because atmospheric tracer transport is intrinsicallya three-dimensional (3D) process, parameterization of the 3D

transport in a 1D vertical transport model is tricky. Mostchemistry models for solar system planets (e.g., Allenet al. 1981; Krasnopolsky & Parshev 1981; Yung et al. 1984;Nair et al. 1994; Moses et al. 2005; Lavvas et al. 2008; Zhanget al. 2012; Moses & Poppe 2017; Wong et al. 2017) andexoplanets (e.g., Line et al. 2011; Moses et al. 2011; Huet al. 2012; Tsai et al. 2017) adopt a 1D chemical-diffusionapproach. Most haze and cloud formation models on planetsand brown dwarfs assume the vertical particle transportbehaves in a diffusive fashion as well (e.g., Turcoet al. 1979; Ackerman & Marley 2001; Helling et al. 2008;Gao et al. 2014, 2017; Lavvas & Koskinen 2017; Powellet al. 2018). This type of 1D framework originated in the Earthatmospheric chemistry literature but has gradually faded outsince the 1990s because it is not sufficient to simultaneouslyexplain the distributions of all the species from observations(Holton 1986) and when the 3D distributions of multipletracers are well determined by satellite observations fromspace. At the moment, this 1D framework is still very useful inplanetary atmospheric studies. Although only a crude approx-imation, 1D models have succeeded in explaining the observedglobal-averaged vertical profiles of many chemical species andparticles on planets (e.g., Titan by Yung et al. 1984 and Zhanget al. 2010a, Jupiter by Moses et al. 2005, Venus by Zhanget al. 2010b, 2012 and Gao et al. 2014, Pluto by Wonget al. 2017 and Gao et al. 2017).In the 1D chemical-diffusion framework, the strength of the

vertical diffusion is characterized by a parameter called eddydiffusivity or eddy mixing coefficient. The 1D effective eddydiffusivity Kzz is often determined empirically by fitting theobserved vertical tracer profiles. Figure 1 shows that Kzz on

The Astrophysical Journal, 866:1 (17pp), 2018 October 10 https://doi.org/10.3847/1538-4357/aada85© 2018. The American Astronomical Society. All rights reserved.

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different planets could exhibit vertical profiles and magnitudevarying across more than eight orders of magnitude. Thisvariation implies different vertical transport efficiencies fromplanet to planet. Since all 3D dynamical effects have beenlumped into a single eddy diffusivity, the specific dynamicalmixing mechanisms that lead to a particular vertical profile ofKzz are often obscure. If the atmosphere is convective, thenusing the traditional Prandtl mixing length theory (e.g.,Prandtl 1925; Smith 1998; Bordwell et al. 2018), one canformulate Kzz as a product of a convective velocity and atypical vertical length scale in a turbulent medium. But thisformalism fails when the atmosphere is stably stratified. In thelow-density middle and upper atmosphere such as Earth’smesosphere, the vertically propagating gravity waves couldbreak and also lead to a strong vertical mixing of the chemicaltracers. Lindzen (1981) parameterized the eddy diffusivity fromthe turbulence and stress generated in breaking gravity and tidalwaves (also see the discussion in Strobel 1981; Strobelet al. 1987). In a stratified atmosphere, such as Earth’sstratosphere, tracer transport is subjected to both large-scaleoverturning circulation and vertical wave mixing (Hunten 1975;Holton 1986), but their relative importance depends on altitudeand many other factors and may differ from planet to planet.

One of the conventional assumptions in the existingframework used in current planetary models is that all tracers,in spite of their different chemical lifetimes or particlemicrophysical/settling timescales, are simulated using thesame eddy diffusivity profile Kzz. The tracer distribution inthe real atmosphere is controlled by 3D dynamical andchemical/microphysical processes. Therefore, a coupling feed-back between the chemistry and vertical transport is expected.Actually it has been noticed in the Earth community (e.g.,Holton 1986), in the presence of meridional (latitudinal)transport in the stratospheres, the derived effective eddydiffusivity as a global-mean transport coefficient could have astrong dependence on the tracer lifetime, and thus its chemical/

microphysical sources and sinks. Although there are several 3Dchemical-transport simulations in planetary atmospheres withsimplified chemical and cloud schemes (e.g., Lefèvre et al.2004, 2008; Cooper & Showman 2006; Marcq & Lebonnois2013; Parmentier et al. 2013; Charnay et al. 2015; Leeet al. 2016; Drummond et al. 2018; Lines et al. 2018), athorough understanding of the physical basis of global-meanvertical tracer transport and Kzz using both analytical theory and2D and 3D numerical simulations is still lacking.Here we aim to reexamine this conventional 1D diffusion

framework. We wish to achieve a more physically basedparameterization of Kzz from first principles. Our study will bepresented in two consecutive papers. In Paper I (the currentpaper), we will construct a first-principles theory of Kzz in a 3Datmosphere and numerically investigate the behaviors of Kzz onfast-rotating planets using a 2D chemical-transport model. InPaper II (Zhang & Showman 2018), we will specifically focuson 3D chemical tracer transport on tidally locked exoplanetsand the associated Kzz using a 3D general circulation model(GCM). We primarily focus on stratified atmospheres such asthe stratosphere on solar-system planets or the photospheres onhighly irradiated exoplanets where most of the chemical tracersand haze/clouds are observed, and that are expected to bestably stratified (e.g., Fortney et al. 2008; Madhusudhan &Seager 2009; Line et al. 2012).In the following sections of this paper, we will first elaborate

on the underlying physics of global-mean tracer transport andconstruct a first-principles estimate of Kzz. Then we will use a2D chemical-transport model to study a 2D meridionalcirculation system on fast-rotating planets and the behaviorsof the associated Kzz under several typical scenarios and Kzz

regimes. We conclude this study with several key statementsand a brief discussion on the effect of vertically propagatinggravity waves on the vertical transport of tracers.

2. Theoretical Background

2.1. Nature of the Problem

In a stratified atmosphere, tracers tend to be mixed upwardby the large-scale overturning circulation if there is acorrelation on an isobar between tracer abundance and verticalvelocity: if the tracer abundance is high where the verticalvelocity is upward, or if the tracer abundance is low where thevertical velocity is downward, tracer is mixed upward(Figure 2). This implies that the net vertical mixing of tracerover the globe depends crucially on horizontal variations of thetracer on isobars and on their correlation with the verticalvelocity field (Holton 1986). If the tracer distribution is initiallyhorizontally uniform across the globe with a vertical gradient ofthe mean tracer abundance, vertical wind transport willnaturally produce tracer perturbations on an isobar that arecorrelated with the vertical velocity field (Figure 2). Forexample, if the initial tracer mixing ratio is higher in the loweratmosphere and lower in the upper atmosphere, the upwellingbranch of the overturning circulation will transport the higher-mixing-ratio tracers upward and the downwelling branch willtransport the lower-mixing-ratio tracers downward from theupper atmosphere. As a result, the tracers on an isobar will bemore abundant in the upwelling branch than in the down-welling branch, exhibiting a positive correlation with thevertical velocity field (Figure 2). After this horizontal tracer

Figure 1. Vertical profiles of eddy diffusivity in typical 1D chemical models ofplanets in and out of the solar system. Sources: Zhang et al. (2012) for Venus;Allen et al. (1981) for Earth; Nair et al. (1994) for Mars; Li et al. (2015) forTitan; Wong et al. (2017) for Pluto; Moses et al. (2005) for Jupiter, Saturn,Uranus, and Neptune; Moses et al. (2013) for GJ436b; and Moses et al. (2011)for HD189733b and HD209458b. For HD209458b, we show eddy diffusivityprofiles assumed in a gas chemistry model (dashed line, Moses et al. 2011) andthat derived from a 3D particulate tracer-transport model (solid line, Parmentieret al. 2013).

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distribution is established, the upwelling branch will transporthigher-mixing-ratio tracers across an isobar and the down-welling branch will transport lower-mixing-ratio tracers acrossthe same pressure level—the upward and downward tracerfluxes do not cancel out. When averaged over the globe, therewill be a net upward tracer flux, resulting in an effectiveupward tracer transport in the global-mean sense.

Several other processes act to enhance or damp thosehorizontal tracer perturbations on the isobar. Horizontalmixing/diffusion due to eddies and waves (e.g., breaking ofRossby waves) normally smooth out the tracer variation acrossthe globe (e.g., Holton 1986; Yung et al. 2009; Friedson &Moses 2012). Horizontal advection due to the mean flow mayincrease or decrease the horizontal tracer variations, dependingon the correlation between the horizontal velocity conv-ergence/divergence and the tracer distribution on the isobar.The distribution of tracer sources and sinks due to nondyna-mical processes such as chemistry3 also plays an importantrole. In a simplified picture, those nondynamical processescould be assumed to relax the tracer distribution back to thechemical-equilibrium distribution that the tracers would have inthe absence of dynamics, which could be either uniformlydistributed across the globe or with significant variationsdepending on the local sources and sinks (Marcq &Lebonnois 2013). It is expected that the 1D effective eddydiffusivity Kzz depends on the magnitude of the verticalvelocity, the chemical timescale of the species, the horizontaltransport timescale, and the horizontal variation of the tracerdistribution under chemical equilibrium.

Holton (1986) first quantified these effects in the Earth’satmosphere with a scenario that envisions the vertical mixing isaccomplished by a meridional circulation in a 2D (latitude–pressure) framework. The derived eddy diffusivity exhibits a

strong dependence on the circulation strength, tracer chemicallifetime, and horizontal mixing. Holton showed that thespecies-dependent eddy diffusivity might help simultaneouslyexplain the vertical profiles of several species in the strato-sphere of Earth, whereas the species-independent eddydiffusivity could not. Here we generalize the 2D theory fromHolton (1986) to a 3D atmosphere so that it can also be appliedto other planets that are not as zonally symmetric as the Earth.

2.2. Governing Equation of Vertical Tracer Transport

Here we include the tracer advection by 3D atmosphericdynamics and tracer chemistry via a simplified chemicalscheme to study the global-mean tracer transport. Based onthese investigations, we will achieve an analytical parameter-ization of the 1D effective eddy diffusivity Kzz. We will alsodemarcate different atmospheric regimes in terms of the tracerchemical lifetime and horizontal tracer distribution underchemical equilibrium.First, we start from a general 3D tracer-transport equation:

c= ( )D

DtS, 1

where S is the net sources/sinks of the chemical tracer withmixing ratio χ. In principle, isentropic coordinates are moreappropriate for discussion of the tracer transport (Andrewset al. 1987). But for simplicity, here we just adopt the log-pressure coordinate { }x y z, , . The coordinates are defined as

l f=x a sin and f=y a , where a is the planetary radius, λ islongitude, and f is latitude. Here, º - ( )z H p plog 0 is the log-pressure where H is a constant reference scale height, p ispressure, and p0 is the reference pressure in the pressurecoordinate. = ¶ ¶ + + ¶ ¶·uD Dt t w zh is the materialderivative. = ¶ ¶ ¶ ¶( )x y,h is the horizontal gradient atconstant pressure. = ( )u u v, is the horizontal velocity atconstant pressure, where u is the zonal (east–west) velocity andv is the meridional (north–south) velocity. w=Dz/Dt is thevertical velocity. In the log-pressure coordinates, Equation (1)becomes:

cc

c¶¶

+ +¶¶

=· ( )ut

wz

S. 2h

The continuity equation is:

+¶¶

=-· ( ) ( )u ez

e w 0. 3z H z Hh

Combining the above two equations, we get the flux form ofthe tracer-transport equation:

cc c

¶¶

+ +¶¶

=-· ( ) ( ) ( )ut

ez

e w S. 4z H z Hh

Here we define an eddy-mean decomposition = + ¢A A Awhere A represents any quantity. A is the globally averagequantity at constant pressure and ¢A is the deviation from themean, or the “eddy” term. Taking the global average of thecontinuity Equation (3) to eliminate the the horizontaldivergence term, we obtain =w 0 for each atmospheric levelif we assume that the global-mean vertical velocity w vanishesat top and bottom boundaries. Globally averaging Equation (4)

Figure 2. Illustration of tracer transport with a large-scale circulation. Theorange solid lines indicate constant tracer mixing ratio surfaces. The tracermixing ratio is higher in the lower atmosphere. Gray lines show horizontal andvertical wind transport with vertical velocity w. Δχis the deviation of localtracer mixing ratio from the horizontally averaged mixing ratio on an isobar(dashed).

3 Hereafter, we use the generic term “chemistry” to represent anynondynamical processes that affect the tracer distribution, such as chemicalreactions in the gas and particle phase, haze and cloud formation, or otherphase transition processes.

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and using =w 0, we obtain:

cc

¶¶

+¶¶

¢ =-( ) ( )t

ez

e w S . 5z H z H

This is the global-mean vertical tracer-transport equation. Itstates that the evolution of the global-mean tracer mixing ratioc is related to its global-mean vertical eddy fluxes. Based onEquation (5), c cannot be solved directly unless we establish arelationship between the mean value and the eddy flux c¢w . Aconventional assumption is the “flux-gradient relationship” thatlinks the eddy tracer flux to the vertical gradient of the meanvalue (e.g., Plumb & Mahlman 1987) by introducing a 1D“effective eddy diffusion” Kzz such that:

cc

¢ » -¶¶

( )w Kz

. 6zz

If Equation (6) is valid, the global-mean tracer-transportEquation (5) can be formulated as a vertical diffusion equation:

c c¶¶

-¶¶

¶¶

=-( ) ( )t

ez

e Kz

S . 7z H z Hzz

This is the widely used 1D chemical-diffusion equation. Weneed to estimate the eddy flux in Equation (6) to solve for Kzz.Subtracting both Equations (5) and (3), of which Equation (3)is multiplied by c from Equation (4), we obtain:

cc

cc c

¶ ¢¶

+ ¢ +¶¶

+¶¶

¢ - ¢

= ¢

-· ( ) [ ( )]

( )

ut

wz

ez

e w w

S .8

z H z Hh

Now we need to estimate the horizontal tracer variation χ′along an isobar. In terms of the tracer lifetime tc and horizontaltracer distribution under chemical equilibrium c0, the behaviorof Kzz can be categorized into three typical regimes (Figure 3):(I) a short-lived tracer with uniform distribution of chemical-equilibrium abundance, (II) a short-lived tracer with non-uniform distribution of chemical-equilibrium abundance, and(III) a long-lived tracer whose lifetime has long been comparedwith the transport timescale.

2.3. Regime I: Short-lived Tracers with Uniform ChemicalEquilibrium: Diffusive Case

For short-lived tracers, we can estimate χ′ from the verticalgradient of c under the following four assumptions.(i) We neglect the temporal variation (time evolution) of the

χ′ in statistical steady state because we focus on the time-averaged behavior in this study. The first term on the left-handside of Equation (8) can be neglected.(ii) We neglect the complicated eddy term (the last term in

the left-hand side of Equation (8)) that could be much smallerthan the third term. In other words, we assume that thedeviation of the eddy flux from its global mean is smaller thanthe local vertical transport of the global-mean tracer. As shownlater in numerical simulations, this is generally valid in the casewhere the deviation of the tracer mixing ratio from the meanis small and the material surface is not significantly distorted(χ′ is small), or where the vertical gradient of χ′ is smallcompared with the mean tracer gradient. For long-lived tracers(Regime III), this assumption is not valid.(iii) We approximate the horizontal tracer eddy flux term

c c t ¢ » ¢· ( )uh d, where td is the characteristic dynamicaltimescale in the horizontal mixing processes. In general, underdifferent situations, this divergence term could act in anadvective way, or in a diffusive way, motivating two possibleways of formulating the dynamical timescale. In the advection,the dynamical transport timescale t » L Ud h adv, where Lh isthe horizontal characteristic length scale and Uadv is thehorizontal wind speed. In the diffusive case, the dynamicaltimescale t » L Dd h

2 , where D is the effective horizontal eddydiffusivity. The horizontal length scale Lh is determined bydominant flow patterns in the atmosphere (see detaileddiscussions on page 9 in Perez-Becker & Showman 2013).For tracers transported by a global-scale circulation pattern(e.g., equator-to-pole meridional circulation), Lh is usuallytaken as the planetary radius a. Note that in this linearrelaxation approximation, we have assumed that horizontaldynamics always reduces the horizontal variation of the tracer.This generally holds true if the horizontal tracer field is notcomplicated. Some exceptions will be discussed in thenumerical simulation in later sections.(iv) For simplicity, we consider a linear chemical scheme

that relaxes the tracer distribution toward local chemicalequilibrium c0 in a timescale tc:

c ct

=-

( )S . 90

c

In general, the chemical-equilibrium tracer distribution c0depends on many local factors, such as temperature, abun-dances of other species, photon fluxes, and precipitating ionfluxes. It is expected that photochemical species will exhibit c0that varies between the equator and poles. These effects mighthave more pronounced influences on the horizontal distribu-tions of c0 on tidally locked exoplanets than on solar systemplanets. If the species advected upward from the deepatmosphere with a uniform thermochemical source, c0 isassumed constant along an isobar. Without losing generality,here we consider a non-uniform horizontal distribution of thechemical equilibrium: c c c= + ¢0 0 0, where c0 is the globalmean of c0, which is only a function of pressure, and c¢0 (whichcan be a function of longitude and latitude as well as pressure)is the departure of equilibrium tracer abundance from its globalmean. Inserting c0 into Equation (9), we obtain the globally

Figure 3. Atmospheric regimes of global-mean tracer transport as a function oftracer chemical lifetime and horizontal tracer distribution under chemicalequilibrium χ0.

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averaged chemical source/sink term c c t= -( )S 0 c and thedeparture c c t¢ = ¢ - ¢( )S 0 c.

With the assumptions (i)–(iv), Equation (8) can be written:

c ct

c c

t¶¶

+¢=

¢ - ¢( )w

z. 10

d

0

c

We solve for χ′:

ct c

t t¢ =

- + ¢

+

c¶¶

-

- - ( )w

. 11z c1

0

d1

c1

This expression for the 3D distribution of c¢ is qualitativelysimilar to the previous 2D model result in Holton (1986) with aphotochemical source (cf. his Equation (19)). However, Holton(1986) mainly focused on a special case (c¢ = 00 ) withoutfurther elaborating on the physical meaning of the generalexpression Equation (11). Here we explicitly point out that theglobal-mean vertical tracer transport is composed of twophysical processes: a diffusion process and a non-diffusiveprocess. Based on Equation (11), the global-mean verticaltracer flux c¢w is:

ct t

c c

t t¢ =

-+

¶¶

+¢

+- - - ( )ww

z

w

1. 12

2

d1

c1

0

d1

c

The first term in Equation (12) implies diffusive behavior,because this term’s contribution to the tracer vertical flux isproportional to the vertical gradient of the mean tracerabundance. But the second term does not depend on the meantracer gradient, suggesting a non-diffusive behavior. Instead,the second term originates from the correlation between theequilibrium tracer distribution and the vertical wind field. It canbe neglected if the tracer under chemical equilibrium is more orless uniformly distributed across the globe. However, if theequilibrium tracer distribution is significantly non-uniform (forinstance, if, in chemical equilibrium, the equator and poleshave strongly differing chemical abundances), the conventional“eddy diffusion” framework breaks down because the non-diffusive process might dominate the vertical tracer transport inthe global-mean sense. To elaborate on the physics here, wenow further discuss these two cases: a uniformly distributedtracer under chemical equilibrium and a non-uniform case.

If the equilibrium tracer distribution is uniformly distributed(i.e., c¢ = 00 ), the non-diffusive term in Equation (12) vanishes.In this special case of short chemical lifetime (among otherassumptions), mean tracer transport can be regarded as adiffusive process and the eddy diffusivity Kzz can beparameterized based on the relationship between eddy tracerflux to the global-mean tracer gradient based on Equations (12)and (6):

t t»

+- - ( )Kw

. 13zz

2

d1

c1

This result is also consistent with that from a zonal-mean 2Dmodel in Holton (1986) with a uniform chemical-equilibriumabundance. The parameterized Kzz depends on circulationstrength, chemical lifetime of the tracer and horizontaltransport/mixing timescale in the atmosphere. It is expectedthat different chemical tracers are subjected to different eddydiffusivity strength, which has not been considered to date in1D chemical models of planetary atmospheres.

If c¢ = 00 , the horizontal distribution of χ′, deviation of thetracer mixing ratio from the mean on an isobar, is correlated

with the horizontal distribution of vertical velocity w at thesame pressure level (Equation (11)):

ct t

¢ =-

+

c¶¶

- - ( )w

. 14z

d1

c1

If the tracer has a large, uniform chemical-equilibriumabundance in the deep atmosphere but is photochemicallydestroyed in the upper atmosphere—for example, sulfurdioxide on Venus (Zhang et al. 2012), methane on Jupiter(Moses et al. 2005), or water on hot Jupiters (Moses et al.2011)—the mean tracer gradient c¶ ¶z is negative, andtherefore χ′ is positively correlated with w (Equation (14)).This implies that c¢ >w 0, i.e., the tracer is transportedupward. On the other hand, if the tracer source is in the upperatmosphere, for example, chemically produced species that areuniformly distributed under chemical equilibrium, or oxygenspecies with uniform fluxes from comets into the upperatmospheres of giant planets (Moses & Poppe 2017), the meantracer gradient c¶ ¶z is generally positive and χ′ is antic-orrelated with w. This implies that c¢ <w 0, i.e., the tracer istransported downward. In both cases, the c¢ - w correlationpatterns result in a net tracer transport away from their sourceregions.From Equation (14), if the circulation is stronger, i.e., w is

larger, the horizontal variation of the tracer mixing ratio will belarger. If the chemical loss or the horizontal mixing due toadvection or diffusion is stronger, i.e., td and tc are smaller, thetracer variation is smaller. In other words, a large-scalecirculation enhances the tracer variation on isobars, whereaschemical relaxation and horizontal tracer mixing homogenizesthe tracer distribution on isobars. The resulting eddy diffusivityis larger if the circulation is stronger, the tracer chemicallifetime is longer, or the horizontal mixing timescale is longer.We emphasize that Kzz should depend on the chemical

lifetime tc—an important relationship that all current 1Dmodels have ignored. To further elaborate on this, we considertwo different tracers, one with short tc and another with long tc,that both exist in an atmosphere with a specified 3Datmospheric circulation. For concreteness, imagine that thebackground tracer abundance of both tracers decreases upward,with the same background gradient for both tracers. Because ofthe advection, for both tracers, the tracer abundance on anisobar will be greater in upwelling regions and smaller indownwelling regions, implying an upward flux of the tracer inboth cases. However, when tc is short, the chemistry verystrongly relaxes the abundance toward equilibrium, whereaswith long tc, this relaxation is weak. This implies that, instatistical equilibrium, the deviation of the actual tracerabundance from chemical equilibrium, c¢, is greater when tcis long than when it is short. Since the circulation is the same inthe two cases, the upward tracer flux c¢w is greater when tc islong and smaller when tc is short. Given the definition of eddy

diffusivity cc

= - ¢¶¶

K wz

zz from Equation (6), we thus have

the situation where the tracer with short tc has a smaller Kzz

than the tracer with large tc—even though the atmosphericcirculation, by definition, is precisely the same for the twotracers. Indeed, for this simple situation, the value of Kzz wouldgo to zero in the limit t 0c , because in that case the tracer isconstant on isobars, so there is no net correlation betweenvertical velocity and tracer abundance, implying that theupward flux of the tracer is zero.

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2.4. Regime II: Short-lived Tracer with Nonuniform ChemicalEquilibrium: Non-diffusive Component

The horizontal distribution of tracer equilibrium abundancecould be significantly non-uniform, i.e., c¢ ¹ 00 . This couldoccur if there is a local plume source of the species, forexample, volcanic eruption on Earth (Self et al. 1993),convective injection of sulfur species to the middle atmosphereof Venus (Marcq et al. 2013), and impact debris from incomingcomets on Jupiter (Friedson et al. 1999). A more common caseis a photochemically produced species like ozone on terrestrialplanets or ethane on giant planets, where the incomingultraviolet solar flux changes with the solar angle, leading todifferent photochemical equilibrium abundances at differentlatitudes (e.g., Moses & Greathouse 2005). An extremeexample is the tidally locked exoplanets, on which differentchemical-equilibrium states are expected between the perma-nent dayside and nightside for a tracer whose chemistrycritically depends on factors such as temperature, incomingphoton, and ion fluxes. For example, formation of condensedhaze/cloud particles favor the colder nightside than the warmerdayside (Powell et al. 2018). For another example, speciesproduced by photochemistry or ion chemistry are expected tohave a larger equilibrium abundance on the dayside than on thenightside.

In these situations, the dependence of Kzz on the circulationand tracer chemistry is more complicated. The eddy tracer fluxdue to the non-diffusive process, i.e., the second term on theright-hand side of Equation (12), cannot be neglected. Thehorizontal distribution of χ′ might not be strongly correlatedwith the horizontal distribution of vertical velocity w at thesame pressure level (Equation (11)). The correlation term c¢w 0could have a significant effect on Kzz (Equation (12)).Physically speaking, in the case with uniform chemical-equilibrium abundance, atmosphere circulation will shape theinitially homogeneous tracer distribution toward a correlationpattern that c¢w has the opposite sign as the backgroundgradient c¶ ¶z. This implies that the tracer is mixed down thevertical gradient of the horizontal-mean tracer abundance. Inthe case of the non-uniform chemical-equilibrium case, c0might correlate or anticorrelate with the vertical velocitypattern, causing an additional contribution to the net verticaltracer transport in an essentially non-diffusive way. Whetherthis contribution will enhance or reduce the net vertical mixingefficiency depends on the correlation as well as the verticalgradient of the mean tracer (Equation (12)).

If we still adopt the traditional “eddy diffusion” framework,inserting the global-mean vertical tracer flux c¢w(Equation (12)) into the definition of Kzz (the flux-gradientrelationship Equation (6)), we can approximate the non-diffusive behavior and estimate Kzz for tracers with non-uniform chemical equilibrium:

t t

c

t tc

»+

-¢

+¶¶- - -

-⎛⎝⎜

⎞⎠⎟ ( )K

w w

z1. 15zz

2

d1

c1

0

d1

c

1

The second term on the right-hand side represents inherentlynon-diffusive behavior, because it has a dependence on thevertical gradient of the mean tracer c¶ ¶z. This implies thatthe eddy tracer flux (Equation (6)) has a term that does notscale linearly with the background vertical tracer gradient,which violates the fundamental assumption of a diffusivesystem that the flux scales linearly with the tracer gradient.

Also, unlike the first term on the right-hand side ofEquation (15), the second term can have either sign, dependingon the sign of the correlation between w and the anomalies onisobars of the chemical-equilibrium abundance, c¢0. Note thatEquation (15) is no longer a closed expression for Kzz becauseit depends on the vertical gradient of the global-mean tracermixing ratio, a quantity that we require Kzz to solve for. Onecould imagine that an iterative process between the chemical-diffusion simulation and updating Kzz might lead to a finalsteady state of the system.We emphasize that the Kzz expression with the non-diffusive

correction needs to be used cautiously in 1D chemical-diffusionsimulations and can only be used when the non-diffusivecontribution is not dominant. If the second term dominates andis negative, the predicted Kzz can be negative in somesituations. For example, if the tracer lifetime is very short,t 0c , Equation (15) becomes:

cc

» - ¢ ¶¶

-⎛⎝⎜

⎞⎠⎟ ( )K w

z. 16zz 0

1

In this limit, the diffusive term vanishes and the non-diffusive term dominates. Vertical tracer mixing is significantlycontrolled by the correlation between the chemical-equilibriumdistribution and the vertical velocity distribution. If thecorrelation is positive, i.e., the tracer is more abundant in theupwelling region than in the downwelling region due tochemistry, and if the tracer is produced at the top, i.e., thevertical gradient of the mean tracer mixing ratio is positive, thederived Kzz is negative (Equation (16)). A negative effectiveeddy diffusivity does not make sense physically as it suggeststhat the tracer is mixed toward its source. The reason is simplybecause the global-mean vertical tracer transport is essentiallynon-diffusive in this case.For a given vertical gradient of the global-mean tracer, Kzz

becomes larger if the circulation is stronger, the tracer chemicallifetime is longer, and the horizontal mixing timescale islonger. This trend is consistent with that in the case withuniform chemical-equilibrium abundance. Interestingly, as thechemical timescale becomes longer, the diffusive term getslarger and the non-diffusive term becomes smaller, perhapsbecause the longer-lived tracers are more controlled by thedynamical transport so that the non-uniform chemical equili-brium has less effect. Thus the non-diffusive correction inEquation (15) should work better for species with relativelylong timescales.

2.5. Regime III: Long-lived Tracers

If the tracer chemical lifetime is long and the tracer is almostinert, the material surfaces (tracer contours) are usuallydistorted significantly.4 The above discussion for short-livedtracers could be violated since the assumption (ii) inSection 2.2 might no longer be valid. It is expected that the3D distribution of such a “quasi-conservative” tracer issignificantly controlled the atmospheric dynamics. Due to thecomplicated dynamical transport behavior, no simple analytical

4 If the tracer is absolutely inert with a very long lifetime, it will becompletely homogenized over the globe by the horizontal transport. But in thisstudy we do not investigate this type of real conservative, well-mixed tracers.The “long-lived tracer” in our study stands for the species with a significantlylong chemical lifetime so that the atmospheric dynamics greatly shapes itsmaterial surface, but the horizontal variation of the tracer distribution is still notsmall.

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theory of the global-mean vertical tracer transport exists, andthus the corresponding Kzz is not generally known.

Although it is expected that there are some non-diffusiveeffects in this regime, the diffusive framework may still beuseful. If we still apply the Kzz theory in Section 2.3 to thisregime and let t ¥c , the non-diffusive contribution vanishes(Equation (12)). In this case, the influence of the chemical-equilibrium tracer distribution is negligible and Equation (15)is reduced to Equation (13). One might expect Kzz to approachits asymptotic value in Equation (13):

t» = ˆ ( )K w wL , 17zz2

d v

where we have introduced a vertical transport length scalet= ˆL wv d and w is the root mean square of the vertical velocity

=ˆ ( )w w2 1 2. Here the vertical transport timescale is assumedto be the horizontal tracer mixing timescale td due to thecontinuity equation. Equation (17) is in a similar form as thatfrom the mixing length theory although there is no convectiveor small-scale mixing due to wave breaking in our theory.Using Lv, the effective eddy diffusivity (Equation (13)) forshort-lived tracers with uniform chemical equilibrium can alsobe represented as:

t t=

+ -

ˆ ( )KwL

1. 18zz

v

d c1

The magnitude of the change in Kzz depends on the ratio of thetimescales between the dynamical and chemical processes:t td c. In the long-lived tracer regime where t ¥c , theeffective eddy diffusivity Kzz approaches Equation (17).

The vertical transport length scale Lv in our Kzz theorycannot be arbitrarily chosen. It critically depends on theatmospheric dynamics, specifically the vertical velocity w andthe horizontal dynamical timescale td. If the dynamicaltimescale td for horizontal tracer mixing is equal to the globalhorizontal advection timescale a/U, where a is approximatelythe planetary radius, through continuity td should be approxi-mately ˆH w. Then Lv would be equal to the pressure scaleheight H. If the horizontal mixing timescale is longer (orshorter) than the horizontal advection timescale, Lv is thenlarger (or smaller) than H by that same factor. Using thevertical velocity from GCMs and simply assuming Lv is H,some previous models (e.g., Lewis et al. 2010; Moses et al.2011) estimated the eddy diffusivities on exoplanets based onEquation (17). Those estimates were much larger than the eddydiffusivity derived based on 3D passive tracer simulations(Parmentier et al. 2013).

There are two reasons. First, Kzz estimated fromEquation (17) is the maximum eddy diffusivity one can obtainfrom Equation (18). For short-lived tracers, the effective eddydiffusivity should be smaller than that from Equation (17).Second, in the long-lived tracer regime, tracers are significantlycontrolled by atmospheric dynamics and the horizontaldynamical timescale td might be different from the globalhorizontal advection timescale a/U. Thus the vertical char-acteristic transport length scale Lv in Equations (17) and (18)could be different from H. This has also been noted in thestudies of tracer transport in the convective atmospheres. Forexample, Smith (1998) investigated dynamical quenching ofchemical tracers in convective atmospheres and found that thevertical transport length scale in the traditional mixing lengththeory should depend on the chemical tracer equilibrium

distribution as well as the chemical and dynamical timescales.Recent work by Bordwell et al. (2018) explored the chemicaltracer transport using nonrotating local convective box models.They found that using the chemical scale height as the verticaltransport length scale, which is usually smaller than the scaleheight H, leads to a better prediction of the chemical quenchinglevels.We reiterate that the assumptions used to derive

Equation (17) likely break down in the regime of long-livedtracers, so it may be that the qualitative dependencies impliedin Equation (17) are not rigorously accurate when the tracersare long-lived. In our theory we have dropped the last term inthe left-hand side in Equation (8), which might becomeimportant in the long-lived regime. As we will demonstrate inthe numerical simulations later, this nonlinear eddy term couldpotentially enhance or decrease the global tracer mixingefficiency. We also emphasize that, although we adopt thediffusive framework here in the long-lived tracer regime, thetracer transport in this regime may not always behavediffusively. As we will also show later in the numericalsimulations, the diffusive assumption could break down insome cases when the tracer material surface is distortedsignificantly and non-diffusive effects are substantial.In sum, in this section we have developed an approximate

analytical theory for the 1D global-mean tracer transport in a3D atmosphere. We demarcated three atmospheric regimes interms of the tracer chemical lifetime and horizontal tracerdistribution under chemical equilibrium. The underlyingphysical mechanisms governing the global-mean vertical tracertransport in the three regimes are different. The traditionalchemical-diffusion assumption is mostly valid in the firstregime but could be violated in the second and third regimes.We predicted the analytical expression of the 1D effective eddydiffusivity Kzz for the global-mean vertical tracer transport. Kzz

depends on both atmospheric dynamics and tracer chemistry.Crudely speaking, if the atmospheric dynamics is fixed, Kzz

roughly scales with the tracer chemical lifetime tµKzz c(Equation (13)) when the tracer lifetime is short (Regime I) andapproaches a constant value when the tracer lifetime is long(Regime III, Equation (17)). We also found that in the short-lived tracer regime (Regime I), Kzz roughly scales with thevertical velocity square µ ˆK wzz

2 (Equation (13)), but in thelong-lived tracer regime (Regime III), it scales with root meansquare of the vertical velocity µ ˆK wzz (Equation (17)). Next,we will perform a series of numerical experiments toquantitatively verify our theoretical arguments. We willprimarily focus on stratified atmospheres here.

3. 2D Simulations on Fast-rotating Planets

Now we consider numerical simulations of tracer transporton a fast-rotating planet on which the tracer is uniformlydistributed with longitude (but not necessarily with latitude).Most of the planetary atmospheres in the solar system are closeto this situation. This is essentially a 2D tracer transportproblem that can be studied using a 2D zonally symmetricmodel. In our numerical simulations, we will only studypassive tracers, i.e., no radiative feedback from the tracer to thedynamics.2D zonal-mean tracer transport has been extensively

discussed in the Earth literature (e.g., Holton 1986). But untilnow there has not been a thorough and specific study on theglobal-mean eddy diffusivity using 2D numerical simulations

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with chemical tracers. On a fast-rotating planet, when averagedzonally, the meridional circulation that transports the tracer(like the one shown in Figure 2) should be considered as the“Lagrangian mean circulation,” which can be approximated bya Transformed Eulerian Mean (TEM) circulation (Andrews &McIntyre 1976), and called the “residual mean circulation.”Another formalism of the zonal-mean tracer transport intro-duced by Plumb & Mahlman (1987) used the “effectivetransport velocity.” The difference between the two circulationformalisms is usually small in Earth’s stratosphere and vanishesif the waves are linear, steady, and adiabatic (Andrewset al. 1987).

In both frameworks, the zonal-mean eddy tracer fluxes canbe parameterized as diffusive fluxes using a symmetric“diffusion tensor” with four parameters (diffusivities): Kyy2D,Kzz2D, Kyz2D, and Kzy2D (Andrews et al. 1987). Kyz2D and Kzy2Dare negligible in isentropic coordinates (Tung 1982). As herewe mainly focus on the stratified atmosphere where theinclination between the isobars and isentropes is usually small,we can ignore Kyz2D and Kzy2D in 2D chemical tracer-transportsimulations in this study (e.g., Garcia & Solomon 1983; Shiaet al. 1989).

A 3D model would naturally produce small-scale eddies thatwould cause mixing, and if the eddies were fully resolved, noparameterization of diffusive fluxes would be needed in thetracer transport. But in our 2D framework that ignores any rolefor such eddies, even for a planet with a great degree of zonalsymmetry at large scales (like Earth or Jupiter), we have toparameterize the horizontal and vertical diffusivities Kyy2D andKzz2D as a separately included process. In principle, Kyy2D andKzz2D can be estimated from the Eliassen–Palm flux (Andrewset al. 1987) in a 3D model. But for our purpose of studying theresponse of the chemical tracer distribution to the dynamics andthe global-mean vertical tracer mixing, we just prescribe thediffusivities and the circulation.

In this study, we prescribe the temperature structure andcirculation pattern and hold them constant with time. Thegoverning equation of a 2D chemical-transport system usingthe coordinates of log-pressure and latitude, can be written as(Shia et al. 1989, 1990; Zhang et al. 2013b):

* *c c c

ff

c

c

¶¶

+¶¶

+¶¶

-¶¶

¶¶

-¶¶

¶¶

=-

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ ( )

tv

yw

z yK

y

ez

e Kz

S

1

coscos

, 19

yy

z H z Hzz

2D

2D

where f = y a is latitude and a is the planetary radius. H is thepressure scale height. The chemical source and sink term Sfollows the linear relaxation scheme of Equation (9).

Residual circulation velocities are v* and w* in themeridional and vertical directions, respectively. For a 2Dcirculation pattern, we can introduce a mass streamfunctionψ such that:

*p r f

y= -¶¶

-( ) ( )va

ez

e1

2 cos20az H z H

0

*p r f

y=

¶¶

( )wa y

1

2 cos, 20b

0

where r0 is the reference density of the atmosphere at log-pressure z=0. With prescribed distributions of ψ, Kyy2D, andKzz2D, we solve the governing equations using the Caltech/JPL

2D kinetics model (for numerics, refer to Shia et al. 1990). Weuse the Prather scheme for the 2D tracer advection(Prather 1986). The model has been rigorously tested againstmultiple exact solutions under various conditions (Shiaet al. 1990; Zhang et al. 2013b).In this work, we adopted two different meridional circulation

patterns (see Section 4 in Zhang et al. 2013b). The circulationyA is an equator-to-pole pattern and circulation yB is pole-to-pole, corresponding to planets with low obliquity and highobliquity, respectively:

y p r f f= h ( )a w e2 sin cos 21az HA

20 0

2

y p r f= h ( )a w e2 cos . 21bz HB

20 0

2

But we do not investigate the seasonal change of the circulationpattern in this study. The above circulation patterns areassumed steady with time in our simulations.The mass streamfunctions and vertical velocities of the two

circulation patterns are shown in Figure 4. For both circula-tions, the area-weighted global-mean vertical velocity scale isabout g hw e z H

0 , where γ is an order-unity prefactor originatingfrom the global average.5

In this 2D chemical-advective-diffusive system, both themeridional advection and horizontal eddy diffusion contributeto the horizontal tracer mixing, and both the vertical advectionand vertical eddy diffusion contribute to the vertical tracertransport. The 1D effective eddy diffusivity Kzz in this systemcan be analytically predicted following the procedure intro-duced in Section 2. But since here we have to treat explicitlythe eddy diffusion terms Kyy2D and Kzz2D originating fromzonal-mean 2D dynamics, we provided a detailed derivation ofKzz in this 2D system in the Appendix. We show that thevertical eddy diffusion by Kzz2D in Equation (19) can be treatedas an additive term in the global-mean effective eddydiffusivity Kzz.In the 2D framework, the Kzz for the situation of non-

uniform chemical-equilibrium mixing ratio c0 can be expressedas (see the Appendix for more details):

gg t

g c

t t gc

» ++ +

-D ¢

+ +¶¶

h

h

h

h

- - -

- -

-⎛⎝⎜

⎞⎠⎟ ( )

K Kw e

K a w e H

w e

K a w e H z1, 22

zz zz

z H

yyz H

z H

yyz H

2D

202 2

2D2

01

c1

0 0

c 2D2

c 01

01

where c0 is the global mean of the non-uniform chemical-equilibrium mixing ratio c0. cD ¢

0 is the root mean square of the

deviation c¢0 over the globe. For a cosine function of c0,

c cD ¢ » 0.280 0 . Here we have also assumed the horizontaltransport length scale ~L ah and vertical transport length scale

~L Hv . If c0 is uniform, Kzz can be reduced to:

gg t

» ++ +

h

h- - - ( )K Kw e

K a w e H. 23zz zz

z H

yyz H2D

202 2

2D2

01

c1

In the second term on the right-hand side, the three terms in thedenominator correspond to horizontal tracer diffusion due toparameterized effects of eddies and waves, horizontal traceradvection by zonal-mean flow, and tracer chemistry, respectively.

5γ is 2 5 for yA and 2 3 for yB from the global integrations of

Equations 21(a) and (b), respectively. Thus the effective transport is a bitstronger using the circulation pattern B.

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We use Jupiter’s parameters in these 2D simulations but withan isothermal atmosphere of 150 K from 3000 Pa to about0.2 Pa for simplicity. For the circulation patterns, we adopt

= - -w 10 m s05 1 and h = 0.5. Given a pressure scale height H

of about 25 km, the vertical advection timescale changes from109 s at the bottom to about 107 s at the top. The vertical profileof the advection timescale is similar to the empirical verticalmixing timescale derived from H Kz

2 , where Kz is theempirical eddy diffusivity in the 1D stratospheric chemicalmodel on Jupiter (Moses et al. 2005, also see Figure 1). Thehorizontal advection timescale in our model is about 108 s at100 Pa, comparable to the vertical circulation timescale due tothe continuity constraint. This horizontal transport timescale isalso similar to that inferred from Voyager and Cassiniobservations (Nixon et al. 2010; Zhang et al. 2013a).

We designed five 2D experiments (Table 1) and simulatednine tracers in each experiment. In all cases, we set the verticaldiffusion coefficients Kzz2D zero because here we mainlyinvestigate the influence of the large-scale overturning circula-tion, horizontal mixing and chemical source/sink on theglobal-mean vertical tracer transport in this study. According toEquation (23), the influence of Kzz2D in the 2D simulation istrivial because it can just be treated as an additive term in Kzz.We also tested different horizontal diffusional coeffi-cients Kyy2D.Experiments I–IV are simulations with chemical tracers with

uniform chemical-equilibrium mixing ratios. Experiment Vfocuses on the chemical tracer transport with non-uniformchemical equilibrium. Experiment I (Figure 4) is the standardcase with an equator-to-pole streamfunction yA and a constant

Figure 4. Latitude–pressure maps of experiments I (left) and II (right). First row: vertical winds (color) and mass streamfunctions (contours, in units of -10 Kg s13 1).Starting from the second row we show volume mixing ratio maps of tracers with chemical timescales of 107 s, 108 s, 109 s, and 1010 s from top to the bottom,respectively.

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horizontal eddy diffusivity = -K 10 m syy2D2 1. This is a “deep

source” case in which tracers are advected upward from thedeep atmosphere at 3000 Pa. The vertical profile of theequilibrium tracer mixing ratio ceq is assumed to be a power-law function of pressure c = -( ) ( )p p p10eq

50

1.7, where=p 3000 Pa0 is the pressure at the bottom boundary. ceq

starts from 10−5 at the bottom and decreases toward 10−12 atthe top of the atmosphere. A linear chemical scheme(Equation (9)) is applied to each tracer with a relaxationtimescale tc that varies from 107 to 1011 s. The chemicaltimescale of the ith tracer is assumed to be t = +10 i

c6.5 0.5 s.

Experiment II is the same as Experiment I but with adifferent circulation pattern yB. Experiment III is also the sameas Experiment I but with a “top chemical source.” Theequilibrium tracer mixing ratio, while still uniform across thelatitude, is large at the top (10−5) and small at the bottom(10−12). In Experiments I to III, the horizontal eddy diffusiontimescale is ~ ´a K 5 10yy

22D

14 s. The diffusive transport ismuch less efficient compared with the horizontal advection andcan be neglected (timescale of –10 107 9 s). We designedExperiment IV using the streamfunction yA but horizontaleddy diffusivity Kyy2D enhanced to -10 m s6 2 1. In this case, thediffusive timescale is about ´5 10 s9 , comparable to thecirculation timescale (but the Kzz2D is still zero). In this setup,we can test the influence of the horizontal eddy mixing to theglobal-mean vertical tracer transport. Experiment V is similarto Experiment III with a “top chemical source” but the tracerchemical-equilibrium mixing ratios are not uniformly distrib-uted with latitude. The non-uniform chemical-equilibriummixing ratio is assumed to be a cosine function of the latitude:c f c f=( ) ( )p p, cos0 eq . This setup is closer to the realisticphotochemical production in planetary atmospheres where thephotochemical photon flux changes with latitude.

We discretized the atmosphere into 35 latitudes in thehorizontal dimension, corresponding to 5° per grid cell.Vertically, the log-pressure grid is evenly spaced in 80 layersfrom 3000 Pa to about 0.2 Pa. The time step in the simulationsis 105 s. We ran the simulations for about 1013 s to ensure thespatial distributions of the tracers have reached the steady state.The tracer abundances were averaged over the last 109 s foranalysis. We tested the model with different vertical andhorizontal resolutions to confirm that the simulation results arerobust.

3.1. Results: Simulations with Uniform c0 (Experiments I–IV)

The steady-state results in Experiments I and II are shown inFigure 4. Although the final tracer distributions are differentbetween the two experiments due to different circulationpatterns, some common behaviors exist. The short-lived tracers

are uniformly distributed across latitude. As the tracer chemicallifetime increases, the effect of the circulation becomesstronger. In the upwelling region, tracers with higher mixingratios are transported upward from their deep source; while inthe downwelling region, tracers with lower mixing ratios aretransported from the upper atmosphere. As a result, the tracermixing ratio is higher in the upwelling region and lower in thedownwelling region on an isobar. The final latitudinalvariations of the short-lived tracers follow the pattern of thevertical velocity (Figure 4 and Equation (14)). However, if thetracer chemical timescale is very long (Figure 4, bottom row),the tracers tend to be homogenized by the circulations.We numerically derive Kzz based on the simulation results

and the flux-gradient relationship (Equation (6)) and comparewith the analytical prediction (Equation (23)). First, weaveraged the tracer distributions with latitude in an area-weighted-mean fashion. The vertical profiles of the global-mean tracer mixing ratios are shown in Figure 5 for ExperimentI and Figure 6 for Experiment II, respectively. The verticalprofiles of short-lived tracers are close to the chemical-equilibrium profile but that of the long-lived tracers are almostwell mixed and “quench” to the lower atmosphere valuesbecause the global circulations efficiently smooth out theirvertical gradients. We then derived the 1D effective eddydiffusivity Kzz by equating the eddy diffusive flux to the global-mean net vertical flux of the tracers (Equation (6), Figures 5and 6). Our analytical prediction based on Equation (23)matches the numerical results well.Kzz increases from the bottom toward the top of the

atmosphere because the vertical velocity is larger and transportis stronger in the upper atmosphere. Our theory predicts thatKzz roughly scales with the square of the vertical velocity

µ ˆK wzz2 in the short-lived tracer regime (Regime I,

Table 12D Simulation Cases in This Study

Experiment Streamfunction

Kyy2D

( -m s2 1 )TracerSource

Latitudinal Dis-tribution of c0

I yA 10 Deep UniformII yB 10 Deep UniformIII yA 10 Top UniformIV yA 106 Deep UniformV yA 10 Top Nonuniform

Figure 5. Vertical profiles of the global-mean volume mixing ratio (left) andderived effective eddy diffusivity (right) from Experiment I. Different colorsfrom cold (blue lines) to warm (red lines) represent tracers with differentchemical timescales from short to long, ranging from 107 s to 1011 s. Theprescribed equilibrium tracer mixing ratio profile is shown in the dashed line inthe left panel, which is nearly on top of the mixing ratio profile of the veryshort-lived tracer (dark blue solid line, see the upper left corner of the leftpanel). The predicted eddy diffusivity profiles based on Equation (23) areshown by dashed lines in the right panel. The solid lines are derived from thesimulations. The thick gray line indicates the empirical eddy diffusivity profileused in current photochemical models in Jupiter’s stratosphere (Moseset al. 2005, Moses & Poppe 2017).

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Equation (13)) and µ ˆK wzz in the long-lived tracer regime(Regime III, Equation (17)). The vertical profiles of the Kzz

follow the scaling very well (Figure 5). The theory alsopredicted that Kzz should be a strong function of chemicaltimescale (Equation (13)). This is also confirmed by thenumerical simulations. Kzz can increase by more than a factorof 1000 from short-lived tracers to long-lived tracers (Figures 5and 6). This increasing trend is well predicted by our analyticaltheory Equation (23). Longer-lived tracers are more dynami-cally controlled than chemically controlled and therefore havebetter correlation with the circulation pattern, leading to alarger global-mean effective vertical transport. As we pointedout in Section 2, if the atmospheric dynamics is fixed (i.e.,passive tracer transport), Kzz scales with the tracer chemicallifetime tµKzz c (Equation (13)) when the tracer lifetime isshort (Regime I) and approaches a constant value when thetracer lifetime is long (Regime III, Equation (17)). Therefore,for tracers with very long chemical timescale, Kzz is insensitiveto the chemical timescale (also see Equation (23)). Thedependence of Kzz on tc is clearly seen in Figure 7.

In Experiments I and II, horizontal diffusive transport viaKyy2D is small compared with the wind advection. Whent ¥c , the effective eddy diffusivity (Equation 21(a))approaches » hK Hw ezz

z H0 . The analytical Kzz converges to

a profile in Figure 5, but does not match the numerical Kzz

exactly. There may be greater disparity in Experiment II(Figure 6) where the numerical Kzz exhibit wavy fluctuations.In the long-lived tracer regime, the derived diffusivitydecreases with chemical timescale in some pressure rangesand even becomes negative at some pressure levels. Thisreveals a drawback in our theory in Section 2. As noted inSection 2.5, when the chemical lifetime is too long comparedwith the circulation timescale, the material surfaces of tracersare distorted significantly. The expression of Kzz might nolonger be valid because the last term (eddy term) in the left-hand side of Equation (8), c c¢ - ¢¶

¶-[ ( )]e e w wz

zz , is compar-

able to or even larger than the third term (mean tracer term)c¶

¶w

zand thus it cannot be neglected. As illustrated in Figure 8,

for a short-lived tracer with t = 10 sc7 , the eddy term is much

smaller than the mean tracer term across the latitude andEquation(23) is a good prediction of Kzz. But for a long-livedtracer with t = 10c

10 s, the eddy term is comparable to themean tracer term. The eddy transport is complicated in thisregime and may lead to some wavy features in the numericalKzz in Figures 4 and 5, which our current analytical theory isunable to capture although the theoretical prediction is stillwithin a factor of 2–5 in Experiment I and can be as large as afactor of 10 in Experiment II.Note that the theoretical prediction in the long-lived tracer

regime generally underestimates the eddy mixing from thenumerical simulations. In this regime, the theoretical

» ˆK wLzz v (Equation (17)). Here we have assumed that thehorizontal dynamical timescale td is equal to the globaladvection timescale a/U and thus the vertical transport lengthscale »L Hv through continuity. The discrepancy between theanalytical and numerical eddy diffusivities implies that theactual horizontal dynamical timescale is larger than a/U butthe mechanism is not clear. A detailed future investigation isneeded.In Experiment II (Figure 4, third row, t = 10 sc

9 case), theoverturning circulation transports the high-concentration

Figure 6. Same as Figure 5 but from Experiment II. Note that some curves(e.g., the orange line) break apart in the long-live tracer regime. This is becausethe material surface is distorted so large that the derived Kzz is negative in themiddle part.

Figure 7. Kzz as a function of tracer chemical timescale for all four experimentsat two typical pressure levels, 80 Pa (filled circles) and 200 Pa (open circles).The solid and dashed lines are the predictions from Equation (23).

Figure 8. Important tracer tendency terms in Equation (8) for a short-livedtracer (upper panel, τc=107 s) and a long-lived tracer (lower panel,τc=1010 s) in Experiments I. The black lines represent the term c¶

¶w

zand

the red are c c¢ - ¢¶¶

-[ ( )]e e w wzz

z .

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The Astrophysical Journal, 866:1 (17pp), 2018 October 10 Zhang & Showman

tracers from the deep atmosphere in the southern hemisphereall the way to the top of the northern hemisphere. These tracersare then mixed downward in the northern hemisphere above100 Pa. The global-mean mixing ratio profile of these tracersalso shows a local shallow minimum at around 100 Pa (yellowline in Figure 6). This type of local minima is also commonlyfound in global-mean tracer mixing ratio profiles in thesimulations of convective atmospheres (see Figure 6 inBordwell et al. 2018). At the pressure levels right above thislocal minimum, because the tracer mixing ratio increases withaltitude but the global-mean tracer transport is still upward, theglobal-mean tracer flux is against the local vertical gradient ofthe tracer. As a result, the predicted eddy diffusivity Kzz isnegative above 100 Pa (Figure 6). As we have discussed inSection 2.4, this type of “negative eddy diffusivity” phenom-enon probably indicates that the global-mean vertical tracertransport in this case does not behave diffusively. Therefore,the diffusive assumption could break down in the long-livedtracer regime when the material surfaces are significantlydistorted.

When a uniform chemical source is located at the top(Experiment III), the horizontal distribution of the tracer isanticorrelated with the vertical velocity distribution (Figure 9),i.e., the tracer abundance is higher in the downwelling regionand lower in the upwelling region, as measured on isobars. Thisdoes not alter our theory of global-mean vertical mixing inSection 2. As before, the derived effective eddy diffusivitiesincrease with tracer chemical timescale and approachesconstant in the long-lived tracer regime (Figure 7). If weenhance the horizontal diffusion via Kyy2D (Experiment IV), thediffusion tends to smooth out the horizontal gradient of thetracer, leading to a weaker correlation between the tracerdistribution and the vertical velocity. The latitudinal distribu-tions of tracers (Figure 9) are flatter in this case compared tothat in Experiment I. The Kzz is thus smaller (Figure 7) but stillincreases with the chemical timescale. The trend is consistentwith other experiments and our theory. As seen in Figure 7, thedecrease of Kzz with a larger Kyy2D can also be predicted inEquation (23).

3.2. Results: Simulations with Nonuniform c0 (Experiment V)

The cases with non-uniform c0 (Regime II) behave quitedifferently from that with uniform c0 (Regime I). Thenumerical simulations in Experiment V are shown inFigure 10. The prescribed c0 follows a cosine function oflatitude. The equilibrium mixing ratio is higher at the equatorand lower at the poles. For the short-lived species, the generalpatterns of the final tracer distributions roughly follow thedistribution of c0. But in the atmosphere above 1 Pa, where thetransport is very efficient, the low-mixing-ratio tracers areadvected upward from the lower atmosphere at the equator,leading to a smaller tracer abundance at low latitudes than atmid-latitudes. The global-mean tracer mixing ratio profilesroughly follow the global-mean chemical-equilibrium tracerprofile (Figure 11).

As the tracer lifetime increases, the global-mean tracermixing ratio profile becomes more vertical with pressure due tostronger dynamical mixing (Figure 11). In those cases,meridional circulation greatly shapes the tracer distributionstoward the pattern with lower-mixing-ratio tracers at theequator and higher-mixing-ratio at the poles (Figure 10). Thelatitudinal trend of the tracer on an isobar is opposite that of

the prescribed chemical-equilibrium tracer distribution c0. Ingeneral, the patterns of longer-lived tracers look similar to thatin Experiment III, where the tracer source is also from the topatmosphere but c0 is flat with latitude in that case. However,there is a significant difference between the Experiments III andV. There are two mid-latitude “tongues” sinking from the topatmosphere in Experiment V, while they are missing inExperiment III, where the tracer mixing ratio is high from themid-latitudes all the way to the poles. The existence of the mid-latitude tracer maxima in Experiment V suggests strongdownwelling of low-mixing-ratio tracers from the top atmos-phere in the polar region where the chemical-equilibrium tracermixing ratio is low. The downward tracer fluxes dilute thehigh-mixing-ratio tracers that are transported from the middlelatitudes to the polar region, resulting in local maxima(“tongues”) that are concentrated at mid-latitudes.Our theory in Section 2 predicts that non-uniform c0 could

introduce a negative component in the global-mean eddydiffusivity Kzz. This effect is more pronounced for short-livedspecies. The numerically calculated Kzz are shown in Figure 11.The derived Kzz become negative below some pressure levelsfor short-lived species with chemical lifetimes smaller than108 s. This non-diffusive effect is primarily due to the non-uniform c0, which is less important when the tracer lifetimebecomes longer and when the tracer distribution substantiallydeviates away from the chemical equilibrium. For long-livespecies, the non-diffusive effect vanishes, and the derived Kzz

profile generally agrees with that from Experiments I, II, and IIIwith uniform c0. Using our analytical formula of Kzz with anon-diffusive correction term (Equation (22)), we can generallyreproduced the negative Kzz values for short-lived species aswell as the positive values for long-lived species (Figure 11).

4. Conclusion and Discussion

The central assumption in the 1D framework is that verticaltracer transport acts in a diffusive manner in a global-meansense. Eddy diffusivity is a key parameter in this frameworkand is normally constrained by fitting the model to observedtracer profiles. However, the physical meaning of thisempirically determined quantity is usually not elucidated, andthere have been few attempts in the planetary literature toestimate it from first principles or show systematically how itshould vary from planet to planet. In this study, we investigatedsome of the fundamental processes that are lumped into—andcontrol—this single quantity. We generalized the pioneeringtheoretical work from Holton (1986) for a 2D Earth model to a3D atmosphere and explicitly derived the diffusivity expressionfrom first principles for specific situations. We performedtracer-transport simulations in a 2D chemical-diffusion-advection model for rapidly rotating planets using a simplechemical source/sink scheme. By deriving the 1D eddydiffusivity from the globally averaged vertical transport flux,we showed that the simulation results in 2D agree with ourtheoretical predictions. Therefore, this study can serve as atheoretical foundation for future work on estimating orunderstanding the effective eddy diffusivity for global-meanvertical tracer transport.The general take-home message from our investigation is

that interaction between the chemistry and dynamics isimportant in controlling the 1D vertical tracer transport. Ourwork demonstrates that the global-mean vertical tracertransport crucially depends on the correlations between the

12

The Astrophysical Journal, 866:1 (17pp), 2018 October 10 Zhang & Showman

vertical velocity field and the tracer horizontal variations,which is significantly modulated by atmospheric circulation,horizontal diffusion and wave mixing, and the local tracerchemistry or microphysics. Importantly, this correlation is alsocontrolled by the chemistry itself, which implies that—even fora given atmospheric circulation—the vertical mixing rates andeffective eddy diffusivity can differ from one chemical speciesto another. In general, we found that the traditional assumptionin current 1D models that all chemical species are transportedvia the same eddy diffusivity breaks down. Instead, the 1Deddy diffusivity should increase with tracer chemical lifetimeand circulation strength but decrease with horizontal mixingefficiency due to breaking of Rossby waves or other horizontalmixing processes. Our analytical theory including these effectscan explain the 2D numerical simulation results over a wideparameter space. This physically motivated formulation ofeddy diffusivity could be useful for future 1D tracer-transportmodels.

We emphasize that the conventional “diffusive” assumptionof the global-mean vertical tracer transport does not alwayshold. In this study, we demarcated three regimes in terms of thetracer lifetime and horizontal tracer distribution under chemicalequilibrium (Figure 3). Only in Regime I, with a short-livedtracer with uniform distribution of chemical-equilibriumabundance, is the traditional diffusive assumption mostly valid.In the other two regimes, with a tracer with non-uniformdistribution of chemical-equilibrium abundance (Regime II)and with the tracer lifetime that is significantly long comparedwith the transport timescale (Regime III), the global-meanvertical tracer transport could be largely influenced by non-diffusive effects. Non-diffusive effects could result in anegative effective eddy diffusivity in the traditional diffusiveframework, either for the short-lived species in Regime II or forthe long-lived species in Regime III. A negative diffusivitydoes not physically make sense and might be difficult toincorporate into 1D models. For relatively long-lived species inRegime II (but not long enough to reach Regime III), we

Figure 9. Same as Figure 4 but for Experiments III (left) and IV (right).

13

The Astrophysical Journal, 866:1 (17pp), 2018 October 10 Zhang & Showman

provided a simple derivation to capture the non-diffusiveeffects, which might be useful for future 1D models.

Our detailed analytical and numerical analysis concludesseveral key points:

(1) Larger characteristic vertical velocities contribute to alarger global-mean tracer mixing. Because verticalvelocities tend to increase with height, we find that theeddy diffusivity generally increases with height as well,although vertical variations of chemical timescale couldcomplicate this picture if they are sufficiently large. Forshort-lived tracers in Regime I, µ ˆK wzz

2 and for thelong-lived tracers in all regimes, µ ˆK wzz if the non-diffusive effect is not significant.

(2) Efficient horizontal eddy mixing due to breaking ofRossby waves or other wave processes will smooth outthe horizontal variations of tracer and thus decrease the

global-mean eddy diffusivity. But the horizontal traceradvection due to the mean flow can either increase ordecrease the eddy diffusivity, depending on the eddytracer flux convergence/divergence induced by the meanflow, which fundamentally depends on the correlationbetween the horizontal mean flow and the horizontaltracer variations.

(3) Global-mean eddy diffusivity depends on the tracersources and sinks due to chemistry and microphysics.In an idealized case with linear chemical relaxation, weshowed that the effective eddy diffusivity increases withthe chemical relaxation timescale (i.e., chemical lifetime).In Regime I, Kzz∝τc. When the chemical lifetime is verylong (Regime III), the effective eddy diffusivity reachesits asymptotic value—a product of vertical wind velocityand vertical transport length scale.

(4) In Regime I, short-lived species exhibit a similar spatialpattern as that of the vertical velocity field (as viewed onan isobar). But if the equilibrium chemical field hashorizontal variations (Regime II), the resulting tracerdistribution is complicated. In Regime III, the pattern of along-lived tracer is largely controlled by the atmosphericdynamics. In this case, the correlation between verticalvelocity and tracer fields (on isobars) breaks down, andthe vertical profile of the tracer abundance differssubstantially from its chemical-equilibrium profile. Thehorizontal variation of the long-lived tracers is generallysmaller than that of the short-lived tracers due tohorizontal mixing. If the tracer lifetime is sufficientlylong (i.e., t ¥c ), the horizontal tracer field is expectedto be completely homogenized across the globe, but wedid not simulate this physical limit in this study.

(5) The diffusive assumption is generally valid in Regime Iand the effective eddy diffusivity is always positive usingour idealized chemical schemes. But in Regime II, thereis a strong variation in the equilibrium chemical field.Non-diffusive effects are important in Regime II if thereis a good correlation between the equilibrium tracerfield and the vertical velocity field. In some situations,tracers could be vertically mixed toward the source in a

Figure 10. Latitude–pressure maps of experiments V with non-uniform c0.First row: vertical winds (color) and mass streamfunctions (contours, in units of

-10 Kg s13 1). Starting from the second row, we show volume mixing ratio mapsof tracers with chemical timescales of 107 s, 108 s, 109 s, and 1010 s from top tothe bottom, respectively.

Figure 11. Same as Figure 5 but from Experiment V with non-uniform χ0. Thepredicted eddy diffusivity profiles based on Equation (22) with non-diffusivecorrection are shown by dashed lines in the right panel.

14

The Astrophysical Journal, 866:1 (17pp), 2018 October 10 Zhang & Showman

global-mean sense, leading to a negative eddy diffusivity.As the chemical lifetime increases in Regime II (but notlong enough to reach Regime III), the non-diffusive effectcould become less significant.

(6) Non-diffusive behavior could occur in Regime III whenthe tracer chemical lifetime is much longer than theatmospheric dynamical timescale. In this regime,the tracer material surface is distorted significantly andthe global-mean tracer profile is complicated. Forexample, in the case where the tracer source is in thedeep atmosphere, the global-mean tracer profile couldexhibit a local minimum in the middle atmosphericlayers. The derived effective eddy diffusivity will benegative above the local minimum, suggesting a strongnon-diffusive effect.

(7) We derived the analytical species-dependent eddydiffusivity for tracers with uniform chemical equilibrium(Equation (13)). We also provided a simple non-diffusivecorrection for long-lived tracers with non-uniformchemical equilibrium (Equation (15)). The dynamicaltimescale in our theory depends on the horizontal lengthscale Lh and/or vertical length scale Lv of the tracertransport. Using the pressure scale height H as Lv, thetheoretical predictions generally agree with our 2Dnumerical simulations. For long-lived species, the actualLv appears larger than H in our 2D simulations.

(8) A widely accepted assumption in current 1D chemicalmodels of planetary atmospheres—species-independenteddy diffusion, which assumes a single profile of verticaleddy diffusivity for all species—is generally invalid.Using a species-dependent eddy diffusivity in our theorywill lead to a more realistic understanding of global-meantracer transport in planetary atmospheres.

In this paper, we only focused on fast-rotating planets wherea zonal-symmetric assumption generally holds true. But forplanetary atmospheres with significant zonal asymmetry, suchas the atmospheres on tidally locked exoplanets, a 3D tracer-transport model is necessary to test our theory derived in thisstudy. In a subsequent paper (Paper II, Zhang & Show-man 2018), we will focus on the tracer transport on tidallylocked planets using a GCM and simple tracer schemes anddemonstrate that our analytical theory can also be applied tothat regime.

We should note that, in this study, we only consideredidealized chemical schemes that relax the tracer distributiontoward a specified chemical equilibrium over a specifiedtimescale. Other types of sources/sinks, for instance, due tocloud particle setting or due to more complex gas-phasechemistry have yet to be explored. The global-mean verticaltransport behaviors of those tracers merit investigation in thefuture. Future work could also study the seasonal effect of thetracer transport for high-obliquity planets where the circulationpatterns significantly change with time.

In our simulations, we prescribed the meridional circulationand eddy diffusivities to study the response of the tracerdistributions, but this approach prevents us from investigatingthe detailed dynamical interaction between the waves/eddiesand the tracers. Waves and eddies also have strong dynamicaleffects on zonal jets that influence the tracer transport. Forexample, the polar jets driven by waves and eddies on Earthcould act as potential vorticity barriers for tracer transport(Cammas et al. 1998). A 3D GCM relevant for stratospheres of

fast-rotating planets is needed in the future to explore theeffects of small-to-regional-scale waves and eddies on tracertransport. Furthermore, besides the large-scale upwelling anddownwelling transport, waves might also contribute to thevertical mixing in the stratified atmosphere. In the highatmosphere, such as the mesosphere on Earth or thermosphereon Jupiter, gravity waves could lead to a strong tracer transportand tracer mixing. Lindzen (1981) first proposed that thegravity wave breaking, including the breaking of tidal waves,will significantly mix the tracers vertically. Strobel (1981) andSchoeberl & Strobel (1984) proposed a parameterizationincluding both gravity wave breaking and tracer mixing bylinear waves. Strobel et al. (1987) also found that the effectivevertical eddy diffusivity due to gravity waves depends on thetracer chemical lifetime. It is expected that the total verticaleddy diffusivity would be a sum of both large-scale mixing andgravity wave mixing. Further investigation is needed so that 1Dvertical tracer transport can be understood in a coherent 3Dchemical-transport framework from the deep, turbulent atmos-phere to the upper, low-density atmosphere including con-vective mixing, large-scale circulation, and mixing fromatmospheric wave processes, such as breaking of Rossbywaves and gravity waves.

This research was supported by NASA Solar SystemWorkings Grant NNX16AG08G to X.Z. and A.P.S. Wededicate this work to Dr. Mark Allen (1949–2016), one ofthe founders of the Caltech/JPL kinetics model. We thank Y.Yung, R. Shia, and J. Moses for providing the eddy diffusionprofiles in their chemical models. Some of the simulations wereperformed on the Stampede supercomputer at TACC throughan allocation by XSEDE.

AppendixDerivation of 1D Effective Eddy Diffusivity in a 2D

Chemical-advective-diffusive System

The governing tracer-transport equation of a 2D chemical-advection-diffusive system using the coordinates of log-pressure and latitude is (Equation (19) in the text, Shiaet al. 1989, 1990; Zhang et al. 2013b):

* *c c c

ff

c

c

¶¶

+¶¶

+¶¶

-¶¶

¶¶

-¶¶

¶¶

=-

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ ( )

tv

yw

z yK

y

ez

e Kz

S

1

coscos

, 24

yy

z H z Hzz

2D

2D

where χ is the tracer mixing ratio, f=y/a is latitude, and a isthe planetary radius. º - ( )z H p plog 0 is the log-pressurecoordinate, where H is a constant reference scale height, p ispressure, and p0 is the pressure at the bottom boundary. Kyy2D

and Kzz2D are the horizontal and vertical eddy diffusivities,respectively. Residual circulation velocities are v* and w* in themeridional and vertical directions, respectively. A massstreamfunction ψ can be introduced such that:

*p r f

y= -¶¶

-( ) ( )va

ez

e1

2 cos25az H z H

0

*p r f

y=

¶¶

( )wa y

1

2 cos, 25b

0

15

The Astrophysical Journal, 866:1 (17pp), 2018 October 10 Zhang & Showman

where ρ0 is the reference density of the atmosphere at log-pressure z=0. The v* and w* given in Equation (25) naturallysatisfy the continuity equation:

* *f

f¶¶

+¶¶

=-( ) ( ) ( )y

v ez

e w1

coscos 0. 26z H z H

Combining Equations (24) and (26), we obtain the flux formof the tracer-transport equation:

* *c

ff c c

ff

c

c

¶¶

+¶¶

+¶¶

-¶¶

¶¶

-¶¶

¶¶

=

-

-

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( ) ( )

( )

t yv e

ze w

yK

y

ez

e Kz

S

1

coscos

1

coscos

. 27

z H z H

yy

z H z Hzz

2D

2D

Here we also define the eddy-mean decomposition= + ¢A A A , where A is the latitudinally averaged quantity at

constant pressure and A′ is the eddy term. Taking the latitudinalaverage of Equation (26) to eliminate the meridional transportterm (the v* term), we obtain * =w 0 at isobars if we assume thelatitudinal-mean vertical velocity *w vanishes at top and bottomboundaries. We average Equation (27) and both meridionaladvection and diffusion terms vanish. Using * =w 0, we obtain:

*c

cc¶

¶+

¶¶

¢ -¶¶

¶¶

=- -⎛⎝⎜

⎞⎠⎟( )

( )t

ez

e w ez

e Kz

S .

28

z H z H z H z Hzz2D

Based on the “flux-gradient relationship,” we can introducean eddy diffusivity Kw to approximate the vertical transportterm associated with w* (the second term in the left-hand side)such that:

*cc

¢ » -¶¶

( )w Kz

. 29w

Then the 1D global-mean tracer-transport equation(Equation 28) can be formulated as a vertical diffusionequation:

c c¶¶

-¶¶

¶¶

=-⎛⎝⎜

⎞⎠⎟ ( )

te

ze K

zS , 30z H z H

zz

where we define the total 1D effective eddy diffusivity= +K K Kzz w zz2D. Therefore, the Kzz2D vertical eddy diffusion

term in Equation (28) can be treated as an additive term in thetotal global-mean effective eddy diffusivity Kzz.

In order to analytically derive Kw, we subtract bothEquations (28) and (26) multiplied by c from Equation (27):

*

*

*

cf

f cf

fc c

c cc

¶ ¢¶

+¶¶

¢ -

´¶¶

¶ ¢¶

+¶¶

+¶¶

´ ¢ - ¢ -¶ ¢¶

= ¢-

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

( )

( )

t yv

yK

yw

ze

z

e w w Kz

S

1

coscos

1

cos

cos

. 31

yyz H

z Hzz

2D

2D

To solve this equation for χ′, we made four assumptions thatare similar to those in Section 2.3.

(i) We neglect the first temporal evolution term in thestatistical steady state.

(ii) We neglect the complicated eddy term (the fifth term) byassuming the vertical transport of the tracer eddy χ′ ismuch smaller than that of the latitudinal-mean tracer (c,the fourth term). Here we have also assumed that thevertical diffusion of χ′ (the Kzz2D term) is much smallerthan the vertical advection of the mean tracer.

(iii) We approximate the horizontal tracer eddy flux terms(second and third terms) in a linear relaxation form. InSection 2.3, we have decomposed the horizontal mixingprocesses in a 3D atmosphere into the tracer advection byzonal-mean flow and the tracer diffusion by horizontaleddies and waves. In the zonal-mean 2D system, thesecond and third terms in the left-hand side ofEquation (31) correspond to the advection and diffusionprocess, respectively. For the advection term, we have:

*f

f cct

¶¶

¢ »¢( ) ( )

yv

1

coscos , 32

adv

where *t » ˆL vadv h , where Lh is the horizontal lengthscale and *v is the meridional velocity scale. From thecontinuity equation (Equation (26)), we can also show

*t » ˆL wadv v , where Lv is the vertical length scale and*w is the vertical velocity scale. Similarly, the diffusive

term can be approximated as:

ff

c ct

-¶¶

¶ ¢¶

»¢⎛

⎝⎜⎞⎠⎟ ( )

yK

y

1

coscos , 33yy2D

diff

where t » a Kyydiff2

2D. Here we have made an additionalassumption that the positive tracer anomaly (χ′>0) isassociated with the tracer eddy flux divergence (i.e., thesign of the left-hand side is positive in this situation) andthe negative anomaly is associated with the tracer eddyflux convergence. This is generally a good assumption aslong as the material surface of the tracer is not verycomplicated. For example, if the horizontal tracerdistribution can be approximated by a second-degreeLegendre polynomial as in Holton (1986), one can showthat our above argument holds true.

(iv) We also consider a linear chemical scheme that relaxes thetracer distribution toward local chemical equilibrium χ0 in atimescale τc. After the mean-eddy decomposition, we obtainthe latitudinal mean of the chemical source/sink term

c c t= -( )S 0 c and the departure c c t¢ = ¢ - ¢( )S 0 c.See Section 2.3 for details.

With the above assumptions, Equation (31) can be simplifiedas:

*c c

tct

c c

t¶¶

+¢

+¢

=¢ - ¢

( )wz

. 34adv diff

0

c

We solve for χ′:

*c

t c

t t t¢ =

- + ¢

+ +

c¶¶

-

- - - ( )w

. 35z c1

0

diff1

adv1

c1

16

The Astrophysical Journal, 866:1 (17pp), 2018 October 10 Zhang & Showman

Inserting Equation (35) in (29), we solve for Kw and finallyobtain the analytical expression for Kzz:

*

*

t t t

c

t t t tc

» ++ +

-¢

+ +¶¶

- - -

- -

-⎛⎝⎜

⎞⎠⎟ ( )

K Kw

w

z1. 36

zz zz2D

2

diff1

adv1

c1

0

diff1

c adv1

c

1

In our 2D cases, if the chemical-equilibrium abundance χ0 isnot uniformly distributed across the latitude, the magnitude of χ0′,cD ¢

0, can be approximated by the root mean square of c¢0 over theglobe. One can show that, for a cosine function of χ0,c cD ¢ » 0.280 0 from the area-weighted global integration. The

vertical gradient of c can be approximated by that of the global-mean chemical-equilibrium mixing ratio c0 . Given the specificmeridional circulation patterns in Equation (20a), the area-weighted global-mean vertical velocity scale is * g= hw w e z H

0 .If the tracers are transported by global-scale circulation patterns,we can assume that the horizontal transport length scale Lh is theplanetary radius a and vertical transport length scale Lv is thepressure scale height H, we predict Kzz for our 2D cases(Equation (22) in the text):

gg t

g c

t t gc

» ++ +

-D ¢

+ +¶¶

h

h

h

h

- - -

- -

-⎛⎝⎜

⎞⎠⎟ ( )

K Kw e

K a w e H

w e

K a w e H z1. 37

zz zz

z H

yyz H

z H

yyz H

2D

202 2

2D2

01

c1

0 0

c 2D2

c 01

01

If the chemical-equilibrium abundance c0 is uniformlydistributed across the latitude, c¢0 is zero and thus the last termin the right-hand side of Equation (36) vanishes. Kzz becomes(Equation (23) in the text):

gg t

» ++ +

h

h- - - ( )K Kw e

K a w e H. 38zz zz

z H

yyz H2D

202 2

2D2

01

c1

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