submitted to the ApJPreprint typeset using LATEX style emulateapj v. 01/23/15

GLOBAL-MEAN VERTICAL TRACER MIXING IN PLANETARY ATMOSPHERESI: THEORY AND FAST-ROTATING PLANETS

Xi Zhang1 and Adam P. Showman2

1Department of Earth and Planetary Sciences, University of California Santa Cruz, CA 95064 and2Department of Planetary Sciences and Lunar and Planetary Laboratory, University of Arizona, AZ 85721

submitted to the ApJ

Abstract

Most chemistry and cloud formation models for planetary atmospheres adopt a one-dimensional(1D) diffusion approach to approximate the global-mean vertical tracer transport. The physical un-derpinning of the key parameter in this framework, eddy diffusivity Kzz, is usually obscure. Here weanalytically and numerically investigate vertical tracer transport in a 3D stratified atmosphere andpredict Kzz as a function of the large-scale circulation strength, horizontal mixing due to eddies andwaves and local tracer sources and sinks. We find that Kzz increases with tracer chemical lifetimeand circulation strength but decreases with horizontal eddy mixing efficiency. We demarcated threeKzz regimes in planetary atmospheres. In the first regime where the tracer lifetime is short comparedwith the transport timescale and horizontal tracer distribution under chemical equilibrium (χ0) isuniformly distributed across the globe, global-mean vertical tracer mixing behaves diffusively. Butthe traditional assumption in current 1D models that all chemical species are transported via thesame eddy diffusivity generally breaks down. We show that different chemical species in a single at-mosphere should in principle have different eddy diffusion profiles. In the second regime where traceris short-lived but χ0 is non-uniformly distributed, a significant non-diffusive component might leadto a negative Kzz under the diffusive assumption. In the third regime where the tracer is long-lived,global-mean vertical tracer transport is also largely influenced by non-diffusive effects. Numericalsimulations of 2D tracer transport on fast-rotating zonally symmetric planets validate our analyticalKzz theory over a wide parameter space.Subject headings: planetary atmospheres - eddy mixing - tracer transport - methods: analytical and

numerical

1. INTRODUCTION

The spatial patterns of atmospheric tracers such aschemical species, haze and cloud particles are signifi-cantly influenced by atmospheric transport. For mostplanets in the solar system, the horizontal variationsof tracers are usually much smaller than their verticalvariations. For extra-solar planets, the horizontal dis-tributions of tracers are not well resolved by observa-tions. The global-mean vertical tracer distribution isthe primary focus in most chemistry and cloud models.Vertical transport by large-scale atmospheric circulationand wave breaking leads to a tracer profile that devi-ates from its chemical or microphysical equilibrium. Oneexample is the “quenching” phenomenon in the atmo-spheres of giant planets and close-in exoplanets wherethe gas tracer concentrations (e.g., CO on Jupiter) inthe upper atmosphere are not in their local chemicalequilibrium, but instead are dominated by strong ver-tical tracer mixing from the deeper atmosphere. As aresult, the observed tracer abundance in the upper at-mosphere is “quenched” at an abundance correspondingto the chemical-equilibrium abundance at the locationwhere the tracer chemical timescale is equal to the ver-tical mixing timescale. This disequilibrium “quenchingeffect” has a significant influence on the interpretation ofobserved spectra of planetary atmospheres (e.g., Prinn &Owen 1976, Prinn & Barshay 1977, Smith 1998, Cooper& Showman 2006, Visscher & Moses 2011).

1 Correspondence to be directed to xiz@ucsc.edu

Investigation of global-mean vertical tracer distribu-tion could be simplified to a one-dimensional (1D) prob-lem. However, because atmospheric tracer transport isintrinsically a three-dimensional (3D) process, parame-terization of the 3D transport in a 1D vertical transportmodel is tricky. Most chemistry models for solar systemplanets (e.g., Allen et al. 1981, Krasnopolsky & Parshev1981, Yung et al. 1984, Nair et al. 1994, Moses et al. 2005,Lavvas et al. 2008, Zhang et al. 2012, Wong et al. 2017,Moses & Poppe 2017) and exoplanets (e.g., Moses et al.2011, Line et al. 2011, Hu et al. 2012, Tsai et al. 2017)adopt a 1D chemical-diffusion approach. Most haze andcloud formation models on planets and brown dwarfs as-sume the vertical particle transport behaves in a diffu-sive fashion as well (e.g., Turco et al. 1979, Ackerman &Marley 2001, Helling et al. 2008, Gao et al. 2014, Lav-vas & Koskinen 2017, Powell et al. 2018). This type of1D framework originated in the Earth atmospheric chem-istry literature but gradually faded out since the 1990’sbecause it is not sufficient to simultaneously explain thedistributions of all the species from observations (Holton1986) and when the 3D distributions of multiple tracersare well determined by satellite observations from space.At the moment, this 1D framework is still very useful inplanetary atmospheric studies. Although only a crudeapproximation, 1D models have succeeded in explain-ing the observed global-averaged vertical profiles of manychemical species and particles on planets.

In the 1D chemical-diffusion framework, the strengthof the vertical diffusion is characterized by a parameter

2 Zhang & Showman

10-2 100 102 104 106 108 1010

Eddy Diffusivity (m2 s-1)

104

102

100

10-2

Pre

ssu

re (

Pa

)

VenusEarthMarsTitanPlutoJupiterSaturnUranusNeptuneGJ436bHD189733bHD 209458b

Fig. 1.— Vertical profiles of eddy diffusivity in typical 1D chem-ical models of planets in and out of the solar system. Sources:Zhang et al. (2012) for Venus, Allen et al. (1981) for Earth, Nairet al. (1994) for Mars, Li et al. (2015) for Titan, Wong et al. (2017)for Pluto, Moses et al. (2005) for Jupiter, Saturn, Uranus and Nep-tune, Moses et al. (2013) for GJ436b and Moses et al. (2011) forHD189733b and HD209458b. For HD209458b, we show eddy dif-fusivity profiles assumed in a gas chemistry model (dashed, Moseset al. 2011) and that derived from a 3D particulate tracer transportmodel (solid, Parmentier et al. 2013).

called eddy diffusivity or eddy mixing coefficient. The 1Deffective eddy diffusivity Kzz is often determined empir-ically by fitting the observed vertical tracer profiles. Kzz

on important planets exhibit vertical profiles and magni-tude varying across more than eight orders of magnitude(Fig. 1), implying different vertical transport efficienciesfrom planet to planet. Since all 3D dynamical effectshave been lumped into a single eddy diffusivity, the spe-cific dynamical mixing mechanisms that lead to a partic-ular vertical profile of Kzz are often obscure. If the atmo-sphere is convective, then using the traditional Prandtlmixing length theory (e.g., Prandtl 1925, Smith 1998,Bordwell et al. 2018), one can formulate Kzz as a productof a convective velocity and a typical vertical length scalein a turbulent medium. But this formalism fails when theatmosphere is stably stratified. In the low-density mid-dle and upper atmosphere such as Earth’s mesosphere,the vertically propagating gravity waves could break andalso lead to a strong vertical mixing of the chemical trac-ers. Lindzen (1981) parameterized the eddy diffusivityfrom the turbulence and stress generated in breakinggravity and tidal waves (also see discussion in Strobel1981, Strobel et al. 1987). In a stratified atmosphere suchas Earth’s stratosphere, tracer transport is subjected toboth a large-scale overturning circulation and the verti-cal wave mixing (Hunten 1975, Holton 1986), but whoserelative importance depend on altitude and many otherfactors and may differ from planet to planet.

One of the conventional assumptions in the existingframework used in current planetary models is that alltracers, in spite of their different chemical lifetimes orparticle microphysical/settling timescales, are simulatedusing the same eddy diffusivity profile Kzz. The tracerdistribution in the real atmosphere is controlled by 3Ddynamical and chemical/microphysical processes. There-fore a coupling feedback between the chemistry and verti-cal transport is expected. Actually it has been noticed inthe Earth community (e.g., Holton 1986), in the presence

of meridional (latitudinal) transport in the stratospheres,the derived effective eddy diffusivity as a global-meantransport coefficient could have a strong dependence onthe tracer lifetime, and thus its chemical/microphysi-cal sources and sinks. Although there are several 3Dchemical-transport simulations in planetary atmosphereswith simplified chemical and cloud schemes (e.g., Lefevreet al. 2004, Lefevre et al. 2008, Marcq & Lebonnois 2013,Stolzenbach et al. 2015, Cooper & Showman 2006, Par-mentier et al. 2013, Charnay et al. 2015, Lee et al. 2016,Drummond et al. 2018, Lines et al. 2018), a thorough un-derstanding of the physical basis of global-mean verticaltracer transport and Kzz using both analytical theoryand 2D and 3D numerical simulations is still lacking.

Here we aim to reexamine this conventional 1D diffu-sion framework. We wish to achieve a more physicallybased parameterization of Kzz from first principles. Ourstudy will be presented in two consecutive papers. InPaper I (the current paper), we will construct a first-principles theory of Kzz in a 3D atmosphere and numer-ically investigate the behaviors of Kzz on fast-rotatingplanets using a 2D chemical-transport model. In PaperII (Zhang & Showman 2018), we will specifically focus on3D chemical tracer transport on tidally locked exoplan-ets and the associated Kzz using a 3D general circulationmodel (GCM). We primarily focus on stratified atmo-spheres such as the stratosphere on solar-system plan-ets or the photospheres on highly irradiated exoplanetswhere most of the chemical tracers and haze/clouds areobserved, and which are expected to be stably stratified(e.g., Fortney et al. 2008, Madhusudhan & Seager 2009,Line et al. 2012).

In the following sections of this paper, we will firstelaborate the underlying physics of tracer transport in aglobal-averaged sense and construct a first-principles es-timate of Kzz. Then we will use a 2D chemical-transportmodel to study a 2D meridional circulation system onfast-rotating planets and the behaviors of the associatedKzz under several typical scenarios and Kzz regimes. Weconclude this study with several key statements and abrief 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 mixedupward by the large-scale overturning circulation if thereis a correlation on an isobar between tracer abundanceand vertical velocity: if the tracer abundance is highwhere the vertical velocity is upward, or if the tracerabundance is low where the vertical velocity is down-ward, tracer is mixed upward (Fig. 2). This impliesthat the net vertical mixing of tracer over the globe de-pends crucially on horizontal variations of the tracer onisobars and on their correlation with the vertical veloc-ity field (Holton 1986). If the tracer distribution is ini-tially horizontally uniform across the globe with a verti-cal gradient of the mean tracer abundance, vertical windtransport will naturally produce tracer perturbations onan isobar that are correlated with the vertical velocityfield (Fig. 2). For example, if the initial tracer mix-ing ratio is higher in the lower atmosphere and lower inthe upper atmosphere, the upwelling branch of the over-

Vertical Tracer Mixing in Planetary Atmospheres Part I 3

+Δχ

-Δχ+ω

-ω

Longitude/Latitude

Pressure

Fig. 2.— Illustration of tracer transport with a large-scale circu-lation. The orange solid lines indicate constant tracer mixing ratiosurfaces. Tracer mixing ratio is higher in the lower atmosphere.Gray lines show horizontal and vertical wind transport with ver-tical velocity w. ∆χ is the deviation of local tracer mixing ratiofrom the horizontally averaged mixing ratio on an isobar (dashed).

turning circulation will transport the higher-mixing-ratiotracers upward and the downwelling branch will trans-port the lower-mixing-ratio tracers downward from theupper atmosphere. As a result, the tracers on an iso-bar will be more abundant in the upwelling branch thanin the downwelling branch, exhibiting a positive corre-lation with the vertical velocity field (Fig. 2). Afterthis horizontal tracer distribution is established, the up-welling branch will transport higher-mixing-ratio tracersacross an isobar and the downwelling branch will trans-port lower-mixing-ratio tracers across the same pressurelevel—the upward and downward tracer fluxes do notcancel out. When averaged over the globe, there will bea net upward tracer flux, resulting an effective upwardtracer 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., breakingof Rossby waves) normally smooth out the tracer vari-ation across the globe (e.g., Holton 1986, Yung et al.2009, Friedson & Moses 2012). Horizontal advection dueto the mean flow may increase or decrease the horizontaltracer variations, depending on the correlation betweenthe horizontal velocity convergence/divergence and thetracer distribution on the isobar. The distribution oftracer sources and sinks due to non-dynamical processessuch as chemistry2 also plays an important role. In asimplified picture, those non-dynamical processes couldbe assumed to relax the tracer distribution back to thechemical equilibrium distribution that the tracers wouldhave in the absence of dynamics, which could be eitheruniformly distributed across the globe or with a signifi-cant variations depending on the local sources and sinks(Marcq & Lebonnois 2013). It is expected that the 1Deffective eddy diffusivity Kzz depends on the magnitude

2 Hereafter we just use the generic term “chemistry” to representany non-dynamical processes that affect the tracer distribution,such as chemical reactions in the gas and particle phase, haze andcloud formation, or other phase transition processes.

of the vertical velocity, chemical timescale of the species,horizontal transport timescale and the horizontal vari-ation of the tracer distribution under chemical equilib-rium.

Holton (1986) first quantified these effects in theEarth’s atmosphere with a scenario that envisions thevertical mixing is accomplished by a meridional circula-tion in a 2D (latitude-pressure) framework. The derivededdy diffusivity exhibits a strong dependence on the cir-culation strength, tracer chemical lifetime and horizon-tal mixing. Holton showed that the species-dependenteddy diffusivity might help simultaneously explain thevertical profiles of several species in the stratosphere ofEarth, whereas the species-independent eddy diffusivitycould not. Here we generalize the 2D theory from Holton(1986) to a 3D atmosphere so that it can also be appliedto other planets that are not zonally symmetric as theEarth.

2.2. Governing Equation of Vertical Tracer Transport

Here we include the tracer advection by 3D atmo-spheric dynamics and tracer chemistry via a simplifiedchemical scheme to study the global-mean tracer trans-port. Based on these investigations we will achieve an an-alytical parameterization of the 1D effective eddy diffu-sivity Kzz. We will also demarcate different atmosphericregimes in terms of the tracer chemical lifetime and hor-izontal tracer distribution under chemical equilibrium.

First we start from a general 3D tracer transport equa-tion:

Dχ

Dt= S (1)

where S is the net sources/sinks of the chemical tracerwith mixing ratio χ. In principle, isentropic coordinatesare more appropriate for discussion of the tracer trans-port (Andrews et al. 1987). But for simplicity, here wejust adopt the log-pressure coordinate {x, y, z}. The co-ordinates are defined as x = aλ sinφ and y = aφ, wherea is the planetary radius, λ is longitude and φ is lati-tude. Here, z ≡ −H log(p/p0) is the log-pressure whereH is a constant reference scale height, p is pressure andp0 is the reference pressure in the pressure coordinate.D/Dt = ∂/∂t + ~u · ∇h + w∂/∂z is the material deriva-tive. ∇h = (∂/∂x, ∂/∂y) is the horizontal gradient atconstant pressure. ~u = (u, v) is the horizontal veloc-ity at constant pressure where u is the zonal (east-west)velocity and v is the meridional (north-south) velocity.w = Dz/Dt is the vertical velocity. In the log-pressurecoordinates, Eq. (1) becomes:

∂χ

∂t+ ~u · ∇hχ+ w

∂χ

∂z= S. (2)

The continuity equation is:

∇h · ~u + ez/H∂

∂z(e−z/Hw) = 0. (3)

Combining the above two equations, we get the fluxform of the tracer-transport equation:

∂χ

∂t+∇h · (χ~u) + ez/H

∂

∂z(e−z/Hwχ) = S. (4)

Here we define an eddy-mean decomposition A =

4 Zhang & Showman

A + A′ where A represents any quantity. A is the glob-ally average quantity at constant pressure and A′ is thedeviation from the mean, or the “eddy” term. Takingthe global average of the continuity Eq. (3) to eliminatethe the horizontal divergence term, we obtain w = 0 foreach atmospheric level if we assume the global-mean ver-tical velocity w vanishes at top and bottom boundaries.Globally averaging Eq. (4) and using w = 0, we obtain:

∂χ

∂t+ ez/H

∂

∂z(e−z/Hwχ′) = S. (5)

This is the global-mean vertical tracer transport equa-tion. It states that the evolution of global-mean tracermixing ratio χ is related to its global-mean vertical eddyfluxes. Based on Eq. (5), χ cannot be solved directly un-less we establish a relationship between the mean valueand the eddy flux wχ′. A conventional assumption is the“flux-gradient relationship” that links the eddy tracerflux to the vertical gradient of the mean value (e.g.Plumb and Mahlman 1987) by introducing a 1D “effec-tive eddy diffusion” Kzz such that:

wχ′ ≈ −Kzz∂χ

∂z. (6)

If Eq. (6) is valid, the global-mean tracer transportequation (5) can be formulated as a vertical diffusionequation:

∂χ

∂t− ez/H ∂

∂z(e−z/HKzz

∂χ

∂z) = S. (7)

This is the widely-used 1D chemical-diffusion equation.We need to estimate the eddy flux in Eq. (6) to solvefor Kzz. Subtracting both Eq. (5) and Eq. (3) that ismultiplied by χ from Eq. (4), we obtain:

∂χ′

∂t+∇h·(χ′~u)+w

∂χ

∂z+ez/H

∂

∂z[e−z/H(wχ′−wχ′)] = S′.

(8)Now we need to estimate the horizontal tracer varia-

tion χ′ along an isobar. In terms of the tracer lifetime τcand horizontal tracer distribution under chemical equi-librium χ0, the behavior of Kzz can be categorized intothree typical regimes (Fig. 3): (I) a short-lived tracerwith uniform distribution of chemical equilibrium abun-dance, (II) a short-lived tracer tracer with non-uniformdistribution of chemical equilibrium abundance, and (III)a long-lived tracer whose lifetime is long compared withthe transport timescale.

2.3. Regime I: Short-lived Tracers with UniformChemical Equilibrium: Diffusive case

For short-lived tracers, we can estimate χ′ from thevertical gradient of χ under following four assumptions.

(i) We neglect the temporal variation (time evolution)of the χ′ in statistical steady state because we focus onthe time-averaged behavior in this study. The first termon the left hand side of Eq. (8) can be neglected.

(ii) We neglect the complicated eddy term (the lastterm in the left hand side of Eq. (8)) that could bemuch smaller than the third term. In other words, weassume that the deviation of the eddy flux from its globalmean is smaller than the local vertical transport of the

Short-lived

Uniform

0χ

Non-uniform

0χ

II

I

III

Long-lived

di!usive

di!usive +

non-di!usive

di!usive +

non-di!usive

Fig. 3.— Atmospheric regimes of global-mean tracer transportas a function of tracer chemical lifetime and horizontal tracer dis-tribution under chemical equilibrium χ0.

global-mean tracer. As shown later in numerical simula-tions, this is generally valid in the case where the devi-ation of the tracer mixing ratio from the mean is smalland 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-livedtracers (Regime III), this assumption is not valid.

(iii) We approximate the horizontal tracer eddy fluxterm ∇h ·(χ′~u) ≈ χ′/τd where τd is the characteristic dy-namical timescale in the horizontal mixing processes. Ingeneral, under different situations, this divergence termcould act in an advective way, or in a diffusive way, mo-tivating two possible ways of formulating the dynami-cal timescale. In the advection, the dynamical transporttimescale τd ≈ Lh/Uadv where Lh is horizontal character-istic length scale (usually taken as the planetary radiusa) and Uadv is the horizontal wind speed. In the diffusivecase, the dynamical timescale τd ≈ L2

h/D where D is theeffective horizontal eddy diffusivity. Note that in this lin-ear relaxation approximation, we have assumed that hor-izontal dynamics always reduces the horizontal variationof the tracer. This generally holds true if the horizontaltracer field is not complicated. Some exceptions will bediscussed in the numerical simulation sections later.

(iv) For simplicity, we consider a linear chemicalscheme which relaxes the tracer distribution towards lo-cal chemical equilibrium χ0 in a timescale τc:

S =χ0 − χτc

. (9)

In general, the chemical equilibrium tracer distributionχ0 depends on many local factors, such as temperature,abundances of other species, photon fluxes and precip-itating ion fluxes. It is expected that photochemicalspecies will exhibit χ0 that varies between the equatorand poles. These effects might have more pronouncedinfluences on the horizontal distributions of χ0 on tidallylocked exoplanets than on solar system planets. If thespecies advected upward from the deep atmosphere witha uniform thermochemical source, χ0 is assumed con-stant along an isobar. Without losing generality, here

Vertical Tracer Mixing in Planetary Atmospheres Part I 5

we consider a non-uniform horizontal distribution of thechemical equilibrium: χ0 = χ0 + χ′0, where χ0 is theglobal-mean of χ0 which is only a function of pressure,and χ′0 (which can be a function of longitude and latitudeas well as pressure) is the departure of equilibrium tracerabundance from its global mean. Inserting χ0 into Eq.(9), we obtain the globally averaged chemical source/sinkterm S = (χ0−χ)/τc and the departure S′ = (χ′0−χ′)/τc.

With the assumptions (i-iv), Eq. (8) can be written:

w∂χ

∂z+χ′

τd=χ′0 − χ′

τc. (10)

We solve for χ′:

χ′ =−w ∂χ

∂z + τ−1c χ′0

τ−1d + τ−1c. (11)

This expression for the 3D distribution of χ′ is qualita-tively similar to the previous 2D model result in Holton(1986) with a photochemical source (c.f., his Eq. 19).However, Holton (1986) mainly focused on a special case(χ′0 = 0) without further elaborating the physical mean-ing of the general expression Eq. (11). Here we explicitlypoint out that the global-mean vertical tracer transportis composed of two physical processes: a diffusion pro-cess and a non-diffusive process. Based on Eq. (11), theglobal-mean vertical tracer flux wχ′ is:

wχ′ =−w2

τ−1d + τ−1c

∂χ

∂z+

wχ′01 + τ−1d τc

. (12)

The first term in Eq. (12) implies diffusive behavior,because this term’s contribution to the tracer verticalflux is proportional to the vertical gradient of the meantracer abundance. But the second term does not dependon the mean tracer gradient, suggesting a non-diffusivebehavior. Instead, the second term originates from thecorrelation between the equilibrium tracer distributionand the vertical wind field. It can be neglected if thetracer under chemical equilibrium is more or less uni-formly distributed across the globe. However, if the equi-librium tracer distribution is significantly non-uniform(for instance, if, in chemical equilibrium, the equator andpoles have strongly differing chemical abundances), theconventional “eddy diffusion” framework breaks downbecause the non-diffusive process might dominate thevertical tracer transport in the global-mean sense. Toelaborate the physics here, we now further discuss thesetwo cases: a uniformly distributed tracer under chemicalequilibrium and a non-uniform case.

If the equilibrium tracer distribution is uniformly dis-tributed (i.e., χ′0 = 0), the non-diffusive term in Eq. (12)vanishes. In this special case of short chemical lifetime(among other assumptions), mean tracer transport canbe regarded as a diffusive process and the eddy diffusivityKzz can be parameterized based on the relationship be-tween eddy tracer flux to the global-mean tracer gradientbased on Eq. (12) and Eq. (6):

Kzz ≈w2

τ−1d + τ−1c. (13)

This result is also consistent with that from a zonal-mean 2D model in Holton (1986) with a uniform chemi-

cal equilibrium abundance. The parameterized Kzz de-pends on circulation strength, chemical lifetime of thetracer and horizontal transport/mixing timescale in theatmosphere. It is expected that different chemical trac-ers are subjected to different eddy diffusivity strength,which has not been considered to date in 1D chemicalmodels of planetary atmospheres.

If χ′0 = 0, the horizontal distribution of χ′, deviationof the tracer mixing ratio from the mean on an isobar,is correlated with the horizontal distribution of verticalvelocity w at the same pressure level (Eq. 11):

χ′ =−w ∂χ

∂z

τ−1d + τ−1c. (14)

If the tracer has a large? uniform chemical equilibriumabundance in the deep atmosphere but is photochemi-cally destroyed in the upper atmosphere—for example,sulfur dioxide on Venus (Zhang et al. 2012), methaneon Jupiter (Moses et al. 2005) or water on hot Jupiters(Moses et al. 2011)—the mean tracer gradient ∂χ/∂z isnegative, and therefore χ′ is positively correlated withw (Eq. 14). This implies that wχ′ > 0, i.e., the traceris transported upward. On the other hand, if the tracersource is in the upper atmosphere, for example, chem-ically produced species which are uniformly distributedunder chemical equilibrium, or oxygen species with uni-form fluxes from comets into the upper atmospheres of gi-ant planets (Moses & Poppe 2017), the mean tracer gra-dient ∂χ/∂z is generally positive and χ′ is anti-correlatedwith w. This implies that wχ′ < 0, i.e., the tracer istransported downward. In both cases, the χ′ − w corre-lation patterns result in a net tracer transport away fromtheir source regions.

From Eq. (14), if the circulation is stronger, i.e, w islarger, the horizontal variation of the tracer mixing ratiowill be larger. If the chemical loss or the horizontal mix-ing due to advection or diffusion is stronger, i.e., τd andτc are smaller, the tracer variation is smaller. In otherwords, a large-scale circulation enhances the tracer varia-tion on isobars, whereas chemical relaxation and horizon-tal tracer mixing homogenizes the tracer distribution onisobars. The resulting eddy diffusivity is larger if the cir-culation is stronger, the tracer chemical lifetime is longer,or the horizontal mixing timescale is longer.

We emphasize that Kzz should depend on the chemi-cal lifetime τc—an important relationship that all current1D models have ignored. To further elaborate this, weconsider two different tracers, one with short τc and an-other with long τc, that both exist in an atmosphere witha specified 3D atmospheric circulation. For concreteness,imagine that the background tracer abundance of bothtracers decreases upward, with the same background gra-dient for both tracers. Because of the advection, for bothtracers, the tracer abundance on an isobar will be greaterin upwelling regions and smaller in downwelling regions,implying an upward flux of the tracer in both cases. How-ever, when τc is short, the chemistry very strongly relaxesthe abundance toward equilibrium, whereas with long τc,this relaxation is weak. This implies that, in statisticalequilibrium, the deviation of the actual tracer abundancefrom chemical equilibrium, χ′, is greater when τc is longthan when it is short. Since the circulation is the samein the two cases, the upward tracer flux wχ′ is greater

6 Zhang & Showman

when τc is long and smaller when τc is short. Given thedefinition of eddy diffusivity Kzz = −wχ′/∂χ∂z from Eq.(6), we thus have the situation where the tracer withshort τc has a smaller Kzz than the tracer with largeτc—even though the atmospheric circulation, by defini-tion, is precisely the same for the two tracers. Indeed,for this simple situation, the value of Kzz would go tozero in the limit τc → 0, because in that case, the traceris constant on isobars, so there is no net correlation be-tween vertical velocity and tracer abundance, implyingthat the upward flux of tracer is zero.

2.4. Regime II: Short-lived Tracer with Non-uniformChemical Equilibrium: Non-diffusive Component

The horizontal distribution of tracer equilibrium abun-dance could be significantly non-uniform, i.e., χ′0 6= 0.This could occur if there is a local plume source of thespecies, for example, volcanic eruption on Earth (Selfet al. 1993), convective injection of sulfur species to themiddle atmosphere of Venus (Marcq et al. 2013), andimpact debris from incoming comets on Jupiter (Fried-son et al. 1999). A more common case is photochemi-cally produced species like ozone on terrestrial planets orethane on giant planets, where the incoming ultravioletsolar flux changes with the solar angle, leading to differ-ent photochemical equilibrium abundances at differentlatitudes (e.g., Moses & Greathouse 2005). An extremeexample is the tidally locked exoplanets, on which dif-ferent chemical equilibrium states are expected betweenthe permanent dayside and nightside for a tracer whosechemistry critically depends on factors such as tempera-ture, incoming photon and ion fluxes. For example, for-mation of condensed haze/cloud particles favor the coldernightside than the warmer dayside (Powell et al. 2018).For another example, species produced by photochem-istry or ion chemistry are expected to have a larger equi-librium abundance on the dayside than on the nightside.

In these situations, the dependence of Kzz on the cir-culation and tracer chemistry is more complicated. Theeddy tracer flux due to the non-diffusive process, i.e.,the second term in the right hand side of Eq. (12),cannot be neglected. The horizontal distribution of χ′

might not be strongly correlated with the horizontal dis-tribution of vertical velocity w at the same pressure level(Eq. 11). The correlation term wχ′0 could have a sig-nificant effect on Kzz (Eq. 12). Physically speaking, inthe case with uniform chemical equilibrium abundance,atmosphere circulation will shape the initially homoge-neous tracer distribution toward a correlation patternthat wχ′ has the opposite sign as the background gradi-ent ∂χ/∂z. This implies that the tracer is mixed downthe vertical gradient of the horizontal-mean tracer abun-dance. In the case of non-uniform chemical equilibriumcase, χ0 might correlate or anti-correlate with the verti-cal velocity pattern, causing an additional contributionto the net vertical tracer transport in an essentially non-diffusive way. Whether this contribution will enhance orreduce the net vertical mixing efficiency depends on thecorrelation as well as the vertical gradient of the meantracer (Eq. 12).

If we still adopt the traditional “eddy diffusion” frame-work, inserting the global-mean vertical tracer flux wχ′

(Eq. 12) into the definition of Kzz (the flux-gradient re-

lationship Eq. 6), we can approximate the non-diffusivebehavior and estimate Kzz for tracers with non-uniformchemical equilibrium:

Kzz ≈w2

τ−1d + τ−1c− wχ′0

1 + τ−1d τc(∂χ

∂z)−1. (15)

The second term on the right hand side represents in-herently non-diffusive behavior, because it has a depen-dence on the vertical gradient of the mean tracer ∂χ/∂z.This implies that the eddy tracer flux (Eq. 6) has aterm that does not scale linearly with the backgroundvertical tracer gradient, which violates the fundamentalassumption of a diffusive system that the flux scales lin-early with the tracer gradient. Also, unlike the first termin the right hand side of Eq. (15), the second term canhave either sign, depending on the sign of the correlationbetween w and the anomalies on isobars of the chemical-equilibrium abundance, χ′0. Note that Eq. (15) is nolonger a closed expression for Kzz because it dependson the vertical gradient of the global-mean tracer mix-ing ratio, a quantity that we require Kzz to solve for.One could imagine that an iterative process between thechemical-diffusion simulation and updating Kzz mightlead to a final steady state of the system.

We emphasize that the Kzz expression with the non-diffusive correction needs to be used cautiously in 1Dchemical-diffusion simulations and can only be used whenthe non-diffusive contribution is not dominant. If thesecond term dominates and is negative, the predictedKzz can be negative in some situation. For example,if the tracer lifetime is very short, τc → 0, Eq. (15)becomes:

Kzz ≈ −wχ′0(∂χ

∂z)−1 (16)

In this limit, the diffusive term vanishes and the non-diffusive term dominates. Vertical tracer mixing is signif-icantly controlled by the correlation between the chemi-cal equilibrium distribution and the vertical velocity dis-tribution. If the correlation is positive, i.e., tracer is moreabundant in the upwelling region than the downwellingregion due to chemistry, and if the tracer is produced atthe top, i.e., the vertical gradient of the mean tracer mix-ing ratio is positive, the derived Kzz is negative (Eq. 16).A negative effective eddy diffusivity does not make sensephysically as it suggests the tracer is mixed towards itssource. The reason is simply because the global-meanvertical tracer transport is essentially non-diffusive inthis case.

For a given vertical gradient of global-mean tracer, Kzz

becomes larger if the circulation is stronger, the tracerchemical lifetime is longer, and the horizontal mixingtimescale is longer. This trend is consistent with that inthe case with uniform chemical equilibrium abundance.Interestingly, as the chemical timescale becomes longer,the diffusive term gets larger and the non-diffusive termbecomes smaller, perhaps because the longer-lived trac-ers are more controlled by the dynamical transport sothat the non-uniform chemical equilibrium has less ef-fect. Thus the non-diffusive correction in Eq. (15) shouldwork better for species with relatively long timescales.

2.5. Regime III: Long-lived Tracers

Vertical Tracer Mixing in Planetary Atmospheres Part I 7

If the tracer chemical lifetime is long and the traceris almost inert, the material surfaces (tracer contours)are usually distorted significantly 3. The above discus-sion for short-lived tracers could be violated since theassumption (ii) in Section 2.1.1 might no longer be valid.It is expected that the 3D distribution of such a “quasi-conservative” tracer is significantly controlled the atmo-spheric dynamics. Due to the complicated dynamicaltransport behavior, no simple analytical theory of theglobal-mean vertical tracer transport exists, and thus thecorresponding Kzz is not generally known.

Although it is expected that there are some non-diffusive effects in this regime, the diffusive frameworkmay still be useful. If we still apply the Kzz theory inSection 2.3 to this regime and let τc → ∞, the non-diffusive contribution vanishes (Eq. 12). In this case,the influence of the chemical equilibrium tracer distribu-tion is negligible and Eq. (15) is reduced to Eq. (13).One might expect Kzz to approach its asymptotic valuein Eq. (13):

Kzz ≈ w2τd = wLv (17)

where we have introduced a vertical transport lengthscale Lv = wτd and w is the root-mean-square of thevertical velocity w = (w2)1/2. Here the vertical trans-port timescale is assumed as the horizontal tracer mixingtimescale τd due to the continuity equation. Eq. (17) isin the similar form of that from the mixing length the-ory although there is no convective or small-scale mixingdue to wave breaking in our theory. Using Lv, the effec-tive eddy diffusivity (Eq. 13) for short-lived tracers withuniform chemical equilibrium (Section 2.3.1) can also berepresented as:

Kzz =wLv

1 + τdτ−1c

. (18)

The magnitude of the change in Kzz depends on the ratioof the timescales between the dynamical and chemicalprocesses: τd/τc. In the long-lived tracer regime whereτc → ∞, the effective eddy diffusivity Kzz approachesEq. (17).

The vertical transport length scale Lv in our Kzz the-ory cannot be arbitrarily chosen. It critically depends onthe atmospheric dynamics, specifically the vertical veloc-ity w and the horizontal dynamical timescale τd. If thedynamical timescale τd for horizontal tracer mixing isequal to the global horizontal advection timescale a/Uwhere a is approximately the planetary radius, throughcontinuity τd should be approximately H/w. Then Lvwould be equal to the pressure scale height H. If thehorizontal mixing timescale is longer (or shorter) thanthe horizontal advection timescale, Lv is then larger (orsmaller) than H by that same factor. Using the verticalvelocity from general circulation models and simply as-suming Lv is H, some previous models (e.g., Lewis et al.2010, Moses et al. 2011) estimated the eddy diffusivities

3 If the tracer is absolutely inert with a very long lifetime, itwill be completely homogenized over the globe by the horizontaltransport. But in this study we do not investigate this type of realconservative, well-mixed tracers. The “long-lived tracer” in ourstudy stands for the species with a significantly long chemical life-time so that the atmospheric dynamics greatly shapes its materialsurface, but the horizontal variation of the tracer distribution isstill not small.

on exoplanets based on Eq. (17). Those estimates weremuch larger than the eddy diffusivity derived based on3D passive tracer simulations (Parmentier et al. 2013).

There are two reasons. First, Kzz estimated from Eq.(17) is the maximum eddy diffusivity one can obtain fromEq. (18). For short-lived tracers, the effective eddy diffu-sivity should be smaller than that from Eq. (17). Second,in the long-lived tracer regime, tracers are significantlycontrolled by atmospheric dynamics and the horizontaldynamical timescale τd might be different from the globalhorizontal advection timescale a/U . Thus the verticalcharacteristic transport length scale Lv in Eq. (17) and(18) could be different from H. This has also been notedin the studies of tracer transport in the convective at-mospheres. For example, Smith (1998) investigated dy-namical quenching of chemical tracers in convective at-mospheres and found that vertical transport length scalein the traditional mixing length theory should dependon the chemical tracer equilibrium distribution as wellas the chemical and dynamical timescales. Recent workby Bordwell et al. (2018) explored the chemical tracertransport using non-rotating local convective box mod-els. They found that using the chemical scale heightas the vertical transport length scale, which is usuallysmaller than the scale height H, leads to a better predic-tion of the chemical quenching levels.

We reiterate that the assumptions used to derive Eq.(17) likely break down in the regime of long-lived tracers,so it may be that the qualitative dependencies impliedin Eq. (17) are not rigorously accurate when the trac-ers are long-lived. In our theory we have dropped thelast term in the left hand side in Eq. (8), which mightbecome important in the long-lived regime. As we willdemonstrate in the numerical simulations later, this non-linear eddy term could potentially enhance or decreasethe global tracer mixing efficiency. We also emphasizethat, although we adopt the diffusive framework here inthe long-lived tracer regime, the tracer transport in thisregime may not always behave diffusively. As we willalso show later in the numerical simulations, the diffu-sive assumption could break down in some cases whenthe tracer material surface is distorted significantly andnon-diffusive effects are substantial.

In sum, in this section we have developed an approx-imate analytical theory for the 1D global-mean tracertransport in a 3D atmosphere. We demarcated threeatmospheric regimes in terms of the tracer chemical life-time and horizontal tracer distribution under chemicalequilibrium. The underlying physical mechanisms gov-erning the global-mean vertical tracer transport in thethree regimes are different. The traditional chemical-diffusion assumption is mostly valid in the first regimebut could be violated in the second and third regimes.We predicted the analytical expression of the 1D effectiveeddy diffusivity Kzz for the global-mean vertical tracertransport. Kzz depends on both atmospheric dynamicsand tracer chemistry. Crudely speaking, if the atmo-spheric dynamics is fixed, Kzz roughly scales with thetracer chemical lifetime Kzz ∝ τc (Eq. 13) when thetracer lifetime is short (regime I) and approaches to aconstant value when the tracer lifetime is long (regimeIII, Eq. 17). We also found that in the short-lived tracerregime (regime I), Kzz roughly scales with the the ver-

8 Zhang & Showman

tical velocity square Kzz ∝ w2 (Eq. 13), but in thelong-lived tracer regime (regime III), it scales with root-mean-square of the vertical velocity Kzz ∝ w (Eq. 17).Next, we will perform a series of numerical experimentsto quantitatively verify our theoretical arguments. Wewill primarily focus on stratified atmospheres here.

3. 2D SIMULATIONS ON FAST-ROTATING PLANETS

Now we consider numerical simulations of tracer trans-port on a fast-rotating planet on which the tracer is uni-formly distributed with longitude (but not necessarilywith latitude). Most of the planetary atmospheres in thesolar system are close to this situation. This is essentiallya 2D tracer transport problem that can be studied usinga 2D zonally symmetric model. In our numerical simula-tions, we will only study passive tracers, i.e., no radiativefeedback from the tracer to the dynamics.

2D zonal-mean tracer transport has been extensivelydiscussed in the Earth literature (e.g., Holton 1986). Buttill now there has not been a thorough and specific studyon the global-mean eddy diffusivity using 2D numeri-cal simulations with chemical tracers. On a fast-rotatingplanet, when averaged zonally, the meridional circulationthat transports the tracer (like the one shown in Fig.2) should be considered as the “Lagrangian mean circu-lation”, which can be approximated by a TransformedEulerian Mean (TEM) circulation (Andrews & McIntyre1976), and called the “residual mean circulation”. An-other formalism of the zonal-mean tracer transport in-troduced by Plumb & Mahlman (1987) used the “effec-tive transport velocity”. The difference between the twocirculation formalisms is usually small in Earth’s strato-sphere and vanishes if the waves are linear, steady, andadiabatic (Andrews et al. 1987).

In both frameworks, the zonal-mean eddy tracer fluxescan be parameterized as diffusive fluxes using a sym-metric “diffusion tensor” with four parameters (diffusiv-ities): Kyy2D, Kzz2D, Kyz2D and Kzy2D (Andrews et al.1987). Kyz2D and Kzy2D are negligible in isentropic co-ordinates (Tung 1982). As here we mainly focus on thestratified atmosphere where the inclination between theisobars and isentropes is usually small, we can ignoreKyz2D and Kzy2D in 2D chemical tracer transport simu-lations in this study (e.g., Garcia & Solomon 1983, Shiaet al. 1989).

A 3D model would naturally produce small-scale eddiesthat would cause mixing, and if the eddies were fully re-solved, no parameterization of diffusive fluxes would beneeded in the tracer transport. But in our 2D frameworkthat ignores any role for such eddies, even for a planetwith a great degree of zonal symmetry at large scales (likeEarth or Jupiter), we have to parameterize the horizon-tal and vertical diffusivities Kyy2D and Kzz2D as a sep-arately included process. In principle Kyy2D and Kzz2D

can be estimated from the Eliassen-Palm flux (Andrewset al. 1987) in a 3D model. But for our purpose of study-ing the response of the chemical tracer distribution to thedynamics and the global-mean vertical tracer mixing, wejust prescribe the diffusivities and the circulation.

In this study, we prescribe the temperature structureand circulation pattern and hold them constant withtime. The governing equation of a 2D chemical trans-port system using the coordinates of log-pressure andlatitude, can be written as (Shia et al. 1989, Shia et al.

1990, Zhang et al. 2013b):

∂χ

∂t+ v∗

∂χ

∂y+ w∗

∂χ

∂z− 1

cosφ

∂

∂y(cosφKyy2D

∂χ

∂y)

−ez/H ∂

∂z(e−z/HKzz2D

∂χ

∂z) = S

(19)

where φ = y/a is latitude and a is the planetary radius.H is the pressure scale height. The chemical source andsink term S follows the linear relaxation scheme of Eq.(9).

Residual circulation velocities are v∗ and w∗ in themeridional and vertical directions, respectively. For a2D circulation pattern, we can introduce a mass stream-function ψ such that:

v∗ = − 1

2πaρ0 cosφez/H

∂

∂z(e−z/Hψ) (20a)

w∗ =1

2πaρ0 cosφ

∂ψ

∂y(20b)

where ρ0 is the reference density of the atmosphere atlog-pressure z = 0. With prescribed distributions of ψ,Kyy2D and Kzz2D, we solve the governing equations us-ing the Caltech/JPL 2D kinetics model (for numerics,refer to Shia et al. 1990). We use the Prather scheme forthe 2D tracer advection (Prather 1986). The model hasbeen rigorously tested against multiple exact solutionsunder various conditions (Shia et al. 1990, Zhang et al.2013b).

We adopted two different meridional circulation pat-terns (see Section 4 in Zhang et al. 2013b). The circula-tion ψA is an equator-to-pole pattern and circulation ψBis pole-to-pole, corresponding to planets with low obliq-uity and high obliquity, respectively:

ψA = 2πa2ρ0w0eηz/H sinφ cos2 φ (21a)

ψB = 2πa2ρ0w0eηz/H cos2 φ. (21b)

The mass streamfunctions and vertical velocities of thetwo circulation patterns are shown in Fig. 4. For bothcirculations, the area-weighted global-mean vertical ve-locity scale is about γw0e

ηz/H where γ is an order-unitypre-factor originating from the global average.4

In this 2D chemical-advective-diffusive system, boththe meridional advection and horizontal eddy diffusioncontribute to the horizontal tracer mixing, and both thevertical advection and vertical eddy diffusion contributeto the vertical tracer transport. The 1D effective eddydiffusivity Kzz in this system can be analytically pre-dicted following the procedure introduced in Section 2.But since here we have to treat explicitly the eddy dif-fusion terms Kyy2D and Kzz2D originating from zonal-mean 2D dynamics, we provided a detailed derivation ofKzz in this 2D system in the Appendix. We show thatthe vertical eddy diffusion by Kzz2D in Eq. (19) can betreated as an additive term in the global-mean effectiveeddy diffusivity Kzz.

In the 2D framework, the Kzz for the situation of non-uniform chemical equilibrium mixing ratio χ0 can be ex-

4 γ is 2/√

5 for ψA and 2/√

3 for ψB from the global integrationsof Eq. (21a) and (21b), respectively. Thus the effective transportis a bit stronger using the circulation pattern B.

Vertical Tracer Mixing in Planetary Atmospheres Part I 9

TABLE 12D Simulation cases in this study.

Experiment Streamfunction Kyy2D (m2 s−1 ) Tracer source Latitudinal distribution of χ0

I ψA 10 Deep UniformII ψB 10 Deep UniformIII ψA 10 Top UniformIV ψA 106 Deep UniformV ψA 10 Top Non-uniform

−90° −60° −30° 0° 30° 60° 90°Latitude (Degree)

103

102

101

100

Pre

ssu

re (

Pa

)

° ° ° ° ° ° °

3

2

1

0

° ° ° ° ° ° °

3

2

1

0

10−14

10−12

10−10

10−8

10−6

−90° −60° −30° 0° 30° 60° 90°Latitude (Degree)

103

102

101

100

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ssu

re (

Pa

)

° ° ° ° ° ° °

3

2

1

0

10−14

10−12

10−10

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10−6

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100

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ssu

re (

Pa

)

° ° ° ° ° ° °

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2

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0

10−14

10−12

10−10

10−8

10−6

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100

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ssu

re (

Pa

)

° ° ° ° ° ° °

3

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0

10−14

10−12

10−10

10−8

10−6

−90° −60° −30° 0° 30° 60° 90°

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100

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ssu

re (

Pa

)

° ° ° ° ° ° °

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1

0

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10−12

10−10

10−8

10−6

−90° −60° −30° 0° 30° 60° 90°

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ssu

re (

Pa

)

° ° ° ° ° ° °

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ssu

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Pa

)

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0

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−90° −60° −30° 0° 30° 60° 90°

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ssu

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Pa

)

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ssu

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Pa

)

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1

0

−0.003

−0.002

−0.001

0.000

0.001

0.002

0.003

−10.00−5.0

0−

1.0

0−

.50

−0.50

−0.1

0

−

−0.0

5

05

0.0

50.1

0

0

0.50

5.0

010

.00

−90° −60° −30° 0° 30° 60° 90°

103

102

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100

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ssu

re (

Pa

)

° ° ° ° ° ° °

3

2

1

0

−0.003

−0.002

−0.001

0.000

0.001

0.002

0.003

0.0

5

0.0

5

0.1

0

0.1

0

0.5

0

0.50

1.0

0

1.00

5.00

10.0

0

50.00

Wind (m s−1 Wind (m s−1))

Fig. 4.— Latitude-pressure maps of experiments I (left) and II (right). First row: vertical winds (color) and mass streamfunctions(contours, in units of 1013 Kg s−1). Starting from the second row we show volume mixing ratio maps of tracers with chemical timescalesof 107 s, 108 s, 109 s and 1010 s from top to the bottom, respectively.

10 Zhang & Showman

pressed as (See Appendix for more details):

Kzz ≈ Kzz2D +γ2w2

0e2ηz/H

Kyy2Da−2 + γw0eηz/HH−1 + τ−1c

− γw0eηz/H∆χ′0

1 + τcKyy2Da−2 + τcγw0eηz/HH−1(∂χ0

∂z)−1

(22)where χ0 is the global-mean of the non-uniform chemi-cal equilibrium mixing ratio χ0. ∆χ′0 is the root-mean-square of the deviation χ′0 over the globe. For a cosinefunction of χ0, ∆χ′0 ≈ 0.28 χ0. Here we have also as-sumed the horizontal transport length scale Lh ∼ a andvertical transport length scale Lv ∼ H. If χ0 is uniform,Kzz can be reduced to:

Kzz ≈ Kzz2D +γ2w2

0e2ηz/H

Kyy2Da−2 + γw0eηz/HH−1 + τ−1c.

(23)In the second term on the right hand side, the three termsin the denominator correspond to horizontal tracer diffu-sion due to eddies and waves, horizontal tracer advectionby zonal-mean flow and tracer chemistry, respectively.

We use Jupiter’s parameters in these 2D simulationsbut with an isothermal atmosphere of 150 K from 3000Pa to about 0.2 Pa for simplicity. For the circulationpatterns, we adopt w0 = 10−5 m s−1 and η = 0.5. Givena pressure scale height H of about 25 km, the verticaladvection timescale changes from 109 s at the bottom toabout 107 s at the top. The vertical profile of the advec-tion timescale is similar to the empirical vertical mixingtimescale derived from H2/Kz where Kz is the empiricaleddy diffusivity in the 1D stratospheric chemical modelon Jupiter (Moses et al. 2005, also see Fig. 1). The hori-zontal advection timescale in our model is about 108 s at100 Pa, comparable to the vertical circulation timescaledue to the continuity constraint. This horizontal trans-port timescale is also similar to that inferred from Voy-ager and Cassini observations (Nixon et al. 2010, Zhanget al. 2013a).

We designed five 2D experiments (Table 1) and sim-ulated 9 tracers in each experiment. In all cases, weset the vertical diffusion coefficients Kzz2D zero becausehere we mainly investigate the influence of the large-scaleoverturning circulation, horizontal mixing and chemicalsource/sink on the global-mean vertical tracer transportin this study. According to Eq. (23), the influence ofKzz2D in the 2D simulation is trivial because it can justbe treated as an additive term in Kzz. We also testeddifferent horizontal diffusional coefficients Kyy2D.

Experiments I-IV are simulations with chemical tracerswith uniform chemical equilibrium mixing ratios. Ex-periment V focuses on the chemical tracer transportwith non-uniform chemical equilibrium. Experiment I(Fig. 4) is the standard case with an equator-to-polestreamfunction ψA and a constant horizontal eddy dif-fusivity Kyy2D = 10 m2 s−1. This is a “deep source”case in which tracer is advected upward from the deepatmosphere at 3000 Pa. The vertical profile of equilib-rium tracer mixing ratio χeq is assumed to be a power-law function of pressure χeq(p) = 10−5(p/p0)1.7 wherep0 = 3000 Pa is the pressure at the bottom boundary.χeq starts from 10−5 at bottom and decreases towards

10-12 10-10 10-8 10-6 10-4

Mean mixing ratio

1000

100

10

1

Pre

ssu

re (

Pa

)

10-3 10-2 10-1 100 101 102 Eddy diffusivity (m2 s-1)

1000

100

10

1

w2

w

Fig. 5.— Vertical profiles of the global-mean volume mixing ratio(left) and derived effective eddy diffusivity (right) from ExperimentI. Different colors from cold (blue) to warm (red) represent tracerswith different chemical timescales from short to long, ranging from107 s to 1011 s. The prescribed equilibrium tracer mixing ratioprofile is shown in the dashed line in the left panel. The predictededdy diffusivity profiles based on Eq. (23) are shown in dashed inthe right panel. The solid lines are derived from the simulations.The thick gray line indicates the empirical eddy diffusivity pro-file used in current photochemical models in Jupiter’s stratosphere(Moses et al. 2005, Moses & Poppe 2017).

10-12 10-10 10-8 10-6 10-4

Mean mixing ratio

1000

100

10

1

Pre

ssu

re (

Pa

)

10-3 10-2 10-1 100 101 102 Eddy diffusivity (m2 s-1)

1000

100

10

1

Fig. 6.— Same as Fig. 5 but from Experiment II. Note thatsome curves (e.g., the orange line) break apart in the long-livetracer regime. This is because the material surface is distorted solarge that the derived Kzz is negative in the middle part.

10−12 at the top of the atmosphere. A linear chemicalscheme (Eq. 9) is applied to each tracer with a relaxationtimescale τc that varies from 107 to 1011 s. The chemicaltimescale of the ith tracer is assumed as τc = 106.5+0.5i

s.Experiment II is same as Experiment I but with a dif-

ferent circulation pattern ψB . Experiment III is alsosame as Experiment I but with a “top chemical source”.The equilibrium tracer mixing ratio, while still uniformacross the latitude, is large at top (10−5) and small atthe bottom (10−12). In Experiments I to III, the hori-zontal eddy diffusion timescale is a2/Kyy2D ∼ 5 × 1014

Vertical Tracer Mixing in Planetary Atmospheres Part I 11

107 108 109 1010 1011

Chemical timescale (s)

0.001

0.010

0.100

1.000

10.000

100.000

Ed

dy d

iffu

siv

ity (

m2 s

-1)

IIIIIIIV

τc

Keff

Fig. 7.— Kzz as a function of tracer chemical timescale for allfour experiments at two typical pressure levels, 80 Pa (filled circles)and 200 Pa (open circles). The solid and dashed lines are thepredictions from Eq. (23).

s. The diffusive transport is much less efficient com-pared with the horizontal advection and can be neglected(timescale of 107 − 109 s). We designed Experiment IVusing the streamfunction ψA but horizontal eddy diffu-sivity Kyy2D enhanced to 106 m2 s−1. In this case, thediffusive timescale is about 5× 109 s, comparable to thecirculation timescale (but the Kzz2D is still zero). Inthis setup we can test the influence of the horizontaleddy mixing to the global-mean vertical tracer trans-port. Experiment V is similar to Experiment III witha “top chemical source” but the tracer chemical equi-librium mixing ratios are not uniformly distributed withlatitude. The non-uniform chemical equilibrium mixingratio is assumed to be a cosine function of the latitude:χ0(p, φ) = χeq(p) cosφ. This setup is closer to the realis-tic photochemical production in planetary atmosphereswhere the photochemical photon flux changes with lati-tude.

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 80layers from 3000 Pa to about 0.2 Pa. The time step inthe simulations is 105 second. We ran the simulationsfor about 1013 s to ensure the spatial distributions of thetracers have reached the steady state. The tracer abun-dances were averaged over the last 109 s for analysis.We tested the model with different vertical and horizon-tal resolutions to confirm that the simulation results arerobust.

3.1. Results: Simulations with Uniform χ0

(Experiments I-IV)

The steady state results in Experiments I and II areshown in Fig. 4. Although the final tracer distributionsare different between the two experiments due to differ-ent circulation patterns, some common behaviors exist.The short-lived tracers are uniformly distributed acrosslatitude. As the tracer chemical lifetime increases, the ef-fect of the circulation becomes stronger. In the upwellingregion, higher-mixing-ratio tracers are transported up-ward from their deep source; while in the downwellingregion, lower-mixing-ratio tracers are transported fromthe upper atmosphere. As a result, the tracer mixing

-50 0 50

-4×10-17

-2×10-17

0

2×10-17

4×10-17

-50 0 50Latitude (degree)

-1×10-15

-5×10-16

0

5×10-16

1×10-15

Fig. 8.— Important tracer tendency terms in Eq. (8) for a short-lived tracer (upper panel, τc = 107 s) and a long-lived tracer (lowerpanel, τc = 1010 s) in Experiments I. The black lines represent the

term w ∂χ∂z

and the red are ez ∂∂z

[e−z(wχ′ − wχ′)].

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 ofthe vertical velocity (Fig. 4 and Eq. 14). However, ifthe tracer chemical timescale is very long (Fig. 4, bot-tom row), the tracers tend to be homogenized by thecirculations.

We numerically derive Kzz based on the simulationresults and the flux-gradient relationship (Eq. 6) andcompare with the analytical prediction (Eq. 23). First,we averaged the tracer distributions with latitude in anarea-weighted-mean fashion. The vertical profiles of theglobal-mean tracer mixing ratios are shown in Fig. 5for Experiment I and Fig. 6 for Experiment II, respec-tively. The vertical profiles of short-lived tracers are closeto the chemical equilibrium profile but that of the long-lived tracers are almost well mixed and “quench” to thelower atmosphere values because the global circulationsefficiently smooth out their vertical gradients. We thenderived the 1D effective eddy diffusivity Kzz by equat-ing the eddy diffusive flux to the global-mean net verticalflux of the tracers (Eq. 6, Fig. 5 and Fig. 6). Our analyt-ical prediction based on Eq. (23) matches the numericalresults well.Kzz increases from the bottom towards the top of the

atmosphere because the vertical velocity is larger andtransport is stronger in the upper atmosphere. Our the-ory predicts that Kzz roughly scales with the square ofthe vertical velocity Kzz ∝ w2 in the short-lived tracerregime (regime I, Eq. 13) and Kzz ∝ w in the long-lived tracer regime (regime III, Eq. 17). The verticalprofiles of the Kzz follow the scaling very well (Fig. 5).The theory also predicted that Kzz should be a strongfunction of chemical timescale (Eq. 13). This is alsoconfirmed by the numerical simulations. Kzz can in-crease by more than a factor of 1000 from short-livedtracers to long-lived tracers (Fig. 5 and Fig. 6). Thisincreasing trend is well predicted by our analytical the-ory Eq. (23). Longer-lived tracers are more dynamicallycontrolled than chemically controlled and therefore havebetter correlation with the circulation pattern, leadingto a larger global-mean effective vertical transport. Aswe pointed out in Section 2, if the atmospheric dynamics

12 Zhang & Showman

−90° −60° −30° 0° 30° 60° 90°Latitude (Degree)

103

102

101

100

Pre

ssu

re (

Pa

)

° ° ° ° ° ° °

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

is fixed (i.e., passive tracer transport), Kzz scales withthe tracer chemical lifetime Kzz ∝ τc (Eq. 13) when thetracer lifetime is short (regime I) and approaches to aconstant value when the tracer lifetime is long (regimeIII, Eq. 17). Therefore, for tracers with very long chemi-cal timescale, Kzz is insensitive to the chemical timescale(also see Eq. 23). The dependence of Kzz on τc is clearlyseen in Fig. 7.

In Experiments I and II, horizontal diffusive transportvia Kyy2D is small compared with the wind advection.When τc → ∞, the effective eddy diffusivity (Eq. 21)approaches Kzz ≈ Hw0e

ηz/H . The analytical Kzz con-verges to a profile in Fig. 5, but does not match the nu-merical Kzz exactly. There may be greater disparity inExperiment II (Fig. 6) where the numerical Kzz exhibitwavy fluctuations. In the long-lived tracer regime, thederived diffusivity decreases with chemical timescale insome pressure ranges and even becomes negative at somepressure levels. This reveals a drawback in our theory inSection 2. As noted in Section 2.2.2, when the chem-ical lifetime is too long compared with the circulationtimescale, the material surfaces of tracers are distorted

significantly. The expression of Kzz might no longer bevalid because he last term (eddy term) in the left handside of Eq. (8), ez ∂∂z [e−z(wχ′ − wχ′)], is comparable toor even larger than the third term (mean tracer term)

w ∂χ∂z and thus it cannot be neglected. As illustrated in

Fig. 8, for a short-lived tracer with τc = 107 s, the eddyterm is much smaller than the mean tracer term acrossthe latitude and Eq. (23) is a good prediction of Kzz.But for a long-lived tracer with τc = 1010 s, the eddyterm is comparable to the mean tracer term. The eddytransport is complicated in this regime and may lead tosome wavy features in the numerical Kzz in Figs. 4 and5, which our current analytical theory is unable to cap-ture although the theoretical prediction is still within afactor 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-livedtracer regime generally underestimates the eddy mixingfrom the numerical simulations. In this regime, the the-oretical Kzz ≈ wLv (Eq. 17). Here we have assumedthat the horizontal dynamical timescale τd is equal to

Vertical Tracer Mixing in Planetary Atmospheres Part I 13

the global advection timescale a/U and thus the verticaltransport length scale Lv ≈ H through continuity. Thediscrepancy between the analytical and numerical eddydiffusivities implies that the actual horizontal dynamicaltimescale is larger than a/U but the mechanism is notclear. A detailed future investigation is needed.

In Experiment II (Fig. 4, third row, τc = 109 scase), the overturning circulation transports the high-concentration tracers from the deep atmosphere in thesouthern hemisphere all the way to the top of the north-ern hemisphere. These tracers are then mixed downwardin the northern hemisphere above 100 Pa. The global-mean mixing ratio profile of these tracers also shows alocal shallow minimum at around 100 Pa (yellow line inFig. 6). This type of local minima is also commonlyfound in global-mean tracer mixing ratio profiles in thesimulations of convective atmospheres (see Fig. 6 in Bor-dwell et al. 2018). At the pressure levels right abovethis local minimum, because the tracer mixing ratio in-creases with altitude but the global-mean tracer trans-port is still upward, the global-mean tracer flux is againstthe local vertical gradient of the tracer. As a result, thepredicted eddy diffusivity Kzz is negative above 100 Pa(Fig. 6). As we have discussed in Section 2.3.2, thistype of “negative eddy diffusivity” phenomenon probablyindicates that the global-mean vertical tracer transportin this case does not behave diffusively. Therefore, thediffusive 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 traceris anti-correlated with the vertical velocity distribution(Fig. 9), i.e., the tracer abundance is higher in the down-welling region and lower in the upwelling region, as mea-sured on isobars. This does not alter our theory of global-mean vertical mixing in Section 2.2.1. As before, thederived effective eddy diffusivities increase with tracerchemical timescale and approaches constant in the long-lived tracer regime (Fig. 7). If we enhance the horizontaldiffusion via Kyy2D (Experiment IV), the diffusion tendsto smooth out the horizontal gradient of tracer, leadingto a weaker correlation between the tracer distributionand the vertical velocity. The latitudinal distribution oftracers (Fig. 9) are flatter in this case compared thanthat in Experiment I. The Kzz is thus smaller (Fig. 7)but still increases with the chemical timescale. The trendis consistent with other experiments and our theory. Asseen in Fig. 7, the decrease of Kzz with a larger Kyy2D

can also be predicted in Eq. (23).

3.2. Results: Simulations with Non-uniform χ0

(Experiment V)

The cases with non-uniform χ0 (Regime II) behavequite different from that with uniform χ0 (Regime I).The numerical simulations in Experiment V are shownin Fig. 10. The prescribed χ0 follows a cosine function oflatitude. The equilibrium mixing ratio is higher at equa-tor and lower at poles. For the short-lived species, thegeneral patterns of the final tracer distributions roughlyfollow the distribution of χ0. But in the atmosphereabove 1 Pa where the transport is very efficient, thelow-mixing-ratio tracers are advected upward from thelower atmosphere at equator, leading to a smaller tracer

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Fig. 10.— Latitude-pressure maps of experiments V with non-uniform χ0. First row: vertical winds (color) and mass streamfunc-tions (contours, in units of 1013 Kg s−1). Starting from the secondrow we show volume mixing ratio maps of tracers with chemicaltimescales of 107 s, 108 s, 109 s and 1010 s from top to the bottom,respectively.

abundance at low latitudes than at mid-latitudes. Theglobal-mean tracer mixing ratio profiles roughly followthe global-mean chemical equilibrium tracer profile (Fig.11).

As the tracer lifetime increases, the global-mean tracermixing ratio profile becomes more vertical with pressuredue to stronger dynamical mixing (Fig. 11). In thosecases, meridional circulation greatly shapes the tracerdistributions towards the pattern with lower-mixing-ratio tracers at equator and higher-mixing-ratio at poles(Fig. 10). The latitudinal trend of the tracer on an isobaris opposite to the prescribed chemical equilibrium tracerdistribution χ0. In general the patterns of longer-livedtracers look similar to that in Experiment III, where thetracer source is also from the top atmosphere but χ0 isflat with latitude in that case. However, there is a sig-

14 Zhang & Showman

10-12 10-10 10-8 10-6 10-4

Mean mixing ratio

1000

100

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1

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1000

100

10

1

Fig. 11.— Same as Fig. 5 but from Experiment V with non-uniform χ0. The predicted eddy diffusivity profiles based on Eq.(22) with non-diffusive correction are shown in dashed in the rightpanel.

nificant difference between the Experiments III and V.There are two mid-latitude “tongues” sinking from thetop atmosphere in Experiment V while they are missingin Experiment III, where the tracer mixing ratio is highfrom the mid-latitudes all the way to the poles. The ex-istence of the mid-latitude tracer maxima in ExperimentV suggests strong downwelling of low-mixing-ratio trac-ers from the top atmosphere in the polar region wherethe chemical equilibrium tracer mixing ratio is low. Thedownward tracer fluxes dilute the high-mixing-ratio trac-ers that are transported from the middle latitudes to thepolar region, resulting local maxima (“tongues”) that areconcentrated at mid-latitudes.

Our theory in Section 2 predicts that non-uniform χ0

could introduce a negative component in the global-meaneddy diffusivity Kzz. This effect is more pronounced forshort-lived species. The numerically calculated Kzz areshown in Fig. 11. The derived Kzz become negativebelow some pressure levels for short-lived species withchemical lifetime smaller than 108 s. This non-diffusiveeffect is primarily due to the non-uniform χ0, which is lessimportant when the tracer lifetime becomes longer andwhen the tracer distribution substantially deviates awayfrom the chemical equilibrium. For long-live species, thenon-diffusive effect vanishes, and the derived Kzz profilegenerally agrees with that from Experiments I, II and IIIwith uniform χ0. Using our analytical formula of Kzz

with a non-diffusive correction term (Eq. 22), we cangenerally reproduced the negative Kzz values for short-lived species as well as the positive values for long-livedspecies (Fig. 11).

4. CONCLUSION AND DISCUSSION

The central assumption in the 1D framework is thatvertical tracer transport acts in a diffusive manner in aglobal-mean sense. Eddy diffusivity is a key parameterin this framework and is normally constrained by fit-ting the model to observed tracer profiles. However, thephysical meaning of this empirically determined quan-tity is usually not elucidated, and there have been fewattempts in the planetary literature to estimate it from

first principles or show systematically how it should varyfrom planet to planet. In this study we investigated someof the fundamental processes that are lumped into—andcontrol—this single quantity. We generalized the pio-neering theoretical work from Holton (1986) for a 2DEarth model to a 3D atmosphere and explicitly derivedthe diffusivity expression from first principles for specificsituations. We performed tracer transport simulationsin a 2D chemical-diffusion-advection model for rapid-rotating planets using a simple chemical source/sinkscheme. By deriving the 1D eddy diffusivity from theglobally averaged vertical transport flux, we showed thatthe simulation results in 2D agree with our theoreticalpredictions. Therefore this study can serve as a theo-retical foundation for future work on estimating or un-derstanding the effective eddy diffusivity for global-meanvertical tracer transport.

The general take-home message from our investigationis that interaction between the chemistry and dynamicsis important in controlling the 1D vertical tracer trans-port. Our work demonstrates that the global-mean verti-cal tracer transport crucially depends on the correlationsbetween the vertical velocity field and the tracer horizon-tal variations, which is significantly modulated by atmo-spheric circulation, horizontal diffusion and wave mixing,and the local tracer chemistry or microphysics. Impor-tantly, this correlation is also controlled by the chemistryitself, which implies that—even for a given atmosphericcirculation—the vertical mixing rates and effective eddydiffusivity can differ from one chemical species to an-other. In general, we found that the traditional assump-tion in current 1D models that all chemical species aretransported via the same eddy diffusivity breaks down.Instead, the 1D eddy diffusivity should increase withtracer chemical lifetime and circulation strength but de-crease with horizontal mixing efficiency due to breakingof Rossby waves or other horizontal mixing processes.Our analytical theory including these effects can explainthe 2D numerical simulation results over a wide parame-ter space. This physically motivated formulation of eddydiffusivity could be useful for future 1D tracer transportmodels.

We emphasize that the conventional “diffusive” as-sumption of the global-mean vertical tracer transportdoes not always hold. In this study we demarcated threeregimes in terms of the tracer lifetime and horizontaltracer distribution under chemical equilibrium (Fig. 3).Only in regime I, a short-lived tracer with uniform distri-bution of chemical equilibrium abundance, is the tradi-tional diffusive assumption mostly valid. In the other tworegimes, tracer with non-uniform distribution of chemicalequilibrium abundance (regime II) and the tracer lifetimesignificantly long compared with the transport timescale(regime III), the global-mean vertical tracer transportcould be largely influenced by non-diffusive effects. Non-diffusive effects could result in a negative effective eddydiffusivity in the traditional diffusive framework, eitherfor the short-lived species in regime II or for the long-lived species in regime III. A negative diffusivity does notphysically make sense and might be difficult to incorpo-rate into 1D models. For relatively long-lived species inregime II (but not long enough to reach regime III), weprovided a simple derivation to capture the non-diffusiveeffects, which might be useful for future 1D models.

Vertical Tracer Mixing in Planetary Atmospheres Part I 15

Our detailed analytical and numerical analysis con-cludes several key points:

(1) Larger characteristic vertical velocities contributeto a larger global-mean tracer mixing. Everything elsebeing equal (in particular, if there are no huge variationsin chemical timescales with height), this then impliesthat, if the vertical velocity increases with height, theeddy diffusivity also increases with the height in the at-mosphere. For short-lived tracers in regime I, Kzz ∝ w2

and for the long-lived tracers in all regimes, Kzz ∝ w ifthe 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 theglobal-mean eddy diffusivity. But the horizontal traceradvection due to the mean flow can either increase or de-crease the eddy diffusivity, depending on the eddy tracerflux convergence/divergence induced by the mean flow,which fundamentally depends on the correlation betweenthe horizontal mean flow and the horizontal tracer vari-ations.

(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 reaches itsasymptotic value–a product of vertical wind velocity andvertical transport length scale.

(4) In regime I, short-lived species exhibit a similarspatial pattern as the vertical velocity field (as viewedon an isobar). But if the equilibrium chemical field hashorizontal variations (regime II), the resulting tracer dis-tribution is complicated. In regime III, the pattern of along-lived tracer is largely controlled by the atmosphericdynamics. In this case, the correlation between verti-cal velocity and tracer fields (on isobars) breaks down,and the 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 to hor-izontal mixing. If the tracer lifetime is sufficiently long(i.e., τc → ∞), the horizontal tracer field is expected tobe completely homogenized across the globe, but we didnot simulate this physical limit in this study.

(5) The diffusive assumption is generally valid inregime I and the effective eddy diffusivity is always posi-tive using our idealized chemical schemes. But in regimeII, there is a strong variation in the equilibrium chemicalfield. Non-diffusive effects are important in regime II ifthere is a good correlation between the equilibrium tracerfield and the vertical velocity field. In some situations,tracers could be vertically mixed towards the source in aglobal-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 IIIwhen the tracer chemical lifetime is much longer thanthe atmospheric dynamical timescale. In this regime,the tracer material surface is distorted significantly andthe global-mean tracer profile is complicated. For exam-ple, in the case where the tracer source is in the deep

atmosphere, the global-mean tracer profile could exhibita local minimum in the middle atmospheric layers. Thederived effective eddy diffusivity will be negative abovethe local minimum, suggesting a strong non-diffusive ef-fect.

(7) We derived the analytical species-dependent eddydiffusivity for tracers with uniform chemical equilibrium(Eq. 13). We also provided a simple non-diffusive cor-rection for long-lived tracers with non-uniform chemicalequilibrium (Eq. 15). The dynamical timescale in ourtheory depends on the horizontal length scale Lh and/orvertical length scale Lv of the tracer transport. Usingthe pressure scale height H as Lv, the theoretical predic-tions generally agree with our 2D numerical simulations.For long-lived species, the actual Lv appears larger thanH in our 2D simulations.

(8) A widely accepted assumption in current 1-D chemical models of planetary atmospheres—species-independent eddy diffusion, which assumes a single pro-file of vertical eddy diffusivity for all species—is generallyinvalid. Using a species-dependent eddy diffusivity inour theory will lead to a more realistic understanding ofglobal-mean tracer transport in planetary atmospheres.

In this paper, we only focused on fast-rotating plan-ets where a zonal-symmetric assumption generally holdstrue. But for planetary atmospheres with significantzonal asymmetry, such as the atmospheres on tidallylocked exoplanets, a 3D tracer-transport model is nec-essary to test our theory derived in this study. In a con-secutive paper (Paper II, Zhang & Showman 2018b), wewill focus on the tracer transport on tidally locked plan-ets using a GCM and simple tracer schemes and demon-strate that our analytical theory can be also applied tothat regime.

We should note that, in this study we only consideredidealized chemical schemes that relax the tracer distri-bution toward a specified chemical equilibrium over aspecified timescale. Other types of sources/sinks, for in-stance, due to cloud particle setting or due to more com-plex gas-phase chemistry have yet to be explored. Theglobal-mean vertical transport behaviors of those tracersmerit investigation in the future.

In our simulations we prescribed the meridional circu-lation and eddy diffusivities to study the response of thetracer distributions, but this approach prevents us frominvestigating the detailed dynamical interaction betweenthe waves/eddies and the tracers. Waves and eddies alsohave strong dynamical effects on zonal jets that influencethe tracer transport. For example, the polar jets drivenby waves and eddies on Earth and Saturn could act aspotential vorticity barriers for tracer transport. A 3DGCM relevant for stratospheres of fast-rotating planetsis needed in the future to explore the effects of small-to-regional-scale waves and eddies on tracer transport.Furthermore, besides the large-scale upwelling and down-welling transport, waves might also contribute to the ver-tical mixing in the stratified atmosphere. In the highatmosphere such as the mesosphere on Earth or thermo-sphere on Jupiter, gravity waves could lead to a strongtracer transport and tracer mixing. Lindzen (1981) firstproposed that the gravity wave breaking including thebreaking of tidal waves will significantly mix the tracersvertically. Strobel (1981) and Schoeberl & Strobel (1984)proposed a parameterization including both gravity wave

16 Zhang & Showman

breaking and tracer mixing by linear waves. Strobel et al.(1987) also found that the effective vertical eddy diffusiv-ity due to gravity waves depends on the tracer chemicallifetime. It is expected that the total vertical eddy dif-fusivity would be a sum of both large-scale mixing andgravity wave mixing. Further investigation is needed so

that 1D vertical tracer transport can be understood ina coherent 3D chemical-transport framework from thedeep, turbulent atmosphere to the upper, low-density at-mosphere including convective mixing, large-scale circu-lation, and mixing from atmospheric wave processes suchas breaking of Rossby waves and gravity waves.

APPENDIX

DERIVATION 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 (Eq. 19 in the text, Shia et al. 1989, Shia et al. 1990, Zhang et al. 2013b):

∂χ

∂t+ v∗

∂χ

∂y+ w∗

∂χ

∂z− 1

cosφ

∂

∂y(cosφKyy2D

∂χ

∂y)− ez/H ∂

∂z(e−z/HKzz2D

∂χ

∂z) = S (A.1)

where χ is the tracer mixing ratio, φ = y/a is latitude and a is the planetary radius. z ≡ −H log(p/p0) is the log-pressure coordinate, where H is a constant reference scale height, p is pressure and p0 is the pressure at the bottomboundary. Kyy2D and Kzz2D are the horizontal and vertical eddy diffusivities, respectively. Residual circulationvelocities are v∗ and w∗ in the meridional and vertical directions, respectively. A mass streamfunction ψ can beintroduced such that:

v∗ = − 1

2πaρ0 cosφez/H

∂

∂z(e−z/Hψ) (A.2a)

w∗ =1

2πaρ0 cosφ

∂ψ

∂y(A.2b)

where ρ0 is the reference density of the atmosphere at log-pressure z = 0. The v∗ and w∗ given in Eq. (A.2) naturallysatisfy the continuity equation:

1

cosφ

∂

∂y(cosφ v∗) + ez/H

∂

∂z(e−z/Hw∗) = 0. (A.3)

Combining Eq. (A.1) and Eq. (A.3), we obtain the flux form of the tracer transport equation:

∂χ

∂t+

1

cosφ

∂

∂y(cosφ v∗χ) + ez/H

∂

∂z(e−z/Hw∗χ)− 1

cosφ

∂

∂y(cosφKyy2D

∂χ

∂y)− ez/H ∂

∂z(e−z/HKzz2D

∂χ

∂z) = S.(A.4)

Here we also define the eddy-mean decomposition A = A + A′ where A is the latitudinally averaged quantity atconstant pressure and A′ is the eddy term. Taking the latitudinal average of Eq. (A.3) to eliminate the meridionaltransport term (the v∗ term), we obtain w∗ = 0 at isobars if we assume the latitudinal-mean vertical velocity w∗

vanishes at top and bottom boundaries. We average Eq. (A.4) and both meridional advection and diffusion termsvanish. Using w∗ = 0, we obtain:

∂χ

∂t+ ez/H

∂

∂z(e−z/Hw∗χ′)− ez/H ∂

∂z(e−z/HKzz2D

∂χ

∂z) = S. (A.5)

Based on the “flux-gradient relationship”, we can introduce an eddy diffusivity Kw to approximate the verticaltransport term associated with w∗ (the second term in the left hand side) such that:

w∗χ′ ≈ −Kw∂χ

∂z. (A.6)

Then the 1D global-mean tracer transport equation (Eq. A.5) can be formulated as a vertical diffusion equation:

∂χ

∂t− ez/H ∂

∂z(e−z/HKzz

∂χ

∂z) = S (A.7)

where define the total 1D effective eddy diffusivity Kzz = Kw +Kzz2D. Therefore, the Kzz2D vertical eddy diffusionterm in Eq. (A.5) can be treated as an additive term in the total global-mean effective eddy diffusivity Kzz.

In order to analytically derive Kw, we subtract both Eq. (A.5) and Eq. (A.3) multiplied by χ from Eq. (A.4):

∂χ′

∂t+

1

cosφ

∂

∂y(cosφ v∗χ′)− 1

cosφ

∂

∂y(cosφKyy2D

∂χ′

∂y)+w∗

∂χ

∂z+ez/H

∂

∂z[e−z/H(wχ′−w∗χ′−Kzz2D

∂χ′

∂z)] = S′. (A.8)

To solve this equation for χ′, we made four assumptions that are similar to those in Section 2.3.(i) We neglect the first temporal evolution term in statistical steady state.

Vertical Tracer Mixing in Planetary Atmospheres Part I 17

(ii) We neglect the complicated eddy term (the fifth term) by assuming the vertical transport of the tracer eddy χ′

is much smaller than that of the latitudinal-mean tracer (χ, the fourth term). Here we have also assumed the verticaldiffusion of χ′ (the Kzz2D term) is much smaller than than 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 mixing processes in a 3D atmosphere into the tracer advection byzonal-mean flow and the tracer diffusion by horizontal eddies and waves. In the zonal-mean 2D system, the secondand third terms in the left hand side of Eq. (A.8) correspond to the advection and diffusion process, respectively. Forthe advection term, we have:

1

cosφ

∂

∂y(cosφ v∗χ′) ≈ χ′

τadv(A.9)

where τadv ≈ Lh/v∗ where Lh is the horizontal length scale and v∗ is the meridional velocity scale. From the continuityequation (Eq. A.3), we can also show τadv ≈ Lv/w∗, where Lv is the vertical length scale and w∗ is the vertical velocityscale. Similarly, the diffusive term can be approximated as:

− 1

cosφ

∂

∂y(cosφKyy2D

∂χ′

∂y) ≈ χ′

τdiff(A.10)

where τdiff ≈ a2/Kyy2D. Here we have made an additional assumption that the positive tracer anomaly (χ′ > 0) isassociated with the tracer eddy flux divergence (i.e., the sign of the left hand side is positive in this situation) and thenegative anomaly is associated with the tracer eddy flux convergence. This is generally a good assumption as long asthe material surface of the tracer is not very complicated. For example, if the horizontal tracer distribution can beapproximated by a second-degree Legendre polynomial as in Holton (1986), one can show that our above argumentholds true.

(iv) We also consider a linear chemical scheme which relaxes the tracer distribution towards local chemical equilibriumχ0 in a timescale τc. After the mean-eddy decomposition, we obtain the latitudinal mean of the chemical source/sinkterm S = (χ0 − χ)/τc and the departure S′ = (χ′0 − χ′)/τc. See Section 2.3 for details.

With above assumptions, Eq. (A.8) can be simplified as:

w∗∂χ

∂z+

χ′

τadv+

χ′

τdiff=χ′0 − χ′

τc. (A.11)

We solve for χ′:

χ′ =−w∗ ∂χ∂z + τ−1c χ′0

τ−1diff + τ−1adv + τ−1c. (A.12)

Inserting Eq. (A.12) in Eq. (A.6), we solve for Kw and finally obtain the analytical expression for Kzz:

Kzz ≈ Kzz2D +w∗2

τ−1diff + τ−1adv + τ−1c− w∗χ′0

1 + τ−1diffτc + τ−1advτc(∂χ

∂z)−1. (A.13)

In our 2D cases, if the chemical equilibrium abundance χ0 is not uniformly distributed across the latitude, themagnitude of χ′0, ∆χ′0, can be approximated by the root-mean-square of χ′0 over the globe. One can show that, fora cosine function of χ0, ∆χ′0 ≈ 0.28 χ0 from the area-weighted global integration. The vertical gradient of χ canbe approximated by that of the global-mean chemical equilibrium mixing ratio χ0. Given the specific meridionalcirculation patterns in Eq. (20), the area-weighted global-mean vertical velocity scale is w∗ = γw0e

ηz/H . Assumingthat the horizontal transport length scale Lh is the planetary radius a and vertical transport length scale Lv is thepressure scale height H, we predict Kzz for our 2D cases (Eq. 22 in the text):

Kzz ≈ Kzz2D +γ2w2

0e2ηz/H

Kyy2Da−2 + γw0eηz/HH−1 + τ−1c− γw0e

ηz/H∆χ′01 + τcKyy2Da−2 + τcγw0eηz/HH−1

(∂χ0

∂z)−1. (A.14)

If the chemical equilibrium abundance χ0 is uniformly distributed across the latitude, χ′0 is zero and thus the lastterm in the right hand side of Eq. (A.13) vanishes. Kzz becomes (Eq. 23 in the text):

Kzz ≈ Kzz2D +γ2w2

0e2ηz/H

Kyy2Da−2 + γw0eηz/HH−1 + τ−1c. (A.15)

ACKNOWLEDGEMENTS

This research was supported by NASA Solar System Workings Grant NNX16AG08G to X.Z. and A.P.S.. We dedicatethis work to Dr. Mark Allen (1949-2016), one of the founders of the Caltech/JPL kinetics model. We thank Y. Yung,

18 Zhang & Showman

R. Shia, and J. Moses for providing the eddy diffusion profiles in their chemical models. Some of the simulations wereperformed on the Stampede supercomputer at TACC through an allocation by XSEDE.

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