Inhomogeneous cloud evaporation, invariance, and Damköhler number



[1] An idealized eddy-diffusivity model of entrainment mixing and evaporation is presented consisting of a continuous Eulerian relative humidity field and discrete Lagrangian droplets. Lagrangian droplet trajectories are calculated using a Brownian motion with commensurate diffusivity and a Lagrangian velocity decorrelation that is consistent with turbulent mixing. Using this new model, we revisit Jensen and Baker (1989)'s (JB) classic study of inhomogeneous mixing. Analysis of the governing dynamical and microphysical equations for the simplified geometry used by JB indicates that droplet spectral dispersion is a function of only three parameters: the ratio of the mixing and droplet evaporation timescales (the Damköhler number, Da), the fraction of entrained air and the critical entrainment fraction that produces a cloudless but saturated well-mixed final state. These dependencies are encapsulated in four invariance theorems for isobaric mixing and evaporation that we derive and that establish a set of self-similar solutions for general entrainment scenarios that include sedimentation. A comparison of the predictions of our new model with JB's results verifies the acknowledged “homogeneous evaporation bias” in their model formulation and the exclusion of the large Da limit in JB's study. A full investigation of the JB entrainment scenario using our new model reveals a transition from homogeneous to inhomogeneous mixing in the droplet spectral dispersion at Da ≈ 5 and extreme inhomogeneous mixing behavior in the limit Da → ∞. The PDFs of the time-integrated Lagrangian subsaturation, equation imageint, in the inhomogeneous regime are observed to asymptote to a equation imageint−1 law; implications of this finding for subgrid cloud modeling are discussed.

1. Introduction

[2] The evolution of the cloud droplet size distribution is controlled by the local subsaturation experienced by droplets along Lagrangian trajectories. During entrainment and mixing, individual droplets can experience widely different subsaturation environments, a process termed “inhomogeneous mixing” by Baker et al. [1980]. Observational studies indicate spectral broadening to smaller sizes and show large jumps in subsaturation across the clear-cloudy air interface over distances smaller than a few centimeters [Siebert et al., 2006; Haman et al., 2007; Burnet and Brenguier, 2007]. The difference in the droplet size distribution before and after entrainment mixing is therefore a measure of the turbulent Lagrangian mixing statistics, in contrast to state variables like liquid water density which are path-independent at equilibrium.

[3] The evolution of droplet size in centimeter scale filaments during entrainment mixing impacts large scale cloud properties, most notably cloud solar reflectivity and precipitation efficiency. Droplet spectral evolution thus plays a key role in the response of clouds to aerosol forcing and the first and second indirect effects, respectively [Lohmann and Feichter, 2005]. The relationship between Lagrangian mixing statistics, evaporation inhomogeneity, and large-scale properties is revealed in the following back-of-the-envelope calculation of cloud response.

[4] Consider the response of shortwave optical depth, τNr2, to an evaporative change in cloud liquid water density, ρlNr3, where N is droplet concentration and r is radius. In the pure homogeneous mixing limit all droplets experience the same subsaturation during evaporation such that (∂/∂ρl)homo ∼ (∂/∂r3)N, while in the extreme inhomogeneous limit, those droplets that mix with entrained air evaporate completely such that (∂/∂ρl)inhomo ∼ (∂/∂N)r. Optical depth thus responds to the nature of mixing and evaporation in the following specific ways:

equation image

which can be compared to the Twomey (1st indirect) response (∂lnτ/∂lnN)ρl = 1/3. These simple relations suggest that centimeter scale Lagrangian mixing statistics affect the albedo of entraining clouds, and idealized sensitivity studies using prescribed droplet spectral response verify this dependence of large-scale cloud properties on (parameterized) small-scale mixing behavior [Chosson et al., 2004; Grabowski, 2006].

[5] Compared to numerous studies that exploit simplified Lagrangian parcel models to study the effect of entrainment on cloud properties, very few studies have addressed the inhomogeneous mixing problem using simplified models and idealized geometries that enable a complete determination of the relevant parametric dependencies [Baker et al., 1980, 1984; Baker and Latham, 1982; Jensen and Baker, 1989; Su et al., 1998]. Jensen and Baker [1989] (hereinafter referred to as JB) develop a 1D turbulent eddy mixing model and assess the impact of (1) entrained air fraction ϕsub and (2) turbulent mixing intensity on the predicted droplet spectral width. This case study is a reference against which other models can be compared, although the authors raise the concern that their model suffers from a “homogeneous evaporation bias” (JB, p. 2828).

[6] Idealized studies like JB are typically performed using particular values for the initial temperature and moisture of clear and cloudy air and for initial droplet size and concentration. As a result, the sensitivity of the model predicted behavior to this choice of initial conditions and the generality of the model results is unknown. This inhibits the comparison of results from different modeling studies based on differing initial conditions. We address this deficiency in section 2.2 where we present four invariance theorems for cloud mixing and evaporation. An invariance is a rescaling of the dynamical and microphysical variables by a set of factors {λ1, λ2,…} such that the system evolution remains self-similar. Identification of the invariant transforms of coupled equations provides the same information as a formal nondimensionalization but emphasizes the scaling relationship between various fields. These invariance theorems establish that only three initial microphysical conditions are germane to the JB reference study: the initial droplet spectrum, the critical entrainment fraction, ϕcrit, that produces a cloudless but saturated well-mixed final state, and the evaporation timescale.

[7] As discussed by JB (p. 2816), two separate microphysical timescales influence the droplet mixing and evaporation inhomogeneity, the relative humidity timescale, τRH, and the droplet size timescale,τR. For the JB reference case, these timescales are approximately equal. This allows the Damköhler number to be uniquely identified as [Jeffery and Reisner, 2006b]

equation image

where is τeddy the mixing timescale and the “reactive” response of the system is given, equivalently, by τreact = τRτRH. Ignoring sedimentation, the JB model results are thus most effectively expressed as a function of the single nondimensional number Da at given {ϕsub, ϕcrit}.

[8] In section 5 we compare the behavior of JB's turbulent eddy mixing model with results from a new eddy-diffusivity model of cloud mixing and evaporation developed in section 4. In this approach, Lagrangian droplet trajectories are modeled using a Brownian motion with an ensemble-averaged droplet transport that is diffusive and commensurate with the eddy-diffusion of relative humidity. Additionally, the Lagrangian velocity decorrelation time, an additional degree of freedom in this modeling approach, is chosen to be consistent with turbulent mixing at large Reynolds number. We verify JB's homogeneous evaporation bias and demonstrate that our new model reproduces both the homogeneous (Da → 0) and extreme inhomogeneous (Da → ∞) mixing limits in a simplified 1D geometry. We present in section 5 a complete set of cloud mixing and evaporation experiments that encompass a wide range of Da values and that can be used as a reference to compare against other modeling approaches. Section 6 contains a summary.

2. Invariance and Evaporation Timescales

[9] We begin with the isobaric evolution equation for RH [Pruppacher and Klett, 1997, chap. 13], derived through a linearization of the Clausius-Clayperon equation and valid in the limit T′ ≪ T* where T*RvT2/Lv is in the range 13–17 C°, and thermodynamic symbols are described in the notation.

equation image

where (l/dt)evap in the source term is the change in liquid water density, ρl, due to evaporation, D/Dt = ∂/∂t + u(x, t) · − κ∇2 is the total advective-diffusive derivative, u(x, t) is a velocity field and κ is the molecular diffusivity of water vapor and temperature, assumed equal. The factor F is given by

equation image

and is a weak function of temperature, air density and RH. Consistent with the approximation T′ ≪ T*, the quantities {F, ρs} in equation (1) are replaced with the averaged quantities {〈F〉, 〈ρs〉} where 〈·〉 represents a suitable space and time average. This substitution enables the direct space-time integration of equation (1) over a closed volume containing both cloudy and clear (environmental) air. The resulting relation expresses conservation of water and heat during advective-diffusive mixing in a closed volume:

equation image

where 〈·〉x represents a spatial average. For notational convenience we introduce the subsaturation Ssub ≡ 1 − RH, a strictly positive quantity for the isobaric mixing and evaporation considered here. Of great relevance to this analysis is the critical subsaturation, 〈S〉crit, defined such that the final well-mixed state of the system is ρl = 0 and Ssub = 0:

equation image

〈S〉crit is thus the minimum average subsaturation that evaporates all of the initial cloud liquid water.

[10] We now introduce the evolution equations for our system which, by virtue of equation (1), is specified by the continuous Eulerian scalar field Ssub(x, t) and a number of discrete droplets labeled i = 1,2,3… of radius Ri and Lagrangian position Xi(t):

equation image
equation image

where N(x, t) is the droplet concentration field, a ≈ 2 μm is an accommodation length, 〈⋯∣x, tr represents an Eulerian average over droplet radius r ≥ 0 in the volume element at (x, t) and G is a function of temperature and density similar in form to F [Pruppacher and Klett, 1997, chap. 13]:

equation image

Note that G and ρs have been replaced by space-time averages in equation (2b). For the time being, the droplet velocity equation image is left unspecified.

[11] Equations (2a) and (2b) contain an implicit continuum assumption that droplet growth in each volume element is driven by a single Ssub value and described by the mean-field result (2b). This continuum limit is uniformly valid for clouds, requiring only that droplet sizes be much smaller than the Kolmogorov scale. Note that equations (2a) and (2b) do not assume a complete or continuous droplet size spectrum in each element.

2.1. Nondimensionalized Equations

[12] Consider the following nondimensionalized variables: Ssub normalized by its initial maximum value, equation image ≡ Ssub/Smax where Smax ≡ max(S(x, 0))x, and N normalized by its initial in-cloud value, equation imageN/Nc where Nc is an average in-cloud droplet number density. The value of Nc determines the fraction of entrained environmental air, ϕsub, via

equation image

where 〈·〉i is an average over all drops in the system. This relation is of importance because it relates the Eulerian average 〈·〉x to the discrete Lagrangian average 〈·〉i. This division of the system into clear and cloudy components must satisfy (1 − ϕsub)Nc = 〈Nx.

[13] In these new variables, equation (2) is written

equation image
equation image


equation image
equation image

and we have used equation imageF〉〈G−1 ≈ 1. The timescales, τE and τL, are similar in form but are distinct Eulerian and Lagrangian representations, respectively, of τRH and, hence, the time dependence of equation image. By comparison, the droplet size timescale of the ith drop is τR(Ri) = 〈S〉crit[Smax(1 − ϕsub)]−1τL(Ri). Both timescales are regularized by a in the limit {r, Ri} → 0.

[14] Inspection of equations (4)(6) reveals that the accommodation length considerably complicates the coupled evolution of {equation image, Ri}. In the limit of large initial droplet sizes Ria, we expect the a dependence of mixing and evaporation to be second order. Taking a → 0, equations (4)(6) simplify considerably and are given by

equation image
equation image


equation image
equation image

Here we have introduced the conditional quantity Rj(tXj(t) = x) defined as those droplets j = 1, 2, 3… that obey Xj(t) = x. This conditional quantity relates Eulerian and Lagrangian averaging according to 〈⋯(tXj(t) = x)〉j = 〈⋯(x, t) 〉r.

[15] It is important to emphasize that the equation sets (4)(6) and (7)(9) do not contain any new information that is not contained in equation (2). Nor are they necessary for the proceeding analysis. The value of this nondimensionalization is that functions equation image, equation image and normalized droplet radius are explicitly scale-independent such that the scale dependence is explicitly isolated in τE, τL, D/Dt, 〈S〉crit and Smax.

2.2. Invariance

[16] In this section we state four invariance theorems for isobaric mixing and evaporation. An invariance is a rescaling by the set {λ1, λ2,…} that leaves the three nondimensionalized microphysical quantities: equation image(x, t), equation image(x, t) and appropriately normalized droplet radius, unchanged. To determine this set, we first note that the parameters {x, t, equation imageF−1ρs〉}, Lagrangian variables {equation imagei, Ri} and space-time Eulerian fields {u, N, Ssub} are observed to span a very large range of values; each variable is assigned a multiplicative scaling factor λi that increases (λi > 1) or decreases (λi < 1) the variable's value. The rescaled fields are valid if the evolution equations remain satisfied, and this constraint leads to algebraic relationships among the scaling factors. Since the microphysical parameters {Dv, a, κ} are roughly constant, they are not rescaled. Note that the entrained fraction ϕsub is a normalized function of the system's initial spatial structure and is invariant under x rescaling. The proofs of the following theorems follow from direct substitution of the rescaled quantities into equations (4) or (7).

[17] The {λ1, λ2,…} rescaling is closely related to nondimensionalization; an invariant transformation can be recast as a relationship among characteristic quantities that keeps a set of nondimensional numbers constant. Given a characteristic mixing timescale τeddy, velocity U, system length L, eddy-diffusivity κe, and relative humidity timescale τRH which is a measure of τE(x), the following nondimensional numbers are relevant to the theorems of this section: the mixing number Nmix = eddy/L (Nmix = κeτeddy/L2 for eddy-diffusive transport), the Reynolds number (defined in terms of molecular diffusivity κ) Re = L2/τeddyκ), the Damköhler number Da = τeddy/τR and the evaporation number Nevap = 〈S〉crit/Smax which is the ratio of the droplet radius and RH timescales.

2.2.1. Theorem 1

[18] A necessary condition for invariance of equation (4) at fixed R, and equation (7) with rescaled R, is constant evaporation number Nevap = 〈S〉crit/Smax. Remark

[19] The evolution of droplet radius and subsaturation are coupled by the ratio of their respective timescales; this ratio must be maintained for the system evolution to remain self-similar. Since the accommodation length in equations (5) and (6) affects τE and τL in a slightly different manner, it is technically possible to vary Nevap and keep the proportionality of τE and τL fixed. This possibility is excluded in theorem 1 by keeping R fixed in equation (4). Corollary 1

[20] For the special case of entrained air of constant subsaturation, Senv, a necessary condition for invariance of equation (4) at fixed R, and equation (7) with rescaled R, is constant ϕcrit where ϕcrit is the critical entrainment fraction. Proof

[21] By definition, entrainment of a volume fraction ϕcrit of Senv air completely evaporates 〈ρlcrit)〉x:

equation image

Noting that (1 − ϕsub)〈ρlcrit)〉x = (1 − ϕcrit)〈ρlsub)〉x, we can rewrite 〈S〉crit as

equation image

Using Smax = Senv we find

equation image

Corollary 1 follows noting that ϕsub is a geometric parameter that is invariant under rescaling.

2.2.2. Theorem 2

[22] The following transformations preserve τE, τL and the evaporation number Nevap = 〈S〉crit/Smax. Theorem 2(A)

[23] Equations (4a) and (4b) are invariant under the rescaling

equation image Theorem 2(B)

[24] Equations (7a) and (7b) are invariant under the rescaling

equation image Remark

[25] The rescaling of theorem 2(A) preserves the water vapor density ρv = Ssubequation imageρs〉 multiplied by the small factor 〈F−1. Theorem 2(B) is valid when droplet sedimentation is negligible since the droplet velocities equation imagei, which are not rescaled, are implicitly R-independent. The invariance of theorem 2(B) spans a range of cloud types and cloud liquid water densities which rescale as λ12. For the particular case of invariant relative humidity (λ2 = 1), an invariant entraining cloud with larger droplets is warmer and has fewer droplets.

2.2.3. Theorem 3

[26] The following space-time transformations preserve the velocity field u, the droplet velocities equation imagei, and the nondimensional numbers {Nmix, Da, Nevap}: Theorem 3(A)

[27] Defining D/Dt = ∂/∂t + u(x, t) · as purely advective transport, and assuming that the Eulerian and Lagrangian velocities are arbitrary functions of (x, t), the normalized microphysical fields equation image, equation image and the set of all droplet radii Ri, the rescaling

equation image

leaves the (λ1x, λ1t) space-time rescaled microphysical fields described by equations (4a) and (4b) invariant. Theorem 3(B)

[28] Defining D/Dt = ∂/∂tequation imageeequation image2 as modeled turbulent transport with eddy-diffusivity, κe, and assuming that κe and Lagrangian droplet velocities are arbitrary functions of (x, t), the normalized microphysical fields equation image, equation image and the set of all droplet radii Ri, the rescaling

equation image

leaves the (λ1x, λ1t) space-time rescaled microphysical fields described by equations (4a) and (4b) invariant. Remark

[29] In these transformations, the Reynolds number dependence of the velocities ∝ λ1 is not preserved and molecular diffusivity κ = 0. However, an arbitrary Ri-dependent droplet sedimentation is allowed. These space-time rescalings allow us to consider a larger version of a given entraining cloud. For the particular case of invariant relative humidity (λ2 = 1), an invariant larger cloud is colder and has fewer droplets.

2.2.4. Theorem 4

[30] The following space-time transformations preserve the Reynolds number, Re, in addition to the nondimensional numbers {Nmix, Da, Nevap}: Theorem 4(A)

[31] Defining D/Dt = ∂/∂t + u(x, t) · equation imageequation image2 as the Reynolds number−dependent total advective-diffusive derivative, and assuming that the Eulerian and Lagrangian velocities are arbitrary functions of (x, t), and the normalized microphysical fields equation image, equation image, the rescaling

equation image

leaves the (λ11/2x, λ1t) space-time rescaled microphysical fields described by equations (7a) and (7b) invariant. Theorem 4(B)

[32] At fixed liquid water density, the rescaling of theorem 4(A) reduces to

equation image Theorem 4(C)

[33] Assuming Stokes terminal velocity equation imagei · equation image = equation imageRi2 for the vertical component of the Lagrangian droplet velocities, a modified form of theorem 4(A) is given by

equation image Remark

[34] In theorems 4(A) and 4(B), the velocity and droplet size rescaling are considered to be inconsistent with sedimentation, and thus the droplet velocities equation imagei are specified to be R-independent. In theorem 4(C) this restriction is dropped and a particular sedimentation dependence is considered. The space-time rescaling of theorem 4(B) predicts that a larger version of a given entraining cloud at fixed ρl advected by a slower velocity field will have fewer larger droplets, with the size of the drops scaling proportionally to the size of the cloud. The space-time rescaling of theorem 4(C) predicts that a larger version of a given entraining cloud with sedimenting droplets advected by a slower velocity field will have fewer smaller droplets, and thus smaller ρl.

2.3. Evaporation Timescales

[35] For the special case of entrained air of constant subsaturation, Senv, equation (4b) becomes

equation image


equation image

from equations (3) and (10).

[36] The constant timescales τRH and τR that characterize the system evolution are thus defined naturally as

equation image

This definition of τR is equivalent to the expression τR = (1/3)〈Gρw/(Dvρs〉)〈(Rj + a)2jSenv−1 inferred from equation (2b).

[37] The variable ϕcrit, or equivalently Nevap, is a measure of the relative droplet evaporation efficiency. Larger ϕcrit values imply greater relative liquid water content relative to the environmental subsaturation and, hence, slower relative evaporation. In fact, since 〈ρs〉Senv is approximately the entrained water vapor density, Nevap has the simple interpretation as the ratio of in-cloud liquid water density to environmental water vapor density, with an additional small multiplicative factor 〈F〉.

[38] As a first step in analyzing the relative impact of {τRH, τR} on inhomogeneous mixing we consider the case ϕcrit ≈ 1/2 such that τRτRH and we can unambiguously define the microphysical timescale. In analogy with Jeffery and Reisner [2006b], we label this timescale τreact since τreact singularly represents the “reactive” response (both R and RH) of the system. This case was first treated by Jensen and Baker [1989]; in section 5 we generalize their mixing scenario with the aid of the theorems derived in this section.

3. Jensen and Baker's [1989] Turbulent Eddy Mixing Model

[39] JB provide a reference study which we will first revisit before developing a new modeling approach in section 4. JB develop a simple 1D formulation of isobaric turbulent mixing and evaporation based on Broadwell and Breidenthal [1982]'s formulation of the evolution of filament thickness λ in a turbulent flow:

equation image

where ε is the kinetic energy dissipation rate. JB resolve only the 1D vertical internal structure of a single filament which is described by temperature T(z), water vapor mixing ratio qv(z), and a population of Lagrangian droplets Ri(Zi(t),t) where Zi is vertical Lagrangian position. The filament is initially composed of cloud (Ssub = 0, N = Nc) and clear air (Ssub = Senv, N = 0) occupying disjoint regions of space. The temperature and vapor fields diffuse molecularly and the droplets sediment and evaporate. As λ decreases, the effect of molecular diffusion and sedimentation on the droplet spectrum becomes more pronounced; a well-mixed final state is achieved when λ reaches the microscale (κ3/ε)1/4 with molecular diffusivity κ.

[40] A peculiar feature of the JB model is that the filament's internal fields do not evolve as λ decreases from its initial value down to a meter (JB, p. 2820). Thus the internal variance var(Ssub) remains unchanged by turbulent mixing during this period, in contradistinction to mean-field theoretical results and Oboukhov-Corrsin scaling for the inertial-convective subrange which predict a cascade of scalar variance to small scales and a constant (λ-independent) scalar dissipation

equation image

which is enhanced by turbulent mixing. JB justify this behavior by arguing that the temporal evolution of the average droplet spectrum is described by a single representative filament (eddy) which uniformly decreases in size without breaking, internally deforming or mixing with its surroundings. This claim is fundamentally inconsistent with eddy-diffusive behavior which is considered to represent ensemble averaged turbulent transport. Indeed, the phenomenology of turbulent mixing rests firmly on the fundamental principle that mixing efficiency increases with increasing stirring scale λ(0), a property not shared by the JB model.

[41] JB find that their model produces only weakly inhomogeneous mixing behavior for ε in the range 2.5 × 10−3 to 5 × 10−2 m2s−3 and conclude that “the sense of all the approximations we have made is to bias our model results toward homogeneous evaporation.” This bias is consistent with the λ(0) independence displayed by their model. Assuming that the JB model actually selects for a constant molecular mixing length of about λ* ≈ 10 cm, we find τeddy = (λ*2/ε)1/3 ≈ 1 s, τR ≈ 1 s (JB, p. 2816) and a Damköhler number of near unity, consistent with strongly homogeneous mixing.

4. Eddy Diffusivity Model

4.1. Model Description

[42] In the spirit of Jensen and Baker [1989]'s simple 1D Eulerian-Lagrangian model of turbulent mixing and evaporation, we extend the 1D eddy-diffusivity model used in Jeffery and Reisner [2006b] to include Lagrangian droplet trajectories and radii. A related application of Lagrangian droplet modeling in two-dimensional cloud simulations is pursued by M. Andrejczuk et al. (The impacts of carbon activation mechanisms on a stratus deck: Does aerosol number matter more than type?, submitted to Journal of Geophysical Research, 2007). Specifically, our model solves equation set (2) with total eddy-diffusive derivative D/Dt = ∂/∂t − κe2 and eddy-diffusivity κe. Following Jeffery and Reisner [2006b] and JB, the initial system state is divided into a continuous fraction ϕsub of entrained clear air (Ssub = Senv, N = 0), disjoint from cloudy air (Ssub = 0, N = Nc). The droplet concentration field N(xi, t) at numerical grid cell i is constructed from the conditional summation Σjxi−1/2Xj(t) < xi+1/2 weighted by Nc and the inverse of the initial droplet density per grid cell.

[43] In our approach, the droplets are also assumed to eddy-diffuse using Brownian motion corresponding to the same diffusivity, equation imagee, used to transport Ssub. Sedimentation is not included. The Langevin equation for each Viequation imagei is [Risken, 1989]

equation image

where τe is the Lagrangian decorrelation time and Γ(t) is a Langevin force with zero mean and δ correlation

equation image

where τA is the acceleration renewal time and σA2 is the acceleration variance. In our numerical approach, the independent Gaussian random variable Γ is drawn each time step, Δt = τA.

[44] Einstein's relation states that κe = τeσV2 with [Risken, 1989]

equation image

This gives us a single degree of freedom to choose {τe, σA} for a given κe, although this choice should not appreciably influence the net droplet transport. Standard turbulence phenomenology suggests that both τe and σV scale with the outer (system) scale L according to τe ∼ ε−1/3L2/3 and σV ∼ ε1/3L1/3; consistent with this Kolmogorov scaling we choose τeσV = L/5.

4.2. Model Invariance

[45] One of the goals of this study is to compare and contrast the present model with the JB model for their mixing and evaporation experiments. The theorems of section 2.2 establish which parameter values used by JB are germane and which are inconsequential. JB studied the mixing and evaporation of cloudy air with initial droplet density 464 cm3, droplets of initial size ≈12.5 μm with little initial spectral width (variance), and entrained fraction ϕsub of environmental air with subsaturation ≈0.97. Of central importance, JB state on p. 2821 that ϕcrit = 0.58 for their cloud evaporation experiments.

[46] For a given initial field spatial structure (presently described by the single parameter ϕsub), corollary 1 states that constant ϕcrit is a necessary condition for the existence of invariant solutions, a condition satisfied in JB's model simulations. For eddy-diffusive transport and including droplet sedimentation, these invariant solutions are described by the rescaling of theorem 3(B) and preserve the numbers {Nmix, Da, Nevap}. Thus we conclude that only 3 microphysical quantities used by JB are germane to a model comparison: ϕcrit = 0.58, the initial droplet spectrum f(R, 0), and τRτRH. In contrast, four microphysical parameters are immaterial: the temperature and water vapor of entrained air, the initial cloud temperature, and the initial droplet number concentration. (Small microphysical differences between equation (2b) and JB's equation (16) are assumed inconsequential: JB's solute terms and the specification of a.)

[47] In the present model, advection-diffusion is modeled by a simple eddy-diffusive process with constant diffusivity such that a single timescale defines the strength of the modeled turbulent mixing. Following Jeffery and Reisner [2006b], we define the eddy mixing timescale, τeddy, as

equation image

where χ = κe〈∣∂Ssub/∂x2x is the scalar dissipation rate and τefold is the 〈Ssubxe-folding time. This adaptive definition of τeddy, diagnosed a posterior for each model experiment, is a better estimate of τeddy than the simple (constant Nmix) scaling L2κe−1 which is independent of boundary conditions and the spatial structure of the initial fields [Jeffery and Reisner, 2006b].

[48] In the JB scenario, ϕcrit is fixed at the single value 0.58 such that τRHτR by equation (12) and, as a result, the Damköhler number for the present model is uniquely defined:

equation image

where, following historical precedent [Baker et al., 1980; Siebert et al., 2006; Burnet and Brenguier, 2007] and without loss of generality, we take τreactτR. Thus in summary, we state that for the mixing and evaporation experiments of JB uniquely defined by the set {ϕcrit = 0.58, ϕsub, f(r)}, and for a given definition of τeddy (which fixes Nmix), the present model results are a function of the single parameter Da. We conclude this section by emphasizing the self-consistency of our analysis: substitution of the rescaling of theorem 3(B) into equation (12) for τRH and equation (14) for τeddy leaves Da invariant.

5. Results

[49] Motivated by the unconventional nature of JB's turbulent eddy mixing model, we repeat their cloud mixing and evaporation experiments with the following specific objectives: (1) Investigate inhomogeneous mixing predicted by the present eddy-diffusivity model as a function of Da when τRHτR and Da is uniquely defined. (2) Demonstrate that a simple 1D model formulation can overcome JB's “homogeneous evaporation bias” (JB, p. 2828) and predict extreme inhomogeneous mixing in the large Da limit. (3) Document a complete set of cloud evaporation experiments that can be used as a reference to compare against alternative (and more sophisticated) modeling approaches.

5.1. Spectral Broadening

[50] A comparison of the in-cloud droplet spectral width σr (the standard deviation of RRi > 0) for ϕcrit = 0.58 predicted by the present eddy-diffusivity (ED) model for Da ∈ {0.5, 1, 2} and by JB's model is shown in Figure 1. Results from the ED model are shown for two different boundary conditions: no-flux boundary conditions (solid line) and cyclic boundary conditions (dashed line) also used by JB. Our initial spectrum is unimodal with single radius 12.5 μm, consistent with the narrow initial spectrum shown in Figure 9 of JB. We additionally note that two of JB's calculated results include the effects of droplet sedimentation, which is not included in our model. Following JB, σr is plotted against initial cloud fraction F ≡ 1 − ϕsub.

Figure 1.

Comparison of the in-cloud (∀Ri > 0) spectral standard deviation, σr, predicted by (1) ED model with no-flux boundaries (solid line), (2) ED model with cyclic boundaries (dashed line), and (3) JB model with cyclic boundaries (crosses indicate ε = 5 × 10−2 m2s−3 with sedimentation, pluses indicate ε = 2.5 × 10−3 m2s−3 no sedimentation, and asterisks indicate ε = 2.5 × 10−3 m2s−3 with sedimentation). Critical subsaturation is F = 0.42; values for Da ∈ {0.5, 1, 2} are inset in lines.

[51] The comparison in Figure 1 reveals a number of important differences in the behavior of the two models. Firstly, the JB model produces a much shallower trend of spectral broadening with decreasing F for F < 0.85 than the ED model with Da ∈ {0.5, 1, 2}. This is consistent with JB's acknowledged “homogeneous evaporation bias.” It is also consistent with JB's model formulation which requires longer mixing times as F decreases, effectively decreases Da since longer times imply smaller λ(t) and stronger molecular mixing. Secondly, while the JB and ED model results shown are consistent for F > 0.85, the comparison highlights the inability of the JB model to explore mixing scenarios at large Damköhler number.

[52] The ED model results shown in Figure 1 also indicate that the model predictions are insensitive to the choice of boundary conditions. This behavior is consistent with a comparison of Jeffery and Reisner [2006b, Figures 2 and 3] which demonstrates that the diagnosis of τeddy using (14) is robust for different initial and boundary conditions. In the rest of this section, we show results using only the no-flux boundary conditions which best represent a single propagating and entraining cloud front.

[53] The impact of larger Da values on spectral width for ϕcrit = 0.58 is shown in Figure 2a, plotted against the normalized abscissa Fnorm ≡ 1 − ϕsubcrit. Figure 2a reveals two regimes of distinct Damköhler number dependence. For Da less than ≈5, σr increases with increasing Da as the system evolution becomes more front-like (less well mixed) and droplets experience a growing range of subsaturations. However for Da > 5, the trend reverses and σr actually decreases with increasing Da. This transition marks the onset of inhomogeneous mixing in which droplets at the cloud edge evaporate completely leaving behind a narrow spectrum composed of a fewer number of large drops. The ability of our ED model to predict this important transition from homogeneous to inhomogeneous mixing demonstrates that the inhomogeneous evaporation limit can be captured in a simple 1D modeling framework. In contrast, the JB model, which by its formulation is limited to small Da, does not reproduce the transition to inhomogeneous mixing.

Figure 2.

Calculated spectral width as a function of Fnorm predicted by the ED model for different Da: (a) σr(∀Ri > 0) and (b) σrall(∀Ri ≥ 0). Two groups of lines are shown: increasing line width (dashed line) corresponding to Da ∈{0.5, 0.8, 1.2, 1.9, 3}, respectively, and increasing line width (solid line) corresponding to Da ∈ {6, 12, 24, 49, 100}, respectively. Also shown in Figure 2b is the extreme inhomogeneous mixing behavior (dotted line) calculated from equation (15).

[54] While the results of Figure 2 are encouraging, the present prediction of inhomogeneous mixing also has a bias, stemming from weaknesses in the eddy-diffusive formulation of turbulent transport. The small-scale statistics of a passive scalar in a turbulent velocity field are highly anisotropic in the direction of a mean gradient, with the skewness of the scalar derivative, the third-order structure function and other higher-order and three-point moments exhibiting anomalous behavior [Mydlarski and Warhaft, 1998]. This anisotropy is inconsistent with eddy-diffusion and is caused by ramp-cliff structures of the scalar that align themselves with the mean gradient. The presence of sharp cliffs in subsaturation at the entrainment interface, combined with droplet sedimentation and inertial effects, should lead to a greater flux of droplets into subsaturated air, and hence greater evaporative homogeneity. We anticipate that the ED is thus somewhat biased toward inhomogeneous mixing, in contradistinction to the homogeneous bias of the JB model.

[55] The asymptotic approach of the present model to the “extreme inhomogeneous limit” [Baker et al., 1980] where σr → 0 as Da → ∞ is clearly revealed in the system-averaged spectral width σrall defined as the standard deviation of droplet radii averaged over all drops including those of zero size. This statistic is plotted as a function of Fnorm in Figure 2b for the Damköhler numbers shown in Figure 2a. In contrast to σr, σrall increases monotonically with increasing Da and appears to asymptote to a constant value as Da → ∞. This asymptote is the extreme inhomogeneous limit which we evaluate analytically.

[56] Consider the fractional decrease in liquid water density during mixing and evaporation, θ ≡ ρl(∞)/ρl(0). From the definition of 〈S〉crit, it is easy to show that 1 − θ = ϕsubSenv/〈S〉crit which, upon substitution into equation (11), gives

equation image

Note that θ cannot be reduced to a function of the single parameter Fnorm. The extreme inhomogeneous limit is, by definition, a decrease in droplet number with no change in droplet size such that the initial spectrum f(r) = δ(rr0) evolves to f(r) = (1 − θ)δ(0) + θδ(rr0). Thus σr = 0 and

equation image

The extreme inhomogeneous mixing line calculated from equation (15) is shown in Figure 2b and does appear to accurately describe the Da → ∞ limit of σrall.

[57] Increasing evaporation inhomogeneity is associated with a decrease in droplet number [Baker et al., 1980]. Figure 3a is reproduced from JB (their Figure 7) and shows the final system-averaged droplet number density, Nf, as a function of Fnorm predicted by the JB model for the same three experiments discussed previously. Because of dilution with the fraction ϕsub of entrained air, purely homogeneous mixing falls along the pure dilution line that intersects Nf = 0 at F = 0. Figure 3a shows that very little total droplet evaporation is produced by the JB model, which predicts Nf values that fall along the dilution line until ϕsub reaches ϕcrit. This behavior can be contrasted with the predictions of the ED model shown in Figure 3b for the same Da values used previously. The ED model exhibits a monotonic decrease in Nf with increasing Da that extends from homogeneous mixing along the pure dilution line at Da = 0.5 and asymptotically approaches the extreme inhomogeneous line which intersects Nf = 0 at Fs = 1 − ϕcrit.

Figure 3.

Comparison of the final, well-mixed droplet number density Nf as a function of F predicted by the (left) JB and (right) ED models. Figure 3 (left) is a reproduction of Figure 7 in JB with the symbols (crosses, pluses and asterisks) as per Figure 1. ED model results are for increasing Da values as indicated in the figure and with Da ∈ {0.5, 0.8, 1.2, 1.9, 3,6, 12, 24, 49, 100} as per Figure 2.

[58] The predictions of the ED model for the present experiments are summarized in Figure 4 which shows contour plots of σr (Figure 4a) and the fraction of not evaporated, i.e., finite size, droplets (Figure 4b) as a function of Fnorm and Da. These two contour plots reiterate the central features of the ED model results: (1) For 0.2 < Fnorm ≤ 1, σr is maximal near Da ≈ 5 which marks the transition from homogeneous (Da < 5) to inhomogeneous (Da > 5) evaporation, and (2) the fraction of not evaporated drops decreases monotonically with decreasing Fnorm and increasing Da. These two plots thus summarize the behavior of the ED model of section 4 which is an archetypal 1D formulation of cloud mixing and evaporation (eddy-diffusion is the de facto turbulent scalar closure) and Lagrangian droplet trajectories (Brownian motion is the de facto Lagrangian analog of eddy-diffusion). The results of Figure 4 therefore provide a reference behavior against which more sophisticated modeling approaches can be compared.

Figure 4.

Contour plots of (a) spectral width in μm and (b) the fraction of not evaporated droplets in the ED model as a function of Fnorm and Da. Also shown in Figure 4a is a contour of the maximum spectral dispersion for a given Fnorm (dashed line).

5.2. equation imageint and Droplet Size PDFs

[59] A plot of the probability density function (PDF) of the normalized Lagrangian subsaturation integral, equation imageint,i, is shown in Figure 5 where

equation image

and, as discussed previously, τreactτRHτR. The PDFs for five different mixing scenarios are plotted with F = 0.6 (Fnorm ≈ 0.31) held constant and Daequation image {0.44, 1.5, 6.2, 28, 96}. Figure 5 reveals a dramatic broadening of P(equation imageint) with increasing Da as expected from inhomogeneous evaporation where individual droplets experience drastically different subsaturated environments during mixing. Figure 5 also demonstrates that the equation imageint-PDF asymptotes to the analytic form P(equation imageint) ∼ equation imageint−1 as Da → ∞, first observed by Jeffery [2004] and Jeffery and Reisner [2006a] using a very different modeling approach.

Figure 5.

PDF of the normalized Lagrangian subsaturation integral equation imageint, defined in equation (16), predicted by the ED model at F = 0.6 (Fnorm ≈ 0.31) and for Daequation image {0.44, 1.5, 6.2, 28, 96}. The shape of the PDF at large Da agrees well with the theoretical estimate P(equation imageint) ∼ equation imageint−1 (squares); at small Da the PDF converges to a δ function at the single equation imageint value for a well-mixed system.

[60] This analytic behavior is easily explained by the linear nature of the microphysical source term in equation (2a). The PDF equation corresponding to nondimensionalized evolution equations of the form of (2a), i.e.,

equation image

is [Jeffery and Reisner, 2006b]

equation image

where τ = t/τeddy. Assuming that for small subsaturations at large Da, the integral radius conditioned on equation image, equation imageNequation imagerequation imagerequation imageequation image, is given by the initial in-cloud value, and hence is independent of equation image, then this equation reduces to

equation image

where the “PDF mixing” term is equation image(1) by our nondimensionalization. In the limit Da → ∞, the evolution of equation image is mixing limited [Jeffery and Reisner, 2006b] and hence the microphysical source term (2nd term on rhs of (17)) must also be equation image(1). This implies P(equation image) asymptotically approaches equation image−1 in the large Da limit. By the very nature of this limit, equation image(Xi, t) along the Lagrangian trajectory Xi is highly correlated in time since the system is far from the well-mixed state. Thus the statistics of equation imageint inherent the statistics of equation image as demonstrated by the P(equation imageint) ∼ equation imageint−1 behavior shown in Figure 5.

[61] Snapshots of Ssub(x) at the 〈Ssubxe-folding time (τefold), shown in Figure 6 and corresponding to the same five experiments used to generate the Figure 5 PDFs, confirm the transition from evaporation limited to mixing limited behavior as Da → ∞. For Da = 0.44, rapid mixing and entrainment of the environmental air increases Ssub, producing a uniform subsaturation and well-mixed cloudy air at t = τefold. In contrast, the mixing limited system evolution at Da = 96 is characterized by a sharp Ssub front that propagates across the domain.

Figure 6.

Snapshots of the subsaturation field, Ssub(x), at the 〈Ssubxe-folding time, τefold, predicted by the ED model for the same simulations shown in Figure 5. A transition from evaporation limited to mixing limited behavior is observed with increasing Da. Also shown is the initial Ssub profile indicating disjoint cloudy (left) and clear (right) air.

[62] A plot of the in-cloud droplet spectra, f(rr > 0), corresponding to the same five experiments used to generate the Figure 5 PDFs, are shown in Figure 7. To aid in presentation, the maximum value of f(r) is truncated for Da ∈{28, 96}. The homogeneous mixing value rh ≈ 10 μm is also indicated. A broad range of spectral shapes is observed with a spectral peak that decreases monotonically toward the homogeneous value as Da → 0. For example, at Da = 0.44 the spectral peak is shifted to a larger size of ≈10.5 μm while broadening to smaller sizes extends down to 8 m. In contrast, for Da ∈ {28, 96} the spectral peak asymptotes to the initial droplet size. Results shown for F = 0.7 by JB (their Figure 9) indicate that this asymptotic inhomogeneous mixing behavior is not predicted by their model.

Figure 7.

In-cloud droplet size spectrum, f(rr > 0), predicted by the ED model for the same simulations shown in Figure 5. The figure ordinate is truncated at 1.5 μm−1 to improve readability. Also shown is the homogeneous mixing radius rh. For Da ∈ 28, 96, arg maxrf is the initial droplet radius; this inhomogeneous mixing behavior is not reproduced by the JB model.

5.3. Narrow Γ Distribution

[63] To conclude this section, we shown results from the ED model for ϕcrit = 0.58 with an initial droplet spectrum described by a Γ distribution of shape parameter 14. These final simulations verify that the qualitative conclusions of the previous subsections are still valid with a broader initial droplet spectrum.

[64] The spectral broadening predicted with Γ-distributed droplets is shown in Figure 8 as a function of Fnorm, and can be directly compared with Figure 2a; identical Da values and line styles are used in both Figures 2a and 8. The pure homogeneous mixing line is also shown. Although the spectral dispersion is shifted to larger values, Figures 2a and 8 demonstrate qualitatively similar behavior. For the Γ-distributed droplets, the transition from homogeneous to inhomogeneous behavior occurs at Da ≈ 3 compared to Da ≈ 5 observed in Figure 2a. In contrast to Figure 2a, strongly inhomogeneous mixing produces σr values significantly less than purely homogeneous evaporation for Fnorm > 0.1.

Figure 8.

Calculated in-cloud spectral width σr(∀Ri > 0) as a function of Fnorm predicted by the ED model for Γ-distributed initial droplets with shape parameter 14. The Da values and line styles are identical to Figure 2a. Also shown is the spectral dispersion produced by purely homogeneous mixing (dotted line).

[65] These conclusions are reiterated in Figures 9a and 9b which show contour plots of σr and the fraction of not evaporated droplets for the Γ-distributed model, and which can be compared directly to Figures 4a and 4b. In particular, comparison of Figures 9b and 4b reveals a striking qualitative similarity, with the Γ-distributed model producing a consistently greater amount of total droplet evaporation for a given (Fnorm, Da).

Figure 9.

Contour plots of (a) spectral width in μm and (b) the fraction of not evaporated droplets as a function of Fnorm and Da predicted by the ED model for Γ-distributed initial droplets with shape parameter 14. Qualitatively similar behavior is seen in Figure 4 for a unimodal initial droplet distribution (infinite shape parameter).

6. Summary

[66] The study of cloud microphysics is confounded by the proliferation of Eulerian fields (temperature, density, moisture, velocity, droplet number concentration) and Lagrangian droplet variables (position, velocity, radius) that describe a cloudy system, even while ignoring aerosol physics and collision-coalescence. As recognized as early as Baker et al. [1980], determination of the relevant microphysical and dynamical timescales in the governing equations provides a means of classifying cloud evolution into different regimes of behavior. This approach works well for archetypal cloud entrainment geometries where the relevant timescales are easily identified. For less idealized mixing scenarios, however, the relevant timescales may be complex functions of space and time and are not uniquely determined.

[67] The mathematical formalism of scaling and invariance provides a straightforward way to identify transformations that produce self-similar (invariant) solutions of coupled equations. Four theorems are presented in section 2.2 that are relevant to idealized cloud studies where the proposed rescaling of cloud microphysics does not strongly feedback into the system dynamics; these theorems highlight the following features of isobaric cloud mixing and evaporation.

[68] 1. Invariant microphysical transformations can only exist when the ratio of the space-time-dependent droplet timescale τR(x, t) and the relative humidity timescale τRH(x, t) is everywhere held constant. We have identified a single constant that relates these timescales, the ratio 〈S〉crit/Smax where 〈S〉crit is the maximum average subsaturation that evaporates all of the initial cloud liquid water, and Smax is the maximum initial subsaturation.

[69] 2. For a straightforward linear scaling of space and time at fixed droplet size, theorem 3 provides the rescaling of the key Eulerian microphysical fields {ρv, T, N} that produces invariant cloud evolution. These solutions exhibit decreased number concentration with increasing cloud size. This theorem implies that the magnitudes of the quantities {ρv, T, N} are not germane in idealized cloud studies with buoyancy-independent transport. Instead, only the initial system geometry, the droplet spectrum and the ratio 〈S〉crit/Smax are relevant to model simulations and comparisons. The only drawback to this result is that Reynolds number is not preserved; that is, transport in a 1 m3 cloud system is assumed to scale with transport in a 1 km3 cloud system.

[70] 3. The Reynolds number–independent rescaling λ−1/2u(λ1/2x, λt) is of general relevance for cloud systems where the effect of sedimentation on droplet transport and the effect of accommodation on droplet growth are negligible. In this case, theorem 4(B) provides invariant solutions at fixed ρl that exhibit increased droplet size with increasing cloud size. For the special case that all droplets fall at their Stokes terminal velocity, theorem 4(C) provides invariant solutions that exhibit reduced ρl with increasing cloud size.

[71] For idealized cloud system geometries, constant timescales can be identified that characterize the microphysical evolution and the evaporation inhomogeneity. This line of investigation is initiated by Baker et al. [1980], and in JB the predictions of a simple 1D model of mixing and evaporation are presented for an idealized geometry where the uniform initial in-cloud timescales τRH and τR (no longer functions of time and space) are approximately equal. The mixing and evaporation experiments introduced by JB therefore provide a standard reference case where the microphysical evolution timescale, given by τRHτR, is uniquely defined.

[72] Motivated by the acknowledged “homogeneous evaporation bias” (JB, p. 2828) in JB's modeling approach, we present a new 1D model of mixing and evaporation that employs eddy-diffusive transport of relative humidity (RH) and the corresponding Brownian transport of Lagrangian droplets. Following Jeffery and Reisner [2006b], the eddy mixing time, τeddy, is defined adaptively as an a posteriori diagnosed measure of the time-averaged ratio of RH variance to the scalar dissipation rate. Given that τRHτR for JB's reference experiments the Damköhler number follows, without loss of generality, as Da = τeddy/τR.

[73] A comparison of the behavior of our new eddy-diffusive (ED) model with JB's results indicates that JB's model formulation severely limits their simulated Damköhler numbers to a narrow range of values around unity. Using the ED model to perform the first-ever exploration of the large Da limit in a simple 1D setting, we find evaporative behavior in good agreement with the qualitative phenomenology of Baker et al. [1980]. In particular, a transition is discovered in the behavior of the in-cloud spectral width (drop size standard deviation) at Da ≈ 5. Two distinct regimes of model behavior are observed: homogeneous evaporation (Da < 5) where spectral width increases with increasing Da, and inhomogeneous evaporation (Da > 5) with spectral width decreasing with increasing Da. This transition was not observed in JB's Da-limited model results.

[74] In the inhomogeneous mixing regime, the system evolution is characterized by a sharp RH front whose propagation speed is controlled by the purely diffusive evolution of droplet number density. Evaporation of droplets in the front is a source of RH that produces a dramatic front sharpening with increasing Da [Jeffery and Reisner, 2006b, Figure 7]. Droplets exposed to the subsaturated environment in the narrow frontal region evaporate while those behind the front remain unchanged. In the large Da limit, droplets at the cloud edge evaporate completely and are removed from the calculation of in-cloud spectral broadening, hence the observed decrease in spectral width with increasing Da. When completely evaporated droplets are included in the calculation of the spectral standard deviation, then this new measure of spectral width is seen to increase monotonically with increasing Da.

[75] An analysis of the probability distribution function of the time-integrated Lagrangian subsaturation, equation imageint, suggests a method to incorporate subsaturation inhomogeneity into a cloud model. As first observed by Jeffery [2004] and Jeffery and Reisner [2006a], and verified using the ED model in this study, the PDF of equation imageint, P(equation imageint), asymptotes to the analytic form P(equation imageint) ∼ equation imageint−1 as Da → ∞. This behavior results from the linear proportionality of evaporation and subsaturation; the Ssub−1 form exactly cancels the evaporation source term in the PDF equation for subsaturation Ssub in regions of uniform integral radius. The statistics of equation imageint inherit this analytic dependence since Lagrangian subsaturation histories at large Da are highly correlated in time.

[76] Since the shape of the equation imageint PDF is now known at large Da where subgrid Ssub fluctuations are important, a microphysical scheme could, in principle, be constructed that diagnoses the unresolved subsaturation distribution in each cloudy model grid cell. However, a number of hurdles prevent the straightforward development of such a scheme. Firstly, a complete theory of cloud mixing and evaporation is required that is valid over the whole range of τRH and τR values. The present study exploits the special case τRHτR such that a single Damköhler number characterizes the system evolution. A comprehensive investigation of the regimes τRHτR and τRHτR is required that must contend with separate Damköhler numbers for the τRH and τR timescales. Secondly, a parameterization for subgrid supersaturation requires a diagnostic scheme that provides an estimate of subsaturation variance as a function of subgrid cloud fraction and the relevant timescales τRH, τR and τeddy. We are currently pursuing both of these research objectives.


accommodation length.


heat capacity of air.


diffusivity of temperature.


diffusivity of water vapor.


latent heat of vaporization.


radius of the ith droplet.

〈⋯∣x, tr

average over droplet radius, r, at (x, t).


gas constant of water vapor.


relative humidity, ρv/ρs.


subsaturation, 1 − RH




assumed equal value of DT and Dv.


cloud liquid water density.


saturation vapor density.


water vapor density.


density of water.


[77] This work was funded by a Los Alamos National Laboratory Directed Research and Development Project entitled “Resolving the Aerosol-Climate-Water Puzzle (20050014DR).” We thank Nicole Jeffery for her comments on this manuscript.