Representing turbulent mixing of non-conservative values in Eulerian and Lagrangian cloud models



In many explicit microphysics cloud models the description of turbulent mixing of droplet size distributions (DSDs) treats them as conservative quantities, which leads to changes in DSDs for adiabatic profiles of liquid water. A new approach representing turbulent diffusion of DSD in Eulerian and Lagrangian cloud models is proposed. The approach is an extension of the classical K-theory (used for calculation of turbulent fluxes) to the mixing of non-conservative quantities. The proposed approach takes into account the growth/evaporation of drops as well as nucleation/denucleation in the course of the turbulent mixing. Implementation of the method is illustrated by the analysis of mixing between several different pairs of adjacent Lagrangian parcels: two cloudy parcels, a cloudy parcel with a drop-free one, and two drop-free parcels. The initial values of DSD and other parameters of the parcels are taken from simulations of stratocumulus clouds using the Lagrangian trajectory ensemble model. It is shown that taking into account the non-conservativity of DSD makes the rate and the results of mixing strongly dependent on the mutual locations of the parcels. The effects of mixing increase with the deviation of the vertical profile of liquid water content (LWC) from the adiabatic one. If the upper parcel contains a larger LWC, the effect of mixing on the DSD is significantly weaker than for lower LWC in the upper parcel. The mixing near cloud base and cloud top is more intense. The standard method underestimates the rate of mixing near cloud top and overestimates it near cloud base.

The proposed approach can serve as an efficient tool for investigating the role of mixing in Eulerian and Lagrangian models of stratocumulus and cumulus clouds. Copyright © 2010 Royal Meteorological Society

1. Introduction

The governing equations of most cloud models are obtained using the closure of the first or 1.5 order, according to which the fluxes of subgrid quantities are assumed to be proportional to gradients of mean values (the Prandtl mixing length concept). For subgrid scales corresponding to turbulent ones, the influx of the subgrid fluxes represents the rate of turbulent mixing. Using the Prandtl mixing length concept and the K-theory based on this concept (K being the turbulent coefficient), the mixing rate of a certain quantity a is described by applying operators like equation image in the conservation equations. Strictly speaking, the application of this operator suggests that ā is a conservative value, e.g. the potential temperature in a dry adiabatic process or the liquid water potential temperature in a moist adiabatic process. With this assumption, mixing takes into account the effect of subgrid processes in the models that have the resolution of a few tens to a few hundred metres (depending on the model resolution). This approach represents a parametrization of the net results of a complicated real mixing accompanied by formation of turbulent filaments of different scales and by other dynamic and thermodynamic processes. The purpose of this parametrization is to represent the general effects of small-scale motions on the averaged (model-resolved) fields. For instance, in the case of dry adiabatic processes, the application of such a mixing operator in the vertical direction should lead to a potential temperature that is constant with height.

The evolution of droplet size distribution (DSD) in Large Eddy Simulation (LES) models of stratocumulus clouds, as well as in models of convective clouds with explicit microphysics, is calculated by solving averaged kinetic equations. The turbulent mixing of DSD in such models is also represented by operator equation image (e.g. Takahashi, 1974; Hall, 1980; Khvorostyanov et al, 1989; Kogan, 1991; Feingold et al, 1994, 1996; Kogan et al, 1995; Khairoutdinov and Kogan, 1999; Yin et al, 2000, Stevens et al, 2005; etc.). This approach (that will be referred to hereafter as the standard one) does not take into account the non-conservative nature of DSD and may lead to significant physical errors. For instance, the 2 µm radius droplets near a cloud base at 40–50 m aloft may grow by diffusion to, say, 5 µm radius. If the drop size changes occurring during the turbulent mixing between these two layers are neglected, large drops (coming from above) will artificially appear near the cloud base, while small drops will appear at the distance of several tens of metres above cloud base (coming from below). Such an approach to mixing leads to an artificial broadening of the droplet spectrum and to an artificial acceleration of raindrop formation (see e.g. Kogan and Mazin, 1981; Khain et al, 2004). In fact, if the small asymmetry between the rates of drop growth and drop evaporation is neglected (the role of this symmetry was investigated in detail by Korolev (1995)), the vertical mixing between moist parcels that are adiabatic in all other respects, undiluted by environmental air, should not change the DSDs in these parcels.

Note that the non-conservativity of different quantities in the atmosphere manifests itself largely during movements (or mixing) in the vertical direction because temperature as well as pressure changes mostly in the vertical direction, and this is why condensation occurs during ascent (saturated vapour pressure is a function of temperature only). The mixing of a non-conservative quantity like temperature or a conservative quantity like potential temperature in the horizontal direction can be described by one and the same operator equation image.

The necessity to take into account the non-conservativity of DSD while describing turbulent mixing was well recognized in the theory of stochastic condensation, which attributes DSD broadening to turbulent fluctuations of supersaturation (Buikov, 1961, 1963; Levin and Sedunov, 1966; Sedunov, 1974; Mazin and Merkulovich, 2008). Studies by Cooper (1989) and Srivastava (1989) which analyse effects of fluctuating supersaturation on the development of DSD also belong to this line of investigation. Detailed derivations of the equations of stochastic condensation using Reynolds averaging and a careful evaluation of covariances (subgrid fluxes) were performed by Stepanov (1975), Clark (1976), Voloshchuk and Sedunov (1977), Manton (1979), Merkulovich and Stepanov (1977), and Khvorostyanov and Curry (1999a, 1999b). Usually, these covariances are described using the Prandtl mixing length concept and the K-theory based on this concept. Supersaturation and droplet growth rate are treated as stochastic variables and the operator of turbulent diffusion equation image is replaced by the operator equation image where A reflects the non-conservativity of DSD and depends on fluctuations of vertical velocity and supersaturation, as well as on drop size (e.g. Khvorostyanov and Curry, 1999a; Mazin and Merkulovich, 2008).

In most studies dedicated to the theory of stochastic condensation the non-conservativity of DSD leads to the fact that vertical mixing of cloud volumes containing adiabatic liquid water content (LWC) does change the DSD in the volumes. This conclusion is in obvious contradiction with the results that can be obtained by applying the operators equation image for turbulent fluxes or equation image for the influxes.

Using this approach, Khvorostyanov and Curry (1999b) offered an analytical solution of a somewhat simplified kinetic equation for DSD and obtained solutions of the gamma-distribution type.

Most conclusions of the theory of stochastic condensation are obtained under different simplification assumptions and have not been applied in the cloud models (with the exception of the study by Khvorostyanov and Curry (1999b), where a one-dimensional (1-D) model was used). Besides, the studies treating DSD as a non-conservative quantity do not take into account any droplet nucleation/denucleation processes that may occur during subgrid turbulent mixing. Here we refer to denucleation as the process of transition of droplets to haze particles during their evaporation. This simplification is also used in all cloud models with explicit microphysics.

While the Eulerian LES models use the K-theory for parametrization of turbulent mixing (even with the errors mentioned above), no attempts to mix Lagrangian parcels in the Trajectory Ensemble Models (TEM) (where a set of Lagrangian parcels move along air stream lines) have ever been performed (Stevens et al, 1996; Feingold et al, 1998; Harrington et al, 2000; Erlick et al, 2005; Khain et al, 2008; Pinsky et al, 2008; Magaritz et al, 2009). In the Lagrangian parcel model of the cloud-capped boundary layer (Magaritz et al, 2009), there are more than 1300 Lagrangian parcels moving within a turbulent–like flow that interact via droplets settling from one parcel onto adjacent parcels. In other TEM models, there is no interaction between parcels at all, which limits the utilization of the models by the analysis of the droplet diffusion growth alone.

In this study, we propose an approach to the turbulent mixing of DSDs that is conceptually close to the stochastic condensation theory, based on the Prandtl mixing length concept extended to the case of mixing of non-conservative values like DSD. Among the specific features of the proposed approach is (1) taking into account droplet nucleation/denucleation in the course of the turbulent mixing, and (2) taking into account the asymmetry in the rates of the diffusion-induced growth and evaporation of small droplets. This approach is applicable for describing in-cloud mixing, mixing between cloudy volumes and drop-free air volumes, as well as mixing between droplet-free volumes. The method is specially designed to be used in both Eulerian and Lagrangian cloud models.

2. Extension of the Prandtl mixing length approach

To clarify the extension issue, we will start with the original Prandtl mixing length concept and then consider several cases in order of increasing complexity.

Let us consider a background flow with non-uniform spatial distribution of different quantities. Turbulent motions are superimposed on the background flow, so that any quantity a can be represented as a sum of averaged (mean) value ā and turbulent fluctuations a′: a = ā + a′. The change of ā with time is determined by divergence of the turbulent fluxes. For the sake of simplicity, we consider the mixing along the vertical axis z:

equation image(1)

where w′ is the turbulent fluctuation of vertical velocity, and the overbar denotes the averaging.

To find the expressions for turbulent fluxes aw, we will follow the approach proposed by Prandtl (1925) with modifications suggested by Laikhtman (1976). The basic assumptions of the Prandtl approach are the following:

  • When a turbulent vortex arises, its thermodynamic quantities (temperature, humidity, velocity, etc.) are equal to the corresponding mean values in its surroundings.

  • While the vortex moves within the distance l, known as the mixing length (e.g. Stull, 1988), no mixing with the environment takes place.

  • Having moved beyond the mixing length, the vortex immediately mixes with its environment. The turbulent fluctuation is defined as the difference between the corresponding quantities in the vortex and its environment.

These assumptions allow one to determine the expressions for turbulent fluctuations and then for the turbulent fluxes. Although the concept may appear too simple, it is in fact the foundation of the surface and boundary-layer theories (Garratt, 1992), as well as the theory of turbulent jets and a number of other theories. Moreover, the above assumptions form the basis of the description of turbulent mixing in all cloud models and other numerical atmospheric models (with the exception of a few models using high-order closure). Yet, many studies have formulated requirements for a proper application of the Prandtl concept (e.g. Corrsin, 1974; Stull, 1988; Garratt, 1992). According to one of the requirements, the mixing length should be small enough to consider changes of the background fields at this distance as linear ones. This allows one to assume turbulent fluxes to be proportional to gradients of the background values, neglecting the curvatures of the flow on mixing-length spatial scales. As was shown by Khvorostyanov and Curry (1999a, 1999b), a proper utilization of these assumptions allows us to obtain important theoretical results as regards the formation of DSD shape caused by turbulent mixing. As mentioned in the handbook by Laikhtman (1976), the errors in description of each mixing event by means of Prandtl's concept are compensated by averaging over many mixing events.

Let us consider, in order of increasing complexity, several examples where the Prandtl mixing-length concept is applied to calculate turbulent fluxes.

2.1. Turbulent fluxes of conservative values

This mixing case is relevant, for instance, for the potential temperature fluxes in a dry adiabatic process.

The turbulent vortex forms at level z1 = z2l with the averaged quantity ā(z1) (point 1) and moves up reaching level z = z2, where it mixes with the environment. Since a is a conservative value, it does not change during the vortex ascent, and the fluctuation a′ at level z = z2 can be written as:

equation image(2)

where the second and the higher-order terms are omitted (Figure 1(a)). It is clear from Figure 1(b) that turbulent fluctuation at level z = z2, caused by a vortex descending from level z3 = z2 + l, is

equation image(3)
Figure 1.

(a) Formation of turbulent fluctuation at level z = z2 due to turbulent updraughts from level z1 = z2l. (b) Formation of turbulent fluctuations at level z = z2 due to turbulent updraughts from level z1 = z2l and downdraughts from level z3 = z2 + l. For mixing of conservative values, fluctuations in updraughts and in downdraughts are of the same absolute value but have opposite signs. Turbulent fluxes in updraughts and downdraughts are equal.

In (2) and (3), the signs + and − denote values in updraughts and downdraughts, respectively. Since in homogeneous turbulence w = 0, 1/2 of the area is covered by updraughts and 1/2 by downdraughts, thus the turbulent fluxes at level z = z2 can be written as:

equation image(4)

The turbulent flux formed by vortices arising at level z1 = z2l can be written for level z2 as equation image. The same expression describes the turbulent flux transported to level z2 by vortices located initially at level z3 = z2 + l. This is due to the fact that if w′< 0, fluctuations (3) have a different sign, i.e. the fluxes in turbulent updraughts and downdraughts have the same direction: in our case, downward. Hence, the total turbulent flux is

equation image(5)

where K = lw> 0 is the turbulent coefficient. Expression (5) represents the K-theory according to which the turbulent fluxes are proportional to gradients of the mean quantities.

2.2. Turbulent fluxes of a non-conservative value: the symmetric case (Figure 2)

In this case, value a in a moving vortex changes with height. The case is referred to as symmetric because the changes in value a in updraughts and downdraughts take place at the same rate. Such a situation arises, for instance, in the case of mixing of temperature in a dry adiabatic process, where the temperature in the ascending and descending air volumes changes according to the dry adiabat (γa = 9.8 °C/km).

Figure 2.

Scheme of turbulent fluctuations formation for mixing of non-conservative values.

The turbulent fluctuation in turbulent updraughts at level z2 is determined as

equation image(6)

where equation image is the individual derivative (the change per unit of length) related to the non-conservativity of a. The meaning of the term equation image is similar to the term equation image used in the stochastic condensational theory. As in the previous case, fluxes caused by updraughts and downdraughts are equal, so that the net turbulent flux of a can be calculated as

equation image(7)

For instance, if a is the temperature T and the process is dry adiabatic, equation image. Correspondingly, the temperature flux can be written as

equation image(8)

where θ is the potential temperature.

As a result, the equation determining the evolution of temperature can be written as:

equation image(8a)

where Qnc is a non-conservative source of T (for instance, latent heat release).

Expressions (8) and (8a) were used, for instance, in the classical handbook by Matveev (1984) for writing the equation for the atmospheric temperature.

2.3. Turbulent fluxes of non-conservative values: the asymmetric case (Figure 3)

In this case, value a in turbulent updraughts changes differently from that in downdraughts. Such a situation arises, for instance, in the case when updraughts take place along the moist adiabat, while downdraughts take place along the dry adiabat. In this case, turbulent velocity fluctuations at level z2, caused by turbulent updraughts and downdraughts, are different. Correspondingly, the turbulent fluxes in updraughts and downdraughts are different as well. The turbulent flux caused by the ascending vortices at level z = z2 is

equation image(9a)

and the turbulent flux caused by the descending vortices at level z2 is

equation image(9b)
Figure 3.

Scheme for determination of turbulent fluxes (and influxes) in the non-symmetric case. Fluctuation a′(z2)+ is formed at level z2 by turbulent vortices coming from level z1. Fluctuation a′(z2) is formed at level z2 by turbulent vortices coming from level z3.

The total flux at level z2 is equal to

equation image(10)

Note that (5) and (7) are particular cases of (10). Expressions (9a), (9b) and (10) reflect the fact that the fluxes in updraughts and downdraughts are different, thus the mixing procedure should take into account the direction of turbulent velocity fluctuations.

3. Parametrization expressions for turbulent mixing

3.1. Mixing on a regular finite-difference grid

Expression (10) for turbulent fluxes can be used in the Eulerian models with regular finite-difference grids. In this case, the standard box method can be used when the fluxes between the adjacent boxes are calculated, and the changes of quantities within a certain box are calculated by the divergence of the fluxes passing through the box boundaries. This approach is illustrated in Figure 4.

Figure 4.

Illustration of the extension of the Prandtl approach to the regular grid geometry. This figure is available in colour online at

The changes of quantity ā caused by turbulent fluxes (9) can be written as

equation image(11)

In (11), the fluxes are

equation image(12a)


equation image(12b)

3.2. Mixing of two parcels

The Prandtl concept described above can be applied to mixing between two parcels. The real boundaries between two adjacent air volumes within a turbulent flow have a very complicated structure containing many filaments of different scales. In this case, it is difficult to use the concept of the ‘interface’ between the mixing parcels to calculate the fluxes between them. Besides, in many Lagrangian models parcels are represented by points (parcel centres). To simplify the problem, we assume that the parcels to be mixed have the same volumes. Any changes caused by mixing will be attributed to the entire parcel volume.

The scheme of turbulent mixing of two adjacent air volumes is presented in Figure 5.

Figure 5.

Schemes showing the analogy of mixing between two parcels (panel (a)) and between air volumes located at two levels (panel (b)). Quantities in the parcels can be regarded as mean values, while the filaments can be considered as turbulent vortices moving from one level to another. DSD1(1) and DSD2(2) are the drop size distributions in the centres of the corresponding parcels. DSD1(2) is the drop size distribution of the filament of parcel 1 in the centre of parcel 2. DSD2(1) is the drop size distribution of the filament of parcel 2 in the centre of parcel 1. This figure is available in colour online at

Let us assume that, during a turbulent diffusion process, filaments of Parcel 1 (P1) penetrate Parcel 2 (P2) and vice versa (Figure 5(a)). The quantities in the parcels play the role of the surroundings with respect to the quantities in the filaments. Thus, the values within parcels are treated as mean values, and the differences between the values in filaments and those in the parcel can be considered to be turbulent fluctuations in a way similar to that described above. The analogy of this scheme to the Prandtl approach is illustrated in Figure 5(b).

We parametrize mixing by considering the fluxes between the parcel centres to be similar to those accepted in the K-theory. The maximum impact on mixing is caused by vortices of the maximum size, which, in our case, typically have the scale equal to the distance between the parcel centres. Hence, we assume that the mixing length is the distance between the parcel centres (the characteristic length of turbulent filaments is of the order of the mixing length), which is also equal to a parcel's typical size. These considerations are valid, at least, if the distance between the parcel centres falls within the inertial turbulent range. Motions with scales larger than the distance between the parcels transport both parcels together (the process of advection) and do not contribute to their turbulent mixing. This definition of the mixing length is widely used in Eulerian atmospheric models, where the mixing length is assumed equal to the distance between neighbouring grid points (e.g. Skamarock et al, 2005).

Therefore, we apply the approach discussed above using the analogy with turbulent mixing of quantities between two layers (illustrated both in Figure 5(b) and Figure 6). In spite of the fact that the interfaces are fractal, we assume that the effects of turbulent mixing related to the fractal features of the interfaces are parametrized by the turbulent coefficients.

Figure 6.

Scheme of fluxes between two adjacent parcels. Filaments belonging to parcel 1 penetrate parcel 2 creating fluxes F+(1) and F+(2). F+(1) generally is not equal to F+(2). Filaments belonging to parcel 2 create fluxes F(2) and F(1). Indices denote the direction of the flux, while numbers (1) and (2) denote parcels playing the role of the surroundings. This figure is available in colour online at

Note that each parcel can be surrounded by several other parcels. Hence, the flux divergence shown in Figure 4 can be calculated as a result of successive mixing in all the parcel pairs.

The changes in time of non-conservative quantities a within each parcel are determined by the influxes within the corresponding parcels:

equation image(13)


equation image(14)

In (13) and (14), l is assumed to be the distance between the parcel centres, as discussed above.

In our case, F+(1) = 0 andF(2) = 0 because turbulent fluctuations at the point of vortex formation are equal to zero (see the Prandtl hypothesis). In this case:

equation image(15)

We assume that mixing takes place in isotropic turbulence within the inertial subrange, where K(l) = Cε1/3l4/3 (the Richardson law); ε is the turbulent kinetic energy dissipation rate and C = 0.2(Monin and Yaglom, 1975). Solutions of (15) that are valid for the general case (with both conservative and non-conservative values) are presented in appendix A.

In the non-conservative case, the total mass changes with time as shown in appendix A.

The final expressions for conservative values are:

equation image(16)

where equation image and equation image represent the initial conditions and equation image is the characteristic mixing time. In the case when F(1) = − F+(2) (the gain in P1 is equal to the loss in P2), we have

equation image(17)

which expresses the conservation of total quantities (integrated over both parcels).

In particular, in the limit case of a very small separation distance between the parcels (l→0), the solution (15) yields full mixing: τmix→0 and equation image. The opposite limit case is the absence of any mixing when the separation distance between the parcels is larger (l→∞):τmix→∞ and equation image, equation image.

In the general case, solutions (16) and (17) are not valid (for instance, if mixing is accompanied by a net heat release).

A comment on the proposed procedure is required here. Similarly to the Prandtl concept (and the K-theory based on this concept) which calculates turbulent fluxes between two levels (or between two neighbouring grid points in LES models), we calculate fluxes between the parcels without considering any single mixing event. Figures and formulas illustrate the ways of the calculation of these fluxes which exist during all the period of parcel mixing. These fluxes obviously change with time due to changes of thermodynamical values and the distance between parcels etc.

4. The algorithm of the mixing procedure

4.1. Conservative and non-conservative values

The process of mixing can be roughly separated into two processes: the process of mutual penetration of air volumes and the process of their mixing per se when the volumes are in close vicinity. In this paper, we consider mixing in adiabatic processes (dry and moist). It means that quantities such as temperature, humidity, DSD and LWC change within air volumes when these volumes move prior to the mixing itself. It means that all these quantities are non-conservative.

During turbulent mixing that can be accompanied by either condensation or evaporation of drops, two variables can be considered as conservative: the total moisture content equation image (where q is the water vapour mixing ratio, ql is LWC in kg/m3 and ρa is the air density) and the liquid potential temperature equation image, where L is specific latent heat release and cp is the air thermal capacity under constant pressure. These values are used as variables in many atmospheric models. Conservation of the total moisture content is due to the mass conservation law, and conservation of the liquid potential temperature is due to the energy conservation law, when the latent heat taken into account.

The algorithm of mixing is developed in this study to be implemented in cloud models, for example, in the Lagrangian model of stratocumulus clouds (Pinsky et al, 2008; Magaritz et al, 2009), where collisions and drop sedimentation are usually treated separately. Therefore, we neglect the effects of drop collisions and sedimentation on DSD during the mixing. As a result, the total particle concentration (of both aerosols and droplets) is considered as a conservative value.

Equation (16) allows one to calculate the total moisture content and the liquid potential temperature in each parcel. To calculate q, ql and θ (or T) in each parcel, it is necessary first to recalculate the DSD that changes in the course of the mixing process.

4.2. Mixing of DSD

We define DSD in each parcel using a mass grid containing m bins. In view of the considerations discussed above, in order to calculate the DSD in P1 after mixing we need to mix the DSD taken in the centre of P1 (denoted as DSD1(1)) with the DSD belonging to the filament of P2 that has reached the centre of P1 (denoted as DSD2(1)). Similarly, to calculate the DSD in P2 after mixing, we need to mix DSD2(2) which is the DSD in the centre of P2 with the DSD of P1 filament that has reached the centre of P2 (see Figure 5). The problem is to calculate DSD1(2) and DSD2(1), i.e. size distributions in the two parcel filaments. Since the filaments are formed due to turbulent velocity fluctuations superimposed on the ‘mean’ parcel motion velocities (the motion of the parcel centre), the problem is to calculate variations in the drop radii caused by these turbulent velocity fluctuations. To perform such calculations means to determine the variations of drop radii δri (i is the number of size bin), of supersaturation δS and of other parameters, these variations being functions of the distance from a parcel centre. The calculations are performed using the diffusion growth/evaporation equation and the equation for supersaturation. The algorithm of the calculation is presented in appendix B. In particular, variations of LWC can be calculated using the expression:

equation image(18)

where coefficients a1, a2, ci are given in appendix B, ρw is the water density, ni is particle concentration in the ith bin and δql is the deviation of the LWC in the parcel filament that has reached the centre of the counterpart parcel, from the LWC in the centre of ‘the mother parcel’. Expression (18) indicates linear changes in LWC variation with height. This result corresponds well with the linear profile of LWC often observed in stratocumulus clouds.

In the case when LWC is comparatively large, the unity in the denominator of (18) can be neglected, and the formula for variation of the adiabatic LWC height profile becomes as follows:

equation image(19)

Equation (19) follows from the equation for supersaturation (B1) in neglecting the time derivative of supersaturation:

equation image(20)

where W is the vertical velocity. Equation (20) means that an increase in supersaturation caused by vertical motion is fully compensated by the supersaturation loss due to condensation of water vapour on droplets. In this case the changes in LWC do not depend on the DSD shape. Expressions (19) and (20) are valid within clouds well above the supersaturation maximum located near the lifting condensation level.

In cases when LWC is low (like near cloud base or in droplet-free air volumes), the second term in the denominator of (18) can be neglected as compared to the unity, and the formula for variation of the adiabatic LWC becomes:

equation image(21)

Since droplet radii grow with height in updraughts, (21) shows that LWC grows with height nonlinearly, significantly faster than δz. The variation of LWC depends on the DSD shape. This situation takes place near the lifting condensation level and in droplet-free air volumes. Equation (21) takes into account effects of curvature and chemical composition of particles on their growth and corresponds to the changes in the amount of water within particles according to the Kohler theory.

Note that supersaturation within clouds is often evaluated using the equilibrium value equation image, where is the mean drop radius and N is droplet concentration (e.g. Cooper, 1989; Korolev and Mazin, 2003). This expression can be obtained from the equations for supersaturation (20) and for droplet diffusional growth (see (B4) in appendix B) in neglecting the curvature and chemistry terms. These simplifications mean that the equation for equilibrium supersaturation is not applicable for haze particles and the smallest droplets and cannot be used at the cloud boundaries and in droplet-free air volumes, where these terms are important. At the same time, Equation (18) obtained using the full equations for supersaturation and diffusional growth is suitable for any condition.

Note that the proposed approach (as described in appendix B) takes into account some fine effects that are not described in (19). For instance, it takes into account the asymmetry in DSD changes during the ascending and descending of the parcels (Korolev, 1995), as well as during droplet nucleation or evaporation. These effects can be important for explanation of DSD evolution. In addition, our approach allows one to take into account both the effect of diffusion growth/evaporation during mixing and the transformation of aerosols into drops and of drops into wet aerosols (nucleation/denucleation).

The DSD in the filaments are calculated by the shift of the ith bin of the initial DSD by δri.

In a simplified form, the procedure of DSD mixing can be illustrated as shown in Figure 7.

Figure 7.

A simplified scheme of the DSD mixing procedure. This figure is available in colour online at

Mixing between parcels P1 and P2 is performed in two stages. At the first stage, the DSD in parcel P1′ is calculated, P1′ being interpreted as a filament of parcel P1 located at the same level as that of parcel P2. This DSD is obtained during the ascent of parcel P1 by the distance of δz.

DSDs in parcels P2 and P1′ are mixed in the horizontal direction. Since parcels P2 and P1′ are located at the same level, this mixing is performed as outlined by the standard method (as conservative values). The entire non-conservativity of DSD is represented in changes of DSD in parcel P1 during its shift upward by δz. Note that (19) describes the adiabatic increase in LWC during the shift by δz. In the case that the vertical profile of LWC is adiabatic, the DSD in parcel P1′ turns out to be similar to that in P2. It means that the mixing (the second step) of DSDs of parcels P2 and P1′ will not lead to any changes in DSD. Thus, our approach reflects the fact that turbulent mixing within a cloud with adiabatic LWC does not lead to changes in the DSDs. This result was emphasized earlier in the stochastic condensation theory.

4.3. Steps of the algorithm

In summary, the algorithm of mixing consists of the following steps:

  • (1) Calculation of characteristic mixing time as equation image.
  • (2) Mixing of total water content equation image and liquid potential temperature equation image using (16).
  • (3) Calculation of variations of δql and δri for each of the mixing parcels using expressions (B11) and (B12) or full equations (B5)–(B10). The DSD in filaments is calculated by the change of the radius of the ith bin of the initial DSD by δri. Hence, at this step the non-conservative nature of DSD is taken into account.
  • (4) Remapping DSD in parcel filaments on the initial (regular) size grid, on which the DSDs of drops in the parcels are defined. As a result of the remapping, the DSDs in a filament and its surrounding parcels turn out to be determined on the same size grid. The remapping is performed using the Kovetz and Olund (1969) scheme conserving both concentration and mass. The testing we performed shows that the errors in DSD caused by remapping are negligibly small as compared to the changes of DSD caused by mixing.
  • (5) Mixing of DSD in parcels and filaments. Since DSD of a parcel and DSD of its filament are calculated at the same spatial point and are represented (after remapping) on the same size grid, they can be mixed bin-by-bin, i.e. mixing of particles belonging to a bin of the same size can be performed. Accordingly, mixing in each bin is carried out using (16).
  • (6) Determination of ql after mixing in each parcel by integrating of particle masses over all the bins.
  • (7) Calculation of the water vapour mixing ratio and the potential temperature in each parcel as
    equation image

For the Eulerian framework, the mixing procedure is similar to that used for the Lagrangian parcels, with some simplifications: namely, the mixing length is assumed equal to the distance between adjacent grid points (which play the role of parcel centres) as illustrated in Figure 4. The number of neighbours in the Eulerian framework is also given: two in the 1-D case, four in the 2-D case and six in the 3-D case.

5. Examples of parcel mixing

In order to illustrate the changes in the microphysics of parcels caused by mixing, we present several idealized examples of mixing of two adjacent parcels. The initial values of temperature, mixing ratio, liquid water content, and drop and aerosol size distributions were taken from the Lagrangian model of stratocumulus clouds (Pinsky et al, 2008; Magaritz et al, 2009) that was used for simulations of clouds observed during the research flight RF07 in the DYCOMS-2 experiment. As was shown by Pinsky et al(2008), the values of microphysical and thermodynamical quantities, as well as of DSD, produced by the model are quite realistic.

Both wet aerosols (non activated haze particles) and droplets are described in the model (and in the examples) as having the same size distribution and are defined on the same mass grid. These size distributions are calculated on a logarithmic size grid containing 500 bins covering radii ranging from 0.01 µm to 1000 µm. The process of droplet nucleation is treated explicitly by solving the equation of diffusion growth for all particles including non-activated aerosols. The separation in the radius space between non-activated aerosols and nucleated droplets arises automatically as a result of solving the equation for diffusion droplet growth: small non–activated wet aerosols are in equilibrium with the environment and do not grow if supersaturation does not change, while the size of nucleated droplets increases under any positive supersaturation. The mixing is performed in combination with diffusion growth/evaporation in each parcel in order to take into account the feedback effect of the mixing-induced changes of supersaturation on DSD. Calculations of the diffusion growth and mixing are performed using time steps of 0.01 s and 1 s, respectively. The dissipation rate of turbulent kinetic energy is chosen to be ε = 10 cm2 s−3, which is typical of stratocumulus clouds.

To illustrate the difference between the mixing algorithm and the standard one, we compared the rates of DSD changes in adjacent parcels calculated by both methods. The rate of the DSD changes are calculated as equation image, where ni is particle concentration in the ith bin. The rates are calculated for short one-second periods of time during which the parcels can be considered to be adjacent. The net statistical effects of mixing will be investigated in detail using the stratocumulus cloud model with the mixing algorithm included.

Three examples are considered here (sections 5.15.3).

5.1. Mixing between two supersaturated cloudy parcels with different but comparatively wide DSDs

The purpose of this example is to evaluate the rates of DSD variations caused by mixing between two cloudy parcels that are typical of cloud updraughts. This case will be referred to as ‘in-cloud mixing’. Parcel P2 is located 50 m above Parcel P1; LWCs in parcels P1 and P2 are equal to 1.6 gm−3 and 1.76 gm−3, respectively. DSDs in the parcels are shown in Figure 8. The values of LWC in these parcels are comparatively close to the adiabatic values. Accordingly, the rates of DSD variation caused by mixing were calculated using the algorithm proposed in the study and were found to be lower than those calculated using the standard method (Figure 8, panels in the lower row). An especially strong difference in the mixing rates is seen in parcel P1 with a lower LWC. However, as shown below, the rates of DSD changes due to turbulent mixing between these parcels within the cloud layer are significantly lower than changes at the cloud boundaries.

Figure 8.

Example of ‘in-cloud’ mixing. Droplet size distributions are presented in the upper panels, the mixing-induced rates of DSD changes in two cloudy parcels are shown in the lower panels.

5.2. Mixing near cloud boundaries

The purpose of these examples is to demonstrate the effects of mixing between cloudy parcels and their unsaturated environment. In these examples, the rates of DSD variation are calculated in the course of mixing of the two parcels, one of which is a supersaturated cloudy parcel with LWC = 0.31 gm−3 (P1), and the second one is an unsaturated droplet-free parcel (P2) with LWC = 3·10−4 gm−3. Two cases are considered: (1) In the first case, P2 was located 50 m above P1. This kind of mixing can take place between cloudy air and dry air located above the cloud top. This simulation is referred to as ‘cloud top’. (2) In the second case, P2 is located 50 m below P1. This kind of mixing can take place between cloudy air and dry air located below the cloud base and is hereafter referred to as ‘cloud base’. The DSDs in these parcels, as well as the rates of their changes in the ‘cloud top’ and ‘cloud base’ simulations, are presented in Figure 9. Note that in cloud boundary mixing the rate of DSD variations is one–two orders of magnitude higher than in the case of ‘in-cloud mixing’. The substantial difference in the DSDs rates in the ‘cloud top’ and the ‘cloud base’ cases, on the one hand, and the ‘standard’ case, on the other hand, takes place only in an unsaturated parcel with a negligible LWC. In the ‘cloud top’ case the undersaturated parcel obtains droplets with sizes larger than those obtained by the ‘standard’ method (Figure 9, right bottom panel). This is because the filaments of cloudy parcel P1 that penetrated P2 are located higher than the P1 centre and, consequently, contain larger droplets. In the ‘cloud base’ case, the undersaturated parcel obtains droplets with sizes smaller than those obtained by the ‘standard’ method. This is because the filaments of cloudy parcel P1 are located lower than the parcel centre and, consequently, contain smaller droplets that penetrate P2. The differences in the rates of the DSD changes indicate that unsaturated parcels located above the cloud top increase their humidity faster than those located below the cloud base. Mixing near the cloud top turns out to be more intense than mixing near the cloud base (for the same turbulent intensity) because near the cloud top the deviation of the vertical profile of LWC. from the adiabatic one is the maximal. Continuation of the mixing procedure for up to 5–10 min and the corresponding recalculation of all the thermodynamic parameters showed that undersaturation in P2 is replaced by supersaturation and nucleation of new droplets. The specific feature of the standard method is that it is insensitive to the mutual location of mixing parcels, and thus underestimates the effect of turbulent mixing near the cloud top and overestimates this effect near the cloud base.

Figure 9.

Droplet size distributions (upper row) and the mixing-induced rates of the DSD changes (lower row) in the ‘cloud top’ and the ‘cloud base’ examples.

As regards the cloudy parcel, its mixing with droplet-free parcels leads to its dilution. The maximum decrease in DSD takes place in the drop size range where DSD reaches its maximum. This size range does not depend on the DSD in the counterpart parcels, because the latter do not contain droplets. The rate of dilution depends on the differences in the thermodynamic parameters of the mixing parcels, on the intensity of turbulence and on the distance between the parcels.

5.3. Mixing of two undersaturated droplet-free parcels

This type of mixing can take place below the cloud base or in droplet–free zones within a cloud layer. In the example, both parcels contain only haze (wet aerosols). The values of LWC in parcels P1 and P2 are 2.4·10−5 gm−3 and 1.4·10−5 gm−3, respectively; P2 is located 50 m above P1. Figure 10 shows both distributions and the rates of aerosol size distribution changes in the parcels due to mixing. One can see that the method proposed in the study leads to faster changes in the aerosol size distributions because it takes into account adaptation of size of wet aerosol parcels to their equilibrium size, while the standard method does not take this effect into account.

Figure 10.

Droplet size distributions (upper row) and the mixing-induced rates of the DSD changes (lower row) in the ‘haze-mixing’ example.

In a supplementary simulation, isobaric mixing discussed by Gerber (1991) and Korolev and Isaac (2000) was successfully simulated, when mixing of two undersaturated parcels having substantially different initial temperatures leads to formation of supersaturation accompanied by nucleation of droplets in a parcel having initially higher temperature.

5.4. Effect on structure of stratiform cloud

At the end of this section we would like to illustrate the accumulated effect of the mixing on the structure of stratiform cloud. For this purpose, the procedure of mixing between Lagrangian parcels described in the present study was implemented into the Lagrangian trajectory ensemble model of the boundary layer (Pinsky et al, 2008; Magaritz et al, 2009). The fields of LWC simulated by the model in cases when mixing between parcels is not taken into account, as well as when turbulent mixing is included, are shown in Figure 11. The dramatic effect of the mixing on the cloud geometrical structure is clearly seen. For instance, when the mixing effects are taken into account, the fluctuations on the level of cloud base are much lower, which brings the model cloud structure closer to that of real clouds. The horizontal variations of LWC also decrease in the presence of mixing between parcels. At the same time, the values of the horizontally averaged values of LWC within a cloud layer remain nearly the same.

Figure 11.

Fields of LWC simulated in the Lagrangian cloud model (a) in cases of no turbulent mixing (upper panels) and (b) in the presence of turbulent mixing (lower panels). Vertical profiles of horizontally averaged LWC and root-mean-square of LWC are also presented.

6. Discussion and conclusions

A new approach for parametrization of turbulent mixing of DSD in different Eulerian and Lagrangian models is proposed. The method represents an extension of the classical Prandtl mixing length concept (i.e. the K-theory), used for calculation of turbulent fluxes and flux divergence, to mixing of non-conservative quantities. Conceptually, the approach is close to that used in the theory of stochastic condensation. Both approaches use the Prandtl mixing length concept to calculate turbulent fluxes and take into account the non-conservative nature of DSD as well as phase transitions in the course of mixing. In agreement with the conclusions of the theory of stochastic condensation, the proposed method predicts no changes in DSD in cases when these DSD correspond to adiabatic LWC profiles (when the asymmetry between the diffusion-inflicted growth and evaporation is neglected). At the same time, the proposed approach has a number of specific features, namely:

  • It takes into account not only growth/evaporation of drops but also nucleation/denucleation of droplets during the turbulent mixing. The standard treatment of DSDs as conservative values (as well as the studies dealing with the stochastic condensation theory) does not take these effects into account. We, however, suppose that this effect is quite important and allows one to break up the linearity between fluctuations of supersaturation and the vertical velocity, which is one of the most important aspects in the problem of DSD broadening at the stage of diffusion growth (see, for instance, the review by Khain et al(2000)).

  • It considers the process of DSD evolution in the course of the mixing using the Lagrangian approach that enables us to simplify the calculation of droplet fluxes and to avoid explicit differentiation of DSD with respect to drop radii, unavoidable in the Eulerian approach.

  • It allows one to mix DSDs of complicated shapes such as bimodal and multimodal spectra, DSDs with droplet gaps, etc., as well as the spectra of non-activated wet aerosols.

  • It describes turbulent mixing under different thermodynamical conditions (in-cloud mixing, mixing between cloudy and dry air, mixing between two dry-air volumes) using the same equation system.

  • It provides a closure for the equation system used for calculation of thermodynamic characteristics during mixing accompanied with phase transitions and latent heat release.

  • It applies the turbulent coefficient that depends on scales according to the Richardson law.

  • It is able to reproduce some fine effects such as isobaric mixing.

  • It is specially designed to be applied in cloud models within a Lagrangian framework.

The method is illustrated using idealized examples of mixing (for short one-second time periods) of several different pairs of adjacent Lagrangian parcels. The initial DSD and other initial parameters of the parcels are taken from simulations of stratocumulus clouds using the Lagrangian trajectory ensemble model.

The rates of the DSD changes caused by mixing were calculated. It has been shown that:

  • (1) The effects caused by mixing increase as the deviation of LWC profile from the adiabatic one increases. The standard treatment of DSD as a conservative value leads to changes in DSD resulting in an artificial DSD broadening;
  • (2) The first conclusion means that the rate of DSD change depends on the mutual location of parcels (namely, whether LWC in the upper parcel is higher or lower than LWC in the lower one). If the upper parcel contains a larger LWC, the effect of mixing on DSD in both parcels is weaker than in the case when the parcel located above contains a smaller LWC. In the case of the standard treatment of DSD, the results of mixing do not depend on the mutual location of the mixing parcels.
  • (3) Mixing-induced changes in the DSD shape in the adjacent cloudy parcels with comparatively large LWC are comparatively small, being one–two orders of magnitude lower than those near cloud boundaries.
  • (4) Mixing near the cloud top is more efficient than mixing near the cloud base (at the same turbulent intensity rates). The standard approach underestimates the effects of mixing near the cloud top and overestimates those effects near the cloud base.
  • (5) The isobaric mixing between two undersaturated parcels was simulated. This mixing leads to formation of supersaturation and new drop nucleation in case of mixing of initially undersaturated air volumes. The parcel with initially higher temperature becomes a cloudy one. The standard treatment of DSDs as conservative values (typical also of studies developing the stochastic condensation theory) does not allow achieving nucleation during the mixing process.

The process of mixing between Lagrangian parcels described in the present study was implemented in the Lagrangian trajectory ensemble model of the boundary layer (Pinsky et al, 2008; Magaritz et al, 2009). A dramatic effect of the mixing on the cloud's geometrical structure was found. The effects of DSD mixing on cloud microphysics and drizzle formation will be investigated in detail using a statistical analysis of the results obtained with the Lagrangian trajectory ensemble model of a cloud-covered boundary layer. This is the topic of future study.


The study is supported by the Israel Science Foundation (grants 950/07 and 140/07). We express deep gratitude to reviewers for valuable comments and remarks.

Appendix A. Time Evolution of Non-Conservative Quantities during Mixing

We assume that mixing takes place in isotropic turbulence within the inertial subrange where K(l) = Cε1/3l4/3(Richardson's four-thirds law) and C is a constant of about 0.2 (Monin and Yaglom, 1975). In this case, conservation equations (14) for two adjacent parcels can be written as

equation image(A1)

where equation image is the characteristic mixing time corresponding to mixing length l, equation image and equation image. Summing and subtracting Eqs. (A1), one can obtain

equation image(A2)

(A2) have the following solution, obeying the initial conditions equation image and equation image:

equation image(A3)
equation image(A4)

Equation (A3) describes the time evolution of the total concentration of non-conservative quantities in mixed parcels in the asymmetric case. For a conservative quantity (equation image), (A3) gives conservation of the total concentration equation image.

The combination of (A3) and (A4) gives a solution of (A1)

equation image(A5)

In particular limit cases, solution (A5) provides full mixing if l→0, τmix→0 and equation image or lack of mixing if l→∞, τmix→∞ and equation image, equation image.

Appendix B. A Method for Calculating DSD and LWC

1. Governing equations

  • The equation of supersaturation S in adiabatic updraught/downdraught has the form (Prupacher and Klett, 1997):

    equation image(B1)

where W is the updraught velocity, a1 and a2 are coefficients slightly dependent on temperature, ρa is air density and ql is LWC. Integration of (B1) with respect to time gives the following equation

equation image(B1′)

where z is the vertical coordinate.

  • Drop size distribution is introduced using radius bin grid ri, i = 1…m (non-uniform) with the concentration in each bin ni(ri) and the radius of aerosol kernel in each bin rai. The normalization condition is

    equation image(B2)

where N is drop concentration. LWC in our definition can be written as:

equation image(B3)

where ρw is water density.

  • In our analysis, we use the condensation growth equation (Pruppacher and Klett, 1997)

    equation image(B4)

where A, B and F are the coefficients slightly dependent on temperature.

2. Equations for variations

Let us introduce linear variations of quantities δW, δz, δS, δql, equation image. From (B1′), (B2) and (B4), one obtains the equations for variations

equation image(B5)
equation image(B6)
equation image(B7)

From (B5)–(B7), one obtains a linear differential equation with respect to δξi

equation image(B8)


equation image(B9)

The solution of (B8) with the initial condition δξi(0) = 0 is

equation image(B10)

Equation (B10) together with (B6) can be used for linear prediction of DSD and LWC. The value of time t can be evaluated if one assumes the vertical velocity variation δW to be created by turbulent fluctuation within the inertial subrange for isotropic turbulence equation image, therefore equation image. There are two cases of application of (B10). When τi > 0, which corresponds to the case when the drop radius is smaller than the critical radius rcr: equation image, (B10) gives a good linear prediction. When ri > ricr, the growth/evaporation of droplets is fast and it is impossible to apply (B10). However, in this case the second and the third terms on the right-hand side of (B4) are small and can be omitted. Thus, we come to the following equation:

equation image(B11)

With small variations δξi ≪ ξi, it is possible to obtain an analytical solution of LWC variation. Multiplying the left and the right side of (B10) by 2πρwniri and summing both sides with respect to index i, one obtains the equation for LWC variation:

equation image(B12)

where coefficients ci are calculated as:

equation image(B13)

Since δz = δlcosψ, where ψ is the angle between the line connecting the centres of the two parcels and the vertical direction (see Figure 6), the individual derivative used in (15) is equal to

equation image(B14)