## 1. Introduction

Many atmospheric processes 1) can be represented in a numerical model as nonlinear functions of model-predicted variables and 2) are too small to be resolved by the model. Prior authors have noted that such nonlinear, small-scale processes pose a special difficulty for numerical models (e.g., Kogan 1998; Stevens et al. 1998; Stevens et al. 1996; Fowler et al. 1996; Fowler and Randall 1996). Namely, a model errs if it substitutes grid box average values into the nonlinear function corresponding to the small-scale process. To avoid such “averaging errors,” the model needs information about subgrid variability.

Prior authors have pointed out several undesirable consequences of ignoring subgrid variability. For example, Fowler et al. (1996) and Fowler and Randall (1996) parameterized autoconversion of cloud droplets to raindrops in a general circulation model by implementing an autoconversion parameterization developed for models with much smaller grid boxes. To obtain reasonable simulations, they were forced to reduce the threshold in specific liquid water content, *q*_{l}, at which autoconversion begins. The problem, they suggested, was that the grid box average of *q*_{l} may be much smaller than *q*_{l} in the localized areas where precipitation forms. Kogan (1998) performed a large eddy simulation of drizzling stratocumulus and then compared domain-averaged autoconversion, as calculated by the formula of Khairoutdinov and Kogan (2000), with the result of substituting domain-averaged quantities into the Khairoutdinov–Kogan formula. The two results differed markedly. Stevens et al. (1998) hypothesized that neglecting correlation terms in Reynolds-averaged microphysical equations causes some one-dimensional models to spuriously activate cloud droplets at the tops of stratocumulus layers; these authors also used a sophisticated bin-microphysics model to study the influence of averaging errors on computations of drizzle formation. Stevens et al. (1996) suggested that ignoring subgrid variability can lead to spurious supersaturation at cloud edges.

Prior authors have also discussed biases. Here, we define a bias to be an error that always has the same sign. Cahalan et al. (1994) pointed out that ignoring subgrid variability in computations of cloud albedo can lead to the so-called plane-parallel albedo bias. They noted that the bias is a consequence of a theorem known as Jensen’s inequality (Jensen 1906). The origin of biases in radiative transfer applications was also discussed by Newman et al. (1995). The independent work of Rotstayn (2000) and Pincus and Klein (2000) discussed biases in autoconversion and their implications for general circulation models. Stevens et al. (1996) stated that ignoring subgrid variability leads to underpredictions of *q*_{l}*T*

The present paper proves that ignoring subgrid-scale variability leads not only to errors in certain microphysical and thermodynamic quantities, but also to biases (that are systematic). The proof follows directly from Jensen’s inequality, upon noting that these quantities are represented by convex or concave functions.^{1} A bias is more pernicious than an ordinary error because a bias is single-signed, rather than partially self-canceling. Our analysis provides a straightforward way to identify what conditions and parameterizations lead to (systematic) biases.

In section 2 of this paper, we first note that the Kessler autoconversion formula is convex, and review the reason that neglecting subgrid variability leads to a bias when using a convex or concave formula. Then we note that the nonlinear autoconversion parameterization of Khairoutdinov and Kogan (2000) is neither convex nor concave and hence is not associated with a (systematic) bias. To demonstrate that errors associated with the Kessler and Khairoutdinov–Kogan parameterizations can be significant in important cases, in section 3 we examine boundary layer clouds observed during the Atlantic Stratocumulus Transition Experiment (ASTEX) field experiment. In section 4, we prove that if *q*_{l}*q*_{l}*T**q*_{t} and liquid water temperature *T*_{l} are conserved within the volume, then *q*_{l}*T**q*_{l}*T*

We stress that the present paper is not intended to be a critique of parametric formulas such as the Kessler or Khairoutdinov–Kogan autoconversion formulas. However, we do indirectly critique numerical models that ignore subgrid variability.

## 2. Grid box average autoconversion bias

*A*

_{K}

*K*

_{1}

*q*

_{l}

*q*

_{crit}

*H*

*q*

_{l}

*q*

_{crit}

*q*

_{crit}is a critical threshold in

*q*

_{l}below which autoconversion vanishes,

*K*

_{1}is an inverse timescale, and

*H*is the Heaviside step function (Kessler 1969). In the Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model version 5,

*q*

_{crit}= 0.5 g kg

^{−1}and

*K*

_{1}= 10

^{−3}s

^{−1}(Grell et al. 1994). The Kessler formula is displayed as the thick solid line in Fig. 1.

The Kessler formula is convex. Graphically, this means that any line segment (e.g., the dashed line in Fig. 1) that joins two points on the Kessler curve lies entirely on or above the Kessler curve, that is, in the gray shaded region in Fig. 1 (Hiriart-Urruty and Lemaréchal 1993, p. 3). Because the Kessler formula is convex, it turns out that if the grid box average specific liquid water content, *q*_{l}*A*_{K}(*q*_{l}*A*_{K}(*q*_{l})

Before proving this inequality, we illustrate it by a simple example. Consider a grid box that is occupied half by clear air and half by cloudy air with *q*_{l} = 1 g kg^{−1}. The probability density function (pdf) of *q*_{l} for this grid box consists of two equally strong Dirac delta functions, one located at *q*_{l} = 0 and the other at *q*_{l} = 1 g kg^{−1}, as depicted in Fig. 1. According to the Kessler formula, autoconversion occurs in the cloudy half of the grid box but not the clear half. The autoconversion rate, averaged over the grid box, is then *A*_{K}(*q*_{l})^{−3} g kg^{−1} s^{−1}, as shown by the dot in Fig. 1. But if one inserts *q*_{l}^{−1} into the Kessler formula [(1)], then one finds that the autoconversion rate *A*_{K}(*q*_{l}

Inspection of the Kessler autoconversion curve in Fig. 1 indicates the situations in which the underestimate is large. Since the Kessler curve is linear except at *q*_{crit}, there is no bias if *q*_{l} ≥ *q*_{crit} everywhere in the grid box or if *q*_{l} ⩽ *q*_{crit} everywhere. Also, there is no bias if *q*_{l} is uniform throughout the grid box, that is, if the pdf of *q*_{l} is a single delta function. Therefore we can surmise that the Kessler bias is most significant when *q*_{l}*q*_{l} is highly variable.

*f*(Λ

_{1}, . . . ,

_{Ns}

*N*

_{s}scalars. Suppose that 0 <

*λ*< 1 is a fractional weighting and that (Λ

_{1}, . . . ,

_{Ns}

_{1}, . . . ,

_{Ns}

*f*is convex if and only if

*f*

*λ*

_{1}

*λ*

_{1}

*λ*Λ

_{Ns}

*λ*

_{Ns}

*λ*

*f*

_{1}

_{Ns}

*λ*

*f*

_{1}

_{Ns}

_{1}, . . . ,

_{Ns}

_{1}, . . . ,

_{Ns}

*λ*(Dudley 1989, p. 272). A simple criterion for convexity is available in the case that

*N*

_{s}= 1 and

*f*is twice differentiable, namely, that the second derivative of

*f*be everywhere nonnegative:

*d*

^{2}

*f*(Λ

_{1})/

*d*Λ

^{2}

_{1}

*f*is said to be concave when −

*f*is convex (Hiriart-Urruty and Lemaréchal 1993, p. 145). Now define an average over a grid box as

*P*(Λ

_{1}, . . . ,

_{Ns}

_{1}, . . . ,

_{Ns}

_{1}, . . . ,

_{Ns}

*f*(Λ

_{1}, . . . ,

_{Ns}

_{1}

_{Ns}

*f*sometimes underpredicts and never overpredicts the quantity we usually desire, the grid box average of

*f,*that is,

*f*(Λ

_{1},.....Λ

_{Ns})

*concave*function sometimes overpredicts and never underpredicts the quantity we desire. Jensen’s inequality has very general implications. Whenever a convex or concave formula of any sort is applied in a numerical model grid box, and subgrid-scale variability is ignored, a (systematic) bias occurs, relative to what would be obtained if subgrid variability were accounted for. (Of course, the bias may have negligible magnitude in practice.)

To prove that a bias is associated with the Kessler formula, we simply note that this formula is convex. Then Jensen’s inequality implies that *A*_{K}(*q*_{l}*A*_{K}(*q*_{l})

*N,*as well as

*q*

_{l}. Including the effect of

*N*on autoconversion is important for assessing how clouds respond to a change in the concentration of cloud condensation nuclei (Cotton and Anthes 1989, p. 92). When an autoconversion formula depends on two parameters, determining whether the formula is convex or concave becomes subtle. Consider the autoconversion formula of Khairoutdinov and Kogan (2000):

*A*

_{KK}

*K*

_{2}

*q*

^{a}

_{l}

*N*

^{b}

*K*

_{2}= 7.419 × 10

^{13}m

^{−5.37}s

^{−1},

*a*= 2.47, and

*b*= −1.79. A sufficient condition for the convexity of a two-dimensional function is (see Boas 1983, 170–171 and Hiriart-Urruty and Lemaréchal 1993, 190–191)

*a*and

*b.*Therefore, if

*N*were treated as a constant parameter in

*A*

_{KK}, then ignoring subgrid variability in

*q*

_{l}would lead to systematic underestimates; that is,

*A*

_{KK}(

*q*

_{l}

*N*

*A*

_{KK}(

*q*

_{l},

*N*

*b*= −1.79 the second condition implies that

*A*

_{KK}(

*q*

_{l}

*N*

*A*

_{KK}(

*q*

_{l}

*N*)

*a*> 0 and

*b*< 0, as in the Khairoutdinov–Kogan parameterization, then the third condition in (6) holds if and only if

*a*+

*b*> 1. This condition is violated for the values of

*a*and

*b*used by Khairoutdinov and Kogan. Although the Khairoutdinov–Kogan formula would be convex if either

*N*were a constant parameter or

*q*

_{l}were a constant parameter, the formula is not convex when both are variable. For example, consider a grid box for which

*q*

_{l}is proportional to

*N.*Then

*A*

_{KK}∝

*N*

^{0.68}, and

*N*

^{0.68}is a

*concave*function of

*N.*

## 3. Assessment of autoconversion errors using observational data

We have seen that when the Kessler autoconversion formula is used, a systematic bias can result from neglect of subgrid variability. This fact can be deduced solely by noting that the Kessler formula is convex. But to determine whether or not the bias is significant, one needs to examine data. To show that the Kessler bias can be large in important cases, we calculate the biases associated with a few examples of boundary layer clouds from the ASTEX field experiment. We also examine averaging errors associated with inserting mean values into the Khairoutdinov–Kogan parameterization. Using field data to assess biases has an advantage over using numerical data calculated by large eddy simulation models. Namely, observed variability of scalars in boundary layers is often greatest at long wavelengths (Cotton and Anthes 1989, 373–383), but large eddy simulations are still often restricted to horizontal domains of several kilometers or less in length; therefore, a large eddy simulation may not capture the full variability that should be accounted for within a mesoscale (e.g., 50 km) grid box.

ASTEX was an investigation of marine boundary layer clouds near the Azores over the North Atlantic Ocean. An overview of ASTEX can be found in Albrecht et al. (1995) and in a special issue (volume 52, issue 16) of the *Journal of the Atmospheric Sciences.* We examine aircraft data from long, constant-altitude flight legs made by The Met. Office C-130 aircraft in two boundary layer regimes. The first regime is a drizzling stratocumulus layer observed on the night of 12–13 June 1992 during the first Lagrangian intensive operations period (flight A209, denoted DRZ). The second regime is a cumulus-rising-into-intermittent-stratocumulus layer observed on 20 June 1992 during the second Lagrangian intensive operations period (flight A213, denoted CUSCU). We have truncated the legs to 500 s worth of data, so that all legs span approximately the same distance, about 50 km. This is a typical grid box size in a mesoscale model. Linear trends were not removed from the legs.

To quantify autoconversion errors, we use measurements of *q*_{l} and droplet number concentration *N.* Here *q*_{l} is obtained from a Johnson–Williams hot-wire probe and logged at 4 Hz. For the autoconversion calculations, *q*_{l} is averaged to 1 Hz. Droplet concentration *N* was measured with a forward scattering spectrometer probe and averaged to 1 Hz. Following Wood and Field (2000), we assume that when *N* < 5 cm^{−3}, large aerosol may be present but cloud droplets are not. In these clear areas, the Johnson–Williams probe may still record small nonzero values of *q*_{l} because of instrument noise about the zero threshold. Since the Khairoutdinov–Kogan autoconversion formula, *A*_{KK}, approaches infinity for vanishing *N* and nonzero *q*_{l}, we set *q*_{l} and *A*_{KK} to zero when *N* < 5 cm^{−3}, in order to avoid spurious contributions to the autoconversion rate. Cloud fraction is computed as the fraction of time (distance) for which *N* > 5 cm^{−3}.

Below we discuss biases in *q*_{l} and *T.* We take this opportunity to mention the additional measurements required to assess these biases. Temperature is obtained using a Rosemount deiced total temperature sensor logged at 32 Hz. To obtain total specific water content *q*_{t}, we use data from a Lyman-*α* hygrometer logged at 64 Hz and averaged to 32 Hz. For the *q*_{l} and *T* biases, *q*_{l} is interpolated to 32 Hz. However, the Johnson–Williams instrument response time is only about 1 Hz (which corresponds to roughly 100 m). This prevents an estimate of small-scale variability. Further details ofthe instrumentation are contained in Rogers et al. (1995).

Time series of the six legs we shall discuss are shown in Fig. 2. Three of the legs—DRZ-OV1, DRZ-OV2, and CUSCU-OV1—are entirely cloudy, that is, overcast. Three others—CUSCU-PC1, CUSCU-PC2, and CUSCU-PC3—are partly cloudy. The pdfs of *q*_{l} for these legs (along with the Kessler formula) are plotted in Fig. 3. A summary of the characteristics of these legs is listed in Table 1. The biased autoconversion rates in Table 1 were computed by substituting leg-averaged values into the autoconversion formulas. The unbiased autoconversion rates were obtained by computing the autoconversion rate at each sample point along the leg and then averaging. Within the flight legs, the standard deviation in pressure is small, indicating that the aircraft was flying at almost constant altitude.

Figure 4 displays the unbiased *A*_{K}(*q*_{l})*A*_{K}(*q*_{l}*q*_{l} = 0.62 ± 0.070 g kg^{−1}) and therefore the pdf of *q*_{l} lies mostly to the right of the threshold in the Kessler curve *q*_{crit} = 0.5 g kg^{−1} (see Fig. 3). For the most part, the pdf samples a linear region of the Kessler curve, and hence the bias is small. This layer illustrates the point that a strongly drizzling, fairly homogeneous layer has a small or zero Kessler autoconversion bias.

To correct errors in autoconversion due to subgrid variability, Fowler et al. (1996) and Fowler and Randall (1996) suggest predicting cloud fraction and inserting into the Kessler formula the liquid water content averaged over just the cloudy regions, rather than the whole grid box. This remedy can diminish the bias in some partly cloudy layers, but not the ones studied here. For our partly cloudy layers, even the in-cloud average liquid water content is less than the Kessler threshold, so that zero autoconversion is still predicted even when cloud fraction is taken into account. Furthermore, accounting for partial cloudiness cannot aid autoconversion prediction in fully overcast layers, and the bias can be large even when the grid box is entirely cloudy, as illustrated by leg CUSCU-OV1.

*A*(

*q*

_{l},

*N*) is convex, then accounting for cloud fraction removes some of the autoconversion bias but still leaves a residual, systematic bias. To state this mathematically, we define

*q*

_{l}

^{cld}

*q*

_{l}

^{clr}

*N,*and we also let

*C*denote cloud fraction. Then

*q*

_{l}

*C*

*q*

_{l}

^{cld}

*C*)

*q*

_{l}

^{clr}

*N*

*q*

_{l},

*N,*and

*A*are zero in the clear areas. Then one may show that

*CA*(

*q*

_{l}

^{cld}

*N*

^{cld}

*A*(

*q*

_{l}

*N*

*CA*(

*q*

_{l}

^{cld}

*N*

^{cld}

*A*(

*q*

_{l},

*N*)

The Khairoutdinov–Kogan autoconversion rates, *A*_{KK}(*q*_{l},*N*)*A*_{KK}(*q*_{l}*N**A*_{KK}(*q*_{l}*N**A*_{KK}(*q*_{l}, *N*)*A*_{KK}(*q*_{l}*N**A*_{KK}(*q*_{l}, *N*)*N* and *q*_{l} as variables, rather than constant parameters, in the calculation of *A*_{KK}(*q*_{l}, *N*)

Table 1 shows that the Kessler and Khairoutdinov–Kogan autoconversion rates differ greatly, a conclusion also reached in the more detailed comparison of Wood (2000). The discrepancy between the Kessler and Khairoutdinov–Kogan rates for leg DRZ-OV1 greatly exceeds the averaging error in the Khairoutdinov–Kogan formula. Over the coming years, autoconversion formulas are likely to improve and hence differences between them are likely to narrow. Such improvements will not, however, reduce averaging errors or biases, for a given grid box size. Therefore, modelers should strive to account for subgrid variability and thereby reduce averaging errors, in addition to developing physically representative autoconversion formulas.

## 4. Grid box average liquid water content and temperature biases

We now discuss biases that arise in partly cloudy grid boxes when average specific liquid water content, *q*_{l}*T**s.* We use this formula because its convexity can be assessed by inspection. Although most models do not use this particular formula (an exception is Bougeault 1981), they use formulas with similar properties.

*T*

_{l}, which is akin to the linearized liquid water potential temperature:

*L*is the latent heat of vaporization, and

*c*

_{p}is the specific heat at constant pressure.

Next we need to define the variable *s* = *s*(*q*_{t}, *T*_{l}, *p*) = *s*(*q*_{t}*T*_{l}*p*) + *s*′, where *q*_{t} is the total specific water content (including vapor and liquid), and *p* is the pressure. Here *s* is conceptually appealing because it can be written as a function of conserved variables (and pressure); furthermore, when *q*_{t} > *q*_{s}(*T, p*), where *q*_{s}(*T, p*) is the saturation specific humidity, *s* approximates *q*_{l} = *q*_{t} − *q*_{s}(*T, p*). However, when *q*_{t} < *q*_{s}(*T, p*), *q*_{l} = 0, but *s* is negative. Also, in this case *s* ≠ *q*_{t} − *q*_{s}(*T, p*), and so *s* cannot be simply interpreted as the deviation of *q*_{t} from saturation.

*s*can be written (Lewellen and Yoh 1993; see also Sommeria and Deardorff 1977; Mellor 1977):

*e*

_{s}is the saturation vapor pressure over liquid, and

*R*

_{d}and

*R*

_{υ}are the gas constants for dry air and water vapor. When positive,

*s*is an approximation of

*q*

_{l}only because the derivation of

*s*requires that

*q*

_{s}(

*T, p*) be expanded about

*T*=

*T*

_{l}. But since (

*T*−

*T*

_{l})/

*T*

_{l}≪ 1, the approximation is accurate.

*s*about

*s*(

*q*

_{t}

*T*

_{l}

*p*) to first order, and assuming

*s*′:

*β*

_{1}

*q*

_{t}

*β*

_{1}

*q*

_{s}), that is neglected by Lewellen and Yoh. In (13), we have

*s*(

*q*

_{t}

*T*

_{l}

*p*) and

*s*′,

*q*

_{l}at a point within a volume can be approximated as follows (Bougeault 1981):

*H*denotes the Heaviside step function. To summarize, (15) relies on two approximations: that (

*T*−

*T*

_{l})/

*T*

_{l}≪ 1, and that fluctuations in

*T*

_{l}and

*q*

_{t}are small.

Now we are ready to prove that ignoring fluctuations in *s* leads to an underestimation of *q*_{l}*q*_{l} is a convex function of *s*′. The result then follows directly from Jensen’s inequality. Most numerical models do not use the particular formula (15), but all such diagnostic formulas for *q*_{l} are approximately convex in shape.

Equation (15) resembles the Kessler autoconversion formula (1). Hence the bias in *q*_{l} has similar properties to the Kessler bias. For instance, the bias in *q*_{l} disappears if the grid box is either entirely overcast or entirely clear. Also, the bias is largest if the grid box contains clear areas that are very dry and cloudy areas that are very moist, with the grid box average near saturation.

We can also prove that ignoring fluctuations in *s* leads to an underestimate of *T**T*_{l}*q*_{l}*T*

The approximation (15) is valid only when fluctuations in *q*_{t} and *T*_{l} over a grid box are small. When the fluctuations are not negligible, but *p* is approximately constant, then the existence of a bias depends on the convexity of a two-dimensional function *q*_{l}(*q*_{t}, *T*_{l}, *p*). If we make the approximation that *q*_{l} ≅ *s*(*q*_{t}, *T*_{l}, *p*)*H*[*s*(*q*_{t}, *T*_{l}, *p*)], then it turns out that *q*_{l} is neither a convex nor concave function of *q*_{t} and *T*_{l}. Therefore, one can construct cases in which neglecting subgrid variability actually overestimates *q*_{l}*p* = 600 hPa and *q*_{t} = 2.2 g kg^{−1} everywhere, but with *T*_{l} = 254 K in half the volume and *T*_{l} = 255 K in the other half. Then neglecting subgrid variability overestimates *q*_{l}^{−4} g kg^{−1}.

To determine the magnitude of the biases in data, we return to the three partly cloudy ASTEX legs from 20 June 1992 (flight A213, denoted CUSCU), each of which is approximately 50 km in length. Table 2 summarizes the biases for these legs. In this table, *q*_{l} is computed via the formula *q*_{l} = *s*(*q*_{t}, *T*_{l}, *p*)*H*[*s*(*q*_{t}, *T*_{l}, *p*)] and *T* is computed via (8). Averaging is performed as for the autoconversion calculations. The biases are modest but not negligible.

The biases exist for shorter legs as well. To illustrate this, we divide leg CUSCU-PC2 into ∼1 km segments. Most segments are either entirely cloudy or entirely clear and therefore have no bias. But some segments near cloud edges do have large biases. The segments with the four largest biases are listed in Table 3. A buoyancy deficit of the magnitude observed here (∼0.4 K) could significantly lower the maximum cloud-top height attained by individual clouds in a model.

## 5. Decrease of *q*_{l} in an isolated system

*q*

_{l}

As discussed above, the fact that *q*_{l} can be approximated as a convex function, (15), has implications for numerical modeling of clouds because convexity of *q*_{l} leads to biases in *q*_{l}*T**q*_{l} on the physics of mixing. Namely, we prove that as a partly cloudy volume of air homogenizes through mixing, *q*_{l}

*q*

_{l}can be adequately approximated by (15), that is, as a convex function of

*s*′. Furthermore, suppose that

*T*

_{l}and

*q*

_{t}are conserved.

^{2}The time evolution of the pdf of

*s*′ determines the time evolution of

*q*

_{l}

*s*′ evolve? Advection merely rearranges fluid particles and therefore cannot directly alter the pdf of

*s*′ within the volume. Diffusion, however, tends to homogenize

*s*′, that is, bring the pdf of

*s*′ toward a delta function. In this uniform final state,

*q*

_{l}is a convex function, then the minimum possible value of

*q*

_{l}

*s*′ that are consistent with

*q*

_{t}

*T*

_{l}

*q*

_{l}[

*s*(

*q*

_{t}

*T*

_{l}

*p*)]. Hence as time → ∞,

*q*

_{l}

One can go further. If pressure is constant, and *q*_{l}*q*_{l}*q*_{l}*T*_{l}, Eq. (8), we may also conclude that *T**f* of conserved variables with equal diffusivities in an isolated fluid system decreases or remains constant in time.

The physical content of the proof may be summarized as follows. Consider a conserved scalar *ϕ* in an isolated fluid volume and a convex function *f*(*ϕ*). Assume that initially the pdf of *ϕ* is broad because *ϕ* varies strongly within the fluid volume. The broad pdf samples much of the nonlinearity of *f.* Because *f* is convex, regions of anomalously large *ϕ* contribute proportionally more to *f**ϕ.* As time passes, however, diffusion leads to homogenization of *ϕ* and hence a narrowing of the pdf of *ϕ.* Then the pdf samples less of the nonlinearity in *f.* The contributions to *f**ϕ* still outweigh those of small *ϕ,* but by a smaller amount. Hence *f*

A rather direct proof that the average of a convex function *f* of conserved variables decreases in time can be constructed via a simple extension of a derivation in Bilger (1989).^{3} Below we present an alternative proof that emphasizes the fact that, because of diffusion, the pdf of the conserved variables evolves toward a delta function, and that the decrease in *f*

*N*

_{s}scalars, Φ(

**x**,

*t*) = (

*ϕ*

_{1}, . . . ,

*ϕ*

_{Ns}

*ϕ*

_{β}satisfies the advection–diffusion equation,

*u*

_{i}is a component of velocity,

*ρ*is fluid density,

*x*

_{i}is a spatial component, and

*t*is time. We use the Einstein summation convention. We have assumed that the diffusivities

*κ*of all scalars are equal. Second, we assume the fluid system is an isolated material fluid volume

*V*with (constant) mass

*M.*There is no flow or molecular diffusion of scalars through the surface enclosing the element. The bounding surface may be removed to infinity.

*P*(Ψ;

*t*) over a material fluid volume. Here Ψ = (

*ψ*

_{1}, . . . ,

*ψ*

_{Ns}

**x**,

*t*). Then the governing equation of the pdf can be derived following Colucci et al. (1998):

*κ*(∂

*ϕ*

_{γ}/∂

*x*

_{i})(∂

*ϕ*

_{β}/∂

*x*

_{i}) | Ψ〉 is the value of

*κ*(∂

*ϕ*

_{γ}/∂

*x*

_{i})(∂

*ϕ*

_{β}/∂

*x*

_{i}) averaged over those locations where, for all

*α, ϕ*

_{α}(

**x**,

*t*) =

*ψ*

_{α}. When only one scalar is present, it is easy to recognize that the governing pdf equation resembles the diffusion equation in scalar space but with a negative “diffusivity,” −〈

*κ*(∂

*ϕ*

*x*

_{i}

*ϕ*/

*x*

_{i}

*q*

_{l}

*t*⩽ 0. When

*q*

_{t}and

*T*

_{l}vary little across the fluid volume and

*p*is constant, the specific liquid water content averaged over the volume,

*q*

_{l}

*q*

_{l}

*q*

_{t}

*T*

_{l}

*p*are constants over the fluid volume and in time, so is

*s*(

*q*

_{t}

*T*

_{l}

*p*). We now assume that

*s*′ obeys an advection–diffusion equation of the form (17). To do so, we assume that

*q*

_{t}and

*T*

_{l}are conserved and neglect the difference between their diffusivities. Then, taking the partial time derivative of (20) and substituting in the expression (18) for ∂

*P*(

*ψ*;

*t*)/∂

*t,*we find

*P*

_{s′}|

_{ψ=∞}and ∂

*P*

_{s′}/∂

*ψ*|

_{ψ=∞}tend to zero faster than

*ψ*

^{−1}. The second term vanishes at the upper limit if

*P*

_{s′}|

_{ψ=∞}= 0. When these conditions hold,

*q*

_{l}

*s*′ becomes large.

Proofs such as the above are useful because if the conclusion of the proof is violated, then we know at least one assumption must also have been violated. In addition, the proof is useful because Eq. (23) provides a formula for the rate of decrease of *q*_{l}*ψ* = −*s*(*q*_{t}*T*_{l}*p*). Specifically, it depends on the dissipation rate of *s*′ and the value of the pdf of *s*′ at cloud edge. It is clear that *q*_{l}*P*_{s′}[*ψ* = −*s*(*q*_{t}*T*_{l}*p*)] ≠ 0, that is, if the volume is partly cloudy. Given a particular dissipation rate, *q*_{l}*P*_{s′}[*ψ* = −*s*(*q*_{t}*T*_{l}*p*)] is large, that is, if many elements within the volume are at saturation.

The mechanism of the temperature decrease is evaporative cooling. Advection within a partly cloudy parcel brings clear and cloudy fluid particles to adjacent positions. The subsequent diffusive mixing evaporates liquid water, leading to cooling. Because advection requires time, the cooling is not realized instantaneously (Krueger 1993). The amount of cooling at a particular moment in time depends on how much mixing has occurred in the past. Even if *q*_{t}*T*_{l}*q*_{l}*T**q*_{t} and *T*_{l}. A numerical model that diagnoses *T**q*_{t}*T*_{l}

*q*

_{l}

*f*(Ψ) of

*N*

_{s}variables is such a function. Then define the average of

*f*as

*N*

_{s}dimensional space of scalars.

*A*

_{f}decays, we take the partial time derivative of (24), use the governing equation of the pdf [(18)] to substitute for ∂

*P*(Ψ;

*t*)/∂

*t,*and integrate by parts twice. Then we find

*A*

_{ψ}of the

*N*

_{s}dimensional space of scalars Ψ. In this case, we can use the definition of a conditional average (see Colucci et al. 1998) to obtain

*A*

_{f}decreases or stays constant with time if

*f*is twice differentiable, this condition holds for all Φ if and only if

*f*is convex (Hiriart-Urruty and Lemaréchal 1993, p. 190). Therefore, if

*f*is convex,

*A*

_{f}decreases or remains constant in time.

The result (25) has relevance to some fast-chemistry reactions. Suppose a chemical reaction occurs much more rapidly than the timescale for molecular mixing, so that chemical equilibrium prevails at each point in the fluid. Examples of such fast-chemistry reactions in the field of turbulent combustion are discussed by Bilger (1976, 1989) and Libby and Williams (1994). When the chemical and thermal diffusivities can be approximately set equal, and initially the fuel and oxidizer are homogeneous and separated, then the concentrations of reactants and products can be written as functions of a conserved scalar. Sometimes in such reactions the reactants are everywhere convex and the products everywhere concave [as in Fig. 4a of Bilger (1976), Fig. 1 of Bilger (1989), and Fig. 3 of Libby and Williams (1994)]. Then (25) states that the average product concentration cannot decrease nor the average reactant concentration increase, even temporarily. Sometimes a reactant can be approximated, as can *q*_{l}, as a convex, continuous, piecewise linear function *f*(*ψ*_{1}) with a slope discontinuity at a single point (Bilger 1976). In this case, *A*_{f} still decreases or remains constant with time even though *f* does not possess a first or second derivative everywhere. This may be seen by proceeding as in the proof for the decay of *q*_{l}, Eqs. (19)–(23). Likewise, a product of a reaction can sometimes be approximated as a concave function with a slope discontinuity at a single point; if so, the average product concentration increases or remains constant with time.

## 6. Conclusions

Jensen’s inequality allows one to prove that (systematic) biases can arise from neglect of subgrid variability and the use of convex or concave functions (Cahalan et al. 1994). The present paper points out two examples of convex functions: the Kessler autoconversion formula, and an approximate formula for specific liquid water content, *q*_{l}, that is valid if fluctuations in thermodynamic quantities are small. If these functions are used in a numerical model, then neglecting subgrid variability leads to biases in autoconversion rate (particularly for highly variable but weakly drizzling clouds), and biases in *q*_{l}

The biases in *q*_{l}*q*_{l}*prognosing* *q*_{l}*q*_{l}*q*_{l} forms; if the parcel is slightly subsaturated somewhere, *q*_{l} may form rapidly. To calculate the increase in *q*_{l}*q*_{l}

Do biases associated with autoconversion and *q*_{l}^{4} to the biases we have discussed. First, Fowler et al. (1996) and Fowler and Randall (1996) found that autoconversion is reduced in (large) general circulation model grid boxes relative to (small) mesoscale model grid boxes. If large grid boxes encompass more variability than small grid boxes, as one would expect, then according to the arguments in the present paper, underprediction of autoconversion should be more severe in large grid boxes. This is essentially how Fowler et al. (1996) and Fowler and Randall (1996) interpreted their results. Second, Sommeria and Deardorff (1977) performed two simulations of a cumulus layer at fine grid spacing (50 m): one simulation neglected subgrid variability, the other did not. The run that neglected subgrid variability tended to produce less liquid water than the other. Again, this is what one would expect from Jensen’s inequality. Although these two sets of authors noticed underpredictions in their simulations, the present paper deepens our understanding by proving that certain microphysical and thermodynamic biases are systematic and placing them in a general theoretical framework that links them to the notion of convexity.

How then should modelers reduce the biases? Equation (3) shows that the average quantities we desire can be obtained without bias if the pdfs of the relevant quantities are known. Therefore we advocate predicting information about the relevant pdfs using the numerical model, as suggested by Manton and Cotton (1977), Sommeria and Deardorff (1977), and Mellor (1977), for example.

## Acknowledgments

The authors would like to thank the RAF aircrew and MRF staff involved in the planning and execution of the ASTEX field campaign. The authors also thank three anonymous reviewers for their suggestions. V. E. Larson acknowledges a helpful conversation with R. L. Walko and financial support from the National Oceanic and Atmospheric Administration, Contract NA67RJ0152. W. R. Cotton and J.-Ch. Golaz acknowledge support by the National Science Foundation under NSF-WEAVE Contract ATM-9904128.

## REFERENCES

Albrecht, B. A., C. S. Bretherton, D. Johnson, W. H. Schubert, and A. S. Frisch, 1995: The Atlantic stratocumulus transition experiment—ASTEX.

*Bull. Amer. Meteor. Soc.,***76,**889–904.Bilger, R. W., 1976: Turbulent jet diffusion flames.

*Prog. Energy Combust. Sci.,***1,**87–109.——, 1989: Turbulent diffusion flames.

*Annu. Rev. Fluid Mech.,***21,**101–135.Boas, M. L., 1983:

*Mathematical Methods in the Physical Sciences.*2d ed. John Wiley and Sons, 793 pp.Bougeault, P., 1981: Modeling the trade-wind cumulus boundary layer. Part I: Testing the ensemble cloud relations against numerical data.

*J. Atmos. Sci.,***38,**2414–2428.Cahalan, R. F., W. Ridgway, W. J. Wiscombe, T. L. Bell, and J. B. Snider, 1994: The albedo of fractal stratocumulus clouds.

*J. Atmos. Sci.,***51,**2434–2455.Chen, J.-Y., and W. Kollmann, 1994: Comparison of prediction and measurement in non-premixed turbulent flames.

*Turbulent Reacting Flows,*P. A. Libby and F. A. Williams, Eds., Academic Press, 211–308.Colucci, P. J., F. A. Jaberi, P. Givi, and S. B. Pope, 1998: Filtered density function for large eddy simulation of turbulent reacting flows.

*Phys. Fluids A,***10,**499–515.Cotton, W. R., and R. A. Anthes, 1989:

*Storm and Cloud Dynamics.*Academic Press, 883 pp.Dopazo, C., 1994: Recent developments in pdf methods.

*Turbulent Reacting Flows.,*P. A. Libby and F. A. Williams, Eds., Academic Press, 375–474.Dudley, R. M., 1989:

*Real Analysis and Probability.*Wadsworth and Brooks/Cole, 436 pp.Fowler, L. D., and D. A. Randall, 1996: Liquid and ice cloud microphysics in the CSU general circulation model. Part III: Sensitivity to modeling assumptions.

*J. Climate,***9,**561–586.——, ——, and S. A. Rutledge, 1996: Liquid and ice cloud microphysics in the CSU general circulation model. Part I: Model description and simulated microphysical processes.

*J. Climate,***9,**489–529.Grell, G. A., J. Dudhia, and D. R. Stauffer, 1994: A description of the fifth-generation Penn State/NCAR Mesoscale Model (MM5). NCAR Tech. Note NCAR/TN-398 + STR, 138 pp.

Hiriart-Urruty, J.-B., and C. Lemaréchal, 1993:

*Convex Analysis and Minimization Algorithms I.*Springer-Verlag, 417 pp.Jensen, J. L. W. V., 1906: Sur les fonctions convexes et les inégalités entre les valeurs moyennes (in French).

*Acta Math.,***30,**175–193.Kessler, E., 1969:

*On the Distribution and Continuity of Water Substance in Atmospheric Circulation. Meteor. Monogr.,*No. 10, Amer. Meteor. Soc., 84 pp.Khairoutdinov, M., and Y. Kogan, 2000: A new cloud physics scheme in a large-eddy simulation model of marine stratocumulus.

*Mon. Wea. Rev.,***128,**229–243.Kogan, Y. L., 1998: On parameterization of cloud physics processes in mesoscale models. Preprints,

*Conf. on Cloud Physics,*Everett, WA, Amer. Meteor. Soc., 348–351.Kristjánsson, J. E., 1991: Cloud parametrization at different horizontal resolutions.

*Quart. J. Roy. Meteor. Soc.,***117,**1255–1280.Krueger, S. K., 1993: Linear eddy modeling of entrainment and mixing in stratus clouds.

*J. Atmos. Sci.,***50,**3078–3090.Lewellen, W. S., and S. Yoh, 1993: Binormal model of ensemble partial cloudiness.

*J. Atmos. Sci.,***50,**1228–1237.Libby, P. A., and F. A. Williams, 1994: Fundamental aspects and a review.

*Turbulent Reacting Flows,*P. A. Libby and F. A. Williams, Ed., Academic Press, 1–61.Manton, M. J., and W. R. Cotton, 1977: Formulation of approximate equations for modeling moist deep convection on the mesoscale. Atmospheric Science Paper Number 266, Colorado State University, Fort Collins, CO, 62 pp.

Mellor, G. L., 1977: The Gaussian cloud model relations.

*J. Atmos. Sci.,***34,**356–358.Newman, W. I., J. K. Lew, G. L. Siscoe, and R. G. Fovell, 1995: Systematic effects of randomness in radiative transfer.

*J. Atmos. Sci.,***52,**427–435.Pincus, R., and S. A. Klein, 2000: Unresolved spatial variability and microphysical process rates in large scale models.

*J. Geophys. Res.,***105,**27 059–27 066.Rogers, D. P., D. W. Johnson, and C. A. Friehe, 1995: The stable internal boundary layer over a coastal sea. Part I: Airborne measurements of the mean and turbulence structure.

*J. Atmos. Sci.,***52,**667–683.Rotstayn, L. D., 2000: On the “tuning” of autoconversion parameterizations in climate models.

*J. Geophys. Res.,***105,**15 495–15 507.Sommeria, G., and J. W. Deardorff, 1977: Subgrid-scale condensation in models of non-precipitating clouds.

*J. Atmos. Sci.,***34,**344–355.Stevens, B., R. L. Walko, W. R. Cotton, and G. Feingold, 1996: The spurious production of cloud-edge supersaturations by Eulerian models.

*Mon. Wea. Rev.,***124,**1034–1041.——, W. R. Cotton, and G. Feingold, 1998: A critique of one- and two-dimensional models of boundary layer clouds with a binned representation of drop microphysics.

*Atmos. Res.,***47–48,**529–553.Tiedtke, M., 1993: Representation of clouds in large-scale models.

*Mon. Wea. Rev.,***121,**3040–3061.Wood, R., 2000: The validation of drizzle parametrizations using aircraft data. Preprints,

*13th Int. Conf. on Clouds and Precipitation,*Reno, NV, International Commission on Clouds and Precipitation.——, and P. R. Field, 2000: Relationships between total water, condensed water and cloud fraction examined using aircraft data.

*J. Atmos. Sci.,***57,**1888–1905.

Six 50-km ASTEX legs.

Biases in *q _{l}*

*T*

Biases in *q _{l}*

*T*

^{1}

Some authors refer to a convex function as “concave up” and a concave function as “concave down.”

^{2}

In this paper, a variable is said to be conserved if it satisfies the advection–diffusion equation with no source term.

^{3}

The proof can be obtained by substituting Eq. (9) of Bilger (1989) into Eq. (1) of Bilger (1989), integrating over the fluid volume, and then noting that convexity of *f* ensures a decrease in *f**f* is a function of several conserved variables.

^{4}

For example, a possible feedback was noted by Kristjánsson (1991), who postulated that increased latent heating may lead to increased vertical velocity in cloudy updrafts and hence further latent heating.