Does the threshold representation associated with the autoconversion process matter?

Di ﬀ erent ad hoc threshold functions associated with the autoconversion process have been arbitrarily used in atmospheric models. However, it is unclear how these ad hoc functions impact model results. Here systematic investigations of the sensitivities of climatically-important properties: CF (cloud fraction), LWP (liquid water path), and 5 AIE (aerosol indirect e ﬀ ect) to threshold functions have been performed using a 3-D cloud-resolving model. It is found that the e ﬀ ect of threshold representations is larger on instantaneous values than on daily averages; and the e ﬀ ect depends on the percentage of clouds in their transitional stages of converting cloud water to rain water. For both the instantaneous values and daily averages, the sensitivity to the 10 speciﬁcation of critical radius is more signiﬁcant than the sensitivity to the “smoothness” of the threshold representation (as embodied in the relative dispersion of droplet size distribution) for drizzling clouds. Moreover, the impact of threshold representations on the AIE is stronger than that on CF and LWP.

date, the primary foci of both parameterization development (Kessler, 1969;Manton and Cotton, 1977;Liu and Daum, 2004) and sensitivity investigations (Iacobellis and Somerville, 2006) have been on the rate function P 0 .
The threshold function has received little attention. In most global climate models (GCMs) and/or cloud resolving models (CRMs), the threshold behavior has been rep- 10 resented by an ad hoc function of liquid water content or droplet size. It ranges from an all-or-nothing Kessler-type (T = Heaviside function, Kessler, 1969) to a smoother Sundqvist-type (T = exponential function, Sundqvist, 1978;Del Genio et al., 1996) and to a constant Berry-type (T =1, Berry 1968;Beheng 1994). Despite their dramatic differences, these functions have been used arbitrarily, and no systematic investigation 15 has been performed to examine whether or not these different representations exert significant effects on model results.
To fill this gap, this study explores how the climatically important properties, i.e., cloud fraction (CF), liquid water path (LWP), and aerosol indirect effect (AIE) respond to different threshold representations by applying a theoretical threshold function to a 20 3-D cloud-resolving model, ATHAM (Active Tracer High-resolution Atmospheric Model) (Herzog et al., 1998(Herzog et al., , 2003Guo et al., 2007a). Liu et al. (2006a) derived a theoretical threshold function (T ε ) that covers all the existing types of threshold representations. Briefly, T ε is described by

Threshold representation
the normalized incomplete Gamma function; ε the relative dispersion (the ratio of the standard deviation to the mean radius of the cloud droplet size distribution); and x c the ratio of the critical to the mean mass of cloud water (Liu et al., 2006a). Equation (2) indicates that T ε depends on two dimensionless variables: ε and x c , as compared to ad 5 hoc threshold functions which depend only on x c . It should be emphasized that ε controls the "type" of T ε , changing from the Kessler-type to the Berry-type as ε increases from 0 to infinity. This dependence of T ε on ε allows us to systematically examine the effect of the "smoothness" of the threshold function, which has been unknowingly buried in arbitrary uses of ad hoc threshold functions in previous studies (Liu et al., 10 2006a).

Model and case descriptions
ATHAM is a non-hydrostatic, fully compressible atmospheric model. In this study, the 3-D version is adopted.

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Canary Islands. The clean case (denoted as "CLEAN") occurred on 26 June 1997, when the ACE-2 area was under the control of a cyclone that brought in pristine maritime air. The polluted case (denoted as "POLL") occurred on 9 July 1997, when the ACE-2 area was under the influence of the Azores High that brought in anthropogenic pollution from Europe (Verver et al., 2000). For the "CLEAN" and "POLL" cases, the total aerosol number concentrations were 218 cm −3 and 636 cm −3 , respectively (Snider and Brenguier, 2000); and the non sea-salt (nss) sulfate mass concentrations were 0.30 µg m −3 and 2.8 µg m −3 , respectively (Guibert et al., 2003). Note that the detailed model set-up and case descriptions were available in Guo et al. (2007a,b).
In addition to the contrasting aerosol and meteorological conditions, significant pro-10 portions of these clouds were in their transitional stages where precipitation depended critically on the threshold representation, providing a good opportunity to explore the effects of different threshold representations on clouds.
(2). [Note: ε=0, 0.4, and 300 approximately represent the Kessler-type, the Sundqvist-type, and the Berry-type threshold functions, respectively]. The results are shown in Fig. 1, where the results for ε=0 are used as the base cases (black) and the results of sensitivity tests for ε=0.4 and 300 are shown 20 as percentage differences relative to the base cases. For the "CLEAN" case, both CF and LWP reach their maxima in the local early morning and their minima in the local afternoon ( Fig. 1a and 1c). But for the "POLL" case, neither CF nor LWP exhibits a significant diurnal cycle due to a strong large-scale subsidence associated with the Azores High ( Fig. 1b and 1d). The magnitude of the AIE is the largest near local noon EGU "CLEAN" case has a larger AIE than the "POLL" case because the "CLEAN" clouds are deeper and moister (Pawlowska and Brenguier, 2003). Note that CF here is defined as the fraction of cloudy columns, and a cloudy column is a column containing one or more cloudy cells with liquid water mixing ratio >0.01 g/kg. The AIE (shortwave plus longwave) at the top-of-the-atmosphere is estimated by swapping the aerosol and meteorological conditions between the "CLEAN" and "POLL" cases and then calculating the radiative flux difference between the clean and polluted aerosol conditions under the same meteorological background (Guo et al., 2007b). The relative differences in the instantaneous CF, LWP, and AIE between the base cases (ε=0) and the sensitivity tests (ε=300) vary significantly, and can reach up to 10 ∼20%, ∼40%, and ∼60%, respectively (Fig. 1). The maximum of the AIE difference can reach 100%. As expected, the difference between ε=0 and 300 is generally larger than that between ε=0 and 0.4. The effect of ε is stronger for the "POLL" case than that for the "CLEAN" case.
To further explore the underlying physics, Fig. 2 shows the relationship of the relative 15 differences in CF, LWP, and AIE (for ε=0 and 300) as a function of x c . It is clear that larger differences in these three quantities are generally associated with larger values of x c , suggesting that an exact representation of the threshold behavior becomes more important as the autoconversion becomes less efficient. This is expected because all threshold functions gradually approach 1 as x c decreases to 0. The association of a 20 larger difference with a larger x c explains why the effect of ε is stronger for the "POLL" case than for the "CLEAN" case as shown in Fig. 1. For the "CLEAN" case, as daytime heating progresses, the cloud water is depleted so quickly (Fig. 1a and 1c) that the magnitude of x c jumps from <<1 to ∼1. Consequently, the "CLEAN" clouds transform quickly from one stage (with efficient drizzle 25 production) to another stage (between drizzling and non-drizzling). But for the "POLL" case, due to their continental origins and the strong large-scale subsidence (Guo et al., 2007b), these "POLL" clouds tend to precipitate less efficiently. As shown in Fig. 2, the "CLEAN" and "POLL" cases span a wide range of x c from 0.1 to 40, covering con-0 to 300, the daily averaged CF and LWP differ by <3%, whereas the daily averaged AIE differs by up to 15%. The larger difference in the AIE is due to the magnification of the differences in CF and LWP by the stronger insolation in the afternoon.
These differences in the instantaneous and averaged values imply that the influence of the "smoothness" of threshold representations (determined by ε) is scale-dependent: 10 more significant for the instantaneous values than for the daily averages. Furthermore, the effect of ε strongly depends on x c or the percentage of clouds in the transitional stage (with weak drizzle or between drizzling and non-drizzling); and the same is expected to hold true for global averages.
4.2 Sensitivity to the critical radius (r c ) 15 As discussed in Sect. 2, in addition to ε, the theoretical threshold function also depends on x c . Since x c is a function of critical radius (r c ) and r c is a parameter widely used in existing autoconversion parameterizations, the sensitivity to x c can be replaced by the sensitivity to r c .
To better understand the effect of r c and compare it to the effect of ε, we have performed sensitivity tests on r c by applying the analytical r c and by prescribing r c to be 10, 15, 20, and 25 µm. 5 Figure 3 shows the results for the "CLEAN" case with ε=0 (i.e., the Kessler-type threshold function). The results with the analytical r c are shown as the base case (black), and the results with r c =10, 15, 20 µm are shown in percentage differences relative to this base case. Evidently, a larger prescribed r c is associated with smaller CF and LWP. At first glance, this association seems contradictory to the hypothesis that 10 decreased precipitation leads to larger CF and LWP (Albrecht, 1989). Deeper analysis reveals that the smaller CF and LWP are due to the stabilization of boundary layer by precipitation formation. Smaller precipitation with increasing r c results in higher entrainment drying and thereby more efficient depletion of cloud water (Guo et al., 2007b). In the afternoon, the instantaneous CF and LWP (with r c =20 µm) are reduced 15 as much as 60% and 50%, respectively. Consequently, clouds exhibit a more significant diurnal change. This is due to a positive feedback between the cloud geometric radius and entrainment. A smaller cloud radius allows cloudy air to have a larger surface area to mix with drier ambient air, and thus enhances entrainment drying (Blyth et al., 1988).
The relative difference in the instantaneous AIE is even more striking and varies by 20 up to ∼80% in the daytime (Fig. 3c). The magnitude of the AIE tends to decrease with a larger r c , corresponding to the decreased CF and LWP. The variation in the AIE tends to be larger than that in CF or in LWP. The difference between the base case and the sensitivity test with r c =10 µm as was suggested by an observational study (Pawlowska and Brenguier, 2003) is minimal, 25 because the averaged analytical r c is ∼10 µm although it varies from 7 to 16 µm . This good agreement provides observational support for using the analytical r c .
The daily averaged CF, LWP, and AIE are reduced by 20%, 20%, and 40%, respectively, when the prescribed r c =20 µm is used (as compared to the analytical r c ); and 5 Discussion and concluding remarks 5 The sensitivities of the cloud fraction (CF), liquid water path (LWP), and aerosol indirect effect (AIE) to different threshold representations associated with the autoconversion process were systematically examined by applying a theoretical threshold function to a 3-D cloud-resolving model. We have found that 1. the sensitivity to threshold representations is larger for the instantaneous CF, LWP, 10 and AIE than for the corresponding daily averages; 2. the sensitivity depends critically on the critical-to-mean mass ratio of cloud water (x c ), or the percentage of transitional clouds with weak drizzle or between drizzling and non-drizzling; 3. the sensitivity to critical radius (r c ) is more significant than the sensitivity to the 15 "smoothness" of threshold representations as embodied in the relative dispersion of droplet size distribution (ε) for drizzling clouds; 4. the relative differences in the instantaneous CF, LWP, and AIE (for ε=0 and 300) are as large as 20%, 40%, and 100%, respectively. But the daily averages are less sensitive to ε; 20 5. both CF and LWP tend to vary most significantly during the local afternoon when different values of ε and r c are used. The magnification of the variations in CF and LWP by the stronger insolation near local noon leads to a larger variation in the AIE. Therefore, the relative differences in the AIE are larger than those in CF or in LWP. Introduction

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The theoretical threshold function in Eq.
(2) is derived from first principles, so its use would be desirable in modeling studies (Lohmann et al., 2007); but it may be more complex than is warranted in current GCMs especially when the complicated subgrid cloud variation is involved (personal communications with L. Rotstayn and A. Chen, 2007). In order to explore whether the above differences also exist in the threshold 5 functions generally used in GCMs, we have conducted similar sensitivity tests using the generalized Sundqvist threshold function (Liu et al., 2006b), and have obtained similar results (not shown here).
Although this study covers all existing types of ad hoc threshold functions, the effect of ε here should not be considered to be the total effect of ε on clouds and/or aerosol 10 forcing, because the current expression for r c (and thus x c ) does not account for ε .  Fig. 1, but for the relative differences (Rel. Diff.) between the base case with the analytical critical radius r c (Ana. r c ) and the sensitivity tests with the prescribed r c of 10 µm (blue), 15 µm (green), 20 µm (red) for the "CLEAN" case.