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1 JULY 2003 JEFFERY AND AUSTIN 1621 Uniﬁed Treatment of Thermodynamic and Optical Variability in a Simple Model of Unresolved Low Clouds CHRISTOPHER A. JEFFERY AND PHILIP H. AUSTIN Atmospheric Sciences Programme, University of British Columbia, Vancouver, British Columbia, Canada (Manuscript received 16 May 2001, in ﬁnal form 6 January 2003) ABSTRACT Comparative studies of global climate models have long shown a marked sensitivity to the parameterization of cloud properties. Early attempts to quantify this sensitivity were hampered by diagnostic schemes that were inherently biased toward the contemporary climate. Recently, prognostic cloud schemes based on an assumed statistical distribution of subgrid variability replaced the older diagnostic schemes in some models. Although the relationship between unresolved variability and mean cloud amount is known in principle, a corresponding relationship between ice-free low cloud thermodynamic and optical properties is lacking. The authors present a simple, analytically tractable statistical optical depth parameterization for boundary layer clouds that links mean reﬂectivity and emissivity to the underlying distribution of unresolved ﬂuctuations in model thermodynamic variables. To characterize possible impacts of this parameterization on the radiative budget of a large-scale model, they apply it to a zonally averaged climatology, illustrating the importance of a coupled treatment of subgrid-scale condensation and optical variability. They derive analytic expressions for two response functions that characterize two potential low cloud feedback scenarios in a warming climate. 1. Introduction qt q s , is a particularly acute problem (Manabe and Wetherald 1967). Statistical cloud schemes have a long history that The Sommeria–Deardorff–Mellor (SDM) statistical dates back to the pioneering work of Sommeria and cloud scheme introduces a stochastic subgrid variable Deardorff (1977) and Mellor (1977). Large-scale at- s that represents unresolved ﬂuctuations in q s q t and mospheric models typically contain temperature, pres- is assumed to be normally distributed.1 The variance of sure, and total water (vapor liquid) ﬁelds that evolve s, 2, in more sophisticated schemes can be diagnosed s according to prescribed dynamical and thermodynami- from a turbulence model (Ricard and Royer 1993) or cal equations. Traditionally these numerical models from neighboring cells (Levkov et al. 1998; Cusack et would assign, for each ﬁeld, a single average value to al. 1999) but, in practice, is often taken as a prescribed an individual grid cell, thereby ignoring any variability fraction (Smith 1990) of q s . A key assumption in the 2 within the cell. The relative importance of this neglected SDM scheme is that each grid cell is assumed to contain variability is, not surprisingly, scale dependent; for a complete ensemble of s from which the statistics of large-scale climate models with grid spacings of 250 unresolved cloud are calculated, regardless of the size km or greater the unresolved variability can be a sub- of the grid or the time step of the model. For example, stantial fraction of the mean value (Barker et al. 1996). the mean liquid water to some power p, q lp , in the cloudy Furthermore, the relative importance of subgrid vari- region of a cell is given by ability is magniﬁed manyfold by the presence of con- densation, which is a small difference in two relatively qt qs large scalar quantities: the saturation vapor density, q s , q lp Ad 1 p a L (q t qs s) p Ps (s) ds and the cell’s total vapor (or water) density, q t , prior to condensation. Early climate modelers were well aware qt qs that the use of ‘‘all-or-nothing’’ condensation schemes, Ad Ps (s) ds, (1) whereby an individual grid cell is either completely clear or completely cloudy depending on the difference where Ps is the probability distribution function (e.g., Gaussian) of s; the cloud density, Ad , is the fraction of Corresponding author address: Christopher A. Jeffery, Los Ala- mos National Laboratory (NIS-2), P.O. Box 1663, Mail Stop D-436, Los Alamos, NM 87545. 1 This notation differs from Mellor (1997), where s represents ﬂuc- E-mail: cjeffery@lanl.gov tuations in q t q s. 2003 American Meteorological Society 1622 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 grid cell occupied by cloud; and aL 1 is a parameter tion of sky covered by cloud when viewed from below. that accounts for the subadiabatic liquid water proﬁles Thus, for example, an increase in q l (or ) caused by typically observed in layer clouds. In what follows an increasing temperatures does not necessarily imply an overbar is reserved to represent an average over the cloudy increase in R if A c decreases, producing an optically fraction of a cell or column of cells, brackets · represent thicker cloud ﬁeld with smaller cloud fraction. The cou- a spatial average over the entire cell/column, and unre- pling of our statistical approach to surface temperature solved variability in each cell is assumed to be centered allows us to investigate the combined ( A c / T, / T) (i.e., have zero mean). response within an analytic framework. Statistical cloud schemes, in their current form, pro- The ﬁrst theoretical study, and one of the only studies vide complete information about q l but only limited in- to date, that investigates the coupled ( A c , ) response formation on cloud optical properties. This is because of a cloud layer to increasing temperature ( T) while optical depth, , is a vertical integral of q lp (z) from cloud holding, alternatively, both A c and ﬁxed is that by base to cloud top and variability in q l has nonzero spatial Temkin et al. (1975). Temkin et al. (1975) compared correlations produced by turbulence, that is, s(z1 )s(z 2 ) and contrasted the temperature sensitivity of A c at ﬁxed 0. Thus while the SDM scheme does provide grid- , ( A c / T) , with the temperature sensitivity of at column-averaged optical depth , it does not provide ﬁxed A c , ( / T)Ac in a simpliﬁed atmosphere with one or higher-order moments without further assumption. cloud layer and constant surface relative humidity (RH). In section 2 below we link thermodynamic and optical They found ( A c / T) 0 and ( / T)Ac 0 using a variability in the SDM scheme by ﬁrst restricting s and nonstochastic model. These results indicate a negative hence P s to be height independent in low clouds. At the low cloud feedback (LCF), where LCF is deﬁned as the same time, we consider a distribution of cloud-top change in the net downward shortwave ﬂux at the top height ﬂuctuations (z top) that is distinct from a height- of the cloud layer produced by a positive temperature independent P s . The resulting low-dimensional model change. is analytically tractable and requires only the speciﬁ- There have been numerous modeling and observa- cation of the z-independent joint z top -s distribution func- tional studies that suggest a range of values for the sign tion to completely determine P (Jeffery 2001). and magnitude of this cloud radiative response. For ex- Our approach extends the results of Considine et al. ample, a negative low cloud ( ) feedback at high lati- (1997), who showed that normally distributed cloud tudes is also implied by the positive liquid water sen- thickness ﬂuctuations can produce distributions of in- sitivity q l / T found by Somerville and Remer (1984) in Russian aircraft measurements, assuming a negligible tegrated cloud liquid water path (LWP) (or, equivalently cloud thickness sensitivity. In contrast, Schneider et al. in their approximation, optical thickness) that qualita- (1978) suggested that warming leads to increased con- tively match Landsat satellite cloud observations for a vection and vertical q t transport and a resulting atmo- range of cloud fractions. Our approach also builds upon sphere that is unable to increase q t sufﬁciently to main- the recent work of Wood and Taylor (2001), who linked tain constant RH. A relative drying of the lower at- s and P but did not consider z top. Below we derive mosphere with warming implies that global cloud feed- general forms for LWP(s, z top) and (s, z top) in a layer back may not be negative. Support for a positive cloud with horizontally ﬂuctuating cloud top and cloud base. feedback is provided by the global climate model We also adopt a radiation parameterization that incor- (GCM) study of Hansen et al. (1984), who found that porates P into the calculation of longwave and short- clouds contribute a feedback of 1.5 C—nearly 1 C wave ﬂuxes. due to a reduction in low clouds—resulting in a net Our analytic expressions for allow us to combine climate sensitivity double that found in an earlier study ﬂuctuations in s and z top into a single subgrid variable with ﬁxed clouds (Manabe and Stouffer 1979). By 1990, s*, and we examine the radiative response of the sta- all 19 of the GCMs compared by Cess et al. (1990) tistical cloud scheme to changes in the variance of s* predicted a decrease in globally averaged A c with in- and hence the optical thickness distribution P . We creasing temperature, although the sign and magnitude choose a form for P s* suitable for large-scale models of the net cloud feedback varies considerably from mod- and similar to the triangle distribution of Smith (1990), el to model. More recently, Tselioudis et al. (1993) an- adopting his choice for the temperature dependence of alyzed global satellite observations of low cloud and unresolved variability, 2 (T). With this modeled * qs 2 s found a generally negative optical depth sensitivity, / coupling of P to the surface temperature through q s , T, (positive cloud feedback) which increases from the we use the parameterization to investigate the change midlatitudes to the Tropics. A positive low/midlatitude in net mean cloud reﬂectivity for a speciﬁed temperature low cloud ( ) feedback is also implied by the temper- change, given an idealized climatology. ature dependence of satellite observations of liquid wa- Understanding the response of cloud-layer reﬂectivity ter path (Greenwald et al. 1995). to increasing temperature is complicated by the fact that In this article we expand on the approach of Temkin the total reﬂectivity ( R A c R ) of a cloud layer is a et al. (1975) and calculate analytic response functions nonlinear function of and cloud fraction A c : the frac- for our statistical treatment of low cloud optical vari- 1 JULY 2003 JEFFERY AND AUSTIN 1623 ability. We ﬁnd ( A c / T) 0 in contradistinction with {A} H : {A 0} H 0, {A 0} H A is a Heaviside Temkin et al. (1975). This result links the observational bracket and we have used z bot w (q 0 1 q t s) from evidence of a largely negative sensitivity (Tselioudis assumption 2. The key feature of Eqs. (2) and (3) is et al. 1993; Greenwald et al. 1995) with GCM simu- that ﬂuctuations in z bot are deﬁned by inverting q l (z bot , lations that predict a negative A c response (Cess et al. s) 0, whereas the unresolved cloud-top height ﬂuc- 1990) as we discuss in section 4. tuations z top are absorbed into the new subgrid variability The article is organized as follows. In section 2 we s*. Thus our distribution Pz top can, in principle, be com- derive our statistical model and compare the predictions bined with P s to give P s* . This is advantageous because of the model with satellite data taken from Barker et al. unresolved variability of q t , q s , and z top is contained in (1996). The behavior of our scheme is analyzed in sec- the single parameter s* and P s* is z independent. In tion 3 using an idealized zonally averaged climatology analogy with the SDM scheme we assume that s* is and in section 4 we present A c– –T response functions. centered and P s* is known. The rhs of Eq. (2) should Section 5 contains a summary. not be confused with q lp 1(z top , x)a L 1 w 1/(p 1) from Eq. (1) since the statistics of s* generally differ from the statistics of s. 2. Model description Formulation of the shortwave and longwave optical Our model of boundary layer cloud optical variability depths follows from Eq. (2) given the appropriate func- is based on two assumptions: 1) horizontal subgrid var- tional relation # func(q l , . . .) dz. At this point, it iability in the boundary layer of large-scale models ex- is convenient to introduce the cloud thickness h(x) ceeds vertical variability; and 2) cloud liquid water in- z top (x) z bot (x) so that Eq. (2) is simply creases linearly with height above cloud base, that is, z top(x) (a L )p p 1 w q s (z) q0 w z where w 0. Assumption 1 is ac- q lp (z, x) dz h (x). (4) curate for large-scale temperature and moisture ﬂuctu- z bot(x) p 1 ations because the horizontal length of a grid cell in a The longwave optical depth, because it depends primarily climate model is much greater than the boundary layer on the LWP, is strictly given by the ‘‘h 2 model’’: height. However, it does not hold near cloud top where the vertical dependence of s at the cloud boundary is (x) h 2 (x) complex. We overcome this deﬁciency by introducing aL a distribution of unresolved cloud-top height ﬂuctua- {q z q0 s*(x)} H . 2 (5) 2 w t w top tions, P z top, that is distinct from a z-independent P s , al- though s and z top may be correlated. Our second as- In contrast, a number of different formulations exist sumption is well supported both numerically and ex- for the shortwave optical depth. For example, writing perimentally in the literature. in terms of LWP and a constant effective radius (reff ) Given assumptions 1 and 2 we are now in a position recovers the h 2 model. In layer clouds a better approach to calculate the optical statistics of low clouds. For clar- due to Pontikis (1993) is to assume proportionality of ity and brevity, we ﬁrst introduce the notation. The var- the effective and volume-averaged radius, reff q1/3, l iable dependence (x) labels unresolved horizontal var- giving p 2/3 and a 5/3 dependence of on h [Pontikis iability, whereas (z) indicates a vertical dependence, 1993, (Eq. 5)]. Thus we have the ‘‘h 5/3 model’’ for short- which, by assumption 1 is nonstochastic; that is, subgrid wave optical depth: vertical ﬂuctuations are assumed negligible. Further- more, (x) represents unresolved variability in a single 3a L 3 2/ (x) {q z q0 s*(x)} H 3 . 5/ (6) 5 w t w top cell whereas (z) is a continuous dependence that may extend through a column of cells. Note that inherent in Eq. (6) is the approximation that the cloud droplet concentration, N, is independent of a. Linking Ps and Pz top to P s*, an approximation that is likely accurate if the ma- jority of low clouds in a grid cell are non- or weakly Consider the integral of q lp(z, x) from cloud base precipitating. z bot (x) to cloud top z top (x) z top z top(x): The constant of proportionality in Eq. (6) goes as N 1/3 z top(x) (Pontikis 1993; see also appendix C). In what follows we q lp (z, x) dz ignore the effect of unresolved ﬂuctuations in N on the z bot(x) statistics of . This approximation is unlikely to be valid p near large sources of N, for example, major industrial aL w 1 {q t z w top q0 p s*(x)} H 1 , (2) cities, but is justiﬁable elsewhere because the 5/3 moment p 1 of ql acts to magnify ﬂuctuations while the 1/3 moment where of N acts to damp ﬂuctuations. This behavior is illustrated with the following example. Consider the dependence of s*(x) s(x) z (x), w top (3) (Y 0 Y ) on mean value Y 0 and the variance of Y , 1624 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 2 Y , where Y is normally distributed. Writing (Y 0 Y ) from LESs of a cloud-topped boundary layer. Thus we Y0 Y we ﬁnd ( 1/3, 0.15) and ( 5/3, ﬁnd that the variance of s* 0.5) for the range Y Y 0 2 Y . Hence, increasing 2 ql at ﬁxed ql,0 acts to increase (ql,0 ql ) 5/3 as expected q0 but a similar increase in N decreases (N 0 N )1/3 be- 2 s* T { z w top Ri 1 } 2 . (8) T cause of the damping effect of the 1/3 exponent. This result suggests that the normalized variance of N would have to As mentioned previously s (z) is usually assumed to be be 3 to 4 times larger than the normalized variance of ql proportional to q s (z), that is, T constant. Assuming for unresolved droplet number ﬂuctuations to have a com- that ztop and Ri are T independent as well permits a parable effect on the statistics of . particularly simple form for the temperature dependence The moments of can be calculated from P s* in anal- of P s* . ogy with P s in Eq. (1) while A c is given by Recent observational studies (Norris 1998a,b; Bajuk and Leovy 1998; Chen et al. 2000) have indicated the q t q s(z top) importance of cloud type in the analysis of cloud prop- Ac Ps* (s*) ds*. (7) erties; in principle the ratio s* / s could be parameter- ized as a function of cloud type diagnosed from various Intuitively, we might expect the maximum cloud overlap stability and potential energy considerations. On the oth- assumption, A c max(A d ) A d (z top ), to hold for a er hand, in consideration of the poor boundary layer model of cloud variability that ignores vertical varia- vertical resolution of typical GCMs and in the absence tions in s. However, comparing Eqs. (1) and (7) we ﬁnd of knowledge of s* / s , we follow current GCM pa- that it does not hold generally since A c is calculated rameterizations and assume s* s q s in section from P s* and not P s . The failure of the maximum cloud 3. Note that z top is coupled to q s through w in Eq. (3). overlap assumption is due to the independence of z top Since s* is z independent by deﬁnition in our scheme and s in our approach. Note that the usual cell averaged we will use q s (z) at the surface [i.e., q 0 (T)] to evaluate quantities, For example, q l (z), are independent of z top the temperature dependence of s* in section 3. and should be calculated with P s . c. Radiation b. Approximations for Ps and Ps* Currently, most GCMs lack the methodology to in- clude unresolved variability in the calculation of cloud Knowledge of both P s and P z top, and hence P s* , is reﬂectivity (R) or emissivity ( ). This deﬁciency may limited. Cloud ensemble (Xu and Randall 1996), large have important implications for the prediction of global eddy simulations (LES; Cuijpers and Bechtold 1995), cloud feedback discussed above. Through the use of a and observational studies (Larson et al. 2001) provide statistical cloud scheme [e.g., SDM, Eq. (1)], many support for a Gaussian P s . Comparatively less is known modern GCMs couple changes in cloud properties [e.g., about P z top. Ground-based (Boers et al. 1988; Albrecht ( Ac , )] in a changing climate to the distribution of et al. 1990) and space-based (Strawbridge and Hoff the subgrid variability s. But they also use the plane- 1996; Loeb et al. 1998) retrievals of z top suggest a stan- parallel homogeneous (PPH) assumption Rpph R( ), dard deviation of 50–100 m for marine stratus over which decouples the optical properties R and from the typical GCM length ( 100 km) and time ( 2 h) scales. underlying thermodynamic cloud variability. The con- A comprehensive analysis of the shape of Pz top is cur- vexity of the functions relating R and to ensures that rently lacking. the optical bias incurred from the PPH assumption is Observational (Klein and Hartmann 1993; Oreopou- positive, that is, R and are overestimated. To reduce los and Davies 1993; Norris and Leovy 1994; Klein et this bias many current GCMs use an effective optical al. 1995; Bony et al. 1997) studies of the marine bound- depth eff where 0.7 to calculate Rpph (Cahalan ary layer over relatively long timescales suggest that s et al. 1994). Although may be tuned in a particular is largely a function of boundary layer temperature, T, GCM to reproduce the measured radiative stream, this while z top is largely controlled by the jump in potential approach is ad hoc in nature and becomes increasingly temperature ( ) at the top of the boundary layer. De- inaccurate as the climate departs from its present state. ﬁning an interfacial Richardson number (Deardorff Below we present a uniﬁed treatment of the thermo- 1981) dynamic and optical variability of boundary layer clouds Ri (g/T )z top /w* 2 based on the SDM scheme. The utility of Eqs. (5) and (6) is not in the calculation where w * is Deardorff (1974)’s convective velocity of directly but, rather, in providing a methodology to scale, Moeng et al. (1999) estimate the standard devi- include subgrid variability in the reﬂectivity and emis- ation of z top as sivity. We do this by calculating P ( ) from P s* using the equations above and a change of variable; then by z top /z top 0.6 Ri 1 deﬁnition 1 JULY 2003 JEFFERY AND AUSTIN 1625 X X( )P ( ) d , (9) where X R or and P is the distribution of in the cloudy part of the column. This approach is discussed in Considine et al. (1997) and Pincus and Klein (2000), albeit not in the context of a generalized framework of unresolved variability. Unfortunately, the analytic ex- pressions for R and are sufﬁciently unwidely to prevent an analytic evaluation of Eq. (9). Another approach, pioneered by Barker (1996), is to assume an analytically friendly form for P that is both sufﬁciently general to approximate P over a wide range of conditions and that allows a closed-form expression for X . Barker et al. (1996) analyzed satellite data of marine low clouds and found that a generalized dis- tribution, P ( ), closely approximates the observed dis- tribution and allows Eq. (9) to be integrated analytically. The last step in our treatment of unresolved optical variability is to relate P to Eqs. (5) or (6) and P s* . The 2 shape of P is controlled by the parameter / 2, which is a measure of the width of the distribution rel- FIG. 1. Plot of A c vs for the h 2 and h 5/3 models calculated using Eqs. (5)–(7), and P s from appendix B. Landsat data in the range ative to its mean. Barker (1996), who introduced P ( , 6.5 from Table 2 *of Barker et al. (1996) is also shown for com- ), did not relate , and in particular 2 , to the unre- parison. solved thermodynamic variability s. Using the frame- work we have presented thus far, we calculate and 2 (and thus ) using Eqs. (5) or (6), and P s* substituted therefore introduce a modiﬁed triangle distribution (ap- for P s in Eq. (1). Analytic expressions for R ( , ) and pendix B) that is similar to Smith’s scheme (or a Gauss- ( , ) used in our analysis in section 3 are given in ian) at large A c but better reproduces Gaussian behavior Barker (1996) and Barker and Wielicki (1997), respec- at small A c . In particular, our distribution gives P s* tively. 0 for a wider range of s*, | s* | (35/3)1/2 s* , than Smith (1990)’s triangle, | s* | (6) 1/2 s . An expression * for the arbitrary moment is given in appendix B. d. Comparison with Landsat data A comparison of A c versus is shown in Fig. 1 for Our scheme provides a one-to-one relationship be- the h 2 and h 5/3 models calculated using Eqs. (5)–(7), and tween and A c for a given P s* that can be tested against our new triangle distribution (appendix B). Also shown the Landsat satellite data compiled by Barker et al. is a subset, ∈ (0, 6.5), of the Landsat data tabulated (1996). To make such a comparison we need to specify in Table 2 of Barker et al. (1996). Agreement between the distribution P s* . Considine et al. (1997) assumed a the data and both theoretical models is good despite normally distributed h—equivalent to a normally dis- uncertainties in the retrieval of A c from the satellite tributed s* in our formulation—and found that the A c scenes (Barker et al. 1996). It should be noted that ∈ dependence of the LWP distribution predicted by the h 2 (6.5, 12) has a signiﬁcant impact on the calculation of model is consistent with Landsat data. Using the (Gauss- R and and is in close agreement with the model, while ian) Considine model, Wood and Taylor (2001) derived the selected data shown in Fig. 1 emphasize the region an approximate relationship between LWP and LWP for ∈ (0.5, 3). However, it is encouraging that the as- large A c and veriﬁed this relationship using First ISCCP ymptotic behavior near 0.5 predicted by our model (International Satellite Cloud Climatology) Regional is consistent with the data. Experiment data. In appendix A we present exact an- alytic results for LWP and LWP 2 , and hence LWP , that 3. Low cloud radiative feedback using a simple are valid for all A c . climatology A potential disadvantage in assuming a normally dis- tributed s* is that closed-form expressions for nonin- As a demonstration of the behavior of our statistical teger moments, for example, the h 5/3 model, are not cloud scheme, we now consider the coupling of unre- available. The triangle distribution ﬁrst used by Smith solved variability and cloud feedback in the model of (1990) is a computationally efﬁcient surrogate for the section 2 using a zonally averaged climatology. We con- Gaussian distribution that is analytically tractable, but sider only the response of low clouds and we specify a it does not accurately mimic a Gaussian at small A c . We ﬁxed (2 C) surface temperature perturbation. Since the 1626 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 prognosed cloud changes do not feed back into the tem- perature perturbation our model experiments are an open-loop study. While our model neglects the merid- ional structure of cloud amount caused by atmospheric dynamics, for example, the storm tracks, and meridional variations in surface properties or s (Rotstayn 1997), we incorporate what we consider to be the major lati- tudinal dependencies: solar zenith angle, saturation va- por density, and surface albedo. We further assume that droplet number concentration is T independent. In this section and in the spirit of Temkin et al. (1975), we consider three different responses of a zonally av- eraged climatology to a ﬁxed global increase in tem- perature. In our ‘‘observationally constrained ’’ re- sponse (CTobs ) model mean optical depth decreases with increasing T according to a parameterization (appendix D) of the satellite observations of Tselioudis et al. (1993). By specifying the sensitivity and assuming q s (z 0, T) q 0 (T) we then predict A c using s * Eqs. (6) and (7). Following Temkin et al. (1975) we also consider (i) a ‘‘constrained ’’ response (CT) model with constant , and (ii) a ‘‘constrained A c’’ response FIG. 2. Comparison of R and with PPH values Rpph R( ) and (CA) model where A c remains constant and we deter- pph ( ). The convexity of R and ensures that the PPH approx- imation overestimates R and . Shortwave [Eq. (6)], longwave mine from our uniﬁed scheme. In these calculations [Eq. (5)], and A c [Eq. (7)] are all calculated using P s from appendix we do not determine q t and z top independently; for ex- B. Expressions for R and are from Barker (1996) *and Barker and ample, the negative optical depth sensitivity observed Wielicki (1997), respectively. Reﬂectivities are diurnally averaged at by Tselioudis et al. (1993) could result from a decrease equinox. See appendix C for parameter values. in z top despite increasing speciﬁc humidity (Tselioudis et al. 1998). Here our CTobs , CT, and CA model exper- iments represent three possible scenarios of the climate’s mation used by GCMs. The well-documented plane- response to increasing temperatures that may have very parallel albedo (reﬂectivity) bias (Cahalan et al. 1994) complex dynamical origins and spatial structure. We do of Rpph , roughly 0.06 in our model, is visible in the lower not make any claims that either CTobs , CT, or CA is a half of the ﬁgure, but it is overshadowed by the much ‘‘most probable’’ low cloud response. Rather, we hope larger bias of pph that averages near 0.3. Early studies to use these experiments to gain some insight into the of feedback (Temkin et al. 1975; Somerville and Re- relationship between low cloud radiative feedback and mer 1984) assumed that longwave optical properties of the coupled ( A c , ) response. clouds are saturated ( 1 pph ) and as a result, First consider our base-state climatology. Our zonally changes in only affect the cloud’s shortwave proper- averaged model extends over latitudes 60 S to ties. Although, as shown in Fig. 2, is not saturated at 60 N where our ‘‘grid cells’’ encompass one latitudinal global scales in our model, assumptions concerning the band. We assume that our new triangle distribution (ap- behavior of are not signiﬁcant for low cloud radiative pendix B) deﬁnes the shape of the distribution P s* that forcing calculations since the longwave forcing is very represents meridional ﬂuctuations in ‘‘subgrid’’ vari- small. Note that the PPH biases calculated using zonally ability, and that this form is independent of and T. averaged values of and A c are larger than the biases Our base-state climatology is constructed from the A c ( ) associated with a typical, partially cloudy GCM grid measurements of Warren et al. (1988) [See Ramaswamy cell for which the cloud fraction tends to exceed that and Chen (1993) and Kogan et al. (1997) for a similar of our zonal climatology. Figure 2 reiterates that a cou- approach.] Mean optical depth ( ) ∈ (3.5,6.2) is es- pled treatment of thermodynamic and optical variability timated from the satellite measurements presented in can substantially impact the predicted values of low Hatzianastassiou and Vardavas (1999). Parameter values cloud R and in a GCM cloud parameterization. are given in appendix C. By specifying A c , , T( ), and We now turn our attention to the modeled response of q s (T), we solve for the two unknowns s* and q t low cloud properties to warming. Consider a T 2C w z top . We use the more accurate h model for (h 2 globally uniform warming where the sensitivity (T 5/3 model for longwave ). T) is prescribed to be (mostly) negative in the CTobs The mean reﬂectivity and emissivity of our base-state model, (T T) (T) constrains the CT model and climatology, averaged over the diurnal cycle at equinox, Ac(T) Ac(T T) constrains the CA model. The low is shown in Fig. 2 along with the corresponding values cloud feedback predicted by the CTobs , CT, and CA models predicted by the plane-parallel homogeneous approxi- is shown in Fig. 3. As before we deﬁne LCF as the change 1 JULY 2003 JEFFERY AND AUSTIN 1627 ative contribution of the A c response has been shaded in Fig. 3. The shading reveals that the total CTobs feed- back is dominated by the negative A c response, even in the Tropics where the change in is largest (Tselioudis et al. 1993). This behavior is in qualitative agreement with the recent 2 CO 2 GCM experiments of Tselioudis et al. (1998, their Fig. 14), which show a relatively small feedback of 0.2 C compared to the 1.5 C A c feed- back reported with an older version of the same GCM (Hansen et al. 1984). The dominance of A c over feed- back is also in agreement with the regional observational studies of Oreopoulos and Davies (1993) (Tropics) and Bony et al. (1997) (subtropics), which imply that the negative A c response may make a larger contribution to shortwave low cloud feedback than the negative re- sponse. Several other observational studies (Klein and Hartmann 1993; Norris and Leovy 1994; Klein et al. 1995) also provide support for a negative A c response. 4. A c– –T response functions FIG. 3. Plot showing LCF predicted by the CTobs , CT, and CA Observational studies of cloud fraction sensitivity ( Ac / models for a uniform 2 C increase in global mean temperature. Since T) and optical depth sensitivity ( / T) are often used to low clouds cool the earth by reﬂecting solar radiation, less low cloud (CTobs and CT) enhances warming (positive cloud feedback) while provide insight into cloud feedback. As pointed out by more low cloud (CA) buffers the warming (negative cloud feedback). Arking (1991), the information provided by these studies The shaded region indicates the dominant contribution of the A c re- is limited because it is not known which parameters are sponse to the overall CTobs feedback. See appendix D for calculation held ﬁxed and which are allowed to vary. In this section details. we present analytic response functions, ( Ac / T) and ( / T)Ac, for our subgrid-scale cloud parameterization that in the net (positive downward) radiative ﬂux at the top of do not suffer from this deﬁciency. Our use of the termi- the boundary layer. The calculations employ the diurnally nology ‘‘response function’’ is an analogy to response averaged equinox R and predicted by our uniﬁed ap- functions in the theory of thermodynamics, for example, proach and the approximation s* (T T) 1 q 0 (T speciﬁc heat and adiabatic compressibility of an ideal gas. T) where 1 s (T)/q 0 (T). The ﬁgure illustrates that We compare our results with the earlier study by Temkin * the CTobs and CT feedbacks are positive and considerably et al. (1975), discussed in section 1, and assess the impact larger in magnitude than the negative LCF ( feedback) of coarse vertical resolution on a discrete numerical eval- of the CA model. Moreover, in the tropical regime 20 uation of these functions. 20 the CTobs cloud feedback—a mixture of neg- Combining Eqs. (6) and (7), s* 1 qs , w 2 qs ative Ac and response—is as much as 3.5 times as large and Smith’s triangle distribution for subgrid variability as the CT cloud feedback. The CTobs and CT cloud feed- (Smith 1990), we derive backs are also generally larger than the 2 CO 2 forcing of 4 W m 2. However, it is important to emphasize that lnA c 4 1 qs 4 L q (10) LCF is not a top-of-the-atmosphere feedback. In particular, T , 5 s T 5 R T2 modulation of the longwave stream through changes in high cloud properties could enhance or buffer the net cloud ln 2 L feedback. , (11) T 3 R T2 Further analysis (not shown) reveals that LCF is rel- Ac, atively insensitive to the treatment of unresolved optical valid for Ac 0.5, where { 1, 2 }, L is the latent variability; LCF computed using plane-parallel homo- heat of vaporization, R is the gas constant for water vapor, geneous optical properties overestimates the CTobs cloud and recall that qs must be evaluated at some ﬁxed height feedback by 15% and the CA feedback by 35% com- (e.g., at the surface) since s* and w are z independent pared to the predicted values shown in Fig. 3. Thus, by deﬁnition. For Ac 0.5, Eq. (11) remains valid but clearly it is the constrained response of our modeled for Eq. (10) ( lnAc / T) , decays monotonically to zero climate, that is, CTobs vs. CT vs. CA, and not the pa- as Ac → 1. The disappearance of the Ac response as Ac rameterization of optical variability that determines LCF → 1 reﬂects the increasing independence of Ac to small to ﬁrst order. This ﬁnding is consistent with the GCM changes in s* in the limit of vanishing unresolved var- sensitivity study of Rotstayn (1999), among others. iability. Overall the more general result ( lnAc / T) , To further explore the CTobs cloud feedback, the rel- 0 and ( ln / T)Ac, 0 is valid for all Ac. 1628 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 We can interpret Eqs. (10) and (11) as representing ticular GCMs tend to underpredict persistent marine two potential low cloud shortwave feedback scenarios stratocumulus cloud sheets in eastern ocean subsidence in a warming climate demarcated by / T 0 and A c / regions (Browning 1994; Bushell and Martin 1999). T 0, respectively. Let F 0 be the net (positive Typically GCMs have only four to six model levels in downward) shortwave radiative ﬂux reﬂected by the the boundary layer (BL) and the vertical resolution of (unforced) low clouds and T 0 be the thermal forc- these levels usually decreases with height. As a result, ing. Consider the small approximation F A c . Then the top model level in the BL will dominate the discrete Eq. (10) implies LCF (4/5)F T/T*, a positive cloud integration of q lp [Eq. (2)] and hence . In this low feedback, while for Eq. (11), LCF (2/3)F T/T*, a resolution limit the h 5/3 model for shortwave optical negative cloud feedback, where T* R T 2 /L . depth [Eq. (6)] becomes Although we make no claims regarding the likelihood of the two scenarios described by Eqs. (10) and (11), (x) {q t z w top q0 s*(x)} 2/3 z, the difference in sign of Eqs. (10) and (11) leads to a where z is the thickness of the model level centered nontrivial asymmetry between the (A c , ) response and at z top . Computing the low vertical resolution response LCF. Using Eq. (10) we ﬁnd that ( / T) 0 is a functions we ﬁnd that the response [Eq. (11)] remains sufﬁcient condition for a positive LCF while Eq. (11) unchanged while the A c response becomes implies that ( A c / T) 0 is a sufﬁcient condition for a negative LCF. These relations follow from the positive lnA c L coupling between A c and , that is, ( / A c ) T, 0. On 2 , (13) T , 1, z R T2 the other hand, ( / T) 0 and ( A c / T) 0 do not uniquely specify the sign of the LCF. Thus our statistical independent of w . A comparison of Eqs. (10) and (13) approach links the observational evidence of a largely reveals that RH-based implementations of statistical negative sensitivity (Tselioudis et al. 1993; Greenwald cloud schemes in low vertical resolution GCMs tend to et al. 1995; Bony et al. 1997) with GCM simulations overestimate the unresolved low cloud A c response by (Hansen et al. 1984; Wetherald and Manabe 1986; Col- a factor of 2.5 for A c 0.5, compared to the same man and McAvaney 1997; Yao and Del Genio 1999) statistical cloud scheme run at higher vertical resolution. that predict a negative A c sensitivity and a positive LCF. It is important to note that although these GCMs do not 5. Summary explicitly use a statistical cloud scheme, their RH-based grid-cell parameterizations for A d (and hence A c ) are Understanding the complex interaction of clouds, ra- formally analogous to Eq. (1) for A d with 2 s q s (T). 2 diation, and climate is a formidable challenge; the sign It is interesting to compare our negative A c response and magnitude of the global cloud feedback remains a function, Eq. (10), with the result ( A c / T) 0 derived question of concern and debate. In this study we focus by Temkin et al. (1975) using a nonstochastic model. on one facet of the cloud–climate interaction problem, In the Temkin et al. (1975) model, the increase in avail- namely, the relationship between the thermodynamic able liquid water with increasing temperature (recall RH cloud properties A c and and the optical properties R is ﬁxed) is placed in a formerly clear column that there- and within the context of a statistical cloud scheme. by increases A c . In contrast, in our statistical approach We restrict our attention to low clouds where the vertical the cloud thickness decreases in the face of increasing proﬁle of cloud liquid water is linear and where hori- w resulting in a negative A c sensitivity. zontal variability dominates. Assuming a known distri- We extend Eq. (11) to another useful form through bution of unresolved variability that includes cloud-top the approximation ( ln / T)Ac, 2( lnR / T)Ac, valid height ﬂuctuations, we derive a self-consistent and com- for O(5), giving putationally efﬁcient set of equations for A c and the moments of , thereby incorporating subgrid optical ln R 1 L ﬂuctuations into the statistical cloud schemes ﬁrst in- . (12) T 3 R T2 troduced in the 1970s (Sommeria and Deardorff 1977; Ac, Mellor 1977). This uniﬁed treatment of thermodynamic Since LCF is relatively insensitive to changes in we and optical variability is particularly well suited for use can combine Eqs. (10) and (12): in a GCM that incorporates a subgrid-scale turbulence |(LCF) , | scheme (Ricard and Royer 1993). 2.4, When cloud-top height ﬂuctuations and temperature/ |(LCF) A c , | moisture ﬂuctuations are treated as a single random var- which illustrates that, in general, Ac feedback dominates iable, then our model of longwave optical depth (liquid the feedback in this model. The approximate 1:2.4 LCF water path) reduces to the Considine et al. (1997) model ratio is illustrated by the CT and CA models in Fig. 3. if this new random variable is normally distributed. This We can also use our response functions to quantify approach, however, is not always valid. For example, a the effect of low model vertical resolution on LCF. The minimum large-scale lifting condensation level—break- representation of low clouds in GCMs is poor; in par- ing the reﬂection symmetry of cloud-base and cloud- 1 JULY 2003 JEFFERY AND AUSTIN 1629 top height ﬂuctuations—requires that cloud-base and using the h 2 model for LWP [see Eq. (5)], z top 0, cloud-top height ﬂuctuations be treated distinctly (Jef- Gaussian P s , and assuming small s /q c where q c qt fery and Davis 2002). Recent improvements in the re- w z top q 0 . Wood and Taylor (2001) state Eq. (A1) trieval of cloud physical properties using multiple re- is accurate to better than 5% for s /q c 1/2. Below mote sensors (Clothiaux et al. 2000; Wang and Sassen we present analytic expressions for LWP and LWP 2 , 2001) should provide more information on the joint sta- and hence LWP , that are valid for all s /q c . tistics of cloud-base and cloud-top height ﬂuctuations Using Eq. (5) and Gaussian P s we ﬁnd that could, in principle, be incorporated into our treat- * ment of low-cloud optical depth. aL qc s* 2 2 Our uniﬁed approach can also be used to probe the LWP q2 2 Ac 1 e q c / 2 s* 2 w c s* 2 sensitivity of parameterized cloud fraction and optical depth to changes in temperature. The coupled ( A c , aL 2 ) global response of clouds to increasing temperature LWP 2 q4 6q c 2 2 3 4 4 w c 2 s* s* is analogous to the response of an open thermodynamic system. Although the particular thermodynamic trajec- tory that the system follows may be very sensitive to Ac 1 2 2 (q c 3 s* 5q c 3 s* )e q c / 2 s* , (A2) external forcing and boundary conditions, much can be 2 learned by computing response functions where one of the thermodynamic coordinates is ﬁxed along the tra- where A c erfc( q c / 2 s )/2. Expanding (A2) to * jectory. This approach was ﬁrst considered by Temkin fourth order in small s /q c gives * et al. (1975), who found ( A c ) 0 and ( )Ac 0 1/ 2 2 1/ 2 using a nonstochastic model of a simpliﬁed atmosphere 2a L aL s* with one cloud layer and constant surface RH. LWP s* LWP , w 4 w Using our statistical treatment of cloud optical vari- ability, we derive analytic response functions in (Ac, , T) from which (A1) follows approximately. Using Eq. (A2) space that demonstrate the overall dominance of the cloud we ﬁnd that (A1) is accurate to better than 7% for s* / fraction feedback in the model. In contradistinction to Te- qc 1/2. mkin et al. (1975), we ﬁnd ( Ac ) 0. In particular, we A potential disadvantage of Eq. (A2) is that corre- show that the global observational evidence of a largely sponding closed-form expressions for the h 5/3 model are negative optical depth sensitivity presented by Tselioudis not available; the modiﬁed triangle distribution intro- et al. (1993) produces in the model a much stronger neg- duced in appendix B has tractable noninteger moments ative cloud fraction response and therefore a net positive and exhibits Gaussian behavior in close agreement with low cloud feedback. Also we ﬁnd that low model vertical (A2). resolution can cause a signiﬁcant overestimation of the unresolved low cloud Ac response by a factor of around APPENDIX B 2.5. The accuracy of these results rests upon the crucial assumption that low-cloud Ac may be parameterized as a function of only relative humidity, an assumption that is Modiﬁed Triangle Distribution typically made in large-scale models. Improvement in our Our modiﬁed triangle distribution is understanding of the factors that control Ac at large scales is therefore a necessary next step towards the reﬁnement 3 in the formulation of the (Ac, ) response functions intro- 3 5|s| |s| P(s) 1 1 , duced in this work. 2 0 3 0 0 Acknowledgments. We are grateful to Nicole Jeffery 0 s 0 , (B1) for a careful reading of the manuscript. We thank three anonymous reviewers for very thorough and construc- where 2 0 (35/3) 2. Using Eqs. (1) and (B1), cloud s tive comments. This work was supported through fund- fraction A c A d (z top ) is ing of the Modeling of Clouds and Climate Proposal by the Canadian Foundation for Climate and Atmospheric 0 QN 1 Sciences, the Meteorological Service of Canada, and (1 Q N ) 4 (1 Q N )/2 1 QN 0 the Natural Sciences and Engineering Research Council. Ac 1 (1 Q N ) (1 Q N )/2 0 QN 1 4 APPENDIX A 1 1 QN , Gaussian Relations for the h 2 Model where Q N q c / 0 and q c qt w z top q 0 . The - th moment of the cloud liquid water used in the cal- Recently Wood and Taylor (2001) derived 1/ 2 culation of via Eqs. (5) or (6) follows in a similar LWP (2a L w 1 )1/ 2 LWP s (A1) manner: 1630 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 0 QN 1 REFERENCES A c F1 1 1 QN 0 (q c s) Albrecht, B. A., C. W. Fairall, D. W. 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