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Unified Treatment of Thermodynamic and Optical Variability in a Simple Model of Unresolved Low Clouds Austin, Philip H. 2011

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1 JULY 2003 1621J E F F E R Y A N D A U S T I N q 2003 American Meteorological Society 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 Statistical cloud schemes have a long history that dates back to the pioneering work of Sommeria and Deardorff (1977) and Mellor (1977). Large-scale at- mospheric models typically contain temperature, pres- sure, and total water (vapor 1 liquid) fields that evolve according to prescribed dynamical and thermodynami- cal equations. Traditionally these numerical models would assign, for each field, a single average value to an individual grid cell, thereby ignoring any variability within the cell. The relative importance of this neglected variability is, not surprisingly, scale dependent; for large-scale climate models with grid spacings of 250 km or greater the unresolved variability can be a sub- stantial fraction of the mean value (Barker et al. 1996). Furthermore, the relative importance of subgrid vari- ability is magnified manyfold by the presence of con- densation, which is a small difference in two relatively large scalar quantities: the saturation vapor density, qs, and the cell’s total vapor (or water) density, qt, prior to condensation. Early climate modelers were well aware that the use of ‘‘all-or-nothing’’ condensation schemes, whereby an individual grid cell is either completely clear or completely cloudy depending on the difference 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. E-mail: cjeffery@lanl.gov qt 2 qs, is a particularly acute problem (Manabe and Wetherald 1967). The Sommeria–Deardorff–Mellor (SDM) statistical cloud scheme introduces a stochastic subgrid variable s that represents unresolved fluctuations in qs 2 qt and is assumed to be normally distributed.1 The variance of s, , in more sophisticated schemes can be diagnosed2s s from a turbulence model (Ricard and Royer 1993) or from neighboring cells (Levkov et al. 1998; Cusack et al. 1999) but, in practice, is often taken as a prescribed fraction (Smith 1990) of . A key assumption in the2qs SDM scheme is that each grid cell is assumed to contain a complete ensemble of s from which the statistics of unresolved cloud are calculated, regardless of the size of the grid or the time step of the model. For example, the mean liquid water to some power p, , in the cloudypql region of a cell is given by q 2qt s p p21 pq 5 A a (q 2 q 2 s) P (s) dsl d E L t s s 2` q 2qt s A 5 P (s) ds, (1)d E s 2` where Ps is the probability distribution function (e.g., Gaussian) of s; the cloud density, Ad, is the fraction of 1 This notation differs from Mellor (1997), where s represents fluc- tuations in qt 2 qs. 1622 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S grid cell occupied by cloud; and aL , 1 is a parameter that accounts for the subadiabatic liquid water profiles typically observed in layer clouds. In what follows an overbar is reserved to represent an average over the cloudy fraction of a cell or column of cells, brackets ^ · & represent a spatial average over the entire cell/column, and unre- solved variability in each cell is assumed to be centered (i.e., have zero mean). Statistical cloud schemes, in their current form, pro- vide complete information about ql but only limited in- formation on cloud optical properties. This is because optical depth, t, is a vertical integral of (z) from cloudpql base to cloud top and variability in ql has nonzero spatial correlations produced by turbulence, that is, ^s(z1)s(z2)& ± 0. Thus while the SDM scheme does provide grid- column-averaged optical depth ^t&, it does not provide or higher-order moments without further assumption.t In section 2 below we link thermodynamic and optical variability in the SDM scheme by first restricting s and hence Ps to be height independent in low clouds. At the same time, we consider a distribution of cloud-top height fluctuations ( ) that is distinct from a height-z9top independent Ps. The resulting low-dimensional model is analytically tractable and requires only the specifi- cation of the z-independent joint -s distribution func-z9top tion to completely determine Pt (Jeffery 2001). Our approach extends the results of Considine et al. (1997), who showed that normally distributed cloud thickness fluctuations can produce distributions of in- tegrated cloud liquid water path (LWP) (or, equivalently in their approximation, optical thickness) that qualita- tively match Landsat satellite cloud observations for a range of cloud fractions. Our approach also builds upon the recent work of Wood and Taylor (2001), who linked s and Pt but did not consider . Below we derivez9top general forms for LWP(s, ) and t(s, ) in a layerz9 z9top top with horizontally fluctuating cloud top and cloud base. We also adopt a radiation parameterization that incor- porates Pt into the calculation of longwave and short- wave fluxes. Our analytic expressions for t allow us to combine fluctuations in s and into a single subgrid variablez9top s * , and we examine the radiative response of the sta- tistical cloud scheme to changes in the variance of s * and hence the optical thickness distribution Pt . We choose a form for Ps* suitable for large-scale models and similar to the triangle distribution of Smith (1990), adopting his choice for the temperature dependence of unresolved variability, ; (T). With this modeled2 2s qs s* coupling of Pt to the surface temperature through qs, we use the parameterization to investigate the change in net mean cloud reflectivity for a specified temperature change, given an idealized climatology. Understanding the response of cloud-layer reflectivity to increasing temperature is complicated by the fact that the total reflectivity (^R& 5 Ac ) of a cloud layer is aR nonlinear function of t and cloud fraction Ac: the frac- tion of sky covered by cloud when viewed from below. Thus, for example, an increase in (or ) caused byq tl increasing temperatures does not necessarily imply an increase in ^R& if Ac decreases, producing an optically thicker cloud field with smaller cloud fraction. The cou- pling of our statistical approach to surface temperature allows us to investigate the combined (]Ac /]T, ] /]T)t response within an analytic framework. The first theoretical study, and one of the only studies to date, that investigates the coupled (DAc, D ) responset of a cloud layer to increasing temperature (DT) while holding, alternatively, both Ac and fixed is that byt Temkin et al. (1975). Temkin et al. (1975) compared and contrasted the temperature sensitivity of Ac at fixed , (]Ac /]T) , with the temperature sensitivity of att tt fixed Ac, (] /]T) in a simplified atmosphere with onet Ac cloud layer and constant surface relative humidity (RH). They found (]Ac /]T) . 0 and (] /]T) . 0 using att Ac nonstochastic model. These results indicate a negative low cloud feedback (LCF), where LCF is defined as the change in the net downward shortwave flux at the top of the cloud layer produced by a positive temperature change. There have been numerous modeling and observa- tional studies that suggest a range of values for the sign and magnitude of this cloud radiative response. For ex- ample, a negative low cloud (t) feedback at high lati- tudes is also implied by the positive liquid water sen- sitivity ]ql/]T found by Somerville and Remer (1984) in Russian aircraft measurements, assuming a negligible cloud thickness sensitivity. In contrast, Schneider et al. (1978) suggested that warming leads to increased con- vection and vertical qt transport and a resulting atmo- sphere that is unable to increase qt sufficiently to main- tain constant RH. A relative drying of the lower at- mosphere with warming implies that global cloud feed- back may not be negative. Support for a positive cloud feedback is provided by the global climate model (GCM) study of Hansen et al. (1984), who found that clouds contribute a feedback of ø1.58C—nearly 18C due to a reduction in low clouds—resulting in a net climate sensitivity double that found in an earlier study with fixed clouds (Manabe and Stouffer 1979). By 1990, all 19 of the GCMs compared by Cess et al. (1990) predicted a decrease in globally averaged Ac with in- creasing temperature, although the sign and magnitude of the net cloud feedback varies considerably from mod- el to model. More recently, Tselioudis et al. (1993) an- alyzed global satellite observations of low cloud t and found a generally negative optical depth sensitivity, ]t/ ]T, (positive cloud feedback) which increases from the midlatitudes to the Tropics. A positive low/midlatitude low cloud (t) feedback is also implied by the temper- ature dependence of satellite observations of liquid wa- ter path (Greenwald et al. 1995). In this article we expand on the approach of Temkin et al. (1975) and calculate analytic response functions for our statistical treatment of low cloud optical vari- 1 JULY 2003 1623J E F F E R Y A N D A U S T I N ability. We find (]Ac /]T) , 0 in contradistinction witht Temkin et al. (1975). This result links the observational evidence of a largely negative sensitivity (Tselioudist et al. 1993; Greenwald et al. 1995) with GCM simu- lations that predict a negative Ac response (Cess et al. 1990) as we discuss in section 4. The article is organized as follows. In section 2 we derive our statistical model and compare the predictions of the model with satellite data taken from Barker et al. (1996). The behavior of our scheme is analyzed in sec- tion 3 using an idealized zonally averaged climatology and in section 4 we present Ac– –T response functions.t Section 5 contains a summary. 2. Model description Our model of boundary layer cloud optical variability is based on two assumptions: 1) horizontal subgrid var- iability in the boundary layer of large-scale models ex- ceeds vertical variability; and 2) cloud liquid water in- creases linearly with height above cloud base, that is, qs(z) 5 q0 2 Gwz where Gw . 0. Assumption 1 is ac- curate for large-scale temperature and moisture fluctu- ations because the horizontal length of a grid cell in a climate model is much greater than the boundary layer height. However, it does not hold near cloud top where the vertical dependence of s at the cloud boundary is complex. We overcome this deficiency by introducing a distribution of unresolved cloud-top height fluctua- tions, P , that is distinct from a z-independent Ps, al-9z top though s and may be correlated. Our second as-z9top sumption is well supported both numerically and ex- perimentally in the literature. Given assumptions 1 and 2 we are now in a position to calculate the optical statistics of low clouds. For clar- ity and brevity, we first introduce the notation. The var- iable dependence (x) labels unresolved horizontal var- iability, whereas (z) indicates a vertical dependence, which, by assumption 1 is nonstochastic; that is, subgrid vertical fluctuations are assumed negligible. Further- more, (x) represents unresolved variability in a single cell whereas (z) is a continuous dependence that may extend through a column of cells. a. Linking Ps and P to Pt9ztop Consider the integral of (z, x) from cloud basepql zbot(x) to cloud top z top(x) 5 z top 1 (x):z9top z (x)top pq (z, x) dzE l z (x)bot p 21a GL w p115 {q 1 G z 2 q 2 s*(x)} , (2)t w top 0 Hp 1 1 where s (x) 5 s(x) 2 G z9 (x),w top* (3) {A}H : {A , 0}H 5 0, {A $ 0}H 5 A is a Heaviside bracket and we have used zbot 5 (q0 2 qt 1 s) from21G w assumption 2. The key feature of Eqs. (2) and (3) is that fluctuations in zbot are defined by inverting ql(zbot, s) 5 0, whereas the unresolved cloud-top height fluc- tuations are absorbed into the new subgrid variabilityz9top s * . Thus our distribution P can, in principle, be com-9z top bined with Ps to give Ps*. This is advantageous because unresolved variability of qt, qs, and z top is contained in the single parameter s * and Ps* is z independent. In analogy with the SDM scheme we assume that s * is centered and Ps* is known. The rhs of Eq. (2) should not be confused with (z top, x) /(p 1 1) fromp11 21 21q a Gl L w Eq. (1) since the statistics of s * generally differ from the statistics of s. Formulation of the shortwave and longwave optical depths follows from Eq. (2) given the appropriate func- tional relation t ; # func(ql, . . .) dz. At this point, it is convenient to introduce the cloud thickness h(x) 5 z top(x) 2 zbot(x) so that Eq. (2) is simply z (x)top p(a G )L wp p11q (z, x) dz 5 h (x). (4)E l p 1 1 z (x)bot The longwave optical depth, because it depends primarily on the LWP, is strictly given by the ‘‘h2 model’’: 2t(x) ; h (x) aL 2; {q 1 G z 2 q 2 s*(x)} . (5)t w top 0 H2Gw In contrast, a number of different formulations exist for the shortwave optical depth. For example, writing t in terms of LWP and a constant effective radius (reff) recovers the h2 model. In layer clouds a better approach due to Pontikis (1993) is to assume proportionality of the effective and volume-averaged radius, reff ; ,1/3ql giving p 5 2/3 and a 5/3 dependence of t on h [Pontikis 1993, (Eq. 5)]. Thus we have the ‘‘h5/3 model’’ for short- wave optical depth: 2/33aL 5/3t(x) ; {q 1 G z 2 q 2 s*(x)} . (6)t w top 0 H5Gw Note that inherent in Eq. (6) is the approximation that the cloud droplet concentration, N, is independent of s * , an approximation that is likely accurate if the ma- jority of low clouds in a grid cell are non- or weakly precipitating. The constant of proportionality in Eq. (6) goes as N1/3 (Pontikis 1993; see also appendix C). In what follows we ignore the effect of unresolved fluctuations in N on the statistics of t. This approximation is unlikely to be valid near large sources of N, for example, major industrial cities, but is justifiable elsewhere because the 5/3 moment of ql acts to magnify fluctuations while the 1/3 moment of N acts to damp fluctuations. This behavior is illustrated with the following example. Consider the dependence of ^(Y0 1 Y9)a& on mean value Y0 and the variance of Y9, 1624 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S , where Y9 is normally distributed. Writing ^(Y0 1 Y9)a&2sY ; we find (a 5 1/3, b ø 20.15) and (a 5 5/3,a2b bY s0 Y b ø 0.5) for the range sY # Y0 # 2sY. Hence, increasing s at fixed ql,0 acts to increase ^(ql,0 1 )5/3& as expectedq9q ll but a similar increase in sN decreases ^(N0 1 N9)1/3& be- cause of the damping effect of the 1/3 exponent. This result suggests that the normalized variance of N would have to be 3 to 4 times larger than the normalized variance of ql for unresolved droplet number fluctuations to have a com- parable effect on the statistics of t. The moments of t can be calculated from Ps* in anal- ogy with Ps in Eq. (1) while Ac is given by q 2q (z )t s top A 5 P (s*) ds*. (7)c E s* 2` Intuitively, we might expect the maximum cloud overlap assumption, Ac 5 max(Ad) 5 Ad(z top), to hold for a model of cloud variability that ignores vertical varia- tions in s. However, comparing Eqs. (1) and (7) we find that it does not hold generally since Ac is calculated from Ps* and not Ps. The failure of the maximum cloud overlap assumption is due to the independence of z9top and s in our approach. Note that the usual cell averaged quantities, For example, (z), are independent ofq z9l top and should be calculated with Ps. b. Approximations for Ps and Ps* Knowledge of both Ps and P , and hence Ps*, is9z top limited. Cloud ensemble (Xu and Randall 1996), large eddy simulations (LES; Cuijpers and Bechtold 1995), and observational studies (Larson et al. 2001) provide support for a Gaussian Ps. Comparatively less is known about P . Ground-based (Boers et al. 1988; Albrecht9z top et al. 1990) and space-based (Strawbridge and Hoff 1996; Loeb et al. 1998) retrievals of suggest a stan-z9top dard deviation of 50–100 m for marine stratus over typical GCM length (ø100 km) and time (ø2 h) scales. A comprehensive analysis of the shape of P is cur-9z top rently lacking. Observational (Klein and Hartmann 1993; Oreopou- los and Davies 1993; Norris and Leovy 1994; Klein et al. 1995; Bony et al. 1997) studies of the marine bound- ary layer over relatively long timescales suggest that s is largely a function of boundary layer temperature, T, while z top is largely controlled by the jump in potential temperature (Du) at the top of the boundary layer. De- fining an interfacial Richardson number (Deardorff 1981) 2Ri [ (g /T )z Du /w*top where w * is Deardorff (1974)’s convective velocity scale, Moeng et al. (1999) estimate the standard devi- ation of asz9top 21s /z ø 0.6 Riz9 toptop from LESs of a cloud-topped boundary layer. Thus we find that the variance of s * 2 ]q02 21 2s ; s 1 {G z Ri } . (8)s* T w top1 2]T As mentioned previously ss(z) is usually assumed to be proportional to qs(z), that is, sT 5 constant. Assuming that ztop and Ri are T independent as well permits a particularly simple form for the temperature dependence of Ps*.Recent observational studies (Norris 1998a,b; Bajuk and Leovy 1998; Chen et al. 2000) have indicated the importance of cloud type in the analysis of cloud prop- erties; in principle the ratio ss*/ss could be parameter-ized as a function of cloud type diagnosed from various stability and potential energy considerations. On the oth- er hand, in consideration of the poor boundary layer vertical resolution of typical GCMs and in the absence of knowledge of ss*/ss, we follow current GCM pa- rameterizations and assume ss* ; ss ; qs in section3. Note that s is coupled to qs through Gw in Eq. (3).z top Since s * is z independent by definition in our scheme we will use qs(z) at the surface [i.e., q0(T)] to evaluate the temperature dependence of ss* in section 3. c. Radiation Currently, most GCMs lack the methodology to in- clude unresolved variability in the calculation of cloud reflectivity (R) or emissivity (e). This deficiency may have important implications for the prediction of global cloud feedback discussed above. Through the use of a statistical cloud scheme [e.g., SDM, Eq. (1)], many modern GCMs couple changes in cloud properties [e.g., (DAc, D )] in a changing climate to the distribution oft the subgrid variability s. But they also use the plane- parallel homogeneous (PPH) assumption 5 R( ),R tpph which decouples the optical properties R and e from the underlying thermodynamic cloud variability. The con- vexity of the functions relating R and e to t ensures that the optical bias incurred from the PPH assumption is positive, that is, and are overestimated. To reduceR e this bias many current GCMs use an effective optical depth teff 5 x where x ø 0.7 to calculate (Cahalant Rpph et al. 1994). Although x may be tuned in a particular GCM to reproduce the measured radiative stream, this approach is ad hoc in nature and becomes increasingly inaccurate as the climate departs from its present state. Below we present a unified treatment of the thermo- dynamic and optical variability of boundary layer clouds based on the SDM scheme. The utility of Eqs. (5) and (6) is not in the calculation of t directly but, rather, in providing a methodology to include subgrid variability in the reflectivity and emis- sivity. We do this by calculating Pt(t) from Ps* using the equations above and a change of variable; then by definition 1 JULY 2003 1625J E F F E R Y A N D A U S T I N FIG. 1. Plot of Ac vs n for the h2 and h5/3 models calculated using Eqs. (5)–(7), and Ps * from appendix B. Landsat data in the range n # 6.5 from Table 2 of Barker et al. (1996) is also shown for com- parison. X 5 X(t)P (t) dt , (9)E t where X 5 R or e and Pt is the distribution of t 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 e 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 Pt that is both sufficiently general to approximate Pt over a wide range of conditions and that allows a closed-form expression for . Barker et al. (1996) analyzed satellite data ofX marine low clouds and found that a generalized g dis- tribution, Pg (t), 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 Pg to Eqs. (5) or (6) and Ps*. The shape of Pg is controlled by the parameter n 5 2/ ,2t s t which is a measure of the width of the distribution rel- ative to its mean. Barker (1996), who introduced Pg (n, t), did not relate n, and in particular , to the unre-2s t solved thermodynamic variability s. Using the frame- work we have presented thus far, we calculate andt (and thus n) using Eqs. (5) or (6), and Ps* substituted2t for Ps in Eq. (1). Analytic expressions for (n, ) andR t (n, ) used in our analysis in section 3 are given ine t Barker (1996) and Barker and Wielicki (1997), respec- tively. d. Comparison with Landsat data Our scheme provides a one-to-one relationship be- tween n and Ac for a given Ps* that can be tested against the Landsat satellite data compiled by Barker et al. (1996). To make such a comparison we need to specify the distribution Ps*. Considine et al. (1997) assumed a normally distributed h—equivalent to a normally dis- tributed s * in our formulation—and found that the Ac dependence of the LWP distribution predicted by the h2 model is consistent with Landsat data. Using the (Gauss- ian) Considine model, Wood and Taylor (2001) derived an approximate relationship between and sLWP forLWP large Ac and verified this relationship using First ISCCP (International Satellite Cloud Climatology) Regional Experiment data. In appendix A we present exact an- alytic results for and , and hence sLWP, that2LWP LWP are valid for all Ac. A potential disadvantage in assuming a normally dis- tributed s * is that closed-form expressions for nonin- teger moments, for example, the h5/3 model, are not available. The triangle distribution first used by Smith (1990) is a computationally efficient surrogate for the Gaussian distribution that is analytically tractable, but it does not accurately mimic a Gaussian at small Ac. We therefore introduce a modified triangle distribution (ap- pendix B) that is similar to Smith’s scheme (or a Gauss- ian) at large Ac but better reproduces Gaussian behavior at small Ac. In particular, our distribution gives Ps* .0 for a wider range of s * , | s * | , (35/3)1/2ss*, thanSmith (1990)’s triangle, | s * | , (6)1/2ss*. An expression for the arbitrary moment is given in appendix B.lt A comparison of Ac versus n is shown in Fig. 1 for the h2 and h5/3 models calculated using Eqs. (5)–(7), and our new triangle distribution (appendix B). Also shown is a subset, n ∈ (0, 6.5), of the Landsat data tabulated in Table 2 of Barker et al. (1996). Agreement between the data and both theoretical models is good despite uncertainties in the retrieval of Ac from the satellite scenes (Barker et al. 1996). It should be noted that n ∈ (6.5, 12) has a significant impact on the calculation of and and is in close agreement with the model, whileR e the selected data shown in Fig. 1 emphasize the region n ∈ (0.5, 3). However, it is encouraging that the as- ymptotic behavior near n 5 0.5 predicted by our model is consistent with the data. 3. Low cloud radiative feedback using a simple climatology As a demonstration of the behavior of our statistical cloud scheme, we now consider the coupling of unre- solved variability and cloud feedback in the model of section 2 using a zonally averaged climatology. We con- sider only the response of low clouds and we specify a fixed (28C) surface temperature perturbation. Since the 1626 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S FIG. 2. Comparison of and with PPH values 5 R( ) andR e R tpph 5 e( ). The convexity of R and e ensures that the PPH approx-e tpph imation overestimates and . Shortwave [Eq. (6)], longwaveR e t t [Eq. (5)], and Ac [Eq. (7)] are all calculated using Ps * from appendix B. Expressions for and are from Barker (1996) and Barker andR e Wielicki (1997), respectively. Reflectivities are diurnally averaged at equinox. See appendix C for parameter values. 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 ss (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-t 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 D and assumingt ss* ; qs(z 5 0, T) 5 q0(T) we then predict DAc usingEqs. (6) and (7). Following Temkin et al. (1975) we also consider (i) a ‘‘constrained ’’ response (CT) modelt with constant , and (ii) a ‘‘constrained Ac’’ responset (CA) model where Ac remains constant and we deter- mine D from our unified scheme. In these calculationst we do not determine qt and z top independently; for ex- ample, the negative optical depth sensitivity observed by Tselioudis et al. (1993) could result from a decrease 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 response to increasing temperatures that may have very complex dynamical origins and spatial structure. We do not make any claims that either CTobs, CT, or CA is a ‘‘most probable’’ low cloud response. Rather, we hope to use these experiments to gain some insight into the relationship between low cloud radiative feedback and the coupled (DAc, D ) response.t First consider our base-state climatology. Our zonally averaged model extends over latitudes f 5 2608S to 608N where our ‘‘grid cells’’ encompass one latitudinal band. We assume that our new triangle distribution (ap- pendix B) defines the shape of the distribution Ps* that represents meridional fluctuations in ‘‘subgrid’’ vari- ability, and that this form is independent of f and T. Our base-state climatology is constructed from the Ac(f) measurements of Warren et al. (1988) [See Ramaswamy and Chen (1993) and Kogan et al. (1997) for a similar approach.] Mean optical depth (f) ∈ (3.5,6.2) is es-t timated from the satellite measurements presented in Hatzianastassiou and Vardavas (1999). Parameter values are given in appendix C. By specifying Ac, , T(f), andt qs(T), we solve for the two unknowns ss* and qt 1 Gwz top. We use the more accurate h5/3 model for t(h2 model for longwave t). The mean reflectivity and emissivity of our base-state climatology, averaged over the diurnal cycle at equinox, is shown in Fig. 2 along with the corresponding values predicted by the plane-parallel homogeneous approxi- mation used by GCMs. The well-documented plane- parallel albedo (reflectivity) bias (Cahalan et al. 1994) of , roughly 0.06 in our model, is visible in the lowerRpph half of the figure, but it is overshadowed by the much larger bias of that averages near 0.3. Early studiesepph of t feedback (Temkin et al. 1975; Somerville and Re- mer 1984) assumed that longwave optical properties of clouds are saturated ( 5 1 ø ) and as a result,e epph changes in t only affect the cloud’s shortwave proper- ties. Although, as shown in Fig. 2, is not saturated ate global scales in our model, assumptions concerning the behavior of are not significant for low cloud radiativee forcing calculations since the longwave forcing is very small. Note that the PPH biases calculated using zonally averaged values of and Ac are larger than the biasest associated with a typical, partially cloudy GCM grid cell for which the cloud fraction tends to exceed that of our zonal climatology. Figure 2 reiterates that a cou- pled treatment of thermodynamic and optical variability can substantially impact the predicted values of low cloud and in a GCM cloud parameterization.R e We now turn our attention to the modeled response of low cloud properties to warming. Consider a DT 5 28C globally uniform warming where the sensitivity (T 1t DT) is prescribed to be (mostly) negative in the CTobs model, (T 1 DT) 5 (T) constrains the CT model andt t Ac(T) 5 Ac(T 1 DT) constrains the CA model. The low cloud feedback predicted by the CTobs, CT, and CA models is shown in Fig. 3. As before we define LCF as the change 1 JULY 2003 1627J E F F E R Y A N D A U S T I N FIG. 3. Plot showing LCF predicted by the CTobs, CT, and CA models for a uniform 28C increase in global mean temperature. Since low clouds cool the earth by reflecting solar radiation, less low cloud (CTobs and CT) enhances warming (positive cloud feedback) while more low cloud (CA) buffers the warming (negative cloud feedback). The shaded region indicates the dominant contribution of the Ac re- sponse to the overall CTobs feedback. See appendix D for calculation details. in the net (positive downward) radiative flux at the top of the boundary layer. The calculations employ the diurnally averaged equinox and predicted by our unified ap-R e proach and the approximation ss*(T 1 DT) ø l1q0(T 1 DT) where l1 5 ss*(T)/q0(T). The figure illustrates that the CTobs and CT feedbacks are positive and considerably larger in magnitude than the negative LCF ( feedback)t of the CA model. Moreover, in the tropical regime 2208 # f # 208 the CTobs cloud feedback—a mixture of neg- ative Ac and response—is as much as 3.5 times as larget as the CT cloud feedback. The CTobs and CT cloud feed- backs are also generally larger than the 2 3 CO2 forcing of ø4 W m22. However, it is important to emphasize that LCF is not a top-of-the-atmosphere feedback. In particular, modulation of the longwave stream through changes in high cloud properties could enhance or buffer the net cloud feedback. Further analysis (not shown) reveals that LCF is rel- atively insensitive to the treatment of unresolved optical variability; LCF computed using plane-parallel homo- geneous optical properties overestimates the CTobs cloud feedback by 15% and the CA feedback by 35% com- pared to the predicted values shown in Fig. 3. Thus, clearly it is the constrained response of our modeled climate, that is, CTobs vs. CT vs. CA, and not the pa- rameterization of optical variability that determines LCF to first order. This finding is consistent with the GCM sensitivity study of Rotstayn (1999), among others. To further explore the CTobs cloud feedback, the rel- ative contribution of the Ac response has been shaded in Fig. 3. The shading reveals that the total CTobs feed- back is dominated by the negative Ac response, even in the Tropics where the change in is largest (Tselioudist et al. 1993). This behavior is in qualitative agreement with the recent 2 3 CO2 GCM experiments of Tselioudis et al. (1998, their Fig. 14), which show a relatively small feedback of ø0.28C compared to the ø1.58C Ac feed-t back reported with an older version of the same GCM (Hansen et al. 1984). The dominance of Ac over feed-t 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 Ac response may make a larger contribution to shortwave low cloud feedback than the negative re-t sponse. Several other observational studies (Klein and Hartmann 1993; Norris and Leovy 1994; Klein et al. 1995) also provide support for a negative Ac response. 4. Ac– –T response functionst Observational studies of cloud fraction sensitivity (]Ac/ ]T) and optical depth sensitivity (] /]T) are often used tot provide insight into cloud feedback. As pointed out by Arking (1991), the information provided by these studies is limited because it is not known which parameters are held fixed and which are allowed to vary. In this section we present analytic response functions, (]Ac/]T) andt (] /]T) , for our subgrid-scale cloud parameterization thatt Ac do not suffer from this deficiency. Our use of the termi- nology ‘‘response function’’ is an analogy to response functions in the theory of thermodynamics, for example, specific heat and adiabatic compressibility of an ideal gas. We compare our results with the earlier study by Temkin et al. (1975), discussed in section 1, and assess the impact of coarse vertical resolution on a discrete numerical eval- uation of these functions. Combining Eqs. (6) and (7), ss* ø l1qs, Gw ø l2qs and Smith’s triangle distribution for subgrid variability (Smith 1990), we derive ] lnA 4 ]q 4 Lc s y215 2 q 5 2 (10)s 21 2]T 5 ]T 5 R Tyt ,l ] lnt 2 Ly5 , (11) 21 2]T 3 R TyA ,lc valid for Ac # 0.5, where l 5 {l1, l2}, Ly is the latent heat of vaporization, Ry is the gas constant for water vapor, and recall that qs must be evaluated at some fixed height (e.g., at the surface) since ss* and Gw are z independentby definition. For Ac . 0.5, Eq. (11) remains valid but for Eq. (10) (] lnAc/]T) ,l decays monotonically to zerot as Ac → 1. The disappearance of the Ac response as Ac → 1 reflects the increasing independence of Ac to small changes in ss* in the limit of vanishing unresolved var-iability. Overall the more general result (] lnAc/]T) ,l #t 0 and (] ln /]T) $ 0 is valid for all Ac.t A ,lc 1628 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S We can interpret Eqs. (10) and (11) as representing two potential low cloud shortwave feedback scenarios in a warming climate demarcated by ] /]T 5 0 and ]Ac/t ]T 5 0, respectively. Let F , 0 be the net (positive downward) shortwave radiative flux reflected by the (unforced) low clouds and DT . 0 be the thermal forc- ing. Consider the small t approximation F ; Ac . Thent Eq. (10) implies LCF 5 2(4/5)FDT/T * , a positive cloud feedback, while for Eq. (11), LCF 5 (2/3)FDT/T * , a negative cloud feedback, where T * 5 Ry T 2/Ly . Although we make no claims regarding the likelihood of the two scenarios described by Eqs. (10) and (11), the difference in sign of Eqs. (10) and (11) leads to a nontrivial asymmetry between the (Ac, ) response andt LCF. Using Eq. (10) we find that (] /]T)l # 0 is at sufficient condition for a positive LCF while Eq. (11) implies that (]Ac/]T)l $ 0 is a sufficient condition for a negative LCF. These relations follow from the positive coupling between Ac and , that is, (] /]Ac)T,l . 0. Ont t the other hand, (] /]T)l . 0 and (]Ac/]T)l , 0 do nott uniquely specify the sign of the LCF. Thus our statistical approach links the observational evidence of a largely negative sensitivity (Tselioudis et al. 1993; Greenwaldt et al. 1995; Bony et al. 1997) with GCM simulations (Hansen et al. 1984; Wetherald and Manabe 1986; Col- man and McAvaney 1997; Yao and Del Genio 1999) that predict a negative Ac sensitivity and a positive LCF. It is important to note that although these GCMs do not explicitly use a statistical cloud scheme, their RH-based grid-cell parameterizations for Ad (and hence Ac) are formally analogous to Eq. (1) for Ad with ; (T).2 2s qs s It is interesting to compare our negative Ac response function, Eq. (10), with the result (]Ac/]T) . 0 derivedt by Temkin et al. (1975) using a nonstochastic model. In the Temkin et al. (1975) model, the increase in avail- able liquid water with increasing temperature (recall RH is fixed) is placed in a formerly clear column that there- by increases Ac. In contrast, in our statistical approach the cloud thickness decreases in the face of increasing Gw resulting in a negative Ac sensitivity. We extend Eq. (11) to another useful form through the approximation (] ln /]T) ø 2(] ln /]T) validt RA ,l A ,lc c for 5 O(5), givingt ] lnR 1 Ly5 . (12) 21 2]T 3 R TyA ,lc Since LCF is relatively insensitive to changes in wee can combine Eqs. (10) and (12): |(LCF) |t ,l 5 2.4,|(LCF) |A ,lc which illustrates that, in general, Ac feedback dominates the feedback in this model. The approximate 1:2.4 LCFt ratio is illustrated by the CT and CA models in Fig. 3. We can also use our response functions to quantify the effect of low model vertical resolution on LCF. The representation of low clouds in GCMs is poor; in par- ticular GCMs tend to underpredict persistent marine stratocumulus cloud sheets in eastern ocean subsidence regions (Browning 1994; Bushell and Martin 1999). Typically GCMs have only four to six model levels in the boundary layer (BL) and the vertical resolution of these levels usually decreases with height. As a result, the top model level in the BL will dominate the discrete integration of [Eq. (2)] and hence t. In this lowpql resolution limit the h5/3 model for shortwave optical depth [Eq. (6)] becomes 2/3t(x) ; {q 1 G z 2 q 2 s (x)} Dz,t w top 0 * where Dz is the thickness of the model level centered at z top. Computing the low vertical resolution response functions we find that the response [Eq. (11)] remainst unchanged while the Ac response becomes ] lnA Lc y5 22 , (13) 21 2]T R Tyt ,l ,Dz1 independent of Gw. A comparison of Eqs. (10) and (13) reveals that RH-based implementations of statistical cloud schemes in low vertical resolution GCMs tend to overestimate the unresolved low cloud Ac response by a factor of 2.5 for Ac # 0.5, compared to the same statistical cloud scheme run at higher vertical resolution. 5. Summary Understanding the complex interaction of clouds, ra- diation, and climate is a formidable challenge; the sign and magnitude of the global cloud feedback remains a question of concern and debate. In this study we focus on one facet of the cloud–climate interaction problem, namely, the relationship between the thermodynamic cloud properties Ac and t and the optical properties R and e within the context of a statistical cloud scheme. We restrict our attention to low clouds where the vertical profile of cloud liquid water is linear and where hori- zontal variability dominates. Assuming a known distri- bution of unresolved variability that includes cloud-top height fluctuations, we derive a self-consistent and com- putationally efficient set of equations for Ac and the moments of t, thereby incorporating subgrid optical fluctuations into the statistical cloud schemes first in- troduced in the 1970s (Sommeria and Deardorff 1977; Mellor 1977). This unified treatment of thermodynamic and optical variability is particularly well suited for use in a GCM that incorporates a subgrid-scale turbulence scheme (Ricard and Royer 1993). When cloud-top height fluctuations and temperature/ moisture fluctuations are treated as a single random var- iable, then our model of longwave optical depth (liquid water path) reduces to the Considine et al. (1997) model if this new random variable is normally distributed. This approach, however, is not always valid. For example, a minimum large-scale lifting condensation level—break- ing the reflection symmetry of cloud-base and cloud- 1 JULY 2003 1629J E F F E R Y A N D A U S T I N top height fluctuations—requires that cloud-base and cloud-top height fluctuations be treated distinctly (Jef- fery and Davis 2002). Recent improvements in the re- trieval of cloud physical properties using multiple re- mote sensors (Clothiaux et al. 2000; Wang and Sassen 2001) should provide more information on the joint sta- tistics of cloud-base and cloud-top height fluctuations that could, in principle, be incorporated into our treat- ment of low-cloud optical depth. Our unified approach can also be used to probe the sensitivity of parameterized cloud fraction and optical depth to changes in temperature. The coupled (DAc, D ) global response of clouds to increasing temperaturet 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 external forcing and boundary conditions, much can be learned by computing response functions where one of the thermodynamic coordinates is fixed along the tra- jectory. This approach was first considered by Temkin et al. (1975), who found (DAc) . 0 and (D ) . 0tt Ac using a nonstochastic model of a simplified atmosphere with one cloud layer and constant surface RH. Using our statistical treatment of cloud optical vari- ability, we derive analytic response functions in (Ac, , T)t space that demonstrate the overall dominance of the cloud fraction feedback in the model. In contradistinction to Te- mkin et al. (1975), we find (DAc) , 0. In particular, wet show that the global observational evidence of a largely negative optical depth sensitivity presented by Tselioudis et al. (1993) produces in the model a much stronger neg- ative cloud fraction response and therefore a net positive low cloud feedback. Also we find that low model vertical resolution can cause a significant overestimation of the unresolved low cloud Ac response by a factor of around 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 typically made in large-scale models. Improvement in our understanding of the factors that control Ac at large scales is therefore a necessary next step towards the refinement in the formulation of the (Ac, ) response functions intro-t duced in this work. Acknowledgments. We are grateful to Nicole Jeffery for a careful reading of the manuscript. We thank three anonymous reviewers for very thorough and construc- tive comments. This work was supported through fund- ing of the Modeling of Clouds and Climate Proposal by the Canadian Foundation for Climate and Atmospheric Sciences, the Meteorological Service of Canada, and the Natural Sciences and Engineering Research Council. APPENDIX A Gaussian Relations for the h2 Model Recently Wood and Taylor (2001) derived 1/2 21 1/2s ø (2a G ) LWP s (A1)LWP L w s using the h2 model for LWP [see Eq. (5)], 5 0,z9top Gaussian Ps, and assuming small ss/qc where qc 5 qt 1 Gwz top 2 q0. Wood and Taylor (2001) state Eq. (A1) is accurate to better than 5% for ss/qc , 1/2. Below we present analytic expressions for and ,2LWP LWP and hence sLWP, that are valid for all ss/qc. Using Eq. (5) and Gaussian Ps* we find q sa c s*L 2 22 2 21 2q /2sc s*LWP 5 q 1 s 1 A ec s* c5 62G Ï2pw 2aL2 4 2 2 4LWP 5 q 1 6q s 1 3sc c s* s*2 54Gw 21Ac 2 23 3 2q /2sc s*1 (q s 1 5q s )e , (A2)c s* c s* 6Ï2p where Ac 5 erfc(2qc/ ss*)/2. Expanding (A2) toÏ2fourth order in small ss*/qc gives 1/2 1/22a s2a L s*Ls ø s LWP 2 ,LWP s*1 2 1 2G 4Gw w from which (A1) follows approximately. Using Eq. (A2) we find that (A1) is accurate to better than 7% for ss*/qc , 1/2. A potential disadvantage of Eq. (A2) is that corre- sponding closed-form expressions for the h5/3 model are not available; the modified triangle distribution intro- duced in appendix B has tractable noninteger moments and exhibits Gaussian behavior in close agreement with (A2). APPENDIX B Modified Triangle Distribution Our modified triangle distribution is 33 5|s| |s| P(s) 5 1 1 1 2 ,1 21 22s 3s s0 0 0 2s # s # s , (B1)0 0 where 5 (35/3) . Using Eqs. (1) and (B1), cloud2 2s s0 s fraction Ac [ Ad(z top) is 0 Q # 21N 4(1 1 Q ) (1 2 Q )/2 21 , Q # 0N N NA 5c 41 2 (1 2 Q ) (1 1 Q )/2 0 , Q , 1N N N 1 1 # Q , N where QN 5 qc/s0 and qc 5 qt 1 Gwz top 2 q0. The l- th moment of the cloud liquid water used in the cal- culation of via Eqs. (5) or (6) follows in a similart manner: 1630 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S 0 Q # 21N 21A F 21 , Q # 0c 1 Nl (q 2 s) 5c 21A (F 1 F ) 0 , Q , 1c 1 2 N 21A (F 1 F 1 F ) 1 # Q , c 1 2 3 N where l l14s (1 1 Q ) 3 2 5Q 24 1 20Q0 N N NF 5 11 52 l 1 1 l 1 2 26 2 30Q 12 1 20QN N1 1 l 1 3 l 1 4 5(1 1 Q )N2 6l 1 5 l l11 3 3 3s Q 2Q 1 3Q Q 2 9Q 9Q0 N N N N N NF 5 1 12 516 l 1 1 l 1 2 l 1 3 33QN2 6l 1 4 l l14s (21 1 Q ) 3 1 5Q 24 2 20Q0 N N NF 5 13 52 l 1 1 l 1 2 26 1 30Q 12 2 20QN N1 1 l 1 3 l 1 4 5(1 2 Q )N2 .6l 1 5 APPENDIX C Parameter Values Parameter values are q0(T) 5 (1.826 3 109 g m23) exp{2Ry /(Ly T)}, Ry 5 461.5 J K21 kg21, Ly 5 2.5 3 106 J kg21, Gw 5 (4 3 1023 K m21) {Ly /(Ry T 2)}q0(T), aL 5 0.75, and the longwave absorption coefficient is 0.15 g21 m2. The constant of proportionality in Eq. (6) is 2(kp)1/3 (4/3rw)22/3N 1/3 (Pontikis 1993) with param- eter values k 5 1, rw 5 1 g cm23, and droplet number density N 5 200 3 106 m23. 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