Unified Treatment of Thermodynamic and Optical Variability in a

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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


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