Potential Biases In Cloud Feedback Diagnosis

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Potential Biases in Cloud Feedback Diagnosis:
An Energy Balance Model Demonstration
by
Roy W. Spencer1
William D. Braswell2
1
Earth System Science Center, University of Alabama in Huntsville, Huntsville, Alabama
35805, roy.spencer@nsstc.uah.edu
2
Earth System Science Center, University of Alabama in Huntsville, Huntsville, Alabama
35805, danny.braswell@nsstc.uah.edu
September 2007
Abstract
Cloud feedbacks are widely considered to be the largest source of uncertainty in
determining the sensitivity of the climate system to increasing greenhouse gas
concentrations, yet our ability to diagnose them from observations has remained
controversial. Here we use a simple energy balance model to demonstrate that daily
random noise in low cloud cover can cause interannual and decadal time scale
temperature variability that produces a positive bias in cloud feedbacks diagnosed in the
traditional manner from the model output. For those model runs producing monthly
anomaly statistics similar to those measured by satellites, the diagnosed feedbacks have
positive biases generally in the range of -0.6 to -1.3 W m-2 K-1. The amount of bias
depends mostly upon the size of the non-feedback cloud variability relative to the surface
temperature variability. These results suggest that current observational diagnoses of
cloud feedback could be significantly biased in the positive direction.
2
1. Introduction
Our understanding of how sensitive the climate system is to radiative
perturbations has been limited by large uncertainties regarding how clouds feed back on
surface temperature change (e.g. Webster and Stephens, 1984; Cess et al., 1990; Senior
and Mitchell, 1993; Stephens, 2005; Soden and Held, 2006). Feedbacks are traditionally
estimated by regressing time- and area-averaged top of atmosphere (TOA) solar
shortwave (SW) radiative flux changes or thermally emitted infrared longwave (LW)
radiative flux changes against surface temperature changes (Tsfc). The regression slope
of the resulting relationship provides the diagnosed feedback parameter (Y) in W m-2 K-1.
A formalism to estimate cloud feedback parameters was presented by Forster and
Gregory (2006, hereafter FG), whose generalized treatment included both internal and
external sources of variability in TOA radiative fluxes. We are specifically interested in
one of FG’s stated assumptions regarding the possible contamination of diagnosed
feedbacks by internal sources of variability (“X” terms) in the TOA flux that are not the
result of feedback on Tsfc. Specifically, FG state:
“The X terms are likely to contaminate the result for short datasets,
but provided the X terms are uncorrelated to Tsfc, the regression
should give the correct value for Y, if the dataset is long enough.”
While it is true that the processes that cause the X terms are, by FG’s definition,
uncorrelated to Tsfc, the response of Tsfc to those forcings can not be uncorrelated to Tsfc -for the simple reason that it is radiative forcing that causes changes in Tsfc. Of course,
ignoring non-feedback sources of cloud variability as a surface temperature forcing
3
mechanism is not new; all investigators who estimate cloud feedbacks in the traditional
manner make the same assumption, whether it is explicitly stated or not.
Here we address the following question: To what degree could non-feedback
sources of cloud variability contaminate traditional feedback estimates? The potential
origins of such cloud variations are many: changes in atmospheric circulation patterns
such as the Pacific Decadal Oscillation (PDO, e.g. Mantua et al., 1997) or North Atlantic
Oscillation (NAO, e.g. Hurrell, 1995); the controversial theory of cloud modulation by
cosmic rays (Svensmark and Friis-Christensen, 1997); or stochastic variations in cloud
cover (e.g. Battisti et al., 1997) and surface heat fluxes (Hasselman, 1976).
2. Model Description
To explore this issue we developed a simple time-dependent energy balance
model in which the energy balance can be perturbed by specified cloud radiative inputs or
non-cloud Tsfc changes, and in which cloud feedbacks can be turned on or off.
We
will use daily random fluctuations in cloudiness as a non-feedback source of cloud
variability. This random, high frequency radiative forcing causes substantial interannual
and decadal temperature variability in the model which we then analyze in the traditional
manner to see how closely the diagnosed feedbacks match the feedbacks that were
specified in the model.
The model is not meant to provide accurate evaluations of errors in cloud
feedback diagnoses, but instead to provide a simple framework for demonstrating
potential problems in those diagnoses. Despite the model’s simplicity, it is still able to
reproduce basic characteristics of observed interannual climate variability.
4
The model, details of which are included in the Appendix, includes the dominant
modes of vertical energy exchange in the Earth’s energy budget summarized by Kiehl
and Trenberth (1997, hereafter KT). The model has a uniformly mixed “swamp” ocean
25 m deep, roughly consistent with the World Ocean Database 1998 average ocean mixed
layer depth for the latitude band 20°N to 20°S (Monterey and Levitus, 1997). The model
atmosphere is a single, 1000 hPa thick layer which has an upper atmospheric temperature
Tupper that radiates to outer space, and a lower altitude temperature Tlower that radiates
downward to the surface.
We assume a negative evaporative feedback on surface temperature having a
strength roughly equal to the average of the IPCC AR4 climate models (Held and Soden,
2006). All non-radiative heat loss is assumed to be through evaporation, which then
immediately warms the model atmosphere through condensational heating. While the
resulting temperature response of the atmosphere can be partitioned between Tupper and
Tlower to mimic lapse rate feedback, here we will assume no lapse rate feedback; that is,
latent heating will be assumed to cause equal temperature changes in both atmospheric
radiating temperatures. Since the infrared absorptivity of the atmosphere is assumed to
remain constant, there is no water vapor feedback.
Since it is known that high-frequency stochastic forcing can cause substantial low
frequency variability in the climate system (e.g. Hasselman, 1976), we chose random
daily fluctuations in low cloud coverage as a non-feedback source of cloud radiative
input. For simplicity, the low cloud variability was allowed to impact the reflected SW at
TOA, but not the emitted LW. Daily random variability in Tsfc can be specified,
representing any non-cloud source of temperature variability, as this is necessary to test
5
feedbacks in the absence of any other radiative forcings. Finally, the strength of low
cloud SW or high cloud LW feedbacks on surface temperature can be specified
separately or together. For simplicity of the demonstration, here we will allow only the
SW feedbacks to vary; LW cloud feedback will be set to zero.
Since the model is run with a variety of forcings and feedbacks, it is useful to
determine which combinations produce physically realistic behavior. To do this we
compared the time variability of the model output to satellite observations. For reflected
SW variability, we computed the standard deviation of 30-day average anomalies in
NASA’s Terra satellite Clouds and the Earth’s Radiant Energy System (CERES, Wielicki
et al., 1996) reflected SW fluxes, and in the Tropical Rain Measuring Mission (TRMM)
satellite Microwave Imager (TMI, Kummerow et al., 1998) measurements of sea surface
temperatures (Wentz et al., 2000). These measures of variability were computed for the
tropical oceans in the latitude band 20°N to 20°S, during the period March 2000 through
December 2005. This is the same region studied in our composite analysis of tropical
intraseasonal oscillations (Spencer et al., 2007). Averaging the data to monthly time
resolution greatly reduces the sampling errors that arise from incomplete coverage of the
tropics by the satellites on short time scales. The resulting standard deviation of the 30
day averaged SW fluxes was 1.3 W m-2, and of the Tsfc variations was 0.134 deg. C.
3. Model Experiment Results
3.1 Random cloud forcing without cloud feedback
We first demonstrate the simple case where the only specified source of model
variability is daily random fluctuations in low cloud fraction; no cloud feedbacks are
included. The magnitude of the random cloud changes was adjusted to approximate the
6
CERES-observed 30 day SW standard deviation of 1.3 W m-2. This required about +/5% variations around the energy balance cloud fraction of 0.225.
The resulting model Tsfc time series is shown in Fig. 1. The multi-decadal
temperature fluctuations reveal a random walk character, but one that is constrained by
energy balance to an average surface temperature of 288 K. Depending upon the random
number generator seed, the time series seen in Fig. 1 can change considerably, with more
or less amounts of low-frequency variability.
When we plot yearly averaged model output of reflected SW against Tsfc (Fig. 2),
we see a substantial positive feedback parameter of –2.26 W m-2 K-1 -- even though no
cloud feedback was specified. (Note that positive feedbacks have negative values).
Note that there is also considerable scatter in the relationship, with a relatively low
explained variance of 26%. It might be significant that the feedback diagnoses of FG
based upon satellite observations also had low correlations. In contrast, if we specify
any cloud feedback on randomly forced Tsfc variations (not shown), the explained
variance is always very high, over 95%. This suggests that the low explained variance in
FG’s feedback diagnosis could, by itself, be evidence for non-feedback sources of SW
variability.
3.2 Random cloud and SST forcing, with cloud feedback
We now address the case where random cloud noise, random Tsfc noise, and cloud
feedback are all included. The random cloud variability is independent of feedbacks, and
the random Tsfc variability is independent of cloud amount. The results are shown in Fig.
3, where the diagnosed feedback parameter is plotted as a function of the ratio of cloud
noise to Tsfc noise. Each panel in the figure represents averages from a Monte Carlo set
7
of one hundred model realizations, each run with different noise generator seeds. Each
panel in Fig. 3 used the specified fixed cloud noise with Tsfc noise varying along the
abcissa. Cloud noise was relative to an energy balance average cloud fraction of 0.225.
The cloud noise was uniformly distributed, while the Tsfc noise was normally distributed.
The cloud noise was expressed as a Tsfc equivalent before taking the ratio in the abcissa.
The solid lines in Fig. 3 represent specified zero or negative cloud feedbacks,
while the dashed lines represent various positive cloud feedbacks. The “true” feedback
values that were specified in the model correspond to zero cloud noise, i.e. where the
lines of diagnosed feedback intersect the ordinate.
The main conclusion that can be made from Fig. 3 is that the existence of a nonfeedback source of cloud variability leads to a positive bias in the diagnosed cloud
feedbacks, as indicated by the fact that all of the lines slope downward to the right, in the
direction of positive feedback.
The dots in Fig. 3 represent those combinations of model forcings and feedback
that produced monthly standard deviations that were within 20% of the satellite-measured
values of 1.3 W m-2 for SW, and 0.134 deg. C for Tsfc. It is interesting that in the special
case of zero cloud noise the only model runs that are consistent with the satelliteobserved SW and Tsfc variability statistics are those with strongly negative feedbacks.
Most of the dots in Fig. 3 represent positive biases in diagnosed feedbacks of
about -0.5 to -1.5 W m-2 K-1. As an example, in Fig. 4 we see the errors corresponding
to the +/- 4% cloud variability panel, which range from -0.6 to -1.3 W m-2 K-1 . We note
that these errors are a substantial fraction of the positive cloud SW feedbacks diagnosed
by FG from observational data. This raises the question of whether previous
8
observational estimates of positive cloud feedback could have been significantly
contaminated by non-feedback cloud variations.
The range of errors represented by the dots in Fig. 3 is, not surprisingly,
dependent upon a variety of model assumptions. For instance, for assumed ocean mixed
layer depths greater than that assumed here (25 m), somewhat smaller biases in feedback
diagnosis are found. Nevertheless, our results raise the possibility that there are as yet
unidentified levels of positive bias in previous diagnoses of cloud feedback.
4. Conclusions
Our simple energy balance model suggests that non-feedback cloud variability
can corrupt estimates of feedback that are diagnosed in the traditional manner.
Significantly, the corrupting influence is always a bias in the direction of positive
feedback. Note that this is an energetic necessity since, all other things being equal,
decreasing low cloud cover can only cause surface warming, while increasing low cloud
cover can only cause surface cooling.
The issue we have addressed with our simple model is directly related to
Stephens’ (2005) discussion of how the climate system is defined when diagnosing
feedbacks. Stephens noted the overly simplistic nature of the climate system that is
implicitly invoked when feedbacks are diagnosed from the covariability between
observed radiative fluxes and surface temperature. Since it is well known that the
processes that control cloud formation and dissipation are myriad and complex, the
existence of non-feedback sources of cloud variability is not an unrealistic expectation.
That significant positive biases in diagnosed feedback occur in our model with only a few
percent random variability in the average cloud fraction indicates that the non-feedback
9
source of cloud radiative input to the system does not have to be very large for significant
contamination of diagnosed feedbacks to occur.
It should be emphasized that these biases in diagnosed feedback exist independent
of the time scale over which any data averaging is performed; the fundamental problem is
related to what the physical sources are of surface temperature variability.
While we have included neither high cloud feedbacks or forcings, nor water vapor
feedback, the inclusion of these would not change the basic concern: to the extent that
non-feedback cloud radiative input forces surface temperature variability, it causes a
positive bias in feedback diagnoses.
These results hopefully provide some semi-quantitative insight into previously
expressed concerns about the validity of cloud feedbacks diagnosed in the traditional
manner from observational data. They also underscore the need for new methods of
diagnosing cloud feedback, as was advocated by Stephens (2005), one example of which
is the methodology developed by Aires and Rossow (2003).
Acknowledgments
The TMI sea surface temperature product is produced by Remote Sensing
Systems and is sponsored by the NASA Earth Science REASoN DISCOVER Project.
The CERES data were obtained from the NASA Langley Research Center EOSDIS
Distributed Active Archive Center. This research was supported by NOAA contract
NA05NES4401001 and DOE contract DE-FG02-04ER63841.
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APPENDIX
A Time-Dependent Energy Balance Model
This appendix gives the equations defining the energy balance model used for this
paper. The model consists of a swamp ocean and a single atmospheric layer with upper
and lower radiating temperatures. The difference between these temperatures is fixed.
The shortwave fluxes at the top-of-atmosphere  FswTOA  are given by
FswTOA
(1)  S0 ,
(A.1.a)
FswTOA
(2)   S0  f sw , and
(A.1.b)
FswTOA
(3)   S0 1  f sw 1  Asw  
(A.1.c)
where equation (A.1.a) gives the incoming solar flux, (A.1.b) gives the upward solar flux
from cloud reflection, and (A.1.c) gives the upward solar flux from surface reflection. In
these equations, So is the solar constant, f sw is the fraction of short-wave clouds, Asw is the
atmospheric absorptivity for shortwave radiation, and  is the surface albedo.
The longwave fluxes at the top-of-atmosphere ( FlwTOA ) are given by
4
FlwTOA
(1)   Tu Alw 1  f lw  ,
(A.2.a)
4
FlwTOA
(2)   Tcld f lw , and
(A.2.b)
4
FlwTOA
(3)   Ts 1  Alw 1  f lw 
(A.2.c)
11
where equation (A.2.a) gives the upward emission from the atmosphere, (A.2.b) gives the
upward emission from clouds, and (A.2.c) gives the upward emission from the surface.
Here  is the Stefan-Boltzman constant, Tu is the upper atmospheric temperature, f lw is the
long-wave cloud fraction, Alw is the atmospheric absorptivity for longwave radiation, Tcld is
the cloud top temperature, and Ts is the surface temperature.
The shortwave flux at the surface ( Fswsfc ) consists only of the downwelling solar
component
Fswsfc(1)  S0 1  f sw 1  Asw 1    .
(A.3.a)
The longwave fluxes at the surface ( Flwsfc ) are given by
Flwsfc(1)   Tl 4 Alw and
(A.4.a)
Flwsfc(2)   Ts4
(A.4.b)
where (A.4.a) gives the downward emission from the atmosphere and (A.4.b) gives the
upward emission from the surface. Tl is the lower atmospheric temperature.
Our energy balance model includes two feedback terms. The short-wave cloud
feedback is given by
f sw  f sw0   sw Ts  Ts 0 
12
(A.5.a)
and the evaporative feedback by
E  E0  2.2 Ts  Ts 0  ,
(A.5.b)
where E is the evaporative flux,  sw is the short-wave cloud feedback factor, and f sw0 , Ts 0 , Eo
are nominal baseline values for f sw , Ts , E based on the Earth’s average energy balance as
defined by the Trenberth and Kiehl (1997) diagram. Equation (A.5.a) is set up such
that  sw is negative for negative cloud feedback and positive for positive cloud feedback.
The atmospheric heating rate is obtained by summing over the fluxes in and out of
the layer, namely
2
dTl
g  3 TOA 3 TOA 1 sfc


Fsw( i )   Flw( i )   Fsw( i )   Flwsfc( i )  E 


dt c p _ a   i 1
i 1
i 1
i 1

(A.6.a)
where g is the gravitational acceleration, c p _ a is the heat capacity of the atmosphere,
and  is the pressure thickness of the atmosphere. Likewise, the surface heating rate is
given by
2
dTs
1
 1 sfc


F

Flwsfc( i )  E 


sw ( i )

dt
c p _ wd  w  i 1
i 1

(A.6.b)
where c p _ w is the heat capacity of water, d is the depth of the model ocean, and  w is the
density of water.
The upper atmospheric temperature Tu is related to the lower atmopspheric
temperature by a contstant c
Tu  Tl  c.
13
(A.7.a)
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Figure Captions
Fig. 1. Energy balance model output of surface temperature when forced with daily
random variations in low cloud fraction.
Fig. 2. Co-variability between reflected SW versus surface temperature for the model
run shown in Fig. 1. The dots represent one-year averages, which were computed every
day during years 10 – 40 of the model run.
Fig. 3. Diagnosed SW feedback parameter values as a function of the size of cloud noise
relative to Tsfc noise in the time-evolving energy balance model, based upon yearly
averages. The dots represent model combinations that reproduce, to within 20%, the
standard deviation of 30 day tropical oceanic average SW and Tsfc variations as observed
by satellites during 2000 through 2005. See text for other details.
Fig. 4. One of the panels in Fig. 3 (+/- 4% cloud noise), but plotted with the error in
diagnosed feedback on the ordinate.
17
Fig. 1. Energy balance model output of surface temperature when forced with daily
random variations in low cloud fraction.
18
Fig. 2. Co-variability between reflected SW versus surface temperature for the model
run shown in Fig. 1. The dots represent one-year averages, which were computed every
day during years 10 – 40 of the model run.
19
Fig. 3. Diagnosed SW feedback parameter values as a function of the size of cloud noise
relative to Tsfc noise in the time-evolving energy balance model, based upon yearly
averages. The dots represent model combinations that reproduce, to within 20%, the
standard deviation of 30 day tropical oceanic average SW and Tsfc variations as observed
by satellites during 2000 through 2005. See text for other details.
20
Fig. 4. One of the panels in Fig. 3 (+/- 4% cloud noise), but plotted with the error in
diagnosed feedback on the ordinate.
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