Estimating the PPH-bias for simulations of convective and ...€¦ · Estimating the PPH-bias for simulations of convective and stratiform clouds ... The first case study is based

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Atmospheric Research 72 (2004) 317–328

Estimating the PPH-bias for simulations of

convective and stratiform clouds

G. Bauml, A. Chlond, E. Roeckner*

Max-Planck-Institut fur Meteorologie, Bundesstr. 55, 20146 Hamburg, Germany

Received 3 July 2003; received in revised form 26 January 2004; accepted 31 March 2004

Abstract

The plane parallel homogeneous (PPH) approximation is known to generate systematic errors in

the computation of reflectivity and transmissivity of a horizontally inhomogeneous cloud field. This

PPH-bias is determined for two cloud fields, a stratocumulus and a shallow convective cloud scene,

which have been simulated using a cloud resolving model. The independent column approximation

has been applied as reference and a PPH analogue has been interpolated from the original cloud data.

In order to correct for the bias the effective thickness approach (ETA) has been employed. For the

two cloud simulations, the corresponding reduction factors have been determined.

D 2004 Elsevier B.V. All rights reserved.

Keywords: Radiative transfer; Inhomogeneous clouds; PPH bias; Effective thickness approach

1. Introduction

Clouds show variability of liquid water path (LWP) on many different scales (Davis et

al., 1999). In current general circulation models (GCMs) with typical horizontal resolution

of some 100 km only the very coarse structure of these cloud fields can be resolved. On

the sub-grid scale the only distinction is made between clear and cloudy sky. The latter is

assumed to fill the whole vertical extent of the layer and is plane parallel and

homogeneous in the horizontal. This is called plane parallel homogeneous (PPH)

approximation. Due to the non-linear dependency of optical properties like reflectivity

on LWP, neglecting sub-grid scale features leads to systematic errors as a direct

consequence of Jensen’s inequality (Jensen, 1906): Cloud reflectivity is overestimated,

0169-8095/$ - see front matter D 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.atmosres.2004.03.019

* Corresponding author. Tel.: +49-4041173368; fax: +49-4041173298.

E-mail address: [email protected] (E. Roeckner).

G. Bauml et al. / Atmospheric Research 72 (2004) 317–328318

transmissivity underestimated. This PPH-bias has received considerable attention over the

last years. Cahalan et al. (1994) developed the effective thickness approach (ETA) to

correct for this error. Basically, it uses an effective cloud optical thickness in the radiation

computation, which is smaller than the thickness derived from mean cloud properties

directly. However, the reduction factor has to be determined empirically.

The ETA is the most widely applied correction method in current GCMs, because it can

be implemented easily into standard two-stream radiation schemes (see e.g. Roeckner et

al., 2003). However, the reduction factor has to be deduced empirically. In this study, we

use data from a cloud resolving model in order to determine the PPH-bias and find

appropriate reduction factors for convective and stratiform clouds.

A different approach based on probability density function (PDF) of liquid water

content on the sub-grid scale has been introduced by Barker (1996). He assumed the PDF

as a Gamma distribution, while Bauml and Roeckner (submitted) start from a Beta

distribution. For the small domain size of the cloud simulations we use in this study, the

statistics are not good enough for fitting an appropriate distribution, thus, these statistical

schemes cannot be applied.

2. Cloud data

The simulations have been performed with a Cloud Resolving Model (CRM)

developed at the Max Planck Institute for Meteorology in Hamburg. It is based on the

idea of Large Eddy Simulation (LES). LES means that all spatial scales, which represent

the dominant large-scale turbulent motions, are explicitly resolved, while the effects of

smaller scale turbulence on the resolved flow are parameterized. Therefore, the dominant

cloud structures are explicitly calculated. An extensive description of the LES model can

be found in Chlond (1992, 1994) and Muller and Chlond (1996).

The first case study is based on a situation encountered during flight RF06 in the frame

work of the Atlantic Stratocumulus Transition Experiment (ASTEX). The flight path

followed a stratocumulus cloud over the North Atlantic (37jN, 24jW) in its transition

state. An initially horizontally homogeneous cloud layer developed into a decoupled

boundary layer with cumulus penetrating the stratocumulus deck from below. Further

details of the ASTEX experiment and especially of the flight RF06 can be found in de

Roode and Duynkerke (1997) and Duynkerke et al. (1999), respectively.

In their LES study Chlond and Wolkau (2000) tried to simulate the observed cloud

field. The data used for the radiation computation in this work are labeled REFERENCE in

their article. The first 30 min are neglected as LES model spin-up time, leaving 49

snapshots with an time interval of 3 min between them. For the REFERENCE-case the

model domain was 28.8� 3.2� 1.5 km3 with a horizontal resolution of Dx =Dy = 50 m

and a vertical grid spacing of Dz = 25 m. Fig. 1 gives an impression of the evolving cloud

field. Cloud cover was 1 throughout the whole integration.

For the second case study initial and boundary conditions are applied, which have been

derived from data collected during the Atlantic Trade Wind Experiment (ATEX)

(Brummer et al., 1974). This case was used as an intercomparison project of the GCSS

program (Stevens et al., 2001). The experiment took place in the Atlantic northeast trade

Fig. 1. Snapshot of the stratocumulus cloud simulation (ASTEX). Shown are volume surface plot, cut at 10 km,

colored by LWP (top), LWP (middle) and vertical profile of mean liquid water content (right).

G. Bauml et al. / Atmospheric Research 72 (2004) 317–328 319

wind region (near 12jN, 35jW). The model domain for this integration is 6.4� 6.4� 3.0

km3, where the horizontal grid spacing is Dx =Dy = 100 m and the vertical Dz = 20 m.

Output is written every 5 min. The first 2 h of model integration are discarded as model

spin-up time, leaving data of 5-h simulation time. Fig. 2 gives an example of the general

structure of the simulated cloud field. Cloud fraction and liquid water path (LWP) vary

substantially during the simulation.

Fig. 2. Snapshot of the trade wind cumulus simulation (ATEX). Shown are volume surface plot, colored by LWP

(top), LWP (middle) and vertical profile of mean liquid water content (right).



3. Methods

All radiation computations are performed using a two-stream scheme, as it is

implemented in the ECHAM4 climate model (Roeckner et al., 1996). It has been

developed by Fouquart and Bonnel (1980) and has been slightly modified (Morcrette,

1989; Gregory et al., 1998). The simulations shown below are performed assuming no

aerosol loading. Therefore clouds are the only scattering objects (apart from Rayleigh

scattering), which simplifies interpretation. The droplet number density is parameterized

for sea surface conditions (see also Roeckner et al. (1996)) and the ground albedo is set to

zero, i.e. there are no surface effects. Since both cloud simulations are for maritime cloud

systems and the observed ocean albedo is low, these parameter settings are consistent with

LES case studies.

In order to determine the PPH bias, a reference computation accounting for the

horizontal inhomogeneities has to be compared to an analogue PPH calculation. As a

reference we use the independent column approximation (ICA) (Chambers, 1997), i.e. the

radiation code is called for each LES cloud model column individually and the fluxes are

averaged for each scene. The ICA assumes that the net horizontal photon transport is

zero, i.e. there are as many photons leaving a column as entering it through its sides. For

an individual column this assumption has been shown to be completely wrong, but the

average fluxes over a cloud domain are a good measure from an energetic point of view

(Titov and Kasjanov, 1996). The ICA can be implemented quite easily: The data from the

cloud resolving model, as described in the previous section, are read column by column.

The cloud fraction for each pixel i is Aci= 0, when the liquid water mixing ratio is under a

threshold value ql < qlthresh, and Ac

i = 1, if ql>q1thresh. For the following analysis this

threshold has been set to q1thresh = 10� 3 g/kg in accordance with Petch and Edwards

(1999). The effective radius reff and the optical properties s, x¯ and g are computed as

discussed in Roeckner (1995). Since reff is a function of the liquid water content, liquid

water content as well as effective radius are horizontally inhomogeneous fields and both

influence the variability of the optical thickness, single scattering albedo and asymmetry

factor. Setting the effective radius to a constant value of 10 Am does not change the

results noticeably. Therefore, this variabililty can be neglected in this study. A standard

two-stream radiation computation is then performed for each column. Finally, the fluxes

for the individual columns are added up and divided by the number of columns to obtain

the scene averages.

The plane parallel counterpart is constructed by reading in the data of each model layer

and work out the algebraic mean of the atmospheric state variables. The cloud fraction is

determined similarly as for the ICA experiment: All pixels with liquid water mixing ratio

ql>q1thresh are counted as cloudy. The cloud fraction in level i then simply is Ac

i= ncdi/ntot,

where ncdi is the number of cloudy pixels in level i and ntot is the total number of pixels per

level. For partial cloudiness the maximum-random overlap assumption is applied. A single

two-stream radiation computation, identically to the one performed for an individual

column in the ICA, is carried out, immediately supplying the PPH fluxes.

The ETA may be applied easily to the PPH dataset by using an effective optical

thickness seff = vs with the reduction factor v< 1. In order to closely ressemble the

conditions of using the ETA in a GCM we employ the same v for each layer. From the


characteristics of a fractal cloud model, Cahalan et al. (1994) derives a relation between v,the mean of the logarithm of liquid water path (LWP), W, and the mean of the logarithm:

v ¼ expðlogW ÞW

: ð1Þ

However, for broken cloudiness like in the ATEX simulation, Eq. (1) cannot be applied

immediately, because log(W) for W= 0 is not defined. Therefore, we define a threshold for

the lowest LWP that is regarded as cloudy. The threshold for liquid water mixing ratio in

the ICA computation, q1thresh = 10� 3 g/kg, results in a threshold for LWP of

Wthresh = 22� 10� 3 g/m2. A more empirical approach is to systematically vary the

reduction factor from 0 to 1 and compare the results with the ICA calculations, thereby

determining the proper value for v.All radiation computations are performed for an intermediate solar zenith angle of 45j

and thus represent mean values of the dirunal cycle. To investigate the zenith angle

dependence in detail, realistic three dimensional calculations, like Monte Carlo techniques,

would be more appropriate, because they also account for other effects such as side

illumination, etc. (O’Hirok and Gautier, 1998).

4. Results

For both cloud simulations, the reflectivity, transmissivity and absorptivity, which result

from the various methods described above, are collected in Figs. 3 and 4. For an overview

of the labels, see Table 1. Fig. 5 shows the reflectivity as a function of model time. For the

stable ASTEX cloud case the corresponding values are nearly constant for all snapshots.

First, we may identify the PPH-bias by comparing PPH and ICA. For both cloud cases,

obviously, the reflectivity computed with the PPH approximation is larger than the

corresponding ICA value, while the transmissivity is smaller. Comparing the results for

the trade wind cumuli of the ATEX simulation with the ones for the stratocumulus in

ASTEX the PPH-bias is substantially larger in the inhomogeneous ATEX case than in the

rather homogeneous ASTEX cloud: The error is around 0.01 for ASTEX and 0.05 for the

ATEX cloud, i.e. the relative error defined as (RPPH�RICA)/RICA comes close to 100% for

ATEX, while it is only around 5% in the ASTEX case. This could be expected

qualitatively from the variability of the liquid water path displayed in Fig. 2, where the

ASTEX cloud looks very much like a plane parallel homogenous slab of cloud, while the

ATEX clouds are highly variable in shape and thickness with a much broader range of

LWP-values. From Fig. 5 it can be seen that the reflectivities, PPH and ICA, vary from

snapshot to snapshot. Nevertheless, the albedo bias is always of the same order of

magnitude.

We may now try to apply the ETA in order to correct for the PPH-bias. First, we will

use Eq. (1) directly. For the ASTEX stratocumulus cloud this yields reduction factors

between vi= 0.93 and vi = 1.0 for the individual timesteps with a mean of v = 0.94. If wetake all time-steps as representation of a bigger cloud field corresponding to different

stages of a stratocumulus cloud at the same time, resulting in a virtual domain size of

Fig. 3. Reflectivity, transmissivity and absorptivity of the stratocumulus simulation (ASTEX) using different

schemes. All values are computed for an incident zenith angle of 45j and are averages over all time-steps. The

labels are explained in Table 1.


Fig. 4. Same as Fig. 3 for the trade wind cumulus simulation (ATEX).


Table 1

Overview of the radiation computations performed with the CRM datasets

Label Description

ICA ICA computation; full vertical resolution

PPH PPH computation; full vertical resolution

ETA.4 ETA computation; reduction factor v= 0.4ETA.7 ETA computation; reduction factor v= 0.7ETA.9 ETA computation; reduction factor v= 0.9S-ICA ICA computation; single cloud layer

S-PPH PPH computation; single cloud layer

S-ETA.4 ETA computation; single cloud layer, v= 0.4


28� 195 km2, we obtain v = 0.90, which is smaller than that for the individual time-steps

since the overall variability is larger. Whatever value we choose, the reduction factor is

substantially larger than the value of 0.7 suggested by Cahalan et al. (1994) for

stratocumulus. Fig. 3 shows the corresponding radiative properties for v = 0.7 and

v = 0.9 labeled ETA.7 and ETA.9, respectively.

In the case of the broken cloud field of the ATEX simulation the threshold value Wthresh

for cloudy cells has to be used. Since Wthresh is not physically based, it is interesting to

ensure that v does not depend on the choice of this threshold value. Fig. 6 shows the

reduction factor (again the mean over all time-steps). The reduction factor computed by

Eq. (1) is far from being independent of Wthresh. For Wthresh < 10� 4 m� 2 the curve

saturates at v = 0.15. Fig. 7 depicts the reflectivity for the ATEX cloud simulation as a

function of reduction factor. The vertical line marks the ICA value. Clearly, the v = 0.16 is

much too small, but v = 0.42 seems to be more adequate. Similar results have been

obtained by Kogan et al. (1995), who found a reduction factor of v = 0.5 for an LES

simulated cumulus cloud field. In Fig. 4, the values corresponding to v= 0.4 and v = 0.7

Fig. 5. Reflectivity for ATEX cloud simulation as function of model time for PPH, ICA and ETA computation

using v= 0.4.

Fig. 6. Scaling factor v of the ETA computed for the ATEX cloud data using Eq. (1) vs. the threshold valueWthresh

of the LWP. Pixels with W<Wthresh are regarded clear sky.


are shown. The reduction factor of v = 0.4 also reveals good results for each snapshot

scene individually, as can be seen from Fig. 5.

All radiation computations are performed using maximum-random overlap assumption.

For the ASTEX case stratus with cloud cover being 1 in nearly all cloudy layers, the

overlap assumption does not matter. In contrast, for the ATEX case overlap is likely to

influence the radiative properties. In order to avoid this inconsistency between CRM

clouds and GCM simulation, we interpolated the ATEX data to the much coarser vertical

resolution of current GCMs. The levels are 20, 40, 500 and 3000 m. The whole cloud

resides in a single layer in this interpolated data set. Thus, cloud overlap assumptions no

longer have any effect. The corresponding optical properties are shown in Fig. 4 labeled S-

PPH, S-ICA and S-ETA.4. There are only little differences between the full resolution and

interpolated data sets for ICA, PPH and ETA, confirming the finding of the full resolution

Fig. 7. Reflectivity of the ATEX cloud data, averaged over all time-steps, as a function of the scaling factor v for a

solar zenith angle of 45j. The values of the reference ICA computation are marked by a horizontal line. From the

intersection of the ICA line with the ETA curve, the best fit scaling factor can be extracted.


analysis that v = 0.4 corrects for horizontal inhomogeneities quite well. For the sake of

completeness one should mention, that the absorption is of course smaller in the ETA

computation than it is in the PPH case, because the clouds are virtually thinned out. It is

even smaller than, but comparable to the ICA case.

5. Conclusion

Using the data from two cloud resolving simulations, a nocturnal marine stratocumulus

case and a trade wind cumulus field, we determined the PPH-bias by comparing the

independent column approximation and the corresponding plane parallel homogeneous

computations. While the bias in reflectivity and transmissivity is only about 0.01 for the

stratus case, it is nearly 0.05 for the broken cumulus cloud case. Absorption seems only

slightly affected. For both cloud types the effective thickness approximation has been

applied. While for the overcast stratus cloud, a reduction factor of v = 0.9 could be

extracted directly from the variability of the liquid water path, this factor could not be

derived for the broken cloud field in the cumulus simulation. Nevertheless, a reduction

factor of vc 0.4 has been deduced empirically. Therefore, the reduction factor v = 0.7suggested by Cahalan et al. (1994) for stratus clouds, may not be regarded as a

representative value. We demonstrated that v depends strongly on the cloud type. This

should be accounted for when the ETA is implemented into GCMs (Tiedtke, 1996; Bauml

and Roeckner, submitted for publication).

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