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  • Moss, S. J., P. R. A. Brown, D. W. Johnson, D. R. Lauchlan, G. M. Martin, M. A. Pickering, and A. Spice, 1993: Cloud microphysics measurements on the MRF C-130: Working group report. MRF Tech. Note 12, Meteorological Research Flight, 147 pp. [Available from The Meteorological Research Flight, Y46 Building, DERA, Farnborough, Hampshire, GU14 0LX, United Kingdom.].

  • Nicholls, S., J. Leighton, and R. Barker, 1990: A new fast response instrument for measuring total water content from aircraft. J. Atmos. Oceanic Technol.,7, 706–718.

  • Ouldridge, M., 1982: An introduction and guide to the Johnson–Williams liquid water content meter. The Met. Office, 15 Internal Rep. 41, 15 pp. [Available from The Met. Office, London Road, Bracknell, Berkshire RG12 2SZ, United Kingdom.].

  • Pope, V. D., M. L. Gallani, P. R. Rowntree, and R. A. Stratton, 1998:The impact of new physical parametrizations in the Hadley Centre climate model—HadAM3. Climate Dyn.,16, 123–146.

  • Slingo, J. M., 1980: A cloud parameterization scheme derived from GATE data for use with a numerical model. Quart. J. Roy. Meteor. Soc.,106, 899–927.

  • Smith, R. N. B., 1990: A scheme for predicting layer clouds and their water content in a general circulation model. Quart. J. Roy. Meteor. Soc.,116, 435–460.

  • ——, D. Gregory, and C. Wilson, 1990: Calculation of saturated specific humidity and large scale cloud. Unified Model Documentation Paper 29, 7 pp. [Available from The Met. Office, London Road, Bracknell, Berkshire RG12 2SZ, United Kingdom.].

  • Sommeria, G., 1976: Three dimensional simulation of turbulent processes in an undisturbed trade wind boundary layer. J. Atmos. Sci.,33, 216–241.

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  • Sundqvist, H., E. Berge, and J. E. Kristjansson, 1989: Condensation and cloud parameterization studies with a mesoscale numerical weather prediction model. Mon. Wea. Rev.,117, 1641–1657.

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  • WMO, 1996: Guide to Meteorological Instruments and Methods of Observation. 6th ed. WMO-8, WMO.

  • Xu, K.-M., and S. K. Krueger, 1991: Evaluation of cloudiness parameterizations using a cumulus ensemble model. Mon. Wea. Rev.,119, 342–367.

  • ——, and D. A. Randall, 1996: A semi-empirical cloudiness parameterization for use in climate models. J. Atmos. Sci.,53, 3084–3102.

  • View in gallery

    Distributions of total water from a stack of six horizontal runs through a well-mixed boundary layer with a layer of 8/8 stratocumulus during ASTEX (flight A209), together with the total water, liquid water, and temperature profile.

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    Distributions of total water from a stack of seven horizontal runs through a decoupled boundary layer with stratocumulus and penetrative cumulus during ASTEX (flight A213), together with the total water, liquid water, and temperature profile.

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    Cloud fraction plotted against the run-mean total water content qt normalized with the run-mean saturation specific humidity qs(l,i). The approximate error in qt/qs(l,i) is shown for each dataset in the inset. See text for more complete description of the errors. Datasets: FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles).

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    Run-mean relative humidity, (qt/qs(l,i), plotted against run-mean total water content qt divided by the run-mean saturation specific humidity qs(i), for EUCREX data.

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    Cloud fraction plotted against the run-mean condensed water content normalized by qs(l,i). The approximate error in qc/qs(l,i) is of the order of 15% for the ASTEX and FIRE cases and 20% for the EUCREX cases. Datasets: FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles).

  • View in gallery

    Form of the diagnostic cloud fraction prediction scheme from Slingo (1980). Cloud fraction is plotted against relative humidity for high–low-level cloud (solid line) and midlevel cloud (dashed line). The definitions of the cloud types are given in the text.

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    Form of the diagnostic cloud fraction prediction scheme from Smith (1990). Cloud fraction is plotted against the run-mean total water content qt normalized with the run-mean saturation specific humidity qs(l,i). Curves for three values of RHcrit are shown: dashed, RHcrit = 0.7; solid, RHcrit = 0.8; and dotted, RHcrit = 0.9.

  • View in gallery

    Predicted cloud fraction using the Slingo (1980) scheme plotted against observed cloud fraction for FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles) data.

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    Predicted cloud fraction using the Smith (1990) scheme plotted against observed cloud fraction for FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles) data.

  • View in gallery

    Predicted cloud fraction using the Xu and Randall (1996) scheme plotted against observed cloud fraction for FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles) data.

  • View in gallery

    Predicted cloud fraction using the functional fit, FWI, suggested in this paper plotted against observed cloud fraction for FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles) data.

  • View in gallery

    Predicted cloud fraction using the functional fit, FWII, suggested in this paper plotted against observed cloud fraction for FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles) data.

  • View in gallery

    Fig. A1. Specific total water for 4 s obtained from the Meteorological Research Flight total water probe during a run in ice cloud (solid line). The minimum envelope fit to the data is shown as a dashed line.

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    Fig. A2. Run-mean ice water content obtained from the Nevzorov probe plotted against the run-mean estimate of ice water content from the total water probe alone.

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    Fig. B1.

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    Fig. B1 (Continued).

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    Fig. B1 (Continued).

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    Fig. B1 (Continued).

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Relationships between Total Water, Condensed Water, and Cloud Fraction in Stratiform Clouds Examined Using Aircraft Data

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Abstract

Relationships among total water, condensed water, and cloud fraction in boundary layer and cold tropospheric stratiform clouds are investigated using a large observational dataset collected by the U.K. Met. Office C-130 aircraft. Values of the above parameters are estimated using horizontal aircraft runs ranging from 40 to 80 km in length. Boundary layer (warm cloud) data were taken from the Atlantic Stratocumulus Transition Experiment (ASTEX) and First International Satellite Cloud Climatology Project (ISCCP) Research Experiment (FIRE). Free tropospheric (cold cloud) data were taken from the European Cloud and Radiation Experiment (EUCREX). In both warm and cold cloud a single reasonably well-defined relationship exists between the cloud fraction and the total water content (vapor + condensate) when normalized with the saturation specific humidity. A relationship exists between the condensed water content and the cloud fraction when appropriately scaled with the saturation specific humidity. Functional forms fitted to the data are used as comparators to test three existing diagnostic cloud fraction parameterization schemes.

Corresponding author address: Dr. Paul R. Field, Meteorological Research Flight, Building Y46, DERA, Farnborough, Hampshire GU14 6TD, United Kingdom.

Email: prfield@meto.gov.uk

Abstract

Relationships among total water, condensed water, and cloud fraction in boundary layer and cold tropospheric stratiform clouds are investigated using a large observational dataset collected by the U.K. Met. Office C-130 aircraft. Values of the above parameters are estimated using horizontal aircraft runs ranging from 40 to 80 km in length. Boundary layer (warm cloud) data were taken from the Atlantic Stratocumulus Transition Experiment (ASTEX) and First International Satellite Cloud Climatology Project (ISCCP) Research Experiment (FIRE). Free tropospheric (cold cloud) data were taken from the European Cloud and Radiation Experiment (EUCREX). In both warm and cold cloud a single reasonably well-defined relationship exists between the cloud fraction and the total water content (vapor + condensate) when normalized with the saturation specific humidity. A relationship exists between the condensed water content and the cloud fraction when appropriately scaled with the saturation specific humidity. Functional forms fitted to the data are used as comparators to test three existing diagnostic cloud fraction parameterization schemes.

Corresponding author address: Dr. Paul R. Field, Meteorological Research Flight, Building Y46, DERA, Farnborough, Hampshire GU14 6TD, United Kingdom.

Email: prfield@meto.gov.uk

1. Introduction

Large-scale numerical weather prediction and climate models are currently unable to resolve atmospheric motions on scales small enough to provide an accurate treatment of cloud formation and dissipation. Even the most high-resolution numerical weather prediction models typically have horizontal grid-box lengths on the order of a few kilometers, whereas the scales most important in cloud formation and dissipation are generally at least one order of magnitude smaller than this. Thus, there is a need to parameterize cloud fraction and condensate amounts, which are affected by subgrid-scale processes.

Many cloud parameterization schemes are based around assumptions of Gaussian or triangular distributions and were originally intended for use in high-resolution cloud models with grid-box lengths less than a kilometer where the only unresolved motions are those on turbulence scales. There is no firm physical evidence to believe that the same assumptions will apply on the grid-box scale of current numerical weather prediction and climate models. Mesoscale motions can occur over a range of scales smaller than the current large-scale model grid length (Atkinson 1981) and currently little is known about these motions and the subsequent affect on the distribution of water vapor.

Early modeling efforts (e.g., Sommeria 1976) made the assumption that a model grid box is either totally saturated or totally unsaturated. It is clear, however, that within a grid box regions of supersaturated and subsaturated air can exist and that this results in a partial cloudiness. Sommeria and Deardorff (1977) and Mellor (1977) assume that within a grid box parameters such as total water content and saturation water content are normally distributed around their mean values, a step forward that can be used to derive the cloud fraction within a particular grid box. With similar approaches Le Treut and Li (1988) and Smith (1990) describe the subgrid variation in the difference between the total water content and the saturation water content (saturation specific humidity) using a symmetric triangular distribution. The width of the distribution is controlled using a tunable critical relative humidity parameter, the value of which determines the grid-box mean relative humidity at which cloud starts to form.

Another approach to the prediction of cloud fraction in large-scale numerical models is to use cloud-resolving or cloud ensemble models under a wide range of conditions and identify resulting relationships between the cloud fraction and parameters such as cloud liquid water and relative humidity. If such relationships are relatively well defined over a wide enough range of initialization conditions, then cloud ensemble model data can be used to derive “semiempirical” parameterizations for cloud fraction. From such simulations, parameterizations for cloud fraction can be obtained that depend upon mean cloud liquid water in addition to relative humidity and saturation specific humidity (e.g., Xu and Randall 1996). Parameterizations of this type are therefore only applicable in large-scale numerical models that carry a prognostic liquid water variable.

The possibility of experimentally validating cloud fraction parameterizations for use in large-scale numerical models has not yet been fully explored. Ek and Mahrt (1991) used a limited number of aircraft measurements from the Hydrological and Atmospheric Pilot Experiment (André et al. 1988) and observed a reasonably well-defined relationship between the cloud fraction and the total water content (qt) normalized with the saturation specific water content (qs) at the boundary layer top on 10 days. The cloud fraction increased from zero for qt/qs ≈ 0.9 to around 0.8 for qt/qs ≈ 1.1. A relatively large degree of error is attributed to the qt/qs measurements by the authors. This error probably arises from the fact that qt/qs was estimated from below-cloud aircraft runs using the assumption of well-mixed dry adiabatic conditions between the aircraft height and the boundary layer top.

The purpose of this paper is threefold. The primary aim is to investigate relationships among approximate grid-box scale (≈60 km) means of total specific humidity, cloud condensed water content, and the cloud fraction using aircraft data in the boundary layer and in the free troposphere. The second aim is to examine the cloud fraction parameterization schemes of Slingo (1980), Smith (1990), and Xu and Randall (1996) using the aircraft data. The Tiedtke (1993) scheme was not examined due to the prognostic nature of the scheme. The Sundqvist scheme (Sundqvist et al. 1989) was not examined because values for constants in the cloud fraction equation were not explicitly given in the work. In addition, a numerical fit to the observational data is presented and compared to the results of the three parameterizations. Finally, the third aim of this paper is to make available in situ aircraft measurements of cloud fraction and atmospheric water to enable other workers to assess other cloud fraction schemes.

Section 2 describes the experimental procedure, instrumentation, and analytical methods used. In section 3 observational data are presented and possible relationships between the measured parameters are discussed. Section 4 begins with a brief description of the cloud parameterizations of Slingo (1980), Smith (1990), and Xu and Randall (1996) and proceeds to examine them using the aircraft data. The paper concludes with a brief discussion and possible inferences that can be drawn in the light of the findings.

2. Experimental details and instrumentation

a. Observational data used

Observational data were obtained using the Meteorological Research Flight C-130 aircraft, which has been involved in international collaborative experiments to investigate clouds in the boundary layer and the free troposphere. The data in this study are all taken from constant-altitude runs ranging from 30 to 100 km in length, with approximately 80% of the runs being 50–70 km in length (run lengths are given in appendix B).

The boundary layer data are taken from the Atlantic Stratocumulus Transition Experiment (ASTEX; Albrecht et al. 1995) and the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment (FIRE; Albrecht et al. 1988) field campaigns. The ASTEX observational campaign was located in the Azores region of the North Atlantic during June 1992, while FIRE observations were made in the stratocumulus region off the southern Californian coast during June–July 1987. ASTEX synoptic conditions were characterized by a high pressure system often with the presence of a strong subsidence inversion at a height of 1–2 km. Cloud types encountered ranged from reasonably thick stratocumulus to trade wind cumulus fields and all condensed water was in the liquid form. In many cases the stratocumulus layer was decoupled from the surface, which allowed a triggering of cumulus cloud patches in the substratocumulus layer at the top of a well-mixed turbulent layer around 500–800 m above the sea surface. Such cumulus clouds often penetrated the stratocumulus layer above maintaining vertical moisture transport from the sea surface to the stratocumulus layer and causing local thickening (Martin et al. 1995). In contrast the FIRE conditions measured by the aircraft were characterized by a much shallower and well-mixed boundary layer (typically less than 1 km deep) in most cases capped with relatively homogeneous stratocumulus cloud. On only one flight from the FIRE dataset used in this study was the boundary layer observed to become decoupled, allowing the formation of cumulus clouds beneath the stratocumulus cloud base.

The free tropospheric data are taken from the European Cloud and Radiation Experiment (EUCREX) field campaign, which was located in Scotland during September–October 1993. Synoptic conditions were characterized by midlatitude cyclonic systems that transferred moisture throughout the troposphere. Cirrus and altostratus clouds were encountered varying from thin, broken, and multilayered structures to thick frontal cloud. Very little liquid water was observed during the flights; only 5% of the EUCREX runs examined in this study contained more than 50% of the condensate in the liquid phase. A summary of all the flights analyzed is given in Table 1. For the purposes of this study, clouds measured in the boundary layer will generally be referred to as warm clouds; those measured in the free troposphere at heights above the freezing level will generally be referred to as cold clouds.

b. Instrumentation and data analysis

A range of instrumentation is used in the analysis. Total water content (qt = vapor + condensate) is measured using a Lyman-α hygrometer system referenced to out-of-cloud data from a General Eastern 1011B dewpoint–frost point hygrometer. All condensate mass is initially evaporated by means of two honeycomb heating elements before passing the Lyman-α beam. The probe has been shown to have a very short response time, and data are logged at 64 Hz. The absolute accuracy of the total water content measured with the probe is estimated to be approximately 0.3 g kg−1 for the boundary layer data (qt ∼ 6–12 g kg−1) and 0.05 g kg−1 for the free tropospheric data (qt ∼ 0.6–2.5 g kg−1). Due to the resolution of the probe it was decided to use only data with run-averaged total water contents greater than 0.6 g kg−1. Further details of the probe are found in Nicholls et al. (1990).

Liquid water content (ql) was measured using a Johnson–Williams hot-wire probe (Ouldridge 1982). Further details pertaining to the operation of the probe on the Meteorological Research Flight C-130 aircraft can be found in Moss et al. (1993). Data from the probe are logged at 4 Hz, although the estimated time response of the instrument is only around 1 Hz and possibly slower in high liquid water contents. The probe collection efficiency begins to decrease for droplets with radii greater than around 15 μm, falling to around 50% for 250-μm droplets. For the range of droplet sizes encountered in the boundary layer clouds in this study, it has been estimated that the Johnson–Williams probe recovers 90%–100% of the liquid water present and that the liquid water content calibration error is approximately 10%–20%.

Measurement of the ice water content was made with the Lyman-α hygrometer total water probe. Details of the method used for data reduction are presented in appendix A. The accuracy of the ice water content measurements is estimated to be ±0.02 g kg−1.

Temperature was measured using a Rosemount deiced total temperature sensor. Temperature data were corrected for dynamic heating effects using a recovery factor (Harmer et al. 1994). Data from the deiced temperature probe are logged at 32 Hz. The absolute accuracy of the corrected temperature data is estimated to be approximately 0.5 K (Stickney et al. 1994).

The formulas used for the calculation of saturation specific humidity over water and ice are those used by the World Meteorological Organization (e.g., Smith et al. 1990). The error in the derived saturation specific humidity resulting from an instrument error of 0.5 K in temperature rises from around 3% (0.3 g kg−1) in the boundary layer to around 5% (0.02 g kg−1) for the highest aircraft runs (approximately 8000 m) used in this study.

Condensation of vapor to either ice or water produces latent heating and a consequential increase in air temperature. The actual air temperature will therefore be greater than the air temperature prior to condensation, which is known as the liquid water temperature, TL. In large-scale models the liquid water temperature is a basic variable and we have decided to make use of TL in presenting the observations. The equation for TL used in this study is given below (e.g., Smith 1990):
TLTLwcpqlLwLicpqi
where Lw is the latent heat of condensation (vapor ↔ liquid), Li is the latent heat of fusion (liquid ↔ solid), cp is heat capacity of air at constant pressure, ql is the liquid water content, qi is the ice water content, and T is the measured air temperature. In cloud-free air T and TL are clearly identical. In the clouds observed in this study, the difference between T and TL is typically smaller than 1 K, but can in cases of high-condensate content be as high as 2 K. A 1 K increase in temperature causes an increase of approximately 0.5 g kg−1 in the saturation specific humidity in the boundary layer. In this study, all saturated specific humidities are calculated using TL.

c. Measurement of cloud fraction

A general problem arising with all cloud fraction measurements, whether they are taken from aircraft, satellites, celiometers, or the human eye, is that for each method of observation a different definition of cloud fraction is used. The meteorological observer uses the term cloud amount, which is “the amount of sky estimated to be covered by a specified cloud type (partial cloud amount) or by all cloud types (total cloud amount)” (WMO 1996). This measurement differs somewhat from the cloud fraction measurement made by other instruments. For example, a ceilometer defines cloud fraction as the fraction of time for which cloud base can be determined. In this study we define cloud fraction as the fraction of the time (distance) spent in cloud for a given horizontal aircraft run.

Implicit in this analysis is the assumption that data obtained from a single aircraft run are representative of the conditions throughout a two-dimensional area equal to the square of the run length. The extent to which this assumption is true is difficult to assess without prior knowledge of the form of the distribution of cloud throughout the measurement region.

Determination of the presence or absence of cloud in this study was carried out at 1 Hz (equivalent to a measurement averaged over 100 m along the aircraft track) for each aircraft run using cloud droplet concentrations from a forward scattering spectrometer probe (FSSP). A threshold droplet concentration is required to distinguish between cloudy and cloud-free regions. In cloud-free air the FSSP concentration is generally nonzero due to the presence of large aerosol particles. The concentration of large aerosol particles in clear air is always at least one and often more than two orders of magnitude lower than the concentration of water droplets observed in the ASTEX and FIRE clouds in this study. Large aerosol concentrations were of the order 0.3–2.0 cm−3, whereas cloud droplet concentrations were of the order 50–200 cm−3. Thus, by using a threshold concentration of a few drops per cubic centimeter the presence of clouds can be determined. In this study the results were obtained using a threshold value of 5 cm−3.

The sensitivity of the derived cloud fraction upon the threshold droplet concentration was examined by varying the threshold from 2.5 to 7.5 cm−3 for each run. The greatest sensitivity to the change in threshold droplet concentration usually occurred for cloud fractions in the range 0.3–0.7 where the aircraft may be passing into and out of cloud many times during a run. The sensitivity in the cloud fraction to the change in threshold was on occasion as large as 0.3, but for the majority of the cases was less than 0.1. In cases where the cloud fraction is less than around 0.3 or greater than around 0.7 the sensitivity to changes in the threshold liquid water content was significantly reduced.

Cloud fraction for cold clouds was determined in a similar manner to that for boundary layer clouds. A 2D-C optical array probe was used to determine the presence or absence of cold cloud. Qualitatively, time series of particle concentration from the 2D-C and FSSP probes have a similar appearance in cold clouds, but given that the 2D-C has a much larger sample volume and that concentration problems have been encountered when using the FSSP probe in clouds consisting predominantly of ice (Gardiner and Hallett 1985) it was decided that the 2D-C probe was more useful for accurate cloud fraction measurement in cold cloud. A particle concentration threshold of 10 L−1 was chosen. The sensitivity of the cloud fraction measurements to changes in the threshold concentration (from 5 to 15 L−1) was investigated. As with the boundary layer cloud fraction measurements using the FSSP the sensitivity was found to be greatest for partially cloudy runs (cloud fraction range 0.3–0.8), where sensitivities of 0.1–0.2 were observed. Again, the sensitivity was smaller outside this range.

3. Observational results

From each straight and level aircraft run a mean quantity was obtained for each parameter. Analyzing the data to examine the variability of a particular parameter along a given aircraft run proved to be difficult because the probability density functions (PDFs) thus derived were generally complex and often exhibited bimodality and large values of skewness. Only a small fraction of the runs had associated PDFs that one might term Gaussian.

Examples of such non-Gaussian PDFs are shown in Fig. 1 adjacent to profiles of total water content, liquid water content, and temperature. Each PDF is constructed from total water content data from a single run either below, in, or above stratocumulus cloud capping a well-mixed boundary layer. The position of the PDF on the page corresponds to the approximate location in the boundary layer shown in the adjacent profiles. For example, the uppermost PDF is taken from a run just above the inversion marking the top of the stratocumulus layer, the next three are from runs in the cloud layer, and the bottom two are from runs below cloud. Although the boundary layer is well mixed [see deRoode and Duynkerke (1997) for an extensive analysis of this case] the PDFs exhibit some complexity, with those in cloud being bimodal. This could be a result of cloud-top cooling entraining relatively dry air into a field of more moist air. The width of the distributions follows no clear trend with height.

Figure 2 is identical in form to Fig. 1, but in this case the cloud and subcloud layer are decoupled from a well-mixed more moist surface mixed layer. Cumulus clouds are observed (550–950 m) to form at the top of the surface layer below the main stratocumulus deck (1600–1900 m). The corresponding PDFs are complex. The lowest run is through the well-mixed surface layer and the corresponding PDF could be described as Gaussian in form. Above the surface layer a variety of different forms exist with bimodal distributions (e.g., at 677 and 1445 m) and strongly skewed distributions (e.g., at 1076 m). These distributions highlight the complex interactions between different levels within the boundary layer. It is difficult to represent these distributions satisfactorily with simple functions for use in a subgrid parameterization scheme. In the following figures data from the three experiments are plotted on the same axes. ASTEX data are denoted by open triangles, FIRE data by plus signs, and EUCREX data by filled circles. Each point represents a single, straight, and level run. The large difference in ambient temperature between the EUCREX and ASTEX/FIRE experiments resulted in a wide range of total water contents. To allow comparison of the relationships among the total water content, the condensed water content, and the cloud fraction, a water content scaling variable is required. Smith’s (1990) parameterization scheme uses the mean saturation specific humidity qs as a scaling variable, and so it was considered a suitable choice for use in this analysis. For temperatures below 273.15 K, the freezing point of water, the question arises as to whether the saturation specific humidity scaling variable is that taken with respect to ice or to water. In this study we use a weighted mean qs(l,i) of the saturation specific humidity with respect to liquid water qs(l) and that with respect to ice qs(i), with the weighting of each proportional to the liquid water ql and ice qi in the cloud, that is,
i1520-0469-57-12-1888-e2

Figure 3 shows the cloud fraction C as a function of the run-mean total water content qt normalized by the run-mean saturation specific humidity qs(l,i) (for qt > 0.6 g kg−1). Data from the total water content device were unavailable during FIRE and so the total water content in this case is calculated from the sum of the water vapor from the 1011B General Eastern dewpoint hygrometer and the liquid water from the Johnson–Williams probe. The error in qt/qs(l,i) is estimated knowing the instrumental errors in temperature and total water content (section 2), and although this error does vary from run to run, it is approximately 0.03–0.05 for the ASTEX data and 0.06–0.08 for the EUCREX data as denoted in the inset in Fig. 3. For the FIRE data the error in qt/qs(l,i) is estimated to be 0.04–0.06, which is greater than the error for the ASTEX data as a result of the inaccuracy incurred from the combination of the liquid and vapor content for these data.

The results show that there is an increase in cloud fraction from cloud free (C = 0) to fully cloudy (C = 1) as qt/qs(l,i) increases from 0.80–0.85 to 1.00–1.05. It is interesting that although there is considerable scatter in the data, which is due to the combination of atmospheric variablility and the error in qt/qs(l,i), the same broad relationship is found for both warm and cold cloud.

The data are fitted fairly well by a hyperbolic tangent function:
i1520-0469-57-12-1888-e3
which is shown as the thin dotted line in Fig. 3 with the constants A = 17.0 and B = 0.95 representing the best fit to the data from the three experiments. This relationship is henceforth referred to as FWI (Field and Wood I).
The variable qt/qs(l,i) is not identically equal to the mean total water saturation ratio, which is denoted as qt/qs(l,i). The two, using Reynolds averaging, are related by
i1520-0469-57-12-1888-e4

Figure 4 shows a plot of qt/qs(l,i) against (qt/qs(l,i)) for the EUCREX data, which shows that in almost all cases the run-mean relative humidity is almost equal to the normalized total water content qt/qs(l,i) used in this analysis. The product term qtqs(l,i)/qs2(l,i) is, for 87% of the runs, less than 5% of qt/qs(l,i) (see histogram in inset of Fig. 4). The product term qtqs(l,i)/qs2(l,i) for the EUCREX and FIRE data is generally even smaller than for the EUCREX data. Thus, it would make little difference to the form of the relationship in Fig. 3 if the run-mean relative humidity was used rather than qt/qs(l,i) as the choice of independent variable, a fact that might prove useful in cloud schemes where relative humidity is the prognostic water vapor variable.

The relationship between the cloud fraction and the run-mean condensed water normalized with the run-mean saturation specific humidity is shown in Fig. 5 for all three experimental datasets. The data can all be described fairly well by a single relationship.

In an attempt to explain the relationship between cloud fraction and cloud condensate we assume a differential equation for the rate of change of cloud fraction with respect to cloud condensate, ∂C/∂qc. It is further assumed that the rate of change of cloud fraction is directly proportional to the fraction of clear sky remaining, that is,
i1520-0469-57-12-1888-e5
The saturation specific humidity is used as a normalization factor to nondimensionalize the constant K. The clear-sky fraction relation ensures that there is a smaller increase in cloud fraction for a small increase in qc as the cloud fraction approaches 1 (addition of condensed water will tend to cause a larger increase in cloud fraction in a clear sky than an overcast one). Equation (5) can then be integrated with appropriate limits (qc = 0, C = 0) to obtain a relation for cloud fraction as a function of cloud condensate:
CKqcqs

From the best fit of (6) to the observational data of Fig. 5 a value of K = 75.0 was obtained. This functional relationship is denoted by the dotted line in Fig. 5 and is henceforth referred to as FWII (Field and Wood II).

4. Comparison of observations with parameterization schemes

The observational data in this study are used to validate cloud parameterization schemes that are either currently employed or could potentially be employed in large-scale climate and numerical weather prediction models. For this study we choose three schemes that were selected because of the differences in their conception: Slingo (1980), Smith (1990), and Xu and Randall (1996). Slingo (1980) uses GARP Atlantic Tropical Experiment (GATE) data to derive separate cloud parameterizations for low-, mid-, and high-level stratiform cloud. Smith (1990) does not quote as validation of his scheme experimental data of any kind but uses the assumption of subgrid distributions of humidity variables, while Xu and Randall (1996) use results from multiple runs of a cloud ensemble model to derive relationships among cloud condensate, relative humidity (defined as [qtqc]/qs(l,i)), and cloud fraction. Overviews of the three parameterization schemes are presented below, and Table 2 gives a brief summary of the features of the three schemes.

a. The Slingo scheme

The simplest of the three schemes is documented in Slingo (1980, henceforth SL80). Satellite observations of cloud fraction were obtained during GATE (Spackman 1975). Simultaneous thermodynamic observations were obtained from validated surface and upper-air observations from ships, land stations, and radiosondes. Low-level (warm) cloud was defined as that occurring below approximately 800 hPa, midlevel cloud as that occurring in the pressure range 400–800 hPa, and high (cold) cloud as that occurring about 400 hPa. Analysis of the GATE data culminated in three separate cloud fraction parameterizations for each of the three levels with each parameterization depending only upon the relative humidity. The SL80 scheme does not include a parameterization for the cloud condensate amount presumably, as this parameter was generally unavailable from GATE data.

The parameterizations for high-, mid-, and low-level clouds assume the form
i1520-0469-57-12-1888-e7
where RH is the mean relative humidity. For low- and high-level cloud M = 0.80. For midlevel cloud M = 0.65.

Figure 6 shows the relationship between the cloud fraction and the normalized total water content for the SL80 scheme. The solid line is the relationship for low and high clouds and the dashed line is for midlevel cloud. For a grid box that is just saturated (RH = 1.0) the cloud fraction is unity.

b. The Smith scheme

The second scheme to be investigated is the diagnostic parameterization for cloud fraction and condensate used in the U.K. Meteorological Office (UKMO) Unified Model, first presented in Smith (1990, henceforth S90). Variants of the S90 scheme are used in several large-scale numerical models worldwide. The scheme assumes a symmetric triangular distribution for the subgrid variability in total water content (a prognostic variable) from which the cloud fraction and condensed water are derived. The use of distribution functions to account for subgrid variability follows the ideas of Sommeria and Deardorff (1977). The critical-relative humidity RHcrit is defined as the minimum grid-box mean relative humidity for which clouds will form and is essentially left as a free parameter to be specified. Values of RHcrit as a function of height have been derived by comparison of model output with data from Earth’s Radiation Budget Experiment and ISCCP. The configuration that is currently employed in the UKMO Unified Model uses a critical relative humidity that is greatest (90%) close to the surface and that decreases with height in the boundary layer. Above the boundary layer a fixed constant value of the critical relative humidity is used (70%). We base our comparison upon the scheme as it is implemented in the UKMO Hadley Centre climate model (Pope et al. 2000). In S90 the cloud fraction is a function of the total water content and RHcrit:
i1520-0469-57-12-1888-e8
The saturation specific humidity used is taken with respect to water for temperatures above 273.15 K and with respect to ice below this temperature. Figure 7a shows the relationship between qt/qs and the cloud fraction from Eq. (8) for RHcrit = 0.7 (dashed), RHcrit = 0.8 (solid), and RHcrit = 0.9 (dotted). In each case the cloud fraction is zero for qt/qs ⩽ RHcrit and increases to reach 0.5 for qt/qs = 1.

c. The Xu and Randall scheme

Xu and Randall (1996, henceforth XR96) have developed a cloud parameterization scheme in a fundamentally different way from SL80 and S90. Instead of making assumptions about the subgrid variability of total water or similar parameter, XR96 have used repeated realizations from a cloud ensemble model to derive a relationship between cloud fraction, cloud condensate content, and relative humidity that can be used in a diagnostic sense in a large-scale model. Xu and Randall (1996) used the University of California, Los Angeles, cloud ensemble model with a three-phase bulk microphysical parameterization (Krueger et al. 1995) to simulate both boundary layer and tropospheric deep convective clouds. Unlike S90 the XR96 scheme does require additional means for prediction of the cloud condensate amount, and so the scheme is purely for use in predicting the cloud fraction.

The XR96 scheme, extended from initial work by Xu and Krueger (1991), is a modest breakthrough in the problem of predicting cloud fraction in large-scale numerical models for one major reason: the use of cloud ensemble models to determine relationships between cloud water and fraction provides a new and potentially fruitful avenue of research. Unlike S90 the XR96 scheme contains no free parameters and so comparison with observational data is relatively straightforward. The precise mathematical form of XR96 is given by
i1520-0469-57-12-1888-e9
where qc is the grid-box mean condensed water content, qs is the saturation specific humidity, p = 0.25, and qυ is the water vapor content; α0 = 100 and γ = 0.49 are constants determined empirically from the cloud ensemble model data.

The functional relationships of the XR96 scheme are more difficult to depict graphically than for the S90 scheme as there is a dependence of the cloud fraction in XR96 on both the condensed water content and RH. The presence in Eq. (9) of the saturation specific humidity effectively introduces a height dependence into the cloud fraction parameterization since qs decreases with height (through its temperature dependence) almost everywhere in the troposphere.

A possible shortcoming with the XR96 scheme is that only two climatic conditions were simulated: tropical deep convection and mid-Atlantic high pressure boundary layers with strong subsidence inversions. The cold clouds in this study, for example, are predominantly forced by weak frontal uplift rather than strong convective motions. Different cloud water–cloud fraction relationships may exist for such markedly different physical systems.

d. Comparison with observational data

In this section measurements of cloud fraction, total water content, saturation specific humidity, and condensed water content are used to validate the three cloud fraction parameterization schemes described above. The two simple functional fits (FWI, FWII) to the data provided in this study are used as a reference in the comparison. Comparison of the C-130 cloud fraction observations with those parameterized in SL80, S90, XR96, and the simple fits FWI and FWII are shown in Figs. 8, 9, 10, 11, and 12 respectively. Obviously we have the advantage of fitting our relationships optimally to the data presented here and it would be possible to adjust the other schemes in the light of this data. For simplicity we chose to use the schemes in the form they were published. The different symbols represent data from the different experiments (as in Figs. 3 and 5) and the dotted line represents perfect agreement. Table 3 shows, for each experiment, parameterization scheme, and the two functional fits, the rms error between the predictions and the observations for three different cloud fraction bands (0–0.3, 0.3–0.7, 0.7–1). Table 4 shows, for each experiment and each parameterization scheme, the mean difference between the parameterized cloud fraction and the observed cloud fraction for the same three different observed cloud fraction bands.

The bias of the predictions for the schemes is up to 0.25 and can change signs between datasets for the same scheme. Both the Slingo and the Xu and Randall schemes do well in comparison to the functional fits, FWI and FWII, while the Smith scheme displays a consistent underestimate of cloud fraction. The rms differences again indicate that the Xu and Randall scheme compares favorably with the reference fits, but for this aspect the Slingo scheme performs less well. Overall, the Xu and Randall scheme appears to have the least bias and spread, often outperforming the reference fits in the comparisons with the data.

5. Discussion and conclusions

While forecast models are tending to adopt dynamical schemes involving sources and sinks for the prediction of cloud fraction and condensed water content, observations presented here indicate that it is possible to diagnose cloud fraction from normalized total water content or a scaled cloud condensate content to a useful degree for stratocumulus cloud in the FIRE and ASTEX regions and midlevel ice cloud in the EUCREX region. A dynamical–microphysical prediction of cloud fraction is more satisfying in principle, but the increased number of tunable parameters makes this method less tractable until the physics of cloud formation and dissipation is known to a greater degree.

It is evident from Figs. 3 and 5 that cloud fraction is a sensitive function of the parameters used in this study. An important observation to note from Fig. 3 is that a cloud fraction of 1.0 is obtained for a qt/qs(l,i) of approximately 1.05. This suggests that a symmetric unchanging probability distribution function of qt (e.g., S90) is not applicable and the distribution is either strongly skewed (Bougeault 1981) or the formation of cloud forces the distribution to become modified (i.e., narrow). The authors plan to further investigate the distribution of atmospheric water along aircraft runs.

The existing cloud fraction parameterizations were tested in the form they were published. Even though it must be admitted that judicious choice of the tunable parameters associated with these schemes may have improved their performance in relation to the observations, it was found that the Xu and Randall scheme provided the greatest accuracy, comparing well with and even outperforming the reference fits.

It should be borne in mind that any parameterization of cloud fraction with variables such as qt/qs(l,i) will not necessarily be applicable for climatic conditions that differ from those under which the observations were made. Therefore, until the physical reasons for such relationships are fully understood caution should be used in regarding these results as both globally and temporally invariant.

Acknowledgments

The authors wish to thank the staff of Meteorological Research Flight, the C-130 RAF aircrew, and all of the other participants and organizers of the FIRE, ASTEX, and EUCREX projects. The authors also wish to thank Doug Johnson, Roy Kershaw, Steven Cusack, Gill Martin, Mark Webb, and Damian Wilson for providing valuable consultation and support.

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APPENDIX A

Estimating Ice Water Content with the MRF Total Water Probe

We present a method of estimating ice water content in ice clouds that capitalizes on the unique instrument response of the Meteorological Research Flight (MRF) total water probe (Nicholls et al. 1990). Evidence is also provided that suggests that the method appears to be accurate to within 0.02 g kg−1.

For this study the ice water content (IWC) was calculated using a total water probe mounted on the C-130. Brown and Francis (1995) detailed a method that found the IWC by subtracting a water vapor signal from the total water signal when the aircraft was known to be in ice cloud. The water vapor signal was provided by a Lyman-α fluorescence device that was also mounted on the C-130. In this appendix we suggest a very similar method to Brown and Francis, but obtain an estimate of the water vapor in cloud from the total water probe itself. The fluorescence device is in theory a self-calibrating device, but on a proportion of the high-level flights presented here, the calibration did not perform well and it is not thought to be accurate at pressures greater than 400 mb. Therefore, an alternative method was sought to obtain a measure of the water vapor in cloud.

The total water probe on the C-130 is a fast response instrument that measures the specific total water content (TWC) of a parcel of air. The data are recorded at 64 Hz, although the response has a slower response at low absolute total water contents. The method used by the probe is to rapidly convert all of the sampled water to vapor by means of efficient evaporators before measuring the total vapor density with a Lyman-α absorption hygrometer. The probe is not an absolute device and is calibrated against the General Eastern (GE) Dewpoint Hygrometer, which provides mass mixing ratios when operated in clear air.

The typical response of the total water probe in cold clouds is depicted in Fig. A1 (solid line). The spiked nature of the data is immediately obvious and is explained by realizing that as ice crystals enter the probe they are evaporated producing a very localized high value of water vapor density. The relatively low number concentrations of large ice particles means that individual arrivals are discernible. The total water probe time series thus consists of an environmental water vapor background with relatively discrete contributions from individual ice crystals. The assumption is made that the environmental water vapor background is the same as a minimum envelope fitted to the total water signal (dashed line in Fig. A1).

The method by which IWC is recovered from the total water content probe is as follows: first, fit a minimum envelope to the TWC time series using a window size of 10 Hz (which is longer than the length of a peak caused by individual ice particles) and then remove the minimum envelope from the 64-Hz TWC time series to leave just the ice particle signature. The remaining ice particle signature can then be averaged to provide a run-averaged IWC, for example.

A Nevzorov probe (Korolev et al. 1998) that measures total condensed water content was recently fitted to the C-130. An intercomparison was carried out for flight A661 (30 March 1999) between the Nevzorov probes estimate of IWC and the total water probe method of estimating the IWC to assess the acuracy of the total water probe method. Run averages (length 10 km) of IWC from the Nevzorov probe are plotted against the total water probe estimates in Fig. A2. The plot reveals that the total water probe estimate is 0.02 g kg−1 greater than the Nevzorov probe estimate. When this bias is removed the data have a random error of 0.02 g kg−1. Therefore, the IWC for the EUCREX flights was estimated using the total water probe method outlined above with the subtraction of 0.02 g kg−1.

APPENDIX B

Tabulated Observational Data

The aircraft run-mean observational data are presented in Fig. B1 in tabular form for each run from the three experiments (FIRE, ASTEX, and EUCREX). For each run the following information is given: Flt (Flight number), L (run length), T (air temperature), P (pressure), qt (total water content), qliq (liquid water content), qice (ice water content), and C (cloud fraction). The flight numbers for the different projects are H801–H813, FIRE; A203–A215, ASTEX; A279–A290, EUCREX.

Fig. 1.
Fig. 1.

Distributions of total water from a stack of six horizontal runs through a well-mixed boundary layer with a layer of 8/8 stratocumulus during ASTEX (flight A209), together with the total water, liquid water, and temperature profile.

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

Fig. 2.
Fig. 2.

Distributions of total water from a stack of seven horizontal runs through a decoupled boundary layer with stratocumulus and penetrative cumulus during ASTEX (flight A213), together with the total water, liquid water, and temperature profile.

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

Fig. 3.
Fig. 3.

Cloud fraction plotted against the run-mean total water content qt normalized with the run-mean saturation specific humidity qs(l,i). The approximate error in qt/qs(l,i) is shown for each dataset in the inset. See text for more complete description of the errors. Datasets: FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles).

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

Fig. 4.
Fig. 4.

Run-mean relative humidity, (qt/qs(l,i), plotted against run-mean total water content qt divided by the run-mean saturation specific humidity qs(i), for EUCREX data.

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

Fig. 5.
Fig. 5.

Cloud fraction plotted against the run-mean condensed water content normalized by qs(l,i). The approximate error in qc/qs(l,i) is of the order of 15% for the ASTEX and FIRE cases and 20% for the EUCREX cases. Datasets: FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles).

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

Fig. 6.
Fig. 6.

Form of the diagnostic cloud fraction prediction scheme from Slingo (1980). Cloud fraction is plotted against relative humidity for high–low-level cloud (solid line) and midlevel cloud (dashed line). The definitions of the cloud types are given in the text.

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

Fig. 7.
Fig. 7.

Form of the diagnostic cloud fraction prediction scheme from Smith (1990). Cloud fraction is plotted against the run-mean total water content qt normalized with the run-mean saturation specific humidity qs(l,i). Curves for three values of RHcrit are shown: dashed, RHcrit = 0.7; solid, RHcrit = 0.8; and dotted, RHcrit = 0.9.

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

Fig. 8.
Fig. 8.

Predicted cloud fraction using the Slingo (1980) scheme plotted against observed cloud fraction for FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles) data.

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

Fig. 9.
Fig. 9.

Predicted cloud fraction using the Smith (1990) scheme plotted against observed cloud fraction for FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles) data.

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

Fig. 10.
Fig. 10.

Predicted cloud fraction using the Xu and Randall (1996) scheme plotted against observed cloud fraction for FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles) data.

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

Fig. 11.
Fig. 11.

Predicted cloud fraction using the functional fit, FWI, suggested in this paper plotted against observed cloud fraction for FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles) data.

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

Fig. 12.
Fig. 12.

Predicted cloud fraction using the functional fit, FWII, suggested in this paper plotted against observed cloud fraction for FIRE (plus signs), ASTEX (triangles), and EUCREX (filled circles) data.

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

i1520-0469-57-12-1888-fa01

Fig. A1. Specific total water for 4 s obtained from the Meteorological Research Flight total water probe during a run in ice cloud (solid line). The minimum envelope fit to the data is shown as a dashed line.

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

i1520-0469-57-12-1888-fa02

Fig. A2. Run-mean ice water content obtained from the Nevzorov probe plotted against the run-mean estimate of ice water content from the total water probe alone.

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

i1520-0469-57-12-1888-fb102

Fig. B1 (Continued).

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

i1520-0469-57-12-1888-fb103

Fig. B1 (Continued).

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

i1520-0469-57-12-1888-fb104

Fig. B1 (Continued).

Citation: Journal of the Atmospheric Sciences 57, 12; 10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2

Table 1.

Summary of C-130 flights analyzed.

Table 1.
Table 2.

Summary of cloud parameterization schemes validated in this study.

Table 2.
Table 3.

Comparison of cloud parameterization schemes with observational data. Root-mean-square error between predicted and observed cloud fractions for the three cloud fraction categories: 0–0.3, 0.3–0.7, 0.7–1.

Table 3.
Table 4.

Comparison of cloud parameterization schemes with observational data. Mean difference between predicted cloud fraction and measured cloud fraction categorized by experiment for three measured cloud fraction categories: 0–0.3, 0.3–0.7, 0.7–1.

Table 4.
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