Accounting for clouds in sea ice modelsdirectory.umm.ac.id/Data...

Ž .Atmospheric Research 52 1999 77–113www.elsevier.comrlocateratmos

Accounting for clouds in sea ice models

Aleksandr P. Makshtas a,c, Edgar L Andreas b,),Pavel N. Svyashchennikov a, Valery F. Timachev a

a Arctic and Antarctic Research Institute, 38 Bering Street, St. Petersburg 199397, Russian Federationb US Army Cold Regions Research and Engineering Laboratory, 72 Lyme Road, HanoÕer, NH 03755-1290,

USAc Current affiliation; International Arctic Research Center, 930 Koyukak DriÕe, Fairbanks, AK 99775-7335,

USA

Accepted 23 April 1999

Abstract

Over sea ice in winter, the clouds, the surface-layer air temperature, and the long-waveŽ .radiation are closely coupled. Here we use archived data from the Russian North Pole NP

Ž .drifting stations and our own data from Ice Station Weddell ISW to investigate this coupling.Both Arctic and Antarctic distributions of total cloud amount are U-shaped: that is, observed cloudamounts are typically either 0–2 tenths or 8–10 tenths in the polar regions. We fitted these datawith beta distributions and, using roughly 70 station-years of observations from the NP stations,compute fitting parameters for each winter month. Although we find that surface-layer airtemperature and total cloud amount are correlated, it is not straightforward to predict one from theother because temperature is normally distributed while cloud amount has a U-shaped distribution.Nevertheless, we develop a statistical algorithm that can predict total cloud amount in winter fromsurface-layer temperature alone and, as required, produces a distribution of cloud amounts that isU-shaped. Because sea ice models usually need cloud data to estimate incoming long-wave radia-tion, this algorithm may be useful for estimating cloud amounts and, thus, for computing the sur-face heat budget where no visual cloud observations are available but temperature is measured —from the Arctic buoy network or from automatic weather stations, for example. The incominglong-wave radiation in sea ice models is generally highly parameterized. We evaluate fivecommon parameterizations using data from NP-4, NP-25, and ISW. The formula for estimatingincoming long-wave radiation that Konig-Langlo and Augstein developed using both Arctic and¨

) Corresponding author. Tel.: q1-603-646-4436; fax: q1-603-646-4644; E-mail:[email protected]

0169-8095r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0169-8095 99 00028-9

( )A.P. Makshtas et al.rAtmospheric Research 52 1999 77–11378

Antarctic data has the best properties but does depend nonlinearly on total cloud amount. Thisnonlinearity is crucial since cloud distributions are U-shaped while common sources of cloud datatabulate only mean monthly values. Lastly, we therefore use a one-dimensional sea ice model toinvestigate how methods of averaging cloud amounts affect predicted sea ice thickness in thecontext of the five long-wave radiation parameterizations. Here, too, Konig-Langlo and Augstein’s¨formula performs best, and using daily averaged cloud data yields more realistic results than usingmonthly averaged cloud data that have been interpolated to daily values. q 1999 Elsevier ScienceB.V. All rights reserved.

Keywords: Clouds; Sea ice; Surface-layer air temperature

1. Introduction

Clouds play a dominant role in determining short-wave and long-wave radiativetransfer in the atmosphere. Cloud area, height, thickness, and water content, among

Ž .other properties, all influence the radiative fluxes Curry and Ebert, 1990 . For sea icemodeling in polar regions, having an adequate description of the cloud cover isespecially important for estimating the radiative fluxes at the snow or sea ice surfacebecause surface melting and ice growth are quite sensitive to short-wave and long-wave

Ž .radiation Maykut and Untersteiner, 1971; Curry and Ebert, 1990 . In turn, the phasechanges and heat storage occurring in the snow or ice cover set the intensity of the

Žsurface turbulent and radiative fluxes and the conductive flux into the ice Makshtas,.1991a . As a result, in polar regions, there is strong coupling among the sea ice, the

albedo, and the clouds that has been termed ice-albedo and cloud-radiation feedbackŽ .Barry, 1984; Moritz et al., 1993; Curry et al., 1995 .

Despite the intensive development of methods for remotely sensing the atmosphere,the main source of data on the climatic characteristics of Arctic cloudiness is still in situvisual observations. Though polar-orbiting satellites frequently cross the Arctic, as

Ž .Raschke 1987 explains, the small image contrast between the ice cover and the cloudsfor visible wavelengths in summer and for infrared wavelengths in winter makes usingsatellite information difficult for estimating cloudiness in polar regions. Fig. 1 illustratesthis problem. It shows cloud amounts deduced from satellite data, surface-based visual

Ž .observations summarized in a climatic atlas Warren et al., 1988 , and visual observa-Ž .tions on the Russian North Pole NP drifting stations. The figure suggests that,

compared to visual observations, the satellite data overestimate the mean monthly cloudamount in the winter and underestimate it in the summer.

In the last few years, several works have attempted to evaluate the influence of iceŽ . Žcrystals Overland and Guest, 1991; Curry and Ebert, 1992 and Arctic haze Blanchet,

.1989; Zachek, 1996 on the radiation and thermal state of the Arctic atmosphere and theŽ .surface. For example, Curry and Ebert 1992 , building on the cloud statistics of

Ž .Huschke 1969 , estimate the annual variation of total cloud amount by including theeffects of low, medium, and high-level clouds and lower tropospheric ice crystal

Ž .precipitation Fig. 2 . They evaluate the role and quantity of ice crystal precipitation bycomparing the results of calculations from numerical radiation models that do not

( )A.P. Makshtas et al.rAtmospheric Research 52 1999 77–113 79

Fig. 1. The annual variation in monthly averaged total cloud amount over the Arctic Ocean on the basis ofsatellite data, visual observations on the NP drifting stations, and the cloud climatology published by Warren

Ž .et al. 1988 . The satellite and Warren et al. traces derived only from grid points north of 808N. Adapted fromŽ .Moritz et al. 1993 .

Fig. 2. The annual cycle in amounts of low-level, mid-level, high-level, and total clouds and in ice crystalŽ .precipitation in the lower troposphere. Adapted from Curry and Ebert 1992 .


incorporate ice crystal precipitation with measurements of the long-wave radiationŽ .balance or the incoming long-wave radiation. Overland and Guest 1991 also note a

discrepancy between observations and model calculations of incoming long-wave radia-tion and likewise suggest that the missing modeled downward long-wave radiationmight be explained by ice crystal precipitation, ‘‘diamond dust.’’ They did not, however,rule out other explanations such as blowing snow or optically thin clouds. Alternatively,on comparing incoming long-wave radiation measurements with theoretical estimates of

Ž .the effective radiant emittance of the atmosphere, Zachek 1996 shows that thetemporal variability of this radiation is closely connected with the temporal variability ofthe concentration of atmospheric aerosols, especially in February–May, when thisconcentration has its maximum.

Although it is important to continue research on the above-cited phenomena and todevelop methods to account for these in calculations of incoming long-wave radiationwith atmospheric radiation models, here we will consider the more simple and moreconventional characteristics of cloudiness observable visually during standard meteoro-

Ž . Ž .logical observations. These are total n and low n cloud amount, the cloudL

parameters most frequently used in climate research for calculating the radiative fluxes.Ž . Ž .The atlases of Prik 1965 and Gorshkov 1983 , among many others, give the spatial

and temporal variability of several climatic variables in the Arctic Basin based ongeneralized data from polar land stations and Russian drifting stations through NP-7.These climatic data, which include charts and tables of monthly and yearly averages ofthe spatial and temporal distributions of n and n , are still the basis for describingL

Žradiative energy exchange in climatic and prognostic models of Arctic sea ice e.g.,.Hibler, 1979; Parkinson and Washington, 1979; Ebert and Curry, 1993 .

The creation of a complete archive of quality-controlled standard meteorological datacollected on the Russian NP drifting stations within the framework of the Russian–American data rescue project, however, now provides the opportunity for a moreaccurate description of cloudiness and its temporal variability in the central Arctic. The

Ž .National Snow and Ice Data Center National Snow and Ice Data Center, 1996 at theUniversity of Colorado in Boulder recently issued a compact disk containing these data,‘‘Arctic Ocean Snow and Meteorological Observations from Drifting Stations: 1937,1950–1991, Version 1.0’’.

Using this large mass of recently available data, we have confirmed the conclusion ofŽ .Voskresenskii and Bryazgin 1988 that, in the central part of the Arctic Basin, cloud

amount tends to fall in two ranges, 0–2 tenths and 8–10 tenths. In other words, thefrequency distribution of cloud amount in the winter in the Arctic is U-shaped. Our more

Ž .limited analyses of cloud observations in the Arctic during 1 year at NP-4 Fig. 3 and inŽ . Ž .the Antarctic during 3 months on Ice Station Weddell ISW Fig. 4 show the same

Žbimodal distribution. Such histograms can be described with the beta distribution Harr,.1977 . Clearly, the common practice of quantifying cloud amount with only its average

value is, at least for spatial scales less than a thousand kilometers, not correct: Fortypical cloud distributions over sea ice, the mean value is the least likely value.

Our investigation of the correlation between atmospheric surface-layer temperatureand total cloud amount, based on the NSIDC data, has allowed us to develop a methodfor partially reconstructing total cloud amount in winter using only air temperature data.


Fig. 3. The histogram of total cloud amount observed on NP-4 from 1 April 1956 to 15 April 1957.

Conveniently, temperature is the most readily available meteorological parameter; forexample, the Arctic buoy network yields it routinely for much of the Arctic. This

Fig. 4. The histogram of total cloud amount observed during the drift of ISW from 25 February through 29May 1992.


method appears to be potentially useful for remote sensing and numerical modeling,especially when cloud amount is an external parameter of the model.

Another no less important problem in developing climatic sea ice models and usingthem for numerical experiments is adequately accounting for cloud amount in calcula-

Žtions of short-wave and long-wave radiation Doronin and Kheisin, 1975; Curry and.Ebert, 1990 . Presently, about ten empirical formulas exist for estimating incoming

long-wave radiation from measured surface-layer temperature, characteristics of theŽ .clouds i.e., amount and type of clouds , and, sometimes, surface-layer humidity.

Ž . Ž . Ž .Recently, Konig-Langlo and Augstein 1994 , Key et al. 1996 , and Guest 1998¨reviewed some of these parameterizations and evaluated their accuracy in accounting forincoming long-wave radiation in polar regions. The meteorological data they used for

˚Ž . Ž . Žtheir evaluations came from Resolute Canada , Barrow Alaska , Ny-Alesund Spitz-.bergen , Georg von Neumayer Station in Antarctica, and the Weddell Sea.

Here we look further at these parameterizations using data collected on NP-4 andŽ .recent data from the Russian–American drifting station Weddell-1 or ISW when that

Ž .station drifted through the western Weddell Sea in 1992 Andreas et al., 1992 . On ISW,we used both Russian and American instruments to measure the hourly averaged

Žcomponents of the radiation budget during the Antarctic fall and winter February. Ž .through May Claffey et al., 1995 . In the second part of this paper, we compare the

long-wave radiation data from NP-4 and ISW with the most frequently used parameteri-zations for incoming long-wave radiation.

Obviously, one of the major applications for a parameterization of incoming long-waveradiation is for estimating that component of the surface heat budget in models of sea icein polar regions. Therefore, we report on several numerical experiments done with aquasi-stationary, zero-dimensional thermodynamic sea ice model similar to that of

Ž .Semtner 1976 . Our purpose is to study the influence of various methods for describingmeteorological information and various parameterizations for the long-wave radiationbalance on the equilibrium thickness of sea ice in the Arctic.

2. Cloud amount in the central Arctic

Ž . Ž .Prik 1965 and Voskresenskii and Bryazgin 1988 published relatively completedata on the characteristics of cloudiness in the north polar region. On generalizing datafrom Russian polar stations and drifting stations through NP-14, Voskresenskii andBryazgin found marked spatial nonhomogeneity in the frequency distribution of clear

Ž .and overcast skies Table 1 .Table 1 shows that, in winter, overcast skies occur most frequently in the eastern and

western regions of the Russian Arctic coast, probably because of the prevalence of moistair masses originating over the Atlantic and Pacific Oceans. In the central region of theRussian Arctic coast and in the eastern part of the Arctic Basin, where the observationsin Table 1 rely heavily on data from the NP drifting stations, anticyclonic circulation ismore common. As a result and also because these regions are more distant from sourcesof moist air, periods with clear skies are roughly twice as likely in winter than along theeastern and western coasts.


Table 1Ž . Ž .Percent of the time with overcast 8–10 tenths and clear 0–2 tenths skies in various regions of the Russian

Ž .Arctic and in the central Arctic Basin from Voskresenskii and Bryazgin, 1988

Summer Winter

0–2 tenths 8–10 tenths 0–2 tenths 8–10 tenths

Western Coast 13 76 29 60Central Coast 11 80 44 44Eastern Coast 13 70 16 56Central Basin 7 88 40 49

In summer, significant uniformity characterizes cloud amount in the Arctic. Overcastskies occur roughly 80% of the time in all regions and perhaps even more frequently inthe central basin.

The opportunity for a more complete analysis of the temporal and spatial variabilityof cloudiness in the central Arctic is now, however, available with the creation of theCD-ROM archiving the standard meteorological observations from all the Russian NP

Ž .drifting stations National Snow and Ice Data Center, 1996 . For this purpose, we havegenerated, for each drifting station, time series consisting of either 3-hourly or 6-hourlyobservations for each month of the station’s drift. In other words, for each month wehave about 70 time series, spanning 1938 to 1991, that describe intermonth andinterannual variability of total and low cloud amount in the central Arctic. To comple-ment the analysis in Table 1, we averaged these data and present in Tables 2 and 3 theresulting statistics of total and low cloud amount at the drifting stations.

As in Table 1, the statistics in Tables 2 and 3 indicate a basic difference between theŽmodes of total and low cloud amount in winter and summer. In the winter November to

Table 2Ž .Total cloud amounts n, in percent observed on the NP drifting stations. ‘‘Mean’’ is the average percentage of

the observations in the listed category; ‘‘Std’’ is the standard deviation of the data in that category

Total cloud amount in tenths

0–2 3–4 5–6 7–8 9–10

Mean Std Mean Std Mean Std Mean Std Mean Std

Jan 40 15 9 8 5 4 6 5 39 12Feb 37 15 11 8 5 4 7 4 40 13March 35 16 12 7 6 4 8 5 40 13April 34 13 10 7 5 3 7 4 43 13May 17 8 4 3 3 2 5 2 71 9June 8 7 3 2 2 2 5 3 82 11July 6 5 3 2 2 2 5 3 84 9Aug 4 4 2 2 1 1 4 3 88 8Sept 6 4 4 2 2 2 4 3 84 7Oct 13 9 6 4 4 3 7 4 70 10Nov 33 15 9 6 5 5 7 5 46 16Dec 38 14 9 8 5 4 6 5 41 12


Table 3Ž .Low cloud amounts n , in percent observed on the NP drifting stations. ‘‘Mean’’ is the average percentageL

of the observations in the listed category; ‘‘Std’’ is the standard deviation of the data in that category

Low cloud amount in tenths

0–2 3–4 5–6 7–8 9–10


Jan 77 12 1 2 1 2 1 2 18 11Feb 78 14 1 2 1 2 1 2 16 11March 80 12 2 2 2 2 1 1 13 10April 78 14 2 2 1 2 1 2 16 12May 38 11 1 1 2 1 2 2 54 11June 22 11 2 2 2 2 3 2 68 12July 21 12 2 1 2 2 3 2 69 13Aug 18 10 2 1 2 2 3 2 73 11Sept 26 11 2 2 2 2 3 2 65 12Oct 45 13 2 2 3 2 3 2 45 13Nov 68 16 2 3 2 2 1 2 24 13Dec 76 13 1 2 1 1 1 1 19 11

.April , there are two practically equal maxima in the distributions of total cloud amount,one for 0–2 tenths and a second for 9–10 tenths. The U-shaped form of the frequencydistribution is thus quite obvious. Also in winter, cloud amounts in the 3–8 tenths binshave a temporally uniform distribution. Lastly, on comparing the winter total and lowcloud amounts in Tables 2 and 3, respectively, we see a prevalence of middle and upperlevel clouds in winter.

Ž .In summer June to September , overcast skies dominate, there are very fewoccurrences of skies with 3–8 tenths cloud cover, and low clouds constitute most of thecover. This latter observation is likely a consequence of the cold, wet, and practically

Ž .isothermal surface melting snow on sea ice combined with a fairly thin summerŽ . Ž .atmospheric boundary layer ABL Kahl, 1990; Serreze et al., 1992 and little entrain-

ment at the top of the ABL. Overcast skies also result when continental air massesŽ .transform as they advect over melting sea ice Matveev, 1981 . May and October are

short transition seasons between the winter and summer cloud distributions.Next, we use the observations from the drifting stations to consider the interannual

variability of cloud amount in the central Arctic. One feature of the drifting stationsnecessary to consider in this analysis, however, is that they constantly changed position.Although this drifting allowed them to sample wide areas of the Arctic, we worry thattheir observational time series might not be homogeneous as a result. Fortunately, wecan check for homogeneity. Usually, two or three drifting stations worked in the Arcticsimultaneously. By computing correlation coefficients for 3-hourly and daily averagedmeteorological data obtained at the same time on different stations, we found significantcorrelations between stations located in different regions. This seems reasonable in lightof the obvious uniformity of the surface and the distance the central basin is from

Ž .continents and oceans, where advection can foster inhomogeneity Table 1 .


In effect, we could thus assume that all stations in the central Arctic at the same timewere sampling the same cloud distribution. We thus created a time series of cloudobservations covering 1955–1991. But to keep the time series homogeneous, weconsidered data from only one station above 778N each year. We use this series forevaluating trends and making a more correct analysis of cloud characteristics in theArctic. Fig. 5 shows the area on which we focus our analysis, the February position ofall the NP stations, and the locations of the stations that yielded the data year we used inour analyses. The figure confirms that the stations providing our data were randomlydistributed within our area of interest.

As we explained above, total cloud amount and, to a smaller degree, low cloudamount have U-shaped distributions, unlike air temperature, which has a normaldistribution. This means that it is incorrect to describe the characteristics of a time series

Fig. 5. February positions of the NP drifting stations manned between 1938 and 1991. The dashed lineencloses the region under study here. The numbers denote stations that contributed each year’s data for our1955–1991 analyses.


of total or low cloud amount with the arithmetic average, as was done, for example, inŽ .the Gorshkov 1983 atlas. On the basis of Tables 2 and 3, it also seems appropriate to

describe the spatial and temporal variability of Arctic cloudiness just in terms of theŽ . Ž .frequency of clear 0–2 tenths and overcast 8–10 tenths skies. Using these criteria, we

Žfind an appreciable reduction in the frequency of clear skies in winter represented by.February data between 1955 and 1991 but no significant change in the frequency ofŽ .overcast skies Fig. 6 . These observations imply an increase in cloud amount in winter

Ž . ŽFig. 6. Observations for 1955–1991 of the frequency of clear skies 0–2 tenths and overcast skies 8–10.tenths in February on the NP drifting stations indicated in Fig. 5. The dashed curves show the 5-year running

means.


Ž . Ž .Fig. 7. The frequency of clear skies 0–2 tenths and overcast skies 8–10 tenths in July on the NP driftingstations indicated in Fig. 5. The total July global solar radiation on the same drifting stations comes from

Ž .Marshunova and Mishin 1994 . The dashed lines are 5-year running means.


that may be connected with the increase in aerosol pollution in the Arctic atmosphereŽ .Radionov and Marshunova, 1992 which peaks in the late winter and early springŽ .Makshtas et al., 1994; Shaw, 1995 . The increased aerosol concentration simplyprovides more cloud condensation nuclei.

Another explanation could be the decrease in sea level pressure in the central ArcticŽ .since about 1987, documented by Walsh et al. 1996 . This decrease is most pronounced

in the winter and would foster meridional advection of moist air from lower latitudesthat could augment winter cloud amounts.

Fig. 7 shows time series of summer cloud observations and global solar radiationŽ .both represented by July data , the latter coming directly from Marshunova and MishinŽ .1994 . We see here a reduction in the frequency of overcast skies in summer that isconfirmed by a corresponding increase in the global solar radiation measured on thesame drifting stations. These changes may be related to changes in the mode of theatmospheric circulation in the Arctic. On analyzing fluctuations in the Arctic atmo-

Ž .spheric circulation since 1963, Dmitriev 1994 documents a reduction in the frequencyof meridional air exchange and an increase in the frequency of zonal processes. In otherwords, in recent years, fewer warm, moist air masses have been entering the ArcticBasin, and, as a consequence, fewer clouds have been forming in situ.

Ž .This result, at first, sounds contrary to the findings of Walsh et al. 1996 that annualmean sea level pressure in the central Arctic has been lower than average since 1987.But Walsh et al. find no statistically significant pressure decrease in the summer,although the annual decrease is statistically significant. Consequently, perhaps bothanalyses are correct: zonal exchange could be enhanced in summer, while meridionalexchange is enhanced in winter. Clearly, more work needs to be done to document thisvariability.

3. Estimating total cloud amount in the winter

Although the spatial and temporal variability of Arctic clouds are some of the poorestdocumented parameters required for modeling the polar atmosphere and sea ice cover,numerical experiments with atmospheric general circulation models show that these

Ž .models have the highest sensitivity to just these parameters e.g., Cess et al., 1989 . Toimprove and validate regional sea ice models and atmospheric general circulationmodels, it is therefore crucial to develop an adequate description of cloud parametersand their spatial and temporal variability.

One possible way to increase the reliability of cloud descriptions is a method ofstatistical modeling that we have developed based on a correlation analysis using

Ž .surface-layer air temperature T and total cloud amount during the winter. Our analysesof archival data from the NP drifting stations show that the correlation coefficients

Ž . Ž .between T and total cloud amount n and between T and low cloud amount n forL

observations from November to March are, on average, 0.6 with a significance level of0.1 in the central Arctic.

It should be noted that, because on the drifting stations there were 4–8 observationsper day, proximate observations may be correlated. That is, all the paired temperature


and cloud observations may not be independent. To test whether this inherent correlationaffected our analysis, we repeated the correlation analysis twice, first using only oneobservation per day and then using one observation every 5 days. These analyses,however, yielded the same correlation coefficients that we gave above.

We also calculated the correlation coefficients between air temperature and cloudamounts in time series for individual months from November to March. Again, thecorrelation coefficients, on average, exceed 0.6. For October data, we also foundsignificant correlation between temperature and cloud amounts, but the correlation was alittle lower.

Based on the significant correlation between T and n and between T and n , it mightL

seem possible to use a linear equation to estimate 3-hourly cloud amounts from 3-hourlytemperature data. Such a method, however, is not justified when the independent

Ž .variable i.e., temperature is nearly normally distributed while the dependent variableŽ .i.e., cloud amount has a U-shaped distribution because any new random variableobtained with a linear transformation from a normally distributed variable will also benormally distributed. Thus, because we require that any predicted time series of n or nL

have a U-shaped distribution, we cannot use standard statistical modeling methods toexploit the observed high correlation between T and n and between T and n but must,L

instead, use a more complicated statistical algorithm.For predicting cloud amounts using air temperature data, we begin with the explicit

form of the frequency distribution of total cloud amount in the Arctic Basin. This isU-shaped; we fit it with a beta distribution. The probability density function of a beta

Ž . w x Ždistribution, f x , for random variable x in the interval 0,1 is e.g., Harr, 1977;.Aivazyan et al., 1983

G aqbŽ .by1ay1f x s x 1yx for 0FxF1, 1aŽ . Ž . Ž .

G a G bŽ . Ž .

f x s0 otherwise, 1bŽ . Ž .

where G is the gamma function.Ž .Empirical values of the parameters a and b in Eq. 1a can be evaluated from the

sample mean of x, x, and the sample standard deviation, s;

x 1yxŽ .asx y1 , 2aŽ .ˆ 2ž /s

x 1yxŽ .bs 1yx y1 . 2bŽ . Ž .2ž /s

ˆTable 4 shows average values of the a and b parameters for winter calculated fromˆŽ . Ž .Eqs. 2a and 2b using the monthly 3-h series of total cloud amount described above.

In other words, in preparing Table 4, we had about 70 months of data to use inˆcomputing each month’s average a and b values. We did, however, exclude approxi-ˆ

mately 10 monthly values from each set of calculations either because the correlation


Table 4ˆValues of a and b for the beta distributions describing total cloud amount in the winter. ‘‘Mean’’ is the valueˆ

averaged from roughly 70 months of fitted beta distributions; ‘‘Std’’ is the standard deviation of the valuesused to create the means

November December January February March


a 0.24 0.18 0.18 0.16 0.19 0.18 0.23 0.18 0.31 0.21ˆb 0.17 0.11 0.17 0.13 0.19 0.15 0.22 0.15 0.25 0.16

between T and n was weak or because the cloud distribution was not obviouslyU-shaped.

Our method for statistically modeling cloud amount compatible with a beta distribu-˜ ˜tion goes as follows. Let T be the normalized surface-layer temperature. As such, T is a

random variable that is approximately normally distributed with mean 0 and standard˜ ˜Ž .deviation 1. On assuming that the total cloud amount n T is a monotonic function of T ,

˜we know from mathematical statistics that the following expression relates n to TŽ .Ventcel, 1964, p. 263 ff. :

y1˜ ñ T sF F T , 3Ž .Ž . Ž .2 1

˜ ˜Ž . Ž .where F T is the cumulative probability density function of random variable T , F n1 2y1Ž .is the cumulative probability density function of random variable n, and F n is the2

Ž .inverse function of F n .2Ž . Ž . ŽSince F n describes a beta distribution, Eq. 3 can be approximated as Aivazyan2.et al., 1983 :

añ T ( , 4Ž .Ž . ˆ ãqb exp 2w TŽ .ˆ

ˆ ˜Ž .where a and b are the parameters of the beta distribution given in Table 4, and w T isâ function of the normalized air temperature. Appendix A fills in the mathematics on

˜Ž . Ž . Ž .which Eqs. 3 and 4 are based and gives the functional form for w T . In summary,ˆŽ .Eq. 4 yields good results for small values of a and b but leads to significant errors ifˆ

either one of them is 0.5 or more.

Table 5Ž .Values of the bias, random, and total errors when Eq. 4 is used to predict the total cloud amounts observed

on NP-25 during the winter of 1982–1983. ‘‘Bias’’ is defined as the observation minus the prediction

Error November December January February March

Bias y0.19 y0.10 y0.32 y0.10 y0.27Random 0.35 0.47 0.35 0.39 0.34Total 0.40 0.48 0.47 0.40 0.43


We tried this algorithm for predicting total cloud amount using the standard meteoro-logical data obtained on drifting station NP-25, which was above 858N from October1982 to October 1983. Table 5 summarizes the errors in these predictions for the winterof 1982–1983, November through March. In that table, the ‘‘bias’’ error is the averageof the difference between the observation and the prediction, the ‘‘random’’ error is thestandard deviation of this difference, and the ‘‘total’’ error is the root-mean-square ofthis difference. In the table, we see that the errors are rather large; but the algorithm,nevertheless, does let us distinguish between clear and overcast skies, which, as we haveestablished, are the two dominant regimes in the Arctic.

w Ž .xFig. 8. Observed and modeled using Eq. 4 total cloud amounts based on observations and data from NP-25in November 1982.


Figs. 8 and 9 show other tests of our algorithm for estimating total cloud amountusing data from NP-25. Fig. 8 shows histograms of the observed and modeled totalcloud amounts based on data collected on NP-25 in November 1982. Fig. 9 compiles240 consecutive observations of total cloud amount and our simultaneous estimates of

ˆŽ .total cloud amount based on Eq. 4 . The figures show that, using the a and bˆcoefficients averaged from all data between 1955 and 1991 and having observations of

Ž .surface-layer temperature on NP-25, we have managed to capture with Eq. 4 not onlythe U-shaped frequency distribution in total cloud amount but also, to an extent, itstemporal variability.

On the down side, however, both Figs. 8 and 9 suggest that, when observed cloudamount is 0, the algorithm does not do especially well in predicting 0 cloud amount. Toreiterate, though, Fig. 9 confirms that our algorithm definitely distinguishes betweenovercast skies and generally clear skies.

Naturally, using this algorithm to estimate total cloud amount has some limitationssince the observed cloud amount can result from any type of cloud from stratus to cirrus.Since each such cloud type influences the radiation and temperature regime in the loweratmosphere differently, a prediction scheme based on only a single parameter, surface-layer temperature, must ultimately be an oversimplification. Nonstationarity in thephysical processes in the lower atmosphere — caused, for example, by advecting andadjusting air masses or by the slow evolution of the ABL during very stable stratifica-tion — can also distort the results. All of these problems as well as some imperfect

Ž .approximations in Eq. 4 should improve, however, with further work on this algorithm.Ž .Nevertheless, in light of the results shown in Figs. 8 and 9, we think Eq. 4 presents

Fig. 9. Temporal variability of 240 consecutive 3-hourly observations of total cloud amount in November 1982Ž .on NP-25 and our modeled total cloud amount based on Eq. 4 .


interesting prospects as an indirect method for estimating cloud amount in the polarregions, especially during the polar night. With the development of the InternationalArctic Buoy Program, in particular, and the consequent availability of simultaneouslymeasured surface-layer temperatures from various parts of the Arctic Basin, our methodcould provide estimates of cloud amounts with coverage comparable to satellites.

4. Parameterizing the long-wave radiation balance in sea ice models

Long-wave radiation is one of the key processes determining the rate at which sea iceŽ .forms in the polar regions in winter Maykut, 1986; Makshtas, 1991a . This fact has led

to numerous parameterizations for the long-wave radiation balance of snow-covered seaice. These have, in turn, been used to study the climatic significance of processesaffecting ocean–atmosphere interaction in high latitudes, especially with coupledocean–ice models in which the characteristics of the atmosphere are external parametersŽ .e.g., Hibler, 1979; Parkinson and Washington, 1979 .

Ž .The long-wave radiation emitted by a surface F is described by the Stefan–Boltz-up

mann law;

F s´s T 4 , 5Ž .up 0

where T is the surface temperature, ´ is the emittance of the surface, and s is the0

Stefan–Boltzmann constant.Ž .The incoming long-wave radiation from the atmosphere F can be determined bydn

Ž .an appropriate radiative transfer model Kondratyev, 1969; Curry and Ebert, 1992 .Using such models, however, requires data on the distribution of air temperature and

Ž .humidity up to heights of at least 30 km. Therefore, since the works of Brunt 1952 and˚ Ž .Angstrom Geiger, 1965; Matveev, 1969 , F has been parameterized from standard¨ dn

meteorological observations using its empirical dependence on cloud amount and on thetemperature and humidity of the atmospheric surface layer. In these parameterizations,the incoming long-wave radiation is estimated from

F s´ n , T , e s T 4 , 6Ž . Ž .dn )

where ´ is the effective long-wave emittance of the atmosphere, a function of cloud)

Ž . Ž .amount and air temperature T and vapor pressure e at a height of 2 m.Many functional expressions for the effective emittance of the atmosphere have been

published. These are generally based on readily available observations and containempirical coefficients obtained with a variety of temporal averaging methods. Weconsider here the functions used most frequently.

4.1. Brunt’s method

Ž . Ž .In the parameterization of Brunt 1952 e.g., Matveev, 1969 , ´ for a clear sky)

depends only on the water vapor content of the atmosphere and is described by

´ sa qb e1r2 , 7Ž .) B B

where e is the vapor pressure in millibars, and a and b are empirical coefficients.B B


On the basis of observations in middle latitudes, Brunt found a s0.526 andBŽ .b s0.065. These coefficients, however, are not universal; Kondratyev 1969 showsB

that their values change with the measurement site. Below we will show Marshunova’sŽ .1961 confirmation of this site dependence.

Ž .Brunt deduced the long-wave radiation balance BsF yF by introducing aup dn

cloud multiplier. That is, the long-wave radiation balance in the presence of clouds is

BsB 1yc n , 8Ž . Ž .0 B

where n is fractional total cloud amount, B is the long-wave radiation balance for a0

clear sky, and c is the average weighting coefficient for all types of clouds. ForBŽ .latitudes above 608N, c s0.81 Berliand, 1956 .B

4.2. MarshunoÕa’s method

Ž .The parameterization of Marshunova 1961 essentially applies Brunt’s methods tometeorological conditions in the Arctic. The coefficients a and b are different,M M

however, and accounting for the influence of clouds occurs directly in the formula forF :dn

´ s a qb e1r2 1qc n , 9Ž . Ž .Ž .) M M M

where n is again the fractional total cloud amount. Tables 6 and 7 list the a , b , andM M

c coefficients that Marshunova obtained from monthly averaged values of B and nM

observed at several polar stations and on drifting stations NP-3 and NP-4 in 1954–1957.Ž .We see from Tables 6 and 7 that the empirical coefficients in Eq. 9 have clear

Žspatial and temporal variability and are, thus, not universal. The variations in c TableM.7 are especially pronounced. The variability in a , b , and c is likely connectedM M M

with the types of air masses and clouds prevalent in a region, a variability we earlierdocumented in Table 1.

4.3. Maykut and Church’s method

On analyzing 3000 hourly observations of air temperature, humidity, incominglong-wave radiation, and cloud amount collected during a year at Barrow, AK, Maykut

Table 6Ž . Ž .The coefficients a and b for use in Eq. 9 , derived by Marshunova 1961 from observations at variousM M

Arctic stations

a bM M

Tikhaya Bay 0.61 0.073Cape Zhelaniya 0.61 0.073Chetyrekhstolbovoy Island 0.69 0.047Cape Schmidt 0.69 0.047NP-3, NP-4, 1954–1957 0.67 0.050


Table 7Ž . Ž .Monthly averages of the coefficient c for use in Eq. 9 , derived by Marshunova 1961 from observations atM

various Arctic stations

J F M A M J J A S O N D

Tikhaya Bay 0.27 0.29 0.29 0.24 0.24 0.22 0.19 0.19 0.21 0.25 0.26 0.28Cape Zhelaniya 0.29 0.29 0.29 0.24 0.24 0.22 0.19 0.18 0.21 0.22 0.26 0.28Chetyrekhstolbovoy Island 0.27 0.27 0.25 0.24 0.22 0.19 0.16 0.19 0.22 0.25 0.25 0.27Cape Schmidt 0.25 0.25 0.20 0.25 0.24 0.18 0.16 0.19 0.22 0.25 0.27 0.26NP-3, NP-4, 1954–1957 0.30 0.30 0.30 0.28 0.27 0.24 0.22 0.23 0.27 0.29 0.30 0.30

Ž .and Church 1973 developed the following expression for the effective emittance of thepolar atmosphere:

´ s0.7855 1q0.2232 n2.75 , 10Ž . Ž .)

where, as above, n is the fractional total cloud amount. The difference between this andprevious parameterizations is that here the influence of water vapor on incominglong-wave radiation is taken into account indirectly in the empirical coefficients.

4.4. Satterlund’s method

Ž .For parameterizing the effective emittance, Satterlund 1979 offers a function of airŽ .temperature and vapor pressure that Brutsaert 1982 claims describes long-wave

radiation well at low temperatures:

T r2016´ s1.08 1yexp ye , 11Ž . Ž .)

where T is in kelvins and e is in millibars. As with Brunt’s method, Satterlund accountsŽ .for cloud effects by using a multiplier in the long-wave radiation balance as in Eq. 8 .

4.5. Konig-Langlo and Augstein’s method¨

Ž .For effective emittance, Konig-Langlo and Augstein 1994; henceforth, KL&A¨suggest

´ sa qb n3 , 12Ž .) K K

where n is again the fractional total cloud amount, and a and b are empiricalK K

coefficients. As in Maykut and Church’s parameterization, these a and b coefficientsK K

implicitly include humidity effects.KL&A obtained a s0.765 and b s0.22 on the basis of visual observations ofK K

cloud amount and measurements of incoming long-wave radiation with an Eppley PIR˚ XŽpyrgeometer at two polar stations Ny-Alesund, 70856 N, 11856E; and Georg von

X X .Neumayer, 70839 S, 8815 W . KL&A’s ´ differs from Maykut and Church’s by its)

stronger dependence on cloud amount.


4.6. New test of the long-waÕe parameterizations

The Russian NP drifting stations yielded many observations of the long-waveŽ .radiation balance. Marshunova and Mishin 1994 recently described the sensors used

for these observations and their accuracy and tabulated monthly averaged data. For thiswork, we created, from the original archived Russian data, a new data set consisting of3-hourly radiation measurements made on NP-4 in 1956–1957. Together with thesurface-level air temperature data included on the National Snow and Ice Data CenterŽ .1996 CD-ROM, this radiation data lets us test our cloud algorithm and the emittanceformulas described above for the central Arctic.

We also obtained new data for additional tests during the drift of the Russian–ŽAmerican ISW in the western Weddell Sea from February to June 1992 Andreas et al.,

.1992; Claffey et al., 1995 . During this period, in the center of a drifting 1-km-wide icefloe, we made continuous, hourly averaged measurements of the components of thelong-wave radiation budget and the usual meteorological variables with both Russianand American sensors. The radiation measurements, in particular, showed good agree-

Ž .ment among the various instruments Claffey et al., 1995 .Ž .Fig. 10 shows the effective atmospheric emittance, computed from Eq. 6 with T

taken as the 5-m air temperature, for all of our ISW data. The curve is the KL&AŽ .relation, Eq. 12 . Because of the negligible increase in emittance for cloud amounts

between 0 and 5 and the steep increase for cloud amounts of 9 and 10 in Fig. 10, noemittance model that is linear in cloud amount, such as Brunt’s, Marshunova’s, orSatterlund’s, can fit the ISW data as well as nonlinear relations like KL&A’s andMaykut and Church’s.

Ž .Fig. 10. The effective atmospheric emittance, computed from Eq. 6 , based on all the ISW data collected from25 February through 29 May 1992. The error bars show one standard deviation. The line is Konig-Langlo and¨

Ž . Ž .Augstein’s 1994 relation, Eq. 12 .


Table 8 lists some statistics of the measurements for all of the observations made inMay 1992 on ISW and in November 1956 on NP-4. We focus on May at ISW becausewe have data for the entire month and because the short-wave components — whichmight complicate our measuring and interpreting the long-wave radiation — were small.November in the Arctic is similar to May in the Antarctic — late fall. Because cloud

Ž .amounts at ISW and at NP-4 had a U-shaped distributions Figs. 3 and 4 , Table 8 treatsŽ . Ž . Ž .the radiation values for clear 0–2 , partly cloudy 3–7 , and overcast 8–10 skies

separately, regardless of other weather conditions.On studying Table 8, we see that the values of long-wave radiation balance at the two

stations are very similar in both the mean and the standard deviation. We thus infer thatthe long-wave radiation balance is influenced similarly by clouds in both regions andhas similar seasonal values, at least when short-wave radiation is weak or absent. Theseare important points in light of the dependence on region and season of some of thesimple parameterizations described above.

Table 9 shows the results of our tests of the five long-wave parameterizationsdescribed above against both the ISW and NP-4 data. Again, we tabulate bias, random,and total errors, as in Table 5. We see in Table 9 that, of the five incoming long-waveparameterizations that we are considering, the one by Konig-Langlo and Augstein¨Ž .1994 performs best. Comparing the entries in Tables 8 and 9, we see that, for it, thetotal error for the difference between the measured values and those calculated with Eqs.Ž . Ž .6 and 12 does not exceed 5%, even for cloud amounts of 3–7 tenths, where therandom error caused by inaccuracies in the measurements is largest. Five percent isapproximately the experimental error in the ISW F values.dn

Ž .The parameterization of Maykut and Church 1973 perform almost as well asKL&A’s, but the other three parameterizations for F are significantly worse. Thedn

Table 8ŽObserved values of the long-wave radiation balance B and the incoming long-wave radiation F both indn

y2 .W m . The observations are from ISW in May 1992 and NP-4 in November 1956. ‘‘No. Obs.’’ is thenumber of observations for the indicated cloud amount, ‘‘Mean’’ is the average value for that cloud amount,and ‘‘Std’’ is the standard deviation

Ž .Cloud amount tenths ISW NP-4

No. Obs. B F No. Obs. Bdn

0–2 110 22Mean 43 154 35Std 5 13 6

3–7 269 12Mean 24 181 30Std 8 22 12

8–10 363 82Mean y1 225 1Std 7 19 10

0–10 742 116Mean 15 198 11Std 18 34 18


Table 9Values of the bias, random, and total errors in the long-wave radiation balance B and in the incoming

Ž y2 .long-wave radiation F both in W m when the May 1992 data from ISW and the November 1956 datadn

from NP-4 are compared with five different incoming long-wave parameterizations. ‘‘Bias’’ is defined as theobservation minus the parameterization

Method Cloud amount ISW NP-4Ž .tenths Bias Random Total Bias Random Total

Ž .Brunt, Eq. 7 0–2 B y30 3.5 30.2 y46 8 46.7F 30 2.5 30.1dn

3–7 B y27 2.9 27.2 y17 4 22.0F 27 2.5 27.1dn

8–10 B y25 4.4 25.4 y18 10 20.6F 25 3.7 25.3dn

0–10 B y26 4.1 26.3 y23 15 27.6F 26 3.6 26.2dn

Ž .Marshunova, Eq. 9 0–2 B y16 3.0 16.3 y20 7 21.2F 14 2.2 14.2dn

3–7 B y25 4.5 25.4 y3 13 13.3F 22 4.1 22.4dn

8–10 B y37 6.0 37.5 y13 13 18.4F 34 5.5 34.4dn

0–10 B y30 9.5 31.5 y13 13 18.4F 27 8.9 28.4dn

Ž .Maykut and Church, Eq. 10 0–2 B 4 4.2 5.8 y3 7 7.6F y6 3.4 7.0dn

3–7 B y11 6.2 12.6 y3 12 12.4F 8 5.7 9.8dn

8–10 B y18 4.6 18.6 y11 11 15.6F 14 3.7 14.4dn

0–10 B y12 9.2 15.1 y8 11 13.6F 9 8.4 12.3dn

Ž .Satterlund, Eq. 11 0–2 B y17 2.8 17.2 y32 8 33.0F 18 2.4 18.2dn

3–7 B y17 2.6 17.2 y8 14 16.1F 18 2.1 18.1dn

8–10 B y20 4.5 20.5 y14 10 17.2F 20 3.8 20.4dn

0–10 B y18 3.9 18.4 y17 12.5 21.2F 18 3.7 18.4dn

Ž .KL&A, Eq. 12 0–2 B 4 4.2 5.8 y7 8 10.6F y6 3.5 6.9dn

3–7 B y10 5.6 11.5 y6 13 14.3F 8 5.0 9.4dn

8–10 B y11 4.9 12.0 y5 11 12.1F 7 4.6 8.4dn

0–10 B y8 7.3 10.8 y6 11 12.5F 5 6.7 8.4dn


nonlinear dependence on clouds in the KL&A and Maykut and Church formulationsamplifies errors in the observations and produces random errors for these parameteriza-

Fig. 11. Measured incoming long-wave radiation during May 1992 on ISW and predictions of it using theequations of Brunt, Marshunova, Maykut and Church, Satterlund, and Konig-Langlo and Augstein. The upper¨panel is a temporal sequence when the total cloud amount was 8–10 tenths. Satterlund’s scheme producesestimates between the Brunt and Maykut and Church estimates and is, thus, left out of this panel for clarity.The lower panel is for observations when the total cloud amount was 0–2 tenths. Because Maykut andChurch’s scheme produces estimates similar to KL&A’s and because Satterlund’s scheme yields results similarto Marshunova’s, we leave these two traces out of this panel for clarity.


tions in Table 9 that are generally larger than for the other three parameterizations. Butthese random errors are fairly independent of cloud amount and thus, seemingly, resultfrom experimental uncertainties rather than faults in the parameterizations. For the otherthree parameterizations, in contrast, the random errors vary with cloud amount and, thus,reflect shortcomings in these parameterizations.

Fig. 11 shows the temporal variability of the incoming long-wave radiation for Mayobserved on ISW and estimated with the five parameterizations under consideration.Although the estimates may differ from the observed radiation for both the clear-sky andovercast-sky cases, each of the five parameterization schemes does predict temporalbehavior that coincides with that in the experimental data.

On comparing the errors tabulated in Table 9 with the mean values in Table 8, we seeŽ .that the total error in evaluating B for clear skies cloud amounts of 0–2 tenths with the

KL&A parameterizations is above 10% for ISW and about 30% for NP-4. If weŽ .consider all the observations during the month 0–10 tenths , the total error increases to

Žover 70% for ISW and to over 100% for NP-4. Notice, too, that for overcast skies 8–10. Ž .tenths , when the absolute values of B are small Table 8 , even the sign of B is

uncertain. Remember, though, the long-wave radiation balance — whether calculated asthe difference between the incoming and emitted long-wave components, as on ISW, oras measured directly with a single sensor, as on NP-4 — represents a small differencebetween large values of F and F . As a result, it has a large relative error;dn up

parameterizing it is consequently difficult. Nevertheless, data from both ISW and NP-4confirm that KL&A’s is the best among five alternatives for parameterizing thelong-wave radiation balance over sea ice during the polar night for the periods studied.

5. Model sensitivity to the description of long-wave radiation

An equation for the heat budget of the upper surface is a necessary component ofŽprognostic and climatic sea ice models regardless of their complexity Maykut and

Untersteiner, 1971; Parkinson and Washington, 1979; Makshtas et al., 1988; Ebert and.Curry, 1993 . Thus, in contrast with the nonstationary, one-dimensional sea ice model of

Ž .Maykut and Untersteiner 1971 , in which the characteristics of the energy exchangebetween the atmosphere and the ocean were prescribed, in subsequent models the

Ž . Ž .turbulent surface fluxes of sensible H and latent heat H , the long-wave radiations LŽ .balance B , and the short-wave radiation balance are internal parameters of the model.

That is, these are simulated conditions of the sea ice cover and, as such, are prescribedby the parameters of the atmospheric surface layer.

Above, we showed that despite the same input data, calculations of the long-waveradiation at a sea ice surface differ depending on which of several popular parameteriza-tions we use. It is, thus, interesting to consider how applying these parameterizations ina sea ice model might affect the computed equilibrium thickness of the sea ice. For thispurpose, we perform several numerical experiments using the one-dimensional sea ice

Ž .model described by Makshtas 1991b .Ž .This model is a version of the 0-layer Semtner 1976 model and includes a stationary

approximation for the equation of heat conduction and two Neumann boundary condi-


tions. These boundary conditions are nonstationary heat balance equations for meltingsnow or ice on the upper surface and for freezing or melting ice on the under surface of

Ž .the ice slab. Among other parameters, the model prescribes surface temperature T , thes

amount of ice the underside grows or melts, and the amount of melted snow. The timestep is 24 h. The model describes five modes of formation for the snow and ice cover,

Ž .depending on the calculated value of T . 1 When T -08C, the albedo is set at 0.81.s sŽ . Ž . Ž .2 When the snow surface is melting i.e., for T s08C , the albedo is set at 0.60. 3s

When the surface is bare ice and is melting, T is set to y0.18C and the albedo is set atsŽ . Ž .0.51. 4 The model also recognizes a ‘‘support mode’’ Semtner, 1976 when T iss

calculated to be less than y0.18C but, because of absorbed solar radiation, the ice iscomputed to be above 08C. For this mode, T is fixed at y0.18C and the albedo is set ats

Ž .0.51. 5 When all the ice has melted — that is, its thickness is computed to be less than5 cm — T is calculated from a nonstationary heat balance equation for an oceanics

mixed layer 30 m thick and for an albedo set at 0.1.As external model parameters in these simulations, we used daily snow depths and

daily averaged data from the standard 3-hourly meteorological observations fromŽ .October 1982 to October 1983 on NP-25 National Snow and Ice Data Center, 1996 .

That station was above 858 latitude during this period. The main physical parametersused in the model are as follows: densities of snow and sea ice are 320 and 900 kg my3,respectively; latent heats of fusion for snow and sea ice, 3.34=105 and 2.98=105

J kgy1, respectively; and thermal conductivities for snow and sea ice, 0.31 and 2.09W my1 Ky1, respectively. The heat flux from the ocean to the bottom of the sea ice

y2 Žwas assigned the usual value, 2 W m e.g., Maykut and Untersteiner, 1971; Parkinson.and Washington, 1979 .

For calculating the turbulent surface heat fluxes, we use the bulk–aerodynamicŽ .method e.g., Andreas, 1998 ;

H src C U T yT , 13aŽ . Ž .s p Hr r s r

H srL C U Q yQ , 13bŽ . Ž .L v Er r s r

where r is the air density; c , the specific heat of air at constant pressure; L , the latentp v

heat of vaporization or sublimation; U , the wind speed at reference height r; T and Q ,r s s

the temperature and specific humidity of the air at the snow, ice, or ocean surface; and Tr

and Q , the temperature and humidity at height r.r

The crux of the bulk–aerodynamic method is defining the bulk transfer coefficientsŽ .appropriate at height r, C and C . These are e.g., Andreas and Murphy, 1986Hr Er

k 2

C s , 14aŽ .Hr ln rrz yc rrL ln rrz yc rrLŽ . Ž . Ž . Ž .0 m T h

k 2

C s . 14bŽ .Er ln rrz yc rrL ln rrz yc rrLŽ . Ž . Ž . Ž .0 m Q h

Here z , z , and z are the roughness lengths for wind speed, temperature, and0 T Q

humidity, respectively, and L is the Obukhov length, a stability parameter. We took z ,0


z , and z all equal to 1.5 mm. For the semi-empirical stability functions forT QŽ . Ž . Ž . Ž .momentum c and heat c in Eqs. 14a and 14b , we used the ‘‘Dutch’’m hŽ .formulation Andreas, 1998; Jordan et al., 1999 recommended by Launiainen and

Ž . Ž .Vihma 1990 for stable stratification and the result of Paulson 1970 for unstablestratification.

We ran the model with cyclic boundary conditions such that the thicknesses of the iceand snow at the end of a year were used as inputs to the new year with the same 1-yearŽ .i.e., October 1982 to October 1983 NP-25 meteorological data repeating. Computa-tions ended when the ice thickness reached equilibrium.

In modeling sea ice there are usually two main problems. First, as mentioned above,is choosing parameterizations for the main components of the surface heat budget.Second is the external meteorological information. Because the iterative procedure inmodels typically uses a 1-day time step, it is necessary to have daily averaged values ofthe external parameters. Such information can be obtained in two ways. One way is toaverage the 3-hourly meteorological observations for each day. Another is to interpolatethe monthly averaged fields computed from many years of data and tabulated in an atlas,

Ž .Gorshkov 1983 for example, to obtain daily values. The input data that this secondmethod yields are more smooth than data from the first method but less accurate.

For example, interpolating monthly data to daily values must produce a time series ofcloud amount that varies monotonically through the month in the same manner yearafter year. Such a series is of little use for investigating interannual variability.

On the other hand, producing 3-hourly observations for the entire Arctic Basinrequires accurately interpolating daily averaged data measured at nonuniformly dis-tributed drifting buoys and polar stations. We mentioned above that, in principle, dailyaveraged surface temperature, at least, should be available for the entire basin from thesesources. Then, using the method described above, it should also be possible to estimatedaily averaged total cloud amount on an arbitrary grid throughout the basin.

The method for describing cloud amount can, of course, have a significant influenceon the computed long-wave radiation balance in particular. The parameterizations forF and B by Brunt, Marshunova, and Satterlund account for clouds with lineardn

functions. Therefore, there should be no difference in monthly averaged values of Fdn

and B regardless of whether they were obtained from 3-hourly cloud observationsaveraged to daily values or from monthly cloud amounts interpolated to daily values —provided, of course, that the monthly cloud values are based on the same 3-hourlyobservations. In contrast, Maykut and Church’s and KL&A’s parameterizations for Fdn

and B depend nonlinearly on cloud amount. For these, the type of averaging is crucial.Table 10 lists numerical model calculations of the equilibrium sea ice thickness and

related parameters computed using the various parameterizations for long-wave radiationthat we have been discussing and the various ways of representing cloud data. All theother meteorological information used in these numerical experiments, including theincoming short-wave radiation, are daily averages computed from 3-hourly observations.

ŽHere we use the traditional definition of equilibrium ice thickness e.g., Maykut and.Untersteiner, 1971; Semtner, 1976; Curry et al., 1995 : It is the cycle in ice thickness

that results when the same year of forcing data is applied repeatedly until the pattern inmodeled ice thickness no longer changes from year to year.


Table 10For our modeled sea ice cover, using various long-wave parameterizations and the NP-25 data from October

Ž .1982 to October 1983, the maximum and minimum thickness, the amplitude Dh of the annual variation inthickness, and the dates on which melting first begins in the snowpack and cooling in the sea ice begins. Toidentify the different experiments, we use the following shorthand: ‘‘day’’ used daily averaged observations oftotal cloud amounts; ‘‘day, rec’’ used daily averaged total cloud amounts for winter reconstructed from air

Ž .temperature using Eq. 4 ; ‘‘month’’ used monthly cloud data interpolated to daily values; ‘‘month, sum’’ useddaily averaged data for the winter but monthly averaged cloud data interpolated to daily values for thesummer; ‘‘month, win’’ used daily averaged data for the summer but monthly averaged data interpolated todaily values for the winter

Ž . Ž . Ž .Experiment Maximum m Minimum m Dh m Melting Cooling

Ž .1. Brunt, Eq. 7 , day 6.50 5.78 0.72 May 20 Sept 10Ž .2. Marshunova, Eq. 9 , day 4.40 3.45 0.95 May 25 Sept 11

Ž .3. Maykut and Church, Eq. 10 , day 4.96 4.19 0.77 May 24 Sept 10Ž .4. Satterlund, Eq. 11 , day 5.09 4.17 0.92 May 24 Sept 12

Ž .5. KL&A, Eq. 12 , day 3.94 2.93 1.01 May 26 Sept 12Ž .6. KL&A, Eq. 12 , day, rec 3.87 2.85 1.02 May 25 Sept 12Ž .7. KL&A, Eq. 12 , month 4.67 3.83 0.84 May 24 Sept 11Ž .8. KL&A, Eq. 12 , month, sum 4.61 3.77 0.84 May 25 Sept 11Ž .9. KL&A, Eq. 12 , month, win 3.99 2.97 1.02 May 25 Sept 12

Ž .10. Marshunova, Eq. 9 , month 4.36 3.39 0.97 May 29 Sept 11Ž .11. Marshunova, Eq. 9 , month, sum 4.37 3.41 0.96 May 27 Sept 11Ž .12. Marshunova, Eq. 9 , month, win 4.38 3.42 0.96 May 27 Sept 11

We see in Table 10 that the spread in the computed maximum and minimum iceŽ .thicknesses based on the various parameterizations for F i.e., experiments 1–5 isdn

rather large. The seasonal amplitude in ice thickness also varies widely — from 0.72 mwith the Brunt parameterization to 1.01 m with the KL&A parameterization. The

Ž .KL&A parameterization i.e., experiment 5 yields equilibrium ice closest to the usuallyŽ .accepted cycle e.g., Semtner, 1976; Hibler, 1979 . The results of experiment 6 in the

table are also very interesting. Here the cloud amounts reconstructed from temperatureŽ .using Eq. 4 lead to predictions that are virtually identical with experiment 5, which we

judge as the experiment closest to reality.The calculated first day of melting in Table 10, nominally May 24–26, is somewhat

Ž . Ž .earlier than Yanes 1962 his Fig. 1 would predict for 858N, about June 20. But hisrelation tracks the onset of ‘‘intense snow melting’’, while we record the first appear-

Ž .ance of liquid water in the snowpack. A simulation by Jordan et al. 1999 of theseasonal cycle on NP-4, which was within 58 latitude of the North Pole in 1956–1957,also predicts a later date, June 18, for the onset of diurnal melting than we list in Table10. But Yanes’s results suggest melting is delayed by about 8 days for every 58 increasein latitude. Hence, the NP-25 and NP-4 results are fairly compatible.

The ‘‘Cooling’’ column in Table 10 lists the date when the sea ice begins coolingagain after the summer ablation season. The nominal date that we calculate as thebeginning of cooling is September 11–12. From thermocouples embedded in the sea ice

Ž .at NP-4, Jordan et al. 1999 show that in 1956 cooling began at this station on aboutAugust 28. Again, since NP-4 was 58 farther north than NP-25, our modeled date for theonset of cooling is reasonable.


Table 11Incoming long-wave radiation, F , during the winter of 1982–1983 on NP-25. The ‘‘Daily Ave’’ values aredn

the observations averaged each day and then averaged for the month; the ‘‘Reconstructed’’ values are theŽ .individual estimates computed using Eq. 4 , averaged daily, and then averaged for the month; the ‘‘Monthly

Ave, Int’’ values are based on monthly data interpolated to daily values using a parabolic interpolation overthree months. Fig. 12 shows daily values of these various cloud amounts for November 1982. For the

Ž . Ž .computed long-wave radiation values, we used Konig-Langlo and Augstein’s 1994 formula, Eq. 12 .¨Numbers in parentheses are the correlation coefficients between the daily averaged observed values oflong-wave radiation and the respective reconstructed and interpolated values. It is important to point out herethat we calculated the monthly averaged values from daily averages observed on NP-25; these averages thus

Ž .differ from those shown in Gorshkov 1983y2Ž .Incoming long-wave radiation W m

November December January February March

Daily Ave 163 160 137 149 168Ž . Ž . Ž . Ž . Ž .Reconstructed 171 0.84 163 0.86 149 0.97 158 0.94 184 0.89Ž . Ž . Ž . Ž . Ž .Monthly Ave, Int 160 0.93 152 0.88 135 0.97 145 0.96 162 0.88

In experiments 7–12, to evaluate model sensitivity to the method for describing thetemporal variability in cloud amount, we used just the long-wave parameterizations ofKL&A and Marshunova. In the Marshunova parameterization, we used the empirical

Ž .Fig. 12. Temporal variability of total cloud amount for November 1982 on NP-25. The traces show 1 theŽ . Ž .3-hourly cloud observations averaged to daily values, 2 cloud amount estimates based on Eq. 4 using daily

Ž .averaged temperature, and 3 cloud amounts estimated using monthly cloud data interpolated to daily valuesusing a parabolic interpolation over 3 months.


Fig. 13. Temporal variability of the daily averaged surface-layer air temperature and the incoming long-waveŽ . Ž .radiation for November 1982 on NP-25. The computed values of F are based on Eq. 12 using 1 3-hourlydn

Ž . Ž . Ž .cloud observations averaged to daily values, 2 daily averaged cloud data estimated using Eq. 4 , and 3monthly cloud data interpolated to daily values with a parabolic interpolation over 3 months.

coefficients listed in Tables 6 and 7 that derived from observations on the driftingstations. As we hinted above, it is clear in Table 10 that the Marshunova parameteriza-

Ž .tion i.e., experiments 2, 10–12 is less sensitive to the method of obtaining cloudŽ .amounts than the KL&A parameterization experiments 5–9 . Because KL&A’s param-

eterization has a cubic dependence on cloud amount, the three sensitivity experimentspredict maximum ice thicknesses that range over 0.7 m, depending on the method fordetermining cloud amount. We thus reiterate that, because the best model for F wasdn

derived from non-averaged data and depends nonlinearly on cloud amount, sea icemodels employing it will be quite sensitive to the method of handling the cloud data.

In Table 10, experiments 5, 6, and 9 yield practically the same results. Likewise,experiments 7 and 8 produce almost identical ice thicknesses, but these differ essentiallyfrom the results in experiments 5, 6, and 9. Table 11, which shows calculations oflong-wave radiation for each winter month on NP-25 based on the KL&A parameteriza-tion, explains the good agreement among these three cases.

We see in Table 11 and in Fig. 13 that, despite the essential differences in the timeŽ .series of cloud amounts used in the calculations Fig. 12 , the time series of calculated

F values correspond well with each other, both on average and in terms of the lineardn

correlation coefficient. Although this good agreement may, at first, seem paradoxical,the formula used to estimate F explains it. To obtain the long-wave fluxes in Table 11dn

Ž . Ž .and in Fig. 13, we used — from Eqs. 6 and 12 —

F ss T 4 a qb n3 . 15Ž .Ž .dn K K


Thus, the air temperature T dominates the calculation. But remember, the correlationbetween T and n is also high. Consequently, because the surface-layer air temperaturein winter over Arctic sea ice depends largely on long-wave radiation processes, thattemperature, in effect, contains information on cloud amount.

Fig. 14. Temporal variability of the long-wave radiation balance and the sensible heat flux for the middle 10days in May 1992 on ISW. The radiation data are hourly averages; the sensible heat flux values derive from

Ž .profile measurements of wind speed and air and surface temperature Andreas et al., 1992 .


Another process that dictates the close agreement among the equilibrium ice thick-nesses in experiments 5, 6, and 9 is the feedback between the long-wave radiationbalance and the vertical turbulent surface flux of sensible heat, the main components ofthe surface heat budget in winter. That feedback is clear in the data from ISW: Fig. 14shows that an increase in the radiative cooling of the snow surface causes an increase in

Žthe temperature gradient in the atmospheric surface layer and a corresponding increase.in the downward sensible heat flux . Consequently, the traces of long-wave radiation

balance and sensible heat flux are almost mirror images. The one-dimensional sea iceŽ .model of Makshtas and Timachev 1992 also reproduces this feedback between sensible

heat flux and net long-wave radiation.The large differences between the computed ice thicknesses in experiments 5, 6, and

9 and the thicknesses in experiments 7 and 8 result because of the absence of feedbackbetween the sensible heat flux and the net long-wave radiation in summer. During thesummer, the snow and ice surface is near the melting point for long periods and, thus,accommodates changes in the net long-wave radiation by changing phase rather than by

wexchanging sensible heat. Also, the strong dependence of F on cloud amount see Eq.dnŽ . Ž .x Ž .12 or Eq. 15 , which in summer is usually 8–10 tenths Table 2 , sharpens thedifference between the results for the two groups of experiments. For a prescribed

Ž .short-wave radiative flux i.e., one that does not depend on cloud conditions , thereduced variability in total cloud amount in summer resulting from interpolatingmonthly averages to daily values results in greater equilibrium ice thicknesses inexperiments 7 and 8 than those reported for the numerical experiments of Makshtas and

Ž .Timachev 1992 .We thus believe that, for sea ice models driven by atmospheric data, reconstructing

total cloud amount from air temperature provides sufficient accuracy for calculating thesurface heat budget in the winter and, especially, for studying its monthly variabilityŽ .Table 11 . On the other hand, in summer, when the surface temperature is nearlyconstant at the melting point and the feedback between the main components of thesurface heat budget is consequently weaker, models will require better cloud data and amore accurate description of cloud effects on short-wave and long-wave radiation.

6. Conclusions

We summarize our results in the following conclusions.Ž .1 Our analysis of observations made on the NP drifting stations shows that the

frequency distribution of total cloud amount in the Arctic Basin, especially in winter, isU-shaped. We fitted these histograms with beta distributions; Table 4 lists the fittingparameters. Since the mean of a quantity with a U-shaped distribution actually corre-sponds to the least likely value of the quantity, representing Arctic cloud amounts withmonthly averaged values is a flawed approach. Four months of cloud observations overAntarctic sea ice revealed the same U-shaped cloud distribution there.

Ž .2 The series of meteorological observations on drifting stations that worked above778N from 1955 through 1991 suggests reduced frequency of clear skies in winter and of

Ž .overcast skies in summer see Figs. 6 and 7 . Both trends could be the consequence of a


documented shift in the general atmospheric circulation, but additional analyses andmodeling are necessary to say for sure.

Ž .3 We developed an algorithm that has statistically significant ability to predict totalcloud amount in winter from atmospheric surface-layer temperature alone. This algo-rithm yields a frequency distribution for total cloud amount that is U-shaped — as itshould be — although the input air temperatures are normally distributed. Becausemodeling results based on this method compare reasonably well with observations, thealgorithm could allow forecasting and, thus, computations of the surface heat budget forregions where no cloud observations are available. Where satellite data and buoytemperatures are both available, the algorithm could also help reduce errors in interpret-ing cloud cover from the satellite images.

Ž .4 Meteorological data collected on NP-4 and ISW confirm that the method ofŽ .Konig-Langlo and Augstein 1994 for handling total cloud amount in parameterizations¨

of incoming long-wave radiation in polar regions is the best among five popularcandidates. We also plan to test this conclusion with our recent data from SHEBA, the

Žyear-long experiment to study the Surface Heat Budget of the Arctic Ocean Andreas et.al., 1999 .

Ž .5 We have confirmed the value of the KL&A parameterization through numericalexperiments using a one-dimensional thermodynamic sea ice model and an annual cycleŽ .October 1982 to October 1983 of meteorological observations from NP-25. Thismodeling shows that, of the five parameterizations considered, the one from KL&Aproduces estimates of equilibrium sea ice thickness and its seasonal variability closest to

Ž .existing notions about those quantities Table 10 . Associated calculations suggest thatin winter the atmospheric surface-layer temperature is largely an integrated parameterbecause of the essential three-way feedback among air temperature, clouds, and long-wave radiation. As such, with our statistical model, temperature data alone are sufficientfor estimating total cloud amount and, thus, the long-wave radiation budget withsufficient accuracy for sea ice modeling.

Acknowledgements

We thank our colleagues Boris Ivanov of AARI and Kerry Claffey of CRREL fortheir help with sampling and analyzing the Ice Station Weddell data and Terry Tuckerand Don Perovich of CRREL for reviewing the manuscript. We also thank an anony-mous reviewer for a thorough review that helped improve our manuscript. The NationalSnow and Ice Data Center at the University of Colorado, Boulder, provided theCD-ROM containing much of the data from the Russian drifting stations used here. TheU.S. Office of Naval Research supported this research through contractsN0001496MP30005 and N0001497MP30002; the U.S. National Science Foundationsupported it with grants OPP-90-24544, OPP-93-12642, and OPP-97-02025; the U.S.Department of the Army supported it through project 4A161102AT24; and the RussianFund for Fundamental Investigations supported it through projects 96-07-89159 and97-05-65926.


( ) ( )Appendix A. Details of Eqs. 3 and 4

Suppose we have a series T of N surface-layer air temperature measurements. Thei

average temperature is

N1Ts T , A1Ž .Ý iN is1

and the standard deviation is

1r2N1 2s s T yT . A2Ž .Ž .ÝT iNy1 is1

Since the T are approximately normally distributed,i

T yTiT s A3Ž .i

sT

is approximately normal with mean 0 and variance 1. That is, the probability density˜function for T is

12˜ ˜f T s exp yT r2 , A4Ž .Ž . Ž .1 '2p

and the cumulative distribution function is

T X X˜F T s f T dT . A5Ž . Ž .Ž . H1 1y`

We have already discussed the distribution function for total cloud amount n; call thisŽ .f n , where2

ˆG aqbˆŽ .by1ay1ˆf n s n 1yn for 0FnF1, A6aŽ . Ž . Ž .2 ˆG a G bŽ .ˆ Ž .

f n s0 otherwise. A6bŽ . Ž .2

ˆAgain, Table 4 lists appropriate a and b values to use in winter. The cumulativeˆdistribution function for total clouds is thus

nX X

F n s f n dn . A7Ž . Ž . Ž .H2 20

We seek a function that predicts total cloud amount from surface-layer air tempera-˜ture alone. In other words, in terms of T , we want c such that

˜nsc T . A8Ž .Ž .


˜Ž .Based on Tables 1–3 and physical intuition, we can reasonably assume that c T is a˜monotonically increasing function of T. From mathematical statistics, we know we can

Ž .then approximate Ventcel, 1964, p. 263 ff.

˜Ž .c T X X˜ ˜F T ( f n dn sF c T . A9Ž . Ž .Ž . Ž .H1 2 20

Consequently,

y1˜ ñsc T (F F T , A10Ž .Ž . Ž .2 1

y1 Ž .where F is the inverse function of F n .2 2Ž . y1Since F n describes a beta distribution, F must be the inverse of a beta2 2

Ž .distribution. Aivazyan et al. 1983 give

ay1˜ ñ T (F F T ( , A11Ž .Ž . Ž .2 1 ˆ ãqb exp 2w TŽ .ˆ

where

˜ 1r2< <T hql 1 1 5 2˜w T s y y lq y , A12Ž .Ž . ž /ž /ˆh 2ay1 6 3hˆ2by1

y11 1hs2 q , A13Ž .ž /ˆ2ay1ˆ 2by1

˜2T y3ls . A14Ž .

6

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