Jt~L- 1977 J. R. G A R R A T T 915Review of Drag Coefficients over Oceans and Continents J. R. GA~A'r'rCSIRO Division of Atmospheric Physics, A spendale, Victoria, 3195, Australia(Manuscript received 1 September 1976, in final form 29 March 1977) ABSTRACT Observations of wind stress and wind profiles over the ocean reported in the literature over the past 10years are consistent with Charnock's (1955) relation between aerodynamic roughness length (50) and frictionvelocity (u.), viz, zo=etu,2/g, with a=0.0144 and g=9.81 m s-2. They also imply a yon Kgrmgn constant=0.41:t=0.025. For practical purposes Chamock's relation may be closely approximated in the range4< V<21 m s-t by a neutral drag coefficient (referred to 10 m) varying with the 10 m wind speed V (m s-~),either by a power law relation CD~v(10) X 10-=0.51V-'46or a linear form CDN (10) X lfP = 0.75 +0.067 V. Results of recent turbulence sensor comparison experiments suggest that much of the source of datascatter in C~)~q (V) plots and of the systematic differences between data sets is due to calibration uncertaintiesassociated with sensor performance in the field. The effects (if any) of fetch, wind duration and unsteadinessremain obscured in this experimental data scatter. Vertical transfer of momentum over land may be described in terms of an effective roughness length orgeostrophic drag coefficient which incorporates the effects of both friction and form drag introduced by flowperturbation around uneven topographical features. Typically low relief topography and low mountains (peaks <0.5-1 km) require a geostrophlc drag coefficient C~- ~3 X 10% while land surfaces in general require Ca~v ~ 2 X 10-~ for which C~?:(10)~ 10X 10-aand the effective aerodynamic roughness length ~0 (eft) m 0.2 m. The latter values satisfy, very approximately,the requirement of global angular momentum balance.1. Introduction Taylor (1916) reasoned that the horizontal reactionbetween the earth's surface and the moving atmospheremight be represented by a velocity square law involvinga nondimensi6nal factor known as the coefficient ofskin friction. Sutcliffe (1936) later referred to this as"the surface resistance to atmospheric flow", andargued (along with Taylor) that knowledge of thecoefficient would allow the rate at which horizontalmomentum is transferred from atmosphere to surfaceto be calculated. Taylor actually used wind observations made overSalisbury Plain and, utilizing his theory of eddy motionin the atmosphere (Taylor, 1915), derived a coefficientof '~2-3X 10-s. Sutcliffe also used pilot balloon windobservations made over similar countryside and calculated the ageostrophic wind components, yielding acoefficient ~ 4-7 X 10-a. A similar method gave 0.4 X 10-afrom observations over the sea. In the succeeding years comprehensive, observationshave indeed verified that over land, except at low windspeeds (e.g. Deacon, 1957), the square law generallyapplies--over the sea this is still a matter of contention.At the same time, considerable advances made inturbulence theories and the advent of similaritytheories for both the atmospheric surface and boundarylayers, allow the drag coefficient (analogous to Taylor'scoefficient) to be explicitly defined in terms of aerodynamic roughness, atmospheric thermal stabilityand other relevant physical parameters (see Section 2). Many general circulation models of the atmosphereuse a drag coefficient (Cv) formulation by employingC~ explicitly but often in simplified form (see GARP,1974). Others attempt a greater sophistication inboundary layer parameterization (see, e.g., Gadd andKeers, 1970; Delsol et al., 1971), while Deardorff's(1972) comprehensive treatment contains C~> implicitlythrough the aerodynamic roughness length (z0). Thelatter (z0) appears in the integrated forms of the windand temperature profiles, both for the _surface layer(Monin-Oboukhov similarity theory) and the remainderof the boundary layer (deficit formulation, e.g., Clarke,1970). Limited applications of Deardorff's parameterization scheme in a general circulation model have utilized values of z0, for land (from Fiedler and Panofsky,1970; see Deardorff, 1972) of 0.2-0.7 m (see Section 4)916 MONTHLy WEATHER REVIEW Vo~.v~aE105and for sea (from Miyake et al., 1970), of 2-3X10-4 m(see Section 3). In recent years a considerable amount of informationon drag coefficients over land and sea has becomeavailable. In this paper recent observations and indirectdeterminations of CD are reviewed giving rise to anumber of implications in the context of boundarylayer 'processes and considerations of large-scale angularmomentum balance. Thus mean annual zonal stresses determined fromthe drag coefficients and observed mean wind field,when summed for all latitude belts, must be consistentwith the requirement of balance of angular momentumflux over a sufficiently long time (say one-year).Priestley (195 i) considered the ~)roblem in his assessmentof oceanic stresses and Newton (1971a) and La Valleand Girolamo (1975), for instance, studied each termin the momentum balance equation using observationswhere possible. This balance is indeed an importantphysical constraint on the choice of large-scale geostrophic drag coefficient--in practice considerableflexibility is retained for adjusting the magnitude ofthe stress term to achieve balance, mainly due to thelack of reliable mean monthly surface and geostrophicwinds over the entire globe (particularly over remoteocean areas) and uncertainty in thermal wind.2. Relations and definitions In the constant flux layer of the atmosphere adjacentto a horizontally homogeneous surface, the local surfacestress may be written in terms of the surface windV(z)'at height z (Taylor, 1916) as~0=pCMZ(zT, (1)where p is air density and Cz> .may be related to theaerodynamic roughness length z0 and stability parameter ~' (= z/L where L is the Monin-Oboukhov length)through the Monin-Oboukhov similarity theory, viz., CDNCD = (2a) F1 - k-lC~,N'-(i')]2whereC~t- =--- (2b)(lnz/zo)~'k is the yon K~trm~n constant, and ~(~) is given by~(~') = F ,/F.Knowledge of the empirical form of the nondimensionalwind gradient ~(t') '(Dyer and Hicks, 1970; Webb,1970; Businger et al., 1971; Carl et al., 1973; Dyer,1974) allows Co/CD~v to be computed as a function of ~(or a bulk Richardson number) and z/zo (see, e.g.,Deardorff, 1968; Paulson, 1969). The variation ofCo/CD~r shown in Fig. la is based upgn the ~2(~-) relations proposed by Dyer (1974).for two values of z0typical of a rough sea (0.1 cm) and land surface (0.1 m),with k = 0.41 and a reference l~eight z = 10 m. Flux parameterization in numerical models of theatmosphere often requires relating the~ stress to alarge-scale wind (1?), e.g., the surface geostrophic windVc (Lettau, 1959) appropriate to a barotropic atmosphere or the vertically averaged geostrophic windin the presence'of baroclinity Vc (Arya and Wyngaard, 1975; Yamada, 1976). Other wind scales whichhave been suggested for use in the real atmosphereinclude the wind at the ~top of the boundary layer(Melgarejo and Deardorff, 1974), the wind at height0.15 u./f (where u, is the surface friction velocity andf the. Coriolis parameter) (Clarke and Hess, 1974),and the maximum wind in the boundary layer (Clarke, 2.o[ j j I J ~ ~'o : I0 cm-__9_O -I -O'5 OO O'5 1.0 F]:o. ~. V~d~tJon o[ C~/C~ (reference height of [0 m) with stability D~mmeter -for two wlues of ~, b~sed on ompiricM st~biUty functions of Dyer ~nd ~[c~s (~970)and Webb (~970)'.JULY1977 J. R. GARRATT 917CoCGN -~oq20 -Ioo -80 -60 -40 -20 o +2o +4~Fro. lb. Variation of Co/Co~r with stability parameter ~ for Ro= 106 based on results of Clarke (1970) and Deacon (1973a).1970). The stress is then written(3)where Co is the geostrophic drag coefficient whenl?= Vo or ~Vo). In general Co will be a function of1) the surface Rossby number Ro= Vo/f Zo [-as expressed through the boundary layer similarity theory(e.g., Kasanskii and Monln, 1961)'] or l/zo, where lis a height scale, replacing Vo/f (e.g., Yamada, 1976),of particular relevance at low latitudes, 2) boundarylayer stability u=--ku./fL or --kl/L, and 3) baroclinity S= ~Vo/Oz, such thatIn Ro =A (/~,S) -ln Co~+kCo-- cos ao, (4a)where ao=sin-t[-B(/~,S)Col/k'] is the angle betweenthe surface wind and wind vector -~, and A and Bare similarity functions to be determined from observation (e.g., Clarke, 1970; D~acon, 1973a; Clarkeand Hess, 1974; .Melgarejo and Deardorff, 1974;Arya, 1975a; Yamada, 1976). Using Deacon's (1973a)values of A(0,0)=l.9 and B(0,0)=4.7, we show theanalogous variation of Co/Cog with #, for Ro=106in Fig. lb. The dependence of A and B on baroclinityis not so well known, but has been investigated, forexample, by Clarke and Hess (1974) and Arya andWyngaard (1975). For practical application using the surface geostrophic wind, Swinbank (1974) has shown that Eq.(4a) in neutral barotropic conditions may be approximated, in the range 104<Ro< 109 where most observations occur, byCoN = 0.0123 Ro--.14.(4b) Finally, for the stress over the sea, Charnock (1955)argued on dimensional grounds that Zo, in aerodynamically rough flow, should depend only upon u*(and g, the acceleration due to gravity, which isessentially constant) giving--=constant(=a), (5)which was later modified for general flow conditions toinclude the possible effects of viscosity (at low windspeeds), surface tension and water density and viscosity(at high wind speeds)--see Charnock and Ellison(1967). More recently Kitaigorodskii (1968) and.Kitaigorodskii and Zaslavskii (1974) considered thecase of transitional flow, where fetch and wind durationmight be important, such thatz0g---= f(Co/u.), (6)where Co is the phase velocity of the dominant wave,Co/u. then describing the degree of wave development,or wave age. Observations of z0 and C~)N (10) over the sea will beconsidered in the next section.3. Values of drag coefficient over the seaa. Methods Observations of the surface stress over the sea havebeen made by numerous workers over the past fewdecades using a number of techniques. Reviews ofsuch observations, and those in the wind tunnel, havealso been made from time to time but before consideringthese, and most recent observations, we summarizehere the main approaches used in the past. Details ofeach approach have been described in the literature(see, e.g., Deacon and Webb, 1962; Roll, 1965, Chap. 4;Kraus, 1972, Chap. 5).918 MONTHLY WEATHER REVIEW VO~.UME105 In early years empirical knowledge of the wind stresswas based largely on such measurements as: 1) The tilt of a water surface under wind action (VanDorn, 1953) and considered recently in refined form byWieringa (1974). The method determines the stress ona scale of tens of kilometers and is generally reliableonly over a sufficiently deep body of water for steadywind speeds ~> 10-15 m s-~ (Deacon and Webb, 1962).Its major sources of error arise from horizontal temperature gradients ]-such that I-C (10 kin)-~ produces thesame tilt as the stress at 5 m s-~-~ and the existence oftidal and seiche movements (Deacon and Webb, 1962;Roll, 1965, Chap. 4). \ 2) The geostrophic departure of flow in the planetaryboundary layer (e.g., Sheppard and Omar, 1952), laterextended to include radial accelerations in hurricanes(e.g., Miller, 1964). The method determines the stresson a scale of tens to hundreds of kilometers and requiresidentification of a level at which the stress or windgradient tends to zero (e.g., Sheppard et al., 1952;Charnock et al., 1956). Sources of error include theinfluence of accelerations upon the wind profile, thermalwind effects and uncertainties in the pressure field overthe height range considered (e.g., Roll, 1965, Chap. 4). More recently, many observations of the local stresshave been made in the surface layer based on: 3) The wind profile form, from which u, and hencethe stress (=pu,2) may be inferred (e.g., Sheppard et al.,1972), requiring correction for thermal stability (seeFig. la) and for surface drift current '(Sheppard et al.,1972; Hicks, 1972). Interpretation of the wind profilemay be difficult at low heights (4 1 m) due to'the influence of wave motion (Stewart, 1961; Miles, 1965;Phillips, 1966, Chap. 4) lnd at low winds due to thevariation of stress with height [-~-10% decreasefrom the surface to 10 m at V~-3 m s-~--see Deacon(1973b)-]. 4) Measurements of the eddy covariance u'w~ from arigid platform (e.g., Smith and Banke, 1975) or buoy(Zubkovskii and Kravchenko, 1967); in the latter casegross errors will result if suitable correction for platformmotion is not made (Hasse, 1970). Sources of errorarise from the non-Gaussian behavior of u'w" (Stewart,1974), effect of spray on sensor performance (Wieringa,1974) and, at low heights (~-1 m), the possible directinteraction of wind and waves as evidenced in turbulence spectra showing spikes at wave frequencies(e.g., Volkov and Mordukovitch, 1965; Kondo et al.,1972; Davidson and Frank, 1973). The important caseof instrumental error will be discussed below in relationto C~> variability.b. Data sdection Our knowledge of Co:v as of 1970 (approximately)is summarized in Table 1, based on major reviews overthe sea (see, also, Bunker, 1976). In most cases datafrom numerous sources have been considered and-CDNinferred for specific wind speed ranges; for each casethe scatter of data is shown as a percentage variability.The CDN relations are summarized in Fig. 2, thepecked curve being based on Charnock's (.1955) relation[-see Eq. (5)~ with ~= 0.016 (Wu, 1969). Since 1970 many more observations have becomeavailable, mainly based on improved techniques ofmeasuring V(z)' and u'w'. We have assessed CD~ fromeach of 17 publications available, values being basedon the eddy correlation and wind profile methods. Theseare shown in Table 2, together with number of data,data variability being represented by the standarddeviation of individual values about the mean (~)and the wind speed ranges over which measurements TABLE 1. Main reviews of the neutral drag coefficient over the sea showing wind speed range, best estimate of C~(10) (either as aconstant or a function of wind speed), and typical data variability as a percentage of C~(10) value over the wind speed range considered (see Fig. 2). Wind speed range Cry (10) Variability Number of Source (ms-~) (X llY) (%) referencesA. Prlestley (1951) 2.5-12 1.25a ? Not stated strong 2.6*B. Wilson (1960) ~-~1-5 1.42 4-50 '~ 47 9 -20 2.37 4-25C. Deacon and Webb (1962) 2.5-13 1 +O.07V 4-25-50 9D. Robinson (1966) 3 - 8.5 1.8~ 4-30 '/~ 14 2.5-14 1.48~ -4-15E. Wu (1969) 3 -15 0.5 V--~'[.* -4-30 ~ 30 15-21 2.5 J +10F. Hidy (1972) 2 -10 1.5 +30 8Micrometeorological data.Geostrophic departure.Overall variation close to Charnock relation with a =0.016.Actually based on Deacon, 1950. Nature~ 165, p. 173.Quotes Sverdrup et al. (1942) and Munk (1947).JULY 1977 J. R. G A R R A T T 919were made. Sources which gave a single drag coefficientvalue for one wind speed have not been considered(e.g., Stegen et al., 1973), nor have values based on thedissipation technique. These are few in number, butMiyake et al. (1970) found, for eight runs, a mean Coof 1.09 (+0.19)X10-a and 1.35 (q-0.26)X10-a forthe eddy correlation and dissipation methods, respectively, while Pond et al. (1971) found, for 20 runs, 1.52(4-0.26)X10-~ and 1.55 (+0.4)X10-~, respectively.Sources of uncertainty with the dissipation methodhave been discussed by Miyake et al. (1970) and Kraus(1972, Chap. 5). Eighteen sets of eddy correlation and wind profiledata have been considered as follows--individual Coyvalues were obtained from references (see Table 2) 1,7, 10, 11, 13a, 13b, 16a and 16b (see Table 2) and meanCDy (and a) for small V ranges from references 4, 6, 8, 9,12 and 15 allowing calculation of CoN and the standarddeviation of the individual data about the mean at! m s-1 intervals. These are shown as a function of Vin Fig. 3. The wind profile estimates in the range5 < V < 11 m s-1 are strongly biased by the large numberof data in Hoeber's (1969) set [see Kraus (1972)]so that Coy has also been evaluated with his dataexcluded. This has only a marginal effect upon the Coyvalues.CoNXlO: 2'~- I I I 1 0 5 I0 15 20V (ms-~) FIe. 2. Mean curves of CoN (t0) plotted against V (t0 m) forreview sources shown in Table 1. Dashed curve is based onzo =etu.2/g with a =0.016 (Wu, 1969) and k =0.41. Several sets of data have been excluded from theanalysis. These include: 1) the data of Ruggles (11970) because of anomalousCo values and large scatter (his mean value for 2.5< V< 10 m s-~ is somewhat higher than that indicatedby the bulk of data in Fig. 3); 2) the data of Davidson(1974) which have no stability correction (his suggestedneutral value for 6< V< 11.5 m s-1 lies within the dataTABLE 2. Neutral drag coefficient values over the ocean taken from the recent literature for a reference height of 10 m: ec= eddy correlatiqn method; wp= wind profile method., is the standard deviation of n data points about the mean value. Wind speed Number range CON(10) Variability a of dataSource (m s-t) ( X 10a) (%) n Method Platform Comments 2.5-21 0.63 -[-0.066V 30 111 ec Mast1. Smith and Banke (1975)2. Kondo (1975)3. Davidson (1974)4. Wieringa (1974)5. Kitaigorodskii et al. (1973)6. Hicks (1972)7. Paulson et al. (1972)8. Sheppard et al. (1972)9. De Leonibus (1971)i0. Pond et al. (1971)11. Brocks and Krugermeyer (1970)12. Hasse (1970)13. Miyake et al. (1970)14. Ruggles (1970)15. Hoeber (1969)16. Weiler and Burling (1967)17. Zubkovskii and Kravchenko (1967)3-16 1.2 -[-0.025 V 15 -- waves Tower 6-11.5 1.44 ? 114 ec Large buoy 4.5-15 0.62V0.a7 20 126 ec Tower or 0.86 q-0.058V 3-11 0.9 (at 3 m s-1) to ? 29 ec Tower 1.6 (at 11 m s-t) 4-10 0.5V0.a 25 74 ec Tower 2-8 1.32 25 19 wp Large buoy 2.5-16 0.36q-0.1V 20 233 wp Tower 4.5-14 1.14 30 78 ec Tower 4- 8 1.52 20 20 ec Large buoy 3-13 1.18 q-0.016V 15 152 wp Buoy3-11 1.21 20 18 ec Buoya. 4- 9 1.09 20 8 ec Mastb. 4- 9 1.13 20 8 wp Mast2.5-10 1.6 50 276 wp Mast 3.5-12 1.23 20 787 wp Buoya. 2-10.5 1.31 30 10 ec Mastb. 2.5- 4.5 0.90 75 6 wp Mast3- 9 0.72 q-0.12V 15 43 ec BuoyAlso utilizes data of Smith (1973) using thrust and sonic anemometersUtilizes data on wave ampli tudes from Kondo et al. (1973)Does not correct for stability effectsSurface tilt and wp estimates are excludedPlots CoN as a function of ~J,so/pAccepts CoN relation as same as Wu (1969)Uses k =0.40Uses k =0.40Data from North Sea and Bal tic Sea--uses k =0.40 See text on data interpretation See text on data interpretation --uses k =0.40Co anomalies found at a num ber of wind ~peeds--uses k =0.42Data from equatorial Atlantic --uses k~0.40Uses k =0.40wp estimates of u, show low correlation with ec; possible effect of buoy motion920 MONTHLY WEATHER R'EVIEW VOLUblE105$'Ocl~xJo$I-OI I ! ~ I I I I I 16 21 ?1 43 31 63 56 27 19 5 I0 6 I I 3 2 2 18 ~7' 56 84 7'1 51 II 34 18 14 9 6 4 I I I' I I I I I I '1 I 0 2 4 6 8 I0 12 14 16 18 20 2 2 V (m s-~) FIG. 3. Neutral drag coefficient values as a function of wind speed at 10 m height, based onindividual data taken from the recent literature (see Table 2 and text). Mean values areshown for 1 m s-~ intervals based on the eddy correlation method (O) and wind profilemethod (O); wind profile data are also shown (/X) with Hoeber's (1969) [see Kraus (1972)~data included. Vertical bars refer to the standard deviation of individual data for each mean,with the number of data used in each 1 m s-~ interval shown above the abscissa axis: top linerefers to (-), bottom line to (O). The dashed curve represents the variation of Co~(10)' withV based on ~o=au.2/g with a=0.0144.scatter of Fig. 3); 3) the data of Zubkovskii andKravchenko (1967) which are suspect at the higherwinds (see comments of Stewart in Hasse, 1970); and4) the data of Kitaigorodskii et al. (1973) which arenot published in a suitable form for analysis (theirimplied Co2~ values at V ~ 3 and 11 m s-~ overlap thedata in Fig. 3). Finally we note that the C~>N relationof Kondo (1975), based on high-frequency wave amplitude data, agrees well with the collected C~>N data. A number of authors have inferred the surface stressin hurricanes, using the geostrophic departure method,and obtained information on CD at considerably greaterwinds than in Fig. 3. These are summarized in Table 3,together with the wind flume experiments of Kunishiand Imasoto (quoted by Kondo, 1975) and the vorticitymass budget analysis of Ching (1975) using data fromBOMEX over a 14-day period. The individual datahave been averaged in 5 m s-~ intervals, and are shownTaBLg 3. Neutral drag coefficients over the ocean taken from the literature, for hurricane and vorticity-massbudget data analyses. Also included are wind flume data of Kunishi and Imasoto (see Kondo, 1975). Wind speed C~m(i0) range range Source (ms-~) ( X 10a) CommentsA. Miller (1964) 17 -52 1.04.0 (linear)B. Hawkins and Rubsam 23 -41 1.2-3.6(1968) (discontinuous)C. Riehl and Malkus 15 -34 2.5(1961)D. Palm6n and Riehl 5.5-26 1.1-2.1(1957) (linear)E. Kunishi and Imasoto 14 47.5 1.5-3.5(see Kondo, 1975)F. Ching (1975) 7.5- 9.5 1.5Hurricanes Donna and Helene-ageostrophicHurricane Hilda- ageostrophicHeld constant to achieve angular momentum balanceComposite Hurricanedata~ageostrophicWind flume experimentVorticity and mass budget at BOMEX1977 J. R. GARRATT 921 4CD# X I03[ 0 3 6 6 5 2 6 5 I I0 I 5 2 3 .% 2 0 I0 20 30 40 50 V (ms-~) Fro. 4. Mean values of the neutral drag coefficient as a function of wind speed at 10 m height for 5 m s-~intervals, based on individual data from hurricane studies (c), wind flume experiment (e) and vorticity/mass budget analysis (~)--see Table 3. Vertical bars as.in Fig. 3. The number of data containedin each mean is shown below each mean value, and immediately above the abscissa scale. The dashedcurve represents the variation of CDN(10) with V based on zo=otu.2/g with a=0.0144.in Fig. 4; values of C~>N at V~20 m s-t are generallysmaller than those found in Fig. 3, suggesting an underestimate in the stress and emphasizing the uncertainties (possibly by a factor of 2) in inferring stressfrom hurricane data (Kraus, 1972, Chap. 5). However,one particular feature of Fig. 4 is the similarity betweenthe wind flume and hurricane data, and between Ching's(1975) data and that in Fig. 3. The wind speed dependence of CDN has been soughtbased on the prediction of Eq. (5); Eq. (6) suggeststhat the dependence upon other parameters such asfetch and wind duration should be looked for. Perhapsbecause these seem to be of secondary importance only,the majority of the reference sources of CDN data inTable 2 do not classify Ci>N accordingly; there is littleor no quantitative information on fetch, wind duration, sea state, etc., with which we can categorizeCi>N in the present analysis. Their effects (if any) remain obscured in the data scatter of individual datasets (see Table 2) and of overall mean values (seeFig. 3). Kraus (1972, Chap. 5) argued that the data scatter,in part, may be due to the nonlinear nature of Eq. (1)and the effect of wind speed variance, so that interpretation should be in terms of the mean square windspeed V2 (Sutcliffe, 1936), involving a factor 1+ (V'2/V2).This is typically < 1.02 over the sea for time scalesless than about several hours [-see data of Smith andBanke (1975) and discussion in Chap. 2 of Pasquill(1974)~, but may be considerably greater at time scalesof ~1 month (e.g., Clarke and Hess, 1975). It seems likely that much of the data scatter andsystematic differences between data sets in Table 2and Fig. 3 are due to insufficiently long averagingperiods, and calibration uncertainties associated withsensor (and electronics) performance in the field, workover the sea being particularly severe in this respect(Hidy, 1972). Turbulence sensor comparison experiments (Miyake et al., 1971 and Tsvang et al., 1973)emphasize the requirement for an averaging period ofat least 30-60 min, if good representative estimates ofu. and u'w' are to be obtained. Most data sets in Table 2are related to shorter averaging periods, from 10-30rain, for which individual u. and u'wi values may differfrom the long-term statistically steady values by up to10-15%. The experiments also showed differences inu'w' from run-to-run, and between sensor arrays placedside by side, ~ 10-20%, and which arose mainly fromdifficulties in determining absolute calibrations in thefield, instabilities of sensors and electronics and airflowdistortion related to the aerodynamic characteristicsof sonic anemometers. Other sources of error havebeen discussed above. The apparent increase of Ci)N with V appears to beincompatible with the results of numerous investigatorswho found C~ constant (Tables 1 and 2), but may beexplained as follows. First, there is the problem of interpretation of individual data sets (Table 2); for instance,Miyake et al. (1970) give CD~=I.09X10-a and 1.13X 10-a, whereas a linear regression gives a significantincrease of Cl>~ with V (Smith and Banke, 1975).922 MONTHLY WEATHER REVIEW VoL~m~105Brocks and Krugermeyer (1970) and Kraus (1972,Chap. 5) interpret the data of Hasse (1970) as Ci>N=l.21X10-a, whereas again linear regression givesCoat increasing with V (Smith and Banke, 1975). Second, there is the consideration of wind speedrange. The four data sets with the greatest V ranges~references 1, 2, 4, 8 (Table 2)3 all show an increaseof CON with V; overall, this occurs in 11 data sets (sixeddy correlation, four wind profile and one wave amplitude method, including references 12 and 13) having amean V range of 9 m s-~. This compares with 6 m s-~for the remaining eight data sets with "constant"Co~. There is evidently a requirement, taking intoaccount the inherent experimental scatter, of a sufficiently wide wind speed range over which a change ofCo~ with V can be resolved. This requirement is certainly obtained for the collection of data summarizedfor the range 4<V<21 m s-~ in Fig. 3.c. The value of a and power law relations We note that a combination of Eqs. (1), (2b) and(5) yieldslnCoN+ =in -2 lnu, CoN- .so that regression of In CoNq-k/Coar~ on In u (withk =0.41) for the data shown in Fig. 3 gave a correlationcoefficient of 0.92 with an in u coefficient =- 1.96;and by inference, for z= 10 m,a = 1.44X 10-2.This value lies close to that deduced by Wu (1969)of 0.016, and by Smith and Banke (1975) of 0.013used to describe their data. The associated variationof CON(10) with V is shown in Fig. 3 as a dashed curve.Extension of this to higher winds (see Fig. 4) illustratesthe good consistency between turbulence data at lowwinds (V~< 20 m s-~) and those at high winds based onindirect estimates of stress. The variation of CoN with V in the range 4<V<21m s-~ with a=0.0144 can be closely approximated byeither C. oN=0.51X10-a V-.4~ or CoNX10a=0.75+0.067V with V in m s-t. The power law relation isclose to that of Wu (1969), while the linear form issimilar to that of Deacon and Webb (1962). Combining Eqs. (4b) and (5) gives u. and r0 explicitly in terms of V~; viz., in the neutral case u. = 0.0462f---8 V ~ -%which, for latitude 45-, becomes u. =0.021VS~'-s '(7)compared to the 10 m (surface) wind relationu. = 0.023 V~'% (8)Eqs. (7) and (8) combine to give VIVa =0.94Va--'a~which is similar to Frost's (1948) relation with z0 ~ u,~,and gives V/Vo values consistent with Hasse andWagner (1971). Overall there is considerable evidence for an increaseof Cz)N(10) with wind speed, with little evidence ofCoN remaining constant over certain wind speed ranges,with transitions related to changes in sea state. If suchanomalies do exist they are sufficiently small as to beindistinguishable from instrumental sources of datavariability.d. Value of the yon K~rm~n constant The variation of Coat(V) shown in Figs. 3 and 4 as adashed curve, and based on Charnock's relation witha=0.0144, depends upon the choice of k. This has beentaken as 0.41 (Garratt, 1974; Hicks, 1976), while avalue of 0.35 (Businger et al., 1971; Frenzen, 1973;'Tennekes, 1973) would give Co~- smaller by approximately 37%. Comparison of the eddy correlation data with windprofile data summarized in Table 2 provides a directmethod of inferring a value of k through Eq. (2). Inthe wind speed range 3<V<16 m s-~ data have beenselected as in Fig. 3, giving 6 wind profile data sets(n=1200) and 8 eddy correlation data sets (n=369).The wind profile estimates of Co~(10) used k=0.40,so that k has been determined at 1 m s-t intervalsfrom k = O. 4[-Cz~ ~ ( 1 O) ~-]~/[-Co N ( 1 O) ~o~o-]~,giving k=0.414-0.025 from the data as a ~vhole.e. Additional theoretical and experimental support for an increase of Co:v with V Munk (1955) in a theoretical analysis showed thatthe drag of the sea surface should not vary greatlywith fetch [see also the wind-wave measurements ofToba and' Kunishi (1970)-], but is dominated by thehigh-frequency components of the wave motion whichcontribute largely to the mean-square slope, but littleto the mean-square amplitude. By incorporating alinear variation of mean-square slope with wind speedfound over the sea by Cox and Munk (1954) he predicted a linear variation of CoN with V. This is close tothat implied in Charnock's (1955) analysis of Zo(U.)in rough flow. Munk's predictio- was essentially confirmed by wind-water tunnel observations showing anincrease of C~ with wind speed (Francis, 1951;Hameda et al., 1963; Kunishi, 1963) which values,when extrapolated to a 10 m height, gave drag coefficients of similar magnitude to those over the sea(Preobrazenskii, 1969). Except possibly at low windspeeds (V~<3 m s-~), the drag is then substantiallyJULY 1977 J. R. G A R R A T T 923greater than that over a smooth surface (Stewart,1974), although the same author argued that effectssuch as surface tension may act to make C~>N less dependent on wind speed than that predicted by Eq. (5). More recently Kitaigorodskii (1968, 1969) presentedan analysis deriving the dependence of aerodynamicroughness upon the parameter Co/u, which characterizes the "age" of gravity waves. Observational evidencesuggests the functional dependence is weak, althoughthe parameter may have relevance in situations ofrapidly rising and falling seas (most probably associatedwith increasing or decreasing wind speed). For instance,Kitaigorodskii (1969) and Kitaigorodskii et al. (1973)presented data showing that the stress for a sea surfacedominated by swell (large Co/u.) was considerably lessthan when wind-driven gravity waves (much smallerCo/u.) were present. Intuitively one expects the drag to result from wavecomponents whose phase velocity (C) is less than thewind speed (Kitaigorodskii, 1969). Stewart (1974)discussed the data of Barnett et al. (1973) and Dobson(1971) as evidence for momentum transfer into bothshort gravity waves and longer waves with C< V.' The role played by the central region of the wavespectrum seems to be in the ability of the longer wavesto "sweep through" the short waves, which steepenand ultimately break on the forward face of long wavecrests (Longuet-Higgins, 1969). In general terms, thisseems to provide the mechanism by which the dragcoefficient continues to increase with increasing windspeed. Growth of the long waves ultimately producessaturation of the mean-square slope, with wave breaking contributing further to the drag at higher windspeeds (e.g., Toba and Kunishi, 1970).4. Values of drag coefficients over land Direct measurements of surface stress at or nearthe surface have been made by numerous workers[e.g., Sheppard (1947) and Bradley (1968) using dragplates; Mordukovitch and Tsvang (1966)using sonicanemometers~, 'usually over flat, locally homogeneousterrain, during micrometeorological expeditions. Suchobservations, together with simultaneous (surfacelayer) profiles of wind and temperature have generallyconfirmed the Monin-Oboukhov similarity theory,with empirical functions and constants subsequentlydetermined (see Section 2). In these circumstancesthe local z0 is exclusively determined by the eventopography of the field site and the physical andvegetational structure of. the surface. At larger scales, and with the inclusion of uneventopography, the average stress over a large area maybe considered as the sum of a "frictional" component(due to .small-scale features, e.g., grass, trees, etc.)and a "form drag" component due to flow perturbationaround hills and mountain features. The latter mayinclude the contribution from gravity waves initiatedby flow over mountains in a stably stratified atmosphere(e.g. Sawyer, 1959). In this situation a large-scaleroughness parameter is defined which is more realistically associated with the geostrophic drag coefficientthrough the surface Rossby number (itself defined interms of the large-scale pressure field) rather than thesurface wind and low-level drag coefficient. We now consider the data on the aerodynamic roughness (and implied drag coefficients) of natural terrain.a. Vegetation and fiat terrain There are numerous observations which relate thelocal or small scale z0 to the physical characteristics ofvegetation comprising the surface. Various empiricalrelations have been discussed by Sutton (1953), Lettau(1969), Kung (1961), Tanner and Pelton (1960),Sellers (1965, Chap. 10) and Stanhill (1969) and resultsbased on specific observations have been described bySchlichting (1936) [see Schlichting (1968)-], Kutzbach(1961), Marshall (1971), Wooding et al. (1973) andArya (1975b). Garratt (1977) has recently reviewedthe data available and finds, in general, O.02<zo/h<0.20, where h is the height of roughness elements.The geometric mean of 0.06 allows z0 to be determinedfrom h alone to within a factor of 3, or if the secondarydependence upon roughness element density is takeninto account, to within a factor of 2. It is obvious thenthat the variation of z0 over different land surfaces isvery large, covering several orders of magnitude. A practical method for determining z0 over largeareas has been demonstrated by Kung and Lettau(1961) and Kung (1963), based on the zo-h relationand the assumption that the areal average roughnesslength (call this ~0) is determined from the areaweighted average of log z0 values for the differentvegetation types in the area. They argued this on thebasis of a strong relation (in neutral conditions)between the geostrophic drag coefficient and thelogarithm of the surface Rossby number. It may also bederived as an approximate relation from Eqs. (1) and(3) if it is assumed that the areal average stress ~ isgiven by ~= (1/A)Y'. a(i)r(i), iwhere a(i) is the area of the ith surface in a grid squareof total area A (Garratt, 1977). Kung and Lettau (1961) determined the vegetationdistribution (eight classes were distinguished) from anumber of sources, and estimated ~0 for 5- and 10-latitude intervals over three continental land masses.Similarly, Garratt (1977) determined S0 over Australiafor 2.5- squares based on 30 vegetation classes. Their results may be used to determine the overallmean S0 (according to the technique just described)for all land surfaces, S0 being a parameter commonlyused in numerical models of the atmosphere, where the924 MONTHLY WEATHER REVIEW VoLu~ml05 TASLE 4. Annual values of ~0, CDN(10) and CG~ based on the data of Kung and Lettau (1961) and Garratt (1977) for extended landmasses. The area and mean latitude (~) of each land mass is shown, together with the percentage cover of forest and desert. CG~ valueshave been calculated from Eq. (4b) for V~ = 10 rn s-~ and mean latitude ~. Annual Area Percentage cover ~o CDN(iO) C~NLand mass (X107 km~) ~ Forest Desert (m) (X10a) (X10~)North America(70--10-N) 1.97 40- 40 < 5 0.17 10.1 1.89South America(10-N-50-S) 1.76 20- 50 <5 0.81 26.6 2.16Northern Africa(40--10-N) 1.43 25 17 66 0.004 2.7 1.03Southern Africa. (10-N-35-S) 1.62 12.5 39 <5 0.27 12.9 -Europe(70--40-N) 0.41 55 24 0 0.07 6.8 1.73u.S.S.R.(70--50-N) 1.56 60 43 < 5 0.1 7.9 1.83Asia(50--20-N) 2.62 35 12 29 0.014 - 3.9 1.31(20-N-10-S) 0.54' 5 43 < 5 0.85 27.7 -Australia(10--40-S) 0.76 25 16 19 0.05 6.0 1.50effects of topography are not included. For instance,values of ~0 are shown in Table 4 for extensive landmasses to illustrate the variation at large scales (atsmaller scales it is considerably greater), and for comparison of drag coefficient values derived by othertechniques (see later in this Section). Clearly, the highest values of ~0 (approaching 1 m fortropical Asia and South America) are related to extensive areas of dense, tropical rain forest; intermediatevalues [--~10 cm (e.g. North America, Europe and T^sL~ 5. Values of C6N, ~o and CD~(10) taken from the literature for a range of land surfaces (italicized). CoN values (notitalicized) associated with ~0 have been calculated from Eq. (4b)for V,: 10 m s-~ and latitude 45-. Ca~Xl0a so q~te(10)Source Surface ( X 10a) (m) ( X 10a)Fiedler and Panofsky "Plains" 2.2 (1972) "Low Mountains" 2.5 "High Mountains" ~2.6Cressman (1960) Rockies 9.7 Himalayas Appalach'ians 2.7 "Low Relief" 2.7 "Moderately High Mountains" 4.7 "Very High Mountains" 8.2 "Flat land" 1.2 N.H. Average 2.2*Palm~n (1955) General Land 1.8'Start and White(1951) General Land 2.3*Sutcliffe (1936) British Isles 1.5La Valle and Girolamo (1975) General Land 1.64'0.420.~91.426.0* Land surfaces in general, including both frictional and form drag. U.S.S.R.)~ to a mixture of temperate forest and wood lands, meadows and pastures; and lower values [-~-1 cm, (e.g.; northern Africa and central Asia)~ to ex tensive areas of desert. Associated variations in the 10 m drag coefficient and geostrophic drag coefficient are 3-27X10-a and 1.0-2.2x10-~, respectively. The global mean value (for land) of ~0 is 0.08 m, giving' C~(10)=7.2X10-~ and Co~ (for Vo=10 m s-t and latitude 45-) ~- 1.7 X 10-~.b. Topography and the effective roughness In qualitative terms the influence of uneven topography on all scales must be to increase the aerodynamicroughness and hence, for a given geostrophic wind,the total stress. The addition of form drag could beaccounted for by increasing C~ or Co (e.g., Cressman,1960; Lorenz, 1967), using the equations in Section 2to calculate the stress. These are then associated withan effective roughness length ~0(eff) (Fiedler andPanofsky 1972), where ~0(eff) is that value of z0 thathomogeneous, flat terrain would have to give the correctsurface stress over a given area. Typically the compositeCo value would apply over a distance greater than thehorizontal scale of the topographical feature or mountain range in the downstream direction to incorporatethe pressure drop across the feature, and be associatedwith the immediate upwind or downwind Vo of theundisturbed flow. Fiedler and Panofsky used turbulence observationsmade from an aircraft to show that ~0(eff) over "mountainous'' areas was some 2-3 times ~0 for "plains"(see Table 5). Cressman (1960) used a simple formulato calculate form drag which was derived from Sawyer's(1959) treatment of gravity wave momentum transfer1977 J. R. G A R R A T T 925arising from flow perturbation over mountain ridges.This gave a Co of 8.5 X 10-a for the Rockies or Himalayas and 1.5X10-a for the Appalachians, which hecompared with 1.2X 10-a for the frictional (vegetation)drag (based on a literature review) of land surfacesin general. The resultant effective Co values are shownin Table 5, together with values applicable to generalmountainous terrain and to the Northern Hemisphere(land only) as a whole.c. Global angular momentum considerations The large-scale drag coefficient applicable to land andsea surfaces must be consistent with the requirement ofangular momentum balance for the globe over a sufficiently long period (e.g., one year). The balance equation has been studied by a number of authors includingPriestley (1951), Lorenz (1967), Newton (1971a) and,most recently, La Valle and Girolamo (1975), whoused it to determine an optimum value of the geostrophic drag coefficient. They considered 14 latitudebelts in the Northern and Southern Hemispheres,assumed values for the angular momentum flux andpressure torques found in the literature (e.g., Newton,1971a) and deduced annual mean zonal stresses. Usingavailable 4-month average geostrophic wind components they deduced Co (sea) ~ 0.41 X 10-a, Co (land) ~ 1.64X 10-a (Table 5),although these values admittedly gave computedstresses which seemed too small in mid-latitudes. Inhis study, Newton (1971a) considered drag coefficientsrequired to give approximate balance in the 0-30-Nlatitude belt. For land surfaces he chose a value of CD(10 m)=6X10-a deduced by Sutcliffe (1936) for flowover the British Isles, and computed from the ageostrophic method (included in Table 5). Cressman alsoquoted larg~-scale values of Co from Palm~n (1955)and Start and White (1951) based upon meridionalmass transport and angular momentum balanceconsiderations.d. Comparison of values in Tables 4 and 5 Values of Co, z0 and Cl> taken from the literature(described above) are shown italicized in Table 5,with equivalent values of Co derived from z0 (nonitalicized) using Swinbank's relation (4b) with Vo= 10 m s-1 for latitude 45-. We may summarize Table 5as follows: 1) Extremely high mountains (peaks up to 4-5 km)are associated with a geostrophic drag coefficientCo~5-9X10-a. Physical interpretation of the geostrophic drag law over high mountains is, however,extremely difficult because of pressure discontinuitiesacross the mountains not necessarily related to formdrag (Newton, 1971b). 2) Low-relief topography and low mountains (peaksgenerally <0.5-1 km) have Co~ 2-3 .X 10-2. 3) Land surfaces in general (the asterisks in Table 5)have Co~2X 10-a which includes both frictional andform drag. The corresponding value of Cl>jv(10) andz0 may be estimated from Eqs. (1), (2b) and (3) takingV/Vow--0.45 (Deacon, 1973a), or from Eqs. (2b) and(4b) with V~ = 10 m s-1 at latitude 45-, ,to give Cmr (10) -~ 10x 10-2, 5o(eft) ~0.2 m.Comparison with the results summarized in Table 4suggest that form drag contributes ~ 200-/o of the overall surface drag over the land surfaces of the globe.5. General application In many numerical general circulation models ofthe atmosphere in use today surface exchange andboundary layer processes are described in part by theaerodynamic drag and heat transfer laws (see Section2). These parametric relations are quite sensitive to thevalue of z0, so that one may question the use of a singlevalue of z0 for land as the normal practice. In GARP(1974) some 20 models are described of which, for landsurfaces: 1) Four models use values o/z0=0.01 m/or generalapplication over land and sea. This is consistent withtaking the average of the log 50 values for land (50~0.2m) and sea (50~0.001 m), taking into account arealcover of each over the globe. 2) Several use values of a drag coefficient which varybetween 14X10-a, but the reference height is notalways specifically stated. If this is assumed to be thelowest atmospheric level resolved in the model (900 mbin some cases), then the above values are more correctlyrelated to a geostrophic drag coefficient, and agree towithin a factor of 2 with the value found in Section 4. Bhumralkar (1975) has recently reviewed methodsof .boundary layer parameterization used in severaladditional numerical (global) models, and' quotesdrag coefficient values similar to those in GARP(1974). In contrast, Benwell et al. (1971) and La Valleand Girolamo (1975) have described Cz~ in terms ofterrain height which, in a simplistic sense, is an attemptto realistically account for the additional form drag. In a number of models described more fully in theliterature, Values of z0 for land surfaces are used, whichcompare favorably with those deduced in Section 4,viz., 50~0.! m (vegetation only) and 0.2 m (includingform drag). Thus Delsol et al. (1971) took 5o=0.17 mbased on Kung's work (discussed earlier), Corby et al.(1972) chose 50=0.1 m based on geostrophic dragcoefficients reviewed at the time, Deardorff (1972)suggested taking 50=0.2-0.7 m based on work of926 MONTHLY WEATHER REVIEW VOLU~dE105Fiedler and Panofsky (1970) and most recently Bhumralkar (1975)gave values of 0.1 and 0.45 m from hisreview of boundary layer parameterization in numericalmodels. The value of ~0=0.2 m then corresponds to a 10 mdrag coefficient CDs(10)~ 10X10-s ~noting that for adifferent reference level, e.g., 75 m, CDN(75) is reducedto ~5X10-s-] and a geostrophic drag coefficientCGN ~ 2 X 10-~. For use over the sea, GARP (1974) and Bhumralkar(1975) indicated a value of the drag coefficient~l-2X 10-a, corresponding to z0~0.001 m, in generalusage. In contrast Delsol et al. (1971) chose Charnock'srelation, with a=0.032. They found that increasingthe com?lexity of boundary layer description did notproduce "large" effects until after 7-10 days. The mostsubstantial effect was produced by simply varying thesurface drag coefficient with two constant values forland and sea, rather than adopting a single valueeverywhere. Nevertheless, some further improvementwas obtained, particularly in relation to heat flux,using Monin-Oboukhov formulation (their ExperimentC) by varying the land value of Co for greas of differentcharacteristics, and adopting Charnock's relation overthe sea. The results were not conclusive, and indicatedthe need of further investigation into boundary laye?parameterization (see Bhumralkar, 1975).6. Final comments Observations of wind stress and wind profiles overthe ocean reported in the literature over the past 10years or so are consistent with Charnock's (1955)relation z0= au.2/g with a= 0.0144, and suggest takingk, the von KgrmS. n constant, =0.414-0.025. Forpractical purposes2 this relation may be closely approximated over the range 4< V< 21 m s-~ by a neutraldrag coefficient (referred to 10 m) varying with the10 m wind speed V (m s-~), either by a power lawrelation CON (10) X l0s = 0.51 V-,4~,or by a linear form Cos (10) X l0s = 0.75-t-0.067 V.These relations are close to those proposed by Deaconand Webb (1962) (the linear form) and Wu (1969)(the nonlinear form) for wind speeds <15 m s-~,and do not support a constant CoN above 15 m s-1 asdeduced by Wu (1969). Incorporating go(u.) into Swinbank's (1974) empirical relation for' the neutral, barotropic geostrophicdrag coefficient [see Eq. (7)-] gives CGN x l0s = 0.44V~-'~6at latitude 45-, and impliesV(iO)/Vo = 0.94Vo---~,giving values consistent with observation (Hasse andWagner, 1971). The results of recent turbulence sensor comparisonexperiments (Miyake et al., 1971; Tsvang et al., 1973)suggest that much of the source of data scatter inCo(V) plots and of the systematic differences betweendata sets is mainly due to calibration uncertaintiesassociated with sensor (and electronics) performancein the field. There is then a requirement, in a fieldexperiment, of a sufficiently wide wind speed range(zXV>~9 m s-~) over which a change of C~N with V willbe resolved. The effects of fetch, wind duration, windgustiness and wind unsteadiness (if any) remain obscured in this experimental data scatter. Future experiments must provide detailed measurements of theseparameters and of the wave structure, with carefulconsideration given to instrumental performance, ifsuch effects are to be resolved. Over land, vertical momentum transfer over largeareas can probably be evaluated realistically usingeffective drag coefficients or roughness parameters.These must account for both the frictional drag (dueto vegetation, etc.) and form drag introduced by perturbation of the mean flow by uneven topographicalfeatures. Quantitative information at present is sparse,but observations of turbulence from aircraft (Fiedlerand Panofsky, 1972) and consideration of gravity wavemomentum transfer over mountain ridges (Sawyer,1959; Cressman, 1960) suggest that 1) low relieftopography (including mountains) requires a geostrophic drag coefficient C~N ~ 3 X 10-s to account forthe total stress at the surface, and that 2) land surfacesin general require CeN-~2X10-~, for which CON(10)~10X10-s and 20(eff)~0.2 m, with such values satisfying, very approximately, the requirement of globalangular momentum balance (La Valle and Girolamo,1975). Consideration of frictional drag only, based onregional and continental distributions of vegetationand other surface types (Kung and Lettau, 1961;Kung, 1963; Garratt, 1977) gives values of z0 between0.01 and 1 m (approximately) for extended land masses,with an average value ~--0.1 m, for which the corresponding Cos (10) ~ 7 X 10-s and CaN ~ 1.7 X 10-s. The implication is that form drag arising from flowover uneven topography (including mountains) contributes --~20% of the overall surface drag of the landsurfaces of the globe. REFERENCESArya, S. P. S., 1975a: Geostrophic drag and heat transfer relations for the atmospheric boundary layer. Quart. J. Roy. Meteor. Soc., 101, 147-162. , 1975b: A drag partition theory for determining large-scale roughness parameter and wind stress on the Arctic pack ice. J. Geophys. Res., 80, 3447-3454. , and J. C. Wyngaard, 1975: Effect of baroclinicity on wind profile and the geostrophic drag law for the convective planetary bofindary layer. J. Atmos. 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