Mar 20, 1985 ... scale (PSS 78) and an equation of state for seawater (EOS 80) marked a ...
chemical and physical properties of seawater with the aim of.
JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 90, NO. C2, PAGES 3332-3342, MARCH
20, 1985
Physical Properties of Seawater' A New Salinity Scale and Equation of State for Seawater N. P. FOFONOFF
Woods Hole OceanographicInstitution, Massachusetts
Developmentsleadingto the new practicalsalinity scaleand equation of statefor seawater,introduced by the Joint Panel on Oceanographic Tables and Standards (JPOTS), are reviewed. The laboratory measurementsusedto constructthe new empirical formulasfor convertingconductivityratios to salinity and for estimating seawaterdensity from salinity, temperature,and pressurewere made on samplesof standard seawaterto minimize the effectsof compositionalvariations. Measured densitieswere fitted
with a standarddeviationof 0.0036kg/m3 at atmospheric pressureand 0.009 kg/m3 over the oceanic pressurerange.Natural seawaterdensitiescan differ from computeddensityby as muchas 0.05 kg/m3 becauseof variations in the compositionof dissolvedsaltsthat are not reflectedequivalentlyin measured conductivity ratios. Differencesfrom the previous Knudsen-Ekman equation of state are also of comparable magnitude.Values of specificvolume anomaly for geopotentialanomaly computationsfor estimation of geostrophiccurrentsand transport are not significantlymodified. Algorithms adopted by the JPOTS for computing adiabatic lapse rate and potential temperatureare compared with formulas
derivedfrom the newequationof state.Potentialtemperatures agreewithin2 x 10-3øCoverthe oceanic range of temperatureand pressureat salinity of 35.
HISTORICAL
as chlorinity. These recommendationswere made in recognition of the increasing use of conductivity bridges (salinometers) for salinity determination and increasing concern about the adequacy of a chlorinity standard to serve as a conductivitystandard. At the same conference,Reid [1958] presented a comprehensive analysis showing the dependenceof geostrophicand dynamic computationson errors in the equation of state. He computed the sensitivityto thermal expansionand the saline contraction coefficientsto demonstrate that dynamic computations were not limited by inaccuraciesof the coefficients.For other calculations, such as stability or Brunt-Vfiisfilfi frequency,accuratecoefficientsare a necessity. A study of the relationshipsamong chlorinity, conductivity, and density was undertaken at the British National Institute of Oceanographyin 1960 on severalhundred samplesof seawater collected from widely distributed locations over the world oceans. Cox et al. [1962] reported a markedly higher correlation between density and conductivity than between density and chlorinity. These findings tended to confirm the misgivings about the methods for standardizing salinity
BACKGROUND
The completionin 1981 of a seriesof recommendationsand reports by the Joint Panel on Oceanographic Tables and Standards (JPOTS) on the definition of a practical salinity scale(PSS 78) and an equation of state for seawater(EOS 80) marked a milestonein the establishmentof preciseand reproducible standards for salinity, density, and related variables for ocean waters.
BecauseJPOTS introduced a new salinity scale based on constant composition of salts and expressedin terms of an electrical conductivity ratio, consistentalgorithms had to be developedfor other derived or dependentfunctions.Formulas for a few of the more commonly usedphysicalpropertieshave been tabulated by Fofonoff and Millard [1984]. These are discussedin the presentarticle together with a review of someof the considerationsleading to the new definitions and algorithms. The notation
and definitions
are similar to those used
by Fofonoff [1962]. The units and symbolsfollow recommendations of the IAPSO (International Associationfor Physical Sciencesof the Oceans)working group on symbols,units, and nomenclature [IAPSO, 1979]. The evolution of the practical salinity scaleand equation of state was initiated in part by Eckart's [1958] concern about the accuracy of knowledge of fundamental propertiessuch as density and thermal expansion of seawater.His examination of the available
data
led him
to conclude
that
thermal
ex-
pansioncoefficientsof seawaterand evenpure water were not known with acceptableaccuracy.With the cooperation and support of the U.S. National Academyof SciencesCommittee on Oceanographyand the Office of Naval Research,he convened a conferenceto examine the state of knowledge of the chemicaland physicalproperties of seawaterwith the aim of stimulatingand focusingfurther researchin this area. Among the recommendationsincluded in the conferencereport [NASNRC, 1958] were stepsto label and certify CopenhagenStandard Seawater for electrical conductivity and density as well Copyright 1985 by the American GeophysicalUnion. Paper number 4C1335. 0148-0227/85/004C- 1335505.00
measurements
and led to the formation
of a Joint
Panel
on
the Equation of State of Seawater,sponsoredby ICES (International Council for the Exploration of the Sea), IAPSO, SCOR (Scientific Committee on Oceanic Research) and UNESCO (renamedthe Joint Panel on OceanographicTables and Standards (JPOTS) in 1964). The panel's task was to review the relationshipof the equation of state to chlorinity, salinity, electrical conductivity, and refractive index to see if redefinition of any of these properties was necessary.The panel membersconcluded,after reviewingthe data presented by Cox [1962; reprinted in UNESCO, 1976a, Append. 1], that salinity should not be defined in terms of chlorinity becauseof the excessivevariability of its relationship to density. Although some of the variability later proved to be spuriousand a consequenceof the laboratory techniqueused to determine chlorinity [Cox et al., 1967], the scatter consistentlyexceeded the precisionof measurement,supportingthe possibilitythat density could be estimatedmore preciselyfrom conductivity than from chlorinity. Becauseof the better correlation between conductivity and
3332
FOFONOFF: PRACTICE OF OCEANOGRAPHY
density, the initial panel recommendationwas to establishan equation defining salinity in terms of density. Different techniques of measurementcould then be implemented by construction of formulas relating density to a measuredvariable such as electrical conductivity, refractive index, or density itselfi Because density can be measured, whereas salinity cannot, the link to salinity would be through the defining density equation. Salinity must be proportional to chlorinity to retain its conservativeproperty and to have a proportionality factor of 1.80655 to match historical data. Subsequent determinationsof salinity of standard seawater (SSW) would be made from measurementsof density or conductivity but not chlorinity, which would be reported as an independent variable.
The recommendationswere made with the assumption that absolute density and conductivity of SSW could be measured with the required precision.However, technical problems encounteredin constructingequipment to achieve the precision necessaryfor acceptable standards delayed implementation of the recommendations.
Relative
measurements
were considered
and dismissed because the references could not be defined with
sufficientprecision. Distilled water, for example, differed in density, dependingon its origin and method of preparation. The uncertaintiesin density were believedto be as large as 30 parts per million [UNESCO, 1976a, Append. 2]. Similarly, conductivity references,such as potassiumchloride solutions, required temperature regulation that was not readily achievable [Cox, 1962]. As a result, only part of the recommendationscould be implemented.Salinity was definedin termsof the electrical conductivity ratio R•5 at 15øC to seawater having a salinity of 35 [UNESCO, 1966, 1976b]. The reference seawater salinity was not defined separatelyin terms of conductivity ratio or density. Tables to convert conductivity ratio to salinity, based on a regressionformula developedby Cox [1962], were published by NIO-UNESCO [1966] and widely distributed to oceanographers with the understandingthat SSW would be certified for conductivityas soon as the necessaryequipmentwas completed. The new scalewas officially adopted after Cox's death in 1967 and was formally announcedin 1969 [Wooster et al.,
3333
tivity on temperatureand pressure.Also, to have an independent check on the absolute conductivity apparatus, Poisson was encouragedto make some measurementsof electrical conductivity of standard seawatersamplesrelative to potassium chloridesolutionsto comparewith the absolutemeasurements planned at National Institute of Oceanography (U.K.) by Culkin. Poisson was able to report at a subsequentJPOTS meeting [UNESCO, 1976a] that deviations of conductivity equivalentto 0.006 salinity were found for different batchesof SSW. The panel viewed theseresultswith some consternation and called for confirmatory studies.Severallaboratories were asked subsequentlyto measureconductivity of 26 batchesof SSW relative to P64. These studies[Millero et al., 1977] confirmed Poisson'sfindings. Some of the batcheswere high in conductivityby amounts equivalent to 0.004 to 0.007 salinity [UNESCO, 1978]. Poisson reported at the same meeting of JPOTS that potassium chloride solutions could be made with high and reproducible accuracy. Furthermore, the temperature control required for thermostatted baths was well within the capability of commerciallyavailable equipment.Thus the reservationsabout acceptingpotassiumchloride as a conductivity standard were removed. The panel accepted a recommendation to use potassium chloride solution as a standard and outlined a procedureto establishthe salinity scale. The determinationswere to be made on a singlebatch of SSW to ensureconstancyof composition.The scalewas to be related to the 1969 and previous scalesat 35 by the chlorinity factor 1.80655 for that particular batch. For subsequent batches the salinity was to be determined from the conductivity ratio to the referenceKC1 standard and not from chlorinity. Hence the chlorinity factor is not a constant but would vary slightly from batch to batch. The stepsthat followed to define the practical salinity scale are thoroughly documented in the eighth, ninth, and tenth reports of JPOTS [UNESCO, 1978, 1979, 1981c] and collectedbackgroundpapers [UNESCO, 1981b]. In addition to the definition of the salinity scale, the panel encouragedthe development of algorithms for converting in situ measurementsof conductivity ratio to salinity. These algorithms were developed to be consistentwith the new practical salinity.
1969]. Objections to the 1969 scale were raised. Park [1964] had THE PRACTICALSALINITYSCALE(PSS 78) already noted that conductivity of different batches of SSW Prior to the introduction of the 1969 salinity scale,salinity $ measured relative to a single referencebatch varied in excess of the precisionof conductivity measurementsand had called was expressed in terms of chlorinity C1, defined as the chloride-equivalentmassratio of halides to the massof seawafor certificationof conductivityas well as chlorinity for SSW. Other objectionssurfaced.The determinationsof R•5 [Cox ter, by a linear least squaresregressionformula fitted to mass et al., 1967] were made on natural seawatersamplesof vari- ratios of salt residues of nine seawater samples evaporated to able composition.Some midrange salinitieswere obtained by dryness [Knudsen et al., 1902]. The relationship $ = 0.030 + mixing Baltic and Red Sea samples.As the salinity-chlorinity 1.8050 C1 served oceanographers for over six decades. The conversionsassumedfixed composition,the salinitieswere not formula for $, commonly referred to as the Knudsen salinity, accurately conservative.As salt ratios of natural seawatersare incorporated changing composition of seawater with dilution and consequentlywas not conservativewith respect to additime dependent, especially at low salinities, the scale is not accuratelyreproducible.Samplesfrom the same geographical tion or removal of water. As long as salinitieswere computed locations could not be relied upon to yield the same relation- from chlorinities, the Knudsen scale was acceptable.The in-
ship betweenconductivityand chlorinity. The increasinguse of in situ instrumentsmeasuringconductivityforced the useof conversionformulas that were obtained by using standard seawater diluted with distilled water [Brown and Allentoil,
creasing use, during the 1950's, of high-precision electrical conductivity bridges for salinity determinations created the need for new standardsfor salinity measurements. The 1969 scale, proportional to chlorinity to satisfy mass
1966]. These formulas were not consistent with the JPOTS versions,which did not cover the temperature range below 10øC. Some of the problems associatedwith the 1969 scale were addressedin the sixth JPOTS report [UNESCO, 1974] with recommendationsto measurethe dependenceof conduc-
conservation, was chosen to coincide with the Knudsen scale
at salinity of 35, yielding the factor 1.80655 = 1.8050135/(35-0.030)]. Differences between the two scales are less than 0.003 over the salinity range 32-38 and increase to 0.03 at low salinity. As noted earlier, the 1969 scale was
3334
FOFONOFF: PRACTICE OF OCEANOGRAPHY
TABLE 1. Concentrationof PotassiumChloride SolutionsHaving the SameConductivityas Seawater (BatchP79) at Salinity S = 35 (PSS 78) Standard
of Residuals,
g/kg
g/kg
32.4356 32.4358 32.4352 32.4356
3.6 x 10-'• 4.7 x 10-'• (3.4 x 10-'•) 0
Reference Dauphineeet al. [1980a] Poisson[1980a]
CulkinandSmith[1980] S-- 35 (PSS 78)
Deviation
Concentration
Standard
Number of
Deviation
of Mean,
Measurements
g/kg
21 14
0.8 x 10-'• 1.3 x 10-'•
17 (by definition)
(0.8 x 10-'•)
Values in parenthesesare calculatedfrom publishedtable.
deficientin not establishinga salinity referencepoint independent of chlorinity and inconsistentin using seawatersamples of variable composition to relate salinity to electrical conductivity. The fundamental step in constructingthe practical salinity scale PSS 78 consisted of defining a single reference point (S = 35) on the scaleas having the same electricalconductivity as a referencepotassiumchloride (KC1) solution at 15øC and atmosphericpressure.The transition from the previous scale was made by selectinga singlebatch (P79) of SSW and equating the new scaleto the old through the chlorinity relationship S = 1.80655C1 for that particular batch. As the salinity on the practical scale is defined to be conservativewith respectto addition and removal of water, the entire salinity range is accessiblethrough precise weight dilution or evaporation without additional definitions.However, the practical scaleis defined in terms of conductivity ratio and not by its conservative properties. Thus the second part of the definition was constructedby measuring the conductivity ratio to the KC1 standard or an equivalent secondarystandard over the entire range of salinities (1 to 42) of samplesprepared by evaporation or dilution of batch P79 SSW and computing an empirical formula $ -- S(R•5), where R•5 is the conductivityratio at 15øC and atmospheric pressure to the KC1 standard. Thus salinities on the PSS 78 scale are defined by conductivity ratios alone. Salinities determined by any other method would not necessarilycoincide with the PSS 78 scaleand would have to be identified separately. Lewis and Perkin [1981] discuss differencesbetween PSS 78 and previous scalesand provide algorithms and tables to convert existing data to the new
The KC1 referencesolution for S- 35 was establishedby three independentsets of measurementsby Dauphineeet al. [1980a], Poisson[1980a], and Culkin and Smith [1980]. The values obtained and the precision of the determinations are summarized
in Table
1. Both standard
deviations
measurements
of Culkin
and Smith.
The data sets used by Perkin and Lewis [1980] to express the dependenceof conductivityratio on temperatureand pressure are listed in Table 2. All of the measurements
different batches of SSW relative to P79. Batches P73 and P75
yieldeddifferencesof -0.0011 and +0.0005, respectively,between conductivity and chlorinity salinity. As the standard deviation of replicates was 0.0004, the batches were not distinguishablefrom one another in composition. The algorithm for convertingconductivity ratio to salinity is constructedin terms of the conductivity ratio R, definedas R = C(S, t, p)/C(35, 15, 0)
TABLE 2. Precision of EmpiricalFormulasRelatingSalinityandConductivity RatiosRt, rt,and Rv Deviation
of Residuals,
Reference
(Salinity Equivalent)
Number of Observations
SSW Batch
S = S(R15) Dauphineeet al. [1980a] Poisson[ 1980a] Perkin and Lewis [1980]
3.6 x 10 -'• 4 x 10 -'• 4.8 x 10 -'•
116 38 combined fit
P79 P75
189
P79
115 101 combined fit
P75 P75
S = S(t,Rt) Dauphineeet al. [1980b] Poisson [1980b] Bradshawand Schleicher[1980] Perkin and Lewis [1980]
10 x 10 -'• 8 x 10 -'• 6.6 x 10 -'•
Pt-- Pt(t) Dauphineeet al. [1980b] Bradshaw and Schleicher[1980] Perkin and Lewis [1980]
4.4 x 10 -'•
34
P79
3 x 10-'•
36
P73, P75
3.2 x 10 -'•
combined fit
Rv = Rv(R,t,p) Bradshawand Schleicher[1980]
were made
by using PSS 78 and batches P73, P75, and P79 of SSW. Differences in composition between the batches would have negligibleeffectson the temperature and pressurecoefficients of conductivity ratios, as neither are strong functions of salinity. Mantyla [1980] measured the conductivity ratios of 39
scale.
Standard
of residuals
and of the means are computed. The means given by Dauphineeet al. [1980a] and Poisson[1980a] are not significantly different from each other but are significantlydifferent (3 to 4 standard deviations)from the mean value obtained by Culkin and Smith. Thus, although all three determinationsare sufficientlyaccurateto definethe referencean order of magnitude more precisely than required for routine CTD instruments, statisticallysignificantdifferencesdid appear for the absolute
13 x 10 -'•
224
P75
(1)
FOFONOFF: PRACTICE OF OCEANOGRAPHY
where C(S, t, p) is electricalconductivityas a function of salinity S, temperaturet, and pressurep. The ratio is factored into the functions
3335
conductivityratio R is required,givenS, t, and p, the ratio Rt can be found by numerical inversion of (3), and R can be found by solvingthe quadratic equation
(2)
R = rt(t)Rt(S,t)R•,(R,t, p)
R= rtRtR•,= rtRt1+ AR+ B
where
r,(t) = C(35, t, 0)/C(35, 15, 0)
or
Rt(S, t)= C(S, t, 0)/C(35, t, 0)
R = {[(ArtRt -- B)2 + 4rtRtA(B+ C)]•/2 + [ArtRt -- B]}/2A
Rt,(R,t, p)= C(S,t, p)/C(S,t, O)
(6)
JPOTS has not acceptedor recommendeda value for conductivity C(35, 15, 0) to avoid any confusionwith conductivity ratio in the definition of practical salinity. For those applications where absolute conductivity is required, a value of 4.29140 S/m (42.9140 mmho/cm) can be used to convert conductivity ratio to electrical conductivity. This value was obtained by Culkin and Smith[1980] usingabsoluteconductivity equipment at the British Institute of Ocean Sciences. Salinity is given by the function
S=
n: o
an+-
1 + kAt
bn Rtn/2
(3)
with coefficients
ao = +0.0080
bo = +0.0005
a• = -0.1692
b•=
a 2 -- + 25.3851
b2 = -0.0066
a 3 = + 14.0941
b3 :- -0.0375
a4 = -7.0261
b4 = +0.0636
as = +2.7081
b5 = -0.0144
-0.0056
At=t--15
At t = 15øC, (3) reduces to the formula defining the practical salinity scale [-Perkin and Lewis, 1980].
Thefactorsrt andR, aregivenby (4)
n=O
where
Co = +0.6766097
C3 =
C• = +2.00564E-2
C• = + 1.0031E-9
- 6.9698E-7
C2 - + 1.104259E-4 and
Rp=l+
e•p + e2p2 + e3p3 1 + d•t + d2t2 + (d3 + d4t)R C
1+
B+AR
with
eI = + 2.070E-5
d1 -- +3.426E-2
e2 ----6.370E-10
d2 = +4.464E-4
e3 = +3.989E-15
d3 = +4.215E-1
IDEAL OR ABSOLUTE SALINITY
Absolute or ideal salinity, defined simply as the mass fraction of saltsin seawater,is important conceptuallyfor formulating rigorousdescriptionsand preciseassumptionsabout the dynamical and thermodynamical behavior of a seawater system.In its 1978 recommendations,JPOTS proposeduse of the symbol SA to denote absolute salinity to distinguish it from practical salinity S. The practical salinity must be proportional to SA (or s used here) to representconservationor salt mass in seawater.
k = +0.0162
rt -- • Cntn
More detailed descriptions, including FORTRAN programs,check values,limits of validity, and tables are given by Fofonoff and Millard [ 1984].
d,• = -3.107E-3
for temperaturein degreesCelsiusand pressurein decibars. Given a measurementof R, t, and p, salinity is computed by
solving(2) for Rt' Rt = R/(rtRp)and evaluatingS from (3). If
The aggregate variable, salinity s, is useful becausemajor ions are in nearly constant proportions to one another throughout the world oceans.This "law of constant proportions," first enunciated by Marcet [1819] and elaborated and documentedmore quantitatively by Forchhammer[1865] and Dirtmar [1884], appears to hold, with some exceptions, to within experimental error [Culkin, 1965]. However, the error of determining some of the major constituentsseparately is relatively large, so that a rigorous test to the same precisionas conductivitymeasurementsis difficult to carry out. Exceptions to the constancy of composition are found in coastal and enclosedregions,suchas the Baltic Sea, diluted by land drainage and surfacewaters in which biological processescan alter the composition significantly.Culkin and Cox [1966] noted that the concentration of calcium is the major variable component affecting density and conductivity betweensurfaceand deep waters of the open ocean. As real variations in composition are present,the effectsof salt contentin seawatercannot
be completelydescribedby a singlevariable (salinity) within the framework of the hydrodynamic equations.The ideal salinity has limitations that require discussion.The first is that no convenient,inexpensive,and accurate techniquesexist for determining total salt content in seawater.Estimates of salinity, obtained indirectly from measurementsof chlorinity, electrical conductivity and, less commonly, specificgravity or refractive index, yield estimates of salinity on practical scales, such as PSS 78 discussedearlier, whose precise relationships to absolute salinity are unknown. The second is that an advective-diffusive conservation equation cannot be written for salinity alone becausediffusion does not change the salt distribution toward constant composition or uniform salinity. The law of constant proportions is a consequenceof dynamical processesof advection and turbulent mixing rather than a trend to thermodynamical equilibrium. This conclusionis developedfurther by examiningthe conservationequationsfor a multicomponent thermodynamicalsystem. Supposethat a massmi Of the ith solute constituentis present in a mass m of seawaterconsistingof mwwater and ms =
3336
FOFONOFF: PRACTICE OF OCEANOGRAPHY
y',mi of solutes. Themassfractionof solutes or idealsalinitys positive definite function.The fluxeswill be linear functionsof is defined to be
the form
s = ms/(mw + ms)= ms/m
Fi = 5',v,kV(lak-3tw)
(7)
(16)
k
and the partial salinity (concentration)si = m•/m of the ith solute so that
The effectsof diffusionare to diminishthe gradientsof bti - #w at rates that can vary from solute to solute. Caldwell and Eide
s = Z s,
(8) [1981] found variations of a factor of 2 or more in the diffu-
If ;• is the ratio of the ith constituent to the total solutes so
that si = 2is,the law of constantproportionsimpliesthat ;•i is independentof s. A mass conservationequation must hold for each constituent mi. If Pi is the partial densityof the ith constituentand Vi its velocity, the individual conservationequationstake the form
--
ference bt= • ;•bq- btw = bts- btwis about3 to 4 per 1000
q- V•oiVi : 0 (9)
C•pw -- + VpwVw = 0 •t
where the subscriptw denotesthe water or solvent component. The sum of the individual conservationequationsleads to the usualcontinuity equation •p
+ vpv = 0
sivities of major salts in seawater.Although diffusive fluxes must be further constrainedand modified by requirementof balanceof electricalchargesand by variationsof pressureand temperature,it is clear that diffusiondoesnot act to preserve constancyof salt ratios but rather to equilibratepartial chemical potentials. Fofonoff[1962] estimatedthat the salinitygradientrequired to equalizethe pressuredependenceof chemicalpotential dif-
(10)
decibars.As vertical gradientsof this magnitude are not observedin the ocean except over short vertical scales,the diffusive fluxes do not appear to play a significantrole compared with advectiveand turbulent transports.The representationof diffusivefluxesin seawaterproportional to gradientsof salinity is, therefore,misleadingand, exceptfor the smallestscales, incorrect becausethe thermodynamical equilibrium state is not incorporated into the equationsand cannot be in terms of salinityalone.The differentdiffusionratescan becomeimportant if the diffusion time scalesare comparableor exceedthe advective
scales.
provided EQUATIONOF STATEFOR SEAWATER
v = (Z
+ wVw)/
= • siVi+ SwVw are defined as the density p and velocity V of seawater.The concentrationssi of solutesand Swof water are relatedby
•si+Sw=
1
(12)
The individual equations can be rewritten relative to the mean velocity V as
c•pi • + VpiV = -Vp(Vi at
- V) = -VFi
(13)
Natural variability in the compositionof seawaterresultsin fluctuations in the relationship between density and conductivity that exceedthe resolution of measurement.The density of a sampleof natural seawatercannot be specifiedby salinity
alone to better than about 50 x 10-3 kg/m3 unlesscorrections for compositionare made. Brewer and Bradshaw[1975] estimatedthat the effectsof increasedalkalinity, total carbon dioxide, and silica content on the conductivity-density relationship lead to an underestimateof density from conduc-
tivity salinityaloneby about 12 x 10-3 kg/m3 for the deep North Pacific waters. Such a correction to density is equivalent to the differenceof potential densityover 500 m vertically.
Millero et al. [1978] found differences of 17.6 x 10-3 kg/m3 between measured
where
F i = pi(Vi-
V)-
psi(Vi - V)
(14)
representsthe flux or diffusion of the ith solute relative to the total massflux. The driving force for diffusion of the ith component is assumedto be the gradient of its partial chemical potential Vbti.If the flux is denotedby J• = psiV•, the product Ji' V/-ti representsthe rate of energy conversionfor the ith
densities and densities calculated
from con-
ductivity salinity at depthsbelow 1000 m for the North Pacific. Poissonet al. [1981] reported differencesof measured and calculated densities for surface waters ranging from
-6 x 10-3 kg/m3 in the western Indian Ocean to -35 x 10-3 kg/m3 in the Red Sea. A preciseequationof state, reproducibleto better than 10 x 10-3 kg/m3,requiresnot only a precisedefinitionof salinity but also of water itself. Both chemical and isotopic vari-
component associated withtheflux.Thesum(y',Ji'Vl,ti -3-Jw' ations in natural waters and seawatersaffect density at this Vbtw) represents a rate of energyconversion D. As onlythe relative or diffusion fluxes contribute, the conversion can be
interpreted as a dissipationrate of energy D or production of entropy D/T, where T is absolute temperature. Using the Gibbs-Duhem equation [Katchalsky and Curran, 1967] and neglectingthermal effects,the dissipationrate for fluxes F•, relative to the massflux p¾, can be put in the form
D = -- 5',Ji'V].li- Jw'VPw = - Z F,. V(#,-
(iS)
For diffusivefluxesthe entropy production on dissipationis a
level of precision. JPOTS had first to define a salinity scale and to choose a reference pure water of specified isotopic composition before considering candidate formulas for the densityand specificvolume. Proceduresfor definingthe practical salinity scaleand the equation of state were set out in the eighth report [UNESCO, 1978] of the panel. Five data sets were identified as the basis for developing the equation of state, using the general form adopted by Chen and Millero [1976], similar to the earlier equation used by Ekman [1908]. Ekman incorporated the effects of pressurein a mean compressioncoefficient,whereasChen and Millero fitted the mean
FOFONOFF: PRACTICE OF OCEANOGRAPHY
TABLE 3. Coefficientsfor the International Equation of State for Seawater
A
to t• t2 t3 t4 t5
to t• t2 t3 t4
EOS
B
+999.842594' + 6.793952E-2 --9.095290E-3 4-1.001685E-4 - 1.120083E-6 + 6.536332E-9
+8.24493E-1 -4.0899 E-3 + 7.6438 E-5 - 8.2467 E-7 + 5.3875 E-9
80
C
D
- 5.72466E-3 + 1.0227 E-4 - 1.6546 E-6
+4.8314E-4
G
19652.21' + 148.4206 -2.327105 + 1.360477E-2
+54.6746 -0.603459 + 1.09987E-2 -6.1670 E-5
+ 7.944 E-2 + 1.6483E-2
ographic work, provided the same referenceis used consistently. Density differences between seawater and SMOW were computedfrom the data of Millero et al. [1976] and Poisson
et al. [1980]. Polynomialsin temperatureand salinity (half powers)were fitted to the differencesto yield a 1-atmosphere equationwith a standarderror (rmsresiduals)of 3.6 x 10-3 kg/m3 [Millero andPoisson, 1981]. The pressuredependencewas developedas a mean bulk modulus for pure water K•, basedon Kell's [1975] and isothermalcompressibilities near 1 atmosphere from soundspeed
- 5.3009E-4
- 5.155288E-5
to t• t2
+ 3.239908*
+ 2.2838E-3
+ 1.43713E-3 4-1.16092E-4
- 1.0981E-5 - 1.6078E-6
t3
- 5.77905E-7 M
J + 1.91075E-4
measurements of Del Grosso [1970] and Del Grosso and Mader [1972]. The bulk modulus for seawaterwas determined from sound speed measurementsof Millero and Kubinski [1975]. At higher pressuresthe differencesfrom pure water, V(S, t, p)- V(O, t, p), were calculatedfrom measurementsby
N
+ 8.50935E-5' -6.12293E-6 5.2787 E-8
definitionof the liter in 1964.The polynomialfitted to Bigg's valueswasscaledby a constantfactorto adjustthe maximum densityfrom 999.972to 999.975kg/m3 for SMOW. Menache [UNESCO, 1976a,Annex 5] estimatedthat the uncertaintyin the maximum density of SMOW did not exceed3 x 10-3
kg/m3.The uncertainty is of littleconsequence in mostocean-
F
I
Chappuis[1907] and Thiesenet al. [1900] on purewater.Bigg adjusted the earlier values to take into account the re-
E
H
3337
-9.9348E-7 + 2.0816E-8 + 9.1697E-10
Chenand Millero [1977] and BradshawandSchleicher[1976]. The differenceswere then fitted by a polynomial in salinity (half powers),temperature,and pressure[Millero et al., 1980]. The overall standarddeviationfor V(S, t, p) is 9.0 x 10-9
V(S, t, p)= V(S, t, 0)[1 - p/K(S, t, p)]
p(S, t, 0):
I/V(S, t, 0) -- ,4 4- BS 4- CS3/24- DS2 m3/kg.The finalequationof statewasadoptedby JPOTSin K(S, t, p) = E + FS + GS3/2 1981 [UNESCO, 1981a]. Coefficientsfor the completeequa+ (H + IS + JS3/2)P tion are given in Table 3. + (M + NS)p2 DENSITY AND STERIC ANOMALIES where the coefficientsare polynomialsin temperature t. Units: salinity, PSS78; temperature, øC;pressure, bars;density,kg/m3;specifFor most oceanographicusagethe full value of densityor ic volume,m specificvolume is unnecessary.Becausethe maximum vari*Coefficientsmodifiedfor calculationof stericanomaly (Table 4).
ation in magnitudeover the oceanicrangeof salinity,temperature, and pressureis only 7%, a considerableimprovementin or secant bulk modulus, the reciprocal of mean compression, numericalresolutionis achievedby usinganomaliesof specific to the data. volume and density.The specificvolume (steric)anomaly The data sets consist of measurementsmade with a magnetic float densimeterof the densitydifferencebetweenseawater and pure water at atmospheric pressure [Millero et al., 1976] and at elevated pressures [Chen and Millero, 1976]; measurementsof compressionand thermal expansionby Bradshaw and $chleicher [1970, 1976]; and compressibilitycoefficients estimated from sound speedmeasurementsof Chen et al. [1977] and Chen and Millero [1978]. A sixth data set was added [UNESCO, 1979; Poissonet al., 1980] to improve the accuracy of the equation at atmospheric pressureat salinity of 35 and temperatures above 25øC. The revised formula was completed by Millero and Poisson [1981] and recommended by JPOTS in the tenth report [UNESCO, 1981a]. Densities
of seawater
are not determined
relative
to abso-
lute standards of mass and length but rather as differences from a definedpure water standard of known isotopic composition. Crai•t [1961] described a standard mean ocean water (SMOW) with isotopic composition referred to a National Bureau of Standards referencesample. Menache [Girard and Menache, 1972; UNESCO, 1974] proposed that SMOW be used as the density referencewith a provisional value of the
maximumdensityof 999.975kg/m3, and the densityvariation with temperature given by a fifth-order polynomial fitted to Bi•tg's [1967] table of pure water densities.Bigg's table was computed as a weighted mean of measurementsmade by
rS(m3/kg) anddensityanomaly7(kg/m 3)aredefinedby (5(S,t, p)= V(S, t, p)-
V(35, O,p)
7(S,t, p)- p(S,t, p)- 1000.0kg/m3
(17)
where V(35, 0, p) is given by a separateformula (Table 4). Fofonoffand Millard [1984] computedcoefficientsof the difference formulas needed to retain numerical precision in TABLE 4. Coefficientsfor V(35, 0, p) and K(35, 0, p) Coefficient
V(35, 0, p) = V(35, 0, 0)[1 - p/K(35, O,p)]
K(35, 0, p) = E35+ H35p 4- M35P2 AK(S, t, p)= K(S, t, p)-
K(35, 0, p)
= AE354- lkH35P+ AM35P 2 E35 = 4-21582.27 H35 -- 4-3.359406 M35 -- 4- 5.03217E-5
AE35 = - 1930.06 AH35 = -0.1194975 AM35 = + 3.47718E-5
V(35,0, 0)= 972.662039x 10-6 m3/kg p(35,0, 0)= 1028.106331 kg/m3 p(35,0, 0) - p(0,0, 0) -- + 28.263737kg/m3 Units'pressure, bars.
3338
FOFONOFF: PRACTICE OF OCEANOGRAPHY
TABLE 5. Steric Anomaly and Specific Volume Differences BetweenKnudsen-Ekmanand EOS 80 Equationsof State for Seawater
AV,
Ag,
x 10-9 m3/kg
x 10-9 m3/kg
S/t
OøC
34 35 36
- 18.9
- 1.6 0.0 1.4
-43.4
- 1.6 0.0 1.4
-41.6
- 1.6 0.0 1.5
5øC
10øC
15øC
20øC
-0.3 1.2 2.4
0.7 2.5 4.0
0.1 2.0 3.5
4.3 5.9 7.3
6.8 8.6 10.1
6.4 8.2 9.8
3.4 5.1 6.5
5.4 7.1 8.7
5.1 6.9 8.5
variation,thedifferences are2 x 10-9 m3/kgor lessandaffect
p = 0 dbar -2.3 - 1.0 0.1
p34 35 36
2000 dbar -0.3 1.2 2.4
p = 4000 dbar 34 35 36
a nearly constant amount. The difference must be taken into accountfor comparisonof propertieson potential density surfacesand similar graphicalor numericaldisplaysusingat as a variable. Steric anomalies computed from the KnudsenEkman equation of state [Fofonoff, 1962] differ by less than the precisionof measurementfrom EOS 80 values.A comparison is given in Table 5. Over most of the open ocean range of
-0.9 0.7 2.0
dynamic height or geopotential anomaly calculations by about 1 dynamic centimeter (0.1 J/kg) or less. As errors of measurement and random variations produced by internal wavesare greater,the new salinity scaleand equation of state do not significantlyaffect dynamic computationsin agreement with the assessment of Reid [1958].
SPECIFIC HEATC•, Specific heatCpis definedasthe heatin Joulesrequiredto raise the temperature of 1 kg of seawater IøC at constant pressure.
Millero et al. [1973] measuredCp for standardseawater, /xV = V•E- VEoss0
A6 = C•KE -- 6EO sSO
evaluating 6 and 7. Given the two anomalies,the values of specificvolume V and density p are recoverableto higher precision than from the full formula for a typical singleprecisioncomputation using a 32-bit computer word length. The increased numerical resolution is particularly important
in computationof static stability where small vertical gradients of density must be resolved. A significantdeparture from historical usageis taken in the
diluted with pure water or concentratedby evaporation,over a chlorinity range of 0-229/00and a temperaturerange of 535øC.The measurementsagreedwith earlier determinationsto within 2 J/(kgøC),except at low temperatures,where differencesfrom values of Cox and Smith [1959] were as high as 6 J/(kgøC). The formulasgiven by Millero et al. [1973] were converted to salinity by usingthe factor 1.80655consistentwith PSS 78 [UNESCO, 1981c,p. 118] for the 1-atmosphereterms. No direct measurementsof specificheat at elevatedpressure are available for seawater.The pressuredependenceis computed by integrationof the thermodynamicrelationship
definition of densityanomaly 7. Previously,a specificgravity anomaly at [Knudsen,1901], definedas at-- 1000 (p(S, t, O)/p(O,tmax, 0)- 1)
t•C• -r c•p
(19)
Using the acceptedvalue of p(0, t.... 0) = 999.975kg/m3 [UNESCO, 1974], (19) becomes
7(S,t, 0)= 0.999975a,- 0.025kg/m3
63t 2
(20)
C•,(S, t,p)= C•(S, t,O)-
•c• Io •(t+ 273.15) 2Vdp
Inm--
;0
pm/Kndp
TABLE 6. Precisionof Data for SpecificHeat of Seawater
Chlorinity
0
t
5
Interval
Maximum
Number of Values
C(S, t, O)from Millero et al. [1973] variable 22%0 13 10
35øC
Total
52
4
Standard Deviation
0.5 J/(kgøC) at 5øC 0.2 J/(kgøC)at 15-35øC
AC• (from EOS 80) t
p
-2
2
50
50
42øC
1000 bars
23
20
460
0.074 J/(kgøC)
720
0.062 J/(kgøC)
AC2 (from EOS 80) S t
5 0
5 5
p
100
100
40 40øC
8 9
1000 bars
10
Cp(S,t, p) = Cp(S,t, 0) + AC•(0,t, p) + AC2(S , t, p)
(22)
Using V = V0(1- p/K), the derivative can be expressedin termsof V0,K, and derivativeswith respectto temperature.As pressureentersonly in termsof the form pm/K",wherernand n are integers,the entire integral can be evaluatedexplicitly in termsof quadratictrinomial integrationsof the form
The new definition of density anomaly yields valuesof 3' that are numericallylower (the dimensionsare different)than at by
Minimum
(21)
(18)
was used,where tma x is the temperatureof the densitymaximum of pure water at atmospheric pressure.The density anomaly 3,,correspondingto at, is
3,(S,t, O)= p(O,t.... 0)(1+ at/1000)- 1000.0kg/m3
c•2V
(23)
FOFONOFF' PRACTICE OF OCEANOGRAPHY
TABLE 7. Specific heatof Seawater Cp(S,t, p) in J/(kgøC) A
B
4127.4
C
-3.720283
- 7.643575 + 0.1072763
+ 0.1770383 -4.07718E-3
+0.1412855
- 1.38385E-3
- 5.148
placement, is required for computation of static stability and comparisonof water typesin the ocean.From thermodynamic considerationsthe lapse rate is given by 3V
rfs,t,p)=(T/Cp) c3t
E-5
- 2.654387E-3
3339
(25)
and can be computed from the equation of state and specific
+ 2.093236E-5
heatCp. to t• t2 t3 t4
to t• t2 t3 t4
D
E
-4.9592 E-1 + 1.45747E-2 -- 3.13885E-4 +2.0357 E-6 + 1.7168 E-8
+4.9247 E-3 - 1.28315E-4 + 9.802 E-7 +2.5941 E-8 -2.9179 E-10
G
H
+ 2.4931 E-4 - 1.08645E-5 + 2.87533E-7 -4.0027 E-9 + 2.2956 E-11
to t• t2 t3
-2.9558 E-6 + 1.17054E-7 -2.3905 E-9 + 1.8448 E-11
F - 1.2331E-4 - 1.517 E-6
+ 3.122 E-8
I
+9.971E-8
J
K
M
-5.422 E-8 + 2.6380E-9 -6.5637E-11 +6.136 E-13
+5.540 E-10 - 1.7682E-11 + 3.513 E-13
- 1.4300E-12
Bryden [1973] developed a formula for F from Bradshaw and $chleicher's[1970] measurementsof thermal expansionof seawater.
As these data
were a subset of their
A
t3 t4 t5
-4.635 - 1.2660
-7.244
+ (d + KS + MS3/2)p 3
E-7 E-8
to
precision to serve asanindependent check ondirectcompu- t• t2 tationbyfinitedifferencing andnumerical integration of(22). t3 Tablesof specific heatvaluesweregenerated from the exact
formulas and polynomial expressionsfitted to the tables in two steps.The temperatureand pressuredependenceat $ -- 0
wasevaluatedby fittingto tablesof AC• = C,(0, t, p)- C,(0, t, 0).
t4
ts
to t2 t3
AC2= [C,(S, t, p)- C,(S,t, 0)] - It,(0, t, p)- C,(0, t, 0)]
F
E-2 E-4 E-5 E-7 E-9 E-11
t4
so that
to
-4.9799 + 2.0840 -3.5905 +6.4237
E-6 E-7 E-9 E-11
- 5.9233
E- 13
E-9
Standard deviations ofresiduals fromthefit aregivenin Table t2
+4.8665
E-11
- 1.1803
E-12
6. The polynomial fits for AC• and AC2 together with the
+ 1.3351 E-14
polynomials for C,(S,t, 0) givenby Milleroet al. [1973]provide specificheat valuesover the full rangeof salinitytemperature and pressure.The coefficients are givenin Table 7. The polynomialtermsgiving the pressuredependencewere fitted with higherprecisionso that the 1-atmosphere portion can be replacedwithout havingto refit the pressuretermscomputed
fromEOS 80.Tablesof C, aregivenby FofonoffandMillard [1984]. LAPSE RATE F
The adiabatic lapserate F(S, t, p), definedas the changeof temperatureper unit pressurechange for an adiabatic dis-
- 2.926
E-5
+3.965 -2.251 +4.143 -4.234
H
+2.9171E-7 -4.5887E-9
+ 8.3866E-9 -9.8194E-11
K
E-8 E-9 E-11 E-13
- 1.9155E-9
+ 6.5229E-11
M
- 1.1437
t3
+ 4.4614E-6 - 3.7257E-7
J
L
(24)
G
-2.027 E-4 + 9.5596E-6 -2.283 E-7 + 4.133 E-9 - 5.392 E-11 +3.717 E-13
I
The salinity dependencewas obtained by fitting additional terms to a table of
+2.4624 - 9.5979 + 1.8232 - 3.0623 + 3.5496 -2.3005
-6.298 E-3 + 1.3742E-4 +2.4153E-6 - 1.5540E-8
D
E-9
E
[Gradsteynand Ryzhik, 1965]. In computingthe pressuredependencethe expressions were evaluatedexplicitlyin double
C
+0.264117 - 8.4901E-3 +9.5714E-5 - 3.483 E-7
to t• t2
Coefficientsare polynomialsin temperaturet. Units' salinity,PSS 78' temperature,øC; pressure,bars.
B
-4.4017 + 1.159313 - 1.42752E-2 + 1.94206E-4
+ (D+ ES+ FS3/2)p + (G+ HS+ IS3/2)p 2
ADIABATIC
ex-
TABLE 8. Adiabatic Lapse Rate FEosSo
Cp(S,t, p)= A + BS+ CS3/2
C.(S.t. p)= c.(s. t. 0) + Ac•(0. t. p) + AC2(S.t. p)
thermal
pansion data for EOS 80, the agreementwith the new equation can be expected to be good. Caldwell and œide [1980] estimated adiabatic lapse rates in seawater by measuring directly the temperature change produced by rapid changesof pressure.The two methods agree within the estimated uncertainties. However, the accuraciesof the two techniquesare not sufficientto permit a critical comparison. For comparison with the Bryden [1973] and Caldwell and Eide [1980] formulas, tables of F were computed from the thermodynamical equations by using EOS 80 and fitted by standard least squaresregressiontechniqueswith polynomials in S, t, and p. Sufficient terms were included to reduce the
+ 8.0102E-12
N
to
+ 3.5773 E-13
t;
-9.0147
E-15
F(S, t, p) = A + BS + CS3/2+ DS2 + (E + FS + G$3/2+ HS2)p + (I + dS + KS3/2)p 2 + (L + MS)p3 + (N)p• Units: salinity, PSS 78; temperature, øC; pressure, bars; F, 10-4øC/bar.
3340
FOFONOFF: PRACTICE OF OCEANOGRAPHY
TABLE 9.
Comparison of Lapse Rate F at S = 35, t = 0øC
Pressure, bars
FBR-- FEOS 80, øC/bar
0
7.8 x 10 -6
200 400 600 800 1000
4.2 - 0.5 - 5.0 -8.7 - 11.7
gration schemeto evaluate the implicit integral in (26). He showed that the potential temperature is given to a numerical
FCE-- FEOS 80, øC/bar
precisionof 0.1 x 10-3øCfor the maximumintegrationstepof
28.5 x 10 -6
Thus the integration can be carried out efficientlyin a single step, making it unnecessaryto develop separateformulas for potential temperature. A comparison of potential temperatures computed with the Runge-Kutta algorithm by using Bryden's[1973] and Caldwell and Eide's [1980] formulas is
23.5 15.7 8.6 4.8 6.2
10,000 decibars.
made in Table 9.
BR - Bryden[1973]' CE - Caldwelland Eide [1980].
Table 10 contains differencesfrom potential temperatures computed from EOS 80 by using the thermodynamical relationships. Agreement at 10øC is better for the direct fittingerrorsto about0.1 x 10-7øC/dbar.The coefficients are measurements of Caldwelland Eide [1980]. At 0øC,agreement givenin Table 8. Brydenestimatedhis fitting error to be about is better with values obtained from Bryden's [1973] formula. 3.4 x 10- 7øC/dbarandthe uncertainty of theestimateof F to As the differencesbetweenthe Bryden formula and EOS 80 in rangefrom 27 to 53 x 10-vøC/dbaror about2% of F. Caldthe oceanicrange are comparableto the precisionof temperwell and Eide obtaineda fitting error of 5.5 x 10-vøC/dbar ature measurementsmade at sea, no real improvement in accorrespondingroughly to their estimateduncertainty.A comcuracyis gainedby introducinga more preciseformula for F. parison with I'EOS8 o at S = 35 is made in Table 9. The differAs a consequence,Fofonoffand Millard [1984] recommended ences are within the rather conservative limits given by using the simpler Bryden formula for F for computation of Bryden but large compared with the fitting errors. Bryden's potential temperature. valuesare closer to FEOS8O, as expected,but show a similar The changeof salinity scaleto PSS 78 producesnegligible trend with pressureas the Caldwell and Eide residuals. changesin F or potential temperature 0. The maximum differBecausethe differencesin lapseratesyield adiabatictemperencein 0 was estimatedto be 0.3 x 10-3øC, usingBryden's ature changesthat are within the precision of temperature [1973] F for Po = 10,000dbar, p, = 0 dbar.
measurements (2-5 x 10-3øC),Fofonoffand Millard [1984]
selectedthe Bryden [1973] formula for computinglapserates and potential temperatures.The introduction of a new formula would yield insignificantimprovementin accuracyof potential temperaturesand may be in need of further revisionin the near future if the differencesbetween the directly measured lapse rates of Caldwell and Eide [1980] and the computed
BRUNT-V•IS•L•
An element of seawater displaced adiabatically and isentropically to z from its equilibriumlevel z0 in a stratifiedlayer will have a densitydifferenceAp from its surroundingsof Ap = p(So,O(So,to, Po,P), P)- p(S, t, p)
rates are confirmed. POTENTIAL
TEMPERATURE 0
Potential temperatureO(S,t, p, p,) is definedto be the tem-
(27)
and will be acted upon by a restoring buoyancy force F per unit mass,givenby • = --(Ap/p)•
perature an element of seawater would have if moved adia-
batically and with no changeof salinity from an initial pressure Po to a referencepressurep, that may be greater or less than Po.This definition is more general than the classicaldefinition given by Helland-Hansen[1912] and requires an explicit choiceof the referencepressurep,. However,it simplifies the computation of static stability, Brunt-V/•is/•l/•frequency, and potential density or steric anomaly for comparison of
FREQUENCY
(28)
where g is gravity. The mass element will be subjectedto a vertical acceleration 5',where
= + r =
= --N2(z -- Zo)= _N2Az
(29)
and, usingthe hydrostaticrelationshipAp = -pgAz,
water types.
/2=
Under this definitionthe potentialtemperatureis givenby O(So,to, Po,P,) = to +
O(So,to, Po,P), P) dp
(26)
where F is the adiabatic lapserate. Fofonoff [1977] adopted a fourth-order Runge-Kutta inte-
+(g/p)Ap/Az=-g2Ap/Ap
(30)
The Brunt-V•iis/il/i frequencyN is a measure of the stratification and the high-frequencylimit for internal wavesthat can be supportedby the verticalpotential densitygradient.
The dependence of N 2 on thermalexpansionand saline contraction coefficientsis obtained by expanding (27) in a
TABLE 10. Potential Temperature Differences 0--0EOS80 for S= 35 and Reference Pressure p, = 0 bar OøC
10øC
Pressure,
bars
CE, x 10-3øC
BR, x 10-3øC
CE, x 10-3øC
BR, x 10-3øC
200 400 600 800 1000
-5.3 - 9.2 - 11.8 - 13.4 - 15.1
- 1.3 - 1.8 - 1.6 -0.7 + 0.5
- 1.7 - 2.0 - 1.1 +0.1 + 0.8
1.0 2.0 3.1 4.3 5.5
BR - Bryden [1973]' CE - Caldwell and Eide [1980].
FOFONOFF' PRACTICE OF OCEANOGRAPHY
Taylor seriesabout P0 as follows 8p
8p
p(S o,O,p)= p(S o,to,P0)+ •-/A0+•pp Ap+0(Ap 2) (31) 8p
8p
8p
p(S, t,p)=p(S o,to,Po) + • AS+•- At+•pp Ap+0(Ap 2) so that
Brown, N. L., and B. Allentort, Salinity, conductivity and temperature relation of seawater over the range 0 to 50 ppt, report, 60 pp., BissetBerman Corp., San Diego, Califi, 1966. Bryden, H. L., New polynomials for thermal expansion,adiabatic temperature gradient and potential temperature of seawater, Deep Sea Res., 20, 401-408, 1973.
Caldwell, D. R., and S. A. Eide, Adiabatic temperaturegradient and potential temperature correction in pure and saline water: An experimental determination, Deep-SeaRes.,27A, 71-78, 1980. Caldwell, D. R., and S. A. Eide, Soret coefficientand isothermal diffusivity of aqueous solutions of five principal salt constituents of seawater, Deep Sea Res., 28A, 1605-1618, 1981. Chappuis, P., Dilation de l'eau, Tray. Mem. Bur. Int. Poids Mes., 13,
Ap
----
334i
•--•pp+••pp-F +0(Ap)
(32)
The compressibilityterms cancelexactly.
For computationalconvenience the expression for N 2 is convertedto usespecificvolume anomaly, i.e.,
N 2 = --g2Ap/Ap- +p2g2AV/Ap= p2g2AS/Ap = p29218(So, O(So, to,Po,P),P)- 8(S,t, p)]/Ap
(33)
D 1-40, 1907.
Chen, C.-T., and F. J. Millero, The specificvolume of seawaterat high pressures,Deep Sea Res.,23, 595-612, 1976. Chen, C.-T., and F. J. Millero, Speed of sound in seawater at high pressures,J. Acoust.Soc. Am., 62, 1129-1135, 1977. Chen, C.-T., and F. J. Millero, The equation of state of seawater determinedfrom soundspeeds,J. Mar. Res.,36, 657-691, i978. Chen, C.-T., R. A. Fine, and F. J. Millero, The equation of state of pure water determined from sound speeds,J. Chem. Phys., 66, 2142-2144, 1977.
Cox, R. A., Temperature compensation in salinometers, Deep Sea
The use of potential steric anomaly 8 to compute N provides the maximum numerical resolution and reduces the computation to two evaluations of steric anomaly. Fofonoff and Millard [1984] give algorithms for computing p, l/, 8. The procedure can be elaborated by referencinga series of observations over a suitable pressureinterval to a single pressureand fitting a least squarescurve to the potential steric anomalies. Such a method [Bray and Fofonoff, 1981] has the advantage
of verticalaveragingto improvetheestimateof N 2. DISCUSSION
The need to establishstandardsfor measurementof salinity
Res., 9, 504-506, 1962.
Cox, R. A., and N. D. Smith, The specificheat of seawater, Proc. R. Soc. London, Set. A, 252, 51-62, 1959.
Cox, R. A., F. Culkin, R. Greenhalgh,and J.P. Riley, Chlorinity, conductivityand densityof seawater,Nature, 193, 518-520, 1962. Cox, R. A., F. Culkin, and J.P. Riley, The electrical conductivity/chlorinityrelationshipin natural seawater,Deep Sea Res.,14, 203-220, 1967.
Craig, H., Standard for reporting concentrationsof deuterium and oxygen-18in natural waters,Science,133, 1833-1834, 1961. Culkin, F., The major constituentsof seawater,in Chemical Oceanography,vol. 1, edited by J.P. Riley and G. Skirrow, pp. 121-161, Academic, New York, 1965.
Culkin, F., and R. A. Cox, Sodium, potassium,magnesium,calcium and strontiumin seawater,DeepSea Res.,13, 789-804, 1966. Culkin, F., and N. D. Smith, Determination of the concentration of potassiumchloridesolutionhavingthe sameelectricalconductivity, at 15øC and infinite frequency,as standard seawater of salinity 35.0000%0(chlorinity 19.37394%o),IEEE d. Oceanic Eng., OE-5,
with a precisionmatchingthe sensitivityof present-dayconductivity bridgeshas forced abandonmentof scalesbasedon natural seawaters in favor of a more rigidly and precisely definedpractical scalereferencedto a conductivitystandard. The practicalscalewill continueto be useduntil measurement 22-23, 1980. techniquesadvance in precision and resolution beyond the Dauphinee,T. M., J. Ancsin, H. P. Klein, and M. J. Phillips, The electricalconductivityof weight diluted and concentratedstandard presentstandards.As salinity is combinedwith temperature seawater as a function of salinity and temperature, IEEE J. Oceanic and pressureto compute density in many applications,the Eng., OE-5, 28-41, 1980a. relationshipbetweenconductivityand densitywill need to be Dauphinee, T. M., J. Ancsin, H. P. Klein, and M. J. Phillips, The exploredfurther to establishthe range of compositionalvarieffectof concentrationand temperatureon the conductivityratio of ations in the world
oceans and to establish base lines for
potassiumchloridesolutionsto standardseawaterof salinity 35%0
monitoringtime-dependentchangesin the relationship.
(C1 19.3740%o), IEEE d. OceanicEng., OE-5, 17-21, 1980b. Del Grosso, V. A., Sound speed in pure water and seawater, J.
Acknowledgment.This article is made possible by the generous supportof the Officeof Naval ResearchundercontractN00014-76-C0197, NR 083-400, and by the interest and motivation provided by membersof JPOTS and SCOR Working Group 51 on evaluation of
Del Grosso, V. A. and C. W. Mader, Speedof sound in pure water, J. Acoust. Soc. Am., 52, 1442-1446, 1972. Dittmar, W., Report on researchesinto the composition of ocean water collectedby HMS Challengerduring the years 1873-76, Phys.
CTD
data. The invitation
to contribute
to a volume dedicated to R.
O. Reid provided an opportunity to review some of the historical developmentof the new salinityscaleand equationof stateof seawater.
Acoust. Soc. Am., 47, 947-949, 1970.
Chem., 1, 1-251, 1984.
Eckart, C., Propertiesof water, 2, The equationof stateof water and seawaterat low temperaturesand pressures,Am. d. Sci., 256, 225240, 1958.
REFERENCES
Bigg, P. H., Density of water in SI units over the range 0-40øC, Brit. d. Appl. Phys., 18, 521-525, 1967. Bradshaw, A. L., and K. E. Schleicher,Direct measurementsof thermal expansionof seawaterunder pressure,Deep Sea Res., 17, 691706, 1970.
Bradshaw, A. L., and K. E. Schleicher,Compressibility of distilled water and seawater,Deep Sea Res.,23, 583-593, 1976. Bradshaw,A. L., and K. E. Schleicher,Electrical conductivity of seawater, IEEE d. OceanicEng., OE-5, 50-62, 1980. Bray, N. A., and N. P. Fofonoff, Available potential energy for MODE eddies,d. Phys.Oceanogr.,11, 30-47, 1981. Brewer, P. G., and A. L. Bradshaw, The effect of the non-ideal com-
positionof seawateron salinityand density,d. Mar. Res.,33, 157175, 1975.
Ekman, V. W., Die Zusammendrueckbarkeitdes Meerwassersnebst einigen Werten ruer Wasser und Quecksilber,Publ. Circ. Cons. Explor. Met., 43, 1-47, 1908. Fofonoff, N. P., Physical propertiesof seawater,in The Sea, vol. 1, editedby M. N. Hill, pp. 3-30, Interscience, New York, 1962. Fofonoff,N. P., Computationof potentialtemperatureof seawaterfor an arbitrary referencepressure,Deep-SeaRes.,24, 489-491, 1977. Fofonoff, N. P., and R. C. Millard, Jr., Algorithmsfor computationof fundamental properties of seawater, UNESCO, Tech. Pap. Mar. Sci.,44, 53 pp., Paris, 1984. Forchhammer, G., On the compositionof seawaterin the different partsof the ocean,Phil. Trans.R. Soc.London,155,203-262, 1865. Girard, G., and M. Menache, Sur le calcul de la massevolumniquede l'eau, C. R. Acad. Sci. Paris Set. B, 274, 377-379, 1972. Gradsteyn, I. S., and I. M. Ryzhik, Tables of Integrals, Series and Products,1086 pp., Academic,New York, 1965.
3342
FOFONOFF' PRACTICE OFOCEANOGRAPHY
Helland-Hansen, B.,Theoceanwaters,Int. Rev.Ges.Hydrobiol. Hydro•lr.,Suppl.3, 1-84, 1912.
Poisson, A., The concentration of theKC1 solutionwhoseconductivityis thatof standard seawater (35%o) at 15øC,IEEE J. Oceanic
IAPSO,SUNReport, Reportof IAPSOworking grouponsymbols, En•t.,OE-5, 24-28, 1980a. A.,Conductivity/salinity/temperature relationship of diluted unitsandnomenclature in physical oceanography, Publ.Sci.,31,56 Poisson, and concentratedstandardseawater,IEEE J. OceanicEnd7.,OE-5,
pp.,Int. Assoc. Phys.Sci.Oceans, Paris,1979.
41-50, 1980b.
Katchalsky, A.,andP. F. Curran,Nonequilibrium Thermodynamics in A., C. Brunet,andJ. C. Brun-Cottan, Densityof standard Biophysics, 248pp.,HarvardUniversity Press, Cambridge, Mass., Poisson, seawatersolutionsat atmospheric pressure, DeepSea Res.,27A, 1967.
1013-1028, 1980. Kell,G. S., Density, thermalexpansibility andcompressibility of A.,J. Lebel,andC. Brunet,Thedensities of western Indian liquidwaterfrom0 to 100øC; corrections andtables foratmospher-Poisson, surfacewaters,Deep ic pressure andsaturation reviewed andexpressed on the 1968 Ocean,Red Seaand easternMediterranean
SeaRes.,28A, 1161-1172, 1981. temperature scale, d. Chem. En•t.Data,20,97-105,1975. of someerrorsin theequation of stateor in Knudsen, M., Hydro•7raphic Tables, 63 pp.,G. E. C. Gad,Copenha- Reid,R. O., Influence
observations on geostrophic currents,Physicaland chemical
gen, 1901.
properties of seawater, Publ.600,pp.10-29,Nat.Acad.Sci.,Nat. Knudsen, M., C. Forch,andS. P. L. S6rensen, Berichtfiberdie Res.Counc., Wash., D.C., 1958. chemische undphysikalische Untersuchung desSeewassers unddie M., K. Scheel, andH. Diesselhorst, Untersuchungen uberdie Aufstellung derneuen Hydrographischen Tabellen, BandNF 6, pp. Thiesen, thermische Ausdehnung vonfestenundtropfbarflusaigen Korpern, 125-184, Wiss.Meeresunters., Komm.Unter.deutsch. Meere, Kiel, Wiss.Abh.Phys.Reichsanst., 3, 1-70, 1900. 1902. Lewis,E. L., andR. G. Perkin,The practical salinityscale1978; UNESCO,SecondReportof the JointPanelon Oceanographic Tablesand Standards,Tech.Pap. Mar. Sci.,4, 9 pp., UNESCO,
Conversion ofexisting data,DeepSeaRes.,28A,307-328,1981.
Paris, 1966.
Mantyla, A.W.,Electrical conductivity comparisons ofStandard Sea-
UNESCO,SixthReportof theJointPanelon Oceanographic Tables andStandards, Tech.Pap.Mar. Sci.,16,27 pp.,UNESCO,Paris, Marcet,A.,Onthespecific gravity, andtemperature of seawaters, in 1974. different partsof theocean, andin particular seas; withsomeaccount of their salinecontents,Phil. Trans.R. Soc.London,109, UNESCO,SeventhReportof the JointPanelon Oceanographic waterbatches P29to P84,DeepSeaRes.,27A,837-846,1980.
Tablesand Standards,Tech.Pap.Mar. Sci.,24, 61 pp., UNESCO,
161-208, 1819.
Paris, 1976a.
Millero,F. J.,G. PerronandJ.F. Desnoyers, Heatcapacity ofseawa-
Reports of theJointPanelon Oceanographic ter solutions from5øto 35øCand0.5 to 22%ochlorinity,J. Geophys. UNESCO,Collected Tablesand Standards,1964-1969,Tech.Pap.Mar. Sci.,27, 126pp., Res.,78, 4499-4506, 1973.
UNESCO, Paris, 1976b. Millero,F. J.,andT. Kubinski, Thespeed of soundin seawater asa EighthReportof the JointPanelon Oceanographic function of temperature andsalinityat oneatmosphere, J. Acoust. UNESCO, Tables and Standards, Tech.Pap.Mar. Sci.,28, 35 pp.,UNESCO, Soc.Am., 57, 312-319, 1975. Millero,F. J.,A Gonzalez, andG. K. Ward,Thedensity of seawater Paris, 1978. NinthReportoftheJointPanelonOceanographic Tables solutions at oneatmosphere asa function oftemperature andsalin- UNESCO, andStandards, Tech.Pap.Mar. Sci.,30, 32 pp.,UNESCO,Paris, ity, J. Mar. Res.,34, 61-93, 1976.
Millero,F. J.,P. Chetirkin, andF. Culkin,Therelative conductivity 1979. Tables anddensity of standard seawaters, DeepSeaRes.,24, 315-321, UNESCO,TenthReportoftheJointPanelonOceanographic andStandards, Tech.Pap.Mar. Sci.,36, 25 pp.,UNESCO,Paris,
1977.
Millero,F. J.,D. Forsht,D. Means,J. Gieskes, andK. E. Kenyon, 1981a. papers andsupporting dataon thepractical Thedensity of NorthPacific Oceanwaters, J. Geophys. Res.,83, UNESCO,Background salinityscale1978,Tech.Pap.Mar. Sci.,37, 144pp.,UNESCO, 2359-2364, 1978. Paris, 1981b. Millero,F. J.,C.-T.Chen,A. Bradshaw, andK. Schleicher, A new Background papersandsupporting data on the internahigh-pressure equation of stateforseawater, DeepSeaRes., 27A, UNESCO, tionalequation of stateof seawater 1980,Tech.Pap.Mar. Sci.,38, 255-264, 1980.
Millero, F. J.,andA.Poisson, International one-atmosphere equation 192pp.,UNESCO, Paris,1981c. of stateof seawater, DeepSeaRes.,28A,625-629,1981.
Wooster, W. S.,A. J. Lee,andG. Dietrich,Redefinition of salinity,
NAS-NRC, Physical andChemical Properties of Seawater, Conferencereport,Publ.600,202pp.,Nat.Acad.Sci.,Nat.Res.Counc., Wash., D.C., 1958.
NIO-UNESCO,International Oceanographic Tables,118pp.,Nat.
DeepSeaRes.,16, 321-322,1969.
N. P. Fofonoff,Departmentof PhysicalOceanography, Woods HoleOceanographic Institution, WoodsHole,MA 02543.
Inst.Oceanogr. Gt. Brit.,UNESCO,Paris,1966.
Park,K.,Reliability ofstandard seawater asa conductivity standard, DeepSeaRes.,11,85-87, 1964.
Perkin,R. G., andE. L. Lewis,Thepractical salinity scale1978: Fittingthedata,IEEEJ. Oceanic En•t., OE-5,9-16,1980.
(Received October16,1984; accepted October27, 1984.)
