On the thermal expansion of sea water

Deep-Sea Research. 1970. Vol. 17. pp. 637 to 640. Pergamon Press. Printed in Great Britain.

On the thermal expansion of sea water*+ WILTON S*URGES~:

(Received 30 July 1969) Abstract--The thermal expansion of sea water as computed from the presently-accepted equation of state is compared with values based on the recent experimental results of Wilson and Bradley. The new results suggest that the formula in standard use gives values which may be slightly too large in the deep ocean, but the discrepancies are barely within the accuracy of the experimental results. INTRODUCTION

OCEANOGRAPHERS have been concerned for many years about the accuracy of the presently-accepted equation of state for sea water. ECKART (1958) showed that Ekman's experimental data, which form much of the basis for present tables of specific volume, are sufficiently accurate to justify the calculation of specific volume only to about one part in 104 or less, so that the common practice of making calculations to one part in 105 (frequently one in 106) is subject to substantial error. The reader is referred to the discussion by FOFONOFF(1962). WILSON and BRADLEY (1968) have made new determinations of the specific volume of sea water. The purpose of the present note is to compare their experimental results with the standard equation of state, to assess the accuracy of present values of the thermal expansion coefficient. Problems caused by an uncertainty in the equation of state are not restricted merely to applications involving absolute values, such as the calculation of total hydrostatic pressure, but may apply to standard hydrographic calculations in deep water. It is well known that geostrophic calculations require accuracy primarily in the coefficients of thermal expansion and saline contraction, rather than directly in specific volume (see REID, 1959). Eckart's results suggest, however, that the thermal expansion coefficient also is seriously in doubt, and that it may be uncertain by as much as 50~% at low temperatures. Where temperature and salinity both decrease with increasing depth, the decrease in specific volume caused by a temperature decrease may be nearly offset by the associated decrease in salinity. As a result, obviously significant temperature and salinity gradients may lead to very small gradients in specific volume. From consideration of the time-averaged ocean circulation well below the main thermocline, it is clear that average horizontal differences in specific volume as small as a few parts in 105, between positions approximately 1000 km apart, are measurable, real, and significantly greater than the noise level. If the equation of state, particularly the thermal expansion coefficient, is as uncertain as Eckart suggests, even the sign of the horizontal gradient in specific volume may be in question over much of the deep ocean. Before discussing the new data, it should be pointed out that previous work has suggested that the situation is not as bad as Eckart suggests. COCnRANE (1959) showed that adiabatic gradients in isolated deep basins approach values computed from Ekman's results. As the likelihood incleases that a water column is adiabatic (i.e., as depth below the sill increases) the agreement between calculated values and the observations becomes amazingly good. With the exception of depths well below 6000 m, for which Ekman made no measurements, one would conclude from Cochrane's results that the thermal expansion coefficient may be slightly too large, but is uncertain by no more than a few per cent. Further support for such a hopeful attitude is given by CREASE'S (1962) calculations of specific volume from measurements of sound speed. From his Fig. lb, one can see that there may be a systematic error in specific volume in the present tables. From the small difference between his curves for 0°C and 4°C, however, one infers that the uncertainty in the thermal expansion coefficient *Contribution of the Graduate School of Oceanography, University of Rhode Island. J'This study was supported in part by the Office of Naval Research. ++University of Rhode Island, Kingston. 637

638

Shorter Contributions

is less than 270 at normal depths, and reaches 5 ~ only at great depths. Similarly optimistic conclusions may be drawn from the results of NEWTON and KENNEDY(1965). IMPLICATIONS OF NEW DATA The present article compares WILSON and BRADLEY'S(1968) experimental values nearest normal sea water with the corresponding values of specific volume computed from the usual formula to determine whether the recent measurements show the presence of significant errors in the thermal expansion coefficient. Specific volume has been computed from the usual formula, as given by FOFONOFF(1962, p. 10, equations 25, 26) for the values of temperature and pressure shown in Wilson and Bradley's Table 5, for salinity 35"568~o. Their pressures were reduced by 1'013 bar to convert to sea pressure. The values of specific volume computed from the standard formula were multiplied by 1'000027 to make the units comparable, in cma/g, the units used by Wilson and Bradley. The complete coefficient of thermal expansion is computed as A~/:tAt. In the present comparison, only the term A0¢is considered, to show the experimental values directly. It is convenient to consider the difference in specific volume at two temperatures, computed from the usual formula, (A~t)f, minus the equivalent difference determined from the data of Wilson and Bradley, (Aa)wn. Figure 1 shows the values (Act)I - (A~)wB. For comparison, 107o of the total change in specific volume at atmospheric pressure is indicated for each interval. (At the greatest pressure these ranges increase by roughly a factor of 2). The new experimental data are said (WILSON and BRADLEY, 1968, p. 358) to have an uncertainty of 2 × 10-L The differences in Fig. 1 show a certain bumpiness of about twice that uncertainty, as expected. The near mirror-image appearance of the two curves near 600-800 bar suggests that the data points at one temperature, at least, may be in error by more than the expected amount. A similar effect is apparent in the 20-15°C and 15-10°C curves at low pressures. The general trend of the comparison indicates that, for temperatures of 10°C and below, the thermal expansion coefficient as computed from the usual formula is slightly too large. This finding is in agreement with the previous results cited above. The disagreement indicated at the highest pressures is scarcely more than twice the reported standard deviation of the data, but suggests that the error may be approximately I0 ~ . At higher temperatures the error may be somewhat greater. The most surprising result of these comparisom, however, concerns the absolute values of specific 30

20

It) I

IC

,t

o r

X

~

% tj

.

~

.

~

~

25-20C

j

~

.

"

I

"--

~'~-~"

/

/ /

-10

-20

i

-~o

0

1 2

t

I I I 4 6 PRESSURE x 102b0r

l

1 8

) lO

Fig. 1. Differences between experimental determinations of the thermal expansion of sea water: values computed from the data of WILSON and BRADLEY(1968, Table 5) are subtracted from those computed from the usual formula (as given by FOrONOFF, 1962, p. 10) for salinity 35"568 per mille. The curve marked 20-15°C refers to the change in specific volume from 20.05°C to 15.12°C, etc. Vertical bars show 10% of the change in specific volume for each temperature interval at atmospheric pressure

Shorter Contributions 40

639

,N. \.\ "

--

00

o0

",.\ . . \\\.~ '% ,o_

--ay

c

\

-Io I

J/--

\ "\,J

i'~2~oc~ 5.4o8c\'\.\./.,,

-20

0

2

4 PRESSURE

6

8

x 102bar

Fig. 2. Differences between absolute values of specific volume of sea water: experimental values reported by WILSON and BRADLEY (1968, Table 5) subtracted from values computed from the usual formula for salinity 35'568 per mille. volume, rather than the difference from one temperature to the next. The data given in Wilson and Bradley's Table 5 have been subtracted from computed values at the temperatures and pressures shown in the table, and the differences are plotted in Fig. 2. The numerical values on which Figs. 1 and 2 are based, at atmospheric pressure only, are shown in Table 1. The differences between the experimental data of Wilson a n d Bradley and values computed from the usual formula are seen to be as great as 40 × 10-5. The greatest differences are found, unexpectedly, at atmospheric pressure, where the usual formula is not thought to be in substantial error. This finding is consistent with the statement by Wilson and Bradley that the standard deviation of the entire data set from the final equation was 13 × 10-5. This large standard deviation is puzzling, however, inasmuch as it is nearly ten times the supposed accuracy of the single observations. Table 1. Specific volume o f sea water at atmospheric pressure, salinity 35'568 per mille, computed from the usual formula and converted to cruZ/g, compared with experimental values o f WILSON and BgADLEY (1968, Table 5). The columns headed As show the thermal expansion from one temperature to the next.

Temp. (°C)

Usual formula (cm3/g)

1'230

0.97231

A~z (cmS/g × 1 0 5 )

Wilson and Bradley (cmS/g)

0" 270

0"

0.

337

0'

15.122

0'

431

0-

0"

545

0.97675

67

0

104

--10

40 40 30 114

0-

0

515

130 24.877

4

401

114 20"050

36 35

297

94

30 137

0.97652

Formula minus W and B

(cmS/g × 105)

230

67 10"145

h~ 1 __ A ~

0'97195 39

5"408

~

--

7 23

640

Shorter Contributions

The comparison given by Wilson and Bradley in their Table 6, by contrast, suggests that their final formula disagrees with the standard formula by no more than 20 x 10 -s, or half that shown in Fig. 2, and furthermore that the agreement at atmospheric pressure, at an interpolated salinity of 35~o, is better than 5 × 10-5. The final formula given by Wilson and Bradley would be expected to suppress the experimental scatter, although perhaps not to so great an extent. Another reason why the comparison shown by Wilson and Bradley looks better than the comparison here is that they failed to convert the values computed from the Hydrographic Tables from ml/g to cm3/g. This may be seen from the disagreement they indicate, in their Table 6, between the atmospheric-pressure values of Crease and those of the Hydrographic Tables. Crease, of course, used the standard values at atmospheric pressure as the starting point for his calculations. One possible reason for the large discrepancy indicated in Fig. 2 may be a salinity error. For values at atmospheric pressure and at 10.1°C and below, Wilson and Bradley's determinations of specific volume can be made to agree with the presently-accepted tables by assuming that the salinity of their sample was 36"08~ 4- 0.04 at the time of measurement. Although such a large salinity error seems improbable, they reported increases in salinity of 0-1%0 from the start to the end of a run. Newton and Kennedy reported changes as great as 0"3~o, but of opposite sign. The large difference in compressibility indicated in Fig. 2 is inconsistent with the findings of Newton and Kennedy, and of Crease. The lower compressibility is ten times the effect expected from the possible error in salinity discussed above. Possible systematic errors in salinity or pressure, however, would have a relatively small effect on the thermal expansion coefficient computed from this data. CONCLUSION

The experimental data of WILSON and BRADLEY (1968) at salinity approximately 35'530° do not indicate that the thermal expansion coefficient as computed from Ekman's data is in substantial error. The comparison suggests that the present values of thermal expansion coefficient may be slightly too great, by roughly 10 ~ , but the discrepancies are barely within the accuracy of the new experimental results. REFERENCES

COCHRAm~ J. D. (1959) Note on the adiabatic temperature gradient of sea water. In : Physical and chemicalproperties of sea water N.A.S.-N.R..C. Publ. 600, pp. 30-37. CI~EASE J. (1962) The specific volume of sea water under pressure as determined by recent measurements of sound velocity. Deep-Sea Res., 9, 209-213. ECKART CARL (1958) Properties of water, Part II. The equation of state of water and sea water at low temperatures and pressures. Am. J. Sci., 256, 225-240. FOFONOFF N. P. (1962) Physical properties of sea water. In: The Sea 1, M. N. HILL, gen. ed. pp. 3-30. Interscience. NEWTON M. S. and G. C. KENr,rEDY (1965) An experimental study of the P - V - T - S relations of sea water. J. mar. Res., 23, 88-103. REID R. O. (1959) Influence of some errors in the equation of state or in observations on geostrophic currents. In: Physical and chemicalproperties of sea water. N.A.S.-N.R.C. Publ. 600, pp. 10-28. WILSON WAYNE and DAVID BRADLEY(1968) Specific volume of sea water as a function of temperature, pressure and salinity. Deep-Sea Res., 15 (3), 355-363.