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The maximum density points of pure and saline water
Deep-SeaReaem'eh,1978,Vol.25,pp. 175to 181. PergamonPress. Printedin GreatBritain
NOTE
The maximum density points of pure and saline water DOUGLAS R. CALDWELL*
(Received 16 March 1977; in revisedform 5 July 1977; accepted 26 July 1977) Abstract--Measurementsoftbe temperature-pressure-salinitypoints at which the density of saline water is a maximum can be made more directly than previously by determining conditions under which compressive heating vanishes. These measurements show that the density formulas of CHEN and MILLERO (Deep-Sea Research, 23, 595-612, 1976) and the expression for maximum-density temperature, Tin, of GEBHART and MOLLENDORF (Deep-Sea Research, 24, 831-848, 1977) give catimates of T. several tenths of a degree too high for saline water. Disagreement is observed at temperatures below 0 °C where density measurements have not been made, and so may indicate caution should be used in extrapolating the various equations of state to these temperatures. The recent pure-water equation of CSEN, FINe and MILLERO (Journal of Chemical Physics, 66, 2142-2144, 1977) predicts the observed T. within 0.04°C. The dependence of T. on pressure is non-linear at higher pressures. The formula T,. = To- AS-BP(1 + CS)(1 + DP) represents these data for T. as a function of salinity in parts per thousand (S) and pressure in bars (P) with a root-mean-square deviation of 0.044°C if To = 3.982, A = 0.2229, B = 0.02004, C = 0.00376 and D = 0.000402.
INTRODUCTION
REStarTS of determinations of the locus in temperature-pressure-salinity of the points where maximum density is achieved are relevant to the calculation of stability in lakes (EKLUND, 1965) and convective effects in cold ocean waters. A determination independent of density or sound-speed measurements would provide check points for the equation of state. Past determinations of the maximum density laws, being by-products of density determinations (ZAwoRsrd and KEENAN, 1967), are suspect in accuracy because of the difficulty of determining the location of an extremum in a curve; small errors in density lead to large displacements in the maximum-density points. Determinations of thermal expansion (BRADSHAW and SCHLEICHER, 1970; CALDWELL and TUCKER, 1970) should give better results, but these measurements were made only for salinities for which maximum appears above the melting point. In this paper a method is described of finding maximum density points as points where adiabatic expansion or compression can affect no change in the temperature of the fluid. Because a zero-crossing is much easier to find than a maximum, this method should give more accurate results than can be achieved through density measurements and can in principle be applied to any material. * School of Oceanography, Oregon State University, Corvallis, OR 97331, U.S.A.
175
176
DOUGLASR. CALDWELL THE METHOD
The temperature change, AT, of an adiabatically expanded or compressed material can be expressed as
where F is the adiabatic lapse rate or adiabatic temperature gradient, Ap the pressure change in bars, 0 the absolute temperature, V the specific volume, fl the thermal expansion coefficient, and Cp the specific heat at constant pressure. In the following, T is the Celsius Pressure reduced
~
55 bars
(O) - 4- I0 - 3 *C
-0
..................
I0 -3 *C
seconds
-40
i-
~, -3oi "o
-20i
E
-~0 O-
~
~
I 0
Pressure reduction by 55 bars I I0 rain
I
i
2 0 min
3 0 rain
Fig. I. Temperature measured by probe versus time. (a) Initial reaction to pressure reductio,1 Ior conditions near maximum density. Initial cooling is caused by direct effect of pressure on the metal of the probe. (b) Long term reaction to pressure reduction. Here the adiabatic gradient is about 39 millidegrees divided by 55 bars. After the first minute or so. heat from the bath penetrates to the probe.
temperature, P the pressure (guage) in bars, and S the salinity in ')~. At maximum density/3 = 0, but all other quantities in F are slowly varying positive functions of T, P and S. Therefore when F passes through zero fl passes through zero, defining a point of maximum density. Our procedure is to measure F versus pressure for given (S, T) and find the pressure at which F passes through zero. This pressure, with the ambient (S, T), gives the point Pro, Tin, Sin, a maximum density point. We consider S to be a parameter and usually consider Pm as a function of T, or Tmas a function of P. To measure F we use a device constructed for the measurement of F in seawater. It consists merely of a pressure bomb in which a very fine thermistor is placed, pressureprotected by canulla tubing of outside diameter 0.047cm. The bomb (3.75 cm inside diameter, 30cm high) is placed in a bath which holds its temperature constant well within 0.001 °C. Pressure is supplied to the bomb by an oil-filled hand pump. Separation of oil and water is accomplished in a vertical tube beside the bomb with an oil-water interlace (also in the bath). Pressure is measured with a newly-calibrated Texas Instruments Bourdon Tube gauge (accuracy 0.2 bars). A measurement of F requires only a rapid change in pressure followed by a measurement
The maximum density points of pure and saline water
177
of the change in temperature indicated by the thermistor. The thermistor changes temperature because of, successively: (1) compressive heating of the canulla tubing; (2) compressive heating of the water; and (3) heat leaking to the bomb and bath. The measurement is possible, and in fact easy, because the heat exchange between water and transducer occurs much more rapidly than exchange between water and bomb walls. In Fig. 1(a) a case where F is nearly zero is shown. The expansive cooling of the steel is less than 0.001 °C and is lost in the greater heat absorbed by the nearby water in a few seconds. When F ~ 0 [-Fig. 1(b)], the thermistor is cooled by heat absorbed in the nearby water in a few seconds, but heat from the bomb is not felt at all for 30 s or so. After the bomb heat is felt at the transducer, an hour or so is required for equilibrium to be achieved. The value of AT used in calculations is read from a thermistor bridge (with digital readout of its out-of-balance voltage) 15 s after the pressure change. If the pressure is immediately returned to its initial value, little irreversible change will have occurred and we .can proceed with the next measurement in a few minutes. The thermistor was calibrated with a crystal thermometer, which in turn was checked with an ice bath and a triple-point cell. The accuracy of this calibration was about 0.01 °C. Further confirmation of these calibrations was provided by our measurements of the melting point of pure water under pressure, which corresponded well with Bridgman's results (DoRs~,', 1940). Sufficiently small expansions and compressions were used that corrections for the alteration of properties by the changing temperature during the measurement were not required. A fairly linear relation between F and P is found (Fig. 2) so that the value of P for which F = 0 is easily determined. The precision of the measurements was thus about 0.01 °C in temperature and about 0.2 bars in pressure. The major difficulty in these measurements, after a transducer with a sufficiently small thermal mass was contrived, lay in the composition of the sample. The salinity was determined before and after measurements, and changes were sometimes observed, particularly if the sample had been allowed to freeze, as sometimes happened. Quite a bit of data was rejected for this reason, including all where freezing took place. The test of our estimates of S is the consistency of measurements on various samples, considered below. The saline water samples were filtered and diluted seawater taken off the Oregon coast. RESULTS
A total of 67 measurements of maximum density points were used, 36 with pure water, the rest at S = 10.45~/0o(5), 10.46~/0o(4), 20.10%oo(8), 25.16~0o (7), and 29.84~/oo (7) (Table 1). These were fitted by a least-squares method to the following formula:
T,,, = T o - A S - B P ( I where
+ C S ) ( I +DP)
(2)
To = 3.982 A = 0.2229 B = 0.02004 C = 0.00376 D = 0.000402. Calculations of Tm based on this formula are given in Table 2 and displayed in Fig. 2. The root mean square deviation of the data points with respect to this formula was 0.044°C. Much of the root mean square deviation arises from the doubt about the value of salinity. For example the average deviation at 20.10%o is -0.069°C, and all values are negative whereas at 26.16~/oothe average deviation is + 0.042°C and all values are positive.
178
DOUGLAS R. CALDWELL
Table I.
Maximum-density-point data in chronolooical order. S=0.O0°/oo
P
T
P
T
57.6 72.1 116.6 188.9 232.4 302.0 363.4 325.4 270.3
2.84 2.54 1.59 -0.07 -i.i0 -2.77 -4.34 -3.41 -1.97
355.8 335.1 315.1 129.6 78.6 42.1 22.8 4.8 55.2
-4.16 -3.64 -3.17 1.25 2.33 3.11 3.51 3.85 2.87
P 20.0 31.0 9.0 16.2 11.4 0.0 69.3 38.3 24.5
T
P
q'
3.55 3.36 3.84 3.70 3.77 4.02 2.52 3.16 3.46
9.6 10.5 134.5 137.2 311.0 344.1 380.9 328.9 ]6.5
3 74 3 74 1 08 1 08 -3 08 -3 91 -4.89 -3.44 3.61
S~I0.45°/..
S=I0.46°/.~
P
T
P
T
85.8 19.3 132.4 168.6 328.9
-0.17 1.33 -1.21 -2.03 -6.08
43.1 229.6 240.6 42.4
0.80 -3.52 -3.80 0.82
S=20.10°/oo
S=25.16%o
P
T
P
6.9 24.5 39.6 75.2 96.2 112.4 129.3 14.8
-0.70 -i. i0 -i. 43 -2.24 -2.71 -3. Ii -3. 51 -0.85
86.3 75.7 62.9 36.7 20.0 ii.4 56.9
S=29.84°/o~ T
-3.54 -3.32 -2.99 -2.40 -1.99 -i. 82 -2.88
P 23.8 Ii. 7 56.2 62.4 69.0 46.9 38.6
T - 3.21 -2.88 - 3.96 -4.13 -4.31 -3.74 -~.55
Therefore the pressure dependence given in equation (2) is likely to be more reliable than the salinity dependence. These data when compared to the formula of GEBHART and MOLL~qDORF (1977) yield a root mean square deviation of 0.31°C and 0.32°C with respect to T. values derived froln the formula of CHEN and MmLEaO (1976). Surprisingly the data at S > 20%o are close to Tm derived from BRYDEN'S (1973) formula for thermal expansion, which was derived from B~DSH^W and SCHLEICHER'S(1970) thermal expansion data for S > 30%o only. We find a nonlinear dependence of T. on P that shows up best in pure water at very
179
The maximum density points of pure and saline water
Table 2. Temperature o[maxinu~m density calculated ~om/brmula fit to data (pressure in bars, sali.ity in ",.. T . i. °C).
T
P
0
5
I0
0 20 40 60 80 i00 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400
3.98 3.58 3.17 2.75 2.33 1.90 1.46 1.02 0.57 0.ii -0.35 -0.82 -1.29 -1.77 -2.26 -2.76 -3.26 -3.76 -4.28 -4.80 -5.32
2.87 2.46 2.04 1.61 1.18 0.74 0.30 -0.15 -0.61 -1.07 -1.54 -2.02 -2.51 -3.00 -3.49 -4.00 -4.51 -5.02
1.75 1.33 0.91 0.48 0.04 -0.41 -0.86 -1.32 -1.79 -2.26 -2.74 -3.23 -3.72 -4.22 -4.72
I
m
for given S
15
I
20
O. 64 O. 21 -0.22 -0.66 -I.ii -i. 56 -2.02 -2.49 -2.97 -3.45 -3.94 -4.43 -4.93
I
I
-0.48 -0.91 -1.35 -I. 80 -2.26 -2.72 -3.19 -3.66 -4 •15 -4.64
I
25
30
-i. 59 -2.03 -2.48 -2.94 -3.40 -3.87 -4.35 -4.83
-2.70 -3.15 - 3.61 -4.08 -4.55 -5.02
I
.i ~o ¢.3
40
80
120
PRESSURE (bars)
Fig. 2. Adiabatic gradient versus pressure, 2.52°C, pure water. The pressure of maximum density is given by the intersection with the adiabatic gradient = 0 line. Open circles represent expansions from 138 bars, closed circles compressions from 69 bars.
180
DOUGLAS R. CALDWELL 400
3OO
u) r~
..¢#O ~ ,
I00
-4
-2
0
2
4
°C
Fig. 3.
Locus of maximum density points on the pressure- temperature plane with salinity as a parameter, as estimated by equation (2).
400
5OO
fl~
IZ) 2OO
I00
-5
-4
-3
-2
-I
0
I
2
3
4
*C
Fig. 4. Data obtained for maximum-density points lor samples of various salinities. GEBHART and MOLLENDORF 0977) equation; . . . . derived from CHEN and MILL~O (1976) density relation; ............... derived from BF.Vr~N (1973) thermal expansion equation. The Chert and Mfllero carves are shown only when they differ from those of Gebhart and Moltendorf. --
The maximum density points of pure and saline water
181
high pressures (Fig. 3) because the curve is longer. Agreement with ZAWORSKIand KEENAN (1967) is quite good below 300 bars. It was possible to extend measurements in pure water below the melting point, but in saline water freezing set in quite close to the melting point, and the supercooled condition could not be maintained. Comparison of these results with values derived from several recent equations of state is shown in Fig. 4. Systematic differences are seen, particularly for the more saline samples. The formulas of CHEN and MILLERO (1976) and GEBHART and MOLLENDORF (1977) place Tmtoo high. The equation of state for pure water (CHEN, FINE and MILLERO, 1977) derived from sound-speed measurements agrees extremely well with our date. The mean r.m.s, deviation of the pure water data from their equation is only 0.05°C with a slight tendency for their equation to predict Tma few hundredths of a degree lower than our data. If values of thermal expansion coefficient from this pure water formula are incorporated into the formula for saline water of CHEN and MILLERO (1976), agreement is improved at low salinities but not much at higher salinities. This overestimation of T= by even a revised Chen and Millero equation and the fact that it appears as a 'base line shift', indicate that the one atmosphere equation of state of MILLERO, GONZALEZ and WARD (1976), on which the Chen and Millero equation is based, cannot be used to extrapolate expansibilities below its range, 0 to 40°C. It should be remembered that the density and soundspeed measurements used in the fitting of equations of state have not been extended to temperatures below 0°C. Using the formulas below 0°C amounts to an extrapolation, and these formulas contain some rather high order terms in temperature. Acknowledoement--This work was supported by the Oceanography Section, National Science Foundation, under Grant DES-74-17968. REFERENCES BRADSHAW A. and K. E. SCHLEICHER (1970) Direct measurement of thermal expansion of seawater under pressure. Deep-Sea Research, 17, 691-706. BRYDEN H. L. (1973) New polynomials for thermal expansion, adiabatic temperature gradient and potential temperature of seawater. Deep-Sea Research, 20, 401-408. CALDWELL D. R. and B. E. TUCKER (1970) Determination of thermal expansion of seawater by observing onset of convection. Deep-Sea Research, 17, 707-719. CHEN C. T. and F. J. MILLERO (1976) The specific volume of seawater at high pressures. Deep-Sea Research, 23, 595-612. CHEN C. T., R. A. FINE and F. J. MILLERO (1977) The equation of state of pure water determined from sound speeds. Journal of Chemical Physics, 66, 2142-2144. D o R ~ y N. E. (1940) Properties of ordinary water substallce. Reinhold, 673 pp. EKLUND H. (1965) Stability of lakes near the temperature of maximum density. Science, 149, 632-633. GEBHART B. and J. C. MOLLENDORF(1977) A new density relation for pure and saline water. Deep-Sea Research, 831-848. MILLERO F. J., A. GONZALEZ and G. K. WARD (1976) The density of seawater solutions at one atmosphere as a fuction of temperature and salinity. Journal of Marine Research, 34, 61-93. Z^WORSKI R. J. and J. H. KEEN^N (.1967) Thermodynamic properties of water in the region of maximum density. Journal of Applied Mechanics, 34, 478-483.
NOTE
The maximum density points of pure and saline water DOUGLAS R. CALDWELL*
(Received 16 March 1977; in revisedform 5 July 1977; accepted 26 July 1977) Abstract--Measurementsoftbe temperature-pressure-salinitypoints at which the density of saline water is a maximum can be made more directly than previously by determining conditions under which compressive heating vanishes. These measurements show that the density formulas of CHEN and MILLERO (Deep-Sea Research, 23, 595-612, 1976) and the expression for maximum-density temperature, Tin, of GEBHART and MOLLENDORF (Deep-Sea Research, 24, 831-848, 1977) give catimates of T. several tenths of a degree too high for saline water. Disagreement is observed at temperatures below 0 °C where density measurements have not been made, and so may indicate caution should be used in extrapolating the various equations of state to these temperatures. The recent pure-water equation of CSEN, FINe and MILLERO (Journal of Chemical Physics, 66, 2142-2144, 1977) predicts the observed T. within 0.04°C. The dependence of T. on pressure is non-linear at higher pressures. The formula T,. = To- AS-BP(1 + CS)(1 + DP) represents these data for T. as a function of salinity in parts per thousand (S) and pressure in bars (P) with a root-mean-square deviation of 0.044°C if To = 3.982, A = 0.2229, B = 0.02004, C = 0.00376 and D = 0.000402.
INTRODUCTION
REStarTS of determinations of the locus in temperature-pressure-salinity of the points where maximum density is achieved are relevant to the calculation of stability in lakes (EKLUND, 1965) and convective effects in cold ocean waters. A determination independent of density or sound-speed measurements would provide check points for the equation of state. Past determinations of the maximum density laws, being by-products of density determinations (ZAwoRsrd and KEENAN, 1967), are suspect in accuracy because of the difficulty of determining the location of an extremum in a curve; small errors in density lead to large displacements in the maximum-density points. Determinations of thermal expansion (BRADSHAW and SCHLEICHER, 1970; CALDWELL and TUCKER, 1970) should give better results, but these measurements were made only for salinities for which maximum appears above the melting point. In this paper a method is described of finding maximum density points as points where adiabatic expansion or compression can affect no change in the temperature of the fluid. Because a zero-crossing is much easier to find than a maximum, this method should give more accurate results than can be achieved through density measurements and can in principle be applied to any material. * School of Oceanography, Oregon State University, Corvallis, OR 97331, U.S.A.
175
176
DOUGLASR. CALDWELL THE METHOD
The temperature change, AT, of an adiabatically expanded or compressed material can be expressed as
where F is the adiabatic lapse rate or adiabatic temperature gradient, Ap the pressure change in bars, 0 the absolute temperature, V the specific volume, fl the thermal expansion coefficient, and Cp the specific heat at constant pressure. In the following, T is the Celsius Pressure reduced
~
55 bars
(O) - 4- I0 - 3 *C
-0
..................
I0 -3 *C
seconds
-40
i-
~, -3oi "o
-20i
E
-~0 O-
~
~
I 0
Pressure reduction by 55 bars I I0 rain
I
i
2 0 min
3 0 rain
Fig. I. Temperature measured by probe versus time. (a) Initial reaction to pressure reductio,1 Ior conditions near maximum density. Initial cooling is caused by direct effect of pressure on the metal of the probe. (b) Long term reaction to pressure reduction. Here the adiabatic gradient is about 39 millidegrees divided by 55 bars. After the first minute or so. heat from the bath penetrates to the probe.
temperature, P the pressure (guage) in bars, and S the salinity in ')~. At maximum density/3 = 0, but all other quantities in F are slowly varying positive functions of T, P and S. Therefore when F passes through zero fl passes through zero, defining a point of maximum density. Our procedure is to measure F versus pressure for given (S, T) and find the pressure at which F passes through zero. This pressure, with the ambient (S, T), gives the point Pro, Tin, Sin, a maximum density point. We consider S to be a parameter and usually consider Pm as a function of T, or Tmas a function of P. To measure F we use a device constructed for the measurement of F in seawater. It consists merely of a pressure bomb in which a very fine thermistor is placed, pressureprotected by canulla tubing of outside diameter 0.047cm. The bomb (3.75 cm inside diameter, 30cm high) is placed in a bath which holds its temperature constant well within 0.001 °C. Pressure is supplied to the bomb by an oil-filled hand pump. Separation of oil and water is accomplished in a vertical tube beside the bomb with an oil-water interlace (also in the bath). Pressure is measured with a newly-calibrated Texas Instruments Bourdon Tube gauge (accuracy 0.2 bars). A measurement of F requires only a rapid change in pressure followed by a measurement
The maximum density points of pure and saline water
177
of the change in temperature indicated by the thermistor. The thermistor changes temperature because of, successively: (1) compressive heating of the canulla tubing; (2) compressive heating of the water; and (3) heat leaking to the bomb and bath. The measurement is possible, and in fact easy, because the heat exchange between water and transducer occurs much more rapidly than exchange between water and bomb walls. In Fig. 1(a) a case where F is nearly zero is shown. The expansive cooling of the steel is less than 0.001 °C and is lost in the greater heat absorbed by the nearby water in a few seconds. When F ~ 0 [-Fig. 1(b)], the thermistor is cooled by heat absorbed in the nearby water in a few seconds, but heat from the bomb is not felt at all for 30 s or so. After the bomb heat is felt at the transducer, an hour or so is required for equilibrium to be achieved. The value of AT used in calculations is read from a thermistor bridge (with digital readout of its out-of-balance voltage) 15 s after the pressure change. If the pressure is immediately returned to its initial value, little irreversible change will have occurred and we .can proceed with the next measurement in a few minutes. The thermistor was calibrated with a crystal thermometer, which in turn was checked with an ice bath and a triple-point cell. The accuracy of this calibration was about 0.01 °C. Further confirmation of these calibrations was provided by our measurements of the melting point of pure water under pressure, which corresponded well with Bridgman's results (DoRs~,', 1940). Sufficiently small expansions and compressions were used that corrections for the alteration of properties by the changing temperature during the measurement were not required. A fairly linear relation between F and P is found (Fig. 2) so that the value of P for which F = 0 is easily determined. The precision of the measurements was thus about 0.01 °C in temperature and about 0.2 bars in pressure. The major difficulty in these measurements, after a transducer with a sufficiently small thermal mass was contrived, lay in the composition of the sample. The salinity was determined before and after measurements, and changes were sometimes observed, particularly if the sample had been allowed to freeze, as sometimes happened. Quite a bit of data was rejected for this reason, including all where freezing took place. The test of our estimates of S is the consistency of measurements on various samples, considered below. The saline water samples were filtered and diluted seawater taken off the Oregon coast. RESULTS
A total of 67 measurements of maximum density points were used, 36 with pure water, the rest at S = 10.45~/0o(5), 10.46~/0o(4), 20.10%oo(8), 25.16~0o (7), and 29.84~/oo (7) (Table 1). These were fitted by a least-squares method to the following formula:
T,,, = T o - A S - B P ( I where
+ C S ) ( I +DP)
(2)
To = 3.982 A = 0.2229 B = 0.02004 C = 0.00376 D = 0.000402. Calculations of Tm based on this formula are given in Table 2 and displayed in Fig. 2. The root mean square deviation of the data points with respect to this formula was 0.044°C. Much of the root mean square deviation arises from the doubt about the value of salinity. For example the average deviation at 20.10%o is -0.069°C, and all values are negative whereas at 26.16~/oothe average deviation is + 0.042°C and all values are positive.
178
DOUGLAS R. CALDWELL
Table I.
Maximum-density-point data in chronolooical order. S=0.O0°/oo
P
T
P
T
57.6 72.1 116.6 188.9 232.4 302.0 363.4 325.4 270.3
2.84 2.54 1.59 -0.07 -i.i0 -2.77 -4.34 -3.41 -1.97
355.8 335.1 315.1 129.6 78.6 42.1 22.8 4.8 55.2
-4.16 -3.64 -3.17 1.25 2.33 3.11 3.51 3.85 2.87
P 20.0 31.0 9.0 16.2 11.4 0.0 69.3 38.3 24.5
T
P
q'
3.55 3.36 3.84 3.70 3.77 4.02 2.52 3.16 3.46
9.6 10.5 134.5 137.2 311.0 344.1 380.9 328.9 ]6.5
3 74 3 74 1 08 1 08 -3 08 -3 91 -4.89 -3.44 3.61
S~I0.45°/..
S=I0.46°/.~
P
T
P
T
85.8 19.3 132.4 168.6 328.9
-0.17 1.33 -1.21 -2.03 -6.08
43.1 229.6 240.6 42.4
0.80 -3.52 -3.80 0.82
S=20.10°/oo
S=25.16%o
P
T
P
6.9 24.5 39.6 75.2 96.2 112.4 129.3 14.8
-0.70 -i. i0 -i. 43 -2.24 -2.71 -3. Ii -3. 51 -0.85
86.3 75.7 62.9 36.7 20.0 ii.4 56.9
S=29.84°/o~ T
-3.54 -3.32 -2.99 -2.40 -1.99 -i. 82 -2.88
P 23.8 Ii. 7 56.2 62.4 69.0 46.9 38.6
T - 3.21 -2.88 - 3.96 -4.13 -4.31 -3.74 -~.55
Therefore the pressure dependence given in equation (2) is likely to be more reliable than the salinity dependence. These data when compared to the formula of GEBHART and MOLL~qDORF (1977) yield a root mean square deviation of 0.31°C and 0.32°C with respect to T. values derived froln the formula of CHEN and MmLEaO (1976). Surprisingly the data at S > 20%o are close to Tm derived from BRYDEN'S (1973) formula for thermal expansion, which was derived from B~DSH^W and SCHLEICHER'S(1970) thermal expansion data for S > 30%o only. We find a nonlinear dependence of T. on P that shows up best in pure water at very
179
The maximum density points of pure and saline water
Table 2. Temperature o[maxinu~m density calculated ~om/brmula fit to data (pressure in bars, sali.ity in ",.. T . i. °C).
T
P
0
5
I0
0 20 40 60 80 i00 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400
3.98 3.58 3.17 2.75 2.33 1.90 1.46 1.02 0.57 0.ii -0.35 -0.82 -1.29 -1.77 -2.26 -2.76 -3.26 -3.76 -4.28 -4.80 -5.32
2.87 2.46 2.04 1.61 1.18 0.74 0.30 -0.15 -0.61 -1.07 -1.54 -2.02 -2.51 -3.00 -3.49 -4.00 -4.51 -5.02
1.75 1.33 0.91 0.48 0.04 -0.41 -0.86 -1.32 -1.79 -2.26 -2.74 -3.23 -3.72 -4.22 -4.72
I
m
for given S
15
I
20
O. 64 O. 21 -0.22 -0.66 -I.ii -i. 56 -2.02 -2.49 -2.97 -3.45 -3.94 -4.43 -4.93
I
I
-0.48 -0.91 -1.35 -I. 80 -2.26 -2.72 -3.19 -3.66 -4 •15 -4.64
I
25
30
-i. 59 -2.03 -2.48 -2.94 -3.40 -3.87 -4.35 -4.83
-2.70 -3.15 - 3.61 -4.08 -4.55 -5.02
I
.i ~o ¢.3
40
80
120
PRESSURE (bars)
Fig. 2. Adiabatic gradient versus pressure, 2.52°C, pure water. The pressure of maximum density is given by the intersection with the adiabatic gradient = 0 line. Open circles represent expansions from 138 bars, closed circles compressions from 69 bars.
180
DOUGLAS R. CALDWELL 400
3OO
u) r~
..¢#O ~ ,
I00
-4
-2
0
2
4
°C
Fig. 3.
Locus of maximum density points on the pressure- temperature plane with salinity as a parameter, as estimated by equation (2).
400
5OO
fl~
IZ) 2OO
I00
-5
-4
-3
-2
-I
0
I
2
3
4
*C
Fig. 4. Data obtained for maximum-density points lor samples of various salinities. GEBHART and MOLLENDORF 0977) equation; . . . . derived from CHEN and MILL~O (1976) density relation; ............... derived from BF.Vr~N (1973) thermal expansion equation. The Chert and Mfllero carves are shown only when they differ from those of Gebhart and Moltendorf. --
The maximum density points of pure and saline water
181
high pressures (Fig. 3) because the curve is longer. Agreement with ZAWORSKIand KEENAN (1967) is quite good below 300 bars. It was possible to extend measurements in pure water below the melting point, but in saline water freezing set in quite close to the melting point, and the supercooled condition could not be maintained. Comparison of these results with values derived from several recent equations of state is shown in Fig. 4. Systematic differences are seen, particularly for the more saline samples. The formulas of CHEN and MILLERO (1976) and GEBHART and MOLLENDORF (1977) place Tmtoo high. The equation of state for pure water (CHEN, FINE and MILLERO, 1977) derived from sound-speed measurements agrees extremely well with our date. The mean r.m.s, deviation of the pure water data from their equation is only 0.05°C with a slight tendency for their equation to predict Tma few hundredths of a degree lower than our data. If values of thermal expansion coefficient from this pure water formula are incorporated into the formula for saline water of CHEN and MILLERO (1976), agreement is improved at low salinities but not much at higher salinities. This overestimation of T= by even a revised Chen and Millero equation and the fact that it appears as a 'base line shift', indicate that the one atmosphere equation of state of MILLERO, GONZALEZ and WARD (1976), on which the Chen and Millero equation is based, cannot be used to extrapolate expansibilities below its range, 0 to 40°C. It should be remembered that the density and soundspeed measurements used in the fitting of equations of state have not been extended to temperatures below 0°C. Using the formulas below 0°C amounts to an extrapolation, and these formulas contain some rather high order terms in temperature. Acknowledoement--This work was supported by the Oceanography Section, National Science Foundation, under Grant DES-74-17968. REFERENCES BRADSHAW A. and K. E. SCHLEICHER (1970) Direct measurement of thermal expansion of seawater under pressure. Deep-Sea Research, 17, 691-706. BRYDEN H. L. (1973) New polynomials for thermal expansion, adiabatic temperature gradient and potential temperature of seawater. Deep-Sea Research, 20, 401-408. CALDWELL D. R. and B. E. TUCKER (1970) Determination of thermal expansion of seawater by observing onset of convection. Deep-Sea Research, 17, 707-719. CHEN C. T. and F. J. MILLERO (1976) The specific volume of seawater at high pressures. Deep-Sea Research, 23, 595-612. CHEN C. T., R. A. FINE and F. J. MILLERO (1977) The equation of state of pure water determined from sound speeds. Journal of Chemical Physics, 66, 2142-2144. D o R ~ y N. E. (1940) Properties of ordinary water substallce. Reinhold, 673 pp. EKLUND H. (1965) Stability of lakes near the temperature of maximum density. Science, 149, 632-633. GEBHART B. and J. C. MOLLENDORF(1977) A new density relation for pure and saline water. Deep-Sea Research, 831-848. MILLERO F. J., A. GONZALEZ and G. K. WARD (1976) The density of seawater solutions at one atmosphere as a fuction of temperature and salinity. Journal of Marine Research, 34, 61-93. Z^WORSKI R. J. and J. H. KEEN^N (.1967) Thermodynamic properties of water in the region of maximum density. Journal of Applied Mechanics, 34, 478-483.