Hydrodynamic Equations of the Sea
4.6 Heat Conduction Equation
y h ( Y h - 1) C,, = c’a: T
.
(4.37a) An alternative form of this relationship that involves only experimentally observed or calculable quantities on the right-hand side may be obtained from the definitions of the coefficients c2, o r , and Cus:= Cps - Ta$/pap
.
(4.37b)The specific heat at constant pressure and the required derivatives may be obtained from an equation of state, the most recent internationally accepted version of which is given ahead. Numerical values for that quantity are near 3994 J (kg “C)
~’
at T = 20°C and s = 35 psu; the specific heat at constant Volume is at most 2% smaller, with the value of Y h at those environmental conditions being approximately 1.014.We are now in a position to write Eqs. 4.13 and 4.14 in terms of the mea- surable coefficients above; at this point of recasting, we shall also generalize them for time-varying problems and immediately obtain two additional equa- tions that contribute to our goal. In addition, we shall write one, the tem- perature equation, in terms of convective heating rates, DqlDt, rather than the entropy derivative, Dv/Dt, but will retain the latter in the equation for specific volume. These useful equatibns give time derivatives of cr and T i n terms of time variations of other state variables that can be calculated-readily:
-
+ - -
(4.38)Da (YaTT Dv Dp 1 Ds
Dt Cps Dt C’ Dt SH, Dt
DT - 1 Dq CYaTT Dp 1 Ds
Dt C p s Dt Cps Dt sxT Dt
and
+ - - + - - .
(4.39)Equation 4.38 will be used in Chapter 7 in the derivation of the acoustic wave equation, while Eq. 4.39 is used immediately below.
Thermodynamics and Energy Relations 141
Dq a4
- = -
+ u - v q
~t at
where the first term in the q derivative represents the rate of change of heat content and the second, heat convection. The right-hand side includes heat conduction and diffusion, and the heating rate per unit mass due t o time changes and the convergence of the vector radiance, ( S ) (see Eqs. 8.175, 8.177, 9.8, and 9.64); here c, is the speed of light in seawater (see Eq. 8.19 and Table 9.2). Included in Q ( e , p , L , ) are the heating rate per unit volume, the viscous dissipation rate, E , chemical reactions, p, and other sources or sinks not accounted for elsewhere. Values of the scalar K~ for molecular conduction in the sea are of order 0.596 W (m2 K m - ' ) - ' ; the correspond- ing eddy conductivities are orders of magnitude larger (see Appendix Three).
For processes in a source-free, incompressible, constant-salinity ocean, the left-hand side can be written in terms of the temperature as well, using Eqs.
4.31a, 4.38, and 4.39:
(4.41) This relationship, evaluated for constant, isotropic heat conductivity and no current, becomes the usual heat conduction equation:
(4.42) where K(,
=
K~ /pC,, is the molecular thermal diffusivity, which has a val- ue in the ocean of approximately 1.49 x 10'
m 2 s ~ I. In the general case, the more complete expressions (Eqs. 4.39 and 4.40) must be retained.4.7 Specific Volume and Salinity Equations
We have already derived equations for density and salinity changes (Eqs.
3.68 and 3.81, respectively). Recasting the first into a form giving the con- vective derivative for specific volume, we obtain
D a
~ =
Crv-u.
Dt (4.43)
The equivalent form for salinity is slightly more complicated, and makes use of a relationship for the quantity pDy/Dt that also invokes the continui- ty equation. Adding p times the convective derivative of a property, y, to y times the continuity equation, we obtain
p- DY
=
p ( $+ u.vy) +
y(; + v-pu
Dt
(4.44)
which may be regarded as an identity. Applying this to Eq. 3.81 yields
where S represents source and sink terms for salinity as given by Eq. 3.81.
Equations 4.40,4.43, and 4.45 are in forms suitable for use in the enthalpy equation as given by Eq. 4.9, and in the specific volume and temperature rate equations (Eqs. 4.38 and 4.39, respectively).
4.8 Equation of State
We have now assembled all of the equations required to complete our hydrodynamic and thermodynamic system, except for an equation of state of the form
or
a = a ( ~ , T , p )
.
(4.46b)An equation of state of sufficient accuracy to be used in the computation of the oceanic density field (which is needed for the so-called dynamic meth- od of determining large-scale oceanic currents; see Chapter 6) is a very com- plicated one indeed. An internationally agreed upon equation of state (UNESCO, 1981) fits the available density measurements with an accuracy of order 3.5 x over the normally encountered range of oceanic pres- sures, temperatures, and salinities. This equation is of the form
Thermodynamics and Energy Relations 143
where each quantity on the right-hand side, except pressure, is expressed as a polynomial series in s and T, expanded about values for zero salinity and a pressure of 1 bar. The secant or mean compressibility, ( - ( l / a ) ( d a / d p ) ) , is defined as:
where K T ( s , T,p) is the secant or mean bulk modulus, whose reciprocal is the mean of ap (Eq. 4.20). It is a function of s, T , and p , and its variation at s = 35 psu is shown in Fig. 4.8 as a function of temperature and pressure.
The compressibility decreases as temperature, pressure, and salinity increase, which is understandable in terms of the molecular character of matter: As higher pressure squeezes the molecules closer together, or as higher concen- trations of ions interleave between the H20 molecules, still further reduc- tions in volume are increasingly resisted by the crowding.
The relationship of Eq. 4.47 is an improvement on the simplest possible linear equation of state, which is an expansion about zero values for T, p , and s of the type given by CY = ao(l
+
aTT - a,p - ass). We may as- sume a slightly more complicated form for improved accuracy, viz: a! =’a,(l
+
aTT) (1 - ass) ( 1 - ( a p ) p ) . Here the ai’s are given by Eqs. 4.19 to 4.21; p ( s , T , O ) = l/aW (1+
a T T ) ( 1 - ass); and am is the specific vol- ume of pure water. The general form of Eq. 4.47 follows immediately from this latter approximation if we incorporate higher-order terms in T, s, a n d p . We shall not give the complete development of the so-called International Equation of State for Seawater,EOS
80, in the interests of brevity; the reader may refer to Millero et al. (1980), UNESCO (1981), or Fofonoff (1985) for a more comprehensive specification.The equation is most clearly displayed in four parts; the first gives the specific volume in the same form as Eq. 4.47:
The second part gives the density at surface pressure (indicated by 0):
p(s,T,O) = l/a(s,T,O) = A
+
Bs+
Cs3’*+
Ds2.
(4.49b)I I I Temperature, T ("C)
0 10 20 30 I
Fig. 4.8 Isothermal compressibility of seawater as a function of temperature and pressure. [Adapted from Knauss, J., lntroduction to Physical Oceanography (1978)l.
The third gives the secant bulk modulus, KT:
KT (s, T , p ) = E
+
FS+
Gs312+
( H+
Is+
J s 3 9 p+
( M+
N s ) p 2 . (4.49c) The fourth part consists of polynomials A , B,...,
N up to fifth degree in the temperature; these are listed in Table 4.2. The temperature is speci- fied in degrees Celcius; the pressure is in bars, or lo5 Pa; the salinity is in psu; the density is in kg m - 3 ; and the specific volume is in m 3 k g - ' . This equation is sufficiently accurate to determine the density field from observed values of salinity, temperature, and pressure to within a standard error ofThermodynamics and Energy Relations 145
approximately 0.009 kg m - 3 over the entire oceanic pressure range. Since variations in the composition of dissolved salts can lead t o differences in the density of natural seawater of order 0.05 kg m P 3 , the equation appears as accurate as necessary when salinities are determined by electrical conduc- tivities, as they are. For this last reason, the relationship between salinity and conductivity is an important one to specify. It is given as a part of the UNESCO literature, with its inverse, i.e., conductivity as a function of sa- linity, being shown via Eqs. 8.33 to 8.36.
TABLE 4.2 Coefficients for the International Equation of State for Seawater, EOS 80
A B
c
? +
999.842594 +8.24493 x l o - ’ -5.72466 xTI +6.793952 x lo-’ -4.0899 x +1.0227 x
T 2 -9.095290 x +7.6438 x lo-’ -1.6546 x l o p 6
T 3 +1.001685 x -8.2467 x lo-’
? -1.120083 x l o e 6 +5.3875 x T5 +6.536332 x l o p 9
D E F
? +4.8314 x l o p 4 19652.21
+
54.6746T 2 - 2.327105 +1.09987 x
T 3 +1.360477 x -6.1670 x l o p 5
T4
-5.155288 x l o p 5T‘
+
148.4206 -0.603459G H I
TO
+7.944 x f3.239908 +2.2838 x l o p 3TI +1.6483 x +1.43713 x -1.0981 x lo-’
TZ -5.3009 x +1.16092 x l o p 4 -1.6078 x
T 3 -5.77905 x lo-’
J M N
TO
+1.91075 x f8.50935 x lo-’ -9.9348 x lo-’T’ -6.12293 x +2.0816 x lo-*
T 2 +5.2787 x +9.1697 x lo-’’
[Adapted from Fofonoff, N. P . , J. Geophys. Res. (1985).]
Plots of density versus the three independent thermodynamic variables are difficult t o construct, but one particular projection of the equation of state has proven to be very useful for a diagnostic technique called water type anaf- ysis. This is the form shown in Figs. 4.9 and 4.10, i.e., the loci of constant
Principles of Ocean Physics
1.013 X 105 Pa
I
= 4
1
I I I I IL
0 10 20 30
Salinity, s (psu)
0
Fig. 4.9 Temperature-salinity-density diagram for seawater over range of normal variations of Tand s. This is a cross-plot of Fig. 4.5. [Adapted from Dietrich, G;, Ocean- ography, An Introductory View (1968).]
density on a plot giving temperature versus salinity. Figure 4.9 shows values of constant density, or isopycnals, in u, units, where u, is defined as
Or = P - P o o Y (4.50)
and where poo = 1000 kg m - 3 at 4°C is the density of fresh water. Thus uI represents the departure of density from the fresh water value.
The graphs of Figs. 4.9 and 4.10 are known as T-s diagrams, and a ho- mogeneous water type is represented by a single point on the diagrams. A sequence of water samples measured at varying depths will show the kind of variation given in Fig. 4.10, with depth given as a parameter along the line; the variations of water type properties with depth are usually found to lie along a curve that is more or less characteristic of the area of the ocean being sampled. The characteristic T-s relationship, along with analyses of dissolved oxygen and silica, for example, allow the identification of the ori-
Salinity, s (psu)
Fig. 4.10 Temperature-salinity-density diagram for an oceanographic station in the tropical Atlantic at 5"N, 25"W. The parameter is density in constant ut units, or isopyc- nals. The S-shaped curve represents the variation of water type properties as the depth increases from z = 0 m at the upper end to -4370 m at the bottom. Various water types are AAIW, NADW, and AABW (see Fig. 2.22 for legend). The straight line be- tween AAIW and NADW indicates the mixing of these two types has occurred at depths between 900 to 1300 m. Similarly, mixing of NADW and AABW water has occurred over the range from 2000 m to the bottom.
gin of the water mass with reasonable certitude. The T-s curve also possess- es the property that two overlying water masses that are initially homogeneous and thus represented by two distinct points in the T-s plot will, upon verti- cal diffusive mixing, appear on the diagram as points connecting an approx- imately straight line separating the two initial values. The example shown in Fig. 4.10 gives evidence of mixing between Antarctic Intermediate Water, North Atlantic Deep Water, and Antarctic Bottom Water (see Fig. 2.22).
Thus the