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R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1
Ocean Sci., 6, 633-677, 2010
www.ocean-sci.net/6/633/2010/
3.4 Humid air
The Helmholtz function / AV (A, T, p) of humid air,
Eq. (2.7), permits the direct computation of all thermody
namic properties if temperature T, density p and air fraction
A are either given or can be obtained from similar quanti
ties such as the specific humidity, q=l—A or the mixing ra
tio r=(l—A)/A. This does not include properties at given
relative humidity which requires the knowledge of vapour
saturation, i.e. of the phase equilibrium between vapour and
liquid water which is a composite system considered later in
Sect. 5.8. A list of equations for the computation of humid-
air properties from Eq. (2.7) is given in Table S5.
4 Level 3: Functions involving numerical solution of
implicit equations
If quantities other than the natural independent variables of
the three potential functions of Sect. 2 are given, in particular,
if the pressure is known rather that the density of pure water,
or the entropy rather than the temperature of seawater, the
relevant thermodynamic equations must be inverted analyt
ically or numerically. These steps inevitably add larger nu
merical uncertainties to all properties that depend on these in
versions, and hence on the settings chosen for the associated
iteration algorithms. Default values for iteration number or
tolerance are specified in the SIA library routines that should
be appropriate for most purposes; if necessary, they can be
modified by related “set_” procedures of the library (Wright
et al., 2010a). Quantities that require such inversions appear
in the libraries as level-3 procedures. To ensure the stability
and uniqueness of the numerical solutions, initial conditions
must be chosen appropriately. Various empirical functions
are used to provide suitable initial values as discussed in the
appendices referenced in Sect. 4.1-4.3. While the algorith
mic success and speed are sensitive to these choices, the final
quantitative results are, within their numerical uncertainty,
independent of the details of the initial “guess” functions.
Therefore, if desired for certain applications, these auxiliary
functions implemented in the library and described in this pa
per may be replaced by more suitable or effective customised
ones without affecting the correctness of the final results.
4.1 Gibbs functions for liquid water and water vapour
To compute properties of fluid water at given T and P from
its Helmholtz potential, / F (T, p), it is necessary to solve
Eq. (S2.ll),
p(T,P)=gp\ (4.1)
for the density. Except for spurious or unstable numerical so
lutions outside the validity range, Fig. la, there is exactly one
physically meaningful solution at supercritical temperatures.
Depending on the pressure, there may be one or two stable
solutions below the critical temperature, given by the inter
section points of isobars with isotherms illustrated in Fig. 1,
providing the density of liquid water, p w (T, P), and/or of
water vapour, p v (T, P).
Consequently, there cannot exist a single-valued Gibbs
function g(T,P) that fully represents the properties of the
Helmholtz function / F (P,p) of fluid water. Rather, there
are two different Gibbs functions,
g w (T,P) — f F (r,p w ^ + P/p w (4.2)
for liquid water and
g v (r,P) = / F (r,p v ) + P/p v (4.3)
for water vapour, which coincide under supercritical condi
tions. Interestingly, critical conditions can be encountered at
hydrothermal vents in the abyssal ocean (Reed, 2006; Sun et
al., 2008).
To implement the above expressions for the Gibbs func
tions we must determine the liquid and vapour densities cor
responding to the temperature and pressure inputs. This re
quires iterative solution of Eq. (4.1), with considerable care
required to select the appropriate root for each case. De
tails on the iterative numerical method and the conditions
used to initialize the iteration procedure are provided in Ap
pendix Al.
Once the liquid or vapour density of water is computed
from the Helmholtz function / F at given temperature and
pressure, the numerical values of the Gibbs function of wa
ter and its partial derivatives can be computed from the for
mulas of Table S6. The equations given there for water,
g w , Eq. (4.2), apply in an analogous manner to vapour, p v ,
Eq. (4.3), if only the density p of liquid water is replaced by
that of vapour.
Note that the above procedure is required to ensure ar
bitrarily precise consistency between the Gibbs function of
pure water and the corresponding Helmholtz function. As
long as this consistency is demanded, determination of the
Gibbs function and its derivatives requires an iterative nu
merical procedure to determine the density argument of the
Helmholtz function, so no explicit algebraic expression is
possible. Thus, the pure water component of the Gibbs func
tion must be determined at level 3 and it is only at this level
that the Gibbs function for seawater can be completely de
termined. However, once the liquid pure water density is
determined, the corresponding Gibbs potential is fully deter
mined and it can be used in the seawater functions described
in Sect. 4.2 and 4.3.
Finally, note that the library functions listed in Table S2 for
pure fluid water in terms of temperature and density are avail
able as similar functions of temperature and pressure with the
prefix liq_ for liquid water and vap_ for water vapour, re
spectively, rather than with the prefix f lu_ given in Table S2.
