Numerical implementation and oceanographic application of the Gibbs potential of ice

Abstract

The 2004 Gibbs thermodynamic potential function of naturally abundant water ice is based on much more experimental data than its predecessors, is therefore significantly more accurate and reliable, and for the first time describes the entire temperature and pressure range of existence of this ice phase. It is expressed in the ITS-90 temperature scale and is consistent with the current scientific pure water standard, IAPWS-95, and the 2003 Gibbs potential of seawater. The combination of these formulations provides sublimation pressures, freezing points, and sea ice properties covering the parameter ranges of oceanographic interest. This paper provides source code examples in Visual Basic, Fortran and C++ for the computation of the Gibbs function of ice and its partial derivatives. It reports the most important related thermodynamic equations for ice and sea ice properties.

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OSD
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Gibbs potential of ice
R. Feistel et al.
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Abstract
Introduction
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References
Tables
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Ocean Science Discussions, 2, 37–61, 2005
www.ocean-science.net/osd/2/37/
SRef-ID: 1812-0822/osd/2005-2-37
European Geosciences Union
Ocean Science
Discussions
Numerical implementation and
oceanographic application of the Gibbs
potential of ice
R. Feistel1, W. Wagner2, V. Tchijov3, and C. Guder2
1Baltic Sea Research Institute, Seestraße 15, D-18119 Warnem¨ unde, Germany
2Ruhr-Universit¨ at Bochum, Lehrstuhl f¨ ur Thermodynamik, D-44780 Bochum, Germany
3Centro de Investigaciones Te´ oricas, Facultad de Estudios Superiores Cuautitl´ an, UNAM,
M´ exico
Received: 21 December 2004 – Accepted: 31 January 2005 – Published: 3 February 2005
Correspondence to: R. Feistel (rainer.feistel@io-warnemuende.de)
© 2005 Author(s). This work is licensed under a Creative Commons License.
37

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OSD
2, 37–61, 2005
Gibbs potential of ice
R. Feistel et al.
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Abstract
Introduction
Conclusions
References
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Abstract
The 2004 Gibbs thermodynamic potential function of naturally abundant water ice is
based on much more experimental data than its predecessors, is therefore significantly
more accurate and reliable, and for the first time describes the entire temperature and
pressure range of existence of this ice phase. It is expressed in the ITS-90 temperature
scale and is consistent with the current scientific pure water standard, IAPWS-95, and
the 2003 Gibbs potential of seawater. The combination of these formulations provides
sublimation pressures, freezing points, and sea ice properties covering the parameter
ranges of oceanographic interest. This paper provides source code examples in Visual
Basic, Fortran and C++ for the computation of the Gibbs function of ice and its partial
derivatives. It reports the most important related thermodynamic equations for ice and
sea ice properties.
5
10
1. Introduction
Oceanography demands very accurate knowledge of freezing temperatures of seawa-
ter. For instance, global temperature rise, as manifested in actually 1.2◦C increase per
century of northern hemisphere land surface winter temperatures (Hagen and Feistel,
20051), is in general accompanied by much smaller changes of ocean temperatures.
But by freezing and melting processes in the polar and mid latitudes, even a tiny cool-
ing or warming of the sea can nonetheless have significant, nonlinear impacts on the
climate system, e.g. in form of growing or shrinking sea ice covers.
Several freezing point formulae are available for water and seawater. The Unesco
fomula of Millero (1978) is valid for air-saturated seawater at low pressures only (up
to 5MPa=500dBar) and does not take into account compressibilities of seawater and
ice. In the zero salinity limit, it is not consistent with the international equation for the
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20
1Hagen, E. and Feistel, R. : Climatic turning points and regime shifts in the Baltic Sea region:
the Baltic winter index (WIBIX, 1659-2002), Boreal Env. Res., submitted, 2005.
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melting pressure of Wagner et al. (1994) for air-free water which covers more than the
oceanic pressure range up to 100 MPa. None of both formulae, moreover, does agree
with the most accurate measured Clausius-Clapeyron coefficient within its experimen-
tal uncertainty (Feistel and Wagner, 2005).
With the numerical availability of mutually consistent formulations for the chemical
potentials of water (Wagner and Pruß, 2002) and seawater (Feistel, 2003), a Gibbs
function approach to the freezing point computation problem became appealing. Ear-
lier Gibbs functions of ice (Feistel, 1993, 2003; Feistel and Hagen, 1995; Tillner-Roth,
1998) were based on only few experimental data from the vicinity of the normal pres-
sure melting point, and suffered from e.g. the pending significant uncertainty of ice
compressibility. Therefore, a reliable and accurate Gibbs function of ice was desired,
consistent with the corresponding latest descriptions of water, vapour, and seawater,
which can provide freezing points of water and seawater over the entire oceanographic
“Neptunian” pressure range. An explicit freezing point formula derived from this newly
developed Gibbs formulation is given in Jackett et al. (2005).
When water or seawater freezes under natural conditions, crystals of hexagonal,
so-called ice Ih are formed. At high pressures, ice possesses at least twelve other
crystalline phases and two amorphous states, exclusively discovered in the laboratory.
In the current formulation, only the naturally abundant ice phase, ice Ih, is described,
applicable to the entire region of its existence from 0 to 210MPa in pressure und from
0 to 273.16K in temperature.
A detailed derivation of this function is given by Feistel and Wagner (20042, 2005).
It was determined by regression with respect to 339 data points belonging to 26 dif-
ferent groups of experiments, reproducing these data within their particular measure-
ment uncertainties. The new formulation obeys Debye’s cubic law at low tempera-
tures and pressure-independence of the residual entropy. The typical uncertainty of
about 100% in isothermal compressibilities of previous formulae is reduced to less than
5
10
15
20
25
2Feistel, R. and Wagner, W.: A comprehensive Gibbs thermodynamic potential of ice, Proc.
14th Int. Conf. Prop. Water Steam (ICPWS), Kyoto, Japan, submitted, 2004.
39

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1%. The resulting Clausius-Clapeyron slope of the melting curve at normal pressure
is 74.305mK/MPa with an uncertainty of only 0.02%, which is about 100 times more
accurate than the corresponding figures of previous formulae. The melting temperature
at normal pressure is inferred to be 273.152 518±0.000 002K.
The current paper provides numerical implementations of the new Gibbs function of
ice and its derivatives in Visual Basic (VB), Fortran, and C++. These implementa-
tions have been developed independently by the different authors of this paper, and
are subject to specific properties of the particular programming language and coding
rules. To give an example, VB does not support complex arithmetic and data types,
which needed to be implemented explicitly. Thus, the three source code versions differ
from each other in various details, and are therefore described separately one by one
in Sect. 3. These code examples are intended as functioning examples and possible
guides for the development of individual implementations into custom program envi-
ronments. They do not provide any user interface nor support data input or graphical
output.
For reference, a collection of explicit expressions for the partial derivatives of the
Gibbs function, as well as various relations to other thermodynamic properties are
given in Sect. 2. Finally, Sect. 4 presents formulae for sea ice, which additionally
require for their evaluation the Gibbs potential of seawater. Numerical implementations
of the 2003 Gibbs function of seawater in VB, Fortran, and C++ are available from a
companion article earlier in this journal (Feistel, 2004).
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2. Gibbs potential and its derivatives
The thermodynamic Gibbs potential function g(T,P ) described in this paper is the
specific free enthalpy (specific Gibbs energy) of ice, which is equal to the chemical
potential µ(T,P ) of ice. We express absolute temperature T by the dimensionless
variable t=T?Ttwith triple point temperature Tt, and absolute pressure P by p=P?Pt
40
25
with triple point pressure Pt. Triple point data and other constants are reported in

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Table 1.
The functional form of g(T,P ) for ice is given as a complex function of temperature,
?
k=1
with two coefficients being polynomials of pressure,
g(T,P)
gU
= g0− σ0τ · t + τRe
2
?
rk
?
(tk− t)ln(tk− t) + (tk+ t)ln(tk+ t) − 2tklntk−t2
tk
??
,(1)
g0(P ) =
4
?
k=0
g0k· (p − p0)k,r2(P ) =
2
?
k=0
r2k· (p − p0)k.
(2)
5
The unit specific free enthalpy is gU=1J/kg, the dimensionless constant τ is τ=Tt/1K =
273.160 and the dimensionless normal pressure is p0=P0/Pt, given in Table 1. The real
constants g00to g04and σ0as well as the complex constants t1, r1, t2and r20to r22
are given in Table 2.
The complex logarithm ln(z) is the principal value, i.e. it evaluates to imaginary parts
in the interval −π < Im ln(z) ≤ +π. The complex notation used here serves only for
compact writing and easy partial derivatives. The residual entropy coefficient σ0given
in Table 2 is its “IAPWS-95 version” as required for correct phase equilibria determina-
tions between ice and pure water using the IAPWS-95 formulation (Wagner and Pruß,
2002), or seawater (Feistel, 2003). Its alternative value represents the true absolute
physical zero-point entropy of ice, σ0=189.13, which is usually not required for practical
computations, however.
The first derivative of g with respect to its independent variable P , provides density,
ρ, and specific volume, v,
1
ρ= v =
∂P
T
?
dp
41
10
15
?∂g
dg0
dp
?
(3)
20
v
vU
=1
ψ
+τ
ψRe
dr2
?
(t2− t)ln(t2− t) + (t2+ t)ln(t2+ t) − 2t2lnt2−t2
t2
??