Abstract

The thermodynamic consequences of the melting of ice and sea ice into seawater are considered. The International Thermodynamic Equation Of Seawater—2010 (TEOS-10) is used to derive the changes in the Conservative Temperature and Absolute Salinity of seawater that occurs as a consequence of the melting of ice and sea ice into seawater. Also, a study of the thermodynamic relationships involved in the formation of frazil ice enables the calculation of the magnitudes of the Conservative Temperature and Absolute Salinity changes with pressure when frazil ice is present in a seawater parcel, assuming that the frazil ice crystals are sufficiently small that their relative vertical velocity can be ignored. The main results of this paper are the equations that describe the changes to these quantities when ice and seawater interact, and these equations can be evaluated using computer software that the authors have developed and is publicly available in the Gibbs SeaWater (GSW) Oceanographic Toolbox of TEOS-10.

1. Introduction

The International Thermodynamic Equation Of Seawater—2010 (TEOS-10) has been adopted as the international standard for the thermophysical properties of (i) seawater, (ii) ice Ih, and (iii) humid air. The TEOS-10 manual (IOC et al. 2010) summarizes the thermodynamic definitions of seawater, ice Ih, and humid air. The way that the thermodynamic potentials of these three substances were made consistent with each other is described in Feistel et al. (2008), and the scientific background to the announcement of this international standard is summarized in Pawlowicz et al. (2012). The terminology “ice Ih” stands for the ordinary hexagonal form of ice that is the naturally abundant form of ice, relevant for the pressure and temperature ranges found in the ocean and atmosphere [see Fig. 1b of Feistel et al. (2010)].

The temperature at which seawater begins freezing is determined from examining the thermodynamic equilibrium between the seawater and ice phases, with the relevant equilibrium condition being that the chemical potential of water in the seawater phase is equal to the chemical potential of water in the ice phase (Feistel and Hagen 1998; Feistel and Wagner 2005). Here we cast the freezing temperature in terms of the Conservative Temperature of seawater, and expressions are derived for the partial derivatives of the freezing Conservative Temperature with respect to Absolute Salinity and pressure (see  appendix C). Because Conservative Temperature is preferred over potential temperature as a measure of the “heat content per unit mass” of seawater (McDougall 2003; Tailleux 2010; Graham and McDougall 2013), we will concentrate on understanding the melting and freezing of ice in terms of its implications on the changes to the Conservative Temperature of seawater. The adoption by the Intergovernmental Oceanographic Commission of TEOS-10 as the new official definition of the properties of seawater, ice, and humid air involves the transition to publishing in the new oceanographic salinity and temperature variables Absolute Salinity and Conservative Temperature, in contrast to the practical salinity and potential temperature of the 1980 equation of state (EOS-80).

The adiabatic lapse rate of ice is shown to be much greater than that of seawater (often 10 times as large), implying that under isentropic vertical motion, the variation of the in situ temperature of ice with pressure is much larger than for seawater.

In this paper, we consider the quantities that are conserved when ice melts into seawater. Writing equations for these conserved quantities (including enthalpy) leads to closed expressions for the Absolute Salinity and Conservative Temperature of the seawater after the melting or freezing has occurred [see Eqs. (8) and (9) below]. These equations apply at finite amplitude and do not assume the ice and seawater to be near to a state of thermodynamic equilibrium. This approach can be linearized to give an expression for the ratio of the changes in Conservative Temperature and Absolute Salinity when a vanishingly small amount of ice melts into a large mass of seawater. This result of this linearization [see Eqs. (16) and (18) below] is comparable to that of Gade (1979), although our approach is more general because it is based on the rigorous conservation of three basic thermodynamic properties (mass, salt, and enthalpy), so that it applies without approximation at finite amplitude, and we also include the dependence of seawater enthalpy on salinity.

This analysis is extended to the melting of sea ice, which is treated as a coarse-grained mixture of pure ice in which pockets of brine are trapped and the salinity of the pockets of brine is determined by thermodynamic equilibrium between the brine and the surrounding ice. This brine salinity has the same value in all pockets with equal temperatures and pressures, irrespective of their particular sizes, and the TEOS-10 description of the thermal properties of the brine apply up to an Absolute Salinity of 120 g kg−1 [see section 2.6 of IOC et al. (2010)].

The upwelling of very cold seawater (colder than the surface freezing temperature) can lead to supercooling and the formation of small ice crystals called frazil ice, and this process is also examined using the TEOS-10 Gibbs functions of ice Ih and of seawater. Under the assumption that the relative vertical velocity (the Stokes velocity) of frazil can be ignored, we derive expressions for the rate at which the Absolute Salinity and the Conservative Temperature of seawater vary with pressure when frazil is present. Because of their tiny size, frazil ice crystals remain in thermodynamic equilibrium with the surrounding seawater when the parcel undergoes pressure excursions. From the thermodynamic perspective, a frazil ice parcel differs from a sea ice parcel only quantitatively, namely, by their opposite liquid–solid ratios. Properties such as the adiabatic lapse rates of these composite systems can formally be derived from a Gibbs function of sea ice (Feistel et al. 2010). Strictly speaking, the mixture of frazil ice with seawater is a metastable state. It still undergoes a slow process known as Ostwald ripening that minimizes the interface energy between ice and seawater, typically by finally forming a single piece of ice (or a single large brine pocket in the case of sea ice). In the TEOS-10 Gibbs function of sea ice, the interface energy is neglected.

A mixture of seawater and frazil ice has two important properties that are quite different from ice-free seawater. First, the second law of thermodynamics requires that the Gibbs function of seawater is a convex function of salinity, that is, the second derivative of the Gibbs function with respect to Absolute Salinity is positive, [see section A.16 of IOC et al. (2010)], while the Gibbs function of sea ice [gSI; see Eq. (54) below] is linear in sea ice salinity (Feistel and Hagen 1998), that is, . As a consequence of the latter, no irreversible mixing effects occur when two sea ice parcels are in contact at the same temperature and pressure but different sea ice salinities, in contrast to ice-free seawater where entropy is produced when parcels having contrasting salinities are mixed. Second, at brackish salinities (up to about 28 g kg−1) seawater possesses a temperature of maximum density where the adiabatic lapse rate changes its sign (McDougall and Feistel 2003), while sea ice exhibits a density minimum (Feistel and Hagen 1998). Ice has a much larger specific volume than water or seawater, and the freezing process is accompanied by volume expansion, that is, by a large negative thermal expansion coefficient (and lapse rate) of sea ice. This effect is strongest at low salinities and in fact the thermal expansion coefficient of pure water has a singularity at the freezing point. With decreasing temperature and increasing brine salinity, the rate of formation of ice in sea ice gradually decreases to the point where the volume increase caused by the newly formed ice (i.e., by the transfer of water from the liquid to the solid phase) is outweighed by the thermal contraction of the pure phases, ice and brine, so that the total thermal expansion coefficient of sea ice changes its sign and turns positive.

The thermodynamic interactions between ice and seawater described in this paper are first derived as equations between the various quantities and are illustrated graphically in the figures. In addition, the thermodynamic properties of ice Ih and the results from the equations of this paper are available as computer algorithms in the Gibbs SeaWater (GSW) Oceanographic Toolbox (McDougall and Barker 2011) and can be downloaded online (from www.TEOS-10.org).

2. The adiabatic lapse rate and the potential temperature of ice Ih

The adiabatic lapse rate is equal to the change of in situ temperature t experienced when pressure is changed while keeping entropy η (and salinity) constant. This definition applies separately to both ice and seawater (where one needs to keep not only entropy but also Absolute Salinity constant during the pressure change). In terms of the Gibbs functions of seawater and of ice Ih, the adiabatic lapse rates of seawater and of ice are expressed respectively as

 
formula

and

 
formula

where and are the thermal expansion coefficients of seawater and ice Ih, respectively, with respect to in situ temperature. Subscripts of the Gibbs functions and of seawater and ice, respectively, denote partial derivatives, and and are the density and the isobaric specific heat capacity.

The adiabatic lapse rates of seawater and ice are numerically substantially different from each other. The thermal expansion coefficient of ice does not change sign as does that of seawater when it is cooler than the temperature of maximum density, and the specific heat capacity of ice is only approximately 52% that of seawater . Figure 1a shows the ratio of the adiabatic lapse rates of seawater and ice at the freezing temperature, as a function of the Absolute Salinity of seawater and pressure. For salinities typical of the open ocean, the ratio is about 0.1, indicating that the in situ temperature of ice varies 10 times as strongly with pressure when both seawater and ice Ih are subjected to the same isentropic pressure variations.

Fig. 1.

(a) Ratio of the adiabatic lapse rates of seawater and of ice Ih at the freezing temperature. (b) Difference (°C) between the potential temperatures of seawater and of ice for parcels of seawater and ice whose in situ temperature is the in situ freezing temperature.

Fig. 1.

(a) Ratio of the adiabatic lapse rates of seawater and of ice Ih at the freezing temperature. (b) Difference (°C) between the potential temperatures of seawater and of ice for parcels of seawater and ice whose in situ temperature is the in situ freezing temperature.

This substantial difference between the adiabatic lapse rates is also illustrated in Fig. 1b as the difference in the potential temperature of seawater and of ice for seawater and ice parcels that are at the in situ freezing temperature. This difference in potential temperatures can be understood as follows: At every point on the p diagram of Fig. 1b, the in situ freezing temperature is calculated. Imagine now raising both a seawater sample and an ice sample from this pressure to the sea surface. Initially, both samples have the same in situ temperature, namely, the freezing temperature . As the pressure is reduced, the in situ temperatures of both the seawater and ice parcels are reduced (assuming that the seawater salinity is large enough that its thermal expansion coefficient is positive), but the temperature of the ice changes typically 10 times as much as that of the seawater. Thus, the two parcels that have the same in situ temperature have different potential temperatures (referenced to p = 0 dbar) as illustrated in Fig. 1b.

3. Pure ice Ih melting into seawater

a. The freezing temperature

As described in IOC et al. (2010), freezing of seawater occurs at the temperature at which the chemical potential of water in seawater equals the chemical potential of ice . Hence, the freezing temperature is found by solving the implicit equation

 
formula

or equivalently, in terms of the two Gibbs functions

 
formula

The Gibbs function of seawater , defined by Feistel (2008) and IAPWS (2008), is a function of the Absolute Salinity , the in situ temperature t, and the pressure of a seawater parcel p. The Gibbs function for ice Ih is defined by Feistel and Wagner (2006) and IAPWS (2009a) and is summarized in  appendix A. Note that Eq. (3) is valid for air-free seawater. The dissolution of air in water lowers the freezing point slightly; saturation with air lowers the freezing temperature by about 2.4 mK for freshwater and by about 1.9 mK at SA = 35.165 04 g kg−1.

The freezing in situ temperatures derived from Eq. (4) were converted to the Conservative Temperature at which air-free seawater freezes and are shown in Fig. 2a as a function of pressure and Absolute Salinity. To compare these TEOS-10 freezing temperatures to those of EOS-80, the conversion between the practical salinity of EOS-80 and Absolute Salinity of TEOS-10 was made using the conversion factor uPS ≡ (35.165 04/35) g kg−1 (Millero et al. 2008; IOC et al. 2010). It was assumed that the EOS-80 freezing temperatures of Millero and Leung (1976) were of air-saturated seawater. Having calculated the air-free in situ freezing temperature of EOS-80 in this manner, the Conservative Temperature is calculated from the TEOS-10 algorithm gsw_CT_from_t. The resulting differences between the freezing Conservative Temperatures from EOS-80 and TEOS-10 are illustrated in Fig. 2b and are very small at 0 dbar, rising to approximately 10 mK at 1000 dbar and 120 mK at 3000 dbar. We have developed a polynomial approximation for the freezing Conservative Temperature (see  appendix D), and the error in using this computationally efficient polynomial is seen in Fig. 2c to be very small, being no larger than 0.05 mK at the sea surface and no larger than approximately 0.25 mK at other pressures.

Fig. 2.

(a) Conservative Temperature (°C) at which air-free seawater freezes as a function of pressure and Absolute Salinity. (b) Difference between the freezing Conservative Temperature derived from EOS-80 and that of TEOS-10 (contours; mK). (c) Error (mK) in using the approximating polynomial expression of  appendix D for the freezing Conservative Temperature.

Fig. 2.

(a) Conservative Temperature (°C) at which air-free seawater freezes as a function of pressure and Absolute Salinity. (b) Difference between the freezing Conservative Temperature derived from EOS-80 and that of TEOS-10 (contours; mK). (c) Error (mK) in using the approximating polynomial expression of  appendix D for the freezing Conservative Temperature.

b. Finite-amplitude expressions for melting

We now turn our attention to the quantities that are conserved when a certain amount of ice melts into a known mass of seawater. In the following section, we will consider the melting of sea ice that contains pockets of brine, but in this section we consider the melting of pure ice Ih that contains no brine pockets. This section of the paper is appropriate when considering the melting of ice from glaciers or icebergs, because these types of ice are formed from compacted snow and hence do not contain the trapped seawater that is typical of ice formed at the sea surface, namely, sea ice.

The general case we consider in this section has the seawater temperature above its freezing temperature, while the ice, in order to be the stable phase ice Ih, needs to be at or below the freezing temperature of pure water (i.e., seawater having zero Absolute Salinity) at the given pressure level, typically at the sea surface. Note that this condition permits situations in which the initial ice temperature is higher than or equal to that of seawater. In other words, the general case we are considering is not an equilibrium situation in which certain amounts of ice and seawater coexist without further melting or freezing. During the melting of ice Ih into seawater at fixed pressure, entropy increases while three quantities are conserved: mass, salt, and enthalpy. While this process is assumed to be adiabatic it is not isentropic. Because of irreversibility, the freezing process is thermodynamically prohibited in a closed system. To form frazil ice in seawater at fixed pressure, more entropy must be exported from the sample than is produced internally; a typical example being an ice floe that is strongly cooled by the atmosphere.

The conservation equations for mass, salt, and enthalpy during this adiabatic melting event at constant pressure are

 
formula
 
formula
 
formula

The superscripts i and f stand for the initial and final values, that is, the values before and after the melting event, while the subscripts SW and Ih stand for seawater and ice Ih.

The mass, salinity, and enthalpy conservation Eqs. (5)(7) can be combined to give the following expressions for the differences in the Absolute Salinity and the specific enthalpy of the seawater phase due to the melting of the ice:

 
formula
 
formula

where we have defined the mass fraction of ice Ih as . The initial and final values of the specific enthalpy of seawater are given by and , where the specific enthalpy of seawater has been written in two different functional forms: one being a function of in situ temperature and the other being a function of Conservative Temperature. This TEOS-10 terminology, where an overhat adorning a thermodynamic variable (as in ) implies that the variable is being regarded as a function of Conservative Temperature, while an unadorned variable (such as ) implies that the thermodynamic variable is being regarded to be a function of in situ temperature, is used throughout this paper.

The use of Eqs. (8) and (9) is illustrated in Fig. 3, where the mass fraction of ice and the in situ temperature of the ice are varied at fixed values of the initial properties of the seawater at = 35.165 04 g kg−1, , and at p = 0 dbar. Note that these results apply for these finite-amplitude differences of temperature and salinity, and these calculations are accurate because of the existence of the TEOS-10 expressions for the specific enthalpies of seawater and ice Ih. We have not needed to resort to a linearization involving the specific heat capacities to obtain Eqs. (8)(9) and the results of Fig. 3. Clearly, the salinity difference in Fig. 3a is simply proportional to , as is also obvious from Eq. (8), while the (relatively weak) dependence of on is apparent from the plot of in Fig. 3b. Note that at p = 0 dbar, Eq. (9) becomes simply .

Fig. 3.

Change in (a) Absolute Salinity (g kg−1) and (b) Conservative Temperature (°C) when the mass fraction of ice is melted into seawater with initial properties = 35.165 04 g kg−1, , and at p = 0 dbar. These data were obtained from the GSW Oceanographic Toolbox function gsw_melting_ice_into_seawater. There are no contours on the right as the final calculated seawater properties there are cooler than the freezing temperature.

Fig. 3.

Change in (a) Absolute Salinity (g kg−1) and (b) Conservative Temperature (°C) when the mass fraction of ice is melted into seawater with initial properties = 35.165 04 g kg−1, , and at p = 0 dbar. These data were obtained from the GSW Oceanographic Toolbox function gsw_melting_ice_into_seawater. There are no contours on the right as the final calculated seawater properties there are cooler than the freezing temperature.

The conservation of Absolute Salinity and enthalpy when ice Ih melts into seawater is illustrated in Fig. 4a. The final values of Absolute Salinity and enthalpy given by Eqs. (8) and (9) are illustrated in Fig. 4a for four different values of the ice mass fraction . These final values lie on the straight line on the Absolute Salinity–enthalpy diagram connecting and . The fact that the same data do not fall on a straight line on the Absolute Salinity–in situ temperature diagram in Fig. 4b nicely illustrates that temperature is not conserved when melting occurs.

Fig. 4.

(a) Absolute Salinity–enthalpy diagram illustrates Eqs. (8) and (9) that embody the conservation of Absolute Salinity and enthalpy when ice Ih melts into seawater. Initial values of the Absolute Salinity and enthalpy of seawater and of ice Ih are shown by the two solid dots, and the final values of Absolute Salinity and enthalpy of the seawater after the ice has melted are shown by the four open circles (for four different values of the ice mass fraction ). These final values lie on the straight line in this diagram that connects the initial values (the solid dots). (b) The same initial and final data are shown in the Absolute Salinity–in situ temperature diagram. Note that the final points (the open circles) do not lie on the straight line connecting the initial points (the solid dots).

Fig. 4.

(a) Absolute Salinity–enthalpy diagram illustrates Eqs. (8) and (9) that embody the conservation of Absolute Salinity and enthalpy when ice Ih melts into seawater. Initial values of the Absolute Salinity and enthalpy of seawater and of ice Ih are shown by the two solid dots, and the final values of Absolute Salinity and enthalpy of the seawater after the ice has melted are shown by the four open circles (for four different values of the ice mass fraction ). These final values lie on the straight line in this diagram that connects the initial values (the solid dots). (b) The same initial and final data are shown in the Absolute Salinity–in situ temperature diagram. Note that the final points (the open circles) do not lie on the straight line connecting the initial points (the solid dots).

c. The linearized expression for the ratio

Gade (1979) developed a mechanistic model of both the laminar and turbulent diffusion of heat and freshwater between ice and seawater, and using both this model and a much simpler linearized version of the conservation of “heat” [in the appendix of Gade (1979)] was able to derive an expression for the ratio of the changes in temperature and salinity in seawater due to the melting of a vanishingly small amount of ice into seawater. Here we have used the simpler “heat budget” approach, which is formally the conservation of enthalpy, and this led to Eqs. (8) and (9) that hold at finite amplitude when a finite mass fraction of ice melts into seawater. In this subsection, we linearize these equations to find the expressions (15)(18) for the ratio of the changes in salinity and temperature when a vanishingly small mass fraction of ice melts into seawater. It is these Eqs. (15)(18) that correspond to Gade’s key result for this ratio.

The enthalpy difference in Eq. (9) is now expanded as a Taylor series in the differences in Absolute Salinity and temperature, and the first-order terms in these differences are retained, leading to

 
formula

where is the specific heat capacity of seawater, , and is the derivative of the seawater specific enthalpy with respect to Absolute Salinity at constant in situ temperature and constant pressure. By regarding specific enthalpy to be a function of Conservative Temperature in the functional form , the Taylor series expansion of Eq. (9) yields

 
formula

where is the partial derivative of the seawater specific enthalpy with respect to Conservative Temperature at fixed Absolute Salinity, and is the partial derivative of the seawater-specific enthalpy with respect to Absolute Salinity at fixed Conservative Temperature. Expressions for these partial derivatives can be found at Eqs. (B4) and (B5) of  appendix B. Equations (10) and (11) can be rewritten as

 
formula
 
formula

The parentheses on the right-hand side of Eq. (12), , if evaluated at the freezing temperature , is the latent heat of melting (i.e., the isobaric melting enthalpy) of ice into seawater, first derived by Feistel et al. (2010) [see also section 3.34 of IOC et al. (2010)]. Note that at p = 0 dbar, is zero, while is nonzero. Expressions for and in terms of the Gibbs function of seawater are given in  appendix B.

The derivation of the isobaric melting enthalpy in Feistel et al. (2010) and IOC et al. (2010) considered the seawater and ice to be in thermodynamic equilibrium during a slow process in which heat was supplied to melt the ice while maintaining a state of thermodynamic equilibrium during which the temperature of the combined system changed only because the freezing temperature is a function of the seawater salinity. During this reversible process, the enthalpy of the combined system increased due to the heat externally applied. The latent heat of melting is defined to be [from Eq. (3.34.6) of IOC et al. (2010)]

 
formula

The present derivation [i.e., Eqs. (12) and (13)] applies to the common situation when the seawater is warmer than the ice that is melting into it, so that the two phases are not in thermodynamic equilibrium with each other during the irreversible melting process. That is, the seawater temperature may be larger than its freezing temperature, and the ice temperature may or may not be less than its freezing temperature. The guiding thermodynamic principle is that there is no change in the enthalpy of the combined seawater and ice system during the irreversible melting process, because this process occurs adiabatically at constant pressure. When freezing (as opposed to melting) is considered, the second law of thermodynamics implies that spontaneous freezing cannot occur except when the seawater is at the freezing temperature (or in a metastable, subcooled state below that), and there must be some incremental external change (e.g., a decrease in pressure in the case of frazil formation or a loss of heat from the system) in order to induce the freezing.

Taking the limit of melting a small amount of ice into a seawater parcel so that the changes in the seawater temperature and salinity are small, we find from Eq. (12) that the ratio of the changes of in situ temperature and Absolute Salinity is given by

 
formula

while the corresponding ratio of the changes in Conservative Temperature and Absolute Salinity is [from Eq. (13)]

 
formula

where the second lines of these equations have been included to be very clear about how these quantities are evaluated. At p = 0 dbar, these equations become

 
formula

and

 
formula

where the potential temperatures of seawater and ice are both referenced to p = 0 dbar. Note that the potential enthalpy of seawater referenced to p = 0 dbar, , is simply times Conservative Temperature, where the constant “specific heat” = 3991.867 957 119 63 J kg−1 K−1.

Equation (17) is very similar to Eq. (25) of Gade (1979). If we associate Gade’s temperature “at the ice-water interface” [Eq. (10) in Gade 1979] with , then the difference between the values of the enthalpy of seawater and of ice at the freezing temperature can be interpreted as Gade’s latent heat term L, while the difference between the enthalpy of seawater and its enthalpy at the freezing temperature is approximately equal to the term that appears in Gade’s equation. The corresponding difference in the enthalpy of ice at its temperature and the value at the freezing temperature is approximately given by Gade’s term . The terms and in Eqs. (16) and (17), respectively, being the appropriate partial derivatives of seawater enthalpy with respect to Absolute Salinity, were absent in Gade’s approach, but in any case these terms are both small because of the deliberate choice of one of the four arbitrary coefficients of the Gibbs function of TEOS-10.

The use of Conservative Temperature rather than potential temperature means that the slope of the melting process on the diagram involves a simpler expression, especially when the melting occurs at the sea surface at p = 0 dbar [Eq. (18)], where (i) is zero, and (ii) the relevant “specific heat capacity” of seawater [see Eq. (B4) of  appendix B] reduces to the constant , so that the specific enthalpy of seawater is simply multiplied by the Conservative Temperature. Note that the numerator of the middle expression of Eq. (18) is simply the difference between the potential enthalpies of seawater and of ice.

The very simple Eqs. (16) and (18) for the slope of the melting process on the diagram are key results of this paper. These equations are linearizations of the exact Eqs. (8) and (9) whose simplicity and rigor are due to the fact that the first law of thermodynamics guarantees that the total enthalpy of the system is unchanged, as is illustrated in Fig. 4. Note that the right-hand side of Eq. (18) is independent of the Absolute Salinity of the seawater into which the ice melts.

We first illustrate these equations for the ratio of the changes of Conservative Temperature to those of Absolute Salinity by considering the melting to occur very close to thermodynamic equilibrium conditions. If both the seawater and the ice were exactly at the freezing temperature at the given values of Absolute Salinity and pressure, then no melting or freezing would occur. In Fig. 5, we consider the limit as the temperatures of both the seawater and the ice approach the freezing temperature. The ratio from Eq. (16) is shown in Fig. 5a with the seawater enthalpy evaluated at the freezing Conservative Temperature and with the ice enthalpy evaluated at the in situ freezing temperature, at each value of pressure and Absolute Salinity. This ratio is proportional to the reciprocal of Absolute Salinity, so it is more informative to simply multiply by Absolute Salinity; this is shown in Fig. 5b. It is seen that the melting of a given mass of ice into seawater near equilibrium conditions requires between approximately 81 and 83 times as much heat as would be required to raise the same mass of seawater by 1°C.

Fig. 5.

(a) Ratio of the change of Conservative Temperature to that of Absolute Salinity when the melting occurs very near thermodynamic equilibrium conditions from Eq. (16) with the seawater enthalpy evaluated at the freezing Conservative Temperature and with the ice enthalpy evaluated at the in situ freezing temperature at each value of pressure and Absolute Salinity. The values contoured have units of K (g kg−1)−1. (b) Absolute Salinity times the values of (a), that is, it is the right-hand side of Eq. (16), evaluated at equilibrium conditions. (c) The right-hand side of Eq. (15) minus the right-hand side of Eq. (16), both evaluated at equilibrium conditions, illustrating the difference between using in situ vs Conservative Temperature. The quantities contoured in (b) and (c) have temperature units (K). The values contoured in (a) and (b) were evaluated from the function gsw_melting_ice_equilibrium_SA_CT_ratio of the GSW Oceanographic Toolbox.

Fig. 5.

(a) Ratio of the change of Conservative Temperature to that of Absolute Salinity when the melting occurs very near thermodynamic equilibrium conditions from Eq. (16) with the seawater enthalpy evaluated at the freezing Conservative Temperature and with the ice enthalpy evaluated at the in situ freezing temperature at each value of pressure and Absolute Salinity. The values contoured have units of K (g kg−1)−1. (b) Absolute Salinity times the values of (a), that is, it is the right-hand side of Eq. (16), evaluated at equilibrium conditions. (c) The right-hand side of Eq. (15) minus the right-hand side of Eq. (16), both evaluated at equilibrium conditions, illustrating the difference between using in situ vs Conservative Temperature. The quantities contoured in (b) and (c) have temperature units (K). The values contoured in (a) and (b) were evaluated from the function gsw_melting_ice_equilibrium_SA_CT_ratio of the GSW Oceanographic Toolbox.

The corresponding result for the ratio of the changes of in situ temperature and Absolute Salinity near equilibrium conditions can be calculated from Eq. (15), and the difference between and is shown in Fig. 5c. The largest contributor to this difference between Eqs. (15) and (16) is due to the dependence of the specific heat capacity on (i) Absolute Salinity, involving a 6.8% variation over this full range of salinity, and (ii) on pressure, involving a change of 2.2% between 0 and 3000 dbar.

Equation (16) for is now illustrated when the seawater and the ice Ih are not at the same temperature and are not in thermodynamic equilibrium at the freezing temperature. We begin by considering the melting of ice Ih at the sea surface, specifically at p = 0 dbar, when Eq. (16) reduces to Eq. (18); this equation is illustrated in Fig. 6a, which applies at all values of Absolute Salinity. The contoured values of Fig. 6a, , increase as 1.0 times changes in and decrease approximately as times changes in the temperature of the ice.

Fig. 6.

(a) Contours of Eq. (18) for the melting of ice Ih into seawater at p = 0 dbar. The six stars are at the freezing temperatures (t and ) for Absolute Salinity values starting at 5 g kg−1 with increments of 5 g kg−1, up to 30 g kg−1. (b) Difference between contours of Eq. (16) at p = 500 dbar, , and the corresponding ratio of (a) (where the pressure was 0 dbar) at = 35.165 04 g kg−1. The double-starred point is at the freezing temperatures (t and ) at p = 500 dbar and = 35.165 04 g kg−1. (c) The mass fraction of ice , which when melted into seawater at = 35.165 04 g kg−1, at p = 0 dbar, and at the Conservative Temperature given by the vertical axis, results in the final mixed seawater that is at the freezing temperature. This figure has been found from the GSW algorithm gsw_ice_fraction_to_freeze_seawater. The double-starred point is at the freezing temperatures (t and ) at p = 0 dbar and = 35.165 04 g kg−1. (d) Values of at = 35.165 04 g kg−1 for various values of pressure up to 3000 dbar. The quantities contoured in (a) and (b) are temperatures (K), while that of (c) is the unitless mass fraction . The values contoured in (a) and (b) were evaluated from the algorithm gsw_melting_ice_SA_CT_ratio of the GSW Oceanographic Toolbox, the values of (b) were found from the algorithm gsw_ice_fraction_to_freeze_seawater, and those of (d) were found from the algorithm gsw_enthalpy_first_derivatives_CT_exact.

Fig. 6.

(a) Contours of Eq. (18) for the melting of ice Ih into seawater at p = 0 dbar. The six stars are at the freezing temperatures (t and ) for Absolute Salinity values starting at 5 g kg−1 with increments of 5 g kg−1, up to 30 g kg−1. (b) Difference between contours of Eq. (16) at p = 500 dbar, , and the corresponding ratio of (a) (where the pressure was 0 dbar) at = 35.165 04 g kg−1. The double-starred point is at the freezing temperatures (t and ) at p = 500 dbar and = 35.165 04 g kg−1. (c) The mass fraction of ice , which when melted into seawater at = 35.165 04 g kg−1, at p = 0 dbar, and at the Conservative Temperature given by the vertical axis, results in the final mixed seawater that is at the freezing temperature. This figure has been found from the GSW algorithm gsw_ice_fraction_to_freeze_seawater. The double-starred point is at the freezing temperatures (t and ) at p = 0 dbar and = 35.165 04 g kg−1. (d) Values of at = 35.165 04 g kg−1 for various values of pressure up to 3000 dbar. The quantities contoured in (a) and (b) are temperatures (K), while that of (c) is the unitless mass fraction . The values contoured in (a) and (b) were evaluated from the algorithm gsw_melting_ice_SA_CT_ratio of the GSW Oceanographic Toolbox, the values of (b) were found from the algorithm gsw_ice_fraction_to_freeze_seawater, and those of (d) were found from the algorithm gsw_enthalpy_first_derivatives_CT_exact.

d. The influence of pressure on the melting ratio

Considering now the melting process at a gauge pressure larger than 0 dbar, the right-hand side of Eq. (16) is evaluated at p = 500 dbar and = 35.165 04 g kg−1, with the differences between these values and the corresponding values at p = 0 dbar contoured in Fig. 6b. That is, this figure is the difference between the right-hand sides of Eqs. (16) and (18), with the in situ temperature of the ice being converted into the potential temperature of ice before Eq. (18) was evaluated. The large star in this figure represents the equilibrium point. The differences are not large and are about 0.15% of . The differences scale almost linearly with pressure; at 3000 dbar the corresponding differences (not shown) are approximately 6.4 times those illustrated at 500 dbar in Fig. 6b. We will now show that the main reason for the differences is the different specific volumes of seawater and ice.

The ratio of Eq. (16) to (18) is given by

 
formula

and we ask how different is the ratio for melting occurring at a general pressure p compared with using the expression (18), which involves the potential enthalpies of seawater and of ice, but is only 100% accurate for melting at p = 0 dbar. The first term after unity on the right-hand side of the last expression of Eq. (19) is , and this term is illustrated in Fig. 6d at = 35.165 04 g kg−1. This term is responsible for less than one-tenth of the 0.15% differences that we see in Fig. 6b between Eqs. (16) and (18) at 500 dbar. The last term in Eq. (19) involves a combination of enthalpy differences that we can express as follows (with the primed variables being the variable of integration and the use of upper case for the integration variable serves to remind that it must be in SI units of pascals in order to have enthalpy and specific volume in their usual units):

 
formula

where these expressions result from the definition of potential enthalpy and the fact that the pressure derivative of specific enthalpy, under adiabatic and isohaline conditions, is equal to the specific volume υ [this is true of both seawater and ice, that is, and , where the specific enthalpy of seawater is written in the functional form and ]. The last term in Eq. (20) is small so that the dominant contribution is due to the nontrivial difference between the specific volumes of seawater and ice. The observed almost linear dependence on the pressure of Eq. (16) is obvious from the form of Eq. (20).

The conclusion from this comparison between Eqs. (16) and (18) is that as far as evaluating the slope on the diagram of melting of ice into seawater at a general pressure p, very little error is made if the melting is assumed to occur at p = 0 dbar and taking the relevant enthalpy difference to be the difference between the potential enthalpies of seawater and of ice Ih, as in Eq. (18). The error in the slope is 0.15% at p = 500 dbar and 0.9% at p = 3000 dbar.

e. An illustration from the Amery Ice Shelf

Figure 7 shows oceanographic data obtained under the Amery Ice Shelf that illustrate the ratio of the changes in Absolute Salinity and Conservative Temperature, as given by Eq. (16), when the melting of ice occurs. The vertical profile named AM06 begins under the ice at a pressure of 546 dbar and the uppermost 175 m of the vertical profile is shown. The data in the uppermost 50–100 dbar are closely aligned with the ratio given by Eq. (16) (as shown by the dashed line) evaluated at this pressure and with the ice temperature being the freezing temperature at this salinity and pressure. Two freezing lines are shown in Fig. 7b, for pressures of 0 and 578 dbar. Any observations cooler than the freezing temperature appropriate to 0 dbar is evidence of the influence of melting of ice or of heat lost by conduction through the ice. AM06 is located on the eastern side of the ice shelf in an area that is melting, as can be inferred by the presence of ocean water at AM06 that is well above the in situ freezing temperature at the base of the ice shelf. This water is thought to be flowing in a primarily southward direction from the open ocean as it enters the underice cavity. The other CTD profile was taken from borehole AM05, located on the western side of the ice shelf in an area that is refreezing [as is drawn in Fig. 7a] and represents flow that has likely come from deeper in the cavity below the ice shelf than at AM06 (Post et al. 2014) and hence has been in contact with the ice for longer. The upper 50 m or so of this cast is at the freezing temperature of seawater at this pressure. For both casts the data near the upper part of the water column have the ratio of the changes of and in close agreement to the ratio given by Eq. (16), the ratio predicted from melting ice into seawater (dashed lines). The ice temperature that is needed to calculate this ratio for each location has been taken to be the in situ freezing temperature of ice in contact with the seawater at the pressure at the base of the ice shelf. Moreover, in this figure the uppermost 100 m of the AM05 data is approximately related to that of the AM06 data through the ratio of Eq. (16). This would be consistent with the notion that the same fluid is proceeding from AM06 to AM05 without being exposed to significant heat loss to the ice [see Fig. 7a]. The vertical profiles shown in Fig. 7b are the average of several vertical profiles taken over the course of 2 days, and the two locations were drilled within 2 weeks of each other.

Fig. 7.

(a) Sketch of the flow under an ice shelf. An inflow of relatively warm water from the open ocean provides heat to melt the ice shelf. Buoyant freshwater that is released during the melting process rises along the underside of the ice shelf and can become locally supercooled at a shallower depth, leading to the formation of frazil and basal accretion of marine ice. (b) The top 175 m of two CTD profiles taken below the Amery Ice Shelf in East Antarctica at a melt site and at a refreeze site are shown. The warmer and saltier of the two casts is AM06 [see Fig. 1 of Galton-Fenzi et al. (2012)] starting at a pressure of 546 dbar. The large round dot is ocean data very near the ice at 546 dbar, the triangle is 50 dbar deeper, the diamond is 100 dbar deeper, and the star is 150 dbar below the bottom of the ice shelf at this location, indicated by the circle. The other vertical cast (AM05) is typical of refreezing locations. The uppermost 50 dbar of this cast is all at the freezing temperature at this pressure.

Fig. 7.

(a) Sketch of the flow under an ice shelf. An inflow of relatively warm water from the open ocean provides heat to melt the ice shelf. Buoyant freshwater that is released during the melting process rises along the underside of the ice shelf and can become locally supercooled at a shallower depth, leading to the formation of frazil and basal accretion of marine ice. (b) The top 175 m of two CTD profiles taken below the Amery Ice Shelf in East Antarctica at a melt site and at a refreeze site are shown. The warmer and saltier of the two casts is AM06 [see Fig. 1 of Galton-Fenzi et al. (2012)] starting at a pressure of 546 dbar. The large round dot is ocean data very near the ice at 546 dbar, the triangle is 50 dbar deeper, the diamond is 100 dbar deeper, and the star is 150 dbar below the bottom of the ice shelf at this location, indicated by the circle. The other vertical cast (AM05) is typical of refreezing locations. The uppermost 50 dbar of this cast is all at the freezing temperature at this pressure.

4. Sea ice melting into seawater

a. Finite-amplitude expressions for melting

Now we consider the situation where the ice contains a certain fraction of salt, such as occurs when ice is formed by freezing from seawater. We reserve the name “sea ice” for this mixture of pure ice Ih and a small amount of trapped brine that is in thermodynamic equilibrium with the ice Ih at the temperature of the ice Ih . Note that the sea ice that contains pure ice Ih and a small amount of brine is all at the same (°C) temperature . The Absolute Salinity of the trapped brine can be calculated from the thermodynamic equilibrium condition Eq. (4) from knowledge of the pressure and temperature of the sea ice, and this relationship is shown in Fig. 8. The specific enthalpy is evaluated from the specific enthalpy of seawater as , while the enthalpy of the ice Ih is evaluated at the same temperature of the sea ice, namely, . The separate potential enthalpies of seawater and of ice in sea ice are and , respectively.

Fig. 8.

The in situ freezing temperature (°C) of air-free seawater as a function of pressure (dbar) and Absolute Salinity, determined from the equilibrium freezing condition Eq. (4). In the context of sea ice, in situ temperature is the temperature of both the pure ice Ih phase and of the trapped pockets of brine.

Fig. 8.

The in situ freezing temperature (°C) of air-free seawater as a function of pressure (dbar) and Absolute Salinity, determined from the equilibrium freezing condition Eq. (4). In the context of sea ice, in situ temperature is the temperature of both the pure ice Ih phase and of the trapped pockets of brine.

The bulk sea ice salinity , which may be as large as 10 g kg−1 but is more commonly around 3–5 g kg−1, is defined to be the mass fraction of sea salt in sea ice so that

 
formula

The conservation of mass, salt, and enthalpy when the melting of sea ice occurs into seawater at fixed pressure are given by the equations

 
formula
 
formula
 
formula

where , being the mass-weighted summation of the enthalpy of the two components of sea ice, namely, ice Ih and brine. The subscripts SW and Ih indicate seawater and ice, respectively. Note that all of the sea ice is assumed to melt so that the final mass of sea ice is taken to be zero. If the bulk Absolute Salinity of the sea ice is zero, then the mass of brine is also zero, and these equations reduce to those of section 3, that is, the sea ice is in fact ice Ih. A different physical limit occurs when the temperature of the sea ice is such that the brine salinity is equal to . In this limit, the sea ice contains no ice Ih and is actually 100% seawater brine of Absolute Salinity equal to . The various sea ice algorithms in the GSW Oceanographic Toolbox avoid this situation by artificially ensuring that the input in situ temperature of the sea ice always less than the air-free freezing temperature at by at least 10−6°C.

The difference between the final and initial values of the Absolute Salinities of the seawater phase can be found from Eqs. (21) to (23) to be

 
formula

where we have defined the mass fraction of sea ice as . Using Eqs. (21), (22), and (23), we find the following equation for the difference between the final and initial values of specific enthalpy of the seawater phase:

 
formula

The use of Eqs. (25) and (26) is illustrated in Fig. 9 where the mass fraction of sea ice and the in situ temperature of the sea ice are varied at fixed values of the initial properties of the seawater at = 35.165 04 g kg−1, , and at p = 0 dbar, and the sea ice salinity is taken to be = 5 g kg−1. Note that these finite-amplitude calculations are accurate because of the existence of the TEOS-10 expressions for the specific enthalpies of seawater and ice Ih. Clearly, the salinity difference in Fig. 9a is simply proportional to , as is obvious from Eq. (25). The differences in Conservative Temperature achieved in this melting process are shown in Figs. 9b and 9c. When the sea ice is not very cold, is quite sensitive to the sea ice temperature.

Fig. 9.

(a) The change in the Absolute Salinity (g kg−1) and (b) Conservative Temperature (°C) when the mass fraction of sea ice is melted into seawater with initial properties = 35.165 04 g kg−1, = 4°C, and at p = 0 dbar and with = 5 g kg−1. These were obtained from the GSW Oceanographic Toolbox function gsw_melting_seaice_into_seawater, specifying = 5 g kg−1. (c) A zoomed-in plot of (b) with the same axes as Fig. 3b. Contours for ice cooler than −7.6°C are not shown because in this range the Absolute Salinity of the brine pockets in the sea ice exceeds 120 g kg−1.

Fig. 9.

(a) The change in the Absolute Salinity (g kg−1) and (b) Conservative Temperature (°C) when the mass fraction of sea ice is melted into seawater with initial properties = 35.165 04 g kg−1, = 4°C, and at p = 0 dbar and with = 5 g kg−1. These were obtained from the GSW Oceanographic Toolbox function gsw_melting_seaice_into_seawater, specifying = 5 g kg−1. (c) A zoomed-in plot of (b) with the same axes as Fig. 3b. Contours for ice cooler than −7.6°C are not shown because in this range the Absolute Salinity of the brine pockets in the sea ice exceeds 120 g kg−1.

We have not contoured values of or in Fig. 9 for mass fractions of sea ice when this would result in the final seawater value of being less than the freezing temperature. As the temperature of the sea ice is increased and approaches the warmest allowed (which is the freezing temperature of seawater having an Absolute Salinity of = 5 g kg−1), larger mass fractions of sea ice are admissible because the ratio approaches 1.0, and the second term in the first line of Eq. (26) becomes significantly positive and acts against the first negative term in this equation. The maximum sea ice mass fraction that can be melted into seawater with initial properties = 35.165 04 g kg−1, = 1°C, and p = 0 dbar can be calculated implicitly from Eq. (26), and this is shown in Fig. 10. Values of in the region of Fig. 9 in which there are no contours would result in the final seawater being frozen.

Fig. 10.

The maximum sea ice mass fraction that can be melted into seawater with initial properties = 35.165 04 g kg−1, = 1°C, and at p = 0 dbar. Data for this figure were calculated from the GSW Oceanographic Toolbox function gsw_seaice_fraction_to_freeze_seawater, which solves Eq. (26) implicitly using a modified Newton method (McDougall and Wotherspoon 2014).

Fig. 10.

The maximum sea ice mass fraction that can be melted into seawater with initial properties = 35.165 04 g kg−1, = 1°C, and at p = 0 dbar. Data for this figure were calculated from the GSW Oceanographic Toolbox function gsw_seaice_fraction_to_freeze_seawater, which solves Eq. (26) implicitly using a modified Newton method (McDougall and Wotherspoon 2014).

The conservation of Absolute Salinity and enthalpy when sea ice melts into seawater is illustrated in Fig. 11a. The final values of Absolute Salinity and enthalpy given by Eqs. (25) and (26) are illustrated in Fig. 11a for four different values of the sea ice mass fraction . These final values lie on the straight line on the Absolute Salinity–enthalpy diagram connecting and . The fact that the same data do not fall on a straight line on the Absolute Salinity–in situ temperature diagram is illustrated in Fig. 11b. This nicely illustrates that temperature is not conserved when melting occurs.

Fig. 11.

(a) Absolute Salinity–enthalpy diagram illustrates Eqs. (25) and (26) that embody the conservation of Absolute Salinity and enthalpy when sea ice melts into seawater. Initial values of the Absolute Salinity and enthalpy of seawater and of sea ice are shown by two solid dots, and the values of Absolute Salinity and enthalpy of the seawater after the sea ice has melted are shown by four open circles (for four different values of the sea ice mass fraction ). These final values lie on the straight line in this diagram that connects the initial values (the solid dots). (b) The same initial and final data are shown in the Absolute Salinity–in situ temperature diagram. Note that the final points (the open circles) do not lie on the straight line connecting the initial points (the solid dots).

Fig. 11.

(a) Absolute Salinity–enthalpy diagram illustrates Eqs. (25) and (26) that embody the conservation of Absolute Salinity and enthalpy when sea ice melts into seawater. Initial values of the Absolute Salinity and enthalpy of seawater and of sea ice are shown by two solid dots, and the values of Absolute Salinity and enthalpy of the seawater after the sea ice has melted are shown by four open circles (for four different values of the sea ice mass fraction ). These final values lie on the straight line in this diagram that connects the initial values (the solid dots). (b) The same initial and final data are shown in the Absolute Salinity–in situ temperature diagram. Note that the final points (the open circles) do not lie on the straight line connecting the initial points (the solid dots).

b. The linearized expression for the ratio of changes in and

The left-hand side of Eq. (26) is expanded in a Taylor series at a fixed pressure so that to first order this equation becomes [taking specific enthalpy in the functional form ]

 
formula

while considering specific enthalpy in the functional form , we obtain

 
formula

and on using Eq. (25) we find

 
formula

and

 
formula

We will henceforth concentrate on Eq. (30) and the changes in Conservative Temperature rather than the changes in the in situ temperature of Eq. (29).

Dividing Eq. (30) by and taking the limit as these differences tend to zero and using Eq. (25), we find

 
formula

which can be rearranged to be

 
formula

The first property of this melting ratio, Eq. (32), applicable to the melting of sea ice into seawater at any pressure, is that if both the sea ice and the seawater are at the freezing temperature, then the right-hand side of Eq. (32) becomes the same as the expression Eq. (16) for pure ice Ih, namely, , and is independent of the concentration of salt in the sea ice . This pleasingly simple result occurs because in this situation (i) the brine salinity is the same as the Absolute Salinity of the seawater phase, and (ii) the enthalpy of seawater is equal to the enthalpy of the brine. Hence, both terms in the second half of the numerator in Eq. (32) are zero.

Now we restrict attention to the case where the melting of sea ice into seawater is occurring at p = 0 dbar, corresponding to the trapping of brine into an ice matrix only occurring due to the rapid freezing of ice at the sea surface. The melting ratio of sea ice is shown in Fig. 12a as a function of the in situ temperature of the sea ice and the Absolute Salinity of the sea ice when it is melting into seawater with the properties = 35.165 04 g kg−1, = 1°C, and p = 0 dbar. For sea ice that is more than 1°C cooler than the freezing temperature ; the melting ratio is not a particularly strong function of or , but the melting ratio varies very strongly as the freezing temperature is approached. In this limit, the sea ice contains no ice Ih and is simply seawater. Another view of this is shown in Fig. 12b, which is of the full Eq. (31). For the case of pure ice Ih, Eq. (31) at p = 0 dbar is and so is simply proportional to the enthalpy difference between seawater and ice Ih. The values contoured in Fig. 12b represent the effective enthalpy difference that is apparent when sea ice melts into seawater. As in Fig. 12a, the values are highly sensitive to only when the freezing temperature is approached.

Fig. 12.

(a) The melting ratio for sea ice melting into seawater at = 35.165 04 g kg−1, = 1°C, and p = 0 dbar. Data contoured here are calculated according to Eq. (31) and came from the function gsw_melting_seaice_SA_CT_ratio of the GSW Oceanographic Toolbox. (b) Plot of Eq. (31), that is, , which is simply times what is plotted in (a). (c) Plot of Eq. (33), being the ratio of (b) to the corresponding ratio for pure ice Ih. The quantity contoured in (a) has units of K (g kg−1)−1, while that in (b) is temperature (K). Contours for ice cooler than −7.6°C are not shown because in this range the Absolute Salinity of the brine pockets in the sea ice exceeds 120 g kg−1.

Fig. 12.

(a) The melting ratio for sea ice melting into seawater at = 35.165 04 g kg−1, = 1°C, and p = 0 dbar. Data contoured here are calculated according to Eq. (31) and came from the function gsw_melting_seaice_SA_CT_ratio of the GSW Oceanographic Toolbox. (b) Plot of Eq. (31), that is, , which is simply times what is plotted in (a). (c) Plot of Eq. (33), being the ratio of (b) to the corresponding ratio for pure ice Ih. The quantity contoured in (a) has units of K (g kg−1)−1, while that in (b) is temperature (K). Contours for ice cooler than −7.6°C are not shown because in this range the Absolute Salinity of the brine pockets in the sea ice exceeds 120 g kg−1.

We now take the ratio of Eq. (31) at p = 0 dbar with Eq. (18), namely, , which applies to the melting of ice Ih (as opposed to sea ice), obtaining

 
formula

In this form, it is clear that the melting ratio becomes infinite when the seawater salinity is the same as the Absolute Salinity of the sea ice. In other words, the reciprocal of this ratio changes sign as changes sign (which would only occur in unusual circumstances where sea ice was blown into a region of relatively freshwater). In the special case where both the sea ice and the seawater are at the freezing temperature, then and so that the right-hand side of Eq. (31) becomes simply , confirming that Eq. (31) becomes the same as Eq. (18). The influence of on the melting ratio is illustrated in Fig. 12c, which is a contour plot of Eq. (33) for = 35.165 04 g kg−1, , and at p = 0 dbar. It is seen that the melting ratio is strongly affected by the presence of salt in the sea ice only when the sea ice is not very cold.

5. Frazil ice formation through adiabatic uplift of seawater

When seawater at the freezing temperature undergoes upward vertical motion so that its pressure decreases, frazil forms, primarily due to the increase in the freezing temperature as a result of the reduction in pressure. When this mixture of seawater and frazil continues to rise to lower pressures (assisted by the buoyancy provided by the presence of the ice), the frazil will experience a larger change in their in situ temperature than the seawater, simply because the adiabatic lapse rate of ice is much larger than that of seawater (see section 2 above). We will here consider this situation under the assumption that the frazil and the seawater moves together, so ignoring the tendency of the frazil to rise faster than the seawater, driven by the buoyancy of the individual ice crystals. We further assume that the uplift rate is sufficiently small that the in situ temperature of the ice and the seawater are the same at each pressure, this temperature being the freezing temperature. Under these conditions no entropy is produced during the freezing process, that is, this freezing process is reversible and can be reversed by increasing the pressure, leading to the related reversible ice melt. The lack of entropy production occurs as long as the exchange of heat and water between seawater and ice is conducted at mutual equilibrium, that is, at equal temperatures and equal chemical potentials of the two phases.

The previous two sections of this paper considered the irreversible melting of ice into seawater at constant pressure, and thermodynamic equilibrium was not assumed except between the pockets of brine and the surrounding ice Ih of the sea ice matrix. This section is different because (i) it considers the seawater phase and the small frazil ice crystals to be in thermodynamic equilibrium, and (ii) we study the consequences of a change in pressure of the seawater/frazil ice mixture.

We will study the thermodynamics of this process of adiabatic uplift of a seawater–ice mixture via a two-step thought process composed of three stages (Fig. 13). First, we imagine the mixture of preexisting ice and seawater to undergo a reduction in pressure but without any exchange of heat, water, or salt between the two phases. That is, during this first part of the process the mass of ice and the mass of seawater remain constant, and the change in the enthalpy of the ice and the change in the enthalpy of the seawater are only due to the pressure change. During this adiabatic process a (infinitesimal) contrast in in situ temperature will develop between the ice phase and the seawater phase because the adiabatic lapse rate of ice is much larger (by about an order of magnitude) than that of seawater (see Fig. 1a). During the second part of our thought experiment, the ice and seawater phases will be allowed to equilibrate their temperatures and further frazil ice will form so that the temperature of both the ice and seawater phases and the final Absolute Salinity of the seawater phase will be consistent with the freezing temperature at this pressure. This part of our thought experiment occurs at constant pressure and so, from the first law of thermodynamics, we know that enthalpy is conserved.

Fig. 13.

Sketch showing the two-step thought process involved with quantifying the formation of frazil ice Ih by the adiabatic uplift of a seawater parcel that may contain preexisting frazil ice. The step from stage 1 to 2 is undertaken without any exchange of heat or mass between the seawater and ice Ih phases. While the in situ temperatures of the seawater and ice phases are assumed to be identical at stage 1, at stage 2 they are unequal because the adiabatic lapse rate of ice Ih is much larger than that of seawater. The step from stage 2 to 3 is undertaken at constant pressure. In this step further ice forms (as shown by the increase in number of the frazil ice crystals), and at the end of this stage the seawater and ice phases have the same in situ temperature, namely, the freezing temperature appropriate to 1) that pressure and 2) the final value of seawater salinity.

Fig. 13.

Sketch showing the two-step thought process involved with quantifying the formation of frazil ice Ih by the adiabatic uplift of a seawater parcel that may contain preexisting frazil ice. The step from stage 1 to 2 is undertaken without any exchange of heat or mass between the seawater and ice Ih phases. While the in situ temperatures of the seawater and ice phases are assumed to be identical at stage 1, at stage 2 they are unequal because the adiabatic lapse rate of ice Ih is much larger than that of seawater. The step from stage 2 to 3 is undertaken at constant pressure. In this step further ice forms (as shown by the increase in number of the frazil ice crystals), and at the end of this stage the seawater and ice phases have the same in situ temperature, namely, the freezing temperature appropriate to 1) that pressure and 2) the final value of seawater salinity.

Let the mass fraction of ice be ; the mass fraction of seawater in the ice–seawater mixture is then . The total enthalpy per unit mass of the ice–seawater mixture at stage 1 of Fig. 13 is the weighted sum of the specific enthalpies of the two phases, namely,

 
formula

where we have chosen to write the specific enthalpy of seawater in terms of Conservative Temperature in the functional form , while the specific enthalpy of ice is written in the functional form , where the temperature variable is the potential temperature of ice with reference pressure 0 dbar ( is not to be confused with the potential temperature of seawater , because these two potential temperatures are not equal; see Fig. 1b).

In going from stage 1 to 2, both the seawater and ice phases undergo an adiabatic change of pressure that changes their specific enthalpies by and , respectively (here υ and are the specific volumes). Hence, at stage 2 the total enthalpy per unit mass of the ice–seawater mixture is (noting that and that at leading order in the perturbation quantities it is immaterial whether υ and are evaluated at the properties of stage 1 or those of stage 2)

 
formula

In going from stage 2 to 3, the total enthalpy of the mixture is conserved. Hence, we equate the total enthalpies at these two stages, giving

 
formula

For an externally imposed change in pressure, this equation may be regarded as giving the amount of new ice formed due to the adiabatic uplifting of the ice–seawater mixture. The other important constraint that we know is that the ice–seawater mixture is at the freezing temperature at both stages 1 and 3.

The enthalpies and on the right-hand side of Eq. (36) are now expanded in a Taylor series about the values at stage 1, keeping the leading-order terms. The pressure derivatives of these enthalpies, being the specific volumes of seawater and of ice, give terms that cancel with the corresponding terms on the left-hand side of the equation to leading order. The remaining leading-order terms are

 
formula

where . Because the salt always resides in the seawater phase, the product is constant so that

 
formula

which reduces Eq. (37) to

 
formula

One key result is apparent from this equation already, namely, that as the mass fraction of frazil ice tends to zero, Eq. (39) tends to our existing result Eq. (16) for the ratio for the melting of ice Ih into seawater. The present frazil ice relation Eq. (39) for the ratio is, however, simpler (or more restrictive) because the temperatures of both the ice and seawater components are constrained to be at the freezing temperature; the ice temperature cannot be lower than the freezing temperature nor can the Conservative Temperature of the seawater exceed its freezing temperature. Hence, in the limit as the mass fraction of frazil ice tends to zero, as the pressure of a seawater–frazil mixture is changed, the ratio is illustrated by the equilibrium situation of Figs. 5a and 5b.

Returning to the more general situation in which is not vanishingly small, we need to evaluate in terms of differentials of Absolute Salinity and pressure. The partial differential can be written as

 
formula

The in situ temperature of ice Ih can be expressed as a function of the potential temperature of ice Ih and pressure as , so that the total differential of the in situ temperature of ice is

 
formula

This equation applies to any material differentials , and and in particular will apply to the differences between these properties at stages 1 and 3 of our thought process. Hence, we can write

 
formula

But the ice at both stages 1 and 3 is at the freezing temperature , so that can also be expressed as

 
formula

and the partial derivatives here are known functions of the Gibbs functions of ice Ih and seawater [see Eq. (C3) of  appendix C].

Combining Eqs. (42) and (43) and using the result in Eq. (40) gives our desired result for , namely,

 
formula

Substituting this equation into Eq. (39) gives a relationship between only , and , namely,

 
formula

Another relationship between , and can be found from the knowledge that in both stages 1 and 3 the seawater is at the freezing Conservative Temperature, and because is a function of only and , the differences and are related by

 
formula

and expressions for these partial derivatives are Eqs. (C5) and (C6) in  appendix C.

Equations (45) and (46) are two equations in , and from which we can find our desired relations for the ratios of changes in our seawater–frazil ice mixture due to adiabatic uplift, namely, , , and . By eliminating the pressure difference from these two equations we find that

 
formula

The leading terms in both the numerator and denominator, namely, and , are the same as in Eq. (16), which applies to the melting of ice Ih into seawater at fixed pressure, the only difference being that in the present case both the ice and seawater are at the freezing temperature. Equation (47) is plotted in Fig. 14a at = 35.165 04 g kg−1 (actually is plotted). The dependence on the mass fraction of sea ice can be illustrated with the case when is different to the value at by about 7.4%. Most of this sensitivity to comes from the denominator in Eq. (47). Equation (47) is again illustrated in Fig. 15a where we show the contours of at the fixed salinity = 35.165 04 g kg−1. That is, Fig. 15a is simply 35.165 04 g kg−1 times Fig. 14a, so that the quantity contoured in Fig. 15a is in temperature units. As the mass fraction of ice tends to zero, Eq. (47) tends to Eq. (16), so that values of Fig. 15a at are the same as those of Fig. 5a at = 35.165 04 g kg−1. The dependence of on the mass fraction of ice is illustrated in Fig. 15b, which shows the difference relative to the case when .

Fig. 14.

(a) Plot of from Eq. (47) as a function of the ice mass fraction and pressure. (b) Plot of from Eq. (48) as a function of the ice mass fraction and pressure. (c) Plot of from Eq. (49) as a function of the ice mass fraction and pressure. Seawater salinity = 35.165 04 g kg−1. The units of (a) are K (g kg−1)−1, (b) are K (Pa)−1, while (c) are (g kg−1) (Pa)−1. The contoured values were evaluated from the GSW algorithm gsw_frazil_ratios of the GSW Oceanographic Toolbox (www.TEOS-10.org).

Fig. 14.

(a) Plot of from Eq. (47) as a function of the ice mass fraction and pressure. (b) Plot of from Eq. (48) as a function of the ice mass fraction and pressure. (c) Plot of from Eq. (49) as a function of the ice mass fraction and pressure. Seawater salinity = 35.165 04 g kg−1. The units of (a) are K (g kg−1)−1, (b) are K (Pa)−1, while (c) are (g kg−1) (Pa)−1. The contoured values were evaluated from the GSW algorithm gsw_frazil_ratios of the GSW Oceanographic Toolbox (www.TEOS-10.org).

Fig. 15.

(a) Plot of from Eq. (47) as a function of the ice mass fraction and pressure, for = 35.165 04 g kg−1. This is simply 35.165 04 g kg−1 times Fig. 14a. (b) Contour plot of (a) with the values of evaluated at ice mass fraction subtracted at each pressure. (c) The difference between and the corresponding derivative of with pressure at constant Absolute Salinity, (obtained from gsw_CT_freezing_first_derivatives). The contoured values of (c) are in K (Pa)−1 and the seawater salinity was taken to be = 35.165 04 g kg−1. Notice that the numbers contoured here are only a few percent of those of shown in Fig. 14b.

Fig. 15.

(a) Plot of from Eq. (47) as a function of the ice mass fraction and pressure, for = 35.165 04 g kg−1. This is simply 35.165 04 g kg−1 times Fig. 14a. (b) Contour plot of (a) with the values of evaluated at ice mass fraction subtracted at each pressure. (c) The difference between and the corresponding derivative of with pressure at constant Absolute Salinity, (obtained from gsw_CT_freezing_first_derivatives). The contoured values of (c) are in K (Pa)−1 and the seawater salinity was taken to be = 35.165 04 g kg−1. Notice that the numbers contoured here are only a few percent of those of shown in Fig. 14b.

Similarly, by eliminating from Eqs. (45) and (46) we find

 
formula

and when is eliminated from these same two equations we find

 
formula

The variation of Conservative Temperature with pressure under frazil ice conditions from Eq. (48) is plotted in Fig. 14b at = 35.165 04 g kg−1. It is seen that is quite insensitive to the frazil ice mass fraction . This is confirmed in Fig. 15c where we show the difference between and the corresponding derivative of with pressure at constant Absolute Salinity . The variation of Absolute Salinity with pressure under frazil ice conditions from Eq. (49) is plotted in Fig. 14c at 35.165 04 g kg−1. This figure follows, of course, as simply the ratio of the Figs. 14a and 14b.

When no frazil is present in seawater, its Conservative Temperature is unaffected by adiabatic and isohaline changes in pressure, but the in situ temperature changes with pressure according to the adiabatic lapse rate that is usually positive. When frazil is present in seawater, an increase in pressure results in changes in Conservative Temperature as contoured in Fig. 14b. This dependence of the temperature (both Conservative Temperature and in situ temperature) of the frazil–seawater mixture to changes in pressure is rather large and negative compared with the (usually positive) adiabatic lapse rate of seawater that is typically less than one-twentieth of the values shown in Fig. 14b for and is usually of the opposite sign. Another way of stating this is that the adiabatic lapse rate of the frazil–seawater mixture is large and negative when frazil is present, compared with the small and positive adiabatic lapse rate of seawater in the absence of frazil (Feistel et al. 2010).

When is very small, the ratio of Eq. (47) approaches that applicable for freezing at constant pressure, namely, Eq. (16) (in the equilibrium situation where the Conservative Temperature of the seawater phase and the in situ temperature of the ice phase both being at the freezing temperature). This is not true of and of Eqs. (48) and (49). In the denominator of both these equations there is the term

 
formula

which is independent of the ice mass fraction . This term [Eq. (50)] makes about a 2.5% difference to the denominators of Eqs. (48) and (49). This means that as , is about 2.5% larger in magnitude than , both being negative. This term arises because the freezing process of frazil ice formation does not occur at constant Absolute Salinity and from Eq. (46) we see that

 
formula

or

 
formula

This explains the origin of the term [Eq. (50)] as taking account of the dependence of the freezing Conservative Temperature on the change in Absolute Salinity that occurs during frazil ice formation. Equation (52) is a relationship that exists between the expressions for and of Eqs. (47) and (48).

Note that the rate at which the freezing Conservative Temperature changes with Absolute Salinity at fixed pressure is quite different (even different signs) from the corresponding change involving frazil ice as the pressure varies . A typical value of is −0.0583 K g−1 kg, while a typical value of is 2.3 K g−1 kg. By contrast, we have seen that the variation of Conservative Temperature with pressure for frazil ice is only a few percent different to the corresponding change at constant Absolute Salinity . Note from Eq. (38) it follows that the rate of change of the ice mass fraction with pressure is

 
formula

The results of this section of the paper could also be derived by the route presented in Feistel and Hagen (1998) and in Feistel et al. (2010) where a Gibbs function of the mixture of seawater and ice Ih is formed as the weighted sum of the Gibbs functions of seawater and ice Ih as

 
formula

For example, the adiabatic lapse rate of this combination of seawater and frazil ice can be evaluated immediately using this composite Gibbs function as . We have chosen the derivation route above to outline the various processes involved in detail and to maintain the close connection with the “potential” variables, the Conservative Temperature of seawater, and the potential temperature of ice Ih.

6. Conclusions

We have exploited the thermodynamically consistent TEOS-10 definitions of seawater and ice Ih to derive the finite-amplitude relationships of Eqs. (8) and (9) that predict the final enthalpy and Absolute Salinity when a certain mass fraction of ice melts into seawater. We have not had to assume that the two components are in thermodynamic equilibrium during the melting process. In the limit as the mass fraction of melting ice tends to zero, the relative rate is derived at which the Conservative Temperature and Absolute Salinity vary when ice melts into seawater, essentially confirming the earlier linearized derivation of Gade (1979). These results have been illustrated graphically, and they have been extended to the case of sea ice that contains a small amount of trapped brine. The final section of the paper-derived results for the way Conservative Temperature and Absolute Salinity vary with pressure when frazil is present and in thermodynamic equilibrium with the seawater phase. These results of frazil ice assume that the ice crystals are small enough to not move relative to the seawater and to permanently remain in thermodynamic equilibrium with it. At some stage in the evolution of frazil ice, the individual ice crystals will become sufficiently large that the relative vertical motion of the ice crystals cannot be ignored and a more complicated analysis incorporating the Stokes drift of the ice crystals would be needed.

APPENDIX A

The TEOS-10 Gibbs Function of Ice Ih and Its Derivatives

The Gibbs energy of ice Ih, the naturally abundant form of ice, having hexagonal crystals, is a function of temperature [International Temperature Scale of 1990 (ITS-90)] and sea pressure, This Gibbs function was derived by Feistel and Wagner (2006), adopted as an International Association for the Properties of Water and Steam (IAPWS) release in 2006, revised in 2009 (IAPWS 2009a), and adopted by the Intergovernmental Oceanographic Commission as the TEOS-10 official description of ice Ih in marine science. This equation of state for ice Ih is given by Eq. (A1) as a function of temperature, with two of its coefficients being polynomial functions of sea pressure :

 
formula

where is not a function of pressure, while

 
formula

with the reduced temperature , and and are given in Table A1.

Table A1.

Constants used in the TEOS-10 definition of ice Ih (IAPWS 2009a).

Constants used in the TEOS-10 definition of ice Ih (IAPWS 2009a).
Constants used in the TEOS-10 definition of ice Ih (IAPWS 2009a).

If the sea pressure is expressed in decibars, then must also be given in these units as Pt = 0.061 165 7 dbar, so that is unitless. The real constants from to and , the complex constants , , and , and from to are listed in IAPWS (2009a) and in the TEOS-10 manual IOC et al. (2010). The value of was slightly changed in the revised IAPWS ice Ih release (IAPWS 2009a) to improve the numerical consistency with the IAPWS (2009b) release for the fluid phase of water (see Feistel et al. 2008). The complex logarithm in Eq. (A1) is taken as the principal value, that is, it evaluates to imaginary parts in the interval . The complex notation used here has no direct physical basis but serves for the convenience of analytical partial derivatives and for the compactness of the resulting formulae, especially in program code.

Here, we point out some computational efficiencies that can be obtained by mathematically rearranging the terms in the Gibbs function of ice and its derivatives. First, the Gibbs function Eq. (A1) can be evaluated using ⅔ of the number of logarithm evaluations from

 
formula

The same computational efficiency applies to and . Similarly, can be evaluated using half the number of logarithm evaluations as given by the formula in Table 4 of IAPWS (2009a) using the expression

 
formula

and the same computational efficiency applies to . The following expression for the specific enthalpy of ice has a factor of 5 fewer natural logarithms to evaluate compared with the straightforward use of the original expressions for and in :

 
formula

APPENDIX B

Summary of Various Seawater Thermodynamic Relationships

Here we list expressions for several key thermodynamic quantities of seawater, as functions of the TEOS-10 Gibbs function of seawater (IOC et al. 2010; IAPWS 2008; Feistel 2008). Here is the Absolute Salinity measured on the Reference-Composition Salinity Scale of Millero et al. (2008), is the in situ temperature in degrees Celsius (ITS-90), and is sea pressure in decibars (this being the absolute pressure minus 10.1325 dbar).

The specific enthalpy of seawater is given by

 
formula

and the specific isobaric heat capacity is the rate of change of specific enthalpy of seawater with temperature at constant Absolute Salinity and pressure so that

 
formula

where is the Celsius zero point T0 = 273.15 K. The partial derivative of the specific enthalpy of seawater with respect to Absolute Salinity at constant in situ temperature and pressure is

 
formula

where is the relative chemical potential of seawater. These first derivatives of specific enthalpy with respect to in situ temperature and Absolute Salinity and can be evaluated using the GSW Oceanographic Toolbox function gsw_enthalpy_first_derivatives_wrt_t_exact.

Now we consider specific enthalpy to be a function of Conservative Temperature (rather than of in situ temperature), that is, we take and it can be shown that the partial derivative can be written as [from McDougall (2003)]

 
formula

The corresponding partial derivative of specific enthalpy with respect to Absolute Salinity at constant Conservative Temperature is [from McDougall (2003) or appendix 11 of IOC et al. (2010)]

 
formula

Here the potential temperature with respect to zero sea pressure is labeled . The partial derivatives and of Eqs. (B4) and (B5) can be evaluated in terms of using the GSW Oceanographic Toolbox function gsw_enthalpy_first_derivatives_CT_exact.

APPENDIX C

The Variation of the Freezing Conservative Temperature with Salinity and Pressure

As described in section 3.33 of the TEOS-10 manual (IOC et al. 2010), the partial derivatives of the in situ freezing temperature with respect to pressure and Absolute Salinity can be found by forming the total differential of the implicit definition of the freezing temperature:

 
formula

obtaining

 
formula

From Eq. (C2) we find that

 
formula
 
formula

Equations (C1)(C3) are for air-free seawater. When seawater contains dissolved air, Eq. (C3a) becomes

 
formula

where is the saturation fraction of dissolved air in seawater, varying up to 1.0 when the seawater is saturated with dissolved air. This equation is based on our empirical knowledge [see section 3.33 of IOC et al. (2010)] that the influence of on the in situ freezing temperature is .

Regarding Conservative Temperature in the functional form , we have its total differential being

 
formula

and combining this equation with Eq. (C3) we find the desired expressions for the dependence of the seawater freezing Conservative Temperature on Absolute Salinity and on pressure, namely,

 
formula

and

 
formula

The expressions in the numerator and denominator, respectively, of the last term in Eq. (C6) are the latency operator of sea ice of Feistel et al. (2010) operating on specific volume and (minus) entropy, respectively. That is, they are the latent contributions to the specific volume and (minus) entropy of sea ice [see Eqs. (5.23) and (5.25) of Feistel et al. (2010)].

The partial derivatives of Conservative Temperature with respect to Absolute Salinity and in situ temperature have been derived in  appendix A.15 of the TEOS-10 manual, and are

 
formula

and

 
formula

where is the potential temperature of seawater, referenced to p = 0 dbar. The partial derivative of with respect to pressure can be found by differentiating with respect to pressure the entropy equality obtaining [using ]

 
formula

In this way, the dependence of on Absolute Salinity and on pressure can be evaluated from Eqs. (C5) and (C6) from knowledge of the Gibbs functions of seawater and of ice Ih. Equations (C3b), (C3c), (C5), and (C6)(C9) have all been implemented as the functions gsw_t_freezing_first_derivatives and gsw_CT_freezing_first_derivatives in the GSW Oceanographic Toolbox (www.TEOS-10.org).

The vertical gradient of Absolute Salinity with pressure required to have the freezing Conservative Temperature constant is given by

 
formula

APPENDIX D

A Polynomial Expression for the Freezing Conservative Temperature of Seawater

The polynomial expression presented here is a fit to the TEOS-10 freezing temperature over the range in SAp space between 0 and 120 g kg−1 and between 0 and 10 000 dbar (100 MPa). We have chosen to do the polynomial fit for the Conservative Temperature at which seawater freezes rather than the in situ freezing temperature because ocean models will have Conservative Temperature as their temperature variable.

The TEOS-10 Gibbs function for seawater is valid in the ranges 0 ≤ SA ≤ 42 g kg −1 and 0 ≤ p ≤ 10 000 dbar. Additionally, at p = 0 dbar TEOS-10 is valid for thermal and colligative properties for Absolute Salinity up to where a constituent of seawater first saturates and comes out of solution. This typically occurs at an Absolute Salinity of between 90 and 110 g kg−1 (Feistel and Marion 2007; Marion et al. 2009). Technically, we should restrict the range of applicability of our polynomial fit to this area of SAp space plus the line at p = 0 dbar up to the Absolute Salinity of saturation, but the work of Feistel and Marion (2007, see their Fig. 15) suggests that the freezing temperatures calculated using the TEOS-10 Gibbs function at high pressures beyond SA = 42 g kg−1 will not contain gross errors.

In the region of validity of the TEOS-10 Gibbs function, the rms accuracy of the freezing temperature is estimated to be 1.5 mK [see section 6.3, Fig. 4, and Table 7 of Feistel (2008)]. The present polynomial fits the full TEOS-10 freezing Conservative Temperature to within ±0.6 mK over both the valid TEOS-10 SAp range and the extrapolated region. Hence, we conclude that the use of this polynomial is essentially as accurate as the full TEOS-10 approach for calculating the freezing temperature. There is a triangle of data in SAp space at the largest pressures and Absolute Salinities where TEOS-10 does not provide the freezing temperature. This is because the TEOS-10 code returns values for the freezing temperature down to about −12°C. This in situ freezing temperature corresponds approximately to the line in SAp space connecting (50 g kg-1, 10 000 dbar) to (120 g kg−1, 5000 dbar), and the polynomial should not be used if the input Absolute Salinity and pressure lie beyond this line in SAp space. In the GSW Oceanographic Toolbox, this polynomial for freezing Conservative Temperature is available as gsw_CT_freezing_poly.

The polynomial for air-free seawater is

 
formula

where the coefficients are given in Table D1. Note that there are no coefficients with , as the square root of Absolute Salinity does not appear in the TEOS-10 polynomial for the chemical potential of water in seawater . The in situ freezing temperature of air-free pure water at p = 0 dbar is known very accurately (with an uncertainty of only 2 μK) to be 0.002 519°C. In terms of Conservative Temperature this is 0.017 947 064 327 968 736°C [because gsw_CT_from_pt(0,0.002 519) = 0.017 947 064 327 968 736°C], and this is the value of above.

Table D1.

Constants of the polynomial expression, Eq. (D1), for the freezing Conservative Temperature of seawater.

Constants  of the polynomial expression, Eq. (D1), for the freezing Conservative Temperature of seawater.
Constants  of the polynomial expression, Eq. (D1), for the freezing Conservative Temperature of seawater.

If there is dissolved air in seawater, the freezing temperature is lowered. The depression of the in situ freezing temperature of pure water (i.e., SA = 0 g kg−1) at p = 0 dbar is 2.4 mK, while for standard seawater with SA = 35.165 04 g kg−1 the in situ freezing temperature depression is 1.9 mK. The dependence of the freezing potential temperature on dissolved air is normally taken to be a linear function of salinity [section 6.3 of Feistel (2008)]. Now the rate at which Conservative Temperature changes with potential temperature at fixed Absolute Salinity is given by [from Eq. (A.12.3a) of IOC et al. (2010)]

 
formula

showing that is proportional to the specific heat of seawater at p = 0 dbar, and we note that is approximately a linear function of Absolute Salinity [see Fig. 4 of IOC et al. (2010)]. We use this approximate linear variation of with to motivate the expression for in terms of the assumed linear dependence of on Absolute Salinity and dissolved air fraction, obtaining the following final expression for :

 
formula

with a = 0.502 500 117 621 and b = 0.057 000 649 899 720. These two coefficients have been chosen so that the in situ temperature freezing point depression for air-saturated seawater at p = 0 dbar is exactly 2.4 mK at SA = 0 g kg−1 and exactly 1.9 mK at SA = 35.165 04 g kg−1.

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