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

or equivalently, in terms of the two Gibbs functions

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 S_A = 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 u_PS ≡ (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.

• Download Figure
• Download figure as PowerPoint slide

(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.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0253.1

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

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:

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 .

• Download Figure
• Download figure as PowerPoint slide

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.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0253.1

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.

• Download Figure
• Download figure as PowerPoint slide

(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).

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0253.1

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

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

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

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)]

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

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

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

and

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.

• Download Figure
• Download figure as PowerPoint slide

(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.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0253.1

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.

• Download Figure
• Download figure as PowerPoint slide

(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.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0253.1

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

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):

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.

• Download Figure
• Download figure as PowerPoint slide

(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.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0253.1

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.

• Download Figure
• Download figure as PowerPoint slide

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.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0253.1

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

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

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

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:

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.

• Download Figure
• Download figure as PowerPoint slide

(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.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0253.1

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.

• Download Figure
• Download figure as PowerPoint slide

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).

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0253.1

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.

• Download Figure
• Download figure as PowerPoint slide

(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).

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0253.1

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 ]

while considering specific enthalpy in the functional form , we obtain

and on using Eq. (25) we find

and

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

which can be rearranged to be

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.

• Download Figure
• Download figure as PowerPoint slide

(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.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0253.1

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

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.