Melting of Ice and Sea Ice into Seawater and Frazil Ice Formation

Transcription

1 JULY 2014 M C DOUGLL ET L Melting of Ice and Sea Ice into Seawater and Frazil Ice Formation TREVOR J. MCDOUGLL ND PUL M. BRKER School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, ustralia RINER FEISTEL Leibniz-Institut f ur Ostseeforschung, Warnem unde, Germany BEN K. GLTON-FENZI ustralian ntarctic Division, Kingston, and ntarctic Climate and Ecosystems Cooerative Research Centre, University of Tasmania, Hobart, Tasmania, ustralia (Manuscrit received 22 November 2013, in final form 17 March 2014) BSTRCT The thermodynamic consequences of the melting of ice and into seawater are considered. The International Thermodynamic Equation Of Seawater 2010 (TEOS-10) is used to derive the changes in the Conservative Temerature and bsolute Salinity of seawater that occurs as a consequence of the melting of ice and into seawater. lso, a study of the thermodynamic relationshis involved in the formation of frazil ice enables the calculation of the magnitudes of the Conservative Temerature and bsolute Salinity changes with ressure when frazil ice is resent in a seawater arcel, assuming that the frazil ice crystals are sufficiently small that their relative vertical velocity can be ignored. The main results of this aer are the equations that describe the changes to these quantities when ice and seawater interact, and these equations can be evaluated using comuter software that the authors have develoed and is ublicly available in the Gibbs SeaWater (GSW) Oceanograhic Toolbox of TEOS Introduction The International Thermodynamic Equation Of Seawater 2010 (TEOS-10) has been adoted as the international standard for the thermohysical roerties 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 otentials 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 ressure and temerature ranges found Corresonding author address: Trevor J. McDougall, School of Mathematics and Statistics, University of New South Wales, NSW 2052, ustralia. Trevor.McDougall@unsw.edu.au in the ocean and atmoshere [see Fig. 1b of Feistel et al. (2010)]. The temerature at which seawater begins freezing is determined from examining the thermodynamic equilibrium between the seawater and ice hases, with the relevant equilibrium condition being that the chemical otential of water in the seawater hase is equal to the chemical otential of water in the ice hase (Feistel and Hagen 1998; Feistel and Wagner 2005). Here we cast the freezing temerature in terms of the Conservative Temerature Q of seawater, and exressions are derived for the artial derivatives of the freezing Conservative Temerature with resect to bsolute Salinity S and ressure P (see aendix C). Because Conservative Temerature is referred over otential temerature as a measure of the heat content er 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 imlications on the changes to the Conservative Temerature of seawater. The adotion by the Intergovernmental Oceanograhic Commission DOI: /JPO-D Ó 2014 merican Meteorological Society

2 1752 J O U R N L O F P H Y S I C L O C E N O G R P H Y VOLUME 44 of TEOS-10 as the new official definition of the roerties of seawater, ice, and humid air involves the transition to ublishing in the new oceanograhic salinity and temerature variables bsolute Salinity and Conservative Temerature, in contrast to the ractical salinity S P and otential temerature u of the 1980 equation of state (EOS-80). The adiabatic lase rate of ice is shown to be much greater than that of seawater (often 10 times as large), imlying that under isentroic vertical motion, the variation of the in situ temerature of ice with ressure is much larger than for seawater. In this aer, we consider the quantities that are conserved when ice melts into seawater. Writing equations for these conserved quantities (including enthaly) leads to closed exressions for the bsolute Salinity and Conservative Temerature of the seawater after the melting or freezing has occurred [see Eqs. (8) and (9) below]. These equations aly at finite amlitude and do not assume the ice and seawater to be near to a state of thermodynamic equilibrium. This aroach can be linearized to give an exression for the ratio of the changes in Conservative Temerature and bsolute 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 comarable to that of Gade (1979), although our aroach is more general because it is based on the rigorous conservation of three basic thermodynamic roerties (mass, salt, and enthaly), so that it alies without aroximation at finite amlitude, and we also include the deendence of seawater enthaly on salinity. This analysis is extended to the melting of, which is treated as a coarse-grained mixture of ure ice in which ockets of brine are traed and the salinity of the ockets of brine is determined by thermodynamic equilibrium between the brine and the surrounding ice. This brine salinity has the same value in all ockets with equal temeratures and ressures, irresective of their articular sizes, and the TEOS-10 descrition of the thermal roerties of the brine aly u to an bsolute Salinity of 120 g kg 21 [see section 2.6 of IOC et al. (2010)]. The uwelling of very cold seawater (colder than the surface freezing temerature) can lead to suercooling and the formation of small ice crystals called frazil ice, and this rocess is also examined using the TEOS-10 Gibbs functions of ice Ih and of seawater. Under the assumtion that the relative vertical velocity (the Stokes velocity) of frazil can be ignored, we derive exressions for the rate at which the bsolute Salinity and the Conservative Temerature of seawater vary with ressure when frazil is resent. Because of their tiny size, frazil ice crystals remain in thermodynamic equilibrium with the surrounding seawater when the arcel undergoes ressure excursions. From the thermodynamic ersective, a frazil ice arcel differs from a arcel only quantitatively, namely, by their oosite liquid solid ratios. Proerties such as the adiabatic lase rates of these comosite systems can formally be derived from a Gibbs function of (Feistel et al. 2010). Strictly seaking, the mixture of frazil ice with seawater is a metastable state. It still undergoes a slow rocess known as Ostwald riening that minimizes the interface energy between ice and seawater, tyically by finally forming a single iece of ice (or a single large brine ocket in the case of ). In the TEOS-10 Gibbs function of, the interface energy is neglected. mixture of seawater and frazil ice has two imortant roerties 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 resect to bsolute Salinity is ositive, g S S. 0 [see section.16 of IOC et al. (2010)], while the Gibbs function of [g SI ; see Eq. (54) below] is linear in salinity (Feistel and Hagen 1998), that is, g SI SS 5 0. s a consequence of the latter, no irreversible mixing effects occur when two arcels are in contact at the same temerature and ressure but different salinities, in contrast to ice-free seawater where entroy is roduced when arcels having contrasting salinities are mixed. Second, at brackish salinities (u to about 28 g kg 21 ) seawater ossesses a temerature of maximum density where the adiabatic lase rate changes its sign (McDougall and Feistel 2003), while exhibits a density minimum (Feistel and Hagen 1998). Ice has a much larger secific volume than water or seawater, and the freezing rocess is accomanied by volume exansion, that is, by a large negative thermal exansion coefficient (and lase rate) of. This effect is strongest at low salinities and in fact the thermal exansion coefficient of ure water has a singularity at the freezing oint. With decreasing temerature and increasing brine salinity, the rate of formation of ice in gradually decreases to the oint where the volume increase caused by the newly formed ice (i.e., by the transfer of water from the liquid to the solid hase) is outweighed by the thermal contraction of the ure hases, ice and brine, so that the total thermal exansion coefficient of changes its sign and turns ositive. The thermodynamic interactions between ice and seawater described in this aer are first derived as equations between the various quantities and are illustrated grahically in the figures. In addition, the thermodynamic

3 JULY 2014 M C DOUGLL ET L roerties of ice Ih and the results from the equations of this aer are available as comuter algorithms in the Gibbs SeaWater (GSW) Oceanograhic Toolbox (McDougall and Barker 2011) and can be downloaded online (from 2. The adiabatic lase rate and the otential temerature of ice Ih The adiabatic lase rate is equal to the change of in situ temerature t exerienced when ressure is changed while keeing entroy h (and salinity) constant. This definition alies searately to both ice and seawater (where one needs to kee not only entroy but also bsolute Salinity constant during the ressure change). In terms of the Gibbs functions of seawater and of ice Ih, the adiabatic lase rates of seawater G and of ice G Ih are exressed resectively as G5 t S,h 5 t S,Q 5 t S,u 52 g TP g TT 5 (T 0 1 t)at rc (1) and G Ih 5 t 5 t h 52 gih TP u Ih g Ih TT 5 (T 0 1 tih )a tih r Ih c Ih, (2) where a t and a tih are the thermal exansion coefficients of seawater and ice Ih, resectively, with resect to in situ temerature. Subscrits of the Gibbs functions g and g Ih of seawater and ice, resectively, denote artial derivatives, and r and c are the density and the isobaric secific heat caacity. The adiabatic lase rates of seawater and ice are numerically substantially different from each other. The thermal exansion coefficient of ice does not change sign as does that of seawater when it is cooler than the temerature of maximum density, and the secific heat caacity of ice c Ih is only aroximately 52% that of seawater c. Figure 1a shows the ratio G/G Ih of the adiabatic lase rates of seawater and ice at the freezing temerature, as a function of the bsolute Salinity of seawater and ressure. For salinities tyical of the oen ocean, the ratio G/G Ih is about 0.1, indicating that the in situ temerature of ice varies 10 times as strongly with ressure when both seawater and ice Ih are subjected to the same isentroic ressure variations. This substantial difference between the adiabatic lase rates is also illustrated in Fig. 1b as the difference in the otential temerature of seawater and of ice u 2 u Ih for seawater and ice arcels that are at the in situ freezing FIG. 1. (a) Ratio of the adiabatic lase rates of seawater and of ice Ih G/G Ih at the freezing temerature. (b) Difference (8C) between the otential temeratures of seawater u and of ice u Ih for arcels of seawater and ice whose in situ temerature is the in situ freezing temerature. temerature. This difference in otential temeratures can be understood as follows: t every oint on the S diagram of Fig. 1b, the in situ freezing temerature t freezing (S, ) is calculated. Imagine now raising both a seawater samle and an ice samle from this ressure to the sea surface. Initially, both samles have the same in situ temerature, namely, the freezing temerature t freezing. s the ressure is reduced, the in situ temeratures of both the seawater and ice arcels are reduced (assuming that the seawater salinity is large enough that its thermal exansion coefficient is ositive), but the temerature of the ice changes tyically 10 times as much as that of the seawater. Thus, the two arcels that have the same in situ temerature have different otential temeratures (referenced to 5 0 dbar) as illustrated in Fig. 1b.

4 1754 J O U R N L O F P H Y S I C L O C E N O G R P H Y VOLUME 44 FIG. 2. (a) Conservative Temerature (8C) at which air-free seawater freezes as a function of ressure and bsolute Salinity. (b) Difference between the freezing Conservative Temerature derived from EOS-80 and that of TEOS-10 (contours; mk). (c) Error (mk) in using the aroximating olynomial exression of aendix D for the freezing Conservative Temerature. 3. Pure ice Ih melting into seawater a. The freezing temerature s described in IOC et al. (2010), freezing of seawater occurs at the temerature t freezing at which the chemical otential of water in seawater m W equals the chemical otential of ice m Ih. Hence, the freezing temerature t freezing is found by solving the imlicit equation m W (S, t freezing, ) 5 m Ih (t freezing, ), (3) or equivalently, in terms of the two Gibbs functions g(s, t freezing, ) 2 S g S (S, t freezing, ) 5 g Ih (t freezing, ). (4) The Gibbs function of seawater g(s, t, ), defined by Feistel (2008) and IPWS (2008), is a function of the bsolute Salinity S, the in situ temerature t, and the ressure of a seawater arcel. The Gibbs function for ice Ih g Ih (t, ) is defined by Feistel and Wagner (2006) and IPWS (2009a) and is summarized in aendix. Note that Eq. (3) is valid for air-free seawater. The dissolution of air in water lowers the freezing oint slightly; saturation with air lowers the freezing temerature by about 2.4 mk for freshwater and by about 1.9 mk at S g kg 21. The freezing in situ temeratures derived from Eq. (4) were converted to the Conservative Temerature at which air-free seawater freezes and are shown in Fig. 2a as a function of ressure and bsolute Salinity. To comare these TEOS-10 freezing temeratures to those of EOS-80, the conversion between the ractical salinity of EOS-80 and bsolute Salinity of TEOS-10 was made using the conversion factor u PS [ ( /35) g kg 21 (Millero et al. 2008; IOC et al. 2010). It was assumed that the EOS-80 freezing temeratures of Millero and Leung (1976) were of air-saturated seawater. Having calculated the air-free in situ freezing temerature of EOS-80 in this manner, the Conservative Temerature is calculated from the TEOS-10 algorithm gsw_ct_from_t. The resulting differences between the freezing Conservative Temeratures from EOS-80 and TEOS-10 are illustrated in Fig. 2b and are very small at 0 dbar, rising to aroximately 10 mk at 1000 dbar and 120 mk at 3000 dbar. We have develoed a olynomial aroximation for the freezing Conservative Temerature (see aendix D), and the error in using this comutationally efficient olynomial is seen in Fig. 2c to be very small, being no larger than 0.05 mk at the sea surface and no larger than aroximately 0.25 mk at other ressures. b. Finite-amlitude exressions 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 that contains ockets of brine, but in this section we consider the melting of ure ice Ih that contains no brine ockets. This section of the aer is aroriate when considering the melting of ice from glaciers or icebergs, because these tyes of ice are formed from comacted snow and hence do not contain the traed seawater that is tyical of ice formed at the sea surface, namely,. The general case we consider in this section has the seawater temerature above its freezing temerature, while the ice, in order to be the stable hase ice Ih, needs to be at or below the freezing temerature of ure water (i.e., seawater having zero bsolute Salinity) at the given ressure level, tyically at the sea surface. Note that this condition ermits situations in which the initial ice temerature 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 ressure, entroy increases while three quantities are conserved: mass, salt, and enthaly. While this rocess is assumed to be adiabatic it is not isentroic. Because of irreversibility, the freezing rocess is thermodynamically rohibited in a closed system. To form frazil ice in seawater at fixed ressure, more entroy must be exorted

5 JULY 2014 M C DOUGLL ET L from the samle than is roduced internally; a tyical examle being an ice floe that is strongly cooled by the atmoshere. The conservation equations for mass, salt, and enthaly during this adiabatic melting event at constant ressure are m f SW 5 mi SW 1 m Ih, (5) m f SW Sf 5 mi SW Si, and (6) m f SW hf 5 m i SW hi 1 m Ih h Ih. (7) The suerscrits i and f stand for the initial and final values, that is, the values before and after the melting event, while the subscrits SW and Ih stand for seawater and ice Ih. The mass, salinity, and enthaly conservation Eqs. (5) (7) can be combined to give the following exressions for the differences in the bsolute Salinity and the secific enthaly of the seawater hase due to the melting of the ice: (S f 2 Si ) 52m Ih m f S i 52wIh S i SW and (8) (h f 2 h i ) 52w Ih (h i 2 h Ih ) 5 (Sf 2 Si ) S i (h i 2 h Ih ), (9) where we have defined the mass fraction of ice Ih w Ih as m Ih /m f SW. The initial and final values of the secific enthaly of seawater are given by h i 5 h(s i, ti, ) 5 ^h(s i, Qi, )andh f 5 h(s f, tf, ) 5 ^h(s f, Qf, ), where the secific enthaly of seawater has been written in two different functional forms: one being a function of in situ temerature and the other being a function of Conservative Temerature. This TEOS-10 terminology, where an overhat adorning a thermodynamic variable (as in ^h) imlies that the variable is being regarded as a function of Conservative Temerature, while an unadorned variable (such as h) imlies that the thermodynamic variable is being regarded to be a function of in situ temerature, is used throughout this aer. The use of Eqs. (8) and (9) is illustrated in Fig. 3,where the mass fraction of ice w Ih and the in situ temerature of the ice t Ih are varied at fixed values of the initial roerties of the seawater at S i 5 S SO g kg 21, Q i 5 48C, and at 5 0 dbar. Note that these results aly for these finite-amlitude differences of temerature and salinity, and these calculations are accurate because of the existence of the TEOS-10 exressions for the secific enthalies of seawater and ice Ih. We have not needed to resort to a linearization involving the secific FIG. 3. Change in (a) bsolute Salinity (g kg 21 ) and (b) Conservative Temerature (8C) when the mass fraction of ice w Ih is melted into seawater with initial roerties S i 5 S SO g kg 21, Q i 5 48C, and at 5 0 dbar. These data were obtained from the GSW Oceanograhic Toolbox function gsw_melting_ice_ into_seawater. There are no contours on the right as the final calculated seawater roerties there are cooler than the freezing temerature. heat caacities to obtain Eqs. (8) (9) and the results of Fig. 3. Clearly, the salinity difference S f 2 Si in Fig. 3a is simly roortional to w Ih, as is also obvious from Eq. (8), while the (relatively weak) deendence of h Ih on t Ih is aarent from the lot of Q f 2Q i in Fig. 3b. Note that at 5 0 dbar, Eq. (9) becomes simly Q f 2Q i 5 2w Ih (Q i 2 h Ih /c 0 ). The conservation of bsolute Salinity and enthaly when ice Ih melts into seawater is illustrated in Fig. 4a. The final values of bsolute Salinity S f and enthaly hf given by Eqs. (8) and (9) are illustrated in Fig. 4a for four different values of the ice mass fraction w Ih. These final values (S f, hf ) lie on the straight line on the bsolute Salinity enthaly diagram connecting (S i, hi ) and

6 1756 J O U R N L O F P H Y S I C L O C E N O G R P H Y VOLUME 44 Eqs. (15) (18) that corresond to Gade s key result for this ratio. The enthaly difference h f 2 h i in Eq. (9) is now exanded as a Taylor series in the differences in bsolute Salinity and temerature, and the first-order terms in these differences are retained, leading to (t f 2 t i )c 1 (S f 2 Si )h S (Sf 2 Si ) S i (h i 2 h Ih ) 52w Ih (h i 2 h Ih ), (10) FIG. 4. (a) bsolute Salinity enthaly diagram illustrates Eqs. (8) and (9) that embody the conservation of bsolute Salinity and enthaly when ice Ih melts into seawater. Initial values of the bsolute Salinity and enthaly of seawater and of ice Ih are shown by the two solid dots, and the final values of bsolute Salinity and enthaly of the seawater after the ice has melted are shown by the four oen circles (for four different values of the ice mass fraction w Ih ). 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 bsolute Salinity in situ temerature diagram. Note that the final oints (the oen circles) do not lie on the straight line connecting the initial oints (the solid dots). (0, h Ih ). The fact that the same data do not fall on a straight line on the bsolute Salinity in situ temerature diagram in Fig. 4b nicely illustrates that temerature is not conserved when melting occurs. c. The linearized exression for the S Q ratio Gade (1979) develoed 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 simler linearized version of the conservation of heat [in the aendix of Gade (1979)] was able to derive an exression for the ratio of the changes in temerature and salinity in seawater due to the melting of a vanishingly small amount of ice into seawater. Here we have used the simler heat budget aroach, which is formally the conservation of enthaly, and this led to Eqs. (8) and (9) that hold at finite amlitude when a finite mass fraction of ice melts into seawater. In this subsection, we linearize these equations to find the exressions (15) (18) for the ratio of the changes in salinity and temerature when a vanishingly small mass fraction of ice melts into seawater. It is these where c is the secific heat caacity of seawater, c 5 h/ Tj S,,andh S 5 h/ S j T, is the derivative of the seawater secific enthaly with resect to bsolute Salinity at constant in situ temerature and constant ressure. By regarding secific enthaly to be a function of Conservative Temerature in the functional form ^h(s, Q, ), the Taylor series exansion of Eq. (9) yields (Q f 2Q i ) ^h Q 1 (S f 2 Si ) ^h S (Sf 2 Si ) S i (h i 2 h Ih ) 52w Ih (h i 2 h Ih ), (11) where ^h Q 5 h/ Qj S, is the artial derivative of the seawater secific enthaly with resect to Conservative Temerature at fixed bsolute Salinity, and ^h S 5 h/ S j Q, is the artial derivative of the seawatersecific enthaly with resect to bsolute Salinity at fixed Conservative Temerature. Exressions for these artial derivatives can be found at Eqs. (B4) and (B5) of aendix B. Equations (10) and (11) can be rewritten as dtc [ (t f 2 t i )c (Sf 2 Si ) S i (h 2 h Ih 2 S h S ) 52w Ih (h 2 h Ih 2 S h S ) and (12) dq ^h Q [ (Q f 2Q i ) ^h Q (Sf 2 Si ) S i (h 2 h Ih 2 S ^hs ) 52w Ih (h 2 h Ih 2 S ^hs ). (13) The arentheses on the right-hand side of Eq. (12), h 2 h Ih 2 S h S, if evaluated at the freezing temerature t freezing (S, ), is the latent heat of melting (i.e., the isobaric melting enthaly) of ice into seawater, first derived by Feistel et al. (2010) [see also section 3.34 of IOC et al. (2010)]. Note that at 5 0 dbar, ^h S is zero, while h S is nonzero. Exressions for c, h S, ^h Q, and

7 JULY 2014 M C DOUGLL ET L ^h S in terms of the Gibbs function of seawater are given in aendix B. The derivation of the isobaric melting enthaly in Feistel et al. (2010) and IOC et al. (2010) considered the seawater and ice to be in thermodynamic equilibrium during a slow rocess in which heat was sulied to melt the ice while maintaining a state of thermodynamic equilibrium during which the temerature of the combined system changed only because the freezing temerature is a function of the seawater salinity. During this reversible rocess, the enthaly of the combined system increased due to the heat externally alied. The latent heat of melting is defined to be [from Eq. (3.34.6) of IOC et al. (2010)] L SI (S, ) 5 h(s, t freezing, ) 2 hih (t freezing, ) 2 S h S (S, t freezing, ). (14) The resent derivation [i.e., Eqs. (12) and (13)] alies to the common situation when the seawater is warmer than the ice that is melting into it, so that the two hases are not in thermodynamic equilibrium with each other during the irreversible melting rocess. That is, the seawater temerature may be larger than its freezing temerature, and the ice temerature may or may not be less than its freezing temerature. The guiding thermodynamic rincile is that there is no change in the enthaly of the combined seawater and ice system during the irreversible melting rocess, because this rocess occurs adiabatically at constant ressure. When freezing (as oosed to melting) is considered, the second law of thermodynamics imlies that sontaneous freezing cannot occur excet when the seawater is at the freezing temerature (or in a metastable, subcooled state below that), and there must be some incremental external change (e.g., a decrease in ressure 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 arcel so that the changes in the seawater temerature and salinity are small, we find from Eq. (12) that the ratio of the changes of in situ temerature and bsolute Salinity is given by dt h 2 h Ih 2 S S h S ds 5 melting at constant c h(s, t, ) 2 h Ih (t Ih, ) 2 S h S (S, t, ) 5, (15) c (S, t, ) while the corresonding ratio of the changes in Conservative Temerature and bsolute Salinity is [from Eq. (13)] dq h 2 h Ih 2 S S ^hs ds 5 melting at constant ^h Q ^h(s, Q, ) 2 h Ih (t Ih, ) 2 S ^hs (S, Q, ) 5, (16) ^h Q (S, Q, ) where the second lines of these equations have been included to be very clear about how these quantities are evaluated. t 5 0 dbar, these equations become du h S 0 2 h Ih 0 2 S h S (S ds 5, u,0) melting at 50 dbar c (S, u,0) h(s, u,0)2h Ih (u Ih,0)2S h S (S, u,0) 5 c (S, u,0) and dq S ds 5 h 0 2 hih 0 melting at 50 dbar c 0 5Q2 hih (u Ih,0) c 0, (17) (18) where the otential temeratures of seawater u and ice u Ih are both referenced to 5 0 dbar. Note that the otential enthaly of seawater referenced to 5 0 dbar, h 0 5 h(s, u, 0)5 ^h(s, Q, 0), is simly c 0 times Conservative Temerature, where the constant secific heat c J kg21 K 21. Equation (17) is very similar to Eq. (25) of Gade (1979). If we associate Gade s temerature T 1 at the ice-water interface [Eq. (10) in Gade 1979] with t freezing, then the difference between the values of the enthaly of seawater and of ice at the freezing temerature can be interreted as Gade s latent heat term L, while the difference between the enthaly of seawater and its enthaly at the freezing temerature is aroximately equal to the term c (T 2 T 1 ) that aears in Gade s equation. The corresonding difference in the enthaly of ice at its temerature and the value at the freezing temerature is aroximately given by Gade s term c Ih (T ice 2 T 1 ). The terms ^h S and h S in Eqs. (16) and (17), resectively, being the aroriate artial derivatives of seawater enthaly with resect to bsolute Salinity, were absent in Gade s aroach, 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 Temerature rather than otential temerature means that the sloe of the melting rocess on the S Q diagram dq/ds involves a simler exression, esecially when the melting occurs at the sea surface at 5 0 dbar [Eq. (18)], where (i) ^h S (S, Q, 0)

8 1758 J O U R N L O F P H Y S I C L O C E N O G R P H Y VOLUME 44 is zero, and (ii) the relevant secific heat caacity of seawater ^h Q 5 c 0 (T 0 1 t)/(t 0 1 u) [seeeq.(b4) of aendix B] reduces to the constant c 0, so that the secific enthaly of seawater is simly c 0 multilied by the Conservative Temerature. Note that the numerator of the middle exression of Eq. (18) is simly the difference between the otential enthalies of seawater and of ice. The very simle Eqs. (16) and (18) for the sloe of the melting rocess on the S Q diagram dq/ds are key results of this aer. These equations are linearizations of the exact Eqs. (8) and (9) whose simlicity and rigor are due to the fact that the first law of thermodynamics guarantees that the total enthaly of the system is unchanged, as is illustrated in Fig. 4. Note that the righthand side of Eq. (18) is indeendent of the bsolute Salinity of the seawater into which the ice melts. We first illustrate these equations for the ratio of the changes of Conservative Temerature to those of bsolute 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 temerature at the given values of bsolute Salinity and ressure, then no melting or freezing would occur. In Fig. 5, we consider the limit as the temeratures of both the seawater and the ice aroach the freezing temerature. The ratio dq/ds j equilibrium from Eq. (16) is shown in Fig. 5a with the seawater enthaly evaluated at the freezing Conservative Temerature and with the ice enthaly evaluated at the in situ freezing temerature, at each value of ressure and bsolute Salinity. This ratio is roortional to the recirocal of bsolute Salinity, so it is more informative to simly multily dq/ds j equilibrium by bsolute 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 aroximately 81 and 83 times as much heat as would be required to raise the same mass of seawater by 18C. The corresonding result for the ratio of the changes of in situ temerature and bsolute Salinity near equilibrium conditions S dt/ds j equilibrium 5 L SI (S, )/ c (S, t freezing, ) can be calculated from Eq. (15), and the difference between S dt/ds j equilibrium and S dq/ds j equilibrium is shown in Fig. 5c. The largest contributor to this difference between Eqs. (15) and (16) is due to the deendence of the secific heat caacity c (S, t freezing, ) on (i) bsolute Salinity, involving a 6.8% variation over this full range of salinity, and (ii) on ressure, involving a change of 2.2% between 0 and 3000 dbar. Equation (16) for S dq/ds j melting at constant is now illustrated when the seawater and the ice Ih are not at the same temerature and are not in thermodynamic equilibrium at the freezing temerature. We begin by considering the melting of ice Ih at the sea surface, FIG. 5. (a) Ratio of the change of Conservative Temerature to that of bsolute Salinity when the melting occurs very near thermodynamic equilibrium conditions dq/ds j equilibrium from Eq. (16) with the seawater enthaly evaluated at the freezing Conservative Temerature and with the ice enthaly evaluated at the in situ freezing temerature at each value of ressure and bsolute Salinity. The values contoured have units of K (g kg 21 ) 21. (b) bsolute 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 Temerature. The quantities contoured in (b) and (c) have temerature units (K). The values contoured in (a) and (b) were evaluated from the function gsw_melting_ice_equilibrium_s_ct_ratio of the GSW Oceanograhic Toolbox.

9 JULY 2014 M C DOUGLL ET L FIG. 6. (a) Contours of Eq. (18) S dq/ds j melting at 50 dbar 5 (h 0 2 h Ih 0 )/c0 5Q2 h ~ Ih (u Ih )/c 0 for the melting of ice Ih into seawater at 5 0 dbar. The six stars are at the freezing temeratures (t and Q) for bsolute Salinity values starting at5gkg 21 with increments of 5 g kg 21,uto30gkg 21. (b) Difference between contours of Eq. (16) at dbar, S dq/ds j melting at 5500 dbar, and the corresonding ratio of (a) (where the ressure was 0 dbar) at S i 5 S SO g kg 21. The double-starred oint is at the freezing temeratures (t and Q) at dbar and S i 5 S SO g kg 21. (c) The mass fraction of ice w Ih, which when melted into seawater at S i 5 S SO g kg 21,at 5 0 dbar, and at the Conservative Temerature given by the vertical axis, results in the final mixed seawater that is at the freezing temerature. This figure has been found from the GSW algorithm gsw_ice_fraction_to_freeze_seawater. The double-starred oint is at the freezing temeratures (t and Q) at 5 0dbar and S i 5 S SO g kg 21. (d) Values of c 0 / ^h Q (u 2 t)/(t 0 1 t)ats i 5 S SO g kg 21 for various values of ressure u to 3000 dbar. The quantities contoured in (a) and (b) are temeratures (K), while that of (c) is the unitless mass fraction w Ih.The values contoured in (a) and (b) were evaluated from the algorithm gsw_melting_ice_s_ct_ratio of the GSW Oceanograhic 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_enthaly_first_derivatives_ct_exact. secifically at 5 0 dbar, when Eq. (16) reduces to Eq. (18); this equation is illustrated in Fig. 6a, which alies at all values of bsolute Salinity. The contoured values of Fig. 6a, (h 0 2 h Ih 0 )/c0 5Q2 h ~ Ih (u Ih )/c 0, increase as 1.0 times changes in Q and decrease aroximately as c Ih /c0 0:52 times changes in the temerature of the ice. d. The influence of ressure on the melting S Q ratio Considering now the melting rocess at a gauge ressure larger than 0 dbar, the right-hand side of Eq. (16) is evaluated at dbar and S 5 S SO g kg 21, with the differences between these values and the corresonding values at 5 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 temerature of the ice being converted into the otential temerature of ice u Ih before Eq. (18) was evaluated. The large star in this figure reresents the equilibrium oint. The differences are not large and are about 0.15% of (h 0 2 h Ih 0 )/c0 5Q2 h ~ Ih (u Ih )/c 0. The differences scale almost linearly with ressure; at 3000 dbar

10 1760 J O U R N L O F P H Y S I C L O C E N O G R P H Y VOLUME 44 the corresonding differences (not shown) are aroximately 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 secific volumes of seawater and ice. The ratio of Eq. (16) to (18) is given by dq 0 1 ds melting at constant (T dq u) B h 2 hih 2 ^hs C (u 2 t) (T 0 1 t) h 0 2 h Ih (T t) 1 (T 0 1 u) (T 0 1 t) ds melting at 50 dbar [(h 2 h 0 ) 2 (h Ih 2 h Ih 0 ) 2 S ^h S ] (h 0 2 h Ih 0 ), (19) and we ask how different is the S Q ratio for melting occurring at a general ressure comared with using the exression (18), which involves the otential enthalies of seawater and of ice, but is only 100% accurate for melting at 5 0 dbar. The first term after unity on the right-hand side of the last exression of Eq. (19) is (u 2 t)/(t 0 1 t) 5 c 0 / ^h Q 2 1, and this term is illustrated in Fig. 6d at S 5 S SO g kg 21. This term is resonsible 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 enthaly differences that we can exress as follows (with the rimed variables being the variable of integration and the use of uer case P 0 for the integration variable serves to remind that it must be in SI units of ascals in order to have enthaly and secific volume in their usual units): h 2 h 0 2 (h Ih 2 h Ih 0 ) 2 S ^h S 5 ð P 0 [^y(s, Q, 0 ) 2 ~y Ih (u Ih, 0 )] dp 0 ð P 2 S ^y S (S, Q, 0 ) dp 0, (20) 0 where these exressions result from the definition of otential enthaly and the fact that the ressure derivative of secific enthaly, under adiabatic and isohaline conditions, is equal to the secific volume y [this is true of both seawater and ice, that is, y 5 ^h P and y Ih 5 h ~ Ih P, where the secific enthaly of seawater is written in the functional form ^h(s, Q, )andh Ih 5 h(u ~ Ih, )]. The last term in Eq. (20) is small so that the dominant contribution is due to the nontrivial difference between the secific volumes of seawater and ice. The observed almost linear deendence on the ressure of Eq. (16) is obvious from the form of Eq. (20). The conclusion from this comarison between Eqs. (16) and (18) is that as far as evaluating the sloe on the S Q diagram of melting of ice into seawater at a general ressure, very little error is made if the melting is assumedtooccurat 5 0 dbar and taking the relevant enthaly difference to be the difference between the otential enthalies of seawater and of ice Ih, as in Eq. (18). The error in the S Q sloe is 0.15% at dbar and 0.9% at dbar. e. n illustration from the mery Ice Shelf Figure 7 shows oceanograhic data obtained under the mery Ice Shelf that illustrate the ratio of the changes in bsolute Salinity and Conservative Temerature, as given by Eq. (16), when the melting of ice occurs. The vertical rofile named M06 begins under the ice at a ressure of 546 dbar and the uermost 175 m of the vertical rofile is shown. The data in the uermost dbar are closely aligned with the ratio given by Eq. (16) (as shown by the dashed line) evaluated at this ressure and with the ice temerature being the freezing temerature at this salinity and ressure. Two freezing lines are shown in Fig. 7b, for ressures of 0 and 578 dbar. ny observations cooler than the freezing temerature aroriate to 0 dbar is evidence of the influence of melting of ice or of heat lost by conduction through the ice. M06 is located on the eastern side of the ice shelf in an area that is melting, as can be inferred by the resence of ocean water at M06 that is well above the in situ freezing temerature at the base of the ice shelf. This water is thought to be flowing in a rimarily southward direction from the oen ocean as it enters the underice cavity. The other CTD rofile was taken from borehole M05, located on the western side of the ice shelf in an area that is refreezing [as is drawn in Fig. 7a] and reresents flow that has likely come from deeer in the cavity below the ice shelf than at M06 (Post et al. 2014) and hence has been in contact with the ice for longer. The uer 50 m or so of this cast is at the freezing temerature of seawater at this ressure. For both casts the data near the uer art of the water column have the ratio of the changes of S and Q in close agreement to the ratio given by Eq. (16), the ratio redicted from melting ice into seawater (dashed lines). The ice temerature that is

11 JULY 2014 M C DOUGLL ET L FIG. 8. The in situ freezing temerature (8C) of air-free seawater as a function of ressure (dbar) and bsolute Salinity, determined from the equilibrium freezing condition Eq. (4). In the context of, in situ temerature is the temerature of both the ure ice Ih hase t Ih and of the traed ockets of brine. roceeding from M06 to M05 without being exosed to significant heat loss Q to the ice [see Fig. 7a]. The vertical rofiles shown in Fig. 7b are the average of several vertical rofiles 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. n inflow of relatively warm water from the oen ocean rovides heat to melt the ice shelf. Buoyant freshwater that is released during the melting rocess rises along the underside of the ice shelf and can become locally suercooled at a shallower deth, leading to the formation of frazil and basal accretion of marine ice. (b) The to 175 m of two CTD rofiles taken below the mery Ice Shelf in East ntarctica at a melt site and at a refreeze site are shown. The warmer and saltier of the two casts is M06 [see Fig. 1 of Galton-Fenzi et al. (2012)] starting at a ressure of 546 dbar. The large round dot is ocean data very near the ice at 546 dbar, the triangle is 50 dbar deeer, the diamond is 100 dbar deeer, and the star is 150 dbar below the bottom of the ice shelf at this location, indicated by the circle. The other vertical cast (M05) is tyical of refreezing locations. The uermost 50 dbar of this cast is all at the freezing temerature at this ressure. needed to calculate this S Q ratio for each location has been taken to be the in situ freezing temerature of ice in contact with the seawater at the ressure at the base of the ice shelf. Moreover, in this figure the uermost 100 m of the M05 data is aroximately related to that of the M06 data through the S Q ratio of Eq. (16).This would be consistent with the notion that the same fluid is 4. Sea ice melting into seawater a. Finite-amlitude exressions 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 ure ice Ih and a small amount of traed brine that is in thermodynamic equilibrium with the ice Ih at the temerature of the ice Ih t Ih.Notethatthe that contains ure ice Ih and a small amount of brine is all at the same (8C) temerature t 5 t Ih.The bsolute Salinity of the traed brine S brine can be calculated from the thermodynamic equilibrium condition Eq. (4) from knowledge of the ressure and temerature of the, and this relationshi is shown in Fig. 8. The secific enthaly h brine is evaluated from the secific enthaly of seawater as h brine 5 h(s brine, t, ), while the enthaly of the ice Ih is evaluated at the same temerature of the, namely, h Ih 5 h Ih (t, ). The searate otential enthalies of seawater and of ice in are h 0 5 ^h(s, Q) 5 c 0 Q and hih 0 5 hih (u, 0), resectively. The bulk salinity S,whichmaybeaslarge as 10 g kg 21 but is more commonly around 3 5 g kg 21,is defined to be the mass fraction of sea salt in so that

12 1762 J O U R N L O F P H Y S I C L O C E N O G R P H Y VOLUME 44 (m Ih 1 m brine )S 5 m S 5 m brine S brine. (21) The conservation of mass, salt, and enthaly when the melting of occurs into seawater at fixed ressure are given by the equations m f SW 5 mi SW 1 m Ih 1 m brine 5 mi SW 1 m, (22) m f SW Sf 5 mi SW Si 1 m brine Sbrine 5 m i SW Si 1 m ice Ssea, and (23) m f SW hf 5 m i SW hi 1 m Ih h Ih 1 m brine h brine 5 m i SW hi 1 m h, (24) where h 5 (m Ih /m )h Ih 1 (m brine /m )h brine, being the mass-weighted summation of the enthaly of the two comonents of, namely, ice Ih and brine. The subscrits SW and Ih indicate seawater and ice, resectively. Note that all of the is assumed to melt so that the final mass of is taken to be zero. If the bulk bsolute Salinity of the S is zero, then the mass of brine is also zero, and these equations reduce to those of section 3, that is, the is in fact ice Ih. different hysical limit occurs when the temerature of the is such that the brine salinity S brine is equal to S. In this limit, the contains no ice Ih and is actually 100% seawater brine of bsolute Salinity equal to S. The various algorithms in the GSW Oceanograhic Toolbox avoid this situation by artificially ensuring that the inut in situ temerature of the always less than the air-free freezing temerature at (S, ) by at least C. The difference between the final and initial values of the bsolute Salinities of the seawater hase can be found from Eqs. (21) to (23) to be (S f 2 Si ) 52m m f (S i ice 2 Ssea ) SW 52w (S i ice 2 Ssea ), (25) where we have defined the mass fraction of w as m /m f SW. Using Eqs. (21), (22), and(23), wefind the following equation for the difference between the final and initial values of secific enthaly of the seawater hase: (h f 2 h i ) 52w (h i 2 h Ih ) 1 w m brine (h brine 2 h Ih ) m 52w (h i 2 h ). (26) The use of Eqs. (25) and (26) is illustrated in Fig. 9 where the mass fraction of w and the in situ temerature of the t Ih are varied at fixed values of the initial roerties of the seawater at S i 5 S SO g kg 21, Q i 5 48C, and at 5 0 dbar, and the sea ice salinity is taken to be S 5 5gkg 21. Note that these finite-amlitude calculations are accurate because of the existence of the TEOS-10 exressions for the secific enthalies of seawater and ice Ih. Clearly, the salinity difference S f 2 Si in Fig. 9a is simly roortional to w, as is obvious from Eq. (25). The differences in Conservative Temerature Q f 2Q i achieved in this melting rocess are shown in Figs. 9b and 9c. When the is not very cold, Q f 2Q i is quite sensitive to the temerature. We have not contoured values of S f 2 Si or Qf 2Q i in Fig. 9 for mass fractions of when this would result in the final seawater value of Q f being less than the freezing temerature. s the temerature of the t Ih is increased and aroaches the warmest allowed (which is the freezing temerature of seawater having an bsolute Salinity of S 5 5gkg 21 ), larger mass fractions of are admissible because the ratio S /S brine aroaches 1.0, and the second term in the first line of Eq. (26) becomes significantly ositive and acts against the first negative term in this equation. The maximum mass fraction w that can be melted into seawater with initial roerties S i 5 S SO g kg 21, Q i 5 18C, and 5 0 dbar can be calculated imlicitly from Eq. (26), and this is shown in Fig. 10. Values of w in the region of Fig. 9 in which there are no contours would result in the final seawater being frozen. The conservation of bsolute Salinity and enthaly when melts into seawater is illustrated in Fig. 11a. The final values of bsolute Salinity S f and enthaly hf given by Eqs. (25) and (26) are illustrated in Fig. 11a for four different values of the mass fraction w. These final values (S f, hf ) lie on the straight line on the bsolute Salinity enthaly diagram connecting (S i, hi ) and (S, h ). The fact that the same data do not fall on a straight line on the bsolute Salinity in situ temerature diagram is illustrated in Fig. 11b. This nicely illustrates that temerature is not conserved when melting occurs. b. The linearized exression for the ratio of changes in Q and S The left-hand side of Eq. (26) is exanded in a Taylor series at a fixed ressure so that to first order this equation becomes [taking secific enthaly in the functional form h(s, t, )]

13 JULY 2014 M C DOUGLL ET L FIG. 10. The maximum mass fraction w that can be melted into seawater with initial roerties S i 5 S SO g kg 21, Q i 5 18C, and at 5 0 dbar. Data for this figure were calculated from the GSW Oceanograhic Toolbox function gsw_seaice_fraction_to_freeze_seawater, which solves Eq. (26) imlicitly using a modified Newton method (McDougall and Wothersoon 2014). (t f 2 t i )c 1 (S f 2 Si )h S 2w (h i 2 h Ih ) 1 w ice Ssea S brine (h brine 2 h Ih ), (27) while considering secific enthaly in the functional form ^h(s, Q, ), we obtain (Q f 2Q i ) ^h Q 1 (S f 2 Si ) ^h S 2w (h i 2 h Ih ) 1 w and on using Eq. (25) we find (t f 2 t i )c 2w " 2 ice Ssea S brine (h i 2 h Ih ) 2 (S i # Ssea ice S brine (h brine 2 h Ih ) (h brine 2 h Ih ), (28) ice 2 Ssea )h S (29) FIG. 9. (a) The change in the bsolute Salinity S f 2 Si (g kg21 ) and (b) Conservative Temerature Q f 2Q i (8C) when the mass fraction of w is melted into seawater with initial roerties S i 5 S SO g kg 21, Q i 5 48C, and at 5 0 dbar and with S 5 5gkg 21. These were obtained from the GSW Oceanograhic Toolbox function gsw_melting_seaice_into_seawater, secifying S 5 5gkg 21. (c) zoomed-in lot of (b) with the same axes as Fig. 3b. Contours for ice cooler than 27.68C arenot shown because in this range the bsolute Salinity of the brine ockets in the exceeds 120 g kg 21. and (Q f 2Q i ) ^h Q 2w " 2 (h i 2 h Ih ) 2 (S i # Ssea ice S brine (h brine 2 h Ih ). ice 2 Ssea ) ^h S (30)

14 1764 J O U R N L O F P H Y S I C L O C E N O G R P H Y VOLUME 44 We will henceforth concentrate on Eq. (30) and the changes in Conservative Temerature rather than the changes in the in situ temerature of Eq. (29). Dividing Eq. (30) by (S f 2 Si ) and taking the limit as these differences tend to zero and using Eq. (25),we find (S 2 S ) dq h 2 h Ih 2 S ^hs ds 5 melting at constant ^h Q 2 S S brine (h brine 2 h Ih 2 S brine ^h S ), (31) ^h Q which can be rearranged to be dq S ds 5 melting at constant 1 2! Ssea ice S brine (h 2 h Ih Ssea ice 2 S ^hs ) 1 S brine [h 2 h brine 2 (S 2 S brine ) ^h S ] ^h Q 1 2 Ssea ice S!. (32) The first roerty of this melting S Q ratio, Eq. (32), alicable to the melting of into seawater at any ressure, is that if both the and the seawater are at the freezing temerature, then the right-hand side of Eq. (32) becomes the same as the exression Eq. (16) for ure ice Ih, namely, (h 2 h Ih 2 S ^hs )/ ^h Q, and is indeendent of the concentration of salt in the S. This leasingly simle result occurs because in this situation(i)thebrinesalinityisthesameasthebsolutesalinity of the seawater hase, and (ii) the enthaly of seawater is equal to the enthaly 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 into seawater is occurring at 5 0dbar, corresonding to the traing of brine into an ice matrix only occurring due to the raid freezing of ice at the sea surface. The melting ratio dq/ds of is shown in Fig. 12a as a function of the in situ temerature of the sea ice and the bsolute Salinity of the S when it is melting into seawater with the roerties S 5 S SO g kg 21, Q518C, and 5 0 dbar. For that is more than 18C cooler than the freezing temerature t f (S, ); the melting ratio dq/ds is not a articularly strong function of t Ih or S, but the melting ratio varies very strongly as the freezing temerature t f (S, ) is aroached. In this limit, the contains no ice Ih and is simly seawater. nother view of this is shown in Fig. 12b, which is of the full Eq. (31).ForthecaseofureiceIh,Eq.(31) at 5 0dbaris(h 0 2 h Ih 0 )/c0 and so is simly roortional to the enthaly difference between seawater and ice Ih. The values contoured in Fig. 12b reresent the effective enthaly difference that is aarent when melts into seawater. s in Fig. 12a, the values are highly sensitive to S only when the freezing temerature t f (S, ) is aroached. We now take the ratio of Eq. (31) at 5 0 dbar with Eq. (18), namely, S dq/ds j melting at 50 5Q2h Ih 0 /c0, which alies to the melting of ice Ih (as oosed to sea ice), obtaining c 0 (S 2 Ssea ice ) dq ds Ssea ice S brine (h 0 2 h Ih 0 ) melting at 50 (h brine 2 h Ih 0 ) (h 0 2 h Ih 0 ). (33) In this form, it is clear that the melting ratio dq/ds becomes infinite when the seawater salinity is the same as the bsolute Salinity of the. In other words, the recirocal of this ratio ds /dq changes sign as (S 2 S ) changes sign (which would only occur in unusual circumstances where was blown into a region of relatively freshwater). In the secial case where both the and the seawater are at the freezing temerature, then S brine 5 S and h brine 5 h 0 so that the right-hand side of Eq. (31) becomes simly 1 2 S /S, confirming that Eq. (31) becomes the same as Eq. (18). The influence of S on the melting ratio is illustrated in Fig. 12c, which is a contour lot of Eq. (33) for S 5 S SO g kg 21, Q518C, and at 5 0 dbar. It is seen that the melting ratio dq/ds is strongly affected by the resence of salt in the S only whentheseaiceisnotverycold. 5. Frazil ice formation through adiabatic ulift of seawater When seawater at the freezing temerature undergoes uward vertical motion so that its ressure decreases, frazil forms, rimarily due to the increase in the freezing temerature as a result of the reduction in