1. Introduction
Exchanges of heat and freshwater at the surface of the ocean represent the major forcing behind the creation of new water masses. In the polar oceans, the source regions for most of the deep and bottom waters, the processes of exchange are strongly influenced by the presence of ice. If ocean models are to simulate the characteristics and distribution of abyssal waters correctly, they must include an adequate representation of the coupling between ice and ocean. Because melting and freezing entail a transfer of water between the two media, this coupling formally manifests itself in the boundary conditions placed on both the vertical velocity and on the tracer concentrations. However, it is an almost universal practice in numerical modeling of the ocean to regard the sea surface as a permeable, but material, interface—properties may diffuse across it, but no water may be exchanged between the ocean and the overlying medium, be it atmosphere or ice. Huang (1993) criticized such practice, on the grounds that it suppresses the Goldsburgh–Stommel circulation and recommended setting the vertical velocity relative to the upper boundary equal to the precipitation, evaporation, melt, or freeze rate. Here we examine what the assumed nature of the ocean surface implies for the balance equations for tracers under conditions of melting and freezing.
The ice–ocean boundary layer is a region that is distinguished by relatively high gradients in tracer concentrations combined with large changes in the effective diffusivity. It is often the case that the equations used to describe the transport of tracers in the bulk of the oceanic water column are not applicable in the boundary region. This problem can be circumvented by incorporating a simple parameterization of the boundary layer processes into the boundary conditions. Such a formulation invariably entails the diagnosis of conditions at the ice–ocean interface that are distinct from those pertaining at the uppermost model grid point within the sub-ice water column. In the presence of such differences, the advection of meltwater into the bulk of the ocean plays an important role in the balance equations for tracers, but the advective flux is formally absent if the ice–ocean interface is treated as a material surface.
We use a simple, one-dimensional model of the ocean beneath sea ice to demonstrate the role played by meltwater advection in the conservation of tracers. To this model we apply two sets of boundary conditions that differ only in the assumption made about the nature of the ice–ocean interface: material or not. We also discuss the two-dimensional model of thermohaline circulation beneath an ice shelf presented by Hellmer and Olbers (1989) and Hellmer et al. (1998). This model has a material interface at the ice–ocean boundary, but changing the kinematic boundary condition would entail a considerable increase in complexity. Instead, we include the effect of meltwater advection through a modified flux boundary condition and show that this simple fix improves the model performance. Finally, we discuss how the boundary formulations discussed here compare with those used elsewhere.
2. Fundamental equations
Boundary conditions of the form given in (5)–(8) were introduced by Josberger (1983) and McPhee (1983). Mellor et al. (1986) recognized the importance of molecular diffusion within the viscous sublayer in determining the melt/freeze rate, and McPhee et al. (1987) and Steele et al. (1989) investigated expressions for the turbulent transfer coefficients that included explicit parameterizations of this process. The magnitudes of the transfer coefficients used in this study are derived from the results of this pioneering work.
3. A simple model of the ocean beneath sea ice
We force the model with a sinusoidal variation in the atmospheric heat flux having an amplitude of 500 W m−2 over open water and a period of one year. For all other tracers we assume that the flux passing through the leads is zero. The water column thickness (H − h) is initially 50 m and the ocean has an initial salinity of 34.5 psu and an initial temperature at the appropriate salinity-dependent freezing point. The initial ice thickness is arbitrary, as we are only concerned with how the melting and freezing cycle influences the bulk ocean properties. The areal coverage of ice is assumed constant at 90% and the model is run for 10 years. We use a turbulent heat transfer coefficient, γT, of 5 × 10−5 m s−1 and a turbulent salt transfer coefficient, γS, of 0.04γT. These values are appropriate for a moderate friction velocity of 5 × 10−3 m s−1 (McPhee et al. 1987;Holland and Jenkins 1999).
For the purposes of diagnosing the melt/freeze rate and the temperature and salinity at the ice–ocean boundary, we assume that no salt is incorporated into the ice on freezing. We also assume the ice to be sufficiently thick that the heat conducted through it is negligible, which allows us to set the first term on the left-hand side of (7) to zero. The pressure-dependent term in (8) can also be neglected. Note that these assumptions, along with the simple formulations we have used for the heat and salt transfer coefficients, do not affect the form of the equations presented below. They influence only the magnitude of the forcing.
a. With meltwater advection
Despite the simplicity of the model we find some support for such a cycle of ocean properties in observations made during the Arctic Ice Dynamics Joint Experiment. Figure 3 shows mixed layer temperatures and salinities recorded daily between late May 1975 and mid-April 1976 at the Snowbird station (Maykut and McPhee 1995). Many factors not considered in our one-dimensional model, such as drift of the station, horizontal advection of water masses, mixing across the pycnocline, variations in ice concentration, and a variable input of turbulent kinetic energy to the ocean with the passage of weather systems, will all have influenced the observations. Given these complicating factors, the basic shape and phasing of the observed annual cycle in mixed layer properties compare favorably with the model results.
The main qualitative difference between the observations and model results is the lack of any sustained period of supercooling, at levels above the detection threshold of the instrumentation used, in the observational record. A lack of supercooling suggests that the ocean response time is shorter during periods of ice growth than during periods of melting. The reason may be frazil ice production in the water column or fundamentally different behavior within the boundary layer when salt, rather than freshwater, is produced at the phase change interface. Whatever the physical mechanism, we can simulate the effect in the model by increasing the magnitude of the heat and salt transfer coefficients during periods of supercooling. The results are shown in Fig. 4. With the wintertime response of the ocean 20 times faster than before, supercooling is restricted to no more than about 0.02°C, and the peak in supercooling and the transition from melting to freezing are practically coincident with the maximum and the change in sign, respectively, of the atmospheric forcing. Because there is little phase lag between the cycles of atmospheric heat flux and ice growth during half of the year, the drifts in ice thickness and ocean salinity are reduced by a factor of 2.
b. Without meltwater advection
The main differences between (15) and (21) arise because the water advected into or out of the ocean in (15), which is absent in (19), has properties equal to those at the boundary. Advection therefore constitutes a tracer flux that is additional to the diffusive fluxes given by (5). Using (21) we obtain a similar evolution of the ice cover to that obtained with (15), but we find a rapid drift of about 0.1 psu yr−1 toward higher salinity (Fig. 5). The increase in salinity is what we would anticipate for a net ice growth of 15 cm yr−1, and is clearly in error. It is a result of the missing advective fluxes, which are expressed as the products of the melt rate and the difference in tracer concentration between the mixed layer and the ice–ocean interface [Eq. (15)]. The concentration differences are generated by the rejection or uptake of tracers during freezing or melting, so they change sign synchronously with the melt rate (Fig. 6). Hence, the advective fluxes are always of the same sign, and, although they are generally small, their neglect leads to an error that accumulates throughout the annual cycle. With the reaction time of the ocean shortened by a factor of 20 during freezing, the difference in salinity across the boundary layer, and hence the salt flux error, is reduced in winter. The annual drift in salinity is therefore approximately halved (Fig. 7).
Although the net salinity drift is a function of our model setup, in particular the choice of water column thickness, the magnitude of the flux errors are determined only by the nature of the ice–ocean boundary conditions. In the appendix we develop general expressions for the size of the errors and demonstrate that they are approximately proportional to the square of the deviation in water temperature from the freezing point. The errors are most significant for slowly diffusing tracers with small γX, the ratio of the excluded advective to included diffusive fluxes being m/γX. For this reason the salinity error is much more apparent in Figs. 5 and 7 than the temperature error. The solution could not drift from the freezing point line in any case, but the heat flux error alone would act to reduce the thickness of the ice cover and freshen the mixed layer.
4. A model of thermohaline circulation beneath an ice shelf
Hellmer et al. (1998) applied the two-dimensional model to the water column beneath the 90-km floating extension of Pine Island Glacier, a fast-flowing outlet glacier of the West Antarctic Ice Sheet. Employing flux boundary conditions of the form shown in (25), they were able to obtain good agreement between modeled outflow characteristics and observations made at the calving front of the glacier. Figure 8 shows a comparison between the results of this model run and the observations. The results of a second model run, which employed boundary conditions of the form shown in (24), are also shown in Fig. 8. These latter results are clearly inferior.
5. Significance for other models
Boundary formulations that include all the effects of meltwater advection are, to our knowledge, uncommon. However, for conditions of low melt/freeze rate and for integrations over short timescales, the errors caused by nonconservation of tracers are probably small. This is the case with most models of the interaction between ice shelves and the ocean, the early versions of which (Hellmer and Olbers 1989; Scheduikat and Olbers 1990;Jenkins 1991) all used boundary formulations analogous to (21) or (24). Later versions of these models (Nicholls and Jenkins 1993; Jenkins and Bombosch 1995; Hellmer et al. 1998) have been corrected, although it is only in the case of the Pine Island Glacier simulations cited in the previous section where melt rates are ∼10 m yr−1, an order of magnitude higher than in the other examples, that the correction has had a significant impact on the results.
Larger-scale ice–ocean models often use a simpler one-equation formulation in which only the freezing point at the upper level or layer of the model is diagnosed. At each time step the computed temperature is reset to the freezing point and an appropriate amount of ice melted or frozen. Once again, because a separate boundary salinity is not calculated, the salinity balance is almost certainly computed correctly, but we suspect that the much smaller errors in the heat balance are routinely made. Of course, none of the errors discussed in this paper will be apparent in models that employ either surface restoring fluxes or diagnosed flux corrections, which are specifically designed to eliminate drift such as that in Figs. 5 and 7.
6. Closing remarks
Mellor et al. (1986) concluded that for small |m/u∗|, where u∗ is the surface friction velocity, it is possible to neglect the influence of vertical advection in the formulation of boundary conditions at an ice–ocean interface. However, this conclusion was based on an assessment of the likely impact of vertical advection on the structure of the boundary layer. We do not dispute this finding and have implicitly made use of it in our parameterization of diffusion through the boundary layer. Nevertheless, we conclude that meltwater advection does play a role in the balance equations for tracers within a sub-ice water column and that this role may be neglected only if |m/γX| is small. Since γS/u∗ ≈ 4 × 10−4 and the turbulent transfer coefficients for most other dissolved species are of a similar order of magnitude to that of salt, this latter criterion is a much more stringent requirement. We conclude that in general it is not possible to neglect the impact of meltwater advection.
We have presented a formulation of the flux boundary condition at an ice–ocean interface that includes both advective and diffusive fluxes and have demonstrated that inclusion of the advective term is necessary for the conservation of tracers. In practice, the advective fluxes are likely to be excluded only if the application of the boundary conditions involves the explicit diagnosis of the tracer concentration at the ice–ocean interface. There are relatively few models that include this level of sophistication and, although the heat flux error is an exception in that its occurrence is likely in almost all formulations, it is also the least significant of the errors. For other tracers, the commonly used boundary conditions already incorporate both advective and diffusive fluxes, but we believe the foregoing discussion to be of more than academic interest. As computing resources become ever more readily available, more detailed representations of key physical processes are likely to become part of even global-scale models. While our current knowledge of ice–ocean boundary physics is not sufficient to say with confidence that melting and freezing can be calculated more accurately if the boundary salinity is explicitly diagnosed, the inclusion of some additional tracers, in particular stable isotopes, does require the computation of the their boundary concentrations. Also, if the use of a nonmaterial interface at the ocean surface, as recommended by Huang (1993), becomes more commonplace, the distinction between the advective and diffusive parts of the tracer fluxes will become important. If a separate advective boundary condition is applied to the scalar equations, the purely diffusive flux boundary condition (24) must be employed to avoid double-counting of the meltwater advection term. Finally, as efforts are already being made to eliminate the need for flux corrections in coupled ocean–atmosphere general circulation models, it is important to isolate all possible causes of spurious drift in the properties of the ocean.
Acknowledgments
This study would not have been undertaken were it not for the attentive reading and constructive criticism of some of the authors’ earlier work by others. Andreas Bombosch and Gregory Lane-Serff independently pointed out the inconsistency in Jenkins (1991). Chris Garrett, in his role as associate editor for an earlier version of Hellmer et al. (1998), asked a number of perceptive questions about the model behavior. In addressing these we became aware of the possible size of the errors we have described. We gratefully acknowledge this invaluable input. Miles McPhee kindly supplied us with the data from the Snowbird ice station, used to generate Fig. 3, and provided an insightful review of the manuscript. We would like to thank Aike Beckmann, Ralph Timmermann, Keith Nicholls, Chris Doake, and Rupert Gladstone for reading through earlier versions of this paper. Their comments lead to significant clarification of a number of points. Collaborative work that gave rise to this paper was supported by U.S. Department of Energy Grant DE-FG02-93-ER61716 and NASA Polar Programs research Grant NAG5-4028.
REFERENCES
Gade, H. G., 1979: Melting of ice in sea water: A primitive model with application to the Antarctic ice shelf and icebergs. J. Phys. Oceanogr.,9, 189–198.
Hellmer, H. H., and D. J. Olbers, 1989: A two-dimensional model for the thermohaline circulation under an ice shelf. Antarct. Sci.,1, 325–336.
——, S. S. Jacobs, and A. Jenkins, 1998: Oceanic erosion of a floating Antarctic glacier in the Amundsen Sea. Ocean, Ice, and Atmosphere: Interactions at the Antarctic Continental Margin, S. S. Jacobs and R. F. Weiss, Eds., Antarctic Research Series, Vol. 75, Amer. Geophys. Union, 83–99.
Holland, D. M., and A. Jenkins, 1999: Modeling thermodynamic ice–ocean interactions at the base of an ice shelf. J. Phys. Oceanogr.,29, 1787–1800.
Huang, R. X., 1993: Real freshwater flux as a natural boundary condition for the salinity balance and thermohaline circulation forced by evaporation and precipitation. J. Phys. Oceanogr.,23, 2428–2446.
Jenkins, A., 1991: A one-dimensional model of ice shelf–ocean interaction. J. Geophys. Res.,96, 20 671–20 677.
——, and A. Bombosch, 1995: Modeling the effects of frazil ice crystals on the dynamics and thermodynamics of Ice Shelf Water plumes. J. Geophys. Res.,100, 6967–6981.
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——, 1990: Small-scale processes. Polar Oceanography, Part A: Physical Science, W. O. Smith Jr., Ed., Academic Press, 287–334.
——, G. A. Maykut, and J. H. Morison, 1987: Dynamics and thermodynamics of the ice/upper ocean system in the marginal ice zone of the Greenland Sea. J. Geophys. Res.,92, 7017–7031.
——, C. Kottmeier, and J. H. Morison, 1999: Ocean heat flux in the central Weddell Sea during winter. J. Phys. Oceanogr.,29, 1166–1179.
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APPENDIX
How Large Are the Errors?
The salt and heat flux errors given by (A12), (A13), (A14), (A17), (A18), and (A19) are plotted in Fig. A1 for a range of values of friction velocity, thermal driving, and basal temperature gradient. The quadratic dependence on the latter two factors means that the errors are always of the same sign. In the case of (A14) the effects of the errors are counteractive, in that the heat flux error should cause melting and dilution of the mixed layer, while the salt flux error acts to increase the mixed layer salinity. In the case of (A19), both errors serve to accentuate the effects of freezing. We note that however the errors combine and whichever is dominant, the effect is most evident in the computed salinity because the properties of the seawater are constrained to follow closely the liquidus, which has an almost flat trajectory in temperature/salinity space.
Schematic of a one-layer ocean beneath sea ice. The ocean layer is characterized by a single temperature and salinity (T, S) and is forced by an atmospheric heat flux (Qh) that passes through leads. The temperature and salinity at the ice–ocean interface (Tb, Sb) differ from those in the interior, and the differences drive melting and freezing. The sea ice is regarded as a continuum, so the ice–ocean boundary conditions and the atmospheric heat flux are formally applied at the level of the mean ice draft (z = −h). The seabed lies at z = −H.
Citation: Journal of Physical Oceanography 31, 1; 10.1175/1520-0485(2001)031<0285:TROMAI>2.0.CO;2
Results of a ten-year integration of (15): (a) change in ice thickness, (b) temperature, and salinity of the ocean. In (a) the dotted lines indicate the time at which the atmospheric heat flux changes sign in spring (S) and autumn (A). In (b) the stars labeled S (spring), S (summer), A (autumn), W (winter) indicate the times of maximum and zero atmospheric forcing, while the straight line labelled Tf indicates the freezing point at atmospheric pressure.
Citation: Journal of Physical Oceanography 31, 1; 10.1175/1520-0485(2001)031<0285:TROMAI>2.0.CO;2
A 331-day record of temperature and salinity at the Snowbird ice camp. Each dot represents an average value for the upper 25 m of the water column. Data from the first day of each month are circled and labeled. The straight line labeled Tf indicates the freezing point at atmospheric pressure, and the two dashed lines indicate the likely uncertainty in the observed deviation from the freezing point (Maykut and McPhee 1995).
Citation: Journal of Physical Oceanography 31, 1; 10.1175/1520-0485(2001)031<0285:TROMAI>2.0.CO;2
Same as Fig. 2 but with heat and salt transfer coefficients increased by a factor of 20 when the ocean temperature falls below the freezing point.
Citation: Journal of Physical Oceanography 31, 1; 10.1175/1520-0485(2001)031<0285:TROMAI>2.0.CO;2
Results of a ten-year integration of (21): (a) change in ice thickness, (b) temperature, and salinity of the ocean. In (a), the dotted lines indicate the time at which the atmospheric heat flux changes sign in spring (S) and autumn (A). In (b), the stars labeled S (spring), S (summer), A (autumn), W (winter) indicate the times of maximum and zero atmospheric forcing, while the straight line labeled Tf indicates the freezing point at atmospheric pressure.
Citation: Journal of Physical Oceanography 31, 1; 10.1175/1520-0485(2001)031<0285:TROMAI>2.0.CO;2
Illustration of the net dilution induced by advection of meltwater into an ocean layer having a nonmaterial upper boundary:(a) melt rate, (b) salinity difference across the ice–ocean boundary layer, and (c) salinity change that results from the inclusion of meltwater advection on the right-hand side of (15a), defined by dS/dt = Am(H − h)−1(Sb −
Citation: Journal of Physical Oceanography 31, 1; 10.1175/1520-0485(2001)031<0285:TROMAI>2.0.CO;2
Same as Fig. 5 but with heat and salt transfer coefficients increased by a factor of 20 when the ocean temperature falls below the freezing point.
Citation: Journal of Physical Oceanography 31, 1; 10.1175/1520-0485(2001)031<0285:TROMAI>2.0.CO;2
Potential temperature vs salinity diagram comparing observations made at the calving front of Pine Island Glacier (dots) with the output from a two-dimensional model of thermohaline circulation in the sub-ice cavity (Hellmer et al. 1998). The vertical profiles of potential temperature and salinity used here constitute the open boundary of the model domain. Model results obtained using flux boundary conditions defined as in (24) and (25) are indicated by open circles and crosses, respectively. Both versions of the model were tuned to give net melt rates for the glacier that were consistent with observation.
Citation: Journal of Physical Oceanography 31, 1; 10.1175/1520-0485(2001)031<0285:TROMAI>2.0.CO;2
Fig. A1. Salt and heat flux errors induced by the neglect of meltwater advection across an ice–ocean boundary for friction velocities of 0.1, 0.5, 1.0, and 2.0 cm s−1: (a) salt flux errors defined by (A12) (solid line) and by the approximation (dashed line) given in (A14a), (b) heat flux errors defined by (A13) (solid line) and by the approximation (dashed line) given in (A14b), (c) salt flux errors defined by (A17) (solid line) and by the approximation (dashed line) given in (A19a), and (d) heat flux errors defined by (A18) (solid line) and by the approximation (dashed line) given in (A19b).
Citation: Journal of Physical Oceanography 31, 1; 10.1175/1520-0485(2001)031<0285:TROMAI>2.0.CO;2