## Abstract

Significant changes in the state of the Arctic ice cover are occurring. As the summertime extent of sea ice diminishes, the Arctic is increasingly characterized by first-year rather than multi-year ice. It is during the early stages of ice growth that most brine is injected into the oceans, contributing to the buoyancy flux that mediates the thermo-haline circulation. Current operational sea-ice components of climate models often treat brine rejection between sea ice and the ocean similarly to a thermodynamic segregation process, assigning a fixed salinity to the sea ice, typical of multi-year ice. However, brine rejection is a dynamical, buoyancy-driven process and the salinity of sea ice varies significantly during the first growth season. As a result, current operational models may over predict the early brine fluxes from newly formed sea ice, which may have consequences for coupled simulations of the polar oceans. Improvements both in computational power and our understanding of the processes involved have led to the emergence of a new class of sea-ice models that treat brine rejection dynamically and should enhance predictions of the buoyancy forcing of the oceans.

## 1. Introduction

In this article, we review some fundamentals of the thermodynamic growth of sea ice (§2) and illustrate the different magnitudes of buoyancy flux resulting from cooling and brine rejection in polynyas and from consolidated sea ice (§3). We review in §4 some of the fundamental principles governing convection in sea ice, emphasizing the key role of the Rayleigh number in determining the existence and strength of brine drainage. We describe experimental observations of the formation of brine channels and their consequences (§5), highlighting in particular the way in which the salinity and porosity of sea ice decrease as a result of the upwards flow of ocean water that compensates for the downwards flow through brine channels. We culminate in §6 by describing and comparing recent one-dimensional models of brine drainage parametrized by local Rayleigh numbers, which allow profiles of sea-ice salinity to be computed dynamically and can be incorporated at reasonable computational cost into ice components of climate models.

## 2. Thermodynamic growth of sea ice

Sea ice is a mushy layer [1] comprising a porous matrix of essentially pure ice crystals filled with brine. The (bulk) salinity of sea ice *S* is related to the salinity of the interstitial brine *S*_{br} by
2.1where *ϕ*(**x**,*t*) is the local volume fraction occupied by ice crystals.

The simplest model of sea-ice growth assumes a quasi-steady (linear) temperature across a layer of constant and uniform solid fraction and computes its thickness *h*(*t*) from the Stefan condition
2.2where *ρ*_{s} is the density of pure ice, *L* is the latent heat of fusion per unit mass, *k*(*ϕ*) is the thermal conductivity of sea ice, *T*_{f} is the freezing temperature of the ocean, *T*_{s} is the surface temperature of the sea ice in contact with the atmosphere and *t* is time. For constant temperatures at the boundaries, equation (2.2) is readily integrated to give
2.3It appears therefore that thermodynamic growth of sea ice depends on its solid fraction, which depends on its salinity. However, to a good approximation, given that the ice platelets in sea ice are oriented vertically (their *c*-axes are dominantly horizontal), the conductivity of sea ice can be estimated by
2.4where *k*_{s} and *k*_{br} are the conductivities of pure ice and of brine, respectively. Therefore,
2.5which is approximately independent of *ϕ* both because *k*_{s} is about five times larger than *k*_{br} and because *ϕ* tends to be a rather large fraction through most of the sea ice. Physically, one sees from equation (2.3) that there are partially compensating influences of enhanced thermal conductivity competing with increased release of latent heat when the solid fraction is larger. Such compensation also occurs in more detailed models of mushy layers in which the solid fraction varies in time and space.

This very simple illustration shows that if one is simply interested in the thickness of sea ice then an accurate estimation of salinity or brine volume may not be critical. What this model (essentially a zero-layer Semtner model [2], which proposes simplifications of the detailed model of Maykut & Untersteiner [3]) lacks is any account of the heat capacity of sea ice; within the model, the growth and the rates of heat transfer between ocean, ice and atmosphere respond instantaneously to changes in forcing. The model is therefore not generally good at responding with the appropriate phase lag to variations in forcing, whether diurnal, synoptic or seasonal.

The effective heat capacity of sea ice is affected significantly by internal release of latent heat as the solid fraction varies in time. This is seen most readily from the thermal energy equation (in the absence of fluid flow)
2.6in which the role of latent heat as an internal heat source as the solid fraction varies is transparent. The left-hand side of equation (2.6) is the rate of change of enthalpy with time, noting that the specific heat capacity is equal to the derivative of enthalpy with respect to temperature at constant pressure [4]. The mean sensible heat capacity per unit volume is
2.7where *c*_{s} and *c*_{br} are the specific heat capacities of pure ice and of brine, respectively, and *ρ*_{br} is the density of brine. From equation (2.1), relating the solid fraction and the interstitial concentration, and a statement of the freezing-point depression of water caused by salt (the liquidus equation)
2.8in which *T*_{m}=0°C is the freezing temperature of pure water and *m*≈0.05°C ppt^{−1} is the liquidus slope, we can determine that the temperature satisfies a nonlinear diffusion equation
2.9in which the effective heat capacity
2.10Note that an assumption is made that the interior of a mushy layer is in local thermodynamic equilibrium so that the liquidus relationship (2.8) holds throughout the layer [1].

The effective heat capacity is dominated by the term involving the latent heat, which is dependent also on the salinity of the sea ice. In deriving equation (2.9), it has been assumed that the bulk salinity is quasi-steady so that ∂*S*/∂*t*≈0. The one-layer Semtner model of sea-ice growth [2] is essentially a discretized implementation of this diffusion equation with a single grid point at mid depth in the sea ice. Similarly, CICE v. 4 [5] (for example) implements the diffusion equation (2.9) on a small number (often 4) of discrete layers, which could be increased if felt necessary as computing power increases. In both cases, the salinity of the sea ice is prescribed, either as a constant or as a given profile.

These sorts of approaches improve the thermodynamic description of sea-ice evolution but there are important questions to be addressed. The typical estimations of ice salinity of approximately 4 ppt used to determine the effective heat capacity are more typical of relatively thick, late-season or multi-year ice, from which the brine flux is relatively small, than of the early formations of ice in leads, from which the brine flux is more significant. They are also based on observations of ice formed in median ocean salinities (approx. 35 ppt) and may be less appropriate for future scenarios involving a freshening ocean or currently in regions of brackish water such as the Baltic Sea. Ideally, a model of the thermodynamic growth and evolution of sea ice should be robust to changing climate scenarios and of utility regardless of regional variations of ocean salinity. To this end, there have been recent efforts to develop models in which the ice salinity profile is predicted dynamically, which is the focus of this article.

An additional reason for wanting to know or be able to predict changing ice salinity is to determine the buoyancy flux to the ocean, which is dominated in polar regions by brine fluxes associated with ice formation. Some current sea-ice models dissociate the processes of thermodynamic growth and brine rejection, sometimes specifying different ice salinities to determine thermodynamic properties from those used to determine brine fluxes but ideally an internally consistent model should couple these elements.

The convective transport within sea ice plays a broader role within the climate system [6]. For example, convection supplies nutrients to the diverse microbiological community within sea ice and interact with the transport of trace gases through sea ice. There are potential physical feedbacks associated with biogeochemical transport, such as through ice albedo and atmospheric chemistry.

Before describing some new modelling approaches that provide an internal description of brine drainage, we first provide a more general discussion of buoyancy fluxes.

## 3. Buoyancy fluxes

The brine fluxes from growing sea ice make a significant contribution to the buoyancy forcing of the ocean. Considering an annual cycle, it is interesting to contrast the buoyancy flux resulting from cooling the surface waters prior to any ice formation with the buoyancy flux that results once ice begins to form either in an open polynya or in a region of consolidated ice.

Although sea water with salinity above approximately 24 ppt has a monotonic variation of density with temperature above its freezing temperature, the same physics that gives rise to the density maximum of pure water at 4°C gives polar sea water a strongly nonlinear equation of state with a thermal expansion coefficient *α*≈10^{−5}°C^{−1} at its freezing temperature of around −2°C, compared with a value of *α*≈10^{−4}°C^{−1} at 20°C. These values and the following estimates of buoyancy fluxes are only intended to illustrate orders of magnitude. Exact values depend on salinity as well as temperature.

The buoyancy flux to the ocean associated with cooling ice-free surface waters in the absence of newly forming sea ice is
3.1where *g* is the acceleration due to gravity and, here, *c*_{p} is the specific heat capacity of the ocean. This buoyancy flux has a value of approximately 10 μW m^{−3} when the surface heat flux from ocean to atmosphere *F*≈400 W m^{−2} and the ocean is near to its freezing temperature. By contrast, within a polynya, where growing ice crystals are blown downwind and the open ocean is continuously exposed to the cold atmosphere, the buoyancy flux associated with the resulting brine flux is
3.2where *β* is the fractional change in density associated with a change in salinity, and *ρ*_{o} and *S*_{o} are the density and salinity of the ocean. This brine-driven buoyancy flux has a value of about 300 μW m^{−3} for the same heat flux. These typical buoyancy fluxes are illustrated in figure 1.

Once the ice consolidates and forms a columnar structure, it insulates the ocean from the cold atmosphere and the ice–ocean interface has a temperature very close to the freezing temperature of the ocean. The thermal buoyancy flux is then associated with the ocean heat flux, and if the ocean heat flux is approximately 6 W m^{−2}, for example, then the associated buoyancy flux is only about 0.2 μW m^{−3}. On the other hand, brine continues to drain from the columnar sea ice, and leads to a time-varying buoyancy flux, illustrated in figure 1. The figure also illustrates the different prediction of buoyancy flux made if the salinity of the ice is assumed to be fixed at 4 ppt rather than evolving dynamically, as described in §6. For now, we simply note that the prediction made using a dynamically evolving salinity reaches a maximum value within the first day or so that is approximately one-fifth of that from an open polynya.

## 4. Fundamentals of convection in sea ice

The brine in the interstices of sea ice is much saltier and therefore denser than the sea water from which the sea ice formed. It therefore has potential energy that can be released by brine draining from the sea ice into the ocean. It is well known that fluid systems that are statically unstable, having a density that increases with height, can be dynamically stable if their Rayleigh number is insufficiently large. The Rayleigh number characterizes the potential energy available to drive fluid motion relative to the dissipation of that energy both by viscosity and by diffusion of the agent causing the density variations. A parcel of interstitial fluid of size *d* in sea ice has potential energy
4.1relative to the ocean, where *z* is the vertical position of the fluid parcel measured downwards from the surface of the ice.

In the net movement of the parcel downwards and the compensating flow upwards through the porous mushy layer there is dissipation of energy
4.2where *U* characterizes the magnitude of the fluid motion (Darcy velocity) and *τ*∼(*h*−*z*)/*U* characterizes the time over which the exchange of potential energy takes place. The other variables are the pressure difference Δ*p* between the location of the parcel and the ocean, the dynamic viscosity of brine *μ* and the permeability of the porous sea ice *Π*.

The motion can only be sustained if the dissipation is less than the potential energy available, which argues for slow motion releasing the potential energy over a long time. On the other hand, if the movement takes too long then the buoyancy will be dissipated by a combination of diffusion and phase change as follows.

The parcel, which is assumed to be much larger than the pore scale, tends to equilibrate its temperature with its new surroundings on a time scale mediated by thermal diffusion. Because the diffusivity of salt is very much less than that of heat, the parcel will not decrease in salinity as quickly by diffusion but will instead dissolve some ice until its salinity matches the liquidus value corresponding to its temperature. Thus the parcel loses its buoyancy on a time scale determined by thermal diffusion. In order to release this potential energy as kinetic energy of convection, *τ* should not be long compared with the thermal diffusion time *d*^{2}/*κ*.

Bearing in mind the dual constraints of viscous and diffusive dissipation of the potential energy, it is optimal if *τ*∼*d*^{2}/*κ* and there will be convection if , so if
4.3From this expression, we see that larger scale motions (large *d*) are preferred and that the optimal energy transfer is achieved when the fluid motions span the full depth of the domain so that *d*∼*h*−*z* and the Rayleigh number
4.4where *g*′(*z*)=[*ρ*(*z*)−*ρ*_{o}]*g*/*ρ*_{o} is a reduced gravity and *ν*=*μ*/*ρ*_{o} is the kinematic viscosity of the brine.

Note that these fundamental physical principles and the associated scaling analysis apply also to convection in a mushy layer once brine channels have formed, as described below. The downflow is then principally through the channels but must be compensated by upflow through the surrounding mushy layer, which provides the dominant resistance to the convective flow of the system.

Theoretical studies of idealized settings [7] have shown that a motionless state is unstable and convection is predicted in a mushy layer that is open to a fluid region when the Rayleigh number exceeds a critical value of approximately 10. This is comparable with the critical Rayleigh number below which convection with liquid channels shuts down [8]. Note that the thermal diffusivity *κ* in this estimation should be taken to be *k*/*c*, taking account of the enhanced specific heat capacity resulting from latent heat release and the phase-averaged thermal conductivity. Often, using simple, dimensional scaling, a Rayleigh number is initially defined in terms of the thermal diffusivity of the liquid phase [9,10]; however, it can then be shown [7] that it is the parameter grouping that must exceed a critical value for convection to ensue, where *Ω* is the factor by which the specific heat capacity is augmented by latent heat release.

Flow through mushy layers causes dissolution of the solid phase when directed from cooler to warmer regions [11]. Thus, downflow in sea ice during growth has a tendency to form brine channels, which form the principal conduit for brine drainage into the ocean, as described below. Conversely, upflow through the mushy region outside the channels promotes further solidification, which can lead to the confinement of convection near the ice–ocean interface, as described and predicted in §6.

## 5. Experimental observations of brine fluxes from sea ice

Laboratory experiments in which salt water is cooled and solidified from above show clearly the formation and structure of mushy layers of ice, with a porous matrix of dominantly vertical ice platelets punctured by brine channels (figure 2). A study of ice mushy layers using magnetic resonance imaging [12] has revealed the evolution of brine channels and shown that freshening of the layers is associated with the presence of channels. Dense brine emerges from brine channels to form strong plumes injecting salt into the underlying liquid, punching through any putative laminar sublayer in a turbulent parametrization of brine fluxes at the ice–ocean interface. Brine fluxes are rather determined by convection internal to sea ice; a laminar flow down brine channels supplied by upflow from the ocean through the surrounding mushy layer.

Experiments in which the rate of solidification was controlled and steadily growing states were achieved [13] have confirmed that brine channels form once the Rayleigh number is large enough and that convection ceases if the Rayleigh number subsequently decreases sufficiently, while growth continues. The fundamental idea that brine drainage is not directly linked to growth was also demonstrated [10] in experiments in which significant brine drainage only began once the layer of sea ice was sufficiently thick and therefore its associated Rayleigh number was sufficiently large.

A key experimental observation [12] is that brine channels need not occupy the full depth of the mushy layer and that brine drainage is confined to that part of the mushy layer that contains (active) brine channels (figure 3). The interpretation is that in those cases the local Rayleigh number, as defined by equation (4.4), exceeds the critical value only in a region of the mushy layer adjacent to the mush–liquid interface. This idea was also explored in interpretations of field measurements of sea ice [14], where it was found that the local Rayleigh number peaked a few centimetres above the ice–ocean interface and had very small values higher in the layer of sea ice. The peak value of the Rayleigh number was also found to hover around a fairly constant value, much lower than the value that would have been attained had brine drainage and freshening of the sea ice not occurred.

Equation (4.4) shows that the Rayleigh number increases with height above the ice–ocean interface as the potential energy driving convection, proportional to [*ρ*(*z*)−*ρ*_{o}](*h*−*z*), increases but decreases as the permeability *Π*(*z*) decreases in response either to colder temperatures or to lower bulk salinities. For example, the permeability can be estimated as *Π*=*Π*_{0}(1−*ϕ*)^{3}=*Π*_{0}(*S*/*S*_{br})^{3}, as used by [10] and which approximates the expression suggested by Freitag [15]. Then if the bulk salinity *S* is uniform, *Π*=*Π*_{0}(*S*/*S*_{br})^{3} and
5.1when the temperature field
5.2is linear, and we get the profile shown in figure 5*a*. Once convection begins, brine drains through the brine channels to the ocean and the necessary return flow brings relatively fresh water upwards from the ocean into the sea ice where it partially freezes to increase the solid fraction and reduce the permeability. Thus convection in a mushy layer is self-regulating.

The fundamental ideas and observations of the preceding two sections have recently been brought together into new, one-dimensional parametrizations of brine drainage in sea ice.

## 6. Theoretical modelling of brine drainage

Detailed, albeit idealized, two-dimensional, numerical calculations of flow in a mushy layer with brine channels have been made by Chung & Worster [8] and by Wells *et al.* [16]. Both studies analysed a periodic array of channels and determined the brine flux from a single channel as it varies with Rayleigh number. But whereas Chung & Worster left open the question of what determines the spacing between channels and, therefore, the total brine flux, Wells *et al.* introduced the hypothesis that the channel spacing optimizes the total flux. Not only did they show with their numerics that such an optimal flux is achieved at a finite, non-zero and sensible spacing, similar to that seen experimentally, but also they showed that the optimal flux correlates well in a linear relationship with the Rayleigh number [17].

Such calculations are computationally expensive, and a step towards a simpler, mostly analytical model of brine drainage, the so-called Chimney-Active-Passive (CAP) model, was developed by Rees Jones & Worster [18] (figure 4). Within the approximations of the CAP model, including an assumption of large supercritical Rayleigh numbers, they showed analytically that the net brine flux in two dimensions is linearly related to the Rayleigh number. They extended their model to three dimensions, where they found that the linear relationship was maintained approximately. The CAP model also showed (in the idealized setting) that the vertical component of velocity through most of the mushy layer decreased linearly with height above the ocean to zero at the top of the convecting region.

Three models of brine drainage from sea ice have emerged in the last year or so that are based on conceptions of buoyancy-driven flow through brine channels and have the potential to be computationally cheap enough to be incorporated into climate models [19–21]. Indeed, the model of Turner *et al.* [19] is now included as an option within CICE v. 5 [22]. They share the common assumption, based on the observations above of more detailed models, that the brine flux varies linearly with the Rayleigh number. They each begin by computing the local value of the Rayleigh number *R*_{m}(*z*) defined by equation (4.4). There are minor differences according to whether the permeability is taken to be the minimum value or the harmonic mean of *Π*(*z*) between *z* and the ice–ocean interface. The latter is more appropriate in the calculation of the pressure difference in (4.2) but the two estimations not too different given that the harmonic mean value is dominated by the lower values. An example profile of *R*_{m}(*z*) is shown in figure 5*a*. If the Rayleigh number exceeds a critical value *R*_{c} anywhere in the domain then it is assumed that there is a horizontally uniform upwards flow *w*(*z*) providing the return flow for brine draining downwards through brine channels. The continuum profiles of the return flow implied by the three models differ slightly as follows.

Turner *et al.* [19] take the flow rate to be the largest of the flow rates they calculate for layers at or above the layer in question. The dominant character of this choice is illustrated by the expression
6.1where *α*(*z*) is a positive function, bounded away from zero, that measures some of the internal dimensions of the mushy layer as well as its material properties.

Griewank & Notz [20] calculate a local downflow at each level, proportional to the amount by which the local Rayleigh number exceeds a critical value, and sum these for all levels above the level in question to determine the upflow there. This is illustrated by the expression 6.2

Based on their CAP model, Rees Jones & Worster [21] assume a simple linear velocity profile across the convecting region, with its magnitude determined by the maximum value of the Rayleigh number, so that
6.3where *z*_{c} is the minimum value of *z* for which *R*_{m}(*z*)≥*R*_{c}.

In each of these expressions, which are illustrated in figure 5, the term in square brackets is taken to be zero if the value inside the brackets is negative, i.e. if *R*_{m}(*z*)<*R*_{c}. In many scenarios, such as that shown in figure 5, the three expressions lead to broadly similar velocity profiles, in that all are monotonic and confined to a region of the sea ice close to the ice–ocean interface, and all provide for convection to cease once the Rayleigh number is too small. Turner *et al.* estimate the magnitude of convection *α* by equating the dynamic pressure drop along the channel with that driving a vertical return flow through the surrounding mush. Rees Jones & Worster [18] calculate *α* in the idealized setting explored with their CAP model but suggest that in the context of sea ice it is best treated as a tuning parameter [21], as is also done by Griewank & Notz [20]. Although various stability analyses have determined critical Rayleigh numbers in different settings, all three sets of authors treat the constant *R*_{c} as a tuning parameter, which seems appropriate for use in the complex settings of real sea ice.

With the vertical velocity computed as above, the ice salinity is updated using the local salt-conservation equation
6.4while the thermal energy equation (2.9) is integrated in the modified form
6.5where *c*_{w}=*ρ*_{br}*c*_{br}+*ρ*_{s}*L*/[*T*_{m}−*T*(*z*)].

It is important to note that all three models are approximate, and the question is rather whether they are reasonable and convenient parametrizations of brine drainage, having the correct phenomenology. The models of Turner *et al.* and of Rees Jones & Worster are faithful to the physics that convection is a global process (communicated through the pressure field) rather than a sum of local processes. The models of Griewank & Notz and of Rees Jones & Worster are faithful to the physics that *w*(*z*_{c})=0 when *z*_{c}=0, whereas the model of Turner *et al.* gives a non-zero vertical velocity at the ice surface when there is whole-layer convection. On the other hand, the models of Griewank & Notz and of Turner *et al.* have the property that *w* is continuous in time, which may make for easier, less costly calculation, whereas *w* can jump discontinuously in the model of Rees Jones & Worster, which is an inconvenience that potentially reduces the order of convergence of numerical schemes implementing the model, though it may reflect the physics of brine channels switching on and off as conditions affecting the Rayleigh number vary.

All three models capture important dynamics. They can all predict the sort of delay in the onset of brine drainage seen in some experiments [10,21]. This may not be an important delay in natural settings given that the early stages of ice growth in open ocean can involve accumulations of frazil into grease ice or pancake ice, for example, before the consolidated regime begins. On the other hand, the physics causing such a delay, that the Rayleigh number is insufficiently large, can occur in later stages of ice growth as the atmospheric and oceanic heat fluxes vary. Related to that, they can all predict enhancement or re-initiation of brine drainage during periods of warming [20]. And they all adjust their dynamics to different oceanic salinities and should work as well in the Baltic, for example, as in the central Arctic.

A representative calculation of growth is shown in figure 6. For the calculation shown, the ocean has salinity 35 ppt and temperature −2°C. The heat flux from the ocean is calculated from a 4/3-power-law convective parametrization to be approximately 6 W m^{−2}. The surface heat flux to the atmosphere and surface temperature are related simply using a heat-transfer coefficient of 20 W m^{−2}°C^{−1}, with a fixed atmospheric temperature of −20°C. The vertical velocity was calculated using equation (6.3) with *α*=0.003 and *R*_{c}=5. We see that, in this monotonic-growth scenario, the temperature field remains approximately linear, the salinity decreases over the 100 days simulated to a modal value of approximately 6 ppt, while the brine volume also decreases to a modal value of less than approximately 0.1. In consequence of the ever-changing porosity, the maximum Rayleigh number is maintained at a roughly constant size. Many other scenarios have been explored, particularly by Griewank & Notz [20].

The brine flux predicted by this model is shown as a function of time in figure 1. In comparison, the dashed curve shows the prediction of a zero-layer Semtner model with the same boundary conditions, with brine fluxes determined by assuming a fixed ice salinity of 4 ppt. The very close agreement between these two models at late times is coincidental and unexpected given the approximate nature of the examples chosen here for illustration and the fact that the two calculations give rather different predictions (not shown) of the thickness of sea ice as a function of time. However, the correspondence of the functional forms predicted is relevant and indicates that the convective model can be tuned to give similar long-term trends as a fixed-salinity model for a given scenario.

## 7. Discussion and conclusion

Brine drainage from sea ice is a complex process involving interactions between fluid flow in a porous medium and phase change that affects the fluid-mechanical properties of the medium. All of that happens on horizontal scales of millimetres to centimetres, and the internal flows are intrinsically three-dimensional on those scales. An understanding of the underlying physics has led to the recent emergence of one-dimensional models that seem capable of predicting the evolution of mean vertical salinity profiles in sea ice. While enormously more simple and cost-effective than two- or three-dimensional models of the resolution required to capture brine channels, these schemes do increase the computational cost relative to current operational ice components of climate models, so questions arise whether they are feasible given current resources and whether they are worthwhile.

Preliminary calculations [19] suggest that equations (6.4) and (6.5) can be evaluated effectively using as few as four layers in the vertical, with results in reasonable agreement with calculations made using 100 layers in the vertical and with field data [14].

The long-term trends of salt flux versus growth rate evaluated using climatological data to determine ocean and atmospheric heat fluxes [20] suggest that, when averaged over a week, a linear relationship between salt flux and growth rate, as used in current models, gives a good approximation, so the sophistication and cost of evaluating the salinity of sea ice dynamically may not be worthwhile. On the other hand, such a coarse-grained view averages over periods of rapid growth while the ice is thin and periods of rapid drainage during warming events. Such periods may be very important dynamically, particularly perhaps during the early formation of sea ice in leads, for which buoyancy fluxes are localized spatially as well as temporally. Like clouds with respect to heat fluxes in the atmosphere, such localized convective events may create significant counter-gradient salt fluxes, which would need to be taken into account for a proper understanding and quantification of oceanic forcing.

So, while simple segregation models may be sufficient to determine thermal properties of sea ice, the development of convective schemes along the lines presented above seems worthwhile, to give better representations of buoyancy fluxes as they vary in time, to be robust in the face of regional variations of ocean salinity or scenarios of ocean freshening, and to provide estimates of the velocity fields inside sea ice that can transport biogeochemical materials such as nutrients for marine life.

## Acknowledgements

We are very grateful for constructive feedback on an earlier draft from Jerome Neufeld and Andrew Wells.

## Footnotes

One contribution of 9 to a discussion meeting issue ‘Arctic sea ice reduction: the evidence, models and impacts (part 1)’.

- Accepted January 6, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.