Large rifts that open in Antarctic ice shelves are known to be filled by snow accumulations, ice shelf fragments, and sea ice. This work demonstrates how these rifts may also be filled from below, through their interaction with ocean water, by marine ice. The model presented here quantifies both the rate of marine ice accumulation at the top of the rift, which results from melt-driven convection at its sides, and the impact of this process on waters occupying the confined environment of the rift. The results show that such a system could fill the rift with tens of meters of marine ice. In the process the temperature and salinity of the ambient water in the rift evolve through two distinct phases. The first is characterized by rapid change that takes the properties of the ambient water to near-equilibrium values, followed by a second phase of much slower change. The melting rate at the lower part of the walls of the rift emerges as the principal factor influencing many aspects of model behavior. Testing the model against field observations from an Antarctic rift showed it to be robust and successful in reproducing the main observed features, including the presence of a thick layer of supercooled ocean water inside the rift.
 Rifts, defined here as openings that span the entire thickness of the ice shelves where they occur, are common features in floating Antarctic ice masses. While such fractures assume different sizes, this study considers those that are at least several tens of meters wide and deep. Rifts form in regions where the tensile stress exceeds a critical value at which ice failure occurs. Such regions are common where an ice shelf undergoes differential motion near a lateral margin or at a boundary between fast and slow moving ice. Tensile stresses are also generated by a number of mechanisms in the vicinity of an ice front, and here fracture and rift formation are the precursors of iceberg calving.
 Fracture is a critical factor in ice shelf dynamics [Hughes, 1983] and its role in controlling the disintegration of ice shelves has been established. A compelling case in point is the disintegration of the Wordie Ice Shelf in which rifting was responsible for the weakening of the central region of the ice shelf and the resulting retreat of the ice front [Doake and Vaughan, 1991]. In another account of an important calving event, Keys et al.  demonstrated how rifts controlled the calving of the B-9 iceberg (154 by 35 km) at the eastern part of the Ross Ice Shelf front.
 There is growing evidence that rifts may be partially or completely filled with ice accumulating both from above and below. Investigators have long observed that rifts are filled from the top with a heterogeneous mixture of sea ice, snow accumulations and broken fragments of the ice shelf [e.g., Ardus, 1965; Stephenson and Zwally, 1989; Rignot and MacAyeal, 1998; MacAyeal et al., 1998]. On the other hand, the large thickness of material that is often observed in rifts suggests that they could also be filled from below with marine ice. Distinct from sea ice, which forms at the top of the ocean through direct heat loss to the atmosphere, marine ice is a body of ice that forms beneath an ice shelf as a result of the accumulation and subsequent consolidation of frazil ice crystals. Daly  defines frazil ice as fine spicule, plate or discoid crystals which form when heat is withdrawn from a turbulent body of water at its freezing point, thus rendering it supercooled. Thus, when Grosfeld et al.  observed that a giant rift (more than 10 km wide and 100 km long) in the northern part of the Filchner Ice Shelf contained tens of meters of ice, they invoked the possibility that the inside of the rift had provided the environment necessary for ice/ocean interaction to fill it with marine ice. Work by Khazendar et al.  and Tison et al.  provided evidence that marine ice can indeed form within rifts. Laboratory analyses carried out by the authors on a 45-m ice core from the Nansen Ice Shelf revealed that the entire core was composed of marine ice. Yet the core had been retrieved from a zone, near the grounding line, of active sub-ice shelf melting as shown by field data [Frezzotti et al., 2000] and as expected from established sub-ice shelf circulation models. Khazendar et al.  demonstrate that the ice must have formed in one of the many rifts open in the grounding line area.
1.2. Ice Pump Concept
 The term “ice pump mechanism” was used by Lewis and Perkin  to describe the process, driven by the dependence of the freezing point on pressure, through which ice is melted at depth in the ocean and deposited higher in the water column. The depression of the freezing point with increased pressure gives water at the surface freezing point the potential to melt ice at depth. When fresh water ice melts in the ocean, both the temperature and salinity of the resulting mixture are reduced relative to the ambient with opposite effects on density. Greisman  showed that for ambient water of typical oceanic salinity and at a temperature below 20°C the impact of lowered salinity would dominate and a mixture of meltwater and ambient water would rise through the water column. Since the pressure freezing point rises with decreasing depth the cooled mixture of meltwater and ambient water will subsequently approach a state of supercooling when ice production can commence.
 Operating within a rift, this process would melt ice at the lower portions of its sidewalls and deposit it on the upper portions of the walls by direct freezing and, more substantially, by producing frazil crystals that accumulate at the top of the cavity to form marine ice. One of the main objectives of this work is therefore to demonstrate theoretically that rifts that open in Antarctic ice shelves can indeed be filled, at least partially, given the prevailing physical conditions, with a significant amount of marine ice as a result of the ice pump process. Another important objective is to compare the results of this model with observations made within a rift and thus to explain some of the observed properties of the water filling the rift.
1.3. Plume Model
 At the heart of the proposed model, the detailed mechanism for the simulation of melting and freezing processes at the ice/ocean interface is provided by a plume model. The plume model created by Jenkins and Bombosch  for sub-ice shelf circulation lends itself readily to the rift problem since it is the only one to date to include an explicit treatment of frazil ice formation in an ascending plume and its subsequent deposition. This is in contrast to the Jenkins  plume model where supercooling is released exclusively through direct freezing on the bottom of the ice shelf (or the side of the rift in the current problem). Therefore the Jenkins and Bombosch  model, which showed good agreement with field measurements when applied to melting and freezing at the base of the Filchner-Ronne Ice Shelf, is adopted and adapted to the rift problem. The Jenkins  simple one-dimensional model, which was later modified to create the 1995 model, is similar in concept to MacAyeal's  streamtube simulation and its equations are adapted from those conventionally used in the description of turbulent gravity currents. Hence the model envisages a two-component (frazil plus water) turbulent gravity current ascending the interactive (melting and freezing) underside of an ice shelf. In the process, the plume entrains water from a stationary body of ambient ocean water.
 The idea that rifts have plumes rising along their vertical or almost vertical walls is supported by the work of Potter and Paren . The authors analyzed temperature and salinity profiles measured through rifts open in the George VI Ice Shelf. This led them to conclude that the velocity of the water in the middle of the rift, where the profiles were taken, must be downward. Potter and Paren  hence explicitly propose that melting must be occurring at the ice walls of the rift causing upwelling at each side and a compensating down flow in the middle.
1.4. Filling Box Concept
 In the confined space of a rift, a plume will actually be modifying the properties of the water it is entraining, to which it is in turn most sensitive. Therefore a description of the manner by which the ambient water in the rift evolves with time is necessary, and this is inspired by filling box models. The pioneering theoretical and experimental work in this field was done by Baines and Turner . Investigation was later expanded by other workers including Germeles , Manins , and Worster and Huppert . The problem they addressed concerns a plume starting from a small source of buoyancy at the bottom of a box filled with heavier ambient fluid. On its way up, the plume entrains this denser fluid. Since the density of the ambient water is constant the plume will always remain buoyant until the top where it spreads a layer of less dense fluid separated from what is beneath it by a front. As a result, the structure of the ambient fluid is modified. Now that the newly deposited layer is part of the ambient fluid, the plume will subsequently entrain the same ambient fluid as before only up to the level of the front. Above, it will start entraining the relatively less dense liquid deposited earlier. Therefore it arrives at the top of the tank even lighter than before and spreads another layer on top of the first one. It should be stated here that the lateral spread of the plume when it loses its buoyancy is considered to be instantaneous and that its details are not considered in the rift model.
 The aspect of most relevance to the rift problem from the filling box work is the rate of advancement of the new density profile in the ambient water. From the principle of conservation of volume the downward flux of ambient water at any height in the box must equal the upward flux of plume fluid at the same height [e.g., Baines and Turner, 1969].
 One important difference between filling boxes and the rift situation is that in the latter the plume could be laden with frazil crystals. The presence of ice will then be a source of additional buoyancy for the frazil water mixture. Once the plume reaches the top of the water column the crystals are deposited upwardly as a result of their lower density. Consequently, it might turn out that the density of the residual water is actually higher than that of the water beneath. Thus the resulting density structure could be unstable.
2.1. General Description of the Rift Model
 The presentation of the rift model begins with a description of the sequence of events in the course of a standard simulation. This is done with the help of Figure 1, which shows the model configuration.
2.1.1. Input Information
 A model run begins by defining initial conditions for the ambient water temperature and salinity and the spatial proportions of the rift. The model can handle walls that are inclined as well as vertical ones. The initial rift width, provided as part of the input, is at once divided by two and the resulting value is used throughout the simulation. This is equivalent to assuming that there are actually two plumes active in the cavity, one at each wall. Complete symmetry between the two sides is assumed. The far field temperature of the ice is specified as a function of depth within the ice shelf. In all but one of the experiments below, a linear function is adopted with temperature increasing from −20°C at the surface of the shelf to the in situ freezing point at the ice/ocean interface. Finally, a small volume flux is specified as the initial condition for the plume at the lowermost point of the rift's wall. The plume then ascends the wall (Figure 1) increasing its volume flux in the process, mainly through the entrainment of ambient water.
2.1.2. Water Deposition and Layer Displacement
 With the above information provided, the simulation commences and a plume is active during the specified time step Δt. At the end of this period, the plume is allowed to spread its water at the top of the cavity. The temperature and salinity of the new layer are those of the plume at the point where it terminates. As for the thickness of the deposited layer, Δz, it is calculated from the principles of the filling box model. Specifically,
where U is the magnitude of the plume's velocity, D its thickness, ua is the magnitude of the downward velocity of the ambient water and l is the half width of the rift, all at the same depth from the surface z. From (1) a profile of ua with respect to z is defined and the depth reached by a certain front in a given time Δt is evaluated as
2.1.3. Ice Deposition
 The thickness of the layer of accumulated ice Δzice, which is assumed to be equal across the cavity's width at the top, is found from
where C is the concentration of frazil ice by volume at the point where the plume terminates. The deposited thickness is that of “pure” ice with no interstitial water being considered. It should be stressed that the present model does not treat, nor are its parameters influenced by, the consolidation process of the accumulated ice. Marine ice consolidation could affect the processes considered here through the gradual release of brine that was initially retained in the ice. Ultimately, however, the resulting salinity of marine ice is so low as to make it an insignificant sink of salt in the system. Also not included in the model is sea ice formation at the ocean-atmosphere interface, which would influence the composition of the top few meters of a body of marine ice.
2.1.4. Water Deposition and Density Check
 Immediately beneath the deposited ice crystals lies the water layer (Figure 1) also introduced by the plume during the time step. This layer extends from the ice-water interface to the depth of the front calculated using equation (2) and has constant temperature and salinity throughout. Its density is now tested relative to the density of the ambient water beneath it. If instability is discovered, then the new layer and the one below it are mixed. Afterward, the density of the water resulting from the mixing process is verified relative to the one below it and so on until the whole ambient water column has been checked and stabilized. Each time the mixing of two layers is performed, the resulting temperature and salinity are calculated as the mean values weighted by the volumes of the two original layers. If no instability is detected, the new layer remains until water deposited by the plume in the following time step pushes it downward and another new layer is introduced between it and the ice-water interface, now lowered because of new ice deposition.
2.1.5. Modification of Rift Dimensions
 The vertical extent of the cavity is constantly diminishing as a result of ice crystal accumulation at the top. Therefore, once the thickness of deposited ice is calculated for a time step, the vertical dimension of the rift is updated accordingly. Furthermore, the width of the rift at each depth is always being modified as a result of melting and freezing processes taking place at the wall. The computed melt rate m′, when multiplied with the time step Δt, gives the distance by which the wall retreats (in the case of melting) or advances (in the case of direct freezing) at a given depth. This calculation is done at depth intervals of 10 m. The lowest point of the wall, where the integration starts and little melting takes place, is set at the end of each time step to retreat by the same extent as the grid point directly above it. This is done to avoid having a thin spike of ice projecting out from the bottom of the wall, which would be contrary to observations showing rifts as having rather rounded edges there [e.g., Potter and Paren, 1985; King, 1994]. After the sequence of events described above, the plume model is rerun for a new time step with the updated rift dimensions and ambient temperature and salinity used as the input data. This cycle continues for the specified run time.
2.2. Plume Model Equations
Jenkins and Bombosch  thoroughly discuss, and in many cases, derive the equations of their model for the circulation in a sub-ice shelf cavity. They are listed here in their form adapted to the rift model. In constructing the model, Jenkins and Bombosch  make several assumptions. The flow of the plume is supposed to be planar and all properties are taken to be constant over its thickness (except when considering the details of diffusion processes at the ice/water interface). Average values of these properties are calculated when dealing with plume dynamics, the granular structure of the plume being approximated by that of a homogeneous fluid. Seawater is treated as incompressible and Boussinesq. Steady state is assumed.
2.2.1. Density Equations
 The mean density of the plume, ρm, is obtained by summing the densities of its two components, thus:
where ρ is the density of the water fraction of the plume while ρi is the density (considered to be constant) of the ice fraction C. One simplification is to put ρ = ρ0 (where ρ0 = 1030 kg m−3 is a reference seawater density) whenever the mass of the fluid, rather than its weight, is being considered. In this case the mixture density is denoted by ρm•.
 The density of the plume water is given by the relation
where S0 = 34.5 and T0 = −2.0°C are the reference salinity and temperature, respectively. The coefficients are given by βS = 7.86 × 10−4 and βT = 3.87 × 10−5 °C−1.
2.2.2. Plume Equations
 The evolution of five plume properties, specifically its thickness (D), velocity (U), concentration of frazil ice by volume (C), temperature (T) and salinity (S) is calculated from the following five ordinary differential equations.
2.2.3. Mass Conservation Equations
 The mass conservation equations are
where e′ is the entrainment rate, m′ is the melt rate at the ice-ocean interface and f′ is the total melt rate of frazil ice in the plume (all are in terms of thickness of water at the reference density ρ0 per second and are positive when mass is gained by the plume). The rate at which frazil crystals precipitate from the moving plume onto the rift wall, p′, is set to zero. This is a good approximation since the flow is never far from the vertical.
2.2.4. Momentum Conservation Equation
 Conservation of linear momentum for the plume can be expressed as:
where g = 9.81 m s−2 is the acceleration of gravity, θ is the slope of the ice/ocean interface with respect to the horizontal and K = 2.5 × 10−3 is the drag coefficient. The average ambient water density over the thickness of the plume, ρaD, is:
 The thickness of the plume is always small compared with the width of the rift, less than one tenth for most of the trajectory of each plume for most of the time. Equation (1) shows that the velocity of the ambient water is smaller by the same proportion than that of the plume, so the downward velocity of the entrained ambient water is taken to be zero in the momentum relation.
2.2.5. Heat and Salt Balance Equations
 If Ta, Tb and Tc are, respectively, the temperature of the ambient water at the edge of the plume, the temperature of the water in contact with the wall of the rift and the temperature of the water in contact with the ice crystals, and if the depth average of the latter across the plume is TcD, then:
where Ac, the total area of the crystal edges per unit volume of the mixture, is given by:
in which the aspect ratio of frazil discs (thickness/diameter) is ε, the specified radius of frazil crystals is r and their number per unit volume of the mixture is N. The volume of water created by the melting of frazil ice per unit volume of mixture per unit time is denoted by w′, while γTb and γTc are the heat transfer coefficients at the wall of the rift and at the edge of frazil crystals, respectively. They are calculated from:
where Pr = 13.8 is the dimensionless Prandtl number and KT = 1.4 × 10−7 m2 s−1 is the molecular thermal diffusivity of seawater. The expression for γTb is adapted from Kader and Yaglom . The parameters in the salt balance equation are defined in an analogous manner with the Schmidt number, Sc = 2432, replacing Pr in the denominator of equation (3) and KS = 8.0 × 10−10 m2 s−1, the molecular diffusivity of salt, replacing KT in equation (4).
2.2.6. Equations for Melting and Freezing at the Ice/Ocean Interface
 The diffusion of heat and salt toward and away from the interface between the wall of the rift and the plume and the phase changes this induces are governed by:
where the parameters for ice are thermal diffusivity, Ki = 1.14 × 10−6 m2 s−1, temperature, Ti, specific heat capacity, ci = 2009.0 J kg−1 °C−1, and latent heat of fusion, L = 3.35 × 105 J kg−1. The specific heat capacity of seawater is c0 = 3974.0 J kg−1 °C−1. In the equation for salt, Sb represents the salinity at the wall of the rift. The depth below sea level (negative downward) of the point at the interface at which temperature and salinity are being calculated is denoted by zb. Equation (6) is a linearized form of the liquidus relation for calculating the freezing point as a function of the depth and salinity of the water. In this case, for seawater, the liquidus slope, a = −0.0573°C psu−1, the offset of the liquidus, b = 0.0832°C and the depression of freezing point with depth, c = −7.61 × 10−4 °C m−1.
2.2.7. Equations for Melting and Freezing of Suspended Ice Crystals
 Heat and mass exchanges at the suspended crystal/plume water interface are analogous to the ones occurring at the ice shelf/ocean interface:
Solving these equations yields the remaining unknown quantities in the plume equations, namely, TcD, ScD and f′. To render this feasible, Jenkins and Bombosch  use the following approximation:
 Since C is of the order of 10−9 to 10−5 throughout the integration the quantity (1 − C) is replaced everywhere by 1 in the model code.
2.2.8. Ambient Fluid Entrainment Relation
 The entrainment rate is calculated, on the basis of the entrainment assumption, by using the relation
proposed by Bo Pederson , where the entrainment constant is assigned the value E0 = 0.072.
2.2.9. Initial Plume Temperature and Salinity
 A formulation provided by Gade  is used to find the initial temperature and salinity of the nascent plume. Knowing the temperature (above freezing) and salinity of the ambient water, the author derives a relation for the salinity of the thin layer that forms at the interface of a melting slab of ice in ocean water. Using the notation of the current work the equation obtained by Gade  is:
This is the form suitable for the case of an infinitesimal boundary layer. Solving this equation along with that of the liquidus relation for the freezing temperature, Tf(Sin), both the input salinity Sin and the corresponding temperature are found.
3. Results and Discussion
3.1. Observed Data Set
 The work of Orheim et al.  and that of Østerhus and Orheim  provide the most detailed observational data set, against which the model can be tested. Working on Fimbulisen (an ice shelf in Dronning Maud Land, East Antarctica), Orheim et al.  discovered a rift at 71°18.6′S, 00°17.2′E in a fracture zone, known as Jutulgryta, located at the edge of a fast ice streamflowing through a relatively stagnant mass of ice. They measured the temperature and salinity from the top of the water column down to the seafloor and recorded the depth and width of the rift. This information is presented in Figure 2. Østerhus and Orheim  returned 23 months later to collect instruments that had been left in place to monitor water properties within the rift and this allowed them to observe the change in ice thickness over the intervening period.
 The 1990 measurements reveal clear differences between the conditions of the water inside the rift, its lower edge being situated around 300 m below the surface, and the properties of the ocean beneath. Figure 2 displays rather constant profiles inside the rift with salinity values fluctuating between 34.34 and 34.35 and in situ temperatures between −1.98°C and −1.97°C (the top 40 to 50 m, which are located partially in the ice, are probably affected by percolating meltwater and the drilling effort). Once outside the rift, these parameters increase noticeably and concurrently toward the seabed reaching values of 34.38 and −1.89°C. At this temperature, the water at the seabed is at its surface freezing point as can be calculated from equation (6). These latter temperature and salinity values, with no stratification, are taken to be those of the water filling the rift initially in the following standard experiment. This implies that the rift opens instantaneously in the ice shelf filling the resultant cavity with the water that had been beneath the ice shelf. Subsequently, the water column is stratified because of plume action.
 A striking feature of the water column in Jutulgryta is the supercooled layer of water extending from 50 m depth to about 115 m beneath the surface. The accounts of the thickness of ice accumulation in the rift given by Orheim et al.  and Østerhus and Orheim  are less clear. The former authors observed 11.1 m of solid ice underlain by 27 m of slushy ice intercepted with what they describe as thick water layers, which gives a total ice and slush thickness of 38.1 m. The latter authors report that 23 months later they found 40 m of solid ice giving a difference in ice thickness between the two observations of 1.9 m. However, the uncertainties involved, such as the unknown thickness of the interceding water layers and whether the ice consolidated through freezing or compaction, make it difficult to quantify accurately the amount of newly accumulated ice. The observations hence provide only a very rough estimate of the general rate of ice accumulation, of the order of 2 m over 23 months.
3.2. Parameter Choice
 From the data provided by Orheim et al. , the width of the rift at the top can be estimated to be 340 m and its initial vertical extent below sea level is 300 m. These dimensions are used for the following standard experiment. The evolution of the model is followed through runs simulating 50 years, where a year is defined to comprise 12 months of 30 days each. A time step of 5 days is used. The 50-year time span corresponds to a very rough estimate of the age of the rift at Jutulgryta being of the order of decades. This is based on the flow velocity of the ice stream combined with the distance between the grounding line, where the marine ice would have started to form, and the location of the rift when it was observed. Crystal radius is specified to be 1.1 mm, which is near the middle of the crystal size range given by Martin  and at the higher end of what Khazendar et al.  have observed in a marine ice core retrieved from a former rift in the Nansen Ice Shelf.
 The initial plume flux, necessary to start the numerical integration of the model, is set to the small value of 1 × 10−3 m2 s−1. Such a flux would take more than three years to flush the rift cavity completely, much longer than the two weeks or so needed for the first density front introduced by the plume to reach the bottom of the rift. This is the only completely arbitrary parameter since with the currently available data it is not possible to quantify the flux of the water entering the rift. What can be assumed with some certainty is the occurrence of melting at the interface between the bottom surface of the ice shelf and the ocean water, given that in the situation studied here the water is above its in situ freezing point. The meltwater at the interface near to the bottom edge of the rift's wall could then enter the rift and initiate the convective process. In the model, the initial temperature and salinity of the plume water are those calculated from equation (8) above. The effects of varying the initial flux are investigated more fully below, where the chosen value is justified.
3.3. Standard Experiment
Figure 3a shows the evolution in time of the ambient water temperature and salinity. Where the ambient water is stratified, the values shown are those of the topmost layer in the water column. This layer is what the plume has deposited in the time step being considered and usually it occupies most of the cavity (approximately the top 290 m). In this experiment, moreover, after month 206 the ambient water is no longer stratified and the values shown in Figure 3a are those of the whole body of water in the rift. This is an indication that at some earlier stage, water deposited by the plume, once the frazil crystals had separated out, began to generate instability within the water column. The ensuing mixing of adjacent layers subsequently diminished the number of layers present until the stratification disappeared. Furthermore, since the plume has a finite initial volume flux, unlike the pure buoyancy sources in the filling boxes studied by the authors cited above, water layers leave the rift to compensate for the input.
Figure 3b depicts the concentration of frazil crystals by volume in the plume upon reaching the top of the cavity and the cumulative thickness of ice deposited with time. The ambient temperature, salinity and crystal concentration values associated with each month are those present at the end of the last time step of that month.
 The figures reveal that the general behavior of ambient temperature and salinity and the ice fraction follow a distinct pattern that comprises two stages. All three parameters undergo rapid change in the first months before entering a second phase that extends over most of the simulated time during which their change is much slower. The point in time when this transition takes place is henceforth designated as the turning point, and it occurs for all the parameters at around month 17 in this simulation. While temperature and salinity continue to fall beyond the turning point, monthly ice accumulation reverses its initial increasing trend and commences a slow decline. To understand these features, the situation is analyzed in the following sections in terms of the effects of melting and freezing processes induced by the plume during its interaction with the face of the rift and the ambient water. Figure 4 shows the change of the plume thickness along its path during the last time step of the first month. This clearly portrays how small the volume of the plume is relative to that of the cavity. The velocity of the plume at the top of the cavity is 2.54 cm/s, which is much higher than the rise velocity of the crystals (2 mm/s, calculated along the lines described by Gosink and Osterkamp ), justifying the assumption that the crystals are carried along with the plume until it reaches the top of the rift. Also, Figure 5 illustrates the profile of the rift's wall halfway through the integration and at its end. The altered shape of the wall reflects the integrated effect of melting over its lower part and freezing higher up.
3.4. First Months
 The initial simultaneous drop in temperature and salinity in Figure 3a is a result of the predominance of melting at the wall of the rift. This is confirmed by the fraction of ice in the plume, which is very near to zero as can be seen in Figure 3b. Melting, however, creates the conditions for ice generation through its effect of lowering both the ambient temperature and salinity. Hence crystal formation becomes noticeable around month 8, then rapidly picks up in the following 10 months, as manifested in Figure 6. The mechanism behind this fast increase in ice generation is discussed below. In the same 10-month period, the decline in both temperature and salinity decelerates due in part to the release of latent heat and salt during freezing. The main reason however is the dropping rate of melting at the lower section of the rift's wall with decreasing ambient temperature and salinity. As an indication of this, the ambient water during the first time step is 0.21°C above the in situ freezing temperature at depth 286.51 m, the point where melting is most active. By month 18, this value is already reduced to 0.15°C. As the effects of melting and freezing approach an overall balance, the temperature, salinity and ice fraction in the plume begin to stabilize.
 The changes in ambient temperature and those of ambient salinity affect the rate of ice production by the same underlying mechanism. The cooling and freshening of the ambient causes the point at which the liquidus condition (equation (6)) is met to move deeper in the cavity. Above this point the ambient is supercooled as observed in Jutulgryta. Figure 6 illustrates the interplay between ice production and the depth at which the liquidus condition is met, termed the “ambient freezing depth” by Lane-Serff . Moving the latter deeper in the water column during the first few months of the simulation causes a decrease in melting and an earlier onset of freezing in the plume. This is manifested as a slow but accelerating increase in the corresponding volume fraction of ice. Later, between months 12 and 16, decreased melting and ice generation themselves cause the deceleration in the descent of the ambient freezing depth. This in turn slows the increase of the rate of ice production, which levels off at the turning point.
 It is interesting to note in Figure 6 that ice generation between months 12 and 15 continues to accelerate despite the fact that the descent in the ambient freezing depth has already begun to decelerate. The explanation lies in the manner in which the concentration of frazil crystals evolves in the plume. An example of this is depicted in Figure 7, where the ice volume fraction of the top 100 m of the plume's path is shown for the last time step of months 13 and 14. The concentration of crystals in the plume, once generation has commenced, increases exponentially. Freezing in the plume manifests itself by an increasing number of crystals, which are all of the same size. The new crystals therefore add even more ice/water interface surfaces over which supercooling can be released and henceforth. Thus, despite the decreasing difference between the ambient freezing depths of months 13 and 14 compared with the previous months the difference is nevertheless enough to give the plume of month 14 an earlier start in ice generation and a much larger final ice volume fraction.
3.5. Beyond the Turning Point
 The 50-year simulation produced a total accumulated ice thickness of 28.47 m. It takes the ambient temperature to a final value of −1.97°C and the salinity to 34.35. This outcome corresponds with the observed temperature and salinity values. The thickness of marine ice deposited is of the same order as that observed but comparison is uncertain because of the thick layers of water interceding ice layers in the Jutulgryta rift. Furthermore, the monthly ice deposition rate between the turning point and the end of the simulation is about 0.05 m. Over the 23-month period following the turning point, 1.26 m of ice accumulated. This is of the same order as the very rough estimate of 2 m of accumulation over a similar time period, deduced from the Jutulgryta data.
 After the turning point, the most salient feature of the behavior of the parameters under study is the slow and steady decline in their values as shown by Figures 3a and 3b. From the above description of the effects of melting and freezing it might have been expected that the process would be self regulating and, once at the turning point, temperature, salinity and ice fraction would henceforth assume stable values. A related question concerning the temperature, other than its declining tendency, is the final value it reaches. This can be seen more clearly through a thought experiment involving a container of salty water at the surface freezing temperature throughout in which a slab of ice at the same temperature is introduced along one of its sidewalls [Lewis and Perkin, 1986]. An ice pump will be active and ice at depth will dissolve only for the resulting meltwater to rise and freeze again. At the end, all the ice in the system will be in a horizontal slab at the top of the container. The temperature of the water should be at the freezing point of the interface between the ice and the water. This notion suggests that the slow decline in the ambient temperature results from the continuous accumulation of ice crystals at the top of the rift, which slowly pushes down the interface between the ice and the ambient water. However, in the rift situation the depth of the interface sets only a higher limit to the ambient temperature because of one difference from the thought experiment, namely, the continuous influx of fluid into the rift. This also acts to push the ambient temperature and salinity down after the turning point. The initial plume flux contains meltwater (equation (8)) and is hence cooler and fresher than whatever ambient water is present at the bottom of the rift. Ambient water continually leaves the rift to compensate for the inflow, so there is a net loss of heat and gain of fresh water in the rift.
 The discussion of the previous section could imply that ice fraction in the plume should grow with falling temperature and salinity. The opposite trend, however, develops after the turning point. The reason for this, keeping in mind the exponential manner in which ice concentration grows with distance in the plume (Figure 7), is the progressive shortening of the part of the path of the plume where ice generation occurs. In other words, the ambient freezing depth does not descend as fast as the ice/water interface at the top of the cavity (the upper interface is impermeable to the plume as the deposited ice crystals are assumed to consolidate instantaneously). Thus the distance between the two at month 20 is 85.98 m corresponding to 0.065°C of supercooling at the top of the rift while these values for month 600 are 83.43 m and 0.063°C, respectively. The underlying mechanism for the slow decline in ice generation is that, while remaining the dominant process, the extent of melting at the wall of the rift continues to decrease progressively as the entrained ambient water cools and freshens. Less melting with time after the turning point is evident from the fact that the lower part of the rift's wall recedes by 63 cm during the twenty months after month 20 compared with 53 cm during the last twenty months of the simulation. In other words, a lower rate of ablation at the bottom of the rift means that there is less ice being transported to the top of the cavity through the ice pump mechanism.
3.6. Supercooling and Crystal Size
Figure 8 follows the evolution of supercooling along the path of the plume at the last time step of month 600. It shows how plume water at first, as a result of entrainment, quickly warms up toward the ambient temperature until reaching its highest temperature above freezing at around depth 288 m, the point at which the wall of the rift is melting most vigorously. Over a relatively short distance above and below that depth, the effects of melting pull the temperature of the plume back toward the freezing point thus slowing its convergence to the ambient temperature. In the meantime, the temperature of the entrained ambient water itself approaches the in situ freezing point with reduced depth. The two effects combine with the ascent of the plume to make it supercooled around depth 112 m, very close to the depth at which the ambient water also becomes supercooled. This depth is in excellent agreement with the 117 m down to which the water in Jutulgryta was observed to be supercooled (Figure 2). Beyond the ambient freezing depth the supercooling of the plume continues to build up linearly (Figure 8) until the end of its path, with whatever frazil has been produced being not yet enough to slow or reverse the trend. Thus the plume adds a supercooled layer to the ambient at the end of month 600.
 One method of demonstrating the validity of this explanation is to run the simulation using different frazil sizes. Equation (4) shows that the smaller the crystal radius the higher is the heat transfer coefficient at the edge of the crystals. Enhanced heat exchange efficiency leads to faster release of supercooling in the form of new frazil crystals. Consequently, the degree of supercooling in the plume, and that of the water it adds to the ambient, should be smaller, and vice versa with larger crystals. Repeating the standard experiment with a crystal radius set at 0.55 mm demonstrates this clearly. In this simulation, the supercooling in the plume at the ice/water interface at the end of month 600 is down to 0.03°C from its standard experiment equivalent value of 0.06°C (Table 1). The higher efficiency in releasing the supercooling results in a notably thicker layer of accumulated ice measuring 39.77 m. On the other hand, running the same experiment with a crystal radius of 2.2 mm produces only 11.53 m of ice and a corresponding supercooling of 0.11°C at the end of the simulation. In these experiments, the final supercooling levels are already largely determined by the time of the turning point in each case. At that time, the evolution of the supercooling of the plume and the ambient is qualitatively very similar to that in Figure 8. After the turning point, the supercooling level should slowly increase with the cooling and freshening of the ambient. However, this is more than compensated for by the lowering of the ice/water interface, the point where supercooling is being considered, because of ice accumulation.
Table 1. Effect of Crystal Size on Supercooling Levels at the Turning Point and at the End of a 600-month Integration, in Addition to Other Output Results
Crystal Radius, mm
Turning Point, month
Turning Point Supercooling, °C
Output Supercooling, °C
Output Temperature, °C
Output Ice Thickness, m
3.7. Entrainment and Heat Exchange
 The extent of the interaction of the plume with its environment, through ambient water entrainment from the cavity on the one hand and phase changes on the wall of the rift on the other, can be manipulated by varying the empirical coefficients in the equations that govern each of these processes.
 The effect of a change in entrainment is tested by running the standard experiment with the entrainment constant, E0, in equation (7) halved then doubled. The final output numbers do not reveal a dramatic change from those of the standard experiment as can be seen in Table 2. The results show that the lower the entrainment constant tested the faster the turning point is reached (Table 2) and the lower the temperature is at that point, before dropping 0.2°C in each case to the final output temperature shown in the table. This uniform drop in temperature, which requires the same amount of ice to be melted, explains the closeness of the thickness of the final output layer of ice in each case. Conditions at, and leading to, the turning point ensured such outcome. When entrainment is reduced, the volume of fluid spread by the plume at the end of its path is necessarily reduced. Consequently, the lower section of the cavity containing warmer unmodified ambient water is larger. Although only a fraction of the total depth of the rift, this is the section where most melting takes place (Figure 8). As an illustration, after the first time step, the density front in the E0/2 case is at depth 276.91 m and maximum melting occurs at depth 279.16 m compared with 292.34 m and 289.22 m, respectively, in the E0 × 2 case. Hence the plume in the first case entrains warmer ambient water over a longer distance, which more than compensates for its lower rate of entrainment, thus making its maximum melting rate one and a half times higher in comparison. This leads to an earlier commencement of supercooling along the path, the system reaches the turning point sooner and the ice volume fraction stabilizes at a higher level than for the case of a higher entrainment constant. However, the plume in the latter is characterized by higher volume flux (Table 2) because of the enhanced entrainment. Therefore the lower fraction of ice it carries is part of a larger volume and the resultant thickness of ice deposited approaches that in the lower entrainment case. This is borne out by the results where in the E0/2 case the thickness of ice deposited during the last month is 4.31 cm compared to 4.52 cm in the E0 × 2 case.
Table 2. Effect of Entrainment Constant on Turning Point Parameters and Output Results for a 600-Month Run
Turning Point, month
Ice Fraction, ×10−5
Flux, m2 s−1
Output Temperature, °C
Output Ice Thickness, m
E0 × 2
 The effect of the extent of phase changes on the wall of the rift is examined, in 50-year simulations, by halving then doubling the value of the drag coefficient K which appears in equation (3) for the heat transfer coefficients. The results in this case provide more testimony to the inherent tendency of the system toward a state of pseudo-equilibrium. The final output temperature and salinity, and their turning point values, in both the higher and lower drag coefficient simulations are the same as those of the standard experiment. The final thickness of deposited ice shows more pronounced differences, as can be seen from Table 3, with 25.83 m in the K/2 case compared with 31.95 m in the K × 2 case. When the heat transfer efficiency is increased, melting over the lower part of the plume's path becomes more vigorous. In the K × 2 case the lower part of the wall receded by a maximum of 23.69 m in 50 years compared with 12.74 m in the K/2 case. This enhanced melting is accompanied by more freezing as can be seen from the final thickness of deposited ice and the extent of the wall advance at the top of the cavity. This was 11.32 m at the end of the K × 2 run, significantly larger than the 5.36 of the K/2 case. We find that the pseudo-equilibrium values of temperature and salinity are robust results, with the system merely adjusting the amount of ice that must be melted and refrozen to reach these values when the efficiency of heat transfer between plume and the rift wall is altered.
Table 3. Effect of Drag Coefficient K (and Thus That of the Heat Transfer Efficiency) on Output Results for a 600-Month Run
Output Temperature, °C
Final Top Wall Advance, m
Final Bottom Wall Retreat, m
Output Ice Thickness, m
K × 2
 Phase changes at the wall of the rift can also be affected by the far field temperature distribution in the ice shelf. Hence model sensitivity to this factor is tested using an extreme case of a constant far field temperature with depth. A temperature of −10°C is assigned, which corresponds to the approximate mean value of the linear distribution used in the other experiments. The results, summarized in Table 4, are not much different from those of the standard experiment. While the final ambient water temperature and the total amount of accumulated ice are basically the same, the final ambient water salinity shows a slight increase from the standard experiment. The explanation for this increase cannot lie in the amount of ice formed in the plume as the deposited quantities are the same in both experiments. Rather, it is the reduced melting at the lower part of the wall in the constant profile experiment (Table 4) that keeps the salinity higher. The final temperature is determined by the same parameters (crystal size, thickness of deposited ice) in each case, but the total amount of melting is reduced in the constant ice shelf temperature case because the far field ice temperature is lower in the zone of melting. Hence more heat is extracted from the plume to provide the larger amounts of sensible heat required to bring the colder ice to the melting point.
Table 4. Results of the Standard Experiment, With a Linearly Increasing Far Field Temperature Profile, Compared With Those Where the Temperature Profile is Constant With Depth at −10°C
Temperature Profile With Depth
Output Temperature, °C
Final Top Wall Advance, m
Final Bottom Wall Retreat, m
Output Ice Thickness, m
3.8. Initial Plume Flux
 Nine initial plume flux values, extending between 1 × 10−5 and 4 × 10−3 m2/s, were used to run the above experiment for 50 years. Figure 9 shows the effect this parameter has on the total amount of accumulated ice, one of the main interests of this study, which increases significantly with higher initial flux. In order to explain such dependence, the data for the 1 × 10−3 m2 s−1 and 4 × 10−3 m2 s−1 experiments were compared more closely. They reveal that in the higher flux case, while having the same qualitative evolution, ambient temperature and salinity reach the turning point faster, at around month 14, and at lower values for both. The fact that it is faster is a consequence of the higher rate at which ambient water is modified because of the higher volume flux of the plume and its enhanced entrainment of the ambient as suggested by equation (7) above. The lower values assumed by the two parameters at the turning point are accounted for by the faster flushing of the rift by the water contained in the initial plume flux, which is cooler and fresher. They are also due to the higher velocity of the plume which is associated with a higher melting rate, the heat transfer coefficient being a function of velocity (equation (3)). Melting is also enhanced by the higher rate of ambient water entrainment (equation (7)) by the plume. Its increased intake of relatively warm water widens the temperature difference in equation (5). More melting is evident from the fact that the lower part of the wall of the rift had receded by a maximum 18.14 m at the end of the 4 × 10−3 m2 s−1 run compared with 17.45 m in the 1 × 10−3 m2 s−1 case. The more intense melting in the lower part of the plume means that it becomes supercooled earlier and the resulting ice production takes place over a longer distance. Again, the final profile of the walls of the rift provides the evidence for this. Thus, in the 4 × 10−3 m2 s−1 simulation the transition point from where the wall had melted to where direct freezing on it had started to take place occurs somewhere between 130 and 140 m below the surface, 40 m lower than in the 1 × 10−3 m2 s−1 situation. Therefore the ice volume fraction in the plume, and subsequently the final thickness of accumulated ice in the rift, reach higher levels with increased initial flux. In the 4 × 10−3 m2 s−1 simulation the ice fraction in the plume stabilizes around 1.39 × 10−5 m2 s−1 compared with 0.77 × 10−5 in the 1 × 10−3 m2 s−1 situation.
 Among the nine tests all those with initial flux values of 1 × 10−3 m2 s−1 and above reproduced part or all of the temperature and salinity ranges observed in Jutulgryta through the 50-year runs. None reached the lower limit of the observed salinity range, 34.34 to 34.35, while the three highest flux experiments simulated all of the observed temperature range, −1.98°C to −1.97°C. For example, in the 2 × 10−3 m2 s−1 the temperature drops from −1.97°C at month 156 to −1.98°C at month 568 producing 29.32 m of ice in the process while in the 4 × 10−3 m2 s−1 case the drop takes place between months 14 and 250 with 29.59 m of ice deposited. In these three experiments, the higher end of the observed temperature is reached earlier and its range covered more quickly the higher the initial flux is, but it is interesting to note that the total amount of ice produced in each case is almost the same. The similarity of the quantities of produced ice is not surprising as it is the reflection of the amount of ice that needs to be melted in order to cool down, to the same final temperature, comparable volumes of water at the same initial temperature.
 The above discussion shows that there are several values for the initial plume volume flux that can reproduce the ambient temperature and salinity observed at Jutulgryta in simulations of 50 years or shorter. The lowest of those tested, 1 × 10−3 m2 s−1, has been chosen for the experiments of the previous sections.
 The model presented here succeeds to a large extent in reproducing the observations from Jutulgryta and in providing a real insight into the mechanisms responsible for their occurrence. The credibility of this insight is further enhanced by the robustness of the model, which was manifest through the relative stability of its output when several of the important parameters were varied in sensitivity tests. Obviously, better constrained values for some of the empirical parameters would be helpful. Future field measurements similar to those of Potter and Paren  mentioned above could help quantify the volume flux of the plumes rising along the sides of ice shelf rifts.
 In the meantime, the model convincingly accounts for the thick supercooled layer of ocean water observed at Jutulgryta. Such presence is intriguing in more than one respect. First, in its origin, especially in light of the fact that no Ice Shelf Water has been observed beneath the ice shelf at the location that could have been the source of the supercooled water. The other interesting aspect is the nontransient nature of the supercooled water. It would have been expected that ice fragments from the walls of the rift, sand grains and biological organisms would have provided nuclei for the formation of ice and the eventual release of the supercooling. The model shows that plumes in the rift, and the details of their interactions with the walls and the ambient water, are plausible sources for the supercooled layer. Furthermore, the model establishes the persistence of this layer as a dynamic process dependent on the continued supply of supercooled water by the plumes. The validity of these explanations is further supported by the very good quantitative agreement that the calculated level of supercooling, ambient temperature, ambient salinity and deposited ice show to the observed ones. Moreover, the model accounts for the near uniformity of the observed water properties in the rift, in comparison with the ocean water underlying the ice shelf, by the entrainment and mixing affected by the turbulent plumes, the density instabilities occasionally arising from the water they deposit and the outflow of stratified layers from the rift.
 Our results also invite a closer look at the impact of ice in rifts on the ice shelves where they occur. Several authors [Ardus, 1965; Stephenson and Zwally, 1989; Doake and Vaughan, 1991] have ascribed to such ice the role of a “glue” or “cement” that could hold ice shelf blocks together. Pertinent to this question is the work of Rignot and MacAyeal  and MacAyeal et al.  which shows that ice in rifts tends to deform coherently in reaction to ice shelf flow and that it has sufficient mechanical strength and integrity to bind large tabular ice shelf fragments to the coast. They suggested that it is through the melting of the ice in the rifts that the oceanic and atmospheric environments can control, in part at least, the location of Antarctica's ice shelf calving fronts and that the nearer a rift gets to the front, the more melting takes place and the weaker and thinner the ice in the rift becomes as a prelude for calving. Interestingly, their measurements did not show any such weakness of ice in the rifts near the front of the ice shelf and one reason that the authors advanced to explain such discrepancy is the possibility of other processes they did not consider in their analysis being active. It is quite plausible that the thickness and dynamic properties of the ice in rifts and its response to atmospheric and oceanic variations would be different if this glue or cement was the result of filling the rift not only from above but also from below through the process described here. In addition, the model presented here can be a useful tool in assessing how the filling of rifts from below would vary in response to changes in oceanic conditions. The conclusions of this work could easily be extended to the large basal crevasses that open in the undersides of ice shelves.
 Finally, understanding of the consolidation and desalination process of marine ice may be advanced by the results obtained here, which provide information on parameters that include the rate of ice accumulation and the temperature and salinity of the water in which this accumulation and consolidation take place.
 The first author gratefully thanks the Wiener-Anspach Foundation in Brussels for its financial support. The authors gladly acknowledge a European Ice Sheet Modeling Initiative exchange grant. The authors would also like to thank M. G. Worster and J.-L. Tison for informative and stimulating discussions. This paper is a contribution to the Belgian Antarctic Programme (Science Policy Office).