Double Diffusion and False Bottoms

Evidence for the importance of double diffusion during melting comes perhaps unexpectedly from a curiosity of the summer ice pack: false bottoms. These occur when concavities in the ice underside fill with fresh meltwater, which being in contact with seawater well below 0°C, forms a thin layer of ice at the fresh-water/seawater interface. The phenomenon was documented nicely during the summer of 1975, when Arne Hanson maintained an array of depth gauges at the main AIDJEX station Big Bear near the center of the Beaufort Gyre in the Canadian Basin. Figure 6.7 (adapted from Notz et al. 2003) shows time series of bottom elevation (with respect to the upper surface) at seven of Hanson's thickness gauges, three initially deployed in thick ice (BB 4-6) and four others deployed in relatively thinner ice (BB 2,3,7,8). The thicker sites all show steady ablation through the summer, but at the thinner sites there is a significant increase in distance to the ice bottom starting at around day 200 (19 July), and in fact some of the thinner sites showed a net increase in thickness over the summer. Hanson attributed the increased thickness to formation of false bottoms. Notz et al. (2003) investigated the evolution of false

Negative ice thickness

Negative ice thickness

-200

-250

160 170 180 1S0 200 210 220 230 240 250 2S0

Day of 1975

Fig. 6.7 Ice bottom elevation relative to the upper surface from ablation measurements made by A. Hanson during the 1975 AIDJEX project in the western Arctic (Adapted from Notz et al. 2003. With permission American Geophysical Union) (see also colorplate on p. 206)

160 170 180 1S0 200 210 220 230 240 250 2S0

Day of 1975

Fig. 6.7 Ice bottom elevation relative to the upper surface from ablation measurements made by A. Hanson during the 1975 AIDJEX project in the western Arctic (Adapted from Notz et al. 2003. With permission American Geophysical Union) (see also colorplate on p. 206)

bottoms, in both laboratory and natural settings, including a simulation of the AIDJEX observations. The study showed that double diffusion is a critical process in the formation of false bottoms, which in turn may play an important role in maintenance of perennial sea ice.

In an idealized view of the false bottom layer after it has attained an initial finite thickness (Fig. 6.8), we assume horizontal homogeneity, and that the layer of water between the existing multiyear ice and the newly formed false bottom is fresh, with temperature equal to 0°C. We also assume that the thin ice layer is fresh with a linear temperature gradient. The false bottom thus borders on two different water types, and because it sustains a significant positive temperature gradient, there will be downward heat transfer. With these assumptions where h is the thickness of the false bottom layer. The upper surface will migrate upward into the fresh water layer at rate ow, a

Kice (Tup To) KicemSo pwcph pwcph

Multiyear Ice

Fresh Water Layer

Fresh Water Layer

Fig. 6.8 Schematic of a thin layer of fresh ice that forms between meltwater trapped in underice concavities and colder seawater

where the latent heat of ice formation balances the downward heat flux. The lower boundary will also move vertically in response to the combination of q and (w'T The total rate of change of thickness for the false bottom layer is then h = hup - (pw/pice)W0 = -(pw/pice)(q/Qfresh + Wo) (6.12)

The bottom heat and mass balance can again be reduced to a quadratic equation for salinity at the interface

(1 + y)mSo2 + (Tw + Tl - (1 + Y) mSce)So - TwSce - TlSW = 0 (6.13)

Bottom ice elevation relative to a starting point on 9 July 1975 (year day 190) indicates false bottoms at four of Hanson's ice thickness sites (Fig. 6.9). They formed at different times, and there is some indication of multiple layer formation during a relatively calm period from day 2o5 to 21o. But beginning with a period of more rapid ice drift from day 21o to 22o, thickness decreased more or less uniformly at all four sites. Following the modeling approach of Notz et al. (2003), the evolution of a false bottom starting from an initial thickness of 2.5 cm was simulated by solving the combined equations (6.11), (6.12), and (6.13), with u*0 estimated from Rossby similarity (4.19) applied to the ice drift speed and with seawater temperature and salinity interpolated from daily CTD observations. We further specified that the

Fig. 6.9 Upper panel: bottom elevation of "false bottom" thickness gauges relative to their readings on day 190. Box marks the ten-day period chosen for simulation. Bottom panel: Interface friction velocity determined from ice drift relative to geostrophic current, for two values of surface roughness spanning range of estimates for AIDJEX station Big Bear (Adapted from Notz et al. 2003. With permission American Geophysical Union) (see also Colorplate on p. 206)

Fig. 6.9 Upper panel: bottom elevation of "false bottom" thickness gauges relative to their readings on day 190. Box marks the ten-day period chosen for simulation. Bottom panel: Interface friction velocity determined from ice drift relative to geostrophic current, for two values of surface roughness spanning range of estimates for AIDJEX station Big Bear (Adapted from Notz et al. 2003. With permission American Geophysical Union) (see also Colorplate on p. 206)

bulk Stanton number (6.10) be 0.0057, the mean SHEBA value, and Z0 = 6mm, assuming the area around the false bottoms would be similar to the undeformed multiyear ice observed during SHEBA (McPhee 2002).

Model results for thick ice show about 6 cm of bottom ablation over the ten days (Fig. 6.10a), compared with about 14 cm of upward migration of the modeled false bottom (Fig. 6.10b). In each case the model matches Hanson's observations pretty well. The combination of ah = 0.0111 and ah/as = 50 (which provides st* = 0.0057, see Fig. 6.6) was chosen as the combination that minimized the root-mean square error between the model and observations in Fig. 6.10b, for R in the range 35 to 70. If the model is run with R = 1, with ah = 0.0058 (to maintain st* = 0.0057) the results are reasonable for thick ice (Fig. 6.11a) but nonsensical for false bottom migration (Fig. 6.11b). The persistence of false bottoms in the summer pack is thus difficult to explain without invoking fairly strong double diffusion.

In Fig. 6.12a, modeled upward heat exchange between the ocean and thick ice is compared with the downward heat flux from false bottoms for the double-diffusive regime of Fig. 6.10. Because of the relatively large positive temperature gradient

Thick ice ablation, Zg = 0.6 cm, ah= 0.0111 , ah/as = 50.0

210 212 214 216 218 220

False bottom elevation change, ^=0.6 cm, ah= 0.0111 , ah/as = 50.0

0.2

-

--

- Model

a

BB2

0.15

-

+

BB3

x

BB7

0.1

■b

b

Day of 1975

Fig. 6.10 a Bottom ablation under thick ice (gauges BB4-6) compared with model (dashed curves). b Elevation change at false bottom sites compared with model. Model parameters (listed) are the same (Adapted from Notz et al. 2003. With permission American Geophysical Union)

Thick ice ablation, Zg = 0.6 cm, ah= 0.0058 , ah/ag = 1.0

Fig. 6.10 a Bottom ablation under thick ice (gauges BB4-6) compared with model (dashed curves). b Elevation change at false bottom sites compared with model. Model parameters (listed) are the same (Adapted from Notz et al. 2003. With permission American Geophysical Union)

210 212 214 216 218 220

False bottom elevation change, Zg = 0.6 cm, ah= 0.0058 , ah/ag = 1.0

214 216

Day of 1975

Fig. 6.11 As in Fig. 6.10, but for equal heat and salt exchange coefficients, chosen to maintain a realistic Stanton number

Ocean-To-lce Heat Flux

Ocean-To-lce Heat Flux

210 212 214 216 218 220

x -go-3 Effective Aggregate Heat Transfer Coefficient x -go-3 Effective Aggregate Heat Transfer Coefficient

False Bottom Area Fraction

Fig. 6.12 a Time series of heat flux to thick ice (darker shading) and heat flux into the ocean from false bottoms (lighter). Average values are shown at right. b Aggregate Stanton number as a function of areal coverage of false bottoms and fresh water (Adapted from Notz et al. 2003. With permission American Geophysical Union) (see also Colorplate on p. 207)

False Bottom Area Fraction

Fig. 6.12 a Time series of heat flux to thick ice (darker shading) and heat flux into the ocean from false bottoms (lighter). Average values are shown at right. b Aggregate Stanton number as a function of areal coverage of false bottoms and fresh water (Adapted from Notz et al. 2003. With permission American Geophysical Union) (see also Colorplate on p. 207)

across the false bottoms, they represent a source of heat for the upper ocean in summer apart from absorbed solar radiation penetrating into the well mixed layer. Depending on how ubiquitous false bottoms (or any freshwater/seawater interface on the ice undersurface) are during the melt season, they may exert a powerful influence on the total heat exchange between the IOBL and the pack ice. An effective aggregate Stanton number

{St*)eS= pc where Htotai/pcp = (1 — Afb) (w'T')0 + Afb (w'T1)fb and Afb is the area fraction of the undersurface covered by false bottoms or meltwater ponds, includes the combined positive and negative fluxes to thick ice ((w'Tand from false bottoms ((w'T')fb). It falls rapidly with increasing false bottom area. For the AIDJEX simulation, (St*)eS is nearly halved if the area fraction approaches 3/10.

False bottoms and other manifestations of underice melt water may have a significant impact on the mass balance, and even the force balance, of the Arctic pack. Notz et al. (2003) reported estimates of false-bottom area fractions ranging from at least 10% (Jeffries et al. 1995) to over 50% (Hanson 1965). Jeffries et al. suggest that the origin of platelets in the Arctic ice cores they analyzed derived mainly from false bottom formation or an "ice-pump" mechanism, and that underice melt ponds may be more common than had been previously appreciated. Our experience while deploying the SHEBA station in the Beaufort Gyre in September 1997, was that when we drilled the late summer ice, we often encountered multiple layers of liquid meltwater interspersed between thin ice layers, suggestive of several successive cycles of false-bottom formation and migration. There was also a significant difference between establishing hydroholes during fall versus early spring. In the latter, it is often possible to extract ice that is dry to within 10-15 cm of the ice bottom, while in the former, we encountered a "water table" relatively high in the ice column below which lateral water movement appeared to be relatively unrestricted (and made further ice excavation more difficult). The concept of a porous water table that migrates downward from the surface as the ice column warms from above implies that any pre-existing concavities in the ice undersurface will be filled with fresh water regardless of a direct vertical connection to the surface.

False bottoms affect the general ice-albedo feedback issue in two important ways. First, they may substantially delay the transfer of heat from the upper ocean to the ice pack by reducing (St*)eff, which allows the upper ocean to maintain its heat content well past the time when sun angles are high. Second, as fresh water begins to collect in underice concavities early in the melt season, false bottom formation protects the thinnest ice from contact with the warming upper ocean thus delaying exposure of open seawater. The ice-albedo feedback is most effective when the ice/upper ocean system can absorb solar radiation at times near the summer solstice. Both of the false-bottom mechanisms described here tend to retard this timing, hence represent a perhaps important negative feedback in the system. A general thinning of the perennial pack (Rothrock et al. 1999) will mean that summer warming and the presumptive downward migrating "water table" will reach the ice base earlier in the summer, hence reinforcing the mitigating impact of underice melt ponds and false bottoms.

6.7 Freezing—Is Double Diffusion Important?

Mellor et al. (1986) and Steele et al. (1989) showed that if double-diffusive tendencies carry over to freezing in the same way that they apparently affect melting, then there ought to be significant production of supercooled water, because heat would be extracted from the upper ocean faster than salt would be injected. Presumably, the supercooled water would either nucleate in situ and form frazil ice crystals distributed in some way through the IOBL, or would nucleate more or less uniformly on the ice undersurface, regardless of ice thickness. Steele et al. (1989), using exchange parameters inferred from MIZEX measurements, estimated that supercooling and subsequent frazil production could account for as much as half of the ice accretion for thin (20 cm) ice to 30% for 80-100 cm thick ice.

If only one thickness of ice is considered, over time it matters little whether the ice forms from congelation at the immediate interface, or by accretion of frazil crystals drifting up to the interface from below. However, with different ice thicknesses there is a potentially interesting wrinkle. If a significant fraction of total ice production is in the form of frazil crystals, growth of categories near the low end of the thickness distribution will be slower than otherwise, while thick ice will accrete faster. If the growth of thin ice is retarded, its steep temperature gradient (responsible for most of the total heat transfer) will persist for a longer time, with the possibility of more overall heat transfer out of the ocean. Holland et al. (1997) examined this in a modeling study coupling an upper ocean model to an ice model with eight thickness categories. Using the "three-equation" parameterization suggested by McPhee et al. (1987), they found that the equilibrium annual average ice thickness increased by about 10 cm compared with an identical model run that was the same except that the exchange coefficients remained equal. There were substantial differences in modeled basal accretion.

In the multiyear ice pack of the Arctic, observations indicate that neither supercooling nor frazil production is extensive during winter. By examining thin sections in sea ice, it is relatively straightforward to distinguish between columnar ice accreted by congelation with horizontal c-axis orientation versus that from frazil, with more random orientation. Weeks and Ackley (1986) report that frazil accounts for only about 5% of total ice volume in Arctic pack ice and in fast sea ice from both hemispheres. It is found mainly near the surface, produced during initial ice formation. In the Antarctic, frazil-dominated structure is much more common, probably as a result of intense air-sea interaction in the vast marginal ice zones of the Southern Ocean. Over most of the Weddell Gyre, for example, the seasonal ice remains quite thin, often with a bi-modal thickness distribution from rafting by waves. Such conditions are conducive to frazil production.

While not common, supercooled water has been observed beneath the Arctic ice pack. Untersteiner and Sommerfeld (1964) reported supercooling of approximately 4 mK (i.e., water temperature about 0.004 K below its freezing temperature, dependent on salinity and pressure) near ice island ARLIS 2 (a drifting tabular berg) from measurements in water under the adjacent pack ice. They used a differential temperature measurement technique that did not require accurate salinity determination, an important consideration at the time. In that case, the supercooling was possibly attributable to the "ice-pump" effect described by Foldvik and Kvinge (1974) and Lewis and Perkin (1983). In typical pack ice, water in the well mixed IOBL will contact ice at varying pressures, e.g., ridge keels at pressures up to 10 dbar and beyond. Water that is at it freezing temperature at the level of the undeformed ice (say, 2 dbar) would be about 6 mK above freezing at 10 dbar.3 The ice pump occurs when this water melts ice at depth, thus attaining a freezing temperature associated with the pressure where melting occurred. As this water rises following the ice morphology, it will be supercooled relative to its in situ pressure and will deposit ice as it encounters nucleation sites at the ice/water interface. In this way, ice can be transported through the thickness distribution from thicker to thinner categories. The ice pump is especially effective under floating ice shelves where large basal melting near the grounding line is "redeposited" as sea ice at higher levels near the terminus.

3 We use the UNESCO formula for the freezing point of seawater from Millero (1978) as reported by Gill (1982), who points out that the formula fits measurements to an accuracy of ±4mK.

Another possible source of supercooling can arise from differential mixing of salt and heat when there are large horizontal gradients in temperature and salinity. Measurements in 2007 of transient supercooling events in an energetic tidal flow in Fremansundet, Svalbard, were suggestive of this mechanism. The events occurred at different times at two different levels when a front between slightly less saline water from outside the fast ice was advected into the sound by the tide. The water on both sides of the front was within millikelvins of freezing, so that the water advected into the sound was slightly warmer. We interpreted the transient events (each lasting about an hour) as the result of heat mixing more rapidly than salt, so that the incoming water mass dipped below its freezing point in the frontal zone. It is perhaps worth noting that the supercooling north of Svalbard reported by Lewis and Perkin (1983) occurred in a region with strong horizontal gradients in temperature and salinity.

Given several possible sources of supercooling and subsequent frazil production (but a lack of evidence that it occurs extensively under multiyear pack ice), is it possible to examine in isolation the hypothesized mechanism of supercooling associated with double diffusion at the interface during rapid ice growth? We approached this problem as follows. Consider growth in thin ice in seawater at freezing under with the following conditions: u*0 = 5mm s_1, sw = 34psu, Tw = Tf (sw) = — 1.865°C, with upward heat conduction of 20 W m—2 in the ice column, corresponding to a temperature gradient in the ice of about —10 K m— We assume sice = 7psu. First solve the interface equation for salinity (6.9), with no double diffusion, i.e., ah = as = 0.0058 (which was shown above to match the Stanton number constraint, st* = 0.0057). In this case, s0 = 34.067psu, the ice grows at a rate of about 7 mm per day, and under the assumption that ice salinity is 7 psu, this produces a salinity flux (w's% = —1.96 x 10—6psu m s—and an upward heat flux from the water column of 0.4W m—2, which would be difficult to detect by covari-ance measurement. It is easy to confirm that (w'= —m (w's')0, i.e., that heat is extracted from the water at just the rate required to maintain the water at its freezing temperature as salt is added at the surface. Note that none of these quantities are extreme by any measure.

Next solve the problem with identical conditions except that we now let double diffusion operate in the interface control volume at levels used in the false bottom simulation (ah = 0.0111; as = ah/50, see Fig. 6.10). In this case s0 = 34.859, congelation growth is significantly reduced to about 3.3 mm per day, and now the heat flux out of the water column is 10.7W m—2. This is easily measured, and thought experiments like this convinced us that the best way to look for the supercooling effect due to double diffusion at the freezing interface was by measuring upward heat flux in water just below the interface.

From these considerations, we designed a field experiment in a relatively controlled environment offered by fast ice and a gentle tidal flow in Van Mijen Fjord, Svalbard. In March 2001, we occupied a site on smooth fast ice, and installed instrumentation to measure ice characteristics and turbulence 1 m below the ice/water interface (McPhee et al. 2008, in press). Temperature profiles measured during the field project (Fig. 6.13) show the impact of changing surface temperature (there was

Fig. 6.13 Ice temperature profiles on three days in fast ice on VanMijen Fjord, Svalbard (see also Colorplate on p. 207)

little snow), but show a gradient in the lower 10 cm or so of the ice column that would indicate upward heat conduction there of about 21W m-2. Turbulence measurements are summarized in Fig. 6.14, where the turbulence data have been bin averaged according to mean current velocity 1 m below the interface in each 15-min turbulence realizations. A least-squares regression through the origin is quite close to the law of the wall for a hydraulically smooth boundary (Hinze 1975).

The regression lines in Fig. 6.14a and b show a slight correlation between flux magnitude and current speed, but are barely distinguishable from zero at the 90% confidence level. Overall the conditions are not much different from the example presented above, and the measured heat flux is only slightly more than what would be required to keep the well mixed layer near freezing as it became saltier. The lack of much ocean heat flux in this controlled environment is probably the most convincing evidence that during freezing double diffusion is relatively unimportant, in contrast to melting. By applying numerical model using local turbulence closure (described in Chapter 8) to longer term measurements from the VMF experiment, we showed that the exchange coefficient had to be close to unity during the VMF 2001 exercise (McPhee et al. 2008, in press).

x 10

x 10

a :

Hydraulically smooth^

x 10"

x 10"

c

I

-—

----

1— I

1

r ~ - - -

' "f

-

' — — _

0.04

0.05

0.06

0.07

0.08

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Fig. 6.14 Average turbulence measurements during the VMF exercise, binned according to tidal flow velocity. Error bars show ± one standard deviation in the 15-min turbulence realizations for each bin. (a) Friction velocity. Solid line is a least-squares fit through the origin; the dot-dashed curve is the law of the wall for a hydraulically smooth surface. (b) Turbulent heat flux. The solid line is a least squares regression for heat flux against current speed; the light dashed lines are confidence limits for the fit. (c) Same as b except for salinity flux

Why the freezing process should be fundamentally different from melting (in terms of double-diffusive effects in the IOBL) is apparently due to the fact that solidification occurs by the advance of mushy layers in which convection within the ice lattice relieves the double diffusive tendency: see, e.g., Wettlaufer et al. (1997), Notz (2005), and Feltham et al. (2006). During melting, the ice undersurface is observed to be uniformly smooth and the diffusion of salt into the crystal lattice apparently takes on a completely different character.

References

Barenblatt, G. I.: Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University

Press, Cambridge (1996) Feltham, D. L., Untersteiner, N., Wettlaufer, J. S., and Worster, M. G.: Sea ice is a mushy layer.

Geophys. Res. Lett., 33, L14501 (2006), doi: 10.1029/2006 GL026290 Foldvik, A. and Kvinge, T.: Conditional instability of sea water at the freezing point. Deep-Sea

Res., 21, 160-174 (1974) Gill, A. E.: Atmosphere-Ocean Dynamics. Academic, New York (1982)

Hanson, A. M.: Studies of the mass budget of arctic pack ice floes. J. Glaciol., 41, 701-709 (1965) Hinze, J. O.: Turbulence, Second Edition. McGraw-Hill, New York (1975)

Holland, M. M., Curry, J. A., and Schramm, J. L.: Modeling the thermodynamics of a sea ice thickness distribution 2. Sea ice/ocean interactions. J. Geophys. Res., 102, 23,093-23,107 (1997) Ikeda, M.: A mixed layer beneath melting sea ice in the marginal ice zone using a one-dimensional turbulent closure model, J. Geophys. Res., 91, 5054-5060 (1986). Incropera, F. P. and DeWitt, D. P.: Fundamentals of Heat and Mass Transfer, Second Edition. Wiley, New York (1985)

Jeffries, M. O., Schwartz, K., Morris, K., Veazey, A. D., Krouse, H. R., and Cushing, S.: Evidence for platelet ice accretion in Arctic sea ice development. J. Geophys. Res., 100(C6), 1090510914 (1995)

Josberger, E. G.: Sea ice melting in the marginal ice zone, J. Geophys. Res., 88, 2841-2844 (1983) Lewis, E. L. and Perkin, R. G.: Supercooling and energy exchange near the Arctic Ocean surface.

J. Geophys. Res., 88 (C12), 7681-7685 (1983) Maykut, G. A. and Untersteiner, N.: Some results from a time-dependent thermodynamic model of sea ice. J. Geophys. Res., 76, 1550-1575 (1971) Maykut, G. A.: An introduction to ice in polar oceans, Report APL-UW 8510, Applied Physics

Laboratory, University of Washington, Seattle, WA (1985) McPhee, M. G.: Turbulent stress at the ice/ocean interface and bottom surface hydraulic roughness during the SHEBA drift. J. Geophys. Res., 107 (C10) 8037 (2002), doi: 10.1029/2000JC000633 McPhee, M. G., Maykut, G. A., and Morison, J. H.: Dynamics and thermodynamics of the ice/upper ocean system in the marginal ice zone of the Greenland Sea. J. Geophys. Res., 92, 7017-7031 (1987)

McPhee, M. G., Kikuchi, T., Morison, J. H., and Stanton, T. P.: Ocean-to-ice heat flux at the North pole environmental observatory. Geophys. Res. Lett., 30 (24) 2274 (2003), doi: 10.1029/2003GL018580

McPhee, M. G., Morison, J. H., and Nilsen, F.: Revisiting heat and salt exchange at the ice-ocean interface: Ocean flux and modeling considerations, J. Geophys. Res., doi: 10.1029/2007JC004383, in press (2008) Mellor, G. L., McPhee, M. G., and Steele, M.: Ice-seawater turbulent boundary layer interaction with melting or freezing. J. Phys. Oceanogr., 16, 1829-1846 (1986) Millero, F. J.: Freezing point of seawater, in: Eighth Report of the Joint Panel on Oceanographic Tables and Standards, UNESCO Tech. Pap. Mar. Sci. No. 28, Annex 6, UNESCO, Paris (1978) Notz, D.: Thermodynamic and fluid-dynamical processes in sea ice. Ph. D. dissertation, Trinity College (2005)

Notz, D., McPhee, M. G., Worster, M. G., Maykut, G. A., Schliinzen, K. H., and Eicken, H.: Impact of underwater-ice evolution on Arctic summer sea ice. J. Geophys. Res., 108 (C7) 3223 (2003), doi: 10.1029/2001JC001173 Owen, P. R. and Thomson, W. R.: Heat transfer across rough surfaces. J. Fluid Mech., 15, 321334 (1963)

Parkinson, C. L. and Washington, W. M.: A large-scale numerical model of sea ice. J. Geophys.

Res., 84, 311-337 (1979) Rothrock, D. A., Yu, Y., and Maykut, G. A.: Thinning of the Arctic ice cover. Geophys. Res. Lett., 26, 3469-3472 (1999)

Steele, M., Mellor, G. L., and McPhee, M. G.: Role of the molecular sublayer in the melting or freezing of sea ice. J. Phys. Oceanogr., 19, 139-147 (1989) Untersteiner, N. and Sommerfeld, R.: Supercooled water and bottom topography of floating ice. J.

Geophys. Res., 69, 1057-1062 (1964) Weeks, W. F. and Ackley, S. F.: The growth, structure, and properties of sea ice. In: N. Untersteiner

(eds.) The Geophysics of Sea Ice, pp. 9-164. Plenum, New York (1986) Wettlaufer, J. S., Worster, M. G., and Huppert, H. P.: Natural convection during solidification of an alloy from above with application to the evolution of sea ice, J. Fluid Mech., 344, 291-316 (1997)

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Nomenclature h i w0 wp w q

Kice

fresh

Ice growth rate

Isostatically adjusted bottom melt rate Interface velocity due to freshwater percolation Interface vertical velocity (wo + Wp) Heat conduction in the ice column divided by pCp Thermal conductivity of sea ice Thermal conductivity of pure ice (2.04 J m_1 KT1 Ice salinity

Vertical temperature gradient in ice

Proportionality constant in Untersteiners (1961) formula (0.117 J m_1

Latent heat of melting for saline ice divided by Cp Latent heat of pure ice (335.5kJ kg-1) Turbulent exchange coefficients for heat and salt Far-field (well mixed layer) temperature, salinity Temperature, salinity at the ice/water interface Tw — To, Sw — S0

Reynolds number based on friction velocity, surface roughness: u*oZo/v Prandtl number, v/vj Schmidt number, v/vs Stanton number

Stanton no. based on u*o; bulk heat transfer factor Departure of well mixed layer temperature from freezing

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