M O D E L L E D BU L K SA L INI T Y O F G R O W IN G F IRST-Y E A R SE A I C E A ND I MPL I C A T I O NS F O R I C E PR OPE R T I ES IN SPRIN G

Transcription

1 POA 11 Montréal, anada Proeedings of the 1 st International onferene on Port and Oean Engineering under Arti onditions July 1-14, 11 Montréal, anada POA M O D E L L E D BU L K SA L INI T Y O F G R O W IN G F IRST-Y E A R SE A I E A ND I MPL I A T I O NS F O R I E PR OPE R T I ES IN SPRIN G hris Petrih 1,, Pat Langhorne 1, Hajo Eiken 1 Department of Physis, University of Otago, Dunedin, New Zealand Geophysial Institute, University of Alaska Fairbanks, Fairbanks, Alaska, USA A BST R A T Sea ie brine volume ontrols many important seasonally varying properties of sea ie, suh as its strength, dieletri properties, development of meltponds and surfae albedo, and oil entrainment and migration to the ie surfae. The brine volume fration, whih aounts for the bulk of the total porosity in first-year sea ie, depends primarily on the ratio of bulk salinity and ie temperature. While ie temperature and ie growth are readily modelled and an also be obtained from in-situ sensors or remote sensing, modelling or in-situ monitoring of bulk salinity has proved to be more elusive. We present a parameterization for the bulk sea ie salinity of growing ie as a funtion of ie growth rate that is based on onepts of natural onvetion in porous media (mushy layers). The results ompare favourably with field data and more rigorous omputational fluid dynamis simulations and provide a quantitative measure of inreases in bulk salinity with inreasing growth rate. The sensitivity of springtime bulk salinity and fleural strength is disussed based on a modelled ase study for the hukhi/beaufort Sea region. IN T R O DU T I O N Sea ie bulk salinity, the loal mass-weighted salinity of ie rystals, air inlusions, and brine pokets, is a valued quantity beause, in onjuntion with temperature and sea ie density, it an be used to alulate the frational brine volume. ompared with seond-year and multiyear sea ie, first-year sea ie ontains only a small fration of air inlusions, allowing the brine volume to be used to alulate ie physial properties suh as strength, permeability to fluid motion, optial etintion, and eletromagneti signatures (Petrih and Eiken, 1). Unlike temperature, bulk salinity has the onvenient property of dereasing during ie formation to reah a quasi-steady state after a few weeks (Nakawo and Sinha, 1981). Sine the steady state lasts until the onset of melt, it is possible to estimate a wealth of ie properties at any given time during winter and early spring, provided the temperature history is known, and the bulk salinity profile has been determined at a representative time during the growth season. In fat, impliit in the desription of bulk salinity developed in this manusript is the onjeture that the bulk salinity profile itself an be estimated from the temperature history of the ie. The latter notion is not new. For eample, Nakawo and Sinha (1981) related data of steady-state bulk salinity, whih they refer to as stable salinity, to growth rate with a funtional relationship derived for a different segregation proess. A similar approah was followed by Kovas (1996)

2 and Petrih et al. (6) who also fitted a funtion to the relationship between bulk salinity and growth rate, although they were guided in the seletion of their fitting funtions by subjetively apparent empirial relations. Here, we will introdue a steady-state bulk salinity parameterization that, in addition, provides a means of estimating how the steady-state bulk salinity would differ if, for eample, brine visosity, sea ie permeability or oean salinity were different. Different fluid visosities have been found in the presene of miro-algal ativity (Niols et al., 5), systemati permeability differenes may be related to differene in ie fabri (platelet, olumnar, granular ie), and signifiantly different seawater salinities are observed for eample in the Balti Sea and aspian Sea. By measuring steady-state bulk salinity and reversing the bulk salinity parameterization, it may be possible to make inferenes on ie struture and growth onditions. Our derivation of the bulk salinity parameterization is based on signifiant simplifiations to the proess of first-year sea ie growth and desalination as we understand it today (e.g. Petrih and Eiken, 1). We derive a parameterization of the steady-state bulk salinity by parameterizing the proess of desalination and integrating that proess over time. The purpose of presenting the derivation of the steady-state bulk salinity parameterization is to provide hooks for subsequent improvements. Following the derivation, we will briefly present some preditions and omparisons with rigorous omputational fluid dynamis simulations, and an appliation to modelling sea ie properties. The most fundamental parameterization used in the following derivation is that of the brine volume flu inside sea ie. As laid out below, we assume that fluid motion is onfined to ie near the ie water interfae where the porosity is larger than a harateristi porosity (Figure 1). This is in-line with observations of, for eample, o and Weeks (1988) and Arrigo et al. (1993) who plae at.5 and.7, respetively. There are two fundamentally different trains of thought as to why solute should be retained below a harateristi porosity for a partiular set of growth onditions, i.e., perolation theory, and the ritial Rayleigh number riterion in porous medium onvetion. Perolation theory predits that, as a result of pore volume redution, the pore spae separates into disrete, disonneted lusters of pores (Golden et al., 1998, Petrih et al. 6). Naturally, ions would be trapped in these lusters leading to steady-state bulk salinity. In a perolating medium of infinite size, the perolation transition, i.e. the appearane of a luster of infinite size, is known to happen at a well-defined ritial porosity that depends on the pore spae geometry. On the other hand, in porous medium onvetion, the dereasing size of individual pores dereases the permeability to fluid motion, whih eventually redues the porous medium Rayleigh number below the ritial Rayleigh number neessary for natural onvetion inside a porous medium (Notz and Worster, 8). In this ase, ions are trapped in a region that does not partiipate in natural onvetion during desalination, although the pore spae ould still be flushed by fored onvetion. While the distintion between these two mehanisms would have to be made eperimentally, for the purpose of the present parameterization, the relevant observation is that brine movement and desalination ome to a (near) halt at low porosity. In the derivation of the parameterization desribed below, we onsider the growth of olumnar sea ie in quiesent onditions, and in the absene of air inlusions. Further, we assume that the growth rate is approimately onstant at the desalination time sale of a few days to weeks during whih a newly areted ie layer is losing most of its salt due to desalination. This assumption would be appliable in partiular in the dark of winter when temperature flutuations are mostly driven by passing storm systems at the time sale of about a week, and are damped by a snow

3 over. Further, we assume a linear dependene of the freezing point of brine on solute onentration, whih is a good approimation in sea ie above the preipitation of mirabilite at about -8 (Petrih and Eiken, 1), i.e., it is valid for the temperature range in whih most of the desalination takes plae. The parameterization will use solute onentration rather than salinity. In the ase of small brine volume frations typial for old sea ie,, the relationship Sie an be approimated by, where S, and ρ are salinity, onentration and S i density of seawater, respetively, and S ie,, and ρ i are bulk sea ie salinity, brine onentration, and density of freshwater ie, respetively. D ESA L IN A T I O N PA R A M E T E RI Z A T I O N Desription Figure 1. artoon of desalination. Sea ie desalinates beause brine in pores within the ie matri is replaed with less saline fluid from the oean. The bulk of desalination of growing sea ie takes plae in the bottom few entimetres, and we assume for simpliity that a depth h an be defined over whih desalination takes plae at an appreiable rate, while desalination essentially eases above h (Figure 1). In this model, h (m) is defined by a presribed harateristi porosity (volume fration). We present a method to determine h, whih is related to the key harateristis of ie growth. The porosity profile in the ie develops in response to natural onvetion and desalination. We parameterize the vertial flu that effets desalination in terms of the dimensionless Rayleigh number Ra as S Ra where w h, (1) where where γ S is a dimensionless fitting onstant, α is the thermal diffusivity of the ie (m /s), and w is the vertial volume flu (m/s), i.e. w (m/s) denotes the vertial interstitial veloity of brine. A onstant γ S is required sine the Rayleigh number is a generi saling parameter, leaving the interpretation of the magnitude of Ra dependent on the geometry and other partiulars of the system under onsideration. The porous medium Rayleigh number is defined as

4 g h Ra, () where Π (m ) is a harateristi permeability of the onveting system, g (m/s ) is the aeleration due to gravity, μ (kg/(sm)) is the dynami visosity of brine, and Δρ (kg/m 3 ) is the density differene that drives motion (Worster, 199). Ultimately, brine moves beause of a hydrostatially unstable density onfiguration, so the density differene to be onsidered is that of seawater and brine at the level h inside the ie. This an be epressed in terms of the brine onentration gradient / z as h, (3) z where /. 8 is the dependene of brine density on solute onentration, and z is the vertial oordinate, positive being upward (Figure 1). The effet of temperature on brine density does not need to be aounted for epliitly given that, for the purpose of this development, brine is at the freezing temperature and in thermodynami equilibrium with ie, i.e. T m, (4) where T ( ) is the temperature of ie and brine, and m=.54 Km 3 kg -1 is the slope of the liquidus. In general, the bulk solute onentration in a one-dimensional system evolves in the absene of diffusion as w, (5) t z where z (m) is the distane from the ie oean interfae, and t (s) denotes time (Petrih et al., 6). The differential an be separated into w w. (6) t z z As a first-order approimation, we assume that the volume flu within z h is independent of position, i.e. w / z and therefore w. (7) t z Finally, we assume that the ie growth rate, oean and atmospheri heat flu hange slowly at the time sale of desalination (i.e., a few days), and that the temperature profile is linear. In this ase, the temperature profile in the ie an be assumed to be onstant in time and spae and written as T 1 T 1 z, (8) T T z where T is the temperature at the ie oean interfae. Hene, v (m/s) may be interpreted as either rate of movement of the ie oean interfae (i.e., ie growth rate) in negative z-diretion (Eulerian piture), or as rate of apparent movement of a fied point inside the ie away from the interfae, in positive z-diretion (Lagrangian piture). Note that v is defined as v> for growing sea ie (Figure 1). This onludes the narrative of the omponents of the model, and we will proeed by deriving an epliit epression for the steady-state bulk salinity. Solution The governing equations of the model are solute mass onservation, w, (9) t z

5 with the parameterization of volume flu with w given by equations (1) and () S g h where w z, (1) where and the assumption of a linear temperature and brine onentration profile 1 1 z, (11) z where is the brine onentration of the oean ( 34 kgm -3 ). Further, the onstitutive relationship between bulk onentration and porosity is. (1) For notational onveniene we define a veloity sale w g (13) and a length sale 1 1 z. (14) z Sine the growth rate v is onstant at the time sale of desalination, we an reast the solute mass onservation equation (9) into a differential equation for the bulk onentration profile. The passage of time is equivalent to a translation in spae following z v, (15) t and therefore (9) an be written as 1 w 1. (16) z v z Further, substituting (1) for w and using Equations (13) and (14), in the region, we get 1 sw z v h z. (17) Integrating with respet to z, starting at z= with, we obtain a general epression for the bulk salinity at position z h of sw h 1 z. (18) v z Desalination eases at z=h, and we obtain the steady-state bulk onentration sw h 1. (19) v z We will solve this equation for h / z. The brine onentration at h is known from (11), i.e. where h 1, () z ( )/ h/ z is the redued brine onentration at.

6 From the onstitutive relationship (1) we therefore know h 1. (1) z Hene, equation (19) an be rewritten as a quadrati equation h w h whih an be reast in terms of h / z as s 1 1 () z v z h v 1 S w 1 1. (3) z sw v The steady-state bulk sea ie onentration an be obtained from h 1. (4) z We remark at this point that the radiand in (3) is generally larger than 1 at ommon ie growth rates, whih suggests the use of the limit for small growth rates, h 1 v. (5) z w However, we will proeed by using Equation (3). F lu at the ie oean interfae s We postulate that the flu at the ie oean interfae, w (m/s), is proportional to the volume flu within the sea ie effeting desalination, i.e. similar to (1), w F Ra, (6) h where γ F is a dimensionless saling onstant, and with substitutions of Equations (13) and (14), h w F w. (7) z Further, we note that the Rayleigh number defined in Equations () and (3) an be epressed as F ree parameters h w z Ra z. (8) Unless stated otherwise, we will use the following parameters: , sw 4.51 ms, F w 3. 1 ms, where the hoie of was inspired by o and Weeks (1988) and sw and F w are fitted to R ESU L TS Figure a shows the growth-rate dependene of the steady-state bulk solute onentration for different values of. The steady-state salinity inreases with growth rate, i.e., based purely on the advetive proess inluded in this model, ie has the potential to desalinate most effiiently at low growth rates (however, in this ase, h will be large, inreasing the desalination time sale).

7 Figure. (a) Normalized steady-state bulk solute onentration as a funtion of normalized growth rate for different values of, and (b) relative flu at the ie oean interfae. The labels at the right (figure a) and at the top (figures a and b) give eample magnitudes for representative saling onstants. The shaded areas in (a) and (b) indiates the range of steady-state onentrations that is ommonly observed in sea ie grown from oean water. Note that the dependene of the flu at the ie oean interfae on F w is not shown (Equation (7)). The results show a strong dependene on, whih an be partially ompensated by tuning γ s in the alibration proess but may be of physial signifiane for ie of different fabri (see Disussion below). The ie oean interfae flu shown in Figure b is, relative to the growth rate, highest at low growth rates, leading to an inreased ehange of oean water per unit of ie grown as the growth rate dereases. The epeted volume of fluid ehanged between oean and sea ie is of the order of 1 to 1 times the volume of ie grown. The flu at the ie oean interfae is not diretly dependent on, suggesting the distintion that the flu at the ie oean interfae depends on the pore spae near the ie oean interfae, while the steady-state bulk onentration depends on the pore spae at lower porosities, i.e. at in this parameterization. The parameterizations of Equations (3)/(4) and (3)/(7) are ompared with output from the omputational Fluid Dynamis (FD) model of Petrih et al. (6). In the FD model, sea ie growth is treated rigorously in a domain that ontains both liquid and porous sea ie. The fluid veloity in the liquid under the ie was left unonstrained and reahed several entimetres per seond. Unlike in the simulations of Petrih et al. (6), the ie oean interfae was left to evolve freely without assumptions about the porosity profile within the bottom-most ells ontaining ie. Results using either one of two permeability porosity relationships are ompared here, neither one of whih (or any other part of the fluid dynamis simulations) ontains an epliit ut-off omparable to the harateristi porosity. The parameterizations are m, and (33) 9 1 m. (34)

8 Figure 3. omparison of the parameterization derived from this model (solid and dashed lines) with data from omputational fluid dynamis simulations (markers) for (a) bulk salinity and (b) ie oean interfae flu. Marker shapes and olours disriminate individual simulations. Upwards-pointing markers are from simulation using Equation (34), while the remaining 9 1 simulations use Equation (33). Dashed line is for model results with sw 4.51 ms. For referene, the dotted line in (a) is an empirial fit given by Petrih et al. (6). In the FD model ie was grown from an upper surfae held at onstant temperature. Simulation results using different domain sizes, grid sizes, and surfae temperatures are ompared with results from the present model in Figure 3. Looking past the satter intrinsi to the fluid dynamis simulations in Figure 3a, the steady-state bulk solute onentration ompare well in the FD results and parameterization. In fat, they also agree well with a parameterization given by Petrih et al. (6), whih was found to agree with field measurements in Arti landfast sea ie (Petrih and Eiken, 1). Figure 3b shows that the interfae flu depends on the interfae permeability in the FD simulations the same way as suggested by the parameterization. In partiular, a redution of the permeability at the ie oean interfae ( 1) by a fator 1 (Equation (34) vs. (33)) is reprodued by the parameterization if w is redued by a fator 1 (dashed line), whih is epeted from the linear dependene of w on Π (Equation (13)). DIS USSI O N A ND O N L USI O NS Desalination parameterization Based on a highly idealized piture of desalination due to gravity drainage and an ad-ho parameterization of fluid motion inside sea ie, we were able to arrive at parameterizations of the steady-state bulk solute onentration and of the flu at the ie oean interfae. This piture is appliable to desalination during quiesent olumnar growth. Proesses that may be partiularly relevant in highly dynami environments, suh frazil ie aumulation, wave pumping, and pressure sution, were not onsidered (e.g., Feltham et al., ). In addition we foused on gravity drainage as the dominant effet of desalination, omitting movement in response to brine epulsion from individual pokets in response to mehanial fores that should take plae regardless of porosity (Notz and Worster, 8). The most ompelling argument for the utility of the derived parameterization is its agreement with rigorous FD simulations, a seletion of whih were presented here. Unlike other parameterizations presented in the literature to-date, the urrent

9 Figure 4. Modelled snow (dotted line), ie (solid line), and thikness-averaged bulk salinity (dashed line) of sea ie at the end of an average growth season in Barrow, Alaska ( May). parameterization is able to predit steady-state bulk salinity and the flu at the ie oean interfae under a variety of onditions, inluding different oean salinities, fluid visosities and differenes in rystal size and pore struture (through Π and ). This apability may be useful to investigate the fabri-dependene of the pore spae, suh as those found in granular, olumnar, and platelet ie. Impliations for ie strength While the fleural strength of first-year sea ie is primarily a funtion of porosity, porosity depends on bulk salinity and temperature. Timo and Johnston () estimated fleural strength as a funtion of ie temperature and bulk salinity. For temperatures above -5, the strength at any given temperature is strongly dependent on bulk salinity and is approimately inversely proportional to bulk salinity. Based on the present model and 9 average temperature and preipitation data for Barrow, Alaska (71 N), we alulate the epeted average bulk salinity of a slab of growing first-year sea ie as a funtion of the date of initial ie formation. Ie growth is simulated with a simple two-layer quasi-steady state thermal ondution model (one layer snow, one layer ie), negleting solar radiation. The end-of-season ie onditions are evaluated on May, at a time where ie temperatures an be epeted to be anywhere above - 7 and snow melt is just about to start. Figure 4 shows the epeted end-of-season snow depth, ie thikness, and average bulk salinity. The relationship between ie age and thikness is nonlinear, with ie starting to form at the end of November growing thikest. However, the average bulk salinity of growing first-year sea ie is linearly orrelated with ie age, with the oldest ie ehibiting the lowest average bulk salinity. Note that this is not beause older ie had more time to desalinate but rather beause older ie had a larger fration of the ie grow at a lower rate, whih redues bulk salinity. The average bulk salinity spans a range from just above 4 to just under 8 ppt, suggesting that the fleural strength of ie enountered at that time of year varies by a fator two, with the oldest ie being strongest, based on onsiderations of bulk salinity alone. These results an be transferred to floes taking into aount spatial inhomogeneity, for eample in response to snow depth variations, or refrozen raks (Langhorne and Haskell, 4; Petrih et al., 7). A K N O W L E D G E M E N TS The parameterization was developed while P was supported by a postdotoral fellowship through the Foundation of Researh, Siene, and Tehnology, New Zealand. In-depth analysis

10 and omparison with omputational fluid dynamis simulations were performed under support of an International Polar Year presidential postdotoral fellowship of the University of Alaska Foundation. The onstrutive omments of the reviewer are gratefully aknowledged. R E F E R E N ES Arrigo, K. R., Kremer, J. N., and Sullivan,. W., 1993, A simulated Antarti fast ie eosystem, Journal of Geophysial Researh Oeans, 98(4), o, G. F. N., W. F. Weeks, 1988, Numerial simulations of the profile properties of undeformed first-year sea ie during the growth season, Journal of Geophysial Researh Oeans, 93(1), 1,449 1,46. Feltham, D. L., M. G. Worster, and J. S. Wettlaufer,, The influene of oean flow on newly forming sea ie. Journal of Geophysial Researh Oeans, 17, 39, doi:1.19/j559. Golden, K. M., S. F. Akley, and V. I. Lytle, 1989, The perolation phase transition in sea ie, Siene, 8(5397), Kovas, A., 1996, Sea ie. Part I. Bulk salinity versus ie floe thikness, old Regions Researh and Engineering Laboratory Report 96-7, 16 pp. Langhorne, P. J., and T. G. Haskell, 4, The fleural strength of partially refrozen raks in sea ie. In Proeedings of the 14th International Offshore and Polar Engineering onferene, Toulon, Frane, 3 8 May 4, Nakawo, M., and N. K. Sinha, 1981, Growth rate and salinity profile of first-year sea ie in the high Arti, Journal of Glaiology, 7(96), Nihols,. M., J. P. Bowman, and J. Guezenne, 5, Effets of inubation temperature on growth and prodution of eopolysaharides by an Antarti sea ie baterium grown in bath ulture, Applied and Environmental Mirobiology, 71(7), Notz, D., and M. G. Worster, 8, In situ measurements of the evolution of young sea ie, Journal of Geophysial Researh Oeans, 113, 31, doi:1.19/7j4333. Petrih,. and H. Eiken, 1, Growth, Struture and Properties of Sea Ie. In Thomas and Diekmann (eds), Sea Ie, nd ed, Wiley Blakwell, pp Petrih,., P. J. Langhorne, and Z. F. Sun, 6, Modelling the interrelationships between permeability, effetive porosity and total porosity in sea ie, old Regions Siene and Tehnology, 44(), doi:1.116/j.oldregions Petrih,., P. J. Langhorne, and T. G. Haskell, 7, Formation and struture of refrozen raks in land-fast first-year sea ie. Journal of Geophysial Researh Oeans, 11, 46, doi:1.19/6j3466. Timo, G. W. and M. E. Johnston,, Sea ie strength during the melt season. In: Ie in the environment: Proeedings of the 16th IAHR International Symposium on Ie. 6 1 Deember, Dunedin, New Zealand, Worster, M. G., 199, Instabilities of the liquid and mushy regions during solidifiation of alloys. Journal of Fluid Mehanis, 37,