NUMERICAL SIMULATION OF SEA ICE GROWTH AND DESALINATION

Transcription

1 NUMERICAL SIMULATION OF SEA ICE GROWTH AND DESALINATION Chris Petrih 1, Pat J. Langhorne 1, Zhifa Sun 1 ABSTRACT We present a numerial model based on the finite volume method to simulate sea ie growth and desalination as a flow proess through a porous medium with phase hange. To ahieve this we estimate a permeability-porosity relationship for the initial stages of sea ie growth from experimental data for brine drainage. The model generates ie sheets with salinity profiles lose to alulations based on ie growth veloity. INTRODUCTION It is often desirable to know the permeability of sea ie to understand mirosopi proesses inside ie sheets that affet the salt budget during sea ie formation (Worster, 199; Feltham et al, 00), or the meltwater budget during sea ie disintegration (Eiken et al., 00). Diret measurements of the permeability as a funtion of surfae temperature (Ono and Kasai, 1985), porosity (Freitag, 1999; Eiken et al., 00) or mirostruture (Freitag, 1999) are diffiult to obtain and are thus sare. They show a onsiderable amount of satter that is attributed, in part, to sample size effets (Freitag, 1999). They reveal the sensitivity of flow resistane to the rystal struture and history of a sea ie sheet. For large sale modelling purposes it an be desirable to find a simple parameterisation of the permeability of sea ie that aounts impliitly for small sale flutuations (Eiken et al, 00). We present an attempt to find a permeability funtion that is suitable for modelling sea ie growth. Numerial simulations are performed that illustrate the apability of a simple permeability-porosity funtion to generate realisti sea ie sheet salinity profiles during ie growth. MODEL DESCRIPTION We treat sea ie growth as the flow of a Newtonian fluid in a two-dimensional domain that is partly pure liquid, and partly porous ie. The solid matrix of the porous medium is stationary in position, but time variable as governed by the phase hange. The governing equations redue to the Navier-Stokes equations with Boussinesq approximation in the liquid region, while flow in the porous medium is dominated by frition that is expressed through a term after Dary (Brinkman, 1947). Permeability is 1 University of Otago, Physis Department, Dunedin, New Zealand pjl@physis.otago.a.nz

2 treated as a funtion of loal porosity. The porous medium undergoes phase hange, and loal thermodynami equilibrium is assumed. The numerial implementation is based on the finite volume method with a staggered retangular grid (Patankar, 1980). The governing equations of the finite volume method are volume-averaged formulations of the Navier-Stokes equations for a pure liquid. We hose the integration volume to be large enough so that small hanges in position ause only small hanges in average properties. We use a governing set of equations that is valid if all physial properties of the pure liquid and of the pure solid are onstant in time and independent of position. In the momentum equations we apply the Boussinesq approximation, i.e. the density of the liquid is onstant exept in the buoyany term, where it is treated as a funtion of loal temperature and salinity. The volume averaged momentum and mass onservation equations for a no-slip boundary ondition between mirosopi solid and liquid are fu u u fu fu p + fu + fv = + f + f g x f t x y x x x l fv v v fv fv p + fu + fv = + f + f g y f t x y x y l fu, (1) fv, () 1 s f fu fv + + = 0, (3) l t x where u and v are the fluid veloity omponents in the x and y diretions, respetively, l and s are the onstant densities of liquid and solid, is the variable density of the liquid, is the dynami visosity, p the pressure, g and split into x- and y-omponents of the aeleration due to gravity and the permeability of the solid, respetively, and f is the volume fration of the liquid phase. We find from experiene that the solution of the mass and momentum equation (see below) on a staggered grid is stable if we solve for fu and fv. We therefore transform the advetion term on the left hand side of (1) and (). The derivative of the phase fration that emerges is dropped as it appears only in the porous medium and fluid flow in the porous medium is dominated by the Dary frition term. The resulting momentum equations are fu fu fu fu fu p + u + v = + f + f g x f t x y x x x l fu, (4) and fv fv fv fv fv p + u + v = + f + f g y f t x y x y l fv. (5)

3 From volume integration of the transport equation we obtain the onservation equations for heat and mass of solute. They are for heat T t T + lu x T + lv k T = x T + s f L t l, (6) and for solute f C t + C fu x + C fv = C f Γ x C f + C t. (7) In (6) and (7) T is the temperature of solid and liquid, C is the solute onentration in the liquid, L is the latent heat of fusion, and Γ is the solute diffusion oeffiient in the liquid. Solute diffusion through the solid is negleted. The average quantities for the porous medium = f + ( 1 f ) and (8) l s k = f k l l + ( 1 k s f ) s (9) are defined from the speifi heat apaities l and s, and heat ondutivities k l and k s of the liquid and solid, respetively. In the derivation of (6) we assume that the average temperatures of solid and liquid are equal and the same as the temperature at the mirosopi solid-liquid interfae. Further, latent heat is released or absorbed at the mirosopi interfae during phase transition. In the derivation of (7) we assume that the onentration of solute at the mirosopi solid-liquid interfae is equal to the average onentration of solute in the liquid, and that the onentration in the solid is zero. Solute is rejeted into the liquid phase at the mirosopi interfae during the phase transition. We presribe loal thermodynami equilibrium to determine the volume fration f. The volume fration is adjusted until the equilibrium ondition, T ( f ) T ( C( f )) =, (10) F hold, where T F (C) is the freezing temperature as a funtion of solute onentration. We use a step-wise linear freezing point equation for sea ie, fitted to data given by Cox and Weeks (198). Sine on the sale of the finite volume simulation sea ie has a distint transition form the purely liquid phase to the porous medium we inorporate a form of freezing front traking. We restrit ie formation to those omputational ells that either already ontain ie or have a liquid volume fration less than the speified threshold f f = 0.8 (Langhorne and Robinson, 1986) at a minimum of one of their faes. The liquid volume fration at the ell fae is estimated by linear extrapolation from neighbouring ells. Equations (3) to (10) form a oupled set of differential equations that are solved iteratively for eah time step with the SIMPLEC algorithm (Versteeg and Malalasekera, 1995). Disretisation of transient and advetion term is limited to first order shemes in

4 this work, as seond order shemes ould not guarantee stability under all phase transition onditions studied. Sine fluid veloities are quite small, first-order shemes give reasonably aurate results. We allow fluid inflow and outflow at the bottom of the domain. At this open boundary we hose the pressure boundary ondition to enfore mass onservation, impose zero veloity gradient normal to the boundary, and zero veloity parallel to it (Sani and Gresho, 1994). Temperature and solute onentration at the open boundary take on presribed values T OB and C OB for inward flow. The integrity of the algorithm is validated by orretly determining the ritial Rayleigh number for Rayleigh-Bernard onvetion, and by alulating the flow pattern over a bakward faing step in the laminar region of Reynolds number Re=800, where the open boundary is plaed suh as to interset the seond eddy (Sani and Gresho, 1994). PERMEABILITY FUNCTION We seek a parameterisation of the permeability as a funtion of the liquid volume fration f in order to solve the momentum equation. Sine the governing equations are based on the assumption of an interonneted liquid phase, any isolated pokets have to be aounted for impliitly by the permeability funtion. Owing to the presene of drainage systems on all length sales, and the hange of pore struture with age and history of the sea ie sheet (Freitag, 1999) we annot expet to find a permeability funtion of only one parameter that predits the permeability in all irumstanes. Here, we are partiularly onerned with permeability in young sea ie under growth onditions, and find a permeability funtion for that situation. Our primary onern in this paper is the modelling of sea ie desalination. Cox and Weeks (1975) have performed a laboratory study on the initial sea ie desalination proess. They find that, after an initial brine segregation proess, the salinity of the ie sheet ontinuously dereases mainly as a result of gravity drainage. They present data for f<0.7 that we fit with a power law funtion and obtain S ie t where the rate of sea ie desalination, T = psu m s K ( f 0.054), (11) z / t, is proportional to the vertial S ie temperature gradient, T / z. The rate of desalination is zero for f Details of the fit are given in Petrih (004). Assuming the hange in salinity of sea ie is proportional to the solute onentration of sea ie, i.e. Sie = β Cie, and assuming that the temperature gradient is proportional to the onentration gradient of liquid brine, i.e. T / z = α Cb / z, where α is the slope of the liquidus line, we rewrite (11) C ie α 1. t = psu m s K β ( f 0.054) C b z. (1) Comparing this expression to the transient and advetion terms of the solute mass balane equation (7) we obtain an expression for the vertial liquid mass flux during freezing, (f u), as a funtion of instantaneous liquid volume fration f, α β ( fu) = psu m s -1 K -1 ( f ) 1.. (13)

5 We desribe fluid flow during brine drainage by Dary s law, = ( fu) p (14) with an isotropi permeability that is harateristi for the brine drainage proess. Assuming that p is approximately independent of the onditions of growth we obtain an isotropi permeability funtion for sea ie during initial desalination of the form ( f 0.054) 1.. (15) We fit the onstant of proportionality in (15) so that the omputer model yields realisti salinity profiles. In order to define what is realisti, referene salinity profiles are alulated from an expression for the stable distribution oeffiient k eff as a funtion of freezing front veloity v, where Sie v S 0 ms = k eff = 0.46 (16) is a power law fit to the stable salinity data of Nakawo and Sinha (1981) from Arti sea ie. The referene veloity of ms -1 is the average freezing front veloity of that data. The justifiation for fitting a power law originates from the sea ie desalination model of Cox and Weeks (1988). Their model suggests (Petrih, 004) a power law relationship with exponent 0.4 for the range of ie growth veloities ms -1 < v < ms -1. In an anisotropi medium the harateristi permeability in (15) an be understood as the path average of three orthogonal omponents, two horizontal omponents x1, and one vertial omponent y. In one instane Freitag (1999) finds a differene of almost one order of magnitude between the horizontal omponents x1 and x in olumnar sea ie. However, we will ontinue to examine the suitability of an isotropi permeability to model sea ie growth. EXAMPLE CALCULATIONS We use the permeability funtion ( f 0. ) 1. = m 054 (17) for f > 0.054, and = m for f Sea ie growth is simulated in domains of various aspet ratios, 30 to 180 mm wide and 640 to 190 mm high on square grids of ell size 0 x 0 mm, 40 x 40 mm, and 80 x 80 mm. Periodi boundaries (Versteeg and Malalasekera, 1995) are imposed in the horizontal (x-diretion), an open boundary at the bottom and the top is isothermal (ydiretion). The salinity of the water is 36 psu, and the water temperature is initially 1 mk above its equilibrium freezing point. Water adveted through the open boundary is at the same temperature and salinity as the water in the domain at the beginning of the

6 Depth (m) Depth (m) (a) Salinity (psu) (b) Salinity (psu) Figure 1 Example of alulated salinity profiles for surfae temperatures 10ºC (rosses) and 0ºC (dots). Dotted lines are profiles alulated from (16). (a) Superposition of 3 profiles at 10ºC and 5 profiles at 0ºC. The salinity lose to the interfae is not shown for larity. (b) Single examples of ie sheet profiles inluding the ie-water interfae obtained from a domain 180 x 180 mm, grid side 16 x 16. Note the different sales on the y-axes. simulation. Simulations are stopped when the freezing front reahes 40 mm above the open boundary. The details of the ie sheet growth simulation depend on the domain onfiguration. With horizontally periodi boundary onditions horizontal layers of fluid motion develop superimposed on the vertial brine drainage pattern, partiularly in domains with few (e.g. 8) horizontal ells. Wider domains provide more opportunity for the brine to disturb this flow. These layers are artefats of the two-dimensional domain. They are faster moving on fine grids than on oarse grids, probably beause oarse grids inrease loalised mixing and dissipation of momentum. We find that high horizontal flow veloities redue brine drainage in the systems investigated. To keep salinity profiles omparable, we limit horizontal flow veloities to 3 mms -1 by introduing an additional frition soure term in the momentum equation. The hoie of this limit originates in the observation that horizontal veloities of 3 mms -1 are seldom exeeded in simulations with 80 mm grid size. Feltham et al (00) have developed an analytial model of ie growth in the presene of a shear flow and demonstrate that brine expulsion depends on shear stress in the fluid. Salinity profiles do not depend on the overall height of the domain. However, as expeted, the smoothness of the horizontally averaged salinity profiles generally inreases with inreasing number of horizontal ells. Figure 1(a) illustrates the range of satter obtained in alulations with various domain onfigurations mentioned above. Apart from the satter the salinity profiles for ie grown from a onstant temperature surfae math the profiles predited from (16) based on the interfae veloity. The salinity losest to the ie-water interfae is not shown for larity. Figure 1(b) ompares the salinity profile of an ie sheet grown at a surfae temperature of 10ºC with one grown in an otherwise idential domain onfiguration at 0ºC. Both profiles show the lassial C-shape. While the profile of the simulated ie

7 37 psu ie sheet 1.8 m water 4 psu Figure Example alulation of ie growth at 0 ºC surfae temperature after 4 x 10 6 s. Grid size 16 x 16. The freezing front is in the range 0.96 to 1.04 m below the surfae. The salinity of one of the white volumes 00 mm below the surfae of the ie sheet is as high as 54 psu. sheet at 10ºC oinides with the predited profile from (16), the simulated profile at 0ºC is systematially higher than predited by 1 to psu. The latter profile shows deviations from a smooth urve at 0.35 and 0.5 m. An illustration of the nature of these deviations is given in Figure. It is apparent that the horizontal salinity distribution is very heterogeneous, and as suh is qualitatively similar to naturally grown sea ie. Deviations in the alulated average salinity profile an usually be attributed to hannellike features, whih have a salinity distintly above the median of the ie sheet at that height. SUMMARY We have demonstrated that it is possible to model sea ie growth and desalination as flow through a porous medium with phase hange. The dynamis of the entire system is desribed by a single set of governing equations. We have estimated a permeabilityporosity relationship from the observed orrelation between rate of brine drainage, temperature gradient and liquid volume fration f. Calulations with this permeability funtion resemble extrapolated data from the Arti surprisingly well, although zero order approximations are made in the derivation, and the derivation does not aount expliitly for an initial solute segregation (Cox and Weeks, 1988). The inhomogeneity of natural sea ie is resembled, inluding features suh as brine hannels. The form of the permeability funtion (17) orresponds to the power law expression predited by perolation theory (Golden et al, 1998), where exponents of 1. to 1.3 are found for perolation in two-dimensional systems (Berkowitz and Balberg, 199). Unfortunately, the present model alulations are unable to distinguish whether or not it is mere hane that the exponent of equation (17) oinides with that for perolation in a twodimensional system. The suess of the model in ases of simple, quasi onedimensional sea ie growth allows its appliability to be tested in more omplex situations.

8 ACKNOWLEDGEMENTS This researh was funded by Foundation for Researh, Siene and Tehnology, New Zealand. CP was finanially supported by a University of Otago Post-graduate Sholarship. We thank Dr. Greg Leonard and Mr. Craig Purdie for a steady stream of feedbak and suggestions during this projet, and Prof. Hajo Eiken and Ms. Amy Heaton for insightful disussions. REFERENCES Berkowitz, B. and Balberg, I. Perolation approah to the problem of hydrauli ondutivity in porous media. Transport in Porous Media, 9: 3, (199). Cox, G. F. N. and Weeks, W. F. Brine drainage and initial salt entrapment in sodium hloride ie. Cold Regions Researh and Engineering Laboratory Researh Report 345, Hanover, NH, USA. (1975). Cox, G. F. N. and Weeks, W. F. Equations for determining the gas and brine volumes in sea ie samples, Cold Regions Researh and Engineering Laboratory Report 8 30, Hanover, NH, USA. (198). Cox, G. F. N. and Weeks, W. F. Numerial simulations of the profile properties of undeformed first-year sea ie during the growth season. Journal of Geophysial Researh 93: C10, (1988). Brinkman, H. C. A alulation of the visous fore exerted by a flowing fluid on a dense swarm of partiles. Applied Siene Researh A1: 7-34 (1947). Eiken, H., Krouse, H. R., Kadko, D., and Perovih, D. K. Traer studies of pathways and rates of meltwater transport through Arti summer sea ie. Journal of Geophysial Researh 107: C10, 8046, doi: /000JC (00). Feltham, D. L., Worster M. G., and Wettlaufer, J. S. The influene of oean flow on newly forming sea ie. Journal of Geophysial Researh 107: C, /000JC (00). Freitag, J. The hydrauli properties of Arti sea ie Impliations for the small sale partile transport (in German). Berihte zur Polarforshung 35 (1999). Golden, K. M., Akley, S. F., and Lytle, V. I. The perolation phase transition in sea ie. Siene 8: (1998). Langhorne, P. J. and Robinson W. H. Alignment of rystals in sea ie due to fluid motion. Cold Regions Siene and Tehnology 1: (1986). Nakawo, M. and Sinha N. K. Growth rate and salinity profile of first-year sea ie in the high Arti. Journal of Glaiology 7: 96, (1981). Patankar, S. V. Numerial Heat Transfer and Fluid Flow, Hemisphere Publishing Co., New York, NY, USA (1980). Sani, R. L. and Gresho, P. M. Résumé and remarks on the open boundary ondition minisymposium. International Journal for Numerial Methods in Fluids 18: (1994). Ono, N. and Kasai, T. Surfae layer salinity of young sea ie. Annals of Glaiology 6: (1985). Petrih, C. Refrozen raks in sea ie. PhD thesis to be submitted, University of Otago, Dunedin, New Zealand (004). Versteeg, H. K. and Malalasekera, W. An introdution to Computational Fluid Dynamis: The Finite Volume Method. Pearson Eduation Ltd., London, UK (1995). Worster, M. G. Instabilities of the liquid and mushy regions during solidifiation of alloys. Journal of Fluid Mehanis 37: (199).