Jan 20, 1984 - than the degree-days are normally found. Wind, water depth, ... As cooling proceeds, the frazil ice rises toward the surface and ... The last effect of salinity to be mentioned is .... rise velocity,. C the mean volume fraction of ice, S the mean salinity, and T ..... 50. 0. 5. 10. 15. 20. 25. 30. 35. â¢0. /â¢5. 50 C (ø/,â¢). 0.
JOURNALOF GEOPHYSICAL RESEARCH, VOL. 89,NO. C1,PAGES735-744,JANUARY20, 1984
Modeling Supercooling andIceFormation In A Turbulent EkmanLayer ANDERSOMSTEDTAND URBAN SVENSSON Swedish Meteorological andHydrological Institute
A theoretical framework forstudying supercooling andiceformation inturbulent waters isdeveloped. Thebasic ideaisthattheproblem canbedescribed bya boundary layertheory in which buoyancy effects become important because ofsuspended icecrystals. A mathematical model based upontransient Ekmandynamics is formulated andexplored. Themodelis based on theconservation equations for
meanmomentum,heat energy,salinity,and suspended ice concentration in their one-dimensional form.
Theturbulent exchange coefficients are calculated witha two-equation modelof turbulence. Theice nucleation isassumed to besecondary, whichmeans thaticeCrystals areassumed to bealways present in thewaterasa result ofmass exchange withtheatmosphere. In themodel themass exchange istreated as a surface boundary condition fortheiceconcentration equation. Calculated timehistories oftemperature andiceconcentration fordifferent meteorological conditions anddifferent risevelocities arepresented anddiscussed. Theresults arein goodqualitative agreement withfieldandlaboratory measurements. Theimportance of thestrong interaction between theiceformation process andthehydrodynamics of
theboundarylayerisemphasized.
1.
INTRODUCTION
Due to variationsin themeteorological forcing,the timefor iceformationin theseacanbequitedifferentfromoneyearto the other.In the waterssurroundingSwedenthis differenceis on the orderof 2 months,seeClimatological Ice Atlasfor the Baltic Sea, Kattegat, Skagerrakand Lake Vdnern1963-1979 [1982]. The problemof forecasting the time for ice formation
ledthepresentinvestigators to seeka moregeneralmathematicalmodelof theproblem.In theirrecentpaper,Omstedt et al. [1983] demonstratedthat a mathematical model based on
differentialequationsand includingan advancedturbulence
modelwascapableof describing the late autumncoolingin
the BothnianBay. Comparisonswith field data were carried out for 52 consecutive days.The presentstudywill extendthis is a challenging scientific taskwithimmediate practical appli- model by adding an equationfor the concentrationof frazil ice.Thisequationwill,asdescribed below,havea strongcoucationsfor shippingandicebreaking service. salinity,andhydrodynamical equaExistingforecastmodelsare basedon the presentand time- plingto the temperature,
integrated conditions at theseasurface (airtemperature, wind, heatflux,seasurface temperature, etc.).The so-called degreeday methodis normallyemployedin somemodifiedform. Descriptions of suchmodelsfor the Balticmay be foundin Ostman[1950], Rodhe[1952, 1955], Palosuo[1963], and Kiihnel[1967].A comparison betweensomeof thesemodels, as appliedto Lake Champlain(USA), can be foundin Bates oo
tions.
Thepurposeof thestudyis thusto formulateandexplorea mathematicalmodelof the ice formationprocess. This model will be basedon the theoryof a horizontallyuniformboundary layer.Attentionwill be focusedon the generalformulation
of theproblem, withno attempts to improvethedescription of specific processes (e.g.,discshapes, flocs,etc.).
and Brown[1979]. The generalconclusionto be drawn from
In thenextsection a shortdescription of thephysics will be
these model studies is that correlations with other variables
provided. Then, in section 3, the mathematical formulation
Sectionfour presents the resultsof a sensitivthanthedegree-days arenormallyfound.Wind,waterdepth, will be discussed. ity test.Finally,a summaryandsomeconclusions aregivenin andwatertemperature areexamples of suchvariables. To get insightinto the ice formationmechanism,several studiesprovideusefulinformation.Standardtextbooks,such
as Hobbs[1974] and Michel[1978],givebasicice physics. Laboratoryexperiments on supercooling and iceformationin turbulentwatersgiveusefulinformationabout the time histor-
iesof temperature andice formation'see,for example,Car-
section 5.
2.
GENERAL DESCRIPTION
In smalllakes that are not too windexposed, iceformation
beginswith a thin elasticcrustof ice,whichgrowsvertically. This type of ice is callednilas [World Meteorological Or-
stens[1966], Katsarosand Liu [1974], Miiller and Calkins ganization,1970]. In the open sea,where wavesand currents [1978], Hanley and Tsang[1982], and Tsang[1983]. Os- are nearly always present,the initial ice formation starts in terkamp[1977] and Hanley[1978] discussed differentnuclea- quitea differentway.Due to turbulentmixing,iceis initially in theformof smallcrystals, whicharesuspended in tion theories.Martin and Kauffman[1981] and Bauer and produced Martin [1983] treatedthe formationand growth of ice in the water. As coolingproceeds,the frazil ice risestoward the leads.Ashton[1978] and Tsang[1982] reviewedformationof surfaceand formsgreaseice,whichis the soupylayer often ice in freshwaters,Martin [1981] reviewedice formationin found on the sea surfaceat this stageof the ice formation riversand oceans,and WeeksandAckley[1982] reviewedthe process. A schematic viewof theproblem,as studiedin the present growth,structure,and propertiesof seaice. Theabove-mentioned drawbacks of existing forecast models paper,is givenin Figure 1. As alreadymentioned,the math-
ematicalmodelwill be formulatedin boundarylayer terms. The turbulentEkmanlayer indicatedis thusthe boundary layer,whichis influenced by a densitygradient.The densityis
Copyright1984by the AmericanGeophysicalUnion.
a functionof temperature,salinity,and ice concentration. This mixture density,causedby ice particlesand water, is used when buoyancyeffectsare considered. In turn, the scalarsare
Paper number 3C1574. 0148-0227/84/003 C- 1574505.00 735
736
OMSTEDT AND SVENSSON:ICE FORMATION IN EKMAN LAYERS
•.•Rotation Surface heat
flux
I•ssexchange•>
::;'".'"."i";' (---' .'.... 'Turbulence •
l
small ice area exposedto the supercooledwater. The supercooling will thus increaseuntil the time of maximum supercooling; after that, freezingreleasesmore heat than is lost at the surface.Eventuallya quasi-stationarysituationis reached, where the product of exposedice area and supercoolingis proportional to the surfacecooling[Carstens,1966]. After this presentationof the generalfeaturesof the problem,somespecific aspectswill be discussed. Salinity affects the ice formation in several ways, one of which is the temperature of maximum density [Caldwell, 1978]. If the salinity is less than 24.7%0,the temperatureof maximum densityis higher than the temperaturefor freezing. This means that cooling from the temperature of maximum
densitywill causestablestratificationin the boundarylayer.If
the salinityis above 24.7%0,surfacecoolingwill give an unstable layer and a differentmeandensityprofile at the time when ice starts to form. Also the freezing temperature itself is a functionof salinity;the higherthe salinity,the lower the freezz ing temperature[Millero, 1978]. The last effect of salinity to be mentionedis salt rejection.When seawater is cooledbelow its freezingpoint, ice crystalswith no or low salinityform.The freezingprocessis therefore associatedwith a releaseof salt from the freezingwater [Martin, 1981]. Fig. 1. Schematicrepresentationof the model. A rise velocitywill result from the differencein densitybetween ice and water. Actually the existenceof a prevailing vertical ice concentrationprofile is the outcome of a balance coupled to the hydrodynamicsthrough the turbulent diffusi- between downward transport by turbulent diffusion and vities. The approach is analogousto the one used by Adams upward transport by negativesedimentation.The rise velocity and Weatherly[1981] whenstudyingsuspended sediment is a functionof the densitydifference,the turbulenceintensity, stratificationin an oceanbottom boundarylayer. and the shapeand size of the ice particle, and in generalit is The transient nature of the problem is shown in Figure 2. hard to estimate. This has been found in studies on sedimenAssumingthat the wind stressand the surfacecooling is con- tation [see, for example, Graf, 1971], which is an indication stant in time, the surfaceice concentration and temperature that the same may apply to ice crystals.As will be demonwill developas outlined.Afterthe temperature for freezingTs strated later, the magnitudeof the velocity will influencesigis reached,supercoolingstarts.Soon after that time, ice forma- nificantlythe initial ice formation process. tion starts, but initially at a rather low rate becauseof the In general the crystal shapeof frazil ice is reported to be
-
_•_
time
Freezing-
SupercooLing Max.supercooLing
•
-'••-•'••
• Residual supercooLing time
•• •- • • CooLing wi,hou, iceform•ion
Fig. 2. Initialiceformation andsupercooling closeto thesurface. Tyisthefreezing temperature.
OMSTEDTAND SVENSSON: ICE FORMATIONIN EKMAN LAYERS
disclike, measuring 1-4 mm in diameter and 1-100 #m in
737
Nusselt number, defined as the ratio of actual heat transfer to
thickness [Martin,1981].Recent laboratory experiments con- that by conductionalone, is known to vary with the Reynolds ductedwith genuineAtlantic seawater[Tsang, 1983] indicate, however, that seawaterfrazil ice has two other basic crystallographicshapes:two-dimensionalhexagonalstarsand threedimensional sea urchin shapes. Tsang [1983] also observed that the physical properties of frazil ice are related to the salinity of the freezingwater. Seawaterfrazil ice showedmuch less cohesiveness and adhesiveness than freshwater frazil ice.
Flocculation of particles has to be consideredfor highvolume concentrations.Lumley [1978] has estimatedthat particle interaction
becomes noticeable
for volume concentration
above 3%0.For the present problem this means that, due to the physicalpropertiesof frazil ice,larger aggregateswill form, and they will have rise velocitieslarge enoughto overcomethe downward transport by turbulent diffusion. Our presentview is that theselarger particlesform greaseice. Seedingis the term usedfor the massexchangeat the ocean
and Prandtl numbers [see, for example, Pitts and Sissore, 1977]. However, in the present study it will be treated as a constant.
The nucleationis assumedto be of the type called secondary, which means that ice crystals are always present in the water becauseof mass exchangewith the atmosphere.Since the presenceof ice will also influencethe density,a discussion of the equation of state and buoyancyeffectsis also called for. However, for the sake of completeness the full set of equations will be given. 3.2. Mean Flow Equations
Within the assumptionsmade, the mean flow equations take the followingform'
r3 t - r3 z
surface caused by aerosols. In laboratoryexperiments therate and time of seedinghave been found to influence the magnitude of the supercooling.Due to the low rate of supercooling observedin nature, the laboratory studiessuggestthat ice crystalproduction starts as a result of massexchangewith the atmosphere.This type of nucleationis referred to as heterogeneous,in contrast to homogeneous,nucleation,which is the term usedfor nucleation in a pure liquid. When nucleation is due to the presenceof ice crystals,the term secondarynucleation is used. For discussions,see Hanley [1978] and Osterkamp[ 1977]. 3.
MATHEMATICAL
FORMULATION
3.1. Basic Assumptions
The mathematical model employed is an extensionof the model presentedby Oms•edtet al. [1983]; the extensionbeing due to the inclusionof processesrelated to the ice formation. The reader is thereforereferredto their paper for a discussion of basicassumptions,boundaryconditions,turbulencemodel, etc. Verification studiesof the samebasic model may also be found in Svensson[1978, 1979, 1981]. In the present paper, attention will be directed toward the ice concentrationequation and the new sourceterms in the heat energy and salinity equations. The study will restrict its attention to horizontally homogeneous flows, which means that terms containing gradientsin the horizontal plane are neglected.Further, it will be assumed that there is no mean vertical velocity,exceptfor the ice crystal rise velocity.Advectiveeffectsdue to Langmuir circulation and wind- and wave-drivencurrentsare thus neglected. The mathematical
model
is based on the transient
Ekman
dynamicswith buoyancy effectsdue to temperature,salinity, and ice concentration gradients. Turbulent exchange coefficientsare calculatedwith a kinetic energy-dissipationmodel
(1)
c• t - c• z
•zz+ Gs
(2)
•t - •z
+ Gr
(3)
•t -•z vr•
+fV
(4)
•t - •z vr
-fU
(5)
where z is the vertical space coordinate positive upward, t is the time coordinate, f is the Coriolis' parameter, U and V are mean velocitiesin horizontal directions,• is the rise velocity, C the mean volume fraction of ice, S the mean salinity, and T the mean temperature.The kinematic eddy viscosityis denot-
ed by vr, while ac, as, and ar are Prandtl/Schmidtnumbers for ice concentration,salinity, and temperature,respectively. Source/sinktermsin the equationsfor ice concentration,salinity, and temperatureare denoted by Gc, Gs, and Gr, respectively. The mean volume fraction of ice C satisfiesthe inequality 0 • C • 1. The sourceterms, which are due to the freezingand melting of ice, can be derivedby consideringa unit volume of a mixture of water and sphericalice particleswith a givenradiusR•. Conductiveheat transfer q•, through a sphericalsurface,is
givenby Fourierslaw [seePitts andSissore, 1977]'
=
-
'
(Wm-:)
wherekw is the thermalconductivity and • is the icesurface temperature takenas the freezingtemperature in the following calculations. The actualheat flux per unit area of ice, q, betweenthe water ahd the ice,can be expressedas
q = Nu kw(•-
of turbulence.
Someassumption concerning theicecrystals areneeded. A
•zz - Wc •zz+ Gc
D(2R•)-•
(Wm-:)
•
(6)
where Nu is the earlier introduced Nusselt number. Consider-
size distribution is expected,and also different basic crystallographic shapes,but as a first approximation the present study assumesspherical crystals of uniform radius. Ice concentration is assumedto be high enough to representa continuum but low enough to allow neglectof particle interac-
ing the unit volumewith ice concentrationC, it can be shown that the term in the temperatureequationtakesthe form
tion.
water. The heat flux, it is assumed,will be directly related to melting or freezing.This givesthe sourceterm in the ice concentrationequation'
The heat transfer between ice crystals and surrounding water is parameterizedby a Nusseltnumber formulation.The
•
Gr = 3Cq(R,p•%)•
(øCs-x)
(7)
wherep• is the densityof waterand c• the specificheat of
738
OMSTEDTAND $VENSSON'ICE FORMATION IN EKMAN LAYERS
TABLE
1.
Model
Constants
Constant
C•,, constant in theturbulence model C1•, C2•, C3•, ak, a•, at, as, ac, •, fl, Po, T•t,
constantin the turbulencemodel constantin the turbulencemodel constantin the turbulencemodel Prandtl/Schmidtnumber Prandtl/Schmidtnumber Prandtl/Schmidtnumber Prandtl/Schmidtnumber Prandtl/Schmidtnumber constantin the equation of state constantin the equationof state referencedensity temperatureof maximum density
Ts, freezingtemperature f, Nu,
Coriolis' parameter Nusselt number R i, mean ice crystalradius Wo rise velocity kw, thermal conductivity L, latent heat of pure ice Pt, densityof ice
%, specific heatof water
Gc= 3Cq(RiLpi)-I
Value
arTM) 0---•+•(picao Pb =vr•/(_2g (T-O T+--• OS po) oc)(13)
Unit
k•
0.09 1.44 1.92 0.8 1.4 1.3 1.0 1.0 1.0
5.6 10 -6 8.0 10 -'• 1.0 103 2.9
-0.3
1.3 10 -'•
vr = C• •
where k is the turbulent kinetic energy,e its dissipationrate,
Ps is production due to shear, and P• is produc-
oc-2 0/o0 -1
kg m-3 øC oC
S-1
10-3 10-3 3.34 105
9.2 i02 4.217 103
(S-1)
W(moC)-1 Jkg -1 kg m-3 j (kgOC)-1
(8)
whereL is the latent heat of pure ice and Pi the densityof ice. By assumingthat the ice has zero salinity,an expressionfor the source/sinkterm in the equationfor the salinity of the water may be formulatedas
Gs= 3SCq(R,Lpw)'
(s- 1)
tion/destructiondue to buoyancy.For a generaldescriptionof this turbulencemodel the reader is referredto Rodi [1980]. A discussionof the basicversionemployedin the presentpaper may be found in Omstedtet al. [1983]. The differencefrom that paper is that ice concentrationgradientswill now influencethe buoyancyproductionterm P•. 3.4. Equationof State
6.0
0.564
(14)
(9)
An underlyingassumptionof the Pb term is that the ice and water can be regardedas a mixture when consideringbuoyancy effectsin the turbulencemodel. The mixture densityPm can be calculated
from
Pm'--'low'4-C(pi -- Pw)
(15)
The density of water Pw in this temperatureinterval is an almost quadratic function of temperatureand is also linearly dependenton salinity. An approximativeform, also used by Ornstedtet al. [1983], reads
Pw= P0(1 --•(T-
TM)2 + •S)
(16)
where • and fi are constants,TM is temperatureof maximum
density,and Pois a reference density.The temperature of max-
imum density,as well as the freezingtemperature,is a function Equation (1) also containsa term that describesthe vertical of salinity and pressure.In the present study both of these transport due to the rise velocity of ice particles.The formu- temperatureswill be set to constantsadequatefor sea surface lationof thistermis straightforward; theproblemis to assign pressureand a salinity of 5%0,seeTable 1. The relative importance of the ice concentration,salinity, a realistic value to the rise velocity Wo This point will be discussed further in section 4.
and temperatureon densitymay be estimatedby differ-
It was statedin the assumptionsthat the mixture of ice and water is treated as a continuum. This raisesa questioncon-
entiatingthe equationof state:
cerningthe lengthscaleto whichthis assumption holdstrue. A rough estimatecan be made in the followingway: Assume that the ice particle radius is 1 mm and a typical con-
centration1%; a volumeof 10-3 m3 (0.1 x 0.1 x 0.1) will then contain2400 particles,while a volumeof 10-6 m3 (0.01 x 0.01 x 0.01) will containjust two particles.This indicates that the continuum assumption holds true down to a lengthscaleon the order of decimeters. 3.3.
Turbulence Model
The turbulencemodel usedin this paper is basedon turbulent exchangecoefficientscalculatedfrom a kinetic energy-
dissipationmodelof turbulence.The equationscan be derived in exactform from the Navier-Stokesequationsand are thereafter "modelled"to the followingform:
Ot- Oz •zz+ Ps+ P•- •
(10)
Of--OZ •zz + c3eel• --c2e8) (11) & • (v• •)+••(Clees
2+(O gX•2• es=¾T((OU• kkOzJ kOzJ /!
(12)
dam = •-• dC+ •- dS+ OT • (a,- ao)dC + aol•dS- 2•p0(T - T•)dT Using the constantsemployedin the presentstudy, one may
estimatethat changes in the temperatureof IøC canbe stabilized by variationsin the salinityof 0.05%0or by variationsin the ice concentration
of 0.5?/00.As will be demonstrated
in the
next section,steepgradientsin the ice concentrationare found closeto the surface,where salinity and temperatureare fairly uniform. The exercisethus givesa hint about the buoyancy effectof the ice particles. 3.5. Boundaryand Initial Conditions
Surfaceboundary conditionsfor mean flow variablesare specifiedaccordingto vTOT
aTOZ --FN(t)(pøCv)1 VT OS -0 O'SOZ
¾TOC - Fc(t) O'C 02
(17) (18)
(19)
OMSTEDT AND $VENSSON' ICE FORMATION IN EKMAN LAYERS
739
•u
vr• = z,,(t)po-'
(20) so.
= wherez,,(t)and %(0 are wind stresses, FN(t)is net heat flux,
• 30
and Fc(t) is massexchangedue to aerosols. The zero flux condition for salinity is an approximation made in this analysisas precipitationand evaporationgenerate a nonzero flux. The resultsare not particularly sensitiveto
thisapproximation, asdensity gradients arecontrolled mainly by ice concentration. Turbulent kinetic energy k and its dissipation rate • are related to the friction velocityat the surface. At the lower boundary a zero flux condition is used for all
0 TO
*36
*12
.t,B
variables. Initial conditions are given as zero velocity, 0øC temperature, no ice particles,and 5%0salinity. This implies that ice can only start to grow as a result of secondarynucleation represented by the massexchangeF c in (19). 3.6.
!
!
,60
+ 72
+86
i
i
i
•,.
Time
,96 Ihoursl
•
-_-Time
-0,•'•-0,2-
Constants Freezingfemp
The constantsenteringthe formulationare shownin Table 1. Most of these are standard values, which need no discussion,and somewill be discussedbelow. A
constant
that
does need
-0,•-
some discussion is the
Prandtl/Schmidt numberof ice concentration. In the present studyit is assumedto be constantand equal to 1.0, which
-0•-
meansthat the turbulent mixing is as effectiveas for momen-
tum.Thismaybea reasonable assumption for smallcrystals,
/
but it is also well known that large crystalscan rise through the water without beingsignificantlyaffectedby turbulentdiffusion. An interestingdevelopmentwould therefore be to relate the Prandtl/Schmidtnumber to the degreeof interac-
/ Ice growthaccording
to'FNL-1p,-1
tion betweenthe turbulence and the crystals. A large
/
Prandtl/Schmidt number willreduce thediffusive effect and
/
thus correspond tosmall interaction, which istypical forlarge [ 6-
crystals.This will alsoreducethe buoyancyeffectdue to ice concentrationgradients,see (13), sincethe Prandtl/Schmidt number enters this term. This seemsrealistic, since no turbu-
/ /
3
Time forfreezingtemp /
lent energyshouldbe spentif the crystalsare not affectedby • Time the turbulence.However, for the purposeof this paper it is +36 +68 +60 (hours) appropriateto usea constantvalue. Fig. 3. Ice formation, supercooling,and vertically integrated ice Anothervariablethat, as a first approximation,is treatedas thickness for the reference case. a constantis the temperature for freezingTs. Accordingto Weber[1977], the ice growthrate can,however,be consider4. RESULTS AND DISCUSSION ablyreducedasa resultof the depression of thefreezingpoint It is appropriateto start this sectionby recallingthe purcausedby salt rejection. poseof the paper,which is to formulateand explorea math,
,
ematical model of the ice formation process.The equations have been introduced,and in this sectionthe performanceof Equations(1)-(21) form a closedsystemand thusconstitute the model will be analyzed by carrying out a sensitivitytest. the formulation of the mathematical model. This set of equa- First, a referencecasewill be established.Then, the effectsof tions, in their finite differenceform, was integratedforward in variations,mainly in the meteorologicalforcing,will be studtime by usingan implicitschemeand a standardtri-diagonal ied. These variations are brought about by changingthe net heat loss,the wind stress,and the massexchangeone by one matrix algorithm [Svensson, 1978]. 3.7.
Numerical
The numerical
Solution
solutions were tested for and found to be
grid and time stepindependent.This was achievedby a grid expandingfrom the surfacewith a total of 50 grid cellscovering a depth of 100 m. The time stepwas chosento 600 s for cooling down to the freezingtemperatureand to 30 s for further cooling.
around values introduced in the reference case. This is, of course, an idealization, since the meteorological forces are
stronglycoupled,particularlythe wind and the net heat loss throughthe sensibleand latent heat fluxes.Wheneverpossible, the resultswill be comparedin a qualitative manner to field and laboratory experiments.
740
OMSTEDTAND SVENSSON' ICE FORMATIONIN EKMAN LAYERS
0,0 [
1,0 I
/,,95
/,,97
i
i
-'0•S
JJ.T(kg m'1s-1) 10
20
30
•0
I
I
I
•
0
50
• •
2,0 I
3,0 I
/,,99
i
i
-0,/.,.
t.,o I
5,01
i
i
i
-0,3
5,03 i
i
-012
•
0
5
10
15
20
25
30
•..
•
•
I
jl
•
I
5,0 ,. (ko rn'•. I 5,05S(%o) i
-0,1 •
35
•0
I,
i
-0,0T(øC ) /•5
50 C(ø/,•)
•
.... ,.....,..... / ,, /'....
/
,, ,
i
I
ß
I
'
8
!•T
C
T
S
•'•
! ß
/
2o-
i i I
½o
i I
i
•o
I •,1
I
i I
Fig. 4.
Vertical distributionof ice concentrationC, temperatureT, salinity S, mean densityanomaly au, and dynamical eddyviscosity#r. Calculationsaccordingto the referencecase.
4.1. The ReferenceCase The coefficientsentering the model have been given earlier (see Table 1). Some of these can be chosento any number within a certain range and may thereforebe regardedas defining the referencecase.One of theseis the Nusseltnumber Nu, which has been set to 6.0, a reasonable value according to laboratory experimentsconductedby Miiller and Calkins [1978]. The radiusRi and the rise velocity Wcof the particles are also difficult to give adequatevalues.The valueschosen
ness of a continuous ice cover, is shown. The dashed line in the samefigure is the calculatedice cover thickness,assuming that all heat lossfrom the time when the surfacewater temper-
ature equals the temperature of freezing is turned into ice production. The line giving the developmentof the verticallyintegrated ice thicknessshowsonly a very slightslopechangeassociated with nucleation.More pronouncedslope changescan, however, be seenin someof the followingfigures.The two curves in this figure rapidly becomealmost parallel, which shows areR• = 10-3 m and Wc= 10-3 ms-•. For a morecomplete that, soon after the coolinghas passedthe time for maximum supercooling, the ice growth is mainly controlledby the surdiscussionof frazil ice parameters,seeDaly [1982]. Surface boundary conditions are for the reference case face net heat loss. The time lag between the two curves is chosento what is expectedto definean averagesituation.The explainedby noting that the numericalmodel takesthe coolshearstresses •:x=0.0 Nm-2 and •:y=0.169Nm-2 corre- ing of the whole mixed layer into account.This can be respondto a wind speedof 10 ms-1, the heatflux FN= 200 vealedby a closerexaminationof the temperatureprofilesat Wm-2 to typicallate autumnconditions, and the massex- To+ 15 and To+ 36. At the earlier time, which is the time changeFc = 10-9 ms-• to a verylightsnowfall. All surface when the freezingtemperatureis reachedcloseto the surface, boundary conditions are stationary. Initial conditions are a weak stratificationis presentin the mixed layer. In the nugiven as zero velocity,0øC temperature,no ice particles,and merical model, ice production starts when the whole mixed layer is supercooled,which happensat To+ 36, as shownin 5%0salinity. The predicteddevelopmentof ice concentrationand tem- Figure 3. The differencein heat contentsof the water between peraturecloseto the surfaceis shownin Figure 3, whereTois thesetwo timesresultsin the time lag shown. As can be seen,there will be negligibleice concentrationfor an arbitrarily definedtime. Close to the surfacemeans,in this and all figuresto follow, 0.1 m below the surface.Also the almost 24 hours after the freezing temperature has been verticallyintegratedvolume per unit area, given as the thick- reached.Then it will rise rather quickly to 50%0,where the
OMSTEDT AND SVENSSON'ICE FORMATION IN EKMAN LAYERS
calculations are terminated. The reason for terminating the calculationsis that flocculationis expectedto be important for high concentrations,as discussedearlier. There is no estimate, as known to the authors, of the volume concentration for which flocculation becomes an important process.In this study a value has been chosen,quite arbitrarily, which is an order of magnitude larger than the concentrationat which particleinteractionbecomesnoticeable. The developmentof the temperaturecurve follows the observationsmade by, for example,Carstens[1966], exceptthat the time scalesare much larger becauseof differencesin net heat lossand water depth. If the wind ceasedwhen the surface ice concentrationwas closeto 50%0,all suspendedice would rise and form an ice cover of 0.05 m thickness.This is only 50% of the value calculatedwith the simple assumptiondescribedabove.The differenceshowsthe importanceof a realistic descriptionof the dynamicsof the boundarylayer. Typical profiles of temperature,salinity, ice concentration, Sigma rn (mean density anomaly defined accordingto asi = Psi- 1000), and eddy viscosityare shown in Figure 4. The profiles give the distribution at the time when the surfaceice concentrationhas reached45%0(seeFigure 3). Most of the ice is to be found in the top 10 meters; this is also the region where ice production is important, as can be understoodfrom the temperature profile. The increaseof salinity close to the surfaceis also due to ice production or, more precisely,salt rejectionduring the freezingprocess.The asi profile has, due to the presenceof ice, a very sharp gradient closeto the surface.This influencesthe hydrodynamics,sincea strongdamping of turbulencedue to buoyancywill be the result.The eddy viscosityprofile clearly demonstratesthe significanceof this damping.
741
I
I i
/ / / i
I
/
/ / /
/ / /
/ / /
lO
o
o,o
..' ." ---.,xJ',
// // i
1
+12
+2/,
i
i
+ 36 i
+/+B i
•
_ Time
+96 (hours)
+ 60 i
i
•
__Time
Net heat toss
= 100 wm'z - 200 wm-z
............
.•00wm-z
12-
4.2 Effect of SurfaceHeat Flux
The effect of varying the rate of cooling was studied by Carstens[1966], who concludedthat an increasein the rate resultedin the maximum supercoolingincreases,the residual supercoolingincreases,and the risefrom maximumsupercooling to residualsupercoolingrequireslesstime. In Figure 5 it is seen that the mathematicalmodel gives resultswhich are in good agreementwith thesegeneralconclusions.It is interestingto note that the resultingice thicknessis the samefor all three cooling rates. This suggeststhat the ice formation processis selfsimilar when normalizedwith the heat loss.Another way of expressingthis is that the timeintegratedsurfaceheat flux is an integralparameterof interest. This parameteris closelyrelated to the degree-daysused in simplermodels. The line giving the ice thicknessfrom the surfaceheat flux and time of freezinghas beenexcludedfrom Figure 5 because the heat flux is the parametervaried in this test. 4.3 Effect of Wind Stress The influence of the wind stress on the time history is shown in Figure 6. By noting that a strong wind will give a deeper and more turbulent well-mixed layer, and therefore a lower coolingrate, all the resultscan be understood.The variation in the amount of ice kept in suspensionwhen the surface concentration reaches its termination value is particularly striking.The strong wind keepsthe equivalentof a 0.13-m ice coverin suspension, while the value for the weakestwind is an order of magnitude smaller. It should also be pointed out that differencesin turbulence intensityinfluencethe heat transferbetweenthe water and the
Time for freezing femp
_..'"
•__ Time
Fig. 5. Effect of heat loss on supercoolingand ice formation. All parametersexceptnet heat lossare accordingto the referencecase.
ice particles.This effectcannot, however,be modeledwithin the assumptionof a constantNusseltnumber.Carstens[1966] noted that "strong"turbulence(high Reynoldsnumbers)produced lower maximum supercoolingsas comparedto "weak" turbulence(low Reynolds numbers).In the proposedmodel this effect can be modeled by letting the Nusselt number becomea functionof the Reynoldsnumber. 4.4 Effect of SurfaceMass Flux
In laboratoryexperimentsthe surfacemassflux, or seeding, has been shown to influence the rate of supercooling[see Hanley and Tsang,1982]. In Figure 7 an experimentwith the present model is displayed.The range of mass flux values testedcoversconditionsfrom no flux to quite heavy precipitation (about 10 mm/24 hours). In the case of quite heavy precipitationthe coolingrate is influencedby melting.To be noted is the extremely fast ice growth for low surfacemass
742
OMSTEDT AND SVENSSON:ICE FORMATION IN EKMAN LAYERS
the sensitivitystudy. The result is shown in Figure 8. It is believedthat the rise velocity of frazil ice crystalsis in the
range10-4 to 10-2 ms-•. Thedrastically differentbehaviorof
• 3c
Time
o •
0,0 •
i
i
TO
+12
i
i
+24
+ 36
+48
+ 60
+72
+84
(hours)
ents will reduce the turbulence level.
For the lower velocity in Figure 8 a different termination
.
criterion was used. Since the ice thickness had reached almost
Windspeed ..... : 2 ms'1 \
the processfor thesetwo valuespoints out the significanceof the rise velocity.This seemsto be a major problemin models of the kind usedin the presentpaper,sincethe effectof crystal shape and size and turbulenceintensity in the suspending media are known to influenceWo Severalanalogiesto the presentproblem may be found in studiesof sedimentationof solid particles [see, for example, Graf, 1971, or Adams and Weatherly, I981]. Analytical solutions,assuminga constant turbulentdiffusivity,show the samesensitivityto the rise velocity. In the presentstudy the sensitivityis expectedto be evenmore pronounced,sincethe particleconcentrationgradi-
....
= 5 ms-1
.............
--10 ms'1 15ms-1
t
Freezing femp
0
,
TO o,o
9-
i........
+12
i
,
;"' +36
+2•
•
+48
40
i
i
Ice growfh •ccording J
to: FNL 4p,-1
•'
+72
3
/
T•me forf,reezing femp•
-0,1
; ...../.... .
+%
___Time (hours) T•me
i
Massexchange •x•x• = 0 0 ms'l ..... = 10-30ms-1 .... = 10-11 ms-1 • = 10-9 ms-1
............
16
+84
""" '•
= 10-7 ms-1
Freez•nq femp
T! me
+4•4 +• ""-(hours)
Fig. 6. Effect of wind mixing on supercooling and ice formation.All parametersexceptwind stressare accordingto the referencecase.
flux. The surfaceice concentrationgoesfrom zero to its termination valuein roughly 10 hours.The resultsare in qualitative accordancewith the laboratory experimentsby Hanley and Tsang. From temperature curves it was noticed that "Com-
pared to seedingwith many nuclei,seedingwith a smalllump
oficewould increase thetimelagbetween theseeding point
Ice growfh according /
to-FNL4 p,4
/
/
and the minimumtemperature point."
/
The calculated supercooling isbetween hundredths and one
/
tenth ofadegree Celsius, values which canbeexpected during formation of the initial ice in the ocean [Weeks and Ackley, 1982]. 4.5 Effect of Rise Velocity
Tir,,,•freezingfemp
/
,/
/
..'
..."/,•"//
.-' .•
i
/ ..."/
So far, only variationsin the surfaceboundaryconditions T' 0 +1'2 +2G +30 +• +60 +•Z +$4 +•6 have been studied. During the test calculations,however, it Fig. 7. Effect of massexchangeon supercoolingand ice formawasfoundthat the risevelocityWchas sucha largeinfluence tion. All parametersexceptmassexchangeare accordingto the referon the dynamicsof the processes that it requiresinclusionin ence case.
OMSTEDT AND $VENSSON'ICE FORMATION IN EKMAN LAYERS
743
this kind of information are available, but more laboratory and field studies are needed. Model calculations like those
carried out in the presentstudy can usefullyaccompanythese experiments. The tentative conclusionsemerging from the study can be
•c
summarized
/
/
J J
_ T•me
o • +12
+2•,
+36
+•8
+60
+72
+96 (hours)
+8Z•
,
0,0 --,w,
•T•me
Risevetooty
- 10-• ms'1 •
= 10-• m s-1
...........
= 10-2 ms-1
as follows:
1. A sensitivitystudy has shown that the mathematical modelsuggested producesresultsthat are in qualitativeagreement with field and laboratory experiments.Systematicvariations in wind stress,surfaceheat flux, surfacemassexchange, and rise velocity produce effectsin the rate of supercooling and ice production that all seemrealistic. 2. The value of the risevelocityof the ice particlesis found to be most significantfor the developmentof the ice formation process.
3. A strong interaction betweenthe ice formation process and the hydrodynamicsof the boundary layer is established through the buoyancyeffect of ice particles.The interaction is taken into accountin the mathematicalmodel through source terms in the mean flow equationsand buoyancy effectsin the turbulencemodel. This aspectof modeling the ice formation processis emphasized.
Freezingtemp
Acknowledgments.This work is a part of a Swedish-Finnish Winter Navigation ResearchProgramme and has been financed by the SwedishAdministrationof Shippingand Navigation.The authors would also like to thank Lennart Billfalk, Tom McClimans, JSrgen SaMberg, Anders Stigebrandt, and GSsta Walin for valuable commentsand constructivecriticismon an earlierdraft of this paper.
-Oh'
REFERENCES
/ !
/
/
/
/
Adams, C. E., and G. L. Weatherly, Some effectsof suspendedsediment stratificationon an oceanicbottom boundary layer, J. Geophys.Res.,86, 416i-4172, 1981. Ashton, G. D., River ice, Ann. Rev. Fluid Mech., 10, 369-392, 1978. Bates, R., and M-L. Brown, Lake Champlain ice formation and ice free dates and predictionsfrom meteorologicalindicators, CRREL Rep. 79-26, pp. 1-21, U.S. Army Cold Reg. Res.Eng. Lab., Hano-
/
/
/
Icegrowth according to FNL'• p'•
/ /
,
/ /
/
/ /
/
ver, N.H.,
Time for freezing femp ,
,
TO
+12
, ,
+2/,
,
!
+36
+/,8
_ Time
+60
(hours)
Fig. 8. Effect of rise velocityon supercooling and ice formation.All parametersexceptrisevelocityare accordingto the referencecase.
0.15 m after 96 hours,it was decidedthat further time integration would be quite unrealistic.This implies that ice crystals with small rise velocitiescan be kept in suspensionby the turbulence
and accumulate 5.
considerable
amounts of ice.
SUMMARY AND CONCLUSIONS
The objective of this study has been to formulate and explore a mathematicalmodel of the ice formation process.The model is an extensionof the model presentedby Omstedtet al. [1983]; the extension being the inclusion of an ice concentration equation and additional sourceterms in the equationsfor temperatureand salinity. The model providesa new theoreticalframeworkfor studying the ice formation processby utilizing recent advancesin turbulencemodeling and numerical analysis.Further refinementsare neededfor realisticdescriptionsof specificprocesses (ice crystal shapes,rise velocity,etc.). Some studiesthat give
1979.
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OMSTEDT AND SVENSSON' ICE FORMATION IN EKMAN LAYERS
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(Received March21, 1983' revisedAugust28, 1983' acceptedSeptember27, 1983.)
