Basic approaches to dry snow-firn densification modeling
A.N. Salamatin1,4, V.Ya. Lipenkov2,4, J.M. Barnola3, A. Hori4, P. Duval3, T. Hondoh4
1Kazan State University, Russia; 2Arctic and Antarctic Research Institute, St. Petersburg, Russia; 3Laboratoire de Glaciologie et Géophysique de l'Environnement, CNRS-Grenoble, France; 4Institute of Low Temperature Science, Hokkaido University,
Sapporo, Japan
Статья поступила в редакцию 3 мая 2006 г. Представлена главным редактором В.М. Котляковым
Рассмотрены общие вопросы моделирования процесса уплотнения снежно-фирновых отложений на поверхности ледниковых покровов в результате переупаковки ледяных зерен и их пластических деформаций под действием давления нагружения с учетом диффузионного роста межкристаллических контактов.
Notation
a Mean relative bond area (in units of R2) deformed ice grains in Arzt's densification
A Mean relative area of the contact segment bases scheme (Fig. 2)
(in units of R2) Rg Gas constant
b Accumulation rate in ice equivalent s Fraction of the free ice-grain surface not
C Relative slope of the ice particle radial distribu- involved in plastic contacts
tion function (RDF) sb Fraction of grain surface area involved in grain
E / Principal strain rates, j = 1,2,3 bonds
F Rheological function introduced by Eq. (7) t Time
g Gravity acceleration T Temperature (in K)
G Rheological function introduced by Eq. (7) T Principal macroscopic deviatoric stresses,
h Depth j = 1,2,3
К Ice-grain rearrangement rate constant w vertical velocity with respect to ice-sheet surface
P Ice pressure produced by normal components of X Fraction of deviatoric deformations due to ice-
contact forces grain rearrangement
Pi Load pressure (hydrostatic overburden stress) Z Coordination number
P Macroscopic pressure in ice skeleton a Power creep exponent in the ice flow law (13)
Q Activation energy в Dilatancy exponent in Eq. (20)
RR) Mean equivalent-sphere current radius of ice Y Lateral deviatoric strain rates
grains (the radius normalized by its surface size Rc) Г Coefficient in Alley's grain boundary sliding
R', R''Fictitious normalized radii of plastically model (12)
п(п')
X Ц
V
Р Pi
ю
Grain bond thickness
Correction coefficient in Eq. (15)
Fraction of the free grain surface occupied by
extra neck volume due to diffusive ice-mass
transfer (bonding factor)
Kinematic (bulk) viscosity in Eqs. (6)
Dilatancy rate parameter defined by Eq. (3)
Non-linear viscosity in the ice flow law (13)
Grain-bond linear viscosity
Relative density of snow-firn deposits
Density of pure ice
Principal macroscopic stresses, j = 1,2,3 Densification (compression) rate
P r
s 0
Superscripts
Fictitious geometric characteristics of ice grains in Arzt's densification scheme (Fig. 2 and Appendices A and B)
Value at the reference temperature T * at Vostok
Subscripts
Ice-grain plasticity characteristic Ice-grain rearrangement characteristic Ice-sheet surface
Critical point (snow-firn transition)
Introduction
Fresh snow on a dry glacier surface is subjected within a few uppermost meters to different depositional, diagenetic and meteorological processes [10]. Even in cold natural conditions there exists a variety of possible mass-transfer (pressureless sintering) mechanisms (e.g. [32]) such as evaporation-condensation and surface diffusion, which are primarily responsible for formation of initial physical properties [2, 7] and structure [18, 26] of the surface snow deposits. Below 2—3 meters, the snow is a low-density (55-65%-porosity) polydisperse powder compact of rounded and bonded ice crystals (grains). Further on, particle rearrangement [13] and sintering under the overburden pressure are the two principal counterparts of the snow-firn densification (firnification) in dry polar regions leading to formation of bubbly ice due to pore closure in the deeper stratum of 50—150 m. It was shown [20, 21, 32] that plastic deformation (dislocation, power-law creep) dominates in the development of the intergranular contacts for load pressures exceeding 0.01—0.1 MPa. This process of the snow-to-glacier ice transformation, being a fundamental glaciological phenomenon, links paleoclimatic records of ice properties in glaciers to those of atmospheric gases trapped in the ice and determines the close-off depth, ice age, and grain size. It also controls the geometrical properties (size and number) of air inclusions, bubbles and hydrates, in polar ice sheets [4, 30]. Here we focus on basic theoretical approaches to the snow-firn densification modeling.
Mechanical properties of snow and its densification at loading were studied and theoretically interpreted in [1]. Widely used semi-empirical phenomenological firni-fication models resulting in the density-depth (age) relationship were suggested in [28] and later in [17]. An alternative approach, not limited to a considered range of experimental data, is to develop a physical theory linking microstructural changes in the snow compact to its macroscopic behavior during compression. In natural conditions at a glacier surface, the snow-firn densification process is characterized by relatively low strain rates on the order of 10-11-10-9 s-1 with well developed intercrys-talline contacts even at shallow depths [26]. Surface fraction of grain bonds is large [11, 15] with neck-to-grain
radii ratio reaching 0.6-0.7 [8, 9]. In this case it is conventionally accepted after [9] that particle rearrangement, dominating in highly porous snow, is controlled by linear-viscous grain-boundary sliding. Microstructural physical description of the repacking mechanism has been sufficiently well developed [9] to be directly incorporated into the snow-firn densification modeling [14].
The number of bonds per grain (coordination number) in the snow grows with density and restricts intergranular sliding. As a result, the creep of contacting ice grains gradually increases and becomes the only mechanism of firnification at a certain «critical» density and coordination number while particle rearrangement essentially stops. This general scenario is commonly simplified by assumption that the critical-density point just separates two successive zones of densification by grain boundary sliding and by plastic deformation termed, respectively, as «snow» and «firn» stages [14,15]. Usually the transition between the two densification regimes is identified after [13] with the bend on the density-depth profile found at a relative density of about 0.6. However, as suggested in [21-23], this first critical point of sharp decrease in den-sification rates may be just a beginning of the intermediate zone where both particle rearrangement and plasticity work together, while the dislocation creep takes over at higher relative densities of about 0.75. A microstructural physical model for the firn stage of densification by power-law creep was constructed in [15] based on geometrical description of a random dense packing of mono-size ice spheres [16]. It was linked to Alley's model [9] for ice-grain rearrangement in the snow stage by introducing the critical density as a variable (tuning) parameter. Although being rather schematic and simplified, the resulting description [14] of the snow-firn densification process allows direct extension of computational simulations and theoretical predictions to various paleoclimatic and thermodynamic conditions.
In this paper, we continue elaboration of the snow-firn densification model, aiming at a more complete and sophisticated representation of the microstructural picture of ice grain kinematics and stress-strain distribution. In contrast with previous studies, we assume that the first (snow) stage, being dominated by ice-particle restacking,
is simultaneously influenced by gradually increasing plasticity of grains. The firn (consolidated-snow) stage, controlled only by the ice-grain dislocation creep under growing overburden pressure, starts when particle rearrangement ceases at closest dense packing.
General notions and equations
Let us consider the process of the snow-firn densification under the load pressure (hydrostatic overburden stress) pi on a glacier surface in dry, cold climatic conditions. Due to lateral constraints, macroscopically it is a confined vertical (uniaxial) compression. We designate the corresponding component of the strain-rate tensor as E\. The deformations in the two other principal horizontal directions «2» and «3» are equal to zero, and
E&1 = -3<b , E&2 = E&3 = 0 .
The snow-firn densification (compression) rate ra coincides with the lateral deviatoric strain rate y and, by definition,
1 dp p dt
■- 3ш ,
cy [5, 6, 19] of the granular ice structure schematically illustrated in Fig. 1 should be additionally taken into account. This means that uniaxial compression of ice powder as an ensemble of rigid particles can occur only at excess deviatoric strains, Yr > <or, and the total deforma-tional compatibility in the ice compact presumes that the difference Yr - rnr is compensated by extra plasticity compression op - Yp. Although in snow the dilatancy effect is generally small with respect to total deviatoric deformations, it controls the interaction between the two densifica-tion mechanisms. Here we assume a conventional linear kinematic relation between the densification rate by the grain rearrangement rnr and the corresponding deviatoric deformation y .
(1 - ^)Yr
(3)
(1)
where p is the relative density (ice volume fraction) of the ice structure; t is the time, and d/dt is the particle derivative.
The overall macroscopic deformations in the ice compact are the sum of two constituent parts due to (a) rearrangement (sliding) of grains as rigid particles and (b) grain plasticity (dislocation creep) under contact forces. Let us distinguish these two mechanisms by subscripts «r» and «p», respectively, to write
Parameter X determines the rate of dilatancy and, as a function of p, i
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