Introduction

Recent modeling of firn densification and its effects on surface elevation indicates that firn densification near the surface (top several meters) is important in the interpretation of temporal variations in altimetry measurements (Reference Arthern and WinghamArthern and Wingham, 1998; Reference Zwally and JunZwally and Li, 2002). However, densification processes are complicated by the involvement of multiple processes (Reference PatersonPaterson, 1994). Although there is no universal physical model for predicting densification rates in the firn column, several semi-empirical densification models (e.g. Reference Herron and LangwayHerron and Langway, 1980; Reference Spencer, Alley and CreytsSpencer and others, 2001) and physically based models (e.g. Reference Wilkinson and AshbyWilkinson and Ashby, 1975; Reference AlleyAlley, 1987), which cope with the different processes dominant at particular stages of the densification, have been developed. The densification rates in these models are primarily governed by the overburden pressure (or the change of pressure) and the firn temperature. Therefore, these models are forced by the accumulation rate and surface air-temperature variations. Based on laboratory experiments on crystal growth and ice creep, Reference Zwally and JunZwally and Li (2002) introduced a temperature-dependent activation energy and rate constant in their densification–elevation model and applied the model to the Summit of Greenland. That treatment increased the sensitivity of the densification rate to temperature, resulting in density variations compatible with field density profiles and seasonal elevation changes compatible with those observed in radar altimeter data over most of the Greenland ice sheet (Li and others, 2003).

Under conditions such as for inland Antarctica, both accumulation and firn temperature are much lower than in Greenland. The modeled magnitude of density variations quickly reduces with decreasing accumulation rate and temperature. However, a smaller magnitude of variation has not been found either in density profiles or in seasonal elevation variations from satellite altimetry measurements. In fact, the opposite trend of density variations has been observed. Reference Qin and YoungQin and Young (1988a) made a detailed study of density variations based on data measured continuously at 10 cm intervals in a number of firn cores collected from East Antarctica. Together with data collected from Byrd Station, West Antarctica (Reference GowGow, 1968), they found that the density variability systematically increases with decreasing accumulation rate (70–248 kgm^–2 a^–1) and annual mean temperature (–28.2 to –52.2°C) with magnitude greater than 100–200 kgm^–3 in the first 2 m of depth (Fig. 1; Reference Qin and YoungQin and Young, 1988b). It has been recognized that models driven by accumulation and surface temperature cannot describe density variations successfully in the top few meters of firn (e.g. Reference AlleyAlley, 1987, Reference Alley1988). This may imply that other processes that are important to the densification have not been properly considered in the constitutive relation used in the models. Shallow firn is well exposed to airflow, subjecting the firn to extreme temperature gradients. Besides the pressure and temperature, field observations show that the vapor flow driven by the temperature gradient plays an important role in the firn metamorphism associated with large firn-density variations (e.g. Reference Sturm and BensonSturm and Benson, 1997). Vapor transportation in polar firn has been well studied both theoretically and experimentally (e.g. Reference ColbeckColbeck, 1983). Studies have been focused mainly on the extraordinary snow crystal growth that is responsible for depth-hoar formation.

Fig. 1. Range of the density variation υs depth at different sites in the Antarctic ice sheet. The variation is calculated at a depth interval of 1 m.The name of the station, accumulation rate (kg m^–2 a^–1) and annual mean temperature (°C) are indicated. After Reference Qin and YoungQin and Young (1988b).

In this study, we incorporate available vapor-transfer theory into our pre-developed densification model (Reference Zwally and JunZwally and Li, 2002) to examine density variations during the densification in firn to depths of 10m.

Model Physics

The densification model used here is modified from the model developed by Reference Zwally and JunZwally and Li (2002). Based on the semi-empirical densification-rate equation derived by Reference Herron and LangwayHerron and Langway (1980), the densification rate dρ/dt is given by:

where K is solely dependent on temperature and A is the mean accumulation rate, representing the change of overburden pressure. The factor α is approximately equal to 1. The temperature dependence of the densification rate is taken to follow the Arrhenius relation:

According to laboratory experiments for ice crystal growth and deformation (Reference Jacka and LiJacka and Li, 1994), Reference Zwally and JunZwally and Li (2002) incorporated temperature-dependent rate constant K₀(T) and activation energy E(T) into the K(T) function shown by Equation (2). Both parameters have usually been taken to be constants independent of temperature. To account for differences between the rates for processes of densification and crystal growth, they also introduced an empirical constant β in the rate factor K₀(T),

β is treated as an adjustable parameter used to calibrate the modeled density profiles according to observed density profiles.

Although the above treatment increases the sensitivity of the densification rate to temperature, resulting in an amplitude that is compatible with the observed seasonal density variation induced by seasonal temperature variations for most of the Greenland ice sheet (Li and others, 2003), this accumulation- and temperature-forced model predicts very small density variations for low-accumulation, low-temperature conditions, as shown by Figure 2a–c.

Fig. 2. Modeled density profiles (thin solid lines) in low-temperature conditions with various accumulation rates: (a) 24, (b) 100, (c) 200, (d) 400 and (e) 900 kgm^–2 a^–1. Thicker solid lines show the density profiles without the consideration of vapor flow (i.e. R_v = 0). Surface temperatures with annual mean –57°C and maximum –17°C are applied for all model runs. The range of accumulation for the inland Antarctic ice sheet is also indicated, showing decreasing amplitude in the density variation with increasing accumulation rate.

We now incorporate vapor-transport theory to modify the densification rate dρ/dt.When a temperature gradient exists in the snow cover, the gradient causes heat and vapor flow in the direction of lower temperatures. During the summer, surface air temperature is normally higher than firn temperature, and vapor flow transports mass from upper firn layers to lower layers, enhancing the total densification rate. In winter, vapor flows upwards due to the opposite direction of the temperature gradient, reducing the densification rate. Here we assume that any upward vapor flow escapes into the surface atmosphere, so there is no contribution to the firn-layer thickness change. dρ/dt consists of two terms,

The first term, R_a, is due to accumulation rate and temperature given by Equation (1). The second term, R_v, is caused by vapor transport driven by a temperature gradient. The densification rate of a firn layer due to vapor condensation or sublimation can be expressed in terms of the depth gradient of vapor flux J,

The vapor flux J can be expressed as

where D_eff is the coefficient of vapor diffusion in snow, p₀ and T₀ are triple point pressure (610.5 Pa) and temperature (273.1 K), L is latent heat of sublimation (2838610³ J kg^–1) and R is the gas constant for water vapor (461 J K^–1kg^–1). For the value of D_eff, Reference ColbeckColbeck (1993) has indicated that the coefficient of vapor diffusion in snow is more than five times greater than its value in air (22610^–6m² s^–1). We follow this suggestion and set D_eff = 5×22×10^–6m² s^–1 as an approximation. Details of the derivation of Equations (5) and (6) and the parameters used have been given by Colbeck (Reference Colbeck1983, Reference Colbeck1989, Reference Colbeck1990, Reference Colbeck1993) and Reference Sturm and BensonSturm and Benson (1997).

The above equations are coupled with the standard one-dimensional time-dependent heat-transfer equation and solved using a multi-layer numerical model (Reference Zwally and JunZwally and Li, 2002).

The temperature gradients are calculated at 0.01 m depth intervals based on the depth–temperature profile at each time t. A time-step of 10 days was used for the computation, giving the initial thickness of each firn layer as approximately 3%of the annual accumulation.

Results and Discussion

In order to demonstrate the density variations, two cases are selected to run the model. One is a low-temperature, low-accumulation regime that is appropriate to inland Antarctica. A surface air temperature (annual mean –57°C, maximum –17.0°C) approximate to that forVostok (annual mean –57°C, accumulation rate 24 kgm^–2 a^–1; cf. Reference PatersonPaterson, 1994, p.210) is used. The second case is a high-temperature, high-accumulation regime that is closer to Greenland conditions. The temperature at the Summit of Greenland (annual mean –30°C, maximum –2°C, accumulation rate 250 kgm^–2 a^–1; Reference Shuman, Steffen, Box and StearnsShuman and others, 2001) is used. In both cases, accumulation rates vary from 5100 to a maximum of 900 kgm^–2 a^–1. All other parameters are kept constant (Fig. 2) in order to show the accumulation-rate dependence. However, the accumulation rate for the dry-snow zone in Antarctica is essentially 5200 kgm^–2 a^–1 (Reference Giovinetto and ZwallyGiovinetto and Zwally, 2000) and 4100 kgm^–2 a^–1 for most of the Greenland accumulation area (Zwally and Reference Giovinetto and ZwallyGiovinetto, 2000), as indicated in Figures 2 and 3 respectively.

Since the densification law used (Equation (1)) is semi-empirical, the rate constant has to be determined empirically according to observed density profiles (Reference Herron and LangwayHerron and Langway, 1980). Reference Zwally and JunZwally and Li (2002) used β =8 to match the observed density profile for the Summit of Greenland. Recently, Reference Li, Zwally, Cornejo and YiLi and others (2003) examined density profiles at multiple sites over the Greenland ice sheet. They found that β must be dependent on annual mean temperature to allow the best agreement between modeled density profiles and field data. According to their regression relation between β and annual mean temperature, two values of β = 18 and 8 that give the best fits to the field density profiles at Vostok and the Greenland summit are respectively applied to the above two cases. Surface snow density was set to 300 kg m^–3 as a constant. Sinusoidal variation of the surface air temperature during the year is assumed.

Figures 2 and 3 present the modeled density variations. The corresponding profiles modeled without the consideration of vapor flow (R_v β 0) are also included for comparison. Figure 2a–c show striking features for the low-temperature, low-accumulation case. At accumulation rate A = 24 kgm^–2 a^–1 (Vostok; Fig. 2a), vapor flow affects the densification significantly, causing large variations in the density profile. The density ranges from about 200 to 400 kgm^–3 along the general slowly increasing trend. If the vapor flow is not considered, i.e. R_v =0, the corresponding density profile shown by the thick solid line (Fig. 2a) indicates that the variability of the density is approximately zero. The difference is obvious. The variability sharply reduces to 20 kgm^–3 (Fig. 2c) when accumulation rate increases to 200 kgm^–2 a^–1. Further increasing the accumulation rate even to 900 kgm^–2 a^–1seems to have little effect on the magnitude of the variation (Fig. 2d–e).

By contrast, in the high-temperature, high-accumulation case (Fig. 3b–e), with increasing accumulation rate from 100 to 900 kgm^–2 a^–1 the density variation increases from approximately 50 to 200 kgm^–3. Comparing both density profiles in each diagram, the density variations are little affected by vapor flow driven by the temperature gradient. The two curves are almost completely overlapped when the accumulation rate reaches 900 kgm^–2 a^–1 (Fig. 3e). In this case, the densification rate is essentially given by Equation (1). From Equation (1) for the same temperatures, larger accumulation rates A increase the densification rate dρ/dt; resulting in higher firn density during the summer period. In winter, the densification process is much retarded due to the colder temperatures (Reference Li and ZwallyLi and Zwally, 2002). Larger accumulation rates allow the newly deposited firn to travel to greater depth. Thus the lower density of this firn is better preserved since it is not as affected by the next summer’s higher temperatures. It is shown by Figure 3b–e that the minimum peak densities of each seasonal cycle decrease from approximately 470 to 420 kgm^–3 at 10m with increasing accumulation rate from 100 to 900 kgm^–2 a^–1. As a consequence, the density variation increases with increasing accumulation rate.

Fig. 3. Modeled density profiles (thin solid lines) in high-temperature conditions with various accumulation rates: (a) 50, (b) 100, (c) 200, (d) 400 and (e) 900 kgm^–2 a^–1. Thicker solid lines show the density profiles without the consideration of vapor flow (i.e. R_v = 0). Surface temperatures with annual mean –30°Cand maximum –2°Care applied for all model runs. The range of accumulation for the accumulation area of the Greenland ice sheet is also indicated, showing increasing amplitude in the density variation with increasing accumulation rate.

The major cause of the large variability in density in the low-temperature, low-accumulation case (e.g. Fig. 2a) is the time of exposure to temperature gradients. This period depends upon the accumulation rate. At low-accumulation sites, each deposited firn layer remains near the surface and exposed to the extreme temperature gradient for a longer period. The importance of the temperature gradient thus weakens with increasing accumulation rate. Figure 4 shows the calculated temperature-gradient profiles for the two extreme cases (Fig. 2a and 3e). Comparing these two cases, although the accumulation rate and surface temperature are extremely different, the temperature-gradient profiles do not show large differences. As expected, the magnitude of the temperature gradient quickly declines with increasing depth from approximately 12–15Km^–1 near the surface to approximately 1–2Km^–1 around 6 m, then further reduces gradually towards zero at depths 410 m. However the temperature-gradient history experienced by a firn layer varies greatly at different accumulation levels. Figure 5 presents the history of a firn layer since its deposition for three selected accumulation rates (24, 100 and 900 kgm^–2 a^–1). The input surface temperature is taken as for Figure 2. Each curve (a, b, c) in Figure 5 indicates the magnitude of the temperature gradient at the corresponding accumulation level. Clearly, firn at a lower-accumulation site (e.g. curve a) experiences larger temperature gradients for a longer amount of time simply because it travels to shallower depth in comparison with the high-accumulation sites (e.g. curve c).

Fig. 4. Calculated temperature-gradient profiles for the two extreme surface conditions. The accumulation rateA (kgm^–2 a^–1), annual mean temperature Tm (°C) and maximum temperature Tmax (°C) used for the model inputs are indicated in the diagram.

Fig. 5. Modeled magnitude of the temperature-gradient profiles for three selected accumulation rates, showing the decreasing magnitude of temperature gradient experienced by a firn layer with increasing accumulation rate. The accumulation rates used are 24, 100 and 900 kgm^–2 a^–1.The input surface temperature is taken as for Figure 2.

Conclusion

The characteristics of density variations observed from inland Antarctica (Fig. 1) are described by our densification model. The time of exposure to temperature gradients at shallow depths determined by the accumulation rate is essential to the density variations. In a low-accumulation, low-temperature region such as inland Antarctica, the temperature gradient has a strong effect on the densification rate and thus the density variation. In higher-accumulation, higher-temperature conditions such as for the Greenland ice sheet, the temperature gradient becomes less important due to the shorter period of exposure. The accumulation rate and high summer temperature dominate the density variation.