Three cylindrical samples (22 ± 0.5 mm diameter; 50 ± 0.5 mm high) were produced at each of three depths of 20, 40 and 60 m from the same 2017 Summit, Greenland firn core that was studied in Li and Baker (2022a). Both the densities and porosities of these above samples are typical of values in the snow-to-ice transition zone as introduced in Section 1. It is important to note that the reduction in effective stress with increasing depth is evident in samples taken from these three specified depths (Appendix A). Before creep testing, one cylindrical firn samples from each depth was stored at a temperature of -5 ± 0.5, -18 ± 0.5, and -30 ± 0.5 °C for two days to achieve thermal equilibrium (Li and Baker, 2022a). It's also important to note that firn is a heterogeneous material that can have variations in layering, fabric, grain size, and impurity concentration across short distances. Thus, care was taken to extract the three replicate samples from the core at each depth as closely as possible to reduce the variability in their initial conditions.

Three home-built creep jigs were placed in individual Styrofoam boxes in three different cold rooms that were held at temperatures of -5 ± 0.5, -18 ± 0.5 and -30 ± 0.5 °C. Each creep jig consists of an aluminum base plate and three polished aluminum-guide rails passing through linear bearings that hold the upper aluminum loading plate (Fig. 1). A linear voltage differential transducer (LVDT-Omega LD-320: resolution of 0.025 %; linearity error of ±0.15 % of full-scale output), parallel to the three aluminum-guide rails, was located adjacent to the center of the upper plate, and fixed firmly using a screw through the plate (Fig. 1) for measuring the displacement during a test. The displacement was logged every 5 s using a Grant SQ2010 datalogger (accuracy of 0.1 %). Temperatures were logged at 300 s time intervals over the entire test period, using a k-type thermocouple (Omega RDXL4SD thermistor: resolution of 0.1 °C) that was mounted inside each box. In this work, specimens were tested at temperatures of -5 ± 0.2, -18 ± 0.2 and -30 ± 0.2 °C from depths of 20 m (applied stress 0.21 MPa), 40 m (0.32 MPa) and 60 m (0.43 MPa). There are smaller error bars for the temperature of the specimens than the room temperature because the creep jigs were in insulated Styrofoam boxes. The stresses were chosen based on experience from previous tests (Li and Baker, 2022a) in order to give measurable creep rates in a reasonable time.

Each specimen at each depth and temperature combination was scanned using a Skyscan 1172 micro-CT, before and after creep testing. Each micro-CT scan lasted ∼ 2 h. The cubic Volume of Interest (VOI, a side length of 8 mm) was taken from near the center of the firn specimen as conducted in Li and Baker (2022a). The microstructural parameters obtained from the micro-CT data are the SSA, the mean structure thickness of the ice matrix (S.Th), the area-equivalent circle diameter of the pores (ECDa), the total porosity (TP), the closed porosity (CP), and the structure model index (SMI). The SSA (mm⁻¹) is the ratio of the ice surface area to total firn volume (ice plus air) in a VOI analytical element, and is calculated using the hexahedral marching cubes algorithm via CTAn software (Wang and Baker, 2013). It characterizes the thickness and complexity of the firn microstructure. Changes in SSA indicate a change in free energy of the ice surfaces, the decrease of which represents the occurrence of sintering-pressure. The S.Th (mm) is the mean structure thickness of an ice matrix (Hildebrand and Ruegsegger, 1997), which represents the characteristic size of an ice particle in the firn, where the ice particle consists of one or many crystals or grains. It is measured based on the largest sphere diameter that encloses a point in the ice matrix and is completely bounded within solid surfaces. The ECDa (mm) is the diameter of a circle having the same area as the average for all pores in the VOI from the 2-D binary images, indicative of the characteristic size for the void space (Adolph and Albert, 2014). The TP (%) is the ratio of the pore volume, including both open and closed pores, to the total VOI. The CP (%) is the ratio of the volume of the closed pores to the total volume of solid plus closed pores volume in a VOI, while the open porosity (%) is the ratio of the volume of the open pores to the total VOI. The SMI is calculated based on the dilation of a 3-D voxel model (Hildebrand and Ruegsegger, 1997) SMI=6S′×V/S2, where S′ is the change in the surface area due to dilation, and V and S are the object volume and surface area, respectively. It indicates the prevalent ice curvature, negative values of which represent a concave surface, e.g. the hollow air structure surrounded by an ice matrix. The more negative the SMI value, the more spherical the pore. Notably, the micro-CT-derived density of each specimen agrees well with the bulk density measured using the mass-volume approach (Li and Baker, 2021).

Thin sections for optical photographs before and after creep testing were cut from bulk specimens, one side of which was first smoothed with a microtome. This side was then frozen onto a glass plate (100 × 60 × 2 mm) by dropping supercooled gas-free water along its edges. Its thickness was reduced to ∼ 2 mm by a band saw, and finally thinned further to a uniform thickness of ∼ 0.5 mm using a microtome. Images were captured using a digital camera after each thin section was placed on a light table between a pair of crossed polarizing sheets.

Increasing firn density with increasing depth from either of the -5, -18, and -30 °C specimens can be readily recognized by visual inspection of the micro-CT 3-D reconstructions of the firn microstructure (Fig. 2). Correspondingly, the microstructural parameters, with the exception of the CP, changed monotonically with increasing depth at each temperature, e.g. the -30 °C samples increased in density from 591 ± 1.4, to 683 ± 4.2, to 782 ± 1.5 kg m⁻³, decreased in SSA from 4.64 ± 0.04, to 3.3 ± 0.06, to 2.39 ± 0.01 mm⁻¹, and decreased in TP from 35.6 ± 0.05 %, 25.6 ± 0.4 %, to 14.8 ± 0.2 % at 20, 40, and 60 m, respectively (Table 1). These above changes are similar to those previously observed in this firn core (Li and Baker, 2022a), implying that the sintering-pressure mechanism plays a crucial role in the densification of polar firn due to the increasing overburden of snow and firn with increasing depth. However, the microstructures of the samples from the three temperatures at each depth show little variability and do not monotonically change with temperature, e.g. at 20 m depth the -5, -18, and -30 °C samples having densities of 589 ± 1.3, 615 ± 2.5, and 591 ± 1.4 kg m⁻³, and SSAs of 4.74 ± 0.03, 4.51 ± 0.04, and 4.64 ± 0.04 mm⁻¹, respectively (Figs. 2–3; Table 1). Here, despite the -18 °C specimen having a higher density than the two others at -5 and -30 °C, is not possible to conclude that the sintering of firn is not directly related to the temperature. This is likely because a thermal equilibration period of two days in the absence of compression is too short to sufficiently exert the influence of temperature on firn sintering. The microstructural differences seen in these specimens more likely arose from the initial samples themselves, which were anisotropic and heterogeneous even if taken from the same depth, attributed to firn pre-deformation and partial annealing before experiments (Li and Baker, 2022a).

The microstructural evolution is characterized by the microstructural parameters shown in Fig. 3. The largest changes occurred in the -5 °C specimens due to the higher temperature, i.e., the density, S.Th, and CP increased, while the ECDa, TP, SSA, and SMI decreased, indicative of consolidation of the firn after creep. It is important to note that for the 60 m sample tested at -5 °C, there was no change in density, i.e., 790.2 ± 1 kg m⁻³ before creep vs. 790.7 ± 0.9 kg m⁻³ after creep, or TP, i.e., 14.0 ± 0.1 % before creep vs. 13.9 ± 0.1 % after creep. This lack of microstructural change is due to the high initial density, which was close to the firn pore close-off density of ∼ 830 kg m⁻³. Thus, the creep of this sample may involve a transition from firn to bubbly ice, as is also indicated by the increase in CP, which would have made it difficult to compress further. Intriguingly, some of the changes in microstructure observed in the micro-CT 3-D reconstructions from the specimens before and after creep, e.g. the distribution of ice-space, are indistinguishable in Fig. 2. This is presumably due to the relatively large initial particle size, or from radial dilation exceeding the axial compression because of the small strains that occurred at relatively low temperatures.

One exception to the expected microstructural change after creep was the decrease of CP, which was likely due to the measurement uncertainty of the micro-CT (Burr et al., 2018), or radial expansion of the specimen during creep. Another exception was the decrease in density after creep for the -18 °C specimen at 20 m and the -30 °C specimen at 60 m, which arose due to a de-densification effect produced by temperature gradient metamorphism, as confirmed by the increase of both TP and S.Th (Li and Baker, 2022b). The thermal gradient appears to be associated with a fluctuation of 0.2 °C around the test temperature, similar to temperature cycling occurred within firn (Mellor and Testa, 1969; Weertman, 1985), which stems from the thermometer's inherent accuracy as noted in Sect. 2.2 (-5 ± 0.2, -18 ± 0.2 and -30 ± 0.2 °C). In the relatively simple deformation found at ice-sheet dome sites, such as Summit, there is no mechanism to decrease density during compression. At sites closer to the ice sheet margins, cracking due to extension of the ice may cause a localized decrease in density. The rate of firn densification should decrease with increasing depth at a given temperature, due to the decrease of effective stress with increasing depth (Appendix A). As a matter of fact, the density of the –5 °C samples after creep increased by 32, 44, and 0.5 kg m⁻³ for the 20, 40, and 60 m samples, respectively. The 44 kg m⁻³ unexpectedly outnumbers the 32 kg m⁻³, implying that the densification of firn is also affected by other undetermined factors, e.g. the effect of inclusions, in addition to the stress and temperature.

Another way to investigate microstructure changes before and after creep tests is to compare their grain sizes using thin sections. As an example, Fig. 4 shows optical micrographs of thin sections made from the -5 °C sample at 40 m before and after creep to a strain of 19.3 %, where the significant reduction in grain size from 0.8 ± 0.67 to 0.5 ± 0.32 mm implies the occurrence of recrystallization during testing. However, it is also unclear at what strain recrystallization was initiated in each test, as noted in Li and Baker (2022a). Recrystallization occurs frequently at a temperature higher than the homologous temperatures of 0.9 Tm. However, no evidence was found for recrystallization after testing at the relatively cold -18 and -30 °C conditions, probably due to the small creep strains at these relatively low temperatures. The creep mechanisms for these samples, and whether the mechanisms were different at different temperatures, could not be determined from the micro-CT-derived microstructural observations alone, because the micro-CT can only capture the microstructure before and after creep. Instead, plots of both strain vs. time and strain rate vs. strain can be used to elucidate the onset of recrystallization during creep (Sect. 3.3 and 3.4; Ogunmolasuyi et al., 2023).

Figure 5 shows the strain vs. time creep curves. The specimens at -5 °C at 20 m and -18 °C at 20, 40, and 60 m, show decelerating transient creep and quasi-viscous steady-state creep, while the specimens at -5 °C at 40 and 60 m show transient, secondary, and accelerating tertiary creep. Note that the curves from the -30 °C specimens are not easily interpreted due to a large amount of noise arising from both the insufficient resolution of a linear voltage differential transducer (Li and Baker, 2022a) and the very small strains. The transient creep stage may be caused by strain hardening that occurs from the yield point to the ultimate strength (Glen, 1955; Jacka, 1984). The plastic deformation is accommodated by an increase in dislocation density through dislocation multiplication or the formation of new dislocations (Frost and Ashby, 1982; Duval et al., 1983; Ashby and Duval, 1985), which leads to an increase of the firn strength as the dislocations become pinned or tangled, and thus more difficult to move. The initial decrease of creep rate may also be related to the rearrangement of dislocations into a more stable pattern through a dragging mechanism (Weertman, 1983) for the -5 °C specimens. The tertiary creep stage may be associated with strain softening deriving either from the thermally-activated processes at the high homologous temperature approaching the melting point of ice, or from recrystallization (Li and Baker, 2022a). Clearly, the creep rate of firn is sensitive to temperature under constant stress at a given depth, viz., the creep rate increases with increasing temperature (Fig. 5). Incidentally, there is no evidence of the onset of recrystallization in the creep curves themselves despite the thin-section observation that -5 °C specimens clearly underwent recrystallization during creep (Sect. 3.2).

A modified Andrade-like equation ε=βtk+ε0 in Li and Baker (2022a) was used to describe the transient creep behavior of the firn, in which the primary creep was well represented in black dashed lines on the creep curves in Fig. 5. The time exponent k, derived from the above equation, ranges from 0.34–0.69: the data for the -30 °C specimens are excluded since the noise in the results makes them uninterpretable. These k values are also smaller than those from monocrystalline and bicrystalline ice: 1.9 ± 0.5, 1.5 ± 0.2, and 1.3 ± 0.4 (Li and Baker, 2022a and references therein). We also note that the k values from the specimens at -5 °C from 20–60 m (0.68, 0.61, and 0.69), and at -18 °C from 40 m (0.49) are greater than 0.33, while the k value from the -18 °C specimens at 20 m (0.34) and 60 m (0.34) are close to 0.33 that is usually obtained for full-density polycrystal ice. Interestingly, an evident relationship between the density of firn and the k values, regardless of the effect of stress (Li and Baker, 2022a) and temperature, remains unknown. A greater k value signifies faster deformation. The k values derived for firn are generally higher than those for polycrystalline ice, implying that the higher firn deformation rates compared to those of ice (k= ∼ 0.33; Cuffey and Paterson, 2010, and references therein) are likely related to the fewer grain-boundary constraints with more void space in firn (Li and Baker, 2022a; Li et al., 2023). Clearly, the above k values, which increased with increasing temperature (Fig. 5), indicate that deformation is easier because of the lower viscosity at the higher temperature. Thus, k seems to be a state variable with respect to temperature. In addition, k values greater than 0.33 may be related to the decrease of viscosity of the firn specimens (Freitag et al., 2002; Fujita et al., 2014). k values lower than 0.33 observed under constant load and temperature occurred at relatively low effective stresses (Li and Baker, 2022a). The identified trend of steadily declining k values across the temperature range of -5 to -18 °C, however, represents a significant gap in our current understanding, necessitating a dedicated investigation into the microstructural or metamorphic causes. Alternatively, the enhanced cohesion strength in the firn, which resulted from both the ice matrix with higher purity and the stronger bond connection of inter-grains, increases the viscosity of test samples and lowers the k value to less than 0.33.

Figure 6 shows log strain rate vs. strain plots from all the -5 and -18 °C specimens; the -30 °C samples are excluded due to noise. The evolution of the strain rate is characterized more clearly in Fig. 6 than in Fig. 5. Clearly, the strain rate is also a state variable of temperature, where the strain rate increases with increasing temperature for a given strain at a given depth (Fig. 6; Table 2).

The strain rate minimum at the secondary creep stage (SRmin) and the strain at the SRmin for all the -5 and -18 °C specimens are shown in Fig. 6 and Table 2. The SRmin was reached at a strain of 11.8 %, 7.5 % and 2.7 % for the -5 °C specimens from depths of 20, 40, and 60 m, respectively, consistent with strains at the SRmin decreasing with increasing depth at a given temperature in Fig. 9 and Table 4 in Li and Baker (2022a). For the –18 °C specimens, the SRmin occurred over a range of strains from 1.81 %–2.9 % at 20 m, at a fixed strain of 4.1 % at 40 m, and at a strain oscillating between 1.1 % and 1.8 % at 60 m. These values of strain at different SRmin values are different from those usually observed at strains of 0.5 %–3 % for fully-dense ice (Cuffey and Paterson, 2010, and references therein), implying different mechanical behavior between firn and pure ice (Duval, 1981; Mellor and Smith, 1966; Jacka, 1984; Li et al., 1996; Jacka and Li, 2000; Song et al., 2005, 2008; Cuffey and Paterson, 2010). Overall, the strain at the SRmin is greater with lower density and higher temperature, e.g. 11.8 % strain from the -5 °C specimens at 20 m, and 4.1 % strain from the -18 °C specimens at 40 m. This is likely due to the effect of strain hardening on density and temperature (Li, 2023). Additionally, tertiary creep is observed during both quasi-steady state deformation, particularly in the -5 °C specimens at depths of 40 and 60 m, and in the ascending stage, as seen in the -5 and -18 °C specimens at 20 m, along with the -18 °C specimen at 40 m. This mechanical behavior is facilitated by lower firn density, increased effective stress, and elevated creep temperatures. For instance, in the -5 °C specimens at 20 m, strain softening primarily results from recrystallization (Duval, 1981; Jacka, 1984; Jacka and Li, 2000; Song et al., 2005; Faria et al., 2014). Also, the activation of easy slip systems contributes to this process (Jonas and Muller, 1969; Duval and Montagnat, 2002; Alley et al., 2005; Horhold et al., 2012; Fujita et al., 2014; Eichler et al., 2017; Vedrine et al., 2025). It is noteworthy that Jacka and Li (1994) observed that steady-state tertiary ice creep, which is marked by stable grain size, is influenced more by applied stresses than by temperature. This finding suggests that there exists a balance between the activation energies required for grain growth and subdivision at a specific temperature.

Experimental observations of the SRmin are limited, as they only occurred for the -5 and at -18 °C specimens at each depth (Table 2). It is hard to achieve the SRmin for all firn specimens in laboratory environments (Landauer, 1958), especially under low temperatures and stresses such as those from the -30 °C specimens in this work. To this end, we offer the various possibilities of the SRmin using the evidence we have. The value of the apparent activation energy of creep, Qc (kJ mol⁻¹), is equal to the slope of a line fitted ln⁡ε˙ versus 1/T as did in Goldsby and Kohlstedt (1997, 2001), using the Arrhenius relation ε˙=Bσnexp⁡-QcRT, where ε˙ (s⁻¹) is the strain rate, B (s⁻¹ Pa-n) is the material parameter, σ (MPa) is the applied stress, n is the creep (stress) exponent, R (8.314 J mol⁻¹ K⁻¹) is the gas constant, and T (K) is Kelvin temperature. It is noted that under constant-stress conditions, the value of the stress exponent n influences the pre-exponential factor B but does not affect the slope of the Arrhenius plot and therefore the derived activation energy Qc. First, the estimation of Qc is based on only two SRmin values from the -5 and -18 °C samples at each depth (Table 2). Glen-King's model ε˙=Aexp⁡(-Qc/RT)=Bσnexp⁡(-Qc/RT) treats the pre-factor A, material parameter B, and stress exponent n as constants (Glen, 1955; Goldsby and Kohlstedt, 2001). This simplification is valid by using the unifying concept of normalized effective stress. The effective stress captures the complex multi-physical behavior of the two-phase ice-air system, accounting for: (1) The incompressibility of individual ice grains versus the compressibility of the porous ice skeleton, (2) The coupled flow of ice and air; and (3) The interplay between different strain components (axial, radial, volumetric, and true). This framework is grounded in the principles of poromechanics, originally developed for soils and later applied to snow and ice (Gubler, 1978; Hansen and Brown, 1988; Mahajan and Brown, 1993; Chen and Chen, 1997; Lade and De Boer, 1997; Ehlers, 2002; Khalili et al., 2004; Gray and Schrefler, 2007; da Silva et al., 2008; Nuth and Laloui, 2008). The variability in density for the samples from 20 m depth on the mechanical behavior are negligible due to a small difference (up to ∼ 4 %), between samples, which falls within an acceptable error range in previous studies. This is likely related to multiple factors, including the intrinsic properties of the samples, e.g. inclusions (impurities, dust, bubbles, clathrate hydrates), the effects of deformation and partial annealing of firn due to stress distribution and temperature changes during drilling, extraction, transportation, or storage, and the fact that the samples are taken from adjacent parts of the core, and might capture heterogeneous density layers, as well as potential measurement errors associated with the equipment used. The Qc values from the 20, 40, and 60 m specimens were calculated to be 61.4, 87.3, and 102.8 kJ mol⁻¹, respectively (Fig. 7). Based on the three SRmin from the -5 and -18 °C samples at 60 m in this work, and from -10 °C samples at 60 m in Li and Baker (2022a), a Qc value for the 60 m specimen was calculated to be 100.7 kJ mol⁻¹. To see whether or not these above Qc values are reliable, we estimated the activation energy of grain-boundary diffusion/viscosity, Qgbd (kJ mol-1), using the relation K=Dt2-D02/t=kexp⁡(-Qgbd/RT), in an alternative form of Qgbd=-R∂ln⁡K/∂1/T, where K is the observed rate of grain growth (mm² a⁻¹), D02 and Dt2 are the measured mean grain area (mm²) in a firn sample at the onset of the creep (t= 0), and at the end time of the creep (t-year), andk is a constant grain growth factor. The grain growth rates are plotted on a logarithmic scale against the reciprocal of T (Fig. 7). For changes in grain size from the related specimens before and after creep see Table 3. Correspondingly, the Qgbd values calculated were 41.4, 40.8, and 40.9 kJ mol⁻¹ for the specimens at 20, 40, and 60 m, respectively. These Qgbd values are comparable to the values of 40.6 kJ mol⁻¹ obtained in laboratory experiments on polycrystalline ice (Jumawan, 1972), and 42.4 kJ mol⁻¹ from 13 polar firn cores (Cuffey and Paterson, 2010) for grain-boundary self-diffusion of polycrystalline ice. Further, the ratio of Qgbd/Qc is 0.67, 0.47, and 0.4 for the 20, 40, and 60 m specimens, respectively. We noted that the ratio of 0.67 for Qgbd/Qc was recommended by Hobbs (1974) and Cuffey and Paterson (2010). The Qc values calculated using the Arrhenius relation for the 40 and 60 m specimens are likely greater than the actual values, and hence are seemingly less reliable. There is little difference between the two-SRmin-derived Qc value (102.8 kJ mol⁻¹) and the three-SRmin-derived Qc value (100.7 kJ mol⁻¹), implying that these two avenues for calculating Qchave equal utility. Moreover, the above Qgbd values are lower than the 48.6 kJ mol⁻¹ that was inferred by the grain growth rate for firn samples with densities ranging from 320–650 kg m⁻³ from cores drilled at the South Pole, Antarctic (Gow, 1969), which makes a ratio of 0.67 for Qgbd/Qc an unreliable sole-criterion. In short, it is difficult to assess the reliability of both Qc and Qgbd, as discussed above due to their scatter and debates in the current literature. Thus, these Qc values estimated in this work, ranging from 61.4–102.8 kJ mol⁻¹, are reasonable, aligning with the literature range of 44.8–113 kJ mol⁻¹ (Table 4).

A great challenge is the estimation of the Qc using the SRmin including the -30 °C specimens, whose SRmin shows high variability due to the extraordinarily slow strain rate at low temperatures. This difficulty cannot be resolved by extrapolating experimental data (Sinha, 1978; Hooke et al., 1980), e.g. the use of Andrade's law (Glen, 1955). Instead, we turned our focus to studying the relationship between the SRmin and temperature by constraining our data in a wide range of Qc values reported in existing literature presented in Table 4. Clearly, there is a larger scatter of Qc values for firn than for ice. The increase of Qc from mono-crystalline and bi-crystalline to polycrystalline ice implies that the greater the reduction in the constraint from grain boundaries, the greater is Qc. Alternatively, firn creep is easier than that of polycrystalline ice due to either the easier sliding of grains in firn along more directions in the more porous and heterogeneous structure (Sect. 3.3), or the decrease of viscosity associated with inclusions (e.g. Baker and Gerberich, 1979; Goodman et al., 1981) that facilitate the intra- and inter-grain sliding (Salamatin et al., 2009). In principle, Qc of firn should exceed that for polycrystalline ice. Intriguingly, some reported Qc values from firn are less than that for ice, meaning the degree of spatial freedom in the ice-matrix is limited by the topological structure of the firn (Liu et al., 2022). Incidentally, the effective stress of porous materials is determined by not only its porosity, but also other factors, e.g. the microstructural topology (Liu et al., 2022) and the impurity types and concentrations in the firn. However, this issue is beyond the scope of this work. In seeking a conclusion, we evaluated the dependence of creep activation energy on firn density. The data indicate no discernible relationship between these two parameters (Fig. 8). In summary, a Qc for firn, which ranges from 44.8–113 kJ mol⁻¹, is plausible due to the intrinsic nature of natural firn that has a far more complicated and changeable microstructure than ice.

The value of the stress exponent n is determined by plotting the line fit the logarithm relation of the steady-state strain rate, ε˙, versus the effective stress, σ, and is, thus, the slope of this line from the measured SRmins (Table 2). We determined stress exponent (n) values of approximately 0.1 and -1.2 for the -5 and -18° C samples based on observed data, respectively. This result directly contradicts the value of n≈ 4.3 reported from the same Greenland firn core by Li and Baker (2022a). Further, these values fall entirely outside the established range of ∼ 1 to ∼ 7.5 (mean ∼ 4.25 ± 3.25) documented across decades of ice mechanics literature (Glen, 1955; Hansen and Landauer, 1958; Butkovich and Landauer, 1960; Kamb, 1961; Paterson and Savage, 1963; Higashi et al., 1965; Mellor and Testa, 1969; Raymond, 1973; Hooke, 1981; Thomas et al., 1980; Duval et al., 1983; Weertman, 1983,1985; Azuma and Higashi, 1984; Pimienta and Duval, 1987; Budd and Jacka, 1989; Jacka and Li, 1994; Goldsby and Kohlstedt, 2001; Bindschadler et al., 2003; Cuffey, 2006; Chandler et al., 2008; Cuffey and Kavanaugh, 2011; McCarthy et al., 2017; Millstein et al., 2022; Colgan et al., 2023; Li, 2025). The wide range of reported n-values is governed by a complex interplay of deformation mechanisms – including grain boundary sliding, diffusion (lattice and grain boundary), and dislocation processes, e.g. hard-slip-dominated, dislocation-accommodated grain boundary sliding, and grain boundary sliding-limited basal dislocation – across varying stresses, temperatures, crystalographic fabrics, impurity contents, and grain-size-to-sample-size ratios. We attribute the significant discrepancy in these findings to the experimental conditions. The lower temperatures used (down to -30 °C) induce slower strain rates, which prevented the tests from reaching a critical strain rate minimum (SRMin). Therefore, to accurately estimate the activation energy for deformation, it is necessary first to calibrate the SRMin value for all noised samples. A constant stress exponent value of n≈ 4.3 (Li and Baker, 2022a) was used to compute the activation energy. This necessary simplification – an acknowledgement of current methodological limitations rather than a dismissal of the underlying physics – introduces a key uncertainty that highlights the need for future advancements in observational methodology within firn research. To proceed, the post-calibration SRmins for the -5 and -18 °C samples are highlighted in Table 2 (see Appendix B in detail). It is important to note that the stress exponent does not depend on the density of the tested samples, thereby negating any basis for discussing a relationship between the stress exponent and sample density. Instead, variations in stress corresponding to density variations are manifested in the strain rate, ensuring that the derivation of the stress exponent and activation energy remains consistent. From here on we only discuss the applied stress since there is little difference between the effective stress and applied stress for calculating the stress exponent (Li and Baker, 2022a). Based on both the reported range of Qc and the two observed SRmins at -5 and -18 °C, the SRmins for the -30 °C samples are inferred (Table 2), using the Arrhenius relation. Also, based on both the observed and inferred SRmins with the upper and lower bounds (Table 2), a series of fitted functions are then found between the SRmin and the reciprocal of the temperature (°C), 1/Tc: 20 m samples: 1SRMin=-3×10-5/Tc-7×10-7[R2=0.988;PC1(L20)]SRMin=-3×10-5/Tc-2×10-7[R2=1;PC1(U20)]SRMin=-1×10-5/Tc-3×10-7[R2=1;PC2(L20)]SRMin=-9×10-6/Tc-2×10-7[R2=0.987;PC2(U20)]SRMin=-2×10-6/Tc-6×10-8[R2=0.998;PC3(L20)]SRMin=-1×10-6/Tc-3×10-8[R2=0.976;PC3(U20)], 40 m samples: 2SRMin=-2×10-4/Tc-4×10-6[R2=0.988;PC1(L40)]SRMin=-2×10-4/Tc-2×10-6[R2=1;PC1(U40)]SRMin=-6×10-5/Tc-2×10-6[R2=1;PC2(L40)]SRMin=-6×10-5/Tc-1×10-6[R2=0.987;PC2(U40)]SRMin=-1×10-5/Tc-3×10-7[R2=0.998;PC3(L40)]SRMin=-9×10-6/Tc-2×10-7[R2=0.976;PC3(U40)], 60 m samples: 3SRMin=-7×10-4/Tc-2×10-5[R2=0.988;PC1(L60)]SRMin=-6×10-4/Tc-6×10-6[R2=1;PC1(U60)]SRMin=-2×10-4/Tc-7×10-6[R2=1;PC2(L60)]SRMin=-2×10-4/Tc-4×10-6[R2=0.987;PC2(U60)]SRMin=-3×10-5/Tc-1×10-6[R2=0.998;PC3(L60)]SRMin=-3×10-5/Tc-7×10-7[R2=0.976;PC3(U60)], where PC1(L20) and PC1(U20) indicate the lower and upper bound values of the post-calibration SRmins from the 20 m samples (Table 1), and other symbols are similarly formatted, e.g. PC1(L40), PC1(U40), PC1(L60), PC1(U60), and so on. These relationships are plotted in Fig. 9, where the SRmin vs. 1/Tc plots from the three depths are almost the same shape, implying that the SRmin is dependent on the temperature at a constant stress. It is important to note that the average (minimum) strain rate for the secondary creep stage for a given temperature increases with increasing depth/density of the samples (Fig. 9; Table 2). This is opposite to a decrease of the SRmin at a fixed stress and temperature in Fig. 9 and Table 4 in Li and Baker (2022a). These changes in SRmin are irrespective of the stress (Appendix A). The temperature plays a predominant role during firn creep for a given density of sample at a constant stress. An interesting question on firn creep at a specific temperature is whether the SRmin slows down or speeds up with decreasing density of firn. Certainly, natural firn samples raise the complexity in interpreting the firn creep due to the influences both from inclusions (Li and Baker, 2022a and references therein; Li, 2022), and from the topology of the microstructures (Liu et al., 2022). In addition, there is a broad spread of the SRmin at each depth, in which the SRmin varies by several times, even one order of magnitude or more between the different possibilities of post-calibration SRmins (Fig. 9), implying that the microstructure of the sample significantly influences the process of the creep of firn. Moreover, it is hard to generalize a universal formula for predicting the SRmin at temperatures below -30 °C, where the SRmins becomes negative (Fig. 9). Thus, there is a need for an in-depth understanding of the polar firn creep behavior in secondary creep stage.

To illustrate the differences between the Qc values calculated from PC1-SRmin, PC2-SRmin, and PC3-SRmin, we have plotted them in Fig. 7. Interestingly, the Arrhenius plots of the natural logarithm of strain rate with 1000/T (Fig. 7) are similar to those observed by Glen (1955) and Homer and Glen (1978), implying that there is no significant difference in the creep mechanism for a temperature range of -30 to -5 °C (Glen, 1955; Homer and Glen, 1978), where both diffusion via grain-boundary, vacancy or interstitial defects (Barnes et al., 1971; Brown and George, 1996; Nasello et al., 2005; Li and Baker, 2022b), and dislocations contribute to the creep of polar firn.