Following Alley et al. (1986), we assume that grain growth can be described by a relationship of the form 2dp-dop=Kt, where K follows an Arrhenius relation: 3K=Kggexp⁡-QggRT. In these equations do is an initial grain size, p is the grain growth exponent, t is time, Kgg is the grain growth constant, Qgg is the activation enthalpy for grain growth, R is the universal gas constant, and T is temperature. Substituting Eq. (3) into Eq. (2) and differentiating with respect to time allows us to write an expression for the rate of grain growth: 4d˙gg=p-1d1-pKggexp⁡-QggRT. Equation (4) provides a general expression for grain growth; however, the values of the grain growth parameters p, Kgg, and Qgg are not well constrained in natural systems and depend on the presence of microparticles, bubbles, and/or other impurities in the ice (e.g., Alley et al., 1986). In Sect. 2.5 we will describe our approach for estimating these parameters using a combination of laboratory and ice core data.

The wattmeter posits that the rate of grain size reduction d˙red is controlled by the rate of mechanical work and the rate at which this work is dissipated (Austin and Evans, 2007, 2009; Bercovici and Ricard, 2012). Specifically, the rate of mechanical work per unit volume, W˙tot, is defined as 5W˙tot=σε˙, where σ is von Mises equivalent stress and ε˙ is strain rate (assuming that the rate of stress change is negligible over the timescale of grain size evolution). This work rate must be balanced by the rate at which the internal energy of the system, E˙int, increases plus the rate at which energy is dissipated, θ˙irr: 6W˙tot=σε˙=E˙int+θ˙irr. The increase in internal energy can be related to the increase in grain boundary area: 7E˙int=-cγd2d˙red, where γ is the grain boundary energy and c is a geometrical factor (π for spherical grains). The rate of dissipation in Eq. (6) is related to the fraction, λ, of the total work rate that is responsible for increases in internal energy: 8θ˙irr=1-λσε˙. Here we note a difference in the application of the wattmeter to ice compared to crustal and mantle rocks. In most terrestrial minerals, the two primary creep mechanisms are diffusion and dislocation creep. Because grain growth during diffusion creep was shown to be the same as that during static conditions (Karato et al., 1986), the work done by diffusion creep is assumed to be completely dissipated (i.e., λdiff=0), and only dislocation creep leads to grain size reduction. By contrast, under Earth-like pressure and temperature conditions, ice deformation proceeds primarily by a combination of GBS and dislocation creep (Goldsby and Kohlstedt, 2001). Because some fraction of the work done by both GBS and dislocation creep will lead to grain size reduction (i.e., λdisl & λGBS>0), the dissipation rate can be rewritten as 9θ˙irr=1-β1-λGBSσε˙+β1-λdislσε˙, where 10β=W˙dislW˙tot. Here we assume that the total work rate can be expressed as the sum of the contributions from the individual deformation mechanisms: 11W˙tot=W˙disl+W˙GBS=σε˙disl+σε˙GBS. Substituting Eqs. (7) and (9) into Eq. (6) we derive an expression relating the rate of grain size reduction to the total work rate: 12d˙red=λGBS-βλGBS+βλdisld2-cγσε˙ The final grain size evolution equation can then be assembled from Eqs. (1), (4), and (12): 13d˙tot=p-1d1-pKggexp⁡-QggRT-λGBS-βλGBS+βλdisld2cγσε˙. It is often useful to define a steady-state grain size, dss, which occurs when d˙tot=0: 14dss1+p=Kggexp⁡-QggRTp-1cγλGBS-βλGBS+βλdislσε˙. The concept of a steady-state grain size is analogous to that derived by Faria et al. (2014a), with the exception that they assumed creep was governed exclusively by the Glen law and grain size was related to stress only (e.g., Jacka and Li, 1994) rather than to the work rate (Eq. 6). In practice the steady-state grain size may not be achieved if there is insufficient time for grains to fully evolve to be in equilibrium with the surrounding deformation conditions. In these situations, Eq. (13) must be solved and coupled with the governing equations and constitutive relationships.

The values of λGBS and λdisl are uncertain and have not been determined independently. Therefore, for simplicity we assume that λ=λGBS=λdisl, thereby lumping the effects of grain boundary energy, grain geometry, and λ into a single “scaling factor” in Eq. (14) (Austin and Evans, 2009). In the following sections, we vary λ to elucidate the behavior of the model with respect to variations in any of these three parameters. We emphasize that the wattmeter models the rate of change in the internal energy and relates this to the grain size reduction rate (and thus increase in internal energy owing to increase in grain boundary area). A key assumption is that the rate of change in grain size is greater than the rate of change in stress – thus the dislocation density can be considered constant for a given stress (Austin and Evans, 2007, 2009).

Finally, we note that the wattmeter approximates mean grain size, d, as the diameter of a circular grain of ice. Comparing these theoretical values to grain sizes in natural systems can be challenging because grains are irregular and are typically measured in a 2-D cross section (e.g., thin section) through a 3-D sample. In our comparisons to data below, natural grain sizes were estimated using the line intercept technique of Alley and Woods (1996). In this approach, the average distance between grain boundaries along a series of lines through a sample is measured and then scaled by a correction factor of the order of 1 (1.5 for circular grains; Gifkens, 1970) in order to account for the fact that when making a thin section many grains are cut near their edge as opposed to near their center (Gow, 1969). Further, because this approach is also used in the measurement of grain sizes in the derivation of the flow laws (Goldsby and Kohlstedt, 2001), it allows us to compare our calculated grain sizes to ice core data in a self-consistent manner.

To apply the grain size evolution model defined by Eq. (13) to natural systems, we calculate the relative rates of deformation by GBS and dislocation creep. To do so, we formulate a two-mechanism composite flow law that contains additive contributions from each creep mechanism of the form 15ε˙=ε˙GBS+ε˙disl. This composite law has been used to model the rheology of ice satellites (Barr and McKinnon, 2007) and the relative contribution of GBS and dislocation creep in ice sheets (Kuiper et al., 2020). Here the creep mechanisms are assumed to be independent and each term on the right-hand-side of Eq. (15) is expressed as a flow law of the general form 16ε˙i=Aid-miσniexp⁡-QiRT, where Ai is a material constant, mi is the grain size exponent for creep, ni is the stress exponent, and Qi is the activation energy. The subscript i denotes the parameters that depend on the deformation mechanism (e.g., GBS or dislocation creep). We note that Goldsby and Kohlstedt (2001) presented a more complicated composite law that includes a term for creep limited by basal dislocation slip and also a theoretical flow law for diffusion creep. However, extrapolations to grain sizes typical of glaciers and ice sheets demonstrate that neither of these additional creep mechanisms are likely to be important for the flow of terrestrial ice bodies. A list of flow law parameters required to extrapolate Eq. (16) to the full temperature range (up to the melting point) is given in Goldsby and Kohlstedt (2001).

To solve for grain size evolution in ice, we consider two scenarios: (1) deformation in a shear zone under an imposed velocity contrast Vsz and (2) deformation in a 1-D vertical column of ice with an assumed surface slope, α. For the case of a shear zone with no along-strike pressure and/or viscosity gradients, the shear stress, τ, will be constant and a function of only the viscosity and velocity contrast: 17τ=η∂v∂w, where η is the viscosity, v is the velocity parallel to the shear zone, and w is the direction perpendicular to the strike of the shear zone. Integrating Eq. (17) over the width of the shear zone, wo, allows us to write stress in terms of the imposed velocity: 18τ=Vsz∫0wo1ηdw-1. The viscosity can be reformulated from the flow law (Eq. 16) in terms of the stress: 19ησ=A-1σ1-ndmexp⁡QRT In the case of deformation within a column of ice with a zero-slip basal boundary condition, the shear stress is calculated from the surface slope and increases linearly as a function of depth, z, in the ice sheet: 20τ(z)=ρgH-zsin⁡(α). Here ρ is the density of ice, g is the gravitational acceleration, and H is the thickness of the ice sheet. Note that in practice we relate the shear stress to the von Mises equivalent stress in the wattmeter (Eq. 13) and flow law (Eq. 16) through the square root of the second invariant of the stress tensor, which in this geometry reduces to τ=3σ.

Before using the wattmeter to predict grain sizes in natural systems, we must first constrain the grain growth parameters used in the model as they will directly control the balance between grain growth and grain size reduction. As noted above, grain growth rates in ice are highly sensitive to the presence of impurities, both soluble (e.g., bubbles, ions) and insoluble (e.g., dust/microparticles) (Alley et al., 1986). While the expressions for grain size evolution derived above do not explicitly account for the effects of impurities, we can parameterize their effects through their influence on grain growth. To constrain the grain growth parameters (p, Kgg, and Qgg) in Eq. (4), we turn to a combination of laboratory and ice core data. Azuma et al. (2012) measured grain growth rates in laboratory samples both with and without bubbles and found that the grain growth exponent for bubble-free ice was relatively small (p ∼ 2) but was significantly larger (p ∼ 7–9) in ice containing bubbles (Fig. 2). The increase in the grain growth exponent in the presence of bubbles was interpreted to reflect the role of “impurity drag”.

To investigate the applicability of these experimentally derived grain growth rates to natural systems, we compared them to grain sizes in the shallow portions of the GRIP and GISP2 ice cores where recrystallization rates are expected to be small and the increase in grain size with depth dominantly reflects the rate of grain growth (Gow et al., 1997). We use only grain sizes from the depth range between 150 m (∼ 500 years; taken to represent the depth at which the ice is fully compacted) and 300 m (∼ 1500 years; below which grain sizes no longer increase at a constant rate, indicative of the influence of recrystallization). For comparison with the laboratory data, depth was converted to time for the GRIP and GISP2 cores based on the age models of Dansgaard et al. (1993) and Ram et al. (2000), respectively.

Using experiments conducted at the temperature conditions found between 150–300 m depth in the GRIP and GISP2 ice cores (243 K; Hvidberg et al., 1997), we first refit the Azuma et al. (2012) experimental data for the grain growth parameters p and Kgg using the approach of Bons et al. (2001). We find grain growth exponents in the range of 7.1–8.4 for experiments with bubbles and p = 1.8 for the single experiment without bubbles (red and blue curves in Fig. 2). Extrapolating these parameters to timescales applicable to glaciers and ice sheets (e.g., 103–105 years), we show that (1) the grain growth parameters derived for ice with bubbles provide a significantly better fit to the ice core data compared to the grain growth rates for bubble-free ice (compare red vs. blue curves in Fig. 2), and (2) the parameters derived from experiment AL5 provide the best overall fit to the ice core data. However, there is some variability in the experimental data – possibly reflecting differences in bubble content and/or the difficulty in extrapolating grain growth parameters determined on timescales of hours to days in the laboratory to timescales of thousands of years in natural systems. In an attempt to address these issues, we refit the Azuma et al. (2012) data from all three experiments containing bubbles at 243 K (AL5, AM5, and AS5) jointly with the GRIP ice core data. We do not include the GISP2 data in this fit, as we will calculate grain size as a function of depth throughout the entire GISP2 core in Sect. 3.3 below. The joint fit results in a grain growth exponent p of 6.03 ± 0.25 (solid black line in Fig. 2), slightly less than the values derived from the individual laboratory experiments.

Our goal in fitting the grain growth exponent in this way is to derive an “empirical” p value that, in conjunction with the corresponding values of Kgg and Qgg, fits a wide range of observations and can be applied to natural settings. We note that additional parameters have been shown to influence grain growth. For example, Arena et al. (1997) showed that the presence of pores can be thought of as changing the Kgg value in the grain growth law. Further, the evolution of microstructure during deformation (compared to static grain growth) can result in changes in Kgg (Roessiger et al., 2014). Thus, if Kgg varies with the microstructure (bubble size/bubble topology), and this scales with grain size, then Kgg will be proportional to some function of grain size f(d). In our formulation, we essentially lump all these effects into the empirically fit p value, which is mathematically similar to a Kgg term with a power-law relationship to grain size.

Below we use the grain growth parameters derived exclusively both from the experiment AL5 and from the joint fit between the experimental and ice core data (Table 1) in our application of the wattmeter and discuss the influence of the grain growth exponent on the derived effective stress exponent for creep in ice.

We first use the wattmeter to predict grain size in a steady-state shear zone deforming at a fixed strain rate. This setup is analogous to constant strain rate laboratory experiments, such as those by Piazolo et al. (2013) discussed in the following Section. In this end-member, we calculate steady-state grain size by iterating between Eqs. (14) and (18) and assuming the grain growth parameters from our joint fit of the Azuma et al. (2012) experiments and the GRIP ice core data. In practice, we set an initial shear stress and grain size. Using these values, we calculate viscosity and use Eq. (18) to make a new estimate of the shear stress. Based on our new estimate of shear stress and the corresponding strain rate (calculated from the flow law), we use the wattmeter to calculate an updated steady-state grain size (Eq. 14). These new estimates for stress and grain size are then used to recalculate viscosity, which is in turn fed back in Eq. (18) for the next iteration. We continue to iterate in this manner until the shear stress varies by less than 0.1 %.

The result is an estimate of stress and grain size within the shear zone for any imposed strain rate; Fig. 3a and b show these estimates calculated at temperatures of 240 and 265 K, respectively. As noted above, the dominant deformation mechanism in ice is sensitive to both grain size and stress, with higher stresses and larger grain sizes favoring dislocation creep and lower stresses and smaller grain sizes favoring GBS-limited creep (Fig. 1). We illustrate the transition between dislocation and GBS creep (often referred to as the “field boundary”) using a deformation mechanism map (Fig. 3). Here we assume that a deformation mechanism acting in kinetic parallel with other creep mechanisms is the dominant mechanism if it yields the fastest creep rate. By overlaying the stresses and grain sizes predicted from the wattmeter on deformation maps calculated at the corresponding temperature, we show how variations in strain rate lead to a transition in the dominant deformation mechanism (Fig. 3).

The relationship between grain size and stress predicted by the wattmeter does not change significantly as a function of temperature but has a steeper slope compared to either the field boundary or the piezometer (Jacka and Li, 1994). For example, both the 240 and 265 K shear zones predict a transition from dislocation to GBS-limited creep at a grain size of 0.2–0.3 mm and a stress of 1–2 MPa (Fig. 3). By contrast, the strain rate at which the shear zone is predicted to cross the field boundary varies from 3 × 10-9 to 1 × 10-6 s-1 for temperatures of 240 and 265 K, respectively. These results indicate that when grain size is allowed to vary with the evolving deformation conditions, the dominant deformation mechanism will not be strongly affected by variations in temperature, but the strain rate corresponding to a specific grain size (and stress) will vary due to the Arrhenius behavior of creep (Eq. 16).

We also examine the relationship between stress and strain rate in the shear zone, comparing cases with a fixed grain size to those in which grain size evolves according to the wattmeter (Fig. 4a). Consistent with the laboratory experiments shown in Fig. 1, the fixed grain size calculations show a distinct change in slope corresponding to the transition from a stress exponent of n = 1.8 in the GBS-limited creep regime to a value of n = 4 in the dislocation creep regime (Fig. 4b). By contrast, the wattmeter predicts a more subdued change in slope in the GBS-limited field corresponding to a higher effective stress exponent (∼ 2.5) than the lab-derived value of n = 1.8. At higher strain rates and stresses the wattmeter converges to the dislocation creep stress exponent (Fig. 4b). We discuss the origin of these differences in the effective stress exponent in Sect. 4.1.

Piazolo et al. (2013) investigated grain size changes as a function of strain in a series of experiments conducted at different strain rates. These experiments are ideal for benchmarking and calibrating the wattmeter as we can compare the final grain size to the steady-state value in Eq. (14) and also evaluate the evolution of grain size as a function of time (determined from the strain given an imposed strain rate) using Eq. (13). Here we investigate a series of cases using the grain growth parameters from the joint fit of the Azuma et al. (2012) experiments and the GRIP ice core data, as well as those derived exclusively from experiment AL5 (Fig. 5). Further, we vary the fraction of the total work rate that is responsible for increases in internal energy assuming λ=λGBS=λdisl. Following the experimental setup of Piazolo et al. (2013), we assume an initial grain size of 0.5 mm and use Eq. (13) to calculate grain size as a function of strain for the strain rates used in the experiments. Simulations were performed to a strain of 0.2, by which time all cases have achieved a steady-state grain size.

As expected, increasing λ results in a smaller steady-state grain size and a more rapid convergence to the steady-state value with increasing strain (Fig. 5). In general, all cases produce the relative variations in grain size as a function of strain rate shown by the experimental data; however, the grain growth parameters derived from experiment AL5 combined with λ=0.005–0.01 provide better fits to the data (Fig. 5e, f).

We next investigate predictions of the wattmeter for a 1-D vertical column of ice in which stress as a function of depth is controlled by the surface slope and ice density (Eq. 20). This setup is analogous to deformation within a deforming ice body and thus can be directly compared with grain size values derived from ice cores. We first simulate a theoretical 1 km column of ice with a surface slope of 2∘, ice density of 920 kg m-3, and constant temperature of 253 K. We calculate the steady-state grain size, velocity, strain rate, and effective stress exponent as a function of depth assuming λ=λGBS=λdisl=0.01 (Fig. 6a). The effective stress exponent is calculated from the numerical solution using the local gradient in stress and strain rate with depth. Grain size decreases with depth due to the increase in stress lower in the column (which drives recrystallization), while grain growth dominates near the surface. Compared to cases with a constant grain size, grain size evolution produces larger gradients in velocity and strain rate with depth as the fine-grained ice softens near the bed (Fig. 6b, c). Calculations with grain growth parameters derived from either the joint fit of the experimental and ice core data or exclusively from experiment AL5 result in similar grain size profiles, with the joint fit predicting slightly smaller grain sizes and correspondingly higher strain rates at the base of the column.

Further, we explore the sensitivity of our results to the range in λ used in our comparison to the laboratory data (λ=λGBS=λdisl = 0.005–0.015). Although λ is poorly constrained (Austin and Evans, 2007, 2009), these values are in the range determined by applying the wattmeter to recrystallization of quartzite (Tokle and Hirth, 2021) and olivine (Holtzman et al., 2018). In general, we find that the differences in the wattmeter predictions due to the uncertainty in λ (Fig. 6e–h) are smaller than the variations associated with the uncertainty in the grain growth parameters (Fig. 6a–d).

The profiles of velocity and strain rate have a similar functional form to those calculated for a fixed grain size; however, the effective stress exponent varies significantly between the fixed grain size cases and those with grain size evolution. With a fixed grain size, the effective stress exponent varies from neff = 1.8 at the surface to neff = 2.6 to 3.7 at the bed for grain sizes of 1 and 10 mm (Fig. 6d). By contrast, the wattmeter predicts an effective stress exponent that varies from ∼ 2.5 at the surface to ∼ 3 at the bed. This result is insensitive to the choice of λ (Fig. 6h). Thus, similar to the fixed-width shear zone models, the 1-D vertical column predicts effective stress exponents more similar to the Glen law value compared to cases with a fixed grain size.

Finally, we compare the wattmeter predictions to those using a piezometric relationship relating grain size directly to stress (Jacka and Li, 1994). The piezometer predicts significantly larger grain sizes in the shallow portion of the column compared to the wattmeter but reaches similar values near the bed (green curves in Fig. 6). Overall, the piezometer results in smaller strain rates throughout most of the column and a significantly higher effective stress exponent (neff ∼ 3.9), similar to the experimental value for dislocation creep.

To investigate how well the wattmeter predicts grain sizes observed in natural ice cores, we next apply the 1-D vertical column model to grain sizes measured in the GISP2 ice core (Gow et al., 1997) using the linear intercept method (Alley and Woods, 1996). For comparison to GISP2, we assume a column thickness of 3 km and the temperature profile of Clow et al. (1995, 1996), which varies from ∼ 241 K at the surface to 263 K at the bed. Stress is calculated using a constant ice density of 920 kg m-3 (Gow et al., 1997) and a surface slope of 0.11∘ (Hvidberg et al., 1997). We assume λGBS=λdisl=0.01 given the success in using these values to reproduce the Piazolo et al. (2013) experimental data.

One important caveat of the 1-D column models shown in Fig. 6 is that the timescale to reach a steady-state grain size, particularly in the shallow portion of the column where strain rates are small, may be greater than 104-5 years. Thus, to compare our model predictions with the ice core data, where the shallowest ice is the youngest ice, we use the time-dependent formulation in Eq. (13) and calculate grain size as a function of time at each depth assuming a fixed surface slope. The age of the ice at each depth is taken from Ram et al. (2000). Incorporating time dependence into our 1-D column calculations does not change the predicted grain sizes near the base of the column where the ice is sufficiently old for grain size to reach steady state. However, it significantly reduces grain sizes in the shallow part of the column, where the young ice does not have sufficient time to reach steady state (dotted curves, Fig. 7).

Overall, we find a good fit between the grain sizes predicted by the wattmeter and those recorded in the GISP2 ice core. Surface velocities predicted by the wattmeter (∼ 1 m yr-1) are also in agreement with those observed near the GISP2 site (Hvidberg et al., 1997). There is little sensitivity to using the grain growth parameters from the Azuma et al. (2012) AL5 experiment only (red curves, Fig. 7) versus the joint fit to all experiments and the ice core data (blue curves, Fig. 7). The one major deviation between the grain size predictions of the wattmeter and the observed grain sizes occurs at the very base of the core. In this region, observed grain sizes increase up to ∼ 10 mm at the bed, while the wattmeter predicts grain sizes that monotonically decrease to a value of ∼ 2 mm. These deviations are discussed in Sect. 4.2 below.

As illustrated in both the steady-state shear zone models (Fig. 4b) and the simulations of a 1-D column of ice deforming due to a surface slope (Fig. 6b), the wattmeter results in an effective stress exponent that is intermediate between the lab-derived values for dislocation and GBS-limited creep and approaches the n = 3 value of the Glen law. To interpret these results, we reconsider the end-member cases of deformation accommodated solely by either dislocation or GBS-limited creep. In the dislocation creep regime, deformation is not sensitive to grain size (i.e., mdisl = 0 in Eq. 16), and we expect no difference in creep behavior or the effective stress exponent as a function of grain size. By contrast, in GBS-limited creep, strain rate is sensitive to both stress and grain size. Further, the steady-state grain size calculated by the wattmeter will vary as a function of stress and strain rate (Eq. 14). Thus, substituting the expression for steady-state grain size, dss in Eq. (14), into the flow law (Eq. 16) we find that strain rate can be related to stress through an effective stress exponent neff that is proportional to nGBS, mGBS, and the grain growth exponent p: 21neff=nGBS1+p+mGBS1+p-mGBS. Using laboratory-determined values for nGBS and mGBS (Table 1) and the grain growth exponent fit by the laboratory and ice core data (p = 6.2), we find neff for GBS-limited creep is equal to ∼ 2.5. This value corresponds to the effective stress exponent calculated in the shear zone at low stress and strain rate (Fig. 4b) and is higher than the laboratory-derived value at a constant grain size.

We note that this expression for the effective stress exponent is only valid in the limit of steady-state grain size. Processes that limit a change of grain size in the GBS regime will result in a stress exponent closer to the lab-derived value. For example, some experiments have shown that grain growth may be limited during GBS creep (Goldsby and Kohlstedt, 2001; Caswell and Cooper, 2017); moreover, in natural ice impurities may also limit grain growth (e.g., Alley and Woods, 1996). Future experiments under different conditions (e.g., initial grain size, impurity and bubble distribution, deformation mechanism) are necessary to further constrain these effects on grain growth.

Comparison of the constant grain size shear zone models to those using the wattmeter shows that both predict a transition in the effective stress exponent near the field boundary at strain rates of 10-2 to 102 yr-1 (Fig. 4b). When grain size is fixed and does not evolve, neff varies from the lab-derived values for GBS (1.8) and dislocation creep (4) and only coincides with the Glen law (n = 3 ± 0.5) over a narrow range of strain rates (e.g., 3 × 10-1–3 × 10-2 yr-1 for a shear zone temperature of 240 K; Fig. 4b). By contrast, the variation in neff derived for steady-state grain size from the wattmeter varies less dramatically with strain rate and is within the range of 3 ± 0.5 for all strain rates found in natural systems. Thus, the stress dependence of grain size evolution, when coupled to the composite flow law (Eq. 15), provides an explanation for why the effective stress exponent in ice is consistent with the Glen law, even though neither dislocation nor GBS creep have stress exponents of ∼ 3.

Further, when grain size evolves according to the wattmeter, smaller values of the grain growth exponent p will result in larger values of neff (Eq. 21), which becomes infinite when p = (mGBS-1). For example, if p = 2 the effective stress exponent for GBS-limited creep becomes 4.25. In this scenario, neither dislocation creep nor GBS-limited creep would result in an effective stress exponent that is consistent with the Glen law value. As noted above, some observations support a stress exponent > 3 (Budd and Jacka, 1989; Cuffey and Kavanaugh, 2011). These data could be consistent with grain-size-insensitive dislocation creep or grain-size-sensitive creep with a p value around 2, similar to bubble-free ice (Azuma et al., 2012). However, with p = 2 the wattmeter underpredicts the grain size in the laboratory experiments and significantly overpredicts the grain size in the GISP2 ice core. This supports our application of the wattmeter using a p value consistent with the larger grain growth exponents inferred from bubble-rich experiments (Azuma et al., 2012). Intriguingly in the theoretical limit of grain growth in the presence of inclusions (p = 3–4; Evans et al., 2001) the effective stress exponent becomes 3.3–2.9 for steady-state grain size in the GBS regime.

Equation (21) can also be used to predict the effective stress exponent for creep in other geologic materials that undergo grain-size-sensitive creep and whose grain size evolution can be predicted by the wattmeter. For example, Hansen et al. (2012) found that at a constant grain size GBS creep in olivine is described by flow law parameters n = 4.1 and m = 0.73. However, in high-strain experiments when grain size evolution occurred, the effective stress exponent increased to n = 5. Plugging the constant grain size parameters for GBS creep into Eq. (21) and assuming a grain growth exponent of p = 3 for olivine (Karato, 1989), we calculate an effective stress exponent of neff = 5.1, consistent with the experimentally determined value from the Hansen et al. (2012) experiments. This provides additional evidence that the wattmeter can be used to capture the physics of grain-size-sensitive creep.

Ice cores show three primary grain size regimes (e.g., Gow and Williamson, 1976; Herron and Langway, 1982; Thorsteinson et al., 1997): (1) a zone of increasing grain size in the upper several hundred meters of ice, (2) a region of relatively constant to slightly decreasing grain size at intermediate depths, and (3) a zone of rapidly increasing grain size near the bed. These variations have frequently been interpreted in terms of the tripartite paradigm or three-stage model (e.g., Alley, 1988, 1992; De la Chapelle et al., 1998), in which Regime 1 is associated with normal grain growth, Regime 2 reflects a balance between normal grain growth and polygonization, and Regime 3 is attributed to migration recrystallization. The later process reflects a combination of rapid grain boundary migration and the nucleation of new grains when temperatures exceed 263 K (Duval and Castelnau, 1995).

More recent studies (e.g., Faria et al., 2014a) have argued that the tripartite model may be an oversimplification, as other processes besides normal grain growth appear to be operating at shallow depths (Kipfstuhl et al., 2006, 2009). Faria et al. (2014a) refer to the process by which grains coarsen while simultaneously undergoing deformation as “dynamic grain growth”. The wattmeter inherently captures the balance between grain growth and grain size reduction, predicting grain sizes that vary continuously between Regimes 1 and 2. However, as noted above, the wattmeter does not explain the increase in grain size observed in Regime 3 near the base of the GISP2 core (Fig. 7) and other ice cores, such as Byrd (Gow and Williamson, 1976), GRIP (Thorsteinsson, et al., 1997), and Law dome (Li et al., 1998). The reason is that the higher stresses and higher strain rates near the bed promote grain size reduction, which dominates the temperature dependence of grain growth even as ice temperatures approach 263 K. One possible explanation for this discrepancy is that grain growth kinetics change as ice enters the pre-melting regime at temperatures > 263 K due to enhanced grain boundary mobility and the role of migration recrystallization in the formation of new grains (Duval and Castelnau, 1995; Hamann et al., 2007). Further, micro-particles on grain boundaries may become more mobile, possibly reducing their pinning effect and leading to enhanced grain growth (Evans et al., 2001).

As a simple test of this hypothesis, we substituted the bubble-free grain growth kinetics from Azuma et al. (2012) experiment T15 (conducted at 263 K) into the lowermost 200 m of our model for GISP2. The result is to increase grain sizes in the basal ice to ∼ 100 mm. This is approximately an order of magnitude greater than the maximum observed values. However, fabric development can lead to significant weakening in regions of high strain, such as near the bed. Approximating this weakening effect by multiplying the pre-exponential term in the dislocation creep law by an enhancement factor of 10 (Cuffey and Paterson, 2010) and using the bubble-free grain growth kinetics provides a good fit to the observations as shown by the dashed line in Fig. 7. While these results are suggestive, future work on grain growth kinetics in the pre-melting regime and the feedback between grain size evolution and strain rate enhancement due to fabric development is needed to fully explore these effects.

Another caveat of our predictions for grain size is that we have made no attempt to incorporate local-scale heterogeneities in impurity contents. The role of impurities is well known to influence grain size in ice cores on multiple spatial and temporal scales. At the centimeter scale, “forest-fire” bands characterized by high ammonium contents and low electrical conductivities are observed to correlate with local reductions in grain size (e.g., Alley and Woods, 1996). Major climatic transitions, such as that associated with the Holocene and Last Glacial Maximum (LGM), are also seen to correlate with variations in grain size (e.g., Duval and Lorius, 1980; Herron et al., 1985; Gow et al., 1997; Li et al., 1998) and zones of enhanced strain rate (e.g., Fisher and Koerner, 1986). Indeed, Durand et al. (2006) argue that grain growth pinned by a combination of dust, bubbles and clathrates is the dominant control on grain size variability in the Dome Concordia core. While incorporating heterogeneous impurity contents is beyond the immediate scope of this study, the wattmeter provides a framework to include such heterogeneities through the use of variable grain growth parameters tuned for different impurity contents. This further highlights the need for additional grain growth experiments under various impurity contents and temperature conditions.

While the Glen law provides an excellent description of ice flow in many settings, certain systems are characterized by larger strain rates than predicted. In such cases, an ad hoc strain enhancement factor is often incorporated into the pre-exponential term of the Glen law to account for the combined effects of grain size, impurities, fabric development, and shear heating (see Cuffey and Paterson, 2010). For example, matching velocity profiles across ice streams (e.g., Echelmeyer et al., 1994; Jackson and Kamb, 1997) and through Pleistocene ice near the base of the Greenland ice sheet (Dahl-Jensen and Gunderstrup, 1987; Shoji and Langway, 1988; Lüthi et al., 2002; Ryser et al., 2014) often requires enhancement factors in the range of 2–10. Cuffey et al. (2000) attempted to quantify the role of grain size in the enhancement factor based on deformation recorded in Meserve Glacier, Antarctica. The grain size evolution model developed here provides additional constraints on the role of grain size in enhanced flow and strain localization in ice.

To illustrate this point, we model deformation within Drill Site D in fast-moving ice near Jakobshavn Isbrae in western Greenland (Iken et al., 1993; Lüthi et al., 2002). This site experiences surface velocities of ∼ 600 m yr-1, and tiltmeter data indicate enhanced strain rates in temperate ice below the Holocene–LGM transition near the bed. Lüthi et al. (2002) developed a thermomechanical model for deformation in the borehole and found that after incorporating the temperature dependence of ice viscosity, enhancement factors of 1.7–2.6 were required to match the observations in the pre-Holocene ice below 680 m. Although neither grain size nor impurity contents were measured in the Site D core, Lüthi et al. (2002) interpreted the enhanced strain rates to reflect smaller grain sizes associated with higher impurity contents below the Holocene–LGM transition.

In Fig. 8 we apply the wattmeter to model deformation with Site D using the same approach as for the GISP2 core (Sect. 3.3) assuming a surface slope of 2∘, an ice thickness of 830 m, downhole temperatures from Iken et al. (1993), and the age model of Lüthi et al. (2002). Calculated grain sizes vary from ∼ 2 mm near the surface to ∼ 0.5 mm near the bed (Fig. 8a). Comparing the corresponding strain rates to those calculated for a case using a constant grain size of 1 mm, we predicted enhancement factors of 1.9–2.5 in ice below ∼ 700 m depth (Fig. 8c). Further, while there are no constraints on grain size for direct comparison, the surface velocity calculated from our model compares favorably with those observed at the Site D location (Fig. 8b). Thus, without invoking additional pinning effects beyond those incorporated in the grain growth exponents extrapolated from the laboratory and GRIP ice core data (Fig. 2), the wattmeter provides a good match to the available observations.

We stress that these results are not meant to imply that elevated impurity contents have no influence on grain size and deformation rates but simply that first-order variations in these parameters are successfully captured by the wattmeter. Moreover, the enhanced strain rates associated with grain size reduction illustrate the potential importance of grain size evolution on strain localization. Indeed, the extreme strain localization in ice stream margins (e.g., Harrison et al., 1998) may be partially accommodated by grain size reduction, in combination with shear heating (e.g., Suckale et al., 2014; Perol and Rice, 2015). Further, the development of crystal fabric in the shear plane will weaken ice (Duval et al., 1983), resulting in a larger strain rate for the same stress. In the 1-D ice column models, stress is fixed by the surface slope, resulting in a positive feedback in which enhanced fabric development will drive further grain size reduction (due to the enhanced work rate). Future studies that simultaneously measure deformation, grain size, crystal fabric, and impurity contents – ideally in regions of high strain rates – will be critical to improving coupled models of deformation and grain size evolution in ice sheets and glaciers.