In the accumulation zone of glaciers and ice sheets snow is transformed into glacial ice by firn densification. Classically, this process is assumed to solely depend on temperature and overburden pressure, which is controlled by the accumulation rate. However, exceptionally thin firn layers have been observed in the high-strain shear margins of ice streams. Previously, it has been proposed that this firn thinning can be explained by an enhancement of firn densification due to the effect of strain softening inherent to power-law creep. This hypothesis has not been validated, and the greater firn densities in the presence of horizontal strain rates have not yet been reproduced by models. Here, we develop a model that corrects the firn densification rate predicted by classical, climate-forced models for the effect of strain softening. With the model it is confirmed that strain softening dominates the firn densification process when high strain rates are present. Firn densities along a cross section of the Northeast Greenland Ice Stream (NEGIS) are reproduced with good agreement, validating the accuracy of the developed model. Finally, it is shown that strain softening has significant implications for ice core dating and that it considerably affects the firn properties over wide areas of the polar ice sheet, even at low strain rates. Therefore, we suggest that, besides temperature and accumulation rate, horizontal strain rates should generally be considered as a forcing parameter in firn densification modeling.

Firn densification refers to the transformation of snow into glacial ice, which occurs in the uppermost layers of ice sheets and glaciers within their accumulation zones, when old snow, now denoted as firn, is buried under younger snow. The overburden pressure gradually increases and causes densification of the firn. Large-scale ice flow is not considered by firn models even though it is known that ice is a non-Newtonian material where strain reduces the viscosity

Firn densification is conventionally divided into stages where different physical mechanisms dominate. Initially, the Newtonian grain-boundary sliding is dominant for densities of

For a variety of glaciological studies, properties of the firn need to be known. For example, it is essential to know the firn air content for deriving the mass balance of an ice sheet from changes of its surface elevation, measured by satellite altimetry (e.g.,

In many applications, these properties are determined by employing a firn densification model. Over the years, a wide range of models have been developed (e.g.,

While efforts are being made to directly model the physical processes that lead to densification of the firn

The limitation to climatic forcing is not only insufficient for finding firn model tuning parameters that are applicable for the whole Greenland ice sheet

The reduced firn thickness due to strain is explained by two processes: horizontal divergence and strain softening.

Horizontal divergence of velocities causes a simple horizontal stretching and thus a vertical thinning of the firn. This effectively reduces the overburden load. The firn density itself is however not directly affected.

Strain softening was first suggested by

Despite the observational evidence, firn densification models used for polar ice sheets do not capture the effect of strain softening. Based on a constitutive equation for the power-law creep of porous media

In this paper, we aim to model the effect of strain softening in the firn with a different approach. We derive a scale factor that can be applied with any climate-forced firn densification model to correct its predicted firn densification rate for the impact of strain. Our approach is computationally cheap, simple to implement and thereby can extend the application range of the well-established classical firn models even further.

A firn densification model in a Lagrangian formulation expresses the densification rate of a firn layer as a function of external forcing parameters and internal parameters, representing its current state. The external parameters are generally time-variable. In a climate-forced model these are the temperature

Generally, at a specific point in the firn the state of the strain rates is described by the symmetric strain rate tensor

In the case of climate-forced models all components of

Here, we aim to derive the actual densification rate

To simplify the situation, we assume that horizontal divergence only has a negligible influence on the pressure, which is justified for the second firn stage, where already a significant overburden pressure has built up and the pressure component by horizontal compression can be neglected. With this assumption both effects can be taken as being independent of each other, and we can correct for them separately. This we do by first applying the strain softening correction that is derived in the following and then correcting for horizontal divergence subsequently, using the layer-thinning scheme by

For the derivation of the strain softening correction, the above assumption means that

By combining Eqs. (

Firn is a compressible material. Following

While the proportionality factors between stress and strain, also denoted as effective viscosities, in the two constitutive relations generally differ, they are coupled in non-Newtonian materials by the fact that the strain rates of one phase also reduce the effective viscosity of the other.

For firn the two weighting coefficients have been calibrated by

To motivate this approach and derive the scale factor, we look at firn compaction on a microscopic level, where firn is a mixture of solid ice and air. Internal stresses are concentrated at the contact areas between grains, and stress chains form, which carry the overburden load

The deformation of the solid ice matrix can be described by Glen's flow law (See also

Glen's flow law can equivalently be formulated in terms of the strain-rate-dependent effective ice viscosity

In practice, it is difficult to apply Eq. (

To relate the microscopic deformation with the macroscopic firn compaction despite the difficulties, we need to make two simplifying assumptions: we expect that high macroscale deformation and compaction correlate with strong microscale deformation. Therefore, we assume that the microscale effective strain rate in the stress chains is proportional to the macroscopic effective strain rate

The second assumption is that the densification rate is controlled by the strain rate of the load-carrying grains as their deformation must be the limiting factor for the densification. Hence, we assume that the bulk behavior of the firn scales with the behavior of such characteristic grains, as follows:

Applying the second assumption to Eq. (

Equations (

By dividing Eqs. (

Thus, the enhancement from strain softening can be expressed in terms of macroscopic firn strain rates. By expanding the effective strain rates into components, we get

In summary, the variable

The solution of Eq. (

The derivation of the strain softening enhancement above relies on a number of assumptions. We have assumed that in the second stage firn densifies by dislocation creep and that this densification is driven by vertical compression. Other densification processes and horizontal compression are therefore assumed to be negligible, and strain softening and horizontal divergence are assumed to be independent of each other. We have further assumed that the microscale solid ice deformation is the key process limiting the rate of firn densification. To relate this to the macroscale compaction, we assume that the microscale effective strain rate of deformation scales with the macroscale effective strain rate that contains compaction (Eq.

The densification in climate-forced models, such as the HL model, is characterized by an exponential decay towards the density of solid ice. The vertical strain rate

To circumvent this issue, a regularization is introduced. Inspired by

Leaving the perspective of firn densification modeling, the residual strain rate can be associated with the general thinning of firn and ice layers in ice sheets that is induced by the flow of ice towards the ice sheet margins. While the densification part of the vertical strain rate approaches zero, this contribution remains finite. The residual strain rate can be obtained by flow modeling or measured using strain gauge instruments

Empirical climate-forced firn models contain two tuning parameters which represent the dependency and sensitivity of the densification rate to temperature and accumulation rate. They are obtained by tuning the model to firn density measurements from, for example, firn cores, whereby it is assumed that all inter-site density variability can be attributed to the variability of temperature and accumulation rate between the sites. If another process that is driven by a different forcing parameter also affects the densification, its contribution will be implicitly captured by the two tuning parameters for the climatic forcing. Thereby, the additional process is not only inadequately represented, but also the model sensitivity to the climatic forcing will be inaccurate. We refer to the implicit contribution as a tuning bias.

In Sect.

The densification rate output of the climate-forced model therefore actually has to be seen as the densification rate by climatic forcing alone

As given in Eq. (

With this equation the strain softening model has no effect when the effective strain rate input

Correctly determining

Our new model for strain softening (Eq.

For the application of the model, we assume in this paper that the horizontal velocities in the firn are uniform with depth. While, from a theoretical point of view, this assumption is not needed in the strain softening model, it is required due to the lack of internal velocity data. This has two consequences: first, horizontal strain rates (

When the strain softening enhancement is applied to the interior of the Greenland (GrIS) and Antarctic ice sheets (AIS), as we do here, this is justified. In ice sheets vertical shear is predominantly confined to the deepest layers (e.g.,

Although the uppermost firn layers are softer and therefore more likely to entail vertical shear (

The presented model is mainly intended to be applied to the polar ice sheets and not to alpine glaciers, where the

According to the assumption that the horizontal strain rates solely determine the strength of strain softening, their magnitude will, in the following, be expressed by the effective horizontal strain rate

The strain softening scale model is implemented as an optional module in the CFM by

In order to reduce the number of input parameters for the horizontal strain rate from three (

The strain softening model is only applied in the second stage of firn densification for

Within the CFM framework the strain softening model can be executed in combination with any of the implemented climate-forced firn densification models. In the following model experiments, we will use the Herron–Langway model

Temperature evolution is neglected in our model experiments as we aim to assess the general impact of strain softening on firn densification and thereby study processes occurring in the second firn stage, where temperature is approximately stable. At this depth seasonal temperature variations are dampened by heat conduction, and only a recent warming trend remains, which for North Greenland lies on the order of 1

Modeling strain softening requires knowledge about the horizontal strain rates. For the modeling experiments conducted in this paper, horizontal strain rates are computed from surface velocity maps of the GrIS and AIS using the logarithmic strain rate computation method as discussed by

For the GrIS horizontal strain rates are computed from the MEaSUREs Multi-year Greenland Ice Sheet Velocity Mosaic

Before determining the strain rates from the velocity fields, a Gaussian filter is applied to the velocity maps to reduce the impact of processing artifacts in the data, which likely were caused by combining velocity data from different sources for producing these velocity fields. These artifacts appear as a grid structure in the strain rate products and clearly do not represent any physical information, but would lead to an overestimation of the horizontal strain rates, if not removed. For the GrIS velocity data, a Gaussian filter with a standard deviation of

Using the strain rate products, the mean of the effective horizontal strain rates at the locations of the firn cores that were used to tune the HL model is determined, whereby the Little America V site is excluded, as no data exist for this point. We obtain a value of

For validating the strain softening model, firn density data recorded at the NEGIS in the vicinity of the EGRIP ice core site are used. As suggested by

The horizontal strain rates that the firn at these sites has experienced in the past are computed by step-wise backtracing their position according to the velocity field with a monthly resolution and interpolating the computed horizontal strain rate components to these points at every time step.

We force the model with a constant temperature of

As accumulation rate input we use the values derived by

The effect of strain softening on firn densification is studied not only on local but also on ice-sheet-wide scales. For this purpose, the output data of the regional climate model HIRHAM5, forced by the ERA-Interim reanalysis product

In the ice-sheet-wide studies, new surface layers are formed with a density of

In order to understand how strain softening behaves under different dynamic conditions, a sensitivity test is conducted. For the climatic conditions present at the EGRIP site, the firn density and age are modeled for a range of effective horizontal strain rates between

The strain-dependent age profile of the firn is shown in Fig.

The picture is similar for the sensitivity of the firn age at a certain depth and the BCO age, represented in Fig.

In summary, both the BCO depth and the BCO age are strongly affected by high strain rates, but even low strain rates of less than

The firn density profiles at the sites of the NEGIS and the EGRIP S5 2019 firn cores are modeled with the HL model by first considering no strain and subsequently activating the modules for horizontal divergence (not shown), strain softening and the tuning bias correction (TBC).

In Fig.

As the no-strain model already matches the data, the strain softening enhancement leads to an underestimation of the firn thickness. But by this fact, the tuning bias becomes apparent, and the underestimation should not be attributed to the strain softening model but to the underlying HL model.

Although the shear margin firn density data of the S5 2019 firn core are not available, the corresponding modeled firn density profiles are shown in Fig.

As before, these numbers have to be understood in the way that at the S5 site strain softening causes firn thinning of

As a next step, the firn densities recorded by

In Fig.

In the case of no strain, the BCO line represents the impact of the accumulation rate variability alone. It shows that the observed density peaks in the shear margins cannot be attributed to the accumulation pattern. In contradistinction to the observed lower firn thickness, the higher snow accumulation in the shear margin would even promote an increase in the firn thickness.

Similar conclusions need to be drawn for the effect of horizontal divergence, as the corresponding BCO line only differs slightly from the case of no strain. Horizontal divergence merely affects the firn thickness by a few meters, at which no clear trend for a firn thinning can be seen. Instead, firn thickness also increases where velocities are actually converging, as for example the case at the S5 2019 firn core site, indicated by the left vertical line. If on a flat ice sheet velocities diverge at one place, they tend to converge elsewhere. Therefore, horizontal divergence cannot explain a pure firn thinning pattern but always results in an increase in firn thickness nearby. Consequently, horizontal divergence cannot explain the reduced firn thickness in the shear margins of ice streams.

The increased shear margin firn density with the respective lowering of the BCO depth can only be reproduced when strain softening is included in the model. In this case the extent of the density peaks can be reproduced well, which validates the model and supports the idea that the enhanced firn densification rates in high-strain environments are caused by strain softening.

Comparing the strain softening cases, modeled with and without applying the tuning bias correction, it becomes notable that even the small contribution of

Outside the ice stream, where effective horizontal strain rate forcing is even lower than

The main difference between data and model in Fig.

In Fig.

Firn properties along the NEGIS density profile, according to the tuning-bias-corrected strain softening model, as shown in Fig.

Previously, the expected depth of these troughs was estimated by integrating the vertical strain rate caused by horizontal divergence along the flow line, giving a total strain of

The BCO age along the profile is shown in Fig.

The gas enclosed in bubbles at the lock-in depth is younger than the surrounding ice. This introduces a difference between the age of the ice matrix and the gas in an ice core (

To gauge the potential impact of strain softening at this site, we test the sensitivity of the firn age to an exemplary effective strain rate of

Accordingly, strain softening can affect

The decrease in the age difference by strain softening, as well as its variability under stable dynamic conditions, is therefore on the order of the observed time lag between Greenland and Antarctic ice core records and needs to be considered for synchronizing them by the methane (CH

In classical climate-forced densification models the densification rate and thus

Finally, the ice-sheet-wide impact of strain softening on firn densification is studied. For this purpose, a range of steady-state firn density profiles are modeled with the HL model and the strain softening extension, but without the tuning bias correction being applied, to create a data grid that can be used to obtain the approximate change of the BCO depth and BCO age by strain softening at every point on the ice sheet in a computationally efficient manner by interpolation. These profiles encompass various combinations of forcing parameters that cover the range of climatic conditions and effective horizontal strain rates that are present over the GrIS and the AIS according to the multi-year average of the HIRHAM5 output data and the satellite-based velocity field products.

For Greenland, temperature was altered between

Locations with warmer temperatures and lower accumulation rates than given by these input ranges were not modeled, and for the GrIS places with an average annual melt of more than

The BCO depth and BCO age for the input forcing were then determined from the modeled steady-state firn profiles. Local firn properties at every point on the ice sheet were obtained by linear interpolation of the local climatic forcing to the parameter grid. With this approach the ice-sheet-wide contribution of strain softening to firn densification can be studied by comparing the interpolated firn thickness in the cases of no strain to the case when the uncorrected strain softening model is employed.

Figure

Study on the contribution of strain softening to firn densification in terms of the firn thickness over the dry zone of the Greenland ice sheet (GrIS,

However, strain rates derived from remotely sensed velocities are sensitive to the degree of smoothing applied. Spatially uncorrelated velocity noise will lead to a positive bias in the effective strain rate. Smoothing reduces the noise amplitude and will act to lessen this bias. Unfortunately, smoothing will also blur the true strain rate signal, leading to a negative bias in the effective strain rate in high-strain regions. The degree of smoothing is therefore a compromise between reducing noise and not degrading the signal too much. In this paper, we have chosen the degree of smoothing necessary to remove obvious artifacts. However, in the interior regions with very slow flow the true strain rates may be so small that even a tiny remaining noise amplitude can still be a substantial component of the estimated effective strain rate. This is an important caveat when interpreting the continental-scale maps in Fig.

The velocity maps moreover contain spots in the polar hole, where data are missing and where it therefore is not possible to model the effect of strain softening. In combination with the difficulties arising from processing artifacts and noise, this highlights the need for high-quality velocity and strain rate data from remote sensing for modeling the effect of strain softening.

Nonetheless, we see that the strain softening contribution to firn densification matters over wide areas of the ice sheets. This however does not mean that existing firn densification models, which do not consider strain softening, generally overestimate the firn thicknesses, but rather that these models will likely include some contribution of strain softening in the form of a tuning bias, as previously discussed. To highlight this effect, Fig.

As we have seen in Fig.

To take account of this implicit contribution, we introduced the tuning bias correction, which is however only a first-order estimate of the implicit strain softening contribution. We want to stress that it is required to consider all three input parameters – i.e., the accumulation rate, the temperature and the effective horizontal strain rate – during the empirical tuning of a firn model to represent all of them accurately and to capture the sensitivity of firn densification to the three forcing parameters correctly. We leave this to future studies.

We have developed an extension for firn densification models that is capable of correcting the densification rate of any climate-forced firn model for the effect of strain softening. Employing this model, it was studied how strain softening affects firn densification on local and ice-sheet-wide scales.

We found that the sensitivity of firn densification to strain softening is highest at low strain rates and that therefore even low strain rates can affect the firn thickness considerably in areas where forcing by accumulation rate and temperature is weak. In high-strain areas, such as the shear margins of ice streams, a significant acceleration of firn densification by strain softening was modeled, which is in good agreement with observations of lower firn thickness in these areas. As other potential processes, like horizontal divergence or a greater accumulation in the shear margin troughs, could not explain this reduction of firn thickness, our work supports the idea that strain softening is the principal cause. It was further observed that the change of firn thickness resembles the lowering of the surface elevation in the shear margins, which suggests that the shear margin troughs form because of a faster settling of the firn due to strain softening.

Strain softening not only affects the firn thickness but also reduces the age of the firn at the firn–ice transition. According to our model this can lead to a reduction of the BCO age by around

Finally, we demonstrate that strain softening has a substantial effect on firn densification over wide areas of ice sheets, and as a consequence horizontal strain rates should generally be considered in firn densification modeling, because a restriction to climatic forcing parameters results in a misrepresentation of the latter. Our work therefore suggests that besides temperature and accumulation rate the effective horizontal strain rate should also be considered as a relevant forcing parameter in firn densification modeling and that all three parameters should already be considered during the empirical tuning of firn densification models.

The model the extension for strain softening is included in the Community Firn Model after Version 1.1.9 (

The model outputs generated in the presented study are not publicly available, but they are available from the corresponding author upon request. This is because they shall not be taken as best estimates of the corresponding firn properties but were produced to illustrate the relative effect of strain softening on firn densification.

The idea for the strain softening model extension was developed by both authors. FMO carried out the model experiments and wrote the majority of the code and paper, based on his master's thesis. AG acted as a supervisor for the thesis as well as in the preparation of the manuscript.

The contact author has declared that neither they nor their co-author has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We acknowledge the personal support by the German Academic Scholarship Foundation to Falk M. Oraschewski. EGRIP is directed and organized by the Centre for Ice and Climate at the Niels Bohr Institute, University of Copenhagen. It is supported by funding agencies and institutions in Denmark (A. P. Møller Foundation, University of Copenhagen), the US (US National Science Foundation, Office of Polar Programs), Germany (Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research), Japan (National Institute of Polar Research and Arctic Challenge for Sustainability), Norway (University of Bergen and Trond Mohn Foundation), Switzerland (Swiss National Science Foundation), France (French Polar Institute Paul-Emile Victor, Institute for Geosciences and Environmental research), Canada (University of Manitoba) and China (Chinese Academy of Sciences and Beijing Normal University). We further acknowledge the Arctic and Climate Research section at the Danish Meteorological Institute for producing and making available their HIRHAM5 model output. Data from the Programme for Monitoring of the Greenland Ice Sheet (PROMICE) were provided by the Geological Survey of Denmark and Greenland (GEUS) at

This research has been supported by the Open Access Publishing Fund of the University of Tübingen and by the Villum Investigator Project IceFlow (grant no. 16572).

This paper was edited by Florent Dominé and reviewed by two anonymous referees.