The net rate of snow accumulation

Firn is snow that has persisted for at least 1 full year on the surface of a glacier or ice sheet. In the absence of significant surface melting, firn is transformed into glacial ice through dry firn compaction. As it is buried by subsequent snowfall, the vertical load of the overlying material compacts firn until it becomes glacial ice

Particularly important is understanding how the thickness of the firn layer will respond to changes in temperature and the rate of snow accumulation

Less consensus exists regarding the dependence of

Starting from a full dynamic model of firn compaction including grain-size advection and growth,

In this paper we explore the implications of the assumptions described above and elucidate how firn thickness depends on accumulation and grain size in simple firn compaction models. We present and analyse an Eulerian firn compaction model based on

In this section we describe the model equations and boundary conditions, nondimensionalization of the model, and the numerical methods used to solve the equations. We then describe a reduced, steady-state ordinary differential equation (ODE) model that will help us examine the accumulation dependence of steady-state firn thickness.

We consider a one-dimensional, isothermal column of firn and ice. The model describes the coupled spatial and temporal evolution of five properties of the firn, all defined in a bulk sense (i.e. considering a spatial scale much larger than the grain size): porosity, vertical normal stress, vertical velocity, grain size, and age (Table

Model variables and coordinates.

Porosity is defined as

Physical constants.

We follow

Although this has no impact on firn thickness, we include the following evolution for the age of the firn and ice

Finally, we use domain-wide mass conservation (Appendix A) to derive a kinematic condition for the time evolution of the thickness of the domain:

Equations (

We define scales as follows:

Substituting scales into Eq. (

Surface temperatures, scales, and nondimensional parameters corresponding to three climatic settings: (1) high accumulation and surface temperature (e.g. a mountain glacier in a maritime climate), (2) intermediate temperature and accumulation (e.g. near-coastal Antarctica), and (3) low temperature and accumulation (e.g. interior East Antarctica). Ice equivalent: i.e.

Two model parameters,

Table

The dimensionless number

We consider first the dependence of

In contrast,

Given its dependence on

The grain-size saturation parameter

Equations (

All simulations use the following initial conditions:

As well as presenting numerical solutions of the full model, we utilize a simplified, steady-state model consisting of a set of coupled ODEs. The purpose of the ODE model is to allow us to test our numerical solutions of the full model and to act as a starting point for several further simplifications designed to clarify the interdependence of firn thickness, porosity, and grain size.

Equating the time derivatives in Eqs. (

Figure

Porosity closely approaches a steady state (

Plotted over the full-model results in Fig.

Accumulation dependence in the absence of grain-size evolution in three different models. The left panels show porosity

To better understand the accumulation dependence of the thickness of the firn layer, we consider a simple case with uniform and constant grain size,

In all simulations, higher accumulation leads to thicker firn;

Simplifying the ODE model helps to demonstrate this behaviour and will assist with contrasting it to the case when the grain size is allowed to evolve, presented in the next section. We start by ignoring the age equation, which has no effect on the

Ignoring the impact of porosity on

Next we consider how grain-size evolution affects the dependence of firn thickness on accumulation.
Figure

Accumulation dependence of the full model, with a grain size that evolves. The layout is similar to Fig.

In all simulations, just as in the previous section where

The dependence of firn thickness

We turn to the ODE model to understand the dependency of the accumulation sensitivity on surface grain size. Starting with Eq. (

To simplify the model further we assume

Note that we recover Eq. (

Firn thickness

The overbraces in Eq. (

The denominator on the right of Eq. (

While the explanation for the relationship between

All results presented above assumed a linear viscous rheology where compactive strain rates depend linearly on the overburden stress (

Firn thickness

All results presented above have been from steady states where the height of the ice sheet surface, represented in this model by the domain thickness

Figure

Steady-state firn thickness

We have described a firn compaction model that includes grain-size evolution. What distinguishes it from most previous models is that it uses an Eulerian reference frame, following the adaption of the equations of

We used these models to examine how accumulation affects firn thickness through its impact on the competing processes of compaction and advection. An Eulerian reference frame lends itself to this analysis as it allows us to compare terms describing both processes. We first considered the case when grain size is uniform and constant – which is the case considered by most previous firn models

When grain size is kept uniform and constant, increasing accumulation increases downward advection. This is not balanced by the resulting increase in compactive strain rates, and the net effect is that firn thickness increases sub-linearly with accumulation rate (Fig.

We showed that the extent to which grain-size advection counteracts porosity advection increases as the surface grain size is decreased between simulations. Therefore, the sensitivity of firn thickness to accumulation rate increases with the surface grain size in this simple model. This is independent of the stress exponent in the firn constitutive relation and of whether the ice surface height is increasing or decreasing at a steady rate. This is significant because if this relationship manifests in nature, then spatial and temporal variability in surface grain size driven by meteorological conditions will translate into spatial and temporal variability in the sensitivity of firn thickness to accumulation rates. Consideration of this effect could yield improvements to reconstructions of past climate that exploit modelled relationships between accumulation, bubble-close-off depth, and stable-isotope ratios

We also considered the case when the grain size can be assumed to be zero at the surface (i.e. when

An implication of these generally unrecognized assumptions underlying some widely used firn models is that models that include viscous firn compaction and grain-size evolution

While our model relaxes some important assumptions, others remain. We assume that temperature is uniform and constant, firn deforms viscously, air pressure is negligible, no water is present, rheological parameters are uniform and constant, firn grains grow via normal grain growth (with a growth exponent of 2), and firn viscosity is proportional to grain size.

To isolate the effects of accumulation rate on firn thickness we assumed a uniform and constant temperature. However, temperature is a first-order control on firn thickness in this model, through its impact on grain growth and on firn compaction (Sect. 2.3). Surface temperatures vary regionally with climate. This variability would need to be taken into account in any attempt to compare model results to observations

We followed most previous firn models and assumed a viscous firn rheology

Assuming dry firn compaction restricts the applicability of our results to regions where no significant melting takes place. In wet-snow zones the grain-scale processes that control compaction and grain growth will differ significantly from those in dry snow. Moreover, refreezing of meltwater contributes to densification. Understanding how grain growth, compaction, and advection interact to control accumulation dependence in wet-snow zones is beyond our scope but will likely become increasingly important as these regions grow in the future (e.g.

For simplicity, and unlike firn compaction models that aim to accurately simulate porosity profiles, we used a uniform compaction coefficient,

We also assumed that firn compaction is inversely proportional to the square of a characteristic grain size. Complications to this simple description could arise from non-uniform grain sizes (i.e which are inadequately described by a single-valued grain-size variable) or from other compaction mechanisms that do not obey this simple inverse relationship.

We also assumed normal grain growth with an exponent of two (Eq.

We have not explored the complications of multiple compaction regimes, different dependencies of compaction on grain size, and different grain growth exponents or parameterizations. However, our work highlights the importance of doing so because commonly used constitutive relationships inspired by

The thickness of the firn layer in cold, dry accumulation zones is controlled by a competition between downward advection of firn and the compaction of each parcel of firn as it advects. To better understand the controls on advection and compaction, we analysed a simplified model that is closely related to previous models

Future work could extend the model to include additional physics and apply the model to different scenarios. Model extensions could include employing a dynamically evolving temperature and varying rheological parameters between porosity-defined regions of the firn. Additional simulations could explore model response to temporal changes in accumulation rate and temperature. Because the model incorporates accumulation differently than models that have been used for this purpose before

The fact that modelled firn thickness depends on grain size at the surface has potentially significant implications because surface grain size varies in time and space due to meteorological conditions. Ongoing and future work by this team to test this idea further include measuring deformation of firn with known grain size using phase-sensitive ice-penetrating radar

Here we use global mass conservation to derive Eq. (

To take account of the temporally evolving domain length, we employ a change in vertical coordinate.

We normalize the vertical (nondimensional) coordinate,

The model equations are modified to account for the change in coordinates by substituting Eqs. (

The porosity equation (Eq.

These six equations are solved with the method of lines to simulate how the six variables evolve in time and space during simulations. Specifically, the spatial domain is discretized into

To facilitate comparison between the results from simulations with different domain heights, all depths are converted from

In Sect.

Starting with Eq. (

Combining Eqs. (

All the code required to run the model and plot the figures in this paper can be found here:

JK and EC initiated the study, CM advised on model physics, RS and JK developed the model and the code, JK led the modelling and writing, and all authors contributed ideas and discussion and helped write the manuscript.

The contact author has declared that none of the authors has any competing interests.

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

The authors would like to thank the editor and reviewers for constructive comments that improved the paper. The authors would also like to thank Robert Arthern for useful discussions.

This research has been supported by the Directorate for Geosciences (grant nos. OPP 19-35438 and PSU 5861-CU-NSF-8934).

This paper was edited by Elisa Mantelli and reviewed by C. Max Stevens and one anonymous referee.