The formation of surface and basal crevasses as a consequence of deformation in ice sheets plays an influential role in glacial mass balance . Crevasses are predominantly mode-I (tensile) fractures that propagate vertically downwards to the depth at which they stabilize (i.e. maximum crevasse depth), depending on the longitudinal-stress state normal to the fracture surface . Fracture propagation can be further aided by the accumulation of meltwater within surface crevasses, the supply and storage of which can be attributed to supraglacial lakes and firn aquifers . This can trigger a process called hydrofracture, wherein the meltwater in the crevasse exerts additional opening stress on the crevasse walls . If the volume of meltwater is sufficiently large, hydrofracture can cause full-thickness crevasse propagation and lead to large-scale iceberg calving events from ice shelves and to increases in basal sliding rates . Glacial mass loss caused by the calving events represents one of the leading contributors to global sea level rise .

The first study to examine crevasse propagation in glaciers and ice sheets was conducted by , who proposed an analytical zero-stress model. The so-called Nye zero-stress model assumed that crevasses are dry and that ice has no resistance to fracturing; thus, a crevasse will propagate to a depth at which tensile stresses are completely offset by the compressive overburden pressure due to gravitational self-weight. The zero stress was later adapted by to accommodate for the presence of meltwater in crevasses. Nevertheless, this simplistic model agreed well with observational results for fields of closely spaced crevasses , where neighbouring crevasses provide shielding effects in relation to crack tip singularities . However, it underestimates the depths of isolated crevasses, where this shielding effect is not present. Therefore, linear elastic fracture mechanics (LEFM) models were introduced by to model these isolated surface and basal crevasses.

LEFM models capture the effects of the stress singularity (assuming that the resulting plastic zone around the crack tip is sufficiently small) by evaluating the stress intensity factor at the crack tip. The role of crack size and orientation, ice thickness, and applied loading conditions can be captured through the evaluation of the stress intensity factor . Using the principle of superposition, the LEFM models include the contributions from the tensile normal stress, the lithostatic compressive stress, and the meltwater pressure. The LEFM approach has been combined with full-Stokes models to study surface and basal crevasse propagation in Thwaites Glacier, with results agreeing well with NASA's radar penetration depths . In addition, LEFM models have been used to map the vulnerability of Antarctic ice shelf crevasses subject to meltwater-driven hydrofracture, with projections agreeing well with existing fractures being mapped by neural networks . The LEFM approach has been successfully combined with boundary element methods, capturing the interactions between basal and surface crevasses and providing estimates regarding the stability and evolution of crevasse shape . However, while analytical LEFM models are computationally efficient and can be implemented into numerical ice sheet models , they do not account for the role of creep deformation or of depth-varying ice material properties in crevasse propagation .

In recent years, several studies focused on the development of computational approaches for modelling crevasse propagation in grounded glaciers and floating ice shelves. Notably, continuum damage mechanics (CDM) approaches were developed to describe more complex thermo-hydro-mechanical phenomena, but they are computationally more intensive compared to zero-stress and LEFM approaches. Such approaches can capture the effects of viscous deformation and long-term stress states on crevasses and can be readily implemented into Stokes-based ice flow models . Recent studies have developed phase field fracture models that formulate LEFM crack propagation criteria within the CDM framework, thus unifying the two approaches . Additionally, CDM-based models can be useful in assessing the accuracy of LEFM and zero-stress models and in understanding the conditions under which their predictions are valid .

While the above-mentioned LEFM and CDM studies capture a range of mechanical interactions, they assume glacial ice to be a homogeneous material, with the values of mechanical properties taken as constants equal to those of fully consolidated ice. In reality, glacial ice forms from the accumulation of snowfall at the upper surface and undergoes compaction as a result of the overburden pressure, the rate of which is dependent on accumulation rates and surrounding temperatures . Moreover, snowflakes are restructured into smaller ice crystals due to wind, which then deform into more stable, compact crystal arrangements ; this causes large differences in the porosity, density, and strength of these top snow layers, referred to as firn, and the deeper glacial ice. Neglecting firn layers leads to an overestimation of mechanical properties in the upper strata . This is of particular importance for geological media subject to self-gravitational loading because the driving stresses are dependent on the mechanical properties in such layered or vertically graded materials. The main hypothesis of this study is that accounting for these depth-dependent material parameters in the LEFM framework would alter the crevasse penetration depths in glaciers and ice shelves.

The aim of this article is to explore the importance of including firn layer effects on surface crevasse propagation. To this end, we determine the maximum crevasse depths in idealized glaciers and ice shelves assuming two different material conditions: fully consolidated homogeneous ice and vertically graded ice as reported from ice core samples, with depth-varying material properties. We derive the analytical solution for the far-field longitudinal stress σxx with depth-dependent properties, which primarily drives the vertical propagation of mode-I crevasses. Next, we systematically investigate the effect of varying each material property in the unconsolidated firn layers through parametric studies as follows: using only the depth-variant density in Sect. , using only the depth-variant Young's modulus in Sect. , and using both the depth-variant density and depth-variant Young's modulus in Sect. . We then conduct fracture mechanics studies in Sect.  for a grounded glacier using the analytical LEFM models presented in and verify the models' accuracy with the stress-based phase field fracture model of . Through variations in meltwater depth ratios in water-filled surface crevasses and in ocean water heights in marine-terminating glaciers, the conditions under which depth-dependent material properties are influential are explored. In Sect. , these studies are extended to floating ice tongues. Sect.  considers the effects of assuming that crevasses develop rapidly (as linear elastic) versus slowly (as non-linear viscous), and we summarize all the findings in Sect. .

We consider a grounded glacier with an idealized rectangular geometry of height H and length L. We neglect lateral shear and restrict the domain to a flow line near the terminus, with x and z representing the along-flow and vertical coordinates. As the out-of-plane length of the glacier is typically much longer than its length, the plane strain assumption is exploited to reduce the three-dimensional glacier geometry to a two-dimensional (height and length) representation. A visual representation of the geometry and boundary conditions can be found in Fig. . Horizontal displacements are restrained in the far left of the domain to prevent rigid body motion, and a free-slip boundary condition is applied by restraining vertical displacements at the base (represented by rollers). The upper surface is defined as a traction-free surface, representing the atmosphere–ice interface. The load contributions considered are the gravitational self-weight, ocean water pressure at the far-right terminus, and meltwater pressure in the crevasse. Note that, under the small-deformation assumption, this representation of the grounded glacier is identical to that of a floating ice shelf in the far-field region.

The far-field longitudinal stress (based on the long-wavelength approximation) within the grounded glacier or ice shelf was derived for the case of steady-state creep with constant (i.e. depth-independent) and homogeneous material properties by . Recently, this longitudinal stress was derived for compressible linear elasticity by as follows: 1σxx=ν(1-ν)ρigz-12H-12ρsghw2H, where ν=0.35 is the Poisson's ratio, assumed to be equal for firn and ice; g is gravitational acceleration; ρi=917 kgm-3 is the density of fully consolidated glacial ice; ρs=1020 kgm-3 is the seawater density; and hw is the seawater height above the glacier bed. We use the convention that positive σxx corresponds to tensile stresses, creating and propagating crevasses, whereas negative σxx is compressive, thus stabilizing the glacier and preventing the formation of crevasses. This analytical stress will be used throughout this paper to compare between the cases with homogeneous (given by the above equation) and depth-dependent (derived in the remainder of this section) material properties. The Poisson ratio ν used within our results represents ice as a linear elastic compressible solid, which is a common assumption for rapidly propagating cracks. If the crevassing process occurs on a timescale well below the Maxwell timescale, ranging from hours to days depending on the strain rate due to the non-linear viscous nature, the assumption of compressibility would be valid. If, instead, the crevassing process occurs slowly, over the span of weeks, the assumption of incompressibility would be valid; thus, a Poisson ratio of ν=0.5 will allow for the model derived here to be applicable over longer timescales. Results for this incompressible case are given in Sect. , whilst results for a depth-dependent Poisson ratio are presented in Appendix .

As shown in Fig. , ice core sample data from the Ronne Ice Shelf, gathered and presented by , indicate large variations in material properties within the firn and meteoric ice layers forming the upper 150 m of the ice core. The ice core data for density can be fitted using the following exponential equation : 2ρ(z)=ρi-(ρi-ρf)e-(H-z)/D, where ρf=350 kgm-3 is the density of the unconsolidated upper surface firn layers, H is the height of the glacier, z is the vertical coordinate (z=0 at the base of the glacier, z=H at the surface), and D is a constant taken as D=32.5m . This constant gives an indication of the thickness of the firn layer: at the surface z=H, the density of the glacier is equal to that of the unconsolidated firn; at a depth of H-z=D below the surface, the density is in between that of firn and ice at 75% of the density of ice; and at a depth of H-z=2.5D below the surface, the density of the glacier is 95% the density of consolidated ice.

The ice core sample data for elastic modulus from Fig.  can be fitted using a similar function as that used for the density, including the depth-dependent Young's modulus: 3E(z)=Ei-Ei-Efe-H-z/D, where Ei=9.5GPa is Young's modulus for solid ice, Ef=1.5GPa is Young's modulus for unconsolidated upper surface firn layers, and D=32.5m is a tuned constant. Here, we have chosen to use a single constant D to describe the depth variations in both density and Young's modulus. While this is not necessary for the derivation of the analytic solutions, the interpretation of D as a length scale of the firn layer thickness indicates that Young's modulus is directly proportional to density and is inherently related to porosity, similarly to porous metallic foams .

In the remainder of this section, analytical expressions are derived for the longitudinal stresses driving the creation of crevasses. We first consider two cases wherein only the density or Young's modulus is depth-varying so as to isolate and examine the influence of each property on the maximum crevasse penetration depth. For the third and final case, we consider both properties to be depth-varying, which is the real-case scenario.

Figure  shows the analytical solutions of the far-field longitudinal stress σxx in a land-terminating grounded glacier (hw=0m) of height H=125m. Each subfigure includes the longitudinal stress versus depth profiles for depth-varying properties (derived in the previous section) and those of the homogeneous case. In addition to the analytical solutions, numerical results from the finite-element solver COMSOL Multiphysics are plotted for the verification of our analytic expressions. The analytical solutions derived are identical to the numerical results, confirming the correctness of the presented expressions along with the appropriateness of the membrane strain assumption.

Using these stress solutions, we predicted crevasse depths based on an analytical LEFM model, as described in the Appendices. For grounded glaciers with free tangential slip at the base, we use the “double-edge crack” weight functions (refer to Appendix ) as this was shown to yield stress intensity factors that are consistent with those calculated using the displacement correlation method . Specifically, we consider the evolution of an isolated surface crevasse within a grounded glacier of height H=125m, assuming damage to initialize beneath a pre-specified surface crack of depth ds=10m. To further verify the LEFM results, we conducted phase field fracture (PFF) simulations with the same configuration used for LEFM studies but with a pre-existing surface notch of depth ds=10m and width b=2.5m. The PFF model simulations capture the non-linear propagation of the crack driven by the mechanical stress beneath the notch until it reaches a final and/or stabilized crevasse depth. For a full description of this PFF model, we refer the reader to our recent work .

The comparison of the crevasse depth ratios (i.e. the final crevasse depth normalized with the ice thickness) obtained from LEFM and PFF models is shown in Fig. . As the ocean water depth is increased, the glacier is subjected to increased compressive hydrostatic pressure at the glacier terminus, which suppresses crevasse growth. The surface crevasse depth ratio decreases non-linearly with the ocean water depth ratio, which is consistent with previous research works . In each of the depth-varying-property cases in Fig. , we obtained excellent agreement between LEFM and PFF models, which serves as a verification of our LEFM implementation. However, the nuanced differences in crevasse propagation results for each of the cases are discussed in the sections below.

The final set of results entails the propagation of surface crevasses in floating ice shelves. We consider an idealized rectangular ice shelf geometry of variable height H and length L=5000 m. Three types of external loads act on the ice shelf: gravitational self-weight, causing a body force, and ocean water pressure and meltwater pressure in the crevasse, causing surface forces. A Robin-type boundary condition is applied at the base of the ice shelf because the buoyancy pressure is a function of the vertical displacement uz, as given by ρsghw-uz. The far-left terminus is constrained to prevent free body motion; the top surface is considered to be traction-free; and the Neumann boundary condition is applied on the right edge to account for the ocean water pressure, similarly to the grounded-glacier case. A schematic diagram representing the applied boundary conditions is shown in Fig. .

The floatation heights for the buoyancy pressure are found by assuming local hydrostatic equilibrium. For the homogeneous ice case, this can be simplified to the ratio of ice density to ocean water density – that is, hw/H=ρi/ρs≈0.9. However, the inclusion of the depth-varying density profile leads to a reduction in the applied gravitational body force, causing a decrease in the floatation height. We evaluate the reduced floatation heights for each ice shelf thickness by integrating the depth-varying density profile over the entire ice shelf thickness and dividing by the thickness and ocean water density. Thus, we find that hw/H is equal to 0.7560 for H=125 m, 0.8268 for H=250 m, 0.8629 for H=500 m, and 0.8809 for H=1000 m. For deeper ice shelves, the material properties of the firn layer become less significant as its thickness is small relative to ice shelf thickness; therefore, the floatation depth tends towards the homogeneous case for thicker ice shelves.

Because of the buoyancy condition at the base and the dependence on ice shelf deflection, the analytical solutions for longitudinal stress derived in Sect. , , and are not appropriate in the regions closer to the terminus. In , we showed that, faraway from the terminus, the longitudinal stress in the ice shelf agreed well with analytical solutions for grounded glaciers. Therefore, we use finite-element analysis to extract the longitudinal-stress data as a function of the vertical coordinate z at the horizontal position x=4750 m (i.e. 250 m from the ice shelf terminus). The data are fitted to a sixth-order polynomial equation, with coefficients presented in Table  in the Appendix, which defines the stress function σxx(z). Surface crevasses in the far-field region are ignored because the longitudinal stress is significantly compressive, preventing their propagation, regardless of whether the crevasses were filled with meltwater or not. The propagation of surface crevasses close to the terminus (ice–ocean front) is investigated by estimating the final and/or stabilized crevasse depth using LEFM. The LEFM model used in (see Appendix ) is appropriate for floating ice shelves as this was shown to match numerically calculated stress intensity factors using the displacement correlation method .

Crevasse penetration depths versus meltwater depth ratios hs/ds from the LEFM model are presented graphically in Fig. . Penetration depths for the dry crevasse (hs/ds=0) and for hs/ds=0.75 are also reported in Tables  and , respectively, for ease of comparing data points. For the homogeneous ice case, there is minimal influence over the normalized crevasse depth by increasing glacier thickness (blue lines); for example, the maximum difference in penetration depth for the dry crevasse is 0.022H when increasing the ice shelf thickness. Surface crevasses with low levels of meltwater only penetrate a few metres below the surface due to the longitudinal-stress profile being predominantly compressive due to the high ocean water pressure. For hs/ds<0.6, an incremental increase in meltwater does not result in significant crevasse growth. Full-fracture propagation is only observed when crevasses are almost fully filled with meltwater (hs/ds>0.9), where the meltwater pressure is sufficient to overcome the compressive longitudinal stress.

Considering the depth-varying Young's modulus leads to a minor reduction in stabilized crevasse depth (black lines), with no growth occurring beyond the initial notch for meltwater depth ratios of hs/ds<0.6 for H=125 m. Penetration depths match with the homogeneous case for meltwater depth ratios of hs/ds≥0.8 as the crevasse penetrates deeper into the ice strata. The influence of the depth-varying Young's modulus is reduced with increasing ice shelf thickness. This is the most noticeable for the crevasse depths reported in Table  as the percentage difference between the depth-varying modulus and homogeneous cases is reduced to 1 % for H=250 m.

Considering the depth-varying density in the fracture analysis results in surface crevasses propagating deeper into the ice strata (red lines), with a dry crevasse propagating to a depth of 0.250H compared to a depth of 0.099H for homogeneous ice within an ice shelf of thickness H=125 m. This is in contrast to the grounded-glacier case (where crevasse depth decreased) and can be attributed to the reduction in overburden pressure and the reduced buoyancy height specific to ice shelves. The inclusion of meltwater within the crevasse results in an increased penetration depth, and full-thickness fracture is achieved for meltwater depth ratios of hs/ds≥0.8. The largest differences in penetration depth for the depth-varying density compared to in the homogeneous case are observed in Table . For thin ice shelves (H=125 m) and a meltwater depth ratio of hs/ds=0.75, the crevasse propagates approximately 3 times deeper when accounting for the depth-varying density. Similarly to the grounded-glacier case, the influence of the depth-varying density is reduced when the ice shelf thickness increases due to a larger proportion of ice being fully consolidated. However, there are still some differences in penetration depth compared to the homogeneous case for thick ice shelves (H=1000 m), with a percentage difference of 14.0% for the dry crevasse and of 19.2% for hs/ds=0.75.

Including the effects of both depth-varying density and depth-varying modulus highlights that density is the more prominent property influencing surface crevasse propagation in ice shelves. It is observed in Fig.  that the majority of results for depth-varying density and modulus (green lines) overlap with the depth-varying density results (red lines). The exception to this is for dry crevasses in thin ice shelves, where the stabilized penetration depth is 0.194H compared to 0.25H when considering only the depth-varying density.

The above analysis has considered ice to behave as a linear elastic compressible solid, with a Poisson ratio of ν=0.35. This is based on the assumption that crevasse propagation occurs in a rapid brittle manner, such that the cracking occurs on a timescale well below the Maxwell time (typically on the order of hours to days for glacial ice). However, it is more common in the glaciology literature to assume that glacier ice deformed over long timescales well before the nucleation and propagation of crevasses. Also, if the slow development of crevasses is considered, with crevasse depths stabilizing over a span of weeks , then ice should be considered to be a viscous material. The analytical solution see Appendix A in for the stress state of a grounded glacier with incompressible linear elastic rheology and that of one with an incompressible non-linear viscous rheology are the same. Therefore, in this study, we simply set the Poisson ratio to ν≈0.5 (using ν=0.49 to prevent numerical issues) to obtain the background stress. In addition, we conduct finite-element simulations for a grounded glacier, including the viscoelastic contributions (with ν=0.35) of ice flow modelled through Glen's flow law, and extract numerical values of the longitudinal stress. To illustrate that the stress state is only sensitive to the incompressibility assumption and not to the linear elastic or non-linear viscous rheology, we plot the longitudinal-stress profile for a land-terminating (hw=0) grounded glacier considering linear elastic compressibility (ν=0.35), linear elastic incompressibility (ν≈0.5), and a non-linear viscous rheology in Fig. .

Figure  shows that if ice is considered to be linear elastic incompressible (ν≈0.5), the obtained stress solution matches with the steady-state creep stress state derived by for depth-independent density and matches stress profiles obtained through simulations using a visco-elastic rheology. We observe that stresses are more extensional in the upper surface and more compressive at the base when considering incompressibility and that the stress obtained through simulations is independent of ice rheology (Glen's law creep coefficients) once a sufficiently long time has passed. For the homogeneous case, the longitudinal stress varies linearly with depth and is symmetrical about the centre line z=H/2. Similarly to the linear elastic compressible case, the inclusion of depth-dependent density results in a reduction in both the lithostatic stress contribution σzz and the resistive stress Rxx for both material rheologies.

The longitudinal-stress profiles presented in Fig.  are used to drive crevasse propagation in the linear elastic fracture mechanics study. Values of crevasse penetration depth for an isolated dry crevasse in a grounded glacier, subject to different values of ocean water height hw, are presented in Fig. . The solid-line curves represent incompressible ice, whilst the dashed lines represent compressible ice of Poisson ratio ν=0.35. Considering ice to be an incompressible solid leads to deeper crevasse penetration depths compared to linear elastic compressibility, but these crevasses follow a similar trend as that observed for the compressible case. Accounting for the depth-varying properties of the firn layer in grounded glaciers leads to two different regimes of crevasse growth behaviour. If the resistive stress Rxx is large (e.g. a low ocean water height), the firn layer promotes crevasse propagation, and the crevasse penetrates to a slightly greater depth than in the constant-density case; this result is in agreement with the study. In contrast, if the resistive stress is small (e.g. high ocean water height) then the firn layer stunts crevasse propagation, and the crevasse depth would be greater in the constant-density case; this result contradicts the study. We note that, even if the Nye zero-stress criterion is used (assuming densely spaced cracks) instead of using LEFM to consider a single isolated crevasse, this finding of the two different crevasse growth regimes holds true.

Comparing the effects of assuming an incompressible and/or viscous rheology, the percentage difference in penetration depth when considering depth-dependent density for a dry crevasse of ocean water height hw=0.5H is reduced to 4 % compared to 20 % for linear elastic compressibility. The ocean water height required to prevent any development of dry crevasses differs, with values of hw=0.55H being sufficient for compressible depth-dependent density cases, whereas ocean water levels of hw=0.8H are required for the incompressible case. Comparing this to the cases in which no density variations are considered still shows a similar trend, with higher ocean water levels being needed to stabilize crevasses when density variations are not considered.

Finally, we consider water-filled surface crevasses in floating ice shelves of height H=125m and length L=5000m using a non-linear viscous ice rheology. Similarly to the linear elastic compressible case, we consider surface crevasses at the horizontal position x=4750m (250 m from the ice shelf terminus) and extract the longitudinal-stress profiles from the finite-element analysis. We plot the stabilized crevasse depth versus meltwater depth ratio for the non-linear viscous (NLV) rheology in Fig.  along with the results for linear elastic (LE) compressibility (ν=0.35) and incompressibility (ν=0.49).

When comparing the stabilized crevasse depths close to the front, we note that the penetration depth is independent of ice rheology, which is in contrast to the grounded-glacier case. For the homogeneous density, minimal crevasse propagation is observed for meltwater depth ratios below hs/ds<0.6, with full-thickness propagation only occurring when fractures are close to saturation. The inclusion of the depth-dependent density results in deeper crevasse penetration depths, with minimal differences in penetration depth between the linear elastic cases and the non-linear viscous rheology. This likely indicates that, for crevasses close to the front, fracture is driven by the flotation height and the bending stresses due to the floating condition. For depth-dependent density, the reduction in flotation height leads to an increase in tensile stress in the upper surface due to increases in Rxx and increased bending stress. In addition, the lithostatic component of longitudinal stress is reduced, leading to deeper crevasse propagation when including firn density.

We also consider the propagation of an isolated surface crevasse located in the far-field region (x=2500 m) of a floating ice shelf, with results presented in Fig. . As shown previously, for the linear elastic compressible rheology, the stress state is fully compressive for both the homogeneous- and the depth-dependent-density case; thus, no crevasse propagation is observed regardless of the meltwater depth ratio. By contrast, when considering the non-linear viscous rheology of ice, surface crevasses may propagate in the far-field region if there is sufficient meltwater pressure present. Large increases in crevasse penetration depth are observed for meltwater depth ratios greater than hs/ds=0.50, with full-thickness propagation being observed close to crevasse saturation at hs/ds=0.95. Similarly to crevasses near the front, the inclusion of depth-dependent density results in increased crevasse penetration depths compared to the homogeneous-density scenario. Thus, similar conclusions can be drawn for both elastic and viscous rheologies.

An important finding of this paper is that the inclusion of the depth-varying mechanical properties of unconsolidated ice strata results in a reduction in both the lithostatic compressive-stress component and the resistive tensile-stress component. Contrarily to the conventional understanding , we find that accounting for depth-varying density and modulus can lead to an overall reduction in surface crevasse depths in grounded glaciers. This is because, in some scenarios, the reduction in resistive stress can hinder crevasse propagation more than the increase in crevasse propagation resulting from the reduction in lithostatic stress. Thus, our study suggests that firn layers can have a stabilizing effect by curtailing surface crevasse growth in grounded glaciers.

Assuming ice to be a linear elastic compressible material, we find that considering depth-varying Young's modulus has a greater influence on crevasse depths than density in thinner glaciers. For example, considering depth-varying density results in a maximum percentage difference of 20 % in the penetration depth of dry crevasses compared to a maximum percentage difference of 45 % when considering the depth-varying Young's modulus. The largest reductions in crevasse depths are observed in thinner glaciers (depths of approximately 100–150 m), where the stabilizing effects of the firn layers seem to be more prominent. Larger meltwater depth ratios are required to propagate surface crevasses in thinner ice shelves, whereas, in thicker glaciers, the influence of firn density is less (in some cases negligible), and so surface crevasses propagate deeper into the fully consolidated strata. Thus, our study reveals that LEFM models assuming homogeneous ice properties are valid for crevasse depth estimation in thicker glaciers with ice thickness H>250 m.

Accounting for depth-varying density in the floating-ice-shelf case increases the penetration depth of surface crevasses close to the ice–ocean front, with this increase being caused by reductions in buoyancy height and lithostatic compressive stresses. The effect of depth-varying density is dominant in thinner ice shelves, but it can still impact surface crevasse propagation in ice shelves as thick as H=1000 m, although this would be to a lesser extent. For instance, the crevasse depth ratio increases to ds=0.91H (188 % increase compared to homogeneous case) for thin ice shelves (H=125m), whereas a 19 % increase is observed for 1km thick ice (ds=0.45H). Considering the depth-varying Young's modulus in the floating-ice-shelf case reduces surface crevasse depth for low meltwater depths slightly, and the effect becomes less significant in thicker ice shelves. This study suggests that, as the ice shelves become thinner due to increased basal melting in warmer oceans, the effects of the firn layer can make them more vulnerable by allowing deeper crevasse propagation. However, without the presence of any meltwater, full-depth penetration of surface crevasses in ice shelves may not be possible. Therefore, an important aspect to explore in a future study is the effect of the firn layer on basal crevasse propagation in ice shelves, which likely controls rift formation and iceberg calving.

The analytical (closed-form) and numerical (polynomial-fitted) solutions developed in this paper provide a more realistic description of the longitudinal stress in glaciers and ice shelves. The effect of depth-varying firn and/or ice properties could be accounted for in the shallow-shelf approximation (SSA) and could be coupled with LEFM models to potentially enable the prediction of rift propagation. proposed a simple method for estimating crevasse depths in an ice shelf, wherein the longitudinal stress calculated from the full-Stokes model was used to evaluate the stress intensity factors based on an LEFM model. A similar approach can be developed to estimate crevasse depths from shallow-shelf models or remote sensing data. In Appendix , we discuss how the 2D stress fields obtained from a shallow-ice-shelf model can be augmented to include the depth-dependent density and Young's modulus. This stress can then be used within an analytical LEFM model to predict crevasse depth and calving.

We acknowledge that there are a few limitations in the current study. Firstly, we assume our glacier and/or ice shelf geometry to be a 2D rectangle with the free-slip condition at the base, which is highly idealized and does not consider any contributions from frictional sliding at the base for the grounded case or buttressing stresses for floating ice shelves . Another limitation of the current model is that the rate of firn consolidation is assumed to be uniform, with a horizontal position, and is assumed to be unaffected by glacier thickness H. However, firn densification is dependent on environmental factors including accumulation rates, overburden pressure, temperature, and local strain rates . Firn densification near terminus regions or in thinner glaciers potentially results in a thinner firn layer and thus a reduced value of the parameter D, indicating a shorter length scale for the transition between firn and dense ice properties. However, while the findings presented here are all based on the density profiles from the Ronne Ice Shelf , the analytic models (and provided MATLAB code) allow for an easy way to evaluate the impact for specific firn heights. This makes it possible to estimate the impact of including firn properties on the crevasse depth for specific locations.

One final limitation of our analytic models is related to water-filled crevasses. While we investigate the effects of including the effect of firn on the density and Young's modulus, both these effects are (partially) driven by the porosity of the firn. However, when water-filled crevasses are considered, no model is included to account for water leaking from the crevasse into the surrounding firn. For colder ice sheets and deeper crevasses, where the full water contents are surrounded by ice of sub-zero temperatures, this assumption is reasonable: any water that seeps into the surrounding ice or firn will freeze, creating an impermeable ice layer surrounding the crevasse, which will prevent water from permeating further into the firn . As these ice layers are typically very thin, they do not alter the mechanical properties of the ice. However, if more temperate glaciers are considered or if one considers conditions where water-filled crevasses do not penetrate to a considerable depth, the firn and/or ice surrounding the crevasse might not be sufficiently cold to cause this ice layer to form. In such circumstances, the presented model will overestimate the crevasse depths obtained as the saturated firn will reduce the effects of the water pressure within the crevasse by re-distributing this pressure over a larger region surrounding the crevasse.

In this paper, we derived analytical equations for the far-field longitudinal stress, including the effects of surface firn layers, described by depth-varying density and Young's modulus profiles based on field data. These analytic expressions were used to perform fracture propagation studies on isolated air- and/or water-filled surface crevasses in grounded glaciers and ice shelves for the homogeneous (assuming fully consolidated glacial ice) and depth-varying ice cases. The derived analytical equations for the far-field longitudinal stress in grounded glaciers were verified with the stress profiles obtained through finite-element analysis. We also used the phase field fracture model to validate the crevasse depths predicted by the LEFM model. The LEFM model results demonstrate the potentially large impact of including the properties of unconsolidated firn layers within the predictions of crevasse depths. For grounded glaciers, depth-varying firn properties inhibit crevasse propagation, requiring lower ocean water heights and larger meltwater depth ratios for full-depth propagation. In contrast, for floating ice shelves and/or tongues, the consideration of depth-varying firn properties promoted crevasse propagation, in some cases up to 3 times as deep. Thus, with regard to fracture and calving, the firn layer may have a stabilizing effect on grounded glaciers, whereas it may have a destabilizing effect on ice shelves. Overall, this study establishes the importance of including the depth dependence of firn layer material properties in crevasse models using LEFM. Additionally, we propose a simple scheme to integrate our analytical solutions with the stresses obtained from the shallow-shelf approximation, which can allow us to assess the vulnerability of ice shelves to calving, accounting for firn layer effects.