We used PISM's hybrid shallow ice shallow shelf approximation (hybrid SIA + SSA). This is the combination of the shallow-ice (SIA; Hutter, 1983; Mangeney and Califano, 1998) and shallow-shelf approximations (SSA; Bueler and Brown, 2009; Weis et al., 1999), enabling PISM to represent both the vertical deformation and longitudinal stretching of the ice, along with basal sliding. This hybrid SIA + SSA has been applied in other mountain valley-based glacial systems (e.g., Candaş et al., 2020; Golledge et al., 2012; Martin et al., 2022; Seguinot et al., 2018).

The stress balance, and the resulting rate of ice deformation (ϵ˙), is described by the Glen-Paterson-Budd-Lilboutry-Duval flow law (Lliboutry and Duval, 1985). This is the default enthalpy-based flow law within PISM, shown in Eq. (1), 1ϵ˙ij=EAT,ωτn-1τi,j where E is the enhancement factor, A is the ice softness, T is the ice temperature, ω is the liquid water fraction, τ is the stress imposed on the ice, and n is the Glen's flow law exponent. E is implemented for both the SIA and SSA, acting as a multiplier on the ice softness inferred from A. Therefore, higher values are likely to represent softer ice that deforms more readily, while lower values of E represent stiffer ice.

For the sensitivity tests, we changed the parameterisation of E for both the SIA and SSA. Many studies have varied ESIA with values between 1 and 6 (Candaş et al., 2020; Ely et al., 2024; Johnson et al., 2023; Zinck and Grinsted, 2022), and ESSA between 0 and 1.5 (Martin et al., 2022; Seguinot et al., 2018; Yan et al., 2023). We varied both ESIA and ESSA at the same time between 0.2 and 20 (see Table 1). This wider range was used due to previous observations of E for SIA within lab studies have found values between 1.3 and 10.2 (Treverrow et al., 2012), and up to 120 in field studies over the Urymqi Glacier No. 1 in China (Echelmeyer and Zhongxiang, 1987). While no observations of E for the SSA are detailed, modelling studies (as shown above) have used narrower values. By applying an extended range to both E for SIA and SSA, we aim to test whether strongly reduced or enhanced deformation could substantially affect modelled output ice volume, thickness, and velocity, and therefore whether these parameters should be prioritised in future parameter ensembles.

In PISM, the subglacial hydrology and sliding scheme was originally developed for ice-sheet contexts and conceptualises the bed as a deformable layer, to represent subglacial “till” or sediment, that can store water and influence basal resistance (Albrecht et al., 2020). We therefore refer to these parameters collectively as the “T component”, where “T” denotes till-related subglacial properties. The extent to which this subglacial sediment is under ice sheets is unknown, which is also the case for Andean glaciers (Cuffey and Paterson, 2010). Although, thick layers of sediment are common in mountain glacier forefields due to repeated glacier advance and retreat phases, and meltwater reworking (e.g., Carrivick and Heckmann, 2017, Lee et al., 2022). However, the formulation for glacier sliding and hydrology does not require, and should not be interpreted as, sediment to be present everywhere beneath the glacier. The effective pressure and sliding behaviour can equally represent hard-bedded conditions, where subglacial water storage may occur within bedrock cavities rather than within sediments (Cuffey and Paterson, 2010; Zoet and Iverson, 2020). Therefore, while we use the term “till” throughout this study for consistency with PISM terminology and previous studies, it should not be interpreted as implying continuous sediment cover beneath Andean glaciers.

The yield stress of the basal material (τc) in PISM is calculated using the Mohr-Coulomb criterion, which incorporates the till friction angle (ϕ), a parameter influenced by the underlying bed geology (Albrecht et al., 2020; Cuffey and Paterson, 2010). This relationship is partly governed by PISM's subglacial hydrology model. The Mohr-Coulomb criterion used to compute yield stress is given in Eq. (2), 2τc=c0+tan⁡ϕNtill where c0 is the till cohesion that uses a default value of 0 (Schoof, 2006), and Ntill is the effective pressure at the base of the ice within the till layer. For every domain we applied a spatially uniform ϕ. Previously used values of ϕ have generally been within ranges of values 5–45°, derived from lab-based experiments of different till types (Cuffey and Paterson, 2010; Koloski et al., 1989). Lower values of ϕ represent a weak, slippery bed that promotes basal sliding due to a low yield threshold, while higher values represent a strong, rigid bed that increases basal friction and makes sliding harder to occur. The default value in PISM is 30°, while in the sensitivity tests, we varied ϕ between 5 and 45° (see Table 1).

Within Eq. (2), Ntill is determined in part by the hydrology beneath the ice. The hydrological model used here is a non-conserving model (Tulaczyk et al., 2000). This does not allow the conservation of any water above an assigned till water thickness (Wtillmax). This is where Wtill is constrained between 0 and the prescribed Wtillmax value. Any water exceeding this is removed from the till water layer and is not retained. The thickness of the water layer stored within the till is determined by Eq. (3), 3∂Wtill∂t=mρw-C where m is the basal melt rate, ρw is the density of fresh water (1000 kg m⁻³), and C is the till water decay rate that denotes how fast water is evacuated from the till water (Albrecht et al., 2020; Flowers, 2015).

C and Wtillmax are tested in our sensitivity analysis through the T Component. To our knowledge, no previous PISM studies have varied C over mountain glaciers, although PISM ice sheet studies have used values between 1 to 10 mm yr⁻¹ (Albrecht et al., 2020). Higher values of C drain the till faster, analogous to efficient drainage systems, that is likely to cause less sliding overall, while lower valuers of C represent a more inefficient drainage, leading to more water within the till and likely allowing more sliding to occur. Here we vary C between 0.1 and 12 mm yr⁻¹.

For Wtillmax previous PISM mountain glacier studies have used values between 1 to 5 m (Candaş et al., 2020; Žebre et al., 2021). High values of Wtillmax allow more water to be retained within the till at the base of the ice. The Wtillmax is varied between 0.1 and 10 m (see Table 1). These ranges either extend beyond previously used values for mountain glacier modelling studies, or applied ranges that have been used over ice sheet scales, aiming to assess whether either parameter has a substantial influence on modelled output of ice volume, thickness, or basal velocity.

In PISM, basal sliding is represented by relating the basal shear stress (τb) to both the ice velocity (u) and effective pressure (N). A velocity threshold (uthreshold) marks when τb equals the yield stress (τc), and therefore when sliding occurs (Cuffey and Paterson, 2010). Within PISM we used the Zoet and Iverson (2020) slip law, which introduces a regularisation term that enables a smooth transition between the viscous-style Weertman sliding (Weertman, 1957), and the Coulomb-plastic behaviours (Aschwanden et al., 2013), without needing prior knowledge of bed type. The Zoet and Iverson (2020) slip law is expressed in PISM by Eq. (4), 4τb=-τcuu+uthresholdqu1-q Zoet and Iverson (2020) in their equation parameterise q=1/m where m= 5, whereas PISM's default value of q is 0.25 (or where m= 4). While the Zoet and Iverson (2020) slip law is relatively new in PISM, being introduced in v2.0, few PISM studies have utilised it. Those modelling efforts that have used the Zoet and Iverson (2020) slip law, (e.g., the Community Ice Sheet Model or CISM; Lipscomb et al., 2019), have varied it between narrow ranges. These have been at 0.2 (Khan et al., 2022; Moreno-Parada et al., 2023), 0.23 (Maier et al., 2022), or 0.33 (van den Akker et al., 2025; Hoffman et al., 2022; Joughin et al., 2024). Within our sensitivity analysis, q and uthreshold were tested through the S component.

We varied q from 0.05 to 0.95 to maximise the coverage of potential parametrisations of q to the extremes. Due to the use of the Zoet and Iverson (2020) slip law in this study, increasing the value of q can lead to decreased resistance were there are low to moderate basal velocities, or where it is near the onset of sliding, increasing velocities in those areas. Where there are the fastest velocities (i.e., ice flowing into valleys), there is likely to be minimal effect on their velocities due to basal shear stress already being at, or near, to the yield stress.

We also varied uthreshold, a parameter that has seen variation for sensitivity or optimisation within a limited number of studies to our knowledge. Bevan et al. (2023), applied the BISICLES ice sheet model to the Amundsen Sea Embayment in West Antarctica, used values between 20 to 600 m a⁻¹. Due to the uthreshold only being varied over ice sheet settings, we varied the uthreshold between 20 and 200 m a⁻¹ (see Table 1). The maximum value we use takes into account the mountain glacier setting we are studying: mountain glaciers are unlikely to reach the high velocities that are achieved by ice streams (> 200 m a⁻¹). It is anticipated that higher parameter values for uthreshold would lead to sliding occurring over less of the glacier, but higher maximum basal velocities. Higher thresholds will delay the transition to Coulomb-limited sliding, but once the threshold is exceeded, higher sliding velocities will occur. Conversely, decreases in the uthreshold are likely to lead to larger portions of the glacier experiencing sliding, but lower maximum velocities.

We used PISM's default positive degree day (PDD) temperature-index scheme (Calov and Greve, 2005) to generate ice within the domains. This required monthly mean air temperature and yearly precipitation (see Sect. 3.3).

Within the PDD scheme, there is stochastic “white noise” to simulate additional undetermined daily variability, as well as a daily temperature standard deviation that is set by default at 5 °C (Winkelmann et al., 2011). We acknowledge that the treatment of sub-monthly temperature variability within the PDD model can influence simulated melt and is therefore another source of uncertainty within the model (Seguinot, 2013).

We forced the model with a constant present-day climate (see Sect. 3.3), to allow glacial ice to reach steady state with its surrounding climate. As this study is focused on purely parameters that influence ice flow, parameters that affect the PDD model component of PISM, such as degree day factors, were kept at their default values and not varied within this study. Any minor fluctuations in steady-state ice extent associated with the PDD scheme are consistent across the ensemble and do not affect the relative comparison between ice-flow parameter perturbations.

The T component parameter tests investigated the subglacial water decay rate (C), the maximum thickness of subglacial water (Wtillmax), and basal friction angle (ϕ). Summary statistics for the T component tests are presented in Table 5 and Fig. 7. Among all domains when the parameters were varied, the Copiapó domain, being the smallest, exhibited the largest change in simulated ice volume (- 40.5 %). The second largest change (+ 28.3 %) occurred in the Mendoza, Maipo and Rapel domain, the largest and most ice-rich domain.

Varying the subglacial water decay rate (C) between its minimum and maximum values (Table 1) resulted in ice volume changes of - 1.4 % to + 8.6 % respectively across most domains. No change was observed in the Copiapó domain. This may reflect the small glacierised area within this domain, where changes in subglacial hydrological parameters have limited influence on domain-mean outputs. It may also reflect the simplified representation of subglacial hydrology in PISM, which may not fully resolve hydrological variability beneath small mountain glaciers at the model resolution used here. Ice thickness and basal velocity changes across all domains were minor or negligible (e.g., no change in the Copiapó domain, Fig. S25). Minimum C values slightly reduced ice thicknesses (mean: - 1.0 %) and increased velocities (mean: + 1.6 %, - 7.5 % in the Vilcanota domain). Maximum C values increased thickness (mean: + 5.1 %) and decreased velocities (mean: - 14.6 %), reflecting the larger deviation of the maximum (12 mm yr⁻¹) from the default (1 mm yr⁻¹) relative to the minimum (0.1 mm yr⁻¹).

When Wtillmax was varied between its minimum and maximum values (Table 1), it resulted in ice volume changes of between -1.0 % to + 7.7 % across all domains (Fig. 7). No changes were seen across the Copiapó domain. Minimum Wtillmax saw minimal reductions in ice thickness (mean: - 0.2 %) and increases in velocity changes (mean: + 3.3 %), while maximum Wtillmax provided slightly increased ice thickness (mean: + 2.5 %) and reduced ice velocity (- 6.6 %) across all domains (Fig. 8). A stronger reduction of - 13.1 % in ice velocity was identified in the Vilcanota domain with minimum Wtillmax values. This may indicate that, in this domain, reducing the maximum till water thickness, and thus the water storage, slightly increased basal resistance in parts of the glacier bed where sliding occurs. However, the response remains small relative to the effects of the till friction angle (detailed below).

The parameters Wtillmax and C had minimal, to no, impact on simulated ice outputs across all domains (Fig. 7). Similar minor effects of Wtillmax over other valley glacier modelling efforts were reported by Candaş et al. (2020) and Žebre et al. (2021), although they saw greater sensitivity in their output than in our study due to Wtillmax being varied in conjunction with the till effective fraction overburden (δ). No PISM-based studies to our knowledge have assessed sensitivity to C for valley glaciers. However, variations in C have been shown to have an influence over ice sheets settings. Albrecht et al. (2020) details that increasing C from 1 to 10 mm yr⁻¹ can cause PISM to simulate an additional 11 m sea level equivalent (SLE) of meltwater from the Antarctic Ice Sheet over multiple glacial cycle timescales. This stronger influence in ice sheet settings is likely due to the greater role of subglacial hydrology in driving ice streaming, influencing basal resistance and therefore ice discharge (Kazmierczak et al., 2022; Verjans and Robel, 2024). While subglacial hydrology does not affect mountain glaciers to the same extent (Mair et al., 2002), they can affect glacier motion on diurnal time scales (Nienow et al., 2005) which would make modelling their interaction difficult. Our results indicate that these specific subglacial hydrology parameters have a limited impact on modelled ice outputs in our PISM simulations and can be removed from future sensitivity analysis. This may reflect the model resolution used here, that may limit the influence of local basal-topographic variability on sliding behaviour, or the use of the simplified representation of a non-conserving subglacial hydrology (see Sect. 3.1.2) in PISM when applied to small mountain glaciers. Alternative PISM hydrology schemes, such a steady flow or routing model, may produce stronger sensitivity in small, steep glacier catchments, allowing it to better represent the subglacial hydrology.

When ϕ was varied (Table 1), simulated ice volumes saw changes between - 40.5 % with minimum values and up to + 23.4 % with maximum values across all domains. Minimum values of ϕ led to substantial reductions in ice thickness (mean: - 24.5 %) due to increases in ice velocity (mean: + 81.9 %), while maximum ϕ values led to increases in ice thickness (mean: + 19.3 %) and reductions in ice velocities (mean: - 23.3 %) (Fig. 9). The most extreme differences were seen in the Copiapó domain, due to the region incurring the smallest glacier area, and any changes can lead to larger relative (%) changes. The values above represent the domain-mean responses. Spatially, the velocity response to variations in ϕ are not uniform. Although increasing ϕ reduced mean basal velocity across the domains, localised increases in basal velocity occurred in some areas (see Fig. 9 Phi_max). These localised increases likely reflect redistribution of ice flow where changes in basal resistance altered glacier stress balance. Similarly, while reducing ϕ increased domain-mean basal velocity, localised decreases occurred in some areas, likely due to redistribution of flow towards faster-flowing parts of the glacier system.

Among the T component parameters, ϕ accounted for the greatest variance in simulated ice volume (Table 7), with a consistent influence across all domains (Fig. 7). Due to ϕ representing how resistant the subglacial sediment is to shear deformation, lower values represent wet fine sandy sediments promoting more basal sliding, while high values represent coarser dry gravels, or bedrock, reducing basal sliding (Koloski et al., 1989; Cuffey and Paterson, 2010). This therefore led to decreased domain-mean subglacial velocities, with higher values of ϕ leading to overall thicker ice (see Phi_max in Fig. 9), while lower values of ϕ increasing domain-mean velocities leading to overall thinner ice (see Phi_min in Fig. 9). The influence of ϕ over the ice basal velocities is consistent with the intuitive nature of increased values of ϕ increasing basal resistance and thus limiting basal sliding. However, the response was not uniform spatially with ice basal velocities presenting the opposite to the domain-mean pattern within localised areas (i.e., increase ϕ seeing increase localised velocities and vice versa). These are likely due to changes in the ice divides and flow regimes that can lead to subsequent changes in the ice thicknesses and ice velocity. This localised vs. domain-mean effect of the ϕ is consistent with previous studies showing that, even when using spatial uniform parameters, there can still be spatially variable basal conditions, that can strongly influence velocity structure and sliding patterns across the model domain (Gowan et al., 2023; Johnson et al., 2023). In comparison with other studies, while ϕ has not been explicitly varied in previous mountain glacier studies, to our knowledge, when modelling ice sheets ϕ is a key control on ice volume and subsequent ice dynamics (Albrecht et al., 2020; Koldtoft et al., 2021; Lowry et al., 2020). For example, lower ϕ values saw a reduction in modelled LGM volumes of the Antarctic Ice Sheet leading to accelerated retreat, whereas higher ϕ values tended to overestimate present-day ice sheet thicknesses (Albrecht et al., 2020; Lowry et al., 2020). Our findings highlight its importance in mountain glacier settings. Though a uniform ϕ was used here, it likely varies with catchment-specific geology (Bareither et al., 2008; Clarke, 2018), suggesting future studies should tune ϕ regionally to improve accuracy in ice dynamics and volume simulations.

When all T component parameters were varied between their minimum, default, and maximum values, simulated ice volume differed by up to + 40.5 % relative to the default simulations. Across all domains, ice volumes cluster into three distinct groups centred on the minimum, default, and maximum ϕ values, most clearly seen in the Copiapó domain (Fig. 7D) and the Mendoza, Maipo and Rapel (Fig. 7E) domain. While ϕ exerts dominant control over ice volumes, C and Wtillmax cause only minor variations within these groups. The highest volumes occurred when ϕ and other subglacial parameters were set to their maximum values {Allmax}. Pearson correlations (Table 6) confirm the strong overwhelming influence of ϕ on simulated ice outputs, with an average coefficient of 0.94 across all domains. Moreover, ϕ was the only subglacial parameter with a statistically significant effect (p≤ 0.01), underscoring its primary role in controlling model outputs in PISM.

Across both univariate and multivariate parameter tests, ϕ consistently exerted the strongest influence on model outputs among the subglacial parameters. This is due to its role in the Mohr–Coulomb criterion, which governs the pseudo-plastic sliding law and modulates basal resistance (Cuffey and Paterson, 2010). Higher ϕ values increase basal resistance, slowing ice flow and leading to thicker ice, thereby raising total ice volume while having limited effect on ice extent. This relationship is reinforced by the “all max” scenario, which produced the thickest and highest volume ice across nearly all domains.

The S component tests focus on two parameters: the sliding exponent (q) and the velocity threshold (uthreshold). Summary statistics for these tests are presented in Table 7 and Fig. 10. The PISM domains of Copiapó and Mendoza, Maipo and Rapel, representing the smallest and largest glaciers respectively, display the most pronounced responses to parameter variation, with maximum ice volume changes of 44.1 % and 30.0 %, respectively.

When uthreshold was varied between its minimum and maximum values (Table 1) produced ice volume differences of + 17.1 % to - 7.2 % respectively, with an absolute average difference of 6.5 %. Across all domains minimum uthreshold saw increased ice thicknesses (mean: + 9.1 %) and basal velocities (mean: - 19.2 %) (Fig. 11). Maximum uthreshold saw reduced ice thicknesses (mean: - 3.7 %) along with increased basal ice velocities (mean: + 7.8 %).

Variations of the uthreshold, which controls the onset of basal sliding, leads to when the uthreshold is set to lower values, ice flow velocities are decreased, increasing ice thickness and volume. However, while overall flow patterns remain very similar, their intensity shifts with varied uthreshold values. As can be seen in Fig. 11, with decreased uthreshold values overall mean velocities decreased, but small localised areas of increased velocities (∼ 10 to 20 m yr⁻¹) are seen where in the default run saw lower velocities occurred. When the uthreshold is increased, overall mean velocity increased, with areas of already faster flowing ice saw an increase in velocity (∼ 20 m yr⁻¹), with locations of localised slower velocities remaining the same as those in the default. This behaviour is in line with the expected behaviour described in Sect. 3.1.3, whereby increases in the uthreshold delaying the transition to Coulomb-limited sliding that facilitates faster flowing ice. Despite this influence, uthreshold is rarely tested in mountain glacier modelling, with most studies using a fixed 100 m yr⁻¹ value (Martin et al., 2022; Seguinot et al., 2014, 2018). Our results, spanning 20 to 200 m yr⁻¹, show that uthreshold meaningfully affects modelled dynamics and should be included in future sensitivity analyses.

When q was varied between its minimum and maximum values (Table 1), it produced a difference in the ice volume between + 39.6 % and - 42.3 %, with an absolute average difference of 20.4 %. The two largest differences in ice volume detailed before were all seen in the smallest domain of Copiapó (- 42.3 %), the second highest difference is seen in Mendoza, Maipo, and Rapel domain (- 27.7 %), both when q is set to its maximum value. Across all domains when q was set to its minimum, there was an increase in ice thickness (mean: + 18.9 %) and a decrease in ice velocity (mean: - 33.6 %), when set to its maximum there was a decrease in ice thickness (mean: - 21.7 % and an increase in ice velocity (mean: + 75.5 %) (Fig. 12).

Variations of q within PISM exert a clear influence on simulated ice dynamics, due to its role in controlling the non-linearity of the basal sliding law (Zoet and Iverson, 2020). Higher q values suppress fast-flowing regions (e.g., > 25 m yr⁻¹) but enhance sliding in slower-flowing regions, producing a more diffuse velocity field (see qmax in Fig. 12). In contrast, lower q values concentrate flow into narrow corridors, altering ice divides and increasing ice thickness in surrounding slower-flow regions by limiting basal sliding (see qmin in Fig. 12). Among PISM studies, q is the most frequently varied sliding parameter, however, this in within the context of using the default Coulomb sliding model. Using the Coulomb sliding model Candaş et al. (2020) over valley glaciers found that varying q altered ice volume by + 22.6 % at q= 0 and - 26.4 % at q= 1. In ice sheet contexts, effects are mixed with Albrecht et al. (2020) reporting lower q reduced velocity and increased Antarctic volume at the LGM by up to ± 3 m SLE. Over Greenland, Aschwanden et al. (2019) shows that the variance of q parameterization of 0.25 to 1.0 can lead to uncertainties on SLE contributions of 26 %–53 % by 2100, 5 %–38 % by 2200, and 2 %–33 % by 2300. While the Zoet and Iverson (2020) slip law has been not used by other PISM modelling studies, no study to have used the slip law within ice sheet models (e.g., CISM; van den Akker et al., 2025) varied the parameterisation of the sliding exponent extensively. Our findings here support the previous conclusion that q significantly affects modelled ice volumes, particularly in regions dominated by valley-confined dynamic flow (see Sect. 4.2.2). The results, at least for q are the first to be presented using the Zoet and Iverson (2020) slip law. We therefore recommend that future valley glacier modelling studies, especially those focused on mass change, include q in their sensitivity analyses.

When q and uthreshold are varied together between their default, minimum, and maximum values, the largest ice volume difference from the default simulation reaches - 44.1 %, observed in the Copiapo catchment (Fig. 10). Excluding this smallest domain, the maximum difference is - 30.0 % in the Mendoza, Maipo and Rapel domain. While q alone exerts the strongest influence, combining both parameters amplifies their effects {Allmax}. This is particularly evident when both are set to their minimum or maximum values, resulting in greater or lesser increases in ice volume than when varied individually.

The Pearson correlation analysis (Table 8) confirms q as the dominant control on sliding-related sensitivity, with strong correlations across nearly all domains except in the Santa catchment. Although the number of combined simulations is limited, uthreshold still produces noticeable changes in simulated outputs (Fig. 11), but its influence remains secondary to q when both are varied simultaneously. This is likely because they both alter ice velocities, making it easier or more difficult for sliding to occur. This supports that these two parameters should continue to be investigated by future model efforts over mountain glaciers.