Basal topography exerts a critical control on ice-sheet motion at all scales considered , but its influence at the intermediate scale – between the 0.5–25 m scale typically used to determine sliding parametrisations and the 400–4000 m scale typical of ice-sheet model fidelity – is particularly poorly understood. Notably, the ice-motion influence of incised fjord landscapes remains almost entirely unexplored. This landscape characterises the terrestrial western margin of the palaeo Scandinavian Ice Sheet and provides a reasonable proxy for marginal areas of the Greenland Ice Sheet (GrIS) which share geological and topographic similarities e.g., as well as regions of the Antarctic Ice Sheet (AIS) such as the Aurora Subglacial Basin . In western Norway, glacial striations and paleoglacial flow directions show perpendicular ice flow over deep subglacial fjords, as well as in oblique and parallel orientations (Fig. ). The implications for ice motion in these flow scenarios remain unclear.

Here, we focus on Veafjorden close to the city of Bergen, Norway (Fig. ) as a characteristic example with a relief of ∼1300 m, and a narrow ∼2 km width. Striation markings on both sides of Veafjorden confirm near-perpendicular flow across the fjord around 15 kyr ago . The Norwegian fjords were likely incised into pre-existing river valleys and geological weaknesses through erosion as the Scandinavian Ice Sheet grew and shrank, but not while the ice sheet was at its fullest extent . In common with a comparable study from south-west South America , it may be that geological factors, rather than dynamic self-organisation of ice streams , exert a first-order control on fjord orientation.

Ice motion perpendicular to idealised subglacial valleys has previously been modelled in 2D and 3D , with investigating the role of a critical angle in V-shaped valleys to predict the onset of Moffatt eddies (spiralling ice flow that form in topographic hollows). However, these studies did not include temperate ice rheology or rate-dependent resistance for slip at the ice-bed boundary, with simplified 2D topography in the case of and simplified 3D topography and a 2D radar transect profile in the case of . Here, we incorporate these additional processes and consider fast ice motion for a section of the palaeo Scandinavian Ice Sheet over 3D high-resolution digital elevation data. In order to assess the difference in controls on ice motion between this high-resolution topography, and the smoother bed products used to model the GrIS and AIS (BedMachine and BedMap), we create a smoothed representation of Veafjorden (henceforth referred to as the smoothed control topography). This substitute provides the basis to examine the impact of the generally low fidelity elevation models presently in use.

Our ensemble consists of 16 simulations, each with a unique ice flow scenario. The varying parameters are (1) target surface velocity, (2) flow direction – parallel, oblique and perpendicular to the fjord incision (See Fig. ) – and (3) plateau ice thickness, ice thickness measured from the highest point in our domain. The flow direction is varied both to simulate the historical shift in flow angle during deglaciation, and to assess how anisotropic bedrock influence ice dynamics. We also perform 4 comparison simulations with the smoothed topography. The smoothed control topography simulations have perpendicular flow direction while target surface velocity and plateau ice thickness are still varied. By simulating Veafjorden with both the smoothed control topography and the high-resolution topography, we reveal the influence of realistic anisotropic basal conditions and limited bed topography fidelity in ice-sheet models.

Over a single glacial cycle, the orientation of ice motion varies , while the landscape remains effectively immutable. The simulations presented here, and consideration of how these extend to the broader palaeo Scandinavian Ice Sheet, indicate that over anisotropic topography complex basal motion patterns should be expected as the norm rather than the exception. Finally, while our simulations reveal ice dynamics over fjords in great detail, considerations of simple aspects of fjord geometry point towards the influence such landscapes should exert on intermediate, or macro, scale sliding relationships.

We model 3D ice motion over an 8 km by 4 km rectangular domain covering Veafjorden (Fig. ). The Digital Elevation Model (DEM) was constructed at a resolution of 20 m with data from Kartverket (The Norwegian Mapping Authority), combining terrestrial data at 1 m , with the fjord bathymetry interpolated from a 50 m resolution data set . To conform to the model's periodic boundary conditions the inflow and outflow boundaries must match. We achieved this using a tapering algorithm following , with one third of the down-flow length and width of the domain added as a tapering region that then matches the topography at the inflow boundary (Fig. ). We applied a Gaussian filter with a standard deviation of 1.5 to the DEM to remove artifacts and sharp edges which can present model stability issues . Additionally, a smoothed control topography was created for comparison runs by applying a Gaussian filter with a standard deviation of 50 grid cells to the original DEM.

The domain was discretised with gmsh for Elmer/Ice Version 9.0 using a triangular mesh with a representative element side length of 25 m and 20–30 vertical layers with resolution increasing towards the base. The smoothed control topography mesh has a representative side length of 100 m and 15 vertical layers with 500 m plateau ice thickness. In a first stage, the free surface is allowed to vary and the inflow and outflow and two lateral boundaries are matched periodically for velocity, stress, and free-surface position. This simulation stage runs until a steady state is reached and surface elevation no longer varies. In the second stage, the free surface and the inflow velocity fields are fixed and the enthalpy field is allowed to evolve without a periodic boundary condition requirement. Outflow/side boundaries are set at lithostatic pressure/zero flux depending on simulation orientation (Fig. ).

The central equations and boundary conditions follow with minor adjustments to geometric set up. Table  lists all parameter values. We solve the standard Stokes equations for ice flow 1∇⋅u=0 (conservation of mass)2∇⋅τ-∇p=-ρg (conservation of momentum) where u (m a-1) is the velocity vector, τ (MPa) is the deviatoric stress tensor, p (MPa) is ice pressure, ρ (kgm-3) is the ice density and g is the gravity vector described as 3g=[g sin(θ),0,-g cos(θ)] where g=9.81 ms-2 and θ is the domain slope. In simulations, the x axis is oriented perpendicular to the fjord incision, and the y axis along the fjord. The z axis is normal to the horizontal plane, and does not match the g vector. Adjusting the orientation of g removes the requirement for vertical displacement of periodic inflow-outflow boundaries.

Stress is related to strain using the Nye–Glen isotropic flow law : 4ϵ˙=Aτen-1τ where ϵ˙ (s-1) is the strain rate tensor, τe2=12tr(τ2) (MPa) is the effective stress in the ice, n is the flow exponent set to 3. The creep parameter, A (MPa-3a-1), varies depending on whether ice is above or below the pressure melting point Tm. For ice below Tm, A is set using the homologous temperature, Th (K), while ice above Tm, is determined by the water fraction ω: 5A=A1exp⁡Q1RTh,Th≤TlimA2exp⁡Q2RTh,Tlim<Th<Tm(W1+W2ω×100)W3,Th≥Tm and ω<ωlimAmax,ω≥ωlim where Tm(p)=Ttr-γ(p-ptr) and γ (K MPa-1) is the Clausius–Clapeyron constant, and Ttr and ptr are the temperature and pressure triple points for water, respectively. A1 and A2 (MPaa-1) are rate factors, Q1 and Q2 (J mol-1) are activation energies for T≤Tlim and Tlim<T<Tm, respectively, where Tlim is the limit temperature. R (J mol-1) is the gas constant, W1, W2, and W3 (MPaa-1) are water viscosity factors (constant for all simulations) with default values from adapted from . The liquid water fraction limit, ωlim is set to 2.5 % following experiments of . If ωlim is exceeded then A is limited to a maximum value Amax. Very recent studies propose n=1, or different relationships for A for temperate ice , but we leave exploration of these parameters for a future study. Values for these and subsequent parameters are provided in Table .

Within the Elmer/Ice EnthalpySolver , specific enthalpy, H (Jkg-1), is used as the state variable and is related to T and ω as 6H(T,ω)=12CaT2-Tref2+CbT-Tref,H<Hm(p)ωL+Hm,H≥Hm(p) where Ca (J kg-1K-2) and Cb (J kg-1K-1) are enthalpy heat capacity constants, L (J kg-1), is the latent heat capacity of ice, Hm(p)=12CaTm(p)2-Tref2+CbTm(p)-Tref is the specific enthalpy at the pressure melting point, and Tref is the reference temperature.

At each time step, the enthalpy field is evolved until a steady-state is reached and is calculated as ρ∂H∂t+u⋅∇H=∇(κ∇H)+trτϵ˙7(conservation of energy) where trτϵ˙ is the strain heating term and κ (kg m-1s-1) is the enthalpy diffusivity defined as 8κ=κc,H<Hm(p)κt,H≥Hm(p) where κc and κt are enthalpy diffusivities for cold and temperate ice respectively , meaning that water movement within the temperate ice is assumed to be a diffusive process.

The change in the position of the free surface, s, is calculated as 9δsδt+uxδsδx+uyδsδy=uz in the Elmer/Ice FreeSurfaceSolver where ux, uy, and uz are components of u.

The lower boundary velocity is set to zero normal to the surface (impenetrability condition at base): 10u⋅n=0 where n is the normal vector to the bedrock. Basal traction τb is calculated following as: 11τb=CNeub-n+1ub+AsCnNen1nub where C (dimensionless) is a parameter dependant on basal morphology . Ne=pi-pw (MPa) is the effective pressure where pi (MPa) is the ice overburden pressure and pw (MPa) is the subglacial water pressure, with N set at 24 % of pi following . ub (m a-1) is the basal velocity tangential to the ice-bed interface. The flow exponent n is the same as for the strain rate and set as 3. As (m a-1MPa-3) is the average sliding coefficient based on six values from .

Our results demonstrate the strong control that subglacial fjords and their orientation exert on ice-sheet motion (Fig. ). For flow-perpendicular simulations, Moffatt eddies and basal flow reversal occurs for both 500 and 1000 m ice thickness above plateau, but only with high-resolution (i.e. not smoothed) topography. Deep perpendicularly-oriented valleys beneath an ice sheet also significantly impede overlying ice motion – comparing one smoothed control simulation (8Sm10, see Table ) to its high-resolution topography counterpart (8Pe10) yields an increase in slope of 25.8 %, equivalent to a change in area-averaged driving stress from 295.2–424.8 kPa (44 %). Alternative fjord orientations also significantly influence both motion patterns and temperate ice distribution patterns. Here, we cover the following aspects: (Sect. ) perpendicular flow and the associated formation of Moffatt eddies, (Sect. ) parallel flow, and (Sect. ) separation of flow in oblique-orientation simulations. Last, (Sect. ) we compare smoothed-topography control simulations to their high-resolution counterparts.

Our results show that realistic fjord geometries – which are common across the margins of the palaeo-Scandinavian ice sheet and also likely across the present day GrIS, yet which are largely excluded from large-scale ice-sheet models – significantly complicate patterns of ice motion and temperate ice. Comparing smoothed control and high-resolution topography simulations, when held at similar surface velocities by adjusting the domain slope, reveals substantially higher driving stresses in the case of high-resolution topography. This is clear evidence for a strong anisotropic response of ice motion to the alignment of the underlying landscape. Moffatt eddies might seem like dramatic and isolated features with limited influence on overall ice-sheet motion. However, our results and analysis of flow-aligned slope angles in western Norway (Fig. ) suggest that these features are likely widespread in ice motion over incised fjord landscapes.

The complex flow patterns of Moffatt eddies are also dependant on topographic fidelity. The patterns of flow shown in our perpendicular simulations using high-resolution topography (Fig. a and b), including eddies and lateral transport along the fjord hollow, closely resemble the 3D simulations in Fig. 9 of . The idealised topography in these simulations invite ascribing a critical angle for Moffatt eddy formation. However, this is complicated somewhat by our use of high-resolution topography which is not well represented by the triangular geometry found in . Nonetheless, an opening angle perpendicular to the fjord of ∼100° is smaller than the critical opening angle αc=134° for the rheological exponent n=3 in , while the opening angle ∼170° for the smoothed control simulation is above this αc value. Oblique flow simulations have an opening angle of ∼118°, which would predict the formation of Moffatt eddies. However, no eddies are seen in simulations with these settings. This may be explained by the shift in flow direction exerted by the topography (See Fig. d–f) further widening the opening angle. This indicates that a critical value may also have efficacy in predicting Moffatt eddy occurrence in real settings but that low fidelity bed topography products where depressions such as fjords are not resolved are unlikely to be effective indicators of possible Moffatt eddy locations.

Deep fjords furthermore provide a counter-example to suggestions that basal traction should be treated as bounded across all glacier and ice-sheet settings e.g.. In bounded basal traction relationships (i.e. Eq. ) the traction provided by the bed approaches a maximum value for a given effective pressure that is not exceeded. Whereas, an unbounded basal traction relationship shows a continuous increase in basal shear stress with basal velocity (i.e. ). For hard beds, the suggestion of bounded basal traction follows from Iken's bound , usually expressed as 13τbN≤tan(β) where τb is the basal traction and β is the maximum up-slope angle in the mean flow direction. However, Eq. () omits traction tangential to the ice-bed interface itself and ceases to become an effective bound in the situation of very steep surfaces . In the case of ice flow pressing perpendicularly against the near-vertical cliffs present on Veafjorden's western side (Fig. ), there is no theoretical bound to the resistive traction that could be provided by the cliffs to oppose ice motion. If a large-scale ice-sheet model takes a smoothed basal surface rather than the high-resolution topography, or if the discretisation of the model domain does not permit features such as fjords to be accurately resolved (Fig. ), then the shear-band features and large slope values included in our high-resolution simulations will create a situation, at least locally, where sliding is not well-described by a bounded sliding relationship limited by Iken's bound with a low value of β. Depending on the exact parameter choice, this could lead to a model grid cell with a defined bounded sliding relationship being unable to provide a physically realistic amount of basal traction for that grid cell. As we hold basal-slip parameters (Eq. 11) constant in both settings, our simulations allow quantification of the influence of fjord geometry on driving stress, showing a significant driving-stress increase is required for the same surface velocity when a high-resolution fjord is used rather than a smoothed representation (Fig. ).

Classically, cavities may be anticipated to drown out high bed slopes thereby facilitating bounded basal traction . However, while these studies represent non-dimensionalised problems, their focus is not emphatically on large-scale features such as fjords and the idea of a large cavity within Veafjorden is unintuitive given clear hydrological escape pathways to the north and south (Figs.  and ). Determining the configuration and influence of subglacial cavities in a setting such as Veafjorden should be a focus of future research. Nonetheless, for now we suggest that features such as Veafjorden provide major upslope resistance that is (i) not overcome by cavitation and (ii) not captured by the basal boundary position of ice-sheet models. As the bed slope of cliffs and fjords with flow-aligned slopes exceeding 30° are common across western Norway (Fig. ) and at least partially representative of the basal topography of the GrIS, we suggest that ice-dynamic interactions with these features provides a physically based background to the utility of unbounded power-law sliding relationships across the GrIS , which often invoke Weertman sliding , even if the underlying assumptions are not entirely realistic . Given ice sheet responses to changing climate are typically greater for bounded sliding relationships , exploring this problem represents an important area for future research.

Ice motion across fjords may also help to explain features revealed in the surface velocity and surface position of the GrIS. Deep fjords exist across the ice-free margins of the GrIS, with the continuation of these fjords inland beneath the ice evident until BedMachine mapping begins to lose its sharpness . Flow-aware hill-shading of the GrIS surface and analysis of radio-echo flight lines highlight multiple fast-flowing regions where ice flow is inferred to cross subglacial valleys perpendicularly and which appear to align with along-flow surface velocity variations in excess of 100 ma-1 (Fig. ). This reflects the surface velocity variations modelled in this study (Fig. b and c) and indicates a similar topographic control could be at play. Similarly, such a mechanism could be important in East Antarctica where the Aurora Subglacial Basin and Gamburtsev Mountains exhibit multiple subglacial valleys that cross cut the present-day prevailing flow direction . Furthermore, basal traction inversions from both the GrIS and AIS evidence complex banded patterns, hypothesised by to result from pattern-forming instabilities in subglacial water pressure. We suggest that they may in fact reflect varying resistance as a result of subglacial topography, indicating the potential influence of large scale topographic obstacles on basal traction fields derived from relatively smooth bed topography products.

Our results also illustrate the anisotropic resistance to ice-motion provided by real subglacial landscapes (Fig. ), and situations where the basal velocity vector may be non-parallel to its corresponding basal traction vector over a large area (Fig. ) – previously explored in an idealised case by . This implies that basal traction patterns should be expected to vary over the lifecycle of an ice sheet, as drainage basins evolve and ice motion patterns shift during growth and decay. Bulk basal motion may furthermore be misaligned with the applied basal shear stress, which could result in complications for dynamic modelling where fine details are important. We leave direct quantification of the influence of landscape anisotropy on basal sliding relationship anisotropy for future work, but recognise that the impact of basal sliding anisotropy is likely small for most predictive timescales (100–1000 yr) in the event that flow orientations do not shift substantially.

Last, geological evidence for the occurrence of the described flow patterns and Moffatt eddies is presently limited to the plateau striations surrounding Veafjorden . These striation markings indicate near-perpendicular flow across the fjord, but do not provide information about the motion within. Reverse direction striations within the valley are possible, but given subsequent ice-flow reorganisations it is likely that these lineations will have been removed by erosion or overwritten when ice flow switches to follow the fjord orientation.

We show that the orientation of fjords relative to the flow direction has a substantial influence on the ice flow magnitude. Significantly greater (∼41 %–89 %) driving stresses are required to push ice perpendicularly over a fjord compared to smoothed control topography, with a clear anisotropic response to fjord orientation apparent. Steep valley walls also present an obstacle not presently resolved in standard ice-sheet model bed products that may result in unbounded, rather than bounded, traction being appropriate and hence explain the efficacy of the power-law sliding relationships across large parts of the GrIS , which shares basal-topographic similarities with the paleo-Scandinavian Ice Sheet.

At present, it is infeasible and impractical to incorporate this behaviour into large-scale ice-sheet models by increasing resolution and bed-product fidelity. Instead, parameterising the net influence of the behaviour reported here – and that caused by other large-scale topographic obstacles – on basal traction relationships (and basal traction anisotropy) at the more relevant macro scale provides a reasonable pathway for future progress. Doing so may contribute to alleviating uncertainties in predictions of sea level rise , and to reducing the share of uncertainty in basal traction inversions that pertains directly to sliding processes .