Subglacial Conduit Roughness: Insights From Computational Fluid Dynamics Models
Abstract
Flow resistance in subglacial conduits regulates the basal water pressure and sliding speeds of glaciers by controlling drainage efficiency and conduit enlargement and closure. Flow dynamics within subglacial conduits, however, remain poorly understood due to limited accessibility. Here we report the results of the first computational fluid dynamics simulations of flow within a realistic subglacial conduit beneath Hansbreen, a polythermal glacier in Svalbard, Norway. The simulated friction factor is 2.34 ± 0.05, which is around 5 to 230 times greater than values (0.01–0.5) commonly used in glacier hydrological modeling studies. Head losses from sinuosity and cross-sectional variations dominate flow resistance (∼ 94%), whereas surface roughness from rocks and ice features contributes only a small portion (∼6%). Most glacier hydrology models neglect head losses due to sinuosity and cross-sectional variations and thus severely underestimate flow resistance, overestimating the conduit peak effective pressure by 2 times and underestimating the conduit enlargement area by 3.4 times, respectively.
Key Points
- For the first time, flow resistance in a real subglacial conduit is quantified using CFD and structure-from-motion photogrammetry
- Flow resistance in subglacial conduits is dominated by sinuosity and cross-sectional variations, not surface roughness from rocks and ice
- Most glacier hydrology models severely underestimate flow resistance and significantly misrepresent subglacial flow and ice dynamics
Plain Language Summary
Subglacial conduits drain meltwater from polar ice sheets, thus directly regulating the ice sheet sliding speed through basal flow resistance and water pressure inside the conduits. Despite their importance, our understanding of subglacial conduits is extremely limited due to difficulties of observing them and their interiors with either remote sensing or in situ exploration. Simplified models have been proposed for the hydraulics inside these conduits. A key problem in these models is the lack of scientific support in parameterizing the flow resistance. Currently, the resistance is parameterized but has not been validated due to the accessibility issues. To narrow this knowledge gap, we performed three-dimensional computational fluid dynamics simulations based on a millimeter-scale resolution model of an actual subglacial conduit in the Arctic. For the first time we give a direct and physics-based estimation of the flow resistance in an actual subglacial conduit and highlight the important contributions of the cross-sectional variations and longitudinal sinuosity. We further demonstrate the impacts of our simulated flow resistance on subglacial hydrodynamics and ice sheet dynamics.
1 Introduction
Subglacial water pressure exerts an important but poorly understood control on the velocity of glaciers and ice sheets. A growing body of literature has theorized about (Clarke, 2005; Creyts & Schoof, 2009; Röthlisberger, 1972; Weertman, 1972), remotely observed (Hock & Hooke, 1993; Hooke, 1989; Iken & Bindschadler, 1986; Nienow et al., 1998), and modeled (Covington et al., 2012; Creyts & Schoof, 2009; Hewitt, 2013; Mankoff & Tulaczyk, 2017; Schoof, 2010; Werder et al., 2013) glacier hydrological systems, but difficulties in acquiring data in subglacial conduits have limited the understanding of subglacial drainage efficiency and its impacts on ice motion. Consequently, much of our understanding of subglacial hydrological systems has been informed by the results of numerical models. A persistent problem in models of subglacial hydrology is how to parameterize water flow resistance in conduits. Resistance to water flow is related to hydraulic roughness, a loosely defined term encapsulating energy that is dissipated as heat due to turbulence and viscous forces (i.e., headlosses). Because the heat generated by these headlosses is directly responsible for conduit enlargement which further affects the hydrodynamics (e.g., water velocity and water pressure) in subglacial conduits and associated ice sheet dynamics (Bell, 2008), accurate parameterization of roughness and flow resistance is necessary to accurately simulate the evolution of subglacial hydrological systems.
Direct simulations of energy dissipation require resolving scales ranging from the conduit cross-section scale (of the order 1–2 m; Mankoff et al., 2017) to the Kolmogorov scale (of the order 1–2 μm) and require a computational grid number of the order 10^{16} which exceeds the memory capacity of most current computers (Choi & Moin, 2012). Headlosses are therefore commonly parameterized through physical features that contribute to hydraulic roughness. In glacier hydrological models, roughness is usually defined as the difference between the original conduit and a simplified straight pipe, which characterizes the effects of boulders or bedrock protruding into the flow field (defined here as surface roughness). If the height of surface roughness and the hydraulic diameter of conduits are known, hydraulic roughness coefficients such as the Darcy-Weisbach friction factor can be determined from empirical relationships. Relating friction factor to surface roughness height allows friction factors to be calculated as the hydraulic diameters of conduits fluctuate due to melt and creep. In studies of rivers, bedforms and sinuosity, rather than surface roughness, have been shown to dominate flow resistance (Francalanci et al., 2012; Hey, 1988; Millar, 1999; Parker & Peterson, 1980). However, due to lack of data, the impacts of sinuosity and cross-sectional variations on the resistance of subglacial conduits have rarely been considered, which partially explains the discrepancy between the friction factors commonly used in glaciology models and those obtained in field measurements. For example, numerical models of glacier hydrology are typically parameterized with Darcy-Weisbach friction factors in the range of 0.01–0.5 (Boulton et al., 2007; Clarke, 1996; Colgan et al., 2011; Covington et al., 2012; Fowler, 2009; Schoof, 2010; Spring & Hutter, 1981). In contrast, dye-tracing studies suggest that hydraulic roughness values are orders of magnitude higher, such as one dye tracing study in Svalbard found that Darcy-Weisbach friction factors fluctuated between 0.97 and 75 in response to changes of water level in a subglacial conduit (Gulley et al., 2014).
To better quantify the bulk flow resistance in a real subglacial conduit and the contributions of complex surface features (Figure 1), we perform computational fluid dynamics (CFD) simulations over a 10-m-long millimeter-scale resolution 3-D model of an actual subglacial conduit (Figure 1d) beneath Hansbreen (a polythermal glacier in the archipelago of Svalbard; Gulley et al., 2012; Mankoff et al., 2017) and synthetic models (Figure 2a) derived from the actual model. The quantified bulk flow resistance is then fed to a one-dimensional subglacial model (Spring & Hutter, 1981) to evaluate its impacts on subglacial hydrodynamics.
2 Methods
To untangle the effects of different surface features, we processed the conduit surface as follows (Figure 2a). To isolate the effects of sinuosity on flow resistance, we removed the sinuosity from the initial 3-D model surface S_{0}, by straightening its centerline and creating surface S_{1}. To isolate the effects of cross-sectional variations on flow resistance, we removed the cross-sectional variations from the straightened surface S_{1} by subtracting the cross-sectional variations from the surface S_{1} and creating surface S_{2}. Here the cross-sectional variations are defined as the difference between a variable diameter pipe (P_{2}) and a constant diameter pipe (P_{1}), which characterize both the longitudinal shrinking trend and small variations. The diameter of P_{1} is D_{1}= with V_{1} and L_{1} denoting the volume and length of S_{1}. The diameter of P_{2} is calculated by D(x)=2 with A(x) denoting the area of a cross section at location x on surface S_{1} (see Text S1 in the supporting information for details of surface decomposition and the definition of surface subtracting). We categorized the simulations into four groups (0–3) corresponding to surfaces S_{0}, S_{1}, S_{2}, and P_{1} for convenience (see Table 1 for configurations and main results).
Group | Case | Surface | σ_{r} | U_{0} | R_{D} | δ | f_{t} | f_{c}±σ_{f}_{c} | F_{p}/F_{t} | |
---|---|---|---|---|---|---|---|---|---|---|
name | name | name | (m) | (m/s) | 10^{6} | (cm) | (%) | |||
0 | S_{0a} | S_{0} | NA | NA | 1 | 1.34 | 2.19 | NA | 2.46 ± 0.06 | 99.4 |
S_{0b} | S_{0} | NA | NA | 1 | 1.34 | 1.70 | NA | 2.34 ± 0.04 | 99.3 | |
S_{0c} | S_{0} | NA | NA | 1 | 1.34 | 1.32 | NA | 2.34 ± 0.05 | 99.3 | |
S_{0d} | S_{0} | NA | NA | 0.5 | 0.67 | 1.94 | NA | 2.42 ± 0.04 | 99.2 | |
S_{0e} | S_{0} | NA | NA | 1 | 1.34 | 1.94 | NA | 2.41 ± 0.05 | 99.3 | |
S_{0f} | S_{0} | NA | NA | 2 | 2.67 | 1.94 | NA | 2.40 ± 0.05 | 99.4 | |
1 | S_{1a} | S_{1} | 0.21 | 0.16 | 0.5 | 0.66 | 1.65 | 0.135 | 1.95 ± 0.04 | 99.1 |
S_{1b} | S_{1} | 0.21 | 0.16 | 1 | 1.33 | 1.65 | 0.135 | 1.94 ± 0.04 | 99.2 | |
S_{1c} | S_{1} | 0.21 | 0.16 | 2 | 2.65 | 1.65 | 0.135 | 1.94 ± 0.05 | 99.3 | |
2 | S_{2a} | S_{2} | 0.19 | 0.14 | 1 | 1.33 | 1.48 | 0.123 | 0.142 ± 0.003 | 96.4 |
3 | S_{3a} | P_{1} | 0 | 0 | 1 | 1.33 | 1.27 | 0.0111 | 0.0103 ± 0.00004 | 0 |
S_{3b} | P_{1} | 0.027 | 0.02 | 1 | 1.33 | 1.57 | 0.0486 | 0.0467 ± 0.00033 | 0 |
- Note. Symbols denote surface roughness (σ_{r}), transverse scale of a pipe (D_{e}), mean velocity over the pipe inlet (U_{0}), relative roughness (σ_{r}/D_{e}), Reynolds number (R_{D}), and average grid size (δ). Reynolds number is defined as R_{D}=U_{0}D_{e}/ν with ν denoting the kinematic viscosity of water. Average grid size is defined as δ = (V_{i}/N_{c})^{1/3} with V_{i} and N_{c} denoting the total volume and number of grid cells of each computational domain. f_{t} and f_{c} denote the friction factors calculated from existing rough pipe theories and time-averaged friction factor from CFD simulations, respectively. is the 1 standard deviation of the instantaneous friction factor away from the time-averaged value. F_{p}/F_{t} is the ratio of pressure force to the total resistance force. (See values of the above variables in section 2 and Appendices Appendix A and Appendix B; see values V_{i}, N_{c}, and the maximum and minimal values of f_{c} in Table S1). CFD = computational fluid dynamics; NA = not applicable.
Flow resistance greatly impacts the physical properties of subglacial conduit systems, such as melt opening and creep closure rates, discharge, water velocity, and effective pressure. To evaluate the impacts of our simulated friction factor (f_{c}) on conduit dynamics relative to values commonly used, such as 0.1 (Schoof, 2010), we solved the governing equations of the mass, momentum, and energy balance for a simplified one-dimensional circular straight conduit (Spring & Hutter, 1981; see Appendix Appendix C for details of governing equations and parameters). Other parameters typically used in this model include conduit slope angle θ (=3°; Schoof, 2010), ice sheet thickness H_{i} (=1,000 m), the inflow discharge to moulin/lake from upstream rivers Q_{0} (= 3 m^{3}/s; Swift et al., 2005), moulin/lake area A_{L}(=50 m^{2}), conduit length l (=50 km), and initial conduit size S_{0} (=1.41 m^{2}).
3 Results
We show results in the order of Groups 3, 2, 1, and 0 which have increasingly complex surface features. Starting from the two simulations in Group 3 with pipe surface P_{1}, the simulated Darcy-Weisbach friction factor (f_{c}) is around 0.01 and 0.05 for the relative roughness of 0% and 2%, respectively. These values compare well with empirical formulas for pipe flows (see equations B1-B3 and Table 1), which validates our computational model.
In Group 2, one simulation was performed with the surface S_{2}, where both sinuosity and cross-sectional variations have been removed, and only surface roughness remains. The simulated friction factor has a value of 0.142, as illustrated by the time history of case S_{2a} in Figure 2e. This value is about 14 times larger than those for smooth pipe in Group 3. Following the definition of roughness in this paper, we quantified the roughness height on rough surface as the standard deviation of the radius of each cross section (Mankoff et al., 2017). The global roughness height (averaged over all cross sections) was calculated to be σ_{r} = 0.19 m. Assuming empirical formulas is applicable to subglacial conduits; an estimated friction factor f_{t} has a value of 0.123, which is close to the simulated value of 0.142. The good agreement between the two indicates that the roughness defined in this work is able to predict the portion of flow resistance due to surface roughness.
In Group 1 simulations with the surface S_{1}, where only the sinuosity is removed while the surface roughness and cross-sectional variations remain on the surface, the simulated friction factor increased to around 1.94, as illustrated by the time history of case S_{1b} in Figure 2e. This is about 200 times larger than the value with surface P_{1} and about 14 times larger than the value with surface S_{2}. If using the linear superposition principle, which assumes that contributions of different sources of flow resistance are independent (Einstein & Banks, 1950; Leopold et al., 1960), the friction factor due to cross-sectional variations is around 1.8 which is 13 (=1.8/0.142) times of that due to surface roughness.
In Group 0 with the original surface S_{0}, where surface roughness, cross-sectional variations, and sinuosity are all included, the simulated Darcy-Weisbach friction factor is about 2.34 (case S_{0c}) in Figure 2e, which is about 230 and 16 times of the resistance of cases S_{3a} and S_{2a}, respectively. The contribution of sinuosity to flow resistance is about 0.4, calculated by taking the difference between cases S_{0c} and S_{1b} with holding the linear superposition principle.
4 Discussion
4.1 Important Surface Features
With the CFD simulations, we quantify the bulk flow resistance in a realistic subglacial conduit as around 2.34, with cross-sectional variations, sinuosity, and surface roughness contributing around 1.8, 0.4, and 0.14, respectively. Cross-sectional variations therefore contribute the biggest portion in the bulk flow resistance (77% of the total resistance), with sinuosity contributing about 17% and surface roughness only contributing about 6%. This conclusion is made based on the linear superposition principle which assumes no interactions among all surface features. There are other studies only assuming no interactions between surface roughness and cross-sectional variations but considering sinuosity as an enhanced effect (Cowan, 1956; Lane, 2005; Powell, 2014; Yen, 2002). Following this assumption, the total resistance can be partitioned as f=(f_{R}+f_{C})(1+f_{S}) with f_{R}, f_{C}, and f_{S} denoting the resistance attributed to surface roughness, cross-sectional variations, and sinuosity. The friction factor calculated from the original surface S_{0} is around 2.34, while that from surface S_{1} (sinuosity removed) is around 1.94, which means that the enhanced effect is about 21% (=2.34/1.94-1). Removing the sinuosity effect, surface roughness contributes to 6% and cross-sectional variations contribute to 73%. Here the relative importance of surface roughness to cross-sectional variations is unchanged because we still assume no interactions between these two features. While these percentages are only rough estimations, they likely explain the big discrepancy between field-measured friction factors and those typically used in models (Gulley et al., 2014), as models of subglacial conduits have relied on numerical simulators that do not simulate cross-sectional variations or sinuosity impacts on flow resistance which contribute around 94% (=17%+77% or 21%+73%) of the total resistance. As assumptions of interactions between different surface features were made in the resistance partition methods, we performed one simulation using a surface generated by only removing surface roughness, to consider the effects of these assumptions (see Text S3).
It is important to note that there is little sinuosity in this mostly straight 10-m section. As seen in Figure 1a, prior to and after this section there are approximately 90° bends in the subglacial conduit. However, even this minor sinuosity in this section contributes to a flow resistance 3 times of that due to surface roughness. We observe that such flow resistance is attributed to three metrics in sinuosity: transverse oscillation angle θ_{xy}, vertical oscillation angle θ_{xz}, and oscillation distance r_{xy} (see Figure 1d for definitions). For a straight pipe, all these metrics have a value of zero. For a realistic conduit, however, all have nonzero values and contribute to flow resistance. Figure 2f shows how these metrics for this real conduit vary in the longitudinal direction. For conduits with large sinuosity over a longer distance in natural subglacial systems, the flow resistance due to these metrics likely becomes more important. In addition, the cross-sectional variations in this 10-m section contain both a long-term shrinking trend and small-scale local variations. These local variations are more likely uniform across the conduit, but the long-term trend varies a lot along flow direction. Considering the dominant contributions of cross-sectional variations and sinuosity, our work suggests that the next generation glaciology models need to explicitly consider the effects of the long-term trend of the conduit size and the sinuosity metrics. Our CFD results also show that the simulated friction factor is independent of Reynolds number (see Text S4).
4.2 Impacts on Hydrodynamics and Ice Dynamics
By solving the governing equations (Appendix Appendix C) for a simplified one-dimensional straight circular subglacial conduit with our simulated friction factor 2.34 and the commonly used value 0.1, we highlight the impacts of parameterized flow resistance on conduit enlargement, subglacial hydrodynamics, and associated ice sheet dynamics. For the case simulated here, the results show that the conduit evolution can be divided into two stages: the initial adjusting stage (from the beginning to the first peak of conduit properties; see vertical dashed lines in Figure 3) and the quasi-equilibrium stage (the time after the first peak). During the initial adjusting stage, the conduit is less developed and cannot accommodate all inflow to the moulin (Q/Q_{0}<1; see Figure 3c), resulting in water overspilling the moulin and creating a constant pressure head system. Constant pressure head conditions remain until enlargement of the subglacial conduit begins lowering moulin water levels. The timescale for initial lowering of moulin water levels is 6.7 and 1.7 days for the friction factor value of 2.34 and 0.1, respectively, which means that the use of a smaller friction factor (0.1) underestimates the initial adjusting stage by about 4 times (=6.7/1.7).
At the quasi-equilibrium stage (t> 6.7 days), the conduit has fully developed and its drainage is mainly constrained by inflow discharge to the moulin (Figure 3c). At this stage, all conduit properties are approximately constant despite small amplitude oscillations and the mean ice melting rate due to viscous dissipation is approximately constant (Figure 3b). Simulations parameterized with the smaller friction factor 0.1 underestimated the average conduit cross-sectional area by 3.4 times (=3.4/1.0, Figure 3d) and overestimated the water velocity by 3.5 times (=3.2/0.92, Figure 3e) compared to those with the friction factor 2.34. The use of the smaller friction factor 0.1 also underestimates the moulin water level and effective pressure fluctuation period by about 1.6 times (1.6 days with f=2.34 vs. 1.0 day with f=0.1; Figures 3a and 3b) and overestimates the effective pressure peak value by about 2 times (Figure 3f).
While our CFD modeling provides new insights on subglacial conduit roughness, flow resistance in real conduits is probably even higher than what we have calculated here and likely much higher than values commonly employed in subglacial hydrological models (Boulton et al., 2007; Clarke, 1996; Colgan et al., 2011; Covington et al., 2012; Fowler, 2009; Schoof, 2010; Spring & Hutter, 1981). For a longer subglacial conduit with larger sinuosity and more frequent cross-sectional size changes, the friction factor is expected to increase many times than the friction factor of 2.34 used in our models. Indeed, friction factors determined from dye-tracing studies indicate that friction factors can be as high as 75 (Gulley et al., 2014). While we assumed a fixed hydraulic roughness value in our modeling, friction factors in real subglacial conduits are likely to be time varying. Friction factors could be especially high during the early stages of conduit enlargement when the ice roof is much closer to the conduit floor.
Regardless of the cause, our results demonstrate that diurnal fluctuating amplitudes in moulin water level, and hence effective pressure, are attenuated with elevated conduit flow resistance (Figures 3a and 3f). Because ice sliding speeds are inversely proportional to effective pressure (Bindschadler, 1983; Fowler, 1986), lower effective pressures in rougher conduits suggest that the ability of channelization to decrease sliding speeds of glaciers and ice sheets may be diminished in natural systems. The influence of hydraulically rough conduits on effective pressure and ice sliding speeds should be explored further, as increased channelization of subglacial drainage is widely regarded to act as a buffer against future meltwater-based acceleration of the Greenland Ice Sheet (Nienow et al., 2017; Sole et al., 2013). Further research should be conducted when larger-scale 3-D models of subglacial conduits become available.
Acknowledgments
This research was supported by the National Science Foundation Award OPP 1503928. The CFD simulations were performed on the HPCs at the Institute for CyberScience at the Pennsylvania State University. Conduit geometric data and CFD simulation cases are available at https://doi.org/10.18739/a23z3w.
Appendix A: Computation Model
The simulation cases for each group are designed as follows. Group 0 has six cases with S_{0a}–S_{0c} designed to test mesh independence and S_{0d}–S_{0f} designed to test the effects of Reynolds number. Group 1 also contains three cases to test the effects of Reynolds number. The change of the Reynolds number is through the changing of mean velocity U_{0}. In Group 2, only one simulation was performed to calculate flow resistance purely due to roughness as defined in this work. For Group 3, two simulations, one with a smooth wall and another with a rough wall with a prescribed roughness height, were performed based on the volume equivalent pipe P_{1}. The computational domain and mesh generation are shown in Figures 2b and 2c (case S_{2a} is chosen as an example). The turbulent inflow condition was generated using an internal mapping approach where the flow variables on the mapped inflow boundary were first interpolated to the inflow boundary (denoted by and ) and then were adjusted to satisfy the target mean velocity U_{0}. If we denote the average values of the interpolated flow variables over the inflow boundary by and , then we adjusted the flow variables as follows: if , , otherwise, . This adjusting is also applied to the turbulent kinetic energy. The boundary conditions for turbulent properties on the inflow and outflow are zero gradient except for the turbulent kinetic energy. We set a no-slip condition on the wall of the pipe, which numerically imposes a velocity = 0 and a pressure = on the wall. Here denotes the pressure at the center of the first cell away from the wall boundary at the previous time step. When a wall function is also applied, a water density divided wall shear stress τ_{b} is also imposed in the tangential direction of the wall boundary. Therefore, a no-slip boundary constrains the solution by adding two extra force terms, a pressure force due to and wall force . Here S_{b}, , and denote the area and the normal and tangential directions of a boundary face. If we use y_{0} and to denote the distance from this boundary face to the center of the nearest cell and its tangential velocity component, the wall shear stress can be determined by a logarithmic law with , , and (see the form of G in Spalding, 1961). In this way, the logarithmic law of the wall is imposed as a boundary condition.
The open source CFD platform OpenFOAM was used for the simulations (CFDDirect, 2017). In OpenFOAM, the governing equations were solved using the finite volume method. The unsteady (temporal) terms were discretized with a backward (implicit) Euler scheme; the convection terms were discretized with a central differencing scheme (termed as Gauss linear scheme in OpenFOAM). A corrected version of the Gauss linear scheme was chosen to discretize the diffusion terms. The linearizations of velocity-velocity (advection term in equation 2) and velocity-pressure coupling were achieved through the Pressure-Implicit with Splitting of Operators (PISO) algorithms (Issa, 1985). PISO is composed of one predictor step and multiple corrector steps. At each time step, the predictor is first implemented by solving the momentum equation with a guessed pressure field (usually the pressure at an old time) and computing predicted mass fluxes at cell faces. The subsequent corrector step contains two types of corrections: a pressure correction achieved by solving a Poisson equation (taking a divergence over equation 2) with the previously updated velocity and fluxes and a velocity correction achieved by solving the momentum equation with the corrected pressure. Such a corrector step was repeated multiple times until the continuity equation is satisfied, followed by solving the turbulent kinetic energy equation (equation A1) and then the time step marches to the next one. It is worthy to note that the velocity-velocity coupling is linearized by replacing one velocity by its value at previous iteration step, and the velocity-pressure coupling is decoupled by solving the momentum equation and Poisson equation separately. The momentum prediction and correction and the turbulent kinetic energy equation were solved with a preconditioned biconjugate gradient method, while the Poisson equation was solved with a generalized geometric-algebraic multigrid method. The computational meshes were generated using the snappyHexMesh tool in OpenFOAM. The initial time step was set as 0.0001 s and allowed to adjust during simulation to not exceed the max Courant number of 0.4. The most time-consuming part of the simulations is the pressure correction. Each time step has about 15–20 corrections, and each correction solves about 1–3 million (the same as the computational cell number) linear equations. To demonstrate the simulated flow field, Figure 2d shows the longitudinal velocity component in half of the S_{2a} case domain. A grid convergence study is discussed in Text S2 and Figure S2.
In terms of the non-dimensional scales, it is convenient to define the conduit length and the diameter of a pipe whose volume equating to that of the original conduit V_{0} or straight-up conduit V_{1}. With this method, for Group 0, the longitudinal length scale is L_{0} = 10 m, the transverse scale (diameter) is D_{0} = 2R_{0} = = 1.34 m, where V_{0} = 14.13 m^{3}. For Groups 1 and 2, the longitudinal length scale is L_{1} = 10.45 m due to straightening, and thus the transverse scale is D_{1} = 2R_{1} = = 1.33 m, where V_{1} = 14.53 m^{3}. Here we see that the straightening affects little in the conduit non-dimensional scales for this 10 m section.
Appendix B: Semiempirical Resistance Formulas
The key for applying equations B2 and B3 to rough conduits is to define a proper hydraulic roughness height k_{s}. However, for the roughness fields in subglacial conduits, the relative roughness is likely to be higher than 5% (Mankoff et al., 2017). In addition, the roughness distribution is very different from those for the derivation of empirical formulas and the Moody diagram. For a subglacial conduit, the roughness is not even well defined. Using the detailed CFD results, we found that the applicability of the empirical formulas and the Moody diagram is limited to estimate the friction contribution to small-scale roughness and sinuosity. They are not applicable to subglacial conduits with large cross-sectional variations.
Appendix C: One-Dimensional Subglacial Conduit Model
The ice melting rate M_{f} is determined by an energy balance between the latent heat of fusion of ice and the heat generated by viscous friction. The friction heat generation is proportional to the wall shear stress . Here ρ_{w} (=1,000 kg/m^{3}), ρ_{i} (=917 kg/m^{3}), and L (=3.34 × 10^{5} J/kg) are the water density, ice density, and the latent heat of fusion per unit mass of ice, respectively. a_{0} = is a constant for circular shape. K_{0} has a value of 4.5 × 10^{−25} Pa^{−3} · s^{−1} in this work. P_{i} is the ice pressure calculated as P_{i}=ρ_{i}gH_{i} with g = 9.81 m^{2}/s. H_{i} is the ice sheet thickness. The difference between ice pressure and water pressure is commonly termed as effective pressure N = P_{i}−P_{w}. θ is the conduit slope angle. Dimensional analysis has shown that the second term in the right-hand side of equation C2 is less than 5% of the other terms (Fowler, 2009), and thus, we dropped this term for simplicity and stability of our numerical algorithms, while keeping all other terms to capture the necessary physics.
The conduit system is similar to a pipe flow which is driven by the pressure difference between the two ends. Thus, we use a time-varying pressure boundary condition at the conduit entrance (x = 0) and the conduit outlet (x = l). We assume a zero effective pressure at outlet, and thus, the fixed water pressure at the outlet is . The pressure at the entrance is determined by a lake conduit or moulin conduit system where the entrance pressure is calculated as where V_{0} denotes the volume at the beginning of each time step, A_{L} is the lake area (assumed constant), and Q_{0} is the upstream discharge to the lake or moulin conduit system. Δt is the time step size, and Q_{out} is calculated as the product of the conduit cross-sectional area and velocity at the entrance. The boundary conditions for conduit size and water velocity are set as zero gradient at the conduit entrance and outlet. Initially, the water pressure at the entrance is set as P_{w0}=ρ_{w}gH_{0} with H_{0}=V_{0}/A_{0}. We set a fixed area A_{0} = 50 m^{2} and 1×10^{4} m^{2} to represent typical moulin size and lake size, respectively. To investigate the effects of ice thickness, we consider ice thickness H_{i}=1,000 m and initial entrance water depth H_{0}=900 m. The measured diurnal upstream discharge to moulins and lakes in Greenland Ice Sheet is in the range 0–10 m^{3}/s (Palmer et al., 2015; Smith et al., 2017); we choose an input discharge, Q_{0} = 3 m^{3}/s, in this range (Swift et al., 2005). We set the initial water pressure as a linear distribution with P_{w0} and P_{i} at the two boundaries. The initial condition for cross-sectional area and velocity is uniformly distributed with the value of 1.41 m^{2} and 1 m/s, respectively. This initial conduit size for the smaller friction factor 0.1 is too big so that the conduit quickly adjusted to a proper size of 0.55 m^{2} (see Figure 3d) after a short time (t<0.3 day; see Figures 3b and 3d).
With these boundary and initial conditions, we first solve the mass and momentum conservation equations (equations C2 and C3) with the known value of cross-sectional area from the previous time step. The solution of the two coupled equations for velocity and pressure is through the PISO scheme in CFD (Issa, 1985). Then we update the mass balance equation for ice (equation C1). We solve the above equations with the finite volume method, and the time derivatives are discretized with a second-order Crank-Nicolson method. The spatial derivatives for pressure and cross-sectional area are discretized with the second-order central differencing scheme (termed as Gauss linear scheme in OpenFOAM), while the advection term of flow velocity is discretized by a linearized upwind scheme. The initial time step is set as Δt = 0.001 s but allowed to automatically adjust up to Δt = 1 s. The grid resolution is set as Δx = 50 m. A solver named conduitFoam for solving the governing equations with these boundary/initial conditions has been implemented in the open source platform OpenFOAM (CFD Direct, 2017).