Subglacial hydrology has long been an area of interest, initially in the context of geomorphology, groundwater, and surface hydrology from alpine glaciers and more recently in the context of its influence on ice sheet dynamics. Below is a brief and selective summary of previous subglacial hydrology modeling work motivated by glacier sliding. We direct readers to Flowers (2015) for a comprehensive review of the full subject history, recent advancements, and current challenges.

The first major efforts to quantitatively model subglacial hydrology began in the 1970s. Shreve (1972) described a system of arborescent subglacial channels, and Röthlisberger (1972) formulated equations for semicircular channels melted into the base of the ice sheet in a state of equilibrium between melt opening and creep closure. Nye (1973) expanded the work of Röthlisberger to consider channels incised into bedrock or subglacial sediments and more fully developed the equations into models for explaining outburst floods (Nye, 1976). In a different approach, Weertman (1972) considered subglacial drainage through a water sheet of approximately uniform thickness. In the following decade, different plausible drainage configurations were also proposed, such as a system of “linked cavities”, spaces that open behind bedrock bumps as a result of glacier sliding (Walder, 1986; Kamb, 1987). By the mid-1980s, it was recognized that the major components of subglacial hydrology could be classified as either efficient (channels or canals) or inefficient (thin sheets, flow through porous till, or distributed systems of linked cavities, often represented in continuum models as a sheet). While channels themselves emerge as a result of self-organized selective growth from a linked cavity system, a clear distinction between these two subsystems was established.

Since 2000, a renewed surge of interest in subglacial hydrology has been sparked as mass loss increases from glaciers and ice sheets and sea level rise is increasingly perceived as an imminent reality, generating a flurry of new observations and modeling advances. Although the effects of surface melt on ice sheet dynamics are not yet entirely understood (e.g., Clarke, 2005; Joughin et al., 2008), observations have reinforced the fact that surface meltwater significantly influences flow behavior in alpine glaciers and ice sheets (e.g., Mair et al., 2002; Zwally et al., 2002; Bartholomaus et al., 2008; Howat et al., 2008; Shepherd et al., 2009; Bartholomew et al., 2010, 2012; Hoffman et al., 2011; Sundal et al., 2011; Meierbachtol et al., 2013; Andrews et al., 2014). Along with more detailed observations, several efforts were made in the early 2000s to accurately simulate subglacial hydrology. Some of these studies treated the subglacial system as a water sheet of uniform thickness (e.g., Flowers and Clarke, 2002; Johnson and Fastook, 2002; Creyts and Schoof, 2009; Le Brocq et al., 2009). Arnold and Sharp (2002) presented a model with both distributed and channel flow, but only one configuration could operate at a time. Kessler and Anderson (2004) introduced a model using discrete drainage pathways that could transition between distributed and channelized modes, and Flowers et al. (2004) used a combination of a distributed sheet in parallel with a network of efficient channels. Schoof (2010) developed a 2-D network of discrete conduits that could behave like either channels or cavities and found that with sufficiently large discharge an arborescent network of channel-like conduits would form, although the resulting geometry was highly dependent on the rectangular grid used. Hewitt (2011) developed a model that used a water sheet to represent evolving linked cavities averaged over a patch of bed (an effective porous medium) coupled to a single channel.

More recent studies tied together key elements of subglacial drainage to form increasingly realistic 2-D models. Hewitt (2013) introduced a linked-cavity continuum sheet integrated with a structured channel network. In that model, channels open by melt, while the distributed sheet opens only by sliding over bedrock bumps (neglecting opening by melt from dissipative heat). Melt from dissipative heat contributes only to opening in channels. Werder et al. (2013) presented a model that involves water flow through a sheet (representative of averaged linked cavities) along with channels that are free to form anywhere along edges of the unstructured numerical mesh, exchanging water with the surrounding distributed sheet. Approaching the problem in a different way, Bougamont et al. (2014) reproduced seasonal ice flow variability through the hydromechanical response of soft basal sediment in lieu of simulating the evolution of a subglacial drainage system. To capture broad characteristics of subglacial drainage without resolving individual elements, de Fleurian et al. (2014) employed a 2-D dual-layer porous medium model, and Bueler and van Pelt (2015) formulated equations for a 2-D model that combines water stored in subglacial till with linked cavities. To help explain observations of high water pressure in late summer and fall, recent observations and modeling efforts have highlighted the importance of representing hydraulically isolated or “weakly connected” regions of the bed (Hoffman et al., 2016; Rada and Schoof, 2018) and addressed the problem by facilitating seasonal changes in the hydraulic conductivity (Downs et al., 2018).

A common theme in the subglacial hydrology modeling literature is a distinction between channelized (efficient) and sheetlike (inefficient or distributed) drainage systems or components. In most existing 2-D models, either only one of these forms is considered, or else slightly different equations are applied to coupled channel and sheet components. For the sheetlike system, these models only consider opening (i.e., growth of the sheet thickness) due to sliding over bedrock bumps, disregarding opening by melting of the upper ice surface. Melt is generated by the thermal energy obtained from dissipated mechanical energy (commonly referred to as energy loss or head loss). However, these models redirect the generated thermal energy into adjacent channel components that are allowed to melt and grow. Channel components are allowed to form in prespecified locations or to evolve along the edges of sheetlike elements, as in Werder et al. (2013). The main reason that most of these models disregard melt opening in the sheetlike system is to avoid the unstable behavior that has been found to occur when it is included, leading to unstable growth in which the melt opening rate exceeds the closure rate, sparking channelization (Hewitt, 2011) or driving initiation of glacial floods (Schoof, 2010). The transition to a channelized state has been described elegantly in previous work (e.g., Walder, 1986; Kamb, 1987; Schoof, 2010; Hewitt, 2011; Schoof et al., 2012; Werder et al., 2013; Hoffman and Price, 2014).

In reality, the subglacial hydrologic system is comprised of a wide array of drainage features, of which the sheet and channel are two end-members. Imposing a sharp distinction between the treatment of the melt opening term and dividing the governing equations between different model components may not allow for the full array of drainage features to arise. It is also a bit artificial to redirect the opening by melt in sheetlike elements to nearby channels. In the model formulation described in this paper, a single set of governing equations is applied over the entire domain, including the melt opening term everywhere. In our formulation, the hydraulic transmissivity of the subglacial domain is allowed to vary spatially and temporally, allowing for a continuum of drainage features. We also account for laminar, turbulent, and intermediate flow regimes based on an experimentally verified flow law for rough-walled rock fractures (Zimmerman et al., 2004). The gap thickness of each computational element in a discretization of the governing equations is allowed to evolve flexibly, and sequential elements with high gap growth rates typically link up to produce channelized features. The ability to represent coexisting turbulent, laminar, and intermediate regimes appears to be a promising approach to overcoming the previously mentioned instability that occurs when the melt generated by mechanical energy dissipation is retained in the sheet system equations. Even with the melt opening term included everywhere in the domain, we are able to generate steady and transient drainage configurations that include channel-like efficient drainage pathways. Our model does not aim to simulate every individual cavity or specific channel cross section, but rather captures the homogenized effects of these elements on a discrete mesh. As we demonstrate in Sect. 3, although the resolution of subglacial geometry in our approach is mesh and grid sensitive, the patterns of simulated basal water pressure and effective pressure (which are most relevant for calculating sliding velocities in ice dynamics models) are relatively robust with coarse resolutions (∼400 m).

The SHAKTI model is based upon governing equations that describe the conservation of water and ice mass, the evolution of the gap height, water flux (approximate momentum equation for water velocity integrated over the gap height), and internal melt generation (approximate energy equation for heat produced at the bed). All variables used in the equations are summarized in Table 1, with constants and parameters summarized in Table 2.

In general, a complete set of governing equations for subglacial hydrology models should include acceleration terms in the momentum equation, and advection and in-plane conduction terms should be included in the energy equation. The most general form of the conservation equations for subglacial hydrology would be a multidimensional extension of the equations described by Spring and Hutter (1981) and Clarke (2003), with augmentation to account for opening by sliding. Our model formulation and most existing subglacial hydrology models typically neglect the acceleration terms in the momentum equation and employ an approximate energy equation in which all dissipated mechanical energy is locally used to produce melt; the equations presented here should be viewed as an approximation to the more general equations.

The water mass balance equation is written as ∂b∂t+∂be∂t+∇⋅q=m˙ρw+ie→b, where b is subglacial gap height, be is the volume of water stored englacially per unit area of bed, q is basal water flux, m˙ is basal melt rate, and ie→b represents the input rate of surface meltwater from the englacial to subglacial system. This water balance assumes that the subglacial gap is always filled with water and that water is incompressible.

Evolution of the gap height (subglacial geometry) involves opening due to melt and sliding over bumps on the bed, as well as closing due to ice creep:

∂b∂t=m˙ρi+βub-A|pi-pw|n-1(pi-pw)b, where A is the ice flow-law parameter, n is the flow-law exponent, pi is the overburden pressure of ice, pw is water pressure, β is a dimensionless parameter governing opening by sliding, and ub is the magnitude of the sliding velocity. Equation (2) may be viewed as a generalized ice mass balance equation augmented to consider opening by sliding. In most existing 2-D models that include both channel and distributed sheetlike drainage components (e.g., Werder et al., 2013), melt opening is typically considered “channel opening” and opening by sliding over bumps on the bed is considered “cavity opening”, with the different terms applied to the appropriate components within the model. Our model differs from other existing models in that we include both opening terms everywhere in the domain, similar to the conduit model of Schoof (2010). The opening by sliding parameter β is a function of typical bed bump height (br) and bump spacing (lr), as well as local gap height (so that opening by sliding only occurs where the gap height is less than the typical bump height). In defining β, we follow Werder et al. (2013). β|b<br=(br-b)lrβ|b≥br=0

The horizontal basal water flux (approximate momentum equation) is described based on equations developed for flow in rock fractures (e.g., Zimmerman et al., 2003; Rajaram et al., 2009; Chaudhuri et al., 2013): q=-b3g12ν(1+ωRe)∇h, where g is gravitational acceleration, ν is kinematic viscosity of water, ω is a dimensionless parameter controlling the nonlinear transition from laminar to turbulent flow, Re is the Reynolds number, and h is hydraulic head defined as h=pw/(ρwg)+zb (where zb is bed elevation). Note that the dimensions of the basal water flux are m2 s-1, i.e., a flow rate per unit width, obtained as an integral of the velocity profile across the gap thickness. The momentum Eq. (5) is approximate in the sense that acceleration terms are neglected and the flow is approximated as a locally plane shear flow. Equation (5) is a key piece of our model formulation in that it allows for a spatially and temporally variable hydraulic transmissivity in the system and facilitates the representation of the simultaneous coexistence of laminar, transitional, and turbulent flow in subregions of the domain. Many existing subglacial hydrology models prescribe a hydraulic conductivity parameter and assume the flow to be turbulent everywhere. Equation (5) has been employed extensively for modeling flow in rock fractures, especially in the laminar flow regime (ωRe≪1), wherein it is commonly referred to as the local cubic law. The extension of the local cubic law to transitional and turbulent flows, by incorporating a Reynolds number dependence as in Eq. (5), has also been employed in previous work on rock fractures (Zimmerman et al., 2004; Rajaram et al., 2009; Chaudhuri et al., 2013) and was experimentally verified by Zimmerman et al. (2009).

In the laminar flow regime, Eq. (5) derives from assuming locally plane Poiseuille flow and integrating the Stokes equations twice across the gap thickness to obtain qlam=-b3g12ν∇h, where ν is the kinematic viscosity of water. The definition of the Reynolds number follows the precedent in fracture literature using the gap height b as a characteristic length scale: Re=|v|bν=|q|ν, where v is the average velocity across the gap. Note that for laminar flow, the flux in Eq. (6) is proportional to the hydraulic gradient ∇h. The flux equation in the laminar regime (Eq. 6) is modified to allow for transition to a turbulent regime by introducing the additional term in the denominator to account for Reynolds number dependence. For fully developed turbulent flow with a high Reynolds number (ωRe≫1), the magnitude of the flux q given by Eq. (5) is proportional to the square root of the magnitude of the hydraulic gradient. |qturb|2=b3g12ω|∇h|

Equation (8) is analogous to the Darcy–Weisbach equation with a constant (i.e., not dependent on Reynolds number) friction factor for flow in ducts. For intermediate Reynolds numbers, Eq. (5) captures a nonlinear dependence between flux and hydraulic gradient that is in between the linear and square root dependences corresponding to laminar and turbulent flow regimes. The parameter ω controls the Reynolds number at which the deviation from the linear dependence becomes significant and is also related to the friction factor. For example, with ω=0.001, ωRe is of order 10 at Re=10000, representing the value at which the friction factor becomes independent of the Reynolds number. For comparison, in pipe flow, fully developed turbulent flow with a constant friction factor occurs at Re∼10000 in very rough pipes (relative roughness >0.02).

Internal melt generation is calculated through an energy balance at the bed: m˙=1L(G+|ub⋅τb|-ρwgq⋅∇h-ctcwρwq⋅∇pw), where L is latent heat of fusion of water, G is geothermal flux, ub is the ice basal velocity vector, τb is the stress exerted by the bed onto the ice, ct is the change in pressure melting point with temperature, and cw is the heat capacity of water. Melt is therefore produced through a combination of geothermal flux, frictional heat due to sliding, and heat generated through internal dissipation (whereby mechanical kinetic energy is converted to thermal energy) minus the heat consumed or released in maintaining the water at the pressure melting temperature in the presence of changing water pressure. We note that this form of the energy equation assumes that all heat produced is converted locally to melt and neglects the advective transport and storage of dissipative heat. We assume that the ice and liquid water are isothermal and consistently at the pressure melting point temperature. These assumptions may not be strictly valid under certain real conditions that may have interesting heat transfer implications, such as heat advection (Clarke, 2003), supercooling (Creyts and Clarke, 2010), or where meltwater enters a system of cold ice (below the pressure melting point), but we leave these potential model extensions for future work. As mentioned previously in Sect. 1.2, Werder et al. (2013) and similar models do not include the internal dissipation term in their sheetlike drainage components, but assign any melt from dissipation to contribute to opening in the nearest channel component.

For the sake of versatility, we also include an option to parameterize storage in the englacial system (note that this is not necessary for numerical stability; we use zero englacial storage in the example simulations presented in Sect. 3 of this paper). Following Werder et al. (2013), the englacial storage volume is defined as a function of water pressure: be=evρwgh-ρwgzbρwg=ev(h-zb), where ev is the englacial void ratio (ev=0 for no englacial storage).

Equations (1), (2), (5), and (9) are combined to form a parabolic, nonlinear partial differential equation (PDE) in terms of hydraulic head, h. ∇⋅-b3g12ν(1+ωRe)∇h+∂ev(h-zb)∂t=m˙1ρw-1ρi+A|pi-pw|n-1(pi-pw)b-βub+ie→b With no englacial storage (ev=0), Eq. (11) takes the form of an elliptic PDE.

Defining a hydraulic transmissivity tensor, K=b3g12ν(1+ωRe)I, Eq. (13) can be written more compactly as ∇⋅(-K⋅∇h)+∂ev(h-zb)∂t=m˙1ρw-1ρi+A|pi-pw|n-1(pi-pw)b-βub+ie→b. Although we employ an isotropic representation of the hydraulic transmissivity tensor in Eq. (12), our model formulation can be readily generalized to incorporate anisotropy. The source terms on the right side of the Eq. (13) and the conductivity depend on h, as a result of which Eq. (13) is nonlinear, and solving for h requires iterative methods.

Boundary conditions can be applied as either prescribed head (Dirichlet) conditions or as flux (Neumann) conditions. To represent land-terminating glaciers, we typically apply a Dirichlet boundary condition of atmospheric pressure at the edge of the ice sheet: hfront=zb. To represent marine-terminating glaciers, the outlet boundary condition can be set to the overlying fjord water pressure. Prescribed flux boundary conditions are imposed on the other boundaries of the subglacial drainage domain: ∇hbound=f, where f can be set to represent no flux (f=0) or a prescribed flux, which can be constant or time varying.

In our current formulation, there is no lower limit imposed on the water pressure; this means that unphysical negative pressures can be calculated in the presence of steep bed slopes, as in Werder et al. (2013). While suction and cavitation may occur in these situations, the flow most likely transitions to free-surface flow with the subglacial gap partially filled by air or water vapor. At high water pressure, we restrict the value to not exceed the ice overburden pressure, which would in reality manifest as uplift of the ice or hydrofracturing at the bed. These extreme “underpressure” and “overpressure” regimes are important situations that have been considered in other studies (e.g., Tsai and Rice, 2010; Hewitt et al., 2012; Schoof et al., 2012), but are quite complex in 2-D and remain to be addressed carefully in future developments.

The overall computational strategy employed is semi-implicit with an implicit backward Euler discretization of Eq. (13) to solve for the head field (h), combined with an explicit treatment of Eq. (2) for the evolution of the gap height (b). Within each time step, the nonlinear Eq. (13) is solved using Picard iteration to obtain the head (h) field. From h, we calculate pw, q, Re, and m˙ to be used in the subsequent iteration (in each iteration, pw, q, Re, and m˙ are lagged from the previous iteration). Once the Picard iteration has successfully converged to a solution for h, the gap height geometry (b) is then updated explicitly based on basal gap dynamics using Eq. (2) to advance to the next time step. A schematic of this numerical procedure is presented in Fig. 1. Due to the explicit treatment of Eq. (2), there is a time step limitation, which will be discussed further in Sect. 4.

SHAKTI is implemented within ISSM, an open source ice dynamics model for Greenland and Antarctica developed by NASA's Jet Propulsion Laboratory and University of California at Irvine (Larour et al., 2012; http://issm.jpl.nasa.gov, last access: 14 July 2018). ISSM uses finite-element methods and parallel computing technologies, and includes sophisticated data assimilation and sensitivity analysis tools, to support numerous capabilities for ice sheet modeling applications on a variety of scales. The SHAKTI hydrology model solves the equations presented above in a parallel architecture using linear finite elements (i.e., P1 triangular Lagrange finite elements), which can be based on a structured or unstructured mesh. The source code is written in C++ and we rely on data structures and solvers provided by the Portable, Extensible Toolkit for Scientific Computation (PETSc; http://www.mcs.anl.gov/petsc, last access: 14 July 2018). The user interface in MATLAB is the same as for other solutions implemented in ISSM designed to facilitate model setup and post-processing (see Documentation; https://issm.jpl.nasa.gov/documentation/hydrologyshakti/, last access: 14 July 2018). The iterative solution of Eq. (13) for hydraulic head employs the direct linear solver MUMPS in PETSc in each iteration, but other solvers provided by PETSc could be easily tested in future work.

Model inputs include spatial fields of bed elevation, ice surface elevation, initial hydraulic head, initial basal gap height, ice sliding velocity, basal friction coefficient, typical bed bump height and spacing, englacial input to the bed (which can be constant or time varying and can be spatially distributed or located at discrete points to represent moulin input), and appropriate boundary conditions. Parameters that can either be specified or rely on a default value are geothermal flux, the ice-flow-law parameter and exponent, and the englacial storage coefficient.

Model outputs include spatiotemporal fields of hydraulic head, effective pressure, subglacial gap height (the effective geometry representative of an entire element), depth-integrated water flux, and “degree of channelization” (the ratio of opening by melt in each element to the total rate of opening in that element by both melt and sliding). Head and effective pressure are calculated at each vertex on the mesh; gap height, water flux, and degree of channelization are calculated over each element (these quantities are based on the head gradient). Instructions for setting up, running a simulation, and plotting outputs can be found in the SHAKTI model documentation (https://issm.jpl.nasa.gov/documentation/hydrologyshakti/  last access: 14 July 2018) and in an example tutorial (https://issm.jpl.nasa.gov/documentation/tutorials/shakti/, last access: 14 July 2018).

In this first example, we consider a 1 km square, 500 m thick tilted ice slab with a surface and bed slope of 0.02 along the x direction. Steady input of 4 m3 s-1 is prescribed at a single moulin at the center of the square (x=500 m, y=500 m). Water pressure at the outflow (left edge of the domain, x=0) is set to atmospheric pressure, with zero flux boundary conditions at the other three sides of the domain. All other constants and parameters are as described in Table 2. We use an unstructured triangular mesh with a typical edge length of 20 m (with 4004 elements). The model is run to a steady configuration (steady state is reached by 12 days) starting from an initial gap height of 0.01 m. A channelized drainage pathway emerges from the moulin to the outflow, with higher effective pressure (i.e., lower head and water pressure), larger gap height, and higher basal flux than its surroundings (Fig. 2). The degree of channelization metric also indicates a value close to 1 (indicating that opening by melt dominates opening by sliding) within the channelized drainage path. Note that the precise configuration of the channelized pathway is somewhat influenced by the unstructured mesh. Mesh sensitivity will be examined below in Sect. 3.2.

Scripts for running this example are included as a tutorial in ISSM (https://issm.jpl.nasa.gov/documentation/tutorials/shakti/, last access: 14 July 2018) and can serve as a template for more sophisticated simulations. Run times will vary by machine and number of processors, but to run this simulation on 24 processors for 30 days with a time step of 1 h, the entire simulation has a run time of approximately 38 s.

For the next example, we consider a rectangular domain 10 km long and 2 km wide, with a flat bed (zb=0 everywhere) and parabolic surface profile with a minimum thickness of 300 m and a maximum of 610 m. Ten moulins are located at arbitrarily chosen locations in the domain, each with a steady input of 10 m3 s-1. The model is run to 365 days with a time step of 1 h (steady state is reached before 50 days), starting from an initial gap height of 0.01 m. The resulting steady distributions shown in Fig. 3 on five different meshes show a clear channelized drainage structure. Rather than each moulin forming a unique channel to the outflow, the moulin inputs influence each other, warping the pressure field and forming arborescent efficient pathways that combine downstream. For this specific arrangement of moulin inputs, a single principal drainage channel emerges. The unique drainage configuration that evolves in a particular circumstance and setting is affected by many factors, including bed topography, ice thickness, sliding velocity, meltwater input location, and input intensity.

The exact configuration of self-organizing channels also depends to some extent on the mesh. The five unstructured meshes used in this example have typical edge lengths ranging from 50 m (12 714 elements) to 400 m (205 elements). Using an unstructured mesh reduces bias in the channel direction compared to a structured mesh, but the orientation and size of the elements still affect the resulting geometry. Most subglacial hydrology models that resolve individual channels are mesh dependent (e.g., Werder et al., 2013). The different cases shown in Fig. 3 provide a qualitative view of the dependence of channelization structure on mesh size. Specifically, the gap height field on the coarsest mesh does not show a clear channel, and a well-defined narrow channel is evident for larger distances upstream from the outflow boundary as the mesh is refined. The general structure of the channel is quite similar in the two finest meshes, but differences in alignment persist due to the unstructured nature of the mesh. From the viewpoint of coupling to ice motion and sliding calculations, the subglacial head and effective pressure fields obtained from the subglacial hydrology model are most important. The head and effective pressure fields shown in Fig. 3 are much smoother than the gap height field and appear to show less sensitivity to the mesh size. To evaluate this sensitivity further, Fig. 4 presents quantitative plots of the mean head and effective pressure (averaged in the y direction) for the five meshes. Across much of the domain, they converge remarkably well, but diverge slightly in the region of significant channelization.

Next we consider a transient example involving a seasonal input cycle of meltwater, with input distributed uniformly across a rectangular domain 4 km long and 8 km wide. The bed is flat (zb=0 everywhere). The ice surface follows a parabolic profile, with ice thickness ranging from 550 m at x=0 to 700 m at x=4 km and is uniform across the y direction. We begin with an initial subglacial gap height of 0.01 m perturbed with random variations drawn from a normal distribution with a standard deviation of 1 %. The purpose of these random variations in the initial gap height is to serve as triggers for potential instability and channelization, which is an important phenomenon in subglacial hydrologic systems (Walder, 1986; Kamb, 1987; Schoof, 2010; Hewitt et al., 2011). Even in nature, the gap height is unlikely to be uniform and the ubiquitous irregular variations in the gap height and bedrock surface will act as natural perturbations to initiate instabilities and channelization. As the ice slides over bedrock, abrasion processes may also serve to generate irregularities. In the literature on the self-organized formation of dissolution channels in rock fractures in karst formations (e.g., Cheung and Rajaram, 2002; Szymczak and Ladd, 2006; Rajaram et al., 2009), it has been established that under conditions that lead to self-organized channel formation, the specific nature of the initial random variations does not influence the structure and spacing of the channels; rather it serves as a trigger for the initiation of channels. In unstructured meshes, it is also possible for mesh-related asymmetries to introduce perturbations that can serve as triggers for this instability. In stable regimes, however, the same perturbations will not produce channelization.

The model is first run with steady distributed input of 1 m a-1 in a spin-up stage with a time step of 1 h (steady state achieved in 4 days). After a steady configuration is achieved, a cycle of meltwater input variation is imposed and run for 1 year (365 days), also with a time step of 1 h. Seasonal meltwater input in m a-1 is approximated by a cosine function between 0.4 and 0.7 a (days 146 and 255). ie→b=-492.75×cos⁡(2π/0.3(t-0.4))+493.75

This yields a maximum meltwater input at the peak of the summer of 986 m a-1, with a winter minimum of 1 m a-1 and annual mean input of 149 m a-1. The peak melt input corresponds to approximately 1000 m3 s-1 for the entire domain. Note that the values used here are unrealistically high and are designed intentionally to show stable behavior of the system across a variety of input magnitudes, even when subjected to extreme forcing. Figure 5 shows time series plots of this “seasonal” input forcing over one full annual cycle, with the corresponding minimum, mean, and maximum gap height and head. Snapshots of the subglacial hydrologic variable fields at intervals through the annual cycle are shown in Fig. 6, and an animation of this simulation is included in the Supplement. As melt increases, the maximum gap height increases, corresponding to growth of the subglacial system and emergence of self-organized efficient channels. The maximum gap height increases with increasing meltwater input until the peak of the melt season, then decreases simultaneously as melt input decreases (note that we use zero englacial storage in this simulation, so there is no lag due to water storage in the system). The hydraulic head initially increases with increased input (meaning an increase in subglacial water pressure as additional water is added to the system), then decreases as efficient low-pressure channels form, then increases again as melt starts to decrease and the channels collapse. We hold the sliding velocity constant, but in reality ice sheet sliding velocity generally increases with increased water pressure (i.e., lower effective pressure) and decreases with lower water pressure. With two-way coupling between the subglacial system and ice dynamics (e.g., Hoffman and Price, 2014; Koziol and Arnold, 2018), the sequence of hydraulic head or basal water pressure variation seen here would likely result in a mid-to-late summer decline in sliding velocity, after which the sliding velocity would increase again. Subsequently, as melt input decreases to the winter minimum, the hydraulic head decreases to low values, which would correspond to a decrease in sliding velocity. As shown in Fig. 6 for the early and late parts of the year, the system essentially behaves as a one-dimensional system because the melt inputs are not large enough to take the system into a regime in which channelization can occur. During the melt season when inputs increase substantially, self-organized, regularly spaced channels emerge, seen in Fig. 6 as having lower heads than their immediate surroundings in the y direction. These channelized structures collapse and disappear entirely as the meltwater input drops off and returns to the winter minimum. The simulation results shown here demonstrate the ability of our modeling framework to represent both stable regimes, in which the subglacial system takes on a relatively smooth quasi-one-dimensional configuration, and unstable regimes with self-organized efficient pathways when high meltwater inputs and discharge trigger the transition to channelization.

To examine mesh dependence in this case of self-organized channelization, Fig. 7 presents gap height and head distributions on three unstructured meshes with typical edge lengths of 50, 100, and 200 m. At 100 m resolution, the channelization effects are obvious, with similar spacing as on the finer 50 m mesh. At 200 m resolution, the channels are still apparent but the head and effective pressure fields are more smoothed than with the finer meshes, especially in the upstream portions of the domain. In the early and late parts of the cycle, the behavior obtained with different mesh sizes is in good agreement for sheetlike drainage. The mesh dependence is evaluated more quantitatively in Fig. 8 with y-averaged quantities for day 1 (sheetlike drainage everywhere), day 200 (peak melt input and extreme channelization), and day 250 (near the end of the melt input cycle as channelization collapses). We see that the solutions obtained with different mesh resolutions converge well for sheetlike drainage, but they show some variation with channelization. These local differences are more pronounced in the quantities calculated over elements (gap height and degree of channelization), while differences are relatively small in the smooth pressure distributions calculated at mesh vertices.

This paper is intended to present a description of the SHAKTI model formulation with illustrative simulations under simple scenarios. Application to real glaciers remains for upcoming work, but we wish to clearly address the limitations of the model and acknowledge challenges faced by this and other subglacial hydrology models.

Time stepping is an important factor in numerical models of the highly transient subglacial hydrologic system, such as SHAKTI. To illustrate the influence of time step size, Fig. 10 presents the evolution of maximum head in the single-moulin example (see Sect. 3.1 and Fig. 2) for different time step sizes. In this example, the model converges properly to the same steady configuration for time step sizes dt=0.25 h to dt=3 h. Note that as the time step increases to about 3 h, small but stable fluctuations are seen. With dt=4 h, however, the model never converges to the solution, but instead enters a large systematic oscillation between incorrect values. For larger time steps than dt=4 h, the nonlinear iteration itself has difficulty converging and the amplitude of the oscillations becomes very large with water pressure exceeding ice overburden pressure, which is accompanied by very large dissipation rates. Difficulties in convergence during numerical solutions of nonlinear PDEs with larger time steps is a well-known issue in a variety of contexts. The appropriate time step size is dependent on various parameters specific to a simulation such as topography, ice thickness, and meltwater input rates. Due to the highly nonlinear nature of the equations, it is unfortunately not straightforward to establish a time step criterion for stable model behavior. As a general guideline we suggest conducting an initial test with a time step of 1 h and adjusting accordingly. We plan to implement adaptive time stepping in future developments of SHAKTI. Note that the time steps required in subglacial hydrology models are typically much smaller than the time steps frequently used in long-term ice dynamics simulations, which may be on the order of years or decades. Although it is desirable to maintain longer time steps in subglacial hydrology models, the essential physics operates on much smaller timescales and using a smaller time step of the order of hours may be unavoidable. Coupling with ice sheet models may rely on spatiotemporally integrated basal water and effective pressures.

We calculate basal gap height over each element, which means that the geometry is dependent on mesh size. It is not our aim to necessarily capture each individual cavity or channel cross section, but rather to obtain the effective geometry over each element and its effect on the pressure field, which has an important influence on ice sheet sliding velocity. In Sect. 3.2–3.3, we examined mesh sensitivity in example simulations (see Figs. 3 and 7). With very large elements (kilometer scale), the effects of channelized drainage may be smoothed out. For large-scale simulations, a variable mesh should be used with coarser resolution in the ice sheet interior away from the margins and finer resolution at lower elevations at which the bulk of meltwater is produced and enters the subglacial system (in which channelized networks are likely to form and sliding velocities are higher). The typical edge length scale should be selected according to the particular application depending on the resolution of bed topography, sliding velocities, modeling goals, and practical concerns of computing power. As a rough guideline to capture the formation of channelization in decent detail, we suggest an edge length of 150 m or less in the domain area of most interest (e.g., the few kilometers nearest the terminus of a glacier).

As stated in Sect. 2.2, the current formulation does not handle high water pressures that exceed overburden (we cap water pressure at overburden pressure and do not represent uplift) or low water pressures at which the system would transition to free surface flow (we assume the subglacial gap is always filled with water and allow unphysical negative water pressures to be calculated in the presence of steep slopes). The sample simulations presented in Sect. 3 do not involve either of these extreme pressure ranges in their solutions, so the results included here are unaffected by the upper limit imposed on water pressure or by allowing negative water pressures in lieu of transitioning to a partially filled system.

The examples in Sect. 3 do not involve complex bed topography, which is beyond the scope of this initial model description paper. The model has been successfully tested on real ice and bed geometry, however, and results will be included in forthcoming work.

Under thick ice with low meltwater input, the nonlinear iteration may have trouble converging to a head solution, entering a stable oscillation. This can frequently be resolved by decreasing the time step and/or employing under-relaxation to help the nonlinear iteration converge.

The SHAKTI model is not currently coupled to ice dynamics in a two-way manner. We prescribe a constant ice sliding velocity, and this sliding velocity does not evolve according to the influence of subglacial water pressure. With this one-way coupling, we are able to infer only qualitatively how the ice velocity would be affected by the changing subglacial system. In upcoming work, we plan to implement two-way coupling with the ice dynamics of ISSM to test different sliding laws and the behavior of the fully coupled system.