Subglacial hydrology has a strong influence on glacier and ice sheet dynamics, particularly through the dependence of sliding velocity on subglacial water pressure. Significant challenges are involved in modeling subglacial hydrology, as the drainage geometry and flow mechanics are constantly changing, with complex feedbacks that play out between water and ice. A clear tradition has been established in the subglacial hydrology modeling literature of distinguishing between channelized (efficient) and sheetlike (inefficient or distributed) drainage systems or components and using slightly different forms of the governing equations in each subsystem to represent the dominant physics. Specifically, many previous subglacial hydrology models disregard opening by melt in the sheetlike system or redistribute it to adjacent channel elements in order to avoid runaway growth that occurs when it is included in the sheetlike system. We present a new subglacial hydrology model, SHAKTI (Subglacial Hydrology and Kinetic, Transient Interactions), in which a single set of governing equations is used everywhere, including opening by melt in the entire domain. SHAKTI employs a generalized relationship between the subglacial water flux and the hydraulic gradient that allows for the representation of laminar, turbulent, and transitional regimes depending on the local Reynolds number. This formulation allows for the coexistence of these flow regimes in different regions, and the configuration and geometry of the subglacial system evolves naturally to represent sheetlike drainage as well as systematic channelized drainage under appropriate conditions. We present steady and transient example simulations to illustrate the features and capabilities of the model and to examine sensitivity to mesh size and time step size. The model is implemented as part of the Ice Sheet System Model (ISSM).

One of the significant consequences of contemporary climate change is rising sea level. A large component of sea level rise is the transfer of ice from glaciers and ice sheets into the ocean via melt, runoff, and iceberg calving (Church et al., 2013). Future ice dynamics remain a major uncertainty in sea level rise predictions involving many uncertain factors, including basal lubrication and effects on sliding velocities from subglacial drainage (e.g., Church et al., 2013; Shannon et al., 2013).

Although massive outlet glaciers of West Antarctica may be on the verge of irreversible collapse in the next 200 to 1000 years (Joughin et al., 2014; DeConto and Pollard, 2016), the Greenland ice sheet is currently the single largest contributor to sea level rise (Shepherd et al., 2012). Considering the substantial amount of water held in this frozen reservoir, it is important to improve understanding of its behavior, including the subtleties of its drainage, which affects ice velocity through sliding. Since 1990, many Greenland outlet glaciers have displayed dramatic accelerations and frontal retreats, yielding substantial changes on the rapid timescale of decades or years (Joughin et al., 2010). Other glaciers, however, have accelerated less rapidly or even decelerated over the same period (McFadden et al., 2011), and the mechanisms driving these contrasting responses are still not entirely understood. The recent accelerations observed in marine-terminating outlet glaciers, which exhibit some of the greatest accelerations and are highly sensitive to changes in terminus conditions, may be in response to changing ocean temperatures (Nick et al., 2009; Rignot et al., 2010; Andresen et al., 2012), but their diverse behaviors have been found to depend on more factors than ocean temperature alone, such as bed topography and subglacial discharge distribution (Slater et al., 2015; Rignot et al., 2016). In land-terminating glaciers, the observed accelerations are likely driven largely by water inputs to the ice sheet from the surface via crevasses and moulins, similar to alpine glaciers (e.g., Anderson et al., 2004; Bartholomaus et al., 2008). Meltwater inputs have been shown to drive variation in ice velocities on the Greenland ice sheet (e.g., Zwally et al., 2002; Bartholomew et al., 2012), as well as seasonal changes in the efficiency of the subglacial drainage system (e.g., Bartholomew et al., 2010; Chandler et al., 2013; Cowton et al., 2013; Andrews et al., 2014).

The hydrology of meltwater on the surface, within, and beneath glaciers and ice sheets should ideally be viewed and modeled as a complex system of processes considering the interconnectedness of surface mass balance, meltwater retention, discharge at the ice margin, and feedbacks between hydrology and ice dynamics (e.g., Rennermalm et al., 2013; Nienow et al., 2017). Water delivered to the bed through englacial conduits drives basal sliding, which has important effects on flow in some regions (Vaughan et al., 2013), and year-round sliding can occur with temperate bed conditions (Colgan et al., 2011). Increased meltwater input to the bed, however, does not necessarily imply increased basal sliding, contrary to what might seem intuitive. For example, as meltwater input increases, water pressure under the ice increases, leading to enhanced basal lubrication and higher sliding velocity (Zwally et al., 2002). But with sustained meltwater input over a melt season, more efficient drainage channels can develop, decreasing the water pressure (Schoof, 2010). Characteristics of individual outlet glaciers such as bed topography, ice geometry, surface temperature, and other factors all play into the intricate choreography of the seasonal evolution of the subglacial drainage system and its influence on ice velocity. Subglacial hydrology models have had success in simulating realistic drainage behavior, but challenges still remain.

The goal of this modeling effort is to see if a single set of governing equations can produce systematic, self-organized channelization where it should occur. In this paper, we describe the model formulation of SHAKTI (Subglacial Hydrology and Kinetic, Transient Interactions), which allows for flexible evolution of the subglacial drainage system configuration and flow regimes using a single set of governing equations over the entire domain. The model aims to represent the complex interactions due to (kinetic) movement of ice and water and (transient) changes in the subglacial system through time. We hope this unified formulation may be used to facilitate an exploration of the conditions under which different drainage system types form and persist and the flow regimes experienced in different areas of a domain. With upcoming application to actual glaciers, this type of model could provide useful insights into the seasonal evolution of real subglacial drainage systems and their influence on mass loss from the Greenland ice sheet, with the potential for broader application to Antarctica and alpine glaciers.

The paper is structured as follows: in Sect. 1.1–1.2, we provide a brief summary and review of historical and recent subglacial hydrology modeling progress to put our model in context. We then present the model's governing equations and the numerical framework in Sect. 2, with illustrative simulations to demonstrate key model features and capabilities in Sect. 3 and a discussion of implications and model limitations in Sect. 4.

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 (

This flexible subglacial hydrology model can handle transient meltwater inputs, both spatially distributed and localized, and allows the basal water flux and geometry to evolve according to these inputs to produce flow and drainage regimes across the spectrum from sheetlike to channelized. The subglacial drainage system is represented as a sheet with variable gap height, and we employ a flux formulation based on fracture flow equations. Channelized locations are not prescribed a priori, but can arise and decay naturally as reflected in the self-organized formation of connected paths of large gap height (calculated across elements) and lower water pressure (calculated at vertices) than their surroundings. In contrast, previous models allow efficient channels to arise along element or grid edges and calculate a specific cross-sectional channel area (e.g., Schoof, 2010; Hewitt et al., 2013; Werder et al., 2013).

The parallelized, finite-element SHAKTI model is currently implemented as
part of the Ice Sheet System Model (ISSM; Larour et al., 2012;

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.

Variables used in model equations.

Constants and parameters.

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

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:

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):

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

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

Internal melt generation is calculated through an energy balance at the bed:

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:

Equations (1), (2), (5), and (9) are combined to form a parabolic, nonlinear
partial differential equation (PDE) in terms of hydraulic head,

Defining a hydraulic transmissivity tensor,

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:

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 (

Schematic of the computational procedure used to solve the model equations.

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;

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
(

To demonstrate the capabilities of SHAKTI, here we present simple illustrative simulations that highlight some of its features. These test problems are designed to show the formation of sheetlike and channelized drainage in the context of different input scenarios (steady input, transient input, moulin point inputs, and distributed input) in simple model domains. We explore the mesh dependence of the model for the more complex examples in Sect. 3.2 and 3.3, with further discussion of this and other limitations included below in Sect. 4.

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

Steady configurations of hydraulic head, effective pressure, gap
height, depth-integrated basal water flux, and degree of channelization for
steady input of 4 m

Scripts for running this example are included as a tutorial in ISSM
(

For the next example, we consider a rectangular domain 10 km long and 2 km
wide, with a flat bed (

Steady-state distributions resulting from steady input of
10 m

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

Mean head and effective pressure (averaged in the

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 (

The model is first run with steady distributed input of 1 m a

This yields a maximum meltwater input at the peak of the summer of
986 m a

Seasonal cycle of distributed meltwater input over one annual cycle, with gap height and head evolution time series. As meltwater input increases, the maximum gap height increases, then decreases simultaneously with the decrease in input. As meltwater input increases, the head increases, then decreases as efficient drainage pathways are established (corresponding to lower water pressure in the efficient pathways and lower head in the unchannelized upstream regions as shown in Fig. 6). As melt decreases, mean head increases again as the efficient pathways start to collapse, then decreases as melt returns to the winter minimum.

Seasonal evolution with distributed meltwater input as shown in
Fig. 5 on a 4 km by 8 km domain over one full annual cycle. Self-organized
efficient drainage pathways form from the outflow (left edge of the domain)
as melt input increases, persist through the melt season, and collapse again
as melt input decreases, returning to a steady sheet configuration. The
efficient pathways show lower head (i.e., higher effective pressure) than
their surrounding areas in the

Mesh dependence shown for the transient example with distributed input (see Sect. 3.3 and Figs. 5 and 6) with typical element edge lengths of 50, 100, and 200 m.

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

Mesh dependence shown with

The flexible geometry and flow regimes of the SHAKTI model allow for various
drainage configurations to arise naturally. We conserve mass and energy in
all parts of the domain, in contrast to several existing models that neglect
the role of melt opening in sheetlike drainage systems or redistribute
dissipated mechanical energy in the sheet system to adjacent channels.
Previous studies found that with similar equations, including the melt term
in a distributed system leads to an instability and runaway growth, which
initiates channelization (Schoof, 2010; Hewitt, 2011). In our formulation,
even including melt from internal dissipation, we are able to achieve stable
configurations of subglacial geometry, basal water flux, and pressure fields
with steady and transient input forcing. Channelized pathways with lower
water pressure than their surroundings form from moulin inputs (Figs. 2 and
3) as well as self-organized configurations with high distributed melt input
(Fig. 6). A feature of our formulation that contributes to this behavior is
the way we calculate the basal water flux (approximate momentum equation,
Eq. 5), which allows for a transient, spatially variable transmissivity that
transitions naturally between laminar and turbulent flow regimes locally,
while allowing both types of flow regime to coexist in the model domain, as
well as flow that exhibits attributes along the wide transition between
laminar and turbulent flow. To illustrate this behavior more clearly, Fig. 9
presents the distribution of the Reynolds number through the initiation of
channelization for days 145–175 of the transient example in Sect. 3.3. On
day 145 (just before the onset of increased melt input; see Fig. 5), the
Reynolds number is low throughout the domain (the maximum Reynolds number is
only about 70), corresponding to laminar flow. On day 155, the Reynolds number
has increased, particularly near the outflow at the left, transitioning into
the turbulent regime in much of the domain with

Reynolds number evolution during the onset of channelization in the transient example with distributed input (see Sect. 3.3 and Figs. 5 and 6). Initially, the entire domain has a low Reynolds number corresponding to laminar flow. As the meltwater input increases, the Reynolds number transitions into the turbulent regime and becomes clearly higher in the self-organized channelized structures than in the surrounding sheetlike regions. Note that the color scale is different for each plot.

The transient example in Sect. 3.3 illustrates one possible pattern of idealized seasonal evolution of the subglacial drainage system, in which channels emerge with increased melt and collapse to a sheetlike system again in the winter. The higher water pressure during the melt season would imply increased sliding velocity in a two-way coupled system, with a decrease in mid-to-late summer with well-established channelized drainage, followed by an increase as the efficient system initiates its shutdown and a decrease as meltwater input returns to the background winter rate. This seasonal pattern is reminiscent of observations of some Greenland outlet glaciers (Moon et al., 2014), and subglacial hydrology may indeed play a key role in shaping the seasonal velocity behavior of some glaciers, both land-terminating and marine-terminating. In future work on real glacier topography, we aim to investigate other velocity signatures, such as those that experience an annual minimum velocity in the late melt season, which is thought to be a result of highly efficient channel development (Moon et al., 2014), or those with high winter sliding velocities, which may be indicative of hydraulically isolated or poorly connected regions of the bed that maintain high water pressure through winter (e.g., Hoffman et al., 2016; Downs et al., 2018; Rada and Schoof, 2018). To accurately capture the influence of transient sliding velocities on the evolution of subglacial hydrology, two-way coupling between subglacial hydrology and ice dynamics is important.

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.

Maximum head evolution to illustrate time step dependence for the
steady simulation with a single-moulin input (see Sect. 3.1 and Fig. 2). For
d

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 d

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.

In this paper, we presented the SHAKTI model formulation with simple illustrative simulations to highlight some of the model features under different conditions. The model is similar to previous subglacial hydrology models, but employs a single set of “unified” governing equations over the entire domain, including opening by melt from internal dissipation everywhere, without imposing a distinction between channelized or sheetlike systems. The geometry is free to evolve; efficient, low-pressure channelized pathways can and do form as the subglacial system adjusts and facilitates transitions between different flow regimes. We find that with high meltwater input (via moulins or distributed input), self-organized channelized structures emerge with higher effective pressure (i.e., lower water pressure) than their surrounding areas. As meltwater input decreases, these channelized drainage structures collapse and disappear.

To understand the overall mass balance and behavior of glaciers and ice sheets, it is crucial to understand different observed seasonal velocity patterns and the corresponding enigmatic drainage systems hidden beneath the ice. Combined with advances in remote and field-based observations and the modeling of other processes involved in the hydrologic cycle of ice sheets and glaciers (such as surface mass balance, meltwater percolation and retention, and englacial transport of water), subglacial hydrology modeling may help close a gap in ice dynamics models to inform predictions of future mass loss and sea level rise. Forthcoming work will focus on the application of the SHAKTI model to real glaciers and coupling the model to an ice dynamics model (ISSM, into which SHAKTI is already built).

The SHAKTI model is freely available as part of the open source
Ice Sheet System Model (ISSM), which is hosted in a subversion
repository at

The supplement related to this article is available online at:

HR and AS formulated the model equations. AS wrote the stand-alone versions of the finite-volume and finite-element models. MM built the parallel model into ISSM and assisted AS with further model development. AS performed simulations and compiled the paper with contributions from HR and MM.

The authors declare that they have no conflicts of interest.

This work was primarily supported by a NASA Earth and Space Science
Fellowship award (NNX14AL24H) to Aleah Sommers. A version of this model was
originally presented in a 2010 proposal by Harihar Rajaram and Robert
Anderson. We thank Robert Anderson for his continued encouragement. Special
thanks to Matthew Hoffman for many helpful conversations about subglacial
hydrology modeling, to Basile DeFleurian and Mauro Werder for including our
model in the Subglacial Hydrology Model Intercomparison Project (SHMIP, de Fleurian et al., 2018;