Ice streams on the Siple Coast, West Antarctica, have a complex history of flow because their basal motion is governed by time-varying basal conditions. Although the mechanical interaction between ice and till is well established, very little is known about the potential effect of regionally scaled water transport in a basal water system, which has only recently become apparent. To investigate the combined effect of hydrological and mechanical processes, we developed the Hydrology, Ice and Till model, in which ice flow is coupled to a Coulomb-plastic till layer and a basal water system consisting of discrete conduits. When the model is applied to Kamb Ice Stream (KIS), results confirm that it is capable of oscillating between fast and stagnant modes of flow. We show that when subglacial conduits are disregarded or do not extend to the grounding line, the oscillatory behavior of the ice stream is governed by the basal thermal regime. When conduits extend to the grounding line, the modelled ice stream oscillation period is increased, peak speeds are reduced, and oscillations may ultimately cease if the volume of water supplied is sufficiently high. Three different hydrological states characterize the behavioral patterns of ice flow and these states are distinguished by conditions at the grounding line. Modelled ice stream velocities were found to oscillate with fast and slow periods typically lasting a few hundred years, although varying according to hydrological activity. Our results indicate that KIS could reactivate this century, given its hydrological setting and ~170 years of stagnation.
 Ice streams and outlet glaciers drain over 90% of ice transported from the interior of the West Antarctic Ice Sheet to the surrounding ocean and ice shelves [Morgan et al., 1982]. The flow of these glaciers was originally thought to be mainly influenced by long-term climatic change and the mass balance of their large drainage basins, but is now known to be variable on a much shorter timescale at least in part due to the strong influence of basal conditions on ice stream flow [Hulbe and Fahnestock, 2007; Catania et al., 2012). As such, ice stream dynamics can have a significant impact on contemporary sea level change [Pritchard et al., 2009, 2011; Rignot et al., 2008]. For instance, in the Amundsen Sea Embayment, the net annual ice loss is about 90 Gt mainly due to the acceleration of Pine Island Glacier, whereas about 34 Gt of ice is gained annually along the Siple Coast (Figure 1) [Rignot et al., 2008], where ice streams have a complex but well established history of flow [Hulbe and Fahnestock, 2007; Catania et al., 2012]. Kamb Ice Stream (KIS) stopped almost completely about 170 years ago [Retzlaff and Bentley, 1993], causing a switch from negative to positive mass balance in this region [Joughin and Tulaczyk, 2002]. The neighboring Whillans Ice Stream (WIS) slowed down by 23% in 1979–1997 [Joughin et al., 2002] and is now slowing down by about 1% per year [Joughin et al., 2005], possibly to become stagnant within this century [Bougamont et al., 2003a]. Less dramatic, but nonetheless important, is the 6% slowdown of MacAyeal Ice Stream and the 5% speed up of Bindschadler Ice Stream in 1997–2009 [Scheuchl et al., 2012], which shows that the Siple Coast ice streams are not flowing at a steady pace. This unsteady flow is seemingly consistent with inferred past flow: WIS apparently stagnated 850 years ago and reactivated 350 years later, while MAIS stopped 800 years ago and reactivated 150 years later [Hulbe and Fahnestock, 2007; Catania et al., 2012].
 The fast flow of the Siple Coast ice streams is facilitated by a weak and relatively thin layer of basal till [Blankenship et al., 1986; Engelhardt et al., 1990], the mechanical properties of which evolve according to the amount of meltwater being stored in its pores at a given time [Kamb, 2001]. Tulaczyk et al. [2000a, 2000b] showed that meltwater has a strong effect on ice flow and that ice streams underlain by weak till beds may be inherently unstable because of recurring switches between stagnant andfast modes of flow. This Undrained Plastic Bed (UPB) model introduced time-varying mechanical properties of the basal till layer and a coupling to the overlying ice flow through the basal heat budget and the rate of basal melting. Numerical ice flow models implementing the UPB were subsequently able to reproduce and explain key features of past flow including the stagnation of KIS [Bougamont et al., 2003b; Christoffersen and Tulaczyk, 2003a] and the slowdown of WIS [Bougamont et al., 2003a]. Omitted in these models was the presence of a basal hydrological system, which has subsequently become apparent from the discovery of active subglacial lakes [Gray et al., 2005; Wingham et al., 2006; Fricker et al., 2007]. Although the hydrological system of the Siple Coast ice streams is still not fully identified, itis clear that it is present and that it may influence ice flow [Raymond, 2000], e.g., if meltwater produced in the deep interior of the ice sheet is transported downstream to regions where it can weaken subglacial till. The potential significance of this is underscored by the observed 0.6-m-thick water body at the ice-till interface behind the stagnant trunk of KIS, confirming the presence of a basal hydrological system which might influence the timing of reactivation of this ice stream [Vogel et al., 2005].
 Here, we combine a two dimensional numerical ice flow model based on the UPB model with a physically based hydrological model adapted from Ng  to investigate how changes in subglacial water fluxes might affect the flow of KIS. Whereas earlier modeling work assumed basal hydrology to be distributed and represented as a thin water film [Le Brocq et al., 2009], we favor the channelized model as it appears more consistent with observations [Engelhardt and Kamb, 1997; Kamb, 2001]. We are able to reproduce oscillatory motion with periods of activity influenced by hydrological as well as mechanical processes at the bed. We find ice stream flow dynamics are sensitive to the basal condition near the grounding line (GL) and that three hydrological states can be distinguished based on whether or not subglacial conduits are present at this location.
2 Model Numerics
 The model developed for this study is called the Hydrology, Ice and Till (HIT) Model, and consists of a 2D flowline model that couples hydrology, ice thermodynamics and till rheology. The model is discussed in more detail in Baker . Constants used in the model are listed in Table 1.
Table 1. Table of Constants
Till mixing coefficient, s–1
1 × 10-9
Reference parameter for A, kPa− 3a− 1
9.302 × 107
Amount the initial bed elevation
is out of isostatic equilibrium, m
Till mixing coefficient, s m-1
1 × 105
Empirical constant for A, Kka
Coefficient of compressibility
Till mixing coefficient, m s–1
1 × 10− 5
Heat capacity of water, J kg− 1 K− 1
Void ratio at N0
Acceleration due to gravity, m s–2
Geothermal heat flux, W m-2
Sediment grain size, m
5.0 × 10–5
Numerical integral for Kn
Thermal conductivity of ice
J m-1 a-1 K-1
6.6 × 107
Empirical constant for A
Hydraulic conductivity of till, m s–1
1 × 10-10
Latent heat of ice, J kg-1
3.35 × 105
Glen's flow law exponent
Reference normal effective pressure, Pa
Activation energy for creep, kJ mol-1
Universal gas constant, J mol-1 K
Reference temperature for A, °C
Transverse velocity, m a-1
Grain settling velocity, m a-1
Melting point depression, C Pa-1
–9.8 × 10–8
Melting point change per
unit change in pressure, C Pa-1
7.4 × 10-8
Surface roughness term, m
3 × 10-3
Relaxation time of the asthenosphere, a
Thermal diffusivity of ice, m2 s-1
Atmospheric lapse rate, C m-1
Dynamic viscosity of a fluid, Pa s
1.787 × 10-3
Density of ice, kg m-3
Density of the mantle, kg m-3
Density of water, kg m-3
Density of solid till particles, kg m-3
Back pressure, Pa
Internal angle of friction in till, °
2.1 Ice Thermodynamics Component
 The ice thermodynamics of the HIT model are principally based on previous theoretical work by van der Veen  and Tulaczyk et al. [2000b] and on numerical modelling by Bougamont et al. [2003a, 2003b]. The horizontal axis of the HIT model, x (m), is directed along the flowline in the direction of ice flow, with x = 0 at the onset and a node spacing of 2.5 km (Figure 1). The transverse horizontal axis, y (m), extends across the width, W, of the glacier and assumes symmetry about the centerline, with y = 0 at the centerline and y= ± W/2 at the margins. For all calculations except ice temperature, the vertical axis, z (m), is elevation in relation to sea level. For ice temperature, the vertical axis, zt (m), is positive upward from the base of the ice stream.
 Ice stream velocity, u (m s–1), is the sum of ice deformation, ud, and basal motion, ub, associated with deformation of till near the ice base [Engelhardt and Kamb, 1998]. Internal ice deformation velocity is obtained by integrating Nye's generalization of Glen's flow law, assuming a vertically averaged longitudinal stress, as [van der Veen, 1987]:
where A is a flow parameter (Pa-3s-1), H is the ice thickness (m), is the depth-averaged longitudinal stress (Pa), h is the surface elevation of the ice (m) and τ1 = D + τd, where and τd is the gravitational driving stress (Pa), given by:
where ρi is the density of ice, g is the acceleration due to gravity and the surface slope, ∂ h/∂ x, is negative in this coordinate system.
 Flow parameter A is given by the modified Arrhenius relation that allows for an exponential increase in A with temperature, determined by Hooke :
where A0 is a reference parameter that is a function of the structural state of the ice (but not of pressure or temperature), Q is the activation energy for creep, RG is the universal gas constant, CA is an empirical constant, T is the ice temperature (°C), Tr is a reference temperature and ka is an empirical constant.
 The basal velocity is based on the work of Raymond  and Tulaczyk et al. [2000b], modified to include the gradient in longitudinal stress. When flow parameter n = 3, ub at the centreline is given by:
at the centreline, where τb is the basal shear stress (Pa). For an ice stream experiencing motion due to the deformation of Coulomb-plastic till τb can be estimated by:
where τf is the till yield strength (Pa) (equation (16)).
 The depth-averaged longitudinal stress is calculated using the vertical mean approach of van der Veen , given by:
where D1 = 2H ∂ τ1/∂ x and longitudinal stress in the y-direction is neglected. Equation (6) is solved by numerical iteration.
 The change in ice thickness with time, t (s), is given by a conservation of ice volume [Nye, 1959; Paterson, 1994; van der Veen, 1999]:
where v is velocity in the y direction (m s–1), is the accumulation rate at the surface (m s–1), is a width-averaged basal melt (or, if negative, freeze-on) rate (m s–1) and is the average ice flux (m s–1). The first term in the parentheses on the right hand side (RHS) of equation (7) is lateral input of ice from both sides of the flowline. The second and third terms in the parentheses on the RHS of the equation are ice gain from accumulation at the surface and ice loss from basal melting (or gain from basal freezing), respectively. The last term in the parentheses on the RHS of the equation is the ice flux divergence. The average ice flux, , is the sum of a depth-averaged ice deformation velocity (m s–1) and a width-averaged basal sliding velocity (m s–1). The depth-averaged ice deformation velocity at the centerline is given by [van der Veen, 1987]:
 This is calculated by vertically integrating equation (1). Similarly, the width-averaged basal sliding velocity is obtained by integrating equation (4) over the ice stream width, the result of which simplifies to .
 Glacial isostatic adjustment in the model is given by [Huybrechts, 1993; Pattyn, 2006]:
where bz is the bedrock elevation (m), Θb is the relaxation time of the asthenosphere, ρm is the density of the mantle and b0 is the isostatically adjusted bedrock elevation (m) if ice is removed, given by:
where beq = binit + biso is a bed elevation and Hinit is the initial ice thickness, which are assumed to be in isostatic equilibrium with one another. binit is the initial bed elevation and biso is the amount the initial bed elevation is out of isostatic equilibrium, which is by about 40 m on the Siple Coast [Parizek and Alley, 2004].
 We use an atmospheric lapse rate, λ, to determine the change in surface temperature, Th (°C), in time. For basal sliding processes to be active, the bed must be at the pressure melting point [Engelhardt and Kamb, 1998]. Therefore, the basal temperature, Tb (°C), is given by Tb = ρigHγ, where γ is the approximate melting point depression due to pressure [Hooke, 2005]. We use a column model put forward by Budd  and Budd et al.  [also see Hooke, 2005] to determine the vertical ice temperature profile. This approach results in a change in temperature with time given by:
where κ is the thermal diffusivity of ice (m2 s-1), (m s–1) is the vertical velocity at the surface of the ice, where is the melt rate at the ice base, and (m s–1) is the vertical velocity at the base of the ice. The first term on the RHS of equation (11) represents diffusion, the second vertical advection and the third the horizontal advection of heat. The vertical temperature profile is initialized using [Zotikov, 1986; Hughes, 1998]:
where erf is the error function and is the Peclet number.
2.2 Subglacial Till Component
 The subglacial till component of the HIT model includes the evolution of the till layer and subglacial water fluxes within it. This is based on the UPB model of Tulaczyk et al. [2000b], which assumes that water at the ice-till interface can be incorporated into pore spaces of the till thereby changing its void ratio and hence shear strength. Bougamont et al. [2003b] took this theoretical approach and added a basal till layer with vertical Darcian water fluxes to show that an increase of basal shear strength, associated with dewatering of the till layer by basal freeze-on, might have caused the stagnation of KIS. A related study by Christoffersen and Tulaczyk [2003b] used a model with coupled flows of water, heat and solutes to explain why basal freeze-on causes dewatering of tills.
 In the HIT model we assign x- and y-directions within the till layer that correspond to those of the ice above (Figure 2). We define s as the coordinate along the xz-plane, following the bed slope. In the vertical direction we define zs as being vertical upward from the bed (6 vertical till nodes). A till cell has a volume determined by the distance in the s-direction between nodes, the conduit spacing, Csp (defined in section 2.3), and the till column thickness, Zv (m), which was initially set to 4 m everywhere (till assumed to be 40% solids). This is within the range of till thicknesses found beneath the Siple Coast ice streams [Kamb, 2001; Whillans et al., 2001].
 In addition to elevation changes, water flow in the till is determined by overburden, water and effective pressures (stresses normal to the bed). The overburden pressure, Po (Pa), is the downward force due to the weight of overlying ice and till, given by:
where Pi = ρig H is the ice overburden pressure and Pt is the till overburden pressure (Pa), given by:
where φ is the till porosity, ρs is the sediment (till) density, ρw is the water density, Z′v is the thickness of a till cell (m) and χ is the summation index. Summation in equation (14) is over the layers above a given node (η) in the till column.
 The difference between Po and the till water pressure, Pw (Pa), is referred to as the effective pressure, N (Pa). Changes in void ratio of till, et, caused by the till's changing water content are used to calculate the associated change in N in the till [Scott, 1963]:
where et = φ/(1 − φ), e0 is the void ratio at the reference normal effective stress, N0, and Cc is the dimensionless coefficient of compressibility. e0 and Cc depend on the compressibility of the till. We assume an e0 of 0.6 and a Cc of 0.25, which is consistent with values obtained from till samples [Tulaczyk et al., 2001].
 As geophysical observations indicate that deformation of till beneath WIS is concentrated at the ice-till interface [Engelhardt and Kamb, 1998] and is best described by Coulomb-plastic slip [Tulaczyk et al., 2000b], the till yield strength is given by the Mohr-Coulomb criterion:
where ct is the apparent cohesion of the till, Nmin is the minimum effective pressure in a given till column (Pa) and ϕ is the internal angle of friction. The cohesion term is often neglected in numerical models. However, we retain it here at a value of 1000 Pa, as estimated by Kamb . Note that we use the minimum effective stress within a till column, as this gives the minimum strength of the till where failure will occur [Truffer et al., 2000].
 Subglacial melting at the ice-till interface occurs if the conductive heat loss is smaller than the sum of the heat generated by friction and the geothermal heat flux. If the opposite is true, then freezing occurs at the ice base. The basal melt rate, (m s–1), is given by [Lingle and Brown, 1987]:
where G is the geothermal heat flux, K is the thermal conductivity of ice, dT/dzt is the basal temperature gradient (negative) (°C m-1) and LH is the latent heat of ice. The geothermal heat flux was set to 0.07 Wm-2, a value from Siple Dome (Engelhardt, 2004) that is commonly assigned for this region [Raymond, 2000; Bougamont et al., 2003b; Christoffersen and Tulaczyk, 2003b; van der Veen et al., 2007].
 Water fluxes through till, qξ (m s–1), can be described by Darcy's Law [Paterson, 1994], here with the addition of a till mixing component, fm (m s–1):
where ξ is replaced by either zs for a vertical flux or s for a horizontal flux, Kt is the hydraulic conductivity coefficient of the till and Φ is the hydraulic potential in the till (Pa), given by [Shreve, 1972]:
where Zvc is the till thickness (m) above the point of consideration, where Zvc = 0 at the ice till interface and Zvc = Zv at the base of the till. We assume a homogeneous till and an isotropic value of Kt, as nodata are available to constrain the potential anisotropy. A flux is defined as positive when it is directed downward for vertical fluxes and downstream for horizontal fluxes. fm is given by:
where Z′s is the thickness of solids in a till cell (m) and fmix is the till mixing rate (s-1):
where am, bm and cm are mixing coefficients (Table 1). Equation (20) results in till mixing that is insignificant for velocities below 30 m a-1 and high for velocities above 100 m a-1, values that are similar to those used in previous work by Bougamont et al. [2003b] and Christoffersen and Tulaczyk [2003a]. We include the till mixing term so that sediment immediately beneath the ice stream can deform in a manner similar to that reported by Engelhardt and Kamb  and to match observed porosities [Kamb, 2001]. Previous studies in which a mixing term was also used include Bougamont et al. [2003b], Christoffersen and Tulaczyk [2003a] and Bougamont et al. . See Bougamont et al.  for a detailed discussion. Till mixing facilitates movement of water down into the till during till deformation [Engelhardt and Kamb, 1998; Tulaczyk et al., 1998], whereas the Darcy flux predominantly moves water up in the till due to the gradient in hydraulic potential.
 The volume of a till cell, V' (m3) is composed of fluid and solid components [Craig, 2004). As the volume of solids in a till cell is held constant in time, V' only changes with water content according to:
where qa, qb, qu and qd (ms-1) are the water fluxes into and out of a given cell from above, below, upstream and downstream, respectively. The change in till volume translates to a change in till thickness, neglecting any strain in the x- and y-directions. The porosity of the till cell is then φ = V′f/V′, where V′f (m3) is the fluid volume of the till cell.
 To account for limits in till compressibility, we define the maximum and minimum till porosity values. Till porosities below KIS range from 0.28 to 0.60 [Kamb, 2001]. In the HIT model minimum and maximum porosities of the till were set to 0.2 and 0.6, respectively. The assumed lower till porosity limit represents a state in which the grains can no longer rearrange themselves into a more compact configuration, while the upper limit can be interpreted as the porosity at the liquid limit, which occurs when the till loses its granular structure and hence its shear strength [Tulaczyk et al., 2000a].
2.3 Subglacial Conduit Component
 The subglacial conduit system in the HIT model consists of a system of elliptical conduits located at the ice-till interface. The theoretical framework of this conduit system is based on Ng , adapted to include both laminar and turbulent water fluxes and modified to use the conduit wall melt rate put forward by Shreve  and Hooke .
 In the HIT model it is assumed that there is an odd number of conduits across the ice stream and that they are equally spaced (at distance Csp (m)) across its width. Subglacial conduits in the model are elliptical pipes that are incised both up into the ice and down into the till (Figure 3). The size and shape of each section of a conduit is described by its cross-sectional area, its length and its eccentricity. Although in reality subglacial conduits are unlikely to be perfectly symmetrical, as ice scours over sediment of mixed clast sizes, symmetrical geometry is a reasonable approximation as long as roughness is accounted for in the associated water flux equations. The hydraulic radius, rh (m), of a conduit isgiven by rh = Ac/pc, where Ac is the cross-sectional area (m2) and pc is the wetted perimeter of a conduit (m). As the exact solution of an elliptical wetted perimeter cannot be expressed in a simple algebraic way, we employ Ramanujan's first approximation of an elliptical perimeter, given by [Ramanujan, 1914]:
where a is the semimajor conduit axis (m) and b is the semiminor conduit axis (m) (Figure 4). The relationship between a and b can be expressed through the conduit eccentricity, which is set to 0.995  (equivalent to a ~10:1 ratio for a:b) in line with estimates by Engelhardt and Kamb  (conduit width 1 m, conduit height 0.1 m).
 In a canal-type conduit system, the effective pressure in a conduit, Nc (Pa), is based on a relationship of the form [Walder and Fowler, 1994], where Qc is the volumetric conduit flux rate (m3 s-1) and mc is a positive exponent. The negative sign in front of mc reflects the fact that water fluxes in a canal-type conduit increase with water pressure. Here, we use the approach of Ng , modified to calculate Nc for each conduit section by:
where Φc is the hydraulic potential in a conduit, given by:
where zc is the elevation of the conduit in relation to sea level (m) and Pwc is the water pressure in a conduit (Pa) (Pwc = Po − Nc), and where Kn () is given by:
where gs is the representative sediment grain size, fc is a friction factor (dimensionless), vs is a constant grain settling velocity, I is a numerical integral (Table 1), k = ρwCwΔmp (dimensionless), Cw is the heat capacity of water, Δmp is the change in the melting point per unit of pressure and Qs is the volumetric sediment flux (m3 s-1). Due to issues of numerical stability the hydraulic potential gradient term in equation (24) is approximated by − ∂ Φc/∂ s = ρigsinα, where sinα is the ice surface slope. This equates to a basic hydraulic gradient, imposed by topography [Ng, 2000]. The value of Qs is also unknown, but experimentation reveals that an appropriate approximation is given by Qs = Qc/500. This results in .
 The water flux through a conduit, qc (qc = Qc/Ac) (m s–1), depends on whether water flows are laminar or turbulent and on the shape of the conduit. For both laminar and turbulent flow we use the Darcy-Weisbach equation, given by [White, 1991; Lewis and Boose, 1995; Clarke, 2005]:
where μ is the dynamic viscosity of a fluid. For turbulent flow fc is given by the Swamee-Jain relation:
where ϵr is a surface roughness term and Re is the Reynolds number for an elliptical pipe, given by [Stephenson, 1976]:
 We use qc from the previous time step to determine Re and a transitional scheme where we take the higher value for fc of the turbulent (equation (28)) and laminar cases. The laminar solution is found by setting the laminar and turbulent flux relations in equation (27) equal to one other to solve for fc. This results in a smooth transition between laminar and turbulent flow equations.
 The size of a subglacial conduit increases with melting and decreases with ice creep. Melting occurs in conduits due to friction resulting from water movement past the ice walls, with the melt rate, (m s-1), given by [Shreve, 1972; Hooke, 2005]:
 The creep rate, (m s–1), acting on the upper conduit wall is given by [Ng, 2000]:
 Using equations (30) and (31), we obtain the change in the semiminor axis b as:
where Shp is a shape factor that allows for the fact that the conduit is elliptical, given by Shp = pc/(2πa). We employ the shape factor to ensure that the volume of water lost or gained in the conduit as a result of melting is equal to the change in conduit volume. We assume that the change in the upper and lower halves of a conduit are equal, a simplification that is necessary to maintain the elliptical shape of the conduit.
 We assume that a conduit can only exist when the top layer of the till is at maximum porosity. This is when the effective pressure at the top of the till layer is close to zero. Once a conduit exists, water from it will always preferentially be added to the top till layer should it drop below maximum porosity. This means that the freeze process empties a conduit before it removes water from the till. A conduit will then reform only if the porosity of the top till layer once again reaches its maximum value. The physical premise of this assumed hydrological setting is one where ice stream flow is controlled by the basal shear strength of the till, as demonstrated by Tulaczyk et al. [2000a], but with a modulating effect on the latter from an active hydrological system, as has been observed [Carter et al., 2007; Fricker et al., 2007]. The presence of conduits results in a decreased ice-till contact area. As such, there is less friction at the ice base and we reduce the supported basal shear stress by a ratio of the conduit width to the total conduit spacing.
 Water generation, transport and loss within a subglacial water system can be thought of as a water balance, where:
where Vw is the volume of water stored in the conduit (m3), Vu is the volume of water entering the conduit from upstream (m3), Vd is the volume of water flowing downstream out of the conduit (m3), is the volume of melt water generated in the conduit (m3) and Vt is the volume of water flowing into or out of the till (m3) (including any basal melting at the ice-till interface that does not infiltrate into the till).
 The three conduit geometries that can occur in a conduit section are: (1) elliptical cone; (2) elliptical cylinder; and (3) elliptical frustum, with the conduit volume, Vc (m3), given by:
where c and d are the downstream semimajor and semiminor axes, respectively (Figure 4). For an elliptical cone a >0 and c = 0 when a conduit is open upstream and closed downstream. If the conduit is closed upstream and open downstream, a = 0 and c > 0. For an elliptical frustum the above solution only applies when a > c. If a< c, a should be interchanged with c and b should be interchanged with d in the above equation.
 If Vc < Vw we assume that the conduit is pushed open to accommodate surplus water in a similar manner to hydraulic jacking. To accommodate the water we open the conduit on the upstream and/or downstream side, preferentially opening the side with the smaller cross-sectional area. In cases where a conduit was not filled to capacity the melt rate was reduced and the creep rate enhanced to reduce the size of the conduit. To achieve this the percentage of the conduit that was not full was subtracted from the melt rate and added to the creep rate. This was necessary as melt and creep rate equations assume a full conduit.
2.4 Boundary Conditions and Stability
 The longitudinal stress equation (equation (6)) requires boundary conditions at the onset and GL of the ice stream. At the onset we assume that the gradient of is zero. At the GL ice shelf dynamics result in a of [Thomas, 1977]:
where σb is back pressure (Pa) arising from grounded parts of the ice shelf and shear along the ice shelf sides. The first component of the RHS of equation (35) is the weight induced spreading stress, which ‘sucks’ iceout of the grounded ice sheet [van der Veen, 1987]. This is partly compensated by the second component of the RHS of equation (35), the pressure exerted by sea water, and by the third component of the equation, the back pressure. Back pressure at the GL was set to 2 kPa, which is in the range expected for the GL of KIS with the Ross Ice Shelf [Alley et al., 2004].
 At the onset, the average ice flux was set to 200 m3 s-1, which is equivalent to 50 m a-1 at modern day ice stream width and thickness values. The gradient in hydraulic potential at the onset was determined from the water flux entering the conduit, given by:
where qc is prescribed at the onset. There is a prescribed water flux of 6 × 10-6 m a-1 into the till at the onset and a no flow condition at the base of the till column.
 Model stability is assumed when the surface velocity changes by less than 5% in 5000 years. When oscillations occurred in the model results, we required that the model reach a state where the amplitude and period of oscillations in the surface velocity changed by no more than 10% from one oscillation to the next. The time step of all model runs is 3 hours.
3 Kamb Ice Stream Model Setup
 The model flowline for KIS consists of 299 nodes, which extend from the top of the northernmost upper tributary of the ice stream to its GL (Figure 1). The ice stream centerline and width were determined using RADARSAT-1 Antarctic Mapping Project AMM-1 SAR Image Mosaic of Antarctica, Interferometric Synthetic Aperture Radar (InSAR) ice flow velocities from Joughin et al.  and GL positions for the Siple Coast ice streams produced by Horgan and Anandakrishnan  from ICESat data. Bed and surface elevations of KIS are from BEDMAP Project data [Lythe et al., 2001] (Figure 5). Where possible, higher-resolution data from the original sources of BEDMAP were used. We assume that KIS has 101 conduits equally spaced across its width, which results in a conduit spacing of 295.5 m at the onset and 1267.0 m at the GL. Sensitivity tests on the number of conduits across the width of the ice stream are included as part of our analysis.
 KIS has two principal upper tributaries, which join to form the main trunk of the ice stream. While the modelled flowline follows the northern tributary, ice and water fluxes from the southern tributary were added at the ice stream junction. We assume that the ice flux from the second tributary is equal to that of the modelled tributary. Fluxes of water were also assumed to flow through the till and through conduits from the second ice stream tributary. The flux of water at the junction was taken to be 35% of the upstream flux multiplied by a width ratio (width at the junction divided by width one node upstream of the junction). This is assumed from the fact that the modelled tributary is longer, originates in a much deeper trough, and is likely to exhibit higher basal melt rates than the southern tributary [Joughin et al., 2003]. All fluxes were assumed to be evenly distributed across the ice stream width.
 As direct surface temperature measurements across Antarctica are sparse, we use 25-year (1980-2004) averaged mean annual temperatures from a regional atmospheric climate model (version 2) (RACMO2/ANT), specially adapted for use over Antarctica [van den Broeke, 2008] (horizontal resolution of 55 km). Accumulation rates are from Arthern et al. , which are estimated to have an effective resolution of 100 km.
 We present the results of 4 tests on KIS to determine the effect of the subglacial drainage system configuration on ice flow variations and ice stream stability. In test KIS1 there were no subglacial conduits, with water in excess of the maximum till porosity at the ice-till interface assumed to no longer interact with the till layer. In this case we assume that water in excess of what can be stored in the till is carried away in a basal water system, the presence of which has no effect on the flow of ice. Tests KIS2-KIS4 included prescribed conduits, with different prescribed water fluxes at the ice stream onset, ranging from 1.4 × 10-2 to 4.7 km3 a-1 (for 101 conduits) (Table 2). This spans the range of volumetric flux rates estimated by satellite laser altimetry for subglacial lakes beneath WIS, where discharge from subglacial lake Engelhardt near the GL is ~0.7 km3 a-1 [Fricker et al., 2007] and discharge from subglacial lake Conway farther upstream is ~0.6 km3 a-1 [Fricker and Scambos, 2009]. In comparison to the prescribed water fluxes at the onset, the total amount of melt water produced (averaged over a 2000-year period) was 4.2 × 10-4, 3.7 × 10-4, 2.9 × 10-5 and 2.7 × 10-5 km3 a-1 for tests KIS2-KIS4, respectively.
Table 2. Prescribed Volumetric Flux (Q) at the Onset and Amplitude and Period of Surface Velocity at the Grounding Line
 Tests KIS1-KIS3 experienced oscillations in surface velocity, while test KIS4 achieved a surface velocity that was constant in time (Figures 6 (a-d)). Periods of oscillation of the surface velocity for tests KIS1-KIS3 were 240, 275 and 535 years, respectively (Table 2). The surface velocities of tests KIS1 and KIS2 exhibited similar patterns, with amplitudes of around 1000 m a-1 and mean surface velocities of around 450 m a-1. Test KIS3 had a lower amplitude of oscillation (680 m a-1), but a higher mean surface velocity (~ 500 m a-1). It took 139, 110 and 393 years for the surface velocity to increase from its minimum to its maximum value at the GL in tests KIS1-KIS3, respectively. The remainder of the oscillation period was the time it then took to return to its minimum flow velocity. In total, the surface velocity of tests KIS1-KIS3 at the GL operated under conditions of lower than mean ice flow for 145, 118 and 349 years in a given period of oscillation, respectively, and otherwise experienced faster than mean ice flow.
 The differences in magnitude and period of the surface velocity were due to basal conditions at the GL of the ice stream. In tests KIS1 and KIS2 oscillations in the surface velocity were controlled by thermal processes at the GL, as the conduit did not extend to the GL in either run. During periods of basal melting the till porosity increased sufficiently to lower the basal shear stress, resulting in an increase of the surface velocity and a reduction in the ice thickness. Changes in the basal shear stress and the basal temperature gradient then reduced the melt rate until a state of freezing was reached. This basal freezing reduced the porosity of the till again, which led to a higher basal shear stress and a reduction in the surface velocity. The cycle was completed when the melt rate was sufficiently increased to produce melt water again.
 In test KIS1 subglacial water flow was limited to horizontal and vertical movement of water within the till because conduits were excluded. Water at the ice-till interface moved down into the till or was taken out of the upper till layer depending on whether basal ice melted or formed by accretion. In Figure 6 (i,l) areas with high melt rates (corresponding to thicker ice) have thicker, higher porosity till. In test KIS2, water taken up by the till and required for basal freezing left insufficient water available for the conduit to extend to the GL (Figure 6 (j,m,q)). In comparison to test KIS1, the till layer in test KIS2 had a much larger water content because water was added at the onset and because water was able to transfer downstream in conduits. The period of oscillation of test KIS2 was 35 years shorter than that of test KIS1 because the additional water in the subglacial system at the ice-till interface resulted in a higher porewater content in the till. This caused the till to be at its minimum porosity for a shorter time, whereas it was above minimum porosity for about the same time as in test KIS1 (Figure 6 (e,f)).
 In test KIS3 water availability in conduits was sufficient to enable water to be transported all the way to the GL during part (310 years) of the oscillation cycle, with the conduit closing at the GL for the remainder of the cycle (225 years) (Figure 6 (k,n)). In test KIS3 there was no basal melting at the GL, as the increase in water transportation from upstream raised the till porosity and reduced basal friction. Till porosity at the GL oscillated between the minimum and maximum values set in the model (Figure 6g). Conduit water fed the basal freeze process, enabling the conduit to extend to the GL during periods of low basal freezing (~0.005 m a-1) and to 62.5 km upstream of the GL in periods of high basal freezing (~0.01 m a-1) (Figure 6r). The increased rates of freezing caused by the additional influx of water at the onset is a surprising, but important outcome, as it demonstrates how basal melt water not only lubricates fast flow, but also modulates the basal thermal regime. In this case, additional water promotes freezing, despite a higher mean velocity compared to tests KIS1 and KIS2 (Table 2). This dynamic effect is clearly important for the entrainment and transfer of basal sediment [Bougamont and Christoffersen, 2012]. The increased porewater availability in test KIS3 maintained the till porosity above its minimum for ~320 years longer per oscillation period than in tests KIS1 and KIS2, resulting in oscillations in the surface velocity with a longer period and lower amplitude.
 Comparing Figures 6 (a-c), it is evident that the surface velocity increased faster during ice stream speed-up in tests KIS1 and KIS2 than in test KIS3. This is because the speed-up in tests KIS1 and KIS2 was initiated at the GL due to meltwater generated changes in the till porosity, whereas in KIS3 the speed-up was initiated inland as the subglacial conduit extended toward the GL. In tests KIS1 and KIS2 the basal shear stress at the GL decreased as porosity increased, increasing the surface velocity. This led to an increase in the surface slope upstream of the GL, which resulted in an increase in the surface velocity there. The higher surface velocity increased friction at the bed, which generated more meltwater and enhanced speed-up further inland. In contrast, in test KIS3 an increase in the basal water supply caused the conduit to extend toward the GL. This lowered the basal shear stress, increasing the surface velocity inland of the GL, which reduced the surface slope downstream. Thus, at the GL the basal shear stress remained high while the driving stress was reduced. It was not until the conduit extended to the GL that the basal shear stress there was reduced, by which time the surface slope was lower. The slower increase in the surface velocity in test KIS3 led to a lower amplitude, but higher mean, surface velocity at the GL (Table 2).
 In test KIS4 the water flux at the onset was increased further, resulting in a permanent conduit at the GL (Figure 6 (o,s)). As the top of the till layer at the GL was maintained at a high porosity (Figure 6h), the basal shear stress did not vary significantly and oscillations in the surface velocity ceased (Figure 6d). The resulting surface velocity was higher than the mean values in tests KIS1-KIS3 (Table 3). Any further increase in the water flux at the onset would only serve to increase the size of the subglacial conduit. This would have very little effect on the surface velocity, as a wider conduit reduces a basal shear stress that is already very low.
Table 3. Modelled Surface Velocity, Basal Temperature and Basal Temperature Gradient of KIS Over a 2000 Year Period
Surface Velocity (m a-1)
Basal Temperature (°C)
Basal Temperature Gradient (10-2°C m-1)
 In order to determine the influence of the number of conduits and the conduit spacing on model results, we performed four sensitivity tests (Table 4). If we maintain the flux into each individual conduit, but double the number of conduits (halving the conduit spacing) (test KISC3-A), we have a total influx of twice as much water to the ice stream. This leads to a situation similar to that of test KIS4, where oscillations in the surface velocity no longer occur because the top till layer at the GL is permanently at its maximum porosity (Figure 7). If we maintain the flux into each individual conduit, but halve the number of conduits (doubling the conduit spacing) (test KISC3-B), then the total influx into KIS is halved. In this case the reduction in porewater availability to the GL results in a situation similar to that of test KIS2, where oscillations in the surface velocity are driven by the basal thermal condition at the GL. If we double (test KISC3-C) or halve (test KISC3-D) the number of conduits, but scale the volumetric water flux into each conduit so that the total water influx into the ice stream remains the same as in test KIS3, then there is very little change in the magnitude or period of the surface velocity or till porosity.
Table 4. Prescribed Volumetric Flux (Q) at the Onset, Number of Conduits and Period of Surface Velocity Oscillations for Sensitivity Tests on Conduit Spacing
Number of Conduits
Q at the Onset (1 Conduit) (km3 a-1)
Total Q for All Conduits (km3 a-1)
Period of Surface Velocity (years)
1.9 × 10− 3
1.9 × 10− 3
1.9 × 10− 3
9.5 × 10− 4
3.8 × 10− 3
 Results of the HIT model show that changes in the surface velocity of an ice stream are driven by changes in the till porosity, and hence till yield strength, at the GL. Porewater at the GL has two main sources: (a) local thermally induced basal melting; and (b) water transported from upstream through a subglacial conduit system. Lateral water flows through the porous till layer are small (of the order of 10-13 m a-1) and are not considered a principal water source. This results in three modelled hydrological states at the GL, where there is: (1) no conduit system; (2) a transient conduit system; and (3) a permanent conduit system. In the first hydrological state, oscillations in the surface velocity are generated by thermal conditions at the base of the ice stream. This was the case in tests KIS1 and KIS2, where regardless of the differing upstream hydrological system, till porosity at the GL was dictated by water generated by friction at the ice base, the geothermal heat flux and the basal temperature gradient (equation (17)). In this state the HIT model produced oscillations in the surface velocity of KIS that had an amplitude of the order of a 1000 m a-1 and a period of 200 - 300 years. Oscillation periods of the order of hundreds of years are predominantly due to the ice geometry of the model, especially the widening of the main trunk of KIS to ~128 km at the GL. Sensitivity tests conducted on a simplified ice stream geometry found that ice streams that are wider at the GL have shorter oscillation periods and have higher amplitude and lower mean surface velocities than narrower ones [Baker, 2012].
 In the second hydrological state, oscillations in the surface velocity were controlled by the volume of water transported to the GL from upstream, in addition to the thermal processes of the first state. There is spatial and temporal variability in conduit extent, with the conduit extending to the GL for only part of an oscillation cycle, as was evident in test KIS3. In this case, the HIT model produced oscillations in the surface velocity of KIS with an amplitude of the order of a 700 m a-1 and a period of 500 years. In the third hydrological state, the amount of water available at the GL is sufficient to maintain the upper till layer at its maximum porosity and a steady surface velocity. As such, there is a permanent conduit and any further increase in water transportation has little effect on ice flow dynamics.
 The period of oscillation found in tests KIS1-4 ranges from 200 to 500 years for KIS, with time periods with lower than mean surface velocity ranging between 120 and 350 years. While these periods of oscillation can only be viewed as indicative, it is likely that the ice stream will reach higher than mean surface velocities again within the next 180 years, given that KIS has been subject to stagnated flow for the last ~170 years. If KIS is in hydrological state 1, as hypothesised above, this reactivation may be already underway and KIS may reach higher than mean surface velocities again this century.
 The fact that there is little difference between tests KIS1 and KIS2 suggests that when an ice stream is in hydrological state 1, modelling conduit water fluxes is of small consequence to ice dynamics. However, when an ice stream has conduits that extend to the GL (hydrological state 2 or 3), the transportation of water to the GL has great significance. It is feasible that such ice streams have a more persistent basal water system either due to increased basal melting or from upstream water sources. For example, Rutford Ice Stream, which flows at speeds of 300-400 m a-1 [Barrett et al., 2009], has experienced no significant changes in surface velocity over the last 25 years [Gudmundsson and Jenkins, 2009]. This could mean that Rutford Ice Stream is currently in hydrological state 3.
 Although the bed and basal zone of KIS and its adjacent ice streams were investigated in numerous boreholes drilled to the bed [Engelhardt and Kamb, 1997; Kamb, 2001], direct studies did not conclusively identify a specific type of water system. The latter has only become apparent due to the detection of active subglacial lakes, which show that water is transferred laterally beneath the Siple Coast ice streams (Carter et al., 2007, 2009; Fricker et al., 2007; Fricker and Scambos, 2009; Smith et al., 2009). However, the way in which water is transported is still unknown. Although the Siple Coast ice streams are thought to have extended out onto the continental shelf in the past [Conway et al., 1999], there is no evidence of melt water features in the offshore geological record [Domack et al., 1999; Shipp et al., 1999]. This may indicate that the Siple Coast ice streams exhibit basal conditions in line with those we find in hydrological state 1, where conduits do not extend to the GL. Thus, oscillations in flow are controlled by thermal processes, rather than the rate of water supply to the bed. The apparent disconnect between an obviously active contemporary basal water system [e.g., Fricker et al., 2007] and the absence of evidence of melt water deposits on the Ross Sea floor [e.g., Shipp et al., 1999] may be a result of erosion or re-working of melt water-related material deposited upstream of the GL, that is, when the GL retreats.
 The findings presented above are in good agreement with observations of basal temperature and the basal temperature gradient. Modelled basal ice temperatures were between -3.4 and -3.0 °C at the onset and between -0.95 and -0.65°C at the GL (Table 3). Basal temperature gradients were found to be between -0.088 and -0.024 °C m-1. Around the area of UpC Camp borehole observations measured basal temperatures that ranged from -1.38 to -0.81°C and basal temperature gradients that ranged from -0.071 to -0.055°C m-1, with an average gradient of -0.062°C m-1 [Engelhardt, 2004]. In the same approximate location, the model predicts basal temperatures of between -1.50 and -1.07°C and gradients of between -0.065 and -0.055°C m-1. A comparison of modelled and InSAR-measured surface velocities shows that results from the stagnant part of an oscillation cycle are consistent with the modern flow of about 10 m a-1 and that the results from the active period are consistent with the modern flow of the neighboring active ice streams [Rignot et al., 2011].
 Sensitivity tests were conducted on the number of conduits spanning the width of KIS. Results show that it is the total influx of water to the ice stream that is important, rather than the volumetric flux to individual conduits. Doubling the total water flux entering the ice stream moved conditions from hydrological state 2 in test KIS3 to hydrological state 3 in test KIS3-A, while halving the total water influx at the onset caused the ice stream to change to hydrological state 1 in test KIS3-B. Where the total influx of water to the ice stream was unchanged (test KIS3-C,D), there was very little effect on ice stream dynamics from changes in the assumed number of conduits across the ice stream. This is in agreement with findings from tests KIS1-4 and also shows that changes to other parameters that affect the water flux above the GL, such as the 35% water flux assumed at the junction of the second upper ice stream tributary, would need to be large to change the ice stream's hydrological state.
 The HIT model is limited in that it neglects lateral variability and that it assumes till uniformity. As a 2D flowline model, it does not account for lateral changes in ice thickness and other variables. This also leads to the assumption that subglacial conduits form in a straight line from the onset to the GL, which ignores topographically induced meandering and interactions between neighboring conduits. Although the assumed symmetry of conduits obviously limits the spatial representation of the basal water system, we note that the number of conduits across the ice stream has little effect on the modelled ice flow. The subglacial till in the model is assumed to be spatially uniform and temporally constant in volume, neglecting any difference in deposition and erosion. It is more likely that till thickness and composition are spatially and temporally heterogeneous, which would have implications for till porosities. Despite these simplifications, the model does replicate appropriate magnitudes of the surface velocity and ice stream geometry of KIS.
 It is well known that the flow of ice streams is modulated by the physical properties of a subglacial till layer [Tulaczyk et al., 2000b], and that water flows within a basal drainage system beneath some Antarctic ice streams [Gray et al., 2005; Fricker et al., 2007; Fricker and Scambos, 2009; Smith et al., 2009; Fricker et al., 2010; Wingham et al., 2006]. The potentially important effects from interactions between a water system and till properties are, however, unknown. To obtain a first-order understanding of the influence of subglacial hydrology on ice stream flow we developed the HIT model, which couples ice thermodynamics, subglacial till dynamics and subglacial conduits. The results are significant in that they show that shifts between active and stagnant modes of flow are generated by porewater-induced changes in the till yield strength at the GL of an ice stream. Porewater is mainly derived from local thermally induced basal melting and/or water transported from upstream through a subglacial conduit system. This confirms the theoretical model of Tulaczyk et al. [2000b], which stated that internal oscillations in ice stream dynamics are thermally driven, but goes one step further by showing that water that is not generated locally must also be considered.
 The model suggests that there are three important hydrological states at the GL of an ice stream. These are: (1) where conduits do not extend to the GL and porewater availability is principally dictated by melting or freezing of the ice base; (2) where porewater availability from basal melting or freezing is augmented by upstream water fluxes through conduits that extend to the GL for part of a cycle; and (3) where porewater availability for basal melting or freezing is increased by upstream water fluxes through conduits that permanently extend to the GL. The three states affect the dynamics of ice stream flow by modifying the water balance at the GL.
 In hydrological state 1 (tests KIS1 and KIS2) the porosity of the till is dependent on local thermal processes at the GL, which determine the amount of melting and freezing that occurs there. In hydrological state 2 (test KIS3) the transportation of water from upstream during periods of high basal melting inland of the GL raises the till porosity and, through resulting changes in the basal shear stress, modifies the local basal melt rate and surface velocity. In hydrological state 3 (test KIS4) the same processes occur as in state 2, but with a permanent conduit at the GL. This results in till porosity values that remain at or near the maximum value and a steady (rather than oscillating) surface velocity. Any additional influx of water that occurs once the conduit has extended to the GL will have little impact on ice stream dynamics, as the basal shear stress is already low. The fact that surface velocities at the GL are in line with minimum values found in tests KIS1 and KIS2 suggests that the ice stream is currently in hydrological state 1. Results of tests KIS1-KIS3 are in agreement with observations and modelling by Hulbe and Fahnestock , Christoffersen et al.  and Catania et al. , which suggest that cycles of ice stream flow occur over hundreds of years.
 The cessation of the modelled conduit system halfway down the ice stream in hydrological state 1 (tests KIS1-KIS2) may explain the apparent disconnect between the fact that water is observed to flow in a hydrological system beneath WIS and KIS and that there is very little evidence of hydrological systems in the offshore geological record [Domack et al., 1999; Shipp et al., 1999]. If KIS is in hydrological state 1, which is most likely according to this study, and the periods of oscillation found by the model are indicative of its dynamics, its current stagnation for ~ 170 years suggests that the ice stream will reactivate in the coming century.
 This study was funded by a PhD scholarship, awarded to NvdW by Cambridge Commonwealth Trust and Trinity College Cambridge, and research grant NE/E005950/1 awarded to PC and MB by the UK Natural Environmental Research Council. The authors thank Bob Bindschadler, Michiel van den Broeke, Sasha Carter, Helen Fricker, Huw Horgan and Ted Scambos for advice and data. The authors are also grateful to Gary Clarke and two anonymous reviewers as well as editors Bryn Hubbard and Martin Truffer, all of whom helped improve the manuscript.