Notes on Glacier Hydrology McCarthy Summer School, 2016 · Notes on Glacier Hydrology McCarthy Summer School, 2016 Gwenn Flowers Department of Earth Sciences Simon Fraser University

Notes on Glacier Hydrology

McCarthy Summer School, 2016

Gwenn FlowersDepartment of Earth Sciences

Simon Fraser UniversityBurnaby, BC, CANADA

[email protected]

Note: Some of these course notes draw heavily from/plagiarize an open-access reviewpaper (Flowers, 2015). Please see the original for full details.

Contents

1 Introduction 3

2 Description of glacier drainage 4

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Elements of the subglacial drainage system . . . . . . . . . . . . . . . . . . 4

2.2.1 Sheets and films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.3 Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.4 Sedimentary canals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.5 Porous flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 A conceptual framework for subglacial drainage 6

4 Theory 7

4.1 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.2 Hydraulic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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4.3 Water velocity, flux and discharge . . . . . . . . . . . . . . . . . . . . . . . . 10

4.3.1 Laminar flow in a sheet . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.3.2 Laminar flow in a pipe . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3.3 Turbulent flow in a pipe . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3.4 Turbulent flow in a sheet . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3.5 Turbulent flow in cavities . . . . . . . . . . . . . . . . . . . . . . . . 13

4.3.6 Laminar flow in a porous medium . . . . . . . . . . . . . . . . . . . 13

4.4 Opening and closure terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.4.1 Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.4.2 Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.4.3 Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.5 Other governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.5.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Summary 16

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1 Introduction

Water was recognized to facilitate ice flow by some of the earliest pioneers of modernglaciology (see review by Clarke, 1987). The foundations of our conceptual models ofglacier drainage were largely laid in the 1960’s, ’70’s and ’80’s, with some work focused onglaciers as storage reservoirs in the hydrological cycle, and other work focused on the basaldrainage system and its influence on glacier and ice-sheet dynamics (see Walder, 2010,for a review). The latter motivated the mathematical description of individual “drainageelements”, which form the building blocks of the subglacial drainage system as we stillunderstand it. These elements include water films and sheets (e.g. Weertman, 1972; Walder,1982), cavities formed in the lee of bedrock obstacles (e.g. Lliboutry, 1968; Iken, 1981;Walder, 1986; Kamb, 1987), ice-walled conduits (e.g. Shreve, 1972; Rothlisberger, 1972),canals in sediment (e.g. Walder and Fowler, 1994; Ng, 2000a) and bedrock channels (Nye,1973). Spatial and temporal transitions between elements that comprise fast, efficientor channelized drainage versus slow, inefficient or distributed drainage have long beenunderstood to play an important role in ice dynamics (e.g. Muller and Iken, 1973).

A proliferation of studies over the last decade from the Greenland and Antarctic icesheets implicate subglacial water in driving rapid ice-flow variations detected many hun-dreds of metres above at the ice surface. The discovery of filling and draining subglaciallakes in Antarctica (e.g. Gray and others, 2005; Wingham and others, 2006; Smith andothers, 2009), and their association with persistent fast-flow features (e.g. Bell and others,2007; Fricker and others, 2007; Langley and others, 2011) and transient acceleration (e.g.Stearns and others, 2008), has revealed unexpected dynamism in this subglacial systemisolated from the surface. In Greenland, measurements of seasonal (e.g. Zwally and others,2002; Sole and others, 2011; Joughin and others, 2013) and short-term (e.g. Das and others,2008; Hoffman and others, 2011) variations in outlet-glacier flow have prompted researchinto surface-to-bed hydrological coupling (e.g. van der Veen, 2007; Tsai and Rice, 2010;Stevens and others, 2015) and the role of surface meltwater in overall ice-sheet dynamics(e.g. van de Wal and others, 2008; Cowton and others, 2013; Meierbachtol and others,2013).

These notes focus on glacier hydrology with a strong bias toward the basal drainagesystem and the formulation of basal hydrology models for the purpose of coupling withmodels of ice flow. Notably absent is any consideration of polythermal bed conditions andcontent related expressly to the basal hydrology of Antarctica. For reviews of the obser-vational basis of glacier hydrology theory and modelling, see Hooke (1989), Hubbard andNienow (1997) and Fountain and Walder (1998) for temperate glaciers, Hodgkins (1997),Irvine-Fynn and others (2011) for polythermal glaciers and Chu (2014) for Greenland.Rothlisberger and Lang (1987), Jansson and others (2007) and Walder (2010) include the-oretical contributions, and Lang (1986) and Hock and Jansson (2005) focus on surface

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hydrology and runoff prediction.

2 Description of glacier drainage

2.1 Overview

Water flows over, in and under glaciers in a variety of ways, with some hard-to-discoverfraction of surface water making its way to the base of the glacier in environmental settingswhere abundant surface water is produced. This excludes much of the Antarctic ice sheet,but includes the periphery of the Greenland ice sheet and other polythermal and temperateice masses. Where surface water accesses the glacier bed, it is often considered to supplya much greater volume of water than water sourced from the bed itself as basal melt.Though glacier ice is permeable along grain boundaries, it is often treated as having noprimary porosity. Rather, surface water enters the glacier through macroscopic portalssuch as crevasses and moulins. Moulins are formed by flowing water itself and often exploitstructural weaknesses such as crevasses. Water accesses the bed of thick, cold ice (suchas is found in Greenland) through moulins and crevasse fields, in addition to throughshort-lived fractures that can form at the base of supraglacial lakes. Glaciers can play asignificant role in modulating the runoff from glacierized basins, acting as storage elementsfor precipitation, with different time constants associated with water release from snow,firm and the englacial/subglacial drainage system.

2.2 Elements of the subglacial drainage system

Numerous subglacial drainage-system structures have been envisioned based on deglaciatedlandscapes (e.g. Walder and Hallet, 1979) or measurements of proglacial stream variables(e.g. Collins, 1979; Raiswell, 1984) or injected tracers (e.g. Kamb and others, 1985). Mea-surements of basal water pressure (e.g. Mathews, 1964; Lappegard and others, 2006) andother water quality indicators (e.g. Tranter and others, 2002) have been used to studydrainage-system structure and dynamics (e.g. Fudge and others, 2008). Direct visual ob-servations are rare, though borehole imagery (e.g. Harper and Humphrey, 1995) and explo-ration of englacial/subglacial cavities and tunnels (e.g. Anderson and others, 1982; Gulleyand others, 2009) have furnished some important and surprising information (e.g. Vincentand others, 2010).

Common to the mathematical descriptions of many of the elements is a considerationof (1) the mechanisms of element opening (e.g. by melting of the ice roof, removal of finesfrom unconsolidated sediment, erosion of bedrock, glacier sliding over bedrock obstacles)

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and closure (e.g. by ice creep, sediment deformation, freeze-on of supercooled water),(2) whether water flow through the elements is laminar or turbulent, (3) the relationshipbetween water discharge and effective pressure and (4) the stability of the drainage systemto perturbations in discharge. Below is brief qualitative sketch of each of the commonelements.

2.2.1 Sheets and films

The water sheet or film is among the oldest and conceptually simple descriptions of basaldrainage, often associated with Weertman (1957) as part of his theory of sliding by rege-lation and enhanced creep around bedrock obstacles. The film is a thin water layer every-where separating ice and bed. Films in excess of of several millimetres are predicted tobe unstable (Walder, 1982) unless supported by intermittent clasts at the bed (Creyts andSchoof, 2009).

2.2.2 Channels

Conduits formed partially or entirely within glacier ice represent the canonical “fast” orefficient drainage element. Englacial and subglacial drainage through a three-dimensionalchannelized system was articulated by Shreve (1972), while Rothlisberger (1972) describedthe physics of the ice-walled conduits that now bear his name (“R-channels”). Channelsevolve through the competition of melt opening and creep closure (detailed later). In steadystate, there is a weak inverse relationship between discharge and the pressure gradient,consistent with observations of a declining baseline in basal water pressure over the meltseason and a late summer minimum sliding speed. Channels also feature prominently inthe description of glacier outburst floods, or jokulhlaups.

2.2.3 Cavities

Separation of the ice and bed during the sliding process produces water-filled cavities in thelee of bed obstacles (e.g. Lliboutry, 1968; Iken, 1981; Fowler, 1986). A system of “linkedcavities” is the classical picture of the slow, inefficient or distributed subglacial drainagesystem (Kamb, 1987), where steady-state water pressure increases with discharge. Cavitiesopen by sliding and close by creep, with viscous dissipation of heat in the flowing watercontributing to melt enlargement of the cavities roofs and their connecting passageways.Cavitation can produce both positive and negative feedbacks on sliding, depending on thevolume and mobility of water stored at or near the bed.

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2.2.4 Sedimentary canals

In the presence of an unconsolidated glacier bed, water may flow in broad low canalsthat are sediment-floored and ice-roofed (e.g. Shoemaker, 1986; Walder and Fowler, 1994;Ng, 2000a). In principle, the ice roofs may behave like R-channels, but canals are oftenclassified as distributed (slow, inefficient) drainage elements. Various approaches are usedto account for till deformation and sediment transport in these systems, from neglectingboth to including canal closure by till creep (e.g. Alley, 1992; Fowler and Walder, 1993) toexplicitly modelling sediment discharge (Ng, 2000b).

2.2.5 Porous flow

Treating the subglacial drainage system as a porous medium—essentially a confined ground-water aquifer—has a long history (Lingle and Brown, 1987, e.g.) and has been useful ininterpreting borehole observations from sediment-rich beds (e.g. Stone and Clarke, 1993).Descriptions of drainage system opening and closure are usually restricted to till dilation orsimply omitted. Darcy’s law governs water flow through a porous medium, with hydraulicconductivity or permeability often loosely interpreted for the subglacial system, while theMohr-Coulomb criterion is used to relate till shear strength and effective pressure in thesesystems (see Clarke, 2005, for a review).

3 A conceptual framework for subglacial drainage

Mathematical models of glacier drainage require several common ingredients. Conservationof mass gives rise to the continuity equation,

∂h

∂t+∇ · q = b, (1)

written here for an incompressible fluid with an areally averaged water volume or waterdepth h [L], water flux q [L2 T−1] and source/sink term b [L T−1]. Potential water sourcesand sinks,

b = bs + be + ba + bb, (2)

arise from melt, rain or runoff from the ice surface bs, water produced through englacial pro-cesses such as strain heating be, groundwater recharge/discharge in underlying or adjacentaquifers ba and basal water production or consumption bb. In-situ basal water productioncommonly includes melt due to the geothermal flux Qg/(ρi L) and the frictional heat fluxassociated with glacier sliding Qf/(ρi L), with ρi the density of ice and L the latent heatof fusion.

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Conservation of linear momentum can be used to derive an expression for fluid velocityu, from which flux q can be computed by integrating velocity over flow depth h. Morecommonly however, models simply adopt empirical expressions for flux of the followinggeneral form:

q = −K hα (∇φ)β , (3)

where K is a rate factor, ∇φ is the fluid potential gradient and exponents α and β dependon the conceptual model of the drainage system, including whether flow is laminar orturbulent. For α = β = 1, (3) reduces to Darcy’s law for laminar flow, while α = 3/2and β = 1/2 are typical for turbulent flow through a sheet or cavity system. In its mostgeneral form, the rate factor Kij(x, y, z, t) is a tensor field, allowing for both anisotropy andheterogeneity. For pipe-like rather than sheet-like subglacial drainage, an expression fordischarge Q [L3 T−1] rather than flux q [L2 T−1] would be used, with a flow cross-section Sreplacing the equivalent flow depth h. In this case, K might be a function of the hydraulicradius and wetted perimeter of the flow element, as well as its surface roughness.

The fluid potential φ is the total mechanical energy per unit volume of fluid requiredto move the fluid from one state to another, where the states differ by a pressure pw andan elevation z:

φ = pw + ρw g z, (4)

where ρw is the density of water and g is the gravitational acceleration. Gradients inhydraulic potential, ∇φ, are the primary drivers of water flow. Some models require arelationship between h and pw to close the equations. Other models set φ = ρi g H+ρw g z,with ice thickness H, such that fluid potential is defined only by ice geometry.

Finally, models that allow the drainage system itself to evolve require additional equa-tions of the form

∂h′

∂t= opening − closure, (5)

where h′ is a measure of the effective drainage-system capacity, such that h = h′ impliessaturation. The opening and closure terms are specific to the conceptual model. Ice-walledconduits are the most common example of evolving drainage elements, where opening isdescribed by melting of the conduit walls and closure by the inward creep of ice.

4 Theory

Below we examine some of the ingredients of drainage models in more detail.

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4.1 Continuity

In glacier hydrology models, statements of mass conservation, or continuity, are oftenwritten in terms of water volume, under the assumption that water is incompressible.Additional terms are needed when including compressibility, which can be important inconfined aquifer flow.

For a 1-D subglacial water sheet of thickness h = h(x, t) that varies in time t and inspace x, the volume per unit width of water in a sheet of length l is

V =

∫ l

oh(x, t) dx. (6)

The rate of change of total water volume must be equal to the sum of sources and sinks,and can be written using the material derivative (see notes on continuum mechanics):

dV

dt=

∫ l

o

(∂h

∂t+∂(v h)

∂x

)dx =

∫ l

o(b+m) dx, (7)

where v is the fluid velocity, b represents the rate of water supply (from external sourcessuch as the glacier surface or a groundwater system), m represents in-situ water production(e.g. from basal melt), and both b and m can be positive (sources) or negative (sinks).Defining the water flux in the standard way as q = v h, we can write the local form of thewater balance as

∂h

∂t+∂q

∂x= b+m. (8)

This should look familiar as a 1-D expression of Equation 1.

In the case of a conduit (i.e. a pipe) of cross-section S = S(s, t) we use the conservationof mass, instead of volume, as the phase change of the conduit walls will become important.The mass of water in a section of conduit of length l is

M = ρw

∫ l

oS(s, t) ds, (9)

using the flow-following coordinate s. By analogy to the procedure for the water sheet, wetake the full time derivative of Equation 9 (making use of the material derivative), set itequal to the sum of sources and sinks, and write the local form of the water balance as

∂S

∂t+∂Q

∂s=b+m

ρw, (10)

with discharge Q = v S. In this case, b could be incoming water from a moulin, or exchangeof water with the surrounding drainage system; m would be dominated by melt of the

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conduit walls (or freeze-on if negative) which requires additional physics to understand.Note that b and m in Equation 10 have dimensions of M L−1 T−1, instead of L T−1 as inEquation 8. Sources/sinks in the conduit case can be thought of as mass melting/freezingrates per unit length of conduit.

4.2 Hydraulic potential

Equation 4 is the starting point for many subglacial drainage calculations. Water flowsin response to two primary drivers: pressure gradients and gravity. The fluid potentialincludes contributions from both, with gravity represented in the elevation potential termρw g z and pressure potential in the term pw. Equation 4 is closely related to the concept ofhydraulic head in groundwater hydrology: h = φ/(ρw g) = Ψ + z, where Ψ is the pressurehead and z is the elevation head.

A common way to determine the direction of water flow in or under a glacier is tobegin with the fluid potential gradient:

∇φ = ∇pw + ρw g∇z, (11)

where ∇ is the gradient operator, and spatial gradients in ρw are assumed negligible. Ifwe restrict the analysis to water flow at the glacier bed, z = zb, and make the assumptionthat water pressure is some fixed fraction k of the ice overburden pressure, we can write

∇φ = k∇pi + ρw g∇zb. (12)

Writing pi = ρi g H and defining ice thickness as the difference in surface and bed elevations,H = zs − zb, yields

∇φ = k ρi g∇zs + (ρw − k ρi) g∇zb. (13)

In an ice-free landscape, or for the hypothetical condition that pw ≈ 0, k is set to zero, andwater flows reassuringly downhill:

∇φ = ρw g∇zb. (14)

In subglacial hydraulic potential analysis, k is almost always taken to be 1, giving

∇φ = ρi g

[∇zs +

(ρw − ρi)ρi

∇zb]. (15)

Equation 15 is the source of statements such as “The glacier surface slope is ten times moreimportant than the bed slope in determining the direction of water flow.” Shreve (1972)is most commonly cited as the source for such statements. Inserting ρw = 1000 kg m−3

and ρi = 900 kg m−3 results in (ρw−ρi)ρi

∼0.1, thus a factor of ∼9 between the coefficients of

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surface and bed slope. Assuming instead ρi = 917 kg m−3 gives (ρw−ρi)ρi

∼0.09, and a factorof ∼11 between the two coefficients.

Though seemingly crude, the above analysis often works well in predicting approximatelocations of outlet rivers and the paths of subglacial floods or jokulhlaups (e.g. Bjornsson,2002). It works best at large spatial scales, but in considering only glacier geometry it failsto capture the potentially important influence of the distribution of water sources/sinks(e.g. moulins) on drainage direction (e.g. Werder and others, 2013). In assuming k = 1,spatial and temporal fluctuations in water pressure are neglected. Though values of k otherthan 1 and 0 are sometimes presented, it is unclear that there is a good physical basis forsuch an assignment.

4.3 Water velocity, flux and discharge

For a water film or sheet, flux is defined as q = v h, where v is the mean velocity andh is the film or sheet thickness. In the case of a pipe or conduit, we more commonlydefine discharge, Q = v S, where S is the cross-sectional area of the conduit. It is inthe prescription of velocity (and therefore flux or discharge) that various drainage systemelements or types differ.

4.3.1 Laminar flow in a sheet

We make use of the conservation of momentum (Newton’s second law) to describe thewater velocity. We start with a form of the Navier-Stokes equation:

ρwdv

dt= −∇φ+ µ∇2v + ρw g, (16)

with water density ρw, velocity v and dynamic viscosity µ. Assuming steady state (noacceleration) and a simplified form for the velocity vector v = (vx(z), 0, 0) in a horizontal1-D system,

µd2vxdz2

=dφ

dx. (17)

Integrating twice with respect to z and applying boundary conditions vx(z = 0) = vx(z =h) = 0 yields

vx(z) =dφ

dx

h2

2µ

(z2

h2− z

h

). (18)

Note that for laminar flow, the velocity scales with the fluid potential gradient. The fluxis computed by integrating the velocity with depth:

qx =

∫ h

0vx(z) dz =

dφ

dx

h2

2µ

∫ h

0

(z2

h2− z

h

)dz = −dφ

dx

h3

12µ. (19)

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The depth-averaged water velocity is then simply

vx =qxh

= −dφdx

h2

12µ. (20)

4.3.2 Laminar flow in a pipe

For water flow in a tube or pipe, we introduce a flow-following coordinate s and write theNavier Stokes equation in radial coordinates as

µ

r

d

dr

(rdv

dr

)= −dφ

ds, (21)

again for steady state horizontal flow. Integrating twice with respect to r and applyingboundary conditions gives

vs(r) = −dφds

(R2 − r2)4µ

(22)

for pipe radius R. Laminar flow occurs only at low Reynolds numbers,

Re =ρw v d

µ, (23)

where d is a characteristic length scale, say the thickness of a water sheet or radius of apipe. Re = 2300 is often considered the critical value at which the laminar-to-turbulenttransition occurs for pipe flow.

4.3.3 Turbulent flow in a pipe

Turbulent pipe flow is one of the most commonly used formulations in the descriptionof glacier drainage (e.g. Rothlisberger, 1972). The governing equations are based on abalance between the driving and resistive forces for water flow. The driving force is relatedto the total fluid potential gradient (pressure and elevation gradients) and the resistiveforce arises from the stress between the fluid and the surrounding pipe wall. The latter isusually parameterized using one of several formulations from engineering hydraulics.

For a pipe of radius R, the driving force per unit length is related to the fluid potentialgradient ∂φ

∂s and the cross-sectional area of the pipe π R2, while the resistive force per unitlength is related to the wall stress τwall and wall perimeter 2π R. Equating these termswith the appropriate signs gives

−∂φ∂s

=2

Rτwall, (24)

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where the formulation of wall stress must be specified in order to solve for velocity. Thetwo most common prescriptions of wall stress employ the Gauckler–Manning–Strickler for-mulation or the Darcy–Weisbach formulation. We will use the latter for reasons describedin Clarke (2003):

τwall =1

8fR ρw v

2, (25)

where fR is a friction factor that describes wall roughness and v is the mean velocity. Asa function of the Manning roughness n′,

fR =8 g n′2

R1/3H

, (26)

where RH is the hydraulic radius equal to πR/2(π+2) for a channel of circular cross-sectionand R/2 for a channel of semi-circular cross-section. Substituting 25 into 24 and solvingfor velocity gives

v = −(

4R

fr ρw

)1/2 ∂φ

∂s

∣∣∣∣∂φ∂s∣∣∣∣−1/2, (27)

where the slightly cumbersome notation ensures that the sign of the velocity is oppositethat of the potential gradient, and the magnitude of the velocity scales with the squareroot of the magnitude of the fluid potential gradient as expected for turbulent flow.

4.3.4 Turbulent flow in a sheet

The turbulent flow formulation for a pipe is readily adapted to a sheet by writing velocity(here in 1-D) in terms of hydraulic radius RH :

v = −(

8RHfr ρw

)1/2 ∂φ

∂x

∣∣∣∣∂φ∂x∣∣∣∣−1/2. (28)

The hydraulic radius is defined as the ratio of flow cross-section to wetted perimeter. Fora thin sheet (h� w), RH ≈ h/2 and

v = −(

4h

fr ρw

)1/2 ∂φ

∂x

∣∣∣∣∂φ∂x∣∣∣∣−1/2. (29)

qx = v h = − 2h3/2

(fr ρw)1/2∂φ

∂x

∣∣∣∣∂φ∂x∣∣∣∣−1/2. (30)

The above may differ from other formulations by a factor like√

(π + 2)/π, though wewon’t dwell on this as the friction factor fR is usually poorly enough constrained thatthese differences would be absorbed in any model tuning.

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4.3.5 Turbulent flow in cavities

Turbulent flow in cavities is expressed in different ways, depending on assumptions ofcavity geometry and the treatment of the system (e.g. individual cavities versus a systemof interconnected cavities). The tendency is to use discharge Q and cross-sectional areaS in describing flow through individual cavities, but to use flux q and an effective heightor thickness h in describing flow through a system of cavities. Formulations for individualcavities closely resemble turbulent flow through pipes (see e.g. Walder, 1986; Kamb, 1987;Schoof, 2010), while formulations for cavity systems more closely resemble turbulent flowin a sheet (see e.g. Hewitt, 2011; Werder and others, 2013).

4.3.6 Laminar flow in a porous medium

Darcy’s law is used to describe laminar flow in a porous medium, with velocity (in 1-D)

vx = −K∂φ

∂x, (31)

where K is hydraulic conductivity (here expressed as a scalar), and flux

qx = −K h∂φ

∂x. (32)

This gets more interesting in two and three dimensions and when hydraulic conductivityis anisotropic. Using index notation (see notes on continuum mechanics):

qi = −Kij h∂φ

∂xj. (33)

4.4 Opening and closure terms

4.4.1 Channels

The rate of energy supply (power) per unit length of conduit, due to the release of total(gravitational plus pressure) potential energy, is

P = −∂φ∂s

Q. (34)

In the presence of a pressure gradient, some power per unit length of conduit is consumedor released in keeping the water at the pressure melting point:

Pm = −ct ρw cw(−∂pw∂s

)Q, (35)

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where ct is the pressure-melting coefficient and cw is the specific heat capacity of water.The above can be worked out by considering a volume of fluid that travels from one pressurestate to another, each associated with a different melting temperature.

We define for convenience the quantity γ = ct ρw cw and write the net power per unitlength of conduit available for melting as the sum of Equations 34 and 35:

Pnet = P + Pm = Q

[−∂φ∂s− γ

(−∂pw∂s

)]. (36)

Recalling the definition of fluid potential φ we can separate pressure gradient and elevationgradient terms:

Pnet = Q

[−ρw g

∂zb∂s− (1− γ)

(−∂pw∂s

)]. (37)

The first term is the power per unit length of conduit available for wall melting fromthe loss of gravitation potential energy. The second term is the power per unit lengthof conduit available for wall melting from the loss of pressure potential, less the powerconsumed or released in keeping the water at the pressure-melting temperature. The valueof γ is approximately 0.3, meaning the energy associated with pressure-melting effects canbe non-negligible (e.g. Werder, 2016).

If we define the mass rate of conduit wall melting per unit length as m, the volume rateof melting per unit length is m/ρi, and we can write an expression for the rate of conduitenlargement by wall melting:

∂S

∂t

∣∣∣∣melt opening

=m

ρi=Pnet

ρi L=

Q

ρi L

[−ρw g

∂zb∂s− (1− γ)

(−∂pw∂s

)]. (38)

Note that m in Equation 38 is equivalent to m in the continuity equation for channels(Equation 10).

The tendency for conduits to enlarge by melt opening is counteracted by creep closure.For a circular conduit, we use radial coordinates to write the expression

1

R

dR

dt= A

(pi − pwn

)n, (39)

where R is conduit radius, A is Glen’s flow-law coefficient and n the flow-law exponent.Noting S = πR2 and using the chain rule, we can write the closure in terms of S:

∂S

∂t

∣∣∣∣creep closure

= 2S A

(pi − pwn

)n, (40)

or in a more compact form using the effective pressure N = pi − pw:

∂S

∂t

∣∣∣∣creep closure

= 2S A

(N

n

)n. (41)

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This form of the closure relation applies to a channel of circular cross-section, and is alsocommonly used for channels semi-circular in cross-section. Other closure relations havebeen proposed for different cross-sectional shapes (e.g. Hooke and others, 1990).

One can solve for the steady-state discharge and cross-sectional area of a conduit bysetting Equations 38 and 40 equal, and using Q = v S and an expression for turbulent flowvelocity (e.g. Equation 27). For a flat bed, steady-state channel discharge is found to beweakly but inversely related to the pressure gradient:

Q ∝(−∂pw∂s

)−7. (42)

This result can be used to illustrate an important property of steady state Rothlisbergerchannels. Steeper pressure gradients are associated with smaller channels, implying atendency for large channels to capture water from small ones. This property leads to apositive feedback that makes channels inherently unstable in the presence of an adequatewater supply (Nye, 1976). Under steady-state conditions and for a typical valley glaciergeometry, the above would lead to an arborescent drainage system whereby very few largechannels are fed by a network of smaller tributaries (Rothlisberger, 1972; Shreve, 1972).

4.4.2 Cavities

In principle, melt opening of cavity roofs occurs in the same way as described for conduits.This is, in fact, what is hypothesized to drive the transition from a distributed cavity-dominated (inefficient) to a channelized (efficient) drainage system. In unified models (e.g.Schoof, 2010), where a single drainage element can evolve into a cavity or channel, meltopening is prescribed in much the same way as in Equation 38. In models that distinguisha cavity system from a channel or network of channels, melt opening is rarely applied tocavities. Instead, cavities are considered to open by sliding, a process for which there isstrong observational evidence (e.g. Iken and others, 1983). The increase in conduit cross-sectional area due to basal sliding over obstacles can be approximated as

ub h′ (43)

where is ub the basal sliding speed and h′ is the obstacle height. Creep closure of the cavityroof is often written in an identical fashion to that of a channel.

4.4.3 Sheets

Opening and closure terms are often neglected when the basal drainage system is concep-tualized as a water sheet or film. Complex and uncertain ice-roof geometry provides one

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deterrent. A notable exception is the work of Creyts and Schoof (2009) for a clast-supportedwater sheet, where both regelation and creep are included in the closure term but its formrequires prescription of the distribution and sizes of supporting clasts. Sheet opening dueto elastic flexure by “hydraulic jacking” can be an important process in the presence offloods (e.g. Bjornsson, 2002; Stevens and others, 2015) and has been parameterized in somemodels (e.g. Pimentel and Flowers, 2011).

4.5 Other governing equations

4.5.1 Temperature

Though most treatments of basal hydrology assume fully temperate conditions, and makeassumptions of instantaneous heat transfer between flowing water and the surrounding ice,conservation of energy can be used to derive an equation for fluid temperature:

∂Tw∂t

= −v ∂Tw∂s

+1

ρw cw S

(Pwet τwall v −m

(L+ cw(Tw − Ti)−

v2

2

)), (44)

with water temperature Tw, ice temperature Ti, wetted perimeter Pwet and other variablesas defined above. Such an equation makes it possible to advect heat in the flowing water, aseemingly more physically motivated approach than assuming instantaneous heat transfer(see Clarke, 2003). Observations of jokulhlaups, however, suggest that heat transfer duringconduit formation is more rapid than is accounted for in the standard model, and thus thatthe assumption of instantaneous heat transfer may be more appropriate than allowing foradvection. The flaw is likely in the assumption of a uniform distribution of heat in thecross-section of flow, leading to an underestimate of the temperature gradient near theconduit walls and thus the wall melting rate.

5 Summary

This is overdue and I’ve run out of time. The next iteration of these notes will

• includes figures and diagrams

• include some content related to supraglacial and englacial drainage

• address the role of glaciers in the hydrologic cycle

• touch on the hydrology of the Antarctic ice sheet and ice streams

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• lay out typical sets of governing equations for modelling basal drainage

• introduce applications!

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