CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
S-1
Text S1
S1. Sheet model
We envision a rough bed of protrusions (Fig. 3d) that span the water and partially
support the weight of the overlying ice. A fraction of water depth H will be occupied by
protrusions and make the average water depth less than the maximum depth. To calculate
the change in water depth, we assume protrusions are spherical sediment grains with a
size distribution equivalent to a deformation till. We employ fractal scaling of grain sizes
with Ns,j as the number density of the jth size class defined as
Ns,j
l2
:= 0 2 = N0
πRj
Rj
R0
−a
,
(S1)
where l02 is a bed area with edge length l0 , Rj is the average radius of the jth protrusion
size at H = 0, N0 is a reference number density and R0 is a reference grain size. There is
exactly one protrusion of radius R0 in area l02 . We set fractal index a = 3, which implies
that each grain size class occupies the same volume of the till. This value of a is close
to measured values [Fischer and Hubbard , 1999; Hooke and Iverson, 1995; Khatwa et al.,
1999]. To discretize grain sizes, we employ the Φ-scale commonly used to classify loose
sediments. In terms of grain radius R, Φ is defined as
2R = 2−Φ × 0.001 m.
(S2)
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
S-2
We construct grain size classes with average radius Rj by setting Rj = 2−(Φj +1) × 0.001 m
with Φj = −9, −8, . . . , 8. The largest size class is R1 = 0.256 m, and our grain size
distribution includes coarse gravel, cobbles, and boulders. Assuming that each grain size
is exposed at zb as a hemisphere into the sheet [Creyts, 2007, Appendix B]. For any change
in water depth, there is a linear transformation using equations (S1) and (S2) to obtain
the mean water depth from the maximum water depth or vice versa.
S1.1. Closure velocity
In equation (7b), the closure rate is the sum of a creep component wc , and a regelation component wr . This formulation is similar to a hard-bed sliding formulation [e.g.,
Weertman, 1964; Paterson, 1994, Chap. 7] except that the direction is normal rather
than tangential to the bed.
We take the closure velocity as
∂H
βK Ae,j Si,j
= −wi = − A|σe,j |n−1 σe,j le,j +
σe,j ,
∂t
ρ L Ve,j Ss,j
|
{z
} |i
{z
}
creep
(S3a)
regelation
where A is the flow law coefficient, σe,j is the incremental effective stress on the jth
protrusion size, le,j is the length scale separating protrusions in the jth protrusion size,
K is an average thermal conductivity of ice and protrusions, Ae,j and Ve,j are the average
cross sectional area and volume, respectively, of a protrusion of the jth size class that
penetrates into ice, and Si,j and Ss,j are the fraction of the unit area occupied by ice
and sediment, respectively. Both creep and regelation have been studied extensively and
equation (S3a) recasts previous work [e.g., Nunn and Rowell , 1967; Nye, 1967, 1953]. We
assume that the velocity is continuous over all grain sizes,
wj = wj+1 = . . . = wN = wi .
(S3b)
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
S-3
If the local stress driving closure is not the far-field effective stress, then a simplified
relationship for the stress at the bed is required. In this case, Creyts and Schoof [2009]
use a linear stress recursion to divide the stresses among the different protrusion sizes
σe,j = σj − σj+1 ,
(S4a)
where σe,j is an incremental effective stress and σj is a stress on a protrusion size. The
difference in stress between the jth protrusion size class and the j + 1th size class drives
ice closure. For the largest and smallest protrusion sizes, the first stress is σ1 = σi and
last stress is σN +1 = pw . Summation over all protrusion sizes yields the overall effective
pressure,
N
X
σe,j = pe .
(S4b)
j=1
Where the ice overburden stress and water pressure are known, only the configuration
of protrusions along the bed is necessary to define the terms le,j , Si,j /Ss,j , Ae,j , and Ve,j .
Grains are assumed to be hemispherical at the glacier bed. For the jth size class, bed
protrusions are spaced distance lj apart and thus protrude above the datum H = 0 at
zb . All size classes that remain in contact with the ice roof must then have Rj ≥ H and
contact cross-sectional area πrj where rj2 = Rj2 − H 2 . For H = 0, Rj = rj exactly. The
effective length scale is
le,j =
q
lj2 − πrj2 ,
(S5a)
where exactly one protrusion is in area lj2 . The ratio of the ice to protrusion area is
Pj−1
Si − k=1 Ss,k
Si,j
=
=
Ss,j
Ss,j
1−
Pj−1 πrk2
k−1 2
lk
.
2
πrj
lj2
(S5b)
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
S-4
Finally, from assumption that the protrusions are hemispheres, the effective volume and
area are, respectively,
Ve,j =
π
2Rj3 − Hrj2 ,
3
Ae,j =πrj2 .
(S5c)
(S5d)
For comparison, Creyts and Schoof [2009] assumed Ae,j /Ve,j ∼ 1/rj .
With equation (S3a), the bed distribution equations (S1) and (S2), and the geometrical
assumptions (equations (S5a–d)) we can now solve for the closure relationship for a range
of effective pressures. These solutions are shown in Figure S1. Because solutions require
a Newton-Raphson iterative solver, we do not compute the solutions at every time step
in the numerical solution of the hydrodynamic equations (3a–d) for the sheet. Rather,
we make a lookup table and interpolate from known solutions to intermediate points.
This lookup table introduces a maximum error of 1.2% relative to exact solution of the
equations [see Creyts, 2007, App. F]. This error is insignificant.
S2. Heat flow between water and ice
For turbulent water flow, heat must be transferred from a well-mixed interior through a
boundary layer to an ice wall. The magnitude of conductive heat flux q across a laminar
hydraulic boundary layer can be formulated as,
q = −Kw ∆T δ,
(S6)
δ is the thermal boundary layer thickness and ∆T is a temperature difference driving heat
transfer [e.g., Schlichting, 1979]. It is common to multiply (S6) by a Nusselt number Nu
to obtain a flux that includes convection in the well mixed portion of the flow,
q = −NuKw ∆T ℓ,
(S7)
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
where ℓ is a characteristic length scale for heating,
(
4Rh channel,
ℓ=
H
sheet.
S-5
(S8)
The Nusselt number is a ratio of total heat transfer to conductive heat transfer. The melt
rate is then the heat flux to the wall divided by the latent heat of fusion,
m = qPi L
(S9)
Substituting q from equation (S7) with the appropriate length scale into (S9) yields equations (8a) and (8b).
There are numerous Nusselt number correlations that take the form
Nu = CRea Prb ,
(S10a)
where a, b, and C are constants [Bird et al., 1960; Schlichting, 1979; Hutter , 1982]. The
Reynolds number Re, and Prandtl number Pr, are
(
4Rh |u|ρw µ
channel,
Re =
H|u|ρw µ
sheet,
Pr = µcw Kw
(S10b)
(S10c)
where µ is the viscosity of water.
In glaciology, the Dittus-Boelter relation for smooth-walled tubes [McAdams, 1954,
p. 219],
Nu = 0.023Re4/5 Pr2/5 ,
(S11a)
has found wide use in models of outburst floods [e.g., Clarke, 2003; Mathews, 1973; Nye,
1976; Spring and Hutter , 1981]. This equation is less reliable in the transition region,
defined approximately as Re = 2100–10 000 [Bird et al., 1960, p. 401]. Despite this
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
S-6
shortcoming, equation (S11a) appears to be the most relevant above the laminar flow
regime.
Within the laminar flow regime, we use
1/3
Nu = 1.86 Re Pr ℓ ds
(S11b)
[Bird et al., 1960, p. 399] with ℓ from equation (S8) . Assuming H/ds = 0.01 gives a
Nusselt number of approximately 6 for the laminar case and 27 using equation (S11a). In
general, water in the subglacial system flowing below a Reynolds number of about 2100
will not have a strong effect on melt or accretion because laminar heat transfer is slow
relative to turbulent transfer. While neither is ideal because of the rough boundaries of a
subglacial system, we employ (S11a) and (S11b) in equation (3).
S3. Porosity of the accreted ice
Porosity of accreted ice is not well-defined but will likely be a function of ice formation
mechanism and flow conditions. To our knowledge, direct observations of subglacial ice
actively accreting do not exist. Observations made at glacier termini are often used as
proxies for the subglacial environment. Frazil ice forms at Matanuska Glacier in water
that ascends vents adjacent to the terminus and flocculates to form variable density crystal
aggregates [Lawson et al., 1998]. Other glacier termini show similar crystal aggregate formation [e.g., Roberts et al., 2002; Tweed et al., 2005].Even though water that ascends steep
local features may not be representative of a subglacial environment, frazil formation is a
viable mechanism of accreting ice and likely dominates where glaciohydraulic supercooling occurs. Accretion ice could also form from a mushy layer that grows downward from
the overlying ice [e.g., Worster , 1997]. Depending on conditions, the porosity in mushy
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
S-7
layers can be high. Unrelated closed ice conduit experiments showed that accretion ice
is not very porous [Gilpin, 1981]. However, in these experiments pressure gradients and
water velocities were much higher than and probably not analogous to most subglacial
environments.
In the absence of an exact relationship for the evolution of the ice porosity, ni becomes
an adjustable parameter that primarily affects the mass balance, equations (3a) and (3b).
We conservatively choose porosity as 0.05 (Table 1). In the numerator of the final term
on the right hand side of (3b), the first term (1 − ni )ρi is the amount of ice melted, and
the second term ni ρw is the amount of water added from pore space as ice melts.
S4. Synthetic sections
Synthetic examples of glacier sections provide idealized water system geometry particularly with respect to surface to bed slope ratios. Longitudinal sections interpreted from
field radar data may not necessarily be representative of a glacier hydraulic section. Here,
we provide a formulation for longitudinal sections used in simulations.
The section has downstream coordinate t x = x0 at the terminus and an upstream value
x = xl . Subglacial water flows from xl to x0 . The vertical coordinate z, is a height above
an arbitrary datum. The glacier ice thickness Zi , is
Zi = zr − zb ,
(S12)
where zr is the ice surface elevation and zb is the elevation of the ice–bed interface.
As shown in Figure S2, the synthetic section has three regimes,

zl
for xu < x ≤ xl ,



C (x − x )2cb + z
for xw < x ≤ xu ,
x
u
l
zb =
2cb −1

2Cx cb (xw − xu )
(x − xw )



2cb
+Cx (xw − xu ) + zl
for x0 ≤ x ≤ xw .
(S13)
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
S-8
where the coefficient Cx is an arbitrary real number and cb is an arbitrary positive integer
that determines how rapidly a slope transition occurs for an overdeepening. The first
regime is a horizontal entrance of the water from xl to xu at height zb = zl . From xu
to xw , there is a parabolic transition where the base turns upwards from horizontal to a
constant slope. The slope is chosen to ensure that dzb /dx is smooth at xw because water
flow along this horizon is dependent on a smooth hydraulic potential. From xw to x0 ,
the slope of the subglacial horizon is constant, giving a wedge shape. The surface of a
synthetic section has a constant slope such that
zr =
(Zl + zl ) − (Z0 + z0 )
x + Z0 + z0 ,
xl − x0
(S14)
where Zl and Z0 are the ice thicknesses at xl and x0 , respectively. If zl , xu , and Z0 are
known, then one of xw , zw or Zl is necessary to solve equations (S13) and (S14) for the
longitudinal profile.
Where a specific ratio is required between the surface slope and the slope of the wedge
such that tan αb = −R tan αr , then the location of xw needs to be found with respect to
the third equation in (S13). For cb = 1 and a known z0 , xw takes the value
1
xw = x0 + (xu − x0 ) +
(zl − z0 )
Cx
2
21
.
(S15)
Substituting this equation into the second equation of (S13) gives zw . Rewriting equation
(S14) with the ratio R yields
1
zr =
R
z0 − zw
x0 − xw
x + z0 + Z0 .
(S16)
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
S-9
If the upstream ice thickness Z0 is known and cb = 1, then xw takes the value
1
x0 − xl
xw =x0 +
+
(xl − x0 (R + 1) + Rxu )2
R
R
1/2
2
R
+
(Z0 − Zl )
.
Cx
(S17)
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
S - 10
References
Bird, R. B., W. E. Stewart, and E. N. Lightfoot (1960), Transport phenomena, 1st ed.,
John Wiley and Sons, New York.
Clarke, G. K. C. (2003), Hydraulics of subglacial outburst floods: new insights from the
Spring-Hutter formulation, J. Glaciol., 49 (165), 299–313.
Creyts, T. T. (2007), A numerical model of glaciohydraulic supercooling: thermodynamics
and sediment entrainment, Ph.D. thesis, Univ. of B.C., Vancouver, B.C., Canada.
Creyts, T. T., and C. G. Schoof (2009), Drainage through subglacial water sheets, J.
Geophys. Res., 114, F04008, doi:10.1029/2008JF001215.
Fischer, U. H., and B. Hubbard (1999), Subglacial sediment textures: character and
evolution at Haut Glacier d’Arolla, Switzerland, Ann. Glaciol., 28, 241–246.
Gilpin, R. R. (1981), Ice formation in a pipe containing flows in the transition and turbulent regimes, J. Heat Transf., 117 (2), 363–368.
Hooke, R. L., and N. R. Iverson (1995), Grain-size distribution in deforming subglacial
tills: role of grain fracture, Geology, 23 (1), 57–60.
Hutter, K. (1982), A mathematical model of polythermal glaciers and ice sheets, Geophys.
Astro. Fluid, 21 (3–4), 201–224.
Khatwa, A., J. K. Hart, and A. J. Payne (1999), Grain textural analysis across a range of
glacial facies, Ann. Glaciol., 28, 111–117.
Lawson, D. E., J. C. Strasser, E. B. Evenson, R. B. Alley, G. J. Larson, and S. A. Arcone
(1998), Glaciohydraulic supercooling: a freeze-on mechanism to create stratified, debrisrich basal ice: I. field evidence, J. Glaciol., 44 (148), 547–562.
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
Mathews, W. H. (1973), Record of two jökulhlaups, in International Association of Scientific Hydrology Publication 95, 95, pp. 99–110, IASH.
McAdams, W. H. (1954), Heat transmission, 3rd ed., McGraw-Hill, New York.
Nunn, K. R., and D. M. Rowell (1967), Regelation experiments with wires, Philos. Mag.,
16 (144), 1281–1283.
Nye, J. F. (1953), The flow law of ice from measurements in glacier tunnels, laboratory
experiments and the Jungfraufirn, Proc. Roy. Soc. Lond. A Mat., 219, 477–489.
Nye, J. F. (1967), Theory of regelation, Philos. Mag., 16 (144), 1249–1266.
Nye, J. F. (1976), Water flow in glaciers: jökulhlaups, tunnels, and veins, J. Glaciol.,
17 (76), 181–207.
Paterson, W. S. B. (1994), The physics of glaciers, 3rd ed., Pergamon, Tarrytown, NY.
Roberts, M. J., F. S. Tweed, A. J. Russell, Ó. Knudsen, D. E. Lawson, G. J. Larson, E. B.
Evenson, and H. Björnsson (2002), Glaciohydraulic supercooling in Iceland, Geology,
30 (5), 439–442.
Schlichting, H. (1979), Boundary Layer Theory, 7th ed., McGraw-Hill, New York.
Spring, U., and K. Hutter (1981), Numerical-studies of jökulhlaups, Cold Reg. Sci. Technol., 4 (3), 227–244.
Tweed, F. S., M. J. Roberts, and A. J. Russell (2005), Hydrologic monitoring of supercooled meltwater from Icelandic glaciers, Quaternary Sci. Rev., 24, 2308–2318, doi:
10.1016/j.quascirev.2004.11.020.
Weertman, J. (1964), The theory of glacier sliding, J. Glaciol., 5 (39), 287–303.
Worster, M. G. (1997), Convection in mushy layers, Annu. Rev. Fluid Mech., 29, 91–122.
S - 11
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
Table S1.
S - 12
Parameters used to create the synthetic glacier sections in Figure 4
Threshold Above-
Flat-
Below-
threshold bedded threshold
Panel in Figure 4
(A)
(B)
(C)
e
e
e
g
Outlet location:
x0
m
0.00
0.00
0.00
0.00
Inlet location:
xl
m
250.00
250.00
250.00
250.00
Start of overdeepening:
xu
m
150.00
150.00
n/a
150.00
Start of wedge:
xw
m
107.59
122.84
n/a
107.59
Elevation at outlet:
z0
m
21.85
14.82
0.00
21.85
Elevation at inlet:
zl
m
0.00
0.00
0.00
0.00
Outlet ice thickness:
Z0
m
0.00
0.00
0.00
0.00
Inlet ice thickness:
Zl
m
46.86
46.86
46.86
35.99
[unitless]
-1.70
-0.85
0.00
-3.40
Surface slope:
tan αr [unitless]
0.10
0.13
0.19
0.06
Bed slope:
tan αb [unitless]
0.17
0.11
0.00
0.17
Transition coefficient:
cb
[unitless]
1.0
1.0
1.0
1.0
Coefficient:
Cx
m1−2cb
0.002
0.002
0.002
0.002
Bed to surface slope ratio: R
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
0.25
10
R1
Effective pressure (m i.e.)
20
30
40
50
S - 13
−5
x 10
4
3.5
2.5
R2
−1
0.15
3
Velocity (m s )
1e−006
5e−007
1e−007
Water depth (m)
0.2
2
0.1
1.5
R3
1
0.05
R4
R6
0
0.5
R5
0.1
0.2
0.3
Effective pressure (MPa)
0.4
0.2
0
Figure S1. Closure velocities for the sheet as a function of effective pressure and water
depth. The first six grain sizes in the closure scheme (R1 –R6 ) are shown for reference.
Maximum downward velocities occur in the upper right of the figure where contour spacing
converges. Maximum values are approximately 4.3 × 10−5 m s−1 for this range of effective
pressure and water depth. Most closure velocities plotted are less than 0.2 × 10−5 m s−1
as indicated by the compressed color scale. Contour intervals are 0.01 × 10−5 m s−1 .
CREYTS AND CLARKE: SUBGLACIAL SUPERCOOLING
Height (m)
xl
xu
xw
S - 14
x0
40
−z0
20 Ice
−z
0
l
Bed
−20
250
Figure S2.
200
150
100
Distance upstream from terminus (m)
50
0
Parameters used to create the synthetic sections using the same section as
Figure 4g. Subglacial water flows from left to right along the line at the interface of the
ice and substrate. Parameters are given in Table 2. The aspect ratio is 1 : 1.