Journal of Glaciology (2017), 63(240) 670–682
doi: 10.1017/jog.2017.33
© The Author(s) 2017. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.
org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Subglacial flood path development during a rapidly rising
jökulhlaup from the western Skaftá cauldron, Vatnajökull, Iceland
BERGUR EINARSSON,1 TÓMAS JÓHANNESSON,1 THORSTEINN THORSTEINSSON,1
ERIC GAIDOS,2 THOMAS ZWINGER3
1
Icelandic Meteorological Office, Reykjavík, Iceland
Department of Geology & Geophysics, University of Hawai’i, Honolulu, Hawai’i, USA
3
CSC – IT Center for Science Ltd., Espoo, Finland
Correspondence: Bergur Einarsson <bergur@vedur.is>
2
ABSTRACT. Discharge and water temperature measurements in the Skaftá river and measurements of
the lowering of the ice over the subglacial lake at the western Skaftá cauldron, Vatnajökull, Iceland,
were made during a rapidly rising glacial outburst flood (jökulhlaup) in September 2006. Outflow
from the lake, flood discharge at the glacier terminus and the transient subglacial volume of floodwater
during the jökulhlaup are derived from these data. The 40 km long initial subglacial path of the
jökulhlaup was mainly formed by lifting and deformation of the overlying ice, induced by water pressure
in excess of the ice overburden pressure. Melting of ice due to the heat of the floodwater from the subglacial lake and frictional heat generated by the dissipation of potential energy in the flow played a
smaller role. Therefore this event, like other rapidly rising jökulhlaups, cannot be explained by the
jökulhlaup theory of Nye (1976). Instead, our observations indicate that they can be explained by a
coupled subglacial-sheet–conduit mechanism where essentially all of the initial flood path is formed
as a sheet by the propagation of a subglacial pressure wave.
KEYWORDS: glacier hydrology, jökulhlaups (GLOFs), subglacial lakes
1. INTRODUCTION
Jökulhlaups in the river Skaftá from western Vatnajökull
occur at 1–2 year intervals with volumes of 0.05–0.4 km3
and maximum discharge of 50–3000 m3 s–1 (Björnsson,
1977; Zóphóníasson, 2002; unpublished data from the
Icelandic Meteorological Office). The floods originate from
two subglacial lakes below 50–150 m deep and 1–3 km
wide depressions (cauldrons) in the ∼450 m thick surrounding ice cap. Together, the depressions drain ∼50 km2 of the
ice cap (Pálsson and others, 2006) (Fig. 1). The jökulhlaups
travel ∼40 km subglacially before they emerge at the terminus of the outlet glacier Skaftárjökull.
Jökulhlaups in Skaftá reach maximum discharge in 1–3
days and typically recede in 1–2 weeks (Björnsson, 2002)
(Fig. 2). They are on the ‘rapidly rising’ side of the spectrum
of ‘rapidly rising’ to ‘slowly rising’ jökulhlaups (Einarsson and
others, 2016). The hydrograph of slowly rising jökulhlaups,
such as most floods from Grímsvötn subglacial lake in
Vatnajökull, gradually rises to maximum discharge in 1–3
weeks and recedes in <1 week, often in only 1–3 days
(Björnsson, 2002) (Fig. 2). Jökulhlaups from subglacial and
marginal lakes at other locations in Iceland may rise even
more rapidly than jökulhlaups in Skaftá and can reach
maximum discharge in less than half a day (Thórarinsson,
1974; Sigurðsson and Einarsson, 2005; Jónsson and
Þórarinsdóttir, 2011). The hydrographs of slowly rising
jökulhlaups are reasonably well explained by the theory of
conduit-melt–discharge feedback, developed by Nye
(1976). The rapid initial increase of the hydrograph during
jökulhlaups in Skaftá is difficult to explain with Nye’s
theory without invoking implausibly high temperature for
the lake water (Björnsson, 1992), and many aspects of the
dynamics of rapidly rising jökulhlaups are still unresolved.
As rapidly rising jökulhlaups have a fast discharge increase,
they may be extremely dangerous since warning times for
response are short.
The fundamental reasons that govern whether a jökulhlaup
develops as a rapidly rising or slowly rising flood are not fully
understood. The predominant discharge development mechanism appears to be different for these two types of floods.
Hydraulic uplift of the glacier, caused by water pressure
exceeding glacier overburden pressure in a propagating subglacial pressure wave, is likely to be an important component
in the flood path formation for rapidly rising jökulhlaups
(Björnsson, 2002; Jóhannesson, 2002; Flowers and others,
2004; Roberts, 2005; Einarsson and others, 2016). Flood
path formation by coupled subglacial sheet of water and
conduit flow has been used for modelling of rapidly rising
jökulhlaups (Flowers and others, 2004). The formation of
the initial sheet has been proposed to be caused by ice
dam flotation near the subglacial lake (Björnsson, 2002,
2010; Flowers and others, 2004; Sugiyama and others,
2008) and hydraulic jacking along the flood path (Flowers
and others, 2004). Positive feedback between discharge
and melting in the sheet leads to the creation of conduits
that carry an increasing proportion of the water as the flood
develops. The subglacial flood path of slowly rising
jökulhlaups is, on the other hand, thought to be mainly
formed by melting by frictional heat released in the flow
by dissipation of potential energy and the initial heat of
the source water (Nye, 1976; Spring and Hutter, 1981,
1982). Recent research on jökulhlaups from Grímsvötn
indicates that lifting of the glacier may also play a role in
flood path formation of some slowly rising jökulhlaups
(Björnsson, 2010; Magnússon and others, 2011; Einarsson
and others, 2016).
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Subglacial flood path development during a rapidly rising jökulhlaup
Fig. 1. (a) The Skaftá cauldrons and subglacial lakes in the western
Vatnajökull ice cap and the upper part of the watershed of the Skaftá
river. The inferred subglacial paths of jökulhlaups (dotted lines) and
locations of instruments described in the text are shown. (b) A
hillshade of a DEM of the cauldron area measured by lidar
(acronym for ‘light detection and ranging’) in 2010.
The discharge development of a jökulhlaup seems to
depend on both local conditions at each site and the initial
conditions for each particular flood. Both rapidly rising and
slowly rising jökulhlaups can originate from the same
source location. As an example, the 1861, 1892, 1938 and
the November 1996 jökulhlaups from Grímsvötn in
Vatnajökull were all of the rapidly rising type (Thórarinsson,
1974; Björnsson, 2002), whereas over 20 jökulhlaups from
Grímsvötn in the decades from 1940 to 2010 were slowly
rising (Thórarinsson, 1974; Björnsson, 2002; Sigurðsson
and Einarsson, 2005; unpublished data from the Icelandic
Meteorological Office). Different drainage mechanisms for
different outbursts from the same source location have likewise been identified for Gornersee in Switzerland (Huss
and others, 2007).
Subglacial water flow and variations in subglacial water
pressure have attracted increasing attention in recent years
as a likely cause of the large variations in ice flow velocities
that have been observed on the main outlets of the
Greenland ice sheet and some of the ice streams of
Antarctica (e.g. Rignot and Kanagaratnam, 2006; Fricker
and others, 2007; Stearns and others, 2008; Doyle and
others, 2015). Subglacial accumulation of water has been
observed or inferred at many locations on large and small
glaciers and found to be associated with substantial increases
in ice flow velocities (e.g. Iken and Bindschadler, 1986;
Fudge and others, 2009; Magnússon and others, 2011).
Jökulhlaups provide one of the best opportunities to study
the response of the subglacial hydraulic system to large
and sudden variations in water flow, and lessons learned
from studies of jökulhlaups may be useful for understanding
variations in basal sliding and ice flow in glaciers and ice
sheets in general (Bell, 2008).
To gain understanding of the energy balance, heat dissipation and flood path formation and development in rapidly
rising jökulhlaups, a campaign to monitor the Skaftá cauldrons and the Skaftá river was initiated in 2006. This paper
reports our results on the subglacial hydrology of the
September 2006 jökulhlaup from the western Skaftá cauldron. Outflow from the subglacial lake and transient
storage of water in the subglacial flood path are derived
and used, together with water temperature measurements
in the subglacial lake and near the terminus, to shed light
on the dynamics of the flood path development for this
type of flood. The emptying of a cylindrically symmetric subglacial lake is simulated with the full-Stokes ice-dynamic
model Elmer/Ice (Gagliardini and others, 2013) to deduce a
relationship between water volume in the lake and iceshelf elevation for calculations of outflow from the lake and
for analysing the size of the subglacial lake in relation to
the observed dimensions of the ice-surface depression.
2. DATA AND METHODS
Fig. 2. Comparison of the hydrographs of a large rapidly rising
jökulhlaup in 2002 (solid curve) and the small jökulhlaup in 2006
with a rapid initial rise, which is the subject of this paper (dashed
curve), both from the western Skaftá cauldron. The hydrograph of
a typical slowly rising jökulhlaup from Grímsvötn in 1986 (dotted
curve) is also shown. The hydrograph of the Grímsvötn jökulhlaup
is based on discrete discharge measurements which are shown as
dots. The rapid rise of the hydrograph of the 2006 jökulhlaup is
more clearly visible in Figure 6 which shows the same discharge
curve with the vertical scale expanded.
The ice shelf covering the western Skaftá cauldron was penetrated by a hot water drill in June 2006 (Thorsteinsson and
others, 2007). (The term ‘ice shelf’ is used in this paper to
describe the ice cover overlying the subglacial lakes as the
weight of the ice is to a large extent supported by flotation.
The ice shelves are, however, fundamentally different from
the large ice shelves at the margins of ice sheets and glaciers
in, for example, Antarctica, Greenland and Canada.) A temperature sensor was deployed at the bottom of the lake and
connected with a cable to a data logger on the surface
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Subglacial flood path development during a rapidly rising jökulhlaup
logged data at 15 seconds intervals for 5 minutes at 08:00
every morning. The average of the measurements from
each 5 minutes logging interval provided one GPS location
per day during the period 14 September to 25 October.
The data were processed with the Trimble Geomatics
Office software using base station data from the stations
GFUM (Grímsfjall, length of baseline ∼20 km) and SKRO
(Skrokkalda, length of baseline ∼37 km). Based on fluctuations in the measurements between adjacent days during
time periods of slow vertical movements before and after
the jökulhlaup, the relative accuracy of the measured
surface elevation is estimated to be <1 m.
Fig. 3. Schematic drawing of the western Skaftá cauldron showing
the ∼300 m thick floating ice shelf penetrated by a drillhole into
the ∼100 m deep subglacial lake and the scientific instruments
deployed in the lake and on the glacier surface in the campaign in
2006. Note that the vertical scale is exaggerated five-fold.
(Fig. 3). A differential GPS instrument was placed at the
centre of the cauldron, and a water temperature logger was
placed in the Skaftá river, 3 km downstream of the port
where the river emerges from beneath the ice (Fig. 1).
2.1. Proglacial discharge
The proglacial discharge in Skaftá was measured at the
hydrometric station at Sveinstindur, 25 km downstream of
the glacier margin. The discharge at the river outlet at the
glacier terminus was back-calculated with flood routing
using the 1-D HEC-RAS hydraulic model for unsteady flow in
open channels (Jónsson, 2007). In addition to the jökulhlaup
component, the discharge measured at Sveinstindur includes
the normal discharge from the glacier and from several
relatively small tributaries. These additional discharge components were estimated, based on a comparison with river
discharge records for similar weather conditions, and subtracted from the measured discharge to yield a discharge
estimate for the flood originating from the cauldron
(Einarsson, 2009). The uncertainty in this discharge estimate
arises from the uncertainty of the discharge measurement at
Sveinstindur and the uncertainties of the estimates for base
flow and for rain and melt-induced runoff from the
watershed during the period 26 September to 2 October. The
uncertainties of the base flow (±5 m3 s–1) and rain and melt
runoff (±10 m3 s–1) are relatively large for this small
jökulhlaup in 2006, compared with the uncertainty of discharge measurement at Sveinstindur (±2%; see Jónsdóttir
and others, 2001). This leads to ±12 m3 s–1 uncertainty in
the jökulhlaup discharge and ±20% uncertainty in the estimated flood volume and rate of outflow from the subglacial
lake because subtraction of base flow and melt-induced
runoff increases the relative uncertainty in the final results.
2.2. Glacier surface elevation over the subglacial lake
The elevation of the ice shelf varies in response to changes in
the volume of water in the subglacial lake. The elevation of the
surface of the ice shelf was measured near the centre of the
cauldron with a Trimble 4000SE GPS receiver equipped
with a Trimble 4000ST L1 Geodetic antenna. The receiver
2.3. The geometry of the cauldron and the subglacial
lake
The difference of two 5 m × 5 m DEMs of the cauldron
between filled and empty stages was used to deduce the
geometry of the part of the subglacial volume that is
emptied out in jökulhlaups. A DEM based on aerial synthetic
aperture radar (SAR) measurements from August 1998 was
made of western Vatnajökull by Magnússon (2003). A
jökulhlaup with a peak discharge of 149 m3 s–1 and total
volume of 0.116 km3 drained from the western cauldron in
September 1998, <1 month after the aerial survey, so the
western cauldron was nearly filled at the time of the
survey. A second DEM of the Skaftá cauldron area was
made with airborne lidar in July 2010 (Jóhannesson and
others, 2013). The western cauldron was at a low water
level at the time of the lidar survey, as a jökulhlaup with a
peak discharge of ∼410 m3 s–1 and total volume of 0.190
km3 drained the western cauldron in June 2010, ∼1 month
before the survey.
Our inferred geometry of the subglacial volume is not
based on direct measurements immediately before and
after the 2006 flood and may be affected by changes in ice
thickness in the 12 years between the SAR and lidar icesurface measurements in 1998 and 2010 when seven other
jökulhlaups are recorded (Zóphóníasson, 2002; unpublished
data from the Icelandic Meteorological Office). The shape of
the ice surface of the western cauldron shortly before a
jökulhlaup is flat and smooth and is expected to be very
similar for different floods. The general size and shape of
the cauldron at the end of a jökulhlaup has been monitored
from many reconnaissance flights during and after the floods
and is also found to be rather similar for the events that have
been observed (O. Sigurðsson, personal communication,
2016). The geometry of ice thickness changes over the subglacial lake in a jökulhlaup cycle are therefore expected to
be rather similar between cycles. The general lowering of
the glacier surface due to a negative mass balance since
1998 is accounted for by subtracting the elevation difference
in the adjacent area unaffected by the cauldron subsidence.
The subglacial lake is not fully emptied out in jökulhlaups so
that large parts of the ice shelf do not touch the underlying
bedrock at the end of the floods. Small-scale transient
changes due to melting at the underside of the ice shelf,
caused by possible spatial variations in the subglacial geothermal activity, are, therefore, not expected to affect the
inferred geometry of the volume that is emptied out in
jökulhlaups. The inferred geometry of the subglacial
volume is affected by a hump near the centre of the cauldron.
The hump is most likely an ice-dynamic thrust phenomenon
connected to ice flow into the cauldron and therefore does
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Subglacial flood path development during a rapidly rising jökulhlaup
not represent the shape of the subglacial lake. This effect is
small compared with other uncertainties in the estimation
of the geometry of the lake.
2.4. The hypsometry of the subglacial lake
The water volume in the subglacial lake is not a simple function of the lake geometry and water level as for lakes in
bedrock basins because the geometry of the lake may
change with the ice-shelf elevation. A hypsometric curve,
i.e. lake water volume as a function of ice-shelf elevation,
for the lowering of the ice shelf was calculated with the
full-Stokes ice-dynamic model Elmer/Ice (Gagliardini and
others, 2013) by simulating the emptying of a cylindrically
symmetric subglacial lake below an ice cover with the
approximate dimensions of the western Skaftá cauldron.
The question to be addressed by the simulation is whether
the subglacial water body maintains a similar shape as the
ice shelf is lowered and the lake is emptied or whether the
lake geometry changes substantially due to internal shear
within the ice shelf near the lake edge such that the grounding line at the lateral boundary moves inward as the shelf
lowers. These two possibilities correspond to different relationships between the rate of lowering of the ice shelf and
the rate of outflow from the subglacial lake, leading to different trajectories between the two known points on the
volume–ice-shelf-elevation curve that are determined by
the observed total outflow and surface lowering. Our modelling is simplistic and meant to capture the main physical processes that determine the shape of the hypsometric curve but
not the 2006 event in detail. The model setup is, thus, based
on a simplified geometry of the western cauldron and general
characteristics of jökulhlaups released from there and the
final results are scaled to the observed volume and ice
shelf lowering in the 2006 event. The physical basis of our
model formulation is in principle similar to the analytical
model of cauldron subsidence during jökulhlaups developed
by Evatt and Fowler (2007) but there are differences in the
geometrical and physical assumptions as will be further
described in Section 4.
The Skaftá cauldrons and ice flow in their vicinity are
close to being cylindrically symmetric in geometry (Figs 1
and 3). A cylindrically symmetric model configuration was
chosen because the detailed geometry of the ice-flow basin
away from the subglacial lake, which deviates from cylindrical symmetry, is not expected to influence the form of
the calculated hypsometric curve to a significant degree.
We take the overall shape and size of the subglacial lake
as given, based on the inferred geometry of the volume
emptied out in the 2010 jökulhlaup. This shape is dynamically determined over many jökulhlaup cycles and it is outside
the scope of this paper to derive and analyse this timedependent geometry by modelling. The modelled lake is
assumed to have a convex shape and maximum depth of
∼100 m at the centre at the start of a jökulhlaup and to be
covered with an ice shelf with surface geometry based on
the 1998 SAR DEM (see Fig. 4, which explains the notation
used to describe the model geometry). The glacier and the
lake are underlain by flat bedrock in the model set-up.
The dynamic and kinematic boundary conditions at the
ice surface and the ice/bed interface are formulated with a
stress-free upper surface, as is customary in ice flow
models of this kind (e.g. Gagliardini and others, 2013), and
a constant uniform positive surface mass balance bs. The
stress-free upper surface is assumed to have a smooth geometry, ignoring the dynamic effect of surface crevasses that are
observed to be formed in a concentric pattern during the subsidence of the ice shelf. The lidar measurements in 2010,
shortly after a jökulhlaup, showed crevasse depths up to
40 m, which should be considered a minimum as the lidar
is not likely to have reached to the bottom of the deepest crevasses. Crevasses formed over a timescale of several days
during a jökulhlaup may be estimated to be on the order of
100 m deep (Cuffey and Paterson, 2010, Eqn (10.6),
p. 449). As the surrounding glacier is ∼450 m thick, the crevasses may have some effect on the ice dynamics but
neglecting them is not likely to have a dominating effect on
our results considering other simplifications in our modelling
specification.
The kinematic boundary condition at the bottom of the
ice takes geothermal melting of ice, mg, into account
within the radius of the geothermal area rg. The dynamic
boundary condition at the ice/bed interface assumes
Weertman sliding, ub ¼ C τ bm 1 τ b , where ub is the bed-parallel velocity component, τb is the basal shear stress and C
and m are parameters.
Water pressure in the lake, pw, is assumed to be hydrostatic:
pw ¼ ρw g ðzw
zÞ;
ð1Þ
where zw is the piezometric height of the lake, and the
dynamic boundary condition at the ice/water interface is formulated in terms of the water pressure that is set equal to the
(negative of the) surface normal stress in the ice at the shearstress-free bottom of the ice shelf. The position of the grounding line, where the ice/water interface meets the ice/bed
interface, is part of the solution and evolves with time. Its
Fig. 4. The Elmer/Ice computational finite-element mesh for the cylindrically symmetric model of a glacier on a flat bed overlying a subglacial
lake with dimensions corresponding to the western Skaftá cauldron. The figure explains the notation used to define the geometry of the model:
the elevation of the ice surface, zs, and the bottom of the ice, zb, the time-dependent piezometric water level of the subglacial water lake, zw,
the radial distance to the grounding line, rl, the radius of the geothermal area, rg, the radius to the ice divide at the boundary of the cauldron ice
flow basin with the surrounding ice cap, rd and the ice-surface elevation at the ice divide, zd. A jökulhlaup is released when zw reaches a
critical level z1 and terminated when zw reaches z2.
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Subglacial flood path development during a rapidly rising jökulhlaup
position at each time step is determined by solving a contact
problem. At each node, the normal force exerted by the ice
on the bedrock is compared with the water pressure at that
location and the ice allowed to separate from the bed in
case the water pressure is higher (see Gagliardini and
others, 2013).
The lake is assumed to be hydraulically connected to the
atmosphere through water passageways such as crevasses or
moulins. The volume of water in the lake and surrounding
waterways, Vw, is the sum of (i) the lake volume, taken to
be the integral of the elevation of the bottom of the ice
shelf, zb, over the area of the lake, and (ii) the water
volume in the passageways between the lake and the atmosphere, which we assume can be expressed as water level in
the passageways, zw, multiplied with their effective area, S,
by analogy with the traditional way to express ground
water storage in terms of a storage coefficient (Fetter,
2001). Therefore, variations in Vw, may be expressed as
dVw
¼
dt
Z
rl
0
dzb
dzw
2 π r dr þ S
¼ qs þ qf þ qg
dt
dt
qj ;
ð2Þ
where the flux components qs, qf, qg and qj on the right-hand
side are inflow into the lake due to surface melting and rainfall on the surface, inflow of geothermal fluid, basal geothermal melting and rate of outflow from the lake during
jökulhlaups, respectively. This equation is solved for zw
with explicit forward time stepping in such a way that the
flux components, as well as simulated changes in the iceshelf geometry, force changes in water pressure through
adjustment of the piezometric height of the lake that
induce further changes in the ice-shelf geometry through
the dynamic boundary condition at the ice/water interface
in a feedback loop. Each time step of the modelling involves
an update of the mesh, calculation of the area where the
glacier is grounded and an update of the piezometric
height of the lake based on the previous geometry and the
flux components. Changes in normal pressure at the
ice/water interface due to variations in the piezometric
height lead to stresses and strains in the ice, and the resulting
ice motion drives changes in the volume of the subglacial
water body that subsequently lead to changes in the piezometric height and grounding line position in the next time
step. Because of the very different timescales for filling and
emptying of the lake between and during jökulhlaups, variable-length time stepping was implemented with time steps
switching between the order of hours between jökulhlaups
and the order of minutes during jökulhlaups.
Surface mass balance over the ice flow basin, bs, the rate
of geothermal melting of ice, mg and the rate of inflow of geothermal fluid from the geothermal system, qf, are estimated
from mass and energy conservation assuming long-term
balance of total precipitation over the ice flow basin with
area Ai ¼ π rd2 , bottom geothermal melting over an area Ag ¼
π rg2 and outflow in jökulhlaups when averaged over many
jökulhlaup cycles as in Jóhannesson and others (2007). The
flux components qs and qg in Eqn (2) are assumed to be constant in time, ignoring the seasonal mass-balance cycle,
which is not expected to be important for the emptying of
the lake. The rate of jökulhlaup outflow, qj, is assumed to
rise linearly with time from zero to a constant discharge qk
over a time period tr and fall linearly to zero over a time
period tf when the piezometric water level in the subglacial
lake rises or falls to the thresholds z1 and z2, respectively
(see Fig. 4). Table 1 gives the values used for model geometry
and physical constants in the simulations.
Several model parameters are not well constrained by
available observations or theory, in particular the parameters
describing the geothermal melting, Ag (and consequently
mg), the timescales for the rise and fall of the jökulhlaup discharge, tr and tf, the effective area of the assumed hydraulic
connection with the atmosphere, S, as well as the sliding
parameters C and m. The numerical values of the parameters
Ag, tr, tf and S as well as the grid resolution were varied by a
Table 1. Parameters defining an idealized, cylindrically symmetric model for the western Skaftá cauldron
Parameter
Value
Comment
2
Ai
zd
Ag
rl
bs
ds
mg
qs
qf
qg
qk
z1
z2
tr
tf
S
20 km
500 m
π km2
700 m
2.22 m w.e. a−1
0.4 m w.e. a−1
17.8 m w.e. a−1
0.25 m3 s–1
0.36 m3 s–1
1.8 m3 s–1
100 m3 s–1
400 m
320 m
12 hours
12 hours
0.1 km2
Area of ice flow basin (Pálsson and others, 2006) (rd = 2.52 km)
Ice divide elevation
Area of geothermal melting (rg = 1 km)
The initial radius to the grounding line
Surface mass balance
Sum of rain and (absolute value of the) surface ablation
Melting of ice at bottom of the ice shelf or glacier within radius rg
Discharge corresponding to rain and surface ablation
Discharge corresponding to inflow of geothermal fluid
Discharge corresponding to geothermal melting
Discharge during a jökulhlaup (maximum value after initial rise and final fall)
Piezometric water level in the lake at the start of a jökulhlaup
Piezometric water level in the lake at the termination of a jökulhlaup
Time period for the rise of discharge at the start of a jökulhlaup
Time period for the fall of discharge at the end of a jökulhlaup
‘Storage coefficient’ (effective area) of an assumed hydraulic connection of the subglacial lake
with the atmosphere
A
2.4 × 10−24 s−1 Pa−3
n
C
m
3
2.3 × 10−21 s1/3 m−1/3 Pa
3
Parameter in Glen’s flow law (e_ ij ¼ Aτ nij , where e_ and τ are strain rate and deviatoric stress,
respectively)
Exponent in Glen’s flow law
Parameter in Weertman’s sliding law
Exponent in Weertman’s sliding law
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Subglacial flood path development during a rapidly rising jökulhlaup
factor of 2–100 to test the sensitivity of the model, and no
significant changes in simulation results occurred. For computational reasons, the adopted values of S are large compared with the expected magnitude of water passageways
through bottom channels, englacial cavities and crevasses.
Varying S by two orders of magnitude made little difference
to model results, indicating that this parameter only affects
the resulting hypsometric curve to a small degree.
The resulting hypsometric curve shows an approximately
linear relation between ice-shelf elevation and lake water
volume (Fig. 5, dashed curve), indicating that the subglacial
water body maintains a similar shape as the ice shelf is
lowered. The simulated hypsometric curve was scaled to fit
the observed values for total outflow volume and surface
lowering during the 2006 jökulhlaup, allowing the calculation of outflow discharge as a function of time based on the
measured lowering of the ice shelf during the flood.
2.5. Floodwater temperature and energy available for
subglacial melting
Continuous measurements of water temperature in Skaftá
river were made at Fagrihvammur, 3 km downstream of the
glacier margin (Fig. 1), during the September/October 2006
jökulhlaup, using a Starmon mini temperature recorder
with an accuracy of ± 0.1°C. The measurements, as well as
an evaluation of potential warming over the 3 km distance
from the glacier margin to the measurement site, are
described by Einarsson (2009). Discrete measurements of
floodwater temperature at the glacier margin in jökulhlaups
in Skaftá in August and October 2008 and October 2015
were made with a high-precision RBR thermometer, with
an accuracy of ± 0.005°C. Measurements of water temperature in the subglacial lake of the western cauldron in June
2006 are described by Jóhannesson and others (2007).
The energy available for melting of ice along the subglacial flood path until a particular point in time after the
floods starts may be used to calculate an upper bound on
the volume of the flood path that can have been created by
Table 2. Physical parameters used in melt volume calculations
Parameter
Value
Comment
g
ρi
ρw
L
cw
9.82 m s–2
910 kg m–3
1000 kg m–3
3.34 × 105 J kg−1
4.22 × 103 J kg−1 K−1
Acceleration of gravity
Density of ice
Density of water
Latent heat of fusion
Heat capacity of water
melting at that time
Vmelt ¼
cw ρw Vl Tl þ gρw Vl ΔH
Lρi
cw ρw Vs Ts
;
ð3Þ
where g is the acceleration of gravity and ρw, cw, ρi and L are
the density and the heat capacity of water, and the density
and latent heat of fusion of ice, respectively (Table 2). Tl
and Ts are the temperature of the floodwater in the lake
and at the glacier snout. ΔH is the elevation difference
between the water level in the subglacial lake and the
glacier snout (815 m for the flood path from the western
Skaftá cauldron, Einarsson, 2009). Finally Vl and Vs are the
volumes of water released from the subglacial lake and at
the glacier snout up to the time in question. The first term
in the numerator on the right-hand side of the equation is
the available thermal energy due to the initial heat of the
water in the lake, the second term is the available potential
energy and the third term is the thermal energy left in the
water at the glacier snout. The kinetic energy in the water
flow at the glacier snout is neglected as it is small compared
with the other terms. This volume estimate is an upper bound
as the potential energy component corresponds to flow all
the way from the lake to the outlet and the thermal energy
corresponding to the deviation of the subglacial floodwater
temperature from the freezing point upstream from the
snout is included in the estimate of the available energy.
3. RESULTS
3.1. Proglacial discharge
Fig. 5. Hypsometric curves for the subglacial lake below the western
Skaftá cauldron showing lake volume as a function of the elevation
of the cauldron centre. The result of the Elmer/Ice modelling (dashed
curve) and a scaled curve that fits the observed flood volume vs. iceshelf lowering in the September/October 2006 jökulhlaup (solid
curve) are shown.
A rapidly rising jökulhlaup from the western Skaftá cauldron
emerged at the glacier margin on 27 September 2006, reaching a maximum discharge close to 100 m3 s–1 in ∼2 days. It
had the typical form of small jökulhlaups from the western
cauldron, with a relatively flat discharge maximum for ∼6
days, and receded in ∼4 days. The back-calculated flood discharge at the glacier terminus (Fig. 6, solid curve), after subtraction of the base flow, reached its maximum of ≈97 m3 s–1
in the afternoon of 2 October. As the estimation of the timevarying base flow is uncertain, the true discharge maximum
might have occurred in the interval 29 September to 2
October, but the maximum value of ≈100 m3 s–1 is fairly
accurate as the discharge peak was broad and flat.
As for other jökulhlaups from the Skaftá cauldrons, the
rapid rise in flood discharge during the first 2 days cannot
be described by classic jökulhlaup theory (Björnsson, 1977,
1992; Sigurðsson and Einarsson, 2005). Using model parameters based on the geometry of the flood path and
channel roughness nm (Manning’s roughness) in the range
0.01–0.1 s m–1/3, as reported for subglacial conditions by
Cuffey and Paterson (2010), the classic theory implies that
the discharge should increase by a factor of 2 in the range
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676
Fig. 6. Back-calculated discharge of jökulhlaup water (solid curve)
at the glacier terminus during the jökulhlaup in September/
October 2006 after subtracting base flow from the glacier and
tributary rivers upstream of the Sveinstindur hydrometric station.
The relative elevation of the cauldron centre, as measured with
GPS, is also shown (dashed curve) as well as the calculated
outflow from the subglacial lake (dash-dotted curve) derived from
the lowering of the ice surface elevation using the hypsometric
curve for the subglacial lake.
of 1–7 days for discharge between 20 and 80 m3 s–1. The discharge, however, increased from ∼20 to ∼40 m3 s–1 in
approximately half a day and from ∼40 to ∼80 m3 s–1 in
less than a day. The form of the discharge variation is also
far from the nearly exponential concave shape typical for
slowly rising jökulhlaups (Fig. 6).
3.2. The geometry of the cauldron, the subglacial lake
and the adjacent flood path
Inward movement of the grounding line in the modelled lowering of the ice shelf (Fig. 7) is minimal, ∼40 m and the water
depth is reduced in approximately the same proportion
everywhere during outflow. The lake shape for different
stages is therefore similar, leading to the approximately
linear hypsometric curve shown in Figure 5. This indicates
that the shelf has a substantial internal strength and shear
stress support at the sides. We find that the water pressure
at the bottom of the ice shelf in the centre of the cauldron
falls below the ice overburden (positive effective pressure
at the top of the subglacial cavity) by 0.2–0.4 MPa during
Fig. 7. Lake and surface geometry during a jökulhlaup modelled with
Elmer/Ice. The geometry is drawn at daily intervals with darker colour
as time progresses. The lake and surface geometries 1 month after the
end of the jökulhlaup are also drawn (dashed curves).
Subglacial flood path development during a rapidly rising jökulhlaup
the main outflow phase. This imbalance is compensated by
a bridging shear stress of 0.15–0.3 MPa at the margins of
the ice shelf. The modelling, furthermore, shows considerable thickening of the ice shelf during subsidence, which
also affects the time-dependent shape of the lake (Fig. 7).
This thickening from 305 to 320 m is caused by increased
ice flow towards the centre during subsidence. The ice
flow is increased both because of increased surface slope
of the cauldron and shear softening of the glacier ice. The
shear softening is caused by higher shear stresses in the ice
shelf as support from the water pressure in the lake is
reduced. The linear shape of the hypsometric curve is,
thus, a net result of complex ice dynamics where both
lateral support and vertical extension play a role. The reciprocal of the slope of a straight line, fitted to the hypsometry
(Fig. 5), is the ‘effective’ area of the lake, if it were being
emptied in a piston-like manner. This area, 1 km2, is
smaller than the 1.5 km2 initial area of the modelled lake
and relatively constant during lowering. The lowering of
the surface of the cauldron affects an area substantially
larger than the subglacial lake so that the surface lowering
at the lake edge is ∼20% of the maximum lowering at the
centre. The modelled surface lowering is more than 1 m at
a radius of 1050 m, 1.5 times the initial radius of the lake.
The subtraction of the SAR and lidar DEMs from 1998 and
2010 indicates a subglacial water body with a smooth,
approximately cylindrically symmetrical shape (Fig. 8). The
area that subsided is ∼2.5 km in diameter, with maximum
depth of ∼90 m near the centre of the cauldron. The elongated
shapes that appear on the flanks of the central water cupola are
due to surface crevasses represented in the 2010 lidar DEM
that were formed or enhanced during the July 2010
jökulhlaup. These shapes are thus not surface topography
expressions of variations in subglacial lake depth. The modelled lowering of the ice shelf indicates that the margins of
the ice surface depression with comparatively little difference
in elevation between 2010 and 1998 (Fig. 8) are formed by
ice dynamics during the lowering of the ice shelf and by ice
flow into the cauldron over the month that elapsed from the
jökulhlaup in June 2010 to the lidar survey in July (Fig. 7).
The true width of the subglacial lake before a jökulhlaup
may thus be expected to be smaller than shown in Figure 8.
Melting over the subglacial flood path has created an
elongated depression in the outlet area of the cauldron,
visible as the extension to the southwest from the area of
the western cauldron in Figure 1b. The depression is
deeper shortly after a jökulhlaup (as in 2010) than shortly
before a jökulhlaup (as in 1998) and has been observed to
be bounded by thin belts of narrow crevasses on each side
after jökulhlaups (O. Sigurðsson, personal communication,
2008), indicating subsidence of the glacier surface along
the depression. The ice surface elevation difference indicates
that an ice volume ∼10 m high, a few hundred metres wide
and ∼3 km long, which takes the shape of a ridge in Figure 8,
is melted as the depression is formed during the initial stage
of a jökulhlaup. Due to smoothing caused by ice dynamics
the true geometry of the subglacial volume could be narrower and higher.
3.3. The water level and the depth of the subglacial
lake
The water level in the borehole was at 1488 m a.s.l. at the
time of the instrument set-up in June 2006, which is ∼5 m
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677
Subglacial flood path development during a rapidly rising jökulhlaup
lake after the 2006 jökulhlaup than after the 2010
jökulhlaup.
3.4. Outflow from the subglacial lake
The outflow from the subglacial lake deduced from the GPS
measurements of the lowering of the ice shelf is shown in
Figure 6. The outflow reached a maximum of ∼100 m3 s–1
in ∼4 days, at about the same time that appreciable flood discharge started at the terminus. The outflow then receded in
∼4 days.
The travel time of the subglacial flood wave from the cauldron to the terminus is in the range 29–62 hours. This wide
range arises both from an uncertainty in the timing of the
start of outflow from the lake and from the terminus. There
is a ±12 hour uncertainty of the timing of the start of
outflow from the lake as the GPS recorder in the cauldron
only recorded elevations once per day. The onset of
outflow at the terminus is also not well determined, as the
diurnal discharge variation at the outlet masked the start of
discharge increase there. The mean speed of the front of
the subglacial flood wave along the 40 km path from the
cauldron is estimated at 0.2–0.4 m s–1.
3.5. Transient volume of subglacial floodwater
Fig. 8. (a) A hillshade of the difference between the adjusted 1998
SAR DEM and the 2010 lidar DEM. (b) A contour map of the
inferred depth of the subglacial water body emptied in the 2010
flood, contour interval of 5 m. The data are smoothed with a 100
m × 100 m window.
higher than the level corresponding to flotation of the central
part of the shelf. This deviation from floating balance means
that an excess pressure of ∼0.05 MPa was acting on the underside of the ice shelf near the centre (Einarsson, 2009). The ice
shelf rose slowly by 12 m from early June to the triggering of
the jökulhlaup. It then fell by 67 m in 11 days and by ∼55
m in the 6 days of most rapid decline (Fig. 6, dashed curve).
Measurements of water depth and ice shelf lowering at the
location of the June 2006 borehole show that the lake was
∼125 m deep just before the jökulhlaup while the lowering
of the ice shelf during the jökulhlaup was only 67 m,
leaving ∼60 m of water in the lake at the location of the borehole. Thus, the lake was not completely drained by the flood.
The elevation of the centre of the western cauldron according
to the 2010 lidar DEM (adjusted to account for the general
lowering of the glacier in this area between 2006 and
2010) was ∼16 m lower than after the jökulhlaup in 2006.
This indicates that more water remained in the subglacial
Figure 9 shows the estimated volume of floodwater in the subglacial lake and the cumulative volume of the jökulhlaup discharge at the glacier margin during the September/October
2006 jökulhlaup, as well as the transient volume of water
stored in the subglacial pathway. The volume of water in
the subglacial pathway is estimated by subtracting the cumulative volume of flood discharge at the terminus from the
cumulative volume of floodwater released from the subglacial lake. The subglacial water volume is considerable
relative to the total flood volume of 5.3 × 10−2 km3. It
reached a maximum of ∼3.3 × 10−2 km3 on 30 September,
2 days after the outflow from the lake reached its maximum,
and was already ∼1:4 × 10 2 km3 before any outflow
started at the glacier margin. For flow speed equal to or
larger than the mean propagation speed of the subglacial
flood front and flood path length of 40 km, one finds that a
flood path with a volume of less than half the inferred subglacial water volume is needed to carry the flood discharge at
Fig. 9. Volume of floodwater in the subglacial lake (solid curve),
cumulative volume of the flood discharge at the glacier terminus
(dash-dotted curve), the estimated volume of water stored in the
subglacial flow path (dashed curve) and calculated amount of melt
due to friction in the flow and initial heat of the floodwater (dotted
curve). Dates are day/month of 2006.
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678
the time of maximum subglacial water storage. This will be
elaborated below.
3.6. Floodwater temperature and thermodynamics
The measurements of floodwater temperature in Skaftá
during the September/October 2006 jökulhlaup indicate
that the outburst water was within 1°C from the melting
point as it emerged from beneath the ice cap. The uncertainty
of up to 1°C is caused by frictional heating and heat
exchange with the atmosphere along the ∼3 km long river
path from the terminus to the thermometer location.
Measurements of floodwater temperature at the glacier
margin in jökulhlaups in Skaftá in August and October
2008 and October 2015 showed floodwater temperatures
at the melting point within ± 0.01°C. The measurements
were carried out at a discharge close to the maxima of 240
and 1290 m3 s–1 for the August and October jökulhlaups in
2008, respectively, and close to the maximum of ∼3000
m3 s–1 for the 2015 jökulhlaup. The predicted outflow temperatures for these events, according to the heat transfer
equation in the unsimplified jökulhlaup theory of Nye
(1976) as formulated in Jóhannesson (2002), are 0.05–0.8°
C and 1.0–2.5°C for the August and October jökulhlaups in
2008, respectively, and 1.8–3.4°C for the 2015 jökulhlaup.
These calculations are based on the geometry of the flood
path and channel roughness nm (Manning’s roughness) in
the range 0.01–0.1 s m–1/3, and also account for the initial
heat stored in the lake water.
The vertical temperature profile through the 100 m deep
subglacial lake in June 2006 (Jóhannesson and others, 2007)
showed that the bulk of the water was close to 4.7°C, with a
distinct ∼10 m deep layer at 3.5°C at the bottom (both
temperatures are higher than the temperature corresponding
to maximum density of water at the pressures encountered in
the lake). Temperature measurements at different depths in
the subglacial lake below the eastern Skaftá cauldron over
several months in 2007 showed little variation with time in
the lake water temperature, which was close to 4.0°C at
most depths (unpublished data from the Icelandic
Meteorological Office). The lake water temperature in the
western cauldron during the September/October 2006
jökulhlaup may thus be assumed to have been close to
4.5°C.
Most of the initial heat in the lake water is presumably lost
during the first few kilometres of the flood path. This is indicated by the few hundred metres wide and up to ∼10 m deep
depression that stretches ∼3 km southwestward from the
western cauldron, along the assumed location of the flood
path, that is more prominent shortly after jökulhlaups than
shortly before the floods, as described above (Figs 1b and
8). The volume change of the depression between the 1998
DEM and the 2010 DEM is ∼ 1 10 2 km3 . Bearing in
mind that the heat transfer in subglacial water flow may be
assumed to be very rapid (Jóhannesson, 2002; Jarosch and
Zwinger, 2015) and comparing this to the volume of ice
melted by the heat content in 0.190 km3 of 4.5°C warm
water discharged out of the cauldron in the 2010
jökulhlaup, which is 1.2 × 10−2 km3, it seems likely that
most of the initial heat in the 2010 jökulhlaup floodwater
was lost in the flow along this part of the path. These
numbers should of course be considered to represent order
of magnitudes estimates as the volume of the depression is
determined from measurements of the glacier separated by
Subglacial flood path development during a rapidly rising jökulhlaup
many jökulhlaup cycles. We expected similar heat loss to
have taken place in the 2006 jökulhlaup, and therefore the
floodwater may be assumed to have been near the melting
point for most of the length of the flood path.
4. DISCUSSION AND CONCLUSIONS
Our measurements of the lowering of the ice shelf in the
western Skaftá cauldron during a jökulhlaup, along with discharge measurements of the proglacial flood discharge in
Skaftá river, allow estimation of the transient subglacial
storage of floodwater. The inferred subglacial storage and
the measured discharge, along with information on floodwater temperatures in the subglacial reservoir and close to
the glacier margin, may be used to draw conclusions about
the formation of the subglacial flood path.
At the time of maximum subglacial storage, 4.3 × 10−2
km3 of water had been released from the subglacial lake
(Fig. 9). The initial heat contained in this volume of water
with a temperature of ∼4.5°C and heat formed by potential
energy dissipation in the flow down the subglacial flood
path (corresponding to a temperature rise of ∼2.0°C) are sufficient to melt at most ∼3:4 × 10 3 km3 of ice (Eqn (3)), if all
available thermal energy is used in melting, and considering
the potential energy corresponding to the entire elevation difference from the lake to the snout. This is <∼10% of the
volume of water in the subglacial pathway and subglacial
storage along the path, which thus must have been formed
mainly by mechanical processes. The ratio of the volume
of ice that could have been melted at any particular point
in time during the entire rising phase of the jökulhlaup to
the subglacial water volume at the same time is also on the
order of 10%. Melting therefore cannot have been the
main process responsible for the formation of the initial
subglacial flood path. Other processes, such as lifting of the
ice, hydraulic fracturing as well as viscous and elastic
deformation induced by water pressure higher than overburden pressure, must therefore be the main processes responsible for the propagation of the initial jökulhlaup flood
wave and the formation of the subglacial path. This is in
agreement with earlier findings of Björnsson (1992, 2002),
Jóhannesson (2002), Flowers and others (2004) and
Einarsson and others (2016) for rapidly rising jökulhlaups,
and is in contrast to classic jökulhlaup theory (Nye, 1976)
where the flood path is formed by a feedback mechanism
between water flow and conduit enlargement. Similar transient water storage created by high subglacial water pressure
and glacier lifting has been observed for other rapidly rising
jökulhlaups at Gornersee, Switzerland (Huss and others,
2007; Werder and others, 2009), and at Hidden Creek
Lake, Alaska (Bartholomaus and others, 2008, 2011). The
substantial volume of floodwater that spreads subglacially
during the initial phase of the jökulhlaup, ∼ 1:4 × 10 2 km3 ,
before any outflow has started at the terminus, also shows
that outflow from the subglacial lake does not require direct
throughflow of water extending from the lake to the terminus.
It may be assumed that essentially all the subglacial
volume of the jökulhlaup path is formed during each individual flood. Preexisting channels, incised into the bedrock or
conduits in the ice formed in prior jökulhlaups must be icefilled since air- or water-filled subglacial cavities in the
form of an incipient flood path with a substantial extension
are unlikely to be sustained under the glacier between
jökulhlaups. According to Nye’s (1953) analysis of the
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679
Subglacial flood path development during a rapidly rising jökulhlaup
contraction of a subglacial cylindrical cavity with the creep
parameter for temperate ice from Cuffey and Paterson
(2010), the radius of an air-filled cavity under 400 m of ice
will decrease to <10% of its initial value in less than a
week. Low and broad air-filled cavities collapse even
faster. Water-filled conduits with flowing water will contract
and adjust to available water sources between jökulhlaups.
This adjustment is not instantaneous but will take place
over days to weeks, depending on the position within the
glacier and the amplitude and rate of change of water input
(Cuffey and Paterson, 2010). Finally, a substantial water
body of stagnant water at the glacier bed is only stable in
local depressions in the hydraulic potential (Björnsson,
1988), such as at the cauldrons between jökulhlaups, and
such conditions are not present along the flood path of the
Skaftá jökulhlaups (Magnússon, 2003).
Due to the strength of the overlying glacier, the part of the
flood path that is initially formed by mechanical processes
can be expected to be broad and flat and therefore the
flow there will be in the form of a sheet. This part of the
path is formed by lifting of the ice through elastic and
viscous deformation and hydraulic fracturing induced by
water pressure higher than the ice overburden pressure. It
is, therefore, unlikely to have a horizontal width smaller
than a few ice thicknesses.
Measurements of floodwater temperatures in Skaftá for
discharges on the order of 240–3000 m3 s–1 show that the
jökulhlaup water is at or very close to the melting point
when it emerges at the glacier terminus. Essentially all
initial heat of the lake water and heat formed in the flood
path by dissipation of potential energy has thus been lost
from the floodwater. As pointed out by Björnsson (1992),
Jóhannesson (2002) and Clarke (2003), this indicates much
more effective heat transport from the floodwater to the surrounding ice walls of the subglacial flood path than is consistent with the heat transfer mechanism assumed in the classic
unsimplified jökulhlaup theory of Nye (1976), and later
developments of this theory by Spring and Hutter (1981,
1982) and Fowler (1999). A better physical understanding
of subglacial water flow is clearly needed to explain this
very efficient heat transfer (Jóhannesson, 2002; Werder and
Funk, 2009; Jarosch and Zwinger, 2015).
Magnússon and others (2007) observed an increase in
surface speed in an 8 km wide area on Skaftárjökull for a
small jökulhlaup from the eastern cauldron in 1995, except
for the uppermost 6 km of the flood path near the cauldron
where no speed-up was observed. They suggested that the
1995 jökulhlaup was drained in a conduit along this uppermost part of the flood path and as a sheet farther down in
agreement with the interpretation presented here for the
2006 jökulhlaup. Narrow flood paths with width <500 m
extending from both cauldrons are indeed indicated by the
elongated surface depressions near the cauldrons described
earlier. Although melting driven by the initial heat of the
floodwater can only create ∼10% of the subglacial water
storage, this melting can play an important role over a
limited part of the flood path length. A plausible mechanism
for the initiation of the flood therefore appears to be (i)
melting of a conduit along the first ∼3 km of the path,
driven by rapid release of the initial heat of the lake water,
and (ii) formation of a sheet-like flood path farther downglacier by lifting and ice deformation due to a propagating
subglacial pressure wave (Jóhannesson, 2002; Einarsson
and others, 2016). Such a sheet-like initial subglacial flood
path can rapidly develop conduits to become a highly efficient waterway (Jóhannesson, 2002; Flowers and others,
2004; Einarsson and others, 2016) allowing faster rise of discharge than the conduit-melt–discharge feedback (Nye,
1976) (Fig. 6, solid curve). The formation of a pressure
wave requires an efficient pressure connection between
two locations of the flood path where subglacial water pressure near ice overburden at the upper location leads to pressure higher than ice overburden at the (lower) location farther
down the path. In the case of the Skaftá cauldrons, the
conduit formed by the release of the initial heat near the cauldron may provide this connection with a small potential gradient needed to drive the water flow. This situation may be an
essential component in the formation of a subglacial pressure
wave in jökulhlaups at this location. The propagating pressure wave would then not be formed at the source lake but
some kilometres downstream of it, explaining the shift from
conduit flow to sheet flow inferred for the 1995 jökulhlaup
by Magnússon and others (2007).
We find that only part of the subglacial water storage is
needed to carry the discharge of the flood, even for low estimates of the flow speed, particularly during the initial rise of
the flood discharge. This may be interpreted as initial storage
in subglacial reservoirs that do not contribute much to the
transportation of floodwater. Such transient subglacial
storage of floodwater has been documented for a rapidly
rising jökulhlaup in 2004 from Gornersee in Switzerland
(Huss and others, 2007; Werder and others, 2009). Up to
half of the volume of this flood is reported to have been
stored subglacially and they suggest that this storage is
related to lateral spreading of the floodwater and uplift of
the glacier. This is not a unique feature of rapidly rising
jökulhlaups, as lateral spreading and glacier uplift has also
been observed for slowly rising jökulhlaups from
Grímsvötn and Gornersee, where a propagating subglacial
pressure wave was not observed (Huss and others, 2007;
Magnússon and others, 2007; Werder and others, 2009;
Magnússon and others, 2011; Einarsson and others, 2016).
A rise in the ratio of subglacial volume needed to carry the
discharge of the flood to the total volume of subglacial water
may be inferred from our data. This indicates a development
towards more efficient subglacial water flow and/or release
of water from subglacial storage to the main flood path
over the course of the 2006 jökulhlaup. This may be interpreted as a development towards effective conduit flow
from ineffective initial sheet flow formed in the wake of a
subglacial pressure wave.
Our estimate for the travel speed of the subglacial flood
front in 2006 (0.2–0.4 m s–1) is faster than the propagation
speed of a small jökulhlaup from the eastern Skaftá cauldron
in October 1995 (<0.06 m s–1) estimated by Magnússon and
others (2007, there is a typo in Magnússon’s paper where this
speed is given as 0.6 m s–1). Magnússon and others’ (2007)
estimation is based on detection of ice surface speed-up
due to the jökulhlaup by SAR satellite imagery before the
flood front reached the ice margin. Similar propagation
speeds as our estimate for the 2006 jökulhlaup have also
been estimated for a small jökulhlaup from the western cauldron in August 2008 and a large jökulhlaup from the eastern
cauldron in October 2008, 0.1–0.3 and 0.4–0.6 m s–1,
respectively (Einarsson and others, 2016). These travel
speed estimates are considerably slower than the speed of
the flood front of the rapidly rising jökulhlaup from
Grímsvötn in November 1996 (1.3 m s–1) (Björnsson, 2002)
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680
but similar to the speed of a flood front of a rapidly rising
jökulhlaup from Hidden Creek Lake, Alaska, in July 2006
(0.4 m s–1) (Bartholomaus and others, 2011). The speed of
the subglacial flood front for rapidly rising jökulhlaups therefore seems to vary between different locations due to differences in flood path geometry, and large subglacial floods
appear to propagate faster than small floods at the same
location.
There are indications that the speed of the subglacial
water flow increases at later stages for jökulhlaups in Skaftá
after the proglacial discharge has peaked. This development
may also be interpreted as filling of lateral storage of floodwater during the initial phase of the flood that may later
release subglacially stored water into the main flood path.
Earthquake tremors, indicating boiling within the geothermal
system below the subglacial lake, due to the pressure release
accompanying the emptying of the lake, were observed
during a jökulhlaup from the eastern cauldron in 2002. A
time series for the concentration of suspended material in
the jökulhlaup waters at the gauging station at Sveinstindur
displayed a peak believed to result from the same boiling
event. The timing of the earthquake tremors and the peak
in the suspended sediments were used to estimate a speed
for the subglacial water flow, ∼0.8 ± 0.1 m s–1 (O.
Sigurðsson, personal communication, 2008). This is 2–4
times the speed estimated here for the initial phase of the
2006 jökulhlaup and 1.3–2 times the speed estimated for
the 2008 October jökulhlaup.
The derived shape of the lake at a full stage (Fig. 8) resembles the theoretically predicted shape of subglacial lakes
below a surface cauldron with a slope of the ice/water interface at the top of the lake that is opposite to the slope of the
ice surface and an order of magnitude larger (Björnsson,
1975, 2002). The derived shape does not reflect the full
depth and extent of the water body, as some water was left
in the lake after the jökulhlaups in 2006 and 2010. For
similar ice-shelf thickness in 2010 to that in 2006, ∼40 m
of water would have been left at the cauldron centre. The
resolution of bedrock data is insufficient to determine
whether this water is located in a bedrock depression or
retained by the closing of the lake seal. The decline of the discharge after the flood peaks varies between jökulhlaups, and
the final elevation of the ice shelf after jökulhlaups varies by
tens of metres (unpublished data from the Icelandic
Meteorological Office and the Institute of Earth Sciences at
the University of Iceland). This indicates that the closing of
the seal depends delicately on some (subglacial) conditions
that vary from event to event. Termination of jökulhlaups
before the water level drops below the bedrock threshold is
also observed at Grímsvötn (Björnsson, 1974) and has been
assumed to be caused by conduit closing because of ice
deformation exceeding melting of conduit walls (Björnsson,
1974; Nye, 1976) or settling of a flat-based ice dam onto a
smooth bedrock (Björnsson, 1974). Some jökulhlaups, for
example at Gornersee, continue, however, until the source
lake is empty (Werder and others, 2009).
The hypsometric curve for a subglacial lake during lowering in a jökulhlaup depends on the strength of the overlying
ice shelf, which determines to what extent the shelf is carried
by floating or shear forces. Our simulations show that the
shelf has considerable shear strength, due to shear stresses
induced by the subsidence that are caused by vertical
shear straining over the entire shelf except at the centre,
and the local force balance therefore deviates from floating
Subglacial flood path development during a rapidly rising jökulhlaup
equilibrium. The resulting hypsometric curve is nearly
linear (Fig. 5) and not concave as it would be for a lake
under an ice shelf mainly carried by floating. Our results
show that ice-surface depressions caused by the emptying
of subglacial lakes are considerably larger than the footprint
of the corresponding water body at the glacier bed; in the
case of the western Skaftá cauldron, modelled ice-surface
subsidence >1 m is found over an area more than twice
that of the subglacial lake. This result may be relevant for
time-varying subglacial water bodies at other locations.
Our magnitude for the pressure difference and lateral
shear stress, 0.2–0.4 and 0.15–0.3 MPa, respectively, may
be crudely compared with the analytical results of the cauldron subsidence model of Evatt and Fowler (2007). Their
model predicts ∼0.15 and ∼0.2 MPa for the pressure difference and shear stress, respectively, when the model parameters have been adapted to the spatial scale of the
western Skaftá cauldron, assuming ice flow parameters for
temperate ice and a similar outflow magnitude as for the
2006 jökulhlaup. The model of Evatt and Fowler is based
on 2-D geometry, rather than cylindrical geometry, and the
water outflow is calculated by Nye’s (1976) theory for
slowly rising jökulhlaups, whereas we employ an outflow
variation corresponding to a rapidly rising flood. Considering
these differences in assumptions and the formulation of the
models, there is overall agreement between the models on
the dynamics of the cauldron subsidence.
Many of the most well-known rapidly rising jökulhlaups,
such as the jökulhlaups from the Katla volcano (Tómasson,
1996) and the 1996 Grímsvötn jökulhlaup, are dramatic
and up to more than three orders of magnitude larger than
the jökulhlaup described in this paper. The relatively small
size of the rapidly rising jökulhlaups from the Skaftá cauldrons therefore indicates that the total water volume (i.e.
flood size) is not the key factor determining the rapidity of
flood release. Considering that some large and small
jökulhlaups in Iceland in modern times were rapidly rising
(Björnsson, 2002; Jóhannesson, 2002; Sigurðsson and
Einarsson, 2005; Einarsson and others, 2016), the type of
large jökulhlaups at the end of the last glaciation in Iceland
and elsewhere must be considered an open question. A fundamental understanding of the conditions that determine the
development of subglacial floods, in particular whether they
develop rapidly by lifting of the overlying ice or over a longer
time through a feedback between discharge and ice melting
in a conduit, is therefore required for an improved understanding of prehistoric jökulhlaups. Theoretical studies of
the palaeohydraulics of jökulhlaups from Lake Missoula,
Montana, USA (Clarke and others, 1984), and Lake
Agassiz, North America (Clarke and others, 2004), are
based on the assumption that these floods were of the
slowly rising type and controlled by the melt-discharge-feedback in a conduit. The possibility of a sheet-like flood from
Lake Agassiz is deemed highly unlikely by Clarke and
others (2005) as the inflow into the lake was not rapid and
as it is not likely that the critical conditions for ice-dam
lifting would be reached over large areas of a dam resting
on an irregular bed and variously attacked by iceberg
calving. Our example of a rapidly rising flood with an
initial sheet-like flood path from the slowly filling subglacial
lake below the western Skaftá cauldron shows that rapid
inflow is not an essential condition for the release of such
floods. Our interpretation also indicates that a flood path
that starts as a conduit near the source lake can develop as
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681
Subglacial flood path development during a rapidly rising jökulhlaup
a sheet farther downstream if a propagating subglacial pressure wave is formed.
ACKNOWLEDGEMENTS
The Icelandic Research Fund, the Landsvirkjun (National
Power Company of Iceland) Research Fund, the Icelandic
Road Administration, the Kvískerja fund and the Iceland
Glaciological Society provided financial and field support,
which made this study possible. A map of the subglacial topography along the route of jökulhlaups from the Skaftá cauldrons and the inferred paths of the jökulhlaups shown in
Figure 1 was made available by Helgi Björnsson and
Finnur Pálsson at the Institute of Earth Sciences of the
University of Iceland. Halldór Geirsson at the Icelandic
Meteorological Office provided GPS base data from
Grímsvötn and Skrokkalda. We thank Olivier Gagliardini
for assistance with the Elmer/Ice model calculations. The
Icelandic Coast Guard provided helicopter transportation to
the western cauldron in November 2006. This material is
based upon work supported in part by the National
Aeronautics and Space Administration through the NASA
Astrobiology Institute under Cooperative Agreement
NNA04CC08A issued through the Office of Space Science.
Thomas Zwinger was supported by the Nordic Centre of
Excellence, ‘eScience Tools for Investigating Climate
Change at High Northern Latitudes’ (eSTICC) funded by
NordForsk. We thank Vilhjálmur S. Kjartansson, Gunnar
Sigurðsson, Hlynur Skagfjörð Pálsson and Mary Miller for
assistance during field operations. We thank Gwenn
Flowers and two anonymous reviewers of an earlier version
of the manuscript and Geoffrey W. Evatt and an anonymous
reviewer of this manuscript for detailed and constructive
comments that helped us improve the paper and Ken
Moxham for help with the English language. This publication
is contribution No. 4 of the Nordic Centre of Excellence
SVALI, ‘Stability and Variations of Arctic Land Ice’, funded
by the Nordic Top-level Research Initiative (TRI).
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