Available online at www.sciencedirect.com
Global and Planetary Change 59 (2007) 217 – 224
www.elsevier.com/locate/gloplacha
Tide-induced perturbations of glacier velocities
Robert H. Thomas ⁎
EG&G Services, NASA/Wallops Flight Facility, Bldg N-159, Wallops Island, VA 23337, USA
Centro de Estudios Cientificos, Avenida Arturo Prat 514, Valdivia, Chile
Available online 9 January 2007
Abstract
Recent observations showing substantial diurnal changes in velocities of glaciers flowing into the ocean, measured at locations
far inland of glacier grounding lines, add fuel to the ongoing debate concerning the ability of glaciers to transmit longitudinal-stress
perturbations over large distances. Resolution of this debate has major implications for the prediction of glacier mass balance,
because it determines how rapidly a glacier can respond dynamically to changes such as weakening or removal of an ice shelf.
Current IPCC assessment of sea-level rise takes little account of such changes, on the assumption that dynamic responses would be
too slow to have any appreciable effect on ice discharge fluxes. However, this assumption must be questioned in view of
observations showing massive increases in glacier velocities following removal of parts of the Larsen Ice Shelf, Antarctica, and of
others showing diurnal velocity changes apparently linked to the tides.
Here, I use a simple force-perturbation model to calculate the response of glacier strain rates to tidal rise and fall, assuming
associated longitudinal-force perturbations are transmitted swiftly far inland of the glacier grounding line. Results show reasonable
agreement with observations from an Alaskan glacier, where the velocity changes extended only a short distance up-glacier.
However, for larger Antarctic glaciers, big velocity changes extending far upstream cannot be explained by this mechanism, unless
ice-shelf “back forces” change substantially with the tides.
Additional insight will require continuous measurement of velocity and strain-rate profiles along flow lines of glaciers and ice
shelves. An example is suggested, involving continuous GPS measurements at a series of locations along the centre line of Glaciar
San Rafael, Chile, extending from near the calving front to perhaps 20 km inland. Tidal range here is about ± 0.8 m, which should
be sufficient to cause a variation in ice-front velocity of ± 2 cm h− 1 about its average value of 75 cm h− 1, assuming local seawater
depth of 150 m and glacier thickness of 200–400 m.
© 2006 Elsevier B.V. All rights reserved.
Keywords: glacier; ice shelf; force model; tide; ice velocities
1. Introduction
There is a growing body of evidence for tidal influence on glacier flow Walters and Dunlap, 1987;
O'Neel et al., 2001; Doake et al., 2002; Anandrakrishnan et al., 2003; Bindschadler et al., 2003a,b. Velocity
variations can be quite large (± 50%) with highest values
⁎ Tel.: +1 757 824 1405; fax: +1 757 824 1036.
E-mail address: [email protected].
0921-8181/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.gloplacha.2006.11.017
generally, but not in every case, during the lower part of
the tidal cycle. A simple explanation for these variations
is that, at low tide, the back force caused by seawater
pushing the glacier upstream is less than at high tide. But
the magnitude of some of the variations is surprising.
For Whillans Ice Stream, flowing from West Antarctica
into the southeast corner of the Ross Ice Shelf, velocities
increased from about zero to more than 1 m h− 1 in
minutes or less, paced by tidal oscillations (Bindschadler et al., 2003a). This was explained as evidence of
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R.H. Thomas / Global and Planetary Change 59 (2007) 217–224
stick-slip in till beneath the glacier, modulated by the
changing resistance offered by the ocean undergoing
tidal rise and fall of about 1 m (Bindschadler et al.,
2003a). Although this explanation provides a good
match with the observed timing of glacier accelerations,
it requires the period between slip events in the absence
of tides to be almost exactly 12 h, which appears to be
remarkably fortuitous.
The observations of tidal variability of glacier velocity have all been made on glaciers that move quite
rapidly (hundreds to thousands of meters per year), and
where motion is primarily by basal sliding, longitudinal
stretching, and shear past glacier margins. Substantial
changes in glacier velocity imply similar changes in all
of these processes, with the changes in stretching rates
extending for a considerable distance upstream. This
suggests that comparatively small force perturbations
associated with the changing tide can affect glacier
behavior far inland of ice that is directly exposed to the
ocean. If so, this has importance beyond the tidal
response of glaciers; it implies that these glaciers may
show far bigger responses to larger force perturbations
associated, for instance, with weakening of their floating
ice tongues and ice shelves. Here, I apply a simple forceperturbation model to some of the observations in order
to investigate upstream glacier behavior during a tidal
cycle, and to assess whether observed velocity changes
are consistent with glacier response to tidally-induced
changing longitudinal forces.
2. Theory
The following is a brief summary of a force-perturbation model presented by Thomas (2004), where the
model is applied to ongoing acceleration and thinning of
one of the world's fastest glaciers, Jakobshavn Isbrae in
Greenland. Fast glaciers move primarily by sliding over
their beds, by shearing past their sides, and by
longitudinal stretching, with all of these three processes
occurring simultaneously. This means that there is very
little shearing in vertical planes, so that the longitudinal
strain rate (ε· x) can be approximated by an equation
similar to that derived for ice shelves:
ε:x ∼kð0:5ρgH−PÞn
ð1Þ
(Thomas, 1973a) with ρ = ice density, g = gravity
acceleration, k determined by ice hardness and the shape
of the strain-rate tensor, n ∼3 for conditions typical on
glaciers, H = ice thickness, and P = total backpressure
acting on the glacier at x, resulting from back forces
caused by downstream basal and lateral drag and forces
exerted by seawater on any floating extension. Changes
(Δ) in any of the terms on the right hand side of this
equation then result in a change in ·ε.x and hence glacier
velocities, expressed by:
x ∼k′ð0:5ρgH′−P′Þ3 ¼ k′ð0:5ρgH−P
ε′
þ 0:5ρgΔH−ΔPÞ3
where post-perturbation values are primed. Writing the
back pressure as P = Φ / H and approximating ΔP ∼ ΔΦ /
H, where Φ is the back force per unit width, this can be
simplified to:
x ∼k′ð−ΔΦ=H þ ð1−ΔH=HÞ:ð εx =kÞ1=3
ε′
þ ρgΔHÞ3
ð2Þ
During a tidal cycle, ice-thickness changes caused by
strain-rate changes should be very small, so ΔH / H ≪ 1.
Then, assuming k ′ ∼ k, Eq. (2) becomes:
Δ εx ∼kðð ε:x =kÞ1=3 −ΔΦ=H þ ρgΔHÞ3 − εx
ð3Þ
∼3ðk ε2x Þ1=3 ðρgΔH−ΔΦ=HÞ
ð4Þ
for small perturbations.
The term k is:
k ¼ θ=B3
ð5Þ
where B is the depth-averaged ice stiffness parameter,
and θ = (1 + λ + λ 2 + γ 2 + ϕ 2 ) / (2 + λ) 3 with λ = ·εy / ·εx,
γ = ·ε.xy / ·εx, ϕ = ·εxz / ·εx, ·εy is lateral strain rate, ·ε.xy and ·εxz
are shear strain rates in the horizontal and vertical
planes. Here, I assume shear strain rates are small compared to longitudinal, as should be the case along a
glacier centre line and for fast-moving ice with low basal
shear stresses, so that θ ∼ (1 +λ + λ2) / (2 + λ)3 = 0.125 for
λ = 0, and θ = 0.111 for λ = 1.
The back force at distance x along the glacier, F(x),
is:
FðxÞ ¼ FgðxÞ þ Fb ðxÞ þ Fm ðxÞ þ Fw þ Fs
w=2
¼ ∫Lx ∫−w=2 ðρi gHtanβ þ τb Þdy þ 2 ∫H
τ
dz
dx
m
0
w=2
þ ∫−w=2 0:5ρw gD2 dy þ Fs
ð6Þ
where x = L at the grounding line, w = glacier width,
ρi = ice density, and g = acceleration due to gravity. The
first term on the right hand side is the compressive
force (Fg) associated with the net upstream component
R.H. Thomas / Global and Planetary Change 59 (2007) 217–224
219
and, assuming no change in Bm and w,
Δτm ≈Bm ð4=ðwV 2 ÞÞ1=3 ΔV =3
ð10Þ
Substituting Eq. (10) into Eq. (7):
ΔΦ≈ΔLτb þ LðΔτb
þ 2HBm ð4=ðw4 V 2 ÞÞ1=3 ΔV =3Þ þ ΔFs =w
þ ρw gDΔD
ð11Þ
and, from Eq. (4):
Fig. 1. Forces acting on a glacier flowing into an ice shelf or floating
tongue.
of weight forces acting on the glacier as it flows over a
basal slope β, taken positive uphill in the direction of
motion (Fig. 1). The remaining terms are the resisting
force (Fb) caused by basal drag (τb), and that (Fm)
caused by marginal shear (τm) of the glacier past its
sides, the back force (Fw) exerted on the glacier at its
grounding line by depth D of sea or lake water with
density ρw, and additional back force (Fs), transmitted
to the grounding line by any floating extension, associated with areas where this extension runs aground or
shears past its sides. Assuming that conditions are
uniform across the glacier, the back-force reduction per
unit glacier width, caused by tidal water-level and associated changes in basal and marginal shear stresses,
is:
ΔΦ≈ΔLðτb þ 2Hτm =wÞ þ LðΔτb
þ 2HΔτm =wÞ þ ρw gDΔD þ ΔFs =w
ð7Þ
with ΔFs representing changes in ice-shelf back forces
associated with velocity changes, and assuming
negligible changes in H and w and that no calving
occurs from the floating part of the glacier. For tidal rise
and fall, D is seawater depth at the grounding line, ΔD
is determined by the state of the tide, and ΔL is
determined by bedrock slope near the grounding line.
Centre-line ice velocities on fast glaciers can be
approximated as:
V ≈wðτm =Bm Þ3 =4
ð8Þ
Δ εx = εx ∼3ðk= εx Þ1=3 ðρgΔH−ðΔLτb þ LðΔτb
þ 2HBm ð4V =w4 Þ1=3 ΔV =3V Þ þ ΔFs =w
þ ρw gDΔDÞ=HÞ
ð12Þ
This indicates that: glacier acceleration is favored by
low tides (negative ΔD); fractional changes in longitudinal strain rates (and therefore velocity) should be
largest for glaciers with low strain rates; and that velocity changes should be damped by associated changes in
thickness and marginal and basal shear stresses. The
effects of changes in ice-shelf back pressure are not
clear, but it appears likely that they would favor acceleration at high tide (with slight un-grounding of ice
rumples), thus also damping the direct influence of the
tide expressed in the final term of Eq. (12). This damping would be further enhanced by grounding-line advance at low tide if the basal shear stress is high, and by
increased shear along ice-shelf margins as velocities
increase. However, despite these various damping effects, observations show velocity changes on some glaciers far inland of the grounding line, apparently linked
to the tidal cycle. In the next section, I shall briefly
review these, and discuss whether they are compatible
with the implications of Eq. (12).
3. Observations of tide-induced glacier velocity
changes
Precise measurement of glacier velocities over short
periods has been substantially improved by GPS and has
resulted in several recent observations of diurnal and
semi-diurnal changes in ice velocity that appear to be
related to the tides. These observations include ice
shelves, tidewater glaciers, and Antarctic glaciers both
close to their grounding lines and far inland.
where Bm is the stiffness parameter for marginal ice.
The marginal shear stress is:
3.1. Leconte Glacier
τm ≈Bm ð4V =wÞ1=3
The near-terminus speed of this Alaskan glacier
fluctuates by about 5% (∼ 1.5 m d− 1 compared to a total
ð9Þ
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R.H. Thomas / Global and Planetary Change 59 (2007) 217–224
of 27 m d− 1), with the highest speed at low water in a
tidal range varying between about 2 and 6 m (O'Neel
et al., 2001). Longitudinal strain rates are extremely
high, averaging almost 4 a− 1 for the most seaward
500 m. The velocity fluctuation is damped to 1 / e of the
ice-front value about 500 m up-glacier (O'Neel et al.,
2001). Consequently, the strain-rate fluctuations are
largely confined to this region, where the average total
fluctuation is 1.5 / 500 = 0.003 d− 1 ∼ ± 0.55 a− 1. This
tidewater glacier has no floating extension, so that L = 0
at the calving front and ΔFs = 0. Neglecting feedback
effects from basal and marginal shear as velocities
change, and assuming that ΔL and ΔH are zero, Eqs. (3)
and (11) reduce to:
Δ εx ∼ðð εx Þ1=3 þ ðρw gDΔDθ1=3 Þ=HBÞ3 − εx
ð13Þ
Assuming lateral ≪ longitudinal strain rates, and with
H ∼ 300 m, D ∼ 250 m, ΔD equal to the full tidal range
of ± 3 m (O'Neel et al., 2001), and B appropriate to
temperate ice ∼ 200 kPa a1/3 (Paterson, 1994, Table 5.2),
this yields Δε·.x ∼ ± 0.5 a− 1 which is almost identical to
the average strain-rate fluctuation estimated above.
However, this very close agreement is largely fortuitous
in view of the many approximations involved.
Eq. (12) can be used to estimate how far inland (Lm)
the velocity fluctuations will propagate before tidal perturbations are completely damped by changes in marginal shear stress, by setting Δε·.x / ·εx = 0:
Lm ∼ρw gDΔD=ðΔτb þ 1:06HBm ΔV =ðVw2 Þ2=3 Þ
ð14Þ
With values of D and ΔD assumed above, Bm∼
200 kPa a1/3, appropriate to temperate ice, the average
thickness over Lm of H ∼ 500 m, w ∼ 1 km, V ∼ 27 m
d− 1 ∼ 104 m a− 1, and ΔV ∼ ± 250 m a− 1 at the ice front,
the average value of ΔV within distance Lm is
approximately ± 125 m a − 1 , Eq. (14) gives
Lm ∼ 2.6 km if Δτb = 0. This is about double the distance
observed (O'Neel et al., 2001), and good agreement
would require a basal shear response also, of about
Δτb ∼ ± 11 kPa. Again, considering the approximations
made in obtaining these estimates, the agreement is
reasonable but by no means sufficient to confirm the
theory. In the next example, I consider ice that is moving
far more slowly, with quite low strain rates.
3.2. Brunt Ice Shelf
This Antarctic ice shelf on the eastern side of the
Weddell Sea is a fringing ice shelf (i.e. not embayed),
and is fed by ice flowing from Dronning Maud Land,
including the quite active Stancomb Wills Glacier flowing into its eastern side. Recent observations show that
the western part of the ice shelf currently moves almost
800 m a− 1 (Doake et al., 2002), at more than double its
speed during the 1960s (Thomas, 1973b). These measurements were made to the west of an area where the
ice shelf is pushed over shoaling seabed to form the
slow-moving McDonald Ice Rumples.
This part of the Brunt Ice Shelf extends only about
50 km seaward from the coast, where ice flows into the
ice shelf in a northwest direction, approximately perpendicular to the grounding line (Thomas, 1973b). As
the ice moves seaward, however, its flow direction shifts
to the west, so that ice near the ice front, where measurements were made, moves almost due westward.
This shift in flow direction is caused by the very active
Stancomb Wills Glacier, which pushes westward the
slower ice near McDonald Ice Rumples. Consequently,
doubling of the measured velocities over the last three
decades indicates a substantial increase in Stancomb
Wills velocities also, or progressive un-grounding of
McDonald Ice Rumples to allow the ice shelf to slide
more easily over the shoaling seabed. Such un-grounding would require the ice shelf to be thinning, and/or
substantial erosion of underlying seabed.
Tidal fall and rise should also result in a cycle of
partial grounding/un-grounding of the ice rumples, with
high tides favoring higher velocities. This is opposite to
the effect of water pressure on the velocity of tributary
glaciers, which favors higher speeds at low tide. The
recent observations (Doake et al., 2002) show a strong
semi-diurnal variation in ice-shelf motion, with maximum westerly velocity about 2 h after low tide, and
velocity variation typically ± 200 to 300 m a− 1 from the
long-term average. There is also a correlation between
transverse (north–south) motion and tidal height, with
100 to 200 m a− 1 northwards or southwards during
rising and falling tides respectively. Thus, the transverse
motion is northward at the time of maximum westward
velocity, consistent with increased influence from
Stancomb Wills Glacier as the rising tide lifts the ice
shelf. Strain-rate measurements made near McDonald
Ice Rumples also show a tidal influence (Doake 2000),
but it is unlikely that such fluctuations can explain the
observed velocity changes. Brunt Ice Shelf strain rates
are typically 0.002 a− 1 or less (Thomas, 1973b), and
would have to fluctuate by 100% or more over most of
the ice shelf to explain the velocity changes.
The fact that velocity maxima occur before high tide
suggests that increased back pressure on the glacier from
the rising water counters, to some extent, the effects of
partial un-grounding of McDonald Ice Rumples. Tide-
R.H. Thomas / Global and Planetary Change 59 (2007) 217–224
induced changes in water pressure should favor higher
glacier strain rates, and velocities, at low tide. I assume
that ΔL = L = 0 near the grounding line, ΔH = 0, and that
B ∼ 500 kPa a1/3 , similar to that inferred for the Brunt Ice
Shelf (Thomas, 1973c). Then, Eq. (12) gives an estimate
of strain-rate change induced solely by changes in back
force from water pressure and from the ice shelf:
Δ εx = εx ∼−3ðk= εx Þ1=3 ðΔFs =w
þ ρw gDΔD=HÞ
ð15Þ
At the grounding line, ice thickness (H) is in
hydrostatic equilibrium with water depth (D), so that
D / H ∼ 0.9, with lower values inland, and tidal range is
about ± 1 m (Doake et al., 2002). Longitudinal strain
rates (ε· x) measured on the ice sheet south east of McDonald Ice Rumples (where ice velocity V ∼ 200 m a− 1)
are typically 0.01 a− 1 (Thomas, 1973a,b,c), but could be
as high as 0.1 a− 1 on Stancomb Wills Glacier (V ∼ 1 km
a− 1). Although strain rates may have increased since
these early measurements, as suggested by velocity
increases near McDonald Ice Rumples, solutions of
Eq. (15) are comparatively insensitive to such changes.
Assuming ΔFs = 0 and θ ∼ 0.1, Δε·x / ·ε.x ∼ ± 12% south
east of McDonald Ice Rumples and ∼ ± 5% on the
glacier, and even if such strain-rate changes also apply
far inland, this represents speed changes of only 24 to
50 m a− 1, with maximum values at low tide. In reality,
·x / ·ε.x should fall subinland from the grounding line, Δε
stantially as D / H decreases and any increase in velocity
probably causes basal shear stresses also to increase.
Consequently, nearly all the observed velocity fluctuation is probably caused by partial grounding/un-grounding of McDonald Ice Rumples and associated changes in
Fs. If, as suggested above, the velocity increase since the
early 1960s by about 400 m a− 1 was caused by partial
un-grounding of the ice rumples, then the tidal velocity
fluctuations represent about half of the total increase.
This implies that the velocity increase could have resulted from thinning of the rumples and/or erosion of
underlying seabed by about 2 m.
Available data are insufficient to determine whether
the tidal fluctuations in velocity result simply from a
westerly shift in the flow direction of the Stancomb
Wills floating ice tongue as the tide rises, or whether the
glacier also accelerates.
221
grounding line, and at locations 40 km and 80 km
upstream, show strong tidal modulation of the velocity
(Anandrakrishnan et al., 2003), with speeds ranging
from + 50% during a falling tide to − 50% during a rising
tide. The measurements can also be plotted as longitudinal strain rates between stations (Fig. 2), which
reach a maximum during falling tides and minimum
during rising tides. The 40-km station was on a nearhorizontal “ice plain” in the mouth of the ice stream
seaward of its shearing margins. This ice plain is pushed
seaward by the ice stream, with probably little resistance
from basal shear. This lends support to the assumption
of ice-shelf-like dynamics implicit in Eq. (1). Fig. 2
· x ∼ 0.003 a− 1 over the 40-km inland of the
shows Δε
grounding-line station, where the average strain rate
·ε.x ∼ 0 (Anandrakrishnan et al., 2003), and substitution
of these values in Eq. (13) yields
ρw gDΔD=H∼F170 kPa m−1
assuming ΔFs = 0, θ ∼ 0.12, and B ∼ 600 kPa a1/3, similar to that inferred for the Ross Ice Shelf (Thomas,
1973c). For a water depth (D) of 750 m at the grounding
line and average ice thickness of 1000 m over the 40 km,
this would require the tidal range to be ± 23 m if ΔFs = 0,
which is far higher than the ± 1.3 m inferred from a tidal
model for the period of the observations (personal communication from L. Padman, Nov. 2003). Although this
calculation is extremely sensitive to errors in ·ε.x (e.g. for
·ε.x = 0.003 a− 1, the required tidal range drops to ± 6 m), it
does indicate that longitudinal-stress perturbations
caused by tidal changes in sea level cannot explain
most of the observed changes in velocity and strain
rates. Ice Stream D flows into the Ross Ice Shelf far
from its calving front, and it is quite possible that tidal
3.3. Ice Stream D
This large ice stream flows into the Ross Ice Shelf,
southwest of Roosevelt Island, at a speed of about 700 m
a− 1. Detailed measurements on floating ice near the
Fig. 2. Longitudinal strain rates for Ice Stream D, over the 40 km
immediately upstream of the grounding line (inferred from Anandrakrishnan et al., 2003) and tides beneath nearby Ross Ice Shelf (personal
communication from L. Padman, Nov. 2003). The broken line
represents tidal rise and fall.
222
R.H. Thomas / Global and Planetary Change 59 (2007) 217–224
rise and fall affects the ice shelf and its interactions with
its margins and Roosevelt Island sufficiently to alter iceshelf back forces (Fs). Moreover, tide-induced currents
along the ice-shelf bottom surface may also contribute to
ΔFs.
4. Discussion
Of the above three examples, only LeConte Glacier
shows reasonable quantitative agreement between observed velocity changes and estimated values assuming
the changes to be caused by tide-induced longitudinalstress perturbations transmitted up-glacier. In both of the
other cases, glaciers flow into ice shelves that may also
be affected by tidal motion. We might expect tidal
effects on ice-shelf back force to be dominated by
grounded ice rises and ice rumples, where tidal rise most
probably reduces compressive forces between the floating and grounded ice. This would result in maximum
velocities at high tide rather than near low tide as
observed. But the situation is further complicated by the
probability that ice-shelf back forces are also modified
by changes in tidal currents beneath the ice shelf (Doake
et al., 2002). Alternatively, tidal rise and fall may somehow be affecting properties of till beneath the ice
(Anandrakrishnan et al., 2003) sufficiently to alter the
basal shear stresses far inland. If so, this would imply a
remarkably sensitive dependence of glacier velocity on
probably quite subtle basal changes. Moreover, a major
determining factor of the motion of most ice streams is
shear between the glacier and its margins, and this
would tend to subdue the effects of changes in basal
conditions.
An additional possibility is the direct influence of
lunar gravity on forces that determine glacier motion,
equivalent to a slight tilting of the local geoid. At Antarctic latitudes, the Moon is low above the horizon,
favoring a northward “pull” from its gravity, with a
timing of maximum impact dependent on glacier flow
direction. Moreover, the Moon's passage from east to
west might explain observations of a “tidal” variation in
glacier lateral motion (e.g. Anandrakrishnan et al.,
2003). The Moon's gravitational attraction on Earth is
approximately 30 mN m− 1 of ice. For the simple case of
a uniform-thickness glacier flowing towards the rising
or setting Moon, this represents a longitudinal stress on
the glacier of S ∼ 30 Pa for length (l) up-glacier from the
ice front of 1 km. Under conditions when the local ocean
surface is tilted with respect to the current geoid, this
also applies to ice shelves. The resulting stress increases
progressively up-glacier if the longitudinal force is fully
transmitted by the ice, which is the case for floating ice
shelves, and may be well approximated for glaciers
flowing over low-friction beds. For parts of the Ross and
Filchner–Ronne ice shelves and their tributary ice
streams, l N 500 km and S N 15 kPa, which is of similar
magnitude to stress perturbations caused by tideinduced changing water pressure.
Key information needed to investigate these options
would be hourly (or less) measurements of longitudinal
strain-rate profiles on ice shelves and glaciers through
many tidal cycles, and extending far enough inland to
quantify the decay of tidal velocity perturbations with
distance from the sea. In the next section, I suggest such
a measurement program on one of the larger Chilean
glaciers.
5. Proposed measurements on Glaciar San Rafael
This is a tidewater glacier flowing from the Hielo
Patagonico Norte into a tidal lagoon connected to the
Pacific Ocean. It is about 40 km long and 2 km wide at
its calving front, where it moves at more than 7 km a− 1
(Rignot et al., 1996). Its response to tidal changes could
be monitored by a series of continuously-recording GPS
receivers planted along the glacier centre line, operating
over a period of several tidal cycles. These should extend from as near the calving front as practicable to
perhaps 30-km inland, with a few km separation between stations. Resulting measurements should give a
direct estimate of time-varying strain rates between stations, which could be compared to both local tides and
passage of the Moon.
In order to estimate the strain-rate changes consistent
with possible tide-induced changes in longitudinal stresses, I assume that longitudinal strain rates are far larger
than lateral strain rates, so that θ ∼ 0.125, and the ice is
temperate, with B ∼ 200 kPa a1/3. Ice thickness has not
been measured on Glaciar San Rafael, but at the equilibrium line (1200 m above sea level) it was estimated to
be between 200 and 475 m (Rignot et al., 1996) depending on the ratio between basal-sliding and icedeformation velocities. The glacier is grounded at its
calving front, where H ∼ 180 m and D ∼ 150 m, with a
tidal range of ΔD ∼ ± 0.8 m. Because there is no floating
extension ΔFs = 0, and Eq. (13) becomes:
Δ εx ∼ðð εx Þ1=3 F3=HÞ3 − εx
ð16Þ
which was solved for a range of values for ·ε.x and H, to
·x) on the
give the plots shown in Fig. 3. Strain rates (ε
−1
glacier are about 2.4 a within 1 km of the calving ice
front, decreasing to an average of 0.6 a− 1 over the next
5 km, with very low values further inland (Rignot et al.,
R.H. Thomas / Global and Planetary Change 59 (2007) 217–224
223
6. Conclusions
Fig. 3. Strain-rate changes calculated for Glaciar San Rafael, for a
range of ice thicknesses and longitudinal strain rates, indicated by the
numbers (a− 1) to the left of the curves. Likely conditions on the glacier
near the ice front are shown by the star, and further inland by the bold
line.
1996). Assuming ice thickness is approximately 200 m
near the calving ice front and is 200–400 m farther
inland, Fig. 3 shows estimates of strain-rate perturba· x ∼ ± 0.08 a− 1
tions associated with low and high tides Δε
over the region within 1 km of the calving ice front
(marked by the star), and Δε·.x ∼ ± 0.016 a− 1 over the
5 km farther up-glacier (marked by the bold curve). This
implies that velocity changes by approximately ± 2 cm
h− 1 at the calving front during tidal fall and rise, and
about half this rate 1-km inland, progressively decreasing to zero over the next 5 km.
A series of velocity measurements along the glacier
should reveal any velocity changes that might be related
to the tides. Preferably, this would include measurements very close to the calving front, at approximately
1-km intervals for 5 km upstream, and 5-km intervals for
another 10–20 km. Intense crevassing near the ice front
militates against continuous GPS measurements here,
but it might be possible to install markers into the
surface from a helicopter, for frequent survey by theodolite. Indeed, in April, 2004, after this paper was originally submitted, Gino Casassa and I made theodolite
observations of prominent ice pinnacles within 1 km of
the ice front from a station on the north side of the
glacier. Unfortunately, results were inconclusive because successful observations were limited to 2 d during
a period when daylight restricted observations to just
part of a tidal cycle each day. A more rigorous effort
would be best attempted during mid summer, with
artificial markers lowered onto the badly-crevassed part
of the glacier by helicopter, and GPS measurements
further upstream. Such a survey should include transverse velocity profiles in order to quantify the negative
feedback associated with marginal stresses as glacier
velocities change.
The analysis presented here addresses the possibility
that tide-related velocity perturbations on glaciers may
represent glacier responses to changing water pressure
on the glacier's floating extensions or calving fronts as
the tide rises and falls. This would require the associated
changes in longitudinal forces acting on the glacier to be
transmitted up-glacier, much as they are in floating ice
shelves. Of the three examples considered here, only
LeConte Glacier in Alaska shows reasonable agreement
with the theory, and here the velocity perturbations do
not extend very far up-glacier. For the other two examples, large ice shelves are included in the glacier systems, and observed velocity changes are far larger than
would be expected from the comparatively small force
perturbations associated with the effects of changing
water depth during tidal rise and fall. Moreover, in at
least one case, they extend far inland from the grounding
line. The velocity changes may represent glacier responses to bed conditions (and presumably marginal
conditions) that somehow change in response to the tide,
but the associated mechanisms are far from clear. Perhaps a more likely explanation is tidal influence on the
ice shelves causing substantial changes in ice-shelf back
forces. This could range from modification of ice-shelf
interaction with its margins, ice rumples, and ice rises, to
changes in the effects of ocean drag beneath the ice shelf
in response to changing tidal currents (Doake et al.,
2002). The direct influence of the lunar gravity on
glaciers is an added effect that has not to my knowledge
previously been considered, probably because it seems
reasonable to assume that it would be very small. However, for large ice shelves the associated forces become
appreciable if the ocean locally is tilted, and this aspect
of the problem deserves further attention.
Acknowledgements
Most of this work was completed while I was visiting
the Centro de Estudios Cientificos de Chile (CECS), and
I thank the director Claudio Bunster, Chief Glaciologist
Gino Casassa, and many others for their hospitality. I
also thank Laurence Padman for providing results from
his tidal model for Ice Stream D and the Brunt Ice Shelf,
and for numerous helpful discussions, and Serdar
Manizade for improving the text and preparing the
Figures. I am particularly grateful to Shin Sugiyama and
an anonymous referee for providing thorough and
thoughtful reviews and several suggestions that have
substantially improved the paper. Funding support was
provided by CECS and NASA's ICESat Project.
224
R.H. Thomas / Global and Planetary Change 59 (2007) 217–224
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