Supplementary Information:
A 3-D coupled ice sheet – sea level model
applied to Antarctica through the last 40 ky
Natalya Gomez a, David Pollardb, Jerry Mitrovicaa
a. Earth and Planetary Sciences, Harvard University. 20 Oxford Street,
Cambridge, MA 02138, USA.
b. Earth and Environmental Systems Institute, Pennsylvania State University,
2217 Earth-Engineering Science Building, University Park, Pennsylvania,
16802, USA.
1. Sea Level Primer
In this section we will review the relevant physics of post-glacial sea-level
change on different timescales using two scenarios for ice-sheet retreat, one
characterized by widespread ice loss and the other by local ice loss. The
simulations are performed using the sea-level model described in Section 2.2,
with no coupling to the ice-sheet model. In each scenario, we alter the ice
geometry rapidly in a single, 200 year time step at the start of the simulation and
then hold the ice distribution constant for 10 ky. Sea-level changes are predicted
every 200 years after the ice-melting event.
To begin, consider the widespread ice loss across the Antarctic Ice Sheet
(AIS) depicted in Figure S1. This ice geometry was constructed by taking the
difference in ice distribution between 11 ka and 10 ka predicted in a simulation
with the coupled ice sheet-sea level model. There is an associated loss of
grounded ice of ~90,000 km3, equivalent to ~1.78 m of globally average sea-level
rise (after account is taken of water inundation into Antarctic marine sectors freed
of ice at 10 ka).
Figure S2a show a global map of the sea-level change immediately
following the melting event and Figure S2b shows the same field focused on
Antarctica. On this short timescale, deformation of the Earth can be considered
purely elastic. The pattern of sea-level change is dominated by a large sea-level
fall in the vicinity of ice melt (i.e., the near-field) that peaks an order of magnitude
higher than the global average (i.e., greater than ~13 m) and a sea-level rise at
distance from the ice sheet (i.e., the far field) that reaches up to ~30% higher
than the global average. The local sea-level fall is due largely to gravitational
and deformational effects. In particular, the ice mass exerts a gravitational
attraction on the surrounding water. As the ice retreats, this attraction weakens
and water migrates away, leading to a drawdown of the sea surface height within
~2000 km of the ice sheet, and a (nearly) progressive increase in sea level at
greater distance from the ice sheet. In addition to this gravitational effect, the
land beneath the ice sheet elastically rebounds when freed of the weight of the
ice which also contributes significantly to the predicted sea-level fall in the near
field. This uplift will expel water out of the region, accentuating the far-field sealevel rise. Ocean loading and as a signal due to load-induced perturbations in
Earth’s rotation vector also contribute to the spatial variability of the sea-level
change (see, e.g., Gomez et al., 2010; Mitrovica et al. 2011). The former is the
reason that the peak far-field sea-level rise occurs in the middle of ocean basins,
while the latter is responsible for some of the azimuthal dependence in the sealevel prediction.
Figures S2c and S2d show the sea-level change predicted over the 10 ky
period following the instantaneous sea-level response plotted in Figures S2a and
S2b. (Note, once again, that there is no change in ice geometry during this time
period.) In this case, the sea-level change is strongly dominated by viscous
deformation of the solid Earth in response to the unloading of ice and loading of
melt water (with associated gravitational and rotational effects). Mitrovica and
Milne (2002) have summarized the various physical effects that contribute to this
pattern of sea-level change and we review these effects here. There is an
ongoing, large amplitude sea-level fall in regions that experienced ice loss as the
solid surface viscously rebounds towards isostatic equilibrium. The nearby
surrounding areas experience a sea-level rise as subsidence occurs at the
periphery of regions of ice loss. The location and amplitude of the peripheral
subsidence is sensitive to both the geometry of ice loss and the Earth model. In
the far-field, there is a broad region of relatively small amplitude sea-level fall due
largely to the migration of water towards the Antarctic to fill the accommodation
space created within peripheral regions experiencing subsidence. This process is
known as ocean syphoning (Mitrovica and Milne, 2002). There is additional
spatial variability in the far-field associated with the viscous effects of meltwater
loading, including a tilting of the crust at continental margins. This so-called
continental levering (Nakada and Lambeck, 1989; Mitrovica and Milne, 2002) is
reflected in a thin offshore zone of sea-level rise; the accommodation space
created by this deformation also contributes to ocean syphoning (Mitrovica and
Milne, 2002).
In Figure S3, we plot time series of the total sea-level change (blue
curves), as well as the radial displacements of the sea surface (green curves)
and solid surface (red curves) relative to their positions at the start of the
simulation. Each column corresponds to a different site, as labeled on the maps
in Figure S2. These sites sample, from left to right in Figure S3, the far-field,
near-field and zone at the periphery of the location of ice melting (labeled ‘A’, ‘B’
and ‘C’, respectively, on the maps in Figure S2).
Within the far-field (Figure S3a), the majority of the initial sea-level jump is
due to a rapid rise in the sea surface as meltwater is added to the ocean.
Subsequently, there is a gradual fall in sea level as a drop in the sea surface
height outpaces the ocean load-induced subsidence of the solid surface. For the
site within the region of ice loss (Figure S3b), there is an instantaneous sea-level
fall of 19.4 m immediately following the melt event. This is comprised of a 6.2 m
drop in the sea surface due to the loss of gravitational attraction toward the ice,
and a 13.2 m rise of the solid surface due to elastic crustal rebound. The
subsequent sea-level fall (~170 m in 10 ky) is almost entirely due to viscous
relaxation of the solid Earth towards isostatic equilibrium. There is also a small
sea surface height increase at this location due to the gradual relaxation of the
initial gravitational perturbation as the crust rebounds. Finally, the site at the
periphery of the melt event (Figure S3c) initially experiences a sea-level fall due
to a combination of gravitational and (elastic) deformational effects.
Subsequently, as the region experiences crustal subsidence due to the adjacent
ice unloading, and sea surface uplift due to the weakening of the initial
gravitational perturbation, the site experiences a monotonic sea-level rise.
Next, we consider results (Figure S4) for the localized melting scenario
shown in Figure S1 (inset), in which only a small sector near the Ross Ice Shelf
experiences ice loss. In this case, the melt event involves a change in ice
volume of ~11,000 km3, equivalent to a global average sea-level rise of 0.11 m
when accounting for the inundation of marine sectors freed of ice. The physical
effects active in this scenario are the same as those in the first scenario. Note,
for example, that while the volume of meltwater is an order of magnitude smaller
than in the scenario of Figure S1, the sea-level fall within the region of melting is
significant on both short and long timescales. For example, the site labeled ‘B’ in
Figure S4b experiences an initial sea-level fall of 9 m, comprised of a 2.5 m drop
in the sea surface height and a 6.5 m (elastic) uplift of the solid surface. Over the
subsequent 10 ky, sea level falls an additional 80 m at this site.
The results described in this supplement highlight the strong geographic
variability and temporal complexity in sea-level change following an ice melting
event. This complexity becomes even more pronounced as the ice sheet evolves
and the loci of melting vary in space and time. In the coupled model simulations
described in the main text, we consider ice-sheet retreat through the entire
deglaciation phase. In these coupled simulations, at any given time during the
ice-sheet retreat, the total predicted sea-level change is a combination of a short
timescale signal governed by the total ice melt and a viscous signal associated
with the full history of ice melting.
2. Sea Level Theory for no-IOG calculations
In the main text, we perform simulations with the coupled ice sheet – sea
level model in which we turn off the effects of ice-ocean gravitational attraction.
We call these simulations “noIOG” for “no ice-ocean gravity”. The purpose of
these simulations is to isolate and assess the impact of the treatment of seasurface height deformation in the sea-level model. The “noIOG” simulations
accounts for changes in elevation of the solid Earth in the same way as the fully
coupled model, but assumes that the sea surface height rises and lowers
uniformly as global ice volumes decrease and increase (as in the uncoupled
model).
In the “noIOG”, the change in topography is given by the following modified
version of equation (B17) in Kendall et al. (2005):
𝑒𝑢𝑠
∗
ΔR(θ, φ, t 𝑗 ) = − ∑ 𝑇𝑙 𝐸𝑙 [𝜌𝐼 Δ𝐼𝑙𝑚
(𝑡𝑗 ) + 𝜌𝑊 Δ𝑆𝑙𝑚
(𝑡𝑗 )]𝑌𝑙𝑚 ( 𝜃, 𝜑)
𝑙,𝑚
𝑗−1
𝑒𝑢𝑠
∗ (𝑡 )
− ∑ 𝑇𝑙 ∑ 𝛽(𝑙, 𝑡𝑛 , 𝑡𝑗 )[𝜌𝐼 δ𝐼𝑙𝑚
𝑛 + 𝜌𝑊 δ𝑆𝑙𝑚 (𝑡𝑛 )]𝑌𝑙𝑚 ( 𝜃, 𝜑)
𝑙𝑚
𝑛=0
− Δ𝑆 𝑒𝑢𝑠 (𝑡𝑗 ),
where we zero out the love numbers associated with deformation of the sea
surface equipotential, such that equations (B13) and (B14) in Kendall et al.
(2005) become:
𝐸𝑙 = −ℎ𝑙
and
𝛽(𝑙, 𝑡𝑛 , 𝑡𝑗 ) = ∑𝐾
𝑘=1
− 𝑟𝑘𝑙
𝑠𝑘𝑙
𝑙
[1 − 𝑒 −𝑠𝑘(𝑡𝑗−𝑡𝑛) ].
Δ𝑆 𝑒𝑢𝑠 (𝑡𝑗 ) is the global average, or “eustatic” sea-level change from the start of
the simulation to time 𝑡𝑗 . It is identical to the change in sea surface height
adopted in uncoupled simulations. That is,
𝜌
∆𝑆 𝑒𝑢𝑠 (𝑡𝑗 ) = − 𝜌 𝐼
𝑊
𝑔𝑎𝑓
∆𝐼00 (𝑡𝑗 )
𝐶00 (𝑡𝑗 )
𝐶(𝜃, 𝜑, 𝑡𝑗 ),
𝑔𝑎𝑓
where ∆𝐼00 is the (0,0) spherical harmonic component of the change in ice
grounded above floatation, and 𝐶 is the ocean function, defined by
1,
𝐶(𝜃, 𝜑, 𝑡𝑗 ) = {
0,
𝑡𝑜𝑝𝑜𝑔𝑟𝑎𝑝ℎ𝑦 < 0 𝑎𝑛𝑑 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑔𝑟𝑜𝑢𝑛𝑑𝑒𝑑 𝑖𝑐𝑒
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒.
𝑒𝑢𝑠
𝑒𝑢𝑠
∆𝑆𝑙𝑚
(𝑡𝑗 ) and 𝛿𝑆𝑙𝑚
(𝑡𝑗 ) represent spherical harmonic decompositions of this
eustatic sea-level change from the start of the simulation to time 𝑡𝑗 and over the
time increment from 𝑡𝑗−1 to 𝑡𝑗 , respectively. All other variables and parameters
are as defined in Kendall et al. (2005).
References:
Gomez, N., Mitrovica, J.X., Tamisiea, M.E., Clark, P.U., 2010. A new projection
of sea level change in response to collapse of marine sectors of the Antarctic Ice
Sheet. Geophys. J. Int. 180, 623–634.
Kendall, R.A., Mitrovica, J.X., Milne, G.A., 2005. On post-glacial sea level – II.
Numerical formulation and comparative results on spherically symmetric models,
Geophys. J. Int., 161, 679–706.
Mitrovica, J.X., Gomez, N., Morrow, E., Hay, C., Latychev, K., Tamisiea, M.E.,
2011. On the Robustness of Predictions of Sea-Level Fingerprints. Geophys J.
Int. 187, 729-742.
Mitrovica, J.X., Milne, G., 2002. On the origin of late Holocene sea-level
highstands within equatorial ocean basins. Quat. Sci. Rev. 21, 2179-2190.
Nakada, M., Lambeck, K., 1989. Late Pleistocene and Holocene sea-level
change in the Australian region and mantle rheology. Geophys. J. 96, 497-517.
Figure Captions:
Figure S1: Patterns of ice loss considered in the Supplement. Thickness of
grounded ice loss (in meters) over the course of a 200 year model melt phase
involving widespread retreat over the Antarctic Ice Sheet. Inset – Area outlined in
gray shows the thickness of grounded ice loss (in meters) over the course of a
200 year model melt phase involving localized ice retreat within a small region
near the Ross Ice Shelf.
Figure S2: Initial and subsequent (long term) patterns of sea-level change
following the ice loss scenario shown in Figure S1. (S2a, S2b) Sea-level
change immediately following the rapid loss of grounded ice depicted in Figure
S1, plotted globally (S2a) and local to the Antarctic (S2b). (S2c, S2d) Total sealevel change in the 10 ky period following the initial sea-level perturbation, plotted
globally (S2c) and locally (S2d). No ice loss occurs during this latter 10 ky time
period. Labels A, B and C refer to the locations of the time series plotted in
Figure S3.
Figure S3: Total change in sea level and the surfaces bounding sea level at
specific sites following the ice loss scenario in Figure S1. (S3a) Change in
sea level (blue), sea surface (green) and crustal (red) heights as a function of
time computed for the far-field (subtropical North Atlantic) site labeled “A” in
Figure S2a. Sea surface and crustal heights are defined relative to the center of
the Earth. (S3b) same as Figure S3a, except for the site within the region of ice
loss (western Ross Ice Shelf) labeled “B” in Figure S2b. (S3c) same as in Figure
S3a, except for the site at the periphery of the ice loss region (outer Ross Sea)
labeled “C” in Figure S2.
Figure S4: Initial and subsequent (long term) patterns of sea-level change
following the ice loss scenario shown in Figure S1 (inset). Format as in
Figure S2. This scenario involves ice loss localized to an area near the Ross Ice
Shelf.
Figure S1:
Figure S2:
Figure S3:
Figure S4: