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**The Cryosphere**
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- Abstract
- Introduction
- Estimating the sea-level contribution
- Effect of bedrock changes
- Correcting for bedrock changes
- Density correction
- Externally forced sea-level variations
- Ice-sheet modelling example
- Discussion and conclusions
- Data availability
- Author contributions
- Competing interests
- Acknowledgements
- Financial support
- Review statement
- References
- Supplement

**Brief communication**
05 Mar 2020

**Brief communication** | 05 Mar 2020

Brief communication: On calculating the sea-level contribution in marine ice-sheet models

^{1}Institute for Marine and Atmospheric research Utrecht, Utrecht University, Utrecht, the Netherlands^{2}Laboratoire de Glaciologie, Université Libre de Bruxelles, Brussels, Belgium^{3}Earth and Climate Cluster, Faculty of Science, Vrije Universiteit Amsterdam, Amsterdam, the Netherlands^{4}Geosciences, Physical Geography, Utrecht University, Utrecht, the Netherlands

^{1}Institute for Marine and Atmospheric research Utrecht, Utrecht University, Utrecht, the Netherlands^{2}Laboratoire de Glaciologie, Université Libre de Bruxelles, Brussels, Belgium^{3}Earth and Climate Cluster, Faculty of Science, Vrije Universiteit Amsterdam, Amsterdam, the Netherlands^{4}Geosciences, Physical Geography, Utrecht University, Utrecht, the Netherlands

**Correspondence**: Heiko Goelzer (h.goelzer@uu.nl)

**Correspondence**: Heiko Goelzer (h.goelzer@uu.nl)

Abstract

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Estimating the contribution of marine ice sheets to sea-level rise is complicated by ice grounded below sea level that is replaced by ocean water when melted. The common approach is to only consider the ice volume above floatation, defined as the volume of ice to be removed from an ice column to become afloat. With isostatic adjustment of the bedrock and external sea-level forcing that is not a result of mass changes of the ice sheet under consideration, this approach breaks down, because ice volume above floatation can be modified without actual changes in the sea-level contribution. We discuss a consistent and generalised approach for estimating the sea-level contribution from marine ice sheets.

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How to cite.

Goelzer, H., Coulon, V., Pattyn, F., de Boer, B., and van de Wal, R.: Brief communication: On calculating the sea-level contribution in marine ice-sheet models , The Cryosphere, 14, 833–840, https://doi.org/10.5194/tc-14-833-2020, 2020.

1 Introduction

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Model simulations of past and future ice-sheet evolution are an important
tool to understand and estimate the contribution of ice sheets to sea level
at different timescales (e.g. de Boer et al., 2015; Nowicki et al., 2016).
The mass balance of ice sheets is controlled by mass gain and loss at the
upper, lower and lateral boundaries by melting or sublimation, by
accumulation and freeze-on, and by discharge of ice into the surrounding
oceans. The sea-level contribution from an ice sheet can in principle be
estimated through these different mass balance terms but is in practice
typically based on changes in one prognostic variable, *ice thickness*, and considering
corrections for the ice grounded below sea level (e.g. Bamber et al., 2013).
However, complications arise, especially for longer timescales, when
isostatic adjustment of the bedrock is considered. The discussions in this
communication apply for ice-sheet models that include some form of a
glacio-isostatic adjustment (GIA), but that are not coupled to the sea-level
equation. While other examples exist (e.g. Gomez et al., 2013; de Boer et
al., 2014), the models considered here typically account strictly for uplift
or sinking of the bedrock beneath or proximal to an ice sheet but do not
include other (global) effects, such as sea-level changes due to changes in
Earth's rotation and regional sea-level change due to changes in the Earth's
gravitational field. However, the effect of mass changes from other ice
sheets may be included in a simplified form using an external sea-level
forcing. Such forcing is decoupled from mass changes of the ice sheet itself
and prescribes sea-level changes in the model domain with the aim to capture
its effect on ice floatation. The aim of this paper is to propose an approach
to accurately estimate the contribution of the ice sheet in such a model to
global-mean geocentric sea-level rise (see Gregory et al., 2019).

In our own ice-sheet modelling experience and from exchange with colleagues in different groups, it is not always clear how the sea-level contribution should exactly be calculated and what corrections need to be applied. This goes hand in hand with a lack of documentation and transparency in the published literature on how the sea-level contribution is estimated in different models. With this brief communication, we hope to stimulate awareness and discussion in the community to improve on this situation. We caution that it is very possible that the proposed solutions or equivalent approaches are already in use in several models, since the fundamental ideas have already been laid out (e.g. Bamber et al., 2013; de Boer et al., 2015) and are straightforward to implement. Our aim here is to provide concrete guidelines and a central reference of best practices for ice-sheet modellers.

We describe in the following how to calculate the sea-level contribution for a situation without bedrock changes (Sect. 1), the effect of bedrock changes and how to account for them (Sects. 2 and 3), a density correction (Sect. 4), and modifications required when the model is forced by external sea-level changes (Sect. 5). We conclude with a realistic modelling example (Sect. 6) and a discussion (Sect. 7).

2 Estimating the sea-level contribution

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If changes in the bedrock elevation due to isostatic adjustment are zero or very small, e.g. for centennial timescale simulations (e.g. Nowicki et al., 2016), the sea-level contribution of an ice sheet is typically computed from changes in total ice volume above floatation:

$$\begin{array}{}\text{(1)}& {V}_{\mathrm{af}}=\sum _{n}\left({H}_{n}+min({b}_{n},\mathrm{0}){\displaystyle \frac{{\mathit{\rho}}_{\mathrm{ocean}}}{{\mathit{\rho}}_{\mathrm{ice}}}}\right){\displaystyle \frac{\mathrm{1}}{{k}_{n}^{\mathrm{2}}}}{A}_{n},\end{array}$$

where *H* is ice thickness, *b* is bedrock elevation (negative if below sea
level), and *ρ*_{ice}=910 kg m^{−3} and *ρ*_{ocean}=1028 kg m^{−3} are the densities of ice and ocean water, respectively.
The sum is over the number *n* of grid cells (elements) of an (un-)structured
grid with area *A*_{n}. The unitless map scale factor *k* is applied when the
model grid is laid out on a projected horizontal coordinate system, which is
often the case for polar ice-sheet models (Snyder, 1987; Reerink et al.,
2016). Below, we will often simplify the discussion in order to examine the
interplay between ice-sheet thickness, bedrock elevation, and sea level for
a single column, which can be conceptualised as the values occurring in any
single model grid cell or element (in map view). In that framework, we will
refer to the limit ice thickness required for the ice to start floating as
the *floatation thickness*, which is determined by the local bedrock elevation and sea level.
*V*_{af} of a column of ice grounded below sea level may be interpreted as
the amount of ice volume that has to be removed to reach the floatation
thickness and for the column to start to float. This considers that floating
ice is in hydrostatic balance with the surrounding water and assumes that
the ice does not contribute to sea-level changes when melted. In reality,
however, densities of sea water and melted land ice (freshwater) differ
slightly, which is often neglected. An associated density correction is
discussed below (Sect. 4). For ice grounded on land above sea level, *b*>0
and ${V}_{\mathrm{af}}=H\phantom{\rule{0.125em}{0ex}}{A}_{n}\cdot \frac{\mathrm{1}}{{k}^{\mathrm{2}}}$ .

To estimate the ice volume in global sea-level equivalent (SLE_{af}, m),
the total *V*_{af} has to be converted into the volume it will occupy when
added to the ocean assuming a seawater density *ρ*_{ocean}=1028 kg m^{−3} and divided by the ocean area *A*_{ocean} of typically 3.625×10^{14} m^{2} (Gregory et al., 2019).

$$\begin{array}{}\text{(2)}& {\mathrm{SLE}}_{\mathrm{af}}={\displaystyle \frac{{V}_{\mathrm{af}}}{{A}_{\mathrm{ocean}}}}{\displaystyle \frac{{\mathit{\rho}}_{\mathrm{ice}}}{{\mathit{\rho}}_{\mathrm{ocean}}}}\end{array}$$

*A*_{ocean} is assumed to be constant here, but on longer timescales this
is not necessarily correct. Estimating changes in *A*_{ocean} correctly
would require a fully coupled model of global ice sheet–GIA–sea level (e.g.
Gomez et al., 2013; de Boer et al., 2014). The actual sea-level contribution
of the modelled ice sheet (SLC) is typically calculated relative to a
reference value, often the present-day (modelled) configuration or the
configuration at the start or end of an experiment.

$$\begin{array}{}\text{(3)}& {\mathrm{SLC}}_{\mathrm{af}}=-({\mathrm{SLE}}_{\mathrm{af}}-{\mathrm{SLE}}_{\mathrm{af}}^{\mathrm{ref}})\end{array}$$

Note that the minus sign in front of the parentheses in Eq. (3) is necessary
since SLE_{af} is a function of *V*_{af}, for which an increase over time is
associated with a drop in sea level.

Depending on the amount of ice grounded below sea level, estimating the
sea-level contribution instead from the entire grounded ice volume
*V*_{gr} (Eq. 4) can lead to considerable biases
and is only shown for comparison here.

$$\begin{array}{}\text{(4)}& {\mathrm{SLC}}_{\mathrm{gr}}=-\left[{\displaystyle \frac{{V}_{\mathrm{gr}}}{{A}_{\mathrm{ocean}}}}{\displaystyle \frac{{\mathit{\rho}}_{\mathrm{ice}}}{{\mathit{\rho}}_{\mathrm{ocean}}}}-{\left({\displaystyle \frac{{V}_{\mathrm{gr}}}{{A}_{\mathrm{ocean}}}}{\displaystyle \frac{{\mathit{\rho}}_{\mathrm{ice}}}{{\mathit{\rho}}_{\mathrm{ocean}}}}\right)}^{\mathrm{ref}}\right]\end{array}$$

3 Effect of bedrock changes

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In this section we discuss additional considerations that are required when
the model includes a GIA component that simulates bedrock changes. When
changes in bedrock elevation occur under the ice, *V*_{af} cannot always be
used without a correction as basis for sea-level calculations, because
isostatic uplift or lowering can modify *V*_{af} without actual sea-level
contribution. Figure 1 illustrates this problem for
a single ice column with an uplift of the bedrock elevation (left to right
in each panel), where the bars indicate the bedrock and ice for different
possible configurations. In case A, bedrock is already above sea level (i.e.
*V*_{af} includes all ice) and the vertical upward displacement has no
apparent influence on the grounded configuration. In case B, ice is
displaced upwards with the bedrock, the floatation thickness decreases and
some of the ice is “transformed” into ice above floatation. In case C a
transition from floating to grounded ice occurs, and in case D ocean water
is displaced by the rising bedrock.

The problem of how to interpret these changes in sea-level contribution in the
presence of bedrock changes is further illustrated by an evolution of one
grid box in time (Fig. 2a). If we compare between
*t*_{1} and *t*_{4} and only look at the ice column, we could assume that
there was no net sea-level contribution since the ice is just starting to
float (*t*_{1}) or floating (*t*_{4}) in both cases. However, following the
evolution through *t*_{2} and *t*_{3} gives rise to another interpretation.
At *t*_{1} the ice is just starting to float with a low bedrock elevation.
The bedrock then rises (*t*_{2}) and subsequently ice is lost, for example by
surface melting (*t*_{3}). Finally, more ice is lost, for example by basal melting,
and the ice is floating at *t*_{4}. From *t*_{1} to *t*_{2}, ice is merely
displaced by the bedrock, but the actual sea-level contribution occurs
between *t*_{2} and *t*_{3} and equals the ice above floatation in *t*_{2}
and (by construction) also the bedrock displacement between *t*_{1} and
*t*_{4}.

The differences in sea-level contribution from *t*_{1} to *t*_{4} must be
independent from the interpretation of what happened between *t*_{1} and
*t*_{4}. Hence, bedrock changes have to be taken into account below the ice
and in proximity to the ice sheet. The way bedrock changes impact sea level
is through changes in the volume of the ocean basins. That is, as bedrock is
uplifted, ocean basin volume decreases, leading to a positive sea-level
contribution and vice versa.

4 Correcting for bedrock changes

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Based on the discussion in the previous section, here we propose an approach to correct the sea-level estimate for bedrock changes. Under floating ice and ice-free ocean, rising bedrock displaces ocean water and directly leads to a sea-level rise proportional to the bedrock elevation change. The additional sea-level contribution could be calculated from changes in the volume of the ocean water:

$$\begin{array}{}\text{(5)}& {V}_{\mathrm{ocean}}=\sum _{n}\left(max(-{b}_{n},\mathrm{0})-{H}_{n}{\displaystyle \frac{{\mathit{\rho}}_{\mathrm{ice}}}{{\mathit{\rho}}_{\mathrm{ocean}}}}\right){\displaystyle \frac{\mathrm{1}}{{k}_{n}^{\mathrm{2}}}}{A}_{n},\end{array}$$

where the term in brackets is the difference between lower ice boundary and bedrock for grid cells containing floating ice and the ocean depth where no ice is present.

However, while bedrock changes under grounded ice have no impact on the
estimated ocean volume, they do modify the amount of *V*_{af}, which requires
an additional correction. Consider an ice column near floatation but grounded
below sea level at *b*_{0}, with a height above floatation *h*_{af}=0 (e.g.
*t*_{1} in Fig. 2a). When the bedrock rises by a
certain amount Δ*b* (e.g. transition *t*_{1} to *t*_{2} in
Fig. 2a), the ice is lifted and *h*_{af} (in metres
of ice equivalent) increases by

$$\begin{array}{}\text{(6)}& \mathrm{\Delta}{h}_{\mathrm{af}}=\left({\displaystyle \frac{{\mathit{\rho}}_{\mathrm{water}}}{{\mathit{\rho}}_{\mathrm{ice}}}}\right)\mathrm{\Delta}b.\end{array}$$

If the sea-level contribution was only computed from differences in total
ice volume above floatation (SLE_{af}), this would be incorrectly
recorded as a sea-level lowering. Furthermore, if the bedrock was lifted to
or above sea level, the final change in *h*_{af} would equal the ice
thickness and

$$\begin{array}{}\text{(7)}& \mathrm{\Delta}{h}_{\mathrm{af}}=H=-\left({\displaystyle \frac{{\mathit{\rho}}_{\mathrm{water}}}{{\mathit{\rho}}_{\mathrm{ice}}}}\right){b}_{\mathrm{0}},\end{array}$$

where *b*_{0} is the initial bedrock elevation (e.g. at *t*_{1} in
Fig. 2a).

In order to consider corrections for bedrock changes under grounded ice,
floating ice and ice-free ocean consistently, we chose to modify the ocean
volume estimate to incorporate bedrock changes. Note that we assume in the
following that all bedrock adjustment occurs within the ice-sheet model
domain. We suggest replacing the ocean volume calculation above with an
estimate of the *potential* ocean volume (*V*_{pov}), i.e. the volume between bedrock
and sea level if all ice were instantaneously removed:

$$\begin{array}{}\text{(8)}& {V}_{\mathrm{pov}}=\sum _{n}max\left(-{b}_{n},\mathrm{0}\right){\displaystyle \frac{\mathrm{1}}{{k}_{n}^{\mathrm{2}}}}{A}_{n},\end{array}$$

which requires no distinction anymore between grounded and floating ice. However, we have ignored the density difference between ocean water and freshwater, which we will treat separately below.

To convert a change in potential ocean volume to a sea-level contribution,
*V*_{pov} has to be divided by the ocean area of typically 3.625×10^{14} m^{2}:

$$\begin{array}{}\text{(9)}& {\mathrm{SLC}}_{\mathrm{pov}}=-\left[{\displaystyle \frac{{V}_{\mathrm{pov}}}{{A}_{\mathrm{ocean}}}}-{\left({\displaystyle \frac{{V}_{\mathrm{pov}}}{{A}_{\mathrm{ocean}}}}\right)}^{\mathrm{ref}}\right].\end{array}$$

5 Density correction

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In this section we discuss the correction necessary to deal with the small
difference between freshwater (melted ice) and saline ocean water
densities. Transitions of ice below and above floatation and the associated
sea-level change can occur both due to ice mass changes and due to bedrock
changes, processes associated with a different density (*ρ*_{water} vs
*ρ*_{ocean}). While changes in *V*_{af} due to bedrock adjustment and
cavity changes are recorded in ocean water equivalent, we must assume that
changes in ice-sheet mass ultimately contribute to the ocean with a density
of freshwater (*ρ*_{water}=1000 kg m^{−3}). So far, we have
calculated all changes in ocean water column equivalent, so now we will
apply a density correction for all changes in ice thickness (above and below
floatation).

$$\begin{array}{}\text{(10)}& {\displaystyle}& {\displaystyle}{V}_{\mathrm{den}}=\sum _{n}{H}_{n}\left({\displaystyle \frac{{\mathit{\rho}}_{\mathrm{ice}}}{{\mathit{\rho}}_{\mathrm{water}}}}-{\displaystyle \frac{{\mathit{\rho}}_{\mathrm{ice}}}{{\mathit{\rho}}_{\mathrm{ocean}}}}\right){\displaystyle \frac{\mathrm{1}}{{k}_{n}^{\mathrm{2}}}}{A}_{n}\text{(11)}& {\displaystyle}& {\displaystyle}{\mathrm{SLC}}_{\mathrm{den}}=-\left[{\displaystyle \frac{{V}_{\mathrm{den}}}{{A}_{\mathrm{ocean}}}}-{\left({\displaystyle \frac{{V}_{\mathrm{den}}}{{A}_{\mathrm{ocean}}}}\right)}^{\mathrm{ref}}\right]\end{array}$$

The density ratio *ρ*_{water}∕*ρ*_{ocean} implies that the
correction amounts to ∼3 % of the ice volume grounded
at or below sea level.

Finally, to calculate changes in global-mean sea level due to ice-sheet changes, contributions from ice volume above floatation, potential ocean volume and density correction are added:

$$\begin{array}{}\text{(12)}& {\mathrm{SLC}}_{\mathrm{corr}}={\mathrm{SLC}}_{\mathrm{af}}+{\mathrm{SLC}}_{\mathrm{pov}}+{\mathrm{SLC}}_{\mathrm{den}}.\end{array}$$

6 Externally forced sea-level variations

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For long-term ice-sheet simulations, it is common to force ice-sheet models
with prescribed variations in (global) sea level, e.g. representing changes
in the Northern Hemisphere ice sheets when solely simulating the Antarctic
ice sheet. For a glacial–interglacial transition the external sea-level
forcing (ESLF) may have an amplitude of more than 100 m and can drive
transitions between floating and grounded ice in the model. In the framework
of such simulations, the calculation of sea-level contributions from the ice
sheet must be re-considered, because changes in ESLF imply changes in
*V*_{af} of the modelled ice sheet.

We illustrate the implied changes again with a schematic view of one ice
column changing over time (Fig. 2b). From *t*_{1}
to *t*_{2}, the sea level (horizontal solid line) is increased with respect
to the starting value (horizontal dashed line) at constant bedrock elevation
and ice thickness. Consequently, the geometry in the model column changes
from just grounded to floating ice (with no sea-level contribution from the
ice sheet itself). From *t*_{2} to *t*_{3} the sea level is lowered, such
that some ice that was floating in *t*_{2} is transformed into ice above
floatation. At *t*_{4}, now with combined bedrock change and sea-level change
of the same magnitude relative to *t*_{1}, the ice is just grounded on the
lowered bedrock. Calculating the sea-level contribution as described above
in Eq. (12) would indicate a change of the
contribution from *t*_{1} to *t*_{2} and *t*_{3}. However, since these
changes in sea level are externally forced, they should not directly contribute to
the calculated ice-sheet sea-level contribution itself. For example, the
additional volume under the floating ice at *t*_{2} occurs because the ice
is lifted by the additional, externally forced seawater. Equally, the
additional ice above floatation created in *t*_{3} is merely a consequence of
the lower sea level. Hence, *V*_{af} has to be corrected to calculate SLC in
this case.

This problem can be resolved by calculating changes in *V*_{af} and
*V*_{pov} for the constructed case where sea level is fixed and ESLF has no
direct impact on the results. Practically, Eqs. (1)
and (8) can be modified to compensate for changes in
*b*_{n} that occur solely due to ESLF by corresponding changes in an
arbitrary reference level *z*_{0}, e.g. taken as present-day sea level, that
is time-constant in the absolute reference frame but changes with ESLF (Eqs. 13, 14). In other words,
the term (*b*_{n}−*z*_{0}) is constant with respect to changes in ESLF.

$$\begin{array}{}\text{(13)}& {V}_{\mathrm{af}}^{\mathrm{0}}=\sum _{n}\left({H}_{n}+min({b}_{n}-{z}_{\mathrm{0}},\mathrm{0}){\displaystyle \frac{{\mathit{\rho}}_{\mathrm{ocean}}}{{\mathit{\rho}}_{\mathrm{ice}}}}\right){\displaystyle \frac{\mathrm{1}}{{k}_{n}^{\mathrm{2}}}}{A}_{n}\end{array}$$

$$\begin{array}{}\text{(14)}& {V}_{\mathrm{pov}}^{\mathrm{0}}=\sum _{n}max\left({z}_{\mathrm{0}}-{b}_{n},\mathrm{0}\right){\displaystyle \frac{\mathrm{1}}{{k}_{n}^{\mathrm{2}}}}{A}_{n}\end{array}$$

The density correction in Eq. (10) remains unchanged, leading with Eqs. (3) and (9) to the corrected sea-level contribution

$$\begin{array}{}\text{(15)}& {\mathrm{SLC}}_{\mathrm{corr}}^{\mathrm{0}}={\mathrm{SLC}}_{\mathrm{af}}^{\mathrm{0}}+{\mathrm{SLC}}_{\mathrm{pov}}^{\mathrm{0}}+{\mathrm{SLC}}_{\mathrm{den}}.\end{array}$$

With this approach, ESLF can be applied for its effect on the floatation condition in the ice-sheet model without contaminating the calculation of the sea-level contribution. Note that Eqs. (13)–(15) also hold for the case where ESLF is not a spatially uniform value.

7 Ice-sheet modelling example

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Figure 3 illustrates differences in estimated
sea-level contributions for an Antarctic ice-sheet simulation with a model
that includes a simplified GIA component and external sea-level forcing
(Pattyn, 2017). We have first applied a typical glacial–interglacial
experiment (e.g. Golledge et al., 2014; Pollard et al., 2016; Albrecht et
al., 2020) over the last 120 kyr (Fig. 3a) with
the prescribed external sea-level change (based on sea-level reconstructions
by Bintanja and van de Wal, 2008, and Lambeck et al., 2014) as a dominant forcing.
Atmospheric forcing is produced by perturbing present-day surface
temperatures (RACMO2; Van Wessem et al., 2014) with a spatially constant
temperature anomaly following ice-core reconstructions from EPICA Dome C
(Jouzel et al., 2007), while correcting surface temperatures for elevation
changes (e.g. Huybrechts, 2002). The second part of the experiment
(Fig. 3b) continues from the present-day
configuration and shows the response to an extreme basal melt forcing
applied under floating ice shelves. In this schematic forcing scenario,
present-day melt rates are multiplied by a constant factor of 200, resulting
in melt rates of up to 100 m yr^{−1} in the Weddell and Ross sea sectors.
This extreme melt forcing is not meant to represent a plausible scenario; it
only serves to simulate a rapid removal of all floating ice shelves, leading
to a retreat of the ice sheet (Pattyn, 2017; Nowicki et al., 2013).

Various SLC corrections and estimates are calculated against the initial
configuration in Fig. 3a (120 kyr BP) and against
the present-day configuration in Fig. 3b, c. The
sea-level contribution calculated from changes in ice volume above floatation
(SLC_{af}) includes signatures of bedrock and (in the past)
externally forced sea-level changes. In the future retreat scenario
(Fig. 3b), SLC_{af} is too low compared to our
corrected estimate (${\mathrm{SLC}}_{\mathrm{corr}}^{\mathrm{0}}$) mainly because ice volume above
floatation is “created” by bedrock uplift. This effect of isostatic
adjustment on SLC_{af} is exemplified by the steadily decreasing
SLC_{af} towards the end of the experiment, while ${\mathrm{SLC}}_{\mathrm{corr}}^{\mathrm{0}}$
remains near constant (due to compensating SLC_{pov}). Accounting for
density differences between ocean water and freshwater (SLC_{den}) corrects
an additional, but smaller underestimation of SLC_{af}. The proposed
method (${\mathrm{SLC}}_{\mathrm{corr}}^{\mathrm{0}}$) is identical to (SLC_{corr}) for the future
period (Fig. 3b), where no external sea-level
forcing is applied, and results in an estimate of the sea-level contribution
well above SLC_{af}.

In the paleo-simulation (Fig. 3a), SLC_{af} is biased by both bedrock
changes and external sea-level changes. Since SLC_{pov} is calculated in
a fixed domain that includes grounded and floating ice and ice-free ocean
areas, it is influenced by ice and ocean water loading. In a glaciation
scenario with a growing (Antarctic) ice load and decreasing global sea level
(Fig. 3a, before 15 kyr BP), the correction
SLC_{pov} is a combination of a subsiding bedrock under the ice sheet
(negative SLC_{pov}) and a rising ocean floor in response to reduced
water loading (positive SLC_{pov}). We reiterate that the global ocean area
*A*_{ocean} is assumed to be constant here. Although not fully separable, we
have estimated the contribution of the two effects by calculating
SLC_{pov} within and outside of the glacial ice mask (see Supplement
Fig. S1). Both effects are of similar magnitude in our setup, but
SLC_{pov} is slightly dominated by the changing ocean floor outside of
the ice mask after periods of rapid sea-level forcing change. In addition,
during ice-sheet growth, the negative sea-level excursion in SLC_{af} is
exaggerated with increasing amplitude of the external sea-level forcing (compare
SLC_{af} and ${\mathrm{SLC}}_{\mathrm{af}}^{\mathrm{0}}$). The proposed method
(${\mathrm{SLC}}_{\mathrm{corr}}^{\mathrm{0}}$) results in an estimate of the negative sea-level
contribution in the past of smaller amplitude compared to SLC_{af} and
shows that the magnitude and notably the timing of the Last Glacial Maximum
low stand are subject to considerable biases in SLC_{af} (Fig. 3c).
The relative bias in SLC_{af} is larger for stronger ice-sheet retreat
(not shown). Accounting for all grounded ice (SLC_{gr}) would lead in
all cases to the largest excursions in negative and positive sea-level
contribution, due to ice grounded below the water level that should mostly
be replaced by seawater. Differences between the different approaches to
calculate SLC become important after 2–3 kyr, roughly corresponding to the
shortest response time of bedrock adjustment in the model.

8 Discussion and conclusions

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We have presented a unified approach to calculate the sea-level contribution from a marine ice sheet simulated by an ice-sheet model. The formulation notably corrects for changes in ice volume above floatation in the presence of bedrock changes and external sea-level forcing. In this unified approach, sea-level contributions arise from changes in the ice volume above floatation and potential ocean volume, while changes in external sea-level forcing are corrected for.

When bedrock changes in response to ice loading changes occur under ice that is grounded (below sea level), changes in potential ocean volume compensate for changes in ice volume above floatation, resulting in a near-zero net sea-level contribution as should be expected. Under floating ice (or open ocean), changes in volume above floatation are always zero, but bedrock changes imply ocean depth changes that lead to differences in the sea-level contribution (i.e. due to changes in ocean basin volume). The combination of changes in ice volume above floatation and potential ocean volume leads to a generalised formulation that is consistent across changes from floating to grounded ice and vice versa.

The region over which ice thickness changes and potential ocean volume changes are calculated must be fixed in time for the comparison and may contain the entire model grid (as done here) or a reasonable subset. It should include all locations that potentially see ice thickness and/or bedrock changes during a simulation. For models with local isostatic adjustment, the region could be the glacial ice mask for paleo-simulations and the observed present-day sheet-shelf mask for future simulations dominated by retreat. For non-local isostatic models, the footprint would have to be extended.

In all calculations we have ignored any effects that arise from water storage in lakes on land, for example, and we also did not consider the equation of state of seawater, which implies a non-linear dependence of density on salinity and temperature.

Data availability

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Data availability.

The SLC time series in Figs. 3 and S1 are available online at: https://doi.org/10.5281/zenodo.3692702 (Goelzer, 2020).

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/tc-14-833-2020-supplement.

Author contributions

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Author contributions.

HG conceived the project and developed the SLC corrections with assistance of VC. VC performed and analysed the model experiments. HG wrote the manuscript with assistance of all authors.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

We would like to thank the two reviewers Stephen Price and Rupert Gladstone for their constructive suggestions that helped to improve the manuscript and the editor Ben Galton-Fenzi for guiding the publication process.

Financial support

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Financial support.

Heiko Goelzer has received funding from the programme of the Netherlands Earth System Science Centre (NESSC), financially supported by the Dutch Ministry of Education, Culture and Science (OCW) under grant no. 024.002.001. Violaine Coulon was funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S. – FNRS) with a F.R.S. – FNRS Research Fellowship. Bas de Boer is funded by the SCOR Corporate Foundation for Science.

Review statement

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Review statement.

This paper was edited by Ben Galton-Fenzi and reviewed by Stephen Price and Rupert Gladstone.

References

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Short summary

In our ice-sheet modelling experience and from exchange with colleagues in different groups, we found that it is not always clear how to calculate the sea-level contribution from a marine ice-sheet model. This goes hand in hand with a lack of documentation and transparency in the published literature on how the sea-level contribution is estimated in different models. With this brief communication, we hope to stimulate awareness and discussion in the community to improve on this situation.

In our ice-sheet modelling experience and from exchange with colleagues in different groups, we...

The Cryosphere

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