**Research article**
12 Jun 2018

**Research article** | 12 Jun 2018

# Antarctic sub-shelf melt rates via PICO

Ronja Reese Torsten Albrecht Matthias Mengel Xylar Asay-Davis and Ricarda Winkelmann

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**Ronja Reese et al.**Ronja Reese Torsten Albrecht Matthias Mengel Xylar Asay-Davis and Ricarda Winkelmann

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^{1}Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, P.O. Box 60 12 03, 14412 Potsdam, Germany^{2}Los Alamos National Laboratory, P.O. Box 1663, T-3, MS-B216, Los Alamos, NM 87545, USA^{3}University of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Germany

^{1}Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, P.O. Box 60 12 03, 14412 Potsdam, Germany^{2}Los Alamos National Laboratory, P.O. Box 1663, T-3, MS-B216, Los Alamos, NM 87545, USA^{3}University of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Germany

**Correspondence**: Ricarda Winkelmann (ricarda.winkelmann@pik-potsdam.de)

**Correspondence**: Ricarda Winkelmann (ricarda.winkelmann@pik-potsdam.de)

Received: 19 Apr 2017 – Discussion started: 27 Jun 2017 – Revised: 20 Nov 2017 – Accepted: 09 Feb 2018 – Published: 12 Jun 2018

Ocean-induced melting below ice shelves is one of the dominant drivers for
mass loss from the Antarctic Ice Sheet at present. An appropriate
representation of sub-shelf melt rates is therefore essential for model
simulations of marine-based ice sheet evolution. Continental-scale ice sheet
models often rely on simple melt-parameterizations, in particular for
long-term simulations, when fully coupled ice–ocean interaction becomes
computationally too expensive. Such parameterizations can account for the
influence of the local depth of the ice-shelf draft or its slope on melting.
However, they do not capture the effect of ocean circulation underneath the
ice shelf. Here we present the Potsdam Ice-shelf Cavity mOdel (PICO), which
simulates the vertical overturning circulation in ice-shelf cavities and thus
enables the computation of sub-shelf melt rates consistent with this
circulation. PICO is based on an ocean box model that coarsely resolves ice
shelf cavities and uses a boundary layer melt formulation. We implement it as
a module of the Parallel Ice Sheet Model (PISM) and evaluate its performance
under present-day conditions of the Southern Ocean. We identify a set of
parameters that yield two-dimensional melt rate fields that qualitatively
reproduce the typical pattern of comparably high melting near the grounding
line and lower melting or refreezing towards the calving front. PICO captures
the wide range of melt rates observed for Antarctic ice shelves, with an
average of about 0.1 m a^{−1} for cold sub-shelf cavities, for
example, underneath Ross or Ronne ice shelves, to 16 m a^{−1} for
warm cavities such as in the Amundsen Sea region. This makes PICO a
computationally feasible and more physical alternative to melt
parameterizations purely based on ice draft geometry.

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Dynamic ice discharge across the grounding lines into floating
ice shelves is the main mass loss process of the Antarctic Ice Sheet.
Surrounding most of Antarctica's coastlines, the ice shelves themselves lose
mass by ocean-induced melting from below or calving of icebergs
(Depoorter et al., 2013; Liu et al., 2015). Observations show that many Antarctic ice
shelves are thinning at present, driven by enhanced sub-shelf melting
(Paolo et al., 2015; Pritchard et al., 2012). Thinning reduces the ice shelves'
buttressing potential, i.e., the restraining force at the grounding line
provided by the ice shelves
(Dupont and Alley, 2005; Gudmundsson et al., 2012; Thomas, 1979), and can thereby
accelerate upstream glacier flow. The observed acceleration of tributary
glaciers is seen as the major contributor to the current mass loss in the
West Antarctic Ice Sheet (Pritchard et al., 2012). In particular, the recent
dynamic ice loss in the Amundsen Sea sector
(MacGregor et al., 2012; Mouginot et al., 2014) is associated with high melt
rates that result from inflow of relatively warm circumpolar deep water (CDW)
in the ice-shelf cavities
(D. M. Holland et al., 2008; Hellmer et al., 2017; Jacobs et al., 2011; Pritchard et al., 2012; Schmidtko et al., 2014; Thoma et al., 2008).
Also in East Antarctica, particularly at Totten glacier, as well as along the
Southern Antarctic Peninsula, glacier thinning seems to be linked to CDW
reaching the deep grounding lines
(Greenbaum et al., 2015; Wouters et al., 2015). An appropriate
representation of melt rates at the ice–ocean interface is hence crucial for
simulating the dynamics of the Antarctic Ice Sheet. Melting in ice-shelf
cavities can occur in different modes that depend on the ocean properties in
the proximity of the ice shelf, the topography of the ocean bed and the
ice-shelf subsurface (Jacobs et al., 1992). Antarctica's ice-shelf
cavities can be classified into “cold” and “warm” with typical mean melt
rates ranging from 𝒪(0.1–1.0) m a^{−1} in “cold”
cavities as for the Filchner-Ronne Ice-Shelf and
𝒪(10) m a^{−1} in “warm” cavities like the one
adjacent to Pine Island Glacier (Joughin et al., 2012). For the “cold”
cavities of the large Ross, Filchner-Ronne and Amery ice shelves, freezing to
the shelf base is observed in the shallower areas near the center of the ice
shelf and towards the calving front (Moholdt et al., 2014; Rignot et al., 2013).

Since the stability of the ice sheet is strongly linked to the dynamics of the buttressing ice shelves, it is essential to adequately represent their mass balance. A number of parameterizations with different levels of complexity have been developed to capture the effect of sub-shelf melting. Simplistic parameterizations that depend on the local ocean and ice-shelf properties have been applied in long-term and large-scale ice sheet simulations (Favier et al., 2014; Joughin et al., 2014; Martin et al., 2011; Pollard and DeConto, 2012). These parameterizations make melt rates piece-wise linear functions of the depth of the ice-shelf draft (Beckmann and Goosse, 2003) or of the slope of the ice-shelf base (Little et al., 2012). Other models describe the evolution of melt-water plumes forming at the ice-shelf base (Jenkins, 1991). Plumes evolve depending on the ice-shelf draft and slope, sub-glacial discharge and entrainment of ambient ocean water. This approach has been applied to models with characteristic conditions for Antarctic ice shelves (Holland et al., 2007; Lazeroms et al., 2018; Payne et al., 2007) and for Greenland outlet glaciers and fjord systems (Beckmann et al., 2018; Carroll et al., 2015; Jenkins, 2011). Interactively coupled ice–ocean models that resolve both the ice flow and the water circulation below ice shelves are now becoming available (De Rydt and Gudmundsson, 2016; Goldberg et al., 2012; Seroussi et al., 2017; Thoma et al., 2015). There is a community effort to better understand effects of ice–ocean interaction in such coupled models (Asay-Davis et al., 2016). However, as ocean models have many more degrees of freedom than ice sheet models and require for much shorter time steps, coupled simulations are currently limited to short time scales (on the order of decades to centuries).

Here, we present the Potsdam Ice-shelf Cavity mOdel (PICO), which provides
sub-shelf melt rates in a computationally efficient manner and accounts for
the basic vertical overturning circulation in ice-shelf cavities driven by
the ice pump (Lewis and Perkin, 1986). It is based on the earlier work of
Olbers and Hellmer (2010) and is implemented as a module in the Parallel Ice
Sheet Model (PISM:
Bueler and Brown, 2009; Winkelmann et al., 2011).^{1} Ocean temperature and salinity at the depth of
the bathymetry in the continental shelf region serve as input data. PICO
allows for long-term simulations (on centennial to millennial time scales)
and for large ensembles of simulations which makes it applicable, for
example, in paleo-climate studies or as a coupling module between ice-sheet
and Earth System models.

In this paper, we give a brief overview of the cavity circulation and melt physics and describe the ocean box geometry in PICO and implementation in PISM in Sect. 2. In Sect. 3, we derive a valid parameter range for present-day Antarctica and compare the resulting sub-shelf melt rates to observational data. This is followed by a discussion of the applicability and limitations of the model (Sect. 4) and conclusions (Sect. 5).

PICO is developed from the ocean box model of Olbers and Hellmer (2010),
henceforth referred to as OH10. The OH10 model is designed to capture the
basic overturning circulation in ice-shelf cavities which is driven by the
“ice pump” mechanism: melting at the ice-shelf base near the grounding line
reduces salinity and the ambient ocean water becomes buoyant, rising along
the ice-shelf base towards the
calving front. Since the ocean temperatures on the Antarctic continental
shelf are generally close to the local freezing point, density variations are
primarily controlled by salinity changes. Melting at the ice-shelf base hence
reduces the density of ambient water masses, resulting in a haline driven
circulation. Buoyant water rising along the shelf base draws in ocean water
at depth, which flows across the continental shelf towards the deep grounding
lines of the ice shelves. The warmer these water masses are, the stronger the
melting-induced ice pump will be. The OH10 box model describes the relevant
physical processes and captures this vertical overturning circulation by
defining consecutive boxes following the flow within the ice-shelf cavity.
The strength of the overturning flux *q* is determined from the density
difference between the incoming water masses on the continental shelf and the
buoyant water masses near the deep grounding lines of the ice shelf.

The coefficients in the equation of state (EOS), the turbulent
exchange velocities for heat and salt, are taken from
Olbers and Hellmer (2010). We linearized the potential freezing temperature
equation with a least-squares fit with salinity values over a range of
20–40 PSU and pressure values of 0–10^{7} Pa using Gibbs
SeaWater Oceanographic Package of TEOS-10 (McDougall and Barker, 2011). All
values are kept constant, except for ${\mathit{\gamma}}_{\mathrm{T}}^{*}$ and *C*, which
vary between experiments. The values of these two parameters are the best-fit
from Sect. 3.1.

As PICO is implemented in an ice sheet model with characteristic time scales much slower than typical response times of the ocean, we assume steady state ocean conditions and hence reduce the complexity of the governing equations of the OH10 model. Using this assumption facilitates adaptive box adjustment to grounding line migration, especially since PICO transfers the box model approach into two horizontal dimensions. We assume stable vertical stratification; OH10 found that a circulation state for an unstable vertical water column, which would imply a high (parametrized) diffusive transport between boxes, only occurs transiently (Olbers and Hellmer, 2010). This motivates neglecting the diffusive heat and salt transport between boxes which is small under these conditions. Without diffusive transport between the boxes, some of the original ocean boxes from OH10 become passive and can be incorporated into the governing equations of the set of boxes used in PICO. We explicitly model a single open ocean box which provides the boundary conditions for the boxes adjacent to the ice-shelf base following the overturning circulation, as shown in Fig. 1. In order to better resolve the complex melt patterns, PICO adapts the number of boxes based on the evolving geometry of the ice shelf. These simplifying assumptions allow us to analytically solve the system of governing equations, which is presented in the following two sections. A detailed derivation of the analytic solutions is given in Appendix A. In Sect. 2.3, we describe how the ice-model grid relates to the ocean box geometry of PICO. The system of equations is solved locally on the ice-model grid, as described in Sect. 2.4. Table 1 summarizes the model parameters and typical values.

## 2.1 Physics of the overturning circulation in ice-shelf cavities

PICO solves for the transport of heat and salt between the ocean boxes as
depicted in Fig. 1. Although box *B*_{0}, which is located at depth
between the ice-shelf front and the edge of the continental shelf, does not
extend into the shelf cavity, its properties are transported unchanged from
box *B*_{0} to box *B*_{1} near the grounding line. The heat and salt balances
for all boxes in contact with the ice-shelf base (boxes *B*_{k} for $k\in \mathit{\{}\mathrm{1},\mathrm{\dots}n\mathit{\}}$) can be written as

The local application of these equations for each ice model cell is described
in Sect. 2.4. Since we assume steady circulation,
the terms on the left-hand side are neglected. For the box *B*_{k} with volume
*V*_{k}, heat or salt content change due to advection from the adjacent box
*B*_{k−1} with overturning flux *q* (first term on the right-hand side) and
due to advection to the neighboring box *B*_{k+1} (or the open ocean for
*k*=*n*; second term). Vertical melt flux into the box *B*_{k} across the
ice–ocean interface with area *A*_{k} (third term) and out of the box (fourth
term) play a minor role and are neglected in the analytic solution of the
equation system employed in PICO (a detailed discussion of these terms is
given in Jenkins et al., 2001). The melt rate *m*_{k} is negative if
ambient water freezes to the shelf base. The last term represents heat and
salt changes via turbulent, vertical diffusion across the boundary layer
beneath the ice–ocean interface. The parameters *γ*_{T} and
*γ*_{S} are the turbulent heat and salt exchange velocities which
we assume, following OH10, to be constant.

The overturning flux *q* > 0 is assumed to be driven by the density
difference between the ocean reservoir box *B*_{0} and the grounding line box
*B*_{1}. This is parametrized as in OH10 as

where *C* is a constant overturning coefficient that captures effects of
friction, rotation and bottom form stress.^{2} The circulation strength
in PICO is hence determined by density changes through sub-shelf melting in
the grounding zone box *B*_{1}. From there, water follows the ice-shelf base
towards the open ocean assuming the overturning flux *q* to be the same for
all subsequent boxes. Ocean water densities are computed assuming a linear
approximation of the equation of state:

where ${T}_{*}=\mathrm{0}$ ^{∘}C, ${S}_{*}=\mathrm{34}$ PSU and *α*, *β*
and *ρ*_{*} are constants with values given in
Table 1.

## 2.2 Melting physics

Melting physics are derived from the widely used 3-equation model
(Hellmer and Olbers, 1989; Holland and Jenkins, 1999), which assumes the presence of a
boundary layer below the ice–ocean interface. The temperature at this
interface in box *B*_{k} is assumed to be at the in situ freezing point
*T*_{bk}, which is linearly approximated by

where *p*_{k} is the overburden pressure, here calculated as static-fluid
pressure given by the weight of the ice on top. At the ice–ocean interface,
the heat flux from the ambient ocean across the boundary layer due to
turbulent mixing, ${Q}_{\mathrm{T}}={\mathit{\rho}}_{\mathrm{w}}{c}_{\mathrm{p}}{\mathit{\gamma}}_{\mathrm{T}}\left({T}_{bk}-{T}_{k}\right)$, equals the heat flux due to
melting or freezing ${Q}_{\mathrm{Tb}}=-{\mathit{\rho}}_{\mathrm{i}}L{m}_{k}$. Neglecting heat
flux into the ice, the heat balance equation thus
reads

where $\mathit{\nu}={\mathit{\rho}}_{\mathrm{i}}/{\mathit{\rho}}_{\mathrm{w}}\sim \mathrm{0.89}$, $\mathit{\lambda}=L/{c}_{\mathrm{p}}\sim \mathrm{84}$ ^{∘}C. We obtain the salt flux boundary
condition as the balance between turbulent salt transfer across the boundary
layer, ${Q}_{\mathrm{S}}={\mathit{\rho}}_{\mathrm{w}}{\mathit{\gamma}}_{\mathrm{S}}\left({S}_{bk}-{S}_{k}\right)$, and reduced salinity due to melt water input, ${Q}_{\mathrm{Sb}}=-{\mathit{\rho}}_{\mathrm{i}}{S}_{bk}{m}_{k}$,

To compute melt rates, we apply a simplified version of the 3-equations model (Holland and Jenkins, 1999; McPhee, 1992, 1999) which allows for a simple, analytic solution of the system of governing equations. It has been shown that this formulation yields realistic heat fluxes (McPhee, 1992, 1999). This simplification is used only for melt rates, we nevertheless solve for the boundary layer salinity which is central to the solution of the system of equations as detailed in Appendix A. Melt rates are given by

with ambient ocean temperature *T*_{k} and salinity *S*_{k} in box *B*_{k}. Here,
we use the effective turbulent heat exchange coefficient
${\mathit{\gamma}}_{\mathrm{T}}^{*}$. The relation between *γ*_{T} and
${\mathit{\gamma}}_{\mathrm{T}}^{*}$ is discussed in the
Appendix A.

## 2.3 PICO ocean box geometry

PICO is implemented as a module in the three-dimensional ice sheet model PISM as described in Sect. 2.4. Since the original system of box-model equations is formulated for only one horizontal and one vertical dimension, it needed to be extended for the use in the three-dimensional ice sheet model. The system of governing equations as described in the previous two sections is solved for each ice shelf independently. PICO adapts the ocean boxes to the evolving ice shelves at every time step.

For any shelf *D*, we determine the number of ocean boxes *n*_{D} by
interpolating between 1 and *n*_{max} depending on its size and geometry
such that larger ice shelves are resolved with more boxes. The maximum number
of boxes *n*_{max} is a model parameter; a value of 5 is suitable for the
Antarctic setup, as discussed further in Sect. 3.2. We
determine the number of boxes *n*_{D} for each individual ice shelf *D* with

where rd( ) rounds to the nearest integer. Here, *d*_{GL}(*x*,*y*) is
the local distance to the grounding line from an ice-model grid cell with
horizontal coordinates (*x*, *y*), *d*_{GL}(*D*) is the maximum distance
within ice shelf *D* and *d*_{max} is the maximum distance to the grounding
line within the entire computational domain.

Knowing the maximum number of boxes *n*_{D} for an ice shelf *D*, we next
define the ocean boxes underneath it. The extent of boxes
${B}_{\mathrm{1}},\mathrm{\dots},{B}_{{n}_{D}}$ is determined using the distance to the grounding line
and the shelf front. The non-dimensional relative distance to the grounding
line *r* is defined as

with *d*_{IF}(*x*,*y*) the horizontal distance to the ice
front. We assign all ice cells with horizontal coordinates $\left(x,y\right)\in D$ to box *B*_{k} if the following condition is
met:

This leads to comparable areas for the different boxes within a shelf, which
is motivated in Appendix B. Thus, for
example, the box *B*_{1} adjacent to the grounding line interacts with all ice
shelf grid cells with $\mathrm{0}\le r\le \mathrm{1}-\sqrt{\left({n}_{D}-\mathrm{1}\right)/{n}_{D}}$.
Figure 3 shows an example of the ocean box areas for Antarctica.
PICO does currently not account for melting along vertical ice cliffs, as, for
example, the termini of some Greenland outlet fjords.

## 2.4 Implementation in the Parallel Ice Sheet Model

PICO is implemented in the open-source Parallel Ice Sheet Model (Bueler and Brown, 2009; Winkelmann et al., 2011). In the three-dimensional, thermo-mechanically coupled, finite-difference model, ice velocities are computed through a superposition of the shallow approximations for the slow, shear-dominated flow in ice sheets (Hutter, 1983) and the fast, membrane-like flow in ice streams and ice shelves (Morland, 1987). In PISM, the grounding lines (diagnosed via the flotation criterion) and ice fronts evolve freely. Grounding line movement has been evaluated in the model intercomparison project MISMIP3d (Feldmann et al., 2014; Pattyn et al., 2013).

PICO is synchronously coupled to the ice-sheet model, i.e., they use the same adaptive time steps. The cavity model provides sub-shelf melt rates and temperatures at the ice–ocean boundary to PISM, with temperatures being at the in situ freezing point. PISM supplies the evolving ice-shelf geometry to PICO, which in turn adjusts in each time step the ocean box geometry to the ice-shelf geometry as described in Sect. 2.3.

PICO computes the melt rates progressively over the ocean boxes,
independently for each ice shelf. Since the ice-sheet model has a much higher
resolution, each ocean box interacts with a number of ice-shelf grid cells.
PICO applies the analytic solutions of the system of governing equations
summarized in Sect. 2.1 and
2.2 locally to the ice model grid as detailed
below. Model parameters that are varied between the experiments are the
effective turbulent heat exchange velocity ${\mathit{\gamma}}_{\mathrm{T}}^{*}$ from the
melt parametrization described in Sect. 2.2 and
the overturning coefficient *C* described in
Sect. 2.1. The two parameter values are
applied to the entire ice sheet.

Input for PICO in the ocean reservoir box *B*_{0} is data from observations or
large-scale ocean models in front of the ice shelves. Temperature *T*_{0} and
salinity *S*_{0} are averaged at the depth of the bathymetry in the continental
shelf region. In box *B*_{1} adjacent to the grounding line, PICO solves the
system of governing equations in each ice grid cell (*x*, *y*) to attain the
overturning flux *q*(*x*,*y*), temperature *T*_{1}(*x*,*y*),
salinity *S*_{1}(*x*,*y*), and the melting *m*_{1}(*x*,*y*) at its
ice–ocean interface (given by the local solution of
Eqs. 3, A12,
A8 and 8,
respectively). The model proceeds progressively from box *B*_{k} to box
*B*_{k+1} to solve for sub-shelf melt rate ${m}_{k+\mathrm{1}}\left(x,y\right)$,
ambient ocean temperature ${T}_{k+\mathrm{1}}\left(x,y\right)$, and salinity
${S}_{k+\mathrm{1}}\left(x,y\right)$ (given by the local solution of
Eqs. 13,
A13 and
A8, respectively) based on the previous solutions
*S*_{k} and *T*_{k} in box *B*_{k} and conditions at the ice–ocean interface.
PICO provides the boundary conditions *T*_{k} and *S*_{k} to box *B*_{k+1} as the
average over the ice-grid cells within box *B*_{k}, i.e.,

and analogously for *S*_{k}, where 〈 〉 denotes the average.

The overturning is solved in Box *B*_{1} and given by $q=\langle q\left(x,y\right)\phantom{\rule{0.25em}{0ex}}\text{with}\phantom{\rule{0.25em}{0ex}}\left(x,y\right)\phantom{\rule{0.25em}{0ex}}\text{in}\phantom{\rule{0.25em}{0ex}}{B}_{\mathrm{1}}\rangle $.
Melt rates in box *B*_{k} are computed using the local overburden pressure
*p*_{k}(*x*,*y*) in each ice-shelf grid cell that is given by the weight
of the ice column provided by PISM, i.e.,

This reflects the pressure dependence of heat available for melting and leads to a depth-dependent melt rate pattern within each box. The implications for energy and mass conservation are discussed in Sects. 3.2 and 4.

We apply PICO to compute sub-shelf melt rates for all Antarctic ice shelves
under present-day conditions. Oceanic input for each ice shelf is given by
observations of temperature (converted to potential temperature) and salinity
(converted to practical salinity) of the water masses occupying the sea floor
on the continental shelf (Schmidtko et al., 2014), averaged over the time
period 1975 to 2012. Water masses within an ice-shelf cavity originate from
source regions: neglecting ocean dynamics, we approximate these by averaging
ocean properties on the depth of the continental shelf within regions that
are chosen to encompass areas of similar, large-scale ocean conditions.
Oceanic input is given for 19 basins of the Antarctic Ice Sheet, which are
based on Zwally et al. (2012) and extended to the attached ice shelves and
the surrounding Southern Ocean (Fig. 2). For each ice shelf,
temperature *T*_{0} and salinity *S*_{0} in box *B*_{0} are obtained by averaging
the basin input weighted with the fractional area of the shelf within the
corresponding basin. Figure 2 shows the basin-mean ocean
temperature (shadings and numbers) and salinity (numbers) used.^{3}

Here, we use ${n}_{max}=\mathrm{5}$ from which PICO determines the number of ocean boxes in each shelf via Eq. (9). Figure 3 displays the resulting extent of the ocean boxes for Antarctica, ordered in elongated bands beneath the ice shelves. For the large ice-shelf cavities of Filchner-Ronne and Ross, the ice–ocean boundary is divided into five ocean boxes while smaller ice shelves have two to four boxes (see Table 2). Introducing more than five ocean boxes has a negligible effect on the melt rates, as discussed in Sect. 3.2.

To validate our model, we run diagnostic simulations with PISM+PICO based on
bed topography and ice thickness from BEDMAP2 (Fretwell et al., 2013),
mapped to a grid with 5 km horizontal resolution. Diagnostic
simulations allow us to assess the sensitivity of the model to the parameters
*C* and ${\mathit{\gamma}}_{\mathrm{T}}^{*}$ and to the number of boxes *n*_{max} as well
as the ice model resolution. Transient behavior is exemplified in a
simulation starting from an equilibrium state of the Antarctic Ice Sheet
forced with ocean temperature changes, see Video S1 in the Supplement and
Sect. 3.3.

The number of boxes for each is ice-shelf is given by *b*_{n},
*T*_{0} (*S*_{0}) is the temperature (salinity) in ocean box *B*_{0}, *T*_{n} (*S*_{n})
the temperature (salinity) averaged over the ocean box at the ice-shelf
front, $\mathrm{\Delta}T={T}_{n}-{T}_{\mathrm{0}}$ and $\mathrm{\Delta}S={S}_{n}-{S}_{\mathrm{0}}$. *m* is the average
sub-shelf melt rate, *m*_{observed} the observed melt rates from
Rignot et al. (2013). *q* is the overturning flux. Unit of temperatures
is ^{∘}C, salinity is given in PSU, melt rates in
m a^{−1} and overturning flux in Sv.

## 3.1 Sensitivity to model parameters *C* and ${\mathit{\gamma}}_{\mathrm{T}}^{*}$

We test the sensitivity of sub-shelf melt rates to the model parameters for
overturning strength $C\in [\mathrm{0.1},\phantom{\rule{0.125em}{0ex}}\mathrm{9}]$ Sv m^{3} kg^{−1} and the
effective turbulent heat exchange velocity ${\mathit{\gamma}}_{\mathrm{T}}^{*}\in [\mathrm{5}\times {\mathrm{10}}^{-\mathrm{6}},\phantom{\rule{0.125em}{0ex}}\mathrm{1}\times {\mathrm{10}}^{-\mathrm{4}}]$ m s^{−1}. These ranges
encompass the values identified in OH10, discussed further in
Appendix A. The same parameters for *C* and
${\mathit{\gamma}}_{\mathrm{T}}^{*}$ are applied to all shelves. We constrain the results
by the following qualitative criteria (1) and (2), as well as the quantitative
constraints (3) and (4), summarized in Fig. 4:

*Criterion (1).* Freezing must not occur in the first box *B*_{1} of any
ice shelf, i.e., the ocean box closest
to the grounding line. Freezing in box *B*_{1} would increase ambient salinity,
and since the overturning circulation in ice-shelf cavities is mainly haline
driven, the circulation would shut down, violating the model assumption *q*>0
(see Sect. 2). As shown in Fig. 4a,
the condition is not met for a combination of relatively high turbulent heat
exchange and relatively low overturning parameters. In such cases, freezing
near the grounding line occurs because of the strong heat exchange between
the ambient ocean and the ice–ocean boundary in box *B*_{1} that cannot be
balanced by the resupply of heat from the open ocean through overturning.

*Criterion (2).* Sub-shelf melt rates must decrease between the first
and second box for each ice shelf. This
condition is based on general observations of melt-rate patterns and on the
assumption that ocean water masses move consecutively through the boxes and
cool down along the way, as long as melting in these boxes outweighs
freezing. As shown in Fig. 4b, this condition is violated for
either high overturning and low turbulent heat exchange or, vice versa, low
overturning and high turbulent heat exchange. An appropriate balance between
the strength of these values is hence necessary for a realistic melt rate
pattern.

If criterion (1) or (2) fails, basic assumptions of PICO are violated. Thus,
we choose the model parameters ${\mathit{\gamma}}_{\mathrm{T}}^{*}$ and *C* such that both
criteria are strictly met. The following quantitative, observational
constraints (3) and (4) compare modeled average melt rates with observations
and thus depend on our choice of valid ranges. We choose Filchner-Ronne Ice
Shelf and the ice shelf adjacent to Pine Island Glacier to further constrain
our model parameters for Antarctica. These shelves represent two different
types regarding both the mode of melting and the ice-shelf size.

*Observational constraint (3).* Average melt rates in Filchner-Ronne
Ice Shelf comply with the classification of a “cold” cavity and lie between
0.01 and 1.0 m a^{−1} (Fig. 4c).

*Observational constraint (4).* For Pine Island Glacier, with “warm”
ocean conditions, average melt rates lie between 10 and 25 m a^{−1}
(Fig. 4d).

Generally, an increase in overturning strength *C* will supply more heat and
thus yield higher melt rates, especially for the large and “cold” ice
shelves like Filchner-Ronne. Larger *C* leads to higher melt rates also in
the smaller and “warm” Pine-Island Glacier Ice Shelf but the effect is less
pronounced. In contrast, the turbulent heat exchange alters melting,
particularly in small ice
shelves, while it might decrease melt rates in large ice shelves with
“cold” cavities. Hence, modeled melting in the Filchner-Ronne Ice Shelf is
dominated by overturning while in the Amundsen region melting is dominated by
turbulent exchange across the ice–ocean boundary layer. For three different
parameter combinations, the resulting spatial patterns of melt rates in the
Filchner-Ronne and Pine Island regions are displayed in Fig. S1 in the
Supplement.

All of the above criteria restrict the parameter space to a bounded set with
lower and upper limits as depicted by the green contour lines in
Fig. 4. We determine a best-fit pair of parameters which
minimizes the root-mean-square error of average melt rates for both ice
shelves. The valid range of model parameters around the best-fit parameters
with *C*=1 Sv m^{3} kg^{−1} and ${\mathit{\gamma}}_{\mathrm{T}}^{*}=\mathrm{2}\times {\mathrm{10}}^{-\mathrm{5}}$ m a^{−1} compares well with parameters found in OH10 and
Holland and Jenkins (1999), discussed further in
Appendix A.

## 3.2 Diagnostic melt rates for present-day Antarctica

Using the best-fit values *C*=1 Sv m^{3} kg^{−1} and
${\mathit{\gamma}}_{\mathrm{T}}^{*}=\mathrm{2}\times {\mathrm{10}}^{-\mathrm{5}}$ m s^{−1} found in
Sect. 3.1, we apply PICO to present-day
Antarctica, solving for sub-shelf melt rates and water properties in the
ocean boxes. This model simulation is referred to as “reference simulation”
hereafter.

The average melt rates computed with PICO range from 0.06 m a^{−1}
under the Ross Ice Shelf to 16.15 m a^{−1} for the Amundsen Region
(Fig. 5). Consistent with the model assumptions, melt rates are
highest in the vicinity of the grounding line and decrease towards the
calving front. In some regions of the large ice shelves, refreezing occurs,
e.g., towards the center of Filchner-Ronne or Amery ice shelves. The melt
pattern depends on the local pressure melting temperature. Thus, melt rates
are highest where the shelf is thickest, i.e., near the grounding lines
within box *B*_{1}. Furthermore, freezing and melting can occur in the same box
determined by local ice-shelf thickness. For the vast majority of ice
shelves, the modeled average melt rates compare well with the observed
ranges derived from Table 1 in Rignot et al. (2013). Exceptions are the
Wilkins and Stange ice shelves (in basin 16) with average modeled melt rates
of 9.50 m a^{−1}, which is much higher than the observed range of
1.46 ± 1.0 m a^{−1}. This is most likely due to the ocean
temperature input for this basin (1.17 ^{∘}C, see Fig. 2)
which is higher than for the basin containing Pine Island located nearby
(0.47 ^{∘}C, basin 14), explaining why the melt rates are of the same
order of magnitude in these basins. Modification of water masses flowing into
the shelf cavities, not captured by PICO, might explain the low observed melt
rates in this area despite the relatively high ocean temperatures.

For all ice shelves, ocean temperatures and salinities consistently decrease
in overturning direction, i.e., from the ocean reservoir box *B*_{0} to the
last box adjacent to the ice front *B*_{n}, as shown in Table 2.
Most shelves contain small areas with accretion with a maximum rate of
−1.22 m a^{−1} for the Amery Ice Shelf, see Table S1 in the
Supplement. No freezing occurs at the Western Antarctic Peninsula nor in the
Amundsen and Bellingshausen Seas. A detailed map of sub-shelf melt rates in
this region as well as for Filchner-Ronne Ice Shelf can be found in the
middle panels of Fig. S1. For the Filchner-Ronne Ice Shelf melt rates vary
between −0.67 and 1.76 m a^{−1} and for Pine Island Glacier, melt
rates range from 12.39 to 21.01 m a^{−1}.

PICO tends to smooth out melt rate patterns, see Fig. S4: for Filchner-Ronne and Ross ice shelves the deviations in observed melt rates (Moholdt et al., 2016) from the average melting are larger than the deviations modeled in PICO. The box-wide averages compare well and are at the same order of magnitude for both ice shelves, except for modeled accretion in the later boxes towards the shelf front which is not reflected in the observations. This disagreement might be explained by the fact that the overturning circulation in the model cannot reach neutral density and detach from the ice-shelf base while flowing towards the shelf front.

Aggregated over all Antarctic ice shelves, the total melt flux is
1718 Gt a^{−1}, close to the observed estimate of
1500 ± 237 Gt a^{−1} (Rignot et al., 2013). Overturning
fluxes in our reference simulation range from 0.02 Sv for the small
Drygalski Ice Shelf to 0.32 Sv for Wilkings, Stange, Bach and George
VI ice shelves. Because these are treated in PICO as one ice shelf and they
have a high input temperature, these ice shelves reach together this high
overturning value. The second highest value of 0.23 Sv is found for
Getz Ice Shelf in the Bellingshausen Sea. These overturning fluxes compare
well with the estimates in OH10.

PICO solves the system of governing equations locally in each ice-model grid
cell and calculates input for each ocean box as an average over the previous
box as described in Sect. 2.4. Due to these model
assumptions and because temperature, salinity, overturning and melting are
non-linear functions of pressure in the first box
(Eq. A12), mass and energy are a priori not perfectly
conserved. In Table S1, we compare (within individual shelves) heat fluxes
into the ice-shelf cavities with the heat flux out of the cavities into the
ocean and the latent heat flux for melting. For the whole of Antarctica, the
deviation in heat flux is 403.63 G J s^{−1} which is equivalent to
2.2 % of the latent heat flux for melting. The per basin deviations are
generally low (< 5 %), except around the large Filchner-Ronne and Ross
ice shelves (68 and 43 %). In PICO we assume *q* to be constant,
neglecting changes due to melt water input along the shelf base. This melt
water input amounts to 3.06 % or less of the overturning flux within ice
shelves, and 0.62 % for the entire continent, discussed in
Sect. 2.1.

Melt rates are strongly affected by changes in the ambient ocean
temperatures, see Fig. 6. The dependence is approximately linear
for high and quadratic for lower ambient ocean temperatures. This
relationship is similar to the one observed in OH10 and as expected from the
governing equations. In Pine Island Glacier, melt rates increase by
approximately 6 m a^{−1} per degree of warming. Changes in the
ice-sheet model resolution have little effect on the resulting melt rates
(Fig. S2). For increasing the maximum number of boxes *n*_{max}, average
melt rates converge to almost constant values for ${n}_{max}\phantom{\rule{0.125em}{0ex}}\ge \phantom{\rule{0.125em}{0ex}}\mathrm{5}$ within
all basins, compare Fig. S3.

## 3.3 Transient evolution of PICO boxes and melt rates

PICO is capable of adjusting to changing ice-shelf geometries and migrating
grounding lines. We demonstrate its behavior as a module in PISM in a
transient simulation. Based on an
Antarctic equilibrium state at 8 km resolution comparable to the state
submitted to initMIP (Nowicki et al., 2016), we run PISM+PICO over time with
a simple temperature forcing applied: starting from equilibrium conditions,
ocean temperatures increase linearly over 50 years until an ocean-wide
warming of 1 ^{∘}C is reached. It is then held constant over the next
250 modeled years. The Video S1 shows the temporal evolution of the ocean
temperature input for the ice shelf adjacent to Pine Island Glacier as well
as the Filchner-Ronne Ice Shelf. The ocean temperature increase enhances the
sub-shelf melting for both ice shelves, especially in the first box.
Ice-shelf thinning reduces buttressing and causes the grounding lines to
retreat with the ocean boxes adjusting accordingly.

PICO models the dominant vertical overturning circulation in ice-shelf
cavities and translates ocean conditions in front of the ice shelves, either
from observations or large-scale ocean models, into physically based
sub-shelf melt rates. For present-day ocean fields and ice-shelf cavity
geometries, PICO as an ocean module in PISM reproduces average melt rates of
the same order of magnitude as observations for most Antarctic ice shelves.
With a single combination of overturning parameter *C* and effective
turbulent heat exchange parameter ${\mathit{\gamma}}_{\mathrm{T}}^{*}$ applied to all
shelves, a wide range of melt rates for the different ice shelves is
obtained, reproducing the large-scale patterns observed in Antarctica. The
results are consistent across different ice-sheet and cavity model
resolutions. Additionally, PICO reproduces the common pattern of maximum melt
close to the grounding line and decreasing melt rates towards the ice-shelf
front, eventually with re-freezing in the shallow parts of the large ice
shelves. The governing model equations are solved for individual grid cells
of the ice sheet model (and not for each ocean box with representative depth
value), which allows spatial variability in the resulting melt rate field at
comparatively smaller scale. PICO can adapt to evolving cavities and is
applicable to ice shelves in two horizontal dimensions. It is hence suited as
a sub-shelf melt module for ice-sheet models.

Yet PICO is a coarse model designed as an ocean coupler for large-scale ice-sheet models. It is based on the OH10 model and hence shares some simplifying assumptions with that model: PICO does not resolve ocean dynamics and it parameterizes the vertical overturning circulation in the ice-shelf cavities which is given by one value for each ice shelf. The underlying box model equations of PICO do not resolve horizontal ocean circulation in the ice-shelf cavities driven by the Coriolis force nor seasonal melt variation due to intrusion of warm water from the calving front during Austral summer. Hence, we do not expect that horizontal variations or small scale patterns of basal melt are accurately captured in PICO. Due to the box model formulation, maximum melting in PICO is found directly at the grounding line and not slightly downstream as seen in the high-resolution coupled ice–ocean simulation by, e.g., De Rydt and Gudmundsson (2016). We find that PICO tends to produce smoother melt rate patterns than observed, though the box-wide averages are in good agreement with observations. The effect on ice dynamics of small-scale differences in melt patterns in relation to the large-scale mean melt fluxes is not well established yet and needs further investigation. Following OH10, meltwater does not contribute to the volume flux in the cavity in PICO, introducing a minor error regarding mass conservation. The relative error regarding mass conservation is however small and below 0.7 % of the total overturning strength in our reference run.

A necessary condition for the box model to work is further the assumption
that melting outweighs accretion in box *B*_{1} which is consistent with the
majority of available observations. In PICO, melt rates show a quadratic
dependency on ocean temperature input for lower temperatures, e.g., in the
Filchner-Ronne basin, and a rather
linear dependency for higher temperatures, e.g., in the Amundsen basin. This
is consistent with the results from OH10 and the implemented melting physics
assuming a constant coefficient for turbulent heat exchange. In contrast,
P. R. Holland et al. (2008) employ a dependency of the turbulent heat exchange
coefficient on the velocity of the overturning circulation, suggesting melt
rates to respond quadratically to warming of the ambient ocean water. Here we
follow the approach taken in OH10.

Differently from the OH10 model and relying on much longer timescales of ice dynamics in comparison to ocean dynamics, PICO assumes the overturning circulation to be in steady state. In their analysis, OH10 find unstable vertical water columns to occur only transiently, and hence for PICO a stable stratification of the vertical water column is assumed. Under these conditions, diffusive transport between the boxes is generally small in OH10 and it is hence omitted in PICO. Because PICO also does not consider vertical variations in ambient ocean density, under-ice flow is prevented from reaching neutral density and detaching from the ice-shelf base on its way towards the shelf front. The spatial pattern of melting closer to the calving front of cold ice shelves may in such cases be not represented well.

PICO input is determined by averaging bottom temperatures and salinities over the continental shelf, this is done for 19 different basins. Hence PICO – as a coarse model – misses the nuances of how ocean currents transport and modify CDW over the regions being averaged. The procedure to determine melt rates in PICO is based on the assumption that ocean water that is present on the continental shelf can access the ice-shelf cavities and reach their grounding lines. This implies for example, that barriers like sills that may prevent intrusion of warm CDW are not accounted for and might explain why PICO melting is too high for the ice shelves located along the Southern Antarctic Peninsula. Such phenomena could be tested by varying the ocean input of PICO by evolving the temperature and salinity outside of the cavity over time. Because of the dependence of sub-shelf melting on the local pressure of the ice column above, the model is not fully energy conserving. For the estimated heat fluxes, the relative error is lower than 2.2 % of the latent heat flux due to sub-shelf melting. Hence, we consider our choice of model simplifications as justified, as it introduces small errors in the heat and mass balances in our reference simulation.

PICO is computationally fast, as it uses analytic solutions of the equations of motion with a small number of boxes along the ice shelf. As boundary conditions for PICO are aggregated based on predefined regional basins, the model can act as an efficient coupler of large-scale ice-sheet and ocean models. For this purpose, heat flux into the ice should be added to the boundary layer melt formulation.

The Antarctic Ice Sheet plays a vital role in modulating global sea level. The ice grounded below sea level in its marine basins is susceptible to ocean forcing and might respond nonlinearly to changes in ocean boundary conditions (Mercer, 1978; Schoof, 2007). We therefore need carefully estimated conditions at the ice–ocean boundary to better constrain the dynamics of the Antarctic ice and its contribution to sea-level rise for the past and the future.

The PICO model presented here provides a physics-based yet efficient approach for estimating the ocean circulation below ice shelves and the heat provided for ice-shelf melt. The model extends the one-horizontal dimensional ocean box model by OH10 to realistic ice-shelf geometries following the shape of the grounding line and calving front. PICO is a comparably simple alternative to full ocean models, but goes beyond local melt parameterizations, which do not account for the circulation below ice shelves. We find a set of possible parameters for present-day ocean conditions and ice geometries which yield PICO melt rates in agreement with average melt rate observations. PICO qualitatively reproduces the general pattern of ice-shelf melt, with high melting at the grounding line and low melting or refreezing towards the calving front. Its sensitivity to changes in input ocean temperatures and model parameters is comparable to earlier estimates (Holland and Jenkins, 1999; Olbers and Hellmer, 2010). The model accurately captures the large variety of observed Antarctic melt rates using only two calibrated parameters applied to all ice shelves.

The ocean models that are part of the large Earth system and global circulation models do not yet resolve the circulation below ice shelves. PICO is able to fill this gap and can be used as an intermediary between global circulation models and ice sheet models. We expect that PICO will be useful for providing ocean forcing to ice sheet models with the standardized input from climate model intercomparison projects like CMIP5 and CMIP6 (Eyring et al., 2016; Meehl et al., 2014; Taylor et al., 2012). Since PICO can deal with evolving ice-shelf geometries in a computationally efficient way, it is in particular suitable for modeling the ice sheet evolution on paleo-climate timescales as well as for future projections.

PICO is implemented as a module in the open-source Parallel Ice Sheet Model (PISM). The source code is fully accessible and documented as we want to encourage improvements and implementation in other ice sheet models. This includes the adaption to other ice sheets than present-day Antarctica.

The PICO code used for this publication is available under Reese et al. (2018). For further use of PICO please refer to the latest version at https://github.com/pism/pism/commits/dev.

The data that support the findings of this study are available from the corresponding authors upon request.

Here, we derive the analytic solutions of the equations system describing the overturning circulation (see Sect. 2.1) and the melting at the ice–ocean interface (see Sect. 2.2).

For box *B*_{k} with *k*>1 we solve progressively for melt rate *m*_{k},
temperature *T*_{k} and salinity *S*_{k} in box *B*_{k}, dependent on the local
pressure *p*_{k}, the area of box adjacent to the ice-shelf base *A*_{k} and the
temperature *T*_{k−1} and salinity *S*_{k−1} of the upstream box *B*_{k−1}.
For box *B*_{1}, we additionally solve for the overturning *q* as explained
below. These derivations advance the ideas presented in the appendix of OH10.
Assuming steady state conditions, the balance
Eqs. (1) and (2) for box
*B*_{k} from Sect. 2.1 are

The heat fluxes balance at the boundary layer interface, i.e., the heat flux across the boundary layer due to turbulent mixing ${Q}_{\mathrm{T}}={\mathit{\rho}}_{\mathrm{w}}{c}_{\mathrm{p}}{\mathit{\gamma}}_{\mathrm{T}}\left({T}_{bk}-{T}_{k}\right)$ equals the heat flux due to melting or freezing ${Q}_{\mathrm{Tb}}=-{\mathit{\rho}}_{\mathrm{i}}L{m}_{k}$, omitting the heat flux into the ice. This yields

where $\mathit{\nu}={\mathit{\rho}}_{\mathrm{i}}/{\mathit{\rho}}_{\mathrm{w}}\sim \mathrm{0.89}$, $\mathit{\lambda}=L/{c}_{\mathrm{p}}\sim \mathrm{84}$ ^{∘}C. Regarding the salt flux balance in the
boundary layer, with ${Q}_{\mathrm{S}}={\mathit{\rho}}_{\mathrm{w}}{\mathit{\gamma}}_{\mathrm{S}}\left({S}_{bk}-{S}_{k}\right)$ at the lower interface of the
boundary layer and “virtual” salt flux due to meltwater input
${Q}_{\mathrm{Sb}}=-{\mathit{\rho}}_{\mathrm{i}}{S}_{bk}{m}_{k}$, we obtain

Inserting Eqs. (A2) and (A3) into Eqs. (A1) yields

Comparing $\left({T}_{bk}-{T}_{k}\right)\ll \mathit{\nu}\mathit{\lambda}\approx \mathrm{75}$ ^{∘}C,
allows us to neglect the last term in the temperature equation. Considering
the last two terms of the salinity equation, we find that ${S}_{k}>\left(\mathrm{1}-\mathit{\nu}\right){S}_{bk}\approx \mathrm{0.1}\phantom{\rule{0.125em}{0ex}}{S}_{bk}$, allowing us to neglect the
terms containing *S*_{bk}, which simplifies the equations to

We use a simplified version of the melt law described by McPhee (1992) and
detailed in Sect. 2.2, which makes use of
Eqs. (6) and (5) in which
the salinity in the boundary layer *S*_{bk} is replaced by salinity of the
ambient ocean water:

Holland and Jenkins (1999) suggest that this simplification requires
${\mathit{\gamma}}_{\mathrm{T}}^{*}$ to be a factor of 1.35 to 1.6 smaller than
*γ*_{T} in the 3-equation formulation for the constant values of
*γ*_{T} ranging from $\mathrm{3}\times {\mathrm{10}}^{-\mathrm{5}}$ to $\mathrm{5}\times {\mathrm{10}}^{-\mathrm{5}}$ m s^{−1} used in OH10. This implies that
${\mathit{\gamma}}_{\mathrm{T}}^{*}$ ranges from $\mathrm{2.2}\times {\mathrm{10}}^{-\mathrm{5}}$ to $\mathrm{3.2}\times {\mathrm{10}}^{-\mathrm{5}}$ m s^{−1}, which is consistent with the parameter range we
derive in Sect. 3.1. We apply this
assumption in the computation of melt rates. For the solution of the
transport Eqs. (A1), it is essential to take the
salinity of the boundary layer *S*_{bk} into account, since otherwise the
salinity transport equation would reduce to ${S}_{k}={S}_{k-\mathrm{1}}$ and the overturning
circulation, which is predominantly haline driven, would be reduced.
Inserting the simplified melt law in Eq. (A4) yields

Replacing $x={T}_{k-\mathrm{1}}-{T}_{k}$, $y={S}_{k-\mathrm{1}}-{S}_{k}$, ${T}^{*}=a{S}_{k-\mathrm{1}}+b-c{p}_{k}-{T}_{k-\mathrm{1}}$, ${g}_{\mathrm{1}}={A}_{k}{\mathit{\gamma}}_{\mathrm{T}}^{*}$ and ${g}_{\mathrm{2}}=\frac{{g}_{\mathrm{1}}}{\mathit{\nu}\mathit{\lambda}}$, we obtain

We simplify the previous equations as follows. Rewriting Eq. (A6) as

and inserting it into Eq. (A7), we obtain

Note that we can divide the first line by *q* since, by the model
assumptions, *q*>0. Because $x={T}_{k-\mathrm{1}}-{T}_{k}\ll \mathit{\nu}\mathit{\lambda}\approx \mathrm{75}$ ^{∘}C, we may approximate

Using this approximation, we may proceed to solve the system of equations.
Since we also need to solve for the overturning *q* in box *B*_{1}, which is
adjacent to the grounding line, a slightly different approach is needed than
for the other boxes, as discussed in the next section.

## Solution for box *B*_{1}

The overturning flux *q* is parameterized as

in the model, see Sect. 2.1. Substituting this equation into Eqs. (A6) and (A7), we obtain

Inserting the approximation for *y* from Eq. (A8)
into the Eq. (A10), we obtain a quadratic
equation for *x*,

with $s={S}_{\mathrm{0}}/\mathit{\nu}\mathit{\lambda}$. Since $as=-\mathrm{0.057}\times {S}_{\mathrm{0}}/\mathrm{74.76}=-\mathrm{0.000762}\times {S}_{\mathrm{0}}\ll \mathrm{1}$, we can omit the last part of the last term,

Rearranging (assuming that $\mathit{\beta}s-\mathit{\alpha}>\mathrm{0}$, which we demonstrate below), we obtain

and hence we obtain the solution

The temperature in the box *B*_{1} near the grounding line is supposed to be
smaller than in the ocean box *B*_{0}, since, in general, melting will occur in
box *B*_{1} and hence *T*_{1}<*T*_{0}, or equivalently $x={T}_{\mathrm{0}}-{T}_{\mathrm{1}}>\mathrm{0}$. Furthermore,
we know that ${g}_{\mathrm{1}}/\left(C{\mathit{\rho}}_{*}\right)={A}_{\mathrm{1}}{\mathit{\gamma}}_{\mathrm{T}}^{*}/\left(C{\mathit{\rho}}_{*}\right)$ is positive, as all factors are positive. Since $\mathit{\alpha}=\mathrm{7.5}\times {\mathrm{10}}^{-\mathrm{5}}$, $\mathit{\beta}=\mathrm{7.7}\times {\mathrm{10}}^{-\mathrm{4}}$ and $s={S}_{\mathrm{0}}/\left(\mathit{\nu}\mathit{\lambda}\right)={S}_{\mathrm{0}}/\mathrm{74.76}\ge \mathrm{0.4}$, it follows that *β**s*>*α*. This means that the first summand of
Eq. (A12) is negative and the second (negative)
solution can be excluded. From here, we use ${T}_{\mathrm{1}}={T}_{\mathrm{0}}+x$ and $y=x{S}_{\mathrm{0}}/\left(\mathit{\nu}\mathit{\lambda}\right)$ to solve for *T*_{1}, *S*_{1}, *m*_{1} and *q*.

## Solution for box *B*_{k}, *k*>1

Now, we give the solution for the other boxes *B*_{k} with *k*>1. By inserting
the approximation for *y* in Eq. (A8) into
Eq. (A6), we can solve for *x* as

The denominator is positive, as all terms are positive, and the sign of the
numerator depends on *T*^{*}. The equation can now be solved for *T*_{k}, and
then Eq. (A8) for *S*_{k} and
Eq. (13) for *m*_{k}.

Here, we want to motivate the rule that determines the extent of the boxes
under each ice shelf. The rule aims at equal areas for all boxes. Assuming a
half-circle with radius *r*=1, we want to split it into a fixed number *n* of
equal-area rings. Generalized to the individual shapes of
ice shelves, we will define
the “radius” of an ice shelf as $r=\mathrm{1}-{d}_{\mathrm{GL}}/\left({d}_{\mathrm{GL}}+{d}_{\mathrm{IF}}\right)$. We define *r*_{1}=1 the outer (grounding-line ward)
radius of the half-circle ring covering an area *A*_{1} and corresponding to
box *B*_{1} adjacent to the grounding line, *r*_{2} as the outer radius of second
outer-most half-ring, etc. The box *B*_{k} is then given by all shelf cells
with horizontal coordinates (*x*, *y*) such that ${r}_{k+\mathrm{1}}\le r\left(x,y\right)\le {r}_{k}$ where ${r}_{n+\mathrm{1}}=\mathrm{0}$ is the center point of the
circle. We can use these to determine the areas ${A}_{n}=\mathrm{0.5}\mathit{\pi}{r}_{n}^{\mathrm{2}},\phantom{\rule{0.25em}{0ex}}{A}_{n-\mathrm{1}}=\mathrm{0.5}\mathit{\pi}\left({r}_{n-\mathrm{1}}^{\mathrm{2}}-{r}_{n}^{\mathrm{2}}\right),\phantom{\rule{0.25em}{0ex}}\mathrm{\dots},\phantom{\rule{0.25em}{0ex}}{A}_{n-k}=\mathrm{0.5}\mathit{\pi}\left({r}_{n-k}^{\mathrm{2}}-{r}_{n-k+\mathrm{1}}^{\mathrm{2}}\right)$. If we require that ${A}_{\mathrm{1}}={A}_{\mathrm{2}}=\mathrm{\dots}={A}_{n}$, then, solving progressively, ${r}_{n-k}=\sqrt{k+\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{r}_{n}$. By our
assumption is *r*_{1}=1, hence $\mathrm{1}={r}_{n-\left(n-\mathrm{1}\right)}=\sqrt{n}{r}_{n}$. This
implies that ${r}_{n}=\mathrm{1}/\sqrt{n}$ and thus ${r}_{n-k}=\sqrt{\frac{k+\mathrm{1}}{n}}$.
Hence, the box *B*_{k} for $k=\mathrm{1},\mathrm{\dots},n$ is defined as
$\mathrm{1}-\sqrt{\left(n-k+\mathrm{1}\right)/n}\le {d}_{\mathrm{GL}}/\left({d}_{\mathrm{GL}}+{d}_{\mathrm{IF}}\right)\le \mathrm{1}-\sqrt{\left(n-k\right)/n}$.

The supplement related to this article is available online at: https://doi.org/10.5194/tc-12-1969-2018-supplement.

All authors designed the model. RW conceived the study. RR, RW, TA, and MM implemented PICO as a module in PISM. RR conducted the model experiments. MM created the Antarctic equilibrium simulation for the Video S1 in the Supplement. RR prepared the manuscript with contributions from all co-authors.

The authors declare that they have no conflict of interest.

Development of PISM is supported by NASA grant NNX17AG65G and NSF grants PLR-1603799 and PLR-1644277. Torsten Albrecht was supported by DFG priority program SPP 1158, project numbers LE1448/6-1 and LE1448/7-1. Matthias Mengel was supported by the AXA Research Fund. Ronja Reese was supported by the German Academic National Foundation, the postgraduate scholarship programme of the state of Brandenburg and the Evangelisches Studienwerk Villigst. The project is further supported by the German Climate Modeling Initiative (PalMod) and the Leibniz project DominoES. Xylar Asay-Davis was supported by the US Department of Energy, Office of Science, Office of Biological and Environmental Research under award no. DE-SC0013038. The authors gratefully acknowledge the European Regional Development Fund (ERDF), the German Federal Ministry of Education and Research and the Land Brandenburg for supporting this project by providing resources on the high performance computer system at the Potsdam Institute for Climate Impact Research.

We thank Frank Pattyn and the anonymous reviewer for their helpful comments
on the manuscript. We are grateful to Constantine Khroulev for his excellent
support in improving the implementation of PICO in PISM. We are very thankful
to Hartmut H. Hellmer and Dirk Olbers for their helpful comments and their
ongoing support in further developing their original ocean box
model.

Edited by: Eric Larour

Reviewed by: Frank Pattyn and one anonymous referee

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^{1}

http://www.pism-docs.org (last access: 15 May 2018)

^{2}

For a more detailed discussion see Olbers and Hellmer (2010).

^{3}

We combine drainage sectors feeding the same ice shelf, e.g., all contributory inlets to Filchner-Ronne or Ross Ice Shelves. We also consolidate the basins “IceSat21” and “IceSat22” (Pine Island Glacier and Thwaites Glacier) as well as “IceSat7” and “IceSat8” in East Antarctica.

- Abstract
- Introduction
- Model description
- Results for present-day Antarctica
- Discussion
- Conclusions
- Code availability
- Data availability
- Appendix A: Derivation of the analytic solutions
- Appendix B: Motivation for geometric rule
- Author contributions
- Competing interests
- Acknowledgements
- References
- Supplement

- Abstract
- Introduction
- Model description
- Results for present-day Antarctica
- Discussion
- Conclusions
- Code availability
- Data availability
- Appendix A: Derivation of the analytic solutions
- Appendix B: Motivation for geometric rule
- Author contributions
- Competing interests
- Acknowledgements
- References
- Supplement