Introduction
The Antarctic Ice Sheet is characterized by vast areas of floating ice at its
margins, comprising ice shelves, both large and small, that buttress the
outflow of ice from inland. The stability of these ice shelves is governed by
a delicate mass balance, consisting of an influx of ice from the glaciers,
iceberg calving at the ice front, snowfall and ablation at the surface, and
basal melting due to oceanic heat exchange in the ice-shelf cavities.
Recent studies suggest that Antarctic ice shelves are experiencing rapid
thinning , an effect which can be
traced back to an increase in basal melting .
This is especially apparent in West Antarctica, where relatively warm water
from the Amundsen and Bellingshausen seas is able to flow into the ice-shelf
cavities and enhance melting from below. Increased basal melt rates and
thinning of ice shelves decrease the buttressing effect, enhancing the ice
flow and associated mass loss from the Antarctic glaciers and ice sheet. The
disintegration of the ice shelves can significantly affect future sea-level
rise, as suggested by recent numerical simulations .
In order to correctly predict the evolution of the ice sheet, it is
necessary to have accurate models of the dynamics of ice shelves in which
basal melting at the interface between ice and ocean plays an important role.
State-of-the-art ice-sheet models for large-scale climate simulations (see
e.g. ) provide a complete description of the flow and
thermodynamics of ice. However, due to the complex nature of the system and
high computational cost of climate simulations, these models inevitably
contain approximations and parametrizations of many physical processes, among
which basal melting is no exception. In particular, it is challenging to
resolve the ocean dynamics within the ice-shelf cavities on a continental
scale, which severely restricts the level of detail possible in basal melt
parametrizations. Most recent simulations (e.g. )
determine the basal melt rate from the
local heat flux at the ice–ocean interface , driven by a
far-field temperature and a number of tuning factors. Others include a
dependence on the thickness of the water column beneath the ice shelf in
order to reduce melting near the grounding line .
As demonstrated by observational data (e.g. ), the basal
melt rates around Antarctica show a complex spatial pattern, which can be
inferred to depend heavily on both the geometry of the ice-shelf base and
the ocean temperature. It is unlikely that a description of basal melt based
on local fluxes at the ice–ocean interface can capture this complex pattern
without being either significantly tuned or used in conjunction with
extremely detailed ocean–shelf–cavity models. On the other hand, the ocean
dynamics and associated melt rates within individual ice-shelf cavities
have been studied in rather high detail in recent years. For example,
showed that basal melt rates obtained from a general ocean
circulation model respond quadratically to changing ocean temperatures. These
studies shed light on the minimal requirements of basal melt
parametrizations, i.e. a non-linear temperature sensitivity, an inherent
geometry dependence corresponding to the unresolved ocean circulation, and a
depth-dependent pressure freezing point, yielding higher melt rates at
greater depths and the possibility of refreezing at lesser depths, closer to
the margins of the ice shelves.
Taking these requirements into account, we develop a more advanced
parametrization for the basal melt rates, based on the theory of buoyant
meltwater plumes, which was first applied to the ice-shelf cavities by
. In this theory, it is assumed that the main physical mechanism
driving the ocean circulation within the cavity is the positive buoyancy of
meltwater, which travels upward beneath the ice-shelf base in the form of a
turbulent plume. Melting at the ice–ocean interface is influenced by the
fluxes of heat and meltwater through the ocean boundary layer, which depend
on the plume dynamics. The upward motion of the plume induces an inflow of
possibly warmer ocean water into the ice-shelf cavity, creating more melt.
Entrainment from the surrounding ocean water affects the momentum and
thickness of the plume as it moves up the ice-shelf base. Depending on the
stratification of the ocean water inside the cavity, the plume may reach a
level of neutral buoyancy from which it is no longer driven upward.
Schematic picture of the plume model. The plume travels upward under
the ice-shelf base along the path X with speed U and thickness D while
being influenced by melting and entrainment. Note that, in general, the slope
angle α can vary in the direction of X.
The dynamics of the plume can be captured by a quasi-one-dimensional model of
the mass, momentum, heat and salt fluxes within the plume, as shown
schematically in Fig. . In particular, this work is based
on the plume model of , from which a basal melt parametrization
has recently been derived . This parametrization is
based on an empirical scaling of the plume model results in terms of
ambient ocean properties and the geometry of the ice-shelf cavity. The
geometry dependence is mainly determined by the grounding-line depth and the
slope of the ice-shelf base. The aim of this particular study is to apply the
plume parametrization to a two-dimensional grid covering all of Antarctica in
order to investigate if this type of parametrization is able to give
realistic present-day values and capture the complex pattern of basal melt
rates shown in observations .
In the following section, we describe the details of the plume model and the
basal melt parametrization derived from it (Sect.
and ). An important part of the work is the development of an
algorithm that translates the parametrization from a one-dimensional to a
two-dimensional geometry, as described in Sect. . In
Sect. , we show results from the numerical evaluation of
the (still 1-D) parametrization along flow lines of two well-known Antarctic
ice shelves, namely Filchner–Ronne and Ross. Finally, Sect.
and discuss the application of the 2-D plume
parametrization to the entire Antarctic continent, resulting in a
two-dimensional map of basal melt rates under the ice shelves. Special
attention is given to the construction of an effective ocean temperature
field from observations by inversion of the modelled basal melt rates. The
results are compared with those from simple heat-balance models .
Modelling basal melt
In this section, we start with a description of the basic physics underlying
basal melt models. We summarize the quasi-one-dimensional plume model of
and the development of the plume parametrization
resulting from this model, as shown in previous work.
The main contribution of the current study is the method used to extend this
plume parametrization to two-dimensional input data, which are necessary for use in a
3-D ice-sheet–ice-shelf model.
First of all, we briefly discuss a common feature of many basal melt
parametrizations, namely the dependence on the local balance of heat at the
ice–ocean interface. In its simplest form, this is a balance between the
latent heat of fusion and the heat flux through the sub-ice-shelf boundary
layer, which can be expressed as follows :
ρim˙L=ρwcwγTTa-Tf,
where ρi and ρw are the densities of ice and ocean water, respectively,
m˙ is the melt rate, L is the latent heat of fusion for ice,
cw is the specific heat capacity of ocean water, γT is a turbulent
exchange velocity and Ta is the temperature of the ambient ocean water. In
this model, the melting is driven by the difference between Ta and the
depth-dependent freezing point,
Tf=λ1Sw+λ2+λ3zb,
where Sw is salinity of the ocean water; zb is the depth of the
ice-shelf base; and λ1, λ2 and λ3 are constant parameters.
As explained by , more details can be included in this basal
melt model, e.g. heat conduction into the ice and a balance equation for
salinity (see also Sect. ). Nevertheless, many ice models
contain basal melt parametrizations based on Eq. (1) (see
e.g. ). These models typically use either
constant or temperature-dependent values for γT, leading to a melt
rate that depends either linearly or quadratically on the temperature
difference Ta - Tf. The latter case is consistent with the findings of
, who obtained a similar quadratic relationship from the
output of an ocean general circulation model applied to the ice-shelf
cavities. The non-linearity arose because the exchange velocity γT
in Eq. () was expressed as a linear function of the ocean
current driving mixing across the boundary layer, which is itself a
function of the thermal driving. further explain how this
non-linear temperature dependence is related to the input of meltwater with
an associated decrease in salinity and increase in buoyancy.
Hence, the exchange velocity plays an important role in correctly determining
the heat balance at the ice–ocean interface or, more precisely, the heat
transfer through the ocean boundary layer beneath the ice shelves. However, a
local heat-balance model as expressed by Eq. (1) is too
simplistic to capture the effects of the ocean circulation on the basal
melting, e.g. those depending on the ice-shelf geometry. The plume model and
parametrization discussed in the remainder of this section are considered the
next step in modelling the physics for general ice-shelf geometries without
having to rely on full ocean circulation models, for which there are also
insufficient input data to obtain a universal Antarctic solution.
Plume model
The parametrization used in this study is based on the plume model developed
by . Here we summarize the key assumptions and physics behind
this model. The ice-shelf cavity is modelled by a two-dimensional geometry
(Fig. ), in which the ice-shelf base has a (local) slope
given by the angle α. This geometry is assumed to be uniform in the
direction perpendicular to the plane and constant in time and can be seen as
a vertical cross section along a flow line of the ice shelf. We can define a
coordinate X along the ice-shelf base with slope grounding line (X = 0) and
moving up along the ice-shelf base due to positive buoyancy with respect to
the ambient ocean water.
The situation depicted in Fig. essentially yields a
two-layer system of the meltwater plume with varying thickness D, velocity U,
temperature T and salinity S lying above the ambient ocean with
temperature Ta and salinity Sa. As explained in , the
typically small values of the slope angle α allow us to consider
conservation of mass, momentum, heat and salt within the plume in a
depth-averaged sense. Moreover, as the plume travels upward in the direction
of X, it is affected by entrainment (at rate e˙) of ambient ocean
water, as well as the fluxes of meltwater (with melt rate m˙) and heat
at the ice–ocean interface (with temperature Tb and salinity Sb). These
considerations yield the following quasi-one-dimensional system of equations
for (D, U, T, S) as a function of the coordinate X along the shelf base,
denoting the balance of mass, momentum, heat and salt within the plume:
dDUdX=e˙+m˙,dDU2dX=DΔρρ0gsinα-CdU2,dDUTdX=e˙Ta+m˙Tb-Cd1/2ΓTUT-Tb,dDUSdX=e˙Sa+m˙Sb-Cd1/2ΓSUS-Sb,
where g is the gravitational acceleration, Cd is the (constant) drag
coefficient, Δρ = ρa - ρ is the difference in density between
plume and ambient ocean, and Cd1/2ΓT and Cd1/2ΓS are the
turbulent exchange coefficients (Stanton numbers) of heat and salinity at the
ice–ocean interface. The above formulation makes explicit the linear
dependence of the turbulent exchange velocities on the ocean current
(γT = Cd1/2ΓTU, γS = Cd1/2ΓSU). The
system of Eq. (2) is closed using suitable expressions for
the entrainment rate e˙, an equation of state ρ = ρ(T, S), the
balance of heat and salt at the ice–ocean interface, and the liquidus
condition. The expression for the entrainment rate is assumed to have the
following form :
e˙=E0Usinα,
with E0 a dimensionless constant. Hence, the entrainment rate increases
linearly with the plume velocity, is zero for a horizontal ice-shelf base
and grows with increasing slope angle. Furthermore, a linearized equation of
state yields
Δρρ0=βSSa-S-βTTa-T,
where βS is the haline contraction coefficient and βT the
thermal expansion coefficient. The boundary conditions at the ice–ocean
interface are given by
Cd1/2ΓTUT-Tb=m˙Lcw+cicwTb-Ti,Cd1/2ΓSUS-Sb=m˙Sb-Si,Tb=λ1Sb+λ2+λ3zb,
i.e. the first equation balances the turbulent exchange of heat with heat
conduction and latent heat of fusion L in the ice, where cw and
ci are the specific heat capacities of ocean water and ice, respectively, and
Ti is the ice temperature. Similarly, Eq. () is a balance
between the turbulent exchange of salt and diffusion into the ice.
Equation () is the (linearized) liquidus condition that puts the
interface temperature equal to the pressure freezing point at the local
depth zb of the ice-shelf base, which is equivalent to Eq. ().
Equations (2)–(5) form a closed set that can be solved
to obtain the prognostic variables (D, U, T, S) of the plume as a function of
the plume path X, given the ice-shelf draught zb(X) with slope
angle α(X), the ambient ocean properties Ta(z) and Sa(z), and the ice
properties Ti and Si. Of particular interest for the current work,
however, are the ice–ocean interface conditions (Eq. 5), which
essentially determine the melt rate m˙, the key quantity of this
study. In other words, the melt rate is determined by the fluxes of heat and
salt at the interface, which in turn are linked to the development of the
plume. Note that these boundary conditions can be simplified
to only two equations containing the freezing
temperature Tf of the plume, rather than the interface
properties Tb and Sb:
Cd1/2ΓTSUT-Tf=m˙Lcw+cicwTf-Ti,Tf=λ1S+λ2+λ3zb,
where Cd1/2ΓTS is an effective heat exchange coefficient. This
simplified formulation can be used together with the prognostic
Eq. (2) by substituting Tb with Tf in Eq. () (note that
Tb and Tf are not necessarily equal), whereas Sb disappears from
the problem by substituting Eq. () in Eq. (). Strictly
speaking, Eq. (6) is only valid after assuming a constant ratio
ΓT/ΓS of the exchange coefficients, as explained by
, who also show that both Eqs. (5) and (6)
give similar results when used to describe basal melt
rates under Ronne Ice Shelf. Also note the similarity between
Eq. (6) and the simple melt model described by
Eq. (1), with the difference being the inclusion of heat
conduction and the parametrization γT = Cd1/2ΓTSU as
well as the plume variables T and S instead of ambient ocean properties.
Hence, the turbulent exchange in this model is directly determined by the
plume velocity that appears as a prognostic variable.
Without giving further details, we mention that the plume model described
above can be evaluated for different ice-shelf geometries (i.e. vertical
cross sections along flow lines) and different vertical temperature and
salinity profiles of the ambient ocean . In this model,
the general physical mechanism governing the development of the plume is the
addition of meltwater at the ice–ocean interface, which increases its
buoyancy. Changes in buoyancy affect plume speed, and that, combined with its
temperature and salinity, determines the subsequent input of meltwater.
Basal melt parametrization along a flow line
Evaluating the aforementioned plume model for different geometries and ocean
properties leads to a wide variety of solutions for the basal melt rates. The
question arises whether there exists an appropriate scaling with external
parameters that combines these results into a universal melt pattern. Here we
will summarize how such a scaling can be found, leading to the basal melt
parametrization of for the quasi-one-dimensional geometries
along flow lines described in the previous section; more details can be found
in Appendix . It is important to note that the following
derivation is based on simple geometries with a constant basal slope and
constant ambient ocean properties, though the resulting parametrization can
easily be applied to more general cases, as shown in
Sect. . Section will discuss the
extension of this parametrization to more realistic two-dimensional geometries.
Dimensionless melt curve M^(X^) used in the basal melt
parametrization. Higher melt rates typically occur close to the grounding
line with a maximum at X^ ≈ 0.2. A transition from melting
(red) to refreezing (blue) may occur further away from the grounding line,
depending on the position of the ice front. Note that the value of X^
depends on the distance to the grounding line, as well as the temperature
difference Ta - Tf and the local slope α (see
Appendix ). In other words, X^ = 0 corresponds to
the grounding line, but the dimensionless position of the ice-shelf front
depends on the length scale and is not necessarily equal to
X^ = 1.
The basal melt parametrization used in this study consists of a general
expression for a dimensionless melt rate M^ as a function of the
dimensionless coordinate X^ measured from the grounding line
(Fig. ). This dimensionless coordinate is essentially the
vertical distance of the ice-shelf base from the grounding line, scaled by a
temperature- and geometry-dependent length scale l:
X^=zb-zgll,l=f(α)⋅Ta-TfSa,zglλ3,
where zgl is the grounding-line depth and f(α) a slope-dependent
factor. Hence, X^ = 0 corresponds to the grounding line and any shelf
point downstream from the grounding line corresponds to a value 0 < X^ < 1
depending on Ta, Sa, zgl and α. This scaling also
implies that the ice-shelf front is not necessarily located at X^ = 1,
but its location is highly dependent on the input variables. Similarly, the
melt rate is scaled as follows:
M^=m˙M,M=M0⋅g(α)⋅Ta-TfSa,zgl2,
with a different slope-dependent factor g(α) and a constant parameter M0.
The dimensionless curve M^(X^) in Fig.
is now defined by polynomial coefficients that were found empirically from
the plume model results (; Appendix ). In
summary, to obtain the basal melt rate m˙ at any point beneath the
ice shelf, one requires the local depth zb, local slope α,
grounding-line depth zgl, and ambient ocean properties Ta and Sa to
calculate X^ and find the corresponding value on the dimensionless
curve M^(X^), which then has to be multiplied by the physical
scale given in Eq. () (see Appendix for details). The
physical quantities and constant parameters required for evaluating the
parametrization are summarized in Table .
Physical quantities and constant parameters serving as input for the
basal melt parametrization.
External quantities
Units
zb
Local depth of ice-shelf base
m
α
Local slope angle
–
zgl
Depth of grounding line
m
Ta
Ambient ocean temperature
∘C
Sa
Ambient ocean salinity
psu
Constant parameters
Values
E0
Entrainment coefficient
3.6 × 10-2
Cd
Drag coefficient
2.5 × 10-3
Cd1/2ΓT
Turbulent heat exchange coefficient
1.1 × 10-3
λ1
Freezing point-salinity coefficient
-5.73 × 10-2 ∘C
λ2
Freezing point offset
8.32 × 10-2 ∘C
λ3
Freezing point-depth coefficient
7.61 × 10-4 K m-1
M0
Melt-rate parameter
10 m yr-1 ∘C-2
Cd1/2ΓTS0
Heat exchange parameter
6.0 × 10-4
γ1
Heat exchange parameter
0.545
γ2
Heat exchange parameter
3.5 × 10-5 m-1
Although the scaling defined by Eqs. () and () is found in a
purely empirical way, it is possible to derive the various factors
analytically, as sketched in Appendix . The empirical
procedure and the physical meaning of the different factors are outlined in
the following. A general solution to the problem is challenging to find as
there are at least four length scales that determine the plume evolution
. The first governing length scale is associated with the
pressure dependence of the freezing point that imposes an external control on
the relationship between plume temperature, plume salinity and the melt rate.
discussed how this length scale, (Ta - Tf)/λ3,
approximately determines the distribution of melting and freezing beneath an
ice shelf. extended the analysis of by making the
transition point between melting and freezing dependent on the ice-shelf
basal slope, resulting in the length scale Eq. () with slope factor f(α).
The second of these four length scales is associated with the ambient
stratification, which determines how far the plume can rise before reaching a
level of neutral buoyancy. discuss the plume behaviour and
resulting melt rates when this length scale dominates. Critically, with the
pressure dependence of the freezing point assumed to be negligible, as
required in the analysis of , no freezing can occur. The third
length scale can be formulated by comparing the input of buoyancy from
freshwater outflow at the grounding line with the input of buoyancy by
melting at the ice–ocean interface . This length scale
indicates the size of the zone next to the grounding line where the impact of
ice-shelf melting on plume buoyancy can be ignored and conventional plume
theory applied, and it is generally small compared
with typical ice-shelf dimensions. The final length scale is that at which
the Coriolis force takes over from friction as the primary force balancing
the plume buoyancy in the momentum budget. discussed these
length scales in the context of which would take over as the dominant
control on plume behaviour beyond the initial zone near the grounding line
where the initial source of buoyancy dominates and showed the length scale
associated with the pressure dependence of the freezing point,
(Ta - Tf)/λ3, to be most important for
typical ice-shelf conditions.
Hence, we obtain the second factor of the length scale l in Eq. ()
used in the parametrization. However, this length scale contains two more
ingredients. First, as discussed by , the entrainment rate in
the mass conservation Eq. () explicitly depends on the slope
α, whereas the melt rate is only affected indirectly, so there is a
geometrical factor that scales the elevation of the plume temperature above
the local freezing point:
E0sinαCd1/2ΓTS+E0sinα.
This factor gives rise to the slope dependence f(α) in l, which is
essentially an empirically derived scaling of the transition point between
melting and freezing (Appendix ). The second ingredient is
related to the coefficient ΓTS, which appears in f(α)
through the simplified interface conditions (Eq. 6).
retained the more complex melt formulation (Eq. 5) in the plume model
while seeking empirical scalings based on an effective ΓTS. As
discussed by , the factor relating ΓT
and ΓTS is itself a function of the plume temperature, so
expressed the effective ΓTS as an empirical function of ΓT,
Ta - Tf and Eq. () including a constant initial value
ΓTS0 (see Appendix ). When distance along the
plume path is scaled with this slightly more complex factor (see
Eq. ), the melt rates produced by the plume model conform to
a universal form – first rising to a peak at X^ ≈ 0.2 before
falling and transitioning to freezing at X^ ≈ 0.56 (Fig. ).
With the distance along the plume path appropriately scaled, all that remains
is to scale the amplitude of the melt-rate curves produced by the plume
model and find the melt-rate scale M in Eq. (). As in
the appropriate physical scales are (1) the temperature of the ambient ocean
water relative to the freezing point; (2) the factor in
Eq. () scaling the temperature elevation of the plume above
freezing; (3) a factor that scales the plume speed, given by the ratio of
plume buoyancy to frictional drag:
sinαCd+E0sinαCd1/2ΓTSCd1/2ΓTS+E0sinα.
The second term in parenthesis is the factor that scales the plume
temperature relative to the ambient temperature and thus controls plume
buoyancy. It replaces the initial buoyancy flux at the grounding line used in
the scaling of . The final expression includes factors and
powers that are derived empirically (though some theoretical arguments can
be applied, see Appendix ), giving rise to the form of M
with slope factor g(α) in Eq. (). In summary, the result of this
scaling procedure is an approximately universal melt-rate curve, which can
then be represented by a single polynomial expression that is accurate to
about 20 % for melt rates ranging over many orders of magnitude .
Definition of the ice mask. The ice-shelf criterion is that for
uniform ice with density ρi floating on ocean water with
density ρw. The minimum ice thickness used here is
Hi,min = 2 m.
Mask
Type
Criterion
value
0
ice sheet
(ρi/ρw)Hi > -Hbed
1
ice shelf
(ρi/ρw)Hi ≤ -Hbed
2
ocean (no ice)
Hi ≤ Hi,min
Basal melt parametrization in 2-D: effective plume path
As explained in the previous section, an important feature of the basal melt
parametrization is its dependence on non-local quantities, in particular the
grounding-line depth zgl from which the plume originated. Therefore, in
order to apply the parametrization to realistic geometries, one needs to know
for each ice-shelf point the corresponding grounding-line point(s) serving as
the origin of the plume(s) reaching that particular shelf point. For the
quasi-one-dimensional settings considered so far, this is not an issue, since
the plume can only travel in one direction. However, for general ice-shelf
cavities, an arbitrary shelf point can be reached by plumes from multiple
directions, corresponding not only to different values for zgl, but also
to different slope angles α. This means that the plume parametrization
cannot be directly applied to such geometries. An algorithm is needed to
determine effective values for zgl and α. The development of
this algorithm is the main focus of the current work and discussed below.
Schematic of the algorithm for finding the average grounding-line
depth and associated slope angle used by the basal melt parametrization.
(a) Top view of an ice shelf on a horizontal grid. The algorithm
searches in 16 directions on the grid from the shelf point (i, j).
Possible grounded points found in this way are marked by ×.
(b) Vertical slice along the nth direction (e.g. the red dotted
line in a). If the grounded point is higher than the previous shelf
point, the grounding-line depth zn is found by interpolation along the
bed (zn = 12(Hbed,1 + Hbed,2)).
(c) Interpolation along the ice base if the grounded point in the
nth direction is deeper than the previous shelf point
(zn = 12(zb,1 + zb,2)).
As a starting point, we consider the usual topographic data in terms of
two-dimensional fields for the ice thickness Hi, bedrock/seabed
elevation Hbed and elevation of the upper ice surface Hs used
by ice-dynamical models. The following algorithm is valid for any topographic
data on a rectangular grid with any resolution Δx × Δy.
First of all, the topographic data are used to define an ice mask based on
the criterion for floating uniform ice, as shown in Table .
Furthermore, the depth of the ice base is determined to be
zb=Hs-Hi.
In order to apply the basal melt parametrization to these two-dimensional
data, effective values for zgl and α must be determined for every
ice-shelf point (i, j) with basal depth zb(i, j), where the indices
i and j denote the position on the grid. This is done by first searching
for “valid” grounding-line points in 16 directions on the grid, starting
from any shelf point (i, j), as depicted in
Fig. a. Note that we can calculate a local basal slope
sn(i,j) at the point (i,j) in the nth direction as follows:
sn(i,j)=zb(i,j)-zbi+in,j+jninΔx2+jnΔy2,
where (in, jn) denotes a direction vector on the grid,
i.e. (in, jn) = (1, 0) denotes right, (in, jn) = (0, 1) denotes up, etc., and
Δx and Δy denote the horizontal grid size in the x and
y direction, respectively. To determine whether a grounding-line point
found in 1 of the 16 directions is valid for the calculation of the basal
melt, the following two criteria are applied:
Assuming that a buoyant meltwater plume can only reach the point (i, j) from
the nth direction if the basal slope in that direction is positive, the
algorithm only searches in directions for which sn(i, j) > 0.
If the first criterion is met for the nth direction, the algorithm
searches in this direction for the nearest ice-sheet point. More
precisely, the associated direction vector (in, jn) is added to the grid
indices and the mask value in the resulting point is checked. This process is
repeated until either an ice-sheet point, an ocean point or the domain
boundary is encountered. An ice-sheet point found in this way is only
considered to be a valid grounding-line point if it lies deeper than the
original ice-shelf point at (i, j), assuming again that a buoyant meltwater
plume from the grounding line can only go up. The second criterion then
becomes zn(i, j) < zb(i, j), where zn(i, j) is the grounding-line depth
in the nth direction.
Note, however, that in determining the second criterion the depth
difference between the encountered sheet point and the adjacent shelf point
can be considerable, especially for coarser resolutions. In such cases, the
algorithm tries to obtain a better estimate of the true grounding-line depth
in this direction, zn(i, j), by interpolating along either the bed or the
ice base, as shown in Fig. b and c. The two cases shown in
these figures account for either a positive or a negative basal slope beyond
the grounding line. One should note that this additional step assumes the
grounding line to be located halfway between the sheet and shelf points,
which could be improved by more sophisticated interpolation techniques.
Following the above procedure yields for each ice-shelf point (i, j) a set
of grounding-line depths zn and local slopes sn in the directions that
are valid according to the aforementioned two criteria. Keep in mind that not all
directions may yield a (valid) grounding-line point – in particular those
towards the open ocean. Now, in order to determine the effective
grounding-line depth zgl(i, j) and effective slope angle
α(i, j) necessary for calculating the basal melt in the shelf
point (i, j), we simply take the average of the values found for zn
and sn:
zgl(i,j)=1Nij∑validnzn(i,j),tan[α(i,j)]=1Nij∑validnsn(i,j),
where Nij denotes the number of valid directions found for the shelf
point (i, j). On the other hand, if no valid values for zn and sn are
found for a particular shelf point, we take zgl = zb and α = 0,
leading to zero basal melt in that point (see Appendix ).
In summary, the method described above yields two-dimensional fields for the
effective grounding-line depth zgl and effective slope tan(α),
given topographic data in terms of Hi, Hs, and Hbed and a
suitable ice mask, such as the one defined in Table . These
fields, in turn, serve as input for the basal melt parametrization described
in the previous section, together with appropriate data for the ocean
temperature Ta and salinity Sa (discussed in Sect. ).
We thus obtain a complete method for calculating the basal melt for all
Antarctic ice shelves, given the topography and ocean properties, which can
also be used in conjunction with ice-dynamical models. In the following,
however, we use the Bedmap2 dataset to define the
present-day topography of Antarctica and disregard the ice dynamics. More
specifically, the original Bedmap2 data are remapped to a rectangular grid
with grid size Δx = Δy = 20 km, using the mapping package
OBLIMAP 2.0 . The resulting topographic data can be used as
input for the algorithm described here, leading to the fields for zgl
and tan(α) shown in Fig. , which are used for the
basal melt calculations discussed in Sect. . Note that
the mask in Fig. a does not exactly match the Bedmap2
mask because a constant ρi was used in formulation of
Table as is common in many ice-sheet models. This might cause
discrepancies in the position of the grounding line, which, however, are
likely compensated for by the rather coarse resolution. In
Fig. b one can see that the lowest values of zgl are
obtained towards the inland regions of Filchner–Ronne Ice Shelf (FRIS) and Amery Ice
Shelf. The values for the local slope are typically high near the grounding
line and in some places also near the ice front, as shown in Fig. c.
One should note that, although we attempt to directly translate the concept
of a quasi-1-D plume to a multitude of plumes in two dimensions, there are
important physical effects not taken into account by this approach. Most
importantly, a realistic two-dimensional plume has an additional degree of
freedom because it also develops in the cross-flow direction, causing the
width to be a dynamic variable in addition to the thickness D. This can
have significant consequences for the mass budget currently described by
Eq. (). explored the possibility of adding a variable
plume width to the original plume model and showed that
such a 2-D formulation improves the prediction of melt rates for a realistic
ice-shelf geometry compared to the 1-D model. Although this appears to be an
important extension of the plume model that should be taken into account, the
aim of the current work is to explore the capabilities of the original 1-D
plume parametrization in predicting melt rates around Antarctica. The current
approach is meant to be a simple method to parametrize the net circulation
within an ice-shelf cavity as the average effect of multiple plumes, in order
to be applied around the entire ice sheet. Further extensions for obtaining a
2-D plume model are beyond the scope of this work.
Results
Here we present various results obtained by evaluating the basal melt
parametrization described in the previous section. First, we investigate the
main characteristics of the original 1-D parametrization of
Sect. by evaluating it along flow lines of the
Filchner–Ronne and Ross ice shelves. In Sect. and ,
we turn to the full 2-D geometry of Antarctica using
the algorithm described in Sect. – first by constructing an
appropriate effective ocean temperature field from observational data.
Comparison of basal melt parametrizations along flow lines
Topographic data along flow lines for both Filchner–Ronne Ice Shelf
and Ross Ice Shelf are taken from and ,
respectively. These data can be used to determine the quantities zb,
α and zgl necessary for calculating the basal melt with the
parametrization of Sect. . Furthermore, we define a uniform
ambient ocean temperature Ta = -1.9 ∘C + ΔT, where
ΔT is varied between runs, and a constant ambient ocean salinity
Sa = 34.65 psu.
The results of these calculations are shown in
Fig. and compared with those of the full plume model described
in Sect. . Moreover, we compare with two simple basal melt
parametrizations based on Eq. (1), namely the linear (i.e. in
Ta - Tf) parametrization by with constant γT and
the quadratic parametrization by with
γT = κT|Ta - Tf|. Apart from the values listed in
Table , additional model parameters used for these
calculations are given in Table .
Additional model parameters used for evaluating the plume model and
the simple parametrizations described in Sect. . BG2003
refers to and DCP2016 refers to
.
Constant parameters
Values
L
Latent heat of fusion for ice
3.35 × 105 J kg-1
cw
Specific heat capacity of water
3.974 × 103 J kg-1 K-1
ci
Specific heat capacity of ice
2.009 × 103 J kg-1 K-1
βS
Haline contraction coefficient
7.86 × 10-4
βT
Thermal expansion coefficient
3.87 × 10-5 K-1
g
Gravitational acceleration
9.81 m s-2
ρi
Density of ice
9.1 × 102 kg m-3
ρw
Density of ocean water
1.028 × 103 kg m-3
γT
Turbulent exchange velocity (BG2003)
5.0 × 10-7 m s-1
κT
Turbulent exchange coefficient (DCP2016)
5.0 × 10-7 m s-1 K-1
Effective plume paths under the Antarctic ice shelves as calculated
by the algorithm of Sect. using the Bedmap2 topographic data
remapped on a 20 km by 20 km grid. (a) Ice mask according to
Table . (b) The effective grounding-line
depth zgl. (c) The effective slope tan(α).
(d) The difference between local ice-base depth and associated
grounding-line depth,
zb - zgl.
Comparison of the plume model (Sect. ) with the 1-D
basal melt parametrization (Sect. ), as well as the
parametrizations of (BG2003) and (DCP2016),
for flow lines along Filchner–Ronne Ice Shelf (a, c, e, g) and Ross
Ice Shelf (b, d, f, h), both with uniform ocean temperature
Ta = -1.9 ∘C + ΔT and constant
salinity Sa = 34.65 psu. (a, b) Geometry of the
ice-shelf base. (c, d) Melt pattern for
ΔT = 0 ∘C. (e, f) Melt pattern for
ΔT = 0.8 ∘C. (g, h) Melt-rate average along
the flow line as a function of ΔT. Note that the black curve is
nearly identical to the green curve and might appear below it. Also note the
difference in vertical scale between the left and right columns. The
flow-line locations are indicated in
Fig. .
Figure shows that both the current parametrization and the
original plume model yield approximately the same melt-rate patterns as a
function of the horizontal distance from the grounding line. These patterns
roughly correspond to the dimensionless melt curve in
Fig. , i.e. maximum melt near the grounding line and
possibly refreezing further away along the flow line. This is most apparent
in Fig. c, which shows a transition from melting to freezing,
since the relatively deep draught of FRIS allows higher values of the
dimensionless coordinate X^. On the other hand, Fig. d
does not show refreezing because the draught of Ross Ice Shelf is much
shallower. Increasing the ocean temperature (through ΔT) can
significantly enhance basal melt and remove the area of refreezing, as shown
in Fig. e and f. In these cases, additional melt
peaks occur in regions of high basal slope. Moreover, although the general
agreement is good, the discrepancies between the current parametrization and
the plume model are largest when the basal slope changes rapidly, because the
parametrization responds immediately to the change while the full model has
an inherent lag as the plume adjusts to the new conditions. On the whole, we
see that the melt patterns given by the plume parametrization can be quite
complex, while the two simple parametrizations give nearly constant curves
(i.e. independent of the position with respect to the grounding line).
It is interesting to investigate the temperature sensitivity of the four
models in terms of the horizontally averaged melt rate as a function of ΔT,
as shown in Fig. g and h. In the
case of FRIS, the plume model and parametrization are much more sensitive to
the ocean temperature than the two simpler models. However, the average melt
rates for Ross Ice Shelf are rather similar for all four models and all
values of ΔT. Hence, the difference in the temperature sensitivity
depends significantly on the ice-shelf geometry, where the plume
parametrization appears to have a larger potential for capturing diverse melt
values than the simpler models. Note that in both cases the temperature
dependence of the plume parametrization is slightly non-linear, similar to the
parametrization, while the parametrization
has a linear temperature dependence. Following the discussion of
, the temperature dependence of the plume parametrization
should therefore be more realistic than the one of . However,
the quadratic parametrization of tends to significantly
underestimate the melt rates as well, despite its non-linearity. It appears
that the geometry dependence of the plume parametrization is an important
factor for the temperature sensitivity of the calculated basal melt rates. In
Sect. we show that these geometrical effects are
indeed crucial for obtaining realistic melt rates with the 2-D
parametrization, but first we discuss the matter of determining a suitable
input field for the ocean temperature.
Effective ocean temperature
The previous section dealt with the 1-D basal melt parametrization along a
flow line using a uniform ambient ocean temperature for the entire ice-shelf
cavity. While a uniform temperature might appear a reasonable first
approximation for a single ice shelf, it is far from realistic to apply a
single ocean temperature for multiple ice shelves around the entire Antarctic
continent. Therefore, in order to apply the parametrization to the 2-D
geometry defined by Fig. , a suitable 2-D field for the
ocean temperature Ta is required. In principle, the same is true for the
salinity Sa, but we will assume that the horizontal variations in ocean
salinity around Antarctica are so small that the pressure freezing point Tf
is only affected by variations in depth. In the following, we will
therefore take a uniform salinity Sa = 34.6 psu. One should realize
that vertical variations in Sa, which are not accounted for in the current
parametrization, would be important in reality, as discussed in Sect. .
Two problems arise when considering a 2-D ocean temperature field for forcing
the parametrization. First of all, such a field should ideally be based on
observational data, but ocean temperature measurements in the Antarctic
ice-shelf cavities are sparse. A more feasible approach would be to compute
an interpolated field based on ocean temperature data in the surrounding
ocean, which inevitably contains artefacts resulting from the non-uniform
and predominantly summertime sampling. Secondly, even if a complete dataset
of ocean temperatures were available, it would not be immediately clear which
temperatures (i.e. at which depth) are characteristic of the ocean water
reaching the grounding lines (e.g. ). In principle,
detailed knowledge of the bottom topography and the ocean circulation would
be required for this, which goes beyond the scope of the current modelling approach.
In view of these issues, we construct an effective ocean temperature
field with which the current plume parametrization yields melt rates that are
as close as possible to present-day observations, averaged over entire ice
shelves. In other words, this can be regarded as the inverse problem of
computing the unknown ocean temperatures by assuming that the model output
for the melt rates matches the (averaged) observations. For this purpose, we
use the results of , who calculated the area-averaged melt
rates for each Antarctic ice shelf, based on a combination of observational
data and regional climate model output for the different terms in the local
ice-shelf mass balance. Other datasets for recent Antarctic basal melt rates
exist (e.g. ), as well as more recent data for ice-shelf
thinning from which the basal melt rates can be
calculated when combined with the other terms in the mass balance
(e.g. velocity and surface melt rates). These alternative datasets for the
(area-averaged) basal melt rates are expected to be at least of the same
order of magnitude, which we deem sufficient for the purpose of the current
study. Since it is impossible to resolve each individual ice shelf from the
dataset with the currently used 20 km resolution
(Fig. ), we consider a set of 13 ice-shelf groups and
determine the area-averaged basal melt for each group from the data of
. The definition of these groups and the calculated
average melt rates are shown in Fig. . Note that the
shelves have been grouped based on their geographical location but also for
more practical reasons such as the possibility of distinguishing their
boundaries on the 20 km grid.
The 13 groups of ice shelves used for constructing the effective
ocean temperature field. Average melt rates and error estimates (1 SD – 1
standard deviation) for each group are calculated from the data of
for individual ice shelves. Green lines indicate the
approximate positions of the flow lines used in Fig. .
As a starting point for constructing the effective ocean temperature, we
consider the observational data of the World Ocean Atlas 2013 (WOA13,
), which contains a global dataset of (annual mean)
ocean temperatures within a range of depths (0–5500 m). Restricting
ourselves to the temperature data for latitudes south of 60∘ S, we
average the ocean temperatures over depth intervals [z1, z2], where
z1 is the level of the bed (i.e. the deepest level for which data are available)
with the additional constraint z1 ≥ -1000 m and z2 = min{0, z1 + 400 m}.
This results in a relatively smooth 2-D temperature field
containing an inherent dependence on the bottom topography, which can be
considered a first estimate for the ocean water flowing into the ice-shelf
cavities. The depth-averaged temperature field is now remapped on the same
20 km grid as the topography data (see Sect. and
Fig. ) and interpolated using natural-neighbour
interpolation (i.e. a weighted version of nearest-neighbour interpolation,
giving smoother results) to obtain data in the entire domain of interest.
The resulting temperature field, called T0, is shown in
Fig. a. One should note that both the depth-averaging and
the interpolation procedures introduce biases in the resulting field. In
particular, the rather simple interpolation technique also interpolates ocean
temperatures between ice-shelf cavities separated by the continent or
grounded ice, which is not realistic as it propagates temperatures into
cavities that the corresponding ocean water cannot reach. Using the
natural-neighbour interpolation method appears to limit these effects.
However, the details of the resulting field T0 are somewhat arbitrary as
it needs to be adjusted in order to obtain melt rates that agree with the
data of .
(a) Depth-averaged and interpolated ocean temperature,
T0, calculated from annual mean WOA13 data. (b) Effective ocean
temperature Teff = max{T0 + ΔT, -1.9}
constructed from T0 as described in Sect. . The circles
indicate the positions of the sample points in which the values
of ΔT are imposed. The colour of each circle corresponds to the
imposed value of ΔT (same colour scale), ranging from -1.4 to
0.8 ∘C. The full ΔT field is obtained by linearly
interpolating these values.
The aim is now to modify this depth-averaged, interpolated temperature
field T0 in such a way that the basal melt parametrization yields melt rates
close to those shown in Fig. for each ice-shelf group.
As explained earlier, this modification is necessary for eliminating biases
in T0 caused by the sparse observations and numerical interpolation and
also because the flow dynamics of the ocean are not resolved. The field T0
is now modified by adding a 2-D field of temperature differences (ΔT),
which, in turn, is the result of linearly interpolating
individual values of ΔT in 29 carefully chosen sample points, with
ΔT = 0 on the domain boundary. The sample points and values
of ΔT have been determined by trial and error and are certainly not a
unique nor optimal configuration. The points are mainly located in regions
that are most affected by interpolation between strictly separated cavities
(e.g. grounding line of FRIS) or extrapolation of warm open-ocean
temperatures into cavities (e.g. Dronning Maud Land, shelf groups 2 and 3 in
Fig. ). The resulting effective temperature field,
Teff = T0 + ΔT, is shown in Fig. b, which
also indicates the positions of the aforementioned sample points along with
the used values of ΔT in these points. Note that, for technical
reasons explained in Appendix , we have applied a lower
limit to the effective temperature equal to the pressure freezing point at
surface level. With the current choice Sa = 34.6 psu, this implies
Teff ≥ -1.9 ∘C. Comparing Fig. a
and b, we see that the main effect of ΔT is a decrease in the ocean
temperature over most of the continental shelf and most ice-shelf cavities
(in particular for Ross and Amery ice shelves) and a slight increase in the
ocean temperature in West Antarctica and some regions in East Antarctica
(e.g. shelf group 6 in Fig. ). Again, note that the
details in the procedure for calculating T0 and ΔT are somewhat
arbitrary, since increasing one term would require decreasing the other term
in order to obtain similar values for Teff with similar basal melt rates.
Figure shows the basal melt rates computed by the
parametrization using the effective temperature Teff of
Fig. b as forcing. An area-averaged value is obtained for each
of the 13 ice-shelf groups in Fig. and compared with
the observational values from the data. By construction,
the modelled basal melt rates correspond closely to the observational values
and fall within the error estimates. A notable exception is the value for
Filchner–Ronne Ice Shelf, which is 0.32 ± 0.08 m yr-1
according to the observations, whereas the parametrization gives a value just
above 0.5 m yr-1. This discrepancy is caused by the lower bound of
-1.9 ∘C imposed on the effective temperature, whereas in reality
the temperatures can reach values below -2.0 ∘C (e.g. ).
As we can see in Fig. b, the ocean water
below FRIS is almost entirely at this minimum temperature, making it
impossible to further improve the basal melt rate without using unfeasibly
low values for Teff. This rather technical constraint might be
relaxed in various ways, as briefly discussed in Appendix ,
possibly improving the melt rates in very cold cavities.
Area-averaged basal melt rates for each ice-shelf group in
Fig. obtained with the plume parametrization and the
effective temperature field of Fig. b. The modelled melt rates
are plotted against the averaged observational values given in
Fig. . For four important shelf groups, the data points
are explicitly labelled along with the corresponding group number in
Fig. . The horizontal error bar is 1 standard
deviation uncertainty in the observations.
Nevertheless, the plume parametrization in conjunction with the constructed
effective temperature field appears to yield realistic present-day melt rates
for all shelf groups. By construction, the effective temperature shown in
Fig. b contains an inherent dependence on the bottom
topography, with typically lower temperatures above the continental shelves
(and thus in the ice-shelf cavities), while still retaining the spatial
variation in temperature of the surrounding deep ocean (e.g. higher
temperatures for West Antarctica, leading to higher melt rates for ice-shelf
groups 11 and 12 as defined in Fig. ).
Basal melt rates in metres per year with the Bedmap2 topographic data
and the effective temperature field of Fig. b as obtained from
(a) the plume parametrization with additional input parameters from
Fig. and (b) the quadratic parametrization of
.
Comparison of 2-D melt-rate patterns
The effective grounding-line depth and effective slope in
Fig. , the effective ocean temperature in
Fig. b and the assumption Sa = 34.6 psu constitute the full
set of input parameters necessary for evaluating the plume parametrization on
the entire 2-D geometry. The resulting 2-D field of basal melt rates under
all Antarctic ice shelves is shown in Fig. a (note that
these are the same data used for the area-averaged melt rates in
Fig. but now plotted as a spatial field rather than
averaged values over the ice shelves). A general pattern that can be
observed, especially on the bigger ice shelves, consists of regions of higher
melt close to the grounding line and lower melt or patches of refreezing
closer to the ice front, with the latter being most apparent at the ice fronts of
shelf groups 1, 2 and 9. This pattern is a consequence of the underlying
plume model, as shown in Sect. for data along a flow
line. Moreover, the highest melt rates occur in West Antarctica (shelf
groups 11 and 12) and some specific shelves in East Antarctica (shelf groups 6
and 7), where the constructed effective temperature is significantly
higher than elsewhere. The general melt patterns within individual cavities
appear to be in line with observations, e.g. . However, one
should note that the melt pattern shows a greater spatial
variability, with more patches of (stronger) refreezing occurring between
patches of melting (Fig. a). Especially beneath FRIS and
Ross Ice Shelf, the melt pattern appears quite complex and local deviations
from the general pattern can be considerable (Fig. b). These
discrepancies in the current parametrization might have different reasons,
such as the coarse resolution or the fact that we disregard the details of
the ocean circulation within the ice-shelf cavities, as well as effects due
to the Coriolis force and both seasonal and vertical variability in the
temperature and salinity fields.
As Fig. a, but with a logarithmic colour scale
(negative and zero values shown white) and zoomed in on
(a) Filcher–Ronne Ice Shelf (group 1), (b) West Antarctica
including Pine Island and Thwaites (group 11) and (c) Ross Ice Shelf
(group 9).
Basal melt rates in metres per year extracted from the
observational dataset (courtesy of Dr. Jeremie Mouginot):
(a) raw data plotted together with the currently used mask;
(b) difference between the plume parametrization
(Fig. a) and the observations interpolated on the 20 km
grid.
Furthermore, Fig. shows the melt-rate patterns of the
plume parametrization zoomed in on three regions, giving more insight into the
orders of magnitude of the highest melt rates. The high near-grounding-line
melt rates for FRIS have values between 1 and 10 m yr-1, while those for
Ross Ice Shelf appear 1 order of magnitude smaller. On the other hand, the
West Antarctic melt rates shown in Fig. b have values around
10 m yr-1 or more due to the higher ocean temperatures here. It should
be noted, however, that the latter values are still lower than those observed
in the data, where local melt rates close the grounding
line can reach 100 m yr-1, while the average melt rates over the full
area of Pine Island and Thwaites are 16.2 and 17.7 m yr-1, respectively.
For comparison, we also evaluate the quadratic parametrization of
, described in Sect. , using the same
geometric data and the effective temperature field of Fig. b as
input. The resulting basal melt-rate pattern is shown in
Fig. b. Comparing this figure to Fig. a
shows that the quadratic parametrization yields significantly lower melt
rates than the plume parametrization, at least with the current effective
temperature as input. The only visible patches of basal melt are located in
the aforementioned regions where the ocean temperature is high, as well as
near the grounding line of Filchner–Ronne Ice Shelf. Therefore, if the
effective temperature in Fig. b is indeed characteristic of the
true temperatures in the ice-shelf cavities, the quadratic parametrization
would require significant tuning in order to obtain a similar agreement with
observed melt rates as currently found with the plume parametrization. For
completeness, we mention that the linear parametrization of
yields even lower melt rates due to its low temperature sensitivity, as
discussed in Sect. .
To further clarify the differences between the two parametrizations in
Fig. , we have repeated the steps outlined in
Sect. and constructed a second effective temperature field
based on the quadratic parametrization by instead of the
plume parametrization. The resulting temperature field is shown in
Fig. a. Note that the difference between this field and the
one in Fig. b only lies in the values chosen for ΔT and
not in the underlying interpolated observations (T0). For simplicity, the
ΔT values have been imposed on the same sample points as used for
Fig. b. Comparing the two effective temperature fields in
Figs. b and a shows that much higher ocean
temperatures are required for the quadratic parametrization to give realistic
area-averaged melt rates. The ΔT values imposed on the sample points
indicated in Fig. a range from -0.5 to 5.4 ∘C,
while those used for Fig. b range from -1.4 to 0.8 ∘C.
Furthermore, we can calculate the root
mean square values of Teff - T0 over the entire domain
(disregarding the continental points), yielding 0.3 ∘C for
Fig. b and 1.1 ∘C for Fig. a.
Hence, the effective temperature in Fig. b lies closer to the
underlying observational data T0 than the field in Fig. a.
(a) Effective temperature field constructed in a similar
way as Fig. b, but with different values for ΔT
(indicated by the circles and ranging from -0.5 to 5.4 ∘C), chosen
in order to match the melt rates of the quadratic parametrization of
with the data of . (b) Basal melt
rates obtained with the quadratic parametrization of using
the Bedmap2 topographic data and the effective temperature in (a) as
input.
The basal melt rates resulting from the quadratic parametrization and the new
effective temperature field are shown in Fig. b. Clearly,
the higher ocean temperatures cause significantly higher melt rates than
those shown in Fig. b. However, compared with the plume
parametrization in Fig. a, the spatial distribution of
these melt rates is more uniform, showing less prominent melt peaks near
grounding lines and no patches of refreezing. It appears that the quadratic
temperature dependence together with the (slight) depth dependence through
the pressure freezing point Tf (Eq. ) is not sufficient
for obtaining realistic melt rates without significantly increasing the input
ocean temperature, which can be considered equivalent to using different
tuning factors for different ice shelves. On the other hand, the plume
parametrization, containing an additional geometry dependence through the
grounding-line depth and local slope, appears to yield the required melt
rates rather naturally with ocean temperatures constructed in a plausible
way, and it results in a more realistic spatial pattern with highest basal
melt rates near the grounding line as well as areas of refreezing.
Discussion
The plume parametrization in combination with the 2-D algorithm of
Sect. and the effective temperature field of
Sect. is able to capture a more complex spatial pattern of
basal melt rates and a high temperature sensitivity, which is an important
step forward compared to the simpler models based only on
Eq. (1). However, the plume parametrization also relies on
several rather strong assumptions, which we discuss below. First of all, both
the original plume model and the parametrization have a quasi-1-D
formulation, assuming homogeneity in the span-wise direction. Even though we
attempt to translate this formulation to two dimensions with the algorithm in
Sect. , there are undoubtedly errors associated with the
underlying 1-D assumptions. As already discussed in Sect. ,
an important 2-D effect is the additional degree of freedom associated with
the widening of the plume, which influences the plume dynamics and the melt
rates through the mass budget equation .
Furthermore, the current algorithm for finding the plume paths in 2-D is not
unique and more realistic and efficient methods might be possible, e.g. by
extrapolating the plume outward from the grounding line instead of searching
for surrounding grounding-line points from each shelf point. Also, the
current algorithm was developed for the relatively coarse resolution of
20 km × 20 km, which is suitable for use in an ice-sheet model, and takes into
account only the local slope and overall grounding-line depth, whereas
higher-resolution runs might benefit from a different and more precise method. For
example, the current method inevitably includes unrealistic plume paths along
points where the basal slope reverses, which might give problems at higher
resolutions. On the other hand, a higher resolution would also entail a more
rapid variation of the basal slope, potentially causing high melt peaks
(Sect. ) that would be smoother in the original plume
model. This would introduce the need for a smoothing algorithm for higher
resolutions. All in all, the current formulation should be considered as a
relatively simple parametrization of the net circulation within an ice-shelf
cavity, providing non-local features to the basal melt calculation that are
not present in the simpler models. Further work is needed to determine
whether the realism of the current formulation can be improved.
Another very important feature that has been neglected in the derivation is
the vertical variation in the temperature and salinity fields. In reality,
stratification and the existence of different water masses have a crucial
effect on plume buoyancy, e.g. by causing the plume to detach from the
ice-shelf base at levels of neutral buoyancy. In such cases, new plumes are
formed at the detachment depth and the relation between the plume and the
grounding-line depth breaks down, creating multiple modes in the sub-shelf
circulation and associated basal melt . As explained in
Sect. , the current formulation is based on the assumption that
the freezing-point length scale Eq. () is dominant with respect to the length
scale associated with stratification, as well as those associated with
rotation and the initial meltwater flux at the grounding line. This
assumption indeed works well in conjunction with constant values for Ta
and Sa, describing a net circulation for which the buoyancy is
parametrized in terms of Ta - Tf, as shown more precisely in
Appendix . In this framework, the values of Ta and Sa
determine the overall magnitude of plume buoyancy, while the variation along
the plume path is described by the depth-dependence of the freezing
point Tf. This is also the reason why the small horizontal variations in Sa
have only a small effect on the overall buoyancy and can be neglected, as was
done in Sect. . However, for obtaining a fully realistic
melt-rate pattern it will be important to also include the effects of
vertical and seasonal variations in Ta and Sa, e.g. in order to capture
seasonal intrusion of warmer surface waters (mode 3 melting; ).
An important uncertainty in the current study is the construction of the
effective temperature field (Sect. ). In principle, this is
done due to the lack of detailed ocean temperature observations beneath the
ice shelves. One should note, however, that in attempting to eliminate the
biases caused by the sparse data we are also correcting for errors in the
parametrization itself, since the construction is done by constraining the
modelled melt rates. In this respect, the effective temperature field
(or, more precisely, ΔT) should be regarded as part of the modelling
framework. It would be crucial for the complete validation of the model to
perform additional temperature sensitivity studies to see how the plume
parametrization might respond to an evolving ocean. Ideally, this is done in
the context of a coupled ice–ocean model. On the large scales currently
considered, lack of detail within the ice-shelf cavities will likely remain a
problem also when using an ocean general circulation model. Since the current
formulation is based on constant ocean properties within individual cavities,
a method to determine Teff from an ocean model could be
extrapolating the model temperature within a characteristic depth-range at
the ice front and using a (possibly different) ΔT to constrain the
output melt rate, similar to the construction presented here.
On a more technical note, the current construction of Teff was
not based on a sophisticated optimization algorithm, but it is merely a
simple method to determine an essentially spatially variable field directly
from the observations. An alternative method, which might be more consistent
with the derivation of the parametrization, would be to introduce separate
values for the ocean temperature for each individual cavity, as the ambient
temperature in the current context represents the net inflow into the cavity
and not the temperature of meltwater that is produced or mixed locally. On
the other hand, the current method is more generic in the sense that it
removes the need for defining individual cavities in the model once ΔT
(i.e. the constraint on the melt rates) has been determined. It should be
noted that the current method using only 29 sample points might become
problematic in dynamical simulations that include grounding-line retreat.
Hence, in such a context a more sophisticated method might be necessary.
Furthermore, it is not yet clear if a fixed ΔT is a realistic
assumption for an evolving ocean, and introducing the aforementioned
additional variations of Ta and Sa might require different
considerations altogether.
Finally, it is interesting to note the existence of alternative methods for
describing the net circulation within the ice-shelf cavities. A recent
example is a box model that simulates the upward flow under the ice shelf in
a similar quasi-1-D context by describing the fluxes of heat and salt between
a limited number of predefined boxes . This method has
recently been extended to two dimensions and coupled to an ocean model
, yielding Antarctic basal melt patterns similar to the
ones given by the plume parametrization. Both methods are similar in the
sense that they essentially describe the same type of physical process while
not accounting for features such as stratification and 2-D effects, as
discussed above. One could argue that a systematically derived approximation
to the governing equations is preferred over a simple box model. On the other
hand, a box model might be easier to implement and produce similar results in
a more efficient way. A more detailed comparison of these two methods is
beyond the scope of this work.
Conclusions
In this study, we have presented the application of a basal melt
parametrization, based on the dynamics of buoyant meltwater plumes, to all
ice shelves in Antarctica. The physical basis of this parametrization is the
plume model of , which describes the fluxes of mass, momentum,
heat and salinity within a meltwater plume travelling up from the grounding
line along the ice-shelf base. Details of the proposed parametrization have
been discussed in earlier works for idealized
one-dimensional geometries along an ice-shelf flow line. In particular, the
basal melt rate given by the plume model follows a rather universal scaling
law depending on the ice-shelf geometry (basal depth zb, local slope
angle α and grounding-line depth zgl) as well as the ambient ocean
temperature Ta and the pressure freezing point Tf.
Here, the plume parametrization has been tested for two realistic ice-shelf
geometries along a flow line and, for the first time, applied to a completely
two-dimensional geometry covering all the Antarctic ice shelves. The
one-dimensional tests along flow lines of the Filchner–Ronne and Ross ice shelves
(Sect. ) reveal the typical characteristics of the
parametrization, namely higher melt rates near the grounding line and in
regions of high basal slope. Patches of refreezing can occur further away
from the grounding line. Moreover, the plume parametrization exhibits a
non-linear dependence on the ocean temperature, and the increase in melting
resulting from higher ocean temperature is dependent on the ice-shelf
geometry. In contrast, simpler parametrizations based solely on the local
balance of heat at the ice–ocean interface are not able to capture the
complex melt pattern nor the temperature sensitivity.
Applying the essentially one-dimensional plume parametrization to a
two-dimensional geometry is not trivial and, ideally, it would require a
detailed knowledge of both the ice-shelf geometry and the ocean circulation
in the ice-shelf cavities. The method discussed in Sect.
provides a solution to this issue by constructing a field of effective
grounding-line depths and slope angles for each shelf point from topographic
data. The resulting values for zgl and α can be interpreted as
reflecting the average effect of all plumes that reach the shelf point. This
method provides a straightforward way to extend the parametrization from 1-D
to 2-D for a given topography and ice mask, but it is not unique. As
discussed in the previous section, a fully realistic 2-D formulation of the
plume dynamics would require additional considerations.
However, since the temperature sensitivity of the plume parametrization can
be considerable, a more important factor for the two-dimensional model is
finding an ocean temperature field that is characteristic of the ocean water
flowing into the ice-shelf cavities. In this respect, the results in
Sect. and show that the
depth-averaged and interpolated data from observations require a plausible
offset ΔT between -1.4 and 0.8 ∘C in
order to obtain an effective temperature Teff
(Fig. b) with which the plume parametrization gives basal melt
rates close to the present-day observations of . In
contrast, a much higher offset ΔT between -0.5 to
5.4 ∘C is required for obtaining the same melt rates with the
quadratic parametrization of , as shown in
Fig. . The same low temperature sensitivity of the melt
rates from the latter parametrization is also apparent in ,
where different tuning factors in the basal melt parametrization are used for
different sectors along the Antarctic coastline, and in ,
where offsets of 3 and 5 ∘C are added to the ocean
temperature in the Amundsen and Bellingshausen seas (resulting from an ocean
model) in order to obtain the correct present-day basal melt rates and
grounding-line retreat.
All in all, the presented plume parametrization, together with the
constructed effective temperature field, gives reasonable results for the
spatial pattern of present-day basal melt in Antarctica. The inherent
geometry dependence, based on the plume dynamics, gives a more natural
spatial variation that cannot be captured with local heat-balance models, with a
major aspect being the occurrence of refreezing. Of course, the current
discussion only assumes a steady state regarding the ice dynamics and the
ocean temperature. The question remains on how an ice-dynamical model would
behave when coupled to the plume parametrization, both for present-day
forcing and for a varying climate. As a next step, it is important to perform
such transient simulations of an ice model coupled to the plume
parametrization and conduct sensitivity experiments. For such simulations,
the effective temperature in Fig. b, even though it is a
constructed field, can prove to be a valuable reference state to which
temperature anomalies can be added, as briefly discussed in
Sect. . Eventually, coupled ice–ocean simulations
(e.g. ) might benefit from this approach by using both
ocean-model output and this reference state to determine an appropriate
temperature forcing for this type of basal melt parametrizations.