Prognostic experiments
for fast-flowing ice streams on the southern side of the Academy of Sciences
Ice Cap on Komsomolets Island, Severnaya Zemlya archipelago, were undertaken
in this study. The experiments were based on inversions of basal friction
coefficients using a two-dimensional flow-line thermocoupled model and
Tikhonov's regularization method. The modeled ice temperature distributions
in the cross sections were obtained using ice surface temperature histories
that were inverted previously from borehole temperature profiles derived at
the summit of the Academy of Sciences Ice Cap and the elevational gradient of
ice surface temperature changes (about 6.5
There are many relevant diagnostic observations of glaciers available, including digital Landsat imagery and satellite interferometric synthetic aperture radar (InSAR), airborne measurements, borehole ice temperature, and ice surface mass-balance measurements. These observations provide data for prognostic experiments that allow the prediction of future glacier conditions for different climatic scenarios in the future. These experiments can be performed by mathematical modeling, and in this study a two-dimensional ice flow model was applied to predict future conditions of fast-flowing ice streams on the southern side of the Academy of Sciences Ice Cap on Komsomolets Island, Severnaya Zemlya archipelago (Fig. 1; Dowdeswell et al., 2002).
After Dowdeswell et al., 2002, a map of Severnaya Zemlya showing the Academy of Sciences Ice Cap on Komsomolets Island, together with the other ice caps in the archipelago: Rusanov Ice Cap, Vavilov Ice Cap, Karpinsky Ice Cap, University Ice Cap, Pioneer Glacier, Semenov-Tyan-Shansky Glacier, Kropotkin Glacier, and Leningrad Glacier. The inset shows the location of Severnaya Zemlya and the nearby Russian Arctic archipelagos of Franz Josef Land and Novaya Zemlya within the Eurasian High Arctic.
The observations were based on digital Landsat imagery and satellite InSAR
and revealed four drainage basins and four fast-flowing ice streams on the
southern side of the Academy of Sciences Ice Cap (Fig. 2; Dowdeswell et
al., 2002). The four ice streams were 17–37 km long and 4–8 km wide
(Dowdeswell et al., 2002). Bedrock elevations of these areas were below sea
level, and the ice flow velocities attained a value of 70–140 m a
After Dowdeswell et al., 2002, interferometrically corrected derived
ice surface velocities for the Academy of Sciences Ice Cap. The first two
contours are at velocities of 5 and 10 m a
The flow-line profiles of the three ice streams on the southern side of the
Academy of Sciences Ice Cap are shown in Fig. 3. Ice flow in these ice
streams has been simulated with a two-dimensional flow-line higher-order
finite-difference model (e.g., Colinge and Blatter, 1998; Pattyn, 2000,
2002). This model describes an ice flow along a flow line (Pattyn, 2000,
2002). The results of diagnostic experiments undertaken by Konovalov (2012)
show that for the C–C
The inversion of friction coefficients is based on the minimization of the deviation between the observed and modeled surface velocities. A series of test experiments (Konovalov, 2012), in which modeled surface velocities were used as observations in the inverse problem, have shown that the inverse problem for the full 2-D ice flow-line model is ill posed. More precisely, the surface velocity is weakly sensitive to small perturbations in the friction coefficient, and as a result the perturbations appear in the inverted friction coefficients (Konovalov, 2012).
Herein, in a series of prognostic experiments we used the friction coefficient inversions obtained by applying Tikhonov's regularization method, in which Tikhonov's stabilizing functional is added to the main discrepancy functional (Tikhonov and Arsenin, 1977).
The inversions of the friction coefficients were used in the prognostic
experiments for the fast-flowing ice streams. The 2-D prognostic experiments
were numerical simulations with ice thickness distribution changes performed
by the 2-D flow-line thermocoupled model, which
includes (i) diagnostic equations,
(ii) the heat-transfer equation and (iii) the mass-balance equation (Pattyn, 2000,
2002). Here, we present the results of the prognostic experiments performed
for the B–B
The 2-D flow-line higher-order model includes the continuity equation for an
incompressible medium, the mechanical equilibrium equation in terms of stress
deviator components (Pattyn, 2000, 2002), and Glen's flow law (Cuffey and Paterson, 2010):
The boundary conditions and some complementary experiments that were conducted by applying this model were considered in Konovalov (2012). In particular, the technique in which the boundary conditions are included in the momentum equations in Konovalov (2012) was applied in the prognostic experiments considered here.
The inversion of the friction coefficient was conducted using the gradient
minimization procedure for the smoothing functional (Tikhonov and
Arsenin, 1977):
The details of the gradient minimization procedure and the problem of the
regularization parameter choice are discussed in Nagornov et al. (2006) and
Konovalov (2012). In this study the inversions were obtained for the linear
(viscous) friction law based on the experiments implemented in Konovalov (2012),
with inversions for the C–C
The results of thermocoupled prognostic experiments imply that the 2-D
flow-line model includes the heat-transfer equation (Pattyn, 2000, 2002)
In this model it is suggested that the ice surface temperature at the
Academy of Sciences Ice Cap varies with an elevational gradient of
temperature change, which is equal to about 6.5
The boundary condition at the ice base is defined by the geothermal heat
flux and by heating due to the basal friction. It is expressed as (Pattyn,
2000, 2002)
The boundary conditions at the ice (ice shelf) terminus and at the ice shelf
base are defined by sea water temperature, which was considered to be
The ice thickness temporal changes along the flow line are described by the
mass-balance equation (Pattyn, 2000, 2002)
The mass-balance equation requires two boundary conditions at the summit and
at the ice terminus. The first condition at the ice cap summit implies that
In the model the grounding-line position is defined from the hydrostatic
equilibrium (Schoof, 2007; Pattyn et al., 2012; Seroussi et al., 2014).
Because sea water flow under the ice shelf is not considered in the model,
and hence the pressure in Eqs. (10) and (11) from Pattyn et al. (2012)
are equal to hydrostatic pressure, the grounding-line position is
located where
For the first run of the friction coefficient inversions, the linear ice
temperature profile approximation was applied. Specifically, it was assumed
that the ice temperature linearly increased from
The inverted friction coefficient distributions along the B–B
After the first run of the inversions, the ice temperature simulations were
performed for inverted friction coefficients and boundary conditions (4) and
(5). Boundary condition (4) included the temperature history
For the second run of the basal friction coefficient inversions, the modeled temperature distributions were applied (the modeled temperature was defined from Eqs. 3 to 5). The inverted friction coefficients for the (i) linearly approximated ice temperature and (ii) modeled ice temperature are shown in Fig. 6. Generally, the distinctions in the friction coefficients were insignificant, and therefore the ice temperature approximations could be applied in the inverse problem as the first iteration of the ice temperature distribution in the glacier.
The temperature
distributions within
The friction coefficients inverted along the
The main input data along with flow-line profiles for the prognostic
experiments, namely, the surface mass balance, were adopted from Bassford et
al. (2006). Figure 7 shows the elevational mass-balance distribution along
the C–C
The surface mass-balance elevational distribution along the C–C
Despite future warming scenarios not being included in the prognostic
experiments, the modeled ice cap response to the present environmental
impact, which is reflected in the elevational mass-balance distribution
(Bassford et al., 2006), revealed that the ice thickness gradually
diminished along all three flow lines. Figures 8a–10a show the modeled
successive ice surfaces divided into 50-year time intervals for the
B–B
The grounding-line history, i.e., grounding-line retreat or advance,
specifically reflects a growing or diminishing ice mass, i.e., its history
is an indicator of glacier evolution. The grounding line retreated (a) along
the B–B
The results of the prognostic experiments can be treated in the same way, suggesting changes in the friction coefficients. The glacier terminus, which is currently fast flowing and therefore experiencing pressure melting, eventually became frozen to the ground. The ice thickness was insufficient to provide insulation from the cold atmosphere and reduced driving stress and strain heating. Therefore, the basal friction coefficients could change drastically, given the simulated changes in glacier geometry.
The ice flow velocities in the ice streams decrease with time and this trend
diminishes the outgoing ice fluxes in the future. Figure 12 shows the
modeled outgoing ice flux histories, i.e., it shows how the value
There are small peaks that periodically disturb the main historical trends
of the three outgoing ice fluxes. Each peak reflects ice calving at the
ice shelf terminus. Similarly, the ice calving represents a sudden change in
the value of the outgoing ice flux (
In this model, both the ice shelf length and ice shelf thickness at the
terminus were considered to be variables that could satisfy certain
conditions. If the ice shelf length exceeded a value
To investigate the impact of the parameters on the results of the modeling, the parameters were varied in a series of experiments. However, the simulation revealed that at long timescales the mass balance, friction coefficient, and ice temperature had the main impact on the assessment of the grounding-line retreat derived by the modeling.
Numerical experiments conducted with a 2-D model using the randomly perturbed
friction coefficient have revealed that the horizontal surface velocity is
weakly sensitive to the perturbations (Fig. 4 of Konovalov, 2012). Thus,
the perturbations appear on the
Tikhonov's method is based on the application of the stabilizing functional,
which proportionally reduces the effects of perturbations of the
regularization parameter
The reduction in the existing friction coefficient variability is associated with a growing discrepancy between the observed and modeled surface velocities. Thus, the regularization parameter is chosen as the value at which nonexistent perturbations are reduced, but the real variability of the friction coefficient is not completely reduced by the stabilizing functional. The optimal value of the regularization parameter can be defined approximately in the curve, which is the deviation between the observed and modeled surface velocities versus the regularization parameter (Leonov, 1994; Konovalov, 2012).
Evidently, the stabilizing functional narrows down the range of possible
inverted
The two evidently distinguished levels in the inverted friction coefficient
distributions can be explained by changing the physical properties of the
bedrock along the flow lines. Similarly, the large values of the friction
coefficient at km
The modeled ice temperatures at present (Fig. 5) were qualitatively the
same in the three cross sections. There were resembling zones of relatively
cold ice that could be distinguished in the modeled temperatures
approximately in the middle (in the vertical dimension) of each
cross section. These cold ice zones reflected the surface temperature
minimum about 150–200 years ago in the inverted past temperature history
(Nagornov et al., 2005, 2006). This surface temperature minimum corresponds
to an event known as the Little Ice Age. Thus, surface boundary conditions (4),
and diffusive and advective heat transfers were responsible for the
basal ice temperature, which was mainly in the range of
However, note that the heat-transfer model considered here does not account for meltwater refreezing in the subsurface firn layer (Paterson and Clarke, 1978). The numerical experiments undertaken by Paterson and Clarke (1978) revealed that the heat source had a significant impact on the ice temperature profiles due to melt water refreezing, depending on its percolation depth. Thus, the notion that the basal ice temperature is higher than the modeled temperature and could reach the melting point cannot be dismissed.
The modeled grounding-line history for the
The modeled outgoing ice flux history for the
General formulations of the friction laws assume that the appropriate
equations include the effective basal pressure (e.g., Budd et al., 1979;
Iken, 1981; Bindschadler, 1983; Jansson, 1995; Pattyn, 2000; Vieli et al.,
2001). Introduction of the effective pressure in Eq. (2) does not provide a
constant value of the inverted friction coefficient at
Finally, two areas were identified in the bedrock, where basal ice was
frozen to the bed (km
The prognostic experiments reveal that the extent of both ice mass and ice
stream declined with respect to the reference time-independent mass balance
(Bassford et al., 2006). These experiments demonstrated that the grounding
lines would retreat by about 10 km for the three ice streams over a time
period of 500 years and with a steady-state environmental impact, i.e., a
constant elevation-dependent surface mass balance. The ice flow velocities
in the ice streams would decrease with time due to (a) a diminishing of ice
thicknesses (and thus decreasing driving stress) and (b) a retreat of the
grounding lines from the sliding zones toward the zones where ice is frozen
to the bed (inverted friction coefficient distributions are considered to be
time-independent). Thus, the maxima of the ice flow velocities in the ice
streams decreased from
Observations in the Russian High Arctic (Moholdt et al., 2012) have revealed
that over the period between October 2003 and October 2009 the archipelagos
of Franz Josef Land and Novaya Zemlya have lost ice at a rate of
The modeled ice temperatures at present (Fig. 5) are qualitatively the same in the three cross sections. There are resembling zones of relatively cold ice that can be distinguished in the modeled temperatures in the middle of the cross sections. These cold ice zones reflected the surface temperature minimum about 150–200 years ago in the inverted past temperature history (Nagornov et al., 2005, 2006). This surface temperature minimum corresponds to the event known as the Little Ice Age.
The overall outgoing ice flux history (the sum of the outgoing
fluxes for the three ice streams: B–B
The inversions of the friction coefficient performed for the three
cross sections can be interpreted as follows. The two levels that are
evidently distinguished in the inverted friction coefficient distributions
(Fig. 6) can be explained by changing the physical properties of the
bedrock along the flow lines. Similarly, the large values of the friction
coefficient at km
The prognostic experiments conducted with the reference mass balance
(Bassford et al., 2006) show that the grounding line would retreat by about
10 km in the three ice streams over a time period of 500 years. Similarly,
the grounding line would retreat (a) along the B–B
The data presented in this study are available in the following manuscripts:
The authors declare that they have no conflict of interest.
The authors are grateful to J. A. Dowdeswell et al. for the data that have been used in the paper. The authors are grateful to F. Pattyn for the useful comments made regarding the paper. The authors are grateful to T. Dunse and the anonymous referees for reviewing the paper. Edited by: V. Lucarini Reviewed by: T. Dunse and one anonymous referee