the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 04 Feb 2025
Ice/firn age distribution on the Elbrus Western Plateau (Caucasus) inferred from ice flow model
Abstract. The glaciers of Mount Elbrus (Caucasus) contain paleoclimatic and paleoenvironmental information representative of a vast region. The cold conditions and negligible seasonal melting in the near-summit area of Elbrus ensure the excellent preservation of climatic signals. In 2009, a 182.65 meter long ice core was obtained from the Western Plateau (WP) of Elbrus at approximately 5100 ma.s.l. This core was partially dated using chemical stratigraphy (upper part) and radiocarbon dating (basal samples), serving as the basis for numerous paleoreconstructions. However, the ice age distribution within the intermediate part of the core and across the entire glacier on the WP remained unknown. In this work, a three-dimensional steady state full Stokes flow model for a cold glacier with a rheological law accounting for firn densification was applied both in a purely mechanical and in a thermomechanically coupled versions. Using this model, the ice velocity field was simulated in the central part of the WP. Ice age distribution was determined by solving a boundary value problem for the dating equation. All calculations were performed using Elmer/Ice, a finite element software designed for ice dynamics modeling. The model was calibrated by fitting the simulated age-depth relationship to the observed data from the ice core, with ice viscosity used as the primary calibration parameter. This approach provided a series of three-dimensional ice age distributions on the WP for various modeling scenarios. All versions of the model accurately reproduce the ice age according to ice core data to a depth of 140–150 m (130–180 years). Below 150 m, the ice age increases sharply and the dating discrepancies between different modeling scenarios become larger. Overall the modelled ages fell within 68.2 % confidence intervals of the radiocarbon dated near-bottom ice samples which indicated mean radiocarbon ages ranging from 1 to 2 ka. However, the model was unable to resolve the dating of the basalice layer up to 3–4 m thick. Future model improvements should focus on refining basal conditions, including accounting for potential melting, and identifying areas containing the oldest ice.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2024-3955', Anonymous Referee #1, 17 Mar 2025
-
AC1: 'Reply on RC1', Stanislav Kutuzov, 05 Jun 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2024-3955/egusphere-2024-3955-AC1-supplement.pdf
-
AC1: 'Reply on RC1', Stanislav Kutuzov, 05 Jun 2025
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RC2: 'Comment on egusphere-2024-3955', Martin Lüthi, 08 Apr 2025
Dear collegues
This is an interesting and important analysis of the flow behavior of
a firn-covered large glacier, where an ice core for climatic analysis
has been drilled. The paper is nicely written and comprehensive. The
discussion needs some more consideration of the shortcomings of
neglecting bubble close-off and a static climate.ÂMy recommendation is to publish the manuscript after taking into
account the comments below.General comments
- The abstract is quite long and should be considerably shortened. It
 contains repetitions and too many details.- "full Stokes" does not exist (but seems to be marketing jargon of Elmer Ice)
 -> these are just the Stokes equation from fluid mechanics. There exist "reduced" equations, omitting some terms due to scaling arguments.
- "ice age" usually means a geological epoch. Better replace this term
 everywhere with "the age of the ice"- citation style should be adapted: Often \citet or \citep!
- Section 3 is out of place, and is also partly repeated in Section 4.
 Maybe convert to a table, or relate it better to the rest of the manuscript.- also in section 4 and 5 information is not given in a logcial order.
 You should rater describe, in this order:
 - geometry, density, temperature etc
 - numerical approach & solver etc
 - discretization
 - boundary conditions
To obtain a reasoable age of the ice at the base, the air bubble
pressure after close-off has to be taken into account. This was
pioneered by Pimienta, and implemented in e.g. Lüthi & Funk (2000).
Without that effect, the ice at the botton cannot be dated correctly,
and the age is much too old. Also, varying basal melt could easily be
used to control the age of the ice at the bottom. If it is too old in
the model, just slightly increase the melt.Overall, this is a very nice and conclusive study on the important
topic of dating a climatologically relevant ice core. The study has a
few shortcomings, especially wrt. the model. These problems are mostly
discussed, and are largely due to the lack of data to constrain the
model. These include flow velocities and climate data to constrain the
long-term thermal and dynamical evolution of the glacier. I think that
at this stage it is not useful or necessary to implement bubble
close-off in the Elmer Ice code base (but this needs urgently be done
for subsequent studies).ÂThe discussion should not only list, but also clearly work out the
effects of neglecting in the model the effects of bubble close-off,
and of assuming a high and constant basal heat flow, as well as
assuming steady climate conditions.Â
Specific comments
22 If the area is so large and encompasses North Africa and Asia, it
  is not representative for any of those.34 does the part after "while" refer to measurements? Or the model?
39/45/54 and other places "full Stokes" -> Stokes
40 "the accumulation record"
49 "Mt. Dôme du Goûter" -> leave away Mt.
50 " Blank," -> Blanc
61 the glaciated area
62 a stray parenthesis
74 Better put the information on Pleiades in the Acknowledgements (does
  not fit in the main text)79 Here, I was expecting the Pleiades DEM for the surfaceÂ
79 How do you get a 10x10 m bedrock from a radar survey? Was this somehow gridded?
Figure 1, caption: Coordinates are NOT given in UTM, but this would be much more useful!
89 So, this is a triangulation of the domain. How was this done, with
  Triangle? You should also mention that you need this grid for the FE-model.92 Does this mean that you have prismatic elements, which are
  problematic for the incompressible flow due to LBB stability criterion?  Also, the mesh resolution with depth is wildly varying. I can see
  the advantage of extruding a mesh in the vertical, but a TET4 mesh
  would likely be much more suited for the computations at hand.95 This argument is not clear. It would be trivial to shift the
  bedrock up by 5 m, and then retain the exact horizontal position of
  the borehole. This is crucial, since mainly surface slope drives
  ice flow, especially in delicate saddle-like topographies like the
  one modeled in this study. Point a is on a clear slope, while point
  b is not.  Also, radio echo data was likely not correctly interpreted
  (vertical instead of perpendicular to the surface), and depends a
  lot on the velocity model assumed, which depends on the density
  structure & temperature.
 Â
99, Tab 1 "Mathematical model" Â better: "Numerical model" or "Ice flow model"Tab 1: there is way too much information in this table
  at the same time it is totally unclear what type of relations were used.
113 This "full Stokes..." appears here for the 3rd time. Rather
  describe it properly at some point.120ff please show the data and the approximations in a figure.
125 Show the equations that are solved (maybe in an appendix). Not
  everyone wants to look up these equations. Are these the exactly
  same equations? Then why show them. If not: what is different?
130 The derivation of the tensor invariant makes no sense if it is not
  presented in the context of the flow relation used. Either show
  all, or nothing (but show in the appendix what exact equations were used).130 It is confusing using "D" for several things, and also "T".Â
145 This is wrong in the context of a compressible flow relation. "p"
  is a Lagrange multiplier in a purely incompressible flow law. Here
  a compressible law is used, were isotropic and deviatioric parts
  are mixed. See e.g. Gagliardini & Meyssonnier (1997), Lüthi & Funk (2000)
147 This is Glen's flow law for INCOMPRESSIBLE ice. You should write
  down the proper Duva-Crow flow law!150 References for the parametrizations of c and \kappa should be
  given. Is T in K or degC?  Also, these values can be deleted in Tab 1. It is important to
  notice that both thermal quantities are strongly dependent on
  density.154 This equation is wrong, it is k, not \kappa (diffusivity) Sorry, I
  see that \kappa is used for conductivity here, but it is convention
  to use \kappa for diffusivity and k for conductivity.156 these are the conservation of mass equation (the volume is changing!)
165 Are these b.c. varying with depth? Already mention it here, I see Eq (26).
179 What is the rationale for a non-zero vertical velocity, but a zero
  horizontal velocity?198 This shape of the velocity profile is only reasonable below the
  firn-ice transition, as can be seen e.g. in Fig 4a.207 This is not a reduced flow model, you solve the full flow model
  for a time-constant T-distribution.211 The viscosity must be dependent on the density. This is absolutely
  crucial. If you use Duva-Crow it is given by a(phi) and b(phi).213 This section should be part of the Methods section, and most of
  the text appears here for the 4th time.230 What is the role of the flow-enhancement factor? Is this needed at
  all? How is it applied, to firn and ice simultaneously? This would
  lead to densification values that are not compatible with the
  measured densities anymore.Â245 Figure 4C merits some attention. This temperature profile is truly
  exceptional with 18 K temperature difference on only 180 m! Such
  high temperature gradients and heat fluxes are truly remarkable.
  Why are they occurring? Is the assumed basal heat flux of 340
  mW/m2 really realistic? In mountains, the vertical heat flux is
  often much reduced due to topography.251 Write out "Figure" everywhere in the text, and abbreviate it in
  parentheses.255 Since we cannot see anything useful in Fig 4b, please indicate
  what deviation the age of the basal ice has. Is it too old?Â280 wrong parens
280 But they nicely agree for other sites, such as Colle Gnifetti and
  Col du Dome, with similar model setups (Lüthi, Liciulli, Gagliardini, ...).  The problem with the deepest layers is that they might be remnants
  from a time when the geometry of the glacier was very different,
  the ice divides were shifted and the flow regime was altered.  Also, as mentioned above, neglecting the effect of bubble
  close-off (not yet implemented in Elmer-Ice?) strongly affects the
  age at depth.Figure 4b: There is nothing useful to see in this figure. Show the top
  and bottom parts separately. Additionally/alternatively, you could
  use a logarithimc age scale. Also indicate the reference dates
  (volcanoes) with dots.Figure 4c: Please indicate 0 degrees, and also the pressure melting
  temperature at the base. From the plot one cannot determine the
  basal temperature.  For a steady state model with density variation you will
  never get a straight line, since the vertical heat flux is
  constant, but you need a higher gradient in firn. The measured
  temperature profile also looks advective, which should have been
  captured by the model. Finally, the spatially constant vertical
  heat flux b.c. in the model is not realistic in a mountain
  topography (e.g. Lüthi, 2001).Figure 4 caption: what is this stray "vertical a"??
Citation: https://doi.org/10.5194/egusphere-2024-3955-RC2 -
AC2: 'Reply on RC2', Stanislav Kutuzov, 05 Jun 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2024-3955/egusphere-2024-3955-AC2-supplement.pdf
-
AC2: 'Reply on RC2', Stanislav Kutuzov, 05 Jun 2025
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2024-3955', Anonymous Referee #1, 17 Mar 2025
-
AC1: 'Reply on RC1', Stanislav Kutuzov, 05 Jun 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2024-3955/egusphere-2024-3955-AC1-supplement.pdf
-
AC1: 'Reply on RC1', Stanislav Kutuzov, 05 Jun 2025
-
RC2: 'Comment on egusphere-2024-3955', Martin Lüthi, 08 Apr 2025
Dear collegues
This is an interesting and important analysis of the flow behavior of
a firn-covered large glacier, where an ice core for climatic analysis
has been drilled. The paper is nicely written and comprehensive. The
discussion needs some more consideration of the shortcomings of
neglecting bubble close-off and a static climate.ÂMy recommendation is to publish the manuscript after taking into
account the comments below.General comments
- The abstract is quite long and should be considerably shortened. It
 contains repetitions and too many details.- "full Stokes" does not exist (but seems to be marketing jargon of Elmer Ice)
 -> these are just the Stokes equation from fluid mechanics. There exist "reduced" equations, omitting some terms due to scaling arguments.
- "ice age" usually means a geological epoch. Better replace this term
 everywhere with "the age of the ice"- citation style should be adapted: Often \citet or \citep!
- Section 3 is out of place, and is also partly repeated in Section 4.
 Maybe convert to a table, or relate it better to the rest of the manuscript.- also in section 4 and 5 information is not given in a logcial order.
 You should rater describe, in this order:
 - geometry, density, temperature etc
 - numerical approach & solver etc
 - discretization
 - boundary conditions
To obtain a reasoable age of the ice at the base, the air bubble
pressure after close-off has to be taken into account. This was
pioneered by Pimienta, and implemented in e.g. Lüthi & Funk (2000).
Without that effect, the ice at the botton cannot be dated correctly,
and the age is much too old. Also, varying basal melt could easily be
used to control the age of the ice at the bottom. If it is too old in
the model, just slightly increase the melt.Overall, this is a very nice and conclusive study on the important
topic of dating a climatologically relevant ice core. The study has a
few shortcomings, especially wrt. the model. These problems are mostly
discussed, and are largely due to the lack of data to constrain the
model. These include flow velocities and climate data to constrain the
long-term thermal and dynamical evolution of the glacier. I think that
at this stage it is not useful or necessary to implement bubble
close-off in the Elmer Ice code base (but this needs urgently be done
for subsequent studies).ÂThe discussion should not only list, but also clearly work out the
effects of neglecting in the model the effects of bubble close-off,
and of assuming a high and constant basal heat flow, as well as
assuming steady climate conditions.Â
Specific comments
22 If the area is so large and encompasses North Africa and Asia, it
  is not representative for any of those.34 does the part after "while" refer to measurements? Or the model?
39/45/54 and other places "full Stokes" -> Stokes
40 "the accumulation record"
49 "Mt. Dôme du Goûter" -> leave away Mt.
50 " Blank," -> Blanc
61 the glaciated area
62 a stray parenthesis
74 Better put the information on Pleiades in the Acknowledgements (does
  not fit in the main text)79 Here, I was expecting the Pleiades DEM for the surfaceÂ
79 How do you get a 10x10 m bedrock from a radar survey? Was this somehow gridded?
Figure 1, caption: Coordinates are NOT given in UTM, but this would be much more useful!
89 So, this is a triangulation of the domain. How was this done, with
  Triangle? You should also mention that you need this grid for the FE-model.92 Does this mean that you have prismatic elements, which are
  problematic for the incompressible flow due to LBB stability criterion?  Also, the mesh resolution with depth is wildly varying. I can see
  the advantage of extruding a mesh in the vertical, but a TET4 mesh
  would likely be much more suited for the computations at hand.95 This argument is not clear. It would be trivial to shift the
  bedrock up by 5 m, and then retain the exact horizontal position of
  the borehole. This is crucial, since mainly surface slope drives
  ice flow, especially in delicate saddle-like topographies like the
  one modeled in this study. Point a is on a clear slope, while point
  b is not.  Also, radio echo data was likely not correctly interpreted
  (vertical instead of perpendicular to the surface), and depends a
  lot on the velocity model assumed, which depends on the density
  structure & temperature.
 Â
99, Tab 1 "Mathematical model" Â better: "Numerical model" or "Ice flow model"Tab 1: there is way too much information in this table
  at the same time it is totally unclear what type of relations were used.
113 This "full Stokes..." appears here for the 3rd time. Rather
  describe it properly at some point.120ff please show the data and the approximations in a figure.
125 Show the equations that are solved (maybe in an appendix). Not
  everyone wants to look up these equations. Are these the exactly
  same equations? Then why show them. If not: what is different?
130 The derivation of the tensor invariant makes no sense if it is not
  presented in the context of the flow relation used. Either show
  all, or nothing (but show in the appendix what exact equations were used).130 It is confusing using "D" for several things, and also "T".Â
145 This is wrong in the context of a compressible flow relation. "p"
  is a Lagrange multiplier in a purely incompressible flow law. Here
  a compressible law is used, were isotropic and deviatioric parts
  are mixed. See e.g. Gagliardini & Meyssonnier (1997), Lüthi & Funk (2000)
147 This is Glen's flow law for INCOMPRESSIBLE ice. You should write
  down the proper Duva-Crow flow law!150 References for the parametrizations of c and \kappa should be
  given. Is T in K or degC?  Also, these values can be deleted in Tab 1. It is important to
  notice that both thermal quantities are strongly dependent on
  density.154 This equation is wrong, it is k, not \kappa (diffusivity) Sorry, I
  see that \kappa is used for conductivity here, but it is convention
  to use \kappa for diffusivity and k for conductivity.156 these are the conservation of mass equation (the volume is changing!)
165 Are these b.c. varying with depth? Already mention it here, I see Eq (26).
179 What is the rationale for a non-zero vertical velocity, but a zero
  horizontal velocity?198 This shape of the velocity profile is only reasonable below the
  firn-ice transition, as can be seen e.g. in Fig 4a.207 This is not a reduced flow model, you solve the full flow model
  for a time-constant T-distribution.211 The viscosity must be dependent on the density. This is absolutely
  crucial. If you use Duva-Crow it is given by a(phi) and b(phi).213 This section should be part of the Methods section, and most of
  the text appears here for the 4th time.230 What is the role of the flow-enhancement factor? Is this needed at
  all? How is it applied, to firn and ice simultaneously? This would
  lead to densification values that are not compatible with the
  measured densities anymore.Â245 Figure 4C merits some attention. This temperature profile is truly
  exceptional with 18 K temperature difference on only 180 m! Such
  high temperature gradients and heat fluxes are truly remarkable.
  Why are they occurring? Is the assumed basal heat flux of 340
  mW/m2 really realistic? In mountains, the vertical heat flux is
  often much reduced due to topography.251 Write out "Figure" everywhere in the text, and abbreviate it in
  parentheses.255 Since we cannot see anything useful in Fig 4b, please indicate
  what deviation the age of the basal ice has. Is it too old?Â280 wrong parens
280 But they nicely agree for other sites, such as Colle Gnifetti and
  Col du Dome, with similar model setups (Lüthi, Liciulli, Gagliardini, ...).  The problem with the deepest layers is that they might be remnants
  from a time when the geometry of the glacier was very different,
  the ice divides were shifted and the flow regime was altered.  Also, as mentioned above, neglecting the effect of bubble
  close-off (not yet implemented in Elmer-Ice?) strongly affects the
  age at depth.Figure 4b: There is nothing useful to see in this figure. Show the top
  and bottom parts separately. Additionally/alternatively, you could
  use a logarithimc age scale. Also indicate the reference dates
  (volcanoes) with dots.Figure 4c: Please indicate 0 degrees, and also the pressure melting
  temperature at the base. From the plot one cannot determine the
  basal temperature.  For a steady state model with density variation you will
  never get a straight line, since the vertical heat flux is
  constant, but you need a higher gradient in firn. The measured
  temperature profile also looks advective, which should have been
  captured by the model. Finally, the spatially constant vertical
  heat flux b.c. in the model is not realistic in a mountain
  topography (e.g. Lüthi, 2001).Figure 4 caption: what is this stray "vertical a"??
Citation: https://doi.org/10.5194/egusphere-2024-3955-RC2 -
AC2: 'Reply on RC2', Stanislav Kutuzov, 05 Jun 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2024-3955/egusphere-2024-3955-AC2-supplement.pdf
-
AC2: 'Reply on RC2', Stanislav Kutuzov, 05 Jun 2025
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Gleb Chernyakov
Nelly Elagina
Taisiia Kiseleva
Stanislav Kutuzov
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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