the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Fjord circulation induced by melting icebergs
Abstract. In an icebergchoked fjord, meltwater can drive circulation. Downfjord of the ice, buoyancy and rotation lead to an outflowing surface coastal current hugging one side of the fjord with an inflowing counter current below. To predict the structure and evolution of these currents, we develop an analytical model – complemented by numerical simulations – that involve a rectangular fjord initially at rest. Specifically, we (i) start with the socalled Rossby adjustment problem, (ii) reconfigure it for a closed channel with stratification, and (iii) generalize the conventional 'dambreak' scenario to a gradualrelease one that mimics the continual, slow injection of meltwater. Implicit in this description is the result that circulation is mediated by internal Kelvin waves. The analytical model shows that if the total meltwater flux increases (e.g., a larger melange, warmer water, or enhanced ice–ocean turbulence), then circulation strength increases as would be expected. For realistic parameters, a given meltwater flux induces an exchange flow that is ~50 times larger. This factor decreases with increasing water column stratification and vice versa.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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RC1: 'Comment on egusphere20232106', Anonymous Referee #1, 02 Nov 2023
Review of “Fjord circulation induced by melting icebergs”, by Kenneth G. Hughes, (egusphere20232106)
The manuscript describes the development and application of an analytical model of fjord circulation forced by meltwater input at the surface (the meltwater stems from icebergs in the fjord). The build process of the model is described in an easy to follow fashion by considering and assembling different configurations with increasing complexity. The solutions agree reasonably well (considering the approximations implicit to the analytical solution) with highresolution numerical simulations. A parameter study shows expected and not so intuitive results that can explain some observed behaviour in glacial fjords. The manuscript is well written and easy to follow.
As my main critique, the manuscript introduction does not describe the purpose nor the context of the study, so that the manuscript appears to be “just” a model description (“The core contribution of this paper is an analytical model explaining the firstorder dynamics of a fjord’s response to hundreds of melting icebergs.”) as a very well designed and carried out exercise of geophysical fluid dynamics. In the same way the conclusion is a useful and alternative summary of the model steps, but no geophysical/scientific conclusion are draw. Adding context and purpose would make this manuscript an even better paper. That’s why I reccommend minor revisions.
Not as important:
The model description uses some formulae, but it is not always clear (to me) where they come from, or how they were derived. I appreciate the briefness and clarity of the presentation, but the presentation requires that the reader “believes” the text. Maybe more information in an appendix would help?
There are some technicalities, questions, and notes from reading the manuscript are listed below:
page 1
Abstract: no motivation, no concluding remarks about implications
l2: “Downfjord” Is this a proper term? I would use downstream.
page 2
L2425 “over the top 200 m is a few cm/s over the top 200 m”
repetition, also I would not associate “outflow” with velocity units. Maybe “outflow velocity”?
ll27: this paragraph is very different in style from all other paragraphs, consider rewriting
page 4
The icebergs do not seem to move. Is that justified? Later this is described, but it may be useful to do it here already.
l64: (practical) salinity has no units, as it is a ratio of g salt / kg sea water, see UNESCO reports etc. “psu” should not be used (I know that is done commonly, but it’s still not correct). Absolute salinity (according to TEOS10) has units of g/kg.
l73: At this resolution, nonhydrostatic dynamics may start to become important. Why use hydostatic dynamics?
l77: “(designated as scheme 33 in the MITgcm)”, I think the reference for this DST (direct space time) scheme is
Article{hundsdorfer94,
author = {Willem Hundsdorfer and R. A. Trompert},
title = {Method of Lines and Direct Discretization: a
Comparison for Linear Advection},
journal = {Applied Numerical Mathematics},
year = 1994,
volume = 13,
number = 6,
pages = {469490},
doi = {10.1016/01689274(94)900094}
}
or
@Article{hundsdorfer95,
author = {Willem Hundsdorfer and B. Koren and M. van Loon and
J. Verwer},
title = {A Positive FiniteDifference Advection Scheme},
journal = {J. Comp. Phys.},
year = 1995,
volume = 117,
number = 1,
pages = {3546}
}
page 5
ll91, are the initial conditions (especially T=2deg) reasonable? The high melt rate will probably reduce with lower ocean temperatures (and it does according to the parameter study later on).
l97: only small > only over a small?
l99: the exponential function implies Kelvin waves, why should they be the only ones? What about Poincare waves (mentioned in the text) and Rossby waves?
page 6
Figure 2, it took me a while to understand this complicated figure. I can appreciate it’s value, but maybe extend the description in the caption to say that the vertical slices are at the dashed lines in the top view (or put them, where they belong?). The “pseudo”realistic perspective implies something different and confused me (e.g. I thought that (c) and (d) form the sides of the displayed domain of (b) and it took me a little to figure out that (b) is only part of (a), etc.)
page 8
l115: typo in “$x— —z$ domain”?
page 9
l142: “overkill” maybe too colloquial?
page 10
l145: “For an infinitely narrow channel, the tanh term goes to zero”, not clear why. Eq5 and 7 do not have any x in them that can got infinity.
Section 3.2 is physically plausible but no formulae support it.
page 12
The figure needs more explanation, e.g. there’s density in a+e, but pressure (anomaly) in bd, but the caption doesn’t describe this, nor the labels in the figure, …
page 13
l200: Figure 4 > Figure 5?
l209: “where z* is a dummy variable used to avoid ambiguity with the integral’s lower limit”. I guess that’s common practise and need not be described. I have seen usually z’ used for this.
Eq18: integral over z? dz is missing. Not clear
page 15
Fig6 caption: relatively lowmode components. > relatively larger lowmode components?
page 17
l254: “The integer n_E is the maximum n” and the higher modes all destructively interfere?
page 23
l366: “by looking at” > by evaluating (colloquial?)
ll370 maybe it would be a good idea to add the stationary role of the iceberg and neglected drag already in the model setup description? (See earlier comment on page 4)
page 24
l380 Please discuss why the overestimation is reduced for the stronger stratification and weaker melt cases.
page 29
l643: helps brings > helps bring?
Citation: https://doi.org/10.5194/egusphere20232106RC1 
AC1: 'Reply on RC1', Kenneth Hughes, 15 Dec 2023

AC4: 'Reply on AC1', Kenneth Hughes, 15 Dec 2023
Ignore this attachment. I uploaded the wrong file
Citation: https://doi.org/10.5194/egusphere20232106AC4

AC4: 'Reply on AC1', Kenneth Hughes, 15 Dec 2023
 AC2: 'Reply on RC1: corrected', Kenneth Hughes, 15 Dec 2023

AC1: 'Reply on RC1', Kenneth Hughes, 15 Dec 2023

RC2: 'Comment on egusphere20232106', Anonymous Referee #2, 11 Dec 2023
The author compares an analytic linearized fjord circulation model in the presence of an iceberg melange with numerical model simulation. I like this work and urge the author to keep refining their analytical model. I am not sure whether it will find applications in climate models; however, at the very least, it is a valuable tool to understand the outputs of numerical simulations.
I have several comments about the mathematical exposition of the author's work. I do not doubt the correctness, but the model derivation (being analytical) lacks certain rigor (or perhaps I lack understanding of the author's process, either of which merits further explanation).
 In subsections 3.x I am unsure whether the author follows Hermann 1989 or derives original equations. Please specify which part summarizes previous work and which is new. Most of this section reads as a recap of other work, but perhaps the exposition lacks detail. The following points provide specific questions stemming from this concern:
 How does the author derive relation (11)? Is this taken from another paper? If so, which one? Is N (which the author refers to as stratification) the same as buoyancy frequency in equation (3)?
 I do not understand how the author came up with equation (19). It is presented as a statement, not an assumption, and without much justification. Yet, the derivation of Fourier coefficients for velocity depends on it, so please explain more.
 The author comments that they derive equations (23) and (24) from (21) and (22) by xintgration  please show how this happens, as I do not see the relation clearly. If the factor 2L comes from the integration, then are the terms in (21) and (22) spaceindependent? Also, integrated energy will not have the same unit as the original, so perhaps a different notation is required.
 In Equation 25 the author subtracts two sums from each other. As stated, the result of the operation is in my mind is the sum of terms between n_{E} and n_{W} as the summation terms u_{n}(z) are identical in both sums. Perhaps differentiate the notation here to indicate that terms in the n_{E} sum are different than in the n_{W} sum.
 Please include the numerical values of velocity in the figures comparing analytical and numerical results (i.e. Fig. 7, 8, etc). This is important for reproducibility.
 Are the equations (3235) justifiable in the light of the assumption of small perturbations made in Section 3.3?
 It would be interesting to see the results for x<15 to get an idea of how far off the results are from the simulation. Perhaps including results at several locations (including further downstream than 20km) would increase the reader's sense of confidence in where the model can be applied, and where it fails.
Citation: https://doi.org/10.5194/egusphere20232106RC2  AC3: 'Reply on RC2', Kenneth Hughes, 15 Dec 2023
Interactive discussion
Status: closed

RC1: 'Comment on egusphere20232106', Anonymous Referee #1, 02 Nov 2023
Review of “Fjord circulation induced by melting icebergs”, by Kenneth G. Hughes, (egusphere20232106)
The manuscript describes the development and application of an analytical model of fjord circulation forced by meltwater input at the surface (the meltwater stems from icebergs in the fjord). The build process of the model is described in an easy to follow fashion by considering and assembling different configurations with increasing complexity. The solutions agree reasonably well (considering the approximations implicit to the analytical solution) with highresolution numerical simulations. A parameter study shows expected and not so intuitive results that can explain some observed behaviour in glacial fjords. The manuscript is well written and easy to follow.
As my main critique, the manuscript introduction does not describe the purpose nor the context of the study, so that the manuscript appears to be “just” a model description (“The core contribution of this paper is an analytical model explaining the firstorder dynamics of a fjord’s response to hundreds of melting icebergs.”) as a very well designed and carried out exercise of geophysical fluid dynamics. In the same way the conclusion is a useful and alternative summary of the model steps, but no geophysical/scientific conclusion are draw. Adding context and purpose would make this manuscript an even better paper. That’s why I reccommend minor revisions.
Not as important:
The model description uses some formulae, but it is not always clear (to me) where they come from, or how they were derived. I appreciate the briefness and clarity of the presentation, but the presentation requires that the reader “believes” the text. Maybe more information in an appendix would help?
There are some technicalities, questions, and notes from reading the manuscript are listed below:
page 1
Abstract: no motivation, no concluding remarks about implications
l2: “Downfjord” Is this a proper term? I would use downstream.
page 2
L2425 “over the top 200 m is a few cm/s over the top 200 m”
repetition, also I would not associate “outflow” with velocity units. Maybe “outflow velocity”?
ll27: this paragraph is very different in style from all other paragraphs, consider rewriting
page 4
The icebergs do not seem to move. Is that justified? Later this is described, but it may be useful to do it here already.
l64: (practical) salinity has no units, as it is a ratio of g salt / kg sea water, see UNESCO reports etc. “psu” should not be used (I know that is done commonly, but it’s still not correct). Absolute salinity (according to TEOS10) has units of g/kg.
l73: At this resolution, nonhydrostatic dynamics may start to become important. Why use hydostatic dynamics?
l77: “(designated as scheme 33 in the MITgcm)”, I think the reference for this DST (direct space time) scheme is
Article{hundsdorfer94,
author = {Willem Hundsdorfer and R. A. Trompert},
title = {Method of Lines and Direct Discretization: a
Comparison for Linear Advection},
journal = {Applied Numerical Mathematics},
year = 1994,
volume = 13,
number = 6,
pages = {469490},
doi = {10.1016/01689274(94)900094}
}
or
@Article{hundsdorfer95,
author = {Willem Hundsdorfer and B. Koren and M. van Loon and
J. Verwer},
title = {A Positive FiniteDifference Advection Scheme},
journal = {J. Comp. Phys.},
year = 1995,
volume = 117,
number = 1,
pages = {3546}
}
page 5
ll91, are the initial conditions (especially T=2deg) reasonable? The high melt rate will probably reduce with lower ocean temperatures (and it does according to the parameter study later on).
l97: only small > only over a small?
l99: the exponential function implies Kelvin waves, why should they be the only ones? What about Poincare waves (mentioned in the text) and Rossby waves?
page 6
Figure 2, it took me a while to understand this complicated figure. I can appreciate it’s value, but maybe extend the description in the caption to say that the vertical slices are at the dashed lines in the top view (or put them, where they belong?). The “pseudo”realistic perspective implies something different and confused me (e.g. I thought that (c) and (d) form the sides of the displayed domain of (b) and it took me a little to figure out that (b) is only part of (a), etc.)
page 8
l115: typo in “$x— —z$ domain”?
page 9
l142: “overkill” maybe too colloquial?
page 10
l145: “For an infinitely narrow channel, the tanh term goes to zero”, not clear why. Eq5 and 7 do not have any x in them that can got infinity.
Section 3.2 is physically plausible but no formulae support it.
page 12
The figure needs more explanation, e.g. there’s density in a+e, but pressure (anomaly) in bd, but the caption doesn’t describe this, nor the labels in the figure, …
page 13
l200: Figure 4 > Figure 5?
l209: “where z* is a dummy variable used to avoid ambiguity with the integral’s lower limit”. I guess that’s common practise and need not be described. I have seen usually z’ used for this.
Eq18: integral over z? dz is missing. Not clear
page 15
Fig6 caption: relatively lowmode components. > relatively larger lowmode components?
page 17
l254: “The integer n_E is the maximum n” and the higher modes all destructively interfere?
page 23
l366: “by looking at” > by evaluating (colloquial?)
ll370 maybe it would be a good idea to add the stationary role of the iceberg and neglected drag already in the model setup description? (See earlier comment on page 4)
page 24
l380 Please discuss why the overestimation is reduced for the stronger stratification and weaker melt cases.
page 29
l643: helps brings > helps bring?
Citation: https://doi.org/10.5194/egusphere20232106RC1 
AC1: 'Reply on RC1', Kenneth Hughes, 15 Dec 2023

AC4: 'Reply on AC1', Kenneth Hughes, 15 Dec 2023
Ignore this attachment. I uploaded the wrong file
Citation: https://doi.org/10.5194/egusphere20232106AC4

AC4: 'Reply on AC1', Kenneth Hughes, 15 Dec 2023
 AC2: 'Reply on RC1: corrected', Kenneth Hughes, 15 Dec 2023

AC1: 'Reply on RC1', Kenneth Hughes, 15 Dec 2023

RC2: 'Comment on egusphere20232106', Anonymous Referee #2, 11 Dec 2023
The author compares an analytic linearized fjord circulation model in the presence of an iceberg melange with numerical model simulation. I like this work and urge the author to keep refining their analytical model. I am not sure whether it will find applications in climate models; however, at the very least, it is a valuable tool to understand the outputs of numerical simulations.
I have several comments about the mathematical exposition of the author's work. I do not doubt the correctness, but the model derivation (being analytical) lacks certain rigor (or perhaps I lack understanding of the author's process, either of which merits further explanation).
 In subsections 3.x I am unsure whether the author follows Hermann 1989 or derives original equations. Please specify which part summarizes previous work and which is new. Most of this section reads as a recap of other work, but perhaps the exposition lacks detail. The following points provide specific questions stemming from this concern:
 How does the author derive relation (11)? Is this taken from another paper? If so, which one? Is N (which the author refers to as stratification) the same as buoyancy frequency in equation (3)?
 I do not understand how the author came up with equation (19). It is presented as a statement, not an assumption, and without much justification. Yet, the derivation of Fourier coefficients for velocity depends on it, so please explain more.
 The author comments that they derive equations (23) and (24) from (21) and (22) by xintgration  please show how this happens, as I do not see the relation clearly. If the factor 2L comes from the integration, then are the terms in (21) and (22) spaceindependent? Also, integrated energy will not have the same unit as the original, so perhaps a different notation is required.
 In Equation 25 the author subtracts two sums from each other. As stated, the result of the operation is in my mind is the sum of terms between n_{E} and n_{W} as the summation terms u_{n}(z) are identical in both sums. Perhaps differentiate the notation here to indicate that terms in the n_{E} sum are different than in the n_{W} sum.
 Please include the numerical values of velocity in the figures comparing analytical and numerical results (i.e. Fig. 7, 8, etc). This is important for reproducibility.
 Are the equations (3235) justifiable in the light of the assumption of small perturbations made in Section 3.3?
 It would be interesting to see the results for x<15 to get an idea of how far off the results are from the simulation. Perhaps including results at several locations (including further downstream than 20km) would increase the reader's sense of confidence in where the model can be applied, and where it fails.
Citation: https://doi.org/10.5194/egusphere20232106RC2  AC3: 'Reply on RC2', Kenneth Hughes, 15 Dec 2023
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Kenneth G. Hughes
The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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