A theoretical model of Surtseyan bomb fragmentation

Surtseyan eruptions are an important class of mostly basaltic volcanic eruptions first identified in the 1960s, where erupting magma at an air–water interface interacts with large quantities of slurry, a mixture of previously ejected tephra that re-enters the crater together with water. During a Surtseyan eruption, hot magma bombs are ejected that initially contain pockets of slurry. Despite the formation of steam and anticipated subsequent high pressures inside these bombs, many survive to land without exploding. We seek to explain this by building and solving a simplified spherical mathematical model that describes the coupled evolution of pressure and temperature due to the flashing of liquid to vapour within a Surtseyan bomb while it is in flight. Analysis of the model provides a criterion for fragmentation of the bomb due to steam pressure build-up, and predicts that if diffusive steam flow through the porous bomb is sufficiently rapid the bomb will survive the flight intact. This criterion explicitly relates fragmentation to bomb properties, and describes how a Surtseyan bomb can survive in flight despite containing flashing liquid water, contributing to an ongoing discussion in volcanology about the origins of the inclusions found inside bombs.


Comments to the Author(s)
Review of RSPA-2021-0166 on 6 April 2021 The manuscript under review elaborates a mathematical model for Surtseyan bombs, which are a particular type of volcanic ejecta where hot vesicular lava encloses a cool, water saturated chunk of tephra. The manuscript reports microstructure measurements that are used to estimate the permeability, the development of a mathematical model based on physical arguments, the analysis of that model, and results of numerical simulations and analytical approximations. A previous paper by the same authors developed the original model, which made the simplifying approximation of a frozen temperature field. In the present work, the authors couple the temperature and pressure fields. They obtain a result that differs by a ratio of length scales associated with the initial temperature condition, and is potentially large. They conclude by noticing that their model predicts a minimum permeability to avoid fracture, and that there are no observations of bombs with permeability below this minimum.
The manuscript has a lot of nice features, including novel permeability measurements, an appealing physical model that makes reasonable idealisations, and clever analysis in estimating the relevant physical scales and making simplifications. However it also has some significant problems that should be addressed before I can recommend publication.
The first of these is the cursory introduction that doesn't provide a clear motivation for the paper. It gives a brief overview of the physical picture of Surtseyan bombs, but doesn't discuss a motivation for physical/mathematical modelling of these features. This might have taken the form of questions to be addressed and the hypothesis that is inherent in the models (the pressure of flashing is relieved by porous flow of steam when the permeability is large enough). Moreover, the introduction doesn't discuss why previous theoretical work is inadequate in addressing the questions. Finally, the introduction should provide an overview of the manuscript, explaining its organisation and, I suggest, also its key results. Instead the introduction breaks abruptly at a subsection called "Methods" which is really just about the methods of the permeability measurements (and the results) and then comes back to provide some description of the model. The text and figure about permeability measurements should be moved out of the introduction and into their own section, afterwards.
The manuscript also lacks a more detailed discussion of the model and its limitations, the results and their implications. What discussion there is is provided in a very brief conclusion section. The last two sentences seem to be the only reference back to the motivating questions (presumably) that connect the mathematics to the physical volcanology. I discuss my sense of the model limitations below. But there is a lack of connection to the physical volcanology here. What can equation (6.1) and Figure 5 tell us about the eruption process and materials? If nothing, why?
The model development in section 2 seems a reasonable balance between explaining the physics (though see detailed points below) and not getting too bogged down in the details that will later be neglected. The model reduction section, however, seems to move hastily through a series of approximations based on the size of dimensionless parameters, dropping terms without discussion of the physics that they represent. This seems problematic in that the initial conditions then create transients in which some variables (and their derivatives) are much larger than unity. Because of the lack of physical discussion, it is difficult to see which of the neglected terms might become important, and what its qualitative effect would be. While it is reasonable to make simplifications, the authors provide no discussion a posteriori of the consequences of these simplifications, and hence of the limitations of the results.
Two simplifications that concern me are as follows. In eqn (3.8), the term in \delta_3 is neglected. \delta_3 is indeed small, but near the flashing front (where all the interesting action occurs), \partial p/\partial t can be very large. In equation (3.13), the term from the previous equation in \lambda_2\epsilon_4 has been neglected relative to the jump in heat flux. There is only a factor of 10 in the coefficients, but p can be very large near the flashing front. So in both cases (and perhaps others) in seems somewhat unclear whether neglecting the terms can be justified on physical grounds. At the very least, some discussion of this a posteriori should be included.
A lot of attention is then given to the nifty matched asymptotics for the analytical approximation, which in Fig. 3 seems to do a reasonable job of matching some aspects of the solution. But it seems to fail badly in matching the maximum pressure, which is the key physical output of interest. So after reaching figure 5, it is unclear why the asymptotic model is included in so much detail (or at all). What do we learn from it of value? In figure 5a, the mismatch between numerics and the approximation is as big as the difference between the fracture criterion and the numerics. It feels like much of the manuscript is driven by an interest in mathematical analysis rather than an interest in the physical predictions of the model. I find this problematic, given the motivation of the work.
I am rather dubious of the implication that the model is correct because of the lack of observations of permeabilities less than one Darcy. One Darcy is very small for a magma with porosity of 0.35--0.8, even if the pores are bubbles. The measurements were all made on bombs containing inclusions, I think, which is a sampling bias. What if fragments of fractured bombs had permeabilities in the same range?
Detailed comments by page:line(s). (Page numbers are as in the header, not the black box) 3:13. How are "bombs" related to "lump of lava" and "tail of tephra"? Are they the lump? 3:56. What is "Palabos"? 4:12. Can this be described using a more typical form such as Kozeny-Carman? Is this form usual or just convenient? 4:52. So the model must be accurate over times of less than a second. This suggests that the choice of initial condition is paramount. Obviously getting the IC right isn't easy... 6:29. It is a bit hard to see how the averaging was conducted. Is the microscopic pressure equal to the continuum pressure? What about the stess \tau? The usual upscaling would give a dissipation rate of something like K_e u_l^2 for laminar flow (but you haven't yet assumed laminar flow). I think it would be better to jump right to the continuum equations.
(2.5). A better choice of notation to disambiguate thermal conductivity and permeability would be nice, while avoiding K_el, which looks more like a bulk modulus.
(2.8). Is the \beta in this equation the same as the \beta in line 57 of the same page? I think they're different... 6.57. "this equation" --indefinite antecedent. (2.9). Again the dissipation term is problematic. 9.29. "over-estimate" by how much? What effect does this have on other approximations made to simplify the equations? 11.52 (and elsewhere). The Ideal Gas law, not the perfect gas equation? 13.11 Mention use of subscript notation for partial derivatives Fig 3. Needs panel labels and a much clearer caption that refers to the panels. Maybe break into two figures? Needs much more explanation in the text. 15.10. If the mesh size that is needed on the basis of characteristics of the solution is similar to the pore size, then there's a potential problem with the continuum assumptions. This needs to be highlighted. 15.58. typo 17.47. "they" --indefinite antecedent. Confusing sentence. 17.58. It is concerning that the solution for maximum pressure depends on a ratio that can be arbitrarily large. First of all, it is unrealistic to assume that the enclosure is captured in such a way that there is a step-function in temperature. And secondly, if the mathematics doesn't regularise this then I think the model is ill-posed (possibly due to neglecting a "small" term that isn't small near the flashing front at small time).

Review form: Referee 2
Is the manuscript an original and important contribution to its field? Good

Is the paper of sufficient general interest? Marginal
Is the overall quality of the paper suitable? Acceptable Can the paper be shortened without overall detriment to the main message? No Do you think some of the material would be more appropriate as an electronic appendix? Yes

Recommendation?
Major revision is needed (please make suggestions in comments)

Comments to the Author(s)
Thank you for an interesting paper to review and apologies for my lateness.
You have presented some very detailed models of magma-slurry interaction in Surtseyan eruptions, and consider a range of factors that include the permeable escape of vapor and the potential failure of bombs due to pressure accumulation. The questions that you are asking *are* important, and the overall objectives of the paper could be more clearly communicated to better reflect this.
As you will see from the attached annotated .pdf of the paper, I have added some comments and questions to the text. These largely reflect my encouragement for you to build a much stronger bridge with the existing literature, (re)consider some aspects of the modelling, and add a discussion section in which you can emphasize the importance of your model results to our understanding of Surtseyan eruptions.
My major comments/criticisms are that: 1.
Stronger bridges are needed with existing literature on magma-water interactions. Yes, it is an interesting thought experiment to explore whether a Surtseyan bomb will explode due to the high pressure of vaporized entrained steam, but could you please consider and emphasize the importance of this process? How might the physics of magma-water interaction be affected? And the eruption mechanisms? 2.
Please give more information about the geological setting of the studied bombs, including the deposits, bomb sizes, and inferred emplacement mechanism. It is stated that the studied bombs were smaller than typical bombs at the study site, but more information is needed.

3.
A cartoon illustrating processes in Surtseyan eruptions will greatly improve the paper. This could show the water body, vent, crater, slurry, etc, and also zoom in to show small-scale magma-slurry interactions being considered in the models.

4.
Please discuss whether the vesicularity of bombs is entirely pre-fragmentation, or whether some post-fragmentation vesiculation also occurs. Do the current bomb textures faithfully record the moment of slurry ingress? If not, then should the final permeability values be treated with some caution?

5.
Please consider the thermal and physical state of the "host" bomb close to the slurry. Was it above or below the glass transition temperature when the vapor expansion occurred? It was stated that the near-slurry magma looked compressed, and then stated that compression of magma was not considered. I think it is important to consider and discuss whether the slurry vapor expanded when the bomb was still hot enough to viscously deform. If that had happened, then the vapor expansion and pressurisation could hypothetically drive compaction and collapse of the vesicles.

6.
It would be useful to explicitly state whether you consider the pressure dependence of the vaporization temperature of water.

7.
The discussion of mesh size vs model results is somewhat concerning as it shows to what an extent the model set-up affects the outputs. I'd like to see a deeper discussion of this, and the relationship with the vesicle size, which is touched upon. Some of this writing on model set-up could fit best in an appendix.

8.
Have you considered what particle size is present in the slurry? The Fig Please add a discussion section, in which you discuss the relevance of the model results, their implications for physical processes occurring in Surtseyan eruptions, the shortcomings of the modelling approach, prospect for future work, and more. Stronger connection with existing literature would be useful here. The Editor of Proceedings A has now received comments from referees on the above paper and would like you to revise it in accordance with their suggestions which can be found below (not including confidential reports to the Editor).
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When revising your paper please ensure that it remains under 28 pages long. In addition, any pages over 20 will be subject to a charge (£150 + VAT (where applicable) per page). Your paper has been ESTIMATED to be 18 pages pages. The manuscript under review elaborates a mathematical model for Surtseyan bombs, which are a particular type of volcanic ejecta where hot vesicular lava encloses a cool, water saturated chunk of tephra. The manuscript reports microstructure measurements that are used to estimate the permeability, the development of a mathematical model based on physical arguments, the analysis of that model, and results of numerical simulations and analytical approximations. A previous paper by the same authors developed the original model, which made the simplifying approximation of a frozen temperature field. In the present work, the authors couple the temperature and pressure fields. They obtain a result that differs by a ratio of length scales associated with the initial temperature condition, and is potentially large. They conclude by noticing that their model predicts a minimum permeability to avoid fracture, and that there are no observations of bombs with permeability below this minimum.
The manuscript has a lot of nice features, including novel permeability measurements, an appealing physical model that makes reasonable idealisations, and clever analysis in estimating the relevant physical scales and making simplifications. However it also has some significant problems that should be addressed before I can recommend publication.
The first of these is the cursory introduction that doesn't provide a clear motivation for the paper. It gives a brief overview of the physical picture of Surtseyan bombs, but doesn't discuss a motivation for physical/mathematical modelling of these features. This might have taken the form of questions to be addressed and the hypothesis that is inherent in the models (the pressure of flashing is relieved by porous flow of steam when the permeability is large enough). Moreover, the introduction doesn't discuss why previous theoretical work is inadequate in addressing the questions. Finally, the introduction should provide an overview of the manuscript, explaining its organisation and, I suggest, also its key results. Instead the introduction breaks abruptly at a subsection called "Methods" which is really just about the methods of the permeability measurements (and the results) and then comes back to provide some description of the model. The text and figure about permeability measurements should be moved out of the introduction and into their own section, afterwards.
The manuscript also lacks a more detailed discussion of the model and its limitations, the results and their implications. What discussion there is is provided in a very brief conclusion section. The last two sentences seem to be the only reference back to the motivating questions (presumably) that connect the mathematics to the physical volcanology. I discuss my sense of the model limitations below. But there is a lack of connection to the physical volcanology here. What can equation (6.1) and Figure 5 tell us about the eruption process and materials? If nothing, why?
The model development in section 2 seems a reasonable balance between explaining the physics (though see detailed points below) and not getting too bogged down in the details that will later be neglected. The model reduction section, however, seems to move hastily through a series of approximations based on the size of dimensionless parameters, dropping terms without discussion of the physics that they represent. This seems problematic in that the initial conditions then create transients in which some variables (and their derivatives) are much larger than unity. Because of the lack of physical discussion, it is difficult to see which of the neglected terms might become important, and what its qualitative effect would be. While it is reasonable to make simplifications, the authors provide no discussion a posteriori of the consequences of these simplifications, and hence of the limitations of the results.
Two simplifications that concern me are as follows. In eqn (3.8), the term in \delta_3 is neglected. \delta_3 is indeed small, but near the flashing front (where all the interesting action occurs), \partial p/\partial t can be very large. In equation (3.13), the term from the previous equation in \lambda_2\epsilon_4 has been neglected relative to the jump in heat flux. There is only a factor of 10 in the coefficients, but p can be very large near the flashing front. So in both cases (and perhaps others) in seems somewhat unclear whether neglecting the terms can be justified on physical grounds. At the very least, some discussion of this a posteriori should be included.
A lot of attention is then given to the nifty matched asymptotics for the analytical approximation, which in Fig. 3 seems to do a reasonable job of matching some aspects of the solution. But it seems to fail badly in matching the maximum pressure, which is the key physical output of interest. So after reaching figure 5, it is unclear why the asymptotic model is included in so much detail (or at all). What do we learn from it of value? In figure 5a, the mismatch between numerics and the approximation is as big as the difference between the fracture criterion and the numerics. It feels like much of the manuscript is driven by an interest in mathematical analysis rather than an interest in the physical predictions of the model. I find this problematic, given the motivation of the work.
I am rather dubious of the implication that the model is correct because of the lack of observations of permeabilities less than one Darcy. One Darcy is very small for a magma with porosity of 0.35--0.8, even if the pores are bubbles. The measurements were all made on bombs containing inclusions, I think, which is a sampling bias. What if fragments of fractured bombs had permeabilities in the same range?
Detailed comments by page:line(s). (Page numbers are as in the header, not the black box) 3:13. How are "bombs" related to "lump of lava" and "tail of tephra"? Are they the lump? 3:56. What is "Palabos"? 4:12. Can this be described using a more typical form such as Kozeny-Carman? Is this form usual or just convenient? 4:52. So the model must be accurate over times of less than a second. This suggests that the choice of initial condition is paramount. Obviously getting the IC right isn't easy... 6:29. It is a bit hard to see how the averaging was conducted. Is the microscopic pressure equal to the continuum pressure? What about the stess \tau? The usual upscaling would give a dissipation rate of something like K_e u_l^2 for laminar flow (but you haven't yet assumed laminar flow). I think it would be better to jump right to the continuum equations.
(2.5). A better choice of notation to disambiguate thermal conductivity and permeability would be nice, while avoiding K_el, which looks more like a bulk modulus.
(2.8). Is the \beta in this equation the same as the \beta in line 57 of the same page? I think they're different... 6.57. "this equation" --indefinite antecedent. (2.9). Again the dissipation term is problematic. 9.29. "over-estimate" by how much? What effect does this have on other approximations made to simplify the equations? 11.52 (and elsewhere). The Ideal Gas law, not the perfect gas equation? 13.11 Mention use of subscript notation for partial derivatives Fig 3. Needs panel labels and a much clearer caption that refers to the panels. Maybe break into two figures? Needs much more explanation in the text. 15.10. If the mesh size that is needed on the basis of characteristics of the solution is similar to the pore size, then there's a potential problem with the continuum assumptions. This needs to be highlighted. 15.58. typo 17.47. "they" --indefinite antecedent. Confusing sentence. 17.58. It is concerning that the solution for maximum pressure depends on a ratio that can be arbitrarily large. First of all, it is unrealistic to assume that the enclosure is captured in such a way that there is a step-function in temperature. And secondly, if the mathematics doesn't regularise this then I think the model is ill-posed (possibly due to neglecting a "small" term that isn't small near the flashing front at small time).
Referee: 2 Comments to the Author(s) Thank you for an interesting paper to review and apologies for my lateness.
You have presented some very detailed models of magma-slurry interaction in Surtseyan eruptions, and consider a range of factors that include the permeable escape of vapor and the potential failure of bombs due to pressure accumulation. The questions that you are asking *are* important, and the overall objectives of the paper could be more clearly communicated to better reflect this.
As you will see from the attached annotated .pdf of the paper, I have added some comments and questions to the text. These largely reflect my encouragement for you to build a much stronger bridge with the existing literature, (re)consider some aspects of the modelling, and add a discussion section in which you can emphasize the importance of your model results to our understanding of Surtseyan eruptions.
My major comments/criticisms are that: 1. Stronger bridges are needed with existing literature on magma-water interactions. Yes, it is an interesting thought experiment to explore whether a Surtseyan bomb will explode due to the high pressure of vaporized entrained steam, but could you please consider and emphasize the importance of this process? How might the physics of magma-water interaction be affected? And the eruption mechanisms?
2. Please give more information about the geological setting of the studied bombs, including the deposits, bomb sizes, and inferred emplacement mechanism. It is stated that the studied bombs were smaller than typical bombs at the study site, but more information is needed.
3. A cartoon illustrating processes in Surtseyan eruptions will greatly improve the paper. This could show the water body, vent, crater, slurry, etc, and also zoom in to show small-scale magma-slurry interactions being considered in the models. 4. Please discuss whether the vesicularity of bombs is entirely pre-fragmentation, or whether some post-fragmentation vesiculation also occurs. Do the current bomb textures faithfully record the moment of slurry ingress? If not, then should the final permeability values be treated with some caution? 5. Please consider the thermal and physical state of the "host" bomb close to the slurry. Was it above or below the glass transition temperature when the vapor expansion occurred? It was stated that the near-slurry magma looked compressed, and then stated that compression of magma was not considered. I think it is important to consider and discuss whether the slurry vapor expanded when the bomb was still hot enough to viscously deform. If that had happened, then the vapor expansion and pressurisation could hypothetically drive compaction and collapse of the vesicles.
6. It would be useful to explicitly state whether you consider the pressure dependence of the vaporization temperature of water.

Is the paper of sufficient general interest? Acceptable
Is the overall quality of the paper suitable? Good

Recommendation?
Accept with minor revision (please list in comments)

Comments to the Author(s)
The authors have done a nice job of revising the manuscript and responding to reviewer comments. I think their a posteriori checks on the insignificance of terms is an important confirmation of the approximations.
The writing in the manuscript could be better; there are various places where sentences are convoluted and/or too long, or the wording is awkward or repetitive.
Although it is probably inconsequential, their representation of dissipation by Darcy flow is wrong, as I think I mentioned in my previous review. I refer to the unnumbered equation after (3.11), which uses the viscosity times the square of what the authors seem to consider a strain rate. The actual dissipation in porous flow occurs in the Poiseuille-type flow in the pore-throats, at a strain-rate that is something like the microscopic flow speed divided by the pore-throat radius. At the continuum scale, this is given by \mu/k * u.u, where \mu is viscosity, k is permeability, and u.u is the square of the magnitude of the Darcy flux vector. I'm sure this is in a standard text such as Bear's Dynamocs of Fluids in Porous Media. 9-45: Units on the dimensional temperature of 300? 10-7: "expected to be weaker" --actually I would say that the (1-\phi) factor accounts for the fractional area that is load-bearing. 10-13: "restrict to spherical geometry" --wasn't this already done above? 16-56: "We will derive an upper..." --this sentence seems to make no sense (though I think I understand that you are trying to say that the upper bound diverges as the gradient diverges) 18-57: "the smallest data value for permeability of intact bombs..." --this is an important sentence but the logic is not sufficiently clear. Be explicit about the nature of the "consistency". The reviewer(s) have recommended publication, but also suggest some minor revisions to your manuscript. Therefore, I invite you to respond to the reviewer(s)' comments and revise your manuscript. Please note that we have a strict upper limit of 28 pages for each paper. Please endeavour to incorporate any revisions while keeping the paper within journal limits. Please note that page charges are made on all papers longer than 20 pages. If you cannot pay these charges you must reduce your paper to 20 pages before submitting your revision. Your paper has been ESTIMATED to be 22 pages. We cannot proceed with typesetting your paper without your agreement to meet page charges in full should the paper exceed 20 pages when typeset. If you have any questions, please do get in touch.
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10-Aug-2021
Dear Dr Greenbank I am pleased to inform you that your manuscript entitled "A theoretical model of Surtseyan bomb fragmentation" has been accepted in its final form for publication in Proceedings A.
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