Understanding Magma Ascent through Heat Propagation Buoyancy and Thermal Loss in the Lithosphere: a Thermodynamical Model (MagmaLitho v1)

Godano, Cataldo; Gorgone, Matteo; Munafó, Carmelo F.; Carlino, Stefano; Rogolino, Patrizia; Oliveri, Francesco

doi:10.5194/egusphere-2026-3200

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https://doi.org/10.5194/egusphere-2026-3200

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22 Jul 2026

| 22 Jul 2026

Status: this preprint is open for discussion and under review for Geoscientific Model Development (GMD).

Understanding Magma Ascent through Heat Propagation Buoyancy and Thermal Loss in the Lithosphere: a Thermodynamical Model (MagmaLitho v1)

Cataldo Godano, Matteo Gorgone, Carmelo F. Munafó, Stefano Carlino, Patrizia Rogolino, and Francesco Oliveri

Abstract. We introduce a rather simple model for the ascent of the magma through the lithosphere, based on a mix of conduction, rock partial melting, and buoyancy, where we take into account the heat dissipation due to the melting process. Under suitable assumptions, a hyperbolic linear partial differential equation describing the process is derived from a general thermodynamical model whose constitutive equations are compatible with the second principle of thermodynamics. Then, we derive numerically a solution to a physically meaningful initial/boundary condition that can explain many of the experimental observations. More precisely, we suggest a mechanism for the magma emplacement able to explain the different depths of emplacement depending on one of the model parameters. Moreover, this parameter can be tuned to obtain the disappearance of the emplacement, which justifies the occurrence of hot spots (volcanoes forming a plume of molten rocks that rises from deep within the Earth’s lithosphere).

Received: 03 Jun 2026 – Discussion started: 22 Jul 2026

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Cataldo Godano, Matteo Gorgone, Carmelo F. Munafó, Stefano Carlino, Patrizia Rogolino, and Francesco Oliveri

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RC1:
'Comment on egusphere-2026-3200', Anonymous Referee #1, 18 Sep 2026 reply
The manuscript presents an interesting modelling framework for understanding magma ascent through the lithosphere and begins to address the thermal controls on magma propagation. The theoretical framework is potentially valuable, and I appreciate the authors’ efforts to develop a model that captures some of the complexities of magma ascent. However, I found the manuscript difficult to follow in some places, particularly with respect to the interpretation of some of the results and the treatment of changes in melt fraction, density, and viscosity through time within the model. In some instances, the interpretation of model results appears broader than what can be fully supported by the current model formulation, and I think these interpretations would benefit from being more carefully expressed. I also found that several definitions, equations, initial and boundary conditions, and figures would benefit from additional explanation for the manuscript to be accessible to a broad readership. Overall, I think the manuscript would benefit from further clarification of the model assumptions and limitations, as well as a more nuanced discussion of the implication of the results.
Scientific comments
The abstract is somewhat difficult to follow, and I was not always able to identify where all the concepts introduced in the abstract are shown in the manuscript. I suggest revisiting the abstract to ensure that all the concepts are clearly supported in the main text.

Line 5: It would be helpful to clarify what is meant by a ‘physically meaningful initial/boundary condition’. Is this meant to describe a condition that is both a boundary and initial condition, or are two separate types of conditions being referred to?

Line 27-28: I found the statement ‘ crystal content of magma being incompatible with density contrast required for magma propagation in fractures’ difficult to follow. As currently written, it appears to suggest that ascending magma cannot contain crystal cargo while propagating through fractures. However, there is evidence for crystals derived from deeper regions of magmatic systems being transported to the surface, demonstrating that crystals can be present in ascending magmas (see Cashman et al. 2017, Science for a review). Furthermore, numerical modelling has suggested that crystals can be present within a high melt fraction magma body capable of generating sufficient buoyancy for magma propagation (e.g. Booth et al. 2026, JPet). Therefore, I suggest the authors clarify this statement.

Line 39: Could the authors clarify what drives heat propagation in this model? Is this primarily controlled by the temperature gradient through the lithosphere, radiogenic heat production, heat flow from the mantle, or a combination of these processes?

Line 44: There appear to be a number of approximations introduced throughout the manuscript. I would encourage the authors to provide a consolidated list of the principal assumptions and approximations, perhaps in an appendix, together with a brief discussion of their expected implications for the results. This would be easier for readers to assess the applicability and limitations of the model.

Line 51: Could the authors clarify whether the magma batch at the lithosphere-asthenosphere boundary is present only as an initial condition, prior to ascent, or whether a magma source is continuously maintained at this boundary throughout the model?

Line 53: Point 3. Could the authors clarify whether this is intended to describe the buoyant upwelling of the magma batch as a results of newly melted material? It is currently hard to understand.

Lines 58-59: The index, i, does not appear to be defined. Please clarify what this index means.

Lines 62-64: If T_i =T_m, how does the model account for the possible addition of melt volume to the upwelling magma body as a result of partial melting? More generally, if partial melting of surrounding rocks is included, this process could alter both the volume and composition of the magma. Could the authors discuss how these effects are treated? In addition, compositional changes associated with partial melting could affect the ascent velocity. I would appreciate clarification as to whether partial melting can be fully represented in the current formulation with accounting for these effects.

Line 69-70: References to recent experiments here would be helpful, particularly where these experiments provide the physical basis for assumptions made in the model.

Lines 84-85: References should be provided for the experiments conducted during the 20^th century that are mentioned in this sentence.

Line 91-92: There is a relatively abrupt transition between these paragraphs. I would appreciate some additional explanation of why the framework of EIT is introduced at this point.

Line 111: inequality (4) appears before it has been introduced, which made this section difficult to follow. I suggest introducing the inequality earlier or restructuring this section that the reader is familiar with it before it is used.

Line 125: Could the authors clarify whether the lambda introduced here is the same quantity as the lambda used in the heat-flux formation? If these are different quantities, alternative notation may be good.

Line 131: Please explicitly identify the field equations being referred to, preferably by their equation numbers, so that the readers can easily identify them.

Line 145: It was not immediately clear to me how the matrices in these formulation become the same size such that they can be added. I suggest providing the dimensions of the matrices.

Line 181: The model allows pressure to vary with density and entropy, but pressure will also vary with depth through the lithosphere. How is the depth dependence incorporated into the model? Does density also vary with depth?

Line 182: Are the assumptions in equation (9) the constants?

Line 183: Could the authors specify what ‘other coefficients’ are being referred to here? It would be helpful to identify these explicitly.

Line 185: Could the authors discuss more explicitly how the results would change if velocity was able to vary through time in response to changes in viscosity and density? This seem particularly important if partial melting is a part of the model, as the associated compositional change of partial melting could substantially modify magma properties and therefore ascent velocity.

Line 186: Could the authors explain the justification for neglecting the higher-order terms? Is this related to satisfying the second law of thermodynamics?

Line 191: Could the authors explain why the coefficient r_22 is assigned this value?

Line 209-224: I do not disagree with the simplifying assumptions adopted here for the development of this model. However, there are existing models that incorporate temperature, composition, volatile content and changes in magma viscosity during ascent through a dike (e.g. Abdullin et al. 2024 GJI). I suggest that these studies are acknowledged.

Line 225: Density, viscosity and therefore velocity can all depend on magma composition. Since partial melting appears to be represented in the model, compositional evolution may therefore be an important consideration. I suggest explicitly acknowledging this as a limitation and discussing its potential consequences for the model results.

Line 229-230: I am unsure how the results would be expected to change in three dimensions if propagation is restricted to one dimension, other than faster heat loss. If propagation was allowed in multiple directions, I would expect to see substantially different behaviour, e.g. sill formation. Therefore, I am unsure whether this comparison provides a meaningful validation of the present model.

Line 233: Could the authors explain the rationale for selecting a 3-year timescale for non-dimensionalisation? What is this based on?

Line 237 and throughout: It would be helpful to restate the definitions of u and d. More generally I recommend using the variable’s name rather than symbol within the text for easier readability, especially when there are a lot of symbols to keep track of.

Line 243 and figure 2: Could the authors provide a reason for why these initial and boundary conditions have been chosen? How do they compare to natural systems?

Line 251: I would collate the initial and boundary conditions for readability.

Line 259: 1000oC is relatively high for a rock melting temperature. Some rocks can melt at 700C depending on composition, pressure and volatile content. Could the authors provide the source for the selected melting temperature and specify what rock composition is being represented?

Line 261: The statement that thermal diffusivity does not play a significant role appears to be an important conclusion. Could the authors expand on what this means for magmatic systems? And whether there are any conditions where they might expect thermal diffusivity to become important?

Line 263-267: I understand the authors’ argument that differences in ascent velocity could contribute to differences in magma composition. However, they do not more magma composition change. Furthermore, several properties may affect ascent velocity (density and viscosity) and vary with magma composition and temperature, and these effects are not represented in the current model. In addition, volatile content can have a substantial influence on magma properties and ascent behaviour. Therefore, I suggest that this conclusion is presented more cautiously and that the authors discuss the extent to which it can be supported by the current model

Line 278-279: Magma differentiation could affect several properties in this model, including density. I suggest that the consequences of this simplification are more explicitly acknowledged as a limitation.

Line 282: Processes such as reactive flow and compaction may also influence magma differentiation and transport, including at high melt fractions.

Line 284-285: I find the representation that magma ascent through the brittle crust is ‘no longer dominated by buoyancy forces’ to be not entirely true. Although other forces and mechanisms may become more important in the brittle crust, buoyancy can remain an important driving force for magma migration. Buoyancy can contribute to the pressure differences that can drive magma into and through fractures. I suggest that this statement is reconsidered.

Appendix A1: The magma volumes assumed to erupt from the chamber (0.00005-0.05km3, if I’ve understood correctly) appear relatively small compared with estimated critical chamber volumes associated with eruption in Townsend and Huber 2020 (approximately 1-10km3), as well as volumes inferred for lower-crustal magma bodies in Booth et al 2026 (> approx. 1km3). Could the authors explain the rationale for selecting these relatively small magma volumes and discuss how representative they are of natural systems? Also why only 10% of magma will leave the chamber?

Melt fraction does not appear to be explicitly modelled, although the magma body cools during ascent, which would be expected to influence its melt fraction. As melt fraction decreases, the magma may transition into a crystal-rich mush, potentially reaching a point at which sustained ascent becomes difficult or ceases. Could the authors discuss how robust the predicted ascent behaviour is in absence of modelling melt fraction?

Composition does not appear in the model, despite partial melting being present. As stated in previous comments, composition substantially changes viscosity and density. Could the authors discuss how the conclusions depend on the assumption that these properties remain constant through time?

Technical/presentation comments
Abstract: I would suggest removing ‘rather’ from the first sentence, as this reads somewhat informally.

Line 51: ‘Ingredients of our model’ feels informal. I suggest replacing this with ‘fundamental components of our model’ or ‘key components of the model’.

Line 54: Point 4, I am unsure why the word ‘again’ is present.

Line 58: It would be good to reference figure 1 here.

Line 81 and 88: These equations should be numbered as they are referred to later in the manuscript.

Line 81 and 88: Please define all terms in the equation.

Figure 1: I would appreciate a more detailed description of figure 1, particularly an explanation of the relationship between the four panels and how the four panels relate to each other.

Line 120: It is difficult to determine which gradients are being referred to here. Rather than using ‘the latter’, I suggest explicitly naming the relative gradients.

Line 159: ‘some algebra’ is informal. I suggest either replacing this with a more formal description of the maths being performed or reword.

Line 164: The list of equations would be easier to follow if the individual equations were numbered

Line 240: Please number these equations.

Figure 3-5: I understand why the authors have used non-dimensional variables when plotting. However, it would be more impactful if the dimensional variables could be plotted so that a wider audience could understand the results and gather information.

Figure 3- 5: The axis labels are not currently clear. Ensure that the physical meaning and, where relevant, dimensionality of each plotted quantity is apparent.

All figure captions: The figure captions would benefit from substantially more description. At present, they do not always provide enough information for the reader to understand the figures without repeatedly referring to the main text.

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Citation: https://doi.org/10.5194/egusphere-2026-3200-RC1
- AC1: 'Reply on RC1', Cataldo Godano, 01 Oct 2026 reply
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-3200/egusphere-2026-3200-AC1-supplement.pdf
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  Citation: https://doi.org/10.5194/egusphere-2026-3200-AC1
RC2:
'Comment on egusphere-2026-3200', Anonymous Referee #2, 25 Sep 2026 reply
The authors derive, from the principle of non-negative entropy production, a linear partial differential equation that introduces the second time derivative of temperature (hyperbolization of Fourier’s law) together with a mixed time–space derivative treated as a dissipation term. The resulting equation is then applied to the ascent of magma from the asthenosphere–lithosphere boundary toward the shallow crust and the formation of a magma chamber.
A review of the literature shows that the authors are familiar with existing advanced models. Their stated goal is to introduce a simple model that nevertheless captures the essential physics. The main advantage of the approach is that the PDE remains linear. However, with modern numerical methods and current computational power, the solution of nonlinear PDEs in one or even higher dimensions can no longer be regarded as intractable.
Despite these positive aspects, there are several significant issues with the model itself that prevent its application to real magma transport.
Non-negative entropy production leads to a linear equation only after most of the constitutive coefficients are deliberately set to zero. The only reason given in the paper is that the authors “want a linear equation” (l. 183). Why is linearity necessary? What essential physics is lost? Moreover, Eq. (9) specifies only a few of the coefficients, so system (10) does not follow from the stated assignments: e.g. the term r14 q_z in r1 is linear and survives the neglect of higher-order terms unless r14 = 0 is imposed. All coefficient choices should be listed explicitly.

The heat-flux relaxation time τ is identified with the Maxwell viscoelastic relaxation time of the lithosphere, τ = µ/G (Appendix A3). This identification is not derived and is not required by thermodynamics. Heat-flux relaxation in rocks is controlled by phonon scattering and is many orders of magnitude shorter than 10^5 s.

The model assumes constant physical properties (density, viscosity, etc.) for both host rock and magma, whereas these quantities are clearly temperature-dependent. The references to the Boussinesq approximation (l. 209–224) do not justify this: in the Boussinesq approximation the temperature dependence of density is retained in the buoyancy term, whereas here the ascent velocity u is prescribed and independent of T. Temperature is therefore a passive quantity transported at a fixed speed, and the “emplacement” is only a threshold (T < 1000 °C) applied to the solution, not a mechanism that stops the magma. Magma and rocks have a strong temperature dependent viscosity that also need to be considered in the model.

The Stokes ascent velocity is formulated for a buoyant body rising in an infinite viscous medium. In contrast, the conceptual model assumes the formation of only a thin shell of melt around the ascending batch. In this geometry the actual velocity would be orders of magnitude lower. Moreover, the length scales are inconsistent: the Stokes velocity (A1) is computed for R = 50–500 m, the initial condition (14) implies a batch of ≈ 4 km, and the annulus radius r = 2τu is ≈ 20 m.

It is not clear why the density of the first partial melt is assumed to be higher than that of the magma. This depends strongly on crustal composition; the initial melt is usually more silica-rich and therefore lighter.

Even if the molten shell were denser, the configuration would be gravitationally unstable and would trigger Rayleigh–Taylor instabilities, leading to efficient mixing and rapid cooling of the magma body. This process is ignored in the model equation.

It is unclear why horizontal heat loss can be neglected compared with vertical heat transport. The statement that three-dimensional computations do not change the results substantially (l. 229–230) should be supported by showing these results. More fundamentally, the conductive length scale over the ascent time is √(d t) ≈ 10 m (d = 10^-6 m^2 s^-1, t ≈ 2.7 yr), and the thermal wave speed of the hyperbolic equation, √(d/τ) ≈ 3·10^-6 m s^-1, is about 30 times smaller than u. Heat therefore cannot propagate ahead of a kilometre-scale batch and melt the host rock, which is the core mechanism of the model (Sec. 2).

Scaling analysis of the terms in Eq. (13) shows that the equation is advection-dominated to leading order, while the newly introduced hyperbolic terms (T_tt, T_tz) are smaller by roughly three orders of magnitude. This is consistent with the authors’ observation that the results are independent of d (l. 261). The characteristic speeds of Eq. (12) are u ± √(u² + d/τ), i.e. ≈ 2u and ≈ 0, so the solution is essentially the initial profile transported upward.

The “new dissipative term” 2τu T_tz is not related to melting. Cross-differentiation of Eqs. (10c) and (10d) shows that it arises from applying the material derivative (∂_t + u∂_z) twice in the Cattaneo law. The choice r22 = −τρ0u² serves only to cancel the resulting τu² T_zz term; it makes a constitutive coefficient depend on the flow velocity, which is not Galilean invariant. The physical interpretation of this term (l. 199–200, 207–208) is therefore unfounded.

Equation (13) is second-order in time, higher than the original balance-law system. Consequently, an additional initial condition is required. This extra condition is neither specified nor discussed. No boundary condition at z = 1 is given either.

With κ = −2·10^-9 s^-1 the last term of Eq. (12) is a heat sink proportional to T, not the energy source from the asthenosphere described in l. 200–204.

The initial condition (14) does not represent a lithospheric geotherm: it gives ≈ 50 °C at 10 km depth. A lithosphere thickness of 20 km is also unusually thin.

The authors claim that the model explains many physical aspects of magma ascent dynamics, yet no comparison with field data or with more advanced numerical models is provided. The link between magma chemical composition and the model results also remains unclear.
The code availability statement is inaccurate: Mathematica is proprietary software, so the code cannot be run without a licence, and the sentence on data and scripts (l. 298–299) is unclear.
Minor: the viscosity unit should be Pa s, not Pa (Appendix A1, A3); the diffusivity unit should be m^2 s^-1 (Appendix A2); several derivatives in the set X use x instead of z (l. 143); inequality (4) is referred to before it is defined (l. 111).

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Citation: https://doi.org/10.5194/egusphere-2026-3200-RC2
- AC2: 'Reply on RC2', Cataldo Godano, 01 Oct 2026 reply
  
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  Citation: https://doi.org/10.5194/egusphere-2026-3200-AC2

Cataldo Godano, Matteo Gorgone, Carmelo F. Munafó, Stefano Carlino, Patrizia Rogolino, and Francesco Oliveri

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Short summary

On the basis of a a general thermodynamical model we derive a hyperbolic linear partial differential equation describing the ascent of the magma through the lithosphere. The model is based on a mix of conduction, rock partial melting, and buoyancy, and takes into account the heat dissipation due to the melting process.


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