Comment on “Pressure‐to‐Depth Conversion Models for Metamorphic Rocks: Derivation and Applications” by Bauville and Yamato

Bauville and Yamato (2021, https://doi.org/10.1029/2020gc009280) (G‐cubed) propose model‐based methods to convert metamorphic pressures to depths based on the claim that pressure data from global (ultra)high‐pressure([U]HP) rocks challenge the lithostatic assumption and support their model which invokes excessive overpressures. It is argued here that the opposite is true: Natural pressure data are fully consistent with the lithostatic assumption. They reflect selection of (U)HP rocks by accessibility and preservation. The data are however inconsistent with the model predictions of Yamato and Brun (2017, https://doi.org/10.1038/ngeo2852) and Bauville and Yamato (2021, https://doi.org/10.1029/2020gc009280). Furthermore, their model requires critical assumptions that are not justified by the principles of rock mechanics and unsupported by microstructures from (U)HP rocks.


Introduction
and Yamato and Brun (2017) claim that metamorphic pressures from global (ultra)high-pressure ([U]HP) rocks challenge the lithostatic pressure assumption but support their model that invokes excessive overpressures. Bauville and Yamato (2021) propose methods to convert metamorphic pressure data to depth on the basis of the Yamato and Brun model and its development. The purpose of this comment is threefold. First, I contest their interpretation of the natural pressure data and argue that the data are fully consistent with and better explained by common interpretations based on the lithostatic assumption. Second, I point out that their model requires critical assumptions that are not justified by the principles of rock mechanics and are unsupported by microstructures of (U)HP rocks. Finally, I question some concepts and derivation in Bauville and Yamato (2021) related to finite strain deformation, stress rotations, and the Mohr-Coulomb rheology.

Do Pressure Data From (U)HP Rocks Challenge the Lithostatic Assumption and Support a Mechanic Model Invoking Excessive Overpressures?
The mineral assemblages of (U)HP rocks commonly record a "peak" pressure (P p ), which is commonly interpreted by researchers to represent the maximum depth of rock burial (Chopin, 1984;Smith, 1984), and a lower "retrograde" pressure (P r ) interpreted to represent the depth of the initial isothermal decompression (Ernst et al., 2007;Hacker & Gerya, 2013;Powell & Holland, 2010). The pressure drop,   p r ΔP P P, thus corresponds to the amount of exhumation attained by the isothermal decompression. This interpretation assumes that P p and P r are approximately lithostatic (lithostatic assumption, hereafter). In reality, both P p and P r may deviate from the lithostatic values, but the magnitude of deviation is limited by the rock strength, which is likely less than hundreds of MPa for the time scale relevant for (U)HP metamorphism and far below the GPa level lithostatic pressure (e.g., Jiang & Bhandari, 2018).
The pressure data from global (U)HP rocks as compiled in Bauville and Yamato (2021)

COMMENT
depth without actual ascent of the rocks. Bauville and Yamato (2021) argue that there is a linear dependence of P r on p P that requires their model to explain.
Let us examine the plots in Figures 1a and 1c carefully and see if the assumption that p P and P r are lithostatic will lead to great difficulty.
As p P , P r , and ΔP are related by   p r Δ P P P, for each data point in Figure 1a, one can draw a line of unit slope passing the data point and the intercept of the line on the vertical axis is the corresponding r P (Figure 1b). Considering this for all data points in the set, one realizes that all r P are clustered within a narrow strip ( r P , purple-shaded in Figure 1a The same data with error bars plotted in the p P versus r P space. The upper bound of the gray-shaded area is given by  p r P P. No data may plot in this area.  r P corresponds to the range shown in (a). The blue dashed line (  p 3 r P P) corresponds to that of  p 1.5Δ P P in (a). By changing the horizontal stress magnitude or varying the principal stress orientation, varying slopes for the p P versus r P relation (orange dashed lines) are predicted by Bauville and Yamato (2021) and Yamato and Brun (2017). Purple shaded region outlines the domain (U)HP rocks are preserved and accessible. The data are compiled in Bauville and Yamato (2021). See text for more detail. r P . With the lithostatic assumption,  r P corresponds to depths between ∼12 and 50 km. Thus, Figure 1a suggests that although (U)HP rocks in the current data set were formed over a great pressure range (from below 1 GPa to over 4 GPa), corresponding to 35 km and >140 km depth difference, they were exhumed during the isothermal decompression stage to the limited depth range of ∼12-50 km. This depth range may simply represent the interval where (U)HP rocks are preserved after formation at deeper levels and are accessible to our observations. (U)HP assemblages with r P >∼1.3 GPa may have not been preserved and, if preserved, may still be buried and not accessible for observation yet. Thus, the linear trend of the data may simply reflect natural selection of (U)HP rocks by accessibility and preservation, independent of exhumation mechanisms.
A big claim of Yamato and Brun (2017, p. 47) is that the linear relation as shown in Figure 1a challenges the lithostatic assumption but supports their analysis which yielded a model prediction of , where  and C are the friction angle and cohesion respectively. The dashed blue line in Figure 1a,  p 1.5Δ P P , is for    30 and C = 0. However, the data trend has a much shallower slope near unity and a significant positive intercept of 0.52 that are inconsistent with the model prediction.
Perhaps noticing the above discrepancy between data and prediction, Bauville and Yamato (2021) used the p P versus P r plot instead. In the plot of the same data here (Figure 1c), I have used an equal scale for P r and p P to avoid distortion of line slopes. Figure 1c is also fully compatible with the lithostatic assumption. One should note that although in the lithostatic interpretation p P and P r represent two events at different depths, the distribution of p P versus P r cannot be totally random in space because of the following constraints. First, by definition all data must plot above the gray-shaded area whose upper bound is given by the  p r P P line (Figure 1c). Second, as (U)HP rocks are formed in low-temperature and high-pressure settings, they must be exhumed, shortly after formation (Ernst et al., 2007), to shallower depths (corresponding to  r P in Figures 1a and 1c) so that the (U)HP assemblages are preserved. Direct geological observations are also constrained by the accessibility of rock exposures. The  r P interval is consistent with accessible depth range for direct observations. Thus, a greater p P must in general be associated with a greater ΔP, as supported by Figure 1a, for the rocks to reach the  r P interval. Although the exhumation rate for (U)HP rocks varies and may be as fast as the subduction rate (e.g., Parrish et al., 2006;Rubatto & Hermann, 2001), the maximum amount of stage 1 exhumation is always limited by the duration of the exhumation multiplied by the rate of exhumation. This means that an extremely low P r (like 0.5 GPa) associated with a very high p P (like 4.0 GPa) is unlikely, as such a p P and P r pair requires an unreasonable amount of exhumation in stage 1 (Figure 1c). The current data set suggests that 12 km (∼0.3 GPa) might be the shallowest depth to which (U)HP rocks can be exhumed by stage 1 exhumation. With the above constraints considered, the distribution of p P and P r in Figure 1c is fully consistent with P r being independent of p P .
The argument of Bauville and Yamato (2021) that Figure 1c shows a linear dependence of P r on p P is rather far-fetched. To explain the spread of the data, the authors have to (a) invoke a wide range (from zero to several GPa) of differential stress values by a change in horizontal stress magnitude or a rotation of the stress tensor, and (b) propose that the "linear dependence" of P r on p P is represented by a fan area. There are two issues. First, once the stress tensor is rotated, the stress state is non-Andersonian. The assumption by Bauville and Yamato (2021) that    z gz is not justified. Second, as the data are so scattered, the fan area that defines the linear dependence of P r on p P must cover almost the entire p P versus r P space except the gray-shaded area and the area that requires unreasonable amount of exhumation (Figure 1c). It is much simpler to interpret such a wide distribution of the pressure data as demonstrating that P r and p P are independent.

Model Assumptions
The model proposed by Yamato and Brun (2017) which was used and elaborated on by Bauville and Yamato (2021) requires the following assumptions: (a) the rock rheology follows a Mohr-Coulomb plasticity or a Byerlee's frictional behavior, (b) the stress state is close to or at the yield state, and (c) the stress state is Andersonian.
None of these assumptions can be well justified for (U)HP metamorphism. First, Mohr-Coulomb plasticity and Byerlee's frictional behaviors are the rheological responses for the upper brittle lithosphere (Kohlstedt et al., 1995). Such frictional behaviors may occur at greater depth, but only associated with localized, high strain-rate events such as earthquakes (Andersen et al., 2008;Stöckhert, 2002). The pressure data compiled by Bauville and Yamato (2021) and Yamato and Brun (2017) were derived from mineral assemblages that do not represent such events. Tectonic fabrics are common in (U)HP rocks, as noticed by Bauville and Yamato (2021). They reflect large finite strains, consistent with viscous flow over the million-year time scale (Jin et al., 2001;Kohlstedt et al., 1995). Second, stress state close to the yield state at (U)HP depths requires that GPa-level differential stresses (up to 2 times the lithostatic pressure) be sustained for the time scale and P-T condition of (U)HP metamorphism. Such levels of stress are more than an order of magnitude higher than stress estimates for crustal mylonites (e.g., Behr & Platt, 2014;Stipp & Tullis, 2003) and would have caused (U)HP rocks to flow at strain rates many orders of magnitude faster than crustal mylonites (Hirth et al., 2001;Jin et al., 2001;Lu & Jiang, 2019). There is no microstructural evidence from (U)HP rocks that supports this. Third, because (U)HP rocks are rheologically distinct bodies constrained at great depth in the lithosphere, the stress orientations and magnitudes in them are determined by their mechanical interaction with the surrounding lithosphere (Eshelby, 1957;Jiang, 2016;Jiang & Bhandari, 2018), and are unlikely Andersonian. Bauville and Yamato (2021) have used stress and strain terms interchangeably such as using "flattening deformation" for a stress state. This would have been acceptable if one deals with elastic-frictional deformation in isotropic materials because in such conditions the strain is sufficiently small and the principal axes for the stress tensor and for the strain tensor are coincident. However, the authors propose to use the shape of strain ellipsoid obtained from tectonic fabrics to determine the relative magnitudes of principal stresses. This ignores the fact that tectonic fabrics in (U)HP rocks are related to finite strains which accumulate over time in viscous flows and generally by non-coaxial deformation paths (Means et al., 1980). The strain ellipsoid from tectonic fabrics do not have any simple relation to the principal stress directions and relative magnitudes. If the analysis of Bauville and Yamato (2021) is taken to be valid for an infinitesimal deformation, then it is unclear how the analysis can be extrapolated to finite strains accumulated over millions of years of (U)HP metamorphism. Yamato and Brun (2017) considered Andersonian stress state only. Bauville and Yamato (2021) discussed stress rotations at the r P stage in Section 3.2 of their paper. As pointed out above, once the stress state is non-Andersonian, the vertical stress  z is no longer a principal stress and the assumption by the authors that    z gz still holds requires justification. The derivation in Section 3.2 is difficult to follow and it is not clear how Equations 18-20 were derived and then applied to their Figure 7. One notes that the Mohr-Coulomb plasticity, as a constitutive behavior for elastoplastic materials, is coordinate system independent. The orientation of the "yield surface" in a Mohr-circle plot is always measured with respect to the principal stresses. How a rotation of the stress tensor, which amounts to a coordinate system change, should have any effect on the Mohr circle location and size is not clear from their paper. The authors may clarify these points and give more details of how their Equations 18-20 were obtained and applied.

Data Availability Statement
The database used in this paper is available from https://doi.org/10.5281/zenodo.4126862. Acknowledgments I thank A. Yin for reading an early version of this comment and discussion on buoyancy driving of UHP rock exhumation. Review comments from John Platt and Stefan Schmalholz as well as informal follow-up exchanges with Stefan Schmalholz are greatly appreciated. Financial support for research from Canada's Natural Science and Engineering Research Council (NSERC) through a Discovery Grant and China's National Natural Science Foundation (NSFC, grants 41472184 and 41772213) are acknowledged.