The Fluid Dynamics and Relevance of Volcanology

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 When the Earth breathes, society is simultaneously influenced and threatened by the physical manifestation. The study of volcanoes and their processes allows us to prepare for some of the catastrophic natural disasters that ring throughout time. Driving natural disasters such as lahars, pyroclastic density currents, debris flows, and tephra fallout are complex liquids that can be explained by fluid dynamics. While the field of pure physical phenomena can be studied, the volcanologist seeks to use these tools in order to provide an understanding of magmatic evolution and extraction. The complications that arise in the application of fluid dynamical processes within magmatic systems are found in the physical parameters that characterize different magma compositions. Adding to the complexity is the commonly accepted hypothesis that many volcanoes have reservoirs where a silicic magma lies over a more mafic magma with a stark compositional gap. This paper will highlight the fluid dynamics that can explain both the creation and extraction of such a chamber. As many of these investigations were carried out on the order of decades ago, modern case studies will be presented as a synthesis of applied fluid dynamic processes.

How do we know? Geologic Evidence at the Surface

 Our geologic observations in the field allow us to interpret the conditions that were once present to produce what we see today. To the volcanologist, this means interpreting volcanic or plutonic rocks to make an inference about their state prior to eruption. There are two common deposits that lead us to the hypothesis of a layered magma chamber. The first is an inverted stratigraphy where silicic, crystal poor rocks underlie more mafic, crystal rich rocks. The second are plutonic rocks that have more evolved caps and outward boundaries. For the first case, the compositional gap is stark in outcrop and the underlying rocks are thought to represent an evolved portion of melt that resided at the top of a magma chamber. Examples such as Mount Mazama (Bacon and Druitt, 1988), the Bishop Tuff (Blake and Ivey, 1986; Spera, 1984), and the Skargaard intrusion (Sparks et al., 1984) are good representations of both deposits and inspire fluid dynamic analysis for magma chambers. Mount Mazama’s eruptive products are a rhyodacite pumice (70.4% SiO2) followed by an andesitic pumice along with juvenile scoria (48-61% SiO2). The Bishop Tuff has nine distinct units that comprise much of the work for the Long Valley Caldera. The eruptions are systematic in the sense that they display a trend that reduces in silica content and volatile saturation throughout the eruptive history (Wallace and Anderson, 2000). It is noteworthy that these examples all come from calcium-alkaline magma chambers associated with subduction zones. Magma bodies in these tectonic regimes are characterized by high volatile content and alkalic molecules that are pertinent to the discussion of creating a layered magma chamber.

Creating a Heterogenous Magma Chamber

Double Diffusive Convection

 When field observations led to the interpretation that magma chambers stratify while still in the liquid state, double diffusive convection was one of the methods theorized to create a heterogenous chamber. The entire idea is essentially a force balance that occurs at the margins of a magma chamber paired with a horizontal interface that separates 2 or more layers with different mechanical and compositional properties. The force balance is between a negative buoyancy caused by the temperature difference between the magma and its walls in contrast to a positive buoyancy force caused by chemical diffusion. This is highly analogous to the effect that we see in oceans due to thermal and salt gradients. In a magma chamber however, it is a requirement that the diffusion is fast enough to sustain stratified layers in a chamber. Considering that magma is not a Newtonian fluid, rheological properties must be accounted for when modelling. These are often made to be dimensionless for ease of calculation during modeling before being scaled back to relevant values. Transport properties commonly taken into consideration are the viscosity, temperature, and density of the magma along with the chemical diffusivity that can be accounted for using energy, momentum, and the conservation of mass for the ith species. One major problem affecting the transport properties in these systems is the effect of cooling and crystallizing at the wall. This complexity is often averted by means of implementing a “no-slip” equation at the boundary wall, where the wall temperature is constant and no flow occurs. The validity of using this bypass while still getting a good approximation of flow in the chamber is shown by Spera (1982) when back calculating with numerically determined sidewall heat flow. This may seem minorly important, but the condition is adopted for many numerical analyses following Spera’s example. The viscosity inside a thermal boundary layer is calculated as a function of the viscosity in the liquid state, temperature difference between magma and country rock, and crystal content. In this study, a rigorous scaling analysis is employed using the Rayleigh number (Grashof number*Prandtl number = (buoyancy/viscosity forces)*(momentum/thermal diffusivity)) as a function of rheological properties present in a given boundary layer thickness and viscosity contrast across the boundary layer. Note that the Grashof number is set to unity such that when multiplied by the high Prandtl numbers relevant to magma chambers, the Prandtl number retains its identity (essentially multiplying by a fancy 1). Once this was done, Spera applied his model to a space with rectangular dimensions and a constant flux of basalt (i.e. heat) for a source of enthalpy. His most important conclusion in the numerical analysis is the dependence of rheological parameters on apparent viscosity with respect to distance from the wall (Figure 1). We can see by analyzing this array of graphs that the viscous shear stress is highest at the wall as a result of temperature difference and crystallinity. This causes a complete lack of any upward vertical velocity inside the thermal boundary layer. In conjunction with this is the value of the horizontal velocity. With increasing distance from the thermal boundary layer we see that turbulent (high Reynold’s number) flow causes magma to be brought back into convection while the inverse is true where the thermal gradient is low. This is a remarkable method of calculating flow across a gradient with variable viscosity, but it tells us nothing about how we can move molecules across this layer. Although the process is slow, Soret diffusion explains the phenomenon of molecules moving across a thermal boundary layer due to a temperature gradient. The problem is that this process is slow and the diffusion coefficients for each molecular compound is different. In addition, rapid convection must be sustained to account for the thermal fingers leaking from the top layer to be recycled. The assumption of a constant mass and heat flux from basaltic flow saves this model but we know from seismic data this is not the case. Additionally, the thermogravitaional effects from heat loss could overtake a positive chemical buoyancy. A better model of this was done numerically (Spera et al., 1984) and experimentally (Nilson et al., 1984). Using the same dimensionless parameters described above, the Grashof number was not set to unity and instead was used as a function of thermal and chemical density change across a boundary layer. The Lewis number (thermal/mass diffusivity) was then used as a function of the ratio of the two Grashof numbers to show where there is a thermal (negative) or compositionally (positively) driven flow. It was found that in the region of high viscosity, compositional buoyancy dominates over thermal buoyancy (figure 2).  An important point to note here is that up to the point that these papers were released, these models relied on the single diffusion coefficient for the ith species. Thus, the mass flow rates across the thermal boundary layer during Soret diffusion as well as the positive compositional buoyancy rely on the effective binary diffusion coefficient for an individual species (Trial and Spera, 1990). As opposed to using a high EBDC under the assumption that abundant, high EBDC major components control the boundary layer (the most abundant molecules in a crystal mush) Trial and Spera used the interpretation that chemical diffusion occurs at the sidewall (Lowell, 1985) and not within the mush. Thus, the low EBDC of silica cannot be ignored and, by itself, could never diffuse fast enough to create a stably stratified chamber. However, it was shown using Fick’s law that if low EBDC molecules are coupled to a light component such as H2O or alkalic molecules then a stably stratified chamber could be created. In support of showing that Nusselt numbers become independent of large Lewis numbers (i.e. a large thermal diffusivity causes the temperature gradient to behave asymptotically with no initial exponential increase) (Spera et al., 1986), double diffusive convection with EBDC assumptions becomes obsolete and the importance lies in the components that comprise the Dij matrix, where volatiles and alkalis are the primary contributors to diffusion across a boundary layer.

Figure 2: Adopted from Spera, 1984. Top graph shows region of thermal downwelling. Bottom graph shows region of compositional upwelling. Middle graph shows a combination of the two and where each force is dominant.

Figure 1: Adopted from Spera, 1982. Graphs show rheological parameters and velocity with respect to distance from the sidewall.

Convective Fractional Crystallization

 For those of us that are familiar with commonly taught theories of magma chamber dynamics, the prior section should have sounded like a confusing Segway into convective fractionation. An expansion of double diffusive convection beyond simple Soret diffusion is the nature of the crystals themselves. The removal of specific chemical species during crystallization, tied in with the coupled transport of heavy elements due to volatile content explains convective fractionation on a larger scale (Sparks et al., 1984). The filling box model that has been experimentally tested (Turner, 1981) still holds true except that it is generally accepted that there are not multiple convection cells independent of one another, as shown by Nusselt numbers becoming independent of Lewis numbers. The idea of convective fractionation is in stark contrast to the more traditional idea of crystal settling that was originally proposed by Charles Darwin (Sigurdsson, 2000)when he observed cumulate crystals in the Galapagos. For the crystal settling model, there is a sedimentation process in which crystals come to rest at the bottom of the chamber as a function of the Stokes settling velocity for a sphere following nucleation (Martin and Nokes, 1988). One way of analyzing which end member is most likely to characterize a given magma chamber is by relating the volume of a magma chamber (inferred from eruptive volume) with repose time (Spera and Crisp, 1981). It was shown that with increasing eruptive volume, time between eruptions also increases. The correlation implies that large magma chambers take longer to accumulate an eruptible layer of differentiated magma, but they also seem merely consequential in the mathematical sense. Up to this point the emphasis that has been portrayed on the ability for a system to become zoned has been explained by transport properties that are numerically relevant to magma. We must also take into consideration the geometry of a magma chamber.  A large sill like chamber has a smaller surface area that allows for convective fractionation to occur and will undergo more heat loss at the large, sloping roof (de Silva and Wolff, 1985). Using the time dependent accumulation rate derived by Nilson et al. (1985) and the volume of a magma chamber with a given ratio of width to height, de Silva and Wolff showed that the time to create an eruptable layer at the top of a magma chamber was in agreement with that calculated by Spera and Crisp (1981). By both numerical and geometrical analysis, we can infer that smaller cylindrical chambers will rapidly create stratified layers due to convective fractionation (Figure 3).

Chamber Aspect Ratio

Figure 3: Adopted from de Silva and Wolff, 1984. Graph represents that the growth rate of evolved melt is much higher for chambers of small aspect ratios (width/height).

Magma Mixing

 If we neglect batch melting and focus on partial fusion, it is easy to postulate that the emplacement of a dike and shallow magma chamber weakens surrounding country rock and provides a path of least resistance for recharge. By this manner, a more primitive magma can rise into a more evolved melt and a.) create a hybrid magma, b.) pool at the bottom of a floor and begin crystallizing, c.) cause an eruption, or d.) any combination of the above. One important parameter when making these considerations is the Reynolds number (inertial/viscous forces) of the intruding flow. Should the flow be highly turbulent (inertial forces dominate), thermal equilibrium may be reached relatively quickly due to high thermal diffusivity, but the heat flux may change the course of differentiation as shown experimentally by McBirney et al. (1985). The laboratory experiment was done as an analogy to calc-alkaline magma bodies that had already evolved to the point of eutectic cooling at the roof. The high heat flux caused the crystal mush at the roof to begin dissolving and the more dense crystals began settling to the floor. The result was an even more dilute, hotter upper layer with a denser layer at the floor. When carried to the point of floor crystallization, the “magma” at the floor began to crystallize and a double diffusive interface was created between the main body and the bottom layer. The same effect is seen if the flow of the intruding magma has no buoyancy with respect to the original host and simply pools at the floor (Sparks, 1984). If the recharge is laminar (viscous forces dominate) but the density differences are not too far apart, the flow may rise to the top of a magma chamber and cause a compositional gap. It is not a far jump to say that this occurrence quickly undermines Spera’s analysis of timescales to create an eruptible stratified layer as a function of repose time, should we take into consideration that subsequent eruptions were triggered by various methods. Perhaps one of the best compilations of all the above processes is represented by Mount Mazama. The compositional gap between rhyodacite and andesite is attributed to an initial low strontium andesite that underwent convective fractionation along with repeated intrusion of a low Sr parent magma (Bacon and Druitt, 1988; Druitt and Bacon, 1988). Glass analysis was then used to show that a separate, high strontium andesite began intruding the chamber and formed a high strontium cumulate pile on top of the low strontium cumulate pile (figure 4). Convective fractionation continued until the caldera forming eruption occurred.

Figure 4: Adopted from Bacon and Druitt, 1988. Figure shows the envisioned magma chamber prior to the caldera forming eruption at Mt. Mazama.

From Inception to Eruption: Inverting a Zoned Chamber

 Modeling a zoned chamber based on field observations is one thing, but we can never actually view this happening. We need the eruptive mechanism to explain how we can begin to move forward with these ideas in the first place. An important first step for this problem was carried out by Spera (1984) for the cases of eruption from a central vent and a ring fracture in an isoviscous fluid. The case of eruption through a central vent with constant recharge to make up for mass flux is modelled as follows: Navier-Stokes equations are set up to analyze the change in viscous fluid flow with respect to an unchanging two-dimensional space at a given Reynold’s number. This was done by setting up a Poisson equation (an ellipse) based on vorticity (rotation) and stream function that defines the ellipse of magma that will simultaneously meet at the base of the conduit.  Using the Reynold’s number as a function of constant, given mass flux for an experiment then allows the numerical model to render the ellipses that are known as evacuation isochrons (figure 5). The caldera collapse experiment is done much the same except that the kinematic velocity tied into the stream function is based on volume loss from the collapsing roof. The time to reach steady state eruption (spin up time) is then used as a function of Reynold’s number and chamber geometry, simulating the time to reach a steady state eruption. Using these functions, it was qualitatively shown that heterogenous layers of magma will mix in the conduit and can erupt in together in a single event. The problem of relating differing densities and viscosities can be taken a step further by analyzing the relative inertial and viscous forces that define a critical draw down height (Blake and Ivey, 1986a). Scaling the initial force to the viscous force allows for the determination of the correct flow regime (Reynold’s number) in a given situation. Should a critical force be met, the flow will be sufficiently turbulent and a fully sheared profile will develop during draw up through a conduit. If the critical force is not met, then a plug like flow will develop in the laminar regime. The draw down height is then determined as a function of the mass flow rate, gravity, and (if viscous forces are more important) the viscosity of the magma (Figure 6). This was further compounded upon by the pair where they analyzed the effect of water and crystals (Blake and Ivey, 1986b). Contrary to popular belief, high silica rhyolite is now generally accepted to be less viscous than underlying more mafic magma due to high volatile content, particularly water. This, in conjunction with an increased crystallinity was taken into context and the draw up scales were redefined. Though this was done for a magma with a constant gradient, it is explicitly stated that the parameters may be applied to individual magmas should a heterogenous chamber be assumed. The numerical simulations of isochrons and aspects of the experimentally derived draw down heights were then combined to represent isochrons in a stratified chamber. (Trial and Spera, 1992). A dimensionless Archimedes number was introduced, accounting for the composition and density gradients within the melt, along with a basal hydrostatic pressure that must be exceeded as an eruption driver. The velocity of the magma due to overpressure was found to lie within three regimes: the viscous, transitional, and inertial. Once the velocity is driven through the transitional regime a new quasi-steady state is reached and the flow is fully inertial. Relating these exit velocities to the mass discharge seen in eruptions allows for the determination of the flow regime that was present during eruption. Employing the new exit velocities into the dimensionless conservation equations will render a Reynold’s number that denotes which regime the eruption was in. Referencing equations for the driving pressure, which is also used as a function of observed mass discharge, one can find the overpressure that triggered eruption. By this method, it is found that effusive eruptions have an overpressure of ~103 Pa (interestingly, the pressure that the moon puts on Earth to drive tidal forces) whereas ultraplinian eruptions are on the scale of 109 Pa. We can see from isochrons that not all evolved magma is erupted due to the “coning up” of magma directly below the conduit. A period of repose during an eruptive event may allow for the remaining silicic melt to pool underneath the conduit. By stopping and restarting the simulation, Trial and Spera (1992) showed that this can also cause a compositional gap, and our field observations must account for this possibility. Common sense should also tell us that the effect of reservoir shape can influence erupted composition as well. With these taken into consideration, the two-dimensional coordinates used within the simulation can be changed to account for a sloping roof or a “ring fracture” that was represented as two opposite and opposing conduits. In the former case, isochrons extend to the edge of the roof and silicic melt is erupted more efficiently. In the latter, isochrons have a tendency toward lateral extension as opposed to deepening; strikingly in line with what we can infer from a withdrawal layer thickness driven by roof collapse (Blake and Ivey, 1986b).

Figure 5: Adopted from Spera, 1984. Note that the isochrons are time dependent and reliant on the Reynold’s number and kinematic velocity (viscosity/density).

Figure 6: Adopted from Blake and Ivey, 1986. Flow chart represents a method of determining drawdown height based on eruptive mass flow rate, gravity, and viscosity for two different flow regimes.

Decades Later: Combining Various Processes

 When this subfield was still in its infancy, Spera and Crisp (1981) hit the nail on the head when they noted: “There are probably a number of mechanisms whereby compositional zonation of major, minor, and trace elements develop in chambers.” Magmatic differentiation is still accepted as a cause of heterogeneity, but research has since been scaled up appropriately.  Here, three case studies are presented as a synthesis of the various processes that entail different opinions regarding magmatic differentiation. The first was inspired by the question as to the extent of nuclear waste from the Yucca Valley. δ18O values measured from pristine, unaltered phenocrysts were used to show that original melt values are much lower than the whole rock analysis in the SWNVF field (Bindeman and Valley, 2003), showing that meteoric water had indeed altered the rocks present. In addition to this determination, it was found that rapidly generated tuff units (100-150 k.y.) had distinct isotopic signatures that could not be explained by simple fractionation, as the zoning of oxygen isotopes with respect to other trace elements is not ubiquitous and even inversely related with respect to one another (Figure 7).Not even simple AFC can explain this as the amounts required to generate subsequently erupted tuffs are not attainable at the temperature of latites and rhyolites. The stark changes in composition in relatively short time periods represent the rapid introduction and extrusion of sheet like magma bodies with different parental sources. Note the geometry of the chamber is inferred from the fact that mafic pumice is present at the end of each ring fracture eruption and represents a magma chamber that had a geometry that allotted the full evacuation of a chamber. The second case study was carried out for the formation of the Guacha II Caldera (Grocke et al., 2017). Here, variation in trace element geochemistry requires a two stage AFC model. However, AFC alone cannot account for scatter within major elements, which must be explained by magma mixing from various parental magmas. Additionally, linear trends that represent convective fractionation are used to explain a change in chemical composition from one eruptive period to the next. As this paper is restricted to fluid dynamics, a full explanation of the geochemical relationships will not be carried out here. The third investigation was carried out for the Half Dome Granodiorite of the Tuolumne intrusive suite (Coleman et al., 2012). A geochemical transect across the batholith shows a reflection on opposite sides with the youngest mappable contact lying in the center. Given what has been written in this paper, instinct would point to in situ crystallization. However, it was found that trace element data for aplite dikes is significantly different than leucogranites and volcanic rocks, where the latter two have corresponding data. This is indicative that the cyclic leucogranites were created by crystal fractionation at only a fine scale relative to the entire suite, the bulk of which was created by repeated intrusion from various parent magmas. These three case studies with varying interpretations and geologic settings represent how prior efforts of fluid dynamic investigations has given the field of volcanology a more extensive reach than what was achievable decades ago. Fluid dynamic analysis has brought us to the point where not only can we imagine long term evolution of the crust, we can also test these hypotheses. May we continue to push the boundaries of current thinking as we did forty years ago, as to allow for the evolution of science.

Figure 7: Adopted from Bindemann and Valley, 2003. Open symbols represent rhyolitic compositions, filled symbols represent latite compositions, and ties lines connect the compositions from the same event. Variation of isotopic values between tuff units cannot be related to one another and would require more AFC than what can be achieved by these reservoirs.

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