Over the last 2 decades, magma mixing has received renewed attention in petrology (Wilcox, 1999). Traditionally, magma mixing has served as an explanation wherever a suite of igneous rocks has yielded straight-line trends in chemical variation diagrams of major elements (e.g. Wiesmaier et al., 2012). Recently, however, magma mixing has been recognised to produce non-linear mixing trends on a micro-scale due to diffusive fractionation of elements (Perugini and Poli, 2000, 2004; Perugini et al., 2003; De Campos et al., 2010). Straight trends in entire sample suites, in turn, are most likely the result of performing sample analyses in bulk. As in natural magma mixing scenarios, the diffusive fraction of elements is always combined with variable degrees of irreversible and chaotic fluid mechanic interaction (mingling); the term chaotic mixing was coined (Perugini and Poli, 2000; Perugini et al., 2012). Chaotic mixing fundamentally deconstructs magma mixing in these two realms and shows that both separate phenomena (mingling and diffusion) are necessary to achieve efficient hybridisation.

This dual nature of chaotic mixing allows us to investigate how mingling processes pre-determine the efficacy of magma mixing. Mingling may stretch and fold each liquid member, depending on its rheological properties, and thus results in increasingly complex 3-D morphologies in the form of filaments and blobs (see Perugini et al., 2002). Because of the chaotic nature of mingling, de-mingling is precluded and physical mingling is irreversible. This irreversibility is enhanced by diffusion, which starts immediately in the presence of a compositional gradient at magmatic temperatures and smears the boundaries between two fluids. Without diffusion, mingling would be able to generate infinitesimally small, chemically discrete filaments of two intertwined magmas. Because, however, diffusion is initiated at the moment of juxtaposition of the two magmas, infinitesimal filaments do not occur. During magma mingling, diffusion drives a two-fluid system towards chemical homogenisation. This diffusion may or may not run to completion. Its efficacy is highly dependent on the surface / volume ratio of the involved liquids, where high surface / volume ratios of the liquids (or complex 3-D morphologies) exponentially decrease the diffusive length scales (Perugini et al., 2003; De Campos et al., 2011). Because mingling generally increases the surface / volume ratio of the involved magmas, it follows that the efficacy of different mingling mechanisms affects the overall efficacy of magma mixing.

Magmas are mingled in nature through manifold mechanisms, which are fundamentally buoyancy-related. In detail, however, each of the mechanisms has its own dynamics. For example, during (i) conduit flow, the interface of two separate magmas may become instable and produce “streaky mixtures” of the two members. Although buoyancy-driven, conduit flow mixing is related to turbulence in the conduit, which may form at a lower Re number than expected for turbulence in isoviscous flow (Blake and Campbell, 1986). (ii) Vigorous convection in a magma reservoir, in turn, forms through strong T contrasts from transient mafic replenishment and may entrain, stretch, and fold portions of mafic magma by viscous coupling (Huppert et al., 1983, 1984; Snyder and Tait, 1996). (iii) Double-diffusive convection depends on thermal and chemical diffusivities, resulting when “these two components make opposing contributions to the vertical density gradient” in a magma reservoir (Huppert and Turner, 1981). The implications of double-diffusive convection range from small-scale percolation of more felsic melts along the sidewall of a reservoir, to catastrophic overturn when a replenishing mafic magma fractionates and becomes less dense than an overlying layer of magma. (iv) Forced intrusion implies a competent, felsic body of magma (mush), almost solidified but still hot, which is being intruded by a mafic dyke (Pallister et al., 1992; Izbekov et al., 2004). Simultaneous reheating of the felsic magma may lead to collapse of the dyke, because the surrounding felsic magma melts and loses strength to contain the magma in a rigid conduit. In such cases, highly variable quench textures in mafic enclaves may record the diminishing temperature contrast between mafic and felsic members (Wiesmaier et al., 2011).

The role of bubbles in magma mixing has so far been limited to their density-reducing effect on magma-bubble suspensions. Rapid exsolution of volatiles may reduce the bulk density of layers of basaltic magma to become less dense than an overlying layer of felsic magma (e.g. Tait and Jaupart, 1989; Ruprecht et al., 2008). The resulting instable configuration may then lead to a catastrophic overturn and eruption of an entire magma reservoir (e.g. Woods and Cowan, 2009). In chemical engineering, however, bubble mixing is a long-recognised concept. Devices and setups exploiting the buoyancy force of gas in liquids have been employed for example in medical and biochemical applications (e.g. Sánchez Mirón et al., 2004), fining of glass melts, waste water treatment plants and blending of wines and spirits.

The experiment has been carried out with two natural end-member compositions, basalt and rhyolite. Sample material was obtained in the form of blocks (> 30 cm) from the Bruneau–Jarbidge eruptive centre of the Snake River Plain–Yellowstone hotspot track. The rhyolite sample is from Unit V of the Cougar Point Tuff and the basalt sample from Mary's Creek (see Bonnichsen, 1982; Cathey and Nash, 2009). Samples of these units have been used in previous studies (see Morgavi et al., 2013a, b, c). The basalt is of tholeiitic composition, and the rhyolite belongs to the subalkaline series (Table 1).

Both materials were freed from weathered surfaces and crushed. After milling in a laboratory disc mill, the basalt and rhyolite powders were melted and homogenised in a concentric cylinder viscometer (Dingwell, 1986) at 6 and 24 h respectively to ensure that the materials were exempt of crystals and gas bubbles. Both basalt and rhyolite melts were quenched in air at room temperature. The glasses were cored from the crucibles and machined to cylinders that fit into a Pt-crucible of 25 mm diameter.

At room temperature, a cylinder of Snake River Plain (SRP) rhyolite glass was placed above a cylinder of SRP basaltic glass in a Pt-crucible of 25 mm inner diameter (Fig. 1). This experimental charge was placed into a furnace at 1450 ∘C to remelt both glass cylinders. The heating rate was on the order of 5–15K s-1 based on thermal diffusivity calculations. The target temperature was chosen to (a) keep the melts at superliquidus temperatures to avoid complexities associated with crystals and (b) to achieve an appropriate viscosity for the rhyolite. Prior to the experiment, viscosity of the melts was measured by rotational viscometry and, at 1450 ∘C, found to be ca. 100.4 Pa s-1 for the SRP basaltic melt and ca. 104 Pa s-1 for the SRP rhyolitic melt (Morgavi et al., 2013a). As such, the relatively lower viscosity of our experimental rhyolite melt compared to that of a natural, H2O-bearing rhyolite magma at depth, ensures reasonable experimental run times, while maintaining fluid mechanical scalability.

Upon heating and melting, air trapped in the interstices between the glass cylinders and crucible walls contributed to the formation of bubbles. Thin spaces of air were present underneath the basalt, between basalt and rhyolite, and between the crucible wall and both glass cylinders. Bubbles were thus able to form below the melts as well as at the sidewalls, which ensured unconstrained rising paths of bubbles. According to the ideal gas law, the increase in volume of the trapped air from 293 K (20 ∘C) to 1723 K (1450 ∘C) was ca. 6-fold. Oxygen fugacity was held constant in equilibrium with air at 1 bar.

The bubbles ascended into the rhyolite entraining basaltic melt to create vertical filaments (see Manga and Stone, 1995). Stokes' Law calculations indicated ca. 240 min for a bubble of radius 2 mm to rise through the rhyolite. At t=180 min, the crucible was removed from the furnace and cooled in air to room temperature. The quench rate is estimated to range around few degrees per second, similar to studies using comparable procedures (e.g. Chevrel et al., 2015). The post-experimental glassy assemblage was cored from the crucible to yield a sample cylinder of 20 mm diameter. This run product was characterised by microCT and subsequent high-resolution EMP analyses.

MicroCT provided a non-destructive means to characterise the experimental products both qualitatively and quantitatively. MicroCT scanning was performed at IMETUM, Garching, Germany using a General Electric v|tome|x s© device equipped with a microfocal x-ray tube. Altogether 1000 scans were carried out at 80 kV, 250 µA at an exposure time of 333 ms per scan. Each scan was conducted as the average of three individual scans for noise reduction. The beam was moderated by a 0p3va filter for reduced beam hardening. An effective voxel size of 50 µm was achieved. Subsequent reconstruction was conducted with VGStudio MAX©. The resulting stack of tiff files was then converted to a volume file (.vox) using writeVOX, an auxiliary program of our custom-made, MATLAB-based Tomoview software package. The vox file was segmented in Tomoview and quantified for size parameters of subspherical bubbles.

Discs of experimental glass were prepared as electron microprobe mounts. One or several filaments per microprobe mount were analysed for major element concentrations using a Cameca SX100 electron microprobe at LMU Munich. Measurements were carried out at 15 kV acceleration voltage and 20 nA beam current. To counter alkali loss, Scherrer (2012) found a defocused 10 µm beam as the best solution for that instrument. Standards used were: synthetic wollastonite (Ca, Si), periclase (Mg), hematite (Fe), corundum (Al), natural orthoclase (K), and albite (Na). Matrix correction was performed by PAP procedure (Pouchou and Pichoir, 1984). The precision was below 2.5 % for all analysed elements. Accuracy was tested by analysing MPI-DING standard glasses (e.g. Jochum et al., 2000) and is better than 3.0 % for the analysed elements.

The thickness of experimental filaments has been determined for subsequent statistical analysis. In the final experimental charge, the original filament thickness is not preserved, because the onset of diffusion “smears out” the borders of filaments. Therefore, the thickness of filaments has arbitrarily been defined as the distance between the inflection points in the SiO2 vs. distance curves (Fig. 2). As the main component of silicate melts, Si4+ represents a robust and reproducible proxy for filament thickness, not least for its relatively sluggish diffusion (Dingwell, 1990). According to this procedure, filament thicknesses of the nine EMP transects analysed here range from ca. 70 through to 1650 µm. Potential errors from variable diffusivities of a single species in multi-component silicate melts are deemed insignificant due to the short run time of 180 min and the slow diffusivity of silica.

Concentration variance σ2 represents a statistical measure for the degree of homogenisation of a filament and has been applied successfully in previous experimental studies on magma mixing (Morgavi et al., 2012, 2013a, b, c; Perugini et al., 2013). The calculation of σ2 is given by σ2=∑i=1N(Ci-μi)2N, where N is the number of samples, Ci is the concentration of element i, and μ is the mean composition. In order to provide readable numbers, we normalise concentration variance at time t to the initial variance at t=0: σn2=σ2(Ci)tσ2(Ci)t=0.

As a result, a value of 0 for σ2 represents complete homogenisation, whereas a value of 1 equals completely separate end-members.

We calculated σ2 for all EMP transects measured on the experimental run product. The length of the overall transect was set to double the filament thickness (see Sect. 3.5). For transects longer than double the filament thickness, data points were excluded from the calculation of σ2. For transects shorter than double the filament thickness, artificial data points of rhyolite end-member composition have been appended at the end of these transects. This ensured a correct statistical comparability of σ2 of different transects. Notably, this procedure has only been used for the calculation of concentration variance, and all compositional data presented are as measured from EMP analysis.

Model filaments were calculated to demonstrate the correlation of concentration variance with filament thickness in an ideal case. These ideal diffusion gradients have been modelled based on the thin-source problem, using the following equation: C(x,t)=C0e-x2/(4Dt), where x is the distance measured from the interface between end-members, C is the concentration of the diffusant, C0 is the concentration of the diffusant at the interface (x=0), D is the diffusivity, and t is time (after Eq. 37 in Zhang, 2010). The diffusivity D is kept constant in the calculation and has been arbitrarily set to 1×10-11 m2 s-1. For C0, a value of 78 wt% has been used. For each model filament, C was calculated every 15 µm, comparable to the step length of the analytical profiles obtained by EMP in this study. The resulting gradient of compositional variation was used as the gradient part of the model filament (shaded red in Fig. 2b). Filament thickness was then varied by adding end-member data points in the plateau of the diffusion profile. The justification for this is that at time i, a species s will have diffused the same distance away from the filament, irrespective of filament thickness. Thickness of the model filaments was determined following the approach outlined in Sect. 3.5 (see also Fig. 2b). This approach was repeated for diffusion times of 10, 100, 1000, and 10 000 s. A minimum filament thickness results from this, as the diffusion gradient for each diffusion time possesses a finite length that contributes to the overall filament thickness. Finally, concentration variance was calculated for each modelled diffusion profile, and correlated with filament thickness.

After 3-D tomography, the glass cylinder was sectioned parallel to the basalt–rhyolite interface (see Fig. 3a). This permitted us to obtain three cross sections through the vertically oriented filaments of mafic melt. Backscattered electron (BSE) images of these three cross sections are shown in Fig. 4. The uppermost level, SWM01, features a basaltic area of sub-spherical shape of ca. 2.5 mm diameter (Fig. 4a). The intermediate level SWM02 shows a convoluted, but coherent single filament (Fig. 4b). The lowest level SWM03 appears as three or four separate filaments of varying thickness (Fig. 4c). The EMP transects carried out for each cross section are indicated by yellow lines in Fig. 4. The single filament in sample SWM01 required one transect only (SWM01-01), whereas the more convoluted filament structures of SWM02 and SWM03 were characterised by four EMP transects each (see labels in Fig. 4). The compositional profiles of each filament displayed three distinct shapes: bell-shaped, plateau-shaped and multi-peak/irregular shape.

In model filament calculations, concentration variance correlates systematically with filament thickness. In Fig. 6, each curve represents concentration variance vs. filament thickness at a specified diffusion time. Concentration variance, i.e. the degree to which filament with thickness x has equilibrated, correlates non-linearly with filament thickness. At a particular diffusion time, thin filaments will be much more homogenised (lower value of concentration variance σ2) than thick filaments (cf. Morgavi et al., 2013c).

Because the curves possess steeper slopes towards thin filament thickness, thin filaments equilibrate disproportionally faster than thick ones. When we compare curves of different diffusion time, the contrast between thick and thin filaments decreases with increasing diffusion time, but the overall non-linearity remains. For instance, for the curve in Figure 6 with the longest diffusion time modelled (10 000 s), a 8000 µm thick filament shows a σ2 of ∼ 0.8, whereas a filament of 4000 µm has ∼ 0.5. Hence, half the filament thickness allowed diffusive equilibration to complete 50 % instead of 20 % in the same time. By reducing filament thickness by a factor of 2, the efficiency of diffusive equilibration was increased by a factor of 2.5. As the slope of all curves in Fig. 6 show steeper slopes for thinner filaments, the rate of diffusive equilibration is higher for thinner filaments. This means that also the difference in equilibration rate between two filaments is higher when both are relatively thin.

Polynomial regressions allow us to further characterise the model of diffusive equilibration. All calculated regressions have R2 > 0.998. The asymptotic nature of the regression curves implies that its first derivative approaches zero at infinite filament thickness. Large magmatic bodies are thus very inefficiently equilibrated by diffusion alone. The non-linearity of the regression curves is thus evidence for an exponential increase in the rate of equilibration with decreasing filament thickness.

We calculated viscosities of hybrid filaments based on compositions measured by EMP. The initial viscosity contrast of the pure end-member was measured to be a factor of 4×103 (cf. Morgavi et al., 2012). By employing the viscosity model of Giordano et al. (2008), the EMP compositional data were converted into melt viscosities of the filaments. As the most mafic composition of each individual filament varies, the computed minimum viscosities for the filaments range between ca. 101.5 and ca. 103.5 Pa s-1. Under the experimental conditions (1-atm, 1450 ∘C), the rhyolitic end-member possesses a calculated viscosity of ca. 103.8 Pa s-1, compared to a measured 104 Pa s-1. Viscosity contrasts between rhyolitic and hybrid melt thus range between a minimum factor of 3 to in excess of 100.

The imaging results from microCT reveal that multiple bubbles have risen through the rhyolite, dragging portions of basaltic melt upwards with them (see Fig. 3). These sub-vertical elongated filaments are thoroughly hybridised, according to microCT attenuation values and EMP data. Because several hybrid filaments merged into one thicker filament underneath the topmost bubble, bubbles appear to have exploited the same pathway repeatedly. The experimental setup has thus generated two main modes of mass transport. Firstly, advection of basaltic into rhyolitic melt driven by the segregation of bubbles. Secondly, diffusion of ionic species across the boundary between basalt and rhyolite melts. This experiment therefore demonstrates the potential for bubble mixing at viscosity contrasts of up to 4×103.

Several implications arise from comparison of our experiment to the numerical analysis of migration of drops through fluid–fluid interfaces (Manga and Stone, 1995). Based on numerical constraints, Manga and Stone (1995) suggested that the volume of entrained fluid decreases with increasing ratio of upper to lower fluid viscosity γ. As our experiment constitutes a case of high γ (ca. 4×103) the entrained volume of basaltic melt should be low, perhaps insignificant. However, according to the microCT image, significant amounts of basaltic melt have been propagated, which either indicates that the volume of entrained fluid never reaches 0 even at high-viscosity contrasts, or the higher value of β (0.2 in Manga and Stone, 1995, vs. 0.79 in the experiment) causes a relatively higher buoyancy of the bubble in rhyolite to still permit significant volumes of entrained melt.

The time of penetration of a bubble into the upper layer, or breaking the interface between basaltic and rhyolitic melt, scales at a factor of approximately 1/(1+λ) of the ascent rate, where λ is the ratio of bubble to lower fluid viscosity (Rallison, 1984; Kojima et al., 1984). For our experiment, the large viscosity contrast of air and basaltic melt at 1723 K implies a λ of around 2.35×10-5, meaning the ascent rate of bubbles would hardly be affected by the interface and bubbles essentially pass through with nearly no deceleration from the interface (i.e. only the buoyancy change affects the rise speed). This is consistent with Manga and Stone (1995, Fig. 8d therein, λ=10), where the rise speed decreases continually upon entering the higher viscosity upper liquid, but potential interface effects in the form of a rise speed minimum are not detected.

Discrepancies between our experiment and the numerical treatment of Manga and Stone (1995) arise with respect to the Bond number and surface tension. The surface tension of dry silicate melt against a free vapour phase is an order of magnitude higher than the surface tensions employed for the fluids in the analogue experiments of Manga and Stone (1995). In the present case, the bubble Bond number Bo1 is below 1, whereas Manga and Stone (1995) addressed problems of Bo1>5. Smaller Bond numbers imply that bubble shape is surface tension-controlled and deformation of bubbles in the rhyolite is limited. Consistent with that, the measured aspect ratios of bubbles are close to being equant.

Specific to bubble mixing is that successive bubbles may repeatedly advect mafic material into the same filament. A filament may thus experience mafic recharge at irregular frequency. This gives rise to a complex diffusion scenario, because the composition of the filament may be hybridised initially, but reset to a more mafic composition repeatedly. Additionally, as later pulses of basalt may traverse a filament of already hybrid composition, these later pulses may equilibrate at variable rates due to the reduced compositional contrast they experience. As a result, not only the mode of advection is non-conventional (bubbles instead of inherent buoyancy or convection), but also the calculation of diffusion timescales must take into account; additional mafic melt of variable composition may or may not have been added to a filament at later stages.

The compositional variations across the filaments imply a dynamic rate change for bubble mixing. The viscosity contrast from basalt to hybridised filament is smaller than from basalt to pristine rhyolitic melt. Figure 7 shows this “viscosity valley” in the interior of filaments compared to the surrounding rhyolite. Computing the log viscosities, the maximum viscosity contrast γ at t=0 was 4000 and at t=3 h is reduced to ca. 32 for filament SWM01-01 (γi=101.5/100.4). Envisaging a constant flux of bubbles, this means that when the rhyolite has been penetrated by filaments throughout, bubbles will inevitably enter these previously formed filaments for their reduced viscous resistance against the buoyancy forces of a bubble.

The ramifications of low-viscosity channels for the process of bubble mixing are manifold. First of all, higher ascent velocities are expected for bubbles that rise within a filament, as Stokes' Law velocities are proportional to 1/viscosity. Therefore, at the calculated filament viscosity of ca. 101.5 Pa s-1 (compared to 104 Pa s-1 in the rhyolite), the terminal settling speed of a bubble may be up to 300 times faster in a hybrid filament than through pristine rhyolite. This is certainly a maximum estimate, as streamline geometry and the ratio of bubble to filament diameter will also affect rise speed. Notwithstanding, compared to pristine rhyolite, the reduced viscosity of a hybrid filament inside a rhyolite must enhance the rate of bubble ascent significantly. Secondly, the initial ratio of viscosities of upper to lower fluid γ affect how much basaltic melt can be advected. According to Manga and Stone (1995), γ inversely correlates with the amount of lower fluid entrained into the upper liquid. For filament SWM01-01, γ changed from 4×103 down to 13, implying an increase in transported material. When a bubble enters a hybrid filament, it may thus be able to propagate larger amounts of basaltic liquid than upon entering pristine rhyolite. This affects the mixing rate once more, as more material per bubble is being advected. Thirdly, the buoyancy change β that the bubble experiences upon entering the upper fluid will be diminished. Calculation of β using the viscosity of the hybrid filaments implies a reduced buoyancy change (from 0.8 to 0.9), because the hybrid filament has a higher density than the surrounding rhyolite. The higher value of β implies lesser deceleration of the ascent speed caused by the viscosity contrast. Not only does the rise speed within the rhyolite increase, but also the rise speed through the basalt–rhyolite interface.

The last point to consider is filament geometry. The uppermost filament from our experiment SWM-01 shows visual and compositional evidence for its formation by many pulses of mafic melt, which likely caused the disproportionate thickness of this filament. Filaments are thus widening due to addition of new mafic melt by each passing bubble. However, when the repeated advection of basalt through a filament thickens that filament, bubble advection will be enhanced as the diameter of the low-viscosity channel is enlarged. This represents a positive feedback loop, as a widened filament diameter enhances bubble ascent, which in turn widens filaments even more. Additionally, we speculate that recurring bubble advection may create a network of hybrid filaments by coalescence of bubbles and merging filaments.

The acceleration of the rate of mixing may in theory continue until filaments purely consist of mafic end-member, at which point the lowest possible viscosity is established inside the filament. By then a steady state would be reached and new mafic material advected at a constant rate per bubble and bubble size. Once the source reservoir of the mafic end-member is depleted of a free gas phase, mixing induced by bubbles must decelerate until more volatiles are exsolved or external volatile sources are tapped.

We present a method to detect multi-pulse filaments in natural samples. The 3-D visual evidence demonstrates that multi-pulse filaments are present in our experimental run products. Nevertheless, in natural samples, 3-D visual evidence may not be as clear cut as here. We thus aim to constrain the origin of filaments further by a statistical treatment of the compositional profiles obtained from EMP analysis.

In the context of our experimental setup, single-pulse filaments are expected to stretch out from the basalt–rhyolite interface towards the top of the rhyolite, as one bubble moves upwards within the rhyolite. Diffusional equilibration for single-pulse filaments thus ought to shift systematically from high degrees of equilibration at the bottom of a vertical, cylindrical filament to fewer degrees of hybridisation at the top. Such systematic correlation of diffusion-based equilibration with vertical position would be useful for constraining the timescales of filament formation, but is not detected on our experimental filaments. Instead, rheologic and visual constraints indicate that experimental filaments formed by bubble mixing experienced multiple bubbles rising. Such repeated replenishment of multi-pulse filaments with fresh end-member melt must affect the diffusional equilibration from previous pulses, though. Multi-pulse filaments are thus expected to significantly deviate in their diffusional behaviour from single-pulse filaments. This is important because the calculation of magmatic timescales based on diffusion gradients has commonly been based on a single-pulse origin of magmatic filaments, bar other options recognised so far. It is thus vital to be able to distinguish single- from multi-pulse filaments in natural samples.

The characteristics of our experimental charge are advantageous for distinction of single- and multi-pulse filaments, as it is comparable to natural samples in key aspects. Firstly, bubbles were allowed to rise in uncontrolled fashion. The resulting filaments are therefore a result of the conditions in situ within the experimental charge, and mainly controlled through the parameters buoyancy, surface tension and rheology. Secondly, the time evolution of the experiment is uncontrolled except for the total run time. As direct observations during the experiment were inhibited, the formation of filaments has to be constrained a posteriori, comparable to natural samples. Our experimental design thus mimics natural samples in these aspects, while on the other hand has the advantage that pressure, temperature, and initial end-member compositions are known. This allows us to constrain the origin of the filaments generated during the experiment to a degree much further than is possible in natural samples.

Is bubble mixing relevant in nature? Each natural case represents a unique combination of extrinsic and intrinsic properties, all of which influence whether bubble mixing comes into play during a system's magmatic history. In the following, we will lay out scenarios in which volatile content, thermal history, and viscosity contrast may be favourable for bubble mixing. We then conclude with evidence from natural case studies that may show an influence of bubble mixing.