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Release of dissolved volatiles during submarine fire fountaining eruptions can profoundly influence the buoyancy flux at the vent. Theoretical considerations indicate that in some cases buoyant magma can be erupted prior to fragmentation (~75% vesicle volume threshold). Laboratory simulations using immiscible fluids of contrasting density indicate that the structure of the source flow at the vent depends critically on the relative magnitudes of buoyancy and momentum fluxes as reflected in the Richardson number ( Ri). Analogue laboratory experiments of buoyant discharges demonstrate a variety of complex flow structures with the potential for greatly enhanced entrainment of surrounding seawater. Such conditions are likely to favor a positive feedback between phreatomagmatic explosions and volatile degassing that will contribute to explosive volcanism. The value of the Richardson number for any set of eruption parameters (magma discharge rate and volatile content) will depend on water depth as a result of the extent to which the exsolved volatile components can expand.

Interpreting the processes of submarine explosive eruptions has been difficult owing to the rarity of direct observations. Geological evidence suggests that submarine fire fountaining takes place in the marine environment based on the occurrence of fluidal breccias and highly vesicular basaltic clasts that show similarities to their subaerial counterparts [_{2} and H_{2}O). In their model they assume that fragmentation occurs when a gas volume fraction of 0.75 is attained by gas exsolution either within the conduit or right at the vent. An intriguing problem concerning submarine fire fountaining is to verify whether or not such an assumption is always valid. In subaerial eruptions, fragmentation typically occurs within the conduit or vent [

Subaerial fire fountains consist of a spray of hot fluidal clasts and gas ejected at velocities up to about a hundred meters/s forming spectacular fountains a few tens to several hundred meters above the vent [

Subaqueous eruptions of vesiculating magma will be positively buoyant if the bulk magma density is less than that of the surrounding seawater. The bulk density is a function of the mass fraction of exsolved volatiles (the mass fraction of volatiles that have separated from the solution), density of the exsolved volatile and the dense rock density.

where depends on the total volatile mass fraction and the solubility of the volatile component in the magma,

The soluble limit for a given volatile component and magma composition is a function of pressure and temperature

.

We have evaluated conditions of magma buoyancy using the VolatileCalc 1 program to calculate gas solubilities [_{2}, the density was calculated using the ideal gas law and in the case of water, the density of the exsolved component (steam) was taken from tabulated data (ASME Steam tables). As a result of surface tension, the pressure within a gas bubble exceeds the pressure of the surrounding fluid,

where d is the void diameter. For non-spherical bubbles the diameter is calculated from a major and minor axis,. For simplicity in our calculations we have assumed that the pressure in the bubbles is equal to that of the ambient pressure and ignored this surface tension effect, which requires a detailed knowledge of bubble size and the complex interfacial tension relationship.

Bubbly suspensions of uniformly sized bubbles can commonly exist up to a gas volume fraction of 0.85 before fragmentation [

Simple degassing calculations using the equations cited above demonstrate that it is possible for basaltic magma that is vesiculating to attain positive buoyancy relative to seawater prior to reaching gas volume fractions commonly associated with fragmentation (75%).

The range of volatile contents that we evaluated in

fragmentation in the submarine environment [2,16]. Thus submarine fire fountaining may be driven by volatile contents higher than those typically associated with the primary gas content of seafloor basalts.

Subaqueous eruption of magma differs significantly from subaerial conditions owing to the difference in density of the ambient environment into which magma is discharged and the dramatic variations in the rate of heat exchange between magma and water versus magma and air. Under film boiling conditions, an insulating vapor blanket develops between molten magma and surrounding seawater, greatly suppressing heat transfer. As a result, a reasonable first order approximation is to analyze the hydrodynamic flow structure as decoupled from heat transfer until fragmentation occurs. A vertically-directed immiscible discharge is governed by the functional relationship [

where:

Here, D is the vent diameter; U is the vent exit velocity; h is the characteristic height in flow structure; is the vent geometry parameter; d is the characteristic mingling void size; are the bulk and ambient viscosities, respectively; is the interfacial tension between discharged and ambient fluids and f is the characteristic frequency of the flow structure.

The dependent Strouhal number represents a dimensionless characteristic frequency of the flow structure. In the case of a negatively buoyant fountain, St is the frequency at which the fountain collapses and reforms [^{1}. As will be discussed below, Ri is the dominant parameter in determining the flow structure. Reynolds number is the ratio of the inertial to the viscous forces. Weber number is the ratio of the inertial forces to the interfacial tension forces of the overall vent structure. In the case of volcanic eruptions,. The other independent parameters are a discharge vent shape parameter and the viscosity ratio between the discharged and ambient fluid. The bulk viscosity of the magma is dependent on the Capillary number of the magma [

Our analysis of submarine eruption processes draws on experimental results performed on negatively buoyant fountains and jets that were previously reported in [17,18] and additional experiments conducted on positively buoyant jets and plumes. The equipment and procedures, which are briefly described here, are more thoroughly discussed in [17,18]. The experimental setup (

Sample images for positively and negatively buoyant discharges are shown in

In the case of negatively buoyant eruptions, Ri is overwhelmingly the dominant independent parameter of Eq.1 with specific flow behaviors associated with certain threshold values [

square average velocity to account for the increased momentum of laminar flows. Interfacial surface tension stabilizes the interface and leads to a trend of decreasing transitional with increasing (corrected Weber number defined similarly to). Although the effect is relatively minor, viscosity ratios deviating from unity stabilize the interface leading to a trend of a minimum transitional for viscosity ratios of 1.0. Results for noncircular vents are consistent with the data for cylindrical exits if flow parameters are defined in terms of the hydraulic diameter.

The positively buoyant images shown in

While the experimental results demonstrate that the Richardson number is the dominant parameter (with Reynolds number, viscosity ratio and Weber number providing relatively minor contributions in the range of parameters investigated), it is impossible to match the broad range of viscosity ratios and Reynolds numbers that potentially occur in actual eruptions. The viscosity of basaltic magma can range from 10^{2} to 10^{4} poise. In addition, the bulk viscosity can vary greatly as a result of entrained bubbles, depending on the Capillary number [

In order to further investigate the nature of potential submarine flow structures we used an in-house flow solver called NGA [24-26] to perform numerical simulations of positively buoyant discharges. NGA is designed for simulations of turbulent multiphase flows. Using a consistent mass and momentum transport scheme [

The governing equations for a multiphase flow of Newtonian immiscible fluids are conservation of mass and momentum

where U denotes velocity, p pressure, τ the shear stress tensor, F_{B} body force (e.g. gravity), and F_{ST} is the surface tension force.

NGA is a parallel, finite-difference, structured, flow solver with staggered arrangement of variables. It employs the projection method to solve the governing equations in which the temporal integration is done using an iterative, second-order, Crank-Nicolson formulation. A single set of governing equations is solved in the whole numerical domain for simulations of multiphase flows, where the spatial terms are discretized with second-order accurate schemes. At any point in the domain, the fluid properties, namely density ρ and viscosity μ are determined by a scalar indicator function G (described below). To incorporate jump conditions across fluid interfaces, the Ghost Fluid method (GFM) [25,28] is employed.

To model the kinematics and deformation of fluid interfaces, a special type of the level set method, known as the spectrally refined interface (SRI) method [

Although G can be any smooth function, we use a signed distance function here, so the interface is represented by an iso-surface G_{0} = 0.

Here we consider the effects of water depth on the po-

tential style of submarine basaltic fire fountaining eruptions as a result of its influence on the bulk density of the initial erupting mixture (degassing). As a baseline eruption, we have chosen a discharge rate of 200 m^{3}/s of dense rock equivalent similar to magma discharge associated with subaerial fire fountaining events [

As shown in ^{3}, yielding a Richardson number of 0.03 and these conditions would favor energetic fragmentation by volatile degassing. As depth is increased, the reduction in specific volume of the gas phase causes a decrease in velocity and buoyancy. Initially, the dominate effect is the velocity reduction causing the Richardson number to increase. In the baseline case, buoyancy does not increase sufficiently for a transition to plume behavior. The maximum Richardson number is reached at 500 meters, below which the dominant effect on increasing depth is a reduction in buoyancy. At a depth of 1000 meters, the flow becomes negatively buoyant and the flow structure transitions into a negatively buoyant jet. At a depth of 1400 m, R_{i} < –1 and the flow transitions to a stable collapsing fountain. _{2} has a much lower increase in solubility with pressure, leading to shallower slopes of the Richardson number curves (

Vent diameter and flow rate have no effect on the tran-

sition from positive to negative buoyancy (the sign of Ri), but greatly affect the magnitude (Figures 7 and 8). As previously shown, the baseline case never behaves as a plume as a result of the high inertial forces. A reduction in flow rate to 50 cm/s results in a flow that has little chance of mingling at depths in which it is negatively buoyant and it behaves as a plume at most depths in which it is positively buoyant, excepting very shallow eruptions (less than about 20 meters). Likewise increasing the vent diameter has a similar effect.