External surface water influence on explosive eruption dynamics, with implications for stratospheric sulfur delivery and volcano-climate feedback

2

and respectively. Here u is magma ascent rate, A = πa 2 c is the conduit cross sectional area, and ρ is bulk 239 magma density, averaged over liquid and gas phases, where χ v is the volume fraction of bubbles, and ρ v and ρ m = 2400 kg/m 3 are gas and melt densities where M = u/c is the Mach number of the mixture. Below the fragmentation depth c 2 = K/ρ, where K 248 is the bulk modulus of the mixture Above the fragmentation depth, the bulk modulus of the gas phase K v is calculated from the equation of only that fragmentation proceeds to small enough length scales such that permeable gas escape from the 261 pyroclasts is sufficient to ensure that pore-scale pressures equilibrate to the free gas in the conduit at the 262 vent height (Rust and Cashman, 2011).

263
Assuming negligible gas escape or water infiltration through conduit walls, the primary effect of overlying 264 surface water or ice is to modify the pressure boundary condition at the volcanic vent. Above magmatic 265 fragmentation, the gas-pyroclast mixture fluidizes, accelerates, and decompresses towards the conduit exit. 266 If the flow speed remains below the mixture sound speed, c, then the vent exit pressure, p c , must balance 267 the ambient pressure above the vent, p e , which is determined by water depth: 268 p e = ρ e gZ e + p atmo , where ρ e is the density of external water and p atmo is the atmospheric pressure at the water surface. If use an iterative search to find conduit parameters that satisfy the pressure-balanced or choked conditions. 282 We first allow conduit radius to vary to obtain solutions for a "dry" or subaerial vent where no external 283 water is present and the ambient pressure above the vent is equal to atmospheric (Z e = 0). Subaerial vent 284 simulations were run and suitable conduit radii obtained for a range of "control" MER 10 5.5 ≤ Q 0 ≤ 10 9 285 kg/s, and we refer to these subaerial vent scenarios as "control" simulations hereafter. For control scenarios, 286 we seek specifically solutions where choking occurs at the vent exit and thus no conduit flaring is required.

287
This calculation provides a reference conduit radius to use in scenarios with a water layer present above 288 the vent, with water depths 0 < Z e ≤ 500 m. For these hydrovolcanic cases, we then fix the conduit 289 radius to that of the control scenario and find an adjusted conduit MER q c such that the surface pressure The model PSD is first defined explicitly at the vent (z = 0) as a function of the output from the conduit 300 model. We define an initial power-law PSD following Kaminski and Jaupart (1998) and Girault et al.

301
(2014), over the particle size range −10 ≤ φ i ≤ 8. The number of particles N i at size φ i is given by where D 0 is the power-law exponent, N 0 is an arbitrary normalization constant, and subscript i indicates a 303 particle size bin. We choose a default value of D 0 = 2.9. Each size class is assigned an effective porosity 304 value χ i on the basis of an effective particle radius: 305 χ i = χ 0 , r i ≥ r c1 χ i = χ 0 (1 − r c2 /r i ), r c2 ≤ r i ≤ rc1 This is a provisional file, not the final typeset article Here, χ 0 = 0.75 is the porosity threshold for fragmentation, r i is the particle radius for bin i, r c1 = 10 −2 m 306 and r c2 = 10 −4 m. Particles of sufficiently small size have, thus, no effective porosity and densities equal 307 to that of the pure melt phase (ρ s,i = ρ m ). By contrast, the density of larger particles is a strong function 308 of porosity and bubble gas density (Kaminski and Jaupart, 1998). This approach leads to expressions for 309 particle mass fraction in each size bin, n s,i , and the bubble gas mass fraction of each size bin, n b,i : where subscript s denotes the bulk "solids" phase (melt plus bubbles). Figure 3d shows the initial PSD for 313 D = 2.9, accounting for particle density as a function of porosity (light gray line and square symbols). expanding gas prevents pryoclasts inside the jet from interacting with external water (e.g. Kokelaar, 1986). 319 Our decompression model therefore assumes that turbulent entrainment and mixing of external water 320 begins at heights above L d . For L d , we use a modified form of the free decompression condition of Woods 321 and Bower (1995) to find the height at which the jet gas pressure plus dynamic pressure is equivalent to 322 external water hydrostatic pressure: where p is pressure, u d is the speed after decompression, and ρ is density. Subscripts d and e denote 324 properties of the jet mixture after "decompression" and of "external" water, respectively. Assuming the 325 decompression speed is approximately the mixture sound speed (Ogden et al., 2008), using the dusty-gas 326 approximation (Woods and Bower, 1995), where subscript v denotes the "vapor" phase, the free gas volume fraction χ v ≈ 1, and γ is the ratio of We approximate decompression length L d as proportional to the change in jet radius with decompression: where and Here n v is the jet gas mass fraction, and the subscript c indicates properties in the "conduit" prior to overpressure is sufficiently small that turbulent mixing and entrainment can begin. For a pressure-balanced 342 jet (β = 1), this critical height should be immediately above the vent. We note, however, that due to the 343 rapid pressure change with height in the water column, the mixture will continue to expand and decompress, combination of the enthalpy of exsolved gas bubbles and the specific heat of the melt phase: Here where n w and h w are the mass fraction and enthalpy of water (gas and liquid) within the jet mixture. At the 377 decompression length, the total power supplied by the jet is: Where q c is the conduit MER and g = g(ρ − ρ e )/ρ e is the reduced gravity, and the dot notation over E 379 indicates the rate of energy delivered (i.e. power).

380
From an initial value T 0 , the bulk temperature of the jet mixture T is calculated at each solver step Entrainment of ambient fluid into a jet or plume is driven by both radial pressure variations arising 397 from the relatively fast rise of the jet and local shear at the jet boundary (see Figure 1). Entrainment where α is an entrainment coefficient of order 0. to identify the dynamical "crossover height" L X at which fully turbulent plume rise starts and above which 417 Equation 22 holds. Below L X , the flow evolves predominantly in response to the momentum flux supplied.

418
In this regime, drag related to turbulent instabilities, accelerations, overturning motions and mixing is not 419 established and on dimensional grounds the evolving height of the jet Above L X , plume height predominantly governed by a balance between buoyancy and inertial forces is, by The transition height L X occurs where h jet = h BI , which corresponds to where the characteristic time 423 scale t jet = t BI . After algebra we obtain Starting from height z = L D , we assume the thickness a mix of a turbulent mixing layer at the jet boundary 425 develops monotonically over distance L X : above which the radial turbulent mixing is complete and the velocity profile is top-hat or Gaussian, 427 consistent with the assumption of self-similar flow (Morton et al., 1956;Turner, 1986). We then obtain an effective entrainment coefficient, α ef f , by scaling the entrainment coefficient based on the volumetric 429 growth of the mixing layer: Using a similar entrainment parameterization to Mastin (2007b) which accounts for the relative density 431 difference of the ambient and entraining fluid, the rate of water entrainment into the jet is Here, α RT is the Rayleight-Taylor coefficient for buoyancy driven entrainment, B is the weight determining 447 the relative balance between entrainment driven by buoyancy and that driven by shear-induced turbulence, σ 448 is the surface tension at the water-steam interface, and ω ≈ (0.3u) 2 /(2πa) is the average radial acceleration The process of quench fragmentation of pyroclastic particles of various size during MWI is complex. submerging molten basalt into a fresh water tank to constrain the partitioning of thermal energy lost from 471 the melt between that which is transferred from melt to heat external water and that which is consumed 472 irreversibly through fracturing of the melt to generate new surface area and fine ash. At any height above 473 the vent, the total power delivered to entrained external water from the melt is: and ∆E m is the rate of heat loss from the melt phase. The remaining heat loss from the melt i.e. ζ∆Ė m is where ∆Ė w is the power supplied for heating external water by heated water already in the volcanic jet.

486
Although this energy sink is very small for typical magma water mass fractions of 5% at the vent height, where ∆q w,e is the mass flux of entrained water, h w,f is the final enthalpy of the water phase after thermal and known initial mixture temperature for the current step, respectively. To estimate heat transfer to the 497 entrained water phase, we assume that the change in temperature after equilibration T f − T , is sufficiently 498 small at each step that the jet water heat capacity can be approximated as constant for the current step, such where C w is the water heat capacity at temperature T . Substituting 36 into 35 leads to T f can then be used to estimate heat transfer to entrained water ∆h w = h w,f − h e , which is used along 502 with ζ and the PSD to later calculate the specific fragmentation energy, ∆E ss .

503
Since we assume that the energy consumption during quench fragmentation results from the generation specific surface area at each particle bin size assuming spherical particle geometry, where Λ is a scaling parameter accounting for particle roughness, as true particle surface area can potentially 507 exceed that of ideal spherical particles by up to two orders of magnitude (Fitch and Fagents, 2020). We 508 take a default value Λ = 10, and discuss the effects of different choices for Λ in Sections 3.2 and 4. The 509 total surface specific surface area for a given PSD is To simulate the evolution of the PSD by quench fragmentation, we prescribe a representative range 511 of particle sizes produced by thermal granulation based on the fine mode of particle sizes for the (∼ 100 µm) and standard deviation φ σ = 1.46, and is shown in Figure 3a (blue line).

515
The "input" particle sizes (i.e. particles that fragment to produce the fine fraction) are defined according 516 to the available surface area in the coarse fraction (φ < φ µ ). We use the output mean, φ µ as a fragmentation 517 cutoff -particles of this size and smaller are assumed to not participate in quench fragmentation, but 518 can participate in heat transfer to water. This allows the definition of an effective fragmentation energy 519 efficiency as a function of particle size (see Figure 3a, black line), where n sφµ,f is the mass fraction of the mean size bin in the output PSD. Fragmentation efficiency thus 521 quickly reduces to zero as particle sizes approach the mean output size. In addition to the above particle 522 size limitation on fragmentation, we also halt fragmentation once the bulk mixture passes below the glass 523 transition temperature. We define the glass transition lower bound for a hydrous rhyolitic melt using an , we apply the glass transition limit using a smooth-heaviside step 529 function of temperature, where ∆T g is the glass transition temperature range, with typical values of ∼ 50 K (Giordano et al., 2005).

531
Using hs sm to scale ζ with temperature (Figure 3c), Equation 40 becomes: and the effective fragmentation energy efficiency for determining total fragmentation energy from the PSD 533 is 534 The PSD of the coarse particle fraction (i.e. particle sizes that experience mass loss due to quench 535 fragmentation), n si,0 , is calculated as proportional to available particle surface area in each size bin, 536 modified by the fragmentation efficiency ( Figure 3a, red lines): Finally, we define the specific fragmentation energy (per mass of pyroclasts in the jet) and the change in mass of the pyroclast fraction due to gas release from vesicles on fragmentation: where we choose E s = 100 J/m 2 for the particle surface energy for fragmentation (Dürig et al., 2012).

540
The final differential equations for evolution of the PSD, and conservation of water mass, pyroclast mass, 541 momentum, and energy, are respectively Figure 3d shows the evolution of the total PSD during water entrainment and quench fragmentation in the eruption column as a function of fine ash production. Figure 3e shows the total PSD evolution 554 due to particle fallout in the eruption column for a PSD that has been fines-enriched during MWI. The 555 conservation equations for mass of dry air, water vapor, liquid water, and particles are, respectively: where v ε is the entrainment velocity, subscript a denotes properties for dry air, λ = 10 −2 s −1 is a constant and ξ = 0.27 is the particle fallout probability. The equations for vertical momentum and energy are, 562 respectively: where C s and C e are the heat capacities of particles and air, respectively, T e is the ambient air temperature,

571
As described above, our model approach is to simulate eruptions across a parameter space with 10 5.5 ≤ 572 Q 0 ≤ 10 9 kg/s and 0 ≤ Z e ≤ 500 m. In Table 2 we define the Reference scenario which employs default  Table 2), water depths sufficient to cause this pressure-balanced condition usually lead to a weak jet that does not breach the water surface and/or to a steam plume condition (see Section 3.3 and Figure 9 below).  ). In the model, the primary limit for fine ash production is, thus, the height at which water 697 entrainment causes the mixture temperature to become less than the glass transition temperature. For 698 C m = 1250 J/(kg K)) and T 0 = 1123 K, this condition is met where n e 0.12. However, even with 699 this imposed temperature limit for quench fragmentation, Figure 3d shows that the PSD is substantially it is 80% after the glass transition is passed (Figure 3d, black line). Therefore in the absence of the glass 703 transition limit, coarse particles could be fully depleted. In Section 4 we further discuss the consequences of our choice of fragmentation model and the associated key parameters: initial PSD, particle roughness, 705 fragmentation energy efficiency, and glass transition temperature. water begins to entrain and mix into the jet, whereas our decompression length scaling prevents water 745 ingestion for shallower depths (panel (g)). As the water mass fraction increases above about 30%, the 746 water saturation temperature is reached and the column source includes liquid water (panel (j)), increasing 747 its density. Consequently, jet velocity (panel (i)) decreases for greater water depths, and combined with 748 reduced heat content in the particle fraction to generate buoyancy (panel (k)), it becomes impossible 749 for the jet to undergo a buoyancy reversal, and gravitational collapse occurs (panel (a)). Since the vent maintains the choked and overpressured condition until depths greater than the collapse threshold, the 751 collapse condition for the subaerial column is not significantly influenced by changes in conduit conditions 752 with increasing water depth, and is primarily determined by the mass fraction of entrained external water.

753
At the upper limit for water entrainment, once the water mass fraction reaches ∼ 0.7, the heat budget of the 754 pyroclasts is largely exhausted and most of the plume water ( 95% by mass) is in liquid form, resulting in 755 steam plume conditions where the a dense pyroclast jet collapses within at most ∼1 km above the water 756 surface.
757 Figure 9a shows total plume water mass fraction at the base of the subaerial eruption column as a function 758 of MER and water depth for the Reference scenario. For comparison, the vent radius is marked in purple.

759
The shaded light gray region highlights conditions for which stable buoyant plumes form, whereas collapse with increasing water depth (see also Figure 10). A notable feature is that for MER 10 8.3 , the column 767 collapses for the control case with no external water, but becomes a buoyant column for entrained water 768 mass fractions up to ∼ 30%. In addition, low MER eruptions are able to support higher mass fractions of 769 external water without collapse (e.g. n w ≈ 45% for q c = 10 7 kg/s versus n w ≈ 35% for q c = 10 8 kg/s).

770
The relative buoyancy of low MER columns is caused by more efficient entrainment of air at smaller jet 771 radii, as well as entrainment of atmospheric humidity and condensation and latent heat release in the plume.

772
We note that condensation of atmospheric moisture has a more significant impact on buoyancy for smaller  Table 2

856
The combined effects of quench fragmentation followed by sedimentation in the rising column influences 857 both total retained mass of ash in the eruption cloud and the surface area per unit mass of particles. Figure   858 11c shows the fraction of total erupted particle mass remaining in the column at its maximum rise height,

Stratospheric injection of hydrovolcanic eruption columns
production of particle surface area (ash). In addition to modulating the rise of a hydrovolcanic eruption column, the extent of ash production potentially affects also the SO 2 absorption and the heterogeneous 883 nucleation and growth of sulfur aerosols. Thus, we conclude by discussing the co-injection across the 884 tropopause of ash, SO 2 , and water in hydrovolcanic eruption clouds and implications for chemistry, 885 microphysics, and associated climate impacts. for steam plumes is, for example, not significantly affected because the decompression length is very small 910 at these depths (see Figure 8f), the threshold water depth for column collapse and stratospheric injection 911 decreases by ∼20 to 30% (see Figure 9b).

912
The mechanism of decompression length inhibiting water entrainment in our model can be related to in vapor form, further suggesting that these explosive events were drier than is typical of "Surtseyan"-type 32. For Q 0 10 7 , the ratio of jet cross-sectional area to mass flux a 2 /q c is relatively large, resulting in 973 large entrainment rates comparable to those for fully developed plumes (i.e. No-L d -noL T scenario) and 974 consequently shallow water depths for the column collapse and steam-plume conditions. For Q 0 10 7 975 kg/s, as entrained water is vaporized jet density initially decreases, resulting in enhanced Rayleigh-Taylor 976 entrainment and column collapse for slightly shallower depths than the Reference scenario. However, for 977 larger water depths where the jet cools to the water saturation temperature, entrained water remains liquid, 978 jet density increases and radius decreases (see Figure 8, panels (h) and (j)). As a result, q c dominates 979 in Equation 32 for water depths much greater than the threshold for collapse, and entrainment rates are 980 suppressed. The reduced entrainment rates for large MER and deep water layers, in turn, prevent total 981 exhaustion of the particle heat budget such that, in contrast to other scenarios, the steam plume condition 982 occurs for pressure-balanced jets much deeper than the limit for vent choking (c.f. Figure 9b)). As a final 983 remark here, we reiterate that the mechanics of water entrainment exert the greatest control over column 984 rise. Our results underscore, however, that this process is poorly understood and is a key avenue for future 985 work on hydrovolcanism. As implemented, the shear-driven and buoyancy-driven modes govern water

1073
The fragmentation energy efficiency ζ governs the relative partitioning of thermal energy loss from the 1074 melt between that used to heat and vaporize water and that consumed by fragmentation and production 1075 of particle surface area. Choosing a low value for the fragmentation energy efficiency, ζ = 0.05, (Low-ζ 1076 scenario, yellow symbols in Figure 12) reduces the energy consumed by fragmentation per unit mass of 1077 entrained water, resulting in overall less ash production before the glass transition limit is reached. This 1078 scenario has both the lowest total particle surface area after quench fragmentation and a modest change ∼ 2200 m 2 /kg). Consequently, the strongest control on production of ash surface in this scenario is the 1091 minimum particle size that can be produced during quench fragmentation.

1092
The results of the various fragmentation scenarios above reveal an important trade among PSD, particle 1093 roughness, and the consumption of fracture surface energy during quench fragmentation. The primary 1094 effect of the glass transition limit and fragmentation energy efficiency is to determine the energy budget 1095 for fragmentation, whereas particle roughness and surface energy limit the mass of fine particles than 1096 can be produced within a given energy budget. The initial PSD, in turn, determines the mass of "coarse" 1097 particles available with which to generate new fine ash. The mass in this coarse fraction is dependent on the

1135
Comparing total exsolution for small and large water depths (Figure 5f), differences in vapor exsolution 1136 in the conduit model control the glass transition temperature (Figure 3b), which, in turn, governs the heat 1137 budget available for ash production during the quench fragmentation (Figure 11a). This effect is most

Stratospheric Injection in Hydrovolcanic Eruptions and Implications for Sulfate
The estimated fraction of SO 2 delivered to the stratosphere is the fraction of the integrated area of Equation

1190
59 that lies above the tropopause. Events with injection heights close to the tropopause (Q 0 ≈ 3 × 10 6 kg/s 1191 and Q 0 ≈ 3 × 10 7 kg/s in the high and low latitude atmospheres, respectively) show reduced efficiency 1192 of stratospheric delivery of SO 2 for water depths that surpass the decompression length (and therefore 1193 non-zero quantities of external water are entrained). The exceptions are columns in the low-latitude 1194 atmosphere with minor quantities of entrained water (n w ≈ 0.15), which have increased column heights 1195 relative to control scenarios (see Figure 10b). Panels (b) and (c) show the ratio of fine ash mass flux (particle 1196 diameter < 125 µm) at the maximum plume height relative to control simulations. We find that events 1197 with sufficient entrained water to pass the glass transition (and thus maximize production of fine ash in our particles and subsequently removed from the eruption cloud, with an additional 10% lost directly to the 1232 subglacial lake (16% and 5% of the total magmatic sulfur budget, respectively).  hydrovolcanic scenarios relative to control cases. However, the total surface area generated is sensitive 1293 to processes governing particle fallout and to the physics of quench fragmentation (e.g. particle     Figure 1. Summary of eruption processes from conduit to atmospheric dispersal. See text for a description of processes and their relevance for SO 2 transport. See Table 1 for a complete description of symbols. (a) Dynamical processes during a sustained, "dry" Plinian eruption. Inset: illustration of the entrainment process. (b) Summary of processes influenced by surface water interaction during a hydrovolcanic eruption. Processes in lighter gray text are those not considered in this study, but which are relevant to hydrovolcanic eruptions processes and may play a role in stratospheric delivery of SO 2 .  Figure 2. Schematic summary of coupled model, highlighting geometry of the vent and MWI region. The left and right sides are divided between a control scenario with no external water and a scenario with a shallow water layer, respectively. In the hydrovolcanic case, decompression of the erupting jet of gas and pyroclasts is suppressed relative to the dry control scenario (indicated by decompression length L d and radius a d ), and initiation of turbulent mixing with external water results in water entrainment and quench fragmentation. In the water layer scenario shown here, water depth Z e is greater than the decompression length L d but less than the height at which large entraining eddies are fully developed, L d + L X . See Table  1 and Sections 2.2, 2.3.3, and 2.4 for a complete description of symbols and processes. . The "input" PSD, (i.e. those particle sizes from which mass is removed to generate the products of quench fragmentation), is defined on the basis of available surface area in the total PSD coarse fraction. The input PSD therefore evolves from an initial value (solid red line) to a final value (dashed red line) as the total PSD coarse fraction is progressively depleted (see panel (d)). The solid black line shows the fragmentation energy efficiency as a function of particle size, ζ i (Equation 43), which defines the size bins for the "coarse" fraction. (b) Glass transition temperature data from Dingwell (1998) (squares) and curve fit (black line) as a function of concentration of dissolved water in the melt. The grey shaded rectangle shows the range of values in the Reference set of simulations after exit from the vent. (c) Fragmentation efficiency as a function of temperature (equations 41, 42) for T g = 784 K. (d) Evolution of the total PSD during quench fragmentation, from initial power law with no external water (n e = 0, light grey line) to a coarse-depleted PSD after sufficient external water is entrained (n e ≈ 0.12, black line) to cross the glass transition temperature. Note the preferential depletion of particles in the mid-range (−3 φ 2) driven by particle surface area. The reduced mass fraction of coarse particles (φ 2) in the initial PSD is due to the low density of these particles owing to their large porosity (equations 9-11). (e) Further evolution of PSD due to particle fallout, after water breach and during column rise, with preferential fallout of the coarsest fraction (φ −3) and additional enriching of fines.     (h) radius of the vent and jet after initial decompression (at z = L d ) and at the water surface (z = Z e ); (j) velocity of the jet after initial decompression (at z = L d ) and at the water surface (z = Z e ). Column source conditions: (j) vapor and liquid water mass fractions; (k) bulk mixture temperature.  Figure 9. (a) Plume source water mass fraction as a function of MER and water depth, with overlaid thresholds for behavior of the coupled conduit-plume system. The red line marks the threshold for which the vent is choked and overpressured, with pressure-balanced, subsonic jets occurring at deeper depths. The decompression length is equal to water depth at the blue dashed line, which is the depth above which water entrainment begins. Buoyant columns occur within the grey shaded region, with column collapse elsewhere. The steam plume threshold is marked by the solid blue line -failed plumes with only negligibly small amounts of steam reach the water surface for depths greater than this (indicated by the blue arrow). Finally, the solid black line marks the water depth above which decompression length is zero. (b) Variation in the critical MER to reach the tropopause (solid lines) and maximum water depth before plume failure (i.e. only minor steam breach of the water surface, dashed lines) for different simulation scenarios (see Table 2). Black lines are for the Reference scenario (high latitude atmosphere), while blue lines are for the low latitude atmosphere. The remaining colors are for the four scenarios with different water entrainment parameterizations: no mixing length (No-L X , red), no decompression length (No-L d , yellow), neither mixing length nor decompression length (No-L X -no-L X , purple), and the weighted Rayleigh-Taylor entrainment mode (αRT, light blue).  Figure 11. Effects of MWI and sedimentation on particle specific surface area S. (a) Specific surface area, S, immediately after the jet breaches the water surface (Z = Z e ), as a function of c H 2 O , the water mass fraction still dissolved in the melt after conduit exit. Symbols are sized according to MER at the vent and colored according to the mass fraction of entrained external water. The dissolved water content controls the glass transition temperature, T g , which in turn is the primary limiting factor in the model for how much surface area can be generated during quench fragmentation. (b) S at two different heights in the eruption column: at column source, immediately after MWI (Z = Z e , grey symbols), and at the column maximum height (z = Z max , blue symbols) as a function of water mass fraction at column source. Symbol sizes as in (a). An 'x' denotes a collapsing column, a filled circle denotes a column that is buoyant but with Neutral Buoyancy Level (NBL) below the tropopause, and diamonds are columns that are buoyant with NBL at or above the tropopause. Evolution from grey to blue symbols is a result of sedimentation over the rise height of the column.The approximate water mass fraction above which the pyroclasts cool below the glass transition temperature T g is marked with a vertical blue bar. (c) Fraction of particle mass remaining in the column at its maximum rise height as a function of column source water mass fraction. Symbols are sized by MER as in (a) and (b), and colored according to the value of S at maximum column height. Symbol shapes as in (b). The arrow highlights the subset of simulations with NBL above the tropopause and where the column retains increased (relative to "dry" runs) particle mass and specific surface area.  Figure 12. Specific surface area as a function of water mass fraction at the water surface (circles) and height of neutral buoyancy (diamonds) for scenarios with different fragmentation properties. The Reference scenario is shown in blue. Reducing the fragmentation energy efficiency to ζ = 0.05 (Low-ζ scenario, yellow symbols) reduces the amount of energy consumed to generate surface area per unit mass of entrained water, resulting in a smaller increase in S during MWI relative to the Reference scenario. Conversely, a high initial value of the PSD power-law exponent, D = 3.2 (High-D scenario, purple symbols), concentrates initial particle mass in the fine fraction. Because of the fixed particle sizes for output from quench fragmentation used here (see Figure 3), there is relatively little particle mass available to fragment for the creation of new surface area and the relative change in S with water entrainment is small. Finally, increasing the particle particle roughness scale, Λ = 25 (High-Λ scenario, red symbols), results in initially high particle surface area, but also a greater energy requirement to generate new particles of a given size. This scenario results in the highest absolute changes in particle surface area after quench fragmentation and sedimentation, but a smaller relative change than for the Reference scenario. ), as a function of control MER Q 0 and water depth Z e . In all panels, the dashed blue line is threshold water depth for water entrainment (decompression length equal to water depth, L d = Z e ), and the solid blue line is the threshold depth for steam plumes (see Figure 9). Black regions indicate column collapse. (b) Fine ash mass flux to the eruption column maximum height as a ratio of hydrovolcanic (Z e > 0) to control (Z e = 0) simulations, for particle diameters less than 125 µm. Red line outlines simulations with buoyant plumes at spreading heights at or above the tropopause. (c) Water mass flux to the eruption column maximum height as a ratio of hydrovolcanic (Z e > 0) to control (Z e = 0) simulations. Black regions indicate the steam plume regime in panels (b), (c), (e), (f). Panels (a)-(c) are for with a high latitude (Iceland) atmospheric profile (Reference scenario). Panels (d)-(f) are the same as (a)-(c), respectively, but for the low latitude (Equador) atmosphere (Low-lat scenario).