Did steam boost the height and growth rate of the giant Hunga eruption plume?

Abstract

The eruption of Hunga volcano on 15 January 2022 produced a higher plume and faster-growing umbrella cloud than has ever been previously recorded. The plume height exceeded 58 km, and the umbrella grew to 450 km in diameter within 50 min. Assuming an umbrella thickness of 10 km, this growth rate implied an average volume injection rate into the umbrella of 330–500 km³ s⁻¹. Conventional relationships between plume height, umbrella-growth rate, and mass eruption rate suggest that this period of activity should have injected a few to several cubic kilometers of rock particles (tephra) into the plume. Yet tephra fall deposits on neighboring islands are only a few centimeters thick and can be reproduced using ash transport simulations with only 0.1–0.2 km³ erupted volume (dense-rock equivalent). How could such a powerful eruption contain so little tephra? Here, we propose that seawater mixing at the vent boosted the plume height and umbrella growth rate. Using the one-dimensional (1-D) steady plume model Plumeria, we find that a plume fed by ~90% water vapor at a temperature of 100 °C (referred to here as steam) could have exceeded 50 km height while keeping the injection rate of solids low enough to be consistent with Hunga’s modest tephra-fall deposit volume. Steam is envisaged to rise from intense phreatomagmatic jets or pyroclastic density currents entering the ocean. Overall, the height and expansion rate of Hunga’s giant plume is consistent with the total mass of fall deposits plus underwater density current deposits, even though most of the erupted mass decoupled from the high plume. This example represents a class of high (> 10 km), ash-poor, steam-driven plumes, that also includes Kīlauea (2020) and Fukutoku-oka-no-ba (2021). Their height is driven by heat flux following well-established relations; however, most of the heat is contained in steam rather than particles. As a result, the heights of these water-rich plumes do not follow well-known relations with the mass eruption rate of tephra.

Introduction

Shortly before 0300 UTC on 15 January 2022, the underwater caldera known as Hunga volcano (previously Hunga-Tonga Hunga-Ha’apai) in the Kingdom of Tonga began a several-hour-long climactic eruption. From 0407 to 0437 Universal Coordinated Time (UTC), the plume rose from 5 to 58 km above sea level—the greatest plume height on record for any historical eruption (Aubry et al. 2023). Stereo processing of images from the Himawari 8 and GOES 17 geostationary satellites at 10-min intervals produced maps of plume height (Van Eaton et al. 2023) showing an overshooting top that remained above ~50 km altitude for nearly an hour, from 0437 through 0527 UTC, reaching 58 km at 0437 (Carr et al. 2022; Proud et al. 2022). From 0417 to at least 1607 UTC, the cloud top remained above 18 km, and pulses of eruptive activity could be seen as fluctuations in cloud-top height or lightning rates up to 11 h after the eruption intensified (Van Eaton et al. 2023).

The great height suggests an exceptional mass eruption rate (MER). The 58-km plume height was 10–20 km higher than the 1991 Pinatubo plume, which was the next-highest example observed during the past century (Aubry et al. 2021). A simple empirical relationship, log₁₀($\dot{M}$) = 2.83 + 3.54 × log₁₀(H_t) (Aubry et al. 2023, eq. 2) (where $\dot{M}$ is mass eruption rate in kg/s and H_t is height in kilometers above the vent) suggests an average $\dot{M\ }$≈1.2 × 10⁹ kg s⁻¹—greater than any other eruption of the past century (Aubry et al. 2021). Using this formula for eruption rate and integrating for the time period from 0437 to 0527 implies a total erupted volume of 1.2 km³ dense-rock equivalent (DRE), assuming a magma density of 2,500 kg m⁻³. Integrating 12 h of intense eruption yields an erupted volume of 2.8 km³ DRE (5.4 km³ using Eq. (2) of Mastin (2014)) (Table S1).

The growth rate of the umbrella cloud suggests an even larger erupted volume. The umbrella cloud grew from a diameter of about 15 km at 0407 to 450 km by 0527 (Fig. 1a (Gupta et al. 2022; Van Eaton et al. 2023)). This rate of growth is about 60% faster than the observed umbrella-cloud growth during the climactic 1991 eruption of Pinatubo in the Philippines (Fig. 1a). Gupta et al. (2022) used the following relation (Woods and Kienle 1994; Costa et al. 2013) to estimate the volume flux into the umbrella cloud:

$$R={\left(\frac{3\lambda N\dot{V}}{2\pi}\right)}^{1/3}{t}^{2/3},$$

(1)

where R is the umbrella radius, t is the time since the eruption started, N is the Brunt–Väisälä frequency at the umbrella-cloud elevation, λ is a shape factor (Sparks et al. 1997, Chapter 11), and $\dot{V}$ is the volume flux into the umbrella. A fit through the curve of umbrella radius with time during the first 80 min of the eruption (Fig. 1a), using λ = 0.2 (Suzuki and Koyaguchi 2009) and N = 0.023 s⁻¹ (Van Eaton et al. 2023), suggests that $\dot{V}\cong$ 320 km³ s⁻¹. This value is comparable to the value of 500 km³ s⁻¹ obtained by Gupta et al. (2022). As an independent check, we can estimate the average volume flux required to produce an umbrella with a 250 km radius (Fig. 1a) in about 80 min. An umbrella with an assumed thickness of 10 km would have a volume of 1.96 × 10⁶ km³, and an average growth rate of 410 km³ s⁻¹, which is consistent with our estimate of 320 km³ s⁻¹.

Fig. 1

a Umbrella radius (km) versus time (min), for the Hunga cloud (gray circles), and, for comparison, the umbrella cloud from the 15 June 1991 Pinatubo eruption (diamonds). The upper x axis gives time (UTC) on 15 January 2022 for the Hunga cloud. Lower x axis gives time in minutes after 04:14:58 UTC on 15 January 2022, which is inferred to be the time of initiation of the Hunga umbrella based on a best-fit curve through the Hunga data (dashed line) with a form R = a(t-b)^2/3, where a and b are fitting coefficients. The dotted line is a best-fit curve through Pinatubo data. Hunga radius data from Van Eaton et al. (2023). Pinatubo umbrella radius data and best-fit curve taken from Mastin and Van Eaton et al. (2020). Insets at the base of the plot are images of the umbrella at 0427, 0447, and 0507 UTC taken from Fig. 2 of Van Eaton et al. (2023), with lightning shown as blue dots. b Map of isopachs simulated using the Ash3d model (Schwaiger et al. 2012) with umbrella growth (Mastin and Van Eaton 2020), for an eruption having a dense-rock-equivalent (DRE) volume of 0.2 km³ and umbrella-top height of 30 km (Run 14 of Fig. S1). Lines give modeled isopachs, colored dots give measured deposit thicknesses. c Similar isopach map using an erupted volume of 10 km³ DRE and umbrella-top height of 35 km (Run 9 of Fig. S1). Contour values (from outer to inner) are 0.1, 0.3, 1, 3, 10, 30, 100, and 300 kg m⁻², respectively. Black dots are islands. “NA” = Nuku‘alofa. Hunga volcano is indicated by the black cross. Other inputs and model settings are described in Supplement S1. Sample locations and mass loads are given in Table S4

Van Eaton et al. (2023) converted the average umbrella growth rate to a mass eruption rate $\dot{M}$ using the well-established relation (Costa et al. 2013, corrigendum),

$$\dot{M}=\frac{\dot{V}{N}^{5/4}}{C{k}^{1/2}}$$

(2)

where C is an empirical constant estimated as 430 m³ kg^–3/4 s^–3/2 for tropical eruptions (Costa et al. 2013, corrigendum), k is the (dimensionless) radial entrainment coefficient of air ingested into the rising plume, taken as 0.1, and Ṁ is the total mass eruption rate in kg s^–1. The results suggest $\dot{M}$≈5-6 × 10⁹ kg s⁻¹ (Van Eaton et al. 2023). At this rate, 1 h of eruption would produce 7.2–8.6 km³ of erupted material (DRE). This calculation does not consider the material produced during later activity through 1607 UTC or the material potentially injected into a second, lower-level umbrella cloud which became visible after the upper one drifted west (Carr et al. 2022). Nevertheless, the umbrella-cloud-based estimate of Hunga’s erupted volume is greater than the mass emitted in 9 h at Pinatubo (Wiesner et al. 2004; Wiesner et al. 2005).

These calculations are remarkably consistent with volume estimates based on pre- and post-eruption bathymetry around Hunga volcano, which shows a caldera-collapse structure 4 km in diameter and a volume change of about 9 km³ (Clare et al. 2023; Seabrook et al. 2023). Nevertheless, the plume-based estimates of erupted volume described above must be considered with caution, for several reasons.

One issue is that empirical relationships between plume height and eruption rate have not been calibrated using examples higher than the 35–40 km plume at Pinatubo (Sparks et al. 1997; Mastin et al. 2009; Aubry et al. 2023). Nor are any large eruptions through seawater, of size 5 or greater on the Volcanic Explosivity Index (VEI) (Newhall and Self 1982), included in these data. A second issue is that umbrella-growth relations have also been calibrated using smaller eruptions (Suzuki and Koyaguchi 2009; Costa et al. 2013; Pouget et al. 2013). Third, the SO₂ emissions are not consistent with such a large eruption. Satellite-detected SO₂ emissions from Hunga on 15 January 2022 (Carn et al. 2022) are only 0.4–0.5 Tg; much smaller than the 18 Tg emitted by the VEI 6 Pinatubo eruption, and comparable to much smaller (VEI 3-4) eruptions (Carn et al. 2016). Seawater may have converted much of the SO₂ to sulfate aerosols, producing high aerosol optical depths of the downwind cloud (Carn et al. 2022; Rowell et al. 2022; Sellitto et al. 2022).

The fourth issue, which is critical to this study, is that tephra thicknesses on the islands within 60–140 km of Hunga volcano are unexpectedly thin for such a large eruption. At a few dozen locations, tephra fall deposit thicknesses were measured in the field on flat surfaces using a mm-scale ruler within days to weeks after the eruption. Samples were collected from measured areas, dried, and weighed to obtain density and mass per unit area (MPUA, Table S4). The sampling methodology and results are provided in Paredes-Mariño et al. (2022).

The geographic distribution of measurements is too sparse to draw isopachs and estimate erupted volume by direct integration; however, we have simulated tephra transport and deposition using the Ash3d dispersal model (Schwaiger et al. 2012) with wind fields from the Global Forecast System 0.5 degree model (US National Weather Service 2018). In 14 runs, using erupted volumes ranging from 0.1 to 10 km³ DRE, umbrella-top heights from 25 to 35 km, and different grain-size distributions (Supplement S1, Tables S2-S4), modeled MPUA values that agree best with field measurements use an erupted volume of only 0.1–0.2 km³ DRE (Fig. 1b, c; Fig. S1, S2). Averaged over the first hour of the eruption, this would imply a mass eruption rate of 7–14 × 10⁷ kg s⁻¹. Measured MPUA at most sites is about 5–20 kg m⁻², whereas modeled values at those same locations for simulations that use 5–10 km³ DRE erupted volumes are ~100–700 kg m⁻² (Table S4). Thus, the eruptive plume appears to have contained much less tephra, by at least an order of magnitude, than expected based on its height, umbrella growth rate, and duration.

The large discrepancy between caldera-collapse volume and tephra-fall volume suggests that most ejecta rapidly entered the ocean, feeding the large-volume submarine density currents that spread over the sea floor (Clare et al. 2023). This concept is supported by images of the early plume. These include eyewitness photos from Tongatapu (Fig. 2) that show a column base many kilometers across, with turbulent lobes that appear to be rising from the water surface.

Fig. 2

Eyewitness photos of the 15 January 2022 climactic eruption of Hunga volcano in Tonga. a Photo taken at approximately 0422 UTC by Susie Campbell from the northern foreshore of Sopu, Tongatapu, showing a buoyant thermal (outlined in red) rising vertically from the western margin of the main column. b Photo taken at approximately 0423 UTC by Lisiate Tai, from Ha’atafu on the western peninsula of Tongatapu. This image shows the wide base of the column, roughly 30 km diameter, with respect to the umbrella cloud (~90 km diameter) and a large thermal (red arrow) rising up on the exterior margin of the column, presumably from steam generated by ocean-entering density currents. Inset in (b) from Google Earth® shows the locations where these photos were taken and the view directions. Photos used with permission

Here, we suggest that water also played an important role in plume dynamics. The plume was white in visible imagery, suggesting abundant water and/or ice. Intense lightning in the umbrella cloud suggested the presence of liquid water at 20–30 km altitude (Van Eaton et al. 2023); and at least 150 Tg of water (equivalent to 0.15 km³ seawater) were inferred to be injected into the stratosphere (Millán et al. 2022; Vömel et al. 2022; Xu et al. 2022). In addition, the erupted particles have characteristic density, vesicularity, shape, size distribution, and surface fractures suggesting that water influenced magma fragmentation.

We investigate four scenarios to understand the conditions that could have increased plume height while keeping a modest flux of tephra into the high plume:

• 1. Eruption of a conventional volcanic jet through a dry crater (Fig. 3a).

• 2. Addition of cold seawater to a stable (i.e., non-partially collapsing) plume. This process could occur by entrainment along the margins of a sustained jet as it rises through the water column (Fig. 3b) (Cahalan and Dufek 2021; Rowell et al. 2022). This type of water entrainment was noted, for example, during continuous-uprush eruptions at Surtsey (Thorarinsson 1967). Water may also be added to a plume as repeated hydrovolcanic explosions (i.e., induced fuel-coolant interactions (Dürig et al. 2020)) feed steam and tephra into a sustained column.

• 3. Excess gas in the magma body. Intermediate to silicic magmas erupting 0.01–10 km³ may contain 1–6 wt% of exsolved gas, based on discrepancies between petrologic and satellite-based estimates of sulfur emissions (Wallace 2001). We know of no evidence that excess exsolved gas existed in the Tonga magma but consider it in our analysis for completeness.

• 4. The rise of particle-free steam from PDCs or tephra jets entering the sea (Fig. 3c). Wide, billowing plume margins, suggestive of co-PDC ascent, are visible early in the eruption (Fig. 2) (Clare et al. 2023, Fig. S6), and several cubic kilometers of rapidly emplaced submarine density current deposits were mapped on the seafloor (Clare et al. 2023). This scenario differs from (2) by the fact that most magmatic particles that generate the steam do not rise with the plume.

Fig. 3

a Illustration of an idealized vent condition for a dry eruption in which a gas-particle jet exits through a flaring vent and decompresses while air is entrained. The air gradually works its way into the jet core. b One scenario of magma-water mixing, based loosely on Sigurdur Thorarinsson’s description of continuous uprush events at Surtsey (Thorarinsson 1967, p. 43). In this scenario, water enters the jet initially along a boundary layer and works its way into the jet core as it rises. Thermal equilibration of entrained water with the magma-gas mixture causes the mixture density to decrease and the jet diameter to increase. Jet velocity slows down as the entrained water mixes in, resulting in a wider, slower, jet. c A plume of steam rising from pyroclastic density currents mixing with water. Unlike the other scenarios, the plume is buoyant at the base and accelerates as it rises

The Plumeria model

We study the role of water in Hunga’s plume dynamics using the steady-state, one-dimensional model known as Plumeria (Mastin 2007b; Mastin 2014). The model assumes that all phases—air, particles, gas, water vapor, liquid water, and ice—can be treated as a single homogeneous mixture whose properties are weighted sums of individual-phase properties. It assumes that all phases are at thermal equilibrium and moving at the same speed. The model does not consider partial column collapse or buoyant rise of co-PDC material. The appropriateness of some of these assumptions is considered in Supplement S3.

The model approximates the plume as a series of cylindrical control volumes, each characterized by a given flux in mass, momentum, and energy. Changes in mass flux result from the entrainment of air, which is assumed to occur at rates proportional to the plume axial velocity and the crosswind velocity (Mastin 2014). Changes in momentum flux result primarily from buoyancy forces. Changes in total energy flux result from the energy of entrained air. The total energy flux in the plume is the sum of its enthalpy, kinetic energy, and elevation potential energy. At each step in the solution, total energy is calculated explicitly. The elevation potential energy and kinetic energy can be calculated independently and subtracted (Mastin 2007b, eq. B45), leaving a known total enthalpy of the mixture, h_mix. The mixture enthalpy plays a key role in plume dynamics. Our method of calculating it is detailed below.

Plume thermodynamics and phase relations

The mixture enthalpy h_mix is the sum of the enthalpies h_x of the various phases (g = H₂O gas, i = ice, l = liquid water, a = air, p = particles), multiplied by their respective mass fractions m_x:

$${h}_{\textrm{mix}}={m}_v{h}_v(T)+{m}_i{h}_i(T)+{m}_l{h}_l(T)+{m}_a{h}_a(T)+{m}_p{h}_p(T)$$

(3)

The enthalpy of liquid water in the plume is calculated by linear interpolation through a series of data points taken from established steam tables (Haar et al. 1984) over a range of temperatures under saturated conditions (Mastin 2007b, Table B4). Enthalpy of air and water vapor are calculated using temperature-dependent polynomials (Mastin 2007b, eqs. B30-B33). Enthalpies of ice and particles are calculated as simple linear functions of temperature (Mastin 2007b, eq. B34, B35). The partial pressure of water vapor in equilibrium with liquid water at T < 314 K is considered an exponential polynomial of temperature (Mastin 2007b, eq. B19), while the partial pressure of water vapor in equilibrium with ice is taken as a simple log function of temperature (Mastin 2007b, eq. B18).

The mass fraction of particles m_p can be calculated from the influx at the vent (Plumeria neglects particle fallout), and the mass fraction of air m_a can be calculated from the cumulative amount of air entrained up to a given height. The mass fraction of total water (m_H2O = m_l+m_i+m_v) is also known from the gas influx at the vent, from the mass fraction of external water added, and from the water vapor contained in the entrained air (relative humidity is specified in the atmospheric profile). However, the partitioning of this water into ice, liquid, and vapor is unknown a priori and depends on the temperature, pressure (p), and total water content of the plume.

At a given altitude, the equilibrium temperature and mass fraction of hydrous phases are determined by first identifying four different thermal regimes bounded by the following temperatures:

• 1. T_sat, above which all water is in vapor form; m_v = m_H2O, and the partial pressure of water vapor equals or exceeds the atmospheric pressure. With knowledge of p and m_H2O, one can calculate T_sat from the equation the partial pressure of water vapor at saturation (Mastin 2007b, eq. B19 or B20) and, from that, the value of h_mix at T_sat (Eq. 3).

• 2. T_coldwater, below which liquid water begins to freeze. At T = T_coldwater, m_i = 0, m_l = m_H2O-m_v, and m_v is calculated from known values of p and T_coldwater using the equation for partial pressure of water at saturation (Mastin 2007b, eq. B19 or B20) and related equations (Mastin 2007b, eqs. B22, B24). Once the mass fractions have been determined, h_mix at T_coldwater can be calculated using Eq. (3).

• 3. T_freezing, below which water is present only as ice and vapor. At this temperature, m_l = 0, m_i = m_H2O-m_v, and m_v is determined by the equilibrium partial pressure of water vapor with ice (Mastin 2007b, eq. 18).

The value of h_mix obtained from the conservation equations at a given elevation is then compared with mixture enthalpies at each of these temperature boundaries. Once it is determined in which thermal regime the mixture lies, the final mixture temperature and mass fraction of hydrous phases are obtained by repeatedly adjusting temperature, recalculating mass fractions, and recalculating mixture enthalpy until its value agrees with the known value of h_mix.

Note that the value of T_sat is calculated, given p and m_H2O, whereas the values of T_coldwater and T_freezing are specified. The range between these two numbers is where liquid water and ice coexist. This temperature range has been the subject of considerable uncertainty, despite its importance in controlling the growth of clouds and the occurrence of lightning. In the absence of volcanic ash, atmospheric models have commonly assumed that ice and liquid water coexist over subfreezing temperatures from about −10 °C to the homogenous nucleation temperature of ice at −40 °C (e.g., Khairoutdinov and Randall 2003). Volcanic ash, however, forms efficient ice nuclei and several studies suggest that, in the presence of volcanic ash, the range of coexisting ice and liquid water is narrower, around −15 to −23 °C (Durant et al. 2008; Steinke et al. 2011; Bingemer et al. 2012). In Plumeria, we have varied this range over the years as knowledge has evolved; from −10 to −40 (Mastin 2007b) to −7 to −15 (Mastin 2014). The current version assumes a coexisting range between −15 and −23 °C based on the above-mentioned studies. Uncertainty in this temperature range leads to uncertainty in the height range of this coexisting region.

Model inputs and outputs

Inputs to the model are of two types: (1) atmospheric inputs that include a sounding of temperature, pressure, relative humidity, and wind vectors from sea level to the plume top; (2) volcanic inputs that include the initial plume diameter d_e, velocity u_e, temperature of magma particles T_p, weight percent magmatic gas in the magma m_g, the mass fraction of external water m_w added at the vent, and the external water temperature T_w. The water-magma mixture is assumed to be thermally equilibrated prior to ascent.

Modeling strategy

Our strategy is to simulate four types of eruptions, summarized in Table 1 and listed above:

• 1. Dry eruptions that are driven only by magmatic gas (set 1 in Table 1),

• 2. Wet eruptions in which cold seawater and magma mix and thermally equilibrate in the vent region and then ascend in a steady plume (sets 2–4),

• 3. Eruptions driven by excess magmatic gas (sets 5–7), and

• 4. Eruptions in which the plume is supplemented by water vapor at T_w = 100 °C, rising from magma that mixed with seawater but did not ascend with the plume (sets 8–10). We refer simplistically to this water vapor as steam. Although we recognize that white steam clouds contain micro-droplets of liquid water, we ignore the liquid phase for simplicity in the initial conditions for this scenario. We also ignore the possibility that water vapor rising from subaqueous magma could range in temperature well above 100 °C.

Table 1 List of volcanic inputs used in each set of runs. Symbols refer to the number of simulations (N), magma temperature (T_p), mass fraction gas in the magma (m_g), mass fraction external water (m_w), external water temperature (T_w), equilibrated mixture temperature at the vent (T_mix), log mass eruption rate ($\dot{M}$), and exit velocity (u_e). The “v” listed after water temperature in the last three rows indicates that the water is in vapor form. The log mass eruption rate and exit velocity are given as ranges because the value for each simulation was chosen randomly from within those ranges using a uniform probability density function. Summary inputs and outputs from these runs are in Tables S08 through S17

Table 2 Inputs used in simulations illustrated in Fig. 6. Output for these runs is listed in Tables S18–S20

In each of these cases, we run simulations through a range in mass eruption rate of solid particles in the plume (${\dot{M}}_p=\dot{M}\left[1-\left({m}_g+{m}_w\right)\right]$) and examine how added water affects plume height, umbrella growth rate, and other plume characteristics. The second type is similar to wet eruptions analyzed by the 1-D models of Koyaguchi and Woods (1996) and the 2-D modeling of Van Eaton et al. (2012) but is tailored to this particular case by using local atmospheric conditions. We also go beyond previous studies by using a Monte Carlo approach to account for uncertainty in source parameters.

For Atmospheric conditions, we use an atmospheric sounding from the US National Oceanic and Atmospheric Administration (NOAA) Global Forecast System (GFS) 0.5-degree model, with a posting time of 0000 UTC on 15 January 2022 (Fig. S3, Table S5). This sounding extends to about 80 km elevation above sea level. The higher-resolution ECMWF ERA5 model (Hersbach et al. 2018) extends only to about 48 km elevation; however, it agrees closely with the GFS model (Fig. S3, Table S6). For ocean conditions, we consider a seawater temperature of 25 °C, which is close to the mean seawater temperature in the Tonga Islands.¹

Volcanic inputs were varied in a more complicated manner. Some inputs, such as magma temperature (T_p), remained constant for all simulations. Others, such as magma gas content (m_g), mass fraction external water (m_w), and water temperature (T_w), were varied systematically from one set of runs to another. Still others, such as mass eruption rate ($\dot{M}$) and exit velocity (u_e), were randomly sampled during each simulation from a range of values. Overall, we ran ten sets of simulations, 200 simulations per set, varying inputs as shown in Table 1. Explanation of these inputs follows:

• For magma temperature, we use T_p = 1050 °C, which is in the middle of the range of values (~1000–1100 °C) indicated by equilibrium clinopyroxene-melt thermometry of pre-2022 products (Brenna et al. 2022). We assume a magma density of 2500 kg m⁻³, and magma specific heat of 1100 J kg⁻¹ K⁻¹.

• For magma gas content, most sets of simulations use m_g = 3 wt%, which is slightly above the highest values of dissolved water (~2.71 wt%) measured in Hunga matrix glass erupted in 2022 and in melt inclusions in the same products (in clinopyroxene and feldspar) by Synchrotron-Source Fourier Transform Infrared spectroscopy (Jie Wu, Univ. Auckland, written commun., 2023). In sets 5–7, we explore the effects of higher magmatic gas content on plume height. Excess gas, in the form of a separate vapor phase, has been inferred in many magma bodies based on discrepancies between satellite measurements of co-eruptive SO₂ and petrologic estimates of sulfur solubility (Wallace 2001). Pinatubo for example contained at least 5 volume percent exsolved gas at depth in 1991 (Gerlach et al. 1997), and Mount St. Helens about 15 volume percent in 1980 (Wallace 2001). Total exsolved+dissolved gas contents may be up to twice the amount that could be dissolved in the melt and diminish with increasing eruption size (Wallace 2001). The diminution implies that eruptions tap a gas-rich headspace near the chamber top. The gas content of the Hunga magma would have to be truly exceptional to explain the observed height-eruption rate discrepancy. We have nevertheless run some simulations to examine this effect.

• For mass eruption rate ($\dot{\boldsymbol{M}}$), each simulation used a value drawn randomly from a uniform distribution of log₁₀($\dot{\boldsymbol{M}}$), with minimum and maximum values given in Table 1. This range extends well outside the likely range of values at Hunga but allows us to examine trends. The results are plotted and discussed using the mass eruption rate of solid particles $\dot{{\boldsymbol{M}}_{\boldsymbol{p}}}$, which does not include gas or water. We calculate $\dot{{\boldsymbol{M}}_{\boldsymbol{p}}}$ from the total mass eruption rate using the formula ${\dot{M}}_p=\dot{M}\left[1-\left({m}_g+{m}_w\right)\right]$, where $\dot{M}$ is the total mass eruption rate.

• Jet exit velocity (u_e) is an important source of uncertainty. We assume that the Hunga jet, like most large volcanic jets, was choked—i.e., limited by their sound speed of ~80–200 m s⁻¹—at some near-surface constriction (Kieffer 1989). As the jet accelerated above the choke point, many processes, such as shock waves, particle interactions, and turbulent mixing, would have reduced the final exit velocities below the theoretical maximum of several hundred meters per second (Woods and Bower 1995; Ogden et al. 2008) (Fig. 3a). Smaller eruptions have recorded particle-ejection velocities up to about 400 m s⁻¹ (Taddeucci et al. 2017). Thus, in simulations of dry eruptions, we randomly select u_e from the range of 100 to 350 m s⁻¹ using a uniform probability density function. In simulations that add cold seawater (sets 2–4 in Table 1), mixing could occur under one of several scenarios that could either reduce exit velocity (as shown in Fig. 3b) or increase it (as assumed by Koyaguchi and Woods (1996)). Simulations that inject mostly steam into the atmosphere are assumed to rise from PDCs entering water (Fig. 3c) and thus have lower initial ascent velocities. For these simulations, we randomly select u_e from the range of 5 to 30 m s⁻¹.

• Initial diameter d_e was calculated for each simulation using the formula ${d}_e=2\sqrt{\dot{M}/\left({\rho}_e\pi {u}_e\right)}$, where $\dot{\boldsymbol{M}}$ and u_e are assigned as above, and the exit mixture density ρ_e is calculated from the initial conditions. Note that these diameters do not represent the diameter of the choked vent at the ground surface, but some diameter above the ground surface where the plume has equilibrated to atmospheric pressure (Fig. 3).

Results

Effect of adding water to plume

Figure 4a shows the plume top height (H_t) versus the mass eruption rate of solid particles (${\dot{M}}_p$) for sets of runs with 0, 5, 10, and 20% external water added. An analogous plot, using the total mass eruption rate on the x axis, is shown in Fig. S4. For simulations where ${\dot{M}}_p$<~6 × 10⁸ kg s⁻¹, there is little scatter in each family of curves, suggesting that variations in initial velocity and plume radius (which were randomly selected for each run) do not significantly affect H_t. At ${\dot{M}}_p$>~6 × 10⁸ kg s⁻¹, the heights of the wet plumes in Fig. 4a show slightly more scatter than at lower eruption rates, indicating slightly greater sensitivity to initial exit velocity. At ${\dot{M}}_p$ > 10⁹ kg s⁻¹, some plumes underwent total column collapse (symbols along the base of the plot). Their small symbol size indicates lower exit velocities, consistent with the idea that low exit velocity leads to column collapse (Sparks et al. 1978).

Fig. 4

a Plume top height (H_t) above vent level (avl) versus mass eruption rate of solids for mass fractions of water (m_w) of 0, 0.05, 0.10, and 0.20. b Plume height versus eruption rate for mass fractions of magmatic gas of m_g = 0.03, 0.10, 0.20, and 0.50, respectively, with no added water. c Plume height versus eruption rate for a plume with mass fractions of added steam (m_s) of 0, 0.2, 0.5, and 0.9. Red dashed lines demark the range of heights of 50–58 km, reached by the plume during its most intense eruptive phase from about 0437–0527 UTC. Blue vertical dashed lines demark the range of mass flux into the airborne plume (${\dot{M}}_p$), which is calculated to be 7–14 × 10⁷ kg/s using an eruption duration of 1 h and fall deposit volume of 0.1–0.2 km³ DRE derived from Ash3d simulations that reproduce the fall deposit measurements (Fig. 1b). Marker size is proportional to exit velocity of each simulation. Example marker sizes with labeled exit velocities are shown on the left side of (a) and right side of (c). Model outputs are given in Tables S8–S17

Simulations with no external water reach plume top heights above 50 km when ${\dot{M}}_p$> 3 × 10⁹ kg s⁻¹. At mass eruption rates above about 6 × 10⁸ kg s⁻¹, which are relevant to Hunga, the addition of water significantly decreases the height of the stable plumes, requiring even greater ${\dot{M}}_p$ to reach the observed maximum heights of 50–60 km. Moreover, it appears that no stable plumes containing > 20% water can rise to 50 km, no matter how high the mass eruption rate. Thus, the addition of cold water to a stable, vent-derived plume cannot explain Hunga’s high plume. At such high eruption rates, the added water not only decreases plume top height H_t; it also decreases the volume flux at neutral buoyancy elevation V_nbl (Fig. 5a), requiring greater ${\dot{M}}_p$ to grow the umbrella cloud.

Fig. 5

Volume flux at neutral buoyancy level for simulations using variable mass fractions (m_w) of added water (a); variable mass fractions (m_g) of magmatic gas (b); and variable mass fractions (m_s) of added steam (c). The red dashed lines indicate the inferred volumetric flux rate into the umbrella, assumed to be within a factor of two of 320 km³ s⁻¹. The blue dashed lines demark the range of inferred mass eruption rate, as described in the caption to Fig. 4

This drop in height and volume flux when ${\dot{M}}_p$> ~6 × 10⁸ kg s⁻¹ contrasts with the effect at lower eruption rates, where—as pointed out by Koyaguchi and Woods (1996)—added water has little effect on plume height. Koyaguchi and Woods explained this difference by noting that, for small eruptions, the heat lost from the magma to ambient water at low altitudes returns to the plume in the form of latent heat release at higher levels as the water vapor condenses (or freezes). At ${\dot{M}}_p$> ~6 × 10⁸ kg s⁻¹, however, the greater density of the cooler plume reduces ascent velocity and causes the plume to stall before all the surface water can condense and release its latent heat. This trend is apparent in Fig. 6, which shows three plumes of roughly equal height: a dry plume with ${\dot{M}}_p$ = 4.4 × 10⁸ kg s⁻¹, a wet plume with ${\dot{M}}_p$ = 4.4 × 10⁸ kg s⁻¹ and 20 wt% added water, and a wet plume with ${\dot{M}}_p$= 8.3 × 10⁹ kg s⁻¹ and 20 wt% added water. The higher-${\dot{M}}_p$ wet plume is significantly warmer above 10 km elevation (Fig. 6b), contains much greater enthalpy at the plume top (Fig. 6c), and only begins to freeze near the very top of the plume (Fig. 6e).

Fig. 6

Plume profiles for three eruptions that all produce roughly the same plume height: (1) a dry eruption with ${\dot{M}}_p$ = 4.4 × 10⁸ kg s^-1; (2) a wet eruption with ${\dot{M}}_p$ = 4.4 × 10⁸ kg s⁻¹ and mass fraction (m_w) external water of 0.2; and (3) a wet eruption with ${\dot{M}}_p$ = 8.3 × 10⁹ kg s⁻¹ and m_w=0.2. Profiles are ascent velocity (a), plume temperature (b), excess specific enthalpy (defined as the specific enthalpy of the plume mixture (h_mix) minus that of ambient air at that elevation (h_air)) (c), density difference between the plume (ρ_mix) and surrounding air (ρ_air) (d), and mass fraction of liquid water (m_l, solid lines) or ice (m_i, dashed lines) in the plume (e). In the higher-MER wet eruption, note the excess enthalpy high in the plume in c and the formation of liquid and ice at higher altitude in e. Neutral buoyancy elevations (z_nbl) are shown by dashed lines in d. The ambient air temperature in b is taken from the Global Forecast System model output. Inputs used for these simulations are given in Table 2. Model outputs are given in Tables S18–S20

The addition of water to the plume adds mass but does not add significant energy. Because the rise of plumes is driven by the conversion of thermal energy to elevation potential energy, the addition of mass without energy should reduce plume height for the larger, stable plumes. An important caveat to model runs in sets 1–7 (Table 1) is that Plumeria does not account for the secondary rise of co-PDC (“coignimbrite”) plumes, which Van Eaton et al. (2012) showed can substantially increase the overall height of high-intensity, water-rich eruptions. We consider this process in a simplified way for our 1-D modeling using scenario #4 “effect of adding steam” (sets 8–10, Table 1).

Effect of adding gas to the plume

Figure 4b shows how plume height is affected by adding magmatic gas. H₂O gas has a specific heat about four times that of solid particles; thus, the addition of gas (water vapor at magmatic temperature) significantly increases the heat content of the plume. A stable plume containing a mass fraction gas of 0.03 must erupt at ${\dot{M}}_p$> ~3 × 10⁹ kg s⁻¹ to send a plume above about 50 km, while plumes containing 10, 20, or 50% gas by mass require only 2.2 × 10⁹, 1.5 × 10⁹, and 4.6 × 10⁸ kg s⁻¹ to reach this height, respectively. Thus, an erupting mixture containing about 50 wt% magmatic gas could, in principle, drive a 50-km high plume with about a tenth of the solids contained in a 3% gas mixture. However, the addition of 50% gas by mass to an erupted mass the size of Hunga is highly implausible. Large eruptions such as Pinatubo 1991 (Gerlach et al. 1997) or Mount St. Helens 1980 (Wallace 2001) may have contained 5–15 volume percent of exsolved gas. By contrast, a Hunga magma system containing 50% gas by mass would have to contain > 95% gas by mass, assuming for example a depth of 4 km, pressure of 100 MPa, temperature of 1050 °C, and pure H₂O gas with a density of 166 kg m⁻³ (Haar et al. 1984). Such high-volume fractions can be readily dismissed.

Effect of adding steam

Figure 4c shows the effect on plume height of added water vapor (i.e., “steam”) at a temperature of 100 °C. As shown in Fig. 4a, the dry plumes must exceed ${\dot{M}}_p$ = 3 × 10⁹ kg s⁻¹ to reach 50 km elevation. However, the addition of 50 and 90% steam drops the required mass eruption to ${\dot{M}}_p$≈ 9 × 10⁸ and ~8 × 10⁷ kg s⁻¹, respectively. Similarly, a dry plume requires ${\dot{M}}_p$ > ~1.8 × 10⁹ kg s⁻¹ to feed the umbrella cloud at the observed rate of 330 km³ s⁻¹, while plumes containing 50 and 90% steam achieve it when ${\dot{M}}_p$ > 4 × 10⁸ and 5 × 10⁷ kg s⁻¹, respectively (Fig. 5c).

The effects of added steam are also apparent in the modeled plume dynamics (Fig. 7). At its base, a plume with 50 wt% added steam is significantly cooler than a dry plume (Fig. 7b) but contains comparable enthalpy flux at the vent (Table 3). More importantly, the steam-boosted plumes are positively buoyant even at their base (Fig. 7d), causing them to accelerate in the first few kilometers, and their diameter to decrease with elevation (Fig. 7c). This is a significant contrast with dry, stable plumes with a typical jet-thrust region, which are negatively buoyant at their base and must entrain and heat air to attain positive buoyancy (Sparks and Wilson 1976). In stable, jet-driven plumes, the mass eruption rate is proportional to d_e², whereas the rate of air entrainment is proportional to d_e; thus as vents enlarge, jets entrain less air as a fraction of their mass flux, and they tend toward column collapse (Sparks and Wilson 1976). Positively buoyant steam-driven plumes do not suffer this limitation: they can be arbitrarily wide at the base and still rise buoyantly.

Fig. 7

Profiles of properties for a plume with no added water or steam (red), with 50 wt% added steam (blue), and 90% steam (cyan) having ${\dot{M}}_p$ = 2.95 × 10⁹, 9.32 × 10⁸, and 8.57 × 10⁷ kg s⁻¹, respectively. These three runs were chosen because their plume heights are all approximately H_t = 50 km. Variables plotted are upward velocity (a), plume temperature (b), plume radius (c), density difference between the plume (ρ_mix) and the surrounding atmosphere (ρ_air) (d), and the mass fraction of liquid water (m_l, solid lines) or ice (m_i, dashed lines) (e). The red double arrow in e denotes the elevation range in which liquid water and ice coexist in the plume containing 90% steam. The orange double arrow at the left side of e indicates the height range over which lightning was reported by Van Eaton et al. (2023). The ambient air temperature in b is taken from the Global Forecast System model output. Inputs used for these simulations are given in Table 3

Table 3 Initial conditions at the vent for simulations used in Figs. 7 and 8. In these simulations, external water (m_s) is in the form of water vapor (“steam”) at T = 100 °C. At the vent, the enthalpy flux ($\dot{\boldsymbol{M}}\bullet {\boldsymbol{h}}_{\textbf{mix}}$, right column) is calculated by Plumeria given the initial conditions. All other variables below are specified. Output for these runs is listed in Tables S21–S23

Another important finding is that plumes containing both 50 and 90% steam by mass begin to condense liquid water at 4–15 km elevation and begin to develop ice at 26–28 km (Fig. 7e). Our simulations show a region of coexisting liquid water and ice between about 26 and 33 km elevation, consistent with the findings of Van Eaton et al. (2023) using characteristics of Hunga’s volcanic lightning. By contrast, the dry plume never condenses liquid water, only creating ice by deposition freezing at elevations above about 42 km.

Discussion

The 1D model results indicate that a dry plume rising above 50 km in the atmospheric conditions at Hunga volcano would have required a mass flux of particles into the plume > 3 × 10⁹ kg s⁻¹. However, our ash transport simulations reproduce the observed deposit thicknesses using an erupted volume of only 0.1–0.2 km³ DRE, which suggests the actual flux of particles into the plume was more than an order of magnitude lower, ${\dot{M}}_p$ = 7–14 × 10⁷ kg s⁻¹. By the curves in Fig. 4c, we infer that that the Hunga plume could have risen above about 50 km with ${\dot{M}}_p$ = 7–14 × 10⁷ kg s⁻¹ and an addition of about 90% steam by mass. We think this conclusion is robust despite the simplicity of the 1D model (Supplement S3). Rising steam may also account for the later plume after 0537 UTC, which remained above 18 km for more than 12 h with minimal ashfall observed.

Observations of the climactic Hunga eruption suggest that rapid plume rise coincided with turbulent mixing of magma with seawater. The eruption abruptly intensified around 0400 (Vergoz et al. 2022; Van Eaton et al. 2023), followed by column collapse and a second intensification at 0421 (Clare et al. 2023; Purkis et al. 2023). These events were followed within minutes by density currents that swept down the submarine flanks to distances > 100 km at speeds up to 122 km hr⁻¹ (Clare et al. 2023). At the same time, the observed plume top rose from 22 to 58 km between 0417 and 0437, dropped to 43 km at 0447, rose back to 54 km at 0457, and stayed at or above 50 km until 0527. During this period, Plumeria calculations using 90% added steam (Table S23) indicate that the plume would have ascended to its maximum final height in less than 5 min (e.g., Table S23). The slower observed rise time from 0417 to 0437 suggests that flux into the plume was ramping up over this period.

During mixing, heat must have been transferred from magma to water almost as fast as it erupted. The model scenario of a 50-km-high plume containing 90% steam (Table 3) would have injected about 1.6 km³ s⁻¹ steam, with the energy equivalent to about 2.0 × 10⁹ kg s⁻¹ magma, boiling 8.2 × 10⁵ m³ s⁻¹ of seawater (Table S7). The question of how the erupting magma transferred heat to surrounding water so rapidly is a topic of active research. Some of the heat transfer may have occurred at depth during induced molten fuel-coolant interactions (e.g., Dürig et al. 2020). Violent mixing of this type is suggested by tsunami-generating blasts at 0400, 0406, 0418, and 0456 (Purkis et al. 2023), and blasts that were audible from Tongatapu between 0420 and 0430 UTC (Delmar 2022). However, steam generation could have also occurred as turbulent density currents entered the ocean. The currents with densities of ρ = 1000–2000 kg m⁻³, thicknesses of D = ~100 m, and velocities of u = 30–100 m s⁻¹ would have had Reynolds numbers (Re = ρuD/μ) of 10⁹–10¹⁰ (assuming water-like fluid viscosity of μ ≈ 10⁻³ Pa s). At these huge Reynolds numbers, intense turbulence would cause extreme rates of mixing. Turbulent shedding (Mastin 2007a) would have broken magma into fine fragments by repeatedly quenching their outer surfaces and stripping them away.

Figure 8 shows a schematic of the plume as it may have appeared shortly after 0430 UTC, rising to 50 km elevation, fed by a wide column containing 90% steam (Tables 3, S23). We envision that this column arose from both a central, vent-derived plume and surrounding steam from pyroclastic ejecta that was collapsing into the sea and mixing with the surrounding water. As the plume rose, the diameter of its high-velocity core would have narrowed to several kilometers as it accelerated but then widened again as the accelerating velocity increased air entrainment. A ring of lightning about 50–60 km in radius (Fig. 1a, inset) was moving outward through the cloud at about 100 m s⁻¹, propelled by gravity waves, which were visible on the cloud top (Van Eaton et al. 2023, supplement). The 1-D modeling suggests that ascent velocities at 20–30 km altitude may have reached 270–280 m s^-1.

Fig. 8

Conceptual sketch of the plume as it might have appeared shortly after 0430 UTC, as its top rose above 50 km. In profiles to the left, cyan curves show the ascent velocity and plume temperature of a 50-km-high plume containing 90% steam (inputs in Table 3, output in Table S23). The pink dashed curve is the ambient temperature profile. Horizontal and vertical scales are equal

Other steam-boosted plumes

Well-observed, steam-rich plumes have been a common occurrence over the past decade. At Kīlauea on 20 December 2020, for example, fountain-fed lava flowed into the summit caldera lake, boiling off 0.9M m³ of lake water in about 90 min and producing a 13-km-high, ash-poor plume (Cahalan et al. 2023). On 13 August 2021, Fukutoku Oka-no-ba volcano in Japan erupted from a seawater depth of < 70 m, producing a 16-km-high, water-rich eruption column for more than 9 h (Maeno et al. 2022). Most of the 0.1 km³ (DRE) erupted volume at Fukutoku Oka-no-ba accumulated in the near-vent region as a tuff cone. Through 1-D plume modeling, Maeno et al. (2022) were able to account for the high, ash-poor characteristics of the plume if 50 to 95% of the erupted magma transferred its heat to the plume without rising in it. Similarly, through plume modeling, Cahalan et al. (2023) accounted for the 13-km height of the Kīlauea plume using only the heat contained in the rising steam. Many other high (> 10 km) plumes rising from wet vents have been deemed water-rich based on their white color or lack of an ash signal in satellite images. Examples include Rabaul 1994 (Rose et al. 1995), Grímsvötn 2011 (Moxnes et al. 2014), South Sarigan Seamount 2010 (Embley et al. 2014), Bogoslof 2017–2018 (Schneider et al. 2020), Taal 2021 (Balangue-Tarriela et al. 2022; Van Eaton et al. 2022), and Anak Krakatau 2018 (Prata et al. 2020).

Implications for lightning generation

Our finding that a steam-boosted plume rising > 50 km produces a region of coexisting water and ice at 26–32 km is consistent with the mechanism proposed for abundant, high-altitude lightning in the Hunga plume. This plume produced more lightning than any other meteorological event on record (Van Eaton et al. 2023). Unprecedented flash rates above 2600 per minute were detected by global lightning location networks. When plotted on a map, the flashes that occurred from about 0417 through 0507 UTC expanded outward as a donut-shaped ring at nearly 100 m s⁻¹—similar to the rate of outward movement of a gravity wave that pulsed through the cloud (Fig. 1a inset). Van Eaton et al. (2023) estimated that the bulk of this lightning ring took place at about 20–30 km altitude.

The ground-based networks that detected Hunga’s lightning rely on very low-frequency electromagnetic radiation, usually detected at distances of thousands of kilometers from the source. In volcanic plumes, lightning flashes powerful enough to be detected by global networks are typically produced by a charge-separation process involving droplets of supercooled liquid water falling through ice crystals (Behnke et al. in press). Thus, high rates of globally detected volcanic lightning occur when liquid water and ice coexist in the turbulent core of the plume (Van Eaton et al. 2020; Van Eaton et al. 2022).

Meteorological lightning rarely occurs at elevations above about 18–20 km (MacGorman et al. 2017; Van Eaton et al. 2023). Although supercooled liquid water is known to exist in meteorological clouds at temperatures as low as the homogeneous freezing temperature of water (−40 °C) (Rogers and Yau 1989, p. 152), tropopause temperatures are typically cooler than this, even in the tropics. For example, during this eruption, the ambient temperature was about −71° at the tropopause elevation of 16.6 km (Table S6). Within the rising plume, however, temperatures at 28–32 km elevation in our model were −13 to −20° and −20 to −23 °C for plumes containing 50 and 90% steam, respectively. This is within the range where we assume liquid and ice may coexist in the presence of ash particles.

Implications for inferring mass eruption rate from plume height

The height of plumes and the growth rate of umbrella clouds have long been known to correlate with mass eruption rate (Settle 1978; Wilson et al. 1978; Sparks et al. 1997; Pouget et al. 2013; Aubry et al. 2023). This relationship has been used to give us quantitative constraints on ${\dot{M}}_p$ during eruptions, enabling real-time estimates of eruption size, and supporting model forecasts of ash-cloud concentration (Mastin et al. 2009; Beckett et al. 2020; Mastin et al. 2021).

The correlation between H_t and $\dot{M}$ is empirical, but the form of the relationship (H_t = A$\dot{M^B}$, where A and B are fitting coefficients) is based on the theoretical finding that plume height is proportional to the fourth root of buoyancy flux (Morton et al. 1956). Buoyancy flux, in turn, is roughly the rate at which heat is supplied to the plume. Early studies in volcanology that applied this relationship, such as Settle (1978), noted that most heat supplied to volcanic plumes was contained in the solid pyroclasts; hence, plume height is generally thought to correlate with the mass eruption rate of solids into the plume itself. But if most of the erupted mass decouples from the plume and transfers its heat to rising steam, this correlation may be poor.

If heat flux is the dominant factor controlling plume height, then the height of wet, dry, gas-driven, and steam-driven plumes in Fig. 4 might all collapse onto one curve when plotted as a function of, for example, enthalpy flux at the vent. Figure S5 shows such a plot. The curves come closer but do not collapse onto a single line. The reasons why they do not are beyond the scope of this study, but we hypothesize that differences in the amount of heat per unit mass may play a role.

Conclusions

In this paper, we simulate the volume of tephra-fall deposits from Hunga volcano from its 15 January 2022 eruption and show that it is at least an order of magnitude less than expected based on the extreme height and umbrella-growth rate of the plume. The contrast between the high plume and modest deposit volume raises the question of what could have boosted the plume to such heights with so little ash. One-dimensional plume modeling indicates that the addition of cold seawater to a steady, vent-derived plume could not reproduce the observed height; however, the addition of steam rising from pyroclastic density currents mixing with ocean water, combined with intense, near-source magma-water mixing could have boosted the plume height. The large amount of water involved may also explain the extreme rates of volcanic lightning that occurred at high altitudes. We conclude that the boosting of plume height by steam in ash-poor, wet eruptions may be a common but underappreciated phenomenon.