3.2. Rise of a Bubble
 If methane bubbles are released slowly into shallow seawater where hydrate is not stable (e.g.,<537 mbsl, Figure 1c), bubbles rise through water individually or in streams [Brewer et al., 1997; Clark et al., 2000; Leifer et al., 2000]. As a bubble rises, it dissolves in seawater (which is almost always CH4-undersaturated), and expands as the pressure decreases. The dissolution makes the bubble smaller. The expansion makes the bubble larger. Whether the size of the bubble would increase or decrease depends on the interplay between dissolution and expansion. Zhang and Xu  developed a model for the convective dissolution rate of a spherical bubble as it rises buoyantly through seawater, but nonspherical bubbles (those with radius greater than 1.5 mm) can only be roughly treated. For a given depth, there is a critical bubble size, above which bubble size increases, and below which the bubble size decreases as it dissolves and rises. The critical radius for a bubble to survive 50-m rise is 0.9 mm. Bubble rise velocity is typically ≤0.3 m/s.
3.3. Rise of a Bubble With Hydrate Shell
 If a methane bubble is released into deep water where methane gas and water would react to form hydrate (≥537 mbsl, Figure 1c), a thin hydrate shell may form on the bubble. As the bubble with a thin hydrous shell rises, hydrate would dissolve. Furthermore, gas in the bubble would expand, cracking the hydrate shell. Nevertheless, any new contact between CH4 in the bubble and seawater would lead to new hydrate formation. Hence there would be a delicate balance between dissolution of methane hydrate shell, and its reformation, leading to a steady-state CH4 mass loss from the bubble. Since hydrate is more stable than CH4 gas, the solubility of hydrate in seawater is smaller than that of a CH4 bubble. Hence the formation of a hydrate shell slows down the dissolution. Calculation shows that if the initial bubble radius is 3 mm or greater, the bubble with hydrate shell would be able to survive from any depth to shallow water.
3.4. Methane-Driven Ocean Eruptions
 If CH4-bearing water is released to shallow seawater (e.g., <537 mbsl, Figure 1c), methane hydrate is unstable. With a high bubble number density (number of bubbles per unit volume of water), bubbles would rise collectively as a bubbly water plume because bubbly water has a lower overall density than the surrounding water. As the bubbly plume rises, the volume of the gas phase expands due to pressure reduction. Hence, the density of the bubbly water decreases further, leading to more rapid buoyant rise of the bubbly water plume (Figure 2). This strong positive feedback is similar to what happens in CO2-driven lake eruptions [Zhang, 1996], although the smaller solubility of CH4 in water means that the eruption velocity would be smaller under the same saturation pressure.
Figure 2. A schematic diagram for the dynamics of methane-driven eruptions. Rectangles represent methane hydrate crystals and aggregates; circles represent bubbles; double circles represent bubbles with methane hydrate shell.
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 If hydrate- and bubble-bearing pore water oversaturated in CH4 is released to deep water (> 537 mbsl, Figures 1c and 2), methane hydrate is stable, and CH4 gas bubbles may react rapidly with water to form hydrate shells. Although there is always relative motion between hydrate, bubbles and water, the more interesting motion is the rapid rise (such as ≥1 m/s) of the whole parcel of water containing shelled bubbles and hydrate. Since the water parcel is supersaturated in CH4, bubbles and hydrate would grow rather than dissolve. When the water parcel reaches shallow depths where hydrate is unstable, any hydrate shell on bubbles would dissociate rapidly, releasing the bubbles and producing a bubbly plume. Small hydrate crystals (<5 mm radius) reaching this shallow depth would also dissociate rapidly into methane bubbles and water, and become part of the bubbly plume. The ensuing dynamics of the bubbly water plume would follow that of release into shallow water, resulting in an eruption.
 The dynamics of methane-driven oceanic eruptions are modeled below, following the analyses for CO2-driven lake eruptions [Zhang, 1996, 2000]. Since the initial gas phase can be significant at the depth of hydrate dissociation where the bubble plume forms, the mass fraction of the initial gas phase (δ0) is explicitly included in the modeling. Assuming ideal gas law and equilibrium between the gas and liquid phases, and ignoring shallow water entrainment, the density of the gas-liquid mixture ρ can be expressed as follows [Zhang, 2000]:
where ρl is the liquid density, P is the pressure of the gas phase and is the same as the hydrostatic pressure, P0 is the pressure at the initial depth at which hydrate dissociates and the bubble plume forms, T is the temperature, R is the gas constant for CH4 (518.3 J kg−1 K−1), and λ = CCH4liq/CCH4gas is the Ostwald solubility coefficient and is assumed to be constant (λ depends weakly on T and P but the dependence is ignored for analytical solution below). Combining equation 1 and the Bernoulli equation, and integrating, the following equation can be obtained:
where Pexit is the exit pressure at the sea level, u is the velocity of the bubbly flow as it exits the ocean surface, and g is acceleration due to gravity.
 A more realistic model for the dynamics of CH4-driven water eruptions would require consideration of entrainment and disequilibrium. Entrainment is especially important if the volume of release is small. Owing to the simplifying assumptions, equation 2 and the results below should be considered semi-quantitative. Figure 3 shows calculated maximum exit velocity as a function of the initial depth (at which methane hydrate dissociates to methane gas and water) for several δ0 values and with λ = 0.034. Because the Ostwald solubility coefficient for CH4 is smaller than that for CO2 by a factor of about 30, CH4-driven eruptions are much less violent under the same saturation conditions. For example, if initial water depth is 208 m and δ0 = 0, the maximum exit velocity would be only 18 m/s for CH4-driven water eruptions, compared to 89 m/s calculated for CO2-driven Lake Nyos eruption [Zhang, 1996]. However, the oceans are much deeper than lakes and mass and concentration of CH4 in released pore water can be very large. For example, a 2000 km3 landslide [Bugge et al., 1987; Maslin et al., 1998; Rothwell et al., 1998] might release pore water containing ≥1 Gt CH4, 1000 times the volume of CO2 released in the 1986 eruption of Lake Nyos [Kling et al., 1987]. The dissociation of thin hydrate shells and small hydrate crystals can also contribute initial CH4 gas (δ0 > 0) in a large volume of water. Consequently, CH4-driven oceanic eruptions have the potential to achieve greater exit velocities than CO2-driven lake eruptions. Direct measurements show that pore waters in marine sediment may average 1.1 wt% total CH4 [Dickens et al., 1997]. With such a concentration at 500 mbsl, the maximum exit velocity would be 110 m/s. With high exit velocity and large amounts of gas, oceanic eruptions could be very violent.
Figure 3. Calculated maximum exit velocity (using equation 2) as a function of initial depth. Fraction on each curve indicates the initial mass fraction of the gas phase (δ0). The calculation is done for an exit pressure of 1 bar, T = 280 K, and λ = 0.034. The results are semi-quantitative because shallow water entrainment and disequilibrium between the gas phase and water are ignored.
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 In addition to the solubility difference between CO2-driven and CH4-driven water eruptions, another difference between them is that CH4 gas is less dense than air whereas CO2 gas is denser than air. Hence erupted CH4 gas is expected to rise buoyantly into the atmosphere, instead of forming a ground-hugging “ambioructic” flow [Zhang, 1996], which was the killing agents in CO2-driven eruptions. Therefore, CH4-driven eruptions would only impact on those in the direct path of the rising column. Such eruptions would also provide a pathway for CH4 in marine sediment to rapidly enter the atmosphere as a greenhouse gas.
 Methane-driven oceanic eruptions requires major slumps or other major disturbances. Such conditions are more easily met when there was wholesale warming of ocean bottom water. Hence, the thermal maximum at the Paleocene-Eocene boundary would be an optimum time for such eruptions, leading to rapid CH4 transfer to the atmosphere as a climate driver. Furthermore, the magnitude of δ13C excursion (about −3‰ [Dickens et al., 1995; Kennett et al., 2000]) is quantitatively consistent with CH4 reaction with dissolved oxygen in deep seawater, as shown below. The concentration of dissolved O2 is about 0.00022 M and that of HCO3− is 0.0023 M, with a [O2]/[HCO3−] concentration ratio of about 0.1. One mole of CH4 reduces two moles of O2. Hence complete depletion of deep water O2 by oxidation of CH4 would contribute 5% of total dissolved HCO3− in seawater. Assuming an average of δ13C of −60‰ for CH4, and 0‰ for HCO3−, the resulting extent of δ13C is 0.05 × (−60‰), about −3‰ . Locally the excursion can be greater or smaller because both dissolved O2 content and the δ13C value of CH4 may vary. The depletion of dissolved O2 in deep seawater would have major environmental consequences.