One component of the simulation is a wedge of frozen sediment (permafrost) submerged beneath the ocean on the continental shelf of Siberia, left behind from the Last Glacial Maximum, when the shelves were exposed to the frigid atmosphere by lowered sea level (Romanovskii and Hubberten, 2001). The ice is thought to provide a seal to the upward migration of methane gas (Shakhova et al., 2009), especially where ancient fresh groundwater flow produced a layer of very high saturation ice infill, a formation called the Ice Complex in Siberia (Romanovskii et al., 2000), although there are high ice saturations found in the Alaskan Arctic as well (Zimov et al., 2006).

With inundation by the natural sea level rise over more than the last 10 000 years, the permafrost is transiently melting, although the time constant for this is generally long enough for significant frozen volume to remain, especially in shallower waters which were flooded more recently (Khvorostyanov et al., 2008a; Nicolsky and Shakhova, 2010; Romanovskii and Hubberten, 2001; Romanovskii et al., 2004; Shakhova et al., 2009; Taylor et al., 1996). Even overlying water at the freezing temperature can provoke subsurface melting by providing a warmer boundary condition against which geothermal heat establishes the subsurface temperature profile, but with climate warming, the waters could surpass the freezing temperature, allowing heat to flow from above as well as below (Khvorostyanov et al., 2008b).

Elevated methane concentrations have been measured in the water column over the Siberian shelf, even in areas of shallow water where the permafrost should still be strongly intact (Shakhova, 2010; Shakhova et al., 2005). Chemical and isotopic signatures of hydrocarbons adsorbed onto surface sediments indicate a thermal origin (Cramer and Franke, 2005), suggesting that the methane is produced many kilometers deep in the sediment column. The apparent ability of this methane to transverse the barrier of the Ice Complex has been attributed to hypothesized openings in the ice (called “taliks”), resulting from lakes or rivers on the exposed shelf, or to geologic faults (Nicolsky and Shakhova, 2010; Romanovskii et al., 2004; Shakhova et al., 2009).

Dissolved salt in the pore waters can impact the timing of thawing permafrost (Nicolsky and Shakhova, 2010; Shakhova et al., 2009). When sea level drops and exposes the top of the sediment column to the atmosphere and freshwater, the salinity of the subsurface pore waters can be flushed out by hydrological groundwater flow, driven by the pressure head from the elevated terrestrial water table above sea level. The boundary between fresh and salty pore water tends to intersect the sediment surface at the water's edge (Moore et al., 2011). From there, the boundary tends to dip landward, to a depth of approximately 40 m below sea level for every 1 m of elevation of the table water. The ratio of water table elevation to freshwater lens depth is driven by the relative densities of fresh- and salt water, as the fluid seeks an isostatic balance in which the freshwater displaces an equal mass of salt water (Verrjuit, 1968).

The SpongeBOB model has been modified to simulate the processes responsible for these observations. We do not attempt to simulate a detailed outcropping history over a 62-million-year spinup time of the sediment column but rather demonstrate the general process by subjecting the nearly complete sediment column to a one-time sea level lowering, exposing the continental shelf to groundwater forcing (see Sect. A4). After a few million years, the sediment column subsides, due to compaction and the absence of sediment deposition, resulting in a sediment column that has been considerably freshened by the atmospheric exposure. This freshening persists in the model for millions of years because there is no corresponding “salt-water pump” during high sea level stands. This behavior is consistent with the discovery of vast nearly fresh aquifers in currently submerged continental shelf regions around the world (Post et al., 2013), left over from groundwater forcing during the Last Glacial Maximum.

Another component of the simulation is the Yedoma, deposits of wind-blown dust and organic carbon that accumulated on the coastal plains of exposed continental shelves during glacial times (Zimov et al., 2006). The deposits contain a substantial fraction of organic carbon, consisting of grass roots and remains preserved by the freezing conditions. When they thaw, they begin to release CO2 and methane to the atmosphere (Dutta et al., 2006; Schuur et al., 2008; Zimov et al., 2006). Oxidation of the carbon can give off enough heat to accelerate the melting driven by primary climate forcing (Khvorostyanov et al., 2008b).

When the sediment column is exposed to the atmosphere, the pressure field from the variable elevation of the water table (the pressure head) begins to affect the fluid flow. The pressure head for a fluid particle at the depth of the water table varies in the following way: Phead(z)=g∫zzwtρseawaterdz, where zwt is the elevation of the water table, which affects the pressure throughout the fluid column, and the integral of the fluid density allows the pressure at depth to be affected by the salinity and temperature of the water above. The depth of the water table is a prognostic variable in the model. In these simulations, however, the water table remains very close to the sediment surface, as unsaturated soil produced by subsurface flow is quickly replenished by hydrological recharge.

The pressure head acts in concert with the excess pressure Pexcess, as defined by Archer et al. (2012), to drive horizontal Darcy flow through the sediment. The value of Pexcess is determined from the porosity and sediment load of the sediment in each grid box. An assumed sediment rheology is used to calculate the load-bearing capacity of the solid matrix within a given grid cell. Pexcess is calculated by assuming that the load of the solid phase overlying the grid cell that is not carried by the solid matrix must be carried by the Pexcess in the fluid phase.

The horizontal flow is uDarcy,i→i+1=kh,i+kh,i+12μ(Pexcess,i-Pexcess,i+1)+(Phead,i-Phead,i+1)(Δxi+Δxi+1)/2, while the vertical flow in the model is driven only by compaction pressure wDarcy,j→j+1=kv,jμPexcess,j-Pexcess,j+1(Δzj+Δzj+1)/2, where kh,i is the horizontal permeability at horizontal cell index j, kv,j is vertical permeability at vertical index j, μ is the viscosity, and Δx and Δz are cell dimensions. Notes on numerical issues are given in the Supplement.

The model as described so far represents a laterally homogeneous slab, a poor approximation of hydrology above sea level because of the formation of canyons and river networks in a real drained plateau. The depth of the water table in a river canyon is depressed, relative to the surroundings, to the depth of the canyon. The water table is higher in between the canyons because of recharge, and the difference in head drives lateral flow, the canyons acting to drain the sediment column.

The model formulation has been altered to represent these mechanics in a simplified way. Rather than expand the model into the full third dimension, the two-dimensional field of the model is held to represent the sediment column at a hypothetical ridge crest, as altered by an adjacent canyon. The canyon elevation is represented by zcanyon and its width by a scale Δycanyon. A cross-column flow velocity vDarcy,  j is calculated as vDarcy,j=kh,jμ(Phead, canyon-Phead)Δycanyon, where Phead, canyon is the pressure head as a function of depth in the hypothetical canyon, calculated assuming that the water table outcrops at zcanyon and that the temperatures in the sediment column have adjusted to the formation of the canyon such that the near-surface geothermal gradient is the same between the hypothetical canyon and the bulk sediment column. The lateral “drainage” flow (vDarcy,j) drives vertical velocities by continuity.

The horizontal distance scale Δycanyon is somewhat arbitrary and difficult to constrain, given that, in the reality of river networks, the distance to the nearest canyon from any point in the domain is likely to be a function of altitude, distance from the coast, and time. Another poorly resolved factor is the depth of the canyon. In reality, canyons cut into a plateau following a dynamic that erosion is proportional to slope, stopping at sea level. As a simplification the model is set to hold the canyon depth at current sea level throughout the simulation.

In the real fractal geometry of canyons, the spacing between canyons across a plain is similar to the width of the plain (length of the canyons), so the base simulation assumes a canyon width of 100 km, based on the width scale of the continental shelf of more than 100 km.

The ice model is based on an assumption of thermodynamic equilibrium, in which the heat content of the cell is distributed between the pure ice, hydrate, and brine phases, while the salt content is restricted to the brine. Notes on numerical implementation are given in the Supplement Text S2.

In the permafrost zone where ice is present, the salinity of the brine creates an ice freezing point depression that matches the local temperature. This equilibrium salinity is higher than methane hydrate can tolerate, excluding hydrate from thermodynamic stability. For a more detailed examination of the role of the brine salinity in determining the relative stabilities of ice and hydrate, see the Supplement Text S3.

Permafrost formation has several impacts on the methane cycle in the model. Biogenic methanogenesis is assumed to be stopped in the ice fraction of a grid cell (which approaches unity but never reaches it in the model, due to the exclusion of salt into brine). Bubble transport in the model balances bubble production, driven by a small and not very well constrained standing bubble concentration within the pore space. It is generally assumed (Shakhova et al., 2010b) that permafrost inhibits gas transport through the sediment column, both based on sediment column carbon and hydrogen budgets (Hunt, 1995) and on the tight seal provided by the Ice Complex. The seal provided to Arctic lakes, which can drain overnight if the seal is breached, also lends credence to this idea. In the model, this effect was simulated by stopping gas transport completely when a grid cell exceeds 50 % ice fraction (with sensitivity runs assuming 10, 30, 70, and 90 %).

The point of the spinup phase is to generate an initial condition for the glacial cycle simulations. The more usual approach in modeling hydrates is to start with an ad hoc initial condition (Reagan, 2008; Reagan and Moridis, 2009; Reagan et al., 2011). For SpongeBOB the model state at any time is the result of the time history of sedimentation, which is driven by the time-evolving depth of the sea floor and interacts with isostatic adjustment of the crust. The simplest way to generate an initial condition in the model without a startup transient is to spin the model up from bedrock. The duration of the spinup phase is 62 million years, roughly consistent with the timescale since the opening of the Laptev Rift. The first 60 Myr used a relatively coarse resolution, as shown in Fig. 1a. For the glacial–interglacial experiments, the initial condition was interpolated to a higher-resolution grid in the vertical, as shown in Fig. 1b.

When sea level drops such that the surface of the sediment column outcrops to the atmosphere, the pore fluid becomes subject to the pressure head driving it seaward and to freshwater recharge from precipitation. The pressure head forcing and the buoyancy of the sediment fluid column combine to create a mechanism to excavate salinity from the upper sediment column to depths well below sea level. The salinity of the sediment column tends to be ratcheted down by exposure to the atmosphere because there is no comparable advective pump for the reinvasion of seawater when sea level rises.

A “prefreshened” sediment column was constructed by dropping sea level by 120 m and holding it there for millions of years. The sediment column subsides back into the ocean over a few million years, but the fresh imprint of the hydrological flow persists for millions of years (Fig. 2a and Text S4). If the sediment surface never outcrops, the pore salinities remain nearly uniform and marine (Fig. 2b). Particulate organic carbon (POC) concentrations are highest just off the shelf break (Fig. 3) because this is where most of the sediment is deposited, and because the sedimentary material is richest in POC in shallow ocean water depths (Archer et al., 2012). Methane concentration (Fig. 4a) closely mirrors the solubility of dissolved methane, resulting in near-saturation concentrations through most of the model domain (Fig. 4b). The prefreshened (Fr) versus marine (Mr) initial conditions are taken as end member salinity sensitivity runs (see Table 1).

The simplest scenario (SL) varies the sea level while keeping the air and water temperatures time-invariant. The sea level air temperature is maintained at 0 ∘C. This simulation is nearly permafrost-free, with a small exception where the altitude of the sediment surface is much higher than sea level (due to the lapse rate in the atmosphere). There is no deposition of sediment above sea level in this simulation.

Permafrost formation is added in simulation GL, in which the air temperature is reduced to -16 ∘C at sea level, linearly with the glacial sea level fall (Fig. 5b). In the ocean, shelf waters are always -1.8 ∘C, but an interglacial subsurface temperature maximum of 1 ∘C at 200 m decreases to -1.8 ∘C during glacial times.

The deposition of organic-rich sediments when the surface is exposed to the atmosphere (Yedoma: represented as accumulation of 10 m in 100 kyr, with 30 % POC) is added in scenarios SL + LD and GL + LD (LD for land deposition).

The global warming (GW) scenario begins from a high sea level interglacial state and raises the temperature following the climate impact of the “spike and long-tail” time distribution of a slug of new CO2 added to the atmosphere (Archer et al., 2009; Fig. 8). There is a stage of fast atmospheric drawdown as CO2 invades the ocean, but once the ocean, atmosphere, and land surface reach equilibrium (after a few hundred years), the CO2 content of the entire biosphere begins to relax toward an initial “natural” value, on timescales of hundreds of thousands of years, by weathering reactions with carbonate and siliceous solid rocks. The net result is a CO2 drawdown that can be expressed as the sum of several exponential functions in time, with timescales ranging from 102 to 106 years.

Changes in water column temperature are assumed to be equal to those of the atmosphere, following paleoceanographic reconstructions (Martin et al., 2002) and long-term coupled ocean–atmosphere circulation model experiments (Stouffer and Manabe, 2003). The GW scenario imposes this temperature change on the water column, relaxing toward equilibrium with the atmospheric CO2 trajectory with a time constant of 100 years.

The effect of sea level rise is added to create a second global warming scenario, GW + SL. On timescales of thousands of years, the sea level response to changing global temperature is much stronger than the sea level response over the coming century, as prominently forecast by the IPCC. The reconstruction of sea level and global temperature covariation in the geologic past (glacial times to Eocene hothouse) reveals a covariation of 10–20 m ∘C-1 (Archer and Brovkin, 2008). The global warming with sea level scenario assumes an equilibrium sea level response of 15 m ∘C-1 which it relaxes toward with a time constant of 1000 years.

In the “prefreshened” initial condition (Fr), millions of years have elapsed since the previous exposure of the sediment to hydrological forcing, but a core of freshwater remains. Salinities near the sediment surface have grown saltier due to diffusive contact with seawater (Fig. 6, left). A fully marine initial condition (Mr; Fig. 6, right) was initialized from the unfreshened case, in which sea level was held at a fixed value throughout the 65 Myr spinup of the sediment column. The salinities are nearly uniform in this case.

When the sediment surface is re-exposed to the atmosphere during an interval of low sea level, in the absence of ice formation (simulation SL), the surface layer tends to freshen relatively quickly due to the hydrological forcing, although a subsurface salinity maximum persists (Fig. 6c and d). If the air temperatures are cold enough to form ice (simulation GL), surface salinities in the model increase to up to nearly 190 psu in both the prefreshened and pure marine cases (Fig. 6e and f). By the next interglacial period (Fig. 6g and h), ice near the sediment surface has melted enough for near-surface pore waters to reach relatively low salinities.

The effect of the glacial–interglacial sea level and climate forcing on the pressures and flow velocities is shown in Fig. 7. On a spatial scale of the entire model domain (Fig. 7, left), the highest driving pressures are found at the base of the sediment column, underneath the region of maximum sediment accumulation (the depocenter just off the shelf break). Changes in sea level drive large fluctuations in the pressure head (contours) extending to bedrock. In the near-surface continental shelf (Fig. 7, right), the driving pressure variations are dominated by the pressure head, driven by sea level changes. The formation of permafrost (GL, Fig. 7e and f) seals the upper sediment column to fluid flow.

When sea level rises again, in the model configuration including permafrost, there is a strong pulse of downward flow following partial melting of the permafrost (Fig. 7h). It is possible that this flow, which lasts a few thousand years, is an artifact of the elastic model configuration, in which the release of a load (by submergence of the upper sediment column into the ocean) provokes the expansion of pore spaces in the sediment. The anomalous flow, integrated over its duration, could displace the pore fluid by about 40 m, which is less than one grid cell. The model configuration without the sealing effect of permafrost (SL) does not show this pulse of invasive flow on sea level rise.

There are multiple ways in which the glacial cycles of sea level and air and water temperature might impact the flux of methane to the atmosphere. Submergence in the ocean is one modulating factor because the emerging bubbles dissolve in the ocean rather than reaching the atmosphere. Another factor is the deposition of high-POC surface soils during low sea level stands and its exposure to degradation later when the permafrost soils melt. A third factor is permafrost, impeding gas and fluid flow and excluding dissolved methane and salt from ice formation. The impacts of these processes are assessed by comparing the results from model configurations with and without each process in question.

Ice freezes until the salinity of the residual brine brings about a freezing point depression equal to the in situ temperature. A saltier initial sediment column will reach this condition with a lower ice fraction, its melting is accelerated, and its hydrate inventory is lower (Fig. 14). The equilibrium salinity in the permafrost zone is not affected by the salt inventory of the column, only the relative volumes of the solid and fluid phases.

The atmospheric flux increases along with either shallow, biological methane production, driven by deposition of Yedoma, or thermal methane production in the deep sediment column (Fig. 15). Biogenic methane production is too shallow in the sediment column to impact the inventory of methane hydrate (Fig. 11). The timings through the glacial cycles of atmospheric methane emissions from these scenarios parallel each other because they are controlled in common by the transport-blocking effects of permafrost and sediment submergence in the ocean.

When the heat flux is higher, the temperature gradient is steeper, pivoting about the sediment surface temperature, which is set by the ocean. The base of the methane hydrate stability boundary becomes shallower, while the top remains at about the same depth, resulting in a thinning of the stability zone (Fig. 8). The hydrate inventory through the glacial cycles, however, is not much affected, unless the heat flux becomes small enough for hydrate to persist through the glaciations (Fig. 11).

When ice is included as a competing phase, it excludes methane hydrate from the low-pressure, very cold permafrost zone. The hydrate stability zone thins (from above and below in the model: Fig. 8), and the hydrate inventory decreases (Fig. 11). When ice formation is disallowed, the hydrate stability zone approaches the sediment surface during coldest glacial times, but by the time of an interglacially based global warming climate perturbation, the stability zone boundary has retreated to several hundred meters below the sea floor, precluding a sudden hydrate dissolution response to a suddenly warming ocean.

When the ice fraction of the model exceeds a critical threshold, gas migration is blocked. Changing the value of this threshold has a strong impact on the rates of methane emission during glacial versus interglacial times. This process is therefore a high priority for future model refinement.

The chemistry of continental-margin sediments in this model (Archer et al., 2012) showed a strong sensitivity to flow heterogeneity, achieved by increasing the vertical permeability of every fifth grid cell. In the configuration presented here, the impact of the channels is much smaller. The dynamics of this simulation are thermally driven, rather than by sediment deposition driving fluid flow in the continental-margin case. Atmospheric methane fluxes are spikier when the channels are included, but the mean rate is not much changed.

Groundwater flow carries enough methane to be a significant sink during times of low sea level. However, disabling that flow has only subtle impacts on the other aspects of the methane cycle on the shelf. Spikes of methane emission during late deglaciation become somewhat more intense.

The rates of biological and thermal methane production on the Siberian continental shelf are not well constrained by laboratory measurements or field inferences. These rates are treated as tunable model parameters, and the sensitivity studies show that they are important ones to ultimately get right.

Simulating the hot-spot behavior of bubble emission from the sea floor will also require more detailed treatment of the mechanisms by which gas moves around in the sediment column. The model lacks faults and permeable layers that act as transport highways and hydrate depocenters and may concentrate the flow into a hot-spot ebullition region. The model also lacks the ability to episodically “blow out”, producing the sedimentary wipe-out zones observed seismically in the subsurface (Riedel et al., 2002) and the pockmarks at the sediment surface (Hill et al., 2004). The steady-state hydrate inventory in the model is extremely sensitive to the bubble vertical transport spatial scale (Archer et al., 2012), which determines how far a bubble can get through unsaturated conditions before it redissolves. This result demonstrates the importance of gas transport to predicting the methane hydrate or bubble inventories.

On land, the model lacks seasonal melting of surface permafrost and the thaw bulbs underneath lakes and rivers. In the ocean, the fraction of the bubbles which dissolves in the water column depends on the bubble sizes, which depend on the gas emission rate, ultimately driven by details of gas transport in the sediment.

These uncertainties affect the flux of methane to the atmosphere and model predictions of the standing stocks of methane as gas and hydrate in the sediment column.

The model is consistent with observations (Kort et al., 2012), that the total atmospheric methane flux from the Siberian margin is a small fraction of the global flux of methane to the atmosphere and thus represents only a minor climate forcing. The model would have to be pushed very hard (as would the measurements) to fundamentally change this conclusion.

Groundwater flow, driven by the pressure head, provides an advective means of pumping freshwater into the subsurface sediment column that has no counterpart for salty ocean water. The model lacks the mechanism of salt fingering, which can enhance the diffusion of salt from above into a freshwater aquifer (Kooi et al., 2000). However, higher-resolution models of smaller domains that accounted for salt fingering also show a time asymmetry, with faster freshwater invasion on sea level drop than salt invasion on sea level rise (Lu and Werner, 2013; Watson et al., 2010). As the size of the domain increases with increasing sea level change, advective processes such as hydrological flow should become even more dominant over diffusive processes such as salt fingering. The recent discovery of vast freshwater aquifers on global continental shelves (Post et al., 2013), persisting since the time of lowered sea level 20 000 years ago, and the lower-than-marine salinities of the pore waters measured in submerged surface Arctic sediments (summarized by Nicolsky et al., 2012) are also consistent with the existence of a fresh-water hydrological pump which has a significant impact on sediment column salinities.

In the simulations the pore water salinities in the permafrost zone did not depend on the total salt content of the sediment column but only on the temperature (and secondarily on the pressure). A saltier sediment column will end up with a larger volume of brine in equilibrium than a fresher sediment column would have, but the salinities of the brines would be the same.

In the permafrost zone (low temperature and pressure), ice can tolerate higher salinity (lower water activity) than methane hydrate can. As long as there is no kinetic impediment to ice formation, bubbles of methane rising into this zone should encounter brine salinities too high to permit the formation of methane hydrate (Sect. A3).

The methane flux to the atmosphere through the glacial–interglacial cycles is highest during cold times because sea level is low, and it contrasts with the positive feedback response of releasing methane during warm (high sea level) intervals. Atmospheric methane concentrations were lower during glacial times than interglacials, but since the Arctic Ocean is a small fraction of the total methane budget (Sect. 4.2.1), the atmospheric concentration does not necessarily reflect Arctic fluxes.

There is a warming positive feedback in the simulated future from climate warming, with fluxes rising gradually on a timescale of thousands of years. Shakhova et al. (2010b) proposed that 50 Gton C as methane could erupt from the Arctic on a timescale of a few years. However, the thermodynamic exclusion of methane hydrate from the permafrost zone (Sect. 4.2.2) ensures that methane hydrate will be isolated from changes in ocean temperature by ∼ 400 m of mud and ice. A warming perturbation at the sea floor today will not reach this depth for hundreds or thousands of years. A complex model is not really required to conclude that methane hydrate will probably not produce a methane eruption of this scale so quickly.

Could an abrupt methane release arise from the release of trapped bubbles from melting ice? The model actually does produce a glacial cycle in bubble inventory, with changes exceeding 50 Gton over a cycle, apparently driven by methane exclusion from ice formation (Fig. 11). However, the model does not deliver an abrupt release in response to anthropogenic warming for any of its sensitivity studies (Fig. 14). We would get a faster initial response to global warming if the transition from glacial to global warming sediment surface temperatures had not mostly happened thousands of years ago.

The model provides a poor constraint on the standing stock of bubbles or methane hydrate in the sediment column and neglects many of the mechanisms that could come into play in transporting methane quickly to the atmosphere, such as faults, channels, and blowouts of the sediment column. A continuum model such as this one predicts a smooth methane release response to a warming, growing on some e-folding timescale. A world dominated by features that each represent a small fraction of the total methane reservoir will release methane more episodically, but the statistical distribution of the response in time should still show the e-folding timescale of the underlying driving mechanism, the diffusion of heat into the sediment column.

The way to deliver 50 Gton of methane to the atmosphere on a short timescale is for it all to be released from a single geologic feature pent-up by ice. However, 50 Gton of C represents a large fraction of all the traditional natural gas deposits on Earth (about 100 Gton C). The place to look for such a large unstable gas reservoir is in the field, not in this model, but until such a thing is found it remains conjecture.

On timescales of thousands of years and longer, carbon from deep methane hydrates and frozen organics on the Siberian continental shelf could reach the atmosphere–ocean carbon cycle, potentially significantly amplifying the “long-tail” climate impact of anthropogenic carbon release. Methane that is oxidized in the ocean would eventually equilibrate with the atmosphere, so it is much easier for escaping methane to impact the long tail as CO2 than it is to affect the near future as methane.

The potential for future sea level change is much higher on millennial timescales than the forecast for the year 2100 because it takes longer than a century for ice sheets to respond to changes in climate. The model finds that for the future, if sea level changes by tens of meters, as indicated by paleoclimate reconstructions (Archer and Brovkin, 2008), the impact of sea level rise could overwhelm the impact of warming. The dominance of sea level over temperature in the model of this area is due to the dissolution of methane in the water column rather than a pressure effect on hydrate stability, which is generally a weaker driver than ocean temperature in deeper-water settings (Mienert et al., 2005).