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Appendix F — Ocean Heat Budget

This appendix quantifies the thermal energy delivered to the ocean at the newly opened basins, during boiling-mode delivery and the conductive tail that follows. The budget is fixed by observation: the measured heat flux through the new basins, the solidus that marks the base of the solidified column, and the new-basin area together determine how much heat was removed. The delivery mechanism and its three-phase structure follow from the competition between the rate at which heat arrives and the rate at which the water can carry it away. The closing sections take up what becomes of that heat once it leaves the ocean, and what the geometry of where it entered implies about conditions elsewhere.

F.1 Physical Mechanism

Three interfaces matter in what follows and are named separately throughout: the asthenosphere top, where new material arrives from below; the water–rock interface, the top of the ponded magma body where quenching occurs; and the sea surface, where steam leaves to atmosphere.

The continents are the cork. When Pangea splits, the continental plates slide off the underlying mantle, exposing it progressively at each trailing edge. The mantle beneath is under pressure from the thermal dome (Appendix A) and from the subduction pump — plates descending at the consumption fronts displace mantle material upward at the rift. Magma rises, contacts seawater at the surface, quenches, and crystallizes temporarily.

Two density comparisons matter, and they point in opposite directions.

Against the asthenosphere, everything in the pond is buoyant. Liquid basaltic magma (~2,700 kg/m³), crystal mush (~2,850 kg/m³), and fully solid basalt (~3,000 kg/m³) are all lighter than the asthenosphere beneath, which Appendix A puts at 3,171 kg/m³ on its own thermal reference, against a literature range of 3,200–3,250. The ponded body stays at the surface. It does not founder back into the mantle it rose from.

Against the melt it is mixing with, the quench products are denser. Crystallized material at 2,850–3,000 kg/m³ sits on liquid at ~2,700. Quenched crust is unstable where it forms: it sinks back into the pond and is remixed rather than accumulating as a lid. This is why no coherent plate forms while feeding continues.

The mixture ponds as a growing, buoyant, partially molten layer at the surface of the basin, in continuous contact with the ocean. Fresh mantle material continues to arrive from below, pushed by the subduction pump, rising through and over the ponded body. The surface is a turbulent mixer — hot material arriving, quenching against seawater, crystallizing, foundering, remixing — until enough heat is removed for a coherent lithospheric plate to form.

The analogy is Kilauea lava pouring into the ocean, scaled to basin dimensions. Each batch of magma passes through the water–rock interface on its way to becoming solid rock. The heat transfer is not governed by conduction through a static slab. It is governed by the rate at which fresh hot material is processed through the ocean contact surface.

The order of that sequence matters for everything downstream. While the body is being fed, the water carries off sensible heat as fast as it can and does not keep up; the body grows and stays hot. As feeding slows, the sensible heat in the mixing zone runs down and the latent heat takes its turn. When enough of that has gone, the material reaches its solidus and solidifies — from the outside in and from the top down.

What stands there now is a cooled column resting on material that never gave up its heat. How thick the solidified part is, and what temperature profile sits inside it, are not observed. What is observed is the heat flux through it, and that is the quantity F.2 works from. The material below the solidification front does not enter the budget: it is still there, it is still warm, and none of its heat reached the ocean.

F.2 Energy Budget

The total thermal energy in the emplaced material has two components: sensible heat (the temperature of the rock) and latent heat (the energy released during crystallization from liquid to solid).

Per cubic meter of emplaced material:

Sensible heat (T_mantle to T_ocean): ρ × c_p × (T_mantle − T_ocean) = 2,900 × 1,000 × 1,198 = 3.474 × 10⁹ J/m³

Latent heat of crystallization: ρ × L = 2,900 × 400,000 = 1.160 × 10⁹ J/m³

Total: E_per_m³ = 4.634 × 10⁹ J/m³

Parameters:

The depth of the solidified column follows from the measured flux, as an equivalence rather than a measurement. Take a column carrying a linear gradient from T_ocean at the water–rock interface to T_solidus at its base. Its thickness is

H = (T_solidus − T_ocean) · k / q = 55 m per °C

giving H = 57.6 km at the working flux. This is the thickness of the equivalent linear column, not an observed lid. The real body's internal temperature profile is not known, and neither is the depth at which solid gives way to melt within it. What the equivalence buys is a heat content consistent with the observed surface flux, and the profile term in the sensitivity table below is the cost of not knowing the true shape.

No volume calculation and no mass balance enters this. Material below the solidification front is still there and still warm, and none of its heat reached the ocean, so it does not appear in the budget.

Energy budget of the solidified column:

Sensitivity, each term moved alone from the working case:

The solidus barely matters, which is what makes it safe as the terminus. The flux and the interior profile are the two that count. The profile is not observed here and is left to a three-dimensional thermal treatment.

The budget is the whole of this section's output. How long the delivery takes depends on the interface transport history, not on the total alone, and that is the subject of F.4. This partition does not depend on the delivery mechanism, the boiling flux, or the phase structure. Any model of the delivery mechanism must integrate to this total.

F.3 Inputs and Their Sources

The energy budget above rests on eleven inputs. This section states where each comes from and how far it has been checked, because two carry approximations that matter and several are working values with no citation behind them.

Verified against the source:

Named source, value not confirmed against it:

Working values from standard rock-property ranges, no individual citation:

Model quantities, not measurements. The rift length of 20,000 km and the separation of 5,000 km are the model's own geometry, with the new-basin area of 10⁸ km² computed from them. The transport ceiling q_boil is specified in F.4, not derived and not observed.

Two approximations worth naming

Thermal conductivity is held constant across a thousand-degree column. The column runs from 2 °C at the water–rock interface to about 1,050 °C at its base, and basalt conductivity varies with temperature over that span. A single k is a convenience. Since the column thickness goes as ΔT·k/q, any error in the effective value passes proportionally into the thickness and so into the delivered energy.

The delivered energy is a difference between two states and only one of them is observed. The present state is fixed by the measured flux and the solidus. The emplacement state is taken as magma at mantle temperature. The interior temperature profile between them is not observed, and its shape carries ±16% — the largest single term in the budget. It is left to a three-dimensional thermal treatment.

From the budget to the delivery

The budget fixes how much heat left. It says nothing about when, or at what rate. Those are set by a chain of three interfaces, and the limiting step is not where it might first appear.

At the water–rock interface the ponded body heats the water it contacts. Under kilometers of hydrostatic head there is no vapor of consequence there; the heat enters the water column and convects away. This interface moves heat into the ocean, not out of it.

The ocean carries that heat and distributes it. Removal from the system happens at the sea surface, where the pressure is one atmosphere and water leaves as steam. That is the interface with a ceiling on it, and the quantity that matters there is an evaporation rate rather than a boiling flux.

The delivery rate is bounded three ways — by what the contact can pass, by what the water can carry, and by what can evaporate at the surface — and whichever bound is lowest governs at any moment. Working out that competition, and the phase structure it produces, is the subject of the next section.

F.4 Three-Phase Delivery

Heat delivery from the new ocean basins to the ocean proceeds in three phases, governed by the competition between two rates: the advective input, at which thermal energy arrives with new material, and the removal capacity, at which the ocean can carry it away.

Advective input at each time t:

P_adv(t) = v(t) × L_rift × H × E_per_m³ / sec_per_yr

where H is the thickness of the column that solidified, taken from the measured flux as in F.2 rather than from a mass balance. At peak velocity:

12,000 m/yr × 2.0 × 10⁷ m × 57,600 m × 4.634 × 10⁹ J/m³ ÷ 3.156 × 10⁷ s/yr = 2.03 × 10¹⁸ W

This is the rate at which thermal energy enters the participating body as new mantle material is exposed at the cork's trailing edge. It tracks the velocity curve directly and declines exponentially. Material emplaced below the solidification front never gave up its heat and does not enter this term.

Removal capacity at each time t:

P_rem(t) = q_boil × A(t)

where A(t) = L_rift × W(t) is the new-basin surface area, growing as the basin widens.

q_boil is a water-side transport ceiling, not a measured contact flux. The interface itself could deliver far more. Published contact measurements span several distinct regimes:

Two of the four entries — Oddsson and Moitra — are reported from secondary retrieval and have not been checked against the source publications. They are marked rather than dropped, because they bound the violent-transient end and the selection does not rest on them.

Hardee & Dunn is the entry the 10–20 kW/m² selection does rest on, and it is confirmed against the published abstract: heat flux measurements from 2 to 40 kW/m² on degassed basaltic lava at atmospheric pressure, taken with two independent convective heat-flux probes at liquidus and subliquidus temperatures, with the paper singling out 6–15 kW/m² as the sustained subliquidus rate. The band used here straddles the top of that subliquidus range and sits well inside the measured span.

What the paper measures is local convective extraction from melt into a probe at one atmosphere. That is a defensible analogue for sustained magma-side convection at order of magnitude, and not a calibrated basin-scale delivery rate under hydrostatic head.

What limits the basin is the water side rather than the contact. Heat cannot leave faster than the water can carry it, with the sink temperature pinned near saturation under an open top. Three bounds apply — the contact physics, the water's carrying capacity, and the temperature difference at the sink — and whichever is lowest governs. The model sits on the water-side bound: through Phases 1 and 2 the released power equals the removal capacity at every timestep, never the interface capability. The band is applied as a constant, and its time dependence is not resolved here.

Two processes run concurrently and should not be conflated. Solidification is the rock's accounting: the body reaches its solidus at 1,050 °C, releases its latent heat at that plateau, and the solidification front advances as heat continues to leave. Boiling is the water's accounting: the sea surface stays at boiling while enough heat arrives through the water column, and stops when the deliverable heat is spent. Neither event triggers the other.

Phase 1 — Charging (P_adv > P_rem). The basin is narrow and removal capacity small. Energy arrives faster than the ocean takes it away and the body accumulates stored heat. The ocean removes at the maximum the water allows, a small fraction of the input. The body arrives above its solidus and cools toward it; once it reaches the solidus, temperature holds there while the latent heat is extracted.

Phase 2 — Discharge (P_adv < P_rem). As the basin widens and velocity decays, removal capacity overtakes advective input and the ocean draws down stored heat faster than new material supplies it. This is the ice age engine at full power.

Phase 3 — Conductive tail. When the deliverable heat is spent, boiling ends. The system transitions to a solidified plate cooling by conduction. This is not the end of delivery, only of boiling-mode delivery: conduction continues at the residual flux to the present, contributing under a hundredth of a percent of the total. The transition is gradual — the boiling front retreats across the basin over centuries rather than switching off at once.

Phase timing. The release is capped at q_boil × A(t) throughout Phases 1 and 2, and the total is fixed by the budget, so boiling ends where the cumulative release reaches the delivered energy:

These are computed on the velocity profile of Appendix D unchanged, the flux-derived column thickness, and the delivered energy from F.2. The end-of-boiling times contain no thickness term: they solve ∫ q_boil·A(t) dt = E_delivered, with A(t) following from the velocity profile alone. Thickness enters only through E_delivered itself, which sets where the integral stops.

The latent heat matters most in Phase 1 and early Phase 2. At 1.16 × 10⁹ J/m³ it is 25% of the total per unit volume. Released at the solidus, it holds the body at temperature while it is extracted and extends the high-temperature phase.

F.5 Computational Results

The heat delivery was modeled as a time-stepping energy balance at 1-year resolution. Advective input follows the width integral from Appendix D. Removal is capped at the water-side ceiling. Stored energy is a running balance; bulk temperature uses a three-regime map — sensible above the solidus, latent plateau at 1,050 °C, sensible below. The body is treated as isothermal at each step, a simplification discussed in F.7.

Through Phases 1 and 2 the release equals P_rem. Boiling ends when cumulative release reaches E_delivered, not when the isothermal bulk temperature reaches 100 °C.

10 kW/m²

Phase 1→2: year 463. Boiling ends: year 941. Sustained discharge: 478 yr. Isothermal T at the cut: 402 °C.

15 kW/m²

Phase 1→2: year 358. Boiling ends: year 722. Sustained discharge: 364 yr. Isothermal T at the cut: 297 °C.

20 kW/m²

Phase 1→2: year 293. Boiling ends: year 603. Sustained discharge: 310 yr. Isothermal T at the cut: 197 °C.

Validation

Energy added and released balances at every step, and the total released is pinned to the budget of F.2. The flux changes the timing, not the integral.

What this run is not. If boiling is instead stopped when the isothermal bulk temperature first reaches 100 °C, the cut falls later — year 1,131 at 10 kW, 811 at 15, 638 at 20 — and the released energy exceeds E_delivered, reaching 2.35, 2.15 and 1.97 × 10²⁸ J. That is the isothermal over-release. The tables above use the budget pin.

The isothermal temperature at the budget cut is still hundreds of degrees. A linear residual profile stores more heat at depth than a well-mixed body can, which is the same limitation noted in F.7.

F.6 Disposal and the Geometry of Delivery

Two questions follow from the budget, and they are not the same question. The first is global: can the planet shed this much heat without a permanent change of state? The second is local: what conditions does the delivery produce, and where? The second follows from the geometry of where the heat enters, and it depends on the first — if the planet could not shed the energy at all, no amount of distance from the openings would matter.

Throughout this section, new ocean basins means the Atlantic and Indian openings created by the separation — the surfaces in contact with the emplaced body. The remnant ocean, the Pacific-scale basin that was not created by the event, is not a contact surface and does not boil. Heat reaches it only by ocean mixing and atmospheric transport. The ocean is one connected body and distributes what it receives; the boiling is not distributed with it.

This section works at the level of an energy budget and a surface geometry. It does not produce a climate trajectory, and no atmospheric model adequate to that was available for this work.

Whether the heat can leave

The surplus is large against the modern budget. It is not unbounded, and it is concentrated in centuries rather than sustained indefinitely. Four points establish that it has somewhere to go. None requires a climate model.

Emission rises steeply with temperature. Outgoing radiation scales as T⁴. A temporary surplus drives the emitting temperature up until the surplus is discharged, and the temperature then falls back. Energy conservation alone establishes that a finite input is eventually radiated. What it does not establish is at what temperature, or for how long.

The ocean is the first reservoir. Delivery is into the water column, not directly into the atmosphere, and the atmosphere sees only what the sea surface rejects. Ocean heat capacity buffers the pulse, and the water side is already the binding constraint on the delivery rate.

Steam is a working fluid, not a lid. Water that condenses and falls as rain or snow leaves the vapor inventory. A moisture-loaded atmosphere can be wet, stormy and strongly convective without becoming a permanent steam greenhouse, provided condensation and poleward export continue to operate — which is the same transport the ice-sheet mechanism requires.

A runaway greenhouse is a different condition. The classical runaway requires a planet unable to balance absorbed sunlight in steady state, with outgoing radiation capped by a saturated water-vapor atmosphere. A finite multi-century pulse that is rained out and radiated is not that state.

Where the heat enters

The heat does not enter the system uniformly. It enters at the new ocean basins, and the boiling surface is the strip they occupy, whose width follows the velocity profile of Appendix D:

Two features matter. The global load crosses modern outgoing radiation between year 26 at 20 kW/m² and year 54 at 10, solved from the continuous width integral rather than read between the rows above — the interval in which the delivery is small compared to the planet's ordinary throughput is decades, not centuries. And at its widest the boiling surface covers about one-sixth of the planet. That does not change with time; it is a fact about where the openings are.

The remnant ocean is not a contact surface. No new crust forms there, no mantle is exposed at its floor, and it receives no emplacement heat — only leakage at its existing margins. At roughly 1.65 × 10⁸ km² it is nearly twice the area of the new basins at their widest.

The two are connected, and water moves between them. The northern and southern contacts are open, and flow through them is driven from both sides at once. The remnant basin is being consumed while the new basins open: plate goes down at the consumption fronts, the remnant loses area, and the water it held moves into the gap the separation is opening. Evaporation adds a second and continuing draw, since water boiling off the new basins has to be replaced. A share of it returns as rain on the basins themselves, and that share is not determined here, so the net inflow is smaller than the gross evaporation — but it is not nothing, because moisture carried to high latitudes and locked into ice does not come back, and neither does what falls on the remnant ocean or on land. Both effects act in the same direction: net flow at both contacts runs from the remnant ocean into the new basins for as long as the basins are filling and boiling, and hot surface water does not bulk-export the other way.

The ocean is a conduit, not a reservoir. Nearly all of the delivered heat leaves the sea surface as vapor rather than warming the water. Warming the entire ocean by 20 K would store under one percent of the delivered energy, and absorbing the whole budget as sensible heat would require a rise of thousands of degrees, which the boiling cap forbids. There is no large body of hot water anywhere in the system to be mixed — not in the remnant ocean, and not in the new basins either.

Heat therefore reaches the remnant ocean by two indirect routes and not by contact: as latent heat released where moisture evaporated from the new basins condenses and precipitates over it, and by exchange at the contacts, which the inflow direction limits.

The asymmetry is intrinsic to the mechanism rather than imposed on it. The new basins are hot because they are new — their floors are fresh mantle material at emplacement temperature. The remnant basin is cool because it is old, and because nothing is being emplaced beneath it. Nothing in the model was arranged to produce that contrast; it follows from which ocean the cork-pop creates and which one it leaves alone.

What the geometry implies for survivability

The delivery geometry establishes what conditions to expect and where. Two features beyond those above complete the picture, and both follow from the mechanism rather than being imposed on it.

The boiling surface is surface the event created. The new ocean basins are the gap the separation opened; before it, that ground was continental interior. The lethal zone is therefore not pre-existing habitat that the event destroyed but new ground that the event made. Whatever the conditions there, they are conditions in a place that did not previously exist to be inhabited.

The heat leaves the surface where it enters it. Moist convection over a boiling surface is strongly buoyant. The vapor rises, releases its latent heat at the condensation level, and is exported poleward aloft. It does not travel as a surface layer of steam across continents. Ground away from the new basins is not downwind of a sauna; it lies beneath a circulation whose energy was deposited at altitude thousands of kilometers away.

Taken with the confinement and the asymmetry, these establish the expectation. Conditions are severe at and near the new ocean basins and grow milder with distance from them. The remnant ocean stays within reach of its prior state. Continental ground is off the delivery surface entirely, and exposure falls with distance from it, so on a landmass bounded by new basins on more than one side the least-exposed ground is its interior — the positions most nearly central between the openings, furthest from every boiling surface, on thick crust and elevated above them. The early years are the mildest of all, before the openings have widened.

What the physics fixes is the structure: where the heat enters, where it does not, and how it moves once it does. What it does not fix is magnitude. The consequences for the distribution and recovery of life are developed in the Diaspora series, which takes the structure established here as its starting condition.

The moisture flux

Nearly all of the heat delivered at the sea surface leaves as latent heat, so the transport rate converts almost directly to an evaporation rate over the new basins. A surface at 100 °C radiates about 1.1 kW/m² gross and roughly 0.7 kW/m² net against a 288 K sky, which is 4 to 11 percent of the transport band depending on where in it the flux sits. The figures below are therefore upper bounds by a few percent:

Over the full delivery this is 7.9 × 10²¹ kg of evaporation against an ocean of roughly 1.4 × 10²¹ kg (a working value, in the same class as the material properties in F.3) — about six ocean masses, which establishes that the water cycles rather than being consumed. The figure is a throughput, not an inventory, and it is the quantity a climate treatment would take as its input.

The boiling is the weather engine

The steam is not a waste product but the working fluid of the post-catastrophe climate. Extreme evaporation from the new ocean basins produces extreme atmospheric moisture loading, which drives extreme poleward transport, which produces extreme polar snowfall, which builds ice sheets, which lowers sea levels, which opens land bridges. The ice age is not a separate event requiring a separate explanation — it is what the heat budget does on its way out. The circulation is driven by the gradient between the boiling new basins and the cool remnant ocean, a tectonic gradient rather than a solar one. Cloud cover over and downwind of the new basins would raise planetary albedo and reduce absorbed sunlight, lowering the total load and strengthening the ice-age engine, though the magnitude of that shift is not estimated here.

F.7 What This Appendix Does Not Claim

The budget and the delivery structure rest on approximations, bounds, and working values. This section collects them, so that what is claimed can be separated from what is assumed.

The spatial model

The rock body is treated as isothermal. The computational model in F.5 holds it spatially uniform in temperature at each timestep, over a uniform 57.6 km column. The real body has two heat sinks rather than one — the ocean above it and the cold continental landmass at its margins — so solidification advances inward from the edges as well as downward from the water–rock interface, leaving a solidified column thicker at the margins and thinner toward the axis. A spatially resolved three-dimensional model would track that distribution and produce a smoother Phase 2→3 transition than the sharp threshold the isothermal treatment gives.

The cost of that approximation is visible in the results. At the budget cut the isothermal body stands at 197–402 °C depending on flux, which holds less heat than the linear residual gradient the budget assumes, because a well-mixed body cannot store heat at depth the way a column with a gradient does. The energy totals are unaffected — they are pinned by the endpoint constraints, not by the spatial model — but the temperature trajectory the model reports is not the trajectory a resolved treatment would produce.

Terms in the budget that are not measured

The interior temperature profile is not observed, and it is the largest single uncertainty in the budget. Driving it from strongly bottom-weighted to strongly top-weighted moves the delivered energy by ±16%. F.2 takes it as linear. Nothing here determines it, and a three-dimensional thermal treatment is what would.

The measured flux carries the second-largest term. The 50–65 mW/m² band moves the delivered energy by −8% to +20%, because the column thickness goes as 1/q. The working value of 60 sits inside the band but is not privileged within it.

Thermal conductivity is held constant across a thousand-degree column. Basalt conductivity varies with temperature over the range from 2 °C at the water–rock interface to 1,050 °C at the base, and a single k is a convenience. Any error in the effective value passes proportionally into the column thickness and so into the delivered energy.

The solidus, by contrast, barely matters: a 150 °C spread moves the result by ±4%, which is why it is safe as the terminus.

The delivery rate

The transport ceiling is a bound, not a measurement. q_boil stands for what the water side can carry, and the model sits on it through Phases 1 and 2. It is not a measured contact flux, and no experiment reported here measures basin-scale sustained delivery under hydrostatic head.

It is applied as a constant, and it is not one. The time dependence of the transport bound is unresolved. Recovering it requires the fluid dynamics of the buoyant ponding and the geometry of the quench front, which is beyond what is attempted here.

Three cases bracket the timing; they do not trace a curve. The 10, 15 and 20 kW/m² runs show how the phase structure moves with the flux. They do not resolve where within that band the true value sits, and no claim is made as to which case is nearest.

Sources

Hardee & Dunn (1981) is confirmed against the published abstract — the 2–40 kW/m² range, the 6–15 kW/m² subliquidus band, the two convective probes, degassed basaltic lava at atmospheric pressure. The full text, including its figures and tables, has not been opened.

Two of the four flux entries are unverified. Oddsson et al. (2016) and Moitra et al. are reported from secondary retrieval and have not been checked against the source publications. They bound the violent-transient end of the regime table and the selection does not rest on them, but they are not confirmed.

One citation may not carry the number attached to it. Lange, Cashman & Navrotsky (1994) is direct calorimetry on an olivine basalt, and it is the reference given for the latent heat of crystallization. Its abstract reports how latent heat distributes through crystallization rather than a mass-normalized total, so it does not visibly confirm 4.0 × 10⁵ J/kg. That figure is the long-standing working value for basaltic magma.

Several inputs are working values with no individual citation: the basalt density, specific heat, magma temperature, solidus, ocean-floor temperature, latent heat of vaporization, and the mass of the ocean. Each sits inside a standard published range, and none is sourced to a specific measurement. The evaporation totals in F.6 rest on the last two, and they take all of the delivered heat as leaving in latent form, which F.6 notes is an upper bound by the 4 to 11 percent that leaves radiatively. The six-ocean-mass figure is that bound rather than an estimate.

The atmosphere

No atmospheric model was used. F.6 argues at the level of an energy budget and a surface geometry, and no radiative-transfer treatment of a moisture-loaded atmosphere was available for this work.

Consequently there is no climate result here. No global mean temperature, no regional temperatures, no storm regime, no precipitation distribution, and no line drawn between habitable and uninhabitable ground. What F.6 does establish is narrower: that the global energy budget closes, that the delivery is confined to about one-sixth of the surface at its widest, and that heat entering the atmosphere over the new basins is exported aloft rather than spread across the surface. The distribution of conditions those produce is not computed.

The ocean

Ocean temperature evolution is not modeled in detail. The delivered energy constrains the globally averaged warming. Exchange between the remnant ocean and the new basins is argued by direction rather than by volume: the flows are not quantified here, and no rate is claimed for either the filling or the evaporative replacement.

The spatial distribution of warming — concentrated at the new ocean basins, with the remnant ocean receiving heat only by transport rather than contact — is developed in Papers 4 and 5 of the Diaspora Series, which take the thermal structure established here as their starting condition.

F.8 Summary

The ocean heat budget is fixed by observation. Three quantities determine it, and none is adjustable:

Those give an equivalent solidified column of 57.6 km — the thickness of a linear column that would carry the observed flux down to a solidus base, and a bookkeeping construct rather than a measured lid. On that equivalence the heat content at emplacement is 2.671 × 10²⁸ J and the residual still held in the present gradient is 8.76 × 10²⁷ J. The difference, 1.795 × 10²⁸ J, is what the ocean received. No mass balance and no subducted volume enters. Material below the solidification front is still there and still warm, and none of its heat reached the ocean.

The delivery proceeds in three phases, set by the competition between the rate heat arrives and the rate the water can carry it away. Phase 1 accumulates stored heat while the basins are narrow and the removal capacity small, with the body holding at the solidus as its latent heat is extracted. Phase 2 begins when the widening basins overtake the declining advective input and the ocean draws stored heat down faster than new material supplies it — the ice age engine, at a released power of 0.86 to 1.53 × 10¹⁸ W. Phase 3 begins when the deliverable heat is spent and boiling ends; conduction then carries the residual to the present at under a hundredth of a percent of the total.

The heat has somewhere to go, and it does not go everywhere. Averaged over the boiling window the load is five to eight times modern outgoing longwave radiation — large, bounded, and transient rather than a steady state. Emission rises as the fourth power of temperature, the ocean buffers the pulse, and condensation removes vapor from the atmosphere rather than trapping it, so a finite pulse that rains out is not the condition a runaway greenhouse requires. Where it enters is equally constrained: the boiling surface reaches about one-sixth of the planet at its widest and is newly created ground, the remnant ocean is not a contact surface, and heat entering the atmosphere is exported aloft rather than spread across continents.

The mechanism is the Kilauea model at basin scale. Buoyant magma ponds at the surface, quenches against seawater, crystallizes, and is continuously refreshed by the subduction-driven upwelling. Heat transfer is governed by the emplacement rate, which tracks the velocity profile, and by the removal capacity, which tracks the new-basin surface area. The system is self-regulating at the open top: the boiling cap at one atmosphere limits the sea-surface temperature and converts excess thermal energy into atmospheric steam — the working fluid that drives the post-catastrophe climate, the ice age, and the dispersal corridors.

The budget is fixed by three observables and no tunable parameter. The transport ceiling is the one quantity not derived: the 10–20 kW/m² band sits in the sustained near-liquidus regime of published magma–water contact measurements, straddling the top of the 6–15 kW/m² band Hardee & Dunn (1981) report for subliquidus convective extraction and well inside their measured 2–40 kW/m² span. It is applied as a sustained delivery rate rather than a peak quench transient. Because the total is pinned by the budget, the three flux cases differ in timing alone; what they are checked against is the phase structure, which a closed-form integral and an independent time-stepping run reproduce to within a percent.

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© 2026 D. L. White. Licensed under CC BY-ND 4.0. https://creativecommons.org/licenses/by-nd/4.0/

AI Collaboration Disclosure: Calculations in this appendix were performed by Claude (Anthropic) and Grok (xAI), each cross-checking the other, under the direction of D. L. White. The time-stepping results in F.5 are Grok's integration on the current budget.