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Appendix D — Global Plate-Velocity Profile v(t)

This appendix constructs the global-average plate-velocity curve from the physics established in Appendices A, B and C, and then states the quantity a full three-dimensional treatment would be tested against: the early-decay constant required by the observed continental separation.

It does not compare the curve to a target. The velocity history is not known in advance, and the constraints the mechanism must satisfy — total displacement, present-day velocity, elapsed time, and published material properties — are set out in Section 2 of the main text. The parameter ranges behind the figures used here, their sources, and what each one moves are collected in Appendix E.

D.1 Constructing v(t): Three Time-Dependent Components

The global-average plate velocity at any time is the ratio of driving force to effective resistance:

v(t) = F_drive(t) / R_eff(t)

Both are time-dependent, and they are governed by different physics.

Driving force F_drive(t) has two components, each decaying on its own timescale.

Slab-pull is the primary driver. It peaks during the runaway phase and diminishes as the descending slabs complete their transit through the upper mantle and the initial gravitational potential energy is expended. In this treatment slab-pull is held approximately constant through the fast-decay phase — a simplification justified by slabs still actively descending during the first several hundred years — and declines slowly thereafter as descent completes.

Thermal-dome push is the continental insulation anomaly, F_dome ≈ 1.94 TN/m from Appendix A. It begins dissipating once the rift opens and hot sub-continental mantle is exposed, through convective and hydrothermal cooling at the new rift axis. The shallow component (~50 km) cools on a timescale near 1,000 years; the deep component (~100–150 km) persists for 3,000–5,000 years, consistent with thermal relaxation observed at continental rifts and estimated for mantle plume decay (Sleep 2006; Artemieva 2011). Modeled as exponential decay:

F_dome(t) = 1.94 × exp(−t / τ_dome)        τ_dome ≈ 3,000 yr

Effective resistance R_eff(t) recovers in three phases (from Appendix B, Section B.6).

Phase 1 — Fast recovery. Localized shear-zone viscosity rises as grains regrow from 5–20 μm back toward ~500 μm and melt films solidify. This governs the initial rapid deceleration.

Two timescales operate here and they must not be conflated. The grain-growth timescale is τ_grain ≈ 500 years — the time for the grains themselves to recover. The velocity decays considerably faster, because diffusion-creep viscosity scales as the square to the cube of grain size, so a given proportional change in grain size produces a larger proportional change in resistance. The distinction matters because the two quantities are separately compared to other things later in this appendix and in Appendix E, and they are not interchangeable.

Phase 2 — Regime transition. When grain sizes in the localized zones regrow past the 50–100 μm threshold, diffusion creep no longer dominates and the localized pathways lose their conductance advantage. Plate motion must be accommodated instead along the broad margin-interface contact zones where the continental plates ride over older oceanic lithosphere.

The operative resistance ratio is not the viscosity ratio. Viscosity rises from 10¹⁰–10¹¹ Pa·s in the localized zones to the interface value, which a global compilation of seventeen exhumed subduction-interface mélange shear zones places at 1.9 × 10¹⁸ to 2.8 × 10²⁰ Pa·s (Abila, Behr & Ruh 2024). But resistance in a shear zone depends on its thickness as well as its viscosity — at fixed stress, velocity goes as w/η — and the two configurations differ in thickness by many orders of magnitude, from grain-boundary melt films to a mélange zone measured in hundreds of meters to kilometers. The geometric factor works against the viscosity factor and the net is not fixed by either alone.

What follows is therefore an order-of-magnitude statement and not a derivation. The transition is velocity-gated: it occurs when Phase 1 decay brings velocity down to v_crit ≈ 26.7 m/yr, and the velocity entering the margin-interface regime is of order 0.1–1 m/yr — well above the modern 0.05 m/yr and well below Phase 1. A representative value of 0.5 m/yr is used where a single figure is needed. The healing ceiling in D.3 widens accordingly, and no other result depends on where in that band the true value lies.

The transition is represented here as a discontinuity. It plainly is not one — a physical closure of the localized pathways has a finite width, and resolving that width is beyond a one-dimensional treatment. The idealization is harmless where it is used: a transition of even a century's width would add on the order of a kilometer to a displacement budget of five thousand. It is listed in D.4 as one of the things a full model would supply.

Phase 3 — Margin-interface sliding. After the transition, the plates ride over remnant pre-event oceanic lithosphere at the subduction margins. Progressive grain regrowth and fluid consumption in the subduction channel — dehydration reactions, arc volcanism, serpentinization — increase margin drag and the velocity declines toward modern rates.

This treatment does not determine how long that takes. The healing of a broad subduction-channel interface has no direct experimental calibration, and the mechanism described above implies a drag that increases as healing proceeds rather than a fixed characteristic time. What the phase must satisfy is stated in D.3, and it is a ceiling rather than a value.

D.2 The Velocity Curve

The table below is the output of the one-dimensional forward model at its own parameters. Year zero is the onset of the catastrophic event — the moment shear-zone runaway produces surface-observable velocities. The incubation phase is shown as negative time; surface velocities during it are indistinguishable from background tectonic motion.

The incubation figure is the single-segment incubation time from the one-dimensional integration in Appendix B, not this work's estimate of how long the pre-event phase lasted. It appears here only to mark where the model's clock starts.

Two features of the curve are worth naming. The rise from incubation to peak spans five orders of magnitude, and the table resolves it no more finely than the 250-year interval between its last incubation row and the peak; that is the sudden phase. Post-peak, Phase 1 brings velocity down by nearly three orders over about twelve hundred years, and the regime transition drops it by one to two orders more.

The decay in Phase 1 is not a single exponential. Fitting successive intervals of the table gives decay times of 231, 228, 190 and 184 years — the curve steepens as it proceeds. Any single figure quoted for the Phase 1 decay of this model is an average over a changing quantity, and the value depends on the interval chosen. The peak in this run is the model's own output and sits just under the 12 km/yr central case used in D.3 and consumed by Appendix C. Nothing turns on the difference; D.3's table brackets both.

The transition time in this table is the one-dimensional model's own, and it is not this work's estimate of when the transition occurred. The transition is velocity-gated, so it moves with the decay: at the decay constants required by the observed displacement (D.3) it falls between roughly 1,930 and 4,370 years depending on peak velocity, against the ~1,224 years this run produces. The difference has the same origin as the difference in the decay itself, and it is the required values that a full treatment would be tested against.

Phase 3 is shown only as a direction, not as a curve. The velocity declines from the post-transition value of order 0.1–1 m/yr toward the modern 0.05 m/yr, and the endpoint is measured rather than modeled — global average plate velocity is a geodetic observation. When it is reached is not determined here, for the reason given in D.1, and no row is tabulated for it. What can be said about the timing is in D.3.

D.3 The Decay Constant Required by the Observed Displacement

The continents have separated by approximately 5,000 km. The force balance produces a peak velocity. Those two quantities together determine the early-decay timescale: for whatever peak velocity the force balance produces, τ₁ is fixed by the observed displacement and is not available to be chosen. The peak itself is not a single number, so what follows is a relation between the two rather than a value.

For a decay of the form v(t) = v_peak · exp(−t/τ₁), running until velocity reaches the regime-transition value v_crit, the displacement delivered is

D = τ₁ · (v_peak − v_crit) and therefore τ₁ = D / (v_peak − v_crit)

Phase 1 ends at v_crit. The further drop from there to the post-transition velocity is the regime transition itself, represented as a discontinuity, and it delivers no displacement. v_crit is the gate, not the velocity the plates carry into Phase 3.

This is a curve, not a point: one equation in two unknowns. What selects a stretch of it is olivine grain-growth kinetics, which span roughly a factor of three in the experimental literature (Faul & Jackson 2007) and place Phase 1 decay times between 300 and 800 years. Inverting the relation over that band gives peak velocities from 6.3 to 16.7 km/yr:

This is the appendix's principal result, and it is a specification rather than a measurement: any full treatment that reproduces the observed separation must produce an early decay on this timescale, paired with the peak velocity it computes.

The independent check is on the melt fraction, and it is the one that can fail. Nothing so far is a test — the displacement relation defines the curve and the kinetic band selects a stretch of it, so the decay times and the peak velocities above are two statements of the same constraint. A test requires a third quantity, measured independently of both.

The force balance supplies one. Peak velocity depends steeply on the partial melt fraction at the rifting margins, and melt fraction is observed in active rift zones by methods that have nothing to do with grain-growth rates or with continental reconstruction. Reading the force balance backwards, the peaks above correspond to melt fractions of approximately 0.41% to 0.94%. Documented rift-zone melt fractions run from 0.1% to 2.0%. The required window sits inside the observed range and brackets the value this model uses.

That comparison could have come out otherwise. Had the displacement and the kinetics together demanded 5% melt, no rift on Earth would look like the mechanism, and the account would fail on an observation none of its own machinery produced. What the check establishes is narrow and worth stating exactly: the melt fraction the mechanism needs is one that rift zones are observed to have.

It is not a blanket pass either. The required window spans a factor of about two inside a documented range spanning a factor of twenty, so the mechanism is confined to a narrow part of what the literature permits — and confined at both ends, since a slower peak needs melt near 0.4% and a faster one near 0.9%.

The melt-fraction figures are read from the sensitivity points collected in Appendix E, which imply v ∝ f^1.17. They should be confirmed against the force-balance calculation directly rather than taken from that fit.

What constrains Phase 3

The endpoint is observed: global average plate velocity is 0.05 m/yr, measured geodetically. The mechanism must arrive there. When it arrives is not constrained by displacement, because the displacement budget is spent in the early decay — Phase 3 carries on the order of a kilometer of the five thousand.

What remains is one-sided. Modern velocity is observed now, so the margin-interface healing must be fast enough that the plates have already reached it. Writing the approach as a decay from the post-transition velocity v_t with characteristic time τ_heal, that requires

τ_heal ≤ (T_elapsed − t_transition) / ln(v_t / v_modern)

The bound is real but weak. The post-transition velocity is known only to an order of magnitude, the elapsed time is a range, and the transition year moves with the peak. What the bound establishes is a direction rather than a value: healing must be complete, not merely underway, and any treatment that leaves the plates still decelerating today is excluded. Whether this observation or the grain-growth kinetics is the tighter constraint depends on where in those ranges the true values sit, and this work does not determine it.

There is no lower bound. A history that reached modern velocity early and remained there contradicts nothing observable. This appendix therefore states a ceiling and declines to name an arrival, and no result elsewhere in this work depends on one.

D.4 What a Three-Dimensional Treatment Must Produce

The forward model here is one-dimensional. It is asked for two things it can supply — the peak velocity, from the force balance under the cork-popping geometry with published rheology, and the functional form of the early decay, which is grain-growth controlled. It is not asked for total displacement, and the reason is structural: a one-dimensional treatment cannot represent interacting rift arms, return flow, or strain distributed across multiple simultaneous margins, all of which sustain velocity longer than a single-margin calculation can. That shows in its own output, where the decay steepens through the interval rather than holding constant.

The specification a three-dimensional thermo-mechanical model would be tested against is:

1. A peak velocity between 6.3 and 16.7 km/yr, arising from the force balance rather than imposed as a boundary condition, and produced by a melt fraction between roughly 0.4% and 0.9% — inside documented rift-zone values.

2. An emergent early decay constant satisfying τ₁ ≈ D/(v_peak − v_crit) for the peak the model itself generates — 300 to 800 years across the envelope. The relation is the test, not an instruction: a full treatment produces its own decay from its own grain-size evolution, and the question is whether that emergent value matches what the observed displacement requires at the peak the same run produced. Imposing the relation would defeat the purpose.

3. Grain-growth kinetics inside the published factor-of-three spread. Note that this and requirement 2 are not independent — the velocity envelope in requirement 1 is this band inverted through the displacement relation. The independent test is requirement 1's melt fraction.

4. Arrival at modern plate velocity by the present, with margin-interface healing fast enough to satisfy the ceiling in D.3. No particular arrival year is required, and none should be produced as though it were a result.

5. A resolved regime transition. The present treatment represents the closure of the localized pathways as a discontinuity because a one-dimensional calculation cannot resolve its width. The transition has one. Its width and shape are the quantity this work most needs from a three-dimensional run, and they are not recoverable from anything presented here.

Initial conditions — pre-event lithospheric geometry and thermal state — are deliberately left to the modeler. They belong to the discipline that would perform the run, and prescribing them here would substitute this work's judgment for that of the people equipped to make it.

← Return to paper | Appendix E — Parameters, Sensitivities and Limitations →

© 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 Grok (xAI), with drafting and integration by Claude (Anthropic), under the direction of D. L. White.