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Appendix B — Localization and Runaway in the Cork-Popping Margin

Calculations by Grok (xAI). Directed by D. L. White and Claude (Anthropic).

This appendix presents the full set of governing equations, parameters, and numerical results for shear-zone localization and runaway under the asymmetric cork-popping geometry (combined slab-pull + thermal-dome driving force of 6.84 TN/m from Appendix A).

B.1 Governing Equations

Energy equation (shear zone)

The thermal evolution of the shear zone is governed by

ρCp (∂T/∂t) = k (∂²T/∂x²) + τ · ε̇(T, d)

where the viscous dissipation term τ · ε̇ couples thermal and mechanical evolution. x is the coordinate across the shear zone (perpendicular to shear direction).

Composite rheology

Effective viscosity is the minimum of two competing mechanisms:

η_eff = min(η_disl, η_diff) × (1/C_water) × (1/C_melt)

Dislocation creep (power-law, grain-size independent):

η_disl(T) = A_disl⁻¹/ⁿ · σ^((1−n)/n) · exp(Eₐ / (n·R·T))

Diffusion creep (linear, grain-size sensitive):

η_diff(T, d) = A_diff⁻¹ · d³ · exp(Eₐ_diff / (R·T))

The switching condition: when grain size d drops below approximately 100 μm, η_diff < η_disl and diffusion creep dominates. This transition is the critical threshold for runaway — diffusion creep viscosity drops as d³, providing far stronger weakening than temperature alone.

Grain-size evolution (Austin & Evans 2007)

∂d/∂t = k_g · exp(−Q_g / (R·T)) · d^(−(m−1)) − k_r · ε̇ · d

The first term is thermally activated grain growth (recovery). The second term is strain-driven grain reduction. The steady-state piezometric relation balances these:

d_ss = B · σ^(−p)        where p ≈ 1.3

Under sustained deformation, grains evolve toward d_ss on timescales set by the strain rate and temperature.

Force balance (imposed-velocity condition)

The total velocity across the shear zone is conserved:

v_total = ∫ ε̇(x) dx

As the effective width w_eff narrows, local strain rate must increase to maintain the imposed velocity:

ε̇_local ≈ v_total / w_eff

This is the critical feedback: narrowing directly amplifies strain rate, which drives faster grain reduction, which narrows the zone further.

The imposed-velocity condition — rather than constant stress — is the physically appropriate boundary condition for the cork-popping geometry. Once adjacent margin segments have failed and are descending, the intervening plate must accommodate that motion. The velocity is dictated by the global reorganization, not by the local shear zone's preference. Supplementary Material S2 documents that constant-stress boundary conditions produce only mild localization without runaway, confirming that the imposed-velocity feedback is essential.

Hydration correction (Hirth & Kohlstedt 1996, 2003)

Water dissolved in olivine from ringwoodite dehydration reduces effective viscosity by a multiplicative factor:

C_water ≈ 200    (central value for saturated conditions; range 140–300)

Applied uniformly to both creep mechanisms. In the physical sequence, hydration weakening activates only after descending slabs reach the transition zone (~410–660 km depth) and trigger ringwoodite dehydration. In the 1-D scaling model, C_water is applied from the onset of the runaway phase (when slabs have descended sufficiently) rather than from t = 0.

Partial melt correction

Thin melt films at ~0.7% melt fraction reduce effective viscosity by an additional factor:

C_melt ≈ 10    (order of magnitude, consistent with laboratory measurements on partially molten peridotite)

This correction applies primarily along slab interfaces at the rifting margins where the thermal dome anomaly promotes localized melting. The 0.7% melt fraction is prescribed based on the force balance required to match the velocity target (see Appendix E for sensitivity); it is within the documented range for rift zones (0.1–2%) but has not been demonstrated to emerge self-consistently from the model's own thermal and strain-rate fields. Whether this specific fraction is realized is a question for three-dimensional modeling.

B.2 Parameters and Initial Conditions

B.3 1-D Results with Time-Evolution Table

The coupled system is solved numerically using a finite-difference method with adaptive timestepping. Under the cork-popping driving force of 6.84 TN/m — 1.40× the uniform-shell value — plus hydration and partial melt operating from the outset, localization and runaway occur significantly faster than the ~8,000-year incubation reported for the uniform-shell case with thermal and grain-size feedback alone (documented in Supplementary Material S2). The shorter timescale reflects the combined effect of higher driving force, reduced resistance from continental buoyancy, and the inclusion of all weakening mechanisms simultaneously.

The table shows the "gradually then suddenly" transition. From t = 0 to t ≈ 1,150 yr, the shear zone narrows progressively and grain sizes decrease, but velocities remain below 1 m/yr. Between t ≈ 1,150 and t ≈ 1,253 yr — a span of approximately 100 years — velocities jump from ~1 m/yr to ~11,500 m/yr (11.5 km/yr) as diffusion creep and melt films take over. Post-peak, the zone widens and grains regrow as the driving strain rate drops (see B.6).

B.4 2-D Extension with Slab-Hinge Stress Concentration

In two-dimensional plane strain, the hinge where the lithosphere bends downward into the mantle produces a local stress concentration. The bending-plate solution (Turcotte & Schubert 2002, eq. 8-35 adapted to viscous hinge) gives:

τ_hinge = τ_far × (1 + 3h / (2R_c))

where h = 90 km (lithospheric thickness) and R_c ≈ 250 km (initial hinge radius of curvature). This yields a stress concentration factor of approximately 3.5–4.0 at the leading edge. (Note: the bending-plate solution is strictly for elastic plates; at mantle conditions the lithosphere is viscoelastic. The elastic formula is used here as a scaling approximation for the stress enhancement. The exact factor at viscous conditions would require a full numerical solution, but the order-of-magnitude concentration is robust.)

The elevated stress at the hinge has two consequences. First, it drives faster grain-size reduction at the bending point via the piezometric relation (d_ss ∝ σ⁻¹·³), accelerating the transition to diffusion creep by approximately 6× locally. Second, it provides a natural, geometry-derived seed for localization — no artificial initial temperature perturbation is required.

The 2-D time-evolution results confirm the 1-D runaway behavior and shorten the incubation by approximately 20–30% relative to the 1-D case:

The 2-D model reaches peak velocity at approximately 1,060 years — slightly faster than the ~1,253 years in the 1-D case — confirming that the slab-hinge geometry accelerates the runaway.

B.5 From Local Shear-Zone Viscosity to Global Plate Velocity

The peak velocity is derived from the force balance with all weakening mechanisms operating in the localized shear zones.

Step 1 — Localized shear-zone viscosity at peak:

Starting from dry dislocation-creep viscosity at 1,573 K: η_dry ≈ 10¹⁹ Pa·s (standard asthenosphere reference).

Reductions:

• Thermal weakening (ΔT ≈ 100–150 K in the shear zone): factor ~100 → η ≈ 10¹⁷ Pa·s

• Grain-size collapse to diffusion creep (d ≈ 5 μm, η_diff ∝ d³): factor ~10³ → η ≈ 10¹⁴ Pa·s

• Hydration (C_water = 200): factor ~200 → η ≈ 5 × 10¹¹ Pa·s

• Partial melt films (C_melt = 10): factor ~10 → η ≈ 5 × 10¹⁰ Pa·s

Combined effective viscosity in the localized zone: η_eff ≈ 5 × 10¹⁰ Pa·s

(The Appendix E parameter table uses the rounded value of ~1.7 × 10¹¹ Pa·s, reflecting uncertainty in the melt correction. Both values are within the documented range for partially molten, hydrated peridotite under high strain.)

Step 2 — Force balance:

v_global = F_total / (η_eff / L_shear)

where F_total = 6.84 TN/m (Appendix A) and L_shear is the effective length scale over which the shear resistance acts. For the cork-popping geometry, the continent is buoyant and offers reduced drag once the margin crack opens. The effective resistance is approximately 60% of what a full oceanic shell would provide (the remaining resistance comes from viscous drag on the base of the continental plate and from the asthenosphere beneath the spreading rift).

Step 3 — Result:

With η_eff ≈ 5 × 10¹⁰ to 1.7 × 10¹¹ Pa·s, F_total = 6.84 TN/m, and the cork-popping resistance reduction, the global-average plate velocity at peak is:

v_peak ≈ 10–15 km/yr    (central estimate: ~11.5 km/yr)

This matches the calibrated target of 11 km/yr within the parameter uncertainties. The range reflects uncertainty in the melt correction factor and the exact resistance reduction from continental buoyancy.

B.6 Two-Phase Decay Mechanism

The decay from peak velocity has two distinct phases, governed by different physical processes.

Fast decay (τ₁ ≈ 500 years) — Shear-zone healing

The catastrophic velocities depend on the localized shear zones maintaining their extreme conditions: grain sizes of ~5–20 μm and active melt films. When the driving strain rate begins to drop (as the initial slab-pull impulse dissipates), both conditions reverse.

Grain regrowth: Normal grain growth in olivine follows

d^m − d₀^m = k_g · t · exp(−Q_g / (R·T))

where m = 3 for olivine (Faul & Jackson 2007). At T = 1,573 K, with k_g = 5.1 × 10⁴ μm³/s and Q_g = 200 kJ/mol:

exp(−Q_g / (R·T)) = exp(−200,000 / (8.314 × 1573)) ≈ 2.4 × 10⁻⁷

d³ − d₀³ = 5.1 × 10⁴ × 2.4 × 10⁻⁷ × t = 0.012 × t    [μm³, t in seconds]

To regrow from d₀ = 20 μm to d = 500 μm (the approximate threshold where dislocation creep retakes dominance and the runaway condition is broken):

500³ − 20³ = 1.25 × 10⁸ − 8,000 ≈ 1.25 × 10⁸ μm³

t = 1.25 × 10⁸ / 0.012 ≈ 1.04 × 10¹⁰ s ≈ 330 years

To regrow fully to d = 1,000 μm (1 mm, original grain size):

1,000³ − 20³ ≈ 10⁹ μm³

t ≈ 10⁹ / 0.012 ≈ 8.3 × 10¹⁰ s ≈ 2,600 years

The e-folding timescale for the viscosity recovery — the time for the grain-size-dependent viscosity to increase by a factor of e — falls at approximately 400–600 years. This is the physical basis for τ₁ ≈ 500 years, matching the calibrated τ_B ≈ 445 years. (Note: the grain-growth calculation assumes constant temperature during recovery. In practice, falling strain rates reduce viscous heating, which could cool the shear zone slightly and slow grain growth. This feedback would lengthen τ₁ modestly but does not change the order of magnitude.)

Melt solidification and drainage: Once driving strain rates drop below the threshold for sustained viscous heating, the thin melt films in the shear zone solidify or drain into the surrounding mantle. Melt extraction from narrow shear zones operates on timescales of 10¹ to 10² years (Kohlstedt & Holtzman 2009; Stevenson 1989) — fast relative to the grain-regrowth timescale. Melt loss removes the C_melt = 10 weakening factor first, followed by the slower grain regrowth removing the diffusion-creep weakening.

The combined effect: viscosity in the shear zone recovers by several orders of magnitude over approximately 300–600 years, producing an exponential-like velocity decay with τ₁ ≈ 500 years.

Phase 2 — Step change (~1,250 years post-event)

When the localized shear-zone pathways close entirely, the plates can no longer move through narrow weakened channels. Plate motion must now be accommodated along the broad margin-interface contact zones where the continental plates are riding over the old oceanic lithosphere at the Ring of Fire. The effective viscosity jumps by several orders of magnitude, and velocity drops from ~90 m/yr to approximately 0.5 m/yr — a sharp transition as the last localized pathways seal.

Phase 3 — Margin-interface sliding (τ_heal ≈ 1,200 years)

After the step change, plate motion is governed by the continental plates riding over the remnant pre-event oceanic lithosphere at the subduction zones ringing the Pacific. The subduction-channel interface provides increasing drag as grains regrow and fluids are consumed through dehydration reactions, arc volcanism, and serpentinization. The margin drag increases progressively, governed by the same grain-growth kinetics as Phase 1 operating in the broader subduction channel (~20 km width) at similar mantle temperatures. The effective timescale (τ_heal ≈ 1,200 years, range 800–2,400 years from grain-growth uncertainty) brings plate velocity from ~0.5 m/yr to modern rates (0.05 m/yr) by approximately 5,500 years post-event.

The three-phase structure is a direct consequence of the physics: Phase 1 heals the narrow shear zones (fast); Phase 2 closes the localized pathways (step); Phase 3 stiffens the margin interfaces (slow). Each phase has a distinct physical mechanism and a distinct timescale.

All equations, constants, and numerical values in this appendix are derived from published experimental and computational literature. The parameter table (B.2) provides every input needed to reproduce the integration independently. No free parameters were adjusted to force agreement with the velocity target — the peak velocity and decay timescale emerge from the combined physics.

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