======================================================================== UPSIDE-DOWN ROCKS — club model of buoyancy-driven stratigraphic inversion ======================================================================== Anchor paper : Rudjord, J. E. & Huuse, M. (2025), Commun. Earth Environ. 6, 490 doi:10.1038/s43247-025-02398-8 This script : the Science Journaling Club's own simplified model. Not the paper's analysis. Assumptions listed in docstring. numpy available : True g : 9.81 m/s^2 pore brine density : 1030 kg/m^3 quartz grain density : 2650 kg/m^3 opal-A grain density : 2100 kg/m^3 (range 1950-2200) idealised body radius : 500 m sink distance modelled : 200 m ooze interval thickness : 200-500 m sand friction angle : 32 deg ======================================================================== PART 1 — Saturated bulk density vs porosity ======================================================================== rho_bulk = phi*rho_fluid + (1-phi)*rho_grain porosity | SAND (quartz grains) | OOZE (opal-A grains) phi | rho_bulk [kg/m^3] | rho_bulk [kg/m^3] ------------------------------------------------------------------------ 0.30 | 2164.0 | 1779.0 0.35 | 2083.0 | 1725.5 0.40 | 2002.0 | 1672.0 0.45 | 1921.0 | 1618.5 0.50 | 1840.0 | 1565.0 0.55 | 1759.0 | 1511.5 0.60 | 1678.0 | 1458.0 0.65 | 1597.0 | 1404.5 0.70 | 1516.0 | 1351.0 0.75 | 1435.0 | 1297.5 0.80 | 1354.0 | 1244.0 Realistic field values (what we adopt for the rest of the run): sand: phi = 0.40 -> rho_sand = 2002.0 kg/m^3 ooze: phi = 0.65 -> rho_ooze = 1404.5 kg/m^3 density contrast d_rho = 597.5 kg/m^3 (younger sand is the HEAVY one) ratio rho_sand/rho_ooze = 1.425 ======================================================================== PART 1b — Where is the density inversion window? ======================================================================== For each ooze porosity, find the MAXIMUM sand porosity at which the sand is still denser than the ooze. Above that, the stack is stable and boring. Crossover: phi_sand_max = 1 - (rho_ooze - rho_w) / (rho_qtz - rho_w) ooze phi | rho_ooze | max sand phi | unstable? ------------------------------------------------------------------------ 0.45 | 1618.5 | 0.637 | YES 0.50 | 1565.0 | 0.670 | YES 0.55 | 1511.5 | 0.703 | YES 0.60 | 1458.0 | 0.736 | YES 0.65 | 1404.5 | 0.769 | YES 0.70 | 1351.0 | 0.802 | YES 0.75 | 1297.5 | 0.835 | YES Read that table as: at ooze porosity 0.65 the sand can be as loose as phi = 0.769 and still outweigh the ooze it sits on. Real shallow-buried marine sands are ~0.35-0.45. The window is wide open. Sensitivity of d_rho to the opal-A grain density we assumed (sand held at phi=0.40, ooze at phi=0.65): opal grain density | rho_ooze | d_rho ------------------------------------------------------------------------ 1950 | 1352.0 | 650.0 2025 | 1378.2 | 623.8 2100 | 1404.5 | 597.5 2150 | 1422.0 | 580.0 2200 | 1439.5 | 562.5 Inversion survives the whole plausible opal-A range. It is not a knife-edge. ======================================================================== PART 2 — Driving force ======================================================================== Buoyancy force per unit volume f = d_rho * g d_rho = 597.5 kg/m^3 f = 5861.5 N/m^3 (= 5.8615 MPa per km of height) driving pressure across 50 m of ooze : 293074 Pa = 0.293 MPa driving pressure across 100 m of ooze : 586148 Pa = 0.586 MPa driving pressure across 200 m of ooze : 1172295 Pa = 1.172 MPa driving pressure across 500 m of ooze : 2930738 Pa = 2.931 MPa For scale: 1.172 MPa across the 200 m ooze layer is about 11.6 atmospheres of persistent, one-way push. Small. But geology has all the time it needs. ======================================================================== PART 3 — Sink velocity and time, Stokes-type drag ======================================================================== v = 2 * d_rho * g * R^2 / (9 * mu) R = 500 m, d_rho = 597.5 kg/m^3 t = 200 m / v ; Re = rho_ooze * v * (2R) / mu (Stokes valid for Re << 1) mu [Pa s] | v [m/s] | v [m/yr] | t to sink 200 m | Reynolds ------------------------------------------------------------------------ 1e6 | 3.256e+02 | 1.028e+10 | 1.95e-08 yr | 4.57e+02 1e7 | 3.256e+01 | 1.028e+09 | 1.95e-07 yr | 4.57e+00 1e8 | 3.256e+00 | 1.028e+08 | 1.95e-06 yr | 4.57e-02 1e9 | 3.256e-01 | 1.028e+07 | 1.95e-05 yr | 4.57e-04 1e10 | 3.256e-02 | 1.028e+06 | 0.000195 yr | 4.57e-06 1e11 | 3.256e-03 | 1.028e+05 | 0.00195 yr | 4.57e-08 1e12 | 3.256e-04 | 1.028e+04 | 0.0195 yr | 4.57e-10 1e13 | 3.256e-05 | 1.028e+03 | 0.195 yr | 4.57e-12 1e14 | 3.256e-06 | 1.028e+02 | 1.9 yr | 4.57e-14 1e15 | 3.256e-07 | 1.028e+01 | 19.5 yr | 4.57e-16 1e16 | 3.256e-08 | 1.028e+00 | 194.6 yr | 4.57e-18 1e17 | 3.256e-09 | 1.028e-01 | 1.95 kyr | 4.57e-20 1e18 | 3.256e-10 | 1.028e-02 | 19.5 kyr | 4.57e-22 1e19 | 3.256e-11 | 1.028e-03 | 195 kyr | 4.57e-24 1e20 | 3.256e-12 | 1.028e-04 | 1.95 Myr | 4.57e-26 Plausibility gate: the mounds formed inside the Miocene-Pliocene window, so the sinking must complete in roughly 1e3 to 1e7 years to be credible. Faster than ~1e3 yr and we would expect to see the process still running at the modern seabed; slower than ~1e7 yr and it never finishes. mu [Pa s] | t to sink 200 m | verdict ------------------------------------------------------------------------ 1e6 | 1.95e-08 yr | too fast - would still be visible 1e7 | 1.95e-07 yr | too fast - would still be visible 1e8 | 1.95e-06 yr | too fast - would still be visible 1e9 | 1.95e-05 yr | too fast - would still be visible 1e10 | 0.000195 yr | too fast - would still be visible 1e11 | 0.00195 yr | too fast - would still be visible 1e12 | 0.0195 yr | too fast - would still be visible 1e13 | 0.195 yr | too fast - would still be visible 1e14 | 1.9 yr | too fast - would still be visible 1e15 | 19.5 yr | too fast - would still be visible 1e16 | 194.6 yr | too fast - would still be visible 1e17 | 1.95 kyr | GEOLOGICALLY PLAUSIBLE 1e18 | 19.5 kyr | GEOLOGICALLY PLAUSIBLE 1e19 | 195 kyr | GEOLOGICALLY PLAUSIBLE 1e20 | 1.95 Myr | GEOLOGICALLY PLAUSIBLE Plausible viscosity band: 1e17 to 1e20 Pa s (4 decades wide). Context: water 1e-3, honey ~1e1, pitch ~1e8, window glass ~1e18, rock salt ~1e17-1e18, Earth's upper mantle ~1e21 Pa s. So the model needs the liquefied ooze/sand system to behave, on average, somewhere between hot pitch and rock salt. That is not an absurd ask, but it is an unmeasured one, and it is the weakest joint in the argument. Reynolds check at mu = 1e18 Pa s: Re = 4.57e-22 -> deep creeping flow, Stokes OK. Re first drops below 1 at mu = 1e8 Pa s and is astronomically small throughout the plausible band, so the drag law is at least self-consistent even where the geology is not. The two fastest rows (1e6, 1e7 Pa s) have Re > 1 and Stokes does not strictly apply there - they are rejected anyway. Because v ~ R^2, body size matters enormously. At mu = 1e18 Pa s: radius [m] | v [m/yr] | t to sink 200 m ------------------------------------------------------------------------ 50 | 1.028e-04 | 1946 kyr 100 | 4.111e-04 | 486.6 kyr 250 | 2.569e-03 | 77.85 kyr 500 | 1.028e-02 | 19.46 kyr 1000 | 4.111e-02 | 4.866 kyr A 50 m blob is 100x slower than a 500 m one. Being huge is the whole reason these bodies could sink at all within the available time. Small load casts need much weaker sediment to do the same trick. ======================================================================== PART 4 — Liquefaction: the pore-pressure ratio that kills shear strength ======================================================================== Terzaghi: sigma' = sigma_v - u Mohr-Coulomb: tau = sigma' * tan(phi_fric), phi_fric = 32 deg, no cohesion Excess pore pressure ratio r_u = delta_u / sigma'_v0 sigma'_v0 = (rho_sand - rho_w) * g * z (buoyant weight of overburden) tau(r_u) = (1 - r_u) * sigma'_v0 * tan(phi_fric) => tau = 0 exactly at r_u = 1.00. That is the liquefaction threshold. depth z [m] | sigma_v [MPa] | u_hydro [MPa] | sigma'_v0 [MPa] | tau0 [MPa] ------------------------------------------------------------------------ 10 | 0.196 | 0.101 | 0.095 | 0.060 25 | 0.491 | 0.253 | 0.238 | 0.149 50 | 0.982 | 0.505 | 0.477 | 0.298 100 | 1.964 | 1.010 | 0.954 | 0.596 200 | 3.928 | 2.021 | 1.907 | 1.192 400 | 7.856 | 4.042 | 3.814 | 2.383 Extra pore pressure required to reach r_u = 1 (i.e. total liquefaction): depth z [m] | delta_u needed [MPa] | total u at failure [MPa] | u / sigma_v ------------------------------------------------------------------------ 10 | 0.095 | 0.196 | 1.000 25 | 0.238 | 0.491 | 1.000 50 | 0.477 | 0.982 | 1.000 100 | 0.954 | 1.964 | 1.000 200 | 1.907 | 3.928 | 1.000 400 | 3.814 | 7.856 | 1.000 Residual shear strength as pore pressure climbs (z = 50 m): r_u | tau [kPa] | % of drained strength ------------------------------------------------------------------------ 0.00 | 297.92 | 100.0 0.20 | 238.33 | 80.0 0.40 | 178.75 | 60.0 0.50 | 148.96 | 50.0 0.60 | 119.17 | 40.0 0.80 | 59.58 | 20.0 0.90 | 29.79 | 10.0 0.95 | 14.90 | 5.0 0.99 | 2.98 | 1.0 1.00 | 0.00 | 0.0 The headline: shear strength is LINEAR in r_u. Get the pore pressure to 90% of the buoyant overburden stress and the sand keeps 10% of its strength; get to 100% and it keeps none. There is no cliff edge, no critical exotic condition - just a straight line running down to zero. Earthquake shaking does this routinely in the top tens of metres of loose saturated sand. The paper invokes exactly that trigger. ======================================================================== PART 5 — Does the predicted spacing match kilometre-scale mounds? ======================================================================== Two-layer viscous Rayleigh-Taylor, equal thickness b and equal viscosity: lambda_max = 2.568 * b (classical result, Turcotte & Schubert) Take b = the ooze interval thickness reported for the study area. ooze thickness b [m] | lambda_max [m] | lambda_max [km] ------------------------------------------------------------------------ 150 | 385 | 0.39 200 | 514 | 0.51 300 | 770 | 0.77 400 | 1027 | 1.03 500 | 1284 | 1.28 600 | 1541 | 1.54 For the reported 200-500 m ooze interval the model predicts a natural spacing of 0.51-1.28 km. Observed mounds are described as 'several kilometres wide', with a characteristic scale around 1 km. Our number lands in the same decade. That is agreement at the level of 'not obviously wrong', which is all a one-line textbook formula is entitled to claim. It is NOT confirmation. ======================================================================== SUMMARY LEDGER — numbers quoted in the article ======================================================================== sand bulk density (phi=0.40) 2002 kg/m^3 ooze bulk density (phi=0.65) 1404 kg/m^3 density contrast d_rho 598 kg/m^3 density ratio sand/ooze 1.425 max sand porosity for inversion @ ooze 0.65 0.769 buoyancy force per unit volume 5861 N/m^3 driving pressure across 200 m ooze 1.172 MPa plausible viscosity band 1e17 - 1e20 Pa s sink rate at mu = 1e18 Pa s 1.03 cm/yr (= 10.3 m/kyr) time to sink 200 m at mu = 1e18 Pa s 19.5 kyr time to sink 200 m at mu = 1e20 Pa s 1.95 Myr liquefaction threshold r_u = 1.00 (sigma' = 0) delta_u needed at z = 50 m 0.477 MPa drained shear strength at z = 50 m 297.9 kPa RT wavelength for b = 200-500 m 0.51 - 1.28 km All figures in the article are drawn from exactly these numbers. End of run.