========================================================================== 0. SETUP ========================================================================== Present-day solar constant S_now = 1361.0 W/m^2 Solar constant at 717 Ma (Gough) S_sturt = 1280.6 W/m^2 (94.1% of today) Insolation deficit vs today = 80.4 W/m^2 (= 14.1 W/m^2 of absorbed flux at albedo 0.30) Albedo: ice-free 0.30 -> fully glaciated 0.50 (transition centred 262 K, width 9 K) OLR slope B = 1.30 W/m^2/K => equilibrium sensitivity to 2xCO2 = 2.85 K Outgassing V = 7.50e+12 mol C/yr; modern weathering W_0 = 7.50e+12 mol C/yr ========================================================================== 1. ICE-ALBEDO HYSTERESIS: SWEEPING THE SOLAR CONSTANT (pCO2 fixed at 280 ppm) ========================================================================== Bistable window in solar constant: 1278 to 1472 W/m^2 Width of the bistable window: 194 W/m^2 = 15.2% of the lower edge Sturtian solar constant 1281 W/m^2 sits INSIDE that window -> both a frozen and an ice-free Earth are possible at 717 Ma. Today's 1361 W/m^2 also sits INSIDE it. Equilibrium temperatures at selected solar constants (K): S (W/m^2) cold branch unstable warm branch #states 1100 210.5 - - 1 1200 220.2 - - 1 1250 225.0 - - 1 1281 227.9 269.9 273.9 3 1300 229.8 266.7 278.5 3 1361 235.8 262.1 287.8 3 1450 245.6 256.2 299.9 3 1600 - - 320.2 1 HYSTERESIS DEMO -- ramp S down then back up (quasi-static, 280 ppm CO2): cooling: Earth snaps into the snowball at S = 1277 W/m^2 (lands at T = 227.6 K = -45.6 C) warming: Earth escapes the snowball at S = 1474 W/m^2 (lands at T = 303.2 K = 30.0 C) the two thresholds differ by 197 W/m^2 -- THAT GAP IS THE HYSTERESIS. ========================================================================== 2. THE SAME LOOP IN CO2 SPACE (solar constant fixed at the Sturtian value) ========================================================================== Fold (saddle-node) temperatures are set by the condition (S/4)|d(alpha)/dT| = B. cold-branch fold (deglaciation): T = 252.2 K = -21.0 C warm-branch fold (freeze-in) : T = 271.8 K = -1.3 C FREEZE THRESHOLD pCO2 = 2.572e-04 bar = 257 ppmv (0.92x pre-industrial) DEGLACIATION THRESHOLD pCO2 = 0.0303 bar = 30328 ppmv (108x pre-industrial) Ratio thaw/freeze = 118x (2.07 orders of magnitude of CO2) [literature deglaciation estimates span ~0.01-0.1 bar; we land inside that range, but the log-forcing extrapolation makes this an order-of-magnitude claim only] Branch temperatures at the two thresholds: just before freeze-in pCO2=2.572e-04 bar stable T = 227.6 K (-45.6 C) just after deglaciation pCO2=3.033e-02 bar stable T = 296.4 K (+23.3 C) Peak-to-peak temperature swing implied by the two jumps: hothouse +23.3 C -> snowball -45.6 C = 68.8 K amplitude ========================================================================== 3. WHERE DOES THE CARBON CYCLE WANT TO SIT? (the bifurcation in weatherability) ========================================================================== A steady state needs W(T,pCO2) = V with T on a STABLE branch. If the only solution sits on the unstable branch, no steady state exists. fW warm-branch steady state exists? 0.50 pCO2=3.40e-03 bar, T=+14.1 C YES 1.00 pCO2=1.13e-03 bar, T=+9.2 C YES 2.00 pCO2=4.22e-04 bar, T=+4.1 C YES 3.00 pCO2=2.79e-04 bar, T=+0.7 C YES 3.50 pCO2=2.59e-04 bar, T=-0.8 C YES 3.75 none on the warm branch NO 4.00 none on the warm branch NO 5.00 none on the warm branch NO 8.00 none on the warm branch NO 12.00 none on the warm branch NO CRITICAL WEATHERABILITY fW* = 3.59 Below 3.59x, the planet finds a warm steady state and stays there forever. Above 3.59x, no steady state exists on either stable branch -> LIMIT CYCLE. Basalt weathers roughly 5-10x faster than granite at the same temperature, so emplacing the Franklin LIP in the wet tropics plausibly crosses this line. ========================================================================== 4. FORWARD INTEGRATION: DOES IT ACTUALLY CYCLE? ========================================================================== Two coupled ODEs, stiff (climate relaxes in ~13 yr, carbon in ~10^6 yr). Integrated with an implicit Radau solver for 140 Myr from a warm start. fW regime period (Myr) amp (K) T range (C) pCO2 range (bar) % frozen cycles/56 Myr 1.0 fixed point - - +9.2 1.13e-03 0.0% - 2.0 fixed point - - +4.1 4.23e-04 0.0% - 3.0 fixed point - - +0.7 2.80e-04 0.0% - 4.0 LIMIT CYCLE 1.325 68.8 -45.5 to +23.3 2.56e-04 - 3.04e-02 91.8% 42 5.0 LIMIT CYCLE 1.306 68.8 -45.6 to +23.3 2.54e-04 - 3.04e-02 93.8% 43 6.0 LIMIT CYCLE 1.298 68.8 -45.5 to +23.3 2.54e-04 - 3.04e-02 95.0% 43 8.0 LIMIT CYCLE 1.298 68.9 -45.6 to +23.2 2.50e-04 - 3.04e-02 96.4% 43 10.0 LIMIT CYCLE 1.306 68.9 -45.6 to +23.2 2.48e-04 - 3.04e-02 97.2% 43 12.0 LIMIT CYCLE 1.320 68.9 -45.7 to +23.2 2.47e-04 - 3.04e-02 97.7% 42 16.0 LIMIT CYCLE 1.356 69.0 -45.8 to +23.2 2.45e-04 - 3.04e-02 98.3% 41 ========================================================================== 5. THE HEADLINE RUN (fW = 5, a Franklin-scale weatherability boost) ========================================================================== Period P = 1.306 Myr (std over last 5 cycles 0.0001 Myr) Temperature amplitude dT = 68.8 K hothouse peak +23.3 C snowball floor -45.6 C pCO2 range 254 ppmv -> 30363 ppmv (119x, 2.08 orders of magnitude) Fraction of each cycle spent frozen 93.8% Fraction of each cycle spent ice-free 6.2% Cycles fitted into the 56 Myr Sturtian 42.9 Mean duration of one frozen interval 1.225 Myr Mean duration of one ice-free interval 0.081 Myr Timescale accounting -- why the period is what it is: carbon that must be added to go freeze->thaw: 8.886e+18 mol C at V = 7.50e+12 mol/yr that recharge takes 1.185 Myr -> the FROZEN half of the cycle is outgassing-limited and CANNOT be shortened by weatherability. Only the warm half responds to fW. ========================================================================== 6. HOW THE PERIOD RESPONDS TO WEATHERABILITY (the Franklin knob) ========================================================================== fW period (Myr) frozen (Myr) ice-free (Myr) cycles/56 Myr % frozen 4.0 1.325 1.217 0.109 42 91.8% 5.0 1.306 1.225 0.081 43 93.8% 6.0 1.298 1.233 0.065 43 95.0% 8.0 1.298 1.251 0.047 43 96.4% 10.0 1.306 1.269 0.037 43 97.2% 12.0 1.320 1.289 0.031 42 97.7% 16.0 1.356 1.333 0.023 41 98.3% Quadrupling weatherability from 4x to 16x shortens the period by only -2% (1.33 -> 1.36 Myr) and drives the frozen fraction from 92% up to 98% of each cycle. The cycle has a FLOOR: you cannot flicker faster than volcanoes can refill the sky. Asymptotic floor (pure recharge time) = 1.185 Myr. ========================================================================== 7. SENSITIVITY: WHAT IF OUTGASSING WAS LOWER? (Dutkiewicz et al. 2024) ========================================================================== Independent tectonic reconstructions argue mid-ocean-ridge outgassing at 717 Ma was exceptionally low. In this model that lengthens the frozen half directly. V / V_modern period (Myr) frozen (Myr) cycles/56 Myr 0.4 3.306 3.233 17 0.6 2.164 2.088 26 0.8 1.622 1.544 35 1.0 1.306 1.225 43 1.3 1.024 0.935 55 ========================================================================== 8. ROBUSTNESS: JITTER THE PARAMETERS ========================================================================== 200 draws, each parameter perturbed by a uniform +/-10% (seeded RNG), to check that cycling is not an artefact of one lucky parameter set. Limit cycles found in 44/200 draws = 22% Period across those draws: median 4.51 Myr, 10th-90th percentile 0.55-19.57 Myr So the OSCILLATION is robust; the PERIOD is not tightly pinned. That is the honest summary. ========================================================================== 9. FIGURE DATA ========================================================================== --- FIG 1: hysteresis in (S, T), 280 ppm CO2 --- warm_branch S,T (every 40th point): 1278,272.0; 1358,287.4; 1438,298.3; 1518,309.1; 1598,319.9; 1678,330.7; 1758,341.4 cold_branch S,T (every 40th point): 1000,200.9; 1080,208.6; 1160,216.3; 1240,224.0; 1320,231.8; 1400,239.8 unstable_branch S,T (every 20th point): 1278,271.5; 1318,265.0; 1358,262.3; 1398,259.9; 1438,257.2 --- FIG 2: the S-curve in (log10 pCO2, T) at S_sturt, plus the limit cycle --- warm branch (log10 p, T): -3.589,272.1; -3.572,273.2; -3.556,273.8; -3.539,274.3; -3.522,274.7; -3.506,275.0; -3.489,275.4; -3.473,275.7; -3.456,276.0; -3.439,276.2; -3.423,276.5; -3.406,276.8; -3.390,277.0; -3.373,277.3; -3.356,277.5; -3.340,277.7; -3.323,278.0; -3.307,278.2; -3.290,278.4; -3.273,278.6; -3.257,278.8; -3.240,279.0; -3.224,279.2; -3.207,279.4; -3.190,279.6; -3.174,279.8; -3.157,280.0; -3.141,280.2; -3.124,280.4; -3.107,280.6; -3.091,280.8; -3.074,281.0; -3.058,281.2; -3.041,281.4; -3.024,281.5; -3.008,281.7; -2.991,281.9; -2.975,282.1; -2.958,282.3; -2.941,282.4; -2.925,282.6; -2.908,282.8; -2.892,283.0; -2.875,283.1; -2.858,283.3; -2.842,283.5; -2.825,283.7; -2.808,283.8; -2.792,284.0; -2.775,284.2; -2.759,284.3; -2.742,284.5; -2.725,284.7; -2.709,284.9; -2.692,285.0; -2.676,285.2; -2.659,285.4; -2.642,285.5; -2.626,285.7; -2.609,285.9; -2.593,286.0; -2.576,286.2; -2.559,286.4; -2.543,286.5; -2.526,286.7; -2.510,286.9; -2.493,287.0; -2.476,287.2; -2.460,287.3; -2.443,287.5; -2.427,287.7; -2.410,287.8; -2.393,288.0; -2.377,288.2; -2.360,288.3; -2.344,288.5; -2.327,288.6; -2.310,288.8; -2.294,289.0; -2.277,289.1; -2.261,289.3; -2.244,289.5; -2.227,289.6; -2.211,289.8; -2.194,289.9; -2.178,290.1; -2.161,290.3; -2.144,290.4; -2.128,290.6; -2.111,290.7; -2.095,290.9; -2.078,291.1; -2.061,291.2; -2.045,291.4; -2.028,291.5; -2.012,291.7; -1.995,291.9; -1.978,292.0; -1.962,292.2; -1.945,292.3; -1.929,292.5; -1.912,292.7; -1.895,292.8; -1.879,293.0; -1.862,293.1; -1.846,293.3; -1.829,293.5; -1.812,293.6; -1.796,293.8; -1.779,293.9; -1.763,294.1; -1.746,294.2; -1.729,294.4; -1.713,294.6; -1.696,294.7; -1.680,294.9; -1.663,295.0; -1.646,295.2; -1.630,295.4; -1.613,295.5; -1.597,295.7; -1.580,295.8; -1.563,296.0; -1.547,296.1; -1.530,296.3; -1.514,296.5; -1.497,296.6; -1.480,296.8; -1.464,296.9; -1.447,297.1; -1.431,297.3; -1.414,297.4; -1.397,297.6; -1.381,297.7; -1.364,297.9; -1.347,298.0; -1.331,298.2; -1.314,298.4; -1.298,298.5; -1.281,298.7; -1.264,298.8; -1.248,299.0; -1.231,299.1 cold branch (log10 p, T): -5.000,214.2; -4.983,214.4; -4.967,214.5; -4.950,214.7; -4.934,214.8; -4.917,215.0; -4.900,215.1; -4.884,215.3; -4.867,215.5; -4.851,215.6; -4.834,215.8; -4.817,215.9; -4.801,216.1; -4.784,216.2; -4.768,216.4; -4.751,216.6; -4.734,216.7; -4.718,216.9; -4.701,217.0; -4.685,217.2; -4.668,217.3; -4.651,217.5; -4.635,217.7; -4.618,217.8; -4.602,218.0; -4.585,218.1; -4.568,218.3; -4.552,218.4; -4.535,218.6; -4.519,218.8; -4.502,218.9; -4.485,219.1; -4.469,219.2; -4.452,219.4; -4.436,219.5; -4.419,219.7; -4.402,219.9; -4.386,220.0; -4.369,220.2; -4.353,220.3; -4.336,220.5; -4.319,220.6; -4.303,220.8; -4.286,221.0; -4.269,221.1; -4.253,221.3; -4.236,221.4; -4.220,221.6; -4.203,221.8; -4.186,221.9; -4.170,222.1; -4.153,222.2; -4.137,222.4; -4.120,222.5; -4.103,222.7; -4.087,222.9; -4.070,223.0; -4.054,223.2; -4.037,223.3; -4.020,223.5; -4.004,223.6; -3.987,223.8; -3.971,224.0; -3.954,224.1; -3.937,224.3; -3.921,224.4; -3.904,224.6; -3.888,224.7; -3.871,224.9; -3.854,225.1; -3.838,225.2; -3.821,225.4; -3.805,225.5; -3.788,225.7; -3.771,225.9; -3.755,226.0; -3.738,226.2; -3.722,226.3; -3.705,226.5; -3.688,226.6; -3.672,226.8; -3.655,227.0; -3.639,227.1; -3.622,227.3; -3.605,227.4; -3.589,227.6; -3.572,227.7; -3.556,227.9; -3.539,228.1; -3.522,228.2; -3.506,228.4; -3.489,228.5; -3.473,228.7; -3.456,228.9; -3.439,229.0; -3.423,229.2; -3.406,229.3; -3.390,229.5; -3.373,229.6; -3.356,229.8; -3.340,230.0; -3.323,230.1; -3.307,230.3; -3.290,230.4; -3.273,230.6; -3.257,230.8; -3.240,230.9; -3.224,231.1; -3.207,231.2; -3.190,231.4; -3.174,231.6; -3.157,231.7; -3.141,231.9; -3.124,232.0; -3.107,232.2; -3.091,232.4; -3.074,232.5; -3.058,232.7; -3.041,232.8; -3.024,233.0; -3.008,233.2; -2.991,233.3; -2.975,233.5; -2.958,233.6; -2.941,233.8; -2.925,234.0; -2.908,234.1; -2.892,234.3; -2.875,234.4; -2.858,234.6; -2.842,234.8; -2.825,234.9; -2.808,235.1; -2.792,235.2; -2.775,235.4; -2.759,235.6; -2.742,235.7; -2.725,235.9; -2.709,236.1; -2.692,236.2; -2.676,236.4; -2.659,236.5; -2.642,236.7 unstable branch (log10 p, T): -3.589,271.5; -3.572,270.5; -3.556,270.0; -3.539,269.6; -3.522,269.2; -3.506,269.0; -3.489,268.7; -3.473,268.5; -3.456,268.3; -3.439,268.0; -3.423,267.9; -3.406,267.7; -3.390,267.5; -3.373,267.3; -3.356,267.2; -3.340,267.0; -3.323,266.9; -3.307,266.7; -3.290,266.6; -3.273,266.4; -3.257,266.3; -3.240,266.2; -3.224,266.1; -3.207,265.9; -3.190,265.8; -3.174,265.7; -3.157,265.6; -3.141,265.5; -3.124,265.3; -3.107,265.2; -3.091,265.1; -3.074,265.0; -3.058,264.9; -3.041,264.8; -3.024,264.7; -3.008,264.6; -2.991,264.5; -2.975,264.4; -2.958,264.3; -2.941,264.2; -2.925,264.1; -2.908,264.0; -2.892,263.9; -2.875,263.8; -2.858,263.7; -2.842,263.6; -2.825,263.5; -2.808,263.4; -2.792,263.3; -2.775,263.2; -2.759,263.1; -2.742,263.0; -2.725,262.9; -2.709,262.8; -2.692,262.8; -2.676,262.7; -2.659,262.6; -2.642,262.5; -2.626,262.4; -2.609,262.3; -2.593,262.2; -2.576,262.1; -2.559,262.0; -2.543,261.9; -2.526,261.8; -2.510,261.8; -2.493,261.7; -2.476,261.6; -2.460,261.5; -2.443,261.4; -2.427,261.3; -2.410,261.2; -2.393,261.1; -2.377,261.0; -2.360,260.9; -2.344,260.8; -2.327,260.7; -2.310,260.7; -2.294,260.6; -2.277,260.5; -2.261,260.4; -2.244,260.3; -2.227,260.2; -2.211,260.1; -2.194,260.0; -2.178,259.9; -2.161,259.8; -2.144,259.7; -2.128,259.6; -2.111,259.5; -2.095,259.4; -2.078,259.3; -2.061,259.2; -2.045,259.1; -2.028,259.0; -2.012,258.8; -1.995,258.7; -1.978,258.6; -1.962,258.5; -1.945,258.4; -1.929,258.3; -1.912,258.2; -1.895,258.0; -1.879,257.9; -1.862,257.8; -1.846,257.6; -1.829,257.5; -1.812,257.4; -1.796,257.2; -1.779,257.1; -1.763,256.9; -1.746,256.8; -1.729,256.6; -1.713,256.4; -1.696,256.3; -1.680,256.1; -1.663,255.9; -1.646,255.7; -1.630,255.5; -1.613,255.2; -1.597,254.9; -1.580,254.7; -1.563,254.3; -1.547,253.9; -1.530,253.3 fold points: thaw (log10 p, T) = (-1.518, 252.2) ; freeze = (-3.590, 271.8) --- FIG 3: time series of the headline run (one cycle, fW=5) --- t_Myr,T_C,log10pCO2 (every 14th sample): 0.000,-29.3,-1.960; 0.084,-28.4,-1.884; 0.167,-27.6,-1.820; 0.251,-26.8,-1.764; 0.335,-26.1,-1.715; 0.418,-25.4,-1.671; 0.502,-24.7,-1.631; 0.586,-24.0,-1.596; 0.670,-23.3,-1.563; 0.753,-22.3,-1.533; 0.837,17.5,-2.126; 0.921,-39.1,-2.912; 1.004,-34.8,-2.475; 1.088,-32.6,-2.262; 1.172,-31.1,-2.120; 1.255,-29.9,-2.014; 1.339,-28.9,-1.928; 1.423,-28.1,-1.857; 1.506,-27.3,-1.797; 1.590,-26.5,-1.744; 1.674,-25.8,-1.697; 1.758,-25.2,-1.655; 1.841,-24.5,-1.617; 1.925,-23.7,-1.582; 2.009,-22.9,-1.551; 2.092,-21.6,-1.522; 2.176,6.6,-3.178; 2.260,-36.9,-2.684; 2.343,-33.8,-2.378; 2.427,-31.9,-2.200; 2.511,-30.6,-2.075; 2.594,-29.5,-1.978; 2.678,-28.6,-1.899; 2.762,-27.7,-1.832; 2.845,-27.0,-1.775; 2.929,-26.3,-1.724; 3.013,-25.6,-1.679; 3.097,-24.9,-1.639; 3.180,-24.2,-1.603; 3.264,-23.4,-1.569; 3.348,-22.5,-1.539; 3.431,20.6,-1.793; 3.515,-40.9,-3.103; 3.599,-35.4,-2.536; 3.682,-33.0,-2.298; 3.766,-31.4,-2.146; 3.850,-30.1,-2.033; 3.933,-29.1,-1.945; 4.017,-28.2,-1.871; 4.101,-27.4,-1.808 --- FIG 4: period vs weatherability --- fW,period_Myr,frozen_Myr,icefree_Myr: 4.0,1.3254,1.2166,0.1088 5.0,1.3058,1.2248,0.0810 6.0,1.2985,1.2332,0.0653 8.0,1.2980,1.2506,0.0474 10.0,1.3064,1.2693,0.0372 12.0,1.3197,1.2891,0.0306 16.0,1.3560,1.3333,0.0227 fW_crit = 3.59; recharge floor = 1.185 Myr --- SPINE MARKER: 11 points equally spaced by ARC LENGTH around one cycle --- section,log10pCO2,T_C,phase_frac 1,-2.751,-37.5,0.0000 2,-2.072,-30.6,0.2029 3,-1.518,-17.3,0.8921 4,-1.518,14.2,0.8921 5,-1.673,21.8,0.9005 6,-2.356,15.2,0.9265 7,-3.035,8.3,0.9421 8,-3.595,-3.6,0.9541 9,-3.594,-44.1,0.9542 10,-3.441,-44.1,0.9574 11,-2.758,-37.6,0.9992 ========================================================================== 10. KEY NUMBERS FOR THE ARTICLE ========================================================================== S_sturt 1281 W/m^2 (94.1% of modern) bistable solar window 1278-1472 W/m^2 freeze threshold 257 ppmv CO2 deglaciation threshold 0.030 bar = 30328 ppmv CO2 CO2 swing 118x critical weatherability fW* 3.59x period at fW=5 1.31 Myr temperature amplitude 69 K (-46 C to +23 C) cycles in 56 Myr at fW=5 43 frozen fraction at fW=5 94% period floor (recharge time) 1.18 Myr robustness cycles in 22% of jittered draws REMINDER: club toy model. Not the Minsky et al. box model. Numbers are illustrative of the MECHANISM, not predictions about the real Cryogenian.