========================================================================== OCEAN DARKENING -- CLUB LIGHT-ATTENUATION AND LOST-VOLUME MODEL Anchor: Davies, T. W. & Smyth, T. (2025). Darkening of the Global Ocean. Global Change Biology 31(5), e70227. DOI: 10.1111/gcb.70227 This is the club's own simplified model, NOT the paper's analysis. numpy: yes, v2.4.2 | seed: 20250527 ========================================================================== PART 0 -- WHAT THE PAPER REPORTS, IN OUR UNITS -------------------------------------------------------------------------- Kd(490) increased over 75,341,181 km2 (20.82% of ocean) Kd(490) decreased over 37,269,515 km2 (10.30% of ocean) net area darkened minus lit 38,071,666 km2 ratio darkened : brightened 2.02 : 1 photic depth shoaled >10 m 68,402,842 km2 (18.90%) photic depth shoaled >50 m 32,449,129 km2 (8.97%) photic depth shoaled >100 m 9,392,219 km2 (2.60%) For scale: the darkened area is 4.41 times the land area of Russia and 0.51 times the total land area of Earth (148.9 Mkm2). PART A -- BEER-LAMBERT: HOW DEEP IS THE 1% LIGHT LEVEL? -------------------------------------------------------------------------- I(z) = I0 * exp(-Kd * z) -> z_1% = ln(100)/Kd = 4.60517/Kd Kd(490) water type z_1% z_10% z_0.1% (1/m) (m) (m) (m) ------------------------------------------------------------------ 0.020 clearest subtropical gyre 230.3 115.1 345.4 0.030 clear oligotrophic open ocean 153.5 76.8 230.3 0.040 open ocean, low chlorophyll 115.1 57.6 172.7 0.060 open ocean, global average-ish 76.8 38.4 115.1 0.080 productive open ocean 57.6 28.8 86.3 0.120 shelf edge 38.4 19.2 57.6 0.200 coastal, moderate 23.0 11.5 34.5 0.350 coastal bloom 13.2 6.6 19.7 0.600 turbid nearshore 7.7 3.8 11.5 1.000 river plume 4.6 2.3 6.9 Light remaining at fixed depths, as a percentage of the surface: Kd(490) 10 m 25 m 50 m 75 m 100 m 150 m 200 m ------------------------------------------------------------------------ 0.020 81.873% 60.653% 36.788% 22.313% 13.534% 4.979% 1.832% 0.030 74.082% 47.237% 22.313% 10.540% 4.979% 1.111% 0.248% 0.040 67.032% 36.788% 13.534% 4.979% 1.832% 0.248% 0.034% 0.060 54.881% 22.313% 4.979% 1.111% 0.248% 0.012% 0.001% 0.080 44.933% 13.534% 1.832% 0.248% 0.034% 0.001% 0.000% 0.120 30.119% 4.979% 0.248% 0.012% 0.001% 0.000% 0.000% 0.200 13.534% 0.674% 0.005% 0.000% 0.000% 0.000% 0.000% 0.350 3.020% 0.016% 0.000% 0.000% 0.000% 0.000% 0.000% 0.600 0.248% 0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 1.000 0.005% 0.000% 0.000% 0.000% 0.000% 0.000% 0.000% PART B -- THE NONLINEARITY: METRES LOST PER UNIT OF ADDED MURK -------------------------------------------------------------------------- dz/dKd = -ln(100)/Kd^2. The cost of added murk scales as 1/Kd^2. Kd start z_1% start +0.005 1/m +0.010 1/m +10% relative (1/m) (m) lost (m) lost (m) lost (m) -------------------------------------------------------------------- 0.020 230.3 46.05 76.75 20.93 0.030 153.5 21.93 38.38 13.96 0.040 115.1 12.79 23.03 10.47 0.060 76.8 5.90 10.96 6.98 0.080 57.6 3.39 6.40 5.23 0.120 38.4 1.54 2.95 3.49 0.200 23.0 0.56 1.10 2.09 0.350 13.2 0.19 0.37 1.20 0.600 7.7 0.06 0.13 0.70 1.000 4.6 0.02 0.05 0.42 The same +0.005 1/m of added murk costs the clearest gyre 46.1 metres and the river plume 0.0229 metres. Ratio: 2010 to 1. But in RELATIVE terms the nonlinearity vanishes completely. A 10% rise in Kd removes exactly 9.0909% of the euphotic depth at every single value of Kd. Check, across five orders of magnitude: Kd = 0.001 z_1% 4605.170 m -> 4186.518 m loss 9.0909% Kd = 0.010 z_1% 460.517 m -> 418.652 m loss 9.0909% Kd = 0.100 z_1% 46.052 m -> 41.865 m loss 9.0909% Kd = 1.000 z_1% 4.605 m -> 4.187 m loss 9.0909% Kd = 10.000 z_1% 0.461 m -> 0.419 m loss 9.0909% So 'the ocean lost 9% of its lit depth' and 'the ocean lost 50 m of lit depth' are statements about very different water. PART C -- SECCHI DISK LOOKUP TABLE (bring your own rope) -------------------------------------------------------------------------- Kd = 1.7 / Z_secchi -> z_1% = ln(100)*Z_secchi/1.7 = 2.7089 * Z_secchi Secchi Kd(490) z_1% z_10% loses this many m depth (1/m) (m) (m) per 10% more murk -------------------------------------------------------------- 0.5 m 3.4000 1.4 0.7 0.12 1.0 m 1.7000 2.7 1.4 0.25 2.0 m 0.8500 5.4 2.7 0.49 3.0 m 0.5667 8.1 4.1 0.74 5.0 m 0.3400 13.5 6.8 1.23 7.0 m 0.2429 19.0 9.5 1.72 10.0 m 0.1700 27.1 13.5 2.46 15.0 m 0.1133 40.6 20.3 3.69 20.0 m 0.0850 54.2 27.1 4.93 25.0 m 0.0680 67.7 33.9 6.16 30.0 m 0.0567 81.3 40.6 7.39 35.0 m 0.0486 94.8 47.4 8.62 40.0 m 0.0425 108.4 54.2 9.85 50.0 m 0.0340 135.4 67.7 12.31 Rule of thumb worth memorising: euphotic depth is about 2.71 times whatever your Secchi reading was. Round it to 2.7. Sensitivity to the conversion constant (Z_secchi = 20 m): constant 1.40 (Devlin 2008 ) -> Kd = 0.0700 1/m, z_1% = 65.8 m constant 1.44 (Holmes 1970 ) -> Kd = 0.0720 1/m, z_1% = 64.0 m constant 1.70 (Poole & Atkins 1929 ) -> Kd = 0.0850 1/m, z_1% = 54.2 m constant 2.00 (Murray & Markager 2011) -> Kd = 0.1000 1/m, z_1% = 46.1 m Spread across published constants: 46.1 to 65.8 m, a factor of 1.43. This is why a Secchi disk is a screening tool and not a sensor. PART D -- LOST LIT VOLUME FROM THE PAPER'S REPORTED AREAS -------------------------------------------------------------------------- Un-nesting the reported bands into disjoint bands: band area (km2) rep. shoaling lost volume (km3) ---------------------------------------------------------------------- shoaled 10-50 m 35,953,713 30.0 m 1,078,611 shoaled 50-100 m 23,056,910 75.0 m 1,729,268 shoaled >100 m 9,392,219 100.0 m 939,222 ---------------------------------------------------------------------- TOTAL 68,402,842 3,747,102 Central estimate of lit habitat lost, 2003 to 2022: 3747102 km3 ( 3.747 million km3 ) As a fraction of things: of total ocean volume (1.335e+09 km3) 0.2807 % of the epipelagic 0-200 m (7.238e+07 km3) 5.177 % smeared evenly over the whole ocean: a layer 10.35 m thick smeared over just the darkened 21%: a layer 49.74 m thick Rate, treating the change as linear over 19 elapsed years: 197216 km3 per year 1972159 km3 per decade 0.1477 % of ocean volume per decade 540 km3 per day Comparisons, because km3 means nothing to anybody: Lake Superior holds 12,100 km3. We lost 310 Lake Superiors. That is 16.3 Lake Superiors of lit water per year. Mediterranean Sea volume is about 3,750,000 km3; our estimate is 1.00 Mediterraneans. PART D2 -- MONTE CARLO ON THE REPRESENTATIVE DEPTHS -------------------------------------------------------------------------- The only free choice above is where inside each band to put the representative shoaling. Sample it uniformly, 200,000 seeded draws. The top band is open-ended; we cap the sampler at 150 m, which is a guess, and it is the largest single source of spread in the answer. draws 200,000 mean 3,981,544 km3 median 3,981,096 km3 68% interval 3,403,407 to 4,562,414 km3 95% interval 2,942,672 to 5,022,867 km3 our central estimate sits at the 34.5th percentile median as % of ocean volume 0.2982 % median per decade 2,095,314 km3 PART E -- WHAT Kd CHANGE WOULD PRODUCE A 50 m SHOALING? -------------------------------------------------------------------------- Inverting Beer-Lambert at the 1% level. If a patch of water starts at Kd0 and loses exactly 50 m of euphotic depth, the new Kd is Kd1 = ln(100) / ( ln(100)/Kd0 - 50 ) and this is only possible at all if z_1%(Kd0) > 50 m, i.e. Kd0 < 0.09210 1/m. Kd0 z_1% (m) Kd1 needed change in Kd relative change -------------------------------------------------------------------- 0.020 230.3 0.0255 +0.0055 +27.7 % 0.025 184.2 0.0343 +0.0093 +37.3 % 0.030 153.5 0.0445 +0.0145 +48.3 % 0.035 131.6 0.0565 +0.0215 +61.3 % 0.040 115.1 0.0707 +0.0307 +76.8 % 0.050 92.1 0.1094 +0.0594 +118.8 % 0.060 76.8 0.1721 +0.1121 +186.9 % 0.070 65.8 0.2917 +0.2217 +316.7 % 0.080 57.6 0.6088 +0.5288 +661.0 % Read that column of percentages. In the clearest water a 50 m loss needs a Kd rise of about 28%. In water at Kd = 0.08 it needs 661%, which no satellite would call a subtle trend. NOTE: the paper's >50 m shoaling figures are computed against the Calanus light threshold, which sits far below the 1% level, so the Kd changes THEY imply are smaller than this table. This table is a 1%-convention exercise and should be read as such. ========================================================================== HEADLINE NUMBERS USED IN THE ARTICLE ========================================================================== euphotic depth = 2.709 x Secchi depth clearest gyre loses 46.1 m per +0.005 1/m; river plume loses 0.0229 m ratio of those two = 2010 : 1 a 10% rise in Kd always costs 9.09% of euphotic depth lit volume lost (central) = 3,747,102 km3 lit volume lost (MC median) = 3,981,096 km3 68% interval = 3,403,407 to 4,562,414 km3 = 0.2807% of ocean volume, 5.18% of the epipelagic = a layer 10.35 m thick over the entire ocean = 1,972,159 km3 per decade = 310 Lake Superiors, or 16.3 per year ==========================================================================