Coral Polyp Growth in 2D

What you are looking at

One coral colony grows from a small nubbin on a reef-rock plate. Every white-to-brown blob is a 1 cm block of aragonite skeleton and every tentacled glyph is a patch of living polyps on an exposed face. Colour is the live state: golden-brown when the symbiotic algae (zooxanthellae) are healthy, white when they are lost (bleaching), green-grey when dead skeleton is overgrown by algae. Pale tips are the freshest, fastest-growing skeleton. The enlarged polyp cutaway is an illustration of the anatomy and follows the live symbiont and calcification state.

1. Ocean chemistry: aragonite saturation

Coral skeleton is aragonite, CaCO₃. Whether seawater tends to build or dissolve it is the saturation state Ω = [Ca2+][CO32-] / Ksp. The page solves the full carbonate system from your pCO₂, total alkalinity, salinity and temperature: CO2 = K0 pCO2, [HCO3-] = K1CO2/[H+], [CO32-] = K1K2CO2/[H+]2, and finds [H⁺] by bisection so that TA = [HCO3-] + 2[CO32-] + [B(OH)4-] + [OH-] - [H+]. Constants: K₁, K₂ from Lueker et al. (2000), K₀ from Weiss (1974), aragonite K_sp from Mucci (1983), boric acid from Dickson (1990), water from Millero (1995), all on the total pH scale. More CO₂ lowers pH and carbonate ion, so Ω falls; this is ocean acidification.

Inside the polyp the calcifying fluid is not seawater. Calicoblastic cells raise its pH (here by +0.5, literature +0.3 to +0.9) and concentrate carbon (here x1.3), so Ω there is 5 to 20. The readout applies the same equilibrium to that fluid. Acidification does not stop this machinery but makes it cost more energy.

2. Calcification rate

Empirically calcification rises steeply with saturation: G ∝ (Ω - 1)1.5 (Langdon & Atkinson 2005 and many later experiments report exponents of 1 to 2), multiplied by a temperature curve peaking about 1 C below the warmest monthly mean (MMM) and falling quickly above it, by a light factor 0.28 + 0.72 Z tanh(I/300) (light-enhanced calcification needs symbionts Z) and by the living tissue fraction M. It is normalised to 100 % at today’s tropical surface water (420 uatm, 27.5 C). Below Ω = 1 the skeleton would dissolve; the model then stops growth (it does not erode existing skeleton).

Growth is a mass balance: cells added per day = G_ref x relative rate x exposed living area / CaCO₃ per cell, with G_ref = 0.35 umol CaCO₃ cm^-2 h^-1 (whole-colony daily mean) and 1.2 g cm^-3 bulk skeleton density, so one 1 cm cell holds 12 mmol.

3. Heat stress and bleaching

Sea temperature is a seasonal cosine plus your warming and any marine heatwave. As in NOAA Coral Reef Watch, HotSpot = SST - MMM and DHW = (1/7) ∑ HotSpot over the last 84 days, counting only days with HotSpot ≥ 1 C (unit: degree C-weeks). DHW ≥ 4 is the usual threshold for significant bleaching and ≥ 8 for severe bleaching and mortality. Symbionts follow dZ/dt = -kb Z + r Z(1 - Z) with kb = 0.03 (DHWeff/4)1.2 per day; recovery r = 0.02 per day happens only when HotSpot < 1 C. DHW_eff is reduced by water motion (flushing of reactive oxygen) and by shade at depth. If symbionts fall below 25 % the tissue starves with a time constant of 25 days plus up to 90 days of plankton feeding. These rates are phenomenological, tuned so that DHW 4 halves the symbionts in about 3 weeks; real thresholds differ between species and reefs.

4. Branching morphology: diffusion-limited growth

The colony’s shape comes from a dielectric-breakdown model (Niemeyer, Pietronero and Wiesmann 1984; Kaandorp’s coral models). Dissolved carbon and nutrients diffuse from the open water toward the colony, which consumes them: the page solves the Laplace equation ∇2c = 0 by successive over-relaxation with c = 1 on the open boundaries and c = 0 on the skeleton. A surface site is chosen to grow with probability ∝ cη exp(phototropism · height). Protruding tips see more supply and grow faster, which gives branches. η = 1 is classic diffusion-limited aggregation (2D fractal dimension about 1.7), η = 0 is compact Eden growth (dimension 2), larger η gives spikier colonies. In nature thinner boundary layers in strong flow favour sturdier, more compact forms (Kaandorp et al.), which is why the current preset uses a low η.

What to try

1. Press the 2100 preset: Ω drops from about 3.5 to about 2.1 and calcification to about 28 % of today’s; the colony grows slower and stays smaller.
2. Trigger a heatwave in the first summer, then watch HotSpot, DHW and the colour change. Try again with feeding and strong flow.
3. Set warming to +3 C: bleaching every summer, no time to recover, the colony dies and algae take over.
4. Move η from 0 to 2 on a new colony and compare compact blobs with open branches; on the 2D page switch on the supply field to see why tips win.
5. Deepen the colony to 25 m: less light, less calcification, less bleaching.

Honest simplifications

The lattice has 1 cm cells and no real branching mechanics; polyp glyphs are patches, not individual polyps. One symbiont type, one tissue pool and one set of rates stand for a whole species. The carbonate system ignores pressure, nutrients and the pH-scale offsets of a few hundredths. Calcification and bleaching parameters are calibrated, not fitted to a data set. Reef fish are decoration whose number follows the colony’s health. Five checks run live: the chemistry against benchmark seawater and against an independent inverse solution, degree-heating-week arithmetic, the exact logistic solution for symbiont recovery, and the fractal dimension of the growth rule.