In winter the sea is stirred to great depth, so plankton drift out of the light. In spring the surface warms and stirring stops at a shallow depth: suddenly the light is enough, nutrients are plentiful, and the plankton bloom until nutrients run out or grazers catch up. Change the mixing, light, water clarity and grazing and watch the year unfold.
-peak chlorophyll (mg/m3) and its date
-bloom visible (above 30 % of the peak)
-mixed layer first shallower than critical depth
-annual mean chlorophyll (mg/m3)
-mixed layer depth today
-critical depth today
-surface light today / mean in the mixed layer (W/m2)
-what limits growth today
The science
Light. The daily mean sunlight at the top of the atmosphere is Q = (S0/pi)(h0 sin(lat) sin(decl) + cos(lat) cos(decl) sin h0). We keep 75 % for clouds and air, 43 % of it as photosynthetically active light, and 94 % after reflection from the sea. Under water the light fades as I(z) = I0 exp(-k z), with k = k_water + k_c P: plankton shade themselves.
Sverdrup's critical depth. Plankton in a stirred layer of depth h see the average light I_mean = I0 (1 - exp(-k h)) / (k h). A population can grow only while the mean light exceeds a compensation light Ic, where growth equals losses. The critical depth Zcr is the depth at which they just balance, the root of I0 (1 - exp(-k Zcr)) / (k Zcr) = Ic. If the mixed layer is deeper than Zcr, cells spend too long in the dark and the bloom cannot start.
The living model. Three boxes in the mixed layer: nutrient N, phytoplankton P and zooplankton Z, in nitrogen units. dP/dt = mu P - g(P) Z - loss P - exchange P with mu = mu_max N/(K + N) I_mean/(I_mean + I_k), grazing g = g_max (1 - exp(-lambda P)), dZ/dt = 0.7 g Z - m Z^2 - exchange Z, and nutrient taken up by growth and returned by recycling. When the layer deepens, deep water with nutrient N0 and no plankton mixes in (dilution); when it shoals nothing is lost. Integrated with fourth-order Runge-Kutta, with two years of spin-up.
Why the bloom ends. Nutrients are used up and the grazers, which respond with a lag, catch up; in autumn deepening mixing dilutes and darkens the layer again. Chlorophyll is taken as 1.6 mg per mmol N.
Honest limits. A one-box model with a prescribed mixed layer cycle: no horizontal movement, no temperature effect, one phytoplankton and one grazer type, no iron or silica limit, no sinking and no vertical structure inside the layer (so no deep chlorophyll maximum). Respiration and mortality are one rate. The average-light growth rule is an approximation. Every coefficient is a typical, order-of-magnitude value, so use the model for trends and timing, not absolute numbers, and check any value against the literature before quoting it. In the 3D picture plankton are drawn thousands of times too large, and the depth scale is 1 unit = 5 m.