Climate change in a conceptual atmosphere–phytoplankton model

We develop a conceptual coupled atmosphere–phytoplankton model by combining the Lorenz’84 general circulation model and the logistic population growth model under the condition of a climate change due to a linear time dependence of the strength of anthropogenic atmospheric forcing. The following types of couplings are taken into account: (a) the temperature modifies the total biomass of phytoplankton via the carrying capacity; (b) the extraction of carbon dioxide by phytoplankton slows down the speed of climate change; (c) the strength of mixing/turbulence in the oceanic mixing layer is in correlation with phytoplankton productivity. We carry out an ensemble approach (in the spirit of the theory of snapshot attractors) and concentrate on the trends of the average phytoplankton concentration and average temperature contrast between the pole and Equator, forcing the atmospheric dynamics. The effect of turbulence is found to have the strongest influence on these trends. Our results show that when mixing has sufficiently strong coupling to production, mixing is able to force the typical phytoplankton concentration to always decay globally in time and the temperature contrast to decrease faster than what follows from direct anthropogenic influences. Simple relations found for the trends without this coupling do, however, remain valid; just the coefficients become dependent on the strength of coupling with oceanic mixing. In particular, the phytoplankton concentration and its coupling to climate are found to modify the trend of global warming and are able to make it stronger than what it would be without biomass.


S1: Analytic results without mixing
We derive analytic results for the long-term concentration and temperature contrast behavior when mixing is negligible (γ = 0). Numbers in (brackets) refer to the equations of the main part paper, letter S and numbers refer to equations on this Supplamantary Material.

Without seasonality
Let us consider equation (2) and (9) with γ = 0 and A = 0 leading tȯ This equation is found to possess an asymptotically linear behavior With this form, from (4) and (8), we find where is the slope of the time evolution of the temperature contrast. This shows that the temperature contrast, and in particular, the strength of the climate change, becomes influenced by the phytoplankton concentration.
In order to specify the unknown constants S and δ, let us rewrite (S1) as For long times, i.e. t 1/r, the quadratic term dominates on the right hand side which cannot be compensated by anything on the left hand side. The coefficient of the quadratic term should vanish, i.e. αD = S from which, since D = D 0 − βS, as stated in (11,12). Assuming that the linear form is valid not only for very large times, but for intermediate ones, too, the linear terms on both sides should compensate each other. From (S5), this yields After dividing by S and using that αD = S, we find This result provides the constant δ in the long-term dynamics (S2) of phytoplankton.

With seasonality
Let us add now periodic forcing in the form of (4) with (8), yielding We expect the phytoplankton concentration to be driven to oscillate with the same frequency ω but with some phase shift φ. For long times, t 1/r, we assume that the snapshot attractor is of the form with an amplitude B. The corresponding long term behavior of the carrying capacity is In order to fix the constants, we substitute these into (2) to find The vanishing of the quadratic term in t provides the same equation as without periodic forcing, therefore (S6) and (S7) turn out to remain valid. Similarly, from the linear terms (S10) is recovered.
From the product of trigonometric and linear terms, the following equation follows: After an application of trigonometric identities, from the vanishing of the coefficient of t cos(ωt) we find . (S15) From the vanishing of the coefficient of t sin(ωt) from which the concentration amplitude is found to be We have thus been able to determine analytically the snapshot attractor c * (t) in a strongly nonlinear model with a linear drift and periodic forcing. The snapshot attractor in the concentration remains point-like (no internal variability in c), but changes in an oscillatory fashion about a linear growth with an amplitude B determined by all the system parameters.   Temporal slope of temperature contrast in F 0 (t) due to anthropogenic origins c(t) Time-dependent phytoplankton concentration, see Eq.
(2) S Temporal slope of phytoplankton concentration c(t), see Eq. (10) Time-dependent carrying capacity, see Eq. (9) Index () r refers to values in the reference state without anthropogenic effects. Bracket <> indicates ensemble average.