Dense surface blooms of toxic cyanobacteria in eutrophic lakes may lead to mass mortalities of fish and birds, and provide a serious health threat for cattle, pets, and humans. It has been argued that global warming may increase the incidence of harmful algal blooms. Here, we report on a lake experiment where intermittent artificial mixing failed to control blooms of the harmful cyanobacterium Microcystis during the summer of 2003, one of the hottest summers ever recorded in Europe. To understand this failure, we develop a coupled biological-physical model investigating how competition for light between buoyant cyanobacteria, diatoms, and green algae in eutrophic lakes is affected by the meteorological conditions of this extreme summer heatwave. The model consists of a phytoplankton competition model coupled to a one-dimensional hydrodynamic model, driven by meteorological data. The model predicts that high temperatures favour cyanobacteria directly, through increased growth rates. Moreover, high temperatures also increase the stability of the water column, thereby reducing vertical turbulent mixing, which shifts the competitive balance in favour of buoyant cyanobacteria. Through these direct and indirect temperature effects, in combination with reduced wind speed and reduced cloudiness, summer heatwaves boost the development of harmful cyanobacterial blooms. These findings warn that climate change is likely to yield an increased threat of harmful cyanobacteria in eutrophic freshwater ecosystems.

The intriguing impact of physical mixing processes on species interactions has always fascinated ecologists. Here, we exploit recent advances in plankton models to develop competition theory that predicts how changes in turbulent mixing affect competition for light between buoyant and sinking phytoplankton species. We compared the model predictions with a lake experiment, in which the turbulence structure of the entire lake was manipulated using artificial mixing. Vertical eddy diffusivities were calculated from the measured temperature microstructure in the lake. Changes in turbulent mixing of the lake caused a dramatic shift in phytoplankton species composition, consistent with the predictions of the competition model. The buoyant and potentially toxic cyanobacterium Microcystis dominated at low turbulent diffusivity, whereas sinking diatoms and green algae dominated at high turbulent diffusivity. These findings warn that changes in the turbulence structure of natural waters, for instance driven by climate change, may induce major shifts in the species composition of phytoplankton communities.

Deep chlorophyll maxima (DCMs) are widespread in large parts of the world's oceans. These deep layers of high chlorophyll concentration reflect a compromise of phytoplankton growth exposed to two opposing resource gradients: light supplied from above and nutrients supplied from below. It is often argued that DCMs are stable features. Here we show, however, that reduced vertical mixing can generate oscillations and chaos in phytoplankton biomass and species composition of DCMs. These fluctuations are caused by a difference in the timescales of two processes: (1) rapid export of sinking plankton, withdrawing nutrients from the euphotic zone and (2) a slow upward flux of nutrients fuelling new phytoplankton production. Climate models predict that global warming will reduce vertical mixing in the oceans. Our model indicates that reduced mixing will generate more variability in DCMs, thereby enhancing variability in oceanic primary production and in carbon export into the ocean interior.

The FORTRAN program RKC is intended for the time integration of parabolic partial differential equations discretized by the method of lines. It is based on a family of Runge-Kutta-Chebyshev formulas with a stability bound that is quadratic in the number of stages. Remarkable properties of the family make it possible for the program to select at each step the most efficient stable formula as well as the most efficient step size. Moreover, they make it possible to evaluate the explicit formulas in just a few vectors of storage. These characteristics of the program make it especially attractive for problems in several spatial variables. RKC is compared to the BDF solver VODPK on two test problems in three spatial variables.

Ezplizite Runge -K u t t a -Verfahren m-ten Grades werden hergeleitet, f u r die der maximal stabile Zeitschritt pro d u swertung der rechten Seite mit m proportioniert ist, wenn diese Methoden auf semi-diskretisierte parabolisclie Anfangsrandwertprobleme angewandt werden. Dns interne Stabilitatsverhalten dieser Methoden wird mit gleichartigen, cn der Lrterntur aorgeschlagenen Runge-Kutta-Formeln verglichen. Wir zeigen sowohl mittels Analyse uiie durch numeridche ExperL-nt~nte, daJ der sn-Wert der in dieser Arbeit vorgeschlagenen Schemata nicht von internen Instabilitaten beschra,tkt i~i r d .Explicit, m-stage R u n g e -K u t t a methods are derived for which the mnximal stable integrntion step per light hnntl .$id? waluation i s proportional to m when applied to semi-discrete parabolic initial-boundary d u e problenis. Thr inirrnnl stability behaviour of these methods is compared with that of similnr R u n g e -K u t t a methods proposed in the literatior. Both by analysis and by numerical experiments we show that the value of m in the schemes proposed in this popcr 2 P not restricted by internal instabilities. mar BpeMem 3a 06pa60~1iy npaBoii w c m nponopmoHaneH c m , ecm 3TH MeTonbI npnMeHmoTcn H a nony-~H C K~~T H~O B~H H~I X napa60nariec~~x aanav c HawinLi-Imw M Kpaenmm 3navemmm. CpanimuueToi RIIYTpeHHblll xapamep Y C T Q~H B O C T I I BTUX MeTonoB c OnHoponHbmm, npennonarambma B nmepaType ~K C I I~~L I M~I I T O B , w o m -m a v e~~e cxeM npennonaraeMbIx B 3~0 i i cTaTbe He opraHmmaeTca BHyTpeHHmw BblBOnRTCII RRHbIe MeTOHbI TElna P y Ilre-Hy T T a m -O f i CTelleHLl, KOTOPblX M a I E C L l~l a~b H O -y C T O € I a r n h r i i @OpMynaMH TIIlla PyHre-)CyTTa. ~OKa3blBaeTCH C I l O M O U b H ) aHaJlkl3a KBK II C IIQMOUibM BbI~ACJlllTe.JlbIiblXHeyCTO&lHBOCTIIMEl.