B.P. Sommeijer

CHANGES IN TURBULENT MIXING SHIFT COMPETITION FOR LIGHT BETWEEN PHYTOPLANKTON SPECIES

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.

Explicit Runge-Kutta (-Nystro¨m) methods with reduced phase errors for computing oscillating solutions

We construct explicit Runge–Kutta (–Nystrom) methods for the integration of first (and second) order differential equations having an oscillatory solution. Special attention is paid to the phase errors (or dispersion) of the dominant components in the numerical oscillations when these methods are applied to a linear, homogeneous test model. RK(N) methods are constructed which are dispersive of orders up to 10, whereas the (algebraic) order of accuracy is only 2 or 3. Application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced.

How Do Sinking Phytoplankton Species Manage to Persist

Phytoplankton require light for photosynthesis. Yet, most phytoplankton species are heavier than water and therefore sink. How can these sinking species persist? Somehow, the answer should lie in the turbulent motion that redisperses sinking phytoplankton over the vertical water column. Here, we show, using a reaction‐advection‐diffusion equation of light‐limited phytoplankton, that there is a turbulence window sustaining sinking phytoplankton species in deep waters. If turbulent diffusion is too high, phytoplankton are mixed to great depths, and the depth‐averaged light conditions are too low to allow net positive population growth. Conversely, if turbulent diffusion is too low, sinking phytoplankton populations end up at the ocean floor and succumb in the dark. At intermediate levels of turbulent diffusion, however, phytoplankton populations can outgrow both mixing rates and sinking rates. In this way, the reproducing population as a whole can maintain a position in the well‐lit zone near the top of the water column, even if all individuals within the population have a tendency to sink. This theory unites earlier classic results by Sverdrup and Riley as well as our own recent findings and provides a new conceptual framework for the understanding of phytoplankton dynamics under the influence of mixing processes.

Parallel iteration of high-order Runge-Kutta methods with stepsize control

Abstract This paper investigates iterated Runge-Kutta methods of high order designed in such a way that the right-hand side evaluations can be computed in parallel. Using stepsize control based on embedded formulas a highly efficient code is developed. On parallel computers, the 8th-order mode of this code is more efficient than the DOPR18 implementation of the formulas of Prince and Dormand. The 10th-order mode is about twice as cheap for comparable accuracies.

On the Internal Stability of Explicit, m‐Stage Runge‐Kutta Methods for Large m‐Values

Explicit, m-stage Runge-Kutta methods are derived for which the maximal stable integration step per right hand side evaluation is proportional to m when applied to semi-discrete parabolic initial-boundary value problems. The internal stability behaviour of these methods is compared with that of similar Runge-Kutta methods proposed in the literature. Both by analysis and by numerical experiments we show that the value of m in the schemes proposed in this paper is not restricted by internal instabilities.

Diagonally implicit Runge-Kutta-Nystro¨m methods for oscillatory problems

Implicit Runge–Kutta–Nystrom (RKN) methods are constructed for the integration of second-order differential equations possessing an oscillatory solution. Based on a linear homogeneous test model we analyse the phase errors (or dispersion) introduced by these methods and derive so-called dispersion relations. Diagonally implicit RKN methods of relatively low algebraic order are constructed, which have a high order of dispersion (up to 10). Application of these methods to a number of test examples (linear as well as nonlinear) yields a greatly reduced phase error when compared with “conventional” DIRKN methods.

Stability of collocation-based Runge-Kutta-Nystro¨m methods

We analyse the attainable order and the stability of Runge-Kutta-Nyström (RKN) methods for special second-order initial-value problems derived by collocation techniques. Like collocation methods for first-order equations the step point order ofs-stage methods can be raised to 2s for alls. The attainable stage order is one higher and equalss+1. However, the stability results derived in this paper show that we have to pay a high price for the increased stage order.

Population dynamics of sinking phytoplankton in light-limited environments: simulation techniques and critical parameters

Abstract Phytoplankton use light for photosynthesis, and the light flux decreases with depth. As a result of this simple light dependence, reaction-advection-diffusion models describing the dynamics of phytoplankton species contain an integral over depth. That is, models that simulate phytoplankton dynamics in relation to mixing processes generally have the form of an integro-partial differential equation (integro-PDE). Integro-PDEs are computationally more demanding than standard PDEs. Here, we outline a reliable and efficient technique for numerical simulation of integro-PDEs. The simulation technique is illustrated by several examples on the population dynamics of sinking phytoplankton, using both single-species models and competition models with several phytoplankton species. Our results confirm recent findings that Sverdrups critical-depth theory breaks down if turbulent mixing is reduced below a critical turbulence. In fact, our results show that suitable conditions for bloom development of sinking phytoplankton depend on a number of critical parameters, including a minimal depth of the thermocline, a maximal depth of the thermocline, a minimal turbulence, and a maximal turbulence. We therefore conclude that models that do not carefully consider the population dynamics of phytoplankton in relation to the turbulence structure of the water column may easily lead to erroneous predictions.

Stability in linear multistep methods for pure delay equations

Abstract The stability regions of linear multistep methods for pure delay equations are compared with the stability region of the delay equation itself. A criterion is derived stating when the numerical stability region contains the analytical stability region. This criterion yields an upper bound for the integration step (conditional Q-stability). These bounds are computed for the Adams-Bashforth, Adams-Moulton and backward differentiation methods of orders ⩽8. Furthermore, symmetric Adams methods are considered which are shown to be unconditionally Q-stable. Finally, the extended backward differentiation methods of Cash are analysed.

A-stable parallel block methods for ordinary and integro-differential equations

Abstract In this paper we study the stability of a class of block methods which are suitable for integrating ordinary and integro-differential equations on parallel computers. A-stable methods of orders 3 and 4 and A(α)-stable methods with α > 89.9° of order 5 are constructed. On multiprocessor computers these methods are of the same computational complexity as implicit linear multistep methods on one-processor computers.