Francis J. Poulin

Size-structured planktonic ecosystems: constraints, controls and assembly instructions.

Here we present a nutrient–phytoplankton–zooplankton (NPZ) model that has arbitrary size-resolution within the phytoplankton- and zooplankton-state variables. The model assumes allometric scaling of biological parameters. This particular version of the model (herbivorous zooplankton only) has analytical solutions that allow efficient exploration of the effects of allometric dependencies of various biological processes on the models equilibrium solutions. The model shows that there are constraints on the possible combinations of allometric scalings of the biological rates that will allow ecosystems to be structured as we observe (larger organisms added as the total biomass increases). The diversity (number of size classes occupied) of the ecosystem is the result of simultaneous bottom-up and top-down control: resources determine which classes can exist; predation determines which classes do exist. Thus, the simultaneous actions of bottom-up and top-down controls are essential for maintaining and structuring planktonic ecosystems. One important conclusion from this model is that there are multiple, independent ways of obtaining any given biomass spectrum, and that the spectral slope is not, in and of itself, very informative concerning the underlying dynamics. There is a clear need for improved size-resolved field measurements of biological rates; these will both elucidate biological processes in the field, and allow strong testing of size-structured models of planktonic ecosystems.

Parametric instability in oscillatory shear flows

In this article we investigate time-periodic shear flows in the context of the two-dimensional vorticity equation, which may be applied to describe certain large-scale atmospheric and oceanic flows. The linear stability analyses of both discrete and continuous profiles demonstrate that parametric instability can arise even in this simple model: the oscillations can stabilize (destabilize) an otherwise unstable (stable) shear flow, as in Mathieus equation (Stoker 1950). Nonlinear simulations of the continuous oscillatory basic state support the predictions from linear theory and, in addition, illustrate the evolution of the instability process and thereby show the structure of the vortices that emerge. The discovery of parametric instability in this model suggests that this mechanism can occur in geophysical shear flows and provides an additional means through which turbulent mixing can be generated in large-scale flows.

The Influence of Topography on the Stability of Jets

In this article, the effect shelflike topography has on the stability of a jet that flows along the smooth shelf is addressed. The linear stability problem is solved to determine for which nondimensional parameters a shelf can either destabilize or stabilize a jet. These calculations reveal an intricate dependence of growth rate on topography. In particular, the authors determine that retrograde topography (with the shallow water on the left) always stabilizes the jet (in relation to the flat-bottom equivalent), whereas prograde topography (with the shallow water on the right) can either stabilize or destabilize the jet depending on the particular values of the Rossby number and topographic parameters. For Rossby numbers of order 1 and larger, prograde topography is strictly stabilizing. For small Rossby numbers, small-amplitude topography destabilizes whereas large topography stabilizes. The nonlinear evolution of these instabilities is explored to confirm the predictions from the linear theory and, also, to illustrate how stabilization is directly related to fluid transport across the shelf.

Approximating intrinsic noise in continuous multispecies models

In small-scale chemical reaction networks, the local density of molecules is changed by discrete jumps owing to reactive collisions, and through transport. A systematic perturbation scheme is developed to analytically characterize these non-equilibrium intrinsic fluctuations in a multispecies spatially varying system. The method is illustrated on a variety of model systems. In all cases, the continuous approximation method is corroborated with extensive stochastic simulation. As an example of our technique applied to a spatially varying steady state, we demonstrate that a model for embryonic patterning mediated by regulatory mRNA is surprisingly robust to intrinsic fluctuations.

An NPZ Model with State-Dependent Delay Due to Size-Structure in Juvenile Zooplankton

The study of planktonic ecosystems is important as they make up the bottom trophic levels of aquatic food webs. We study a closed nutrient-phytoplankton-zooplankton (NPZ) model that includes size structure in the juvenile zooplankton. The closed nature of the system allows the formulation of a conservation law of biomass that governs the system. The model consists of a system of a nonlinear ordinary differential equation coupled to a partial differential equation. We are able to transform this system into one of delay differential equations where the delay is of threshold type and is state dependent. The system of delay differential equations can be further transformed into one with fixed delay. Using the different forms of the model, we perform a qualitative analysis of the solutions, which includes studying existence and uniqueness, positivity and boundedness, local and global stability, and conditions for extinction. Key parameters that are explored are the total biomass in the system and the maturity level at which the juvenile zooplankton reach maturity. Numerical simulations are also performed to verify our analytical results.

The stochastic Mathieu's equation

In this manuscript, we consider generalizations of the classical Mathieus equation to stochastic systems. Unlike previous works, we focus on internal frequencies that vary continuously between periodic and stochastic variables. By numerically integrating the system of equations using a symplectic method, we determine the Lyapunov exponents for a wide range of parameters to quantify how the growth rates vary in parameter space. In the nearly periodic limit, we recover the same growth rates as the classical Mathieus equation. As the stochasticity increases, the maximum growth decreases and the unstable region broadens, which signifies that the stochasticity can both stabilize and destabilize a system in relation to the deterministic limit. In addition to the classical parametric modes, there is also an unstable stochastic mode that arises only for moderate stochasticity. The power spectrum of the solutions shows that an increase in stochasticity tends to narrow the width of the subharmonic peak and increase the decay away from this peak.

On fully nonlinear, vertically trapped wave packets in a stratified fluid on the f-plane

The ubiquity of solitary and solitarylike internal waves in the coastal ocean has been recognized for some time. Recent theoretical studies of a strongly nonlinear, weakly nonhydrostatic set of layer-averaged model equations have predicted that rotation, for example, on the f-plane, can lead to the decay and subsequent reemergence of internal solitary waves. We reconsider this problem using high resolution numerical simulations of the rotating stratified Euler equations. We find that in certain cases the initial disturbances indeed fission into nonlinear wave packets, with the constituent waves making up the wave packet being, in themselves, nonlinear. However, for typical coastal ocean parameters this only occurs at rotation rates higher than those on Earth on the time scales we are able to simulate. We confirm, using the Dubreil–Jacotin–Long equation, that the vertical structure of the wave-induced currents is well predicted by the fully nonlinear theory of nonrotating internal solitary waves and that w...

Asymptotic Analysis of the Forced Internal Gravity Wave Equation

In this paper an asymptotic analysis of impulsively started, forced linear internal gravity waves is presented. This investigation describes horizontally propagating waves for fluids with constant ...

A closed NPZ model with delayed nutrient recycling

We consider a closed Nutrient-Phytoplankton-Zooplankton (NPZ) model that allows for a delay in the nutrient recycling. A delay-dependent conservation law allows us to quantify the total biomass in the system. With this, we can investigate how a planktonic ecosystem is affected by the quantity of biomass it contains and by the properties of the delay distribution. The quantity of biomass and the length of the delay play a significant role in determining the existence of equilibrium solutions, since a sufficiently small amount of biomass or a long enough delay can lead to the extinction of a species. Furthermore, the quantity of biomass and length of delay are important since a small change in either can change the stability of an equilibrium solution. We explore these effects for a variety of delay distributions using both analytical and numerical techniques, and verify these results with simulations.

Realizing Surface-Driven Flows in the Primitive Equations

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