Is the diatom sex clock a clock?

The unique life cycle of diatoms with continuous decreasing and restoration of the cell size leads to periodic fluctuations in cell size distribution and has been regarded as a multi-annual clock. To understand the long-term behaviour of a population analytically, generic mathematical models are investigated algebraically and numerically for their capability to describe periodic oscillations. Whereas the generally accepted simple concepts for the proliferation dynamics do not sustain oscillating behaviour owing to broadening of the size distribution, simulations show that a proposed limited lifetime of a newly synthesized cell wall slows down the relaxation towards a time-invariant equilibrium state to the order of a hundred thousand generations. In combination with seasonal perturbation events, the proliferation scheme with limited lifetime is able to explain long-lasting rhythms that are characteristic for diatom population dynamics. The life cycle thus resembles a pendulum clock that has to be wound up from time to time by seasonal perturbations rather than an oscillator represented by a limit cycle.

Modelling of sediment transport and morphological evolution under the combined action of waves and currents

Coastal defence structures are often constructed to prevent beach erosion. However, poorly designed structures may cause serious erosion problems in the downdrift direction. Morphological models are useful tools to predict such impacts and assess the efficiency of defence structures for different scenarios. Nevertheless, morphological modelling is still a topic under intense research effort. The processes simulated by a morphological model depend on model complexity. For instance, undertow currents are neglected in coastal area models (2DH), which is a limitation for simulating the evolution of beach profiles for long periods. Model limitations are generally overcome by predefining invariant equilibrium profiles that are allowed to shift offshore or onshore. A more flexible approach is described in this paper, which can be generalised to 3-D models. The present work is based on the coupling of the MOHID modelling system and the SWAN wave model. The impacts of different designs of detached breakwaters and groynes were simulated in a schematic beach configuration following a 2DH approach. The results of bathymetry evolution are in agreement with the patterns found in the literature for several existing structures. The model was also tested in a 3-D test case to simulate the formation of sandbars by undertow currents. The findings of this work confirmed the applicability of the MOHID modelling system to study sediment transport and morphological changes in coastal zones under the combined action of waves and currents. The same modelling methodology was applied to a coastal zone (Costa da Caparica) located at the mouth of a mesotidal estuary (Tagus Estuary, Portugal) to evaluate the hydrodynamics and sediment transport both in calm water conditions and during events of highly energetic waves. The MOHID code is available in the GitHub repository.

Modelling of sediment transport and morphological evolution under the combined action of waves and currents

Coastal defence structures are often constructed to prevent beach erosion. However, poorly designed structures may cause serious erosion problems in the downdrift direction. Morphological models are useful tools to predict such impacts and assess the efficiency of defence structures for different scenarios. Nevertheless, morphological modelling is still a topic under intense research effort. The processes simulated by a morphological model depend on model complexity. For instance, undertow currents are neglected in coastal area models (2DH), which is a limitation for simulating the evolution of beach profiles for long periods. Model limitations are generally overcome by predefining invariant equilibrium profiles that are allowed to shift offshore or onshore. A more flexible approach is described in this paper, which can be generalised to 3D models. The present work is based on the coupling of the MOHID modelling system and SWAN wave model. The impacts of different designs of detached breakwaters and groynes were simulated in a schematic beach configuration following a 2DH approach. The results of bathymetry evolution are in agreement with the patterns found in the literature for several existing structures. The model was also tested in a 3D test case to simulate the formation of sandbars by undertow currents. The findings of this work confirmed the applicability of the MOHID modelling system to study sediment transport and morphological changes in coastal zones under the combined action of waves and currents. The same modelling methodology was applied to a coastal zone (Costa da Caparica) located at the mouth of a mesotidal estuary (Tagus Estuary, Portugal) to evaluate the hydrodynamics and sediment transport in calm water conditions and during events of highly energetic waves.

Ergodic properties of matrix equilibrium states

Given a finite irreducible set of real$d\times d$matrices$A_{1},\ldots ,A_{M}$and a real parameter$s>0$, there exists a unique shift-invariant equilibrium state on$\{1,\ldots ,M\}^{\mathbb{N}}$associated to$(A_{1},\ldots ,A_{M},s)$. In this paper we characterize the ergodic properties of such equilibrium states in terms of the algebraic properties of the semigroup generated by the associated matrices. We completely characterize when the equilibrium state has zero entropy, when it gives distinct Lyapunov exponents to the natural cocycle generated by$A_{1},\ldots ,A_{M}$, and when it is a Bernoulli measure. We also give a general sufficient condition for the equilibrium state to be mixing, and give an example where the equilibrium state is ergodic but not totally ergodic. Connections with a class of measures investigated by Kusuoka are explored in an appendix.