Emergence of steady and oscillatory localized structures in a phytoplankton-nutrient model

Co-limitation of marine phytoplankton growth by light and nutrient, both of which are essential for phytoplankton, leads to complex dynamic behavior and a wide array of coherent patterns. The building blocks of this array can be considered to be deep chlorophyll maxima, or DCMs, which are structures localized in a finite depth interior to the water column. From an ecological point of view, DCMs are evocative of a balance between the inflow of light from the water surface and of nutrients from the sediment. From a (linear) bifurcational point of view, they appear through a transcritical bifurcation in which the trivial, no-plankton steady state is destabilized. This article is devoted to the analytic investigation of the weakly nonlinear dynamics of these DCM patterns, and it has two overarching themes. The first of these concerns the fate of the destabilizing stationary DCM mode beyond the center manifold regime. Exploiting the natural singularly perturbed nature of the model, we derive an explicit reduced model of asymptotically high dimension which fully captures these dynamics. Our subsequent and fully detailed study of this model - which involves a subtle asymptotic analysis necessarily transgressing the boundaries of a local center manifold reduction - establishes that a stable DCM pattern indeed appears from a transcritical bifurcation. However, we also deduce that asymptotically close to the original destabilization, the DCM looses its stability in a secondary bifurcation of Hopf type. This is in agreement with indications from numerical simulations available in the literature. Employing the same methods, we also identify a much larger DCM pattern. The development of the method underpinning this work - which, we expect, shall prove useful for a larger class of models - forms the second theme of this article

A nonlocal eigenvalue problem and the stability of spikes for reaction-diffusion systems with fractional reaction rates

We consider a nonlocal eigenvalue problem which arises in the study of stability of spike solutions for reaction-diffusion systems with fractional reaction rates such as the Sel'kov model, the Gray-Scott system, the hypercycle Eigen and Schuster, angiogenesis, and the generalized Gierer-Meinhardt system. We give some sufficient and explicit conditions for stability by studying the corresponding nonlocal eigenvalue problem in a new range of parameters

Existence and stability of hole solutions to complex Ginzburg-Landau equations

We consider the existence and stability of the hole, or dark soliton, solution to a Ginzburg-Landau perturbation of the defocusing nonlinear Schroedinger equation (NLS), and to the nearly real complex Ginzburg-Landau equation (CGL). By using dynamical systems techniques, it is shown that the dark soliton can persist as either a regular perturbation or a singular perturbation of that which exists for the NLS. When considering the stability of the soliton, a major difficulty which must be overcome is that eigenvalues may bifurcate out of the continuous spectrum, i.e., an edge bifurcation may occur. Since the continuous spectrum for the NLS covers the imaginary axis, and since for the CGL it touches the origin, such a bifurcation may lead to an unstable wave. An additional important consideration is that an edge bifurcation can happen even if there are no eigenvalues embedded in the continuous spectrum. Building on and refining ideas first presented in Kapitula and Sandstede (Physica D, 1998) and Kapitula (SIAM J. Math. Anal., 1999), we show that when the wave persists as a regular perturbation, at most three eigenvalues will bifurcate out of the continuous spectrum. Furthermore, we precisely track these bifurcating eigenvalues, and thus are able to give conditions for which the perturbed wave will be stable. For the NLS the results are an improvement and refinement of previous work, while the results for the CGL are new. The techniques presented are very general and are therefore applicable to a much larger class of problems than those considered here.Comment: 41 pages, 4 figures, submitte

Global Transition Rules for Translating Land-use Change (LUH2) To Land-cover Change for CMIP6 using GLM2

Information on historical land-cover change is important for understanding human impacts on the environment. Over the last decade, global models have characterized historical land-use changes, but few have been able to relate these changes with corresponding changes in land-cover. Utilizing the latest global land-use change data, we make several assumptions about the relationship between land-use and land-cover change, and evaluate each scenario with remote sensing data to identify optimal fit. The resulting transition rule can guide the incorporation of land-cover information within earth system models

The role of peatland degradation, protection and restoration for climate change mitigation in the SSP scenarios

Peatlands only cover a small fraction of the global land surface (∼3%) but store large amounts of carbon (∼600 GtC). Drainage of peatlands for agriculture results in the decomposition of organic matter, leading to greenhouse gas (GHG) emissions. As a result, degraded peatlands are currently responsible for 2%–3% of global anthropogenic emissions. Preventing further degradation of peatlands and restoration (i.e. rewetting) are therefore important for climate change mitigation. In this study, we show that land-use change in three SSP scenarios with optimistic, recent trends, and pessimistic assumptions leads to peatland degradation between 2020 and 2100 ranging from −7 to +10 Mha (−23% to +32%), and a continuation or even an increase in annual GHG emissions (−0.1 to +0.4 GtCO2-eq yr−1). In default mitigation scenarios without a specific focus on peatlands, peatland degradation is reduced due to synergies with forest protection and afforestation policies. However, this still leaves large amounts of GHG emissions from degraded peatlands unabated, causing cumulative CO2 emissions from 2020 to 2100 in an SSP2-1.5 °C scenario of 73 GtCO2. In a mitigation scenario with dedicated peatland restoration policy, GHG emissions from degraded peatlands can be reduced to nearly zero without major effects on projected land-use dynamics. This underlines the opportunity of peatland protection and restoration for climate change mitigation and the need to synergistically combine different land-based mitigation measures. Peatland location and extent estimates vary widely in the literature; a sensitivity analysis implementing various spatial estimates shows that especially in tropical regions degraded peatland area and peatland emissions are highly uncertain. The required protection and mitigation efforts are geographically unequally distributed, with large concentrations of peatlands in Russia, Europe, North America and Indonesia (33% of emission reductions are located in Indonesia). This indicates an important role for only a few countries that have the opportunity to protect and restore peatlands with global benefits for climate change mitigation

Phase Slips and the Eckhaus Instability

We consider the Ginzburg-Landau equation, $\partial_t u= \partial_x^2 u + u - u|u|^2$, with complex amplitude $u(x,t)$. We first analyze the phenomenon of phase slips as a consequence of the {\it local} shape of $u$. We next prove a {\it global} theorem about evolution from an Eckhaus unstable state, all the way to the limiting stable finite state, for periodic perturbations of Eckhaus unstable periodic initial data. Equipped with these results, we proceed to prove the corresponding phenomena for the fourth order Swift-Hohenberg equation, of which the Ginzburg-Landau equation is the amplitude approximation. This sheds light on how one should deal with local and global aspects of phase slips for this and many other similar systems.Comment: 22 pages, Postscript, A

Focal-plane wavefront sensing with the vector-Apodizing Phase Plate

Context. One of the key limitations of the direct imaging of exoplanets at small angular separations are quasi-static speckles that originate from evolving non-common path aberrations (NCPA) in the optical train downstream of the instrument’s main wavefront sensor split-off. Aims. In this article we show that the vector-Apodizing Phase Plate (vAPP) coronagraph can be designed such that the coronagraphic point spread functions (PSFs) can act as wavefront sensors to measure and correct the (quasi-)static aberrations without dedicated wavefront sensing holograms or modulation by the deformable mirror. The absolute wavefront retrieval is performed with a nonlinear algorithm. Methods. The focal-plane wavefront sensing (FPWFS) performance of the vAPP and the algorithm are evaluated via numerical simulations to test various photon and read noise levels, the sensitivity to the 100 lowest Zernike modes, and the maximum wavefront error (WFE) that can be accurately estimated in one iteration. We apply these methods to the vAPP within SCExAO, first with the internal source and subsequently on-sky. Results. In idealized simulations we show that for 107 photons the root mean square (RMS) WFE can be reduced to ~ λ/1000, which is 1 nm RMS in the context of the SCExAO system. We find that the maximum WFE that can be corrected in one iteration is ~ λ/8 RMS or ~200 nm RMS (SCExAO). Furthermore, we demonstrate the SCExAO vAPP capabilities by measuring and controlling the 30 lowest Zernike modes with the internal source and on-sky. On-sky, we report a raw contrast improvement of a factor ~2 between 2 and 4 λ/D after five iterations of closed-loop correction. When artificially introducing 150 nm RMS WFE, the algorithm corrects it within five iterations of closed-loop operation. Conclusions. FPWFS with the vAPP coronagraphic PSFs is a powerful technique since it integrates coronagraphy and wavefront sensing, eliminating the need for additional probes and thus resulting in a 100% science duty cycle and maximum throughput for the target