Large-time behavior of a two-scale semilinear reaction-diffusion system for concrete sulfatation

We study the large-time behavior of (weak) solutions to a two-scale reaction-diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (/cement)-based materials with sulfates. We prove that as $t\to\infty$ the solution to the original two-scale system converges to the corresponding two-scale stationary system. To obtain the main result we make use essentially of the theory of evolution equations governed by subdifferential operators of time-dependent convex functions developed combined with a series of two-scale energy-like time-independent estimates.Comment: 20 page

Energetics of the global ocean: The role of mesoscale eddies

This article reviews the energy cycle of the global ocean circulation, focusing on the role of baroclinic mesoscale eddies. Two of the important effects of mesoscale eddies are: (i) the flattening of the slope of large-scale isopycnal surfaces by the eddy-induced overturning circulation, the basis for the Gent–McWilliams parametrization; and (ii) the vertical redistribution of the momentum of basic geostrophic currents by the eddy-induced form stress (the residual effect of pressure perturbations), the basis for the Greatbatch–Lamb parametrization. While only point (i) can be explained using the classical Lorenz energy diagram, both (i) and (ii) can be explained using the modified energy diagram of Bleck as in the following energy cycle. Wind forcing provides an input to the mean KE, which is then transferred to the available potential energy (APE) of the large-scale field by the wind-induced Ekman flow. Subsequently, the APE is extracted by the eddy-induced overturning circulation to feed the mean KE, indicating the enhancement of the vertical shear of the basic current. Meanwhile, the vertical shear of the basic current is relaxed by the eddy-induced form stress, taking the mean KE to endow the eddy field with an energy cascade. The above energy cycle is useful for understanding the dynamics of the Antarctic Circumpolar Current. On the other hand, while the source of the eddy field energy has become clearer, identifying the sink and flux of the eddy field energy in both physical and spectral space remains major challenges of present-day oceanography. A recent study using a combination of models, satellite altimetry, and climatological hydrographic data shows that the western boundary acts as a “graveyard” for the westward-propagating eddies

Large-time asymptotics of moving-reaction interfaces involving nonlinear Henry's law and time-dependent Dirichlet data

We study the large-time behavior of the free boundary position capturing the one-dimensional motion of the carbonation reaction front in concrete-based materials. We extend here our rigorous justification of the $\sqrt{t}$-behavior of reaction penetration depths by including non-linear effects due to deviations from the classical Henry's law and time-dependent Dirichlet data.Comment: 19 page

Distributed space scales in a semilinear reaction-diffusion system including a parabolic variational inequality: A well-posedness study

This paper treats the solvability of a semilinear reaction-diffusion system, which incorporates transport (diffusion) and reaction effects emerging from two separated spatial scales: $x$ - macro and $y$ - micro. The system's origin connects to the modeling of concrete corrosion in sewer concrete pipes. It consists of three partial differential equations which are mass-balances of concentrations, as well as, one ordinary differential equation tracking the damage-by-corrosion. The system is semilinear, partially dissipative, and coupled via the solid-water interface at the microstructure (pore) level. The structure of the model equations is obtained in \cite{tasnim1} by upscaling of the physical and chemical processes taking place within the microstructure of the concrete. Herein we ensure the positivity and $L^\infty-$bounds on concentrations, and then prove the global-in-time existence and uniqueness of a suitable class of positive and bounded solutions that are stable with respect to the two-scale data and model parameters. The main ingredient to prove existence include fixed-point arguments and convergent two-scale Galerkin approximations.Comment: 24 pages, 1 figur

The vertical structure of the surface wave radiation stress for circulation over a sloping bottom as given by thickness-weighted-mean theory

Previous attempts to derive the depth-dependent expression of the radiation stress have led to a debate concerning (i) the applicability of the Mellor approach to a sloping bottom, (ii) the introduction of the delta function at the mean sea surface in the later papers by Mellor, and (iii) a wave-induced pressure term derived in several recent studies. The authors use an equation system in vertically Lagrangian and horizontally Eulerian (VL) coordinates suitable for a concise treatment of the surface boundary and obtain an expression for the depth-dependent radiation stress that is consistent with the vertically integrated expression given by Longuet-Higgins and Stewart. Concerning (i)-(iii) above, the difficulty of handling a sloping bottom disappears when wave-averaged momentum equations in the VL coordinates are written for the development of (not the Lagrangian mean velocity but) the Eulerian mean velocity. There is also no delta function at the sea surface in the expression for the depth-dependent radiation stress. The connection between the wave-induced pressure term in the recent studies and the depth-dependent radiation stress term is easily shown by rewriting the pressure-based form stress term in the thickness-weighted-mean momentum equations as a velocity-based term that contains the time derivative of the pseudomomentum in the VL framework

A thermo-diffusion system with Smoluchowski interactions: well-posedness and homogenization

We study the solvability and homogenization of a thermal-diffusion reaction problem posed in a periodically perforated domain. The system describes the motion of populations of hot colloidal particles interacting together via Smoluchowski production terms. The upscaled system, obtained via two-scale convergence techniques, allows the investigation of deposition effects in porous materials in the presence of thermal gradients