Hydrodynamic and hydromagnetic stability

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

# Introduction to Steady State Bifurcation Theory # Introduction to Dynamic Bifurcation # Reduction Procedures and Stability # Steady State Bifurcation # Dynamic Bifurcation Theory: Finite Dimensional Case # Dynamic Bifurcation Theory: Infinite Dimensional Case # Bifurcations for Nonlinear Elliptic Equations # Reaction-Diffusion Equations # Pattern Formation and Wave Equations # Fluid Dynamics

Bifurcation Theory and Applications

User's Guide.- General Theory of Phase Transition Dynamics.- Dynamic Transition Theory.- Equilibrium Phase Transition in Statistical Physics.- Fluid Dynamics.- Geophysical Fluid Dynamics and Climate Dynamics.- Dynamical Transitions in Chemistry and Biology.- References .

Phase Transition Dynamics

We study in this article the bifurcation and stability of the solutions of the Boussinesq equations, and the onset of the Rayleigh-Benard convection. A nonlinear theory for this problem is established in this article using a new notion of bifurcation called attractor bifurcation and its corresponding theorem developed recently by the authors in [6]. This theory includes the following three aspects. First, the problem bifurcates from the trivial solution an attractor AR when the Rayleigh number R crosses the first critical Rayleigh number Rc for all physically sound boundary conditions, regardless of the multiplicity of the eigenvalue Rc for the linear problem. Second, the bifurcated attractor AR is asymptotically stable. Third, when the spatial dimension is two, the bifurcated solutions are also structurally stable and are classified as well. In addition, the technical method developed provides a recipe, which can be used for many other problems related to bifurcation and pattern formation.

/pdf/dynamic-bifurcation-and-stability-in-the-rayleigh-benard-2yijbtwhbw.pdf

Dynamic Bifurcation and Stability in the Rayleigh-Benard Convection

The main objective of this article is part of a research program to link the dynamics of fluid flows with the structure of these fluid flows in physical space and the transitions of this structure. To demonstrate the main ideas, we study the two-dimensional Rayleigh-Benard convection, which serves as a prototype problem. The analysis is based on two recently developed nonlinear theories: geometric theory for incompressible flows [T. Ma and S. Wang, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 119, 2005] and bifurcation and stability theory for nonlinear dynamical systems (both finite and infinite dimensional) [T. Ma and S. Wang, World Scientific, 2005]. We have shown in [T. Ma and S. Wang, Commun. Math. Sci., 2(2), 159–183, 2004] that the Rayleigh-Benard problem bifurcates from the basic state to an attractor AR when the Rayleigh number R crosses the first critical Rayleigh number Rc for all physically sound boundary conditions, regardless of the multiplicity of the eigenvalue Rc for the linear problem. In this article, in addition to a classification of the bifurcated attractor AR, the structure of the solutions in physical space and the transitions of this structure are classified, leading to the existence and stability of two different flows structures: pure rolls and rolls separated by a cross the channel flow. It appears that the structure with rolls separated by a cross-channel flow has not been carefully examined although it has been observed in other physical contexts such as the Branstator-Kushnir waves in atmospheric dynamics [G.W. Branstator, J. Atmos. Sci., 44, 2310–2323, 1987] and [K. Kushnir, J. Atmos. Sci., 44, 2727–2742, 1987].

https://www.intlpress.com/site/pub/files/_fulltext/journals/cms/2007/0005/0003/CMS-2007-0005-0003-a003.pdf

Rayleigh Bénard convection: dynamics and structure in the physical space

The main objective of this article is part of a research program to link the dynamics of fluid flows with the structure and its transitions in the physical spaces. As a prototype of problem and to demonstrate the main ideas, we study the two-dimensional Rayleigh-Benard convection. The analysis is based on two recently developed nonlinear theories: geometric theory for incompressible flows [10] and the bifurcation and stability theory for nonlinear dynamical systems (both finite and infinite dimensional) [9]. We have shown in [8] that the Rayleigh-Benard problem bifurcates from the basic state to an attractor A_R when the Rayleigh number R crosses the first critical Rayleigh number R_c for all physically sound boundary conditions, regardless of the multiplicity of the eigenvalue R_c for the linear problem. In this article, in addition to a classification of the bifurcated attractor A_R, the structure and its transitions of the solutions in the physical space is classified, leading to the existence and stability of two different flows structures: pure rolls and rolls separated by a cross the channel flow. It appears that the structure with rolls separated by a cross channel flow has not been carefully examined although it has been observed in other physical contexts such as the Branstator-Kushnir waves in the atmospheric dynamics [1,7].

Rayleigh-Benard Convection: Dynamics and Structure in the Physical Space

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