Abstract:
A model for three-dimensional Rayleigh-B\'{e}nard convection in low-Prandtl-number fluids near onset with rigid horizontal boundaries in the presence of a uniform vertical magnetic field is constructed and analyzed in detail. The kinetic energy $K$, the convective entropy $\Phi$ and the convective heat flux ($Nu-1$) show scaling behaviour with $\epsilon = r-1$ near onset of convection, where $r$ is the reduced Rayleigh number. The model is also used to investigate various magneto-convective structures close to the onset. Straight rolls, which appear at the primary instability, become unstable with increase in $r$ and bifurcate to three-dimensional structures. The straight rolls become periodically varying wavy rolls or quasiperiodically varying structures in time with increase in $r$ depending on the values of Prandtl number $Pr$. They become irregular in time, with increase in $r$. These standing wave solutions bifurcate first to periodic and then quasiperiodic traveling wave solutions, as $r$ is raised further. The variations of the critical Rayleigh number $Ra_{os}$ and the frequency $\omega_{os}$ at the onset of the secondary instability with $Pr$ are also studied for different values of Chandrasekhar's number $Q$.

Abstract:
A detailed study of the Rayleigh-B\'enard convection in two-dimensions with free-slip boundaries is presented. Pseudo-spectral method has been used to numerically solve the system for Rayleigh number up to $3.3 \times 10^7$. The system exhibits various convective states: stationary, oscillatory, chaotic and soft-turbulent. The `travelling rolls' instability is observed in the chaotic regime. Scaling of Nusselt number shows an exponent close to 0.33. Studies on energy spectrum and flux show an inverse cascade of kinetic energy and a forward cascade of entropy. This is consistent with the shell-to-shell energy transfer in wave number space. The shell-to-shell energy transfer study also indicates a local energy transfer from one shell to the other.

Abstract:
We investigate the influence of convective overshoot on stellar evolution models of the thermal pulse AGB phase with M_ZAMS = 3 Msol. An exponential diffusive overshoot algorithm is applied to all convective boundaries during all evolutionary stages. We demonstrate that overshooting at the bottom of the pulse-driven convective zone, which forms in the intershell during the He-shell flash, leads to more efficient third dredge-up. Some overshoot at the bottom of the convective envelope removes the He-H discontinuity, which would otherwise prohibit the occurrence of the third dredge-up for this stellar mass. However, no correlation between the amount of envelope overshoot and the efficiency of the third dredge-up has been found. Increasingly efficient third dredge-up eventually leads to a carbon star model. Due to the partial mixing efficiency in the overshoot region a C13-pocket can form after the third dredge-up event which may be crucial for n-capture nucleosynthesis.

Abstract:
A statistical-mechanical investigation is performed on Rayleigh-B\'enard convection of a dilute classical gas starting from the Boltzmann equation. We first present a microscopic derivation of basic hydrodynamic equations and an expression of entropy appropriate for the convection. This includes an alternative justification for the Oberbeck-Boussinesq approximation. We then calculate entropy change through the convective transition choosing mechanical quantities as independent variables. Above the critical Rayleigh number, the system is found to evolve from the heat-conducting uniform state towards the convective roll state with monotonic increase of entropy on the average. Thus, the principle of maximum entropy proposed for nonequilibrium steady states in a preceding paper is indeed obeyed in this prototype example. The principle also provides a natural explanation for the enhancement of the Nusselt number in convection.

Abstract:
We consider stability of steady convective flows in a horizontal layer with stress-free boundaries, heated below and rotating about the vertical axis, in the Boussinesq approximation (the Rayleigh-Benard convection). The flows under consideration are convective rolls or square cells, the latter being asymptotically equal to the sum of two orthogonal rolls of the same wave number k. We assume, that the Rayleigh number R is close to the critical one, R_c(k), for the onset of convective flows of this wave number: R=R_c(k)+epsilon^2; the amplitude of the flows is of the order of epsilon. We show that the flows are always unstable to perturbations, which are a sum of a large-scale mode not involving small scales, and two large-scale modes, modulated by the original rolls rotated by equal small angles in the opposite directions. The maximal growth rate of the instability is of the order of max(epsilon^{8/5},(k-k_c)^2), where k_c is the critical wave number for the onset of convection.

Abstract:
In this work a deep relation between topology and thermodynamical features of manifolds with boundaries is shown. The expression for the Euler characteristic, through the Gauss- Bonnet integral, and the one for the entropy of gravitational instantons are proposed in a form which makes the relation between them self-evident. A generalization of Bekenstein-Hawking formula, in which entropy and Euler characteristic are related in the form $S=\chi A/8$, is obtained. This formula reproduces the correct result for extreme black hole, where the Bekenstein-Hawking one fails ($S=0$ but $A \neq 0$). In such a way it recovers a unified picture for the black hole entropy law. Moreover, it is proved that such a relation can be generalized to a wide class of manifolds with boundaries which are described by spherically symmetric metrics (e.g. Schwarzschild, Reissner-Nordstr\"{o}m, static de Sitter).

Abstract:
Convective turbulence in a Rayleigh Benard system has shown a marked reluctance to exhibit clear scaling in the energy or entropy spectrum. The recent numerical simulation of Pandey, Verma and Mishra has shown significantly better evidence of scaling in the infinite Prandtl number limit. This prompted us to look at this limit analytically. We find that the inevitable presence of sweeping helps give a very good understanding of the results of Pandey et. al. In the presence of rotation, the Rayleigh number dependence of the Nusselt number shows a strong increase in slope in recent experiments and simulations at finite Prandtl number. In the infinite Prandtl number case we find that this steepening does not occur for any rotation speed and our results satisfy the Doering-Constantin bound.

Abstract:
Rigid actions have zero Rokhlin entropy and nonpositive sofic entropy. Because rigidity is a stable orbit-equivalence invariant, this provides the first example of an essentially free, ergodic, probability-measure-preserving action of the free group that has nonpositive sofic entropy and any essentially free action stably-orbit-equivalent to it also has nonpositive sofic entropy.

Abstract:
We investigate the influence of the thermal properties of the boundaries in turbulent Rayleigh-Benard convection on analytical bounds on convective heat transport. Using the Doering-Constantin background flow method, we systematically formulate a bounding principle on the Nusselt-Rayleigh number relationship for general mixed thermal boundary conditions of constant Biot number \eta which continuously interpolates between the previously studied fixed temperature ($\eta = 0$) and fixed flux ($\eta = \infty$) cases, and derive explicit asymptotic and rigorous bounds. Introducing a control parameter R as a measure of the driving which is in general different from the usual Rayleigh number Ra, we find that for each $\eta > 0$, as R increases the bound on the Nusselt number Nu approaches that for the fixed flux problem. Specifically, for $0 < \eta \leq \infty$ and for sufficiently large R ($R > R_s = O(\eta^{-2})$ for small \eta) the Nusselt number is bounded as $Nu \leq c(\eta) R^{1/3} \leq C Ra^{1/2}$, where C is an \eta-independent constant. In the $R \to \infty$ limit, the usual fixed temperature assumption is thus a singular limit of this general bounding problem.

Abstract:
We consider the Boltzmann equation for a gas in a horizontal slab, subject to a gravitational force. The boundary conditions are of diffusive type, specifying the wall temperatures, so that the top temperature is lower than the bottom one (Benard setup). We consider a 2-dimensional convective stationary solution, which is close for small Knudsen number to the convective stationary solution of the Oberbeck-Boussinesq equations, near above the bifurcation point, and prove its stability under 2-d small perturbations, for Rayleigh number above and close to the bifurcation point and for small Knudsen number.