Stationary statistical properties of Rayleigh‐Bénard convection at large Prandtl number

This is the third in a series of our study of Rayleigh‐Bénard convection at large Prandtl number. Here we investigate whether stationary statistical properties of the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number are related to those of the infinite Prandtl number model for convection that is formally derived from the Boussinesq system via setting the Prandtl number to infinity. We study asymptotic behavior of stationary statistical solutions, or invariant measures, to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number. In particular, we show that the invariant measures of the Boussinesq system for Rayleigh‐Bénard convection converge to those of the infinite Prandtl number model for convection as the Prandtl number approaches infinity. We also show that the Nusselt number for the Boussinesq system (a specific statistical property of the system) is asymptotically bounded by the Nusselt number of the infinite Prandtl number model for convection at large Prandtl number. We discover that the Nusselt numbers are saturated by ergodic invariant measures. Moreover, we derive a new upper bound on the Nusselt number for the Boussinesq system at large Prandtl number of the form which asymptotically agrees with the (optimal) upper bound on Nusselt number for the infinite Prandtl number model for convection. © 2007 Wiley Periodicals, Inc.     

Infinite Prandtl number limit of Rayleigh‐Bénard convection

We rigorously justify the infinite Prandtl number model of convection as the limit of the Boussinesq approximation to the Rayleigh‐Bénard convection as the Prandtl number approaches infinity. This is a singular limit problem involving an initial layer. © 2003 Wiley Periodicals, Inc.     

Asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number

We study asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number. In particular, we show that the global attractors to the Boussinesq system for Rayleigh‐Bénard convection converge to that of the infinite‐Prandtl‐number model for convection as the Prandtl number approaches infinity. This offers partial justification of the infinite‐Prandtl‐number model for convection as a valid simplified model for convection at large Prandtl number even in the long‐time regime. © 2006 Wiley Periodicals, Inc.     

Bifurcation of infinite Prandtl number rotating convection

We consider infinite Prandtl number convection with rotation which is the basic model in geophysical fluid dynamics. For the rotation free case, the rigorous analysis has been provided by Park (2005, 2007, revised for publication) [5], [6] and [25] under various boundary conditions. By thoroughly investigating we prove in this paper that the solutions bifurcate from the trivial solution u=0 to an attractor Σ_R which consists of only one cycle of steady state solutions and is homeomorphic to S¹. We also see how intensively the rotation inhibits the onset of convective motion. This bifurcation analysis is based on a new notion of bifurcation, called attractor bifurcation which was developed by Ma and Wang (2005); see [15].     

On the stability of two-dimensional convection rolls in an infinite Prandtl number fluid with stress-free boundaries

Numerical solutions for two-dimensional convection rolls in a fluid layer of infinite Prandtl number are obtained by the Galerkin method. Stress-free, isothermal boundaries are assumed at the horizontal boundaries of the fluid layer. The stability of the steady solutions with respect to three-dimensional disturbances is analyzed in the Rayleigh number-wave number space. It is found that even for Rayleigh numbers as high as several millions there appears to exist a region of the wavenumber where the convection rolls are stable. This result contrasts with the well known transition to three-dimensional bimodal convection in the presence of no-slip boundaries, but it agrees with simple arguments about the stability of the thermal boundary layers.

---

Zusammenfassung Numerische Lösungen für zwei-dimensionale Konvektionsrollen in einer Flüssigkeitsschicht mit unendlicher Prandtlzahl sind gewonnen worden durch Anwendungen der Galerkin-Methode. Es wurden spannungsfreie isotherme Ränder an den Grenzen der horizontalen Flüssigkeitsschicht angenommen. Die Stabilität der stationären Lösungen bezüglich drei-dimensionaler Störungen wurde im Rayleighzahl-Wellenzahl-Raum analysiert. Es wurde gefunden, daß für Rayleighzahlen bis zu einigen Millionen ein Bereich der Wellenzahl existiert, in dem die Konvektionsrollen stabil sind. Dieses Resultat steht im Gegensatz zu dem wohl bekannten Übergang zu bimodaler Konvektion im Fall der festen Ränder, ist aber im Einklang mit einfachen Betrachtungen über die Stabilität der thermischen Grenzschichten.

     

Influence of helicity on the turbulent Prandtl number: Two-loop approximation

Using the field theory renormalization group technique in the two-loop approximation, we study the influence of helicity (spatial parity violation) on the turbulent Prandtl number in the model of a scalar field passively advected by the helical turbulent environment given by the stochastic Navier-Stokes equation with a self-similar Gaussian random stirring force δ-correlated in time with the correlator proportional to k ^4−d−2ɛ. We briefly discuss the influence of helicity on the stability of the corresponding scaling regime. We show that the presence of helicity increases the value of the turbulent Prandtl number up to 50% of its nonhelical value.     

The limit of the Boussinesq system of Rayleigh-Bénard convection as the Prandtl number approaches infinity

In this paper, the initial layer problem and infinite Prandtl number limit of Rayleigh-Bénard convection is studied by the asymptotic expansion methods of singular perturbation theory and the classical energy methods. For ill-prepared initial data, an exact approximating solution with expansions up to any order are given and the convergence rates O(ɛ ^m+1/2) and the optimal convergence rates O(ɛ ^m+1) are obtained respectively. This improves the result of J.G. SHI.     

The Ulam number of infinite graphs

The following is a conjecture of Ulam: In any partition of the integer lattice on the plane into uniformly bounded sets, there exists a set that is adjacent to at least six other sets. Two sets are adjacent if each contain a vertex of the same unit square. This problem is generalized as follows. Given any uniformly bounded partitionP of the vertex set of an infinite graphG with finite maximum degree, letP (G) denote the graph obtained by letting each set of the partition be a vertex ofP (G) where two vertices ofP (G) are adjacent if and only if the corresponding sets have an edge between them. The Ulam number ofG is defined as the minimum of the maximum degree ofP (G) where the minimum is taken over all uniformly bounded partitionsP. We have characterized the graphs with Ulam number 0, 1, and 2. Restricting the partitions of the vertex set to connected subsets, we obtain the connected Ulam number ofG. We have evaluated the connected Ulam numbers for several infinite graphs. For instance we have shown that the connected Ulam number is 4 ifG is an infinite grid graph. We have settled the Ulam conjecture for the connected case by proving that the connected Ulam number is 6 for an infinite triangular grid graph. The general Ulam conjecture is equivalent to proving that the Ulam number of the infinite triangular grid graph equals 6. We also describe some interesting geometric consequences of the Ulam number, mainly concerning good drawings of infinite graphs.     

Upper bounds of the rates of decay for solutions of the Boussinesq equations

In this paper,upper bounds of the L2-decay rate for the Boussinesq equations are considered.Using the L2 decay rate of solutions for the heat equation,and assuming that the solutions of the Boussinesq equations are smooth,we obtain the upper bounds of L2 decay rate for the smooth solutions and difference between the solutions of the Boussinesq equations and those of the heat system with the same initial data.The decay results may then be obtained by passing to the limit of approximating sequences of solutions.The main tool is the Fourier splitting method.     

Spatial parity violation and the turbulent magnetic Prandtl number

Using the field theory renormalization-group technique in the two-loop approximation, we study the influence of helicity (spatial parity violation) on the turbulent magnetic Prandtl number in the model of kinematic magnetohydrodynamic turbulence, where the magnetic field behaves as a passive vector quantity advected by the helical turbulent environment given by the stochastic Navier-Stokes equation. We show that the presence of helicity decreases the value of the turbulent magnetic Prandtl number and that the two-loop helical contribution to the turbulent magnetic Prandtl number is up to 4.2% of its nonhelical value. This result demonstrates the strong stability of the properties of diffusion processes of the magnetic field in turbulent environments with spatial parity violation compared with the corresponding systems without the helicity.     

Two-Dimensional Infinite Prandtl Number Convection: Structure of Bifurcated Solutions

This paper examines the bifurcation and structure of the bifurcated solutions of the two-dimensional infinite Prandtl number convection problem. The existence of a bifurcation from the trivial solution to an attractor Σ _R was proved by Park (Disc. Cont. Dynam. Syst. B [2005]). We prove in this paper that the bifurcated attractor Σ _R consists of only one cycle of steady-state solutions and that it is homeomorphic to S¹. By thoroughly investigating the structure and transitions of the solutions of the infinite randtl number convection problem in physical space, we confirm that the bifurcated solutions are indeed structurally stable. In turn, this will corroborate and justify the suggested results with the physical findings about the presence of the roll structure. This bifurcation analysis is based on a new notion of bifurcation, called attractor bifurcation, and structural stability is derived using a new geometric theory of incompressible flows. Both theories were developed by Ma and Wang; see Bifurcation Theory and Applications (World Scientific, 2005) and Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics (American Mathematical Society, 2005).     

The number of infinite clusters in dynamical percolation

Summary. Dynamical percolation is a Markov process on the space of subgraphs of a given graph, that has the usual percolation measure as its stationary distribution. In previous work with O. H?ggstr?m, we found conditions for existence of infinite clusters at exceptional times. Here we show that for ℤ ^d , with p>p _c , a.s. simultaneously for all times there is a unique infinite cluster, and the density of this cluster is θ(p). For dynamical percolation on a general tree Γ, we show that for p>p _c , a.s. there are infinitely many infinite clusters at all times. At the critical value p=p _c , the number of infinite clusters may vary, and exhibits surprisingly rich behaviour. For spherically symmetric trees, we find the Hausdorff dimension of the set T ^k of times where the number of infinite clusters is k, and obtain sharp capacity criteria for a given time set to intersect T ^k . The proof of this capacity criterion is based on a new kernel truncation technique. Received: 5 May 1997 / In revised form: 24 November 1997     

Finite dimensional approximation in infinite dimensional mathematical programming

We consider the problem of approximating an optimal solution to a separable, doubly infinite mathematical program (P) with lower staircase structure by solutions to the programs (P(N)) obtained by truncating after the firstN variables andN constraints of (P). Viewing the surplus vector variable associated with theNth constraint as a state, and assuming that all feasible states are eventually reachable from any feasible state, we show that the efficient set of all solutions optimal to all possible feasible surplus states for (P(N)) converges to the set of optimal solutions to (P). A tie-breaking algorithm which selects a nearest-point efficient solution for (P(N)) is shown (for convex programs) to converge to an optimal solution to (P). A stopping rule is provided for discovering a value ofN sufficiently large to guarantee any prespecified level of accuracy. The theory is illustrated by an application to production planning.The work of Robert L. Smith was partially supported by the National Science Foundation under Grant ECS-8700836.     

A note on Prandtl boundary layers

This note concerns nonlinear ill‐posedness of the Prandtl equation and an invalidity of asymptotic boundary layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear illposedness results established by Gérard‐Varet and Dormy and an analysis by Guo and Tice. We show that the asymptotic boundary layer expansion is not valid for nonmonotonic shear layer flows in Sobolev spaces. We also introduce a notion of weak well‐posedness and prove that the nonlinear Prandtl equation is not well‐posed in this sense near nonstationary and nonmonotonic shear flows. On the other hand, we are able to verify that Oleinik's monotonic solutions are well‐posed. © 2011 Wiley Periodicals, Inc.     

Qualitative Properties of Separating Flows in the Prandtl Boundary Layer

We study the behavior of solutions to the system of Prandtl boundary layer equations beyond the separation point of the boundary layer. We obtain conditions on the positive pressure gradient which guarantee the attachment of the boundary layer to the streamlined surface after separation. We prove the possibility of controlling the boundary layer by alternating suction and injection.     

Well-posed infinite horizon variational problems on a compact manifold

We give an effective sufficient condition for a variational problem with infinite horizon on a compact Riemannian manifold M to admit a smooth optimal synthesis, i.e., a smooth dynamical system on M whose positive semi-trajectories are solutions to the problem. To realize the synthesis, we construct an invariant Lagrangian submanifold (well-projected to M) of the flow of extremals in the cotangent bundle T*M. The construction uses the curvature of the flow in the cotangent bundle and some ideas of hyperbolic dynamics.     

Boundary layers associated with the 3-D Boussinesq system for Rayleigh–Bénard convection

We investigate the boundary layer effects of the 3-D incompressible Boussinesq system for Rayleigh–Bénard convection with vanishing diffusivity limit. By adopting the multi-scale analysis and the asymptotic expansion methods of singular perturbation theory, we construct an exact approximating solution for the viscous and diffusive Boussinesq system with well-prepared initial data. In addition, we obtain the convergence result of the vanishing diffusivity limit.     

Continuity in functional differential equations with infinite delay

In this paper, we investigate the continuous dependence of solutions of the functional differential equation with infinite delayx(t)=f(t,x _t ) on initial functions. Endowing the phase space ag-norm as well as a supremum norm, we show that if the equation satisfies a mild fading memory dondition, then the continuity off in respect to the topology induced by the supremum norm can yield the continuity of solutions of the equation in respect to the topology induced by theg-norm which is stronger than the ahead one.This research was supported in part by an NSF grant with number NSF-DMS-8521408.On leave from South China Normal University, Guangzhou, PRC. This research was supported in part by the National Science Foundation of PRC.     

On the Hadwiger number of infinite graphs

LetX be a connected graph with bounded valency and at least one thick end. We show that the existence of certain subgroups of the automorphism group ofX always implies thatX has infinite Hadwiger number.     

The list-chromatic number of infinite graphs

We investigate the list-chromatic number of infinite graphs. It is easy to see that Chr(X) ≤ List(X) ≤ Col(X) for each graph X. It is consistent that List(X) = Col(X) holds for every graph with Col(X) infinite. It is also consistent that for graphs of cardinality ? ₁, List(X) is countable iff Chr(X) is countable.