Limit Of Infinite Prandtl Number Research Articles

Global well-posedness for a rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number

We analyze a three-dimensional rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number, i.e., when the momentum diffusivity is much more dominant than the thermal diffusivity. Consequently, the dynamics of the velocity field takes place at a much faster time scale than the temperature fluctuation, and at the limit the velocity field formally adjusts instantaneously to the thermal fluctuation. We prove the global well-posedness of weak solutions and strong solutions to this model.

Initial-boundary layer associated with the 3-D Boussinesq system for Rayleigh-Benard convection

This article concerns the initial-boundary layer effects of the 3-D incompressible Boussinesq system for Rayleigh-Benard convection with ill-prepared initial data. We consider a non-slip boundary condition for the velocity field and inhomogeneous Dirichlet boundary condition for the temperature. By means of multi-scale analysis and matched asymptotic expansion methods, we establish an accurate approximating solution for the viscous and diffusive Boussinesq system. We also study the convergence of the infinite Prandtl number limit.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/31/abstr.html

Large Prandtl number asymptotics in randomly forced turbulent convection

We establish the convergence of statistically invariant states for the stochastic Boussinesq equations in the infinite Prandtl number limit and in particular demonstrate the convergence of the Nusselt number (a measure of heat transport in the fluid). This is a singular parameter limit significant in mantle convection and for gasses under high pressure. The equations are subject to a both temperature gradient on the boundary and internal heating in the bulk driven by a stochastic, white in time, gaussian forcing. Here, the stochastic source terms have a strong physical motivation for example as a model of radiogenic heating. Our approach uses mixing properties of the formal limit system to reduce the convergence of invariant states to an analysis of the finite time asymptotics of solutions and parameter-uniform moment bounds. Here, it is notable that there is a phase space mismatch between the finite Prandtl system and the limit equation, and we implement methods to lift both finite and infinite time convergence results to an extended phase space which includes velocity fields. For the infinite Prandtl stochastic Boussinesq equations, we show that the associated invariant measure is unique and that the dual Markovian dynamics are contractive in an appropriate Kantorovich–Wasserstein metric. We then address the convergence of solutions on finite time intervals, which is still a singular perturbation. In the process we derive well-posed equations which accurately approximate the dynamics up to the initial time when the Prandtl number is large.

Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit

AbstractThe main purpose of this paper is to study the initial layer problem and the infinite Prandtl number limit of Rayleigh-Bénard convection with an ill prepared initial data. We use the asymptotic expansion methods of singular perturbation theory and the two-time-scale approach to obtain an exact approximating solution and the convergence rates $O(\varepsilon^{\frac{3}{2}})$ and O(ε2).

On the infinite Prandtl number limit in two-dimensional magneto-convection

In this paper, the infinite limit of the Prandtl number is justified for the two-dimensional incompressible magneto-convection, which describes the nonlinear interaction between the Rayleigh–Bénard convection and an externally magnetic field. Both the convergence rates and the thickness of initial layer are obtained. Moreover, based on the method of formal asymptotic expansions, an effective dynamics is constructed to simulate the motion within the initial layer.

Dynamo action in rapidly rotating Rayleigh–Bénard convection at infinite Prandtl number

In order better to understand the processes that lead to the generation of magnetic fields of finite amplitude, we study dynamo action driven by turbulent Boussinesq convection in a rapidly rotating system. In the limit of infinite Prandtl number (the ratio of viscous to thermal diffusion) the inertia term drops out of the momentum equation, which becomes linear in the velocity. This simplification allows a decomposition of the velocity into a thermal part driven by buoyancy, and a magnetic part driven by the Lorentz force. While the former velocity defines the kinematic dynamo problem responsible for the exponential growth of the magnetic field, the latter encodes the magnetic back reaction that leads to the eventual nonlinear saturation of the dynamo. We argue that two different types of solution should exist: weak solutions in which the saturated velocity remains close to the kinematic one, and strong solutions in which magnetic forces drive the system into a new strongly magnetised state that is radically different from the kinematic one. Indeed, we find both types of solutions numerically. Interestingly, we also find that, in our inertialess system, both types of solutions exist on the same subcritical branch of solutions bifurcating from the non-magnetic convective state, in contrast with the more traditional situation for systems with finite inertia in which weak and strong solutions are thought to exist on different branches. We find that for weak solutions, the force balance is the same as in the non-magnetic case, with the horizontal size of the convection varying as the one-third power of the Ekman number (the ratio of viscous to Coriolis forces), which gives rise to very small cells at small Ekman numbers (i.e. high rotation rates). In the strong solutions, magnetic forces become important and the convection develops on much larger horizontal scales. However, we note that even in the strong cases the solutions never properly satisfy Taylor’s constraint, and that viscous stresses continue to play a role. Finally, we discuss the relevance of our findings to the study of planetary dynamos in rapidly rotating systems such as the Earth.

Self-Consistent Field Theory for the Convective Turbulence in a Rayleigh-Benard System in the Infinite Prandtl Number Limit

The kinetic energy spectrum \(E_{u}(k)\) for three dimensional convective turbulence in a Rayleigh-Benard system,where k is the wave vector, was shown to scale as \( k ^{-13/3}\) on heuristic grounds in the recent work of Pandey, Verma and Mishra in the infinite Prandtl number limit. They also presented clear numerical evidence of this scaling. This limit is very similar to the spherical model of critical phenomena and hence amenable to exact treatment in a self-consistent field theory. We find that self-consistency gives \(E_u (k)\propto R^{22/15}k^{-13/3}(R\) is the Rayleigh number) but the inevitable presence of sweeping adds a part which is proportional to \(k^{-7/2}\). This can account for the slight k-dependence of the compensated spectrum of Pandey et al. We also estimate the anisotropy in the spectrum and find that the second order Legendre function has a strength of 15 % relative to the isotropic part. In two spatial dimensions the scaling exponent of the energy spectrum is still 13/3 but the anisotropy is larger.

Bounds on heat transport in rapidly rotating Rayleigh–Bénard convection

The heat transport in rotating Rayleigh–Bénard convection is considered in the limit of rapid rotation (small Ekman number E) and strong thermal forcing (large Rayleigh number Ra). The analysis proceeds from a set of asymptotically reduced equations appropriate for rotationally constrained dynamics; the conjectured range of validity for these equations is Ra ≲ E−8/5. A rigorous bound on heat transport of Nu ⩽ 20.56Ra3E4 is derived in the limit of infinite Prandtl number using the background method. We demonstrate that the exponent in this bound cannot be improved on using a piece-wise monotonic background temperature profile like the one used here. This is true for finite Prandtl numbers as well, i.e. Nu ≲ Ra3 is the best upper bound for this particular setup of the background method. The feature that obstructs the availability of a better bound in this case is the appearance of small-scale thermal plumes emanating from (or entering) the thermal boundary layer. The derived upper bound is consistent with, although significantly higher than the observed behaviour in simulations of the reduced equations, which find at most Nu ∼ Ra2E8/3.

The perturbed problem on the boussinesq system of Rayleigh-Bénard convection

In this paper, the infinite Prandtl number limit of Rayleigh-Benard convection is studied. For well prepared initial data, the convergence of solutions in L∞(0, t;H2(G)) is rigorously justified by analysis of asymptotic expansions.

Scaling of heat flux and energy spectrum for very large Prandtl number convection.

Under the limit of infinite Prandtl number, we derive analytical expressions for the large-scale quantities, e.g., Péclet number Pe, Nusselt number Nu, and rms value of the temperature fluctuations θ(rms). We complement the analytical work with direct numerical simulations, and show that Nu ∼ Ra(γ) with γ ≈ (0.30-0.32), Pe ∼ Ra(η) with η ≈ (0.57-0.61), and θ(rms) ∼ const. The Nusselt number is observed to be an intricate function of Pe, θ(rms), and a correlation function between the vertical velocity and temperature. Using the scaling of large-scale fields, we show that the energy spectrum E(u)(k) ∼ k(-13/3), which is in a very good agreement with our numerical results. The entropy spectrum E(θ)(k), however, exhibits dual branches consisting of k(-2) and k(0) spectra; the k(-2) branch corresponds to the Fourier modes θ[over ̂](0,0,2n), which are approximately -1/(2 nπ). The scaling relations for Prandtl number beyond 10(2) match with those for infinite Prandtl number.

Thermosolutal convection at infinite Prandtl number: initial layer and infinite Prandtl number limit

We examine the initial layer problem and the infinite Prandtl number limit of the thermosolutal convection, which is applicable to magma chambers. We derive the effective approximating system of the Boussinesq system at large Prandtl number using two time scale approach [M. Holmes, Introduction to Perturbation Methods, Springer, New York, 1995, A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, Vol. 9, New York, American Mathematical Society, Providence, RI, 2003]. We show that the effective approximating system is nothing but the infinite Prandtl number system with initial layer terms. We also show that the solutions of the Boussinesq system converge to solutions of the effective approximating system with the convergence rate of O(ϵ).

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-Benard 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(ɛm+1/2) and the optimal convergence rates O(ɛm+1) are obtained respectively. This improves the result of J.G. SHI.

Thermal perturbations caused by large impacts and consequences for mantle convection

We examine the effects of thermal perturbations on a convecting layer of incompressible fluid with uniform viscosity in the limit of infinite Prandtl number, for two upper boundary conditions (free‐ and no‐slip) and heat sources (100% volumetric heating and 100% bottom heating) in 2‐D Cartesian finite element simulations. Small, low‐temperature perturbations are swept into nearby downflows and have almost no effect on the ambient flow field. Large, high‐temperature perturbations are rapidly buoyed and flattened, and spread along the layer's upper boundary as a viscous gravity current. The spreading flow severs and displaces downwellings in its path, and also thins and stabilizes the upper thermal boundary layer (TBL), preventing new instabilities from growing until the spreading motion stops. A return flow driven by the spreading current displaces the roots of plumes toward the center of the spreading region and inhibits nascent plumes in the basal TBL. When spreading halts, the flow field is reorganized as convection reinitiates. We obtain an expression for the spreading time scale, ts, in terms of the Rayleigh number and a dimensionless perturbation temperature (Θ), as well as a size (Λ), and a condition that indicates when convection is slowed at a system‐wide scale. We also describe a method for calculating the heat deposited by shock waves at the increased temperatures and pressures of terrestrial mantles, and supply estimates for projectile radii in the range 200 to 900 km and vertical incident velocities in the range 7 to 20 km s−1. We also consider potential applications of this work for understanding the history of early Mars.

Infinite prandtl number convection

We prove an inequality of the type N≤CR 1/3(1+log+ R)2/3 for the Nusselt number N in terms of the Rayleigh number R for the equations describing three-dimensional Rayleigh–Benard convection in the limit of infinite Prandtl number.

Convection in a rotating spherical fluid shell with an inhomogeneous temperature boundary condition at finite Prandtl number

We examine thermal convection in a rotating spherical shell with central gravity and a spatially non-uniformly heated outer surface at two values of the Prandtl number: Pr=7.0, appropriate for water at room temperature, and Pr=0.7, appropriate for air at standard temperature and pressure, by numerical calculation. Four calculations are performed in a sequence: the onset of convection with homogeneous temperature boundary condition, nonlinear boundary-forced steady convection, stability of the forced steady convection to infinitesimal disturbances, and time stepping of subsequent secondary convection. Unlike our previous study of the infinite Prandtl number limit [J. Fluid Mech. 250, 209 (1993)] inertial terms in the equation of motion for moderate Prandtl numbers play a key role in the dynamics. The effects of an inhomogeneous temperature boundary condition on nonlinear convection are illustrated by varying the wavelength and strength of the imposed boundary temperature. It is shown that even a slight inhomogeneity in the thermal boundary condition can lock azimuthally drifting convection and make it stationary, or modify the normal drifting convection rolls to a vacillating structure. In the infinite Prandtl number case, when inertial forces are absent from the equation of motion, resonance occurs when the wavelengths of boundary forcing and natural convection coincide. Fluid inertia destroys this resonance for finite Prandtl number fluids. The same effect reduces in size the stability region where steady convection is locked to the boundary, and steady convection becomes unstable to time-dependent convection. The period of the secondary convection is close to that obtained with uniform temperature boundaries but the spatial structure is dramatically changed, exhibiting vacillations between the wavelength of the boundary temperature and that of the natural convection.

Surface-tension-driven Bénard convention at infinite Prandtl number

Surface-tension-driven convection in a planar fluid layer is studied by numerical simulation of the three-dimensional time-dependent governing equations in the limit of infinite Prandtl number. Emphasis is placed on the spatial scale of weakly supercritical flows and on the generation of small-scale structures in strongly supercritical flows. The decrease of the size of weakly supercritical hexagonal convection cells that we find is in agreement with experimental results. In the case of high Marangoni number, discontinuities of the temperature gradient are formed between convection cells, producing a universal spectrum E ∼ k−3 of the two-dimensional surface temperature field. The possibility of experimental verification is discussed on the basis of shadowgraph images calculated from the predicted hydrodynamic fields.

Vacillatory convection in a rotating spherical fluid shell at infinite Prandtl number

The temporal and spatial behaviour of three-dimensional convection for a rotating spherical fluid shell in the limit of infinite Prandtl number is investigated. The main features of the instabilities and the bifurcation solutions at two representative Taylor numbers are determined

The route to chaos in thermal convection at infinite Prandtl number: 1. Some trajectories and bifurcations

An important issue is whether thermal convection in the Earth's mantle is chaotic and the mechanisms by which this chaos may be generated. In this paper, we conclude that the high Prandtl number/high Raleigh number thermal convection associated with mantle convection is chaotic, and we document the route to chaos. Deterministic chaos was first exhibited by the Lorenz equations, a three‐mode spatial Fourier expansion of the nonlinear thermal convection equations. This transition to chaos is characterized by the two nontrivial fixed points undergoing a reverse Hopf bifurcation into two unstable limit cycles. However, when the limit of infinite Prandtl number is taken for these equations, the fixed points are stable for all Rayleigh numbers. Is the stability of these fixed points a characteristic of infinite Prandtl number convection or is this an artifact of the truncation? To answer this question, we extend the expansion to 12 modes {n = 0, 1, 2; m = 1, 2, 3, 4} where n is the horizontal wave number and m is the vertical wave number to generate a set of 12 ordinary differential equations. We calculate the loci of roots of this system as a function of Rayleigh number, and we calculate several trajectories. At Rayleigh numbers between Ra = 0 and Ra = 4.140×104 we find pitchfork bifurcations to 14 separate nontrivial branches of solution, and we determine their stability by calculating the eigenvalues of the Jacobian matrix along each branch; we thus find 40 Hopf bifurcations. These fixed points and their bifurcations give excellent quantitative agreement with the trajectories, converging to stable fixed points at and below Ra = 4×104, and oscillating aperiodically at and above Ra = 4.5×104, where many of the Hopf bifurcations occur. Our conclusion is that thermal convection at infinite Prandtl number becomes chaotic by means of symmetry‐breaking pitchfork bifurcations and the appearance of the Hopf bifurcations which produce unstable periodic orbits.

Square-pattern convection in fluids with strongly temperature-dependent viscosity

Three-dimensional numerical solutions are obtained describing convection with a square lattice in a layer heated from below with no-slip top and bottom boundaries. The limit of infinite Prandtl number and a linear dependence of the viscosity on temperature are assumed. The stability of the three-dimensional solutions with respect to disturbances fitting the square lattice is analysed. It is shown that convection in the form of two-dimensional rolls is stable for low variations of viscosity, while square-pattern convection becomes stable when the viscosity contrast between upper and lower parts of the fluid layer is sufficiently strong. The theoretical results are in qualitative agreement with experimental observations.

Nonlinear convection in a layer with nearly insulating boundaries

A general class of solutions is studied describing three-dimensional steady convection flows in a fluid layer heated from below with boundaries of low thermal conductivity. Non-linear properties of the solutions are analysed and the physically realizable convection flow is determined by a stability analysis with respect to arbitrary three-dimensional disturbances. The most surprising result is that square-pattern convection is preferred in contrast to two-dimensional rolls that represent the only form of stable convection in a symmetric layer with highly conducting boundaries. The analysis is carried out in the limit of infinite Prandtl number and for a particular boundary configuration. But it is shown that the results hold for arbitrary Prandtl number to the order to which they have been derived and that other assumptions about the boundaries require only minor modifications as long as their thermal conductance remains low.