Infinite Prandtl Number Convection Research Articles

Dynamics of Density Patches in Infinite Prandtl Number Convection

Dynamics of Density Patches in Infinite Prandtl Number Convection

Structure of bifurcated solutions of two-dimensional infinite Prandtl number convection with no-slip boundary conditions

We consider the two-dimensional infinite Prandtl number convection problem with no-slip boundary conditions. The existence of a bifurcation from the trivial solution to an attractor ΣR was proved by Park [13]. The no-stress case has been examined in [14]. We prove in this paper that the bifurcated attractor ΣR consists of only one cycle of steady state solutions and it is homeomorphic to S1. By thoroughly investigating the structure and transitions the solutions of the infinite Prandtl number convection problem in physical space, we confirm that the bifurcated solutions are indeed structurally stable. We show what the asymptotic structure of the bifurcated solutions looks like. This bifurcation analysis is based on a new notion of bifurcation, called attractor bifurcation, and structural stability is derived using geometric theories of incompressible flows.

Localized subcritical convective cells in temperature-dependent viscosity fluids

Numerical simulations of infinite Prandtl number convection in the stagnant lid regime of temperature-dependent viscosity convection demonstrate the existence of spatially localized, stable convective cells below the critical Rayleigh number (subcritical convection). These solutions are in stark contrast to the usual, supercritical, convective planforms, where convective cells form in the entire layer. The isolated cell has a shape of an axisymmetric dome with an upwelling at the center and thus appears as a very weak plume. Formation of these structures requires subcritical conditions and a localized initial temperature perturbation but does not require any spatial heterogeneity in the material properties or the heat flux. When several localized plumes form, they tend to attract to each other and form stable clusters. This type of subcritical convection may play a role in the formation and longevity of localized features on planetary bodies, including the crustal dichotomy and Tharsis region on Mars and the asymmetric pattern of volcanism on Mercury.

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) [5,6,25] 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 S 1 . 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].

Large Rayleigh number thermal convection: Heat flux predictions and strongly nonlinear solutions

We investigate the structure of strongly nonlinear Rayleigh–Bénard convection cells in the asymptotic limit of large Rayleigh number and fixed, moderate Prandtl number. Unlike the flows analyzed in prior theoretical studies of infinite Prandtl number convection, our cellular solutions exhibit dynamically inviscid constant-vorticity cores. By solving an integral equation for the cell-edge temperature distribution, we are able to predict, as a function of cell aspect ratio, the value of the core vorticity, details of the flow within the thin boundary layers and rising/falling plumes adjacent to the edges of the convection cell, and, in particular, the bulk heat flux through the layer. The results of our asymptotic analysis are corroborated using full pseudospectral numerical simulations and confirm that the heat flux is maximized for convection cells that are roughly square in cross section.

Bound on vertical heat transport at large Prandtl number

We prove a new upper bound on the vertical heat transport in Rayleigh–Bénard convection of the form c Ra 1 3 ( ln Ra ) 2 3 under the assumption that the ratio of Prandtl number over Rayleigh number satisfies Pr Ra ≥ c 0 where the non-dimensional constant c 0 depends on the aspect ratio of the domain only. This new rigorous bound agrees with the (optimal) Ra 1 3 bound (modulo logarithmic correction) on vertical heat transport for the infinite Prandtl number model for convection due to Constantin and Doering [P. Constantin, C.R. Doering, Infinite Prandtl number convection, J. Stat. Phys. 94 (1) (1999) 159–172] and Doering, Otto and Reznikoff [C.R. Doering, F. Otto, M.G. Reznikoff, Bounds on vertical heat transport for infinite Prandtl number Rayleigh–Bénard convection, J. Fluid Mech. 560 (2006) 229–241]. It also improves a uniform (in Prandtl number) Ra 1 2 bound for the Nusselt number [P. Constantin, C.R. Doering, Heat transfer in convective turbulence, Nonlinearity 9 (1996) 1049–1060] in the case of large Prandtl number.

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).

On upper bounds for infinite Prandtl number convection with or without rotation

Bounds for the bulk heat transport in Rayleigh–Benard convection for an infinite Prandtl number fluid are derived from the primitive equations. The enhancement of heat transport beyond the minimal conduction value (the Nusselt number Nu) is bounded in terms of the nondimensional temperature difference across the layer (the Rayleigh number Ra) according to Nu⩽cRa2/5, where c<1 is an absolute constant. This rigorous upper limit is uniform in the rotation rate when a Coriolis force, corresponding to the rotating convection problem, is included.

Multiple scales in mantle convection

The multiple-scales structure of mantle convection remains largely unknown. A 3D model of convection is used to investigate the effect of a viscosity step on the multiple-scales pattern of an infinite Prandtl number convection for Rayleigh numbers ( Ra) from 10 5 to 1.75×10 7. The planform of the large-scale convection is polygonal and the ascending plumes are quite steady even for a Rayleigh number of 1.75×10 7. When the Rayleigh number exceeds a value of 3×10 5, the upper thermal boundary layer becomes unstable. Small-scale instabilities appear away from the plume and they are advected by the large-scale flow. The pattern of these small-scale flows evolves with the Rayleigh number. They form circular-like waves centered on the axis of the plume for Ra=10 6. An increase of the Rayleigh number, above Ra=5×10 6, leads to the generation of cold drops inside the circular waves. Finally, for Ra=1.75×10 7, the circular waves disappear at some distance from the plume axis. Then, the small-scale flow consists of cold drops. As they are advected by the large-scale flow, the cold drops mature inside cold radial ‘gutters’. This superimposition of small-scale instabilities on the large-scale flow clearly results from the viscosity jump. The implications of these models appear more straightforward in the case of Venus than of the Earth due to the absence of moving plates in the model. The existence of small-scale convection around the plumes could explain the presence of volcanoes in the plains around the equatorial highlands of Venus.

The timescales of plume generation caused by continental aggregation

To understand the thermal evolution of the mantle following the aggregation of non-subductable thick continental lithosphere, we study a numerical model in which a supercontinent, simulated by high viscosity raft, HVR, covers a part of the top surface of a convection layer. We model infinite Prandtl number convection either in a three-dimensional (3D) spherical shell, 3D rectangular box (aspect ratios: 8 and 4) or two-dimensional (2D) rectangular box (aspect ratio: 8) and except for the HVR, we specify a constant viscosity. The HVR, which has a viscosity higher than that of its surrounding, is instantaneously placed on the top surface of a well-developed convection layer and its position is fixed. Our results from 3D spherical shell cases with and without phase transitions show the emergence of a large plume characterized by a long wavelength thermal anomaly (a degree one pattern) for a Pangea-like geometry. We analyze the volume averaged temperature under the HVR (=〈 T C〉) the remaining (oceanic) area (=〈 T O〉) and total area (=〈 T M〉) to determine the timescale of plume generation. The difference between 〈 T C〉 and 〈 T O〉(=Δ T CO) and 〈 T M〉 show the existence of two characteristic timescales.Δ T CO exhibits an initial rapid increase and may become constant or continue to gradually increase. Meanwhile, 〈 T M〉 shows a similar behavior but with a longer timescale. We find that these timescales associated with the increase of Δ T CO and 〈 T M〉 can be attributed to the formation of large scale flow (i.e. plume) and response of the whole system to the emplacement of the HVR, respectively. For 3D spherical cases, we find that the timescale of plume generation is 1–2 Gyr, if the Rayleigh number is 10 6. To determine the effects of the viscosity of the HVR, 2D versus 3D modeling and the effects of the internal heating, we have also studied 2D and 3D rectangular box cases. A factor of about two variation exists in the timescale of plume generation. It appears that the timescale becomes greater for a smaller amount of internal heating. This may be attributed to the time-dependent flow caused by the internal heating. For 2D cases, we find that the timescale of the high Rayleigh number (10 7) case is shortened by a factor of three to five when compared to the Ra=10 6 case, which is consistent with the simple boundary layer theory. This may imply that well-developed plumes may arise with the timescale of 0.2 to 0.4 Gyr (for Ra=10 7).

Large eddy simulations of thermal convection at high Rayleigh number

With the vastly improved speed and the new shared-memory architecture of the current massively parallel systems, it is now possible to simulate thermal convection at very high Rayleigh (Ra) number lying in the turbulent regime. However, both the dynamics and the interaction with the turbulence may be simultaneously important. Even when only the large scales are of interest, we cannot simply ignore the smallest scales because of the feedback from strong non-linearities everywhere in the flow. Large eddy simulation (LES), a time-honoured method in engineering fluid mechanics and meteorology, may be the only way to simulate the time-dependent physics in its full complexity, while keeping a reasonable accuracy at the largest scales in highly non-linear geophysical fluid dynamical flows, such as mantle convection in the early Earth, convection inside the Jovian moons and the geodynamo. We have tested here a LES model based on the Smagorinsky assumption for 2-D turbulent convection for a finite Prandtl (Pr) number fluid with Pr = 1, free-slip boundary conditions and an aspect ratio of 3. The subgrid-scale model is only employed for the temperature equation, where the steepest gradients are developed. The same model can also be used for infinite Prandtl number convection. This LES model has been validated by comparison with direct numerical simulation (DNS) for Ra between 108 and 1010 with a grid up to 512 × 1536 points. Statistical properties of the flow based on LES are presented for the probability distribution functions (PDF) in space and also the spectra describing the thermal and kinetic energy distributions for Ra up to 1010.

Generation of plumes under a localized high viscosity lid in 3‐D spherical shell convection

We conduct numerical simulations of incompressible infinite Prandtl number convection in a spherical shell with a single localized high viscosity lid (HVL) on the top surface to understand the possible effects of the continental lithosphere on plume generation. The temperature under the HVL increases rapidly after the emplacement of the HVL on constant viscosity convection with internal and bottom heatings. Later, upwellings at the bottom merge into a large scale flow and a large plume emerges under the HVL. This time‐scale depends on the existence of phase changes. In our model, whose Rayleigh number is 106, the large plume is formed on a time‐scale of 1 Gyr. A simple scaling suggests that this time‐scale may be reduced by a several times, if the Rayleigh number is around 107. Despite the complicated 3‐D thermal structure, it is dominated by an l = 1 pattern controlled by the position of the HVL.

Three-Dimensional Infinite Prandtl Number Convection in a Spherical Shell with Temperature-Dependent Viscosity.

We study the numerical simulations of 3-D inifinite Prandtl number convection with temperature-dependent viscosity in a spherical shell. Control volume method coupled with SIMPLE algorithm is used to solve the basic equations. Both constant and temperature-dependent viscosity cases are studied under the condition that the ratio of the inner to the outer shell radius is 0.55, which is appropriate for mantle convection of the Earth, with only basal heating. For the case of constant viscosity, when the Rayleigh number (Ra) is less than 105, steady state solutions (both cubic and tetrahedral types) are obtained. However, for Ra = 106, we find only a time-dependent solution. For the temperature-dependent viscosity case, we assume that the viscosity changes with temperature exponentially and we can successfully obtain solutions with the viscosity variation up to 10000. For the viscosity contrast of 1000 times and the surface Rayleigh number of Ra0 = 5 × 103, we get the flow patterns which consist of the sheet-like downwelling and small upwellings.

Plume structures in the hard‐turbulent regime of three‐dimensional infinite Prandtl number convection

Numerical simulations of three‐dimensional infinite Prandtl number thermal convection with Rayleigh number (Ra) up to 108 are reported. Convection with Ra higher than 107 is characterized by the appearance of disconnected thermal plumes. The smaller plumes are detached by the currents produced by the larger plumes. The low wavenumber portion of a thermal power spectrum near the boundary layer becomes flat at high Ra, while the spectrum measured in the interior shows a positive slope for low wavenumbers. Differences are found in the thermal spectra between 2D and 3D models.

Infinite Prandtl-number convection in a cube heated from below. Numerical experiment by the pseudospectral method.

Thermal convection of fluids heated from below occurs when the Rayleigh number becomes greater than a critical value. In general, convection gradually evolves to complex, often time-dependent, states as the Rayleigh number increases. The process involves various forms of transition, such as from one convecting pattern to another, steady to time dependent and simply periodic to chaotic oscillation. The present research focuses on an evolution process of infinite Pr convection in a cubical geometry with stress-free boundary conditions. The numerical computations have been carried out up to Ra = 2×105 using an efficient scheme based on the pseudospectral method. With increasing Ra, a convecting pattern starts losing its symmetries at about Ra = 7×104, but does not evolve to time-dependent states. Effects of truncation number on the numerical accuracy are also investigated. For example, a truncation number of N = 12 is adequate at Ra = 104, while N = 16 is needed at Ra = 105. The Nusselt number appears to grow at a rate of Ra1/3 when Ra exceeds 104, suggesting the formation of distinct thermal boundary layers.

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.

Three‐dimensional infinite Prandtl number convection in one and two layers with implications for the Earth's gravity field

The three‐dimensional planform of infinite Prandtl number convection is studied in one and two layers, with a numerical code which combines spectral and finite difference methods. The equations are solved in rectangular cavities supposing mirror symmetry on the sides. The one‐layer experiments serve mainly for testing the code. These experiments reveal the outstanding importance of the initial and boundary conditions in the evolution of the flow pattern. The main emphasis of this paper is on two‐layer convection models, with implications for the Earth's mantle and its gravity anomaly field. In this case, the interface between the two layers is fixed at a given depth, and coupling of the two circulations is ensured by the continuity of velocities and horizontal stress across the interface. The two‐layer cases have been run with Rayleigh numbers up to 130 times the critical value, the thickness of the lower layer being twice that of the upper one. The ratio of the lower layer viscosity to the upper one is gradually increased. When this ratio is small (between 1 and 10), viscous coupling prevails, and bimodal or square cells formed in the lower layer induce similar, but counterrotating cells in the thin upper layer. When the viscosity ratio is high (several hundred), thermal coupling becomes dominant: again, the convective structure of the lower layer is more or less duplicated in the upper layer, but now the flows of both layers rotate in the same sense. In these situations the gravity anomalies are positively correlated with the topography of the upper surface. When the viscosity ratio is moderate (e.g., 25), the flow direction of the rolls in the thin upper layer becomes perpendicular to that observed in the lower layer, leading to a balance between the conflicting effects of the viscous and thermal couplings. This allows the upper rolls to have an aspect ratio close to 1. In this case, gravity is positevely correlated with deflection of the interface, but no more correlation can be seen with surface topography, which is determined by small‐scale flow of the upper layer. These results suggest that the gravity and bathymetry anomalies of the Earth should be either positively correlated or not correlated at all. This has to be contrasted with earlier findings of two‐dimensional modeling [Cserepes and Rabinowicz, 1985] which allow the possibility of having anticorrelation between gravity and topography.

Geoid and topography for infinite Prandtl number convection in a spherical shell

Geoid anomalies and surface and lower‐boundary topographies are calculated for numerically generated thermal convection in an infinite Prandtl number, Boussinesq, axisymmetric spherical fluid shell with constant gravity and viscosity. Calculations were done for heating entirely from below (HFB) and entirely from within (HFW). Convective solutions were obtained for Rayleigh numbers Ra up to 20 times the critical Ra in the HFB case and 27 times critical in the HFW case. Bifurcations that occur with varying Ra are associated with different convective patterns. Boundary topography is positive at upwellings and negative at downwellings. Absolute topography is greater at the poles than at the equator due to poleward compression of the flow field; lower‐boundary topography is greater than at the surface due to radial compression of the flow field. The geoid is primarily dominated by the gravitational potential of surface masses due to surface boundary deformations; hence the geoid parallels surface undulations. In general, boundary deformation increases with increasing cell wavelength. This property may elucidate whether core‐mantle boundary (CMB) topography is due to mantle convection or mantle dregs (chemically dense material) which are only indirectly of dynamic origin, and hence are unlikely to have the same wavelength dependence in its associated topography. The ratios of boundary deformation and geoid to αΔTd (α, coefficient of thermal expansion; ΔT, characteristic temperature drop across the fluid shell; and d, shell thickness) maximize with increasing Ra at approximately 10 times Ra critical and then steadily decrease out to the maximum Ra calculated. Dimensionless geoid and topography in the HFB case are approximately 5 times greater than in the HFW case. At the highest Rayleigh numbers investigated, dimensionless peak‐to‐trough amplitudes of geoid and topography are weak functions of Ra, which lends some justification to estimating the dimensional values of these quantities with Earthlike parameters. For parameter values characteristic of the Earth's mantle, the geoid and upper and lower boundary topograpies are approximately 1 km, 10 km, and 15 km for the HFB case, and approximately 200 m, 2 km, and 3 km for the HFW case, respectively. The HFW values correlate with geophysical data more closely than the HFB values, suggesting a predominance of internal heating in the mantle. Although this result is qualfied by the low Rayleigh numbers and axisymmetry of the model calculations, our study emphasizes that dynamically induced topography and geoid are sensitive to the mode of heating in the Earth's mantle.

Character and stability of axisymmetric thermal convection in spheres and spherical shells

The influence of shell size and mode of heating on the behavior and stability of axisymmetric, infinite Prandtl number convection in a spherical geometry is studied. Heating from within and below features convection onset governed by a self-adjoint system of equations and boundary conditions. For heating only from within or from below, linearized equations and boundary conditions are non-self-adjoint. Identification of the parameter which initiates the departure from self-adjointness, together with the properties of the self-adjoint solution, provide a basis for calculating the heat transfer characteristics of the non-self-adjoint situations. The investigations are an effort to develop a model for heat transfer in planetary interiors. Further development of the technique by modifying the Galerkin method by the introduction of diagonal mode truncation is suggested to permit the consideration of higher values of the Rayleigh numbers, i.e., those more commensurate with terrestrial planet mantles.

Parameterizations for the horizontally averaged temperature of infinite Prandtl number convection

Parameterizations for the horizontally averaged temperature of infinite Prandtl number numerical convection experiments are determined. The types of convection considered result from uniform heating from below, uniform heating from within, and isothermal boundaries with a higher temperature applied to the bottom. The method used is based on fitting simple analytical expressions to suitably non‐dimensionalized and rescaled numerical solutions, thus providing a concise and in most cases accurate description of the horizontally averaged temperature.