Model Of Thermal Convection Research Articles

Thermal plume turbulence

A two-dimensional Boussinesq model of thermal convection is investigated by numerical simulations. The turbulent heat transport is mainly due to thermal plumes. It is shown that their thin, wrinkled interfaces control the statistics of large temperature excursions. Away from the interfaces, the temperature fluctuations are strongly suppressed and the field in those regions is well mixed. Their statistical signature is in the low-order moments of the temperature increments, whose scaling exponents depend linearly on the order.

Zonal flow driven by deep convection in the major planets

The cause of the fast zonal winds on the large gas planets is disputed. To investigate their possible origin by convection in the planets' molecular hydrogen layer, I study simple numerical models of thermal convection in a rotating sphere at low Ekman numbers for Rayleigh numbers up to 100 times critical. With stress‐free boundaries and high Rayleigh number it is possible to reproduce the observed magnitudes of the zonal and non‐zonal flow components at the surfaces of Jupiter and Saturn. While only three flow bands are found for a thick convecting shell, the number increases to seven for a shell with an inner‐to‐outer radius of 0.6.

Magma flow within the Tavares pluton, northeastern Brazil: Compositional and thermal convection

Crystallization coupled with gravity removal of depleted interstitial melt has long been recognized as a mechanism of magma differentiation. Similarly, heat released by synplutonic basaltic magma intrusions has long been recognized as capable of driving convection in granite chambers. Direct evidence of these processes has seldom been described in granites. In the Tavares pluton, we mapped a number of melt extraction structures from pores of a crystal-liquid mush (an effectively solid magma where crystals form an interconnected skeleton) and a variety of flow structures such as (1) meter-scale tear- or mushroom-shaped blobs representing within-chamber diapirs; (2) meter-scale ellipsoids representing frozen thermal plumes of granite, driven by heat released from disrupted diorite intrusions; and (3) ladder dikes and snail structures representing cross sections of several superposed cylindrical magma channels (possibly feeders of diapirs and plume heads). A fundamental feature of the structures in the Tavares pluton is that they are well delineated by mafic schlieren developed at active channel margins. We postulate a new model for the origin of marginal schlieren, which combines shear flow sorting and melt escape from the flowing magma into an effectively solid surrounding mush. Extraction structures (representing melt extraction from mush pores into melt pockets) and schlieren (representing regions where melt escaped into surrounding mush pores) are both favored by magmas that form an interconnected solid framework at low crystal fractions (∼50%), because these mushes are ductile and permeable. Favorable magmas are those with a high wetting angle between melt and solid (∼60°) and a propitious crystal size and shape distribution. We propose a model of compositional and thermal convection that accounts for all described structures.

Three-Dimensional Model of Thermal Convection in an Anisotropic Porous Medium Bounded by Two Horizontal Coaxial Cylinders

The results of calculating the convective flow in cylindrical porous interlayers are presented as functions of the Rayleigh number, the thickness of the interlayer, and the three-dimensional permeability anisotropy for various methods of specifying the temperature on the cylindrical surfaces. The influence of the three-dimensional effects on the flow structure and heat transfer are analyzed. The existence of single-vortex and multivortex convection regimes is established and the conditions of transition to oscillatory and unsteady flow regimes are investigated in terms of the basic parameters.

Early formation and long-term stability of continents resulting from decompression melting in a convecting mantle

The origin of stable old continental cratonic roots is still debated. We present numerical modelling results which show rapid initial formation during the Archaean of continental roots of ca. 200 km thick. These results have been obtained from an upper mantle thermal convection model including differentiation by pressure release partial melting of mantle peridotite. The upper mantle model includes time-dependent radiogenic heat production and thermal coupling with a heat reservoir representing the Earth's lower mantle and core. This allows for model experiments including secular cooling on a time-scale comparable to the age of the Earth. The model results show an initial phase of rapid continental root growth of ca. 0.1 billion year, followed by a more gradual increase of continental volume by addition of depleted material produced through hot diapiric, convective upwellings which penetrate the continental root from below. Within ca. 0.6 Ga after the start of the experiment, secular cooling of the mantle brings the average geotherm below the peridotite solidus thereby switching off further continental growth. At this time the thickness of the continental root has grown to ca. 200 km. After 1 Ga of secular cooling small scale thermal instabilities develop at the bottom of the continental root causing continental delamination without breaking up the large scale layering. This delaminated material remixes with the deeper layers. Two more periods, each with a duration of ca. 0.5 Ga and separated by quiescent periods were observed when melting and continental growth was reactivated. Melting ends at 3 Ga. Thereafter secular cooling proceeds and the compositionally buoyant continental root is stabilized further through the increase in mechanical strength induced by the increase of the temperature dependent mantle viscosity. Fluctuating convective velocity amplitudes decrease to below 10 mma −1 and the volume average temperature of the sub-continental convecting mantle has decreased ca. 340 K after 4 Ga. Surface heatflow values decrease from 120 to 40 mW m −2 during the 4 Ga model evolution. The surface heatflow contribution from an almost constant secular cooling rate was estimated to be 6 mW m −2, in line with recent observational evidence. The modelling results show that the combined effects of compositional buoyancy and strong temperature dependent rheology result in continents which overall remain stable for a duration longer than the age of the Earth. Tracer particles have been used for studying the patterns of mantle differentiation in greater detail. The observed ( p, T, F, t)-paths are consistent with proposed stratification and thermo-mechanical history of the depleted continental root, which have been inferred from mantle xenoliths and other upper mantle samples. In addition, the particle tracers have been used to derive the thermal age of the modelled continental root, defined by a hypothetical closing temperature.

Mantle convection and the dynamic geoid of Mars

This study compares different 3D spherical models of thermal convection in the Martian mantle. The best fit to the Tharsis bulge is obtained if the endothermic phase transition to perovskite occurs closely above the core‐mantle boundary. In this case, a large‐scale flow develops and the associated geoid anomaly explains approximately half of the observed long‐wavelength geoid. The shallower exothermic phase transitions have little influence on the planform of the flow and on the shape of the geoid. Previous studies that included exothermic transitions were biased by effects of a soft lithosphere, which introduces a large‐scale mantle flow even without a deep endothermic phase transition. However, such models predict a continous resurfacing and a dominant geoid low, which is in conflict with observations.

Entropy production and dynamical complexity in a low‐order atmospheric model

AbstractLorenz's classical model of thermal convection is analysed from the standpoint of irreversible thermodynamics. the entropy production, describing how dissipative processes are operating within the system, is expressed in terms of the model variables. It is subsequently evaluated in different parts of the attractor and the values so obtained are compared with various indicators of the complexity of the dynamics, such as the local Lyapunov exponents and a local generalization of the Kolmogorov entropy. As it turns out, dissipative processes are most efficient when the number of locally stable directions increases. In particular, when the local Kolmogorov entropy is close to zero (three stable directions) the system can dissipate over the entire spectrum of values available. Finally, it is shown that the evolution on the attractor does not correspond to any extremal property of the entropy production.

Mode interactions in Rayleigh-Bénard convection

A long-wave model for two-dimensional thermal convection is examined. This model contains both quadratic and cubic nonlinearity unless the problem is symmetrical about the horizontal mid-plane, in which case only the cubic nonlinearity is present. In the asymmetrical case a strong interaction between modes with wavenumbers k and 2 k is responsible for “wavenumber gaps” - regions in parameter space where steady states of a given spatial period cease to exist. In both the symmetrical and asymmetrical cases a 1:3 resonance generates multiple steady solutions of a given spatial period. We calculate steady states and determine their stability. For the asymmetrical case we also solve the initial-value problem and find stable travelling waves, stable modulated travelling waves and stable heteroclinic cycles, as predicted by an analysis of the 1:2 resonance.

A convective model of water flow in Mururoa basalts

Even long after the end of volcanic activity, the background geothermal flux of Mururoa atoll (French Polynesia) maintains fluid convection. We present evidences that interstitial water is continuously renewed in the carbonate platform, as well as in the volcanic basement. In the carbonate rocks, the presence of a karst system allows convective fluxes high enough for the thermal equilibration of the formation with the ocean around. On the contrary, convection in the volcanic basement is, in most places, too slow to cause a measurable disturbance of temperature profiles. Thermal convection models indicate that the average permeability of the volcanic basement cannot be more than a few 10 mD (10 −14 m 2), implying a residence time of more than 10,000 years. The concentration of Sr in porewaters is used as an indicator of the rock/water ratio and of the residence time of the fluid. Considering the measured permeabilities and the estimated rates of reaction, residence times of more than 1 My, corresponding to average permeabilities of less than 10 −16 m 2, are unlikely in the studied upper kilometer of the volcano. However, the extrapolation of the rates of dissolution for basaltic glass measured in the laboratory to in situ conditions apparently leads to overestimate the rates of reaction. Chemically reactive surface area per volume of fluid is a critical parameter in this extrapolation and its value is dependent on the method used to measure it. Although it may not be the only explanation, the discrepancies can be caused by the presence of clays in conduits for fluid flow and as a replacement product of glass. Comparing our results with studies of Quaternary basalts in Iceland, the 10 Ma alteration history of the Mururoa basalt results in a decrease of the permeability of the aquifers by several orders of magnitude, but does not cause a large change of the chemically reactive surface area.

The influence of surface topography on rotating convection

Busse's annulus is considered as a model of thermal convection inside the Earth's liquid core. The conventional tilted base and top are modified by azimuthal sinusoidal corrugations so that the effects of surface topography can be investigated. The annulus rotates rapidly about its axis of symmetry with gravity directed radially inwards towards the rotation axis. An unstable radial temperature gradient is maintained and the resulting Boussinesq convection is considered at small Ekman number. Since the corrugations on the boundaries cause the geostrophic contours to be no longer circular, strong geostrophic flows may be driven by buoyancy forces and damped by Ekman suction. When the bumps are sufficiently large, instability of the static state is dominated by steady geostrophic flow with the convection pattern locked to the bumps. As the bump size is decreased, oscillatory geostrophic flow is possible but the preferred mode is modulated on a long azimuthal length scale and propagates as a wave eastwards. This mode only exists in the presence of bumps and is not to be confused with the thermal Rossby waves which are eventually preferred as the bump height tends to zero. Like thermal Rossby waves, the new modes prefer to occupy the longest available radial length scale. In this long-length-scale limit, two finite-amplitude states characterized by uniform geostrophic flows can be determined. The small-amplitude state resembles Or & Busse's (1987) mean flow instability. On losing stability the solution jumps to the more robust large-amplitude state. Eventually, for sufficiently large Rayleigh number and bump height, it becomes unstable to a long-azimuthal-length-scale travelling wave. The ensuing finite-amplitude wave and the mean flow, upon which it rides, are characterized by a geostrophic flow, which is everywhere westward.

Asymptotic analysis of convective flows in a rotating spheroid for vanishing viscosity and thermal conductivity

The six-dimensional model of viscous fluid thermal convection in a uniformly rotating ellipsoid [1] is studied. The limiting case in which the viscosity and thermal conductivity approach zero while the Prandtl number Pr remains finite is considered. Using both asymptotic and numerical methods, it is shown that for Pr > 2 the attractor is a two-dimensional invariant torus or a limit cycle; the corresponding convective flows are either quasiperiodic, with two basic frequencies, or periodic. It is also shown that resonances on the torus play a dominant role in the breakdown of two-dimensional toruses with increasing viscosity and thermal conductivity.

Dynamics of fine scale variables versus averaged observables in a simplified thermal convection model

The dynamical properties of time‐ and space‐averaged observables of a simplified one‐dimensional thermal convection model are explored and compared to those of the fine scale variables. It is found that averaging reduces the domain of variability of the system, favors persistence of correlations in time and space, and reduces the “effective” dimensionality of the underlying attractor. Furthermore, space and time averages display enhanced predictability, characterized by an error growth rate smaller than the one of the fine scale variables.

Effects of curvature, aspect ratio and plan form in two- and three-dimensional spherical models of thermal convection

Abstract Three-dimensional models of thermal convection in a spherical shell are presented for five different cases, each characterized by a unique ratio, f, of the radii of the inner and outer bounding surfaces. These solutions are compared to comparable two-dimensional solutions in axisymmetric spherical, cylindrical and Cartesian coordinates. All solutions were obtained with a Rayleigh number of 105, stress free, isothermal boundaries and no internal heating in a constant property Boussinesq fluid of infinite Prandtl number. Similarities and differences between three-dimensional and two-dimensional curvilinear models are discussed in terms of scales and stability of the flow patterns, mean radial temperature profiles and heat transport. It is shown that diagnostic statistics such as mean temperature and Nusselt number may be scaled from one degree of curvature to another for both three- and two-dimensional curvilinear models, provided the aspect ratio and plan form of the flow solutions are comparable. The mean temperature is found to be sensitive to curvature and plan form but not to aspect ratio, while the Nusselt number is found to be sensitive to curvature and aspect ratio but not to the plan form of the flow.

From Short-Scale Atmospheric Variability to Global Climate Dynamics: Toward a Systematic Theory of Averaging

Abstract Traditionally, climate is defined by the properties of the averages of the meteorological fields over an appropriate time interval. In this paper the properties of the time-averaged observables of a red noise atmosphere and of a simplified model of thermal convection are investigated both analytically and numerically and are compared to those of the original finescale variables. It is shown that averaging tends to reduce the domain of variability and the attractor dimension favors persistence of initial correlations and enhances predictability. The implications of these findings in the real-world atmosphere are briefly assessed.

Strong spatial interactions with 1:1 resonance: a three-layer convection problem

The resonant interactions between two spatial structures are investigated, using a three-layer thermal convection model as a reference system with translation and reflection symmetry. In the asymptotic limit of a vanishingly small central fluid layer it is shown that when the depth ratio of the outermost fluid layers are equal, two spatially distinct modes of convection can occur with preferred horizontal wavenumber in the ratio 1:1. The normal form equations are derived and the nonlinear temporal evolution of a disturbance consisting of both these modes is considered. It is shown that the multiple bifurcation is a codimension-4 phenomenon and that nonlinear convection resulting from steady state interactions involve both steady mixed mode states, and oscillatory states in the form of standing, travelling and modulated waves. A detailed description of the possible pattern transitions between these states is given.

Effects of the core cooling on the internal dynamics and thermal evolution of terrestrial planets

The effects of thermal coupling of the core and mantle on the thermal evolution and mantle dynamics of terrestrial planets are investigated using three‐dimensional thermal convection models in a spherical shell. Three pairs of models are presented to study mantle models of different viscosities and internal heating. In each pair, one model adopts a constant temperature at the core‐mantle boundary, and the other allows the core to thermally couple to the mantle and cool. In the first and second pairs, the mantle is heated only from below, whereas in the third pair, internal heating is due to radioactive elements. The mantle viscosity of the first and third pairs is 1021 Pa s, while the mantle of the second pair has a viscosity of 1022 Pa s. The thermal coupling of the core and mantle has strong effects on the thermal boundary layer near the core‐mantle boundary but only minor effects on the thermal boundary layer near the surface. It cools the core, substantially weakens the thermal boundary layer near the core‐mantle boundary, and thus reduces the vigor of convection in the mantle. The models with cooling cores result in a substantially colder and relatively less dynamic mantle than the models with a fixed temperature at the core‐mantle boundary which overestimates temperatures in the mantle.

Predictability experiments on a simplified thermal convection model: The role of spatial scales

The amplification of small initial errors in connection with mesoscale atmospheric variability is studied on a simplified thermal convection model. It is found that when the system is perturbed uniformly over the physical space, the initial regime of error dynamics follows a pathway far from the traditional exponential function whose rate is the largest positive Lyapunov exponent. It comprises two stages: (1) a transient decrease of the initial error as a result of the displacement of the system out of its attractor and (2) a superexponential increase whose origin is in the multifractal character of the attractor. Numerical experiments in which the initial error is introduced selectively in a particular scale reveal that perturbations at larger scales grow faster than those at the intermediate and small scales. A qualitative explanation of this unexpected behavior based on intrinsic properties of the dynamics such as the local Lyapunov vectors is advanced.

A comparative study of parameterized and full thermal-convection models in the interpretation of heat flow from cratons and mobile belts

SUMMARY Heat flow from Archean cratons worldwide is typically lower than from younger mobile belts surrounding them. The contrast in heat flow between cratons and mobile belts has been attributed in previous studies to the greater thermal resistance of thicker lithosphere beneath the cratons which impedes the flow of mantle heat through the cratons and forces more mantle heat to escape through thinner mobile belt lithosphere. This interpretation is based on thermal models which employ a parameterized convection algorithm to calculate heat transfer in the sublithospheric mantle. We test this interpretation by comparing thermal models constructed using the parameterized convection scheme with models developed using an algorithm for full thermal convection. We show that thermal models constructed using the two different convection algorithms yield similar surface heat flow and thermal structure to moderate depths within the lithosphere. Therefore, we conclude that the interpretation of the heat-flow observations in terms of thicker lithosphere under Archean cratons than under mobile belts is robust in the sense that surface heat flow is not sensitive to the details of heat transfer within the convecting mantle and how deep mantle heat is delivered to the base of the lithosphere.

The endo-upwelling concept: from geothermal convection to reef construction

Measurements of interstitial waters in deep wells (0–500 m) in Mururoa and in shallow bores (0–35 m) in Tikehau (French Polynesia) have shown high contents of dissolved nutrients and relatively low salinities, implying a deep oceanic origin. Studies of specific thermal fields within these atolls and carbonate platforms have led to models of deep oceanic water circulation by convection resulting from upward geothermal heat flow. Cold, low-salinity, low-pH, high-CO2, nutrient-rich deep oceanic water pervades the permeable atoll, where it loses density by heating related to geothermal heat flow, subsequently rising to seep out through the outer rim. New nutrients are thus continuously provided, enabling the reef-building community (algae+corals) to thrive in optimal conditions for photosynthesis and carbonate precipitation. This “geothermal endo-upwelling” concept is defined and its role in atoll trophic networks is emphasized as a necessary and sufficient process for reef net production and organic matter exportation. The endo-upwelled flow can then be viewed as a key factor for internal diagenesis, as early cementation of reef framework and dolomitization of deep limestone. Generalization of this interstitial, deep seawater circulation is discussed and linked to previously described thermal convection models in Florida or raised atolls.

Study of a model of thermal convection in cylindrical containers

We propose a generalized Swift-Hohenberg model equation that shows different planforms as observed in experiments in convective layers heated from below. We use this model to study the natural planform when some symmetric inhomogeneities are present on the sidewall boundary conditions. A stability analysis in the 1D case allow us to determine when the rolls parallel to the sidewalls remain stable or perturbations perpendicular to these sidewalls can appear. Having this simple analysis in mind we study numerical solutions of the model equation in 2D considering those inhomogeneities. This shows that, even near threshold and for sufficiently high inhomogeneities, a pattern of concentric rolls can be a stable solution. However, it is replaced by a pattern with a different symmetry when these inhomogeneities decay or are suppressed. This is in qualitative agreement with experimental observations.