Showing papers on "Convection published in 1982"

The Dependence of Numerically Simulated Convective Storms on Vertical Wind Shear and Buoyancy

Abstract: The effects of vertical wind shear and buoyancy on convective storm structure and evolution are investigated with the use of a three-dimensional numerical cloud model. By varying the magnitude of buoyant energy and one-directional vertical shear over a wide range of environmental conditions associated with severe storms, the model is able to produce a spectrum of storm types qualitatively similar to those observed in nature. These include short-lived single cells, certain types of multicells and rotating supercells. The relationship between wind shear and buoyancy is expressed in terms of a nondimensional convective parameter which delineates various regimes of storm structure and, in particular, suggests optimal conditions for the development of supercell type storms. Applications of this parameter to well-documented severe storm cases agree favorably with the model results, suggesting both the value of the model in studying these modes of convection as well as the value of this representation i...

A model of the high‐latitude ionospheric convection pattern

Abstract: Mathematical expressions have been constructed that allow the large-scale global convection characteristics of the high-latitude ionosphere to be reproduced. The model contains no discontinuities in the ion convection velocity and as such should be useful in F region chemical models. The number of variables in the model allow such features as the dayside throat and the Harang discontinuity to be modeled. The applicability of the model to magnetospheric physics is limited by the exclusion of large-magnitude small-scale flow features associated with discrete arcs and by the inability of the model to produce separate flow cells at the same local time.

Effects of molecular diffusion and of thermal expansion on the structure and dynamics of premixed flames in turbulent flows of large scale and low intensity

Abstract: To study effects of flow inhomogeneities on the dynamics of laminar flamelets in turbulent flames, with account taken of influences of the gas expansion produced by heat release, a previously developed theory of premixed flames in turbulent flows, that was based on a diffusive-thermal model in which thermal expansion was neglected, and that applied to turbulence having scales large compared with the laminar flame-thickness, is extended by eliminating the hypothesis of negligible expansion and by adding the postulate of weak-intensity turbulence. The consideration of thermal expansion motivates the formal introduction of multiple-scale methods, which should be useful in subsequent investigations. Although the hydrodynamic-instability mechanism of Landau is not considered, no restriction is imposed on the density change across the flame front, and the additional transverse convection correspondingly induced by the tilted front is described. By allowing the heat-to-reactant diffusivity ratio to differ slightly from unity, clarification is achieved of effects of phenomena such as flame stretch and the flame-relaxation mechanism traceable to transverse diffusive processes associated with flame-front curvature. By carrying the analysis to second order in the ratio of the laminar flame thickness to the turbulence scale, an equation for evolution of the flame front is derived, containing influences of transverse convection, flame relaxation and stretch. This equation explains anomalies recently observed at low frequencies in experimental data on power spectra of velocity fluctuations in turbulent flames. It also shows that, concerning the diffusive-stability properties of the laminar flame, the density change across the flame thickness produces a shift of the stability limits from those obtained in the purely diffusive-thermal model. At this second order, the turbulent correction to the flame speed involves only the mean area increase produced by wrinkling. The analysis is carried to the fourth order to demonstrate the mean-stretch and mean-curvature effects on the flame speed that occur if the diffusivity ratio differs from unity.

Low-Gravity Fluid Flows

Abstract: The behavior of fluids in micro-gravity conditions is examined, with particular regard to applications in the growth of single crystals. The effects of gravity on fluid behavior are reviewed, and the advent of Shuttle flights are noted to offer extended time for experimentation and processing in a null-gravity environment, with accelerations resulting solely from maneuvering rockets. Buoyancy driven flows are considered for the cases stable-, unstable-, and mixed-mode convection. Further discussion is presented on g-jitter, surface-tension gradient, thermoacoustic, and phase-change convection. All the flows are present in both gravity and null gravity conditions, although the effects of buoyancy and g-jitter convection usually overshadow the other effects while in a gravity field. Further work is recommended on critical-state and sedimentation processes in microgravity conditions.

Global circulation and temperature structure of thermosphere with high‐latitude plasma convection

Abstract: This paper examines the effect of magnetospheric convection in modifying the diurnal neutral gas temperature distribution and circulation of the thermosphere for equinox conditions, using NCAR's thermospheric general circulation model. Numerical experiments are presented to illustrate the differences in temperature structure and circulation due to (1) solar heating alone, (2) solar heating plus plasma convection with coincident geographic and geomagnetic poles, and (3) solar heating plus plasma convection with displaced poles. The high-latitude plasma convection has an important influence on the global thermospheric structure and circulation. Plasma convection with displaced poles introduces a universal time dependence to the circulation and temperature structure; similar patterns occur in the northern and southern hemisphere, with a 12-hour time difference. Magnetospheric convection drives a largely rotational, nondivergent, double-vortex wind system at F region altitudes that can attain velocities greater than 500 m s−1 during moderate levels of geomagnetic activity. These vortices extend downward into the lower thermosphere. However, the cold low-pressure cyclonic circulation near the dawn terminator is much more pronounced than the warm high-pressure anticyclonic circulation in the evening sector.

Two‐dimensional convection in non‐constant shear: A model of mid‐latitude squall lines

Abstract: Numerical simulations of two-dimensional deep convection are analysed using analytical models extended to include shallow downdraughts and non-constant shear. The cumulonimbus are initiated by low-level convergence created by a finite amplitude downdraught. These experiments have constant low-level shear and differ only in the profile of mid-and upper-level winds. Quasi-steady convenction is produced if the mid- and upper-level flow has small shear and the low-level shear is large. The surface precipitation ismaximized for no intial relative relative flow aloft, if stationary, this storm (P(O)) can give prodigious locilized rainfall; P(O) is the two-dimentisonal equivalent of the supercell. These results are placed in context with previous two-dimensional simulations. Attention is drawn to the similiarity with previous two-dimensional simulations. Attention is drawn to the similarity with squall lines in central and eastern U.S.A. Storm P(O) is analysed by construction of time-averaged fields of streamfunction, vorticity, teperature, and height deviation. The smoothness of these fields suggests a conceptual model of the storm dynamics which involves cooperation between distinct charcteristic flows; an overturning updraught, a jump type updraught, a shallow downdraught, a low-level rotor, and a boundary layer. An idealized analytical model is described by solution of the equations for steady convection. These solutions, for the remote flow, are derived from energy conversation, mass continuity and a momentum budget, and they give relationships between the non-dimensional parameteres of the problem. It is apparent that the convection is a high Froude (or low Richardson) number flow demanding the existence of a cross-storm pressure gradient. Inherent in this idealized model is a vortex sheet between updraught and down-draught and it is considered that the dynamical instability of this sheet is related to complexities in the numerical simulation. Furthermore, these results show that in two-dimensions both non-constant shear and a shallow downdraught are necessary to maintain steady convection.

A Severe Frontal Rainband. Part I. Stormwide Hydrodynamic Structure

Abstract: A narrow cold frontal band of intense precipitation is examined by means of triple Doppler radar and supporting observations. As the band passed through the Central Valley of California, it was accompanied by strong gusty winds, electrical activity, tornadoes and pressure jumps. Part I delineates the stormwide kinematic and thermodynamic structure. A highly two-dimensional pre-frontal updraft of 15–20 m s−1 results primarily from intense planetary boundary layer forcing of a low-level jet by the gravity-current propagation mechanism. Maximum updraft speed occurs at 2.1 km and the maximum radar echo depth is 6.6 km. Diabatic cooling, due to melting hydrometers, is proposed as a likely mechanism for control of gravity current depth and maintenance of density contrast together with synoptic-scale cold air advection. Available convective potential energy is shown to be small and kinetic energy of the environmental vertical wind shear is proposed as a likely source of energy on the updraft scale. Torn...

Liquid Water and Active Resurfacing on Europa.

Abstract: Arguments for recent resurfacing of Europa by H2O from a liquid layer are presented, based on new interpretations of recent spacecraft and earth-based observations and revised theoretical calculations. The heat flow in the core and shell due to tidal forces is discussed, and considerations of viscosity and convection in the interior are found to imply water retention in the outer 60 km or so of the silicates, forming a layer of water/ice many tens of km thick. The outer ice crust is considered to be too thin to support heat transport rates sufficient to freeze the underlying water. Observational evidence for the calculations would consist of an insulating layer of frosts derived from water boiling up between cracks in the surface crust. Evidence for the existence of such a frost layer, including the photometric function of Europa and the deposits of sulfur on the trailing hemisphere, is discussed.

Mantle convection as a boundary layer phenomenon

Abstract: Summary. The boundary layer nature of vigorous thermal convection is explored using high resolution numerical solutions to the governing hydrodynamic equations. These solutions are obtained for a series of idealized models of the Earth’s mantle in which the viscosity is assumed to be constant. A detailed analysis of the local energy balance within the horizontal and vertical thermal boundary layers is presented in terms of which a test of the fundamental assumptions of boundary layer theory is provided. The results of this test have important geophysical consequences since the asymptotic predictions of boundary layer theory have been employed extensively in the context of thermal history modelling. Although boundary layer theory closely predicts the correct power-law behaviour of various quantities it does not determine their absolute values accurately. Vertical advection is shown to play an important role in the energy balance within horizontal boundary layers at all Rayleigh numbers. Horizontal and vertical advection dominate the energy balance within vertical plumes while horizontal diffusion plays a very minor role. When heating is partially from within the fluid, vertical advection into the upper thermal boundary layer can produce significant departures in the thermal structure from that found when heating is entirely from below. For a free upper boundary this results in a relative flattening of the variations of surface topography and heat flow across the convection cells. For a constant velocity upper boundary (similar to plate motion) the bathymetry flattens but the heat flow does not; this result agrees with marine observations. Rigidity of the thermal boundary layer below the upper surface is not included explicitly in the model, and it is not known whether the inclusion of this feature in future models would significantly alter the topographic expression. If not, the observed departure of the oceanic bathymetry from a 6 dependence at old ocean floor ages could be attributed to a small amount of internal heating in a mantle-wide convective circulation.

Saturation Point Analysis of Moist Convective Overturning

Abstract: A unified approach to the thermodynamics of cloudy air, cloud-clear air mixing processes, atmospheric thermodynamic equilibrium structure and instability is formulated, using a new concept: the Saturation Point. This permits the representation of mixing processes and virtual potential temperature isopleths for clear and cloudy air on a thermodynamic diagram (a tephigram is used here), and their comparison with the atmospheric stratification. Illustrative examples will be given for evaporative mixing instability and convective equilibrium structure for stratocumulus, cumulus and cumulonimbus convection and convection in the incipient severe storm atmosphere.

Self‐consistent theory of time‐dependent convection in the Earth's magnetotail

Abstract: A self-consistent theory of time-dependent convection in the earth's magnetotail is developed. In contrast to earlier convection models our approach takes into account that the plasma sheet particles are effectively trapped on closed geomagnetic field lines. The results demonstrate that steady state convection, although possible in principle, is unlikely to occur in the earth's magnetotail. In particular, if particle or energy losses are sufficiently small and if the outer lobe field lines have convex shape, a steady state is impossible in the framework of the present polytropic model. Quantitative models for time-dependent convection are constructed. The time dependence introduces important consequences for energy storage, stability, and spatial dependence of the convection electric field. The results are consistent with a quasi-periodic evolution of the tail, where periods of quasi-static compressional convection are followed by phases of dynamic evolution.

Experimental Investigation of Natural Convection in Partitioned Enclosures

Abstract: Natural convection heat transfer within a two-dimensional, partitioned enclosure of aspect ratio 1 was investigated experimentally using a Mach-Zehnder interferometer. The vertical walls were maintained isothermal at different temperatures, while the horizontal walls and the partitions were insulated. Local and average heat-transfer coefficients were determined for the air and carbon dioxide filled enclosures both with and without partitions for Grashof numbers between 1.7×105 and 3.0×106 . Good agreement was found between the results in the present study for the nonpartitioned enclosure and those previously published. The partitions were found to significantly influence the convective heat transfer. Observations of the interferometric fringes indicated that the core region is unsteady, with the unsteadiness occasionally affecting the flow along the vertical isothermal walls, beginning at Grashof numbers as low as 5×105 .

Modes of mantle convection and the removal of heat from the Earth's interior

Abstract: Thermal histories for two-layer and whole-mantle convection models are calculated and presented, based on a parameterization of convective heat transport. The model is composed of two concentric spherical shells surrounding a spherical core. The models were constrained to yield the observed present-day surface heat flow and mantle viscosity, in order to determine parameters. These parameters were varied to determine their effects on the results. Studies show that whole-mantle convection removes three times more primordial heat from the earth interior and six times more from the core than does two-layer convection (in 4.5 billion years). Mantle volumetric heat generation rates for both models are comparable to that of a potassium-depleted chondrite, and thus surface heat-flux balance does not require potassium in the core. Whole and two-layer mantle convection differences are primarily due to lower mantle thermal insulation and the lower heat removal efficiency of the upper mantle as compared with that of the whole mantle.

Stable regions in the Earth's liquid core

Abstract: Summary. Slow cooling of the whole Earth can be responsible for the convection in the core that is required to generate the magnetic field. Previous studies have assumed the cooling rate to be high enough for the whole core to convect. Here we study the effects of a low rate of cooling by assuming the temperature at the base of the mantle to remain constant with an initially entirely molten, adiabatic core. We argue that, in such a situation, convection would stop at the top of the core, and calculate the consequent thermal evolution. A stable, density stratified layer grows downwards from the core mantle boundary reaching a thickness of 100-1000 km in a few thousands of millions of years. There is some geomagnetic evidence to support belief in the existence of such a stable layer.

Heat-Transfer Augmentation in Rod Bundles Near Grid Spacers

Abstract: Heat-transfer augmentation by straight grid spacers in rod bundles is studied for single-phase flow and for post-critical heat flux dispersed flow. The heat transfer effect of swirling grid spacers in single-phase flow is also examined. Governing heat-transfer mechanisms are analyzed, and predictive formulations are established. For single-phase flow, the local heat transfer at a straight spacer and at its upstream or downstream locations are treated separately. The effect of local velocity increasing near swirling spacer is considered. For post critical heat flux (CHF) dispersed flow, the heat transfer by thermal radiation, fin cooling, and vapor convection near the spacer are calculated. The predictions are compared with experimental data with satisfactory agreement.

Convection in pulsating stars. I. Non linear hydro-dynamics.

Abstract: A nonlinear, nonlocal, time-dependent treatment of convection suitable for use in models of cool giant stars is presented. Local conservation equations plus a diffusive transport equation are used to derive the convective hydrodynamic equations for the case in which turbulent pressure energy, and viscosity cannot be ignored. The effects of convective overshooting, superadiabatic gradients, convection/pulsation interaction, and time dependence enter this treatment in a natural way. Methods of treating turbulent viscosity and acoustic losses are discussed. Also, an efficient computational scheme for computing the derivatives needed for an implicit hydrodynamic code is outlined. Application to RR Lyrae star envelopes will be presented in a companion paper.

Permeability, Underpressures, and Convection in the Oceanic Crust Near the Costa Rica Rift , Eastern Equatorial Pacific

Abstract: In situ permeability and pore pressures were measured 200 m deep beneath the top of the oceanic crust at DSDP site 504B. These measurements have relevance for the transition from convective to conductive heat flow on the south flank of the Costa Rica Rift. Conventional ‘slug’ and constant rate injection tests were made below a hydraulic packer set at various depths in the hole. The packer was first set in a massive flow unit 37 m below the sediment-basement interface. The bulk permeability of the 172.5 m of pillow basalts and basaltic flows below the packer was found to be about 40 millidarcys (4×10−10 cm2). Measurements over 3- and 15-m intervals at the bottom of the hole in an altered pillow zone indicated a bulk permeability of 2–4 millidarcys. These values are thought to be accurate to ±30%. Formation pore pressures were found to be approximately 8 bars (∼2%) below hydrostatic. Interpretation of the data with respect to simple numerical convection models suggests that the transition from convective to conductive heat flow is controlled by the cessation of convective heat transport through the sedimentary layer rather than the cessation of convection in the sediment. Furthermore, the agreement between observed and modeled underpressures implies that hole 504B penetrated an active ocean crustal convection system. The thick sedimentary layer, layers of basal chert, and massive flow basalts above the layer 2A pillow flows apparently form an impermeable lid, effectively isolating the convection system from the ocean.

Thermal Convection in Non-Newtonian Fluids*

Abstract: Publisher Summary This chapter presents the field of thermal convection in non-Newtonian fluids. The thermal convection in external flows of inelastic fluids is relatively well understood in the chapter. Thermal convection in viscoelastic fluids has been studied only in a restricted way. An approximate correlating equation for mixed convection in external flows on non-Newtonian fluids is presented. The chapter explores the theoretical indications to show that there are likely to be tremendous advantages in adding drag-reducing polymers to impart viscoelastic properties to materials undergoing the process of thermal convection under turbulent conditions. The problem of buoyancy-induced secondary flow in heated horizontal tubes appears to be most challenging from a theoretical viewpoint. There are certain equations that correlate heat transfer data empirically, but the situation is not wholly satisfactory. The possibilities of oscillatory convection in viscoelastic fluids appear to exist under certain circumstances. The chapter concludes that thermal convection experiments are difficult to conduct, therefore there appears to be a massive generation of data correlating overall heat transfer coefficients with process variables but practically very little information on the velocity and temperature fields.

The effects of significant viscosity variation on convective heat transport in water-saturated porous media

Abstract: Weakly nonlinear theory and finite-difference calculations are used to describe steadystate and oscillatory convective heat transport in water-saturated porous media. Two-dimensional rolls in a rectangular region are considered when the imposed temperature difference between the horizontal boundaries is as large as 200 K, corresponding to a viscosity ratio of about 6·5. The lowest-order weakly nonlinear results indicate that the variation of the Nusselt number with the ratio of the actual Rayleigh number to the corresponding critical value R/Rc, is independent of the temperature difference for the range considered. Results for the Nusselt number obtained from finite-difference solutions contain a weak dependence on temperature difference which increases with the magnitude of R/Rc. When R/Rc = 8 the constantviscosity convection pattern is steady, while those with temperature differences of 100 and 200 K are found to oscillate.

Compressible convection in a rotating spherical shell. V. Induced differential rotation and meridional circulation

Abstract: We describe solutions for the mean differential rotation and meridional circulation in a compressible, rotating, spherical fluid shell which are induced by the linear, anelastic solutions of Papers II and IV for global convection. Our mean solutions strongly depend on the density stratification, the rotation state, the convective velocity distribution, and the amount of viscous diffusion relative to thermal diffusion. In order to obtain an equatorial acceleration which is large in amplitude relative to the meridional circulation, together with a small equator-pole temperature difference, when the density stratification is large as in the solar convection zone, at least one of two conditions must be met: either the effect of rotation must be large compared to the effects of viscous diffusion and buoyancy, or viscous diffusion must be small relative to thermal diffusion. In either case, the angular velocity increases with depth in the upper part of the convection zone by decreases with depth and is nearly constant on cylinders in the lower part when the global convection extends to deep layers. Global convection, differential rotation, and meridional circulation in the deep layers are similar to those found for the incompressible case, while near the top they are quite different.

The effects of a conducting E layer on classical F region cross‐field plasma diffusion

Abstract: The rate of cross-field plasma diffusion in the F region ionosphere is significantly increased when the magnetic field lines thread a highly conducting E region below. This reduces the lifetime of small-scale F region electron density irregularities in the polar ionosphere where the presence of a highly conducting E region is commonplace. A simple model is developed to describe the effects of a conducting E layer on classical F region plasma diffusion. In the absence of an E region, the difference in ion and electron diffusion rates leads to a charge separation and, hence, to an electrostatic field that retards ion diffusion. When the highly conducting magnetic field lines are tied to a conducting E region, however, electrons can flow along B to reduce the ambipolar diffusion electric field, and ions can proceed perpendicular to B at a rate approaching their own (higher) diffusion velocity. It is shown that the enhanced total diffusion rate that results depends strongly on the height of the F layer and on the ratio of the E to F region Pedersen conductivities. Although the enhanced classical diffusion rate hastens the removal of irregularities once their production source is removed, it is not a strong enough damping mechanism to prevent instabilities from operating routinely in the polar ionosphere. However, the E region probably plays an important role in determining the scale size of the irregularities that are favored. E region ‘images’ may be important for low E region electron densities and small scale sizes, in which case the diffusion rate is lowered. However, if the E region conductivity is high, the presence of images only reduces the F region cross-field plasma diffusion rate by about 25% from the ion rate. We hypothesize that the spectrum of high-latitude plasma density irregularities is controlled at large scales (λ ≳ 10 km) by structured soft electron precipitation and classical diffusion. Smaller scale waves are produced by plasma instabilities operating on the edges of the large scale structures. The generalized instability (including the current convective process) acts to strengthen waves in the intermediate scale size (100 m ≤ λ ≤ 10 km) in regions where the geometry is appropriate or where field-aligned currents are significant. Universal drift waves transfer energy from the intermediate scale to smaller structures but are ineffectual at large scales. The classical diffusion process described herein is applied (in conjunction with a model of irregularity production and convection) to the problem of explaining the morphology of the large scale high-latitude irregularities in a companion paper (Kelley et al., this issue). The anomalous diffusion due to the instabilities mentioned above is also described in more detail.

Thermal boundary layer convection in silicic magma chambers: Effects of temperature‐dependent rheology and implications for thermogravitational chemical fractionation

Abstract: Solution of the nonlinear boundary value problem describing the thermomechanical structure of a boundary layer adjacent to an isothermal cooled vertical wall in a strongly temperature dependent rheological medium is presented. The analysis and boundary conditions chosen are applicable to large Rayleigh number convection in a magma chamber. Numerical solution of the boundary layer equations by parallel shooting have been obtained for viscosity contrast up to 1011. Parameterization of these results allows extrapolation to higher viscosity contrasts. The non-Arrhenius viscosity function μ = μ∞ exp (−a(T∞ - T), where a is the rheological parameter and T is the absolute temperature, is based on experimental data and accounts for the effects of temperature and crystallinity on magma viscosity. The calculations clearly show the importance of explicit consideration of the viscosity-temperature relationship in determining the quantitative features of boundary layer convection. For example, for anhydrous rhyolite (μ∞ = 109 P), the thermal boundary layer thickness (δT), wall heat flux (q0), wall viscous shear stress (τ0), and maximum vertical convective velocity (umax) are 10 m, 700 HFU, 5×10−3 bar, and 2.5 km yr−1 when μ is taken as constant (i.e., a = 0). However, for a viscosity contrast across the thermal layer of 1012 (i.e., a = 0.07 K−1), δT = 120 m, q0 = 350 HFU, τ0 = 0.36 bar, and umax = 400 m yr−1. Extreme caution must be exercised in using results from isoviscous boundary layer theories for the prediction of the thermomechanical and heat transfer parameters of magma chamber convection. For a wall temperature of 600°C (country rock temperature far from intrusion is ∼200°C) and a core temperature of 1000°C and adopting rheological parameters characteristic of rhyolitic magma containing 3 wt % dissolved H2O, we find at distances of 1 and 10 km from the top of the convecting zone viscous shear stresses, τ0 around 6×10−2 and 3×10−2 bar, maximum vertical velocities wmax of 10 and 25 km/yr, boundary layer thicknesses δT of 20 and 50 m, and marginal heat flows q0 of 1200 and 700 HFU, respectively. Transport parameters depend markedly on H2O content; for anhydrous rhyolite with identical boundary conditions q0 ∼ 600 HFU, umax ∼0.5 km yr−1 and τ0 ∼ 10−1 bar at a distance of 1 km along the vertical wall. Maximum vertical convective velocities usually exceed crystal settling rates by several orders of magnitude; crystal settling cannot be important in vigorously convecting chambers except in local regions. The petrological and geothermal implications of the calculations are discussed in terms of an extreme (but plausible) type of magmatic system, the constant enthalpy open magma chamber. In this case, heat losses due to dissipation of magmatic heat to the country rock are precisely balanced by heat input by injection of hot, mafic magma into the roots of the chamber. The requirements of the thermal steady-state chamber permit an estimation of the mass flow rate into the roots of the chamber. Results agree well with known rates of basaltic magmatism along mid-ocean ridges and at intraplate ‘hot spot’ sites. Semi-quantitative evaluation of the magnitude of chemical fractionation in a constant enthalpy magma chamber due to coupling of rapid (km yr−1) vertical convective flow with slow horizontal Soret diffusion across thin marginal thermal layers suggests that on a 106 year time scale, significant chemical gradients can be generated. An analytical approach is suggested for answering the question of whether or not such fractionated melt can maintain its integrity (i.e., not become remixed).

Nonlinear Marangoni convection in bounded layers. Part 2. Rectangular cylindrical containers

Abstract: We consider liquid in a circular cylinder that undergoes nonlinear Marangoni insta- bility. The upper free surface of the liquid is taken to have large-enough surface tension that surface deflections are neglected. The side walls are adiabatic and impenetrable, and for mathematical simplicity the liquid is allowed to slip on the side walls. The linearized stability theory for heating from below gives the critical Marangoni number Mc as a function of cylinder dimensions, surface-cooling condition and Rayleigh number. The steady nonlinear convective states near Mc are calculated using an asymptotic theory, and the stability of these states is examined. At simple eigenvalues Mc the finite-amplitude states are determined. We find th at the Prandtl number of the liquid influences the stability of axisymmetric states, distinguishing upflow at the centre from downflow. Near those aspect ratios corresponding to double eigenvalues Me, where two convective states of linear theory are equally likely, the nonlinear theory predicts sequences of transitions from one steady convective state to another as the Marangoni number is increased. These transitions are determined and discussed in detail. Time-periodic convection is possible in certain cases.

Transition from periodic to chaotic thermal convection

Abstract: The transition to turbulence in Benard convection in a layer of air bounded by rigid conducting walls is studied by numerical solution of the three-dimensional time-dependent Boussinesq equations. The wavy instability of rolls is compared with available experimental and theoretical results. The subsequent transition to chaotic convection is shown to occur for Rayleigh numbers larger than about 9000. The role of symmetry-breaking perturbations in the production of chaos is clarified.