Convection Wavelength Research Articles

Thermodynamics for non-equilibrium pattern formation

The second entropy theory for non-equilibrium thermodynamics is used to show that the optimum structure or pattern of a time-dependent system corresponds to the maximum entropy. A formula for the total entropy of convective heat flow is derived. The Navier-Stokes equations in Boussinesq approximation for straight roll convection are solved by a Fourier expansion technique. Results for the velocity amplitude are in good agreement with previous computations and experimental measurements. For the spontaneous transitions between straight roll states reported in the literature, it is shown that the measured change in convective pattern wave length corresponds to an increase in the entropy.

Pattern formation in square planform containers

Insight is gained into the onset and non-linear development of convection in square planform containers heated from below through a study of the 2D Swift–Hohenberg equation with no-slip boundary conditions. Numerical computation is used to identify the bifurcation structure of steady-state solutions and to track their non-linear development. For planforms whose dimensions are much greater than the characteristic wavelength of convection, a weakly non-linear theory based on the use of multiple-scale matched asymptotic expansions is used to describe the onset of convection in the form of rolls confined to the neighbourhood of the diagonals of the square. Use of Fourier transforms allows roll curvature induced by the corners of the container to be taken into account and a local cross-roll structure is found to occur near the corners. Theoretical predictions are compared with experimental results for Rayleigh–Bénard convection.

The Dynamic Response of Turbulent Dihedral V Flames: An Amplification Mechanism of Swirling Flames

The authors analyze the dynamic response of swirl-stabilized flames submitted to upstream acoustic perturbation. Extensive measurements were performed in an optically accessible single-nozzle gas turbine combustor operating on natural gas-air at inlet temperatures of 200 and 300 °C over a range of equivalence ratios from 0.55 to 0.70, a range of inlet velocities from 60 to 100 m/s, and swirl angles of 30° and 45°. Temporal oscillations of inlet velocity and heat release rate in the whole flame were measured using the 2-microphone method and global OH*, CH*, and CO2* chemiluminescence emission intensities, respectively, whereas spatially resolved measurement of heat release rate was made using time-averaged CH* chemiluminescence flame images. For the dihedral V flames, amplification characteristic of the flame transfer function was observed. This effect is, unlike the amplification mechanism of a small laminar flame, controlled by the relative ratio of the two length scales, disturbance convective wavelength and flame length. The measured transfer functions show resonance-like behavior when a nondimensional number, the ratio of half the convective wavelength to flame length, approaches unity. It was found that the flame geometric properties, specifically flame angle, also play a crucial role in the flame transfer function. The frequency-dependent behavior of swirl-stabilized flames is closely related to eigenfrequency selection processes at limit cycle pressure oscillations.

Temperature beneath continents as a function of continental cover and convective wavelength

Geodynamic modeling studies have demonstrated that mantle global warming can occur in response to continental aggregation, possibly leading to large‐scale melting and associated continental breakup. Such feedback calls for a recipe describing how continents help to regulate the thermal evolution of the mantle. Here we use spherical mantle convection models with continents to quantify variations in subcontinental temperature as a function of continent size and distribution and convective wavelength. Through comparison to a simple analytical boundary layer model, we show that larger continents beget warming of the underlying mantle, with heating sometimes compounded by the formation of broader convection cells associated with the biggest continents. Our results hold well for purely internally heated and partially core heated models with Rayleigh numbers of 105 to 107 containing continents with sizes ranging from that of Antarctica to Pangea. Results from a time‐dependent model with three mobile continents of various sizes suggests that the tendency for temperatures to rise with continent size persists on average over timescales of billions of years.

Long wavelength convection, Poiseuille–Couette flow in the low-viscosity asthenosphere and the strength of plate margins

SUMMARY Mixed heated 3-D mantle convection simulations with a low-viscosity asthenosphere reveal relatively short and long wavelength regimes with different scalings in terms of surface velocity and surface heat flux and show that mantle flow in the lithosphere–asthenosphere region is a Poiseuille–Couette flow. The Poiseuille/Couette velocity magnitude ratio, D/U, allows us to characterize solid-state flow in the asthenosphere and to predict the regime transition. The transition from dominantly pressure-driven Poiseuille flow at shorter wavelengths to dominantly shear-driven Couette flow at long wavelengths depends on the relative strength of lithosphere and asthenosphere and is associated with a switch in the dominant resistance to convective motion. In the Poiseuille regime significant resistance is provided by plate-bending, whereas in the Couette regime most resistance is due to vertical shear in the bulk mantle. The Couette case corresponds to classical scaling ideas for mantle convection whereas the Poiseuille case, with asthenospheric velocities exceeding surface velocities, is an example of a sluggish lid mode of mantle convection that has more recently been invoked for thermal history models of the Earth. Our simulations show that both modes can exist for the same level of convective vigour (i.e. Rayleigh number) but at different convective wavelengths. Additional simulations with temperature- and yield-stress dependent viscosity show consistent behaviour and suggest an association of the regime crossover with the relative strength of plate margins. Our simulations establish a connection between the strength of plate margins, solid-state flow in the asthenosphere and the wavelength of mantle convection. This connection suggests that plate tectonics in the sluggish lid mode is wavelength dependent and potentially more robust than previously envisioned.

Formation of steep‐sided topography from compositionally distinct dense material at the base of the mantle

In this study we use 3‐D, two‐layer, thermochemical convection experiments to explore the development of a thin, compositionally distinct, dense layer at the base of the mantle. We investigate the conditions under which steep‐sided, flat‐topped topography will form on an initially ubiquitous layer as a result of the convecting mantle. Previous authors found that discrete, steep‐sided piles of material form when compressibility, temperature‐dependent viscosity, or a large viscosity increase in the lower layer is assumed. We observe, however, that this topographic style develops in calculations of incompressible convection with layers of constant viscosity. Experiments involving simple convection planforms are used to systematically investigate the factors which encourage or inhibit this topographic style. The shape of the topography that develops on the density interface can be linked to the convection coupling style between the two layers: viscous coupling leads to discrete, steep‐sided piles and thermal coupling to a ubiquitous layer with cusp‐like peaks. The dependence of the topographic style on the Rayleigh number, density ratio, and volume ratio can largely be explained by considering their effect on the convection coupling style. Consequently, we find that it is crucial for the lower layer to be thin compared with the overlying convection wavelength and have a density contrast of the order 1% in order for discrete, steep‐sided piles to develop.

The combined effects of continents and the 660 km-depth endothermic phase boundary on the thermal regime in the mantle

We explore a wide range of parameter space to investigate the dynamical interaction between effects due to surface boundary conditions representing continental and oceanic lithosphere and the endothermic phase boundary at 660 km-depth in simple two-dimensional convection calculations. Phase boundary induced mantle layering is strongly affected by the wavelength of convective flows and mixed surface boundary conditions strongly increase the horizontal wavelength of convection. Our study shows that for mixed cases the effects of the surface boundary conditions dominate the effects of the phase boundary. We show that the calculations with complete continental coverage have the most significantly decoupled lower and upper mantle flows and substantial thermal and mechanical layering. Unlike the free-slip case where the surface heat flow decreases substantially with increasing magnitude of the Clapeyron slope, surface heat flow is shown to be almost independent of the Clapeyron slope for mixed boundary condition cases. Although very different when not layered, models with free and mixed surfaces have very similar planforms with very large aspect ratio flows when run with large magnitudes of the Clapeyron slope. We also calculate the critical boundary layer Rayleigh number as a measure of the thermal resistance of the surface boundary layer. Our results show that the thermal resistance in the oceanic and the continental regions of the mixed cases are similar to fully free and no-slip cases, respectively. We find that, even for purely basally heated models, the mantle becomes significantly subadiabatic in the presence of partial continental coverage. This is due to the significant horizontal advection of heat that occurs with very large aspect ratio convection cells.

Global warming of the mantle beneath continents back to the Archaean

Throughout its history, the Earth has experienced global magmatic events that correlate with the formation of supercontinents. This suggests that the distribution of continents at the Earth's surface is fundamental in regulating mantle temperature. Nevertheless, most large igneous provinces (LIPs) are explained in terms of the interaction of a hot plume with the lithosphere, even though some do not show evidence for such a mechanism. The aggregation of continents impacts on the temperature and flow of the underlying mantle through thermal insulation and enlargement of the convection wavelength. Both processes tend to increase the temperature below the continental lithosphere, eventually triggering melting events without the involvement of hot plumes. This model, called mantle global warming, has been tested using 3D numerical simulations of mantle convection [Coltice, N., Phillips, B.R., Bertrand, H., Ricard, Y., Rey, P. (2007) Global warming of the mantle at the origin of flood basalts over supercontinents. Geology 35, 391–394.]. Here, we apply this model to several continental flood basalts (CFBs) ranging in age from the Mesozoic to the Archaean. Our numerical simulations show that the mantle global warming model could account for the peculiarities of magmatic provinces that developed during the formation of Pangea and Rodinia, as well as putative Archaean supercontinents such as Kenorland and Zimvaalbara.

Depth‐dependent rheology and the horizontal length scale of mantle convection

Numerical simulations show that depth‐dependent viscosity can increase the wavelength of mantle convection. The physical mechanism behind this phenomenon and its robustness with respect to model parameters remain to be fully elucidated. Toward this end, we develop theoretical heat flow scalings for a convecting fluid layer with depth‐dependent viscosity. Bottom and internally heated end‐members are considered. For the former, the viscosity structure consists of a high‐viscosity central region bounded from above and below by horizontal low‐viscosity channels. For internally heated cases, only a surface low‐viscosity channel is present. Theoretical scalings derived from boundary layer theory show that depth‐dependent rheology lowers the lateral dissipation associated with steady state convective rolls, allowing longer aspect ratio cells to form as the viscosity contrast between the channels and the central region is increased. The maximum cell aspect ratio is estimated from the condition that the pressure gradients that drive lateral flow in the channels do not become so large as to inhibit vertical flow into the channels. Scaling predictions compare favorably to results of numerical simulations for steady state cells. As the Rayleigh number driving convection is increased, small‐scale boundary layer instabilities begin to form. This increases lateral dissipation within the channels and the preferred cell aspect ratio decreases as a result. Internally heated simulations show that a near‐surface high‐viscosity layer, an analog to tectonic plates, can suppress these small‐scale instabilities. This allows a low‐viscosity channel to maintain large aspect ratio cells for Rayleigh numbers approaching that of the present‐day Earth.

Weakly nonlinear forced convection patterns in rectangular planform containers

This paper considers the structure of weakly nonlinear steady-state convection patterns in shallow rectangular planform containers heated from below. The lateral dimensions of the container are assumed to be much larger than the characteristic wavelength of convection, and the lateral boundaries are subject to forcing equivalent, for example, to imperfect thermal insulation in the Rayleigh–Benard problem. This has the effect of generating rolls parallel and perpendicular to the lateral boundaries. The resulting patterns are modelled by a coupled pair of nonlinear amplitude equations derived from a phenomenological model of convection introduced by Swift and Hohenberg [Phys. Rev. A15 (1977) 319]. These equations are applicable in the weakly nonlinear limit to a variety of pattern-forming systems such as the Rayleigh–Benard system. Solutions are found using both numerical and asymptotic methods. The boundary imperfection is shown to give rise to some novel effects, including the possibility of patterns containing square cells. More generally, patterns evolve that are dominated by rolls but with transitions to more complex bimodal forms near the edges of the container. The emergence and structure of transition lines, or grain boundaries, is analysed in detail.

Convective heat transfer as a function of wavelength: Implications for the cooling of the Earth

Attempting to reconstruct the thermal history of the Earth from a geophysical point of view has for a long time been in disagreement with geochemical data. The geophysical approach uses parameterized models of mantle cooling. The rate of cooling of the Earth at the beginning of its history obtained in these models is generally too rapid to allow a sufficient present‐day secular cooling rate. Geochemical estimates of radioactive element concentrations in the mantle then appear too low to explain the observed present mantle heat loss. Cooling models use scaling laws for the mean heat flux out of the mantle as a function of its Rayleigh number of the form Q ∝ Raβ. Recent studies have introduced very low values of the exponent β, which can help reduce the cooling rate of the mantle. The present study instead focuses on the coefficient C in the relation Q = C Raβ and, in particular, on its variation with the wavelength of convection. The heat transfer strongly depends on the wavelength of convection. The length scale of convection in Earth's mantle is that of plate tectonics, implying convective cells of wide aspect ratio. Taking into account the long wavelength of convection in Earth's mantle can significantly reduce the efficiency of heat transfer. The likely variations of this wavelength with the Wilson cycle thus imply important variations of the heat flow out of the Earth on a intermediate timescale of 100 Ma, which renders parameterized models of thermal evolution inaccurate for quantitative predictions.

The relaxation of two-dimensional rolls in Rayleigh–Bénard convection

Large aspect ratio, two-dimensional, periodic convection layers containing a Boussinesq fluid of finite Prandtl number bounded by rigid or free horizontal surfaces are investigated numerically. The fluid equations are solved using both a standard pseudospectral and a Fourier integral method for the time evolution of finite initial perturbations, both random thermal perturbations and localized roll disturbances, into a final equilibrium state. The suggestion that a Fourier integral solution method is required to yield roll relaxation, the two-dimensional process increasing the convection wavelength to values larger than critical, is investigated. Roll relaxation is found for both free-slip and no-slip surfaces using either solution method as long as the initial state is chosen to be of the form of a localized roll disturbance. A wide variety of simulations are performed and roll relaxation is found to be independent of the periodic domain length, weakly dependent on the Rayleigh number and dependent upon the magnitude of the initial localized roll disturbances.

A unified model for the prediction of laminar flame transfer functions: comparisons between conical and V-flame dynamics

Transfer functions of premixed laminar flames submitted to incident flow perturbations are envisaged and a unified model is derived analytically. This model, based on a linearization of the G-equation for an inclined flame, includes convective effects of the flow modulations propagating upstream of the flame. It is shown that the flame dynamics is governed by two relevant parameters, a reduced frequency, ω ∗, and the ratio of the flame burning velocity to the mean flow velocity, S L /ῡ, or equivalently the flame angle α with respect to the flow direction. In the limit of low driving frequencies, the flame motion is only controlled by ω ∗ and the unified model reduces to previous kinematic formulations derived for rim stabilized conical flames and V-flames anchored on a central rod. Flame transfer functions for these flame geometries with the different velocity models proposed are derived and limiting cases are examined. In the conical flame case, the low-frequency model gives a good approximation of the gain, but only a fair approximation of the phase. Convective effects are shown to induce an increasing phase lag, while low-frequency models predict a saturation phenomenon. The convective model derived in this article improves results for the gain and the phase which agree with numerical simulations and experiments. It is shown in particular that 1) the correct transfer function phase trend is retrieved and depends on the flame angle α; 2) the reduced cut-off frequency corresponds to a situation where the convective wavelength along the flame front λ = (ῡ cos α)/ f equals the flame length L; and 3) the flame response is weakly affected by the amplitude of these perturbations. In the V-flame case, the low-frequency model yields a good approximation of the phase but does not feature gain values in excess of one found in the simulations. This behavior is correctly predicted by the convective model and is shown to depend on the flame angle α. A V-flame behaves as an amplifier in a certain range of frequencies. It is shown that these types of flames are more susceptible to combustion instabilities than conical flames. The V-flame response is also shown to strongly depend on the amplitude of the fluctuations even for moderate perturbation levels.

Two structural intensity prediction methods in plates excited by turbulent boundary layers

Structural intensity (S-I) fields may be used to identify energy flow paths through a vibrating structure, as well as energy source and sink regions. Boundary layer excitation of structures occurs in numerous aerospace and underwater applications. This study describes two methods of predicting S-I fields in structures excited by turbulent boundary layers. The first prediction method combines well known multiple-input/multiple-output system theory with finite differencing techniques to obtain S-I predictions. The second method uses an analytic computation technique. This technique combines response matrices between applied pressures and response stresses and velocities to form cross spectra matrices with units of power. These cross spectra matrices are proportional to S-I. The two prediction methods are described in detail and a set of case studies incorporating these methods is presented. The case studies are based on a simply supported plate with an attached point damper. The Corcos model of the boundary layer spatial wall pressure correlation is applied to the plate due to its simplicity. The case studies include intensity fields due to flows above, at, and below aerodynamic coincidence (the frequency at which a structural wavelength equals a convective wavelength). [Work supported by The Applied Research Laboratory, Penn State.]

Convective Wavelength Shifts in the Spectra of Late-Type Stars

We present ultra-high resolution spectra for a set of nearby F-G-K stars on, or close to, the main sequence. The wavelength shifts of stellar lines relative to their laboratory wavelengths are measured for more than a thousand Fe I lines per star, finding a clear correlation with line depth. The observed patterns are interpreted as convective blue-shifts that become more prominent for weaker lines, which are formed in deeper atmospheric layers. A morphological sequence with spectral type or effective temperature is apparent. Two K giant stars have also been studied. The velocity span between weak and strong lines for these stars is larger than for the dwarfs and subgiants of similar spectral types. Our results show that convective wavelength shifts may seriously compromise the accuracy of absolute spectroscopic radial velocities, but that an empirical correction may be applied to measured velocities.

Onset of convection in shallow cylindrical containers

This paper considers the onset of convection in shallow cylindrical containers heated from below where the boundary which forms the cross–section of the cylinder is of arbitrary shape. The lateral dimensions of the boundary are assumed to be much larger than the characteristic wavelength of convection, allowing use of the methods of multiple scales and matched asymptotic expansions. A theory is constructed which describes a local onset of convection consisting of rolls confined to the neighbourhood of the diameter of the boundary (that is, the chord of maximum length that spans the boundary). Use of a Fourier transformation allows the curvature of both the boundary and the resulting roll pattern to be taken into account within a weakly nonlinear framework. Explicit solutions are obtained for the limiting case where the boundary curvature is large. Certain features of the nonlinear development of the roll pattern and the manner in which it spreads to other parts of the domain are discussed with reference to particular geometries. The results suggest that the pattern evolves in a way that maximizes the area of convection. One implication of the theory is that the onset of convection in shallow rectangular containers does not occur in the form of rolls parallel to the shorter sides.

On the boundary-layer structure of patterns of convection in rectangular-planform containers

This paper considers the structure of steady-state solutions of the Swift–Hohenberg equation describing convection in shallow rectangular-planform containers heated from below. The lateral dimensions of the planform are assumed to be much larger than the characteristic wavelength of convection. Results are restricted to patterns composed of rolls orthogonal to the sides of the rectangle in which case convection sets in at a critical value of the Rayleigh number in the form of rolls parallel to the shorter sides. This primary bifurcation from the conductive state of no motion produces a solution which subsequently undergoes a secondary bifurcation in which the low-amplitude motion near the shorter sides is replaced locally by cross-rolls perpendicular to the sides. This results in the formation of grain boundaries (or domain boundaries) within the fluid which mark the division between the different roll orientations.With increasing Rayleigh number the grain boundaries approach the sides of the rectangle and a boundary-layer structure is formed. In the present paper the method of matched asymptotic expansions is used to determine this boundary-layer structure and to predict the location of the grain boundaries. An interesting feature of the solution is that the grain boundaries develop significant curvature and bend into the corners of the rectangle, where the local solution is also determined.The results are compared with numerical computations of the secondary solution branch and with previous numerical and experimental work.

Feedback mechanism of low-speed edgetones

A feedback mechanism of low-speed edgetones is analyzed by using the jet edge interaction model in which reaction of the edge is regarded as an array of dipoles. From the jet-edge interation model the surface pressure of the edge and the upstream wave are estimated by assuming the downstream disturbance as a sinuously oscillating flow with a constant convection speed. The surface pressure distribution on the edge is found to increase from zero at the tip to a peak value around a quarter wavelength downstream, which may be regarded as an effective source point of the upstream-propagating sound wave. From the condition that the two wave trains should be phase-locked at the nozzle lip, the phase factorp is found to be in the range, −1/2<p<0, in the phase criterion of the form,h/A+h/λ=n+p, where h is the stand-off distance.A and λ the wavelengths of downstream and upstream respectively, andn the stage number. From the existing experimental data the phase factor is estimated on the basis that the convection wavelength is dependent only upon the mean jet velocity and the frequency, but not upon the stage number. The experimental values are in the same range of the theoretical estimation indicating that the phase factor is not a universal constant but varies in the range, −1/2<p<0, depending on the nozzle-edge configuration and the flow condition.

Convection and flow in porous media. Part 2. Visualization by shadowgraph

We present results for pattern formation at the onset of convection in fluid-saturated porous media obtained by a novel variation on the shadowgraphic technique (modified shadowgraphic technique). Both ordered and disordered media are used, each exhibiting distinct behaviour. Ordered porous media are constructed from grids of overlapping bars. Convective onset in this type of medium is characterized by a sharp, well-defined bifurcation to straight parallel rolls. The orientation of the convection rolls is determined by the number of bar layers, Nb; odd Nb leads to rolls with axes perpendicular to the direction of the top and bottom bars, and even Nb to rolls at 45° to the bars. Disordered porous layers are produced by stacking randomly drilled disks separated by spacers. In this system, we observe a rounded bifurcation to convection with localized convection near convective onset. More specifically, the flow patterns take on one of several different three-dimensional cellular structures after each cycling through convective onset. These observations may be described by two different mechanisms: random spatial fluctuations in the Rayleigh number (Zimmermann et al. 1993), and/or spatial variation in the thermal conductivity on the length scale of the convection wavelength (Braester &amp; Vadasz 1993).

Direct numerical simulation of thermal convection over a wavy surface

The influence of a wavy surface on thermal convection of Rayleigh-Benard type in a Boussinesq fluid is investigated by direct numerical simulations. The surface height varies sinusoidally in one direction. The wave amplitude amounts up to 10% of the fluid layer height and the wavelength equals about the critical wavelength of Rayleigh-Benard convection. The horizontal size of the computational domain equals this wavelength. For isothermal no-slip boundaries, two-dimensional convection sets in at subcritical Rayleigh numbers in close agreement with linear theory. The heat-transfer rate grows almost with the square of the surface-wave amplitude. Convection in a fluid layer over a no-slip surface with prescribed heat flux and an adiabatic free-slip boundary at the top is investigated for supercritical Rayleigh numbers and a Prandtl number of 0.7 in two and three dimensions. Two-dimensional simulations show oscillatory roll convection which becomes almost stationary if the Rayleigh number is of order 7000 or less. The two-dimensional convection is unstable with respect to three-dimensional disturbances and a cross-role pattern evolves even over a surface which is undulated in one direction only. For Rayleigh numbers exceeding about 15 000, the flow becomes turbulent. The results exhibit little sensitivity of the convection to the wavy surface for a 10% amplitude.