Convective Domain Research Articles

Spectral modelling of mantle convection in a non orthogonal geometry: application to subduction zones

We present a two-dimensional numerical convection model, in a parallelogram-shaped domain adapted to the geometry of subduction zones. The Navier–Stokes problem is solved by a pseudo-spectral solver (named projection-diffusion) coupled to the Richardson iterative scheme. In a convective domain, representative of the upper mantle, plate tectonic parameters are fixed by imposing the velocities on the domain boundaries. In particular, the subducting plate, located on one side of the domain box, moves and dips at constant velocity. The model is validated for the zero obliquity configuration for three boundary conditions: first, for free boundaries, second, for a rigid upper boundary; and third, for a rigid upper and lateral boundary. We also report preliminary results of experiments for a non-zero obliquity at a Rayleigh number value of 10 4, a motionless overriding plate, subduction velocities smaller than 4 cm/yr, and subduction dip angles steeper than 60°. Concerning the evolution of the flow and thermal structures with the subduction movement and geometry, the model predicts the occurrence of a recirculation process, characterised by the formation of an extra convection cell above the subducted plate. Recirculation starts at a critical subduction intensity that is dependent on the subduction geometry: the lower the subduction dip angle, the lower the critical subduction velocity. A critical subduction geometry is predicted by the model because the extra convection cell exists even at very low subduction velocities and subduction dip angles less than 65°.

The onset of natural convection in vertical fault planes:consequences for the thermal regime in crystalline basementsand for heat recovery experiments

Very few results are available on free convection in fractured zones, although this is a major mechanism for heat and mass transfer in crystalline rocks. Murphy (1979) has shown, using analytical stability analysis, that the critical Rayleigh number for the onset of free convection in a fracture greatly exceeds the value of 4π2, which is the value for an infinite porous medium, and, even for a subcritical Rayleigh number, convection may occur after a time delay. Murphy proposed that this delayed convection results from a blanketing effect of the fracture induced by the progressive development of a thermal skin inside the fracture walls. The present paper extends Murphy’s results by means of numerical modelling. Our numerical method involves a 2-D computation of convection in the fracture plane, and a 3-D solution of the conduction problem inside the fracture wall. The coupling of the codes is achieved by imposing a common temperature at the mid-fracture plane, together with the conservation of energy at the fracture–wall interface. We use two kinds of initial perturbation, which are assumed to constitute end-members for natural or application cases. For an A-type initial condition the thermal field is disrupted in the fracture only, while for a B-type initial condition the perturbation is introduced in the fracture and in the walls. For a given perturbation wavenumber, three distinct domains can be defined according to the Rayleigh number (R). In the first domain, convection takes place immediately; in the second one, convection starts after a delay; and in the third one, convection is damped. These three domains are therefore termed the instantaneous convection (R > Rs ), delayed convection (Rd < R < Rs ), and conduction (R < Rd ) domains, respectively. It is noteworthy that these three domains are bounded by the same values of the Rayleigh number for both A-type and B-type perturbations. Except for R close to Rd , the time lapse required for the onset of delayed convection is less than 10 conductive times of the fracture width. This time lapse is therefore negligible compared with geological timescales, and Rd can be considered as the critical Rayleigh number for the onset of free convection in a fracture zone. However, the characteristic conductive time of a fracture, which amounts, for example, to 3 years for a 10-m thick fracture, must be considered for thermal recovery experiments. It is suggested that convection may start during exploitation and will be superimposed on the forced flow. We expect that this will enhance the efficiency of thermal recovery.

An element free Galerkin formulation for stable crack growth in an elastic solid

The discrete formulation for stable crack growth in an elastic solid using the Element Free Galerkin (EFG) method based on the moving least-squares approximations has been established. The EFG method provides the ability to successfully model steep gradients, such as those that exist at a crack tip, through the introduction of an additional distribution of nodal points. In this formulation, the inertia force term in the momentum equation is converted into a spatial gradient term by employing the steady state conditions. A convective domain is employed to account for the analysis domain moving at the same speed as the crack front. A number of stable crack growth problems are examined and comparisons between the numerical predictions and analytical solutions are made for the near-tip fields, the crack opening profiles, as well as the global control parameters such as stress intensity factor K and energy release rate G.

The scales of mantle convection

Seismic, topographic and gravity data show that there are two important scales of mantle convection. These are associated with spherical harmonic degrees l =2 and 6. The l =2 pattern corresponds to the pattern of subduction cooling since the breakup of Pangea. Most hotspots and ridges occur in the half of the globe unaffected by this cooling and all large igneous provinces were generated over this part of the mantle. The l =2 distribution of upwellings and downwellings is likely to be a long-lived feature of the Earth; cold regions of the mantle repeatedly attract continents and subduction reinforces the coldness. An l =1 pattern of convection is probably related to supercontinents and their breakup. Degree 6 convection shows up in the spectrum of hotspots and upper mantle tomography and in the correlation of tomography and topography with the geoid. Cratons, with their deep cold keels, control the l =6 pattern, and may even cause it, by their role in establishing lateral temperature gradients and relief at the top of the convecting mantle. Subduction zones reinforce the craton pattern. Downwellings preferentially occur under cratons; upwellings, and hotspots, occur at complementary locations. Supercontinents, and their associated subduction zones ( l =1) constantly assemble and reassemble in the African-Atlantic hemisphere and the continental fragments, in the dispersed state, periodically settle into the polar band of geoid lows ( l =2 and 6) that now includes the Americas, Antarctica, Australia and India. If cratons control the l =6 pattern of convection (and many patterns of l =6 are possible; e.g., sectoral, zonal, checker-board) then upper-mantle convection may reorganize roughly every 30 Ma as cratons move about. The hotspot spectrum is probably related to lithospheric extension as well as to broad upwellings. A third and smaller scale of convection, order 400–1000 km in dimension, is just below the resolution of global tomography but shows up in the gravity field (geoid) and topography. This scale is controlled by the depth of an endothermic phase change which tends to stratify mantle convection, and the thickness of the upper mantle low viscosity zone. Convective domains of this dimension are also implied by the scales of chemical homogeneity, lengths of rifts, ridges and seamount chains, fracture zone spacing and mid-ocean ridge segmentation. The temperature, geochemistry and fertility of these upper-mantle scale domains is controlled by their previous history of subduction, continental insulation or refrigeration, and processing by ridges. Fertile, or volatile-rich, or hot, cells can be mistaken for plumes. Hotspot swells are typically of the dimension that we argue is a characteristic upper mantle scale rather than a deep mantle plume scale. Geochemical domains of various sizes exist in the upper mantle. They are broad-scale features, rather than point sources, as in plume theories. Lithospheric dynamics and geometric focusing, not mantle dynamics, control the dimensions of so-called hotspot eruptives.

Dynamical and thermal structure of a subduction zone: influence of slab geometry on the convective state of the Earth's upper mantle; preliminary results

The aim of this study is to investigate the influences of plate kinematics and of the downgoing slab's geometry on internal dynamics and thermal structure in the vicinity of a subduction zone. A systematic study depending on these parameters has been carried out based on a two-dimensional finite differences code. In a convective domain representing the upper mantle, plate displacements are prescribed by kinematic boundary conditions. In particular, the subducting plate moves and dips at a constant velocity under the slower overriding one. Typical subduction dips (30°, 45° and 70°) have been studied for a realistic Rayleigh number of about 10 5 and for plate velocities around a ‘coupling’ velocity of 2–3 cm year −1. This limit value was inferred from the balance between surface forces and buoyancy forces. For any subduction dip and plate velocities lower than this coupling velocity, plate kinematics variations induce significant effects on thermal and mineralogical patterns. At greater plate velocities, these patterns stabilize. For plate velocities of the order of the coupling velocity, a subduction dip decrease of 40° involves a heating of the downgoing plate of 100 degrees and a depth increase of 20 km of the olivine-spinel phase transition in this plate. In the particular case of a non-moving overriding plate, the balance between this plate and underlying internal dynamics requires a moving plate velocity lower than 2–3 cm year −1 for a lower dip. Moreover, a shallow subduction enhances a lateral large-scale convection under the upper plate even if this plate is motionless, in contrast to the case of a steep subduction. This is consistent with observed back-arc extension at the surface, which is often associated with steep subductions. Our model suggests a spreading zone of between 500 and 1000 km in the upper plate for a subduction velocity lower than 2–3 cm year −1 and a Rayleigh number of about 10 5.

The wave-vector-frequency spectrum of pressure on a smooth plane in turbulent boundary-layer flow at low Mach number

Properties of the wave-vector-frequency spectrum of fluctuating pressure on a smooth planar wall in turbulent boundary-layer flow at low Mach number are reviewed. In the low but incompressive wave-number range, where ω/c≲K≲max(δ−1,U∞/ω), consistent with the Kraichnan–Phillips hypothesis for inviscid flow, the dependence would be expected to be as K2, but no experimental substantiation exists. In a higher subconvective range where δ−1≲K≲ω/U∞, most pertinent experiments suggest that the spectrum is instead wave-number-white. In the acoustic domain a peak is predicted at K=ω/c. A mean-shear contribution proportional to the square of streamwise wave number appears predominant in the convective domain. An explicit model spectrum is specified that conforms in an appropriate domain to the principle of wall similarity and corresponds to boundary-layer velocity spectra that are planar isotropic in a convected frame. The model potentially encompasses the entire inviscid domain, including the acoustic range. An alternative model exhibits a factorable dependence on the wave-number components. In the acoustic range, the model forms are totally unvalidated by experiment. In the convective domain, the state of determination suffices for many applications, but a preference between model forms and assured choice of parameter values awaits further analysis and perhaps certain further measurements. An addendum based on recent work suggests that the wall pressure at low but incompressive wave numbers may be dominated by a wave-vector-white contribution originating in the viscous wall condition.

A semiempirical model for the wave vector‐frequency spectrum of turbulent wall pressure on a smooth planar boundary

The analytical framework and construction of a model of turbulent boundary‐layer pressure on a planar wall by its relation to velocity‐product fluctuations regarded as its sources are reviewed. Expansion for low (subconvective) wavenumbers and low Mach numbers provides a basis for such a trial model of the wall‐pressure spectrum that is potentially satisfactory at all wavenumbers from the acoustic domain to the convective. The proposed form for source cross spectra incorporates the principle of wall similarity, a kinematic assumption about space‐time correlation, possible wall‐normal profiles of vorticity cross spectra, and assumed non‐negativity. The wall‐pressure model is obtained by analytical integration over the source‐spectral profiles in limiting domains followed by convenient rough interpolation. The experimental basis for determination of parameter values of the model is considered by reference to classical wind‐tunnel measurements dominated by the convective wavenumber domain and to more ambiguous ones directed to the subconvective but superacoustic domain. The inadequate state of validation and determination of parameters characterizing the acoustic domain and the domain of pertinence of the Kraichnan‐Phillips theorem is confronted. Likewise, the open but experimentally resolvable question of the joint dependence on wave vector components in the convective domain is recalled.

Evolution of crystal-settling in magma-chamber convection

The motion of negatively buoyant particles embedded within kinematically prescribed convective velocity fields is used to study the evolution of crystal settling in magma chambers. We consider the effects of Stokesian settling velocity, crystal growth, aspect ratio of the convective domain and the site of crystal nucleation on mixing and transport of heavy crystals in this system. The percentage of particles retained in a convecting fluid is very sensitive to the site of nucleation. Particles nucleating along the cold roof or within the wall boundary layers are generally not retained. As a consequence, crystal settling can occur even in rapidly convecting magma chambers. Equilibrium crystal densities are reached relatively quickly, within one overturn, and the distribution of particles deposited on the bottom of the chamber is always asymmetric. This feature may facilitate the formation of primary sedimentary structures in plutons. Laboratory experiments using small, negatively buoyant spheres in a highly viscous fluid with axisymmetric thermal convection confirm the basic predictions of this model.

The estimated level of low-wavenumber pressure generated by non-linear interaction of a compliant wall with a turbulent boundary layer

The effect of deformation of a compliant surface bounding a turbulent flow in producing low-wavenumber pressure and sound radiation via the non-linear convolving of convective turbulent excitation with itself is estimated by application of the formulation by M. S. Howe to the case of a linearly reacting elastic half-space. Requisite spectra of fluctuation products are formed from spectra of linear fluctuations as though the random field had a joint normal distribution. The model spectrum assumed for the turbulent pressure on a rigid wall is representative, at least, of the convective domain. The resulting estimated non-linear contribution to the low-wavenumber surface-pressure spectrum is proportional to ( U /| c t |) 8 , where U is the flow speed and c t the shear-wave speed. In a typical example with | c t | = 5 U , the level lies below the level of the wave-vector-white, scale-independent spectrum with the Martin-Leehey coefficient, which is comparable with presently measured low-wavenumber levels, by 129 dB. More generally, it is indicated that the non-linear contribution at low wavenumbers due to the convective domain is negligible for any usual bounding surface whose free-wave speeds exceed the flow speed. In the contrary case of a bounding inviscid fluid, which does not represent a real flow-bounding surface, the non-linear effect is indicated not to be calculable from the presently available formulation.

Mixed convection plume above a point heat source in a vertical free stream

The mixed convection boundary layer above a point heat source subjected to a vertically upward uniform stream is considered. In the present work a new approach is presented where entire mixed convection domain is described by a single set of equations. The numerical solutions are presented for Prandtl numbers of 0.72, 6.8, 50, 100 and infinity.

Mixed convection over a semi-infinite horizontal plate

Mixed convection over a horizontal plate has been studied by the method of series truncation for the two cases where the buoyancy effects aid and oppose the main flow. From the Navier-Stokes equations in parabolic coordinates the equations for the first truncation are obtained. The numerical results show that the predictions are very accurate for heat transfer over the entire mixed convection domain. The skin friction predictions are accurate only for moderately large buoyancy effects. In the buoyancy dominated flow the heat transfer is over estimated by four percent and skin friction by thirty percent.

Atmospheric Structure and Radar Backscattering in Clear Air

This study summarizes advances in radar backscattering from refractive index irregularities in the clear air and its relationship to atmospheric structure and turbulence. This recent discipline of radar meteorology originated from radar angel studies and the efforts to explain these mysterious echoes. The significance of specular atmospheric radar reflections now appears limited, and consequently this study is devoted entirely to radar backscattering from turbulent fluctuations in the clear‐air refractive index. Two regimes of atmospheric turbulence are distinguished: (1) in the convective domain, backscattering from water vapor fluctuations outlines regular, three‐dimensional clear‐air structures associated with buoyant, moist air parcels; (2) in the stable regime, horizontally stratified clear‐air echoes originate from water vapor and/or temperature perturbations due to wind‐shear‐generated turbulent mixing in zones of enhanced static stability. Theoretical relationships are given between the radar reflectivity and the refractive index microstructure for the general case, for the isotropic case, and for the inertial subrange. The connection between the microstructure and the atmospheric mean fields is illustrated for a simplified situation in the stable regime; this relationship provides insight to the radar backscattering from clear‐air turbulence (CAT). It is concluded that strong CAT will be associated with zones of increased refractive index variability and enhanced radar returns. A brief résumé of experimental work illustrates the application of radar methods in clear‐air research and summarizes characteristic features of clear‐air structures.