A theory of convection: Modelling by two buoyant interacting fluids

Abstract

A new model of convection and mixing is presented. The fluid is envisioned as being composed of two buoyant interacting fluids, called thermals and anti-thermals. In the context of the Boussinesq approximation, pairs of governing equations are derived for thermals and anti-thermals. Each pair meets an Invariance Principle as a consequence of the reciprocity in the roles played by thermals and anti-thermals. Each pair is transformed into an average equation for which interaction terms cancel and another very simple equation linking the two fluid properties. An important parameter of the model is the fraction, f, of area occupied by thermals to the total area. A dynamic saturation equilibrium between thermals and antithermals is assumed. This implies a constant values of f throughout the system. The set of equations is written in terms of mean values and root-mean-square fluctuations, in keeping with equations of turbulence theories. The final set consists of four coupled non-linear differential equations. The model neglects dissipation and can be applied to any convective situations where molecular viscosity and diffusivity may be neglected. Applications of the model to mixed-layer deepening and penetrative convection are presented in subsequent papers.     

An experimental study of baroclinic wave interactions in a two-layer system

Abstract

Baroclinic waves in a rotating two-layer shear flow are observed by measuring fluctuations in the height of the interface using a technique which does not disturb the flow. Decomposition of the shape of the interface into azimuthal Fourier components shows that the wave spectrum is dominated by a single wave number, but other components have significant amplitudes. Nonlinear interactions between these components are isolated by comparing the results with the predictions of a linear stability theory. The observed interactions are reminiscent of those found by Hide, Mason and Plumb (1977) in a different system, thus demonstrating that they are a fundamental property of baroclinic waves.     

Rayleigh wave propagation in a two-layer anisotropic media

Summary The possibility of propagation of Rayleigh waves in an incompressible crust of constant density and rigidity varying exponentially with depth lying on a semi-infinite transversely isotropic base has been discussed in this paper. Frequency equation has been derived and numerical calculations are made. The result obtained in this case is compared with that ofNewlands [3]²) andDutta [4].     

The field of a plane electromagnetic wave at the surface of two-layer anisotropic earth

Резюме На основе анализа электромагнитного поля плоской волны получены формулы (6) и (7) для коэффициентов отражения в случае двухслойной анизотропной Земли, слои которой характеризуются различающимися тензорами электрической проводимости. Коэффициенты отражения, рассчитанные для падающей волны с магнитным полем поляризованным в двух перпендикулярных направлениях, используются для определения главных и дополнительных импедансов в случаях анизотропного пласта на проводящем или непроводящем основаниях (формулы (11) и (12)) и для случая двух анизотропных слоев с совпадающими направлениями главных осей тензора электрической проводимости (формула (13)).

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Address: Boční II, Praha 4-Spořilov.     

Oblique wave scattering by undulating porous bottom in a two-layer fluid: Fourier transform approach

The problem involving scattering of oblique waves by small undulation on the porous ocean bed in a two-layer fluid is investigated within the framework of linearised theory of water waves where the upper layer is free to the atmosphere. In such a two-layer fluid, there exist waves with two different wave numbers (modes): wave with lower wave number propagates along the free surface whilst that with higher wave number propagates along the interface. When an oblique incident wave of a particular mode encounters the undulating bottom, it gets reflected and transmitted into waves of both modes so that some of the wave energy transferred from one mode to another mode. Perturbation analysis in conjunction with Fourier transform technique is used to derive the first-order corrections of velocity potentials, reflection and transmission coefficients at both modes due to oblique incident waves of both modes. One special type of undulating bottom topography is considered as an example to evaluate the related coefficients in detail. These coefficients are shown in graphical forms to demonstrate the transformation of water wave energy between the two modes. Comparisons between the present results with those in the literature are made for particular cases and the agreements are found to be satisfactory. In addition, energy identity, an important relation in the study of water wave theory, is derived with the help of the Green’s integral theorem.     

均匀倾斜层状地层中地震波球面扩散补偿方法研究

在分析地层深度、炮检距以及地层倾角变化对球面扩散补偿因子影响的基础上,总结了不同因素对补偿因子影响的趋势,以及资料处理中常规球面扩散补偿方法的应用条件．实际应用效果表明,该方法能更加合理的保证不同深度地震信息能量的均衡程度,提高资料的可信度。     

Two-scale analysis for environmental dispersion in a two-layer wetland

Studies on environmental dispersion are essential for applications as water management. The two-scale perturbation analysis is applied in this paper to deduce the environmental dispersion model for the typical case of contaminant transport in two-layer wetland flows. The analysis follows the established theoretical framework on the basis of phase average and the concept of Taylor dispersion. By the obtained flow velocity distribution for the two-layer flow, the analytical expression for the environmental dispersivity is deduced and shown to be consistent with previous results by the concentration moment method, while with much simplifications on the expression for ignoring the less concerned time-dependent stage of the dispersivity.     

A model for the relation between spreading rate and oblique spreading

Atwater and Macdonald have suggested that oblique spreading occurs at mid-ocean ridges which spread slowly (half rate less than 3 cm/yr), while the spreading is perpendicular at faster-spreading ridges. This paper explores this relation using the ratio of the power dissipated at ridges to that on transform faults to find the angle of oblique spreading (θ). The dependence of the energy dissipation rate on spreading rate is included in simple models of a ridge and transform. These arguments suggest that θ is related to the half spreading rate approximately by sin θ ～ V^?1.     

A laboratory system for short wave calibration of net pyrradiometers

Summary The system described enables short wave calibration of net pyrradiometers to be carried out independently of weather conditions. It uses a high-pressure 2 Kw Xenon arc as source, with a mirror drum scanner for improved uniformity of irradiation of the target area. Intensity and spectral distribution of the radiation are comparable with those of solar radiation.     

A compact representation of magnetotelluric responses for two-layer models

In this paper the locations where ρ_app = ρ₁ and ? = π/4 and where these parameters reach an extreme value in two-layer magnetotelluric (MT) sounding curves are summarized in an extremely compact form. The key parameters over two-layer models with conductivities σ₁, σ₂ and upper layer thickness h are the real S and α, where S is the conductivity contrast and α is the distance between the observation site and the conductivity interface, normalized to the half skindepth in the first layer. If the impedance components, various resistivity definitions ( ρ_{Re Z}, ρ_{Im Z} and ρ_|Z|, based on different parts of the complex impedance Z ) and the magnetotelluric phase ? are derived as a function of S and α, then the conditions for the apparent resistivity ρ_app and the phase ? are that they either satisfy ρ_app = ρ₁ and ? = π/4 or attain extreme values which can be given in terms of simple algebraic equations between S and α. All equations are valid for observation sites at any depth 0 ≤ z ≤ h in the first layer. The set of equations, presented in a tabular form, may make it possible to determine a layer boundary from the short period part of the sounding curves, in particular the ρ_{Re Z} and the ?_MT curves.     

Spin-up of a two-layer fluid in a rotating cylinder

Abstract

we report the results of experiments on the spin-up of two layers of immiscible fluid with a free upper surface in a rotating cylinder over a wide range of internal Froude numbers. Observations of the evolution of the velocity field by particle tracking indicates that spin-up of the azimuthal velocity in the upper layer take much longer than in a homogeneous fluid. Initially, spin-up occurs at a rate comparable to that of homogeneous fluid but, at high internal Froude number, a second phase follows in which the remaining lative motion decays much more slowly. Quantitative comparison of these measurements to the theory of Pedlosky (1967) shows good agreement.

Visualization of the interface displacement during spin-up detected the presence of transient azimuthal variations in the interface elevation over a wide range of Froude (F), Ekman (E), and Rossby (ε) number. nalysis of the occurrence of the asymmetric variations using the parameter space (Q, F), where Q = E ^1/2/ε, suggested by the baroclinic instability theory and experiments of Hart (1972), showed that the flow was stable for Q > 0.06 with no discernable dependence on F. This result, together with the prediction of Pedlosky's theory that radial gradient of potential vorticity in the two layers have opposite signs, suggests at the baroclinic instability mechanism was responsible for the asymmetries. The location and timing of these instabilities may account for the discrepancies between the observations and the Pedlosky (1967) theory.     

含不同孔隙流体的砂岩地震波速度随压力变化的实验研究

岩石的地震波性质是区域构造研究和浅部地震勘探应用的基础.延长油田是我国重要的油气生产基地之一,但目前仍缺乏地震波性质方面的基础资料.作者利用Autolab2000多功能岩石物性自动测试设备,在0~180MPa及饱含不同孔隙流体(干燥、饱水及饱油)条件下,研究了三种来自延长油田砂岩岩芯的纵波、横波速度.结果表明:三种砂岩的Vp、Vs1和Vs2均随压力增加(或降低)而基本呈对数曲线增大(或减小);干燥、饱水和饱油三种波速间的关系因砂岩类型不同而不同,这主要取决于岩石的有效弹性模量、孔隙流体性质以及岩石的内部结构等;含同种孔隙流体的不同类型砂岩,其Vp、Vs1和Vs2随压力变化的规律主要受岩石孔隙度和粒度的影响;而含不同孔隙流体的同种砂岩,其Vp、Vs1和Vs2随压力变化的规律则主要受控于岩石的有效弹性模量和流体密度.另外,含水或含油饱和度的变化对Vs1和Vs2基本没有影响.实验结果可以为该地区地震资料的解释及与声波测井之间的对比提供重要的基础数据和参考依据.     

A new approach for free vibration analysis of a system of elastically interconnected similar rectangular plates

A new procedure is proposed to ease the analyses of the free vibration of an elastically connected identical plates system with respect to Kirchhoff plate theory. A structure of n parallel, elastically connected rectangular plates is of concern, whereby the motion is explained by a set of n coupled partial differential equations. The method involves a new change in variables to uncouple equations and form an equal system of n decoupled plates, while each is assumed to be elastically connected to...     

Frontal upwelling in a rotating two-layer fluid

Abstract

This paper describes the source-sink driven flow in a two-layer fluid confined in a rotating annulus. Light fluid is injected at the inner wall, while denser fluid is withdrawn at the outer wall. The interface between the immiscible fluids intersects the bottom and thus produces a front. The net transport from the source to the sink is carried by Ekman layers at the bottom and at the interface, and by Stewartson layers at the side walls. A detached Stewartson layer arises at the front, leading to a pronounced upwelling circulation.     

Seismic array evidence for a two-layer core transition zone

Measurements of traveltimes and traveltime gradients for PKP phases recorded at the Warramunga seismic array from distances between 113° and 176° provide evidence for a two-layer transition zone in the earth's core. The basic data consist of paper recordings played out at 40 mm/sec from analog magnetic tape with amplitude gain control and narrow bandpass filters. Measured values of $\text{d}\text{T}\text{dΔ}$ are perturbed by structure beneath the array and it is necessary to correct for this effect by an empirical approach. On the basis of the $\text{d}\text{T}\text{dΔ}$ values and traveltimes, two precursor branches to PKP_DF for Δ < 143° are identified consistently. These continue as branches intermediate to PKP_DF and PKP_AB for Δ > 145°. The corrected $\text{d}\text{T}\text{dΔ}$ values of all phases are smoothed and used to derive a velocity model for the core. The preferred model includes two velocity discontinuities of less than 0.1 km/sec each in the core transition zone to generate the two additional PKP branches. Together with the velocity jump at the outer-inner core boundary, these discontinuities define two layers each a few hundred kilometers thick. It is argued that recent proposals concerning inhomogeneities at the base of the mantle or within the transition zone as the source of PKP precursors do not agree with our observations nor those of other researchers. Inhomogeneities are not excluded but are considered a secondary effect for PKP phases, the primary effect being due to two discontinuous velocity increases in the transition zone.     

A SPH simulation on large-amplitude sloshing for fluids in a two-dimensional tank

Smoothed particle hydrodynamics(SPH) is a mesh-free adaptive Lagrangian particle method with attractive features for dealing with the free surface flow.This paper applies the SPH method to simulate the large-amplitude lateral sloshing both with and without a floating body,and the vertical parametrically-excited sloshing in a two-dimensional tank.The numerical results show that the SPH approach has an obvious advantage over conventional mesh-based methods in handling nonlinear sloshing problems such as violent fluid-solid interaction,and flow separation and wave-breaking on the free fluid surface.The SPH method provides a new alternative and an effective way to solve these special strong nonlinear sloshing problems.     

双层界面重磁联合反演方法研究

Based on the synchronous joint gravity and magnetic inversion of single interface by Pilkington and the need of revealing Cenozoic and crystalline basement thickness in the new round of oil-gas exploration, we propose a joint gravity and magnetic inversion methodfor two-layer models by concentrating on the relationship between the change of thicknessI and position of the middle layer and anomaly and discuss the effects of the key parameters. Model tests and application to field data show the validity of this method.     

Internal circulation in a buoyant two-fluid Newtonian sphere: implications for composed magmatic diapirs

Igneous plutons frequently show chemical zoning. The most commonly documented zoning is with the lighter, more silicic, rocks in the centre of the body and the denser, more basic, rocks in the external zone (normal zoning). Less commonly, some plutons show reverse zoning so that the more basic rocks occupy the centre. Widespread evidence shows that zoning in many plutons is the result of interaction between basic and silicic melts.

This work studies, by means of finite difference numerical models, pluton zoning which is due to internal circulation in diapirs comprising two magmas of different composition. Diapirs are modelled here as buoyant isothermal spheres composed of two Newtonian fluids rising through a Newtonian ambient fluid. Ratios of viscosities and densities of the two fluids were varied and the results demonstrated two different styles of internal circulation in rising spheres. The first style, termed “coupled circulation”, is characterised by continuous overturning of both the fluids in a single cell, evolving through both normal and reverse compositional zoning. The overturns stir the fluids and enhance both magma mingling and mixing. Coupled circulation develops in spheres comprising fluids of similar densities and viscosities. As these properties become increasingly different the internal circulation tends to decouple. “Decoupled circulation”, is characterised by circulation of the fluids in two separate cells. Decoupling stops the overturns between the two magmas so that the diapir preserves a reverse zoning throughout its rise, with the denser fluid occupying the central zone. There is less possibility of magma mingling in diapirs undergoing decoupled circulation. Thus, pairs of magmas of similar properties, such as andesite and rhyolite, are most likely to develop coupled circulation leading to both normal and reverse zoning in diapirs; whereas magmas of very different properties, such as basalt and rhyolite, are most likely to decouple resulting in reverse zonation.

The models indicate that reverse zoning would be the most common internal pluton geometry if zoning were controlled by internal circulation alone. Model diapirs which rise along channels of warm, low viscosity wall-rock (hot Stokes' models) or low viscosity shear zones show an increased tendency towards coupled circulation and more intense mechanical stirring of the magmas.     

Electromagnetic induction of a three-dimensional conductivity inhomogeneity in a two-layer earth

Summary The results of numerical computations of the electromagnetic field induced in a two-layer model of the Earth with a three-dimensional inhomogeneity (a block) in the subsurface layer are given. Several recommendations are given which have enabled the solution of a complicated system of integral equations and the computation of the field at the Earth's surface in an effective way. The analysis of the obtained solution has proved that, in an anomalous electromagnetic field, the field of the horizontal electric dipole, oriented in the direction of the exciting field, is predominant. A number of practically usable diagrams and approximative formulae is given.