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1.
Abstract Small amplitude two-dimensional Boussinesq convection in a plane layer with stress-free boundaries rotating uniformly about the vertical is studied. A horizontally unbounded layer is modelled by periodic boundary conditions. When the centrifugal force is balanced by an appropriate pressure gradient the resulting equations are translation invariant, and overstable convection can take the form of travelling waves. In the Prandtl number regime 0.53 < [sgrave] < 0.68 such solutions are preferred over the more usual standing waves. For [sgrave] < 0.53, travelling waves are stable provided the Taylor number is sufficiently large. 相似文献
2.
Three-dimensional (3-D) numerical simulations of single turbulent thermal plumes in the Boussinesq approximation are used to understand more deeply the interaction of a plume with itself and its environment. In order to do so, we varied the Rayleigh and Prandtl numbers from Ra?~?105 to Ra?~?108 and from Pr?~?0.025 to Pr?~?70. We found that thermal dissipation takes place mostly on the border of the plume. Moreover, the rate of energy dissipation per unit mass ε T has a critical point around Pr?~?0.7. The reason is that at Pr greater than ~0.7, buoyancy dominates inertia and thermal advection dominates wave formation whereas this trend is reversed at Pr less than ~0.7. We also found that for large enough Prandtl number (Pr?~?70), the velocity field is mostly poloidal although this result was known for Rayleigh–Bénard convection (see Schmalzl et al. [On the validity of two-dimensional numerical approaches to time-dependent thermal convection. Europhys. Lett. 2004, 67, 390--396]). On the other hand, at small Prandtl numbers, the plume has a large helicity at large scale and a non-negligible toroidal part. Finally, as observed recently in details in weakly compressible turbulent thermal plume at Pr?=?0.7 (see Plourde et al. [Direct numerical simulations of a rapidly expanding thermal plume: structure and entrainment interaction. J. Fluid Mech. 2008, 604, 99--123]), we also noticed a two-time cycle in which there is entrainment of some of the external fluid to the plume, this process being most pronounced at the base of the plume. We explain this as a consequence of calculated Richardson number being unity at Pr?=?0.7 when buoyancy balance inertia. 相似文献
3.
Introduction The velocity field of surface plate motion can be split into a poloidal and a toroidal parts.At the Earth′s surface,the toroidal component is manifested by the existence of transform faults,and the poloidal component by the presence of convergence and divergence,i.e.spreading and subduc-tion zones.They have coupled each other and completely depicted the characteristics of plate tec-tonic motions.The mechanism of poloidal field has been studied fairly clearly which is related to … 相似文献
4.
Introduction Mantle convection is thought to be the most effective way of heat transportation in the earth and the source of driving lithospheric plates (Elasser, 1971). The velocity field of plate motion can be split into poloidal and toroidal parts, which are corresponding to the vertical (i.e. radial) and horizontal motions, respectively, in global model. The toroidal component is manifested in the existences of transform faults and the poloidal one in the presences of convergence and diver… 相似文献
5.
Gary A. Glatzmaier 《地球物理与天体物理流体动力学》2013,107(2):223-264
Abstract We describe nonlinear time-dependent numerical simulations of whole mantle convection for a Newtonian, infinite Prandtl number, anelastic fluid in a three-dimensional spherical shell for conditions that approximate the Earth's mantle. Each dependent variable is expanded in a series of 4,096 spherical harmonics to resolve its horizontal structure and in 61 Chebyshev polynomials to resolve its radial structure. A semiimplicit time-integration scheme is used with a spectral transform method. In grid space there are 61 unequally-spaced Chebyshev radial levels, 96 Legendre colatitudinal levels, and 192 Fourier longitudinal levels. For this preliminary study we consider four scenarios, all having the same radially-dependent reference state and no internal heating. They differ by their radially-dependent linear viscous and thermal diffusivities and by the specified temperatures on their isothermal, impermeable, stress-free boundaries. We have found that the structure of convection changes dramatically as the Rayleigh number increases from 105 to 106 to 107. The differences also depend on how the Rayleigh number is increased. That is, increasing the superadiabatic temperature drop, δT, across the mantle produces a greater effect than decreasing the diffusivities. The simulation with a Rayleigh number of 107 is approximately 10,000 times critical, close to estimates of that for the Earth's mantle. However, although the velocity structure for this highest Rayleigh number scenario may be adequately resolved, its thermodynamic structure requires greater horizontal resolution. The velocity and thermodynamic structures of the scenarios at Rayleigh numbers of 105 and 106 appear to be adequately resolved. The 105 Rayleigh number solution has a small number of broad regions of warm upflow embedded in a network of narrow cold downflow regions; whereas, the higher Rayleigh number solutions (with large δT) have a large number of small hot upflow plumes embedded in a broad weak background of downflow. In addition, as would be expected, these higher Rayleigh number solutions have thinner thermal boundary layers and larger convective velocities, temperatures perturbations, and heat fluxes. These differences emphasize the importance of developing even more realistic models at realistic Rayleigh numbers if one wishes to investigate by numerical simulation the type of convection that occurs in the Earth's mantle. 相似文献
6.
Joe M. Straus 《地球物理与天体物理流体动力学》2013,107(1):261-281
AbstractTwo upper bounding problems for thermal convection in a layer of fluid contained between perfectly conducting stress-free boundaries are treated numerically. Since the Euler equations resulting from this variational approach are simpler than the Navier-Stokes equations, they allow numerical calculations to be carried out economically to fairly large values of the Rayleigh number. The upper bounding problem formulated by Howard (1963), which yields a Nusselt number independent of Prandtl number, diverges from the correct behavior as the Rayleigh number increases. In hopes of coming closer to results of previous investigations of the Boussinesq equations of motion, a more restrictive upper bounding problem is formulated. For large Prandtl numbers the momentum equation is linearized and is used as an explicit side constraint on the variational problem, thereby forcing the solutions to more closely resemble the solutions of the Boussinesq equations. Numerical calculations at values of the Rayleigh number up to 1.5 × 105 indicate that the additional constraint decreases the upper bound on the Nusselt number; it appears that this upper bound differs by only a multiplicative factor from that calculated from solutions of the full equations of motion and may be a reasonable approximation for large Rayleigh numbers. 相似文献
7.
O. M. Podvigina 《地球物理与天体物理流体动力学》2013,107(4):409-433
Investigation of magnetic field generation by convective flows is carried out for three values of kinematic Prandtl number: P = 0.3, 1 and 6.8. We consider Rayleigh–Bénard convection in Boussinesq approximation assuming stress-free boundary conditions on horizontal boundaries and periodicity with the same period in the x and y directions. Convective attractors are modelled for increasing Rayleigh numbers for each value of the kinematic Prandtl number. Linear and non-linear dynamo action of these attractors is studied for magnetic Prandtl numbers P m ≤ 100. Flows, which can act as magnetic dynamos, have been found for all the three considered values of P, if the Rayleigh number R is large enough. The minimal R, for which of magnetic field generation occurs, increases with P. The minimum (over R) of critical Pm for magnetic field generation in the kinematic regime is admitted for P = 0.3. Thus, our study indicates that smaller values of P are beneficial for magnetic field generation. 相似文献
8.
The emplacement of kimberlites in the North American and African continents since the early Palaeozoic appears to have occurred during periods of relatively slow motion of these continents. The distribution of kimberlites in time may reflect the global pattern of convection, which forces individual plates to move faster or slower at different times. Two-dimensional numerical experiments on a convecting layer with a moving upper boundary show two different regimes: in the first, when the upper boundary velocity is high, heat is transferred by the large-scale circulation and in the second, when the upper boundary velocity is lower, heat is predominantly transferred by thermal plumes rising from the lower boundary layer. For a reasonable mantle solidus, this second regime can give rise to partial melting beneath the moving plate, far from the plate boundaries. The transition between these modes takes place over a small range of plate velocities; for a Rayleigh number of 106 it occurs around 20 mm yr?1. We suggest that the generation of kimberlite magmas may result from thermal plumes incident on the base of a slowly moving plate. 相似文献
9.
Abstract This paper develops further a convection model that has been studied several times previously as a very crude idealization of planetary core dynamics. A plane layer of electrically-conducting fluid rotates about the vertical in the presence of a magnetic field. Such a field can be created spontaneously, as in the Childress—Soward dynamo, but here it is uniform, horizontal and externally-applied. The Prandtl number of the fluid is large, but the Ekman, Elsasser and Rayleigh numbers are of order unity, as is the ratio of thermal to magnetic diffusivity. Attention is focused on the onset of convection as the temperature difference applied across the layer is increased, and on the preferred mode, i.e., the planform and time-dependence of small amplitude convection. The case of main interest is the layer confined between electrically-insulating no-slip walls, but the analysis is guided by a parallel study based on illustrative boundary conditions that are mathematically simpler. 相似文献
10.
M. W. Dunlop B. Lavraud P. Cargill M. G. G. T. Taylor A. Balogh H. Réme P. Decreau K. -H. Glassmeier R. C. Elphic J. -M. Bosqued A. N. Fazakerley I. Dandouras C. P. Escoubet H. Laakso A. Marchaudon 《Surveys in Geophysics》2005,26(1-3):5-55
This paper reviews Cluster observations of the high altitude and exterior (outer) cusp, and adjacent regions in terms of new multi-spacecraft analysis and the geometry of the surrounding boundary layers. Several crossings are described in terms of the regions sampled, the boundary dynamics and the electric current signatures observed. A companion paper in this issue focuses on the detailed plasma distributions of the boundary layers. The polar Cluster orbits take the four spacecraft in a changing formation out of the magnetosphere, on the northern leg, and into the magnetosphere, on the southern leg, of the orbits. During February to April the orbits are centred on a few hours of local noon and, on the northern leg, generally pass consecutively through the northern lobe and the cusp at mid- to high-altitudes. Depending upon conditions, the spacecraft often sample the outer cusp region, near the magnetopause, and the dayside and tail boundary layer regions adjacent to the central cusp. On the southern, inbound leg the sequence is reversed. Cluster has therefore sampled the boundaries around the high altitude cusp and nearby magnetopause under a variety of conditions. The instruments onboard provide unprecedented resolution of the plasma and field properties of the region, and the simultaneous, four-spacecraft coverage achieved by Cluster is unique. The spacecraft array forms a nearly regular tetrahedral configuration in the cusp and already the mission has covered this region on multiple spatial scales (100–2000 km). This multi-spacecraft coverage allows spatial and temporal features to be distinguished to a large degree and, in particular, enables the macroscopic properties of the boundary layers to be identified: the orientation, motion and thickness, and the associated current layers. We review the results of this analysis for a number of selected crossings from both the North and South cusp regions. Several key results have been found or have confirmed earlier work: (1) evidence for magnetically defined boundaries at both the outer cusp/magnetosheath interface and the␣inner cusp/lobe or cusp/dayside magnetosphere interface, as would support the existence of a distinct exterior cusp region; (2) evidence for an associated indentation region on the magnetopause across the outer cusp; (3) well defined plasma boundaries at the edges of the mid- to high-altitude cusp “throat”, and well defined magnetic boundaries in the high-altitude “throat”, consistent with a funnel geometry; (4) direct control of the cusp position, and its extent, by the IMF, both in the dawn/dusk and North/South directions. The exterior cusp, in particular, is highly dependent on the external conditions prevailing. The magnetic field geometry is sometimes complex, but often the current layer has a well defined thickness ranging from a few hundred (for the inner cusp boundaries) to 1000 km. Motion of the inner cusp boundaries can occur at speeds up to 60 km/s, but typically 10–20 km/s. These speeds appear to represent global motion of the cusp in some cases, but also could arise from expansion or narrowing in others. The mid- to high-altitude cusp usually contains enhanced ULF wave activity, and the exterior cusp usually is associated with a substantial reduction in field magnitude. 相似文献
11.
The effects of variable viscosity on flow dynamics within spherical shells are investigated using a finite-element thermal convection model, and preliminary result for cases with relatively low Rayleigh numbers and small viscosity contrasts are reported. These results demonstrate some general effects of viscosity variation on mantle dynamics, and, in particular, the generation of toroidal energy. Since lateral viscosity variations are necessary in the generation of toroidal motion in a thermally driven convective system, it is not surprising our results show that flows with greater viscosity contrasts produce greater amounts of toroidal energy. Our preliminary study further shows that solutions become more time-dependent as viscosity contrasts increase. Increasing the Rayleigh number is also found to increase the magnitude of toroidal energy. Internal heating, on the other hand, appears to lead to less toroidal energy compared wth bottom heating because it tends to produce a thermally more uniform interior and thus smaller viscosity variations. 相似文献
12.
Abstract Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell with and without inhomogeneous temperature anomalies on the top boundary have been carried out using a three-dimensional, time-dependent, spectral-transform code. The spherical shell of Boussinesq fluid has inner and outer radii the same as those of the Earth's liquid outer core. The Taylor number is 107, the Prandtl number is 1, and the Rayleigh number R is 5Rc (Rc is the critical value of R for the onset of convection when the top boundary is isothermal and R is based on the spherically averaged temperature difference across the shell). The shell is heated from below and cooled from above; there is no internal heating. The lower boundary of the shell is isothermal and both boundaries are rigid and impermeable. Three cases are considered. In one, the upper boundary is isothermal while in the others, temperature anomalies with (l,m) = (3,2) and (6,4) are imposed on the top boundary. The spherically averaged temperature difference across the shell is the same in all three cases. The amplitudes of the imposed temperature anomalies are equal to one-half of the spherically averaged temperature difference across the shell. Convective structures are strongly controlled by both rotation and the imposed temperature anomalies suggesting that thermal inhomogeneities imposed by the mantle on the core have a significant influence on the motions inside the core. The imposed temperature anomaly locks the thermal perturbation structure in the outer part of the spherical shell onto the upper boundary and significantly modifies the velocity structure in the same region. However, the radial velocity structure in the outer part of the shell is different from the temperature perturbation structure. The influence of the imposed temperature anomaly decreases with depth in the shell. Thermal structure and velocity structure are similar and convective rolls are more columnar in the inner part of the shell where the effects of rotation are most dominant. 相似文献
13.
本文以板块绝对运动AM1-2模型为边界条件探讨了不同的瑞利数下地幔热对流模型.结果表明,瑞利数小于10000(529.41)时,地幔对流呈现以板块驱动图式,运动的极型场和环型场由板块运动激发,两种场占有差不多相同的功率.当瑞利数增加到接近或略超过最低临界值时(约1.5倍),对流呈现出复杂状态:1.板块运动速率小于下伏地幔对流速率;2.区域性的双层对流环出现;3.对流谱成分发生变化;4.环型场仅在地幔很浅的区域中起作用,而在地幔深部对流图式中影响很小. 相似文献
14.
15.
Convection in the Earth's core is driven much harder at the bottom than the top. This is partly because the adiabatic gradient steepens towards the top, partly because the spherical geometry means the area involved increases towards the top, and partly because compositional convection is driven by light material released at the lower boundary and remixed uniformly throughout the outer core, providing a volumetric sink of buoyancy. We have therefore investigated dynamo action of thermal convection in a Boussinesq fluid contained within a rotating spherical shell driven by a combination of bottom and internal heating or cooling. We first apply a homogeneous temperature on the outer boundary in order to explore the effects of heat sinks on dynamo action; we then impose an inhomogeneous temperature proportional to a single spherical harmonic Y 2² in order to explore core-mantle interactions. With homogeneous boundary conditions and moderate Rayleigh numbers, a heat sink reduces the generated magnetic field appreciably; the magnetic Reynolds number remains high because the dominant toroidal component of flow is not reduced significantly. The dipolar structure of the field becomes more pronounced as found by other authors. Increasing the Rayleigh number yields a regime in which convection inside the tangent cylinder is strongly affected by the magnetic field. With inhomogeneous boundary conditions, a heat sink promotes boundary effects and locking of the magnetic field to boundary anomalies. We show that boundary locking is inhibited by advection of heat in the outer regions. With uniform heating, the boundary effects are only significant at low Rayleigh numbers, when dynamo action is only possible for artificially low magnetic diffusivity. With heat sinks, the boundary effects remain significant at higher Rayleigh numbers provided the convection remains weak or the fluid is stably stratified at the top. Dynamo action is driven by vigorous convection at depth while boundary thermal anomalies dominate in the upper regions. This is a likely regime for the Earth's core. 相似文献
16.
Abstract This paper develops further a convection model that has been studied several times previously as a very crude idealization of planetary core dynamics. A plane layer of electrically-conducting fluid rotates about the vertical in the presence of a magnetic field. Such a field can be created spontaneously, as in the Childress-Soward dynamo, but here it is uniform, horizontal and externally-applied. The Prandtl number of the fluid is large, but the Ekman, Elsasser and Rayleigh numbers are of unit order. In Part I of this series, it was also supposed that the ratio thermal diffusivity diffusivity/magnetic diffusivity is O(1), but here we suppose that this ratio is large. The character of the solution is changed in this limit. In the case of main interest, when the layer is confined between electrically-insulating no-slip walls, the solution is significantly different from the solution when the mathematically simpler, illustrative boundary conditions also considered in Part I are employed. As in Part I, attention is focussed on the onset of convection as the temperature difference applied across the layer is increased, and on the preferred mode, i.e., the planform and time-dependence of small amplitude convection. 相似文献
17.
Numerical experiments have been carried out on two-dimensional thermal convection, in a Boussinesq fluid with infinite Prandtl number, at high Rayleigh numbers. With stress free boundary conditions and fixed heat flux on upper and lower boundaries, convection cells develop with aspect ratios (width/depth) λ? 5, if heat is supplied either entirely from within or entirely from below the fluid layer. The preferred aspect ratio is affected by the lateral boundary conditions. If the temperature, rather than the heat flux, is fixed on the upper boundary the cells haveλ ≈ 1. At Rayleigh numbers of 2.4 × 105 and greater, small sinking sheets are superimposed on the large aspect ratio cells, though they do not disrupt the circulation. Similar two-scale flows have been proposed for convection in the earth's mantle. The existence of two scales of flow in two-dimensional numerical experiments when the viscosity is constant will allow a variety of geophysically important effects to be investigated. 相似文献
18.
《Earth and Planetary Science Letters》2003,205(3-4):107-121
In the last decade, significant progress has been made toward understanding how plate tectonics is generated from mantle dynamics. A primary goal of plate-generation studies has been the development of models that allow the top cold thermal boundary layer of mantle convection, i.e. the lithosphere, to develop broad and strong plate-like segments separated by narrow, weak and rapidly deforming boundaries; ideally, such models also permit significant strike-slip (toroidal) motion, passive ridges (i.e. pulled rather than pried apart), and self-consistent initiation of subduction. A major outcome of work so far is that nearly all aspects of plate generation require lithospheric rheologies and shear-localizing feedback mechanisms that are considerably more exotic than rheologies typically used in simple fluid-dynamical models of mantle flow. The search for plate-generating behavior has taken us through investigations of the effects of shear weakening (‘stick-slip’) and viscoplastic rheologies, of melting at ridges and low-viscosity asthenospheres, and of grain-size dependent rheologies and damage mechanics. Many such mechanisms, either by themselves or in combination, have led to self-consistent fluid-mechanical models of mantle flow that are remarkably plate-like, which is in itself a major accomplishment. However, many other important problems remain unsolved, such as subduction intiation and asymmetry, temporal evolution of plate geometry, rapid changes in plate motion, and the Archaean initiation of the plate-tectonic mode of convection. This paper presents a brief review of progress made in the plate-generation problem over the last decade, and discusses unresolved issues and future directions of research in this important area. 相似文献
19.
We have used a numerical scheme based on higher-order finite differences to investigate effects of adiabatic heating and viscous dissipation on 3-D rapidly rotating thermal convection in a Cartesian box with an aspect-ratio of 221. Although we omitted coupling with the magnetic field, which can play a key role in the dynamics of the Earth's core, the understanding of non-linear rotating convection including realistic thermodynamic effects is a necessary prerequisite for understanding the full complexity of the Earth's core dynamics. The system of coupled partial differential equations has been solved in terms of the principal variables vorticity , vector potential A and temperature T. The use of the vector potential A allows the velocity field to be calculated with one spatial differentiation in contrast to the spheroidal and toroidal function approach. The temporal evolution is governed by a coupled time-dependent system consisting of and T. The equations are discretized in all directions by using an eighth-order, variable spaced scheme. Rayleigh number Ra of 106, Taylor number Ta of 108 and a Prandtl number Pr of 1 have been employed. The dissipation number of the outer core was taken to be 0.2. A stretched grid has been employed in the vertical direction for resolving the thin shear boundary layers at the top and bottom. This vertical resolution corresponds to around 240 regularly spaced points with an eighth-order accuracy. For the regime appropriate to the Earth's outer core, the dimensionless surface temperature T
0 takes a large value, around 4. This large value in the adiabatic heating/cooling term is found to cause stabilization of both the temperature and velocity fields. 相似文献
20.
The residence time of a tracer in a control domain is usually computed by releasing tracer parcels and registering the time when each of these tracer parcels cross the boundary of the control domain. In this Lagrangian procedure, the particles are discarded or omitted as soon as they leave the control domain. In a Eulerian approach, the same approach can be implemented by integrating forward in time the advection–diffusion equation for a tracer. So far, the conditions to be applied at the boundary of the control domain were uncertain. We show here that it is necessary to prescribe that the tracer concentration vanishes at the boundary of the control domain to ensure the compatibility between the Lagrangian and Eulerian approaches. When we use the Constituent oriented Age and Residence time Theory (CART), this amounts to solving the differential equation for the residence time with boundary conditions forcing the residence time to vanish at the open boundaries of the control domain. Such boundary conditions are likely to induce the development of boundary layers (at outflow boundaries for the tracer concentration and at inflow boundaries for the residence time). The thickness of these boundary layers is of the order of the ratio of the diffusivity to the velocity. They can however be partly smoothed by tidal and other oscillating flows. 相似文献