Quasigeostrophic planetary waves in a two-layer ocean with one-dimensional periodic bottom topography

Although the study of topographic effects on the Rossby waves in a stratified ocean has a long history, the wave property over a periodic bottom topography whose lateral scale is comparable to the wavelength is still not clear. The present paper treats this problem in a two-layer ocean with one-dimensional periodic bottom topography by a simple numerical method, in which no restriction on the wavelength and/or the horizontal scale of the topography is required. The dispersion diagram is obtained for a wavenumber range of [?π/L _b , π/L _b ], where L _b is the periodic length of the topography. When the topographic?β?is not negligible compared to the planetary β, the Rossby wave solutions around the wavenumbers which satisfy the resonant condition among the waves and topography disappear and separate into an infinite number of discrete modes. For convenience, each mode is numbered in order of frequency. As topographic height is increased, the high frequency barotropic Rossby wave (mode 1) becomes a topographic mode which can exist even on the f plane, and the highfrequency baroclinic mode (mode 2) becomes a surface intensified mode. Behaviors of low frequency modes are somewhat complicated. When the topographic amplitude is small, the low frequency baroclinic modes tend to be bottom trapped and the low frequency barotropic modes tend to be surface intensified. As topographic amplitude further increases, the relation between the mode number and vertical structure changes. This change can be attributed to the increase of the frequency of the topographic mode with the topographic amplitude.     

Penetrative convection in the upper ocean due to surface cooling

Abstract

A new non-linear model of mixing and convection based on a modelling of two buoyant interacting fluids is applied to penetrative convection in the upper ocean due to surface cooling. In view of simple algebra, the model is one-dimensional. Dissipation is included, but no mean shear is present. A non-similar analytical solution is found in the case of a well-mixed layer bounded below by a sharp thermocline treated as a boundary layer. This solution is valid if the Richardson number, R _i , defined as the ratio of the total mixed-layer buoyancy to a characteristic rms vertical velocity, is much greater than unity. The model predicts a deepening rate proportional to R _i ^?3/4. The thermocline remains of constant thickness, and the ratio thermocline thickness to mixed-layer depth decreases as R _i ^?3/4 as the mixed layer deepens. If the surface flux is constant, the mixed-layer depth increases with time as t ^½. The vertical structure throughout the mixed layer and thermocline is given by the analytical solution, and vertical profiles of mean temperature and vertical fluxes are plotted. Computed profiles and available laboratory data agree remarkably well. Moreover, the accuracy of the simple analytical results presented here is comparable to that of sophisticated turbulence numerical models.     

Penetrative convection and multi-component diffusion in a porous medium

Linear instability and nonlinear energy stability analyses are developed for the problem of a fluid-saturated porous layer stratified by penetrative thermal convection and two salt concentrations. Unusual neutral curves are obtained, in particular non-perfect `heart-shaped' oscillatory curves that are disconnected from the stationary neutral curve. These curves show that three critical values of the thermal Rayleigh number may be required to fully describe the linear stability criteria. As the penetrative effect is increased, the oscillatory curves depart more and more from a perfect heart shape. For certain values of the parameters it is shown that the minima on the oscillatory and stationary curves occur at the same Rayleigh number but different wavenumbers, offering the prospect of different types of instability occurring simultaneously at different wavenumbers. A weighted energy method is used to investigate the nonlinear stability of the problem and yields unconditional results guaranteeing nonlinear stability for initial perturbations of arbitrary sized amplitude.     

Traveling planetary waves in the stratosphere

This paper reviews the theory and observations of some traveling planetary waves in the stratosphere. Two categories of waves which appear prominently in the literature are discussed: westward propagating waves of periods in the range 3–7 days (the 5-day wave) and in the range 10–20 days (the 16-day wave). Although the observations seem to indicate that these waves are waves of the Rossby type (planetary waves), the evidence is less clear regarding (1) the question of whether these waves are forced internal waves or free (resonant) external waves, and (2) the identification of the observed waves with specific theoretical waves of the Rossby type. When recent observations are compared with theory, the evidence seems to favor the notion that the 5-day and 16-day waves of longitudinal wave number 1 may be identified, respectively, with the gravest and next gravest symmetric free Rossby modes. However, the observational evidence seems to be less clear regarding the nature of the 16-day wave than the 5-day wave.     

Penetrative convection in rotating fluids: A model for the base of stellar convection zones

Abstract

To model penetrative convection at the base of a stellar convection zone we consider two plane parallel, co-rotating Boussinesq layers coupled at their fluid interface. The system is such that the upper layer is unstable to convection while the lower is stable. Following the method of Kondo and Unno (1982, 1983) we calculate critical Rayleigh numbers R_c for a wide class of parameters. Here, R_c is typically much less than in the case of a single layer, although the scaling R_c～T^2/3 as T → ∞ still holds, where T is the usual Taylor number. With parameters relevant to the Sun the helicity profile is discontinuous at the interface, and dominated by a large peak in a thin boundary layer beneath the convecting region. In reality the distribution is continuous, but the sharp transition associated with a rapid decline in the effective viscosity in the overshoot region is approximated by a discontinuity here. This source of helicity and its relation to an alpha effect in a mean-field dynamo is especially relevant since it is a generally held view that the overshoot region is the location of magnetic field generation in the Sun.     

On steady and travelling waves over bottom topography in the model of homogeneous flow in a beta-plane channel

Abstract

A spectral low-order model is proposed in order to investigate some effects of bottom corrugation on the dynamics of forced and free Rossby waves. The analysis of the interaction between the waves and the topographic modes in the linear version of the model shows that the natural frequencies lie between the corresponding Rossby wave frequencies for a flat bottom and those applying in the “topographic limit” when the beta-effect is zero. There is a possibility of standing or eastward-travelling free waves when the integrated topograhic effect exceeds the planetary beta-effect.

The nonlinear interactions between forced waves in the presence of topography and the beta-effect give rise to a steady dynamical mode correlated to the topographic mode. The periodic solution that includes this steady wave is stable when the forcing field moves to the West with relatively large phase speed. The energy of this solution may be transferred to the steady zonal shear flow if the spatial scale of this zonal mode exceeds the scale of the directly forced large-scale dynamical mode.     

地幔动力学研究进展--地幔对流

分类而且系统地回顾了地幔对流研究的进展情况.发现虽然静态地幔对流模型(没有考虑地球自转或差异旋转效应)在拟合某些地表观测如重力异常、大地水准面异常以及地下应力场时取得了较满意的结果,但是却对于某些全球尺度的地质特征如0°、180°半球的非对称性显得难以适应,因此建议性地提出了今后地幔对流的可能发展方向.     

Lagging mantle convection,the geoid and mantle structure

In numerical models of convection incorporating migration of a simulated subduction zone, the main descending flow lags far behind the migrating trench, and a geoid low is associated with the main descending flow. This provides physical plausibility for the suggestion by Chase and Sprowl that present very long-wavelength (degree 2–4) geoid lows are associated with Mesozoic trench locations, and suggests further that the present long-wavelength geoid, deep mantle structure and hotspot distribution may be straightforward consequences of plate evolution since the Upper Paleozoic.     

Viscosity distribution in the mantle convection models

Viscosity is a fundamental property of the mantle which determines the global geodynamical processes. According to the microscopic theory of defects and laboratory experiments, viscosity exponentially depends on temperature and pressure, with activation energy and activation volume being the parameters. The existing laboratory measurements are conducted with much higher strain rates than in the mantle and have significant uncertainty. The data on postglacial rebound only allow the depth distributions of viscosity to be reconstructed. Therefore, spatial distributions (along the depth and lateral) are as of now determined from the models of mantle convection which are calculated by the numerical solution of the convection equations, together with the viscosity dependences on pressure and temperature (PT-dependences). The PT-dependences of viscosity which are presently used in the numerical modeling of convection give a large scatter in the estimates for the lower mantle, which reaches several orders of magnitude. In this paper, it is shown that it is possible to achieve agreement between the calculated depth distributions of viscosity throughout the entire mantle and the postglacial rebound data. For this purpose, the values of the volume and energy of activation for the upper mantle can be taken from the laboratory experiments, and for the lower mantle, the activation volume should be reduced twice at the 660-km phase transition boundary. Next, the reduction in viscosity by an order of magnitude revealed at the depths below 2000 km by the postglacial rebound data can be accounted for by the presence of heavy hot material at the mantle bottom in the LLSVP zones. The models of viscosity spatial distribution throughout the entire mantle with the lithospheric plates are presented.     

Finite amplitude instability thresholds in penetrative convection

Abstract

A nonlinear energy stability analysis is presented for the penetrative convection model of Veronis (1963). For top temperatures between 4°C and 8°C the nonlinear stability boundary obtained is very close to the linear one of Veronis and enables a region of possible sub-critical instabilities to be determined.     

Numerical approaches to solid-state convection in planetary bodies

Numerical methods suitable for the calculation of finite-amplitude thermal convection in planetary interiors are reviewed. Three methods of approach are described and compared; namely, finite-difference, finite-element and spectral procedures. Application of each approach to three-dimensional solid-state convection with complex rheology is examined. Finite-difference methods have enjoyed some success to date. However, future developments are likely to favor finite-element methods when rheology and boundaries are complex, and spectral methods when accuracy is paramount and rheology and boundaries are simple.     

Thermochemical convection in the mantle with oceanic crust recirculation

Basalts of mid-ocean ridges are depleted in incompatible elements that have passed into the continental crust. Basalts of hot spots (oceanic islands and igneous provinces) have a chemical composition close to the primary uniform mantle and are even somewhat enriched in incompatible elements. At present, for explaining the reason for this difference, there are different qualitative schemes of differentiation and mixing of substance in the mantle. In the present work, the results of numerical modeling of the two-component thermochemical convection in the mantle are given. They quantitatively demonstrate with which parameters in the mantle the layers of different chemical composition can remain unchanged. Models with different density contrasts and with variable viscosity are examined. The times of the partial mixing of layers depending on the values of these parameters are calculated. For retaining the stratified mantle for two Ga, the density contrast must be more than 2%. If the layer D″ contains a substance of the primary composition, then, its upper boundary can be the place of origin of the plumes that feed the hot spots of the Earth. The enrichment in the incompatible elements and the variety of the chemical composition of hot spots can be explained by the mixing of the substance of the slowly eroded D″ layer and the oceanic crust accumulated in it.     

Dynamical stability of convection cells in the upper mantle

Summary A two-dimensional flow model of an incompressible fluid with constant viscosity has been used to study the changes in the large-scale flow pattern (aspect ratio 4). Implications for convection in the Earth's mantle are discussed.

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On the aspect ratios of two-layer mantle convection

Some dynamic implications of separate convection systems in the upper and the lower mantle of the Earth are investigated. It is shown that the horizontal scale of convection cells in the lower mantle is likely to determine the scale of flow in the upper mantle. This does not preclude the nearly independent realization of convection cells with horizontal dimensions comparable to the depth of the upper mantle.     

Discussion of the new kinematic mantle convection theory

Summary First of all it is shown that the nine fundamental suppositions of the kinematic theory of the mantle convection fit well into the image which is obtained from the mantle by other geophysical results. The question of episodicity of the orogenesis is dealt with in connection with the questions of steadiness and unsteadiness of the convection. Moreover it is shown that the ocean-spreading theory and the kinematic mantle convection theory are compatible with each other in their main aspects and that they complement each other.     

Quasi-free-decay magnetic modes in planetary cores

A new category of hydromagnetic waves in a rotating conducting fluid within a spherical shell geometry is investigated. These quasi-free-decay magnetic modes are based on particular solutions of the induction equation where the magnetic diffusion plays the central role. These solutions, normally only decaying with time, become propagative owing to the combined action of the background magnetic field and the rotation. The amplitude and sign of their azimuthal phase drift strongly depend on morphology and magnitude of the background magnetic field. The validity domain of these quasi-free-decay (QFD)-modes is related to the Elsasser number and is written as Λ???1. It follows that these modes dissipate quickly before propagating out. This restriction falls when the above criterion is no longer fulfilled (Λ?～?1), the corresponding modes evolving towards distorted QFD-modes. A systematic study of these QFD-modes is made in the limit of small Elsasser number (Λ???1), for the different symmetries allowed. Application to the Earth's and other planetary cores is then examined for an Elsasser number up to Λ?≈?O(1), in relation to the geomagnetic secular variation and the frozen-flux approximation.