Two-and three-dimensional linear convection in a rotating annulus

Abstract

An exceptional case to the model-independent theory of Knobloch (1995) is presented, by investigating a rotating cylindrical annulus of height H and side wall radii r _o and r _i, with non-slip, perfectly thermally conducting side walls and thermally insulating stress-free ends. Radial heating permits the possibility of either two- or three-dimensional convective solutions being the preferred mode. An analytical solution is obtained for the two-dimensional case and a numerical solution for the three-dimensional solution, which is also applied to the two-dimensional solution. It is shown that both two- and three-dimensional solutions can be realized depending on the aspect ratio, γ = H/d, where d = r _o-r _i is the thickness of the annulus, the radii ratio λ = r _i/r _o and the rotation rate of the model. For γ = O(1) and λ = 0.4, the preferred convective solution is three-dimensional when the Taylor number, T < 10² and two-dimensional for T > 10². For small aspect ratios, γ ? 1, the preferred mode is two-dimensional for all rotation rates.     

Transition to two-dimensional turbulent convection in a rapidly-rotating annulus

Abstract

We study a semi-analytical model of convection in a rapidly-rotating, differentially-heated annulus with sloping top and bottom lids. Rapid rotation leads to a preservation of relatively simple, two-dimensional (2-D) structure in the experimentally-observed flow, while temporal complexity increases with the Rayleigh number. The model is, therefore, two-dimensional; it exhibits a sequence of bifurcations from steadily-drifting, azimuthally-periodic convection columns, also called thermal Rossby waves, through vacillation and a period-doubling cascade, to aperiodic, weakly-turbulent solutions.

Our semi-analytical results match to within a few percent previous numerical results with a limited-resolution 2-D model, and extend these results, due to the greater flexibility of the model presented here. Two types of vacillation are obtained, which we call, by analogy with classical nomenclature of the baroclinic annulus with moderate rotation rates, amplitude vacillation and tilted-trough vacillation. Their properties and dependence on the problem's nondimensional parameters are investigated. The period-doubling cascade for each type of vacillation is studied in some detail.     

Thermal convection in a twisted horizontal magnetic field

The onset of convection in a layer of an electrically conducting fluid heated from below is considered in the case when the layer is permeated by a horizontal magnetic field of strength B ₀ the orientation of which varies sinusoidally with height. The critical value of the Rayleigh number for the onset of convection is derived as a function of the Chandrasekhar number Q. With increasing Q the height of the convection rolls decreases, while their horizontal wavelength slowly increases. Potential applications to the penumbral filaments of sunspots are briefly discussed.     

Convection in a rotating annulus having sidewalls of height-dependent thermal conductance

Abstract

Thermal convection in a vertically-mounted, rotating annulus of a particular design proposed by Davies and Walin (1977) is investigated. The annulus used in the present study differs from the conventional type in some important aspects: the sidewalls are finitely conducting, and the thermal conductance of the sidewalls is height-dependent. The theoretical model due to Davies and Walin is briefly recounted. The present study aims to verify the theoretical model; we have acquired numerical solutions to the governing Navier-Stokes equations. The numerical results are supportive of the theoretical contentions. The near-linear dependence of the isothermal slope on the parameter D, which is a function of Ω and ΔT, is corroborated within reasonable limits. New data on the vertical and radial structures of the meridional and azimuthal flows are presented. The numerical results also confirm that the shape of the sidewall thickness has a substantial influence on the meridional flow patterns. In the bulk of the interior flow field, the dominant azimuthal flow field and the temperature field are linked by the thermal wind relation.     

Transitions to chaotic thermal convection in a rapidly rotating spherical fluid shell

Abstract

Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell heated from below and within have been carried out with a nonlinear, three-dimensional, time-dependent pseudospectral code. The investigated phenomena include the sequence of transitions to chaos and the differential mean zonal rotation. At the fixed Taylor number T _a =10⁶ and Prandtl number Pr=1 and with increasing Rayleigh number R, convection undergoes a series of bifurcations from onset of steadily propagating motions SP at R=R _c = 13050, to a periodic state P, and thence to a quasi-periodic state QP and a non-periodic or chaotic state NP. Examples of SP, P, QP, and NP solutions are obtained at R = 1.3R _c , R = 1.7 R _c , R = 2R _c , and R = 5 R _c , respectively. In the SP state, convection rolls propagate at a constant longitudinal phase velocity that is slower than that obtained from the linear calculation at the onset of instability. The P state, characterized by a single frequency and its harmonics, has a two-layer cellular structure in radius. Convection rolls near the upper and lower surfaces of the spherical shell both propagate in a prograde sense with respect to the rotation of the reference frame. The outer convection rolls propagate faster than those near the inner shell. The physical mechanism responsible for the time-periodic oscillations is the differential shear of the convection cells due to the mean zonal flow. Meridional transport of zonal momentum by the convection cells in turn supports the mean zonal differential rotation. In the QP state, the longitudinal wave number m of the convection pattern oscillates among m = 3,4,5, and 6; the convection pattern near the outer shell has larger m than that near the inner shell. Radial motions are very weak in the polar regions. The convection pattern also shifts in m for the NP state at R = 5R _c , whose power spectrum is characterized by broadened peaks and broadband background noise. The convection pattern near the outer shell propagates prograde, while the pattern near the inner shell propagates retrograde with respect to the basic rotation. Convection cells exist in polar regions. There is a large variation in the vigor of individual convection cells. An example of a more vigorously convecting chaotic state is obtained at R = 50R _c . At this Rayleigh number some of the convection rolls have axes perpendicular to the axis of the basic rotation, indicating a partial relaxation of the rotational constraint. There are strong convective motions in the polar regions. The longitudinally averaged mean zonal flow has an equatorial superrotation and a high latitude subrotation for all cases except R = 50R _c , at this highest Rayleigh number, the mean zonal flow pattern is completely reversed, opposite to the solar differential rotation pattern.     

横向黏度变化对球层中热对流的影响

本文在常黏度的基础上,加上横向黏度变化,纯粹研究了横向黏度变化对球层中热对流的影响.结果表明,横向黏度变化可以加速或阻碍球层中的热对流,并对球层中热传输方式有调节作用,但是横向黏度变化对球层中热对流的影响会限制在一定的强度范围内,并且难以改变球层中热对流的格局.横向黏度变化所产生的环型场速度占总速度场的比例最多为十几个百分点,难以解释观测到的地表板块运动中的环型场能量大小.     

The patterns of high-degree thermal free convection and its features in a spherical shell

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…     

Numerical simulations of thermal convection in a rapidly rotating spherical shell cooled inhomogeneously from above

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 5R_c (R_c 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.     

Nonlinear free thermal convection in a spherical shell:A variable viscosity model

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 …     

Transitions in thermal convection with strongly temperature-dependent viscosity in a wide box

Numerical models are systematically presented for time-dependent thermal convection of Newtonian fluid with strongly temperature-dependent viscosity in a two-dimensional rectangular box of aspect ratio 3 at various values of the Rayleigh number Ra_b defined with viscosity at the bottom boundary up to 1.6×10⁸ and the viscosity contrast across the box r_η up to 10⁸. We found that there are two different series of bifurcations that take place as r_η increases. One series of bifurcations causes changes in the behavior of the thermal boundary layer along the surface boundary from small-viscosity-contrast (SVC) mode, through transitional (TR) mode, to stagnant-lid (ST) mode, or from SVC mode directly to ST mode, depending on Ra_b. Another series of bifurcations causes changes in the aspect ratio of convection cells; convection with an elongated cell can take place at moderate r_η (10³–10^5.5 at Ra_b=6×10⁶), while only convection of aspect ratio close to 1 takes place at small r_η and large r_η. The parameter range of r_η and Ra_b for elongated-cell convection overlaps the parameter range for SVC and ST modes and include the entire parameter range for TR mode. In the elongated-ST regime, the lid of highly viscous fluid along the top boundary is not literally ‘stagnant’ but can horizontally move at a velocity high enough to induce a convection cell with aspect ratio much larger than 1.     

Magnetic instability in a rapidly rotating cylindrical annulus with a finitely conducting inner core

Abstract

We consider the stability of a toroidal magnetic field B = B(s*) (where (s*,φ,z*) are cylindrical polar coordinates) in a cylindrical annulus of conducting fluid with inner and outer radii s_i and s _o rotating rapidly about its axis. The outer boundary is taken to be either insulating or perfectly conducting, and the effect of a finite magnetic diffusivity in the inner core is investigated. The ratio of magnetic diffusivity in the inner core to that of the outer core is denoted by ηη→0 corresponding to a perfectly conducting inner core and η→∞ to an insulating one. Comparisons with the results of Fearn (1983b, 1988) were made and a good match found in the limits η→0 and η→∞ with his perfectly conducting and insulating results, respectively. In addition a new mode of instability was found in the eta;→0 regime. Features of this new mode are low frequency (both the frequency and growth rate →0 as η→0) and penetration deep into the inner core. Typically it is unstable at lower magnetic field strengths than the previously known instabilities.     

Thermal and isotopic evidence for surface and subsurface water contributions to baseflow in a high Arctic river

Seven longitudinal water temperature tow surveys were conducted to attempt to identify the location of surface and subsurface river water exchanges along the length of the West River at the Cape Bounty Arctic Watershed Observatory, Melville Island, Nunavut, Canada (74°55′ N, 109°35′ W). Water temperature data were collected using a calibrated thermistor with an accuracy of ±0.002 °C (resolution <0.00005 °C) along the river during July 2014 in conjunction with stable water isotope sampling to support the thermal data and to determine the extent of surface water mixing from different sources such as precipitation, snowmelt, and surface/subsurface water contributions to the river. Atmospheric conditions were found to be the main contributor to seasonal temperature variance in the river, whereas tributary inflows and residual channel snow also had important thermal effects to river temperatures. Residual channel snow was a sustained source of cold water during much of the 2014 summer season (June–August) and substantially offset downstream warming. The longitudinal temperature profiles indicate notable changes to the thermal state of the river, which are interpreted to be indicative of subsurface and surface water exchange through inputs of relatively cold or warm water. Broadly, surface inflows were found to provide warmer water relative to the West River, and contributed to downstream warming of the river, along with downstream enrichment of δD and δ¹⁸O. Subsurface inflows provided cooler water relative to the river, and contributed to downstream depletion of δD and δ¹⁸O and downstream cooling of river temperatures. These results demonstrate that localized changes in river temperature, in conjunction with isotopic tracers, can be used to track channel–slope water interactions in Arctic hydrological systems, work previously limited to alpine and temperate settings.     

The catastrophe structure of thermohaline convection in a two-dimensional fluid model and a comparison with low-order box models

Abstract

We impose a surface forcing on the 2D, Boussinesq, thermohaline equations in a rectangular domain, in the form of equatorially symmetric cosine distributions of salinity flux and temperature. This system may be seen as an idealization of the ocean thermohaline circulation on the global scale over intervals of centuries or millenia. Multiple steady states are found numerically. They reflect the competition between the opposite signs of the temperature and salinity-driven equatorially symmetric circulations. There are also pole-to-pole, equatorially asymmetric circulations. In the control space of the temperature and salinity-flux forcing amplitudes, these equilibria form two cusp catastrophes, and transitions between stable equilibria occur through several distinct bifurcations. These catastrophes can be reproduced in simple box models connecting stirred reservoirs through capillary pipes. This steady-state analysis may provide a framework for a better understanding of climatic transitions between different stable regimes of the ocean-atmosphere system.     

Instability of flows near the onset of convection in a rotating layer with stress-free horizontal boundaries

We investigate instability of convective flows of simple structure (rolls, standing and travelling waves) in a rotating layer with stress-free horizontal boundaries near the onset of convection. We show that the flows are always unstable to perturbations, which are linear combinations of large-scale modes and short-scale modes, whose wave numbers are close to those of the perturbed flows. Depending on asymptotic relations of small parameters α (the difference between the wave number of perturbed flows and the critical wave number for the onset of convection) and ε (ε² being the overcriticality and the perturbed flow amplitude being O(ε)), either small-angle or Eckhaus instability is prevailing. In the case of small-angle instability for rolls the largest growth rate scales as ε^8/5, in agreement with results of Cox and Matthews (Cox, S.M. and Matthews, P.C., Instability of rotating convection. J. Fluid. Mech., 2000, 403, 153–172) obtained for rolls with k = k _c . For waves, the largest growth rate is of the order ε^4/3. In the case of Eckhaus instability the growth rate is of the order of α².     

A preliminary study of the effects of some flow parameters in the generation of poloidal and toroidal energies within a 3-D spherical thermal-convective system with variable viscosity

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.     

Combination of CFCs and stable isotopes to characterize the mechanism of groundwater–surface water interactions in a headwater basin of the North China Plain

Mountainous areas are characterized by steep slopes and rocky landforms, with hydrological conditions varying rapidly from upstream to downstream, creating variable interactions between groundwater and surface water. In this study, mechanisms of groundwater–surface water interactions within a headwater catchment of the North China Plain were assessed along the stream length and during different seasons, using hydrochemical and stable isotope data, and groundwater residence times estimated using chlorofluorocarbons. These tracers indicate that the river is gaining, due to groundwater discharge in the headwater catchment both in the dry and rainy seasons. Residence time estimation of groundwater using chlorofluorocarbons data reveals that groundwater flow in the shallow sedimentary aquifer is dominated by the binary mixing of water approximating a piston flow model along 2 flow paths: old water, carried by a regional flow system along the direction of river flow, along with young water, which enters the river through local flow systems from hilly areas adjacent to the river valley (particularly during the rainy season). The larger mixing ratio of young water from lateral groundwater recharge and return flow of irrigation during the rainy season result in higher ion concentrations in groundwater than in the dry season. The binary mixing model showed that the ratio of young water versus total groundwater ranged from 0.88 to 0.22 and 1.0 to 0.74 in the upper and lower reaches, respectively. In the middle reach, meandering stream morphology allows some loss of river water back into the aquifer, leading to increasing estimates of the ratio of young water (from 0.22 to 1). This is also explained by declining groundwater levels near the river, due to groundwater extraction for agricultural irrigation. The switch from a greater predominance of regional flow in the dry season, to more localized groundwater flow paths in the wet season is an important groundwater–surface water interactions mechanism, with important catchment management implications.     

Application of a perfectly matched layer in seismic wavefield simulation with an irregular free surface

Recently, an effective and powerful approach for simulating seismic wave propagation in elastic media with an irregular free surface was proposed. However, in previous studies, researchers used the periodic condition and/or sponge boundary condition to attenuate artificial reflections at boundaries of a computational domain. As demonstrated in many literatures, either the periodic condition or sponge boundary condition is simple but much less effective than the well‐known perfectly matched layer boundary condition. In view of this, we intend to introduce a perfectly matched layer to simulate seismic wavefields in unbounded models with an irregular free surface. We first incorporate a perfectly matched layer into wave equations formulated in a frequency domain in Cartesian coordinates. We then transform them back into a time domain through inverse Fourier transformation. Afterwards, we use a boundary‐conforming grid and map a rectangular grid onto a curved one, which allows us to transform the equations and free surface boundary conditions from Cartesian coordinates to curvilinear coordinates. As numerical examples show, if free surface boundary conditions are imposed at the top border of a model, then it should also be incorporated into the perfectly matched layer imposed at the top‐left and top‐ right corners of a 2D model where the free surface boundary conditions and perfectly matched layer encounter; otherwise, reflections will occur at the intersections of the free surface and the perfectly matched layer, which is confirmed in this paper. So, by replacing normal second derivatives in wave equations in curvilinear coordinates with free surface boundary conditions, we successfully implement the free surface boundary conditions into the perfectly matched layer at the top‐left and top‐right corners of a 2D model at the surface. A number of numerical examples show that the perfectly matched layer constructed in this study is effective in simulating wave propagation in unbounded media and the algorithm for implementation of the perfectly matched layer and free surface boundary conditions is stable for long‐time wavefield simulation on models with an irregular free surface.     

Parameter transferability under changing climate: case study with a land surface model in the Durance watershed,France

Abstract

In physically-based land surface models, the parameters can all be prescribed a priori but calibration can be used to enhance the realism of the simulations in well instrumented domains. In such a case, the transferability of calibrated parameters under non-stationary conditions needs to be addressed, especially in the context of climate change. To this end, we used the Catchment Land Surface Model (CLSM) in the Upper Durance watershed located in the French Alps, which experienced a significant increase in temperature over the last century. The CLSM is forced by a 50-year meteorological dataset of good quality. Four parameters of the CLSM (one related to snow processes and three to soil properties) are calibrated against discharge observations with a multi-objective algorithm. First, the robustness of the CLSM parameterizations is tested by the Differential Split Sample Test (DSST). The simulations show good performances over a wide range of retrospective climatic conditions, except when the parameters are calibrated over a period with a large contribution of snowmelt to annual mean discharge. Then, the use of a climate change scenario reveals that the parameterizations of soil moisture processes in the CLSM are responsible for an increasing dispersion among simulations when facing dry and warm conditions. However, the differences between the simulated changes of river discharge remain very small. This work shows that calibration conveys some uncertainties, but they are moderate in the studied case, and pertain to the most conceptual parameterizations of this physically-based model.     

三维起伏地表条件下地震波走时计算的不等距迎风差分法

三维起伏地表条件下的地震波走时计算技术是研究三维起伏地表地区很多地震数据处理技术的基础性工具.为了获得适应于任意三维起伏地表且计算精度高的走时算法,提出三维不等距迎风差分法.该方法采用不等距网格剖分三维起伏地表模型,通过在迎风差分格式中引入不等距差分格式、Huygens原理及Fermat原理来建立地表附近的局部走时计算公式,并通过在窄带技术中设定新的网格节点类型来获得三维起伏地表条件下算法的整体实现步骤.精度及算例分析表明:三维不等距迎风差分法具有很高的计算精度且能够适应于任意三维起伏地表模型.