The weak taylor state in an αω-dynamo

Abstract

A spherical αω-dynamo is studied for small values of the viscous coupling parameter ε ～ v^1/2, paying attention particularly to large dynamo numbers. The present study is a follow-up of the work by Hollerbach et al. (1992) with their choice of α-effect and Archimedean wind including also the constraint of magnetic field symmetry (or antisymmetry) due to equatorial plane. The magnetic field scaled by ε^1/2 is independent of ε in the solutions for dynamo numbers smaller than a certain value of D _b (the Ekman state) which are represented by dynamo waves running from pole to equator or vice-versa. However, for dynamo numbers larger than D _b the solution bifurcates and subsequently becomes dependent on ε. The bifurcation is a consequence of a crucial role of the meridional convection in the mechanism of magnetic field generation. Calculations suggest that the bifurcation appears near dynamo number about 33500 and the solutions for larger dynamo numbers and ε = 0 become unstable and fail, while the solutions for small but non-zero ε are characterized by cylindrical layers of local maximum of magnetic field and sharp changes of geostrophic velocity. Our theoretical analysis allows us to conclude that our solution does not take the form of the usual Taylor state, where the Taylor constraint should be satisfied due to the special structure of magnetic field. We rather obtained the solution in the form of a “weak” Taylor state, where the Taylor constraint is satisfied partly due to the amplitude of the magnetic field and partly due to its structure. Calculations suggest that the roles of amplitude and structure are roughly fifty-fifty in our “weak” Taylor state solution and thus they can be called a Semi-Taylor state. Simple estimates show that also Ekman state solutions can be applicable in the geodynamo context.     

Non-axisymmetric α2Ω-dynamo waves in thin stellar shells

Linear α²Ω-dynamo waves are investigated in a thin turbulent, differentially rotating convective stellar shell. A simplified one-dimensional model is considered and an asymptotic solution constructed based on the small aspect ratio of the shell. In a previous paper Griffiths et al. (Griffiths, G.L., Bassom, A.P., Soward, A.M. and Kuzanyan, K.M., Nonlinear α²Ω-dynamo waves in stellar shells, Geophys. Astrophys. Fluid Dynam., 2001, 94, 85–133) considered the modulation of dynamo waves, linked to a latitudinal-dependent local α-effect and radial gradient of the zonal shear flow. These effects are measured at latitude θ by the magnetic Reynolds numbers R _α f(θ) and R _Ω g(θ). The modulated Parker wave, which propagates towards the equator, is localised at some mid-latitude θ_p under a Gaussian envelope. In this article, we include the influence of a latitudinal-dependent zonal flow possessing angular velocity Ω_*(θ) and consider the possibility of non-axisymmetric dynamo waves with azimuthal wave number m. We find that the critical dynamo number D _c?=?R _α R _Ω is minimised by axisymmetric modes in the αΩ-limit (Rα→0). On the other hand, when Rα?≠?0 there may exist a band of wave numbers 0?m?m ^? for which the non-axisymmetric modes have a smaller D _c than in the axisymmetric case. Here m ^? is regarded as a continuous function of R _α with the property m?→0 as R _α→0 and the band is only non-empty when m??>1, which happens for sufficiently large R _α. The preference for non-axisymmetric modes is possible because the wind-up of the non-axisymmetric structures can be compensated by phase mixing inherent to the α²Ω-dynamo. For parameter values resembling solar conditions, the Parker wave of maximum dynamo activity at latitude θ_p not only propagates equatorwards but also westwards relative to the local angular velocity Ω_*(θ _p ). Since the critical dynamo number D _c?=?R _α R _Ω is O (1) for small R _α, the condition m ^??>?1 for non-axisymmetric mode preference imposes an upper limit on the size of |dΩ_*/dθ|.     

A modal α2-dynamo in the limit of asymptotically small viscosity

Abstract

A spherical α²-dynamo is presented as an expansion in the free decay modes of the magnetic field. In the limit of vanishing viscosity the momentum equation yields various asymptotic expansions for the flow, depending on the precise form of the dissipation and boundary conditions applied. A new form for the dissipation is introduced that greatly simplifies this asymptotic expansion. When these expansions are substituted back into the induction equation, a set of modal amplitude equations is derived, and solved for various distributions of the α-effect. For all choices of α the solutions approach the Taylor state, but the manner in which this occurs can vary, as previously found by Soward and Jones (1983). Furthermore, as hypothesized by Malkus and Proctor (1975), but not previously demonstrated, the post-Taylor equilibration is indeed independent of the viscosity in the asymptotic limit, and depending on the choice of a may be either steady-state or oscillatory.     

On the frontal condition for finite amplitude α2Ω-dynamo wave trains in stellar shells

Abstract

In a previous paper, Bassom et al. (Proc. R. Soc. Lond. A, 455, 1443–1481, 1999) (BKS) investigated finite amplitude αΩ-dynamo wave trains in a thin turbulent, differentially rotating convective stellar shell; nonlinearity arose from α-quenching. There asymptotic solutions were developed based upon the small aspect ratio ε of the shell. Specifically, as a consequence of a prescribed latitudinally dependent α-effect and zonal shear flow, the wave trains have smooth amplitude modulation but are terminated abruptly across a front at some high latitude θ_F. Generally, the linear WKB-solution ahead of the front is characterised by the vanishing of the complex group velocity at a nearby point θ_f; this is essentially the Dee–Langer criterion, which determines both the wave frequency and front location.

Recently, Griffiths et al. (Geophys. Astrophys. Fluid Dynam. 94, 85–133, 2001) (GBSK) obtained solutions to the α²Ω-extension of the model by application of the Dee—Langer criterion. Its justification depends on the linear solution in a narrow layer ahead of the front on the short O(θ_f—θ_F) length scale; here conventional WKB-theory, used to describe the solution elsewhere, is inadequate because of mode coalescence. This becomes a highly sensitive issue, when considering the transition from the linear solution, which occurs when the dynamo number D takes its critical value D _c corresponding to the onset of kinematic dynamo action, to the fully nonlinear solutions, for which the Dee—Langer criterion pertains.

In this paper we investigate the nature of the narrow layer for α²Ω-dynamos in the limit of relatively small but finite α-effect Reynolds numbers R _α, explicitly ε^½ ? R ² _α ? 1. Though there is a multiplicity of solutions, our results show that the space occupied by the corresponding wave train is generally maximised by a solution with θ_f—θ_F small; such solutions are preferred as evinced by numerical simulations. This feature justifies the application by GBSK of the Dee—Langer criterion for all D down to the minimum D _min that the condition admits. Significantly, the frontal solutions are subcritical in the sense that |D _min| ≤ |D _c|; equality occurs as the α-effect Reynolds number tends to zero. We demonstrate that the critical linear solution is not connected by any parameter track to the preferred nonlinear solution associated with D _min. By implication, a complicated bifurcation sequence is required to make the connection between the linear and nonlinear states. This feature is in stark contrast to the corresponding results for αΩ-dynamos obtained by BKS valid in the limit R _{2 α} ? ε^½, which, though exhibiting a weak subcriticality, showed that the connection follows a clearly identifiable nonbifurcating track.     

Location of Tibetan earthquakes──a nonlinear approach by a simplex optimized technique

Ｌｏｃａｔｉｏｎ　ｏｆ　Ｔｉｂｅｔａｎ　ｅａｒｔｈｑｕａｋｅｓ──ａ　ｎｏｎｌｉｎｅａｒ　ａｐｐｒｏａｃｈ　ｂｙ　ａ　ｓｉｍｐｌｅｘ　ｏｐｔｉｍｉｚｅｄ　ｔｅｃｈｎｉｑｕｅ（赵珠）（丁志峰）（易桂喜／）（王建格）ＬｏｃａｔｉｏｎｏｆＴｉｂｅｔａｎｅａ...     

The nonlinear oil–water two-phase flow behavior for a horizontal well in triple media carbonate reservoir

Carbonate reservoir is one of the important reservoirs in the world. Because of the characteristics of carbonate reservoir, horizontal well has become a key technology for efficiently developing carbonate reservoir. Establishing corresponding mathematical models and analyzing transient pressure behaviors of this type of well-reservoir configuration can provide a better understanding of fluid flow patterns in formation as well as estimations of important parameters. A mathematical model for a oil–water two-phase flow horizontal well in triple media carbonate reservoir by conceptualizing vugs as spherical shapes are presented in this article. A semi-analytical solution is obtained in the Laplace domain using source function theory, Laplace transformation, and superposition principle. Analysis of transient pressure responses indicates that seven characteristic flow periods of horizontal well in triple media carbonate reservoir can be identified. Parametric analysis shows that water saturation of matrix, vug and fracture system, horizontal section length, and horizontal well position can significantly influence the transient pressure responses of horizontal well in triple media carbonate reservoir. The model presented in this article can be applied to obtain important parameters pertinent to reservoir by type curve matching.     

Electric Potential and Fréchet Derivatives for a Uniform Anisotropic Medium with a Tilted Axis of Symmetry

In this paper we develop analytic solutions for the electric potential, current density and Fréchet derivatives at any interior point within a 3-D transversely isotropic medium having a tilted axis of symmetry. The current electrode is assumed to be on the surface of the Earth and the plane of stratification given arbitrary strike and dip. Profiles can be computed for any azimuth. The equipotentials exhibit an elliptical pattern and are not orthogonal to the current density vectors, which are strongly angle dependent. Current density reaches its maximum value in a direction parallel to the longitudinal conductivity direction. Illustrative examples of the Fréchet derivatives are given for the 2.5-D problem, in which the profile is taken perpendicular to strike. All three derivatives of the Green’s function with respect to longitudinal conductivity, transverse resistivity and dip angle of the symmetry axis (dG/dσ_l, dG/dσ_t, dG/dθ₀) show a strongly asymmetric pattern compared to the isotropic case. The patterns are aligned in the direction of the tilt angle. Such sensitivity patterns are useful in real-time experimental design as well as in the fast inversion of resistivity data collected over an anisotropic earth.     

A numerical study of 1-D nonlinear P-wave propagation in solid

IntroductionBecauseoftheextensivedistributionofruptures,micro-cracksandcrystalfracturesintheearth,therelationshipsbetweenthestressandstrainarenolongerlinear,infact,theyarenonlinear.Inordertoinvestigateandusethenonlinearcharacteristicsofsolidmediumintheearth,weshouldconsidertheinfluenceofnonlinearresponseduringthecourseofseismicmodelingandinversion.Thisisoneoftheimportantstudyfieldsthathavebeenpaidgreatattentionsinthere-centyearsintheworld(Minster,etal,1991;ZHANG,TENG,1993).Thenonlinearchar…     

Research on nonlinear R／S method and its application in earthquake prediction

ＲｅｓｅａｒｃｈｏｎｎｏｎｌｉｎｅａｒＲ／ＳｍｅｔｈｏｄａｎｄｉｔｓｐｐｌｉｃａｔｉｏｎｉｎｅａｒｔｈｑｕａｋｅｐｒｅｄｉｃｔｉｏｎＢｉ－ＱｕａｎＷＡＮＧ（王碧泉）；Ｈａｎ－ＭｉｎｇＨＵＡＮＧ（黄汉明）；Ｈｏｎｇ－ＳｈｕｎＦＡＮ（范洪顺）；Ｃｈｕｅｎ...     

Seismic vulnerability assessment of a mixed masonry–RC building aggregate by linear and nonlinear analyses

From the beginning of the twentieth century, and due to the rapid increase of reinforced concrete (RC) usage, mixed masonry–RC buildings have emerged. In Lisbon, Portugal, old mixed masonry–RC buildings appeared between 1920 and 1960, representing the transition period between masonry and proper RC. These buildings are often integrated in blocks, and frequently share the side-walls, implying, thus, the need to assess the seismic vulnerability of building aggregates. The present paper approaches the seismic vulnerability assessment of a specific type of old mixed masonry–RC buildings in Lisbon. The study comprises the analysis of a building, both as an isolated structure and inserted in its aggregate, using two approaches: (1) linear dynamic analysis with SAP2000 and (2) nonlinear static analysis by means of 3Muri/Tremuri software. A comparison of both approaches derives a good matching between the obtained results. However, a nonlinear analysis is required to identify, in an adequate manner, the critical areas of the structure requiring strengthening.     

Experimental study on scour and flow field in a scour hole around a T-shape spur dike in a 90° bend

In this paper results of experimental study on scour and flow field around a T-shape spur dike in a 90 degree channel bend are presented. Experiments were conducted in a laboratory channel to measure the variations of bed topography under a clear water condition. The three dimensional flow field was measured by an Acoustic Doppler Velocimeter in a scour hole due to a T-shape spur dike. It was found that by increasing the Froude number and length of spur dike the amount of scour increases. Increasing the wing length of spur dike decreases the scour. The amount of scour at the upstream of spur dike is much more as compare to that at the downstream of spur dike. By increasing the wing length of spur dike the zone of flow separation expands. As a result the sizes of vortex increase. Details of flow field are also addressed in the paper.     

Application of the value of nonlinear parameters H and ΔH in strong earthquake prediction

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A semi-analytical approach to a nonlinear stress–strain analysis of buried steel pipelines crossing active faults

In the present paper a semi-analytical methodology for a nonlinear stress–strain analysis of buried steel pipelines at active fault crossings is presented and verified. The developed model introduces a number of critical refinements to the existing methodologies which extend the application area of analytical models in pipeline design. In particular, a strike-slip and normal-slip fault crossings can be analyzed taking into account material and large displacement nonlinearities, nonlinear pipe–soil interaction. The proposed model is verified against the results by other authors and numerical results, obtained with the finite element method.     

Reduction of peak ground velocity by nonlinear soil response–Ⅱ:excitation by a P-wave pulse

We study the reduction of peak velocity on the ground surface of a soil valley caused by loss of wave energy by large nonlinear strains and strain localization inside the valley, for excitation by a half-sine P-wave pulse. This study is a follow up to our previous study of out of plane response for excitation by an SH-pulse. In this paper, we consider the inplane response, and assume that the soil material does not support tension, but the normal stress at a point in the soil can be compression(negative) or zero. A point in the soil with zero stress behaves as a stress-free point, it does not transmit normal stress and appears as a crack point. Because of this, along with the nonlinear response associated with compression and shear, the in-plane response in this study is more complex than that of the out-of-plane SH response. We study the interplay of two opposing effects:(i) jump in impedance from a higher value(half-space) to a lower value(valley), which amplifies the linear motions at the free surface of the valley, and(ii) the occurrence of nonlinear zones in the valley, which reduce the motion at the valley surface.     

Solution of a Green’s function for a saturated porous medium in a half-space subjected to a torsional force

Based on the solutions of the Green’s function for a saturated porous medium obtained by the authors,and using transformation of axisymmetric coordinates,Sommerfeld integrals and superposition of the influence field on a free surface,the authors have obtained displacement solutions of a saturated porous medium subjected to a torsional force in a half-space.The relationship curves of the displacement solutions and various parameters(permeability,frequency,etc.)under action of a unit of torque are also given in this paper.The results are consistent with previous Reissner’s solutions,where a two-phase medium decays to a single-phase medium.The solution is useful in solving relevant dynamic problems of a twophase saturated medium in engineering.     

Historical earthquakes and a tsunami in Bohai Sea

Quantitative analysis on seismicity showed that there are several seismic dense zones in Bohai Sea. These seismic dense zones of modern small earthquakes behave prominent NE orientation, although a seismic dense zone with NW direction exists actually. Taking 39°N as a boundary, seismicity in the south is different from that in north of Bohai Sea. Almost all strong earthquakes and seismic dense zones are concentrated in the southern part. Based on archives and seismic dense characteristics, we amended the epicenter of strong earthquakes in 1548 and discussed about magnitude of the earthquake in 1888. Possibility of the event in 173 as a tsunami was discussed. The event in 1597 was doubted as a strong earthquake in Bohai Sea.     

Scour and flow field around a spur dike in a 90° bend

Spur dike is an important element in river training that creates rapid variations in flow field, sediment transport and bed topography. The mechanism of flow and sediment transport in a channel bend is very complex, especially when a spur dike is constructed in a bend. Most of previous investigations on flow behavior and scour around spur dike were carried out in straight channels. In this paper results of experiments on flow field and scour around a spur dike in a 90 degree channel bend are presented. Sand with uniform grain size was used as the bed material. Experiments were conducted for different locations and different lengths of spur dikes at the bend with different values of discharge. The three dimensional flow fields around a spur dike were investigated. The maximum depth of scour was correlated to the Froude numbers, lengths and the locations of spur dike in the bend.     

Scour and flow field around a spur dike in a 90° bend

Spur dike is an important element in fiver training that creates rapid variations in flow field, sediment transport and bed topography. The mechanism of flow and sediment transport in a channel bend is very complex, especially when a spur dike is constructed in a bend. Most of previous investigations on flow behavior and scour around spur dike were carried out in straight channels. In this paper results of experiments on flow field and scour around a spur dike in a 90 degree channel bend are presented, Sand with uniform grain size was used as the bed material, Experiments were conducted for different locations and different lengths of spur dikes at the bend with different values of discharge, The three dimensional flow fields around a spur dike were investigated, The maximum depth of scour was correlated to the Froude numbers, lengths and the locations of spur dike in the bend.     

Statistical distribution of nonlinear random wave height

1 Introduction The probability distribution of ocean wave element is one of the results of application of random process theory to the ocean wave study. A great amount of outcome has obtained in this field[1,2]. In fact, a great deal of research on linear…     

Weakly nonlinear analysis of the α effect

We address mathematical issues raised by the so-called α?effect of dynamo theory, which is a dynamo mechanism arising in conducting flows with small scale fluctuations. Analytical results on the α?effect concern the linear induction equation, and are usually claimed to hold for the whole magnetohydrodynamics (MHD) system, as long as the amplitude of the perturbations is small. We discuss the justification of that claim, in the case of periodic fluctuations of the fields. We show a nonlinear instability result on the MHD system, that predicts dynamo action for a large class of high frequency periodic flows, up to the fully nonlinear regime.