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1.
Abstract

We derive an equation governing the nonlinear propagation of a linearly polarized Alfvén wave in a two-dimensional, anisotropic, slightly compressible, highly magnetized, viscous plasma, where nonlinearities arise from the interaction of the Alfvén wave with fast and slow magnetoacoustic waves. The phase mixing of such a wave has been suggested as a mechanism for heating the outer solar atmosphere (Heyvaerts and Priest, 1983).

We find that cubic wave damping dominates shear linear dissipation whenever the Alfvén wave velocity amplitude δvy exceeds a few times ten metres per second. In the nonlinear regime, phase-mixed waves are marginally stable, while non-phase-mixed waves of wavenumber ka are damped over a timescale kuRe 0|δ vy/vA |?2, Re 0 being the Reynolds number corresponding to the Braginskij viscosity coefficient η0 and vA the Alfvén speed. Dissipation is most effective where β = (vs /vA) 2 ≈ 1, vs being the speed of sound.  相似文献   

2.
Abstract

The two dimensional incompressible MHD equations describing the decay of a random initial velocity field in the presence of a uniform magnetic background field are solved numerically by a Chebyshev spectral method. The nonlinear interactions of standing Alfvén-waves of a given energy are studied for various Reynolds numbers and field strengths of the magnetic background field. Small scale structures are generated by these interactions, which increase the energy dissipation, however, the uniform background field suppresses the production of arbitrary small scales. Thus energy dissipation is found to be insignificant at sufficiently high Reynolds numbers. Anisotropies of the fluctuating field components are also studied. In the temporal evolution they appear first in the magnetic field. This is explained by the conservation of mean square vector potential in the limit of infinite conductivity.  相似文献   

3.
ABSTRACT

Turbulence in the Earth's outer core not only increases all diffusive coefficients, but it can lead to their anisotropic properties. Therefore, the model of rotating magnetoconvection in horizontal plane layer rotating about vertical axis and permeated by homogeneous horizontal magnetic field, influenced by anisotropic diffusivities, viscosity and thermal diffusivity, is advanced by considering the magnetic diffusivity as anisotropic too. The case of full anisotropy, i.e. all coefficients anisotropic, is compared with both the case possessing isotropic diffusion coefficients and the case of partial anisotropy, i.e. mixed case with isotropic and anisotropic diffusive coefficients (viscosity and thermal diffusivity anisotropic and magnetic diffusivity isotropic). The existence and preference of instabilities is sensitive to all non-dimensional parameters, as well as on anisotropic parameter, the ratio of horizontal and vertical diffusivities. Two types of anisotropy, BM (introduced by Braginsky and Meytlis) and SA (stratification anisotropy) are studied. BM as well as SA were applied by ?oltis and Brestenský to the study of the partial anisotropy; this study is extended, in this paper, to full anisotropy cases (full SA and full BM) and it is shown that the style of convection given by the onset of stationary modes is more affected by anisotropic diffusivities in BM than in SA anisotropy. The important influence of strong anisotropies in the Earth's core dynamics is stressed.  相似文献   

4.
Abstract

This paper analyzes the linear stability of a rapidly-rotating, stratified sheet pinch in a gravitational field, g, perpendicular to the sheet. The sheet pinch is a layer (O ? z ? d) of inviscid, Boussinesq fluid of electrical conductivity σ, magnetic permeability μ, and almost uniform density ρ o; z is height. The prevailing magnetic field. B o(z), is horizontal at each z level, but varies in direction with z. The angular velocity, Ω, is vertical and large (Ω ? VA/d, where VA = B0√(μρ0) is the Alfvén velocity). The Elsasser number, Λ = σB2 0/2Ωρ0, measures σ. A (modified) Rayleigh number, R = gβd2/ρ0V2 A, measures the buoyancy force, where β is the imposed density gradient, antiparallel to g. A Prandtl number, PK = μσK, measures the diffusivity, k, of density differences.  相似文献   

5.
Based on the theory of anisotropic elasticity and observation of static mechanic measurement of transversely isotropic hydrocarbon source rocks or rock‐like materials, we reasoned that one of the three principal Poisson's ratios of transversely isotropic hydrocarbon source rocks should always be greater than the other two and they should be generally positive. From these relations, we derived tight physical constraints on c13, Thomsen parameter δ, and anellipticity parameter η. Some of the published data from laboratory velocity anisotropy measurement are lying outside of the constraints. We analysed that they are primarily caused by substantial uncertainty associated with the oblique velocity measurement. These physical constraints will be useful for our understanding of Thomsen parameter δ, data quality checking, and predicting δ from measurements perpendicular and parallel to the symmetrical axis of transversely isotropic medium. The physical constraints should also have potential application in anisotropic seismic data processing.  相似文献   

6.
Introduction Ready and Renkin (1971) were the first to make the research on anisotropy problems in magnetotellurics (MT). The progress in the research is not evident because it is more complex and difficult than isotropic problems. Now, the one-dimensional (1D) anisotropy problems in MT have been well solved, while for the two-dimensional (2D) cases, the numerical solutions have only been obtained for some particular conditions (Ready and Renkin, 1975). As to the three-dimensional (3D) ani…  相似文献   

7.
Many rocks possess electrical properties with a clearly expressed anisotropy. The anisotropic character of the rocks is often overlooked in forming the Fréchet derivatives or sensitivity functions for parameter updating during the inversion of DC resistivity data. In this study we have compared the sensitivity patterns for an isotropic, homogeneous model with that for a transversely isotropic (i.e. anisotropic) model having a tilted axis of symmetry using a pole–pole array. The sensitivity functions are expressed in terms of the derivatives of the electric potential U with respect to the average conductivity σm (geometric mean of the longitudinal and transverse conductivities) and the coefficient of anisotropy λ. Results are plotted in both cross-section form and plan view for various dip and strike angles of the axis of symmetry. The derivative dU/dλ decreases more rapidly than the isotropic value dU/dσ, and shows pronounced asymmetry and weakening of magnitude with increasing dip of the plane of symmetry. The derivative dU/dσm also exhibits the asymmetric pattern (except for vertical and horizontal dip cases). The positive region between the electrodes only extends to a small depth compared to the isotropic derivative, even in the case of a vertical axis of symmetry (VTI medium). The ratio of this anisotropic derivative to the isotropic derivative, when plotted as a function of position and depth shows prominent differences in both the sign and the magnitude of the sensitivities, especially for steep dips and for strongly anisotropic rocks. The plot highlights the dangers of an isotropic assumption. Even for mildly anisotropic rocks (λ < 1.2) the possibility for error in interpretation is considerable. Combined borehole and surface measurements are needed to diagnose anisotropy. Further work is needed to design optimal electrode configurations in anisotropic situations.  相似文献   

8.
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9.
To investigate the characteristics of the anisotropic stratum, a multi‐azimuth seismic refraction technique is proposed in this study since the travel time anomaly of the refraction wave induced by this anisotropic stratum will be large for a far offset receiver. To simplify the problem, a two‐layer (isotropy–horizontal transverse isotropy) model is considered. A new travel time equation of the refracted P‐wave propagation in this two‐layer model is derived, which is the function of the phase and group velocities of the horizontal transverse isotropic stratum. In addition, the measured refraction wave velocity in the physical model experiment is the group velocity. The isotropic intercept time equation of a refraction wave can be directly used to estimate the thickness of the top (isotropic) layer of the two‐layer model because the contrast between the phase and group velocities of the horizontal transverse isotropic medium is seldom greater than 10% in the Earth. If the contrast between the phase and group velocities of an anisotropic medium is small, the approximated travel time equation of a refraction wave is obtained. This equation is only dependent on the group velocity of the horizontal transverse isotropic stratum. The elastic constants A11, A13, and A33 and the Thomsen anisotropic parameter ε of the horizontal transverse isotropic stratum can be estimated using this multi‐azimuth seismic refraction technique. Furthermore, under a condition of weak anisotropy, the Thomsen anisotropic parameter δ of the horizontal transverse isotropic stratum can be estimated by this technique as well.  相似文献   

10.
Tilted transversely isotropic formations cause serious imaging distortions in active tectonic areas (e.g., fold‐and‐thrust belts) and in subsalt exploration. Here, we introduce a methodology for P‐wave prestack depth imaging in tilted transversely isotropic media that properly accounts for the tilt of the symmetry axis as well as for spatial velocity variations. For purposes of migration velocity analysis, the model is divided into blocks with constant values of the anisotropy parameters ε and δ and linearly varying symmetry‐direction velocity VP0 controlled by the vertical (kz) and lateral (kx) gradients. Since determination of tilt from P‐wave data is generally unstable, the symmetry axis is kept orthogonal to the reflectors in all trial velocity models. It is also assumed that the velocity VP0 is either known at the top of each block or remains continuous in the vertical direction. The velocity analysis algorithm estimates the velocity gradients kz and kx and the anisotropy parameters ε and δ in the layer‐stripping mode using a generalized version of the method introduced by Sarkar and Tsvankin for factorized transverse isotropy with a vertical symmetry axis. Synthetic tests for several models typical in exploration (a syncline, uptilted shale layers near a salt dome and a bending shale layer) confirm that if the symmetry‐axis direction is fixed and VP0 is known, the parameters kz, kx, ε and δ can be resolved from reflection data. It should be emphasized that estimation of ε in tilted transversely isotropic media requires using nonhyperbolic moveout for long offsets reaching at least twice the reflector depth. We also demonstrate that application of processing algorithms designed for a vertical symmetry axis to data from tilted transversely isotropic media may lead to significant misfocusing of reflectors and errors in parameter estimation, even when the tilt is moderate (30°). The ability of our velocity analysis algorithm to separate the anisotropy parameters from the velocity gradients can be also used in lithology discrimination and geologic interpretation of seismic data in complex areas.  相似文献   

11.
The western part of the Bohemian Massif (West Bohemia/Vogtland region) is characteristic in the relatively frequent recurrence of intraplate earthquake swarms and in other manifestations of past-to-recent geodynamic activity. In this study we derived 1D anisotropic qP-wave model of the upper crust in the seismogenic West Bohemia/Vogtland region by means of joint inversion of two independent data sets - travel times from controlled shots and arrival times from local earthquakes extracted from the WEBNET seismograms. We derived also simple 1-D P-wave and S-wave isotropic models. Reasons for deriving these models were: (a) only simplified crustal velocity models, homogeneous half-space or 1D isotropic layered models of this region, have been derived up to now and (b) a significant effective anisotropy of the upper crust in the region which was indicated recently by S-wave splitting. Both our anisotropic qP-wave and isotropic P-and S-wave velocity models are constrained by four layers with the constant velocity gradient. Weak anisotropy for P-waves is assumed. The isotropic model is represented by 9 parameters and the anisotropic one is represented by 24 parameters. A new robust and effective optimization algorithm - isometric algorithm - was used for the joint inversion. A two-step inversion algorithm was used. During the first step the isotropic P- and S-wave velocity model was derived. In the second step, it was used as a background model and the parameters of anisotropy were sought. Our 1D models are adequate for the upper crust in the West Bohemia/Vogtland swarm region up to a depth of 15 km. The qP-wave velocity model shows 5% anisotropy, the minimum velocity in the horizontal direction corresponds to an azimuth of 170°. The isotropic model indicates the VP/VS ratio variation with depth. The difference between the hypocentre locations based on the derived isotropic and anisotropic models was found to be several hundreds of meters.  相似文献   

12.
Abstract

The mean field induction equation of Steenbeck, Krause and Rädler (1966) is solved for anisotropic α ik -tensors of varying anisotropy. The attention is restricted to α2-dynamos in spheres with steady axisymmetric magnetic fields. The ratio of the electrical conductivity outside and inside the sphere is varied, but in all cases it is found that a steady dynamo does not exist when the anisotropy of the α ik -tensor exceeds a critical value. Such a critical value does not exist in the exceptional case of the Fermi boundary condition. The results emphasize the important effect of boundaries on the existence of solutions of the dynamo problem.  相似文献   

13.
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14.
Abstract

We consider the mixing of passive tracers and vorticity by temporally fluctuating large scale flows in two dimensions. In analyzing this problem, we employ modern developments stemming from properties of Hamiltonian chaos in the particle trajectories; these developments generally come under the heading “chaotic advection” or “Lagrangian turbulence.” A review of the salient properties of this kind of mixing, and the mathematics used to analyze it, is presented in the context of passive tracer mixing by a vacillating barotropic Rossby wave. We then take up the characterization of subtler aspects of the mixing. It is shown the chaotic advection produces very nonlocal mixing which cannot be represented by eddy diffusivity. Also, the power spectrum of the tracer field is found to be k ? l at shortwaves—precisely as for mixing by homogeneous, isotropic two dimensional turbulence,—even though the physics of the present case is very different. We have produced two independent arguments accounting for this behavior.

We then examine integrations of the unforced barotropic vorticity equation with initial conditions chosen to give a large scale streamline geometry similar to that analyzed in the passive case. It is found that vorticity mixing proceeds along lines similar to passive tracer mixing. Broad regions of homogenized vorticity ultimately surround the separatrices of the large scale streamline pattern, with vorticity gradients limited to nonchaotic regions (regions of tori) in the corresponding passive problem.

Vorticity in the chaotic zone takes the form of an arrangement of strands which become progressively finer in scale and progressively more densely packed; this process transfers enstrophy to small scales. Although the enstrophy cascade is entirely controlled by the large scale wave, the shortwave enstrophy spectrum ultimately takes on the classical k ? l form. If one accepts that the enstrophy cascade is indeed mediated by chaotic advection, this is the expected behavior. The extreme form of nonlocality (in wavenumber space) manifest in this example casts some doubt on the traditional picture of enstrophy cascade in the Atmosphere, which is based on homogeneous two dimensional turbulence theory. We advance the conjecture that these transfers are in large measure attributable to large scale, low frequency, planetary waves.

Upscale energy transfers amplifying the large scale wave do indeed occur in the course of the above-described process. However, the energy transfer is complete long before vorticity mixing has gotten very far, and therefore has little to do with chaotic advection. In this sense, the vorticity involved in the enstrophy cascade is “fossil vorticity,” which has already given up its energy to the large scale.

We conclude with some speculations concerning statistical mechanics of two dimensional flow, prompted by our finding that flows with identical initial energy and enstrophy can culminate in very different final states. We also outline prospects for further applications of chaotic mixing in atmospheric problems.  相似文献   

15.
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16.
Abstract

In this paper we analyse the stationary mean energy density tensor Tij = BiBj for the x 2-sphere. This model is one of the simplest possible turbulent dynamos, originally due to Krause and Steenbeck (1967): a conducting sphere of radius R with homogeneous, isotropic and stationary turbulent convection, no differential rotation and negligible resistivity. The stationary solution of the (linear) equation for Tij is found analytically. Only Trr , T θθ and T φφ are unequal to zero, and we present their dependence on the radial distance r.

The stationary solution depends on two coefficients describing the turbulent state: the diffusion coefficient β≈?u2?τc/3 and the vorticity coefficient γ ≈ ?|?×u|2?τc/3 where u(r, t) is the turbulent velocity and c its correlation time. But the solution is independent of the dynamo coefficient α≈??u·?×u?τc/3 although α does occur in the equation for Tij . This result confirms earlier conclusions that helicity is not required for magnetic field generation. In the stationary state, magnetic energy is generated by the vorticity and transported to the boundary, where it escapes at the same rate. The solution presented contains one free parameter that is connected with the distribution of B over spatial scales at the boundary, about which Tij gives no information. We regard this investigation as a first step towards the analysis of more complicated, solar-type dynamos.  相似文献   

17.
The mathematical formulation of an iterative procedure for the numerical implementation of an ionosphere-magnetosphere (IM) anisotropic Ohm’s law boundary condition is presented. The procedure may be used in global magnetohydrodynamic (MHD) simulations of the magnetosphere. The basic form of the boundary condition is well known, but a well-defined, simple, explicit method for implementing it in an MHD code has not been presented previously. The boundary condition relates the ionospheric electric field to the magnetic field-aligned current density driven through the ionosphere by the magnetospheric convection electric field, which is orthogonal to the magnetic field B, and maps down into the ionosphere along equipotential magnetic field lines. The source of this electric field is the flow of the solar wind orthogonal to B. The electric field and current density in the ionosphere are connected through an anisotropic conductivity tensor which involves the Hall, Pedersen, and parallel conductivities. Only the height-integrated Hall and Pedersen conductivities (conductances) appear in the final form of the boundary condition, and are assumed to be known functions of position on the spherical surface R=R1 representing the boundary between the ionosphere and magnetosphere. The implementation presented consists of an iterative mapping of the electrostatic potential , the gradient of which gives the electric field, and the field-aligned current density between the IM boundary at R=R1 and the inner boundary of an MHD code which is taken to be at R2>R1. Given the field-aligned current density on R=R2, as computed by the MHD simulation, it is mapped down to R=R1 where it is used to compute by solving the equation that is the IM Ohm’s law boundary condition. Then is mapped out to R=R2, where it is used to update the electric field and the component of velocity perpendicular to B. The updated electric field and perpendicular velocity serve as new boundary conditions for the MHD simulation which is then used to compute a new field-aligned current density. This process is iterated at each time step. The required Hall and Pedersen conductances may be determined by any method of choice, and may be specified anew at each time step. In this sense the coupling between the ionosphere and magnetosphere may be taken into account in a self-consistent manner.  相似文献   

18.
The azimuth moveout (AMO) operator in homogeneous transversely isotropic media with a vertical symmetry axis (VTI), as in isotropic media, has an overall skewed saddle shape. However, the AMO operator in anisotropic media is complicated; it includes, among other things, triplications at low angles. Even in weaker anisotropies, with the anisotropy parameter η= 0.1 (10% anisotropy), the AMO operator is considerably different from the isotropic operator, although free of triplications. The structure of the operator in VTI media (positive η) is stretched (has a wider aperture) compared with operators in isotropic media, with the amount of stretch being dependent on the strength of anisotropy. If the medium is both vertically inhomogeneous, i.e. the vertical velocity is a function of depth (v(z)), and anisotropic, which is a common combination in practical problems, the shape of the operator again differs from that for isotropic media. However, the difference in the AMO operator between the homogeneous and the v(z) cases, even for anisotropic media, is small. Stated simply, anisotropy influences the shape and aperture of the AMO operator far more than vertical inhomogeneity does.  相似文献   

19.
The elastic moduli and anisotropy of organic-rich rocks are of great importance to geoengineering and geoprospecting of oil and gas reservoirs. In this paper, we probe into the static and dynamic moduli of the Ghareb–Mishash chalk through laboratory measurements and new analytical approaches. We define a new anisotropy parameter, ‘hydrostatic strain ratio’ (Ω), which describes the differential contraction of anisotropic rocks consequent to hydrostatic compression. Ω depends on the C11, C12, C13 and C33 stiffness constants of a transversely isotropic material, and therefore enables a unique insight into the anisotropic behaviour of TI rocks. Ω proves more sensitive to anisotropy within the weak anisotropy range, when compared with Thomsen's ε and γ parameters. We use Ω to derive static moduli from triaxial compression tests performed on a single specimen. This is done by novel employment of a hydrostatic-deviatoric combination for transversely isotropic elastic stiffnesses. Dynamic moduli are obtained from acoustic velocities measurements. We find that the bedding-normal velocities are described well by defining kerogen as the load-supporting matrix in a Hashin–Shtrikman model (‘Hashin–Shtrikman (HS) kerogen’). The dynamic moduli of the Ghareb–Mishash chalk in dry conditions are significantly higher than the static moduli. The dynamic/static moduli ratio decreases from ∼4 to ∼2 with increasing kerogen content. Both the static and dynamic moduli decrease significantly with increasing porosity and kerogen content. The effect of porosity on them is two times stronger than the effect of kerogen.  相似文献   

20.
Abstract

We study the propagation of nonlinear MHD waves in a highly magnetized plasma cavity. The cavity's moving boundaries generate Alfvén waves, which in turn drive and interact with slow magnetosonic waves. The interacting wave system is analyzed by a Galerkin and multiple-scale analyses leading to simple dynamical equations. When the frequency of the forcing provided by the moving boundaries and that of the fundamental Alfvén eigenmode are close, the cavity behaves like a Duffing oscillator. Application of the Melnikov function theory shows that the Alfvén wave's amplitude undergoes both flip and saddle-node bifurcations as the amplitude and the phase of the boundary forcing vary. Direct numerical integration confirms these results and provides an estimate of the amount of energy dissipated in the bifurcations.  相似文献   

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