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1.
Abstract

We study the nonlinear stability of MHD waves propagating in a two-dimensional, compressible, highly magnetized, viscous plasma. These waves are driven by a weak, shear body force which could be imposed by large scale internal fluctuations present in the solar atmosphere.

The effects of anisotropic viscosity (leading to a cubic damping) and of the nonlinear coupling of the Alfven and the magnetoacoustic waves are analysed using Galerkin and multiple-scale analysis: the MHD equations are reduced to a set of nonlinear ordinary differential equations which is then suitably truncated to give a model dynamical system, representing the interaction of two complex Galerkin modes.

For propagation oblique to the background magnetic field, analytical integration shows that the low-wavenumber mode is physically unstable. For propagation parallel to the background magnetic field the high-wavenumber wave can undergo saddlenode bifurcations, in way that is similar to the van der Pol oscillator; these bifurcations lead to the appearance of a hysteresis cycle.

A numerical integration of the dynamical system shows that a sequence of Hopf bifurcations takes place as the Reynolds number is increased, up to the onset of nonperiodic behaviour. It also shows that energy can be transferred from the low- wavenumber to the high-wavenumber mode.  相似文献   

2.
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.  相似文献   

3.
Abstract

It is shown that the turning surfaces associated with internal waves in a uniformly rotating, density stratified, Boussinesq fluid in the presence of an arbitrary gravitational field are regular points for the governing eigenvalue differential equation. The results are illustrated for two particular examples that have geophysical and astrophysical significance, namely radially directed spherical gravity, and the gravitational field in a rapidly rotating cylinder.  相似文献   

4.
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   

5.
Abstract

In the present paper the behavior of (internal) magneto-acoustic-gravity waves near the cusp resonance in a lossless, compressible, isothermal, stratified, electrically conducting atmosphere that is permeated by a uniform, nearly horizontal magnetic field is re-addressed (Kamp, 1989). The previously analyzed linear conversion of long acoustic-gravity waves into short magneto-acoustic waves that carry off the energy from the resonance region along the magnetic field, is re-analyzed with boundary layer techniques that are based on the smallness of the vertical component of the magnetic field. More specifically the existence of the so-called valve effect for the generated magneto-acoustic mode near the critical level is explicitly demonstrated and shown to be governed by two rivalling effects.  相似文献   

6.
Abstract

In this, the first part of a three part study, we develop a model for the theoretical analysis of 3‐dimensional internal gravity waves in atmospheres with arbitrary distributions of basic temperature, molecular weight, viscosity and conductivity (both eddy and molecular), Newtonian cooling, anisotropic hydromagnetic (ion) drag, and thermal excitation. Attention is given to the physical bases for our models, and a detailed outline is given of the numerical method used to solve the resulting system of equations.

As an application of the above development, we study the ability of five particular gravity waves (chosen for their observed importance in the neighborhood of 90–100 km—four of the waves are simulated tidal modes) to propagate from 90 km into the thermosphere. We choose to define the thermosphere as that portion of the atmosphere above the turbopause (ca. 110 km). Among the most significant results to emerge are the following: (i) the effects of molecular viscosity and conductivity appear to be more significant than the effects of hydromagnetic drag, and (ii) while most waves considered are significantly attenuated somewhere between 90 Ian and the thermosphere, the main semidiurnal tidal mode is not. In fact, semidiurnal temperature oscillations of only a few degrees amplitude at 90 km can give rise to oscillations of over a hundred degrees amplitude in the exosphere.  相似文献   

7.
The paper starts with a discussion of the linear stochastic theory of ocean waves and its various nonlinear extensions. The directional spectrum, with its unique dispersion relation connecting frequency (ω) and wavenumber (k), is no longer valid for nonlinear waves, and examples of $\left( \mathbf{k},\omega\right) The paper starts with a discussion of the linear stochastic theory of ocean waves and its various nonlinear extensions. The directional spectrum, with its unique dispersion relation connecting frequency (ω) and wavenumber (k), is no longer valid for nonlinear waves, and examples of ( k,w)\left( \mathbf{k},\omega\right) -spectra based on analytical expressions and computer simulations of nonlinear waves are presented. Simulations of the dynamic nonlinear evolution of unidirectional free waves using the nonlinear Schr?dinger equation and its generalizations show that components above the spectral peak have larger phase and group velocities than anticipated by linear theory. Moreover, the spectrum does not maintain a thin well-defined dispersion surface, but rather develops into a continuous distribution in ( k,w)\left( \mathbf{k,}\omega\right) -space. The majority of existing measurement systems rely on linear theory for the interpretation of their data, and no measurement systems are currently able to measure the full spectrum in the open ocean with high accuracy. Nevertheless, there exist a few low-resolution systems where data may be interpreted within a minimal assumption of a non-restricted ( k,w)\left( \mathbf{k,}\omega\right) -spectrum. The theory is reviewed, and analyses based on conventional spectral analysis as well as a directional wavelet analysis are carried out on data from a compact laser array at the Ekofisk field in the North Sea. The investigation confirms the strong impact of the second order spectrum below the spectral peak, but is non-conclusive about the off-set in the support of the first order spectrum seen in the dynamical simulations.  相似文献   

8.
Abstract

The simplest model for geophysical flows is one layer of a constant density fluid with a free surface, where the fluid motions occur on a scale in which the Coriolis force is significant. In the linear shallow water limit, there are non-dispersive Kelvin waves, localized near a boundary or near the equator, and a large family of dispersive waves. We study weakly nonlinear and finite depth corrections to these waves, and derive a reduced system of equations governing the flow. For this system we find approximate solitary Kelvin waves, both for waves traveling along a boundary and along the equator. These waves induce jets perpendicular to their direction of propagation, which may have a role in mixing. We also derive an equivalent reduced system for the evolution of perturbations to a mean geostrophic flow.  相似文献   

9.
Abstract

In the context of ageostrophic theory in a homogeneous ocean, a nondimensional number is determined which corresponds to the Ursell number for long gravity waves. It is defined as Q = NL 2/h 3, where N is the amplitude of the wave travelling along the long length-scale direction, L is its length and h (which for gravity waves is the water depth) is given by h=(l 4 f 2/g)1/3. where l is the short length-scale, f the Coriolis parameter and g the acceleration due to gravity. The physical meaning of Q is as follows: if Q? O(1) the free evolution of the wave is linear and weakly dispersive, if Q = O(1) nonlinear and dispersive effects balance out and finally if Q ?O(1) the evolution is nonlinear and non-dispersive. Expressions for the time scales for the development of dispersive and nonlinear effects are also determined. These results apply to topographically trapped waves, namely barotropic continental shelf and double Kelvin waves travelling along a rectilinear topographic variation.  相似文献   

10.
Abstract

Small amplitude oscillations of a uniformly rotating, density stratified, Boussinesq, non-dissipative fluid are examined. A mathematical model is constructed to describe timedependent motions which are small deviations from an initial state that is motionless with respect to the rotating frame of reference. The basic stable density distribution is allowed to be an arbitrary prescribed function of the gravitational potential. The problem is considered for a wide class of gravitational fields. General properties of the eigenvalues and eigenfunctions of square integrable oscillations are demonstrated, and a bound is obtained for the magnitude of the frequencies. The modal solutions are classified as to type. The eigenfunctions for the pressure field are shown to satisfy a second-order partial differential equation of mixed type, and the equation is obtained for the critical surfaces which delineate the elliptic and hyperbolic regions. The nature of the problem is examined in detail for certain specific gravitational fields, e.g., a radially symmetric field. Where appropriate, results are compared with those of other investigations of waves in a rotating fluid of spherical configuration and the novel aspects of the present treatment are emphasized. Explicit modal solutions are obtained in the specific example of a fluid contained in a rigid cylinder, stratified in the presence of vertical gravity, with the buoyancy frequency N being an arbitrary prescribed function of depth.  相似文献   

11.
Abstract

The dynamical nonequilibrium of close-packed flux tubes is driven by the torsion in the individual tubes so that, wherever tubes with the same sense of twisting come into contact, there is reconnection of their azimuthal field components. The reconnection consumes the local torsion, causing the propagation of torsional Alfven waves into the region from elsewhere along the tubes.

The formal problem of the propagation of the torsion along twisted flux tubes is presented and some of the basic physical properties worked out in the limit of small torsion.

It is pointed out that in tubes with finite twisting the propagation of torsional Alfven waves can be a more complicated phenomenon.

Application to the sun suggests that the propagation of torsion from below the visible surface up into the corona is an important energy supply to the corona for a period of perhaps 10–20 hours after the emergence of the flux tubes through the surface of the sun, bringing up torsion from depths of 104km or more. Torsion is continually supplied by the manipulation and shuffling of the field by the convection, of course.  相似文献   

12.
Abstract

The long wave equations governing the flow in alluvial rivers and channels are considered. The linearized equations are re-cast in the form of a single equation of wave hierarchy type as discussed by Whitham (1974). The dynamic and kinematic waves are of third and second order respectively. Behaviour at the wave fronts is considered and a roll-wave type instability is revealed.

For stable flow, the theory is used to make both qualitative and quantitative predictions in the areas of short and long term floods, tidal waves and channel dredging.

The non-uniformity in the quasi-steady theory on bedform development [see, for example, Reynolds (1985)] as the Froude number, F, approaches unity is also discussed, and appropriate scalings are obtained to derive a theory which remains valid when F ～ 1.  相似文献   

13.
Abstract

A general fluid dynamical theory of discrete unstable spiral modes in disk-shaped galaxies is described. This formulation of modes includes a radiation boundary condition and an exact numerical treatment of the Poisson equation. Thus, the modes are maintained by an outward transport of angular momentum, but they may be composed of both leading and trailing waves. A numerical scheme based on this formulation is described, and examples of modes obtained with this scheme are presented. These examples compare favorably with calculations based on the original asymptotic theory of Bertin, Lau, Lin and Mark. The implications of the present formulation of modes in galactic models support the hypothesis of a quasi-stationary spiral structure.  相似文献   

14.
Abstract

In a recent paper, Buchwald (1972a) has shown that besides the kinetic energy and gravitational potential energy usually associated with planetary waves in an ocean of uniform depth it is useful to define also a “spin energy”, associated with the rotation.

The present paper is basically an extension of Buchwald's result to a uniformly rotating β-plane ocean of variable depth. As in the previous work, energy conservation equations are derived and the separate energies shown to be independently conserved over the total volume of the ocean. The time-averaged energies are further shown to be propagated in the direction of the group velocity and to satisfy the equipartition rule.

Unlike Buchwald, however, we need not consider the boundary conditions in order to achieve these results. Furthermore, the use of a more realistic ocean configuration admits the possibility of a multiply connected region in the present of mean currents.

Finally, there is a physical explanation for the appearance of a spin energy in a rotating system.  相似文献   

15.
Abstract

Barotropic instability of weakly non-parallel zonal flows with localized intense shear regions is investigated numerically. The numerical integrations of the linear stability problem reveal the existence of unstable localized wave packets whose spatial structure and eigenfrequencies depend on two parameters which measure the degree of supercriticality and the zonal length-scale of the shear region. The results indicate that the structure of the instability is determined by conditions that ensure the decay of the wave packet at infinity and the transition from long to short waves across a turning point (critical layer) region which is controlled by non-parallel effects. The controlling influence exerted by the weak non-parallel effects on the evolution of the instability underlines the weakness of the parallel flow assumption which can be used locally, away from critical layers, as a diagnostic tool only.  相似文献   

16.
A new interferometric imaging technique has been used at Jicamarca to study the morphology and dynamics of plasma irregularities occurring during spread F conditions. The technique produces two-dimensional, in-beam images of the coherent scatter from these irregularities, instantaneously mapping the structure of the underlying plasma instabilities in the equatorial plane. We present sequences of images depicting backscatter from bottom-type layers, bottomside layers, and topside plumes drifting and rising through the ionospheric regions illuminated by the Jicamarca radar. The high spatial and temporal resolution of the images permits us to observe dynamical phenomena missed by conventional fixed- and steered-beam techniques. The new images show evidence of intermediate- and large-scale plasma depletions bifurcating and pinching off from the bottomside as they ascend. Analytic models are presented to describe these dynamical effects. The images also differentiate between bottom-type and bottomside spread F layers, the former being shown to have much smaller scale primary waves than the latter. Ample evidence of secondary plasma instabilities appear throughout the images. The consequences of these secondary instabilities and of shear flow are discussed.  相似文献   

17.
It is becoming increasingly clear that electron thermal effects have to be taken into account when dealing with the theory of ionospheric instabilities in the high-latitude ionosphere. Unfortunately, the mathematical complexity often hides the physical processes at work. We follow the limiting cases of a complex but systematic generalized fluid approach to get to the heart of the thermal processes that affect the stability of E region waves during electron heating events. We try to show as simply as possible under what conditions thermal effects contribute to the destabilization of strongly field-aligned (zero aspect angle) Farley-Buneman modes. We show that destabilization can arise from a combination of (1) a reduction in pressure gradients associated with temperature fluctuations that are out of phase with density fluctuations, and (2) thermal diffusion, which takes the electrons from regions of enhanced temperatures to regions of negative temperature fluctuations, and therefore enhanced densities. However, we also show that, contrary to what has been suggested in the past, for modes excited along the E0 × B direction thermal feedback decreases the growth rate and raises the threshold speed of the Farley-Buneman instability. The increase in threshold speed appears to be important enough to explain the generation of Type IV waves in the high-latitude ionosphere.  相似文献   

18.
Abstract

An analysis is presented of the propagation of barotropic non-divergent oscillations along the western side of an ocean basin along which the persistent circulation in the basin is strongly intensified and laterally sheared. Because the Rossby number of a western boundary current is near unity, the properties of these waves are strongly affected by the steady circulation pattern. It is shown that for relatively long wavelengths, these waves can travel along the shelf in both directions; however, for a small range of short wavelengths they can only propagate northward and are unstable. Along the southeastern coast of North America, the unstable waves have wavelengths of order 150 km and periods of order 10 days. However, these waves can become stable oscillations in the deeper water northeast of Cape Hatteras. These oscillations are a possible explanation of the initiation of Gulf Stream meanders along the continental rise.  相似文献   

19.
The purpose of the geoscientific Observatory at Schiltach in the Black Forest in SW-Germany is to measure and monitor motions and deformations of the earth in a frequency band as broad as possible. The spectrum of geodynamic signal contains
• - the short period waves (up to 40 Hz) of near earthquakes,
• - the longer period body and surface waves from teleseismic events,
• - the free oscillations of the earth excited by giant earthquakes,
• - the earth tide deformations caused by the gravitational forces of the moon and the sun having periods of 12 h and longer,
• - deformations of the earth in a wide period range from other sources (e.g. atmosphere).
Besides providing high quality data with some continuity using ‘standard’ sensors, the observatory must be considered as an experimental station and research facility where new instruments and techniques can be tested.  相似文献   

20.
The reflection/transmission laws (R/T laws) of plane waves at a plane interface between two homogeneous anisotropic viscoelastic (dissipative) halfspaces are discussed. Algorithms for determining the slowness vectors of reflected/transmitted plane waves from the known slowness vector of the incident wave are proposed. In viscoelastic media, the slowness vectors of plane waves are complex-valued, p = P + iA, where P is the propagation vector, and A the attenuation vector. The proposed algorithms may be applied to bulk plane waves (A = 0), homogeneous plane waves (A ≠ 0, P and A parallel), and inhomogeneous plane waves (A ≠ 0, P and A non-parallel). The manner, in which the slowness vector is specified, plays an important role in the algorithms. For unrestricted anisotropy and viscoelasticity, the algorithms require an algebraic equation of the sixth degree to be solved in each halfspace. The degree of the algebraic equation decreases to four or two for simpler cases (isotropic media, plane waves in symmetry planes of anisotropic media). The physical consequences of the proposed algorithms are discussed in detail. vcerveny@seis.karlov.mff.cuni.cz  相似文献   

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