Some effects of a mean zonal thermocline gradient on planetary equatorial waves

Planetary equatorial waves are studied with the shallow water equations in the presence of a mean zonal thermocline gradient. The interactions between this gradient and waves are represented by three non-linear terms in the equations: one in the wind-forcing formulation in the x-momentum equation, and two for the advection of mass and divergence of the velocity field in the continuity equation. When the mean gradient is imposed but small, these three (linearized) terms will perturb the behavior of the equatorial waves. This paper gives a simple analytic treatment of this problem.The equatorial Kelvin mode is first solved with all three contributions, using a Wentzel-Kramers-Brillouin method. The Kelvin mode shows a spatial or/and temporal growth when the thermocline gradient is negative which is the usual situation in the equatorial Pacific ocean (deep thermocline in the west and shallow in the east). The more robust and efficient contribution comes from the advection term.The single effect of the advection of the mean zonal thermocline gradient is then studied for the Kelvin and planetary Rossby modes. The Kelvin mode remains unstable (damped), while the Rossby modes appear damped (unstable) for a negative (positive) thermocline gradient.     

Nonlinear waves in rotating magnetohydrodynamics

We consider an electrically conducting fluid in rotating cylindrical coordinates in which the Elsasser and magnetic Reynolds numbers are assumed to be large while the Rossby number is assumed to vanish in an appropriate limit. This may be taken as a simple model for the Earth's outer core. Fully nonlinear waves dominated by the nonlinear Lorentz forces are studied using the method of geometric optics (essentially WKB). These waves are assumed to be of the form of an asymptotic series expanded about ambient magnetic and velocity fields which vanish on the equatorial plane. They take the form of short wave, slowly varying wave trains. The first-order approximation is sinusoidal and basically the same as in the linear problem, with a dispersion relation modified by the appearance of mean terms. These mean terms, as well the undetermined amplitude functions, are found by suppressing secular terms in a “fast” variable in the second-order approximation. The interaction of the mean terms with the dispersion relation is the primary cause of behaviors which differ from the linear case. In particular, new singularities appear in the wave amplitude functions and an initial value problem results in a singularity in one of the mean terms which propagates through the fluid. The singularities corresponding to the linear ones are shown to develop when the corresponding waves propagate toward the equatorial plane.     

Hydromagnetic waves in a differentially rotating,stratified spherical shell

Abstract

Investigations of an earlier paper (Friedlander 1987a) are extended to include the effect of an azimuthal shear flow on the small amplitude oscillations of a rotating, density stratified, electrically conducting, non-dissipative fluid in the geometry of a spherical shell. The basic state mean fields are taken to be arbitrary toroidal axisymmetric functions of space that are consistent with the constraint of the ‘‘magnetic thermal wind equation''. The problem is formulated to emphasize the similarities between the magnetic and the non-magnetic internal wave problem. Energy integrals are constructed and the stabilizing/destabilizing roles of the shears in the basic state functions are examined. Effects of curvature and sphericity are studied for the eigenvalue problem. This is given by a partial differential equation (P.D.E.) of mixed type with, in general, a complex pattern of turning surfaces delineating the hyperbolic and elliptic regimes. Further mathematical complexities arise from a distribution of the magnetic analogue of critical latitudes. The magnetic extension of Laplace's tidal equations are discussed. It is observed that the magnetic analogue of planetary waves may propagate to the east and to the west.     

切变基本纬向流中非线性赤道Rossby长波

为了解决观测和理论研究中的一些问题以及更好地了解热带大气动力学 ,有必要进一步研究基本气流的变化对大气中赤道Rossby波动的影响 .本文研究分析基本气流对赤道Rossby长波的影响 ,利用一个简单赤道 β平面浅水模式和摄动法 ,研究纬向基本气流切变中非线性赤道Rossby波 ,推导出在切变基本纬向流中赤道Rossby长波振幅演变所满足的非线性KdV方程并得到其孤立波解 .分析表明 ,孤立波存在的必要条件是基本气流有切变 ,而且基流切变不能太强 ,否则将产生正压不稳定 .     

Periodic and aperiodic dynamo waves

Abstract

In order to show that aperiodic magnetic cycles, with Maunder minima, can occur naturally in nonlinear hydromagnetic dynamos, we have investigated a simple nonlinear model of an oscillatory stellar dynamo. The parametrized mean field equations in plane geometry have a Hopf bifurcation when the dynamo number D=1, leading to Parker's dynamo waves. Including the nonlinear interaction between the magnetic field and the velocity shear results in a system of seven coupled nonlinear differential equations. For D>1 there is an exact nonlinear solution, corresponding to periodic dynamo waves. In the regime described by a fifth order system of equations this solution remains stable for all D and the velocity shear is progressively reduced by the Lorentz force. In a regime described by a sixth order system, the solution becomes unstable and successive transitions lead to chaotic behaviour. Oscillations are aperiodic and modulated to give episodes of reduced activity.     

Boundary layer dissipation and the nonlinear interaction of equatorial baroclinic and barotropic rossby waves

Two-layer equatorial primitive equations for the free troposphere in the presence of a thin atmospheric boundary layer and thermal dissipation are developed here. An asymptotic theory for the resonant nonlinear interaction of long equatorial baroclinic and barotropic Rossby waves is derived in the presence of such dissipation. In this model, a self-consistent asymptotic derivation establishes that boundary layer flows are generated by meridional pressure gradients in the lower troposphere and give rise to degenerate equatorial Ekman friction. That is to say, the asymptotic model has the property that the dissipation matrix has one eigenvalue which is nearly zero: therefore the dynamics rapidly dissipates flows with pressure at the base of the troposphere and creates barotropic/baroclinic spin up/spin down. The simplified asymptotic equations for the amplitudes of the dissipative equatorial barotropic and baroclinic waves are studied by linear theory and integrated numerically. The results indicate that although the dissipation slightly weakens the tropics to midlatitude connection, strong localized wave packets are nonetheless able to exchange energy between barotropic and baroclinic waves on intraseasonal timescales in the presence of baroclinic mean shear. Interesting dissipation balanced wave-mean flow states are discovered through numerical simulations. In general, the boundary layer dissipation is very efficient for flows in which the barotropic and baroclinic components are of the same sign at the base of the free troposphere whereas the boundary layer dissipation is less efficient for flows whose barotropic and baroclinic components are of opposite sign at the base of the free troposphere.     

Quasi-free-decay magnetic modes in planetary cores

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

Rossby波的螺旋斑图

应用描写大气大尺度运动的准地转方程组，求得了大气Rossby波的三维定常流场以及相应的位温场、涡度场和散度场，其中的三维流场构成了物理空间的一个非线性自治动力系统. 研究表明，Rossby波具有     

Resistive wave breaking in the Earth's outer core

The equations for an electrically conducting fluid in cylindrical coordinates are linearized assuming that the inertial terms in the momentum equation can be ignored (small Rossby number), and that the ratio of the Elsasser number and magnetic Reynolds number is one. After these assumptions, the governing equations are linearized about an ambient solution which vanishes at the the equator. Upon assuming large Elsasser and magnetic Reynolds number, the solutions to the linearized equations are approximated by wave trains having very short wave length (relative to the core radius) but which vary slowly (on a scale of the core radius). The period of the waves is much longer than a day but much shorter than the period of the slow hydromagnetic oscillations. These waves are found to be trapped in a region about the equator and away from the axis of rotation. The waves break at a latitudinal wave region boundary, in the sense that the waves become exponentially large in a boundary layer, having as an exponent some positive power of the large azimuthal wave number. This behavior is amplified as the Elsasser number becomes smaller while still remaining relatively large. Waves in more Earth-like parameter regimes are discussed briefly.     

Hydromagnetic waves in a thin rotating spherical shell

We consider an electrically conducting fluid confined to a thin rotating spherical shell in which the Elsasser and magnetic Reynolds numbers are assumed to be large while the Rossby number is assumed to vanish in an appropriate limit. This may be taken as a simple model for a possible stable layer at the top of the Earth's outer core. It may also be a model for the thin shells which are thought to be a source of the magnetic fields of some planets such as Mercury or Uranus. Linear hydromagnetic waves are studied using a multiple scale asymptotic scheme in which boundary layers and the associated boundary conditions determine the structure of the waves. These waves are assumed to be of the form of an asymptotic series expanded about an ambient magnetic field which vanishes on the equatorial plane and velocity and pressure fields which do not. They take the form of short wave, slowly varying wave trains. The results are compared to the author's previous work on such waves in cylindrical geometry in which the boundary conditions play no role. The approximation obtained is significantly different from that obtained in the previous work in that an essential singularity appears at the equator and nonequatorial wave regions appear.     

Generation and maintenance of bisymmetric spiral magnetic fields and interaction with spiral density waves

Abstract

The paper consists of two parts. The first introduces the dynamo equation into a rotating gaseous disk of finite thickness and then searches for its solution for the generation and maintenance of large-scale bisymmetric spiral (BSS) magnetic fields. We determine numerically the dynamo strength and vertical thickness of the gaseous disk which are necessary for the BSS magnetic fields to rotate as a wave over large area of the disk.

Next we present linearized equations of motion for the self-gravitating disk gas under the Lorentz force due to the BSS magnetic fields. Since the angular velocity of the BSS field is very close to that of the spiral density wave, a nearly-resonant interaction is caused between these two waves to produce large-amplitude condensation of gas in a double-spiral way. The BSS magnetic field is considered as a promising agency to trigger and maintain the spiral density wave.     

Evolution of non-linear α 2-dynamos and taylor's constraint

A key non-linear mechanism in a strong-field geodynamo is that a finite amplitude magnetic field drives a flow through the Lorentz force in the momentum equation and this flow feeds back on the field-generation process in the magnetic induction equation, equilibrating the field. We make use of a simpler non-linear?α?²-dynamo to investigate this mechanism in a rapidly rotating fluid spherical shell. Neglecting inertia, we use a pseudo-spectral time-stepping procedure to solve the induction equation and the momentum equation with no-slip velocity boundary conditions for a finitely conducting inner core and an insulating mantle. We present calculations for Ekman numbers (E) in the range 2.5× 10^?3 to 5.0× 10^?5, for?α?=α ₀cos?θ?sin?π?(r?r_i ) (which vanishes on both inner and outer boundaries). Solutions are steady except at lower E and higher values of?α?₀. Then they are periodic with a reversing field and a characteristic rapid increase then equally rapid decrease in magnetic energy. We have investigated the mechanism for this and shown the influence of Taylor's constraint. We comment on the application of our findings to numerical hydrodynamic dynamos.     

Meridional equatorial electrojet current in the American sector

Huancayo is the only equatorial electrojet station where the daytime increase of horizontal geomagnetic field (H) is associated with a simultaneous increase of eastward geomagnetic field (Y). It is shown that during the counter electrojet period when H is negative, Y also becomes negative. Thus, the diurnal variation of Y at equatorial latitudes is suggested to be a constituent part of the equatorial electrojet current system. Solar flares are known to increase the H field at an equatorial station during normal electrojet conditions (nej). At Huancayo, situated north of the magnetic equator, the solar flare effect, during nej, consists of positive impulses in H and Y and negative impulse in Z field. During counter electrojet periods (cej), a solar flare produces a negative impulse in H and Y and a positive impulse in Z at Huancayo. It is concluded that both the zonal and meridional components of the equatorial electrojet in American longitudes, as in Indian longitudes, flows in the same, E region of the ionosphere.     

Axial and equatorial rotations of the Earth's cores associated with the Quaternary ice age

The differential axial and equatorial rotations of both cores associated with the Quaternary glacial cycles were evaluated based on a realistic earth model in density and elastic structures. The rheological model is composed of compressible Maxwell viscoelastic mantle, inviscid outer core and incompressible Maxwell viscoelastic inner core. The present study is, however, preliminary because I assume a rigid rotation for the fluid outer core. In models with no frictional torques at the boundaries of the outer core, the maximum magnitude of the predicted axial rotations of the outer and inner cores amounts to ∼2° year⁻¹ and ∼1° year⁻¹, respectively, but that for the secular equatorial rotations of both cores is ∼0.0001° at most. However, oscillating parts with a period of ∼225 years are predicted in the equatorial rotations for both cores. Then, I evaluated the differential rotations by adopting a time-dependent electromagnetic (EM) torque as a possible coupling mechanism at the core-mantle boundary (CMB) and inner core boundary (ICB). In a realistic radial magnetic field at the CMB estimated from surface magnetic field, the axial and equatorial rotations couple through frictional torques at the CMB, although these rotations decouple for dipole magnetic field model. The differential rotations were evaluated for conductivity models with a conductance of 10⁸ S of the lowermost mantle inferred from studies of nutation and precession of the Earth and decadal variations of length of day (LOD). The secular parts of equatorial rotations are less sensitive to these parameters, but the magnitude for the axial rotations is much smaller than for frictionless model. These models, however, produce oscillating parts in the equatorial rotations of both cores and also in the axial rotations of the whole Earth and outer and inner cores. These oscillations are sensitive to both the magnitude of radial magnetic field at the CMB and the conductivity structure. No sharp isolated spectral peaks are predicted for models with a thin conductive layer (∼200 m) at the bottom of the mantle. In models with a conductive layer of ∼100 km thickness, however, sharp spectral peaks are predicted at periods of ∼225 and ∼25 years for equatorial and axial rotations, respectively, although these depend on the strength of radial magnetic field at the CMB. While the present study is preliminary in modelling the fluid outer core and coupling mechanism at the CMB, the predicted axial rotations of the whole Earth may be important in explaining the observed LOD through interaction between the equatorial and axial rotations.     

Possible configurations of the magnetic field in the outer magnetosphere during geomagnetic polarity reversals

Possible configurations of the magnetic field in the outer magnetosphere during geomagnetic polarity reversals are investigated by considering the idealized problem of a magnetic multipole of order m and degree n located at the centre of a spherical cavity surrounded by a boundless perfect diamagnetic medium. In this illustrative idealization, the fixed spherical (magnetopause) boundary layer behaves as a perfectly conducting surface that shields the external diamagnetic medium from the compressed multipole magnetic field, which is therefore confined within the spherical cavity. For a general magnetic multipole of degree n, the non-radial components of magnetic induction just inside the magnetopause are increased by the factor 1 + [(n + 1)/n] relative to their corresponding values in the absence of the perfectly conducting spherical magnetopause. An exact equation is derived for the magnetic field lines of an individual zonal (m = 0), or axisymmetric, magnetic multipole of arbitrary degree n located at the centre of the magnetospheric cavity. For such a zonal magnetic multipole, there are always two neutral points and n – 1 neutral rings on the spherical magnetopause surface. The two neutral points are located at the poles of the spherical magnetopause. If n is even, one of the neutral rings is coincident with the equator; otherwise, the neutral rings are located symmetrically with respect to the equator. The actual existence of idealized higher-degree (n > 1) axisymmetric magnetospheres would necessarily imply multiple (n + 1) magnetospheric cusps and multiple (n) ring currents. Exact equations are also derived for the magnetic field lines of an individual non-axisymmetric magnetic multipole, confined by a perfectly conducting spherical magnetopause, in two special cases; namely, a symmetric sectorial multipole (m = n) and an antisymmetric sectorial multipole (m = n – 1). For both these non-axisymmetric magnetic multipoles, there exists on the spherical magnetopause surface a set of neutral points linked by a network of magnetic field lines. Novel magnetospheric processes are likely to arise from the existence of magnetic neutral lines that extend from the magnetopause to the surface of the Earth. Finally, magnetic field lines that are confined to, or perpendicular to, either special meridional planes or the equatorial plane, when the multipole is in free space, continue to be confined to, or perpendicular to, these same planes when the perfectly conducting magnetopause is present.Also Honorary Research Associate, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK and Visiting Reader in Physics. University of Sussex, Falmer, Brighton BN1 9QH, UK     

Propagation of magneto-elastic waves in a perfectly conducting plate extending to infinity in vacuum

Summary The effects of a uniform external magnetic field on the propagation of waves in a homogeneous, infinitely conducting flat plate with free boundaries have been studied. It has been found that in general all the three types of waves —P, SV andSH waves—are coupled and the influence may be more pronounced in coupling the symmetric and antisymmetric types of motions in every mode.When the magnetic field is parallel to the plane faces and transverse to the direction of wave propagation, the shear wave polarized parallel to the field is purely elastic whereas the coupledP andS V waves are magnetoelastic and exhibit dispersion strikingly similar to the non-magnetic case, provided the electro-magnetic radiation into the surrounding free space is neglected.The results reported in an earlier communication [1]²) are also confirmed.     

ESF小尺度湍动谱簇射和非线性稳定性理论（Ⅰ）

本文从等离子体二流体方程出发,推导了适于描述赤道扩展F（ESF）层小尺度湍流的二维静电模型方程.对模型方程进行了线性分析,并且导出了描述波波相互作用的三波非线性耦合方程,以此为基础详细讨论了若干情况下的非线性稳定性和能谱簇射过程.结果表明,赤道F层不仅可以存在正能波,而且可能存在负能波;正能波和负能波的相互作用可以导致爆炸型不稳定性;各种可能的三个波（如三个正能波,二个正能波和一个负能波,等等）之间的相互作用可以产生不同类型的非线性不稳定性,导致不同形式的能量簇射,例如能量可以向大波数方向簇射,也可相反,还可能同时向大和小波数方向簇射等。     

具有非线性地形的正压流体中孤立Rossby波的mKdV方程

正压流体中,采用摄动方法将准地转位涡方程推导出地形效应的mKdV方程,得到Rossby波振幅的演变满足地形效应的mKdV方程的结论,说明地形效应是诱导Rossby孤立波的重要因素.     

Thermal and magnetically driven instabilities in a non-constantly stratified fluid layer

Abstract

The linear hydromagnetic stability of a non-constantly stratified horizontal fluid layer permeated by an azimuthal non-homogeneous magnetic field is investigated for various widths of the stably stratified part of the layer in the geophysical limit q→0 (q is the ratio of thermal and magnetic diffusivities). The choice of the strength of the magnetic field B_o is as in Soward (1979) (see also Soward and Skinner, 1988) and the equations for the disturbances are treated as in Fearn and Proctor (1983). It was found that convection is developed in the whole layer regardless of the width of its stably stratified part. The thermal instability penetrates essentially from the unstably stratified part of the layer into the stably stratified part for A ～ 1 (A characterises the ratio of the Lorentz and Coriolis forces). When the magnetic field is strong (A>1) the thermal convection is suppressed in the stably stratified part of the layer. However, in this case, it is replaced by the magnetically driven instability; which is fully developed in the whole layer. The thermal instabilities always propagate westward and exist for all the modes m. The magnetically driven instabilities propagate either westward or eastward according to the width of the stably and unstably stratified parts and exist only for the mode m=1.