Further evidence of normal mode Rossby waves

Further observational evidence of normal mode Rossby waves with higher meridional mode numbers is presented with the aid of global data from the troposphere to the stratosphere over the period November 1979 through April 1986.It is shown, without using ana priori assumption of meridional structure, that the third antisymmetric modes of zonal wavenumbers 1 and 2,i.e., (1,4) and (2,4) modes, substantially exist in the real atmosphere. These modes are, however, easily influenced by the nonuniform background field even in the equinoctial season; amplitude submaxima near the equator are apt to be dubious in the upper stratosphere so that the prototype meridional structure becomes obscure. The period of the (1,4) mode often falls into that of the (1,3) mode, about 16 days. Hence, these two modes cannot be classified simply by their periods, but the separation is made by their meridional structure.An appearance calendar of various modes is also presented for the analysis period. It is found that each mode appears irregularly throughout the year and that the year-to-year variation is fairly large.     

Dispersive effects of zonally varying topography on quasigeostrophic Rossby waves

Abstract

Dispersion of linear quasigeostrophic plane waves in a stratified ocean with bottom topography is discussed. Particular emphasis is given to cases for which zonal gradients in the sea floor height are important. As such, the relative importance of the topographic and planetary β-effects is strongly dependent on wave vector orientation. The magnitude of the topographic slopes considered is chosen such that these two effects (topographic and planetary β) are of comparable importance. In the interest of simplicity, stratification is taken to be independent of depth. The eigenvalue problem which must be solved to find the free modes of oscillation has already been treated in the literature (in fact, Charney and Flierl (1981) have treated the effects of a more realistic stratification). The aim of this note is to more fully expose, primarily by example, several dispersive properties of these free wave modes which have been largely overlooked.     

Resonance induced by Rossby Lee waves

Abstract

The mutual interaction of fields induced by spatially separated potential vorticity sources in a quasi-geostrophic barotropic flow is investigated using the weakly nonlinear approach. It is found that a powerful nonlinear response can be triggered by Rossby lee waves. This resonance phenomenon which dominates all other nonlinear corrections depends on certain global resonance conditions and on the change in the phase of the Rossby lee wave across the distance separating the sources. The response is particularly strong for topographic forcing possessing δ-function characterisitics.     

On the normal mode instability of modons and Wu-Verkley waves

Abstract

The normal mode instability of steady Wu-Verkley (1993) wave and modons by Verkley (1984, 1987, 1990) and Neven (1992) is considered. All these flows are solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere. A conservation law for infinitesimal perturbations to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (1953) average spectral number of the amplitude of an unstable mode must be equal to a specific number that depends on the degree of the solution in its inner and outer regions as well as on spectral distribution of the mode energy in these regions. Some properties of the conditions for different types of modons are discussed. The maximum growth (and decay) rate of the modes is estimated, and the orthogonality of the amplitude of each unstable, decaying, or non-stationary mode to the basic solution is shown in the energy inner product.

The new instability conditions confine the unstable disturbances of the WV wave and modon to a hypersurface in the perturbation space and allow interpretation of their energy structure. They are also useful both in estimating the maximum growth rate of unstable modes and in testing the numerical algorithms designed for the linear stability study.     

Magnetic Rossby waves in a stably stratified layer near the surface of the Earth's outer core

Abstract

The stratification profile of the Earth's magnetofluid outer core is unknown, but there have been suggestions that its upper part may be stably stratified. Braginsky (1984) suggested that the magnetic analog of Rossby (planetary) waves in this stable layer (the ‘H’ layer) may be responsible for a portion of the short-period secular variation. In this study, we adopt a thin shell model to examine the dynamics of the H layer. The stable stratification justifies the thin-layer approximations, which greatly simplify the analysis. The governing equations are then the Laplace's tidal equations modified by the Lorentz force terms, and the magnetic induction equation. We linearize the Lorentz force in the Laplace's tidal equations and the advection term in the magnetic induction equation, assuming a zeroth order dipole field as representative of the magnetic field near the insulating core-mantle boundary. An analytical β-plane solution shows that a magnetic field can release the equatorial trapping that non-magnetic Rossby waves exhibit. A numerical solution to the full spherical equations confirms that a sufficiently strong magnetic field can break the equatorial waveguide. Both solutions are highly dissipative, which is a consequence of our necessary neglect of the induction term in comparison with the advection and diffusion terms in the magnetic induction equation in the thin-layer limit. However, were one to relax the thin-layer approximations and allow a radial dependence of the solutions, one would find magnetic Rossby waves less damped (through the inclusion of the induction term). For the magnetic field strength appropriate for the H layer, the real parts of the eigenfrequencies do not change appreciably from their non-magnetic values. We estimate a phase velocity of the lowest modes that is rather rapid compared with the core fluid speed typically presumed from the secular variation.     

Chaotic mixing of tracer and vorticity by modulated travelling Rossby waves

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.     

The generation of baroclinic Rossby waves by stationary and and translating currents part 1. Stationary current forcing

Abstract

This paper investigates the generation of linear, baroclinic Rossby waves by an imposed current distribution, in a reduced gravity ocean, both with and without an eastern coast. A zonal current is impulsively applied and maintained along the northern edge of the domain of solution. Using Green's function techniques, analytical solutions are found, and these are evaluated for small times. Numerical solutions are obtained for larger times. The upper layer depth field consists of a transient response, due to the sudden application of the current. Maintenance of the current causes a response which is singular along the line of imposed non-zero h _y. The interior field decays with time (this is shown asymptotically). The parameters used are appropriate for the mid-latitude North Pacific, and the results are relevant to sudden transport changes in the North Pacific Current.     

Instability of planetary waves in a high vertical resolution model

Abstract

A high vertical resolution model is used to examine the instability of a baroclinic zonal flow and a finite amplitude topographically forced wave. Two families of unstable modes are found, consisting of zonally propagating most unstable modes, and stationary unstable modes. The former have time scale and spatial structure similar to baroclinic synoptic disturbances, but are localized in space due to interaction with the zonally asymmetric forcing. These modes transport heat efficiently in both the zonal and meridional directions. The second family of stationary unstable modes has characteristics of modes of low frequency variability of the atmosphere. They have time scales of 10 days and longer, and are of planetary scale with an equivalent barotropic vertical structure. The horizontal structure resembles blocking flows. They are maintained by available potential energy of the basic wave, and have large zonal heat fluxes. The results for both families of modes are interpreted in terms of an interaction between forcing and baroclinic instability to create favoured regions for eddy development. Applications to baroclinic planetary waves are also considered.     

The role played by thermal feedback in heated Farley-Buneman waves at high latitudes

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.     

The scattering of Rossby waves from finite abrupt topography

The scattering of first mode linear baroclinic Rossby waves by a top-hat ridge in a continuously stratified ocean, with Brunt-Väisälä frequency that decays exponentially with depth below a surface mixed layer, is the subject of this study. A numerical mode matching technique is used to calculate the transmission coefficients for the propagating modes over the ridge. It is found that the scattered field depends crucially upon the stratification. For example, when the majority of the density variation is confined to a thin thermocline, corresponding to a small e-folding scale, gamma ^?1, for the Brunt-Väisälä frequency, a large amount of the incident wave energy is reflected by a small amplitude ridge. Appreciable energy conversion between the propagating barotropic and baroclinic modes takes place in this case. An asymptotic analysis for a small amplitude ridge is presented that confirms these numerical results. In the limit gamma ^?1→ 0, it is demonstrated that the scattered field in the continuously stratified ocean model differs markedly from the two-layer solution. The latter does not exhibit appreciable reflection of the incident wave energy for a small amplitude ridge. In conclusion, the application of a two-layer ocean model to describe Rossby wave scattering by ridges in place of a continuously stratified model cannot be recommended.     

Finite-amplitude mountain waves in a compressible atmosphere including the effects of dissipation

Abstract

Adiabatic, two-dimensional, steady-state finite-amplitude, hydrostatic gravity waves produced by flow over a ridge are considered. Nonlinear self advection steepens the wave until the streamlines attain a vertical slope at a critical height z_c. The height z_c , where this occurs, depends on the ridge crest height and adiabatic expansion of the atmosphere. Dissipation is introduced in order to balance nonlinear self advection, and to maintain a marginal state above z_c. The approach is to assume that the wave is inviscid except in a thin layer, small compared to a vertical wavelength, where dissipation cannot be neglected. The solutions in each region are matched to obtain a continuous solution for the streamline displacement δ. Solutions are presented for different values of the nondimensional dissipation parameter β. Eddy viscosity coefficients and the thickness of the dissipative layer are expressed as functions of β, and their magnitudes are compared to other theoretical evaluations and to values inferred from radar measurements of the stratosphere.

The Fourier spectrum of the solution for z ≫ z_c is shown to decay exponentially at large vertical wave numbers n. In comparison, a spectral decay law n ^?-8/3 characterizes the marginal state of the wave at z = z_c .     

Stratigraphic filtering of SH waves in a spherical,layered earth

We present a derivation of the formula for filtering a transmitted SH wavelet by shortperiod multiples in a spherically layered Earth. We use a continuous, rather than a discrete formulation and regard the impedance and the velocity as random variables. The mean shear displacement represents the propagating wavelet as modified by short-period multiples. Standard procedures and approximations Lead to the dispersion relation of the mean displacement. To describe the stratigraphic filtering we introduce a complex quantityF such that a wavelet which has travelled a time T is modified by the filter exp {iFT}. The impact of the higher angular harmonic modes is shown to produce a relative enhancement of those modes over the low angular harmonic modes due to fluctuations in the shear-wave propagation velocity. Numerical estimates indicate that the sizes of the apparent attenuation of the mean field and the time delay introduced by the short-period multiples sit squarely in the regime where they produce a nonnegligible distortion of the SH modes of propagation in both phase and amplitude.     

Hydromagnetic waves in a differentially rotating annulus IV. Insulating boundaries

Abstract

The first three papers in this series (Fearn, 1983b, 1984, 1985) have investigated the stability of a strong toroidal magnetic field B_o =B_o(s?)Φ [where (s?. Φ, z?) are cylindrical polars] in a rapidly rotating system. The application is to the cores of the Earth and the planets but a simpler cylindrical geometry was chosen to permit a detailed study of the instabilities present. A further simplification was the use of electrically perfectly conducting boundary conditions. Here, we replace these with the boundary conditions appropriate to an insulating container. As expected, we find the same instabilities as for a perfectly conducting container, with quantitative changes in the critical parameters but no qualitative differences except for some interesting mixing between the ideal (“field gradient”) and resistive modes for azimuthal wavenumber m=1. In addition to these modes, we have also found the “exceptional” slow mode of Roberts and Loper (1979) and we investigate the conditions required for its instability for a variety of fields B_o(s?) Roberts and Loper's analysis was restricted to the case B_o∝s? and they found instability only for m=1 and ?1 <ω<0 [where ω is the frequency non-dimensionalised on the slow timescale τ_x, see (1.5)]. For other fields we found the necessary conditions to be less “exceptional”. One surprising feature of this instability is the importance of inertia for its existence. We show that viscosity is an alternative destabilising agent. The standard (magnetostrophic) approximation of neglecting inertial (and viscous) terms in the equation of motion has the effect of filtering out this instability. The field strength required for this “exceptional” mode to become unstable is found to be very much larger than that thought to be present in the Earth's core, so we conclude that this mode is unlikely to play an important role in the dynamics of the core.     

Internal tides and waves near the continental shelf edge

Abstract

The subject is reviewed from the viewpoints of theory, internal tide and wave structure and their implications.

A wider theoretical context suggests scope for further investigation of natural or nearly-trapped forms above the inertial frequency.

Although internal tides in many locations are observed to have first-mode vertical structure, higher modes are seen offshore from shallow shelf-break forcing and for particular Froude numbers, and may be expected locally near generation. Bottom intensification is often observed where the sea floor matches the characteristic slope. Solitons form from internal tides of large amplitude or at large changes of depth.

Internal tides and solitons are observed also at many sills and in straits, and to intensify in canyons.

Non-linear effects of the waves, especially solitons, include the conveyance of water, nutrients, ‘‘mixing potential'’ etc. away from their source to other locations, and the generation of mean currents. The waves transfer energy and possibly heat between the ocean and shelf, may be a source of medium frequency waves on the shelf (periods of minutes) and can contribute to interior mixing and overturning, bottom stirring and sediment movement.     

On steady and travelling waves over bottom topography in the model of homogeneous flow in a beta-plane channel

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.     

Hydromagnetic waves in rapidly rotating spherical shells generated by poloidal decay modes

Abstract

This paper presents the first attempt to examine the stability of a poloidal magnetic field in a rapidly rotating spherical shell of electrically conducting fluid. We find that a steady axisymmetric poloidal magnetic field loses its stability to a non-axisymmetric perturbation when the Elsasser number A based on the maximum strength of the field exceeds a value about 20. Comparing this with observed fields, we find that, for any reasonable estimates of the appropriate parameters in planetary interiors, our theory predicts that all planetary poloidal fields are stable, with the possible exception of Jupiter. The present study therefore provides strong support for the physical relevance of magnetic stability analysis to planetary dynamos. We find that the fluid motions driven by magnetic instabilities are characterized by a nearly two-dimensional columnar structure attempting to satisfy the Proudman-Taylor theorm. This suggests that the most rapidly growing perturbation arranges itself in such a way that the geostrophic condition is satisfied to leading order. A particularly interesting feature is that, for the most unstable mode, contours of the non-axisymmetric azimuthal flow are closely aligned with the basic axisymmetric poloidal magnetic field lines. As a result, the amplitude of the azimuthal component of the instability is smaller than or comparable with that of the poloidal component, in contrast with the instabilities generated by toroidal decay modes (Zhang and Fearn, 1994). It is shown, by examining the same system with and without fluid inertia, that fluid inertia plays a secondary role when the magnetic Taylor number T_m ? 10⁵. We find that the direction of propagation of hydromagnetic waves driven by the instability is influenced strongly by the size of the inner core.