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1.
We investigate instability of convective flows of simple structure (rolls, standing and travelling waves) in a rotating layer with stress-free horizontal boundaries near the onset of convection. We show that the flows are always unstable to perturbations, which are linear combinations of large-scale modes and short-scale modes, whose wave numbers are close to those of the perturbed flows. Depending on asymptotic relations of small parameters α (the difference between the wave number of perturbed flows and the critical wave number for the onset of convection) and ε (ε2 being the overcriticality and the perturbed flow amplitude being O(ε)), either small-angle or Eckhaus instability is prevailing. In the case of small-angle instability for rolls the largest growth rate scales as ε8/5, in agreement with results of Cox and Matthews (Cox, S.M. and Matthews, P.C., Instability of rotating convection. J. Fluid. Mech., 2000, 403, 153–172) obtained for rolls with k = k c . For waves, the largest growth rate is of the order ε4/3. In the case of Eckhaus instability the growth rate is of the order of α2.  相似文献   

2.
Abstract

Numerical simulations of internal gravity waves-turbulence are carried out for the inviscid, viscous and forced-dissipative two-dimensional primitive equations using the spectral method. Some of the results are compared with the predictions of the eddy damped quasi-normal Markovian (EDQNM) closure for internal waves of Carnevale and Frederiksen, generalized for periodic boundary conditions and possible random forcing and dissipation. The EDQNM reduces to the Boltzman equation of resonant interaction theory in the continuum space limit and as the forcing and dissipation vanish. However, the limit is singular in the sense that as well as conserving total energy, E, and total cross-correlation between the vorticity and buoyancy fields, C, an additional conservation law, viz. z-momentum, Pz , occurs in the limit. This means that the resonant interaction equilibrium (RIE) solution of the Boltzmann equation differs from the statistical mechanical equilibrium (SME) solution of the EDQNM closure.

The statistical stability of the SME and RIE spectra for the primitive equations is tested by integrating the inviscid equations using initial realizations of these spectra with random phases. It is found that E and C are accurately conserved while Pz undergoes large amplitude variations. The approach to equilibrium of initial disequilibrium spectra is monitored by examining the evolution of the entropy. The increase and asymptotic approach to a constant value corresponding to complete chaos is consistent with the behaviour predicted by the EDQNM closure.

For the viscous decay and forced-dissipative experiments, the behaviour of the entropy is also consistent with that predicted by the EDQNM closure. There is approximate equipartition of potential and total kinetic energies throughout the integrations from initial conditions having equal potential and total kinetic energies and as well equal vertical and horizontal energies, but as expected, the ratio of horizontal to vertical kinetic energy increases with time to a value greater than unity.

With Laplacian viscous dissipation and thermal diffusivity, the statistical steady states produced in the forced-dissipative experiments have k?3 power laws for k≧7. A comparison with the power laws for kinetic energy and passive scalar variance produced in a numerical simulation of the two-dimensional passive scalar problem is also presented.  相似文献   

3.
Abstract

A linear analysis is used to study the stability of a rapidly rotating, electrically-conducting, self-gravitating fluid sphere of radius r 0, containing a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance from the rotation axis. The Lorentz force is of a magnitude comparable with that of the Coriolis force and so convective motions are fully three-dimensional, filling the entire sphere. We are primarily interested in the limit where the ratio q of the thermal diffusivity κ to the magnetic diffusivity η is much smaller than unity since this is possibly of the greatest geophysical relevance.

Thermal convection sets in when the temperature gradient exceeds some critical value as measured by the modified Rayleigh number Rc. The critical temperature gradient is smallest (Rc reaches a minimum) when the magnetic field strength parameter Λ ? 1. [Rc and Λ are defined in (2.3).] The instability takes the form of a very slow wave with frequency of order κ/r 2 0 and its direction of propagation changes from eastward to westward as Λ increases through Λ c ? 4.

When the fluid is sufficiently stably stratified and when Λ > Λm ? 22 a new mode of instability sets in. It is magnetically driven but requires some stratification before the energy stored in the magnetic field can be released. The instability takes the form of an eastward propagating wave with azimuthal wavenumber m = 1.  相似文献   

4.
Abstract

The first three papers in this series (Fearn, 1983b, 1984, 1985) have investigated the stability of a strong toroidal magnetic field Bo =Bo(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 Bo(s?) Roberts and Loper's analysis was restricted to the case Bo∝s? and they found instability only for m=1 and ?1 <ω<0 [where ω is the frequency non-dimensionalised on the slow timescale τx, 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.  相似文献   

5.
Nonlinear analysis of two-dimensional steady flows with density stratification in the presence of gravity is considered. Inadequacies of Long's model for steady stratified flow over topography are explored. These include occurrence of closed streamline regions and waves propagating upstream. The usual requirements in Long's model of constant dynamic pressure and constant vertical density gradient in the upstream condition are believed to be the cause of these inadequacies. In this article, we consider a relaxation of these requirements, and also provide a systematic framework to accomplish this. As illustrations of this generalized formulation, exact solutions are given for the following two special flow configurations: the stratified flow over a barrier in an infinite channel; the stratified flow due to a line sink in an infinite channel. These solutions exhibit again closed-streamline regions as well as waves propagating upstream. The persistence of these inadequacies in the generalized Long's model appears to indicate that they are not quite consequences of the assumptions of constant dynamic pressure and constant vertical density gradient in Long's model, contrary to previous belief.

On the other hand, solutions admitted by the generalized Long's model show that departures from Long's model become small as the flow becomes more and more supercritical. They provide a nonlinear mechanism for the generation of columnar disturbances upstream of the obstacle and lead in subcritical flows to qualitatively different streamline topological patterns involving saddle points, which may describe the lee-wave-breaking process in subcritical flows and could serve as seats of turbulence in real flows. The occurrences of upstream disturbances in the presence of lee-wave-breaking activity described by the present solution are in accord with the experiments of Long (Long, R.R., “Some aspects of the flow of stratified fluids, Part 3. Continuous density gradients”, Tellus 7, 341--357 (1955)) and Davis (Davis, R.E., “The two-dimensional flow of a stratified fluid over an obstacle”, J. Fluid Mech. 36, 127–143 ()).  相似文献   

6.
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   

7.
Abstract

The stability of a plane parallel shear flow with the profile U(z) = tanh z is considered in a rotating system with the axis of rotation in the z-direction. The establishment of the basic flow requires a baroclinic state, but baroclinic effects are suppressed in the stability analysis by assuming a limit of high thermal conductivity. It is shown that the strongest growing disturbance changes from a purely transverse form in the limit of vanishing rotation rate to a nearly longitudinal form as the angular velocity of rotation increases. An analytical solution of the stability equation is obtained for vanishing growth rates of the transverse form of the instability. But, in general, the solution of the problem requires numerical integrations which demonstrate that the preferred direction of the wave vector of the instability is towards the left of the direction of the mean flow.  相似文献   

8.
Direct atmospheric greenhouse gas emissions can be greatly reduced by CO2 sequestration in deep saline aquifers. One of the most secure and important mechanisms of CO2 trapping over large time scales is solubility trapping. In addition, the CO2 dissolution rate is greatly enhanced if density-driven convective mixing occurs. We present a systematic analysis of the prerequisites for density-driven instability and convective mixing over the broad temperature, pressure, salinity and permeability conditions that are found in geological CO2 storage. The onset of instability (Rayleigh–Darcy number, Ra), the onset time of instability and the steady convective flux are comprehensively calculated using a newly developed analysis tool that accounts for the thermodynamic and salinity dependence on solutally and thermally induced density change, viscosity, molecular and thermal diffusivity. Additionally, the relative influences of field characteristics are analysed through local and global sensitivity analyses. The results help to elucidate the trends of the Ra, onset time of instability and steady convective flux under field conditions. The impacts of storage depth and basin type (geothermal gradient) are also explored and the conditions that favour or hinder enhanced solubility trapping are identified. Contrary to previous studies, we conclude that the geothermal gradient has a non-negligible effect on density-driven instability and convective mixing when considering both direct and indirect thermal effects because cold basin conditions, for instance, render higher Ra compared to warm basin conditions. We also show that the largest Ra is obtained for conditions that correspond to relatively shallow depths, measuring approximately 800 m, indicating that CO2 storage at such depths favours the onset of density-driven instability and reduces onset times. However, shallow depths do not necessarily provide conditions that generate the largest steady convective fluxes; the salinity determines the storage depth at which the largest steady convective fluxes occur. Furthermore, we present a straight-forward and efficient procedure to estimate site-specific solutal Ra that accounts for thermodynamic and salinity dependence.  相似文献   

9.
Abstract

We investigate the evolution of a parallel shear flow which has embedded within it a thin, symmetrically positioned layer of stable density stratification. The primary instability of this flow may deliver either Kelvin-Helmholtz waves or Holmboe waves, depending on the strength of the stratification. In this paper we describe a sequence of numerical simulations which reveal for the first time the behavior of the Holmboe wave at finite amplitude and clarify its structural relationship to the Kelvin-Helmholtz wave.

The flows investigated have initial profiles of horizontal velocity and Brunt-Vaisala frequency given in nondimensional form by U = tanhζ and N 2=J sech2 RCζ, respectively, in which ζ is a nondimensional vertical coordinate, J is the value of the gradient Richardson number N 2/(dU/dζ)2 at ζ=0, and R = 3. Linear stability theory predicts that the flow will develop Holmboe instability when J exceeds some critical value Jc' and Kelvin-Helmholtz instability when J is less than Jc; Jc being approximately equal to 0.25 when R=3. We simulate the evolution of flows with J=0.9, J=0.45, and J = 0.22, and find that the first two simulations yield Holmboe waves while the third yields a Kelvin-Helmholtz wave, as predicted.

The Holmboe wave is a superposition of two oppositely propagating disturbances, a right-going mode whose energy is concentrated in the region above the centre of the shear layer, and a left-going mode whose energy is concentrated below the centre of the shear layer. The horizontal speed of the modes varies periodically, and the variations are most pronounced at low values of J. If J ζ Jc' the minimum horizontal speed of the modes vanishes and the modes become phase-locked, whereupon they roll up to form a Kelvin-Helmholtz wave as predicted by Holmboe (1962). When J is moderately greater than Jc' the Holmboe wave ejects long, thin plumes of fluid into the regions above and below the shear layer, as has often been observed in laboratory experiments, and we examine in detail the mechanism by which this occurs.  相似文献   

10.
Abstract

This paper explores magnetic equilibria which could result from the kink instability in a cylindrical magnetic flux tube. We examine a variety of cylindrical magnetic equilibria which are susceptible to the kink, and simulate its evolution in a frictional fluid. We assume that the evolution takes place under conditions of helical symmetry, so the problem becomes effectively two-dimensional. The initial cylindrical equilibrium field is specified in terms of its twist function k(r) = B θ/(rBz ) and for a variety of k(r) functions we calculate linear growth rates for the kink instability, assuming that it develops under helical symmetry with pitch τ. We find that the growth rate is sensitive to the value of τ.

We simulate nonlinear evolution of the kink using a Lagrangian frictional code which constrains the field to have helical symmetry of a given pitch τ. Ideal MHD is assumed and the plasma pressure is taken to be small in order to mimic conditions in the solar corona. In some cases the flux tube evolves to a new smooth helically symmetric equilibrium which involves a relatively small change in the maximum electric current. In other cases there is evidence of current-sheet formation.  相似文献   

11.
Abstract

We discuss the propagation of internal waves in a rotating stratified unbounded fluid with randomly varying stability frequency, N. The first order smoothing approximation is used to derive the dispersion relation for the mean wave field when N is of the form N 2 = N o 2(1 + ?μ), where μ is a centered stationary random function of either depth (z) or time (t), N o = constant and O < ?2 ≦ 1. Expressions are then derived for the change in phase speed and growth rate due to the random fluctuations μ; in particular, attention is focused on the behaviour of these expressions for short and long correlation lengths (case μ = μ(z)) and times (case μ = μ(t)). For the case μ = μ(z), which represents a model for the temperature and salinity fine-structure in the ocean, the appropriate statistics of the fluctuations observed at station P (50°N, 145°W) have been incorporated into the theory to estimate the actual importance of the effects due to these random fluctuations. It is found that the phase speed of the mean wave decreases significantly if (i) the wavelength is short compared to g/No 2 or (ii) the wave number vector is essentially horizontal and the wave frequency is very close to N o. Also, the random fluctuations cause a significant growth (decay) in the amplitude of a wave propagating upwards (downwards) through a depth of a few kilometers. However, in the direction of energy propagation, the kinetic energy is conserved. Finally, it is shown that the average effect of the depth dependent fluctuations at station P is to slightly decrease the stability frequency and the magnitude of the group velocity.  相似文献   

12.
Abstract

The normal mode instability of harmonic waves in an ideal incompressible fluid on a rotating sphere is analytically studied. By the harmonic wave is meant a Legendrepolynomial flow αPn(μ) (n ≥ 1) and steady Rossby-Haurwitz wave of set F 1 ⊕ Hn where Hn is the subspace of homogeneous spherical polynomials of the degree n(n ≥ 2), and F 1 is the one-dimensional subspace generated by the Legendre-polynomial P1(μ). A necessary condition for the normal mode instability of the harmonic wave is obtained. By this condition, Fjörtoft's (1953) average spectral number of the amplitude of each unstable mode must be equal to . It is noted that flow αPn (μ) is Liapunov (and hence, exponentially and algebraically) stable to all the disturbances whose zonal wavenumber m satisfies condition |m| ≥ n. The bounds of the growth rate of unstable normal modes are estimated as well. It is also shown that the amplitude of each unstable, decaying or non-stationary mode is orthogonal to the harmonic wave.

The new instability condition can be useful in the search of unstable perturbations to a harmonic wave and on trials of numerical stability study algorithms. For a Legendre-polynomial flow, it complements Kuo's (1949) condition in the sense that while the latter is related to the basic flow structure; the former characterizes the structure of a growing perturbation.  相似文献   

13.
Abstract

This paper is concerned with a three-dimensional spherical model of a stationary dynamo that consists of a convective layer with a simple poloidal flow of the S2c 2 kind between a rotating inner body core and solid outer shell. The rotation of the inner core and the outer shell means that there are regions of concentrated shear or differential rotation at the convective layer boundaries. The induction equation for the inside of the convective layer was solved numerically by the Bullard-Gellman method, the eigenvalue of the problem being the magnetic Reynolds number of the poloidal flow (R M2) and it was assumed that the magnetic Reynolds number of the core (R M1) and of the shell (R M3) were prescribed parameters. Hence R M2 was studied as a function of R M1 and R M3, along with the orientation of the rotation axis, the radial dependence of the poloidal velocity and the relative thickness of the layers for the three different situations, (i) the core alone rotating, (ii) the shell alone rotating and (iii) the core and the shell rotating together. In all three cases it was found that, at definite orientations of the rotation axis, there is a good convergence of both the eigenvalues and the eigenfunctions of the problem as the number of spherical harmonics used to represent the problem increases. For R M1 =R M3= 103, corresponding to the westward drift velocity and the parameters of the Earth's core, the critical values of R M2 are found to be three orders of magnitude lower than R M1, R M3 so that the poloidal flow velocity sufficient for maintaining the dynamo process is 10-20 m/yr. With only the core or the shell rotating, the velocity field generally differs little from the axially symmetric case. However, for R M2 (or R M3) lying in the range 102 to 105, the self-excitation condition is found to be of the form R M2˙R ½ M1=constant (or R M2˙R½ M3=constant) and the solution does not possess the properties of the Braginsky near-axisymmetric dynamo. We should expect this, in particular, in the Braginsky limit R M2˙R?½; M1=constant.

An analysis of known three-dimensional dynamo models indicates the importance of the absence of mirror symmetry planes for the efficient generation of magnetic fields.  相似文献   

14.
A control strategy is proposed for variable damping elements (VDEs) used together with auxiliary stiffness elements (ASEs) that compose a time‐varying non‐linear Maxwell (NMW) element, considering near‐future excitation influence. The strategy first composes a state equation for the structural dynamics and the mechanical balance in the NMW elements. Next, it establishes a cost function for estimating future responses by the weighted quadratic norms of the state vector, the controlled force and the VDEs' damping coefficients. Then, the Euler equations for the optimum values are introduced, and also approximated by the first‐order terms under the autoregressive (AR) model of excitation information. Thus, at each moment tk, the strategy conducts the following steps: (1) identify the obtained seismic excitation information to an AR model, and convert it to a state equation; and (2) determine VDEs' damping coefficients under the initial conditions at tk and the final state at tk+L, using the first‐order approximation of the Euler equations. The control effects are examined by numerical experiments. Copyright © 2000 John Wiley & Sons, Ltd.  相似文献   

15.
Whether in the mantle or in magma chambers, convective flows are characterized by large variations of viscosity. We study the influence of the viscosity structure on the development of convective instabilities in a viscous fluid which is cooled from above. The upper and lower boundaries of the fluid are stress-free. A viscosity dependence with depth of the form ν0 + ν1 exp(?γ.z) is assumed. After the temperature of the top boundary is lowered, velocity and temperature perturbations are followed numerically until convective breakdown occurs. Viscosity contrasts of up to 107 and Rayleigh numbers of up to 108 are studied.For intermediate viscosity contrasts (around 103), convective breakdown is characterized by the almost simultaneous appearance of two modes of instability. One involves the whole fluid layer, has a large horizontal wavelength (several times the layer depth) and exhibits plate-like behaviour. The other mode has a much smaller wavelength and develops below a rigid lid. The “whole layer” mode dominates for small viscosity contrasts but is suppressed by viscous dissipation at large viscosity contrasts.For the “rigid lid” mode, we emphasize that it is the form of the viscosity variation which determines the instability. For steep viscosity profiles, convective flow does not penetrate deeply in the viscous region and only weak convection develops. We propose a simple method to define the rigid lid thickness. We are thus able to compute the true depth extent and the effective driving temperature difference of convective flow. Because viscosity contrasts in the convecting region do not exceed 100, simple scaling arguments are sufficient to describe the instability. The critical wavelength is proportional to the thickness of the thermal boundary layer below the rigid lid. Convection occurs when a Rayleigh number defined locally exceeds a critical value of 160–200. Finally, we show that a local Rayleigh number can be computed at any depth in the fluid and that convection develops below depth zr (the rigid lid thickness) such that this number is maximum.The simple similarity laws are applied to the upper mantle beneath oceans and yield estimates of 5 × 1015?5 × 1016 m2 s?1 for viscosity in the thermal boundary layer below the plate.  相似文献   

16.
Abstract

The linear, normal mode instability of barotropic circular vortices with zero circulation is examined in the f-plane quasigeostrophic equations. Equivalents of Rayleigh's and Fjortoft's criteria and the semicircle theorem for parallel shear flow are given, and the energy equation shows the instability to be barotropic. A new result is that the fastest growing perturbation is often an internal instability, having a finite vertical scale, but may also be an external instability, having no vertical structure. For parallel shear flow the fastest growing perturbation is always an external instability; this is Squire's theorem. Whether the fastest growing perturbation is internal or external depends upon the profile: for mean flow streamfunction profiles which monotonically decrease with radius, the instability is internal for less steep profiles with a broad velocity extremum and external for steep profiles with a narrow velocity extremum. Finite amplitude, numerical model calculations show that this linear instability analysis is not valid very far into the finite amplitude range, and that a barotropic vortex, whose fastest growing perturbation is internal, is vertically fragmented by the instability.  相似文献   

17.
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 zc. The height zc , 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 zc. 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 &Gt; zc 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 = zc .  相似文献   

18.
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?ri ) (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.  相似文献   

19.
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   

20.
Abstract

Magnetic instabilities play an important role in the understanding of the dynamics of the Earth's fluid core. In this paper we continue our study of the linear stability of an electrically conducting fluid in a rapidly rotating, rigid, electrically insulating spherical geometry in the presence of a toroidal basic state, comprising magnetic field BMB O(r, θ)1ø and flow UMU O(r, θ)1ø The magnetostrophic approximation is employed to numerically analyse the two classes of instability which are likely to be relevant to the Earth. These are the field gradient (or ideal) instability, which requires strong field gradients and strong fields, and the resistive instability, dependent on finite resistivity and the presence of a zero in the basic state B O(r,θ). Based on a local analysis and numerical results in a cylindrical geometry we have established the existence of the field gradient instability in a spherical geometry for very simple basic states in the first paper of this series. Here, we extend the calculations to more realistic basic states in order to obtain a comprehensive understanding of the field gradient mode. Having achieved this we turn our attention to the resistive instability. Its presence in a spherical model is confirmed by the numerical calculations for a variety of basic states. The purpose of these investigations is not just to find out which basic states can become unstable but also to provide a quantitative measure as to how strong the field must become before instability occurs. The strength of the magnetic field is measured by the Elsasser number; its critical value c describing the state of marginal stability. For the basic states which we have studied we find c 200–1000 for the field gradient mode, whereas for the resistive modes c 50–160. For the field gradient instability, c increases rapidly with the azimuthal wavenumber m whereas in the resistive case there is no such pronounced difference for modes corresponding to different values of m. The above values of c indicate that both types of instability, ideal and resistive, are of relevance to the parameter regime found inside the Earth. For the resistive mode, as is increased from c, we find a shortening lengthscale in the direction along the contour BO = 0. Such an effect was not observable in simpler (for example, cylindrical) models.  相似文献   

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