Showing papers in "Geophysical and Astrophysical Fluid Dynamics in 1990"

On open boundary conditions for three dimensional primitive equation ocean circulation models

Abstract: An open boundary condition is constructed for three dimensional primitive equation ocean circulation models. The boundary condition utilises dominant balances in the governing equations to assist calculations of variables at the boundary. The boundary condition can be used in two forms. Firstly as a passive one in which there is no forcing at the boundary and phenomena generated within the domain of interest can propagate outwards without distorting the interior. Secondly as an active condition where a model is forced by the boundary condition. Three simple idealised tests are performed to verify the open boundary condition, (1) a passive condition to test the outflow of free Kelvin waves, (2) an active condition during the spin up phase of an ocean, (3) finally an example of the use of the condition in a tropical ocean.

Local turbulence in the Earth's core

Abstract: It is shown that in the Earth's core, where the geodynamo is at work (and is supplied with energy by the prevailing unstable density stratification), a buoyancy instability of a local character exists which is highly supercritical. This instability results in fully developed turbulence dominated by small scale vortices. The influence of the Earth's rotation and of the magnetic field produced by the geodynamo makes this small scale turbulence highly anisotropic. A qualitative picture of this local anisotropic turbulence is devised and the main parameters characterizing it are estimated. Expressions for the turbulent diffusivity are developed and discussed.

Three-dimensional primary instabilities of a stratified, dissipative, parallel flow

Abstract: We demonstrate the existence of a class of dissipative, stratified, parallel shear flows which, as a consequence of linear supercritical instability, evolve directly into three-dimensional flows without the requirement for an intermediate two-dimensional finite-amplitude state. This represents a counter-example to a common misinterpretation of Squire's theorm, namely that the fastest-growing unstable mode of a dissipative parallel shear flow must be two-dimensional.

A shell model for anisotropic magnetohydrodynamic turbulence

Abstract: In this paper, starting from the spectral DIA equations obtained by Veltri et al. (1982), describing the spectral dynamical evolution of magnetohydrodynamic (MHD) turbulence in the presence of a background magnetic field B 0, we have derived an approximate form of these equations (shell model) more appropriate for numerical integration at high Reynolds numbers. We have studied the decay of an initially isotropic state, with an initial imbalance between the energies for the two signs of the cross-helicity. Reynolds numbers up to 105 have been considered. Numerical results show that the nonlinear energy cascade behaves anisotropically in the k-space, i.e. in the spectra there is a prevalence of the wavevectors perpendicular to B 0 with respect to the parallel wavevectors. This anisotropic effect, which is due to the presence of the background magnetic field, can be understood in terms of the so-called ‘‘Alfven effect''. A different source of anisotropy, due to the difference of the energy transfer ...

The tripole : a new coherent vortex structure of incompressible two-dimensional flows

Abstract: Using a contour dynamical algorithm, we have found rotating tripolar V-state solutions for the inviscid Euler equations in two-dimensions. We have studied their geometry as a function of their physical parameters. Their stability was investigated with the aid of contour surgery, and most of the states were found to be stable. Under finite-amplitude perturbations, tripoles are shown to either fission into two asymmetric dipoles or to evolve into a shielded axisymmetric vortex, demonstrating the existence of two new ‘‘reversible transitions'’ between topologically distinct coherent vortex structures. These dynamical results are confirmed by pseudo-spectral simulations, with which we also show how continuous tripolar long-lived coherent vortex structures can be generated in a variety of ways.

Heat transfer and the onset of chaos in a spherical, axisymmetric, anelastic model of whole mantle convection

Abstract: An isochemical uni-phase model of whole mantle convection has been developed in terms of which factors influencing the onset of time dependent chaotic behavior may be assessed. The model is spherical but restricted in generality to the analysis of axisymmetric solutions. In this paper we have employed it to examine the impact of compressibility and sphericity on the nature and onset of time dependence. Particular attention has been given to an examination of the impact that the onset of time dependence has upon the power law relation that connects the heat transfer (represented by the Nusselt number) to the strength of the thermal forcing (represented by the Rayleigh number). In order to obtain these results very extensive numerical simulations were required and the results themselves should be rather useful in the context of models of the thermal history of the planet.

A benchmark comparison of numerical methods for infinite Prandtl number thermal convection in two-dimensional Cartesian geometry

Abstract: A comparison is made between seven different numerical methods for calculating two-dimensional thermal convection in an infinite Prandtl number fluid. Among the seven methods are finite difference and finite element techniques that have been used to model thermal convection in the Earth's mantle. We evaluate the performance of each method using a suite of four benchmark problems, ranging from steady-state convection to intrinsically time-dependent convection with recurring thermal boundary layer instabilities. These results can be used to determine the accuracy of other computational methods, and to assist in the development of new ones.

Solitary waves on conduits of buoyant fluid in a more viscous fluid

Abstract: Fluid of a lower density and viscosity can buoyantly rise through a viscous fluid through conduits that support simple pipe flows. The conduits also support solitary waves which exhibit near soliton behavior. Laboratory experiments on the characteristics of the solitary waves and their interactions have been conducted and compared with theory. The observations of shape and phase speed of individual waves show good agreement with the theoretical predictions. Large amplitude waves traveled slightly faster than the theoretical predictions. The discrepancy is probably due to higher order effects associated with wave slope not accounted for in the theory. Individual wave characteristics (shape, amplitude and speed) were very nearly preserved after collision with another wave. A phase jump of each wave was the main consequence of an interaction. The larger (faster) waves increased in amplitude by an average of 5 percent after collision and their phase speeds decreased by an average of 4 percent. The sm...

Galactic dynamo models without sharp boundaries

Abstract: A new numerical approach is introduced which allows investigation into the conditions for dynamogeneration of axisymmetric and non-axisymmetric large-scale magnetic field modes in galaxy models which are defined by axisymmetric distributions of the α-parameter, the angular velocity and the electrical conductivity The velocity field is assumed to be localized, however, the common assumption of a sharp boundary of the conducting region is dropped The possible anisotropy of the α-tensor is taken into account The critical dynamo numbers (excitation conditions) for different modes are obtained by a direct method The required steady states are attained by the use of an artificial non-linearity Initial test calculations demonstrate the efficacy of this new concept

On the interpretation of Andrews’ theorem

Abstract: Andrews (1984) has shown that any flow satisfying Arnol'd's (1965, 1966) sufficient conditions for stability must be zonally-symmetric if the boundary conditions on the flow are zonally-symmetric. This result appears to place very strong restrictions on the kinds of flows that can be proved to be stable by Arnol'd's theorems. In this paper, Andrews’ theorem is re-examined, paying special attention to the case of an unbounded domain. It is shown that, in that case, Andrews’ theorem generally fails to apply, and Arnol'd-stable flows do exist that are not zonally-symmetric. The example of a circular vortex with a monotonic vorticity profile is a case in point. A proof of the finite-amplitude version of the Rayleigh stability theorem for circular vortices is also established; despite its similarity to the Arnol'd theorems it seems not to have been put on record before.

The magnetic field structures in two nearby spiral galaxies

Abstract: L'analyse des mesures de rotation permettant de determiner la structure a grande echelle du champ magnetique dans le disque des galaxies spirales est utilisee pour deriver l'angle d'attaque et la localisation de la ligne neutre dans le cas d'une configuration bisymetrique du champ magnetique spiral. L'analyse est appliquee aux observations de la galaxie spirale M81. Les observations VLA de l'emission totale et lineairement polarisee de M 81 a λ 20 cm sont presentees

Travelling wave convection in a rotating layer

Abstract: Small amplitude two-dimensional Boussinesq convection in a plane layer with stress-free boundaries rotating uniformly about the vertical is studied. A horizontally unbounded layer is modelled by periodic boundary conditions. When the centrifugal force is balanced by an appropriate pressure gradient the resulting equations are translation invariant, and overstable convection can take the form of travelling waves. In the Prandtl number regime 0.53 < [sgrave] < 0.68 such solutions are preferred over the more usual standing waves. For [sgrave] < 0.53, travelling waves are stable provided the Taylor number is sufficiently large.

On inverse cascades in alpha effect dynamos

Abstract: The nonlinear evolution of alpha effect dynamo instabilities is considered. It is assumed that the small-scale flow is driven by a body force and that MHD equilibrium holds at large scales, which is the case when the large-scale fields depend on a unique Cartesian coordinate. It is shown that when magnetic modes over an extended range of scales are linearly unstable, an inverse cascade develops, mediated by the small-scale velocity and magnetic fields. Beginning with a weak magnetic field, there is a short transient during which the growth is dominated by the linearly most unstable modes: then the magnetic excitation is transferred to larger and larger scales. Eventually the magnetic field saturates in a stationary configuration dominated by the largest available scale.

A unified approach to a class of slow dynamos

Abstract: Dynamo action in a highly conducting fluid with small magnetic diffusivity η is particularly sensitive to the topology of the flow. The sites of rapid magnetic field regeneration, when they occur, appear to be located at the stagnation points or in regions where the particle paths are chaotic. Elsewhere only slow dynamo action is to be expected. Two such examples are the nearly axially symmetric dynamo of Braginsky and the generalisation to smooth velocity fields of the Ponomarenko dynamo. Here a method of solution is developed, which applies to both these examples and is applicable to other situations, where magnetic field lines are close to either closed or spatially periodic contours. Particular attention is given to field generation in the neighbourhood of resonant surfaces where growth rates may be intermediate between the slow diffusive and fast convective time scales. The method is applied to the case of the two-dimensional ABC-flows, where it is shown that such intermediate dynamo action ...

Cyclonic-anticyclonic asymmetry and a new soliton concept for rossby vortices in the laboratory, oceans and the atmospheres of giant planets

Abstract: It is demonstrated in laboratory experiments with rotating shallow water that large scale Rossby vortices, greater than the Rossby-Obukhov radius in size, have dispersive and non-linear properties that are fundamentally different for the two possible polarities. We call this “cyclonic-anticyclonic asymmetry”. This asymmetry manifests itself in the following way: first, anticylones, unlike cyclones, do not undergo the dispersive spreading inherent in a linear wave packet. and therefore, having a considerably longer natural lifetime, are obvious candidates for Rossby solitons; second, dipolar vortices are, because of the comparatively rapid decay of a cyclone, transformed into anticyclonic solitons; third, anticyclones are much more readily generated by zonal flows of the type existing in planetary atmospheres. The evident dominance of anticyclones amongst the long-lived vortices in the atmospheres of giant planets strongly suggests that the cyclonic-anticyclonic symmetry plays a decisive role in t...

Tangential discontinuities and the optical analogy for stationary fields. V, Formal integration of the force-free field equations

Abstract: This paper demonstrates the appearance of tangential discontinuities in deformed force-free fields by direct integration of the field equation ▿ x B = αB. To keep the mathematics tractable the initial field is chosen to be a layer of linear force-free field Bx = + B 0cosqz, By = — B 0sinqz, Bz = 0, anchored at the distant cylindrical surface ϖ = (x 2 + y 2)1/2 = R and deformed by application of a local pressure maximum of scale l centered on the origin x = y = 0. In the limit of large R/l the deformed field remains linear, with α = q[1 + O(l 2/R 2)]. The field equations can be integrated over ϖ = R showing a discontinuity extending along the lines of force crossing the pessure maximum. On the other hand, examination of the continuous solutions to the field equations shows that specification of the normal component on the enclosing boundary ϖ = R completely determines the connectivity throughout the region, in a form unlike the straight across connections of the initial field. The field can escape...

A note on symmetry and stability for fluid flows

Abstract: In this note it is shown that Andrews' theorem, which states that Arnol'd stability implies symmetry, is not special to fluid mechanics, but holds in the general context of classical mechanics with symmetry. More importantly, we emphasise that in Andrews' theorem one should consider stability modulo the symmetry group of solution class. We do this by using the energy-Casimir method to prove the stability of two-dimensional ABC flows, even though they are not symmetric.

Magnetoconvection in rapidly rotating boussinesq and compressible fluids

Abstract: Finite amplitude convection is considered in a plane fluid layer rotating with angular velocity Ω about a horizontal axis in the y-direction. The fluid is electrically conducting, and a uniform horizontal magnetic field of strength B 0 is applied horizontally, in the x-direction, perpendicular to the rotation axis. The gravitational acceleration, g is parallel to Oz. The layer is infinite in y-extent but is bounded laterallyin the y-direction by vertical walls, which may be either stress-free or rigid, i.e. prevent fluid motion. The model is intended to simulate crudely conditions near the equator of the solar convection zone.

Dynamos with a flat α-effect distribution

Abstract: In order to obtain a better insight into the excitation conditions of magnetic fields in flat objects, such as galaxies, we have calculated critical dynamo numbers of different magnetic field modes for spherical dynamos with a flat α-effect distribution. A simple but realistic approximation formula for the rotation curve is employed. In most cases investigated a stationary quadrupole-type solution is preferred. This is a consequence of the flat distribution of the α-effect. Non-axisymmetric fields are in all cases harder to excite than axisymmetric ones. This seems to be the case particularly for flat objects in combination with a realistic rotation curve for galaxies. The question of whether non-axisymmetric (bisymmetric) fields, which are observed in some galaxies, can be explained as dynamos generated by an axisymmetric αω-effect is therefore still open.

Compressible convection in the presence of rotation and a magnetic field

Abstract: The onset of convection in a polytropic atmosphere with rotation and magnetic field is considered in a geometry such that rotation, magnetic field and gravity are mutually perpendicular. The anelastic approximation is used together with the low diffusion magnetogeostrophic approximation. The factors determining the pattern of convection are discussed, with particular reference to the question of whether convection rolls are primarily aligned with the rotation axis or the magnetic field. As in the Boussinesq problem, the global Elsasser number ℬ2/2μρΩη plays a key role. It is shown that the onset of convection in this system cannot be steady, but must occur in the form of travelling waves. In compressible convection the pressure fluctuations contribute to the buoyancy force as well as the temperature fluctuations: in rotating systems these pressure fluctuations are out-of-phase with the temperature fluctuations, and this phase difference gives rise to the travelling waves. Growth rates and frequen...

Resistive instabilities in rapidly rotating fluids: Linear theory of the tearing mode

Abstract: The linear stabilty is examined of a stratified sheet pinch in a rapidly rotating fluid lying hydrostatically in a gravitational field, g, perpendicular to the sheet. The sheet pinch is a horizontal layer of inviscid, Boussinesq fluid of electrical conductivity σ, magnetic permeability μ, and almost uniform density ρ 0 , confined between two perfectly conducting planes z=0,d, where is height. The prevailing magnetic field, B 0 (z), is horizontal; it is unidirectional at each z level, but that direction depends on z. The layer rotates about the vertical with a large angular velocity, Ω:Ω>>V A /d, where V A =B 0 /√(μρ 0 ) is the Alfven velocity. The Elsasser number, Λ=σB 0 2 /2Ωρ 0 , measures σ. A (modified) Rayleigh number, R=gβd 2 /ρ 0 V A 2 , measures the buoyancy force, where β is the imposed density gradient, antiparallel to gravity, g.

Nonlinear dynamics of a stellar dynamo in a spherical shell

Abstract: A simple mean-field model of a nonlinear stellar dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the only nonlinearity retained is the influence of the Lorentz forces on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.'s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.

Modons and monopoles on a γ-plane

Abstract: The gamma plane approximation introduced in this study corresponds to a nonlinear horizontal shallow flow in a plane where, in addition to the familiar linear variation of f (i.e., β), there is a quadratic variation with latitude. Such a plane may have some application to the mesoscale oceanic flow in the immediate vicinity of the North Pole because at the pole the linear gradient (β) vanishes so that the quadratic variation (γ) is the dominant gradient. It is also applicable to the flow near the center of a rotating (laboratory) tank. Exact analytical solutions analogous to the stationary barotropic mid-latitude modons (Stern, 1975) are constructed. First, it is shown that, for a modon situated slightly off the pole (i.e., both β and γ are present) the condition of stationarity (in a resting ocean) takes the form β ∫∫ Ψ dxdy — 2γ ∫∫ yΨ dxdy = 0, where Ψ is the streamfunction and x and y are Cartesian coordinates pointing eastward and northward, respectively. Secondly, it is shown tha...

Maximally-efficient-generation approach in the dynamo theory

Abstract: We propose a method of derivation of global asymptotic solutions of the hydromagnetic dynamo problem at large magnetic Reynolds number. The procedure reduces to matching the local asymptotic forms for the magnetic field generated near individual extrema of generation strength. The basis of the proposed method, named here the Maximally-Efficient-Generation Approach (MEGA), is the assertion that properties of global asymptotic solutions of the kinematic dynamo are determined by the distribution of the generation strength near its leading extrema and by the number and distribution of the extrema. The general method is illustrated by the global asymptotic solution of the α2-dynamo problem in a slab. The nature of oscillatory solutions revealed earlier in numerical simulations and the reasons for the dominance of even magnetic modes in slab geometry are clarified. Applicability of the asymptotic solutions at moderate values of the asymptotic parameter is also discussed. We confirm this applicability u...

Wave hierarchies in alluvial river flows

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.

Tangential discontinuities and the optical analogy for stationary fields IV. High speed fluid sheets

Abstract: It was shown in the previous paper that a sufficiently strong pressure maximum applied to an equilibrium flux surface, by the fields on either side of the surface, produces a gap in the flux surface. The fields on either side make contact through the gap to produce a surface of tangential discontinuity (current sheet). It is shown in the present paper that there is a high speed sheet of fluid and field sliding over the surface of discontinuity when the applied pressure moves slowly across the flux surface. Conditions in the active X-ray corona of the sun suggest that such sheets are generally present, with velocities of the order of 102 km/sec, but with thicknesses too small to be observed. More substantial high speed sheets of fluid may occur in solar flares.

Nonlinear stability for stratified magnetohydrodynamics

Abstract: By extending Arnol'd's Hamiltonian stability techniques to the situation of magnetohydrodynamics, we determine sufficient conditions for nonlinear Lyapunov stability with respect to three dimensional perturbations of axisymmetric steady flows of an electrically conducting, density stratified, Boussinesq, nondissipative fluid. The steady flow is prescribed by an equilibrium condition given by a nonlinear elliptic partial differential equation for the magnetic flux function Y. It is shown that nonlinear stability follows from the positivity of a quadratic form, whose coefficients are given in terms of explicit functions of Y together with certain technical conditions concerning convexity and regularity that are necessary for the application of Arnol'd's techniques. Some specific examples are discussed. The stability criterium for axisymmetric perturbations is exhibited for a simple spherical model for the Earth's fluid core. Explicit Rayleigh type criteria for nonlinear stability are obtained in th...

Spontaneous tangential discontinuities and the optical analogy for static magnetic fields. VI. Topology of current sheets

Abstract: The electric surface current in a tangential discontinuity in a force-free magnetic field is conserved. The direction of the current is halfway between the direction of the continuous fields on either side of the surface of discontinuity. Hence the current sheets, i.e. the surface of tangential discontinuity, have a topology that is distinct from the lines of force of the field. The precise nature of the topology of the current sheet depends upon the form of the winding patterns in the field. Hence, invariant winding patterns and random winding patterns are treated separately. Current sheets may have edges, at the junction of two or more topological separatrices. The current lines may, in special cases, be closed on themselves. The lines of force that lie on either side of a current sheet somewhere pass off the sheet across a junction onto another sheet. In most cases the current sheets extending along a field make an irregular honeycomb. The honeycomb pattern varies along the field if the windin...

Multiple states for quasi-geostrophic channel flows

Abstract: We consider nonlinear baroclinic instabilities of two-layer quasi-geostrophic flow in a rectilinear channel. The full potential vorticity equations are shown to possess a countable infinity of invariant wavenumber sets. Each set is composed of a particular pattern in wavenumber space in which many Fourier modes have zero energy. Solutions with initial conditions confined to a particular wavenumber pattern will remain forever in that pattern. There is also a general asymmetric state with non-zero energy in all wavenumbers. The final state of a long-time evolution calculation depends on initial conditions and internal stability.

Nonlinear galactic dynamos

Abstract: We present trial calculations for a simple nonlinear ‘‘thick disk'’ galaxy dynamo. The nonlinearity is a simple α-quenching. Our strictly axisymmetric solution demonstrates the possibility of a nonlinear interaction between modes of opposite parity. We suggest that a three dimensional model might exhibit a similar persistent interaction between axisymmetric and non-axisymmetric modes.