Showing papers in "Geophysical and Astrophysical Fluid Dynamics in 1985"

Numerical simulations of stellar convective dynamos III. At the base of the convection zone

Abstract: We describe numerical simulations of giant-cell solar convection and magnetic field generation. Nonlinear, three-dimensional, time-dependent solutions of the anelastic magnetohydrodynamic equations are presented for a stratified, rotating, spherical shell of ionized gas. The velocity, magnetic field, and thermodynamic variables are solved simultaneously and self-consistently with full nonlinear feedback. Convection, driven in the outer part of this shell by a superadiabatic gradient, penetrates into the inner, subadiabatic part. Previous dynamic dynamo sjmulations have demonstrated that, when the dynamo operates in the convection zone, the magnetic fields propagate away from the equator in the opposite direction inferred from the solar butterfly diagram. Our simulations suggest that the solar dynamo may be operating at the base of the convection zone in the transition region between the stable interior and the turbulent convective region. There our simulated angular velocity decreases with depth, as it does in the convection zone; but the simulated helicity has the opposite sign compared to its convection zone value. As a result, our simulated magnetic fields in this transition region initially propagated toward the equator. However, due to our limited numerical resolution of the small amplitude helical fluid motions in this dense, stable region, only the initial phase propagation could be simulated, not a complete magnetic cycle.

A mathematical model of magma transport in the asthenosphere

Abstract: We present a mathematical model for the flow of a partial melt through its solid phase. The model is based on the conservation laws of two-phase flow, which reduce to a generalization of porous flow in a permeable medium, when the solid matrix deforms very slowly. The continuity equation for the melt contains a source term (due to melting), which is determined by the energy equation. In addition, the melt fraction is unknown, and a new equation, representing conservation of pore space, is introduced. This equation may also be thought of as a constitutive law for the melt pressure (which is not lithostatic). The model is non-dimensionalized and simplified. Some simple solutions are considered, and it is suggested that the occurrence of high fluid pressures in the solutions may initiate fractures in the lithosphere, thus providing a starting-up mechanism for magma ascent to the surface.

Steady flows at the top of the core from geomagnetic field models: The steady motions theorem

Abstract: It is demonstrated that the steady tangential velocity at the closed surface of a perfect-fluid conductor bounded by a rigid impenetrable exterior can be uniquely determined from knowledge of the normal component of the time-varying magnetic-flux density on the surface. In the context of a simple earth model consisting of an electrically insulating mantle surrounding a perfectly conducting core, the assumption of steady flow provides enough extra information to eliminate the toroidal ambiguity and to allow derivation of a unique global flow at the top of the core from a model of the geomagnetic field.

The dynamics of triple convection

Abstract: In the parameter space of a fluid subject to triple convection, there is a critical hypersurface on which three growth rates of linear theory vanish and all the rest are distinctly negative. When parameter values are chosen to place the system very near to this polycritical condition, the temporal behavior of the system may be complicated and even chaotic. This remark, based on rather general considerations (Arneodo et al., 1984), is here illustrated by an example from GFD (Arneodo et al., 1982): two-dimensional Boussinesq thermohaline convection (or semi-convection) in a planeparallel layer rotating about a vertical axis and subject to mathematically convenient boundary conditions. The treatment is made in terms that show why the results may apply to many fluid dynamical systems or indeed to other kinds of triply unstable systems and, using both amplitude equations and mappings, we discuss the chaos that can arise.

Laboratory observations of secondary structures in kelvin-helmhoitz billows and consequences for ocean mixing

Abstract: Experiments have been made using shadowgraphs to examine the development of secondary structures in Kelvin-Helmhoitz billows at the diffuse interface between two layers of different densities moving in shear at moderate Reynolds numbers and high Prandtl number. The onset of turbulence in billows reported in earlier work resulted from an interaction between the billows and the side walls of the apparatus. Secondary structure within the billows remote from the side walls occurs later and is, in its early stages, well organised. Regular longitudinal bands lying parallel to the mean flow develop near the vertical boundaries of the billows and extend across their widths. The initial development and scale of the spanwise bands are similar to that of the convective rolls predicted to occur in billows by Klaassen and Peltier (1985a) using a numerical model. No longitudinal instability is observed to occur at the same time in the braids between the billows. Fine scale “turbulence” occurs in the billows ab...

A depth-dependent study of the topographic rectification of tidal currents

Abstract: A depth-dependent model for the topographic rectification of tidal currents in a homogeneous rotating fluid is used to examine the dependence of the rectified mean flow on various tidal, topographic and frictional parameters. Friction is parameterized through a vertically-uniform, time-independent vertical eddy viscosity and a bottom stress law applied near the top of the constant stress layer. The model neglects the interaction of mean and tidal currents, assumes uniformity along isobaths, and is closed with the assumption of zero depth-averaged mean flow across isobaths. In the limit of depth-independence, the model reduces to that considered by Huthnance (1973) and Loder (1980) which, for weak friction, favours anticyclonic mean circulation around shallow regions and Lagrangian flow which is significantly reduced from the Eulerian. With the inclusion of vertical structure, the magnitude of the anticyclonic flow is amplified suggesting that depth-independent models may underestimate the along-i...

The effect of Prandtl number on the evolution and stability of Kelvin-Helmholtz billows

Abstract: A two-dimensional numerical model is employed to calculate the nonlinear evolution of Kelvin-Helmholtz billows in fluids with various Prandtl numbers. We present a detailed analysis of the energy b...

Topological invariants of field lines rooted to planes

Abstract: Two open curves with fixed endpoints on a boundary surface can be topologically linked. However, the Gauss linkage integral applies only to closed curves and cannot measure their linkage. Here we employ the concept of relative helicity in order to define a linkage for open curves. For a magnetic field consisting of closed field lines, the magnetic helicity integral can be expressed as the sum of Gauss linkage integrals over pairs of lines. Relative helicity extends the helicity integral to volumes where field lines may cross the boundary surface. By analogy, linkages can be defined for open lines by requiring that their sum equal the relative helicity. With this definition, the linkage of two lines which extend between two parallel planes simply equals the number of turns the lines take about each other. We obtain this result by first defining a gauge-invariant, one-dimensional helicity density, i.e. the relative helicity of an infinitesimally thin plane slab. This quantity has a physical interpr...

Equilibrium of magnetic fields with arbitrary interweaving of the lines of force. I - Discontinuities in the torsion

Abstract: Consideration is given to the static force-free equilibrium of a magnetic field in which all of the lines of force connect without knotting between parallel planes. The field is formed by continuous deformation from an initial uniform field, and is conventiently described in terms of the scalar function psi, which is the stream function for the incompressible wrapping and interweaving of the lines of force. Local compression and expansion of the lines of force is described in terms of the scalar function Phi. Equilibrium in the field requires satisfaction of two independent equations which cannot be accomplished without the full freedom of both psi and Phi. It is shown that discontinuities in the torsional characteristics of the lines occur when psi is predetermined by an arbitrary pattern. Discontinuities in the winding pattern of the lines can lead to discontinuities in the associated current sheets.

Magnetic buoyancy instabilities for a static plane layer

Abstract: Recent calculations suggest that the bulk of the solar magnetic field may be stored in a thin convectively stable region situated between the convection zone proper and the radiative zone. Determining the stability properties of such a field is therefore important with implications for both the generation and escape of magnetic flux. The magneto-Boussinesq equations are used to perform a linear stability analysis of a static plane layer. Several new instability mechanisms are revealed showing previous ideas concerning magnetic buoyancy instabilities to be over simplified. The most important result is that instability may occur even for fields which increase with height. Detailed results are presented for 2 and 3 dimensional motions for a weakly stratified magnetic field together with a simple calculation for the 2 dimensional instability of a strongly varying field.

Some interactions between small numbers of baroclinic, geostrophic vortices

Abstract: Various interactions between small numbers (two and four) of baroclinic, geostrophic point vortices in a two-layer system are studied with attention to the qualitative changes in behavior which occur as size of the deformation radius is varied. A particularly interesting interaction, which illustrates the richness of baroclinic vortex dynamics, is a collision between two hetons. (A heton is a vortex pair in which the constituent vortices have opposite signs and are in opposite layers. The “breadth” of a heton is the distance between its constituent vortices. A translating heton transports heat.) When two hetons, which initially have different breadths, collide, the result is either an exchange of partners, or a “slip-through” collision in which the initial structures are preserved. It is shown here that the outcome is always an exchange, provided the deformation radius is sufficiently small. This strongly contrasts with a collision between pairs of classical, one-layer vortices in which no exchan...

Convection driven by centrifugal bouyancy in a rotating annulus

Abstract: Drift rates and amplitudes of convection columns driven by centrifugal bouyancy in a cylindrical fluid annulus rotating about a vertical axis have been measured by thermistor probes. Conical top and bottom boundaries of the annular fluid region are responsible for the prograde Rossby wave like dynamics of the convection columns. A constant positive temperature difference between the outer and the inner cylindrical boundaries is generated by the circulation of thermostatically controled water. Mercury and water have been used as converting fluids. The measurements extend the earlier visual observations of Busse and Carrigan (1974) and provide quantitative data for an eventual comparison with nonlinear theories of thermal Rossby waves. The measured drift frequencies are in general agreement with linear theory. Of particular interest is the decline of the amplitude of convection with increasing Rayleigh number in a region beyond the onset of convection.

Characteristics of amplitude vacillation in a differentially heated rotating fluid annulus

Abstract: Experimental results are presented on the characteristics and occurrence of amplitude vacillation observed in the regular wave regime of a differentially heated rotating fluid annulus. The amplitude vacillation appears as the periodic oscillation of the amplitude and frequency of the dominant wave component and its harmonics and occurs, in general, adjacent to a transition to the next lowest wavenumber. The frequency spectra of the amplitude vacillations are characterized by the presence of two distinct frequencies, one from the vacillation and one from the drift of the wave; in general the amplitude vacillations observed can be said to be doubly periodic. The wave interference mechanism of amplitude vacillation is examined but no evidence was found which could be unambiguously interpreted in its favour.

On vertical hydromagnetic waves in a compressible atmosphere under an oblique magnetic field

Abstract: We consider vertical magneto-acoustic-gravity waves in an atmosphere under a constant magnetic field, and show that (Figure 1), in addition to the Alfven wave, there exists a hydromagnetic-gravity wave, which is of fourth-order in the general case of an oblique magnetic field, and reduces to a second-order acoustic/magnetosonicgravity wave respectively for a vertical/horizontal magnetic field. A method is developed to solve exactly linear wave equations with certain types of variable (e.g., exponential) coefficients, describing waves of any order and arbitrary frequency in strongly stratified media. The method relies on the transformation of the wave equation into a standard type [equation (11a)], which on inspection is seen to have general properties such as: (i) cut-off frequencies and related wave components (Figure 2, Table I); (ii) the asymptotic amplitude and phase laws holding at high-altitudes (Table II); (iii) the existence and location of critical levels (Figure 3, Table III). The metho...

Hydromagnetic waves in a differentially rotating annulus. III - The effect of an axial field

Abstract: The first two papers in this series (Fearn, 1983b, 1984) are concerned with the linear stability of the toroidal magnetic field B o(s) (where (s, , z) are cylindrical polar coordinates). The field is confined in a cylindrical annulus and the whole system is rapidly rotating about the axis of symmetry. The present study extends the analysis by adding a uniform axial field Bx o. In a non-rotating system this would have a stabilising influence on ideal instabilities. In the rotating system, a small axial field counteracts the stabilising effect of rotation and so has a destabilising influence. For cases of geophysical interest, larger axial fields are stabilising. Resistive instabilities behave in much the same way as their counterparts in a non-rotating system. The essential requirement for instability is k·B o=0 (where k is the wave vector). The addition of an axial field may therefore be stabilising or destabilising.

Subgrid scale stress models for the large-eddy simulation of rotating turbulent flows

Abstract: The modeling of the subgrid scale stresses is considered from a theoretical standpoint with a view toward developing models that are more suitable for the large-eddy simulation of rotating turbulent flows. It is proven, as a rigorous consequence of the Navier-Stokes equations, that such models must be generally invariant under the extended Galilean group and must be frame-indifferent in the limit of two-dimensional turbulence which can be approached in a rapidly rotating framework. Furthermore, it is shown that a significant increase in the rotation rate must be accompanied by a substantial reduction in the energy dissipation rate of the turbulence. Vorticity subgrid scale stress models as well as several other commonly used models are shown to be in serious violation of one or more of these constraints and, hence, are not generally suitable for the description of rotating flows. Alternative models with the correct physical properties are discussed and compared.

Elliptica topographic waves

Abstract: Simple analytical solutions are presented for nondivergent topographic waves (rotational modes) in a certain type of elliptical basin. Under the assumption that the basin's depth contours form a family of confocal ellipses, the governing potential vorticity equation in elliptic cylindrical coordinates reduces to a Cartesian form, independent of the coordinate scale factors. As a consequence, for the exponential depth profile h=e −bξ, where ξ is the radial coordinate, the radial eigenfunctions for elliptically travelling waves in a basin with a partial vertical barrier along the centerline can be expressed in terms of elementary functions. For a lake without a barrier, approximate analytical solutions are obtained by the Rayleigh-Ritz (variational) method. The periods and streamline patterns of the first few modes of the variational solutions are compared with those due to Ball (1965) for an elliptic paraboloid. The gravest mode period of one of the variational solutions has also been computed for...

Instability of a geostrophic front and its energetics

Abstract: The instability of a current with a geostrophic surface density front is investigated by means of a reduced gravity model having a velocity profile with nearly uniform potential vorticity. It is shown that currents are unstable when the mean potential vorticity decreases toward the surface front at the critical point of the frontal trapped waves investigated by Paldor (1983). This instability is identical with that demonstrated by Killworth (1983) in the longwave limit. The cross-stream component of mass flux and the rates of energy conversions among the five energy forms defined by Orlanski (1968) are also calculated. The main results are as follows, (a) The mass flux toward the surface front is positive near the front and negative around the critical point. The positive mass flux near the front does not vanish at the position of the undisturbed surface front, so that the mean position of the front moves outward and the region of the strong current spreads. (b) The potential energy of the mean f...

Convection in an internally heated high Prandtl number fluid: a laboratory study

Abstract: The stability regimes for convection in an internally heated, high Prandtl number, glycerin-based solution have been investigated for the first time by inducing particular planforms under controlled conditions. Down hexagons were found to be stable to about 40 Rac having wavenumbers which are in reasonable agreement with both linear and finite amplitude calculations. Beyond this value of the Rayleigh number, the cells were observed to undergo a transition from flow down to flow up at the center. Most of these new cells are not completely closed and bear some similarity to distorted rolls. They were also found to be time dependent and their horizontal scale did not differ substantially from that associated with the hexagonal planform at lower Rayleigh numbers. However, at about 125 Rac , the onset of two-scale flow was observed—a phenomenon which may be associated with the finite thermal conductivity of the upper boundary.

Finite amplitude instability thresholds in penetrative convection

Abstract: A nonlinear energy stability analysis is presented for the penetrative convection model of Veronis (1963). For top temperatures between 4°C and 8°C the nonlinear stability boundary obtained is very close to the linear one of Veronis and enables a region of possible sub-critical instabilities to be determined.

Magnetic buoyancy instabilities incorporating rotation

Abstract: Plane layer stability analysis is extended to incorporate the effects of uniform rotation. Detailed studies are made of the interchange, or 'axisymmetric modes' and of the undular or wavelike motions, considering both high and low frequency modes. The force due to rotation on the fluid motions are determined. The types of instability relevant to astrophysical problems, are also discussed with attention given to the occurrence of two distinct modes of instability in a bottom heavy field gradient. Some peculiar stability boundaries, which are due to the absence of a dominating force, are described.

On the coupling of fluid dynamics and electromagnetism at the top of the earth's core

Abstract: A kinematic approach to short-term geomagnetism has recently been based upon pre-Maxwell frozen-flux electromagnetism. A complete dynamic theory requires coupling fluid dynamics to electromagnetism. A geophysically plausible simplifying assumption for the vertical vorticity balance, namely that the vertical Lorentz torque is negligible, is introduced and its consequences are developed. The simplified coupled magnetohydrodynamic system is shown to conserve a variety of magnetic and vorticity flux integrals. These provide constraints on eligible models for the geomagnetic main field, its secular variation, and the horizontal fluid motions at the top of the core, and so permit a number of tests of the underlying assumptions.

Macrodynamics of α2 dynamos

Abstract: Two distributions of the α-effect in a sphere are considered. The inviscid limit is approached both by direct numerical solution and by solution of a simpler nonlinear eigenvalue problem deriving from asymptotic boundary layer analysis for the case of stress-free boundaries. The inviscid limit in both cases is dominated by the need to satisfy the Taylor constraint which states that the integral of the Lorentz force over cylindrical (geostrophic) contours in a homogeneous fluid must tend to zero. For a small supercritical range in α, this condition can only be met by magnetic fields which vanish as the viscosity goes to zero. In this range, the agreement of the two approaches is excellent. In a portion of this range, the method of finite amplitude perturbation expansion is useful, and serves as a guide for understanding the numerical results. For larger α, evidence from the nonlinear eigenvalue problem suggests both that the Taylor state exists, and that the transition from small to large amplitud...

Magnetic-flux transport by a convecting layer including dynamical effects

Abstract: The work of Drobyshevski and Yuferev (1974) and Arter (1983b) is continued and extended. Magnetic flux distributions are calculated for Rayleigh-Bknard convection with square planform. The importance of the local structure of the flow-field near stagnation points is emphasised. When the Lorentz force is included motion often takes the form of rolls, but if three-dimensional flow persists the field distribution is essentially kinematic. Implications for dynamo and sunspot theories are discussed. These calculations do not support the use of the flux-tube as conceived by Parker (1979) to model horizontal magnetic field, although this may be due to their idealized nature.

Linear and weakly nonlinear internal wave theories applied to “morning glory” waves

Abstract: Detailed comparisons are made between the predictions of Benjamin's weakly nonlinear theory for internal solitary waves in fluids of great depth, with observational data on solitary wave-type disturbances in the lower atmosphere associated with the “morning glory” phenomenon. It is shown that, while the theory is not wholly unreasonable, neither is it completely satisfactory. In particular, although the calculated wave speeds are generally close to those observed, they are no improvement on those based on linear long wave theory; at the same time the predicted wave half-widths are too large by a factor of two to three. The limitations of the theory appear to be associated with the requirement that wave half-widths are much less than the total fluid depth, a condition not satisfied in the atmospheric case. However, the alternative theory for shallow fluids, based on the Korteweg-de Vries equation is found to be even more unsuitable. Our analyses highlight some of the problems in comparing theory w...

On the derivation of the energy flux of linear magnetohydrodynamic waves

Abstract: New light is shed on the derivation of the energy flux of the linear MHD waves. It is shown that, according to a suggestion of Lighthill, the usual perturbation procedure, which starts from the general expression for the energy flux, need not be supplemented by an averaging procedure. As a result, it is shown that to second order in the wave amplitude, a quantity identifiable as the wave energy flux is conserved. Some of the subtleties inherent in the derivation of the pertubation energy equation are discussed.

Application of the MHD energy principle to magnetostatic atmospheres

Abstract: The MHD energy principle is applied to the stability of a magnetized atmosphere which is bounded below by much denser fluid, as is the solar corona. The two fluids are treated as ideal; the approximation is consistent with the energy principle, and the dynamical conditions that must hold at a fluid-fluid interface are used to show that if vertical displacements of the lower boundary are permitted, then the lower atmosphere must be perturbed as well. However, displacements which do not perturb the coronal boundary can be properly treated as isolated perturbations of the corona alone.

Laboratory experiments in a baroclinic annulus with heating and cooling on the horizontal boundaries

Abstract: Experiments have been performed in a cylindrical annulus with horizontal temperature gradients imposed upon the horizontal boundaries and in which the vertical depth was smaller than the width of the annulus. Qualitative observations were made by the use of small, suspended, reflective flakes in the liquid (water). Four basic regimes of flow were observed: (1) axisymmetric flow, (2) deep cellular convection, (3) boundary layer convective rolls, and (4) baroclinic waves. In some cases there was a mix of baroclinic and convective instabilities present. As a 'mean' interior Richardson number was decreased from a value greater than unity to one less than zero, axisymmetric baroclinic instability of the Solberg type was never observed. Rather, the transition was from non-axisymmetric baroclinic waves, to a mix of baroclinic and convective instability, to irregular cellular convection.

Stability of the subseismic wave equation for the Earth's fluid core

Abstract: The effects of compressibility on the stability of internal oscillations in the Earth's fluid core are examined in the context of the subseismic approximation for the equations of motion describing a rotating, stratified, self-gravitating, compressible fluid in a thick shell. It is shown that in the case of a bounded fluid the results are closely analogous to those derived under the Boussinesq approximation.

Instability and flow over topography

Abstract: The stability of quasi-geostrophic β-plane flow over topography is examined. The approach is to first calculate the stationary, asymmetric response to a uniform zonal current flowing over topography, and then calculate the stability properties of the total, zonally asymmetric, field. Under many circumstances this flow is unstable, barotropically and or baroclinically particularly if the asymmetric flow is of large amplitude. This is demonstrated first using a long-wave approximation, which examines the stability with respect to perturbations of large meridional scale. So-called form-drag instability then ensues. This may be thought of as a special, nonlocal, form of isosceles triad interaction involving the zonal flow interacting with the topography and another “free” mode of topographic scale. For topography consisting of a single Fourier mode, instability then arises only if the zonal current is eastward and exceeds that required for resonance. However, in general other triads exist in which th...