I. A. Eltayeb

Hydromagnetic convection in a rapidly rotating fluid layer

The linear stability of a rotating, electrically conducting viscous layer, heated from below and cooled from above, and lying in a uniform magnetic field is examined, using the Boussinesq approximation. Several orientations of the magnetic field and rotation axes are considered under a variety of different surface conditions. The analysis is, however, limited to large Taylor numbers, T, and large Hartmann numbers, M. (These are non-dimensional measures of the rotation rate and magnetic field strength, respectively.) Except when field and rotation are both vertical, the most unstable mode at marginal stability has the form of a horizontal roll whose orientation depends in a complex way on the directions and strengths of the field and angular velocity. For example, when the field is horizontal and the rotation is vertical, the roll is directed parallel to the field, provided that the field is sufficiently weak. In this case, the Rayleigh number, R (the non-dimensional measure of the applied temperature contrast) must reach a critical value, Rc, which is O(T2/5) before convection will occur. If, however, the field is sufficiently strong [T = O(M4)], the roll makes an acute angle with the direction of the field, and Rc = O(T1/2), i.e. the critical Rayleigh number is much smaller than when the magnetic field is absent. Also, in this case the mean applied temperature gradient and the wavelength of the tesselated convection pattern are both independent of viscosity when the layer is marginally stable. Furthermore, the Taylor-Proudman theorem and its extension to the hydromagnetic case are no longer applicable even qualitatively. Over the interior of the layer, however, the Coriolis forces to which the convective motions are subjected are, to leading order, balanced by the Lorentz forces. The results obtained in this paper have a bearing on the possibility of a thermally driven steady hydromagnetic dynamo.

Overstable hydromagnetic convection in a rotating fluid layer

The effect of the simultaneous action of a uniform magnetic field and a uniform angular velocity on the linear stability of the Benard layer to time-dependent convective motions is examined in the Boussinesq approximation. Four models, characterized by the relative directions of the magnetic field, angular velocity and gravitational force, are discussed under a variety of boundary conditions. Apart from a few cases, the treatment applies when the Taylor number T and the Chandrasekhar number Q (the square of the Hartmann number) are large. (These parameters are dimensionless measures of angular velocity and magnetic field, respectively.) It is shown that the motions at the onset of instability can be of three types. If the Coriolis forces dominate the Lorentz forces, the results for the rotating non-magnetic case are retained to leading order. If the Coriolis and Lorentz forces are comparable, the minimum temperature gradient required for instability is greatly reduced. Also, in this case, the motions that ensue at marginal stability are necessarily three-dimensional and the Taylor-Proudman theorem and its analogue in hydromagnetics are no longer valid. When the Lorentz forces dominate the Coriolis forces, the results obtained are similar to those for the magnetic non-rotating case at leading order. The most unstable mode is identified for all relations T = KQ α , where K and α are positive constants, taking into account both time-dependent and time-independent motions Various types of boundary layers developing on different boundaries are also examined.

On the stability of the D” layer

Abstract The steady solution for the flow in the D” layer given by Stacey and Loper (1983) is generalized and placed on a firmer mathematical foundation. The stability of this flow is then analyzed and a stability criterion is developed. It is found that the stability of the flow is consistent with a lower-mantle viscosity of 0.5–1.0 1023 Pas and a temperature jump of 700–800K across the layer, but if the viscosity is only 3–5 1021 Pas, stability of the flow requires a much lower temperature jump. If the higher value of viscosity is correct and the flow is believed to be close to marginally stable, this argues against a second thermal boundary layer occurring elsewhere in the mantle.

CRITICAL - LEVEL BEHAVIOUR AND WAVE AMPLIFICATION OF A GRAVITY WAVE INCIDENT UPON A SHEAR LAYER.

The properties of reflexion, refraction and absorption of a gravity wave incident upon a shear layer are investigated. It is shown that one must expect these properties to be very different depending upon the parameters (such as the Richardson number Ri , the wavelength normalized by the length scale of the shear and the ratio of the flow speed to the phase speed of the wave) characterizing the interaction of a gravity wave with a shear layer. In particular, it is shown that for all Richardson numbers there is a discontinuity in the net wave-action flux across the critical level, i.e. at a height where the flow speed matches the horizontal phase speed of the wave. When Ri > ¼, this is accompanied by absorption of part of the energy of the incident wave into the mean flow. In addition it is shown that the phenomenon of wave amplification (over-reflexion) can arise provided that the ultimate shear flow speed exceeds the horizontal phase speed of the wave and Ri is less than a certain critical value Ri c ≃ 0·1129, in which case the reflected wave extracts energy from the streaming motion. It is also pointed out that wave amplification can lead to instability if the boundary conditions are altered in such a way that the system can behave like an ‘amplifier’.

ON LINEAR WAVE MOTIONS IN MAGNETIC-VELOCITY SHEARS

The propagation properties of linear wave motions in magnetic and/or velocity shears which vary in one coordinate z (say) are usually governed by a second order linear ordinary differential equation in the independent variable z. It is proved that associated with any such differential equation there always exists a quantity A which is independent of z. By employing A a measure of the intensity of the wave, this result is used to investigate the general propagation properties of hydromagnetic-gravity waves (e.g. critical level absorption, valve effects and wave amplification) in magnetic and/or velocity shears, using a full wave treatment. When variations in the basic state are included, the governing differential equation usually has more singularities than it has in the W.K.B.J. approximation, which neglects all variations in the background state. The study of a wide variety of models shows that critical level behaviour occurs only at the singularities predicted by the W.K.B.J. approximation. Although the solutions of the differential equation are necessarily singular at the irregularities whose presence is solely due to the inclusion of variations in the basic state, the intensity of the wave (as measured by A) is continuous there. Also the valve effect is found to persist whatever the relation between the wavelength of the wave and the scale of variations of the background state. In addition, it is shown that a hydromagnetic-gravity wave incident upon a finite magnetic and/or velocity shear can be amplified (or over-reflected) in the absence of any critical levels within the shear layer. In a Boussinesq fluid rotating uniformly about the vertical, wave amplification can occur if the horizontal vertically sheared flow and magnetic field are perpendicular. In a compressible isothermal fluid, on the other hand, wave amplification not only occurs in both magnetic-velocity and velocity shears but also in a magnetic shear acting alone.

Compositional convection in the presence of rotation

The stability of a compositionally buoyant plume, of circular cross-section, rising in a rotating infinite fluid is investigated. Both plume and fluid have the same non-zero kinematic viscosity, ν, and thermal diffusivity, κ. The growth rate of the instability depends on the Taylor number, Ta (which is a dimensionless number measuring the effect of the Coriolis force relative to the viscous force) and on the thickness, s 0 , of the plume in addition to the Prandtl number, σ(= ν/κ) and the Reynolds number, R (which measures the strength of the forcing). The analysis is restricted to the case of small R. The presence of rotation enhances instability. A simple model of a single interface separating the two parts of an infinite fluid is investigated first in order to isolate the mechanism responsible for the increase in the growth rate with rotation

On the stability of vertical double-diffusive interfaces. Part 3. Cylindrical interface

This is the final part of a three-part study of the stability of vertically oriented double-diffusive interfaces having an imposed vertical stable temperature gradient. In this study, flow is forced within a fluid of infinite extent by a prescribed excess of compositionally buoyant material within a circular cylindrical interface. Compositional diffusivity is ignored while thermal diffusivity and viscosity are finite. The instability of the interface is determined by quantifying the exponential growth rate of a harmonic deflection of infinitesimal amplitude. Attention is focused on the zonal wavenumber of the fastest growing mode

On the stability of vertical double-diffusive interfaces. Part 2. Two parallel interfaces

This is the second part of a three-part study of the stability of vertically oriented double-diffusive interfaces having an imposed vertical stable temperature gradient. In this study, flow is forced within a fluid of infinite extent by a prescribed excess of compositionally buoyant material between two parallel interfaces. Compositional diffusivity is ignored while thermal diffusivity and viscosity are finite. The stability of the interfaces is analysed first in the limit that they are close together (compared with the salt-finger lengthscale), then for general spacing. Attention is focused on whether the preferred mode of instability is varicose or sinuous and whether its wavevector is vertical or oblique. The interfaces are found to be unstable for some wavenumber for all values of the Prandtl number and interface spacing. The preferred mode of instability for closely spaced interfaces is varicose and vertical for Prandtl number less than about 9, sinuous oblique for Prandtl number between 9 and 15 and sinuous vertical for larger Prandtl number. For general spacing each of the four possible modes of instability is preferred for some range of Prandtl number and interface separation, with no clear pattern of preference, except that the sinuous oblique mode is preferred for widely separated interfaces. The growth rate of the preferred mode is largest for interfaces having separations of from 1 to 3 salt-finger lengths.

The stability of a compositional plume rotating in the presence of a magnetic field

A column of finite thickness containing compositionally buoyant fluid is found to rise in an infinite less buoyant fluid in the presence of uniform rotation and magnetic field. The fluids both within and outside the column have the same finite viscosity, ν, thermal diffusivity, κ and magnetic diffusivity, η. The stability of the column to harmonic disturbances of its interfaces, governed by five dimensionless parameters: the Taylor number, τ2 (which measures the ratio of Coriolis to viscous forces), the Chandrasekhar number, Qc (which measures the ratio of Lorenz to viscous forces), the thickness of the plume, x 0, the Reynolds number, Re (which here measures the strength of the forcing) and Bz and ω H (which measure the inclinations of the ambient magnetic field and rotation vector to the vertical respectively), is studied. The order of the growth rate of the instability in terms of Re is determined by rotation. The column is unstable for all values of the parameters τ, Qc, x 0, Bz , ω H when Re is small. The instability is necessarily three-dimensional. It takes the form of a varicose or sinuous mode propagating at an angle to both field and rotation. The presence of the horizontal component of rotation tends to stabilize the system while that of the vertical field tends to destabilize it. The introduction of a magnetic field inclined to the vertical to an inclined rotation model can reverse the role of the horizontal component of rotation by making it enhance the instability. The dependence of the preference of the varicose and sinuous modes on the parameters of the system is illustrated in regime diagrams. The helicity and α-effect of the unstable motions are discussed briefly.

Time-dependent transport of dust

A diffusion model for the time-dependent transport of dust undergoing dispersion and gravitational settlement is studied analytically. A general analytic expression for the dust concentration above the roughness height is derived. The steady state results obtained by previous authors are easily deduced from this expression. The profiles of the concentration are evaluated numerically and are found to be strongly dependent on time, height above the roughness surface, and falling speed.