Hydromagnetic waves in a differentially rotating Annulus I. A test of local stability analysis

Abstract

A cylindrical annulus containing a conducting fluid and rapidly rotating about its axis is a useful model for the Earth's core. With a shear flow U ₀(s)∮, magnetic field B ₀(s)∮, and temperature distribution T _o(s) (where (s, ∮, z) are cylindrical polar coordinates), many important properties of the core can be modelled while a certain degree of mathematical simplicity is maintained. In the limit of rapid rotation and at geophysically interesting field strengths, the effects of viscous diffusion and fluid inertia are neglected. In this paper, the linear stability of the above basic state to instabilities driven by gradients of B ₀ and U ₀ is investigated. The global numerical results show both instabilities predicted by a local analysis due to Acheson (1972, 1973, 1984) as well as a new resistive magnetic instability. For the non-diffusive field gradient instability we looked at both monotonic fields [for which the local stability parameter Δ, defined in (1.4), is a constant] and non-monotonic fields (for which Δ is a function of s). For both cases we found excellent qualitative agreement between the numerical and local results but found the local criterion (1.6) for instability to be slightly too stringent. For the non-monotonic fields, instability is confined approximately to the region which is locally unstable. We also investigated the diffusive buoyancy catalysed instability for monotonic fields and found good quantitative agreement between the numerical results and the local condition (1.9). The new resistive instability was found for fields vanishing (or small) at the outer boundary and it is concentrated in the region of that boundary. The resistive boundary layer plays an important part in this instability so it is not of a type which could be predicted using a local stability analysis (which takes no account of the presence of boundaries).     

Thermal and magnetic instabilities in a rapidly rotating fluid sphere

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 R_c. The critical temperature gradient is smallest (R_c reaches a minimum) when the magnetic field strength parameter Λ ? 1. [R_c 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 > Λ_m ? 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.     

Energy spectra associated with long's model for steady finite-amplitude flow over orography

Abstract

The process of wave steepening in Long's model of steady, two-dimensional stably stratified flow over orography is examined. Under conditions of the long-wave approximation, and constant values of the background static stability and basic flow, Long's equation is cast into the form of a nonlinear advection equation. Spectral properties of this latter equation, which could be useful for the interpretation of data analyses under mountain wave conditions, are presented. The principal features, that apply at the onset of convective instability (density constant with height), are:

i) a power spectrum for available potential energy that exhibits a minus eight-thirds decay, in terms of the vertical wavenumber k _z ^-;

ii) a rate of energy transfer across the spectrum that is inversely proportional to the wavenumber for large k _z ^-;

iii) an equipartition between the kinetic energy of the horizontal motion and the available potential energy, under the longwave approximation, although all the disturbance energy is kinetic at the point where convective instability is initiated. It is also shown that features i) and ii) apply to more general conditions that are appropriate to Long's model, not just the long-wave approximation. Application to fully turbulent flow or to conditions at the onset of shearing instability are not considered to be warranted, since the development only applies to conditions at the onset of convective instability.     

Rotational and magnetic instability in the diffusive tachocline

In this article we study the linear instability of the two-dimensional strongly stratified model for global MHD in the diffusive solar tachocline. Gilman and Fox [Gilman, P.A. and Fox, P., Joint instability of the latitudinal differential rotation and toroidal magnetic fields below the solar convection zone. Astrophys. J., 1997, 484, 439–454] showed that for ideal MHD, the observed surface differential rotation becomes more unstable than is predicted by Watson's [Watson, M., Shear instability of differential rotation in stars. Geophys. Astrophys. Fluid Dyn., 1981, 16, 285–298] nonmagnetic analysis. They showed that the solar differential rotation is unstable for essentially all reasonable values of the differential rotation in the presence of an antisymmetric toroidal field. They found that for the broad field case B _φ～sinθcosθ, θ being the co-latitude, instability occurs only for the azimuthal m?=?1 mode, and concluded that modes which are symmetric (meridional flow in the same direction) about the equator onset at lower field strengths than the antisymmetric modes. We study the effect of viscosity and magnetic diffusivity in the strongly stably stratified case where diffusion is primarily along the level surfaces. We show that antisymmetric modes are now strongly preferred over symmetric modes, and that diffusion can sometimes be destabilising. Even solid body rotation can be destabilised through the action of magnetic field. In addition, we show that when diffusion is present, instability can occur when the longitudinal wavenumber m?=?2.     

Symmetric baroclinic instability for small ekman numbers

Abstract

The stability of a zonal shear flow to symmetric baroclinic perturbations is examined when the Ekman number, E, is asymptotically small. It is assumed, following Antar and Fowlis (1982), that the zonal Row is generated by imposing a constant horizontal temperature gradient γ* at the horizontal boundaries, and by maintaining a constant temperature difference δT* between them. The boundaries are at rest relative to a rotating frame.

Features of the neutral stability curve are determined for several ranges of values of δT/E ^1/3, where δT = δT*/Hγ* and H is the depth of the fluid layer, and all values of the Prandtl number, [sgrave]. In some cases it is possible to determine the whole curve analytically. The most important feature of the results is that the neutral stability curve is closed.

The results are compared to the numerical integrations of Antar and Fowlis (1982). The qualitative features of the solutions are in accord and the quantitative results are, in most cases, as good as can be expected for E only as small as ～ 10^?4. The implications of the results for experimental observations of symmetric baroclinic instability are explored.     

Diffusive destabilisation of the baroclinic circular vortex

Abstract

It is shown that, even for vanishingly small diffusivities of momentum and heat, a rotating stratified zonal shear flow is more unstable to zonally symmetric disturbances than would be indicated by the classical inviscid adiabatic criterion, unless σ, the Prandtl number, = 1. Both monotonic instability, and growing oscillations ("overstability") are involved, the former determining the stability criterion and having the higher growth rates. The more σ differs from 1, the larger the region in parameter space for which the flow is stable by the classical criterion, but actually unstable.

If the baroclinity is sufficiently great for the classical criterion also to indicate instability, the corresponding inviscid adiabatic modes usually have the numerically highest growth rates. An exception is the case of small isotherm slope and small σ.

A single normal mode of the linearized theory is also, formally, a finite amplitude solution; however, no theoretical attempt is made to assess the effect of finite amplitude in general. But, in a following paper, viscous overturning (the mechanism giving rise to the sub‐classical monotonic instability when σ > 1) is shown to play an important role at finite amplitude in certain examples of nonlinear steady thermally‐driven axisymmetric flow of water in a rotating annulus. Irrespective of whether analogous mechanisms turn out to be identifiable and important in large‐scale nature, it appears then that a Prandtl‐type parameter should enter the discussion of any attempt to make laboratory or numerical models of zonally‐symmetric baroclinic geophysical or astrophysical flows.     

Note on the scalar dynamo model

Abstract

Bayly (1993) introduced and investigated the equation (? _t + v·▽-η ▽²)S = RS as a scalar analogue of the magnetic induction equation. Here, S(r, t) is a scalar function and the flow field v(r, t) and “stretching” function R(r, t) are given independently. This equation is much easier to handle than the corresponding vector equation and, although not of much relevance to the (vector) kinematic dynamo problem, it helps to study some features of the fast dynamo problem. In this note the scalar equation is considered for linear flow and a harmonic potential as stretching function. The steady equation separates into one-dimensional equations, which can be completely solved and therefore allow one to monitor the behaviour of the spectrum in the limit of vanishing diffusivity. For more general homogeneous flows a scaling argument is given which ensures fast dynamo action for certain powers of the harmonic potential. Our results stress the singular behaviour of eigenfunctions in the limit of vanishing diffusivity and the importance of stagnation points in the flow for fast dynamo action.     

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.     

Several effects of a baroclinic current on the cross‐stream propagation of inertial‐internal waves†

Several effects of a baroclinic current on inertial‐internal waves at constant frequency are investigated, primarily through use of the method of characteristics. The special case of waves propagating transverse to a baroclinic current is considered. When the slope of an isopycnal is of the same order of magnitude as the slope of the characteristics, appreciable asymmetries are induced in the characteristics, the phase and group velocities, and the solution itself. These asymmetric effects are especially significant for waves at the low frequency end of the passband for free waves. Also, modifications occur to the passband, resulting in anomalously high and low frequency bands. The effective local inertial frequency, σ_f = [f(f+v_x )]^1/2, separates the normal and anomalously low frequency bands. Hence, the low frequency limit of the normal frequency band increases or decreases depending upon whether the horizontal shear in the mean flow is cyclonic or anticyclonic. In the anomalous frequency bands, the slopes of both characteristics have the same sign, causing various refraction and reflection phenomena. If the absolute value of the slope, s, of an isopycnal exceeds its critical value, s_c = effective local inertial frequency/Väisälä‐Brunt frequency, the anomalously low frequency band extends to imaginary frequencies. If s ? 0, the reflection of waves from a boundary is modified, the effective wavelength is increased, and the lines of constant phase are tilted from the vertical. For the general solution, discontinuities in the first‐order partial derivatives of the velocity field occur across certain characteristics. The nonseparable normal modes do not exhibit these discontinuous derivatives, but they only satisfy one of the two pairs of kinematic boundary conditions in rectangular regions.     

On the transition between axisymmetric and non-axisymmetric flow in a rotating liquid annulus subject to a horizontal temperature gradient: Hysteresis effects at moderate Taylor number and baroclinic waves beyond the eady cut-off at high Taylor number

Abstract

The transition between axisymmetric and non-axisymmetric régimes of flow in a rotating annulus of liquid subject to horizontal temperature gradient is known from previous experimental studies to depend largely on two dimensionless parameters. These are Θ, which is proportional to the impressed density contrast Δρ and inversely proportional to the square of the angular speed of rotation ω, and  (Taylor number), which is proportional to ω² /v² where v is the coefficient of kinematic viscosity. At moderate values of , around 10⁷, the critical value of Θ above which axisymmetric flow is found to OCCUT and below which non-axisymmetric fully-developed baroclinic waves (sloping convection) occur, is fairly insensitive to . Though sharp, the transition exhibits marked hysteresis when the upper surface of the liquid is free (but not when the upper surface is in contact with a rigid lid), and it is argued on the basis of the experimental evidence supported by various results of baroclinic instability theory that both the sharpness of the transition and the hysteresis phenomenon are consequences of the combined effects of potential vorticity gradients and viscosity on the process of sloping convection.

We also present some new experiments on fully-developed baroclinic waves, conducted in a large rotating annulus using liquids of very low viscosity (di-ethyl ether), thus attaining values of  as high as 10⁹ to 10¹⁰. The transition from axisymmetric to non-axisymmetric flow is found to lose its sharpness at such high values of , and it is argued that this occurs because viscosity is no longer able to inhibit instabilities at wavelengths less than the so-called ‘Eady short-wave cut-off’, which owe their existence to potential vorticity gradients in the main body of the fluid.     

Long-wave instability of an isolated front

Abstract

The stability of an isolated one-layer reduced gravity front is examined. It is shown that the system is unstable to long-wave disturbances provided merely that a simple condition on the depth profile is satisfied far from the front. The instability does not require the extremum of potential vorticity needed by quasi-geostrophic theory. The instability releases mean kinetic and mean potential energy from the system, but lacking a second layer cannot truly be termed baroclinic instability.     

Nonlinear stability of baroclinic flows over topography

Abstract

A new nonlinear stability criterion is derived for baroclinic flows over topography in spherical geometry. The stability of a wide class of exact three-dimensional nonlinear steady state solutions subject to arbitrary disturbances is established. The resonance condition, at the highest total wavenumber, for the steady state solutions and the stability criteria for baroclinic flow in the absence of topography provide the boundaries of the regions of stability in the presence of topography. The analogous results for flow on periodic or infinite beta planes incorporating non-orthogonal function large scale flows are also discussed.     

Baroclinic instability with variable static stability—a design study for a spherical atmospheric model experiment

Abstract

Exact solutions are obtained for a quasi-geostrophic baroclinic stability problem in which the rotational Froude number (inverse Burger number) is a linear function of the height. The primary motivation for this work was to investigate the effect of a radially-variable, dielectric body force, analogous to gravity, on baroclinic instability for the design of a spherical, synoptic-scale, atmospheric model experiment for a Spacelab flight. Such an experiment cannot be realized in a laboratory on the Earth's surface because the body force cannot be made strong enough to dominate terrestrial gravity. Flow in a rotating, rectilinear channel with a vertically variable body force and with no horizontal shear of the basic state is considered. The horizontal and vertical temperature gradients of the basic and reference states are taken as constants. Consequences of the body force variation and the other assumptions of the model are that the static stability (Brunt-Väisälä frequency squared) and the vertical shear of the basic state flow have the same functional form and that the transverse gradient of the potential vorticity of the basic state vanishes. The solutions show that the stability characteristics of the model are qualitatively similar to those of Eady's model. A short wavelength cutoff and a wavenumber of maximum growth rate are present. Further, the stability characteristics are quantitatively similar to Eady's results for parameters based on the vertically averaged Brunt-Väisälä: frequency. The solutions also show that the temperature amplitude distribution is particularly sensitive to the vertical variation of the static stability. For the static stability and shear decreasing (increasing) with height a relative enhancement of the temperature amplitude occurs at the lower (upper) surface. The other amplitudes and phases are only slightly influenced by the variation. The implication for the Spacelab experiment is that the variable body force will not significantly alter the dynamics from the constant gravity case. The solutions can be relevant to other geophysical fluid flows, including the atmosphere, ocean and annulus system in which the static stability undergoes variation with height.     

The destabilising nature of differential rotation

Abstract

In a rapidly rotating, electrically conducting fluid we investigate the thermal stability of the fluid in the presence of an imposed toroidal magnetic field and an imposed toroidal differential rotation. We choose a magnetic field profile that is stable. The familiar role of differential rotation is a stabilising one. We wish to examine the less well known destabilising effect that it can have. In a plane layer model (for which we are restricted to Roberts number q = 0) with differential rotation, U = sΩ(z)1 _?, no choice of Ω(z) led to a destabilising effect. However, in a cylindrical geometry (for which our model permits all values of q) we found that differential rotations U = sΩ(s)1 _? which include a substantial proportion of negative gradient (dΩ/ds ≤ 0) give a destabilising effect which is largest when the magnetic Reynolds number R _m = O(10); the critical Rayleigh number, Ra _c, is about 7% smaller at minimum than at R_m = 0 for q = 10⁶. We also find that as q is reduced, the destabilising effect is diminished and at q = 10^?6, which may be more appropriate to the Earth's core, the effect causes a dip in the critical Rayleigh number of only about 0.001%. This suggests that we see no dip in the plane layer results because of the q = 0 condition. In the above results, the Elsasser number A = 1 but the effect of differential rotation is also dependent on A. Earlier work has shown a smooth transition from thermal to differential rotation driven instability at high A [A = O(100)]. We find, at intermediate A [A = O(10)], a dip in the Ra_c vs. R_m curve similar to the A = 1 case. However, it has Ra_c ≤ 0 at its minimum and unlike the results for high A, larger values of R_m result in a restabilisation.     

Magnetic instabilities in rapidly rotating spherical geometries. III. The effect of differential rotation

Abstract

We investigate the influence of differential rotation on magnetic instabilities for an electrically conducting fluid in the presence of a toroidal basic state of magnetic field B ₀ = B_MB₀(r, θ)1 _φ and flow U₀ = U_MU₀ (r, θ)1_φ, [(r, θ, φ) are spherical polar coordinates]. The fluid is confined in a rapidly rotating, electrically insulating, rigid spherical container. In the first instance the influence of differential rotation on established magnetic instabilities is studied. These can belong to either the ideal or the resistive class, both of which have been the subject of extensive research in parts I and II of this series. It was found there, that in the absence of differential rotation, ideal modes (driven by gradients of B ₀) become unstable for A_c ? 200 whereas resistive instabilities (generated by magnetic reconnection processes near critical levels, i.e. zeros of B₀) require A_c ? 50. Here, Λ is the Elsasser number, a measure of the magnetic field strength and Λ_c is its critical value at marginal stability. Both types of instability can be stabilised by adding differential rotation into the system. For the resistive modes the exact form of the differential rotation is not important whereas for the ideal modes only a rotation rate which increases outward from the rotation axis has a stabilising effect. We found that in all cases which we investigated Λ_c increased rapidly and the modes disappeared when R_m ≈ O(Λ_C), where the magnetic Reynolds number R_m is a measure of the strength of differential rotation. The main emphasis, however, is on instabilities which are driven by unstable gradients of the differential rotation itself, i.e. an otherwise stable fluid system is destabilised by a suitable differential rotation once the magnetic Reynolds number exceeds a certain critical value (R_m )_c. Earlier work in the cylindrical geometry has shown that the differential rotation can generate an instability if R_m ) ?O(Λ). Those results, obtained for a fixed value of Λ = 100 are extended in two ways: to a spherical geometry and to an analysis of the effect of the magnetic field strength Λ on these modes of instability. Calculations confirm that modes driven by unstable gradients of the differential rotation can exist in a sphere and they are in good agreement with the local analysis and the predictions inferred from the cylindrical geometry. For Λ = O(100), the critical value of the magnetic Reynolds number (R_m )_c Λ 100, depending on the choice of flow U₀ . Modes corresponding to azimuthal wavenumber m = 1 are the most unstable ones. Although the magnetic field B ₀ is itself a stable one, the field strength plays an important role for this instability. For all modes investigated, both for cylindrical and spherical geometries, (R_m )_c reaches a minimum value for 50 ≈ Λ ≈ 100. If Λ is increased, (R_m )_c ∝ Λ, whereas a decrease of Λ leads to a rapid increase of (R_m )_c, i.e. a stabilisation of the system. No instability was found for Λ ≈ 10 — 30. Optimum conditions for instability driven by unstable gradients of the differential rotation are therefore achieved for ≈ Λ 50 — 100, R_m ? 100. These values lead to the conclusion that the instabilities can play an important role in the dynamics of the Earth's core.     

A Note on Standing Internal Inertial Gravity Waves of Finite Amplitude

The effects of finite amplitude are examined in two-dimensional, standing, internal gravity waves in a rectangular container which rotates about a vertical axis at frequency f/ 2. Expressions are given for the velocity components, density fluctuations and isopycnal displacements to second order in the wave steepness in fluids with buoyancy frequency, N , of general form, and the effect of finite amplitude on wave frequency is given in an expansion to third order. The first order solutions, and the solutions to second order in the absence of rotation, are shown to conserve energy during a wave cycle. Analytical solutions are found to second order for the first two modes in a deep fluid with N proportional to sech( az ), where z is the upward vertical coordinate and a is scaling factor. In the absence of rotation, results for the first mode in the latter stratification are found to be consistent with those for interfacial waves. An analytical solution to fourth order in a fluid with constant N is given and used to examine the effects of rotation on the development of static instability or of conditions in which shear instability may occur. As in progressive internal waves, an effect of rotation is to enhance the possibility of shear instability for waves with frequencies close to f . The analysis points to a significant difference between the dynamics of standing waves in containers of limited size and progressive internal waves in an unlimited fluid; the effect of boundaries on standing waves may inhibit the onset of instability. A possible application of the analysis is to transverse oscillations in long, narrow, steep-sided lakes such as Loch Ness, Scotland.     

A class of analytic solutions of the magnetic force-free field equations

Abstract

Formation of electric current sheets in the corona is thought to play an important role in solar flares, prominences and coronal heating. It is therefore of great interest to identify magnetic field geometries whose evolution leads to variations in B over small length-scales. This paper considers a uniform field B ₀[zcirc], line-tied to rigid plates z = ±l, which are then subject to in-plane displacements modeling the effect of photospheric motion. The force-free field equations are formulated in terms of field-line displacements, and when the imposed plate motion is a linear function of position, these reduce to a 4 × 4 system of nonlinear, second-order ordinary differential equations. Simple analytic solutions are derived for the cases of plate rotation and shear, which both tend to form singularities in certain parameter limits. In the case of plate shear there are two solution branches—a simple example of non-uniqueness.     

A three‐dimensional numerical investigation of the idealized planetary boundary layer

Abstract

The nonlinear equations of motion are integrated numerically in time for a region of x‐y‐z space of volume 3h × h × h, where h turns out to be a height slightly above the level where the wind first attains the geostrophic flow direction. Only the ideal case is treated of a horizontal lower boundary, neutral stability, horizontal homogeneity of all dependent mean variables except the mean pressure, and statistically steady state. The resulting flow patterns are turbulent and the eddies transport required amounts of momentum vertically.

Topics which are investigated include the relative directions of stress, wind shear and wind; differences in Ekman wind spirals for the neutral numerical case and a stable atmospheric case; profiles of dimensionless turbulence statistics; effect of allowing the mean density to be either constant or to decrease with height; effect of the wind direction or latitude upon the turbulence intensities; and characteristic structure of the eddies in the planetary boundary layer.     

Thermal convection in differentially rotating systems

Abstract

The onset of convection in a cylindrical fluid annulus is analyzed in the case when the cylindrical walls are rotating differentially, a temperature gradient in the radial direction is applied, and the centrifugal force dominates over gravity. The small gap approximation is used and no-slip conditions on the cylindrical walls are assumed. It is found that over a considerable range of the parameter space either convection rolls aligned with the axis of rotation or rolls in the perpendicular (azimuthal) direction are preferred. It is shown that by a suitable redefinition of parameters, results for finite amplitude Taylor vortices and for convection rolls in the presence of shear can be applied to the present problem. Weakly nonlinear results for transverse rolls in a Couette flow indicate the possibility of subcritical bifurcation for Prandtl numbers P less than 0.82. Heat and momentum transports are derived as functions of P and the problem of interaction between transverse and longitudinal rolls is considered. The relevance of the analysis for problems of convection in planetary and stellar atmospheres is briefly discussed.     

Experiments on instability of columnar vortex pairs in rotating fluid

We present results from a new series of experiments on the geophysically important issue of the instability of anticyclonic columnar vortices in a rotating fluid in circumstances such that the Rossby number exceeds unity. The vortex pair consisting of a cyclonic and an anticyclonic vortex is induced by a rotating flap in a fluid which is itself initially in a state of solid-body rotation. The anticyclonic vortex is then subject to either centrifugal or elliptical instability, depending on whether its initial ellipticity is small or large, while the cyclone always remains stable. The experimental results demonstrate that the perturbations due to centrifugal instability have a typical form of toroidal vortices of alternating sign (rib vortices). The perturbations due to elliptical instability are of the form of sinuous deformation of the vortex filament in the plane of maximal stretching which corresponds to the plane of symmetry for the vortex pair. The initial perturbations in both cases are characterized by a definite wave number in the vertical direction. The characteristics of the unstable anticyclone are determined by the main nondimensional parameter of the flow - the Rossby number. The appearance of both centrifugal and elliptical instabilities are in accord with the predictions of theoretical criteria for these cases.