The stability of dissipative magnetohydrodynamic shear flow in a parallel magnetic field

Abstract

Strong decay bounds are obtained for linearized perturbations to an unbounded, plane Couette flow in a parallel magnetic field. Finite conductivity and molecular viscosity are found to be stabilizing. Those modes decaying most slowly have the form of rolls aligned with the shear flow. The non-aligned rolls decay at an enhanced rate. Stability bounds at finite amplitude are obtained for flows bounded in one direction using energy methods.     

Hydromagnetic waves in rapidly rotating spherical shells generated by poloidal decay modes

Abstract

This paper presents the first attempt to examine the stability of a poloidal magnetic field in a rapidly rotating spherical shell of electrically conducting fluid. We find that a steady axisymmetric poloidal magnetic field loses its stability to a non-axisymmetric perturbation when the Elsasser number A based on the maximum strength of the field exceeds a value about 20. Comparing this with observed fields, we find that, for any reasonable estimates of the appropriate parameters in planetary interiors, our theory predicts that all planetary poloidal fields are stable, with the possible exception of Jupiter. The present study therefore provides strong support for the physical relevance of magnetic stability analysis to planetary dynamos. We find that the fluid motions driven by magnetic instabilities are characterized by a nearly two-dimensional columnar structure attempting to satisfy the Proudman-Taylor theorm. This suggests that the most rapidly growing perturbation arranges itself in such a way that the geostrophic condition is satisfied to leading order. A particularly interesting feature is that, for the most unstable mode, contours of the non-axisymmetric azimuthal flow are closely aligned with the basic axisymmetric poloidal magnetic field lines. As a result, the amplitude of the azimuthal component of the instability is smaller than or comparable with that of the poloidal component, in contrast with the instabilities generated by toroidal decay modes (Zhang and Fearn, 1994). It is shown, by examining the same system with and without fluid inertia, that fluid inertia plays a secondary role when the magnetic Taylor number T_m ? 10⁵. We find that the direction of propagation of hydromagnetic waves driven by the instability is influenced strongly by the size of the inner core.     

Influence of condensation and latent heat release upon barotropic and baroclinic instabilities of vortices in a rotating shallow water f-plane model

Analysis of the influence of condensation and related latent heat release upon developing barotropic and baroclinic instabilities of large-scale low Rossby-number shielded vortices on the f-plane is performed within the moist-convective rotating shallow water model, in its barotropic (one-layer) and baroclinic (two-layer) versions. Numerical simulations with a high-resolution well-balanced finite-volume code, using a relaxation parameterisation for condensation, are made. Evolution of the instability in four different environments, with humidity (i) behaving as passive scalar, (ii) subject to condensation beyond a saturation threshold, (iii) subject to condensation and evaporation, with three different parameterisations of the latter, are inter-compared. The simulations are initialised with unstable modes determined from the detailed linear stability analysis in the “dry” version of the model. In a configuration corresponding to low-level mid-latitude atmospheric vortices, it is shown that the known scenario of evolution of barotropically unstable vortices, consisting in formation of a pair of dipoles (dipolar breakdown) is substantially modified by condensation and related moist convection, especially in the presence of surface evaporation. No enhancement of the instability due to precipitation was detected in this case. Cyclone-anticyclone asymmetry with respect to sensitivity to the moist effects is evidenced. It is shown that inertia-gravity wave emission during the vortex evolution is enhanced by the moist effects. In the baroclinic configuration corresponding to idealised cut-off lows in the atmosphere, it is shown that the azimuthal structure of the leading unstable mode is sensitive to the details of stratification. Scenarios of evolution are completely different for different azimuthal structures, one leading to dipolar breaking, and another to tripole formation. The effects of moisture considerably enhance the perturbations in the lower layer, especially in the tripole formation scenario.     

Linear theory of rotating fluids using spherical harmonics part I: Steady flows

Abstract

It is shown that a systematic development of physical quantities using spherical harmonics provides analytical solutions to a whole class of linear problems of rotating fluids.

These solutions are regular throughout the whole domain of the fluid and are not much affected by the equatorial singularity of steady boundary layers in spherical geometries.

A comparison between this method and the one based on boundary layer theory is carried out in the case of the steady spin-up of a fluid inside a sphere.     

Penetrative convection in rotating fluids: A model for the base of stellar convection zones

Abstract

To model penetrative convection at the base of a stellar convection zone we consider two plane parallel, co-rotating Boussinesq layers coupled at their fluid interface. The system is such that the upper layer is unstable to convection while the lower is stable. Following the method of Kondo and Unno (1982, 1983) we calculate critical Rayleigh numbers R_c for a wide class of parameters. Here, R_c is typically much less than in the case of a single layer, although the scaling R_c～T^2/3 as T → ∞ still holds, where T is the usual Taylor number. With parameters relevant to the Sun the helicity profile is discontinuous at the interface, and dominated by a large peak in a thin boundary layer beneath the convecting region. In reality the distribution is continuous, but the sharp transition associated with a rapid decline in the effective viscosity in the overshoot region is approximated by a discontinuity here. This source of helicity and its relation to an alpha effect in a mean-field dynamo is especially relevant since it is a generally held view that the overshoot region is the location of magnetic field generation in the Sun.     

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.     

Barotropic unstable modes in zonal and meridional channel on the beta-plane

Abstract

The linear stability of a non-divergent barotropic parallel shear flow in a zonal and a non-zonal channel on the β plane was examined numerically. When the channel is non-zonal, the governing equation is slightly modified from the Orr-Sommerfeld equation. Numerical solutions were obtained by solving the discretized linear perturbation equation as an eigenvalue problem of a matrix. When the channel is zonal and lateral viscosity is neglected the problem is reduced to the ordinary barotropic instability problem described by Kuo's (1949) equation. The discrepancy between the stability properties of westward and eastward flows, which have been indicated by earlier studies, was reconfirmed. It has also been suggested that the unstable modes are closely related to the continuous modes discretized by a finite differential approximation. When the channel is non-zonal, the properties of unstable modes were quite different from those of the zonal problem in that: (1) The phase speed of the unstable modes can exceed the maximum value of the basic flow speed; (2) The unstable modes are not accompanied by their conjugate mode; and (3) The basic flow without an inflection point can be unstable. The dispersion relation and the spatial structure of the unstable modes suggested that, irrespective of the orientation of the channel, they have close relation to the neutral modes (Rossby channel modes) which are the solutions in the absence of a basic shear flow. The features mentioned above are not dependent on whether or not the flow velocity at the boundary is zero.     

An experimental study of global instabilities due to the tidal (elliptical) distortion of a rotating elastic cylinder

Abstract

Broad band secondary instability of elliptical vortex motion has been proposed as a principal source of shear-flow turbulence. Here experiments on such instability in an elliptical flow with no shear boundary layer are described. This is made possible by the mechanical distortion in the laboratory frame of a rotating fluid-filled elastic cylinder. One percent ellipticity of a 10 cm diameter cylinder rotating once each second can give rise to an exponentially-growing mode stationary in the laboratory frame. In first order this mode is a sub-harmonic parametric Faraday instability. The finite-amplitude equations represent angular momentum transfer on an inertial time scale due to Reynolds stresses. The growth of this mode is not limited by boundary friction but by detuning and centrifugal stabilization. On average, a generalized Richardson number achieves a marginal value through much of the evolved flow. However, the characteristic flow is intermittent with the cycle: rapid growth, stabilizing momentum transfer from the mean flow, interior re-spin up, and then again. Data is presented in which, at large Reynolds numbers, seven percent ellipticity causes a fifty percent reduction in the kinetic energy of the rotating fluid. In the geophysical setting, this tidal instability in the earth's interior could be inhibited by sub-adiabatic temperature gradients. A near adiabatic region greater than 10 km in height would permit the growth of tidally destabilized modes and the release of energy to three-dimensional disturbances. Such disturbances might play a central role in the geodynamo and add significantly to overall tidal dissipation.     

On the onset of convection in rotating spherical shells

Abstract

The linear problem of the onset of convection in rotating spherical shells is analysed numerically in dependence on the Prandtl number. The radius ratio η=r _i/r _o of the inner and outer radii is generally assumed to be 0.4. But other values of η are also considered. The goal of the analysis has been the clarification of the transition between modes drifting in the retrograde azimuthal direction in the low Taylor number regime and modes traveling in the prograde direction at high Taylor numbers. It is shown that for a given value m of the azimuthal wavenumber a single mode describes the onset of convection of fluids of moderate or high Prandtl number. At low Prandtl numbers, however, three different modes for a given m may describe the onset of convection in dependence on the Taylor number. The characteristic properties of the modes are described and the singularities leading to the separation with decreasing Prandtl number are elucidated. Related results for the problem of finite amplitude convection are also reported.     

Jet instability and the stabilizing effect of topography on jets in two-layer rotating systems

Abstract

Laboratory experiments concerning azimuthal jets in two-layer rotating systems in the absence and presence of bottom topography aligned along the jets have been conducted. The jets were forced by the selective withdrawal of fluid from the upper layer of a two-fluid system contained in a circular dishpan geometry. The principal parameters measured in the experiments were the jet Rossby number, Ro, and a stratification parameter F = r ₁/(λ₁λ₂)^1/2 where r ₁ is the radius of the circular disc used for the selective withdrawal (i.e., r ₁ is the approximate radius of curvature of the jet) and λ₁,λ₂ are the internal Rossby radii of deformation in the upper and lower fluids, respectively.

The no-topography experiments show that for a sufficiently small F, the particular value depending on Ro, the jet is stable for the duration of the experiment. For sufficiently large F, again as a function of Ro, the jet becomes unstable, exhibiting horizontal wave disturbances from modes three to seven. An Ro against F flow regime diagram is presented.

Experiments are then conducted in the presence of a bottom topography having constant cross-section and extending around a mid-radius of the dishpan. The axis of the topography is in the vicinity of the jet axis forced in the no-topography experiments and the crest of the topography is in the vicinity of the interface between the two fluids (i.e., the front associated with the jet). The experiments show that in all cases investigated the jet tends to be stabilized by the bottom topography. Experiments with the topography in place, but with the interface between the fluids being above the topography crest, are shown to be unstable but more irregular than their no-topography counterparts.

Various quantitative measurements of the jet are presented. It is shown, for example, that the jet Rossby number defined in terms of the fluid withdrawal rate from the tank. Q, can be well correlated with a dimensionless vorticity gradient, V_G , across the upper layer jet. This allows for an assessment of the stability characteristics of a jet based on a knowledge of V_G (which can be estimated given a jet profile) and F.     

Three-dimensional solutions of the magnetohydrostatic equations for rigidly rotating magnetospheres in cylindrical coordinates

We present new analytical three-dimensional solutions of the magnetohydrostatic equations, which are applicable to the co-rotating frame of reference outside a rigidly rotating cylindrical body, and have potential applications to planetary magnetospheres and stellar coronae. We consider the case with centrifugal force only, and use a transformation method in which the governing equation for the “pseudo-potential” (from which the magnetic field can be calculated) becomes the Laplace partial differential equation. The new solutions extend the set of previously found solutions to those of a “fractional multipole” nature, and offer wider possibilities for modelling than before. We consider some special cases, and present example solutions.     

On the completeness of inertial wave modes in rotating annular channels

An important unanswered mathematical question in the theory of rotating fluids has been the completeness of the inviscid eigenfunctions which are usually referred to as inertial waves or inertial modes. We provide for the first time a mathematical proof for the completeness of the inertial modes in a rotating annular channel by establishing the completeness relation, or Parseval’s equality, for any piecewise continuous, differentiable velocity of an incompressible fluid.     

Instability of flows near the onset of convection in a rotating layer with stress-free horizontal boundaries

We investigate instability of convective flows of simple structure (rolls, standing and travelling waves) in a rotating layer with stress-free horizontal boundaries near the onset of convection. We show that the flows are always unstable to perturbations, which are linear combinations of large-scale modes and short-scale modes, whose wave numbers are close to those of the perturbed flows. Depending on asymptotic relations of small parameters α (the difference between the wave number of perturbed flows and the critical wave number for the onset of convection) and ε (ε² being the overcriticality and the perturbed flow amplitude being O(ε)), either small-angle or Eckhaus instability is prevailing. In the case of small-angle instability for rolls the largest growth rate scales as ε^8/5, in agreement with results of Cox and Matthews (Cox, S.M. and Matthews, P.C., Instability of rotating convection. J. Fluid. Mech., 2000, 403, 153–172) obtained for rolls with k = k _c . For waves, the largest growth rate is of the order ε^4/3. In the case of Eckhaus instability the growth rate is of the order of α².     

Asymptotic solutions of differential rotation driven by convection in rapidly rotating fluid spheres with the non-slip boundary condition

A strong Coriolis force in a rapidly rotating planet and star not only enforces two-dimensionality of fluid motion driven by thermal instabilities but also generates strong differential rotation even in the vicinity of the onset of instabilities. We derive an asymptotic solution describing convection-driven differential rotation in rotating, self-gravitating Boussinesq fluid spheres with the no-slip boundary condition, taking into account full spherical curvature and being valid for asymptotically small Ekman numbers. For the purpose of validating the asymptotic solution, the corresponding numerical analysis valid for large or small Ekman numbers is also carried out, showing a satisfactory agreement between the asymptotic and numerical solutions.     

应用裂变径迹不同模式约束岩体冷却史的初步探讨—以滇西独龙江岩体为例

在利用磷灰石裂变径迹热年代学方法来约束岩体冷却史的应用中,由于地质条件和磷灰石退火性质的限制,表观年龄往往不能直接代表特定地质事件的时间.利用封闭径迹的长度直方图模式和高程-年龄模式可定性地反映出岩体冷却史是否遭受过干扰.平均径迹长度-年龄(或香蕉图)模式、径迹年龄谱模式以及反演模拟在某种程度上可用来限定地质过程中冷却启动的时间.然而,对于多阶段的冷却史,模式和模拟分析的不确定性仍较显著,对早期时限的定量揭示仍是研究和应用中的一个难点.只有结合多种方法、模式和模拟的运用并考虑地质背景才能较清晰地约束岩体的冷却历史.     

Coherent modes in multi‐scale variability of precipitation over the headwater catchments in the Pearl River basin,South China

This paper studies the coherent modes of multi‐scale variability of precipitation over the headwater catchments in the Pearl River basin in South China. Long‐term (1952–2000) daily precipitation data spatially averaged for 16 catchments in the basin are studied. Wavelet transform analysis is performed to capture the fluctuation embedded in the time series at different temporal timescales ranging from 6 days to 8.4 years. The catchment clusters of the coherent modes are delineated using the principal component analysis on the wavelet spectra of precipitation. The results suggest that as much as 98% of the precipitation variability is explained by only two coherent modes: high small‐scale mode and high seasonal mode. The results also indicate that a large majority of the catchments (i.e., 15 out of 16) exhibit consistent mode feature on multi‐scale variability throughout three sub‐periods studied (1952–1968, 1969–1984, and 1985–2000). The underlying effects of the coherent modes on the regional flood and drought tendency are also discussed.     

Role of barotropic mechanism in the development of a monsoon depression: A MONEX study

The role of barotropic processes in the development of a monsoon depression, formed on 5 July 1979 during MONEX observational period, is studied by considering it as a quasi-geostrophic divergent barotropic instability problem of zonal flow of 3 July 1979 at 700 mb level. Numerical solutions are obtained by initial value approach. The preferred wave has a wavelength of 2750 km, an e-folding time of 4.3 days, a period of 6.5 days and an eastward phase speed of 4.9 ms^–1. Structure of preferred wave is found to be in good agreement with the observed horizontal structure of the depression at 700 mb. Poleward momentum transports are found to predominate over equatorward transports.Parts of this paper were presented at the National Symposium on Early Results of MONEX-1979. 9–12 March 1981, in New Delhi, India.     

Hot springs mediate spatial exchange of heat and mass in the enclosed sediment domain: A stability perspective

In many natural environments, such as in underwater hot springs and hydrothermal vents, thermal gradients are accompanied with changes in the concentration of chemical compounds transported to the seawater, causing the so-called double-diffusive, mixed convection. To study the physical scenarios in such systems, a vertical channel filled with a porous medium saturated with saline water is considered. The motion in the sediment-filled channel is induced by two buoyancy forces and an external pressure gradient, similar to the situation in a vent with an upward flow direction. The fluid flow has been modeled by an extended Darcy model, and the flow instability mechanisms have been studied numerically. The linear stability analysis is performed considering a wide range of Darcy number (Da = 10⁻⁵ -10⁻⁸). The instability boundary curve showed three distinct dynamic regimes: (i) Rayleigh-Taylor (R-T), (ii) log-log non-linear variation, and (iii) log-log linear variation. The domain of different regimes were sensitive to external pressure gradient as well as permeability. Similar to cross-diffusive natural convection in pure viscous fluids, a linear relationship between logarithmic absolute values of critical thermal Rayleigh number (∣Ra_T∣) and solute Rayleigh number (Ra_C) is found in the third regime. Based on the permeability, for any solute Rayleigh number (Ra_C), there existed a minimum value of Reynolds number (Re), below which R-T type of instability appeared. Above this minimum value, the instability was due to two buoyancy forces, known as buoyant instability. Simulations of secondary flow via energy analysis demonstrated the development of complex dynamics at the critical state in all three regimes characterized by transition of multi to uni-cellular structures and vice verse.