Effect of pressure gradients and line-tying on the magnetic stability of coronal loops

Abstract

In this paper we study the stability of an idealised magnetostatic coronal loop, incorporating both the effect of line-tying, due to the dense photosphere, and of pressure gradients. The stability equations may be solved analytically for our particular equilibrium. From the marginally stable case, the critical conditions separating instability from stability are derived. It is found that stretching or twisting a loop eventually makes it kink unstable.     

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.     

On the stability of plane flow with horizontal shear to three-Dimensional nondivergent disturbances

Abstract

Arnold's (1965a) method is used to investigate the stability of a stationary, nonparallel, plane flow, with horizontal shear, to three-dimensional nondivergent disturbances in a Boussinesq fluid. It is shown that, if the fluid is statically stable, the Rayleigh condition is not sufficient to insure inertial stability to all disturbance modes. For channel flow it is possible to establish the sufficiency condition for stability to some of these modes.     

On inviscid stratified parallel flows varying in two directions

Abstract

A stratified parallel flow in a potential force field is investigated. The density, the velocity, and the potential field of the flow are allowed to vary in two directions. Three sufficient conditions are derived for guaranteed stability of the flow. Two are the classical stability conditions in their respective directions. The third, measured by a newly defined Richardson number, is a result of the shear interaction and the pressure balance condition for stability. Like the classical Richardson number which is always positive preceding stability, this new number acts as a constraint on the other two stability conditions. In addition to the above stability criteria, a semi-ellipse theorem is derived for the present flow.     

Non-linear hydrodynamic stability of oceanic flows

Abstract

In this paper an analytical method to study the hydrodynamic stability of simple barotropic, non-divergent flows is discussed. The method is based on the variational approach introduced by Arnold and derived from the Lyapunov stability criteria. In this context, the sufficient condition for the stability of a steady barotropic flow ψ(x,y) is obtained when dP(ψ)/dP_{ψ = ψ}, the derivative of the absolute vorticity P(ψ), is positive definite. In this case, we discuss the effect of higher derivatives dⁿP(ψ)/dψⁿψ_{ψ = ψ} on the non-linear stability. Then we show that some classical examples of oceanic non-divergent flows (i.e. lee waves downstream an Island, steady flows through a Strait, the Fofonoff gyre) are stable to finite-amplitude perturbations.     

The effect of gravity on the stability of a line-tied coronal magnetohydrostatic equilibrium

Abstract

The magnetohydrodynamic stability of a class of magnetohydrostatic equilibria is investigated. The effect of gravity is included as well as the stabilising influence of the dense photospheric line-tying.

Although the two-dimensional equilibria exhibit a catastrophe point, when the ratio of plasma pressure to magnetic pressure exceeds a critical value, arcade structures, with both footpoints connected to the photosphere, become unstable to three-dimensional disturbances before the catastrophe point is reached.

Numerical results for field lines that are open into the solar corona suggest that they are completely stable. Although there is no definite proof of stability, this would allow the point of non-equilibrium to be reached.     

MHD stability of a filament immersed in a solar bipolar magnetic region

Abstract

The stability properties of the filament model of Low (1981) are investigated. We use a two-dimensional formalism based on the magnetohydrodynamic energy principle of Bernstein et al. (1958). For the parameter range observed in quiescent prominences this model describes stable horizontal oscillations with periods of about 3–6 min. In other parameter ranges we find instability which is driven exclusively by compressional effects. The Lorentz force has a continuously stabilizing influence. In addition, it seems that gravity is practically unimportant for the stability state of the equilibrium.     

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.     

Shear flow instabilities in a rotating system

Abstract

The stability of a plane parallel shear flow with the profile U(z) = tanh z is considered in a rotating system with the axis of rotation in the z-direction. The establishment of the basic flow requires a baroclinic state, but baroclinic effects are suppressed in the stability analysis by assuming a limit of high thermal conductivity. It is shown that the strongest growing disturbance changes from a purely transverse form in the limit of vanishing rotation rate to a nearly longitudinal form as the angular velocity of rotation increases. An analytical solution of the stability equation is obtained for vanishing growth rates of the transverse form of the instability. But, in general, the solution of the problem requires numerical integrations which demonstrate that the preferred direction of the wave vector of the instability is towards the left of the direction of the mean flow.     

A note on a paper by derome and dolph

Abstract

Some symmetries of the stability characteristies of a baroclinic zonal current are found, the basic state being that of the non-geostrophic Eady problem. In particular, they lead to a principle of exchange of stabilities.     

A note on the 1967 paper of braginsky

Abstract

Sufficient conditions for stability are established for the magnetic field problem of the Earth's core considered by Braginsky.     

F) Organisations Nationales National Organizations

Abstract

It is suggested that the apparent change in height of the horizon over a lake will be a useful measure of the atmospheric stability, i.e. of the difference in temperature of the water and the air over it.     

The convective stability of a thin viscous disc

Abstract

We prove that the presence of viscosity does not affect stability to axisymmetric convective modes of a thin differentially rotating disc with no thermal conduction but in which viscosity is taken fully into account. In such a case the Schwarzschild criterion is necessary and sufficient for convective stability to local perturbations. In the proof we use a general formulation of local stability analysis, which allows a rigorous demonstration. Restricted particular forms of the viscous stress tensor introduced in the modelling of thin accretion discs may lead to viscous overstabilities. The additional instability found by Elstner et al. (1989) and described by the authors as a correction to the Schwarzschild criterion is a manifestation of these. However, when viscosity is taken fully into account, such instabilities cannot be discussed within the framework of a local analysis, a fully global treatment being required.     

On the growth of disturbances to forced and dissipated barotropic flows

Abstract

We consider the growth of disturbances to large-scale zonally-asymmetric steady states in a truncated spectral model for forced and dissipated barotropic flow. A variant of the energy method is developed to optimize the instantaneous disturbance energy growth rate. The method involves solving a matrix eigenvalue problem amenable to standard numerical techniques. Two applications are discussed. (1) The global stability of a family of steady states is assessed in terms of the Ekman damping coefficient r. It is shown that monotonic global stability (i.e., every disturbances energy monotonically decays to zero) prevails when r≥r_c . (2) Initially fastest-growing disturbances are constructed in the r<r_c regime. Particular attention is paid to a subregion of the r<r_c regime where initially-growing disturbances exist despite stability with respect to normal modes. Nonlinear time-dependent simulations are performed in order to appraise the time evolution of various disturbances.     

A nonlinear stability analysis of convection in a generalized incompressible fluid

Abstract

We establish a nonlinear stability result for convection in a generalized incompressible fluid. Both numerical calculations and an asymptotic analysis are carried out. The linear and nonlinear results are shown to be very close in both cases, implying that the region of possible subcritical instabilities is very small.

During this work I was supported by a research studentship awarded by the Science and Engineering Council of the United Kingdom.     

Some general results for normal mode instability on a sphere

Abstract

The stability properties are described for two general types of zonal mean flows: solid body rotation and a mid-latitude jet. Growth rates are plotted versus zonal wavenumber and mean flow vertical shear in both cases. The structure of the most unstable modes is described and some physical interpretation given.     

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.     

Stability and stable modes of coastal fronts

Abstract

The stability of a single layer, geostrophic front of zero potential vorticity bounded by a vertical coast (wall) is investigated by means of a Rayleigh integral. It is proved that the flow of the density-driven current is stable at all wavenumbers provided the mean velocity of basic flow exceeds fL (where f is the Coriolis parameter and L is the distance between the wall and the free streamline). The frequency of the stable long waves is either zero or super-inertial.     

Resistive instabilities in rapidly rotating fluids: Linear theory of the convective modes

Abstract

This paper analyzes the linear stability of a rapidly-rotating, stratified sheet pinch in a gravitational field, g, perpendicular to the sheet. The sheet pinch is a layer (O ? z ? d) of inviscid, Boussinesq fluid of electrical conductivity σ, magnetic permeability μ, and almost uniform density ρ _o; z is height. The prevailing magnetic field. B _o(z), is horizontal at each z level, but varies in direction with z. The angular velocity, Ω, is vertical and large (Ω ? V_A/d, where V_A = B₀√(μρ₀) is the Alfvén velocity). The Elsasser number, Λ = σB² ₀/2Ωρ₀, measures σ. A (modified) Rayleigh number, R = gβd²/ρ₀V² _A, measures the buoyancy force, where β is the imposed density gradient, antiparallel to g. A Prandtl number, P_K = μσK, measures the diffusivity, k, of density differences.     

Sensitivity of the stability of river flow regimes to small fluctuations in temperature

Abstract

A river regime describes the average seasonal behaviour of flow. This seasonal pattern reflects climatic and physiographic conditions in the basin. An inherent characteristic of a flow regime is its stability, i.e. regularity or irregularity of the seasonal pattern. A temperature rise, as predicted by climatic models, might cause changes in the patterns and stability of river flow regimes. Sensitivity of the stability of flow regimes to small fluctuations in temperature (= ± 1°C) is investigated with the help of historical temperature and flow series for Scandinavia. The concept of entropy is utilized for quantification of the stability of the flow regimes conditioned on temperature which also allows forecasting of possible changes in this stability due to changes in temperature. The study shows that the stability of flow regime types with rain or mixed rain and snowmelt sources of flow formation is already sensitive to small changes in temperature, especially concerning flow minima.