Semicircle Theorem Research Articles

Yang-Lee Zeros, Semicircle Theorem, and Nonunitary Criticality in Bardeen-Cooper-Schrieffer Superconductivity

Yang-Lee Zeros, Semicircle Theorem, and Nonunitary Criticality in Bardeen-Cooper-Schrieffer Superconductivity

Semicircle theorem for streak instability

Howard’s semicircle theorem about the location of eigenvalues for the stability of one-dimensional shear flows is extended to two-dimensional shear flows. The instability of the latter is one of the main ingredients of a self-sustaining nonlinear 3D process that underlies a multitude of non-trivial coherent solutions of the Navier–Stokes equations for shear flows.

Shear flow instabilities in shallow-water magnetohydrodynamics

Within the framework of shallow-water magnetohydrodynamics, we investigate the linear instability of horizontal shear flows, influenced by an aligned magnetic field and stratification. Various classical instability results, such as Høiland’s growth-rate bound and Howard’s semi-circle theorem, are extended to this shallow-water system for quite general flow and field profiles. In the limit of long-wavelength disturbances, a generalisation of the asymptotic analysis of Drazin & Howard (J. Fluid Mech., vol. 14, 1962, pp. 257–283) is performed, establishing that flows can be distinguished as either shear layers or jets. These possess contrasting instabilities, which are shown to be analogous to those of certain piecewise-constant velocity profiles (the vortex sheet and the rectangular jet). In both cases it is found that the magnetic field and stratification (as measured by the Froude number) are generally each stabilising, but weak instabilities can be found at arbitrarily large Froude number. With this distinction between shear layers and jets in mind, the results are extended numerically to finite wavenumber for two particular flows: the hyperbolic-tangent shear layer and the Bickley jet. For the shear layer, the instability mechanism is interpreted in terms of counter-propagating Rossby waves, thereby allowing an explication of the stabilising effects of the magnetic field and stratification. For the jet, the competition between even and odd modes is discussed, together with the existence at large Froude number of multiple modes of instability.

Linear stability of the Couette flow of a vibrationally excited gas. 1. Inviscid problem

Stability of the Couette flow of a vibrationally excited diatomic gas with a parabolic profile of static temperature is studied within the framework of the linear theory. A set of explicit asymptotic estimates are obtained for inviscid disturbances described by a system of linearized equations of two-temperature gas dynamics. It is shown that the first Rayleigh condition (theorem) is satisfied for unstable modes, and the classification of inviscid modes into even and odd modes is valid. A generalized condition of the presence of an inflection point on the velocity profile, which is necessary for disturbances to evolve, is obtained. The sufficient condition in Howard’s semi-circle theorem is refined. Complex phase velocities of two-dimensional even and odd inviscid modes are numerically calculated as functions of the Mach number, degree of excitation of vibrational levels of energy, and characteristic relaxation time. In the Couette flow problem, in contrast to the case of a free shear layer, the growth rate of the most unstable second mode increases with increasing Mach number and tends to a certain limit for which an asymptotic expression in the form of an ordinary differential equation is obtained. The calculated results show that the effect of reduction of the growth rate on the background of the relaxation process is clearly expressed in the range of flow parameters considered.

A Semi-Circle Theorem in Couple-Stress Fluid in the Presence of Rotation in a Porous Medium

The thermal instability of a couple-stress fluid acted upon by uniform vertical rotation and heated from below in a porous medium is investigated. Following the linearized stability theory and normal mode analysis, the paper through mathematical analysis of the governing equations of couple-stress fluid convection with a uniform vertical rotation in porous medium, for the case of rigid boundaries shows that the complex growth rate σ of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle , in the right half of a complex σ-plane, where TA is the Taylor number, Pl is the dimensionless medium permeability of the porous medium and F is the couple-stress parameter, which prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in a rotatory couple-stress fluid in porous medium heated from below. Further, It is established that the existence of oscillatory motions of growing amplitude in the present configuration, depends crucially upon the magnitude of the non-dimensional number , in the sense so long as , no such motions are possible, and in particular PES is valid.

A semi-circle theorem on the onset of couple stress fluid under rotation saturated by a porous medium

In the present paper, the thermal convection in couple stress fluid under rotation saturating a porous medium is considered. By applying linear stability theory and normal mode analysis method, a semi-circle theorem is derived. The mathematical analysis on the onset of thermal convection in a couple-stress fluid under rotation in a porous medium for the case of rigid boundaries shows that the complex growth rate σ of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle in the right half of a complex σ-plane, which prescribes the upper limits to the complex growth of arbitrary oscillatory motions of growing amplitude.

A semi-circle theorem in couple-stress fluid in the presence of rotation

The thermal instability of a couple-stress fluid acted upon by uniform vertical rotation and heated from below is investigated. Following the linearized stability theory and normal mode analysis, the paper through mathematical analysis of the governing equations of couple-stress fluid convection with a uniform vertical rotation, for the case of rigid boundaries shows that the complex growth rate μ of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle in the right half of a complex σ-plane, where T A is the Taylor number and F is the couple-stress parameter, which prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in a rotatory couple-stress fluid heated from below. Further, It is established that the existence of oscillatory motions of growing amplitude in the present configuration, depends crucially upon the magnitude of the non-dimensional number , in the sense so long as , no such motions are possible, and in particular PES is valid.

A note on the stability of inviscid zonal jet flows on a rotating sphere

AbstractThe linear stability of inviscid zonal jet flows on a rotating sphere is re-examined. A semi-circle theorem for inviscid zonal flows on a rotating sphere is proved. It is also shown that numerically obtained eigenvalues of the linear stability problem do not converge well with a spectral method which was adopted in previous studies, due to an emergence of critical layers near the poles. By using a shooting method where the integral path bypasses the critical layers in the complex plane, the eigenvalues are successfully obtained with ${\ensuremath{\sim} }10\hspace{0.167em} \% $ correction of the critical rotation rates compared to those obtained in Baines (J. Fluid Mech., vol. 73, 1976, pp. 193–213).

On the Stability of Three-Dimensional Disturbances in Stratified Flow with Lateral and Vertical Shear

Howard's semi-circle theorem is generalized for three-dimensional disturbances in a non-Boussinesq stratified fluid with a mean flow varying horizontally and vertically. In case of Boussinesq approximation Howard's and Rayleigh's theorem are extended for three-dimensional disturbances in a fluid with a mean flow varying only with respect to the latitude.

A semi-circle theorem for unstable perturbation in tropical cyclone-scale vorticles

In this paper, we discusses the spectrum distribution of baroclinic disturbances in tropical cyclone-scale vortices with the assumption that the basic flow does not vary with height. The semi-circle theorem for the unstable perturbation is obtained and the upper bound of growth rate can be estimated. The upper bound of instability growth rate increases with the increase of the maximum basic-state tangential wind speed, angular speed, inertial parameter, absolute vorticity, radial gradient of absolute vorticity and static stability. The upper bound of instability growth rate is greater for vortices of larger horizontal scale and smaller vertical scale and lower wave-number disturbances.

Stability of viscoelastic shear flows in the limit of high Weissenberg and Reynolds numbers

We consider a limit of the upper convected Maxwell model where both the Weissenberg and Reynolds numbers are large. The limiting equations have a status analogous to that of the Euler equations for the high Reynolds number limit. These equations admit parallel shear flows with an arbitrary profile of velocity and normal stress. We consider the stability of these flows. An extension of Howard’s semicircle theorem can be used to show that the flow is stabilized if elastic effects are sufficiently strong. We also show how to analyze the long wave limit in a fashion similar to the inviscid case.

A new convective instability of the rotating-disk boundary layer with growth normal to the disk

In this paper we extend the familiar concept of spatial instability and growth of disturbances in the downstream direction to include spatial instability and growth in the wall-normal direction. The stability theory of boundary layers has generally been concerned with determining the evolution of disturbances inside a boundary layer (this is where disturbances have their largest amplitudes and can cause a laminar boundary layer to become turbulent). Outside a boundary layer, where the basic flow is uniform, normal-mode disturbances decay exponentially with distance from the wall to satisfy homogeneous boundary conditions. In this paper we present a surprising scenario where an impulsive disturbance, made up of a superposition of these normal modes, nonetheless grows exponentially with distance from the wall. While the usual convective instability with exponential growth in the downstream direction can be efficiently characterized by spatial modes with complex wavenumbers, the new convective instability can be efficiently characterized by modes with exponentially diverging ‘eigenfunctions’ obtained by moving certain branch-cuts in the complex wavenumber plane. The new instability is therefore associated with an interaction between the discrete spectrum and the continuous spectrum. We emphasize, however, that the homogeneous boundary conditions are always satisfied, and that at any finite time exponential growth only occurs over a finite distance from the wall, but this distance increases linearly with time. Interactions between poles and branch-cuts have been found before, but the results presented here provide a physical interpretation for this spectral behaviour. A further curiosity is that some of these divergent modes have been found to violate Howard's semi-circle theorem.

Streaming Instabilities in Protoplanetary Disks

Interpenetrating streams of solids and gas in a Keplerian disk produce a local, linear instability. The two components mutually interact via aerodynamic drag, which generates radial drift and triggers unstable modes. The secular instability does not require self-gravity, yet it generates growing particle density perturbations that could seed planetesimal formation. Growth rates are slower than dynamical, but faster than radial drift, timescales. Growth rates, like streaming velocities, are maximized for marginal coupling (stopping times comparable dynamical times). Fastest growth occurs when the solid to gas density ratio is order unity and feedback is strongest. Curiously, growth is strongly suppressed when the densities are too nearly equal. The relation between background drift and wave properties is explained by analogy with Howard's semicircle theorem. The three-dimensional, two-fluid equations describe a sixth order (in the complex frequency) dispersion relation. A terminal velocity approximation allows simplification to an approximate cubic dispersion relation. To describe the simplest manifestation of this instability, we ignore complicating (but possibly relevant) factors like vertical stratification, dispersion of particle sizes, turbulence, and self-gravity. We consider applications to planetesimal formation and compare our work to other studies of particle-gas dynamics.

Linear Stability of Parallel Inviscid Flows of Shallow Water and Bubbly Fluid

The extensions of Howard's semicircle theorem, Squire's transformation, and Rayleigh's stability criterion are obtained for the model of dispersive shallow water waves and the model of bubbly fluid.

Semi—circle theorem of unstable spectrum distribution of heterotropic perturbation and the upper bound estimation of its growth rate

Based on the theoretical analysis on the unstable spectrum distribution of heterotropic perturbation, the semi–circle theorem of such spectrum distribution is obtained — the spectrum distributed within an upper semi–circle domain with radius R0 and taking origin as its circle center in the complex plane, and the upper bound estimation of growth rate is given simultaneously. It is found that the smaller the horizontal scale of perturbation is, the higher the model top located, the greater the estimated value of upper bound of such growth rate is. Also the increase of vertical shear of wind and latitude have the positive contribution to the increase of such growth rate. Finally when stratified stability decreases, the relative maximal growth rate increases, while the maximal growth rate decreases instead.

Linear stability theory of zonal shear flows with a free surface

We investigate the linear stability of zonal shear flows with a free surface (like the flow in oceans) taking into account the buoyancy effects at the deformed free surface. The necessary and sufficient conditions for the existence of neutral and unstable modes are established, and a semicircle theorem for unstable modes is deduced. An upper bound on the growth rate of an unstable mode is established. We have given a sufficient condition for stability that has a close resemblance with Miles' criterion for stability of a continuously stratified shear flow. The number of distinct unstable modes is shown not to exceed the number of neutral modes. The stability characteristics of smoothly-varying unbounded shear-layer flow profiles for long-wave disturbances are investigated. Finally, stability of a linear zonal flow velocity profile and a zonal Bickley jet-flow profile are investigated. Some of these results are new even in the special barotropic case (with no free-surface effect).

On the instability of magnetohydrodynamic shear flows

We consider the linear stability of an ideal, plane–parallel magnetohydrodynamic shear flow with velocity u = ( U (z),0, 0) and magnetic field B = (B(z), 0,0). We show how Howard's semicircle theorem is modified by the presence of a magnetic field and how this leads to new sufficient conditions for the stability of the flow. The issue of how tight these stability bounds actually are is addressed, by numerical computation, for one particular shear flow.

Sub-inertial dynamics of density-driven flows in a continuously stratified fluid on a sloping bottom. I. Model derivation and stability characteristics

A theory is developed describing the sub–inertial baroclinic dynamics of bottom–intensified density–driven flows within a continuously stratified fluid of finite depth with variable bottom topography. The evolution of the density–driven current is modelled as a geostrophically balanced homogeneous flow, which allows for finite–amplitude dynamic thickness variations and for which the pressure fields in each layer are strongly coupled together. The evolution of the overlying fluid is governed by baroclinic quasi–geostrophic dynamics describing a balance between the production of relative vorticity, the vortex–tube stretching/compression associated with a deforming gravity current height, and the rectifying influence of a background topographic vorticity gradient. The model is derived as a systematic asymptotic reduction of the two-fluid system in which the upper fluid is described by the Boussinesq adiabatic equations for a continuously stratified fluid and a lower homogeneous layer described by shallow–water theory appropriate for an f -plane with variable bottom topography. The model is shown to possess a non–canonical Hamiltonian formulation. This structure is exploited to give a variational principle for arbitrary steady solutions and stability conditions in the sense of Liapunov. The general linear stability problem associated with parallel shear flow solutions is examined in some detail. Necessary conditions for instability are derived. The instability is convective in the sense that it proceeds by extracting the available gravitational potential energy associated with the lower–layer water mass sliding down the sloping bottom. For the normal–mode instability problem, a semicircle theorem is derived. The linear stability characteristics are illustrated by solving the normal–mode equations for a simple linearly varying lower–layer height profile. In the overlying fluid, the unstable normal modes correspond to amplifying bottom–intensified topographic Rossby waves.

Further Results on the Stability of Rapidly Rotating Vortices in the Asymmetric Balance Formulation

Two kinds of wave-activity invariants, analogous to the pseudomomentum and pseudoenergy densities in the quasigeostrophic theory, are derived for both three-dimensional baroclinic and two-dimensional barotropic asymmetric balance models, and applied to examine the stability of asymmetric circular basic-state vortices under general disturbances. Stability theorems obtained in this paper extend the results of Montgomery and Shapiro, which are valid only for normal mode disturbances. The general properties of normal mode disturbances in quasigeostrophic dynamics are generalized to the asymmetric balance models, which include an orthogonality principle, bounds on the angular phase speeds of neutrally stable regular normal modes, and a semicircle theorem bounding the angular phase speeds and growth rates of unstable normal modes.

Linear Stability of the Three-Dimensional Semigeostrophic Model in Geometric Coordinates

Abstract Motivated by the recent work on the stability properties of balanced dynamics, the author investigates in this paper the stability of parallel basic flow in the baroclinic semigeostrophic (SG) model in geometric coordinates. The linearized baroclinic SG equation with a nonconstant Coriolis parameter is presented. Conservation equations for two wave-activity invariants, analogous to the pseudomomentum and pseudoenergy in baroclinic quasigeostrophic dynamics but with the contributions from lateral boundaries, are derived and then used to examine the stability properties of the SG model. It is found that the lateral boundary contributions to the invariants, which were often ignored, are important to the stability mechanism of the SG model. The general properties of normal mode disturbances in the SG model are investigated. Results obtained in this work include an orthogonality relation, the semicircle theorem to bound the phase speed of normal mode disturbances, and an estimate of upper bound on uns...