Curve Of Marginal Stability Research Articles

Flow regimes for a square cross-section cylinder in oscillatory flow

Two-dimensional direct numerical simulation and Floquet stability analysis have been performed at moderate Keulegan–Carpenter number ($KC$) and low Reynolds number ($Re$) for a square cross-section cylinder with its face normal to the oscillatory flow. Based on the numerical simulations a map of flow regimes is formed and compared to the map of flow around an oscillating circular cylinder by Tatsuno & Bearman (J. Fluid Mech., vol. 211, 1990, pp. 157–182). Two new flow regimes have been observed, namely A$^{\prime }$ and F$^{\prime }$. The regime A$^{\prime }$ found at low $KC$ is characterised by the transverse convection of fluid particles perpendicular to the motion; and the regime F$^{\prime }$ found at high $KC$ shows a quasi-periodic feature with a well-defined secondary period, which is larger than the oscillation period. The Floquet analysis demonstrates that when the two-dimensional flow breaks the reflection symmetry about the axis of oscillation, the quasi-periodic instability and the synchronous instability with the imposed oscillation occur alternately for the square cylinder along the curve of marginal stability. This alternate pattern in instabilities leads to four distinct flow regimes. When compared to the vortex shedding in otherwise unidirectional flow, the two quasi-periodic flow regimes are observed when the oscillation frequency is close to the Strouhal frequency (or to half of it). Both the flow regimes and marginal stability curve shift in the $(Re,KC)$-space compared to the oscillatory flow around a circular cylinder and this shift appears to be consistent with the change in vortex formation time associated with the lower Strouhal frequency of the square cylinder.

Central charge anomalies in 2D sigma models with twisted mass

We discuss the central charge in supersymmetric ${\cal N}=2$ sigma models in two dimensions. The target space is a symmetric K\"ahler manifold, CP$(N-1)$ is an example. The U(1) isometries allow one to introduce twisted masses in the model. At the classical level the central charge contains Noether charges of the U(1) isometries and a topological charge which is an integral of a total derivative of the Killing potentials. At the quantum level the topological part of the central charge acquires anomalous terms. A bifermion term was found previously, using supersymmetry which relates it to the superconformal anomaly. We present a direct calculation of this term using a number of regularizations. We derive, for the first time, the bosonic part in the central charge anomaly. We construct the supermultiplet of all anomalies and present its superfield description. We also discuss a related issue of BPS solitons in the CP(1) model and present an explicit form for the curve of marginal stability.

Intersoliton forces in the Wess–Zumino model

The spectrum of supersymmetric domain wall solitons of the Wess–Zumino model is known to be discontinuous across a curve (of marginal stability) in the moduli space of quartic superpotentials. Here we show how this phenomenon can be understood from the behaviour of the long-range inter-soliton force, which we compute by a method due to Manton.

Long range forces and supersymmetric bound states

We consider the long range forces between two BPS particles on the Coulomb branch of N=2 and N=4 supersymmetric gauge theories. The 1/ r potential is unambiguously fixed, even at strong coupling, by the moduli dependence of central charges supported by the BPS states. The effective Coulombic coupling vanishes on marginal stability curves, while sign changes on crossing these curves explain the restructuring of the spectrum of composite BPS states. This restructuring proceeds via the delocalization of the composite state on approach to the curve of marginal stability. Therefore, the spectrum of BPS states can be inferred by analyzing the submanifolds of the moduli space where the long range potential is attractive. This method also allows us to find certain non-BPS bound states and their stability domains. As examples, we consider the dissociation of the W boson and higher charge dyons at strong coupling in N=2 SU(2) SYM, quark-monopole bound states in N=2 SYM with one flavor, and composite dyons in N=2 SU(3) SYM.

Stable non-BPS states on a 3-brane probe

As a novel application of string junctions, we provide evidence for the existence of stable non-BPS dyons with magnetic charge greater than 1 in (the semiclassical regime of) N=2 SU(2) Super-Yang-Mills theory. In addition, we find a new curve of marginal stability. Moduli space is therefore divided into four regions, each containing a different stable particle spectrum.

String junctions and BPS states in Seiberg-Witten theory

We argue that certain BPS states in the D3-brane probe realization of N = 2 SU(2) Super-Yang-Mills theory correspond to multi-pronged strings connecting the D3-brane to the background 7-branes. This provides a physical realization of the decay of these states on the curve of marginal stability, and explains their absence in the strong coupling regime.

Curves of marginal stability, and weak- and strong-coupling BPS spectra in N = 2 supersymmetric QCD

We explicitly determine the global structure of the SL(2, Z ) bundle over the Coulomb branch of the moduli space of asymptotically free N = 2 supersymmetric Yang-Mills theories with gauge group SU(2) when massless hypermultiplets are present. For each relevant number of flavours, we show that there is a curve of marginal stability of the Coulomb branch, diffeomorphic to a circle, across which the BPS spectrum is discontinuous. We determine rigorously and completely the BPS spectra inside and outside the curve. In all cases, the spectrum inside the curve consists of only those BPS states that are responsible for the singularities of the low energy effective action (in addition to the massless abelian gauge multiplet which is always present). The predicted decay patterns across the curve of marginal stability are perfectly consistent with all quantum numbers carried by the BPS states. As a byproduct, we also show that the electric and magnetic quantum numbers of the massless states at the singularities proposed by Seiberg and Witten are the only possible ones.

Influence of liquid bridge volume on the onset of oscillation in floating zone convection I. Experiments

The onset of oscillation in the floating zone convection driven by the gradient of surface tension was experimentally studied, and discussions were concentrated on the influence of liquid bridge volume on the onset of oscillation. Distributions of critical applied temperature difference and frequency depending on the volume of the liquid bridge were obtained, and there was a gap range of liquid volume which separated the curve of marginal stability into two parts for fixed rod diameter and aspect ratio. The results imply that the volume of the liquid bridge is a sensitive critical parameter for the onset of oscillation. The implication on the instability is also discussed in the present paper.

Approximations to the eigenvalue relation for the Orr-Sommerfeld problem

A comprehensive study is made of the eigenvalue relation for the Orr-Sommerfeld problem. One of the major results obtained is a 'first approximation’ to the eigenvalue relation which is uniformly valid along the entire curve of marginal stability. Two derivations of this approximation are given, one based on the use of uniform approximations to the solutions of the Orr-Sommerfeld equation and the other based on the differential equation satisfied by the eigenvalue relation itself. The theory is developed in detail for symmetrical flows in a channel but it is also applicable, with minor modifications, to flows of the boundary-layer type. Near the nose of the marginal curve the error associated with the approximation is of the order of e 3 , where e = (ixRU c )- 1/3 , and as R → ∞ along the upper and lower branches of the marginal curve the errors are of the order of e 12/5 and e 2 ln e respectively. A comparison is also made with four heuristic approximations to the eigenvalue relation, two of which have been widely used in the past, and detailed calculations for plane Poiseuille flow clearly demonstrate the superiority of the uniform approximation.

The Effects of Friction and β on Finite-Amplitude Baroclinic Waves

Abstract A linear and finite-amplitude analysis of the baroclinic instability of a zonal current is presented. Both the β effect and viscosity are included. A small but finite amount of dissipation destabilizes the system, lowering the curve of marginal stability by an O(1) amount. In the limit of vanishing viscosity, steady unstable waves of finite amplitude are discovered for shears which are subcritical with respect to the inviscid criterion.

On the normal modes of parallel flow of inviscid stratified fluid

The overall pattern of normal modes of parallel flow of inviscid stratified fluid is examined. For a given flow and wavenumber the modes are divided into five classes, some of which may be empty: (i) a finite class of non-singular unstable modes; (ii) a conjugate finite class of non-singular damped stable modes; (iii) a finite class of singular stable modes, each of these having a branch point and being the limit of unstable modes; (iv) a discrete class of modified internal gravity waves, these being non-singular stable modes (if the density decreases with height everywhere); (v) a continuous class of singular stable modes. The modified internal gravity waves are described asymptotically for large values of the Richardson number. These asymptotic results are related to and extended by numerical calculations for a sinusoidal basic velocity profile and a Bickley jet. The wave speeds for small values of the Richardson number are found to depend only upon the local behaviour of the mean flow near an overall simple maximum or minimum of the velocity profile. Finally some difficulties in the use of the Howard formula for perturbation at a curve of marginal stability are elucidated.