Critical Perturbation Research Articles

On the generation of instabilities in variable density flow

Interfacial or fingering instabilities have been studied recently in relation to contamination problems where a more dense plume is enclosed by and is moving along in a body of less dense fluid. Instabilities can play an important role in the mixing or dispersion process. Through the use of a variable density flow and transport code, we were able to study how the style of interfacial perturbation controls the pattern of instability development. Whether initial perturbations grow or decay depends mainly on the wavelength of the perturbing function. A critical perturbation wavelength must be exceeded for a perturbation to grow; otherwise the perturbation simply decays. Our work confirms earlier analyses that suggest that all stratified systems are inherently unstable, given some spectrum of the perturbing waves that exceed the critical wavelength. By implication, Rayleigh number stability criteria are inappropriate for evaluating the dense plume problem. Our study also demonstrates how numerical errors in a mass transport code can serve as a perturbing function and lead to the development of instabilities. However, these instabilities are not physically realistic and are essentially uncontrollable because their character depends on the extent to which numerical errors develop, as evidenced by the grid Peclet and Courant numbers.

Stability and evolution of the quiescent and travelling solitonic bubbles

We simulate numerically the dynamics of the bubble-like solitons of the cubic-quintic nonlinear Schrödinger equation. In agreement with earlier predictions, slow moving bubbles have been observed to be unstable, in contrast to rapid bubbles which stabilize above a certain critical velocity. The empirical formula for the critical velocity has been confirmed to a high degree of accuracy. Based on the choice of some critical perturbation, we derive an integral criterion for stability of kinks and bubbles of the general nonlinear Schrödinger equation which provides an explanation of the appearance of the critical velocity. Finally, we follow numerically the nonlinear evolution of the unstable bubbles in one, two and three dimensions.

Interaction of tearing modes with external structures in cylindrical geometry (plasma)

A basic theoretical framework is developed for the investigation of tearing mode interactions in cylindrical geometry. A set of equations describing the coupled evolution of the amplitude and phase of each mode in the plasma is obtained by combining electromagnetic and fluid flow information. Two interactions are investigated in detail as examples. The first example considered is the slowing down of a rotating magnetic island interacting with a resistive wall. Under certain conditions bifurcated steady state solutions are obtained, allowing the system to make sudden irreversible transitions from high rotation to low rotation states as the interaction strength is gradually increased, and vice versa. The second example considered is the interaction of a rotating tearing mode with a static external magnetic perturbation. In general, a rotating island is stabilized to some extent by the interaction, whereas a locked island is destabilized. In fact, a rotating island of sufficiently small saturated width can be completely stabilized. However, once the island width becomes too large, conventional mode locking occurs prior to complete stabilization. The interaction with a tearing-stable plasma initially gives rise to a modification of the bulk plasma rotation, with little magnetic reconnection induced at the rational surface. However, once a critical perturbation field strength is exceeded, there is a sudden change in the plasma rotation as a locked island is induced at the rational surface, with no rotating magnetic precursor. The implications of these results for typical ohmically heated tokamaks are evaluated. The comparatively slow mode rotation in large tokamaks renders such devices particularly sensitive to error-field induced locked modes, and the collapse of mode rotation due to wall interactions

'Silence, like a Lucrece knife': Shakespeare and the Meanings of Rape

Brief allusions to rape occur throughout Shakespeare's work, combining maximum effect with minimum critical perturbation. Issues often appear reassuringly simple: Macbeth's vision of'wither'd murder' (II. 1.52) stealing with 'Tarquin's ravishing strides' (1. 55) depicts rape as an act of unmitigated evil, inflicting irreversible damage on helpless innocence; The Tempest offers no reason for the audience to wish Caliban's attempt on Miranda had succeeded. Complexities are also part of the act; when Henry V threatens the maidens of Harfleur with.'hot and forcing violation' (II. 3. 21), his speech may sound like fair warning from a just prince or a reminder of the moral degradation war can inflict on victors; either way, Shakespeare is in charge. Titania's hint to Bottom that the flowers are 'lamenting some enforced chastity' (A Midsummer Night's Dream, III. 1.209) strikes a subtler chord. Sexual aggressors posing as defenceless targets cut a comic figure, yet Titania retains a half-contemptuous, half-sympathetic awareness that in some quarters, rape is a serious matter. These are still Shakespeare's ironies. He even copes elegantly with non-events, making it clear that Helen's 'fair rape' (Troilus and Cressida, 1. 2. 148) is only seduction. Shakespeare's detailed treatments of rape are less satisfactory. Responses to The Rape of Lucrece and Titus Andronicus suggest he has failed to take intellectual and artistic control. Titus's daughter Lavinia is raped by Demetrius and Chiron, who cut off her hands and tongue, then subject her to a barrage of sick humour. Titus avenges this and other atrocities by killing the rapists and serving their flesh to their mother, Tamora, and stepfather, the Emperor Saturninus, in a pie. During the table-talk, Titus elicits from Saturninus the opinion that a father is justified in killing his ravished daughter. Titus thereupon kills Lavinia; the play moves swiftly to its close with the deaths of Tamora, Titus, and Saturninus, and the election of Lucius Andronicus as Emperor of Rome. Lavinia's prolonged sufferings and perfunctory death make an uncomfortable spectacle. Rather than contemplating Lavinia herself, critics and directors prefer to make her a symbol of 'the destruction of the Roman political order'.' With Lucrece, Shakespeare's own vision seems to falter; it gives Ian Donaldson 'a sense so rare in

The generation of spiral characteristics

When the Lagrangian points L4, LS in a rotating dynamical system become unstable (at a critical perturbation e = e,) the characteristics of some families of orbits bifurcating from the short and long period orbits (SPO and LPO) become spiral. For a given e, slightly larger than e,, an infinity of families of multiplicities n, n + 1, n + 2, n. do not bifurcate any more from SPO or LPO but join each other into a spiral. As e increases this spiral is joined by lower multiplicity families, until the SPO-LPO family itself joins the spiral. Further spirals with the same or different focuses are formed by joining other sequences of families of order n, n + l, n + 2, n, or n, n + 2, n + 4, n... Other spirals are generated at particular values of e, starting and terminating at the same focus or two different focuses. As the perturbation e increases such spiral characteristics join other families, away from the focus. The orbits along the spiral characteristics have an increasing number of loops (either n, n + 1, n + 2, ..., or n, n + 2, n + 4 n.) around, or close to L4. In the first case the loops are along the symmetry axis, passing through L4. In the second case the loops appear in pairs outside the symmetry axis. The focuses correspond to homoclinic, or heteroclinic orbits, spiralling around L4 and /or L5.

Characterization of the Early Ultrastructural and Biochemical Events Occurring in Dichloromethane Diphosphonate Nephrotoxicity

A chelator, dichloromethane diphosphonate (Cl2MDP), used to treat for malignancy-induced hypercalcemia, has nephrotoxic potential. An acute animal model developed to examine the mechanism was used to further characterize the renal effects. NAG enzymuria appears to be an early premonitor of injury. Ultrastructurally, an increase in size and number of protein-containing phagolysosomal reabsorption droplets in proximal convoluted tubules associated with proteinuria precedes advent of tubular cell necrosis indicating these organelles to be a potential target site for Cl2MDP in the kidney. In vitro studies using rabbit cortical tubules and rat brush border membrane vesicle preparations suggest that the renal toxicity is not due to perturbation of phosphate transport or oxidative metabolism. An operational hypothesis emerges indicating that Cl2MDP may be protein bound affecting carrier protein charge facilitating glomerular leakage with tubular accumulation via protein transport. Cl2MDP may induce critical cation perturbation at the subcellular level as the mechanism of cell death.

Critical dynamics and trees

We show that the dependence of the critical perturbation of Hamiltonian systems on the frequency v is related to the statistics of large planar trees, using a complex map for illustration. A critical function K( v) is given by the radii of convergence of power series whose coefficients are represented as sums over trees. A preliminary approximation based on this relation leads to an expression for the critical function as the solution of a transcendental equation, which appears to provide a reasonable approximation to the true function, and to have the right structure near the dense set of singularities.

A complex-plane strategy for computing rotating polytropic models - Efficiency and accuracy of the complex first-order perturbation theory

view Abstract Citations (29) References (30) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A Complex-Plane Strategy for Computing Rotating Polytropic Models: Efficiency and Accuracy of the Complex First-Order Perturbation Theory Geroyannis, V. S. Abstract In this paper, a numerical method is developed for determining the structure distortion of a polytropic star which rotates either uniformly or differentially. This method carries out the required numerical integrations in the complex plane. The method is implemented to compute indicative quantities, such as the critical perturbation parameter which represents an upper limit in the rotational behavior of the star. From such indicative results, it is inferred that this method achieves impressive improvement against other relevant methods; most important, it is comparable to some of the most elaborate and accurate techniques on the subject. It is also shown that the use of this method with Chandrasekhar's first-order perturbation theory yields an immediate drastic improvement of the results. Thus, there is no neeed - for most applications concerning rotating polytropic models - to proceed to the further use of the method with higher order techniques, unless the maximum accuracy of the method is required. Publication: The Astrophysical Journal Pub Date: April 1988 DOI: 10.1086/166188 Bibcode: 1988ApJ...327..273G Keywords: Perturbation Theory; Polytropic Processes; Stellar Rotation; Stellar Structure; Equations Of State; Poisson Equation; Taylor Series; Astrophysics; STARS: INTERIORS; STARS: ROTATION full text sources ADS |

Stochastic electron motion in magnetically insulated diodes

The transition to stochasticity is investigated for electron motion in the field configuration of a magnetically insulated diode. The model system is planar and periodic. The equilibria studied are nonrelativistic with constant electron density across the sheath. This class of equilibria includes the Brillouin flow equilibrium as a limit. Action-angle variables are introduced that facilitate the nonlinear analysis. It is found that small periodic perturbations of the equilibrium potential may lead to stochastic motion of the sheath electrons. It is shown that the critical perturbation amplitude for global stochasticity goes to zero as the Brillouin limit is approached for a variety of perturbations. A global stochasticity diagram is presented that gives a simple characterization of the electron motion in the allowed phase space of a magnetically insulated diode. A piecewise linear map is introduced that allows an efficient study of the electron dynamics in the stochastic regions.

Formation and growth of deep mantle plumes

Summary. If whole mantle convection occurs in the Earth’s mantle, then the core-mantle boundary constitutes the lower boundary layer for mantle convection. This boundary layer appears to be unstable on a small scale, and thus may be a source of plumes of hot matter which penetrate the mantle and occasionally even the lithosphere (producing hot spots). A finite-amplitude numerical code is used to study the formation of such plumes and their growth through the mantle. The plumes are restricted to being two-dimensional sheets rather than cylinders. The initial conditions consist of a steadily convecting mantle, and plumes are produced by introducing a perturbation in the form of either a pulse or a steady stream of heat into the bottom of the mantle. Two main results are obtained: (1) A critical perturbation size has been found for a mantle with a Rayleigh number of lo’. Small perturbations produce plumes which fail to penetrate the mantle, and instead are swept up by the pre-existing convective pattern, while large perturbations succeed in penetrating the mantle and reaching the lithosphere. The critical perturbation size is shown empirically to be proportional to the effective bouyancy and to a factor related to the shape of the perturbation. A perturbed region 150km wide and 60 km deep should produce a successful plume when the temperature perturbation is 200K or more. (2) Deep mantle plumes appear to require on the order of 50-100Myr to penetrate the mantle; episodic plumes on shorter time-scales appear unlikely. A similar time is required for plumes forming in an initially static, uniform temperature mantle.

Unstable combustion of samples of gasfree compositions in the form of rods of square and circular cross section

For samples of gas-free compositions in the form of long rods with various cross-sections, the form of the critical perturbations may be indicated in linear analysis. In nonlinear analysis of the problem, a general mathematical formulation was undertaken. In the instability region close to the stability boundary, it may be established that a combustion process corresponding to the form of the critical perturbation with a finite amplitude and wave number lying closest to k* will be established. If more than one critical perturbation falls within the instability region, there arises the problem of choosing the preferable form of perturbation. In the present paper these questions are studied for samples in the form of long rods of square and circular cross section. Within the framework of the ''narrow-band'' approximation, the forms of critical perturbation are indicated in the case of negligibly small heat loss. In the region of slight supercriticality, numerical calculations of the nonsteady three-dimensional problem are performed. Stable nonsteady combustion conditions corresponding to the minimal values of the rod diameter are found.

Nearly linear mappings and their applications

A simple 2-dimensional mapping is considered, both analytically and numerically, for which all nonlinear effects are of the same order as the perturbations and of the same origin. Properties of the stochastic instability are investigated, taking the beam-beam interaction in a storage ring as an important particular example of a dynamic system that can be modelled with such a mapping. The special case of time-dependent mappings is discussed. It is shown that low-frequency time dependence sharply decreases the critical perturbation strength for the stochastic transition.

Solitary rossby waves over variable relief and their stability. Part II: numerical experiments

The barotropic, quasi-geostrophic vorticity equation describing large scale, rotating flows over zonal relief supports nonlinear permanent form solutions, namely nonlinear topographic Rossby waves. Through an analytical theory, these solutions have been shown to be neutrally stable to infinitesimal perturbations. Numerical algorithms, which necessarily truncate the infinite number of degrees of freedom of any continuum model to a finite number, are capable of reproducing the numerical equivalent of these form-preserving solutions. Moreover, these numerical solutions are shown to preserve their shape throughout the numerical experiment not only in the limit of small amplitude, but also for high amplitude (Rossby number → O(1)). Through numerical simulation, the stability analysis is carried far beyond the analytical limit of infinitesimal perturbations. The solutions maintain their stability in agreement with the analytical theory, up to perturbations having intensities almost of the same order as the solutions themselves. For higher-amplitude perturbations, the solutions break up and typical turbulent behavior ensues. The passage from wave-like to turbulent behavior, upon surpassing a critical perturbation value, can be observed in the sudden loss of phase locking of the permanent solution Fourier modes.

On the stability of space-periodic convective flows in a vertical layer with sinuous boundaries: PMM vol. 43, no. 6, 1979, pp. 998–1007

The problem of convection in a vertical layer with harmonically distorted boundaries is examined by perturbation theory methods for a small amplitude of sinuosity. The solutions obtained are applicable both in the stability region as well as in the supercritical region of the plane-parallel flow. The stability of the solutions found is investigated with respect to a certain class of space-bounded perturbations that are not necessarily space-periodic. The method of amplitude functions [1], generalized to the case of curved boundaries, is used. The Grashof critical number is found as a function of the period of sinuosity and the form of the neutral curve for the space-periodic motions and their stability region are obtained. It is established that if the deformation period of the boundaries is close to the wavelength of the critical perturbation for the plane-parallel flow or is twice as great, then as the Grashof number grows stability loss does not occur and the motion's amplitude changes continuously (cf. [2 — 4]). A comparison is made with the results of the numerical calculation in [5], An attempt was made in [6] to construct a stationary periodic motion in a layer with weakly-deformed boundaries, in the form of series in powers of a small sinuosity amplitude. However, the solution obtained diverges in a neighborhood of the neutral curve of the plane-parallel flow and approximates unstable motion in the supercritical region of the unperturbed problem. Flows under a finite sinuosity amplitude are calculated by the net method in [5] wherein the stability of the flows was investigated as well, but only with respect to perturbations with wave numbers that are multiples of 2π/l, where l is the length of the calculated region.

Convection in a mantle with variable viscosity

Studies are made on typical features of convection current in a thin viscous layer on a deep layer of high viscosity. The streamlines in this case are concentrated but not closed in the upper thin layer. The horizontal velocity is one sign in the upper layer and the other sign in the lower. The horizontal mass flux in the upper layer is balanced by the counter-horizontal mass flux in the lower layer. The aspect ratio of the critical perturbation, referred to the whole layer thickness, is not so different from that in the homogeneous layer case. The aspect ratio, referred to the thickness of the thin upper layer in which the current is concentrated, is large, about 30 in a case studied in the present paper. The result will give a theoretical justification for the existence of very flat convection cells supposed to exist in the mantle. The free surface of the fluid is low (high) where the temperature is lower (higher) and the vertical velocity is downward (upward). This may explain the formations of guyots in the Pacific. A quantitative check of this idea and estimations of physical parameters involved in the problem are made.

Stability of Sheet Pinch

The growth rate of the sheet pinch instability is calculated in the small Larmor radius limit using the method of integration along unperturbed orbits. It is found that the instability is driven by a small group of ``resonant'' electrons which cross and drift about the neutral plane having zero velocity along the field lines. The microinstability is absolute and develops fastest for a critical perturbation wavelength, of the order of several times the thickness of the plasma sheet.