Long-wave Perturbations Research Articles

Soot and smoke aerosol may not warm climate

Soot and smoke aerosol contain black carbon, which absorbs solar radiation. These aerosols may reduce the overall negative climate forcing of anthropogenic aerosols by absorbing radiation that might otherwise be scattered back to space. They may also reduce overall cloudiness, an effect termed the “semidirect” effect, which is thought to enhance climate warming. Here, we evaluate the climate forcing associated with black carbon and other aerosols using the concept of “relaxed forcing,” which is the forcing associated with two simulations using fixed sea surface temperatures. The consideration of longwave perturbations associated with the relaxed forcing leads to a diminished or even negative semidirect effect associated with absorbing aerosols rather than an enhanced warming. The overall forcing depends significantly on the altitude of injection of the aerosols because higher‐altitude injections tend to enhance the negative longwave forcing. In addition, high‐altitude injection of absorbing aerosols can increase cloudiness at lower altitudes where temperatures, in general, may decrease.

Modulation instability of electromagnetic excitations in the Josephson junction in a thin superconducting film

The modulation instability of plane nonlinear electromagnetic waves with finite amplitudes excited in the Josephson junction in a thin superconducting film has been studied within the framework of nonlocal electrodynamics. A dispersion equation for the increment of the wave amplitude perturbation is obtained, and the regions of the modulation instability development for the waves studied are determined. It is shown that, for the longwave amplitude perturbations, the modulation instability of these waves is developed within a finite region of wavevectors 0 <Q<Qv, while the waves with Q≥Qv are stable.

Model of Nonlinear Evolution of Long-Wave Perturbations on an Ideally Conducting Jet with Current in a Longitudinal Magnetic Field. Collision of Magnetized Jets

Within the framework of the magnetohydrodynamic approach, a system of equations is derived for nonlinear evolution of long-wave axisymmetric perturbations on a conducting fluid jet with surface electric current, located along the axis of a conducting solid cylinder in a longitudinal magnetic field. The fluid is assumed to be inviscid, incompressible, and ideally conducting, like the cylinder walls. It is shown that, if the longitudinal field is uniform and the axial flow is shear-free, this system can be either hyperbolic or elliptic-hyperbolic, depending on problem parameters. The boundaries of hyperbolicity and ellipticity regions in the space of solutions are determined. In the hyperbolicity region, equations of characteristics and conditions on them are obtained. The problem of the decay of velocity discontinuity on the jet is considered. Conditions are found for the existence of a continuous self-similar solution in the hyperbolicity region, corresponding to collision of jets.

Convective instability of the thermovibrational flow of binary mixture in the presence of the Soret effect

The convective instability of the thermovibration flow in a plane horizontal layer filled with an incompressible binary gaseous mixture is investigated. The study takes into account the effect of thermal diffusion or the Ludwig-Soret effect. Several instability mechanisms are discussed. To determine the instability threshold with respect to cell and long-wave perturbations, the Floquet theory was applied to the linearized equations of convection formulated in the Boussinesq approximation. We found that regime parametric instability can occur owing to the finite frequency vibrations. The evolution of plane, spiral and three-dimensional disturbances is studied. We demonstrated that, because of the properties of the system, the subharmonic response of plane disturbances to the external periodic action cannot be observed. The instability can be associated only with synchronous or quasiperiodic modes. Depending on the vibration parameters, modulations can stabilize or destabilize the base state. For spiral perturbations the stability boundary does not depend on the amplitude and frequency of vibrations. In the case of long-wave instability we apply the regular perturbation approach with the wavenumber as a small parameter in power expansions. The stability boundaries are found.

A criterion of the linear long wave stability of steady magnetohydrodynamic jet flows of an ideal fluid

The problem of the linear stability of steady axisymmetric shear magnetohydrodynamic jet flows of an inviscid ideally conducting incompressible fluid with a free boundary is investigated. It is assumed that the jet is of unlimited length, there is a longitudinal constant electric current along its surface, and it is directed along the axis of a cylindrical shell with infinite conductivity, such that there is a vacuum layer between its free boundary and the inner surfaces of the shell. The necessary and sufficient condition for the stability of such flows with respect to small axisymmetric long-wave perturbations of special form is obtained by Lyapunov's direct method. Bilateral exponential estimates of the growth of small perturbations are constructed in the case when this stability condition breaks down, where the indices in their exponents are calculated from the parameters of the steady flows and the initial data for the perturbations. An example of a steady axisymmetric shear magnetohydrodynamic jet flow and of the initial small axisymmetric long-wave perturbations imposed on it is given, which, at the linear stage, will evolve in time and space in accordance with the estimates constructed.

Wave Regimes of Viscous Film Flow down a Vertical Cylinder

The flow of a viscous liquid film down a vertical cylinder in the gravity field is considered. In the case of small Reynolds numbers for long-wave perturbations on a cylinder of radius much greater than the film thickness, the problem can be reduced to a single nonlinear equation for the evolution of the film thickness perturbation. For axially symmetric solutions, this equation coincides with the well-known Sivashinsky-Kuramoto equation. The results of a numerical analysis of this equation for three-dimensional stationary traveling solutions of the problem are reported. The effect of the problem parameters on the solution behavior is demonstrated. Soliton type solutions are presented.

Tollmien-Schlichting Waves in a Hypersonic Boundary Layer Flow in the Neighborhood of a Cooled Surface

A nonlinear time-dependent model of the development of longwave perturbations in a hypersonic boundary layer flow in the neighborhood of a cooled surface is constructed. The pressure in the flow is assumed to be induced the combined variation of the thicknesses of the near-wall and main parts of the boundary layer. Numerical and analytic solutions are obtained in the linear approximation. It is shown that if the main part of the boundary layer is subsonic as a whole, its action reduces the perturbation damping upstream and the perturbation growth downstream, while a supersonic, as a whole, main part of the boundary layer creates the opposite effects. An analysis of the solutions obtained makes it possible to conclude that the asymptotic model proposed can describe the three-dimensional instability of the Tollmien-Schlichting waves.

Quasiperiodic solutions of the Korteweg-de Vries equation

Physical premises for the appearance of quasiperiodic solutions of the Korteweg-de Vries (KdV) equation are considered. Such solutions appear near singularities of the KdV equation describing, for example, polarization waves in a ferroelectric substance featuring the first-order phase transition or in an electron beam-waveguide structure system. The presence of a singularity results in that the velocity of longwave perturbations in the system becomes imaginary, which corresponds to the wave propagation in the range of nontransparency. The second harmonic generation is related to modulation of the initial periodic solution at the second (lower) frequency.

Optimal perturbations in a four-vortex aircraft wake in counter-rotating configuration

We consider the instability of two vortex pairs in a counter-rotating configuration. Such configurations model vortex wakes observed behind aircraft with inboard vortices produced at the inboard flap edges and at the tips of the horizontal tail. The instability potential is characterized by an optimal perturbation analysis. This extends the analysis of Fabre &amp; Jacquin (2000) which was restricted to particular stationary configurations, and that of Crouch (1997) which considered co-rotating configurations. A complete mapping of the optimal perturbations is presented. The optimal perturbations grow faster than the Crow (1970) instability. However, they correspond to short-wavelength perturbations mainly affecting the weaker inboard vortices. A possible strategy which consists of forcing a long wavelength is then investigated. Application of both the optimal and the long-wave optimal perturbations to reduction of vortex wake danger is discussed.

Instability of waveguides in a liquid with surface effects

Long surface capillary-gravity waves and waves beneath an elastic plate simulating an ice sheet are considered for a liquid of finite depth. These waves are described by a generalized Kadomtsev-Petviashvili equation containing higher (as compared with the ordinary Kadomtsev-Petviashvili equation) space derivatives. The generalized Kadomtsev-Petviashvili equation has waveguide solutions (waveguides) corresponding to traveling waves which are periodic in the direction of propagation and localized in the transverse direction. These waves result from the instability of uniform (carrier) periodic waves with respect to transverse perturbations. The stability of the waveguides with respect to longitudinal longwave perturbations is studied. The behavior of these perturbations depends on the wavenumber of the carrier periodic wave. Three intervals of wavenumbers corresponding to all the possible types of governing equations are considered.

Screw instability in black hole magnetospheres and a stabilizing effect of field-line rotation

The screw instability of a magnetic field is a mechanism for prohibiting the generation of strongly twisted field lines on large scales. If it can work in black hole magnetospheres, the global axisymmetric structure and the main process of energy release will be significantly influenced. In this paper, we study the instability condition in the framework of the variational principle, paying special attention to the stabilizing effect due to field-line rotation against the screw-unstable modes satisfying the well-known Kruskal-Shafranov criterion. The basic formulation for the stability analysis of rotating force-free fields in Kerr geometry is provided. Then, for practical use of the analytical method, the stationary configuration is assumed to be cylindrical, and we treat the long-wave mode perturbations excited at large distances from the central black hole, where the strength of gravity becomes negligibly small. This allows us to derive clearly a new criterion dependent on the rotational angular velocity of the magnetic field lines, and the implications of the results obtained for magnetic activities of jets driven by a black hole are briefly discussed.

PHASE INSTABILITIES OF DISTORTED HEXAGONAL PATTERNS

We present an analytical study on the stability of distorted hexagonal patterns. From a general amplitude equation we calculate the instabilities with respect to homogeneous and longwave perturbations. The latter lead to the phase equations that permit to determine the stability regions. Slightly squeezed hexagons are locally stable in a full range of distortion angles. The stability regions obtained from the phase equation are similar to those obtained numerically by other authors [Gunaratne et al., 1994].

Quasi-modes in boundary-layer-type flows. Part 1. Inviscid two-dimensional spatially harmonic perturbations

The work, being the first in a series concerned with the evolution of small perturbations in shear flows, studies the linear initial-value problem for inviscid spatially harmonic perturbations of two-dimensional shear flows of boundary-layer type without inflection points. Of main interest are the perturbations of wavelengths 2π/k long compared to the boundary-layer thickness H, kH = ε [Lt ] 1. By means of an asymptotic expansion, based on the smallness of ε, we show that for a generic initial perturbation there is a long time interval of duration ∼ ε−3 ln(1/ε), where the perturbation representing an aggregate of continuous spectrum modes of the Rayleigh equation behaves as if it were a single discrete spectrum mode having no singularity to the leading order. Following Briggs et al. (1970), who introduced the concept of decaying wave-like perturbations due to the presence of the ‘Landau pole’ into hydrodynamics, we call this object a quasi-mode. We trace analytically how the quasi-mode contribution to the entire perturbation field evolves for different field characteristics. We find that over O(ε−3 ln(1/ε)) time interval, the quasi-mode dominates the velocity field. In particular, over this interval the share of the perturbation energy contained in the quasi-mode is very close to 1, with the discrepancy in the generic case being O(ε4) (O(ε4) for the Blasius flow). The mode is weakly decaying, as exp(−ε3t). At larger times the quasi-mode ceases to dominate in the perturbation field and the perturbation decay law switches to the classical t−2. By definition, the quasi-modes are singular in a critical layer; however, we show that in our context their singularity does not appear in the leading order. From the physical viewpoint, the presence of a small jump in the higher orders has little significance to the manner in which perturbations of the flow can participate in linear and nonlinear resonant interactions. Since we have established that the decay rate of the quasi-modes sharply increases with the increase of the wavenumber, one of the major conjectures of the analysis is that the long-wave components prevail in the large-time asymptotics of a wide class of initial perturbations, not necessarily the predominantly long-wave perturbations. Thus, the explicit expressions derived in the long-wave approximation describe the asymptotics of a much wider class of initial conditions than might have been anticipated. The concept of quasi-modes also enables us to shed new light on the foundations of the method of piecewise linear approximations widely used in hydrodynamics.

Nonlinear stability analysis of the thin pseudoplastic liquid film flowing down along a vertical wall

This article investigates the weakly nonlinear stability theory of a thin pseudoplastic liquid film flowing down on a vertical wall. The long-wave perturbation method is employed to solve for generalized nonlinear kinematic equation with free film interface. The normal mode approach is used to compute the linear stability solution for the film flow. The method of multiple scales is then used to obtain the weak nonlinear dynamics of the film flow for stability analysis. It is shown that the necessary condition for the existence of such a solution is governed by the Ginzburg–Landau equation. The modeling results indicate that both subcritical instability and supercritical stability conditions are possible to occur in a pseudoplastic film flow system. The results also reveal that the pseudoplastic liquid film flows are less stable than Newtonian’s as traveling down along the vertical wall. The degree of instability in the film flow is further intensified by decreasing the flow index n.

Stability of hexagonal patterns in Bénard-Marangoni convection.

Hexagonal patterns in Bénard-Marangoni (BM) convection are studied within the framework of amplitude equations. Near threshold they can be described with Ginzburg-Landau equations that include spatial quadratic terms. The planform selection problem between hexagons and rolls is investigated by explicitly calculating the coefficients of the Ginzburg-Landau equations in terms of the parameters of the fluid. The results are compared with previous studies and with recent experiments. In particular, steady hexagons that arise near onset can become unstable as a result of long-wave instabilities. Within weakly nonlinear theory, a two-dimensional phase equation for long-wave perturbations is derived. This equation allows us to find stability regions for hexagon patterns in BM convection.

Mechanisms of interaction between ultrasound and sound in liquids with bubbles: Singular focusing

A two-dimensional interaction between long-wave (sound) and short-wave (ultrasound) pressure perturbations in a rarefied monodisperse mixture of a weakly compressible liquid with gas bubbles is consid- ered. The conditions at which this interaction leads to a singular focusing (an explosive instability) of ultra- sound are determined. A numerical study of the defocusing and the singular focusing in a bubbly liquid is car- ried out. The effect of the long-wave-short-wave resonance on the development of two-dimensional distur- bances is studied. © 2001 MAIK "Nauka/Interperiodica".

Nonlinear Stability Analysis of Thin Viscoelastic Film Flow Traveling Down along a Vertical Cylinder

This paper investigates the weakly nonlinear stability theoryof a thin viscoelastic liquid film flowing down along the outsidesurface of a vertical cylinder. The long-wave perturbation method isemployed to solve for generalized nonlinear kinematic equations withfree film interface. The normal mode approach is first used to computethe linear stability solution for the film flow. The method of multiplescales is then used to obtain the weak nonlinear dynamics of the filmflow for stability analysis. The modeling results indicate that both thesubcritical instability and supercritical stability conditions arepossible to occur in a viscoelastic film flow system. The degree ofinstability in the film flow is further intensified by the lateralcurvature of cylinder. This is somewhat different from that of theplanar flow. The modeling results also indicate that by increasing theviscoelastic effect and decreasing the radius of the cylinder the filmflow can become less stable as traveling down along the verticalcylinder.

Stability of Vibroconvective Flow in an Inclined Layer with Respect to Three-Dimensional Perturbations

The stability of the average flow in a laterally heated fluid layer is investigated. The flow is driven by the joint action of the gravity force and vertical high-frequency vibrations. Three-dimensional longwave perturbations are analytically considered. A comparison with the numerical results obtained earlier shows that this instability mode is the most dangerous over a wide parameter range.

Initiation of Self-Oscillations at Loss of Stability of Spatially-Periodic, Three-Dimensional Viscous Flows with Respect to Long-Wave Perturbations

We study the long-wave asymptotics of a secondary regime formed at loss of stability of a stationary spatially-periodic flow, as one of the periods grows without bound (the wavenumber α vanishes). It is shown that if the main flow, aligned with the long period, is non-zero and certain nondegeneracy conditions are satisfied, then a self-oscillation regime branches off the main solution as the viscosity decreases, both soft and rigid modes of loss of stability being possible. Explicit formulas for the main terms of the asymptotics are derived. Examples of the self-oscillations on particular flows are calculated and the behavior of the fluid particle paths in the self-oscillation “traveling-wave”' regime branching off the shear flow is studied.

Linear Longwave Instability of a Single Class of Steady-State Jet Flows of an Ideal Fluid in the Field of a Self-Electric Current

A necessary and sufficient condition of linear stability of a certain two-parameter class of cylindrical steady-state shear jet MHD flows of an inviscid incompressible ideally-conducting fluid with a free boundary is obtained by the direct Lyapunov method. The magnetic field is induced by a direct current flowing along the jet so that the field linearly depends on the radius. The stability with respect to small axisymmetric longwave perturbations is considered. The perturbations conserve leave the ratio of the distance between a fluid particle and the jet axis to the azimuthal vorticity component unchanged in each fluid particle. Two-sided exponential estimates of the perturbation growth are derived in the case of violation of the stability condition obtained.