The barotropic stability of the mean winds in the atmosphere

1962 ◽  
Vol 12 (3) ◽  
pp. 397-407 ◽  
Author(s):  
Frank B. Lipps
Keyword(s):  
Maximum Amplitude ◽  
Neutral Curve ◽  
Wave Number ◽  
Maximum Value ◽  
Beta Effect ◽  
Neutral Curves ◽  
The Mean ◽  
The Stability ◽  
Approach Zero ◽  
Barotropic Current

This paper considers the stability of a barotropic current on a beta earth. The motion is assumed to be horizontal, non-divergent and barotropic. The current is taken to be of the formU(y)=Asech2by+B. The perturbations are required to approach zero asyapproaches ± ∞. We introduce the non-dimensional wave-numberland a parameter χ, which is a measure of the rotation effect. χ is inversely proportional to β.There are only two kinds of perturbations: symmetric disturbances (those with maximum amplitude aty= 0) and antisymmetric disturbances (those with zero amplitude aty= 0). We find the neutral curve in the (χ,l2)-plane for both types of disturbances. The rates of amplification in the immediate vicinity of the neutral curves are also found. It is seen that the beta effect, which is due to the earth's rotation, tends to stabilize the current. For the symmetric disturbances we find a band of unstable wavelengths when χ > 1/2; and for large χ the estimated curve of the maximum value of the imaginary part of the phase velocity is asymptotic to the lower branch of the neutral curve. The antisymmetric disturbances are more stable than the symmetric disturbances.

On the stability of heterogeneous shear flows. Part 2

1963 ◽  
Vol 16 (2) ◽  
pp. 209-227 ◽  
Author(s):  
John W. Miles
Keyword(s):  
Shear Flow ◽  
Wave Speed ◽  
Neutral Curve ◽  
Wave Number ◽  
Neutral Curves ◽  
Stratified Shear Flow ◽  
Neutral Mode ◽  
The Mean ◽  
The Stability ◽  
Small Disturbances

Small disturbances relative to a horizontally stratified shear flow are considered on the assumptions that the velocity and density gradients in the undisturbed flow are non-negative and possess analytic continuations into a complex velocity plane. It is shown that the existence of a singular neutral mode (for which the wave speed is equal to the mean speed at some point in the flow) implies the existence of a contiguous, unstable mode in a wave-number (α), Richardson-number (J) plane. Explicit results are obtained for the rate of growth of nearly neutral disturbances relative to Hølmboe's shear flow, in which the velocity and the logarithm of the density are proportional to tanh (y/h). The neutral curve for this configuration, J = J0(α), is shown to be single-valued. Finally, it is shown that a relatively simple generalization of Hølmboe's density profile leads to a configuration having multiple-valued neutral curves, such that increasing J may be destabilizing for some range (s) of α.

The onset of convective instability in a triply diffusive fluid layer

1989 ◽  
Vol 202 ◽  
pp. 443-465 ◽  
Author(s):  
Arne J. Pearlstein ◽  
Rodney M. Harris ◽  
Guillermo Terrones
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Fluid Layer ◽  
Neutral Curve ◽  
Interesting Consequence ◽  
Stability Criteria ◽  
Neutral Curves ◽  
The Stability ◽  
Rayleigh Numbers ◽  
Periodic Bifurcation

The onset of instability is investigated in a triply diffusive fluid layer in which the density depends on three stratifying agencies having different diffusivities. It is found that, in some cases, three critical values of the Rayleigh number are required to specify the linear stability criteria. As in the case of another problem requiring three Rayleigh numbers for the specification of linear stability criteria (the rotating doubly diffusive case studied by Pearlstein 1981), the cause is traceable to the existence of disconnected oscillatory neutral curves. The multivalued nature of the stability boundaries is considerably more interesting and complicated than in the previous case, however, owing to the existence of heart-shaped oscillatory neutral curves. An interesting consequence of the heart shape is the possibility of ‘quasi-periodic bifurcation’ to convection from the motionless state when the twin maxima of the heart-shaped oscillatory neutral curve lie below the minimum of the stationary neutral curve. In this case, there are two distinct disturbances, with (generally) incommensurable values of the frequency and wavenumber, that simultaneously become unstable at the same Rayleigh number. This work complements the earlier efforts of Griffiths (1979a), who found none of the interesting results obtained herein.

Viscous and inviscid centre modes in the linear stability of vortices: the vicinity of the neutral curves

2008 ◽  
Vol 603 ◽  
pp. 1-38 ◽  
Author(s):  
DAVID FABRE ◽  
STÉPHANE LE DIZÈS
Keyword(s):  
Axial Flow ◽  
Neutral Curve ◽  
Numerical Results ◽  
Asymptotic Results ◽  
Asymptotic Description ◽  
Vortex Model ◽  
Neutral Curves ◽  
Trailing Vortices ◽  
The Stability ◽  
Wave Limit

In a previous paper, We have recently that if the Reynolds number is sufficiently large, all trailing vortices with non-zero rotation rate and non-constant axial velocity become linearly unstable with respect to a class of viscous centre modes. We provided an asymptotic description of these modes which applies away from the neutral curves in the (q, k)-plane, where q is the swirl number which compares the azimuthal and axial velocities, and k is the axial wavenumber. In this paper, we complete the asymptotic description of these modes for general vortex flows by considering the vicinity of the neutral curves. Five different regions of the neutral curves are successively considered. In each region, the stability equations are reduced to a generic form which is solved numerically. The study permits us to predict the location of all branches of the neutral curve (except for a portion of the upper neutral curve where it is shown that near-neutral modes are not centre modes). We also show that four other families of centre modes exist in the vicinity of the neutral curves. Two of them are viscous damped modes and were also previously described. The third family corresponds to stable modes of an inviscid nature which exist outside of the unstable region. The modes of the fourth family are also of an inviscid nature, but their structure is singular owing to the presence of a critical point. These modes are unstable, but much less amplified than unstable viscous centre modes. It is observed that in all the regions of the neutral curve, the five families of centre modes exchange their identity in a very intricate way. For the q vortex model, the asymptotic results are compared to numerical results, and a good agreement is demonstrated for all the regions of the neutral curve. Finally, the case of ‘pure vortices’ without axial flow is also considered in a similar way. In this case, centre modes exist only in the long-wave limit, and are always stable. A comparison with numerical results is performed for the Lamb–Oseen vortex.

scholarly journals Stability and energy budget of pressure-driven collapsible channel flows

2011 ◽  
Vol 705 ◽  
pp. 348-370 ◽  
Author(s):  
H. F. Liu ◽  
X. Y. Luo ◽  
Z. X. Cai
Keyword(s):  
Pressure Drop ◽  
Kinetic Energy ◽  
Energy Budget ◽  
Neutral Curve ◽  
Channel Flows ◽  
Mean Flow ◽  
Mode 2 ◽  
The Mean ◽  
The Stability ◽  
Pressure Driven

AbstractAlthough self-excited oscillations in collapsible channel flows have been extensively studied, our understanding of their origins and mechanisms is still far from complete. In the present paper, we focus on the stability and energy budget of collapsible channel flows using a fluid–beam model with the pressure-driven (inlet pressure specified) condition, and highlight its differences to the flow-driven (i.e. inlet flow specified) system. The numerical finite element scheme used is a spine-based arbitrary Lagrangian–Eulerian method, which is shown to satisfy the geometric conservation law exactly. We find that the stability structure for the pressure-driven system is not a cascade as in the flow-driven case, and the mode-2 instability is no longer the primary onset of the self-excited oscillations. Instead, mode-1 instability becomes the dominating unstable mode. The mode-2 neutral curve is found to be completely enclosed by the mode-1 neutral curve in the pressure drop and wall stiffness space; hence no purely mode-2 unstable solutions exist in the parameter space investigated. By analysing the energy budgets at the neutrally stable points, we can confirm that in the high-wall-tension region (on the upper branch of the mode-1 neutral curve), the stability mechanism is the same as proposed by Jensen & Heil. Namely, self-excited oscillations can grow by extracting kinetic energy from the mean flow, with exactly two-thirds of the net kinetic energy flux dissipated by the oscillations and the remainder balanced by increased dissipation in the mean flow. However, this mechanism cannot explain the energy budget for solutions along the lower branch of the mode-1 neutral curve where greater wall deformation occurs. Nor can it explain the energy budget for the mode-2 neutral oscillations, where the unsteady pressure drop is strongly influenced by the severely collapsed wall, with stronger Bernoulli effects and flow separations. It is clear that more work is required to understand the physical mechanisms operating in different regions of the parameter space, and for different boundary conditions.

On the stability of viscous flow between parallel planes in the presence of a co-planar magnetic field

1954 ◽  
Vol 221 (1145) ◽  
pp. 189-206 ◽  
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Wave Number ◽  
Neutral Stability ◽  
Energy Relation ◽  
Geometrical Properties ◽  
Electrically Conducting ◽  
Neutral Curves ◽  
The Stability ◽  
The Magnetic Field

Linearized equations are derived which govern the stability of a viscous, electrically conducting fluid in motion between two parallel planes in the presence of a co-planar magnetic field. With one suitable approximation, which restricts the valid range of Reynolds number of the theory, the problem of stability is reduced to the solution of a fourth-order ordinary differential equation. The disturbances considered are neither amplified nor damped, but are neutral. Curves of wave number against Reynolds number for neutral stability are calculated for a range of values of a certain parameter, q , which represents the magnetic effects. For given physical and geometrical properties, the critical Reynolds number above which the flow is unstable rises with the strength of the magnetic field. These results are completely within the range of the approximation mentioned. In addition, an energy relation is derived which illustrates the balance between energy transferred from the basic flow to the disturbances, and that dissipated by viscosity and by the magnetic field perturbations.

The stability of a family of Jeffery–Hamel solutions for divergent channel flow

1966 ◽  
Vol 24 (1) ◽  
pp. 191-207 ◽  
Author(s):  
P. M. Eagles
Keyword(s):  
Reynolds Numbers ◽  
Neutral Curve ◽  
Wave Number ◽  
Finite Limit ◽  
Lower Branch ◽  
Negative Wave ◽  
Wave Velocities ◽  
Critical Reynolds Numbers ◽  
Divergent Channel ◽  
The Stability

A set of Jeffery–Hamel profiles (for radial, viscous, incompressible flow) have been shown by Fraenkel (1962, 1963) to approximate to profiles in certain two-dimensional divergent channels. The stability of a family of these profiles is investigated by a numerical solution of the Orr-Sommerfeld problem. Neutralstability curves are calculated in the (R,k)-planes (where R is the Reynolds number of the basic flow and k is the wave-number of the disturbance), and fairly low critical Reynolds numbers are found. For those profiles that have regions of reversed flow, negative wave velocities are found on the lower branch of the neutral curve, and also it is found that Rk tends to a finite limit as R → ∞ on the lower branch. These unexpected results are further discussed and verified by independent methods. The relation of the calculations to some experiments of Patterson (1934, 1935) is discussed.

scholarly journals The study of bubbles in bitcoin behavior

2019 ◽  
Vol 14 (4) ◽  
pp. 133-142
Author(s):  
Usama Adnan Fendi ◽  
Asem Tahtamouni ◽  
Yaser Jalghoum ◽  
Suleiman Jamal Mohammad
Keyword(s):  
Time Series ◽  
Financial Risk ◽  
Online Communication ◽  
Return Level ◽  
Test Statistics ◽  
Virtual Currency ◽  
Electronic Payments ◽  
Maximum Value ◽  
The Mean ◽  
The Stability

Bitcoin is an online communication system that facilitates the use of virtual currency, including electronic payments. This paper aims at analyzing the behavior of Bitcoin returns as a proposal for future currencies while making a comparison between Bitcoin and other conventional currencies. This paper uses quantitative approach to analyze the time series of Bitcoin and that of other conventional currencies during the period 2010–2018. It uses 1) a descriptive statistics for the weekly returns for Bitcoin which includes the mean, standard deviation, maximum value, minimum value, skewness, kurtosis, and Jarque-Bera normal distribution test statistics, and 2) duration dependence test on Bitcoin weekly returns by extracting the weekly returns for the Bitcoin that behave in irregular way of the general Bitcoin return level through autocorrelation regression, and taking the residuals for this regression as a time series for irregular returns.This paper has confirmed no empirical evidence for the existence of a speculative bubble in the Bitcoin values and returns. In addressing the question of whether Bitcoin can act as a reliable substitute for conventional currencies, the returns based analysis shows a huge difference between the behavior of Bitcoin returns from conventional currency returns when comparing both aspects of level and stability. The paper concluded that bitcoin is more an investment than a currency. This paper represents a significant contribution in the path of financial economics and financial risk management, and represents a contribution to the stability of the financial system around the world and mitigating financial crises.

Effect of an electric field on the stability of contaminated film flow down an inclined plane

2008 ◽  
Vol 595 ◽  
pp. 221-237 ◽  
Author(s):  
M. G. BLYTH
Keyword(s):  
Electric Field ◽  
Strong Electric Field ◽  
Normal Modes ◽  
Neutral Curve ◽  
Arbitrary Parameter ◽  
Perfect Conductor ◽  
Inclined Plane ◽  
Neutral Curves ◽  
Stability And Instability ◽  
The Stability

The stability of a liquid film flowing down an inclined plane is considered when the film is contaminated by an insoluble surfactant and subjected to a uniform normal electric field. The liquid is treated as a perfect conductor and the air above the film is treated as a perfect dielectric. Previous studies have shown that, when acting in isolation, surfactant has a stabilizing influence on the flow while an electric field has a destabilizing influence. The competition between these two effects is the focus of the present study. The linear stability problem is formulated and solved at arbitrary parameter values. An extended form of Squire's theorem is presented to argue that attention may be confined to two-dimensional disturbances. The stability characteristics for Stokes flow are described exactly; the growth rates of the normal modes at finite Reynolds number are computed numerically. We plot the neutral curves dividing regions of stability and instability, and trace how the topology of the curves changes as the intensity of the electric field varies both for a clean and for a contaminated film. With a sufficiently strong electric field, the neutral curve for a clean film consists of a lower branch trapping an area of stable modes around the origin, and an upper branch above which the flow is stable. With surfactant present, a similar situation obtains, but with an additional island of stable modes disjoint from the upper and lower branches.

The neutral curves for periodic perturbations of finite amplitude of plane Poiseuille flow

1969 ◽  
Vol 39 (3) ◽  
pp. 629-639 ◽  
Author(s):  
C. L. Pekeris ◽  
B. Shkoller
Keyword(s):  
Power Series ◽  
Perturbation Method ◽  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Neutral Curve ◽  
Plane Poiseuille Flow ◽  
Mean Flow ◽  
Periodic Perturbations ◽  
Neutral Curves ◽  
The Mean

It is shown that there exist undamped solutions for perturbations of finite amplitude of plane Poiseuille flow, which are periodic in the direction of the axis of the channel. The shift in the ‘neutral curve’ as a function of the amplitude λ* of the disturbance is shown in figure 2. The solution is obtained by a perturbation method in which the eigenfunctions and the eigenvalue c are expanded in power series of the amplitude λ, as shown in (14), (15), (16) and (17). Near the neutral curve for a finite amplitude disturbance, the curvature of the mean flow shows a tendency to become negative (figure 5).

The stability of stratified shear flows

1970 ◽  
Vol 42 (2) ◽  
pp. 367-377 ◽  
Author(s):  
M. R. Collyer
Keyword(s):  
Shear Flows ◽  
Wave Number ◽  
Neutral Stability ◽  
Small Perturbations ◽  
Density Distributions ◽  
Principal Measure ◽  
Neutral Curves ◽  
Parallel Shear Flow ◽  
The Stability ◽  
Arbitrary Velocity

Small perturbations of a parallel shear flow U(z) in an inviscid, incompressible, stably stratified fluid of density ρ(z) are considered, for which the principal measure of stability is the Richardson number, R. For an arbitrary velocity and density profile we discuss the problem of determining whether a curve of neutral stability has adjacent unstable regions in an (α, R) plane, where α is the wave-number of the disturbance. Neutral curves bounding unstable regions are then obtained for a triangular jet flow in conjunction with various density distributions. A comparison is also made between the stability characteristics of jet and shear flows with corresponding density structures.

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