Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown

1999 ◽  
Vol 396 ◽  
pp. 73-108 ◽  
Author(s):  
D. M. MASON ◽  
R. R. KERSWELL
Keyword(s):  
Finite Amplitude ◽  
Complex Conjugate ◽  
Small Scale ◽  
Conjugate Pair ◽  
Weakly Nonlinear ◽  
Elliptical Instability ◽  
Elliptical Flow ◽  
Complex Conjugate Pair ◽  
Generic Instability ◽  
Secondary Instabilities

A direct numerical simulation is presented of an elliptical instability observed in the laboratory within an elliptically distorted, rapidly rotating, fluid-filled cylinder (Malkus 1989). Generically, the instability manifests itself as the pairwise resonance of two different inertial modes with the underlying elliptical flow. We study in detail the simplest ‘subharmonic’ form of the instability where the waves are a complex conjugate pair and which at weakly supercritical elliptical distortion should ultimately saturate at some finite amplitude (Waleffe 1989; Kerswell 1992). Such states have yet to be experimentally identified since the flow invariably breaks down to small-scale disorder. Evidence is presented here to support the argument that such weakly nonlinear states are never seen because they are either unstable to secondary instabilities at observable amplitudes or neighbouring competitor elliptical instabilities grow to ultimately disrupt them. The former scenario confirms earlier work (Kerswell 1999) which highlights the generic instability of inertial waves even at very small amplitudes. The latter represents a first numerical demonstration of two competing elliptical instabilities co-existing in a bounded system.

scholarly journals A New Signal Representation Using Complex Conjugate Pair Sums

2019 ◽  
Vol 26 (2) ◽  
pp. 252-256 ◽  
Author(s):  
Basheeruddin Shah Shaik ◽  
Vijay Kumar Chakka ◽  
Arikatla Satyanarayana Reddy
Keyword(s):  
Complex Conjugate ◽  
Signal Representation ◽  
Conjugate Pair ◽  
Complex Conjugate Pair

On the onset of instability by oscillatory modes in hydromagnetic Couette flow

1967 ◽  
Vol 301 (1467) ◽  
pp. 451-471 ◽  
Keyword(s):  
Couette Flow ◽  
Hartmann Number ◽  
Outer Wall ◽  
Angular Speed ◽  
Complex Conjugate ◽  
Neutral Stability ◽  
Narrow Gap ◽  
Conjugate Pair ◽  
Average Value ◽  
Complex Conjugate Pair

The transition of the onset of instability from stationary modes to oscillatory modes for an incompressible, conducting Couette flow between two coaxial, perfectly conducting, non-permeable, rotating cylinders under the influence of an axially applied magnetic field is considered. Results for three cases are reported. These pertain to flow between (1) a rotating inner wall with a stationary outer wall, (2) counterrotating walls, and (3) corotating walls. It is found that, for high values of the Hartmann number, there may exist some range of convective wavenumbers for which neither of the two lowest modes of axisymmetrical disturbances will become stationary. Within this range, the neutral stability curve is determined by a complex-conjugate pair of oscillatory axisymmetrical modes of equal stability. The oscillatory modes may, in fact, become more critical than the stationary modes. It is demonstrated that the approximation of replacing the angular speed by its average value, combined with the assumption of a narrow gap between the cylindrical walls, eliminates the oscillatory axisymmetrical modes.

Secondary instabilities in rapidly rotating fluids: inertial wave breakdown

1999 ◽  
Vol 382 ◽  
pp. 283-306 ◽  
Author(s):  
R. R. KERSWELL
Keyword(s):  
Three Dimensional ◽  
Finite Amplitude ◽  
Small Scale ◽  
Inertial Waves ◽  
Rotating Fluids ◽  
Initial Excitation ◽  
Geostrophic Flows ◽  
Inertial Wave ◽  
Secondary Instabilities ◽  
Wave Breakdown

Inertial waves are a ubiquitous feature of rapidly rotating fluids. Although much is known about their initial excitation, little is understood about their stability. Experiments indicate that they are generically unstable and in many cases catastrophically so, quickly causing the whole flow to collapse to small-scale disorder. The linear stability of two three-dimensional inertial waves observed to break down in the laboratory is considered here at experimentally small but finite Ekman numbers of [les ]10−4. Surprisingly small threshold amplitudes for instability are found. The results support the conjecture that triad resonances are the generic mechanism for secondary instability in rapidly rotating fluids but also highlight the ability of geostrophic flows to derive energy through a finite-amplitude inertial wave. This latter finding may go some way to explaining the significant mean circulations typically observed in inertial wave experiments.

Poles and Zeros of a Bandpass Filter Obtained by Transformation of a Lowpass Filter

1979 ◽  
Vol 16 (1) ◽  
pp. 39-41
Author(s):  
S. C. Duttaroy
Keyword(s):  
Bandpass Filter ◽  
Damping Ratio ◽  
Complex Conjugate ◽  
Lowpass Filter ◽  
Transformed Roots ◽  
Conjugate Pair ◽  
Complex Conjugate Pair ◽  
Poles And Zeros ◽  
Explicit Expressions

The standard lowpass to bandpass transformation is shown to transform a complex conjugate pair of roots (poles or zeros) to two such pairs having equal damping ratio. Explicit expressions are given for the locations of the transformed roots; these should be useful in active R.C. bandpass realizations.

scholarly journals On Complex Conjugate Pair Sums and Complex Conjugate Subspaces

2019 ◽  
Vol 26 (9) ◽  
pp. 1403-1407
Author(s):  
Shaik Basheeruddin Shah ◽  
Vijay Kumar Chakka ◽  
Arikatla Satyanarayana Reddy
Keyword(s):  
Complex Conjugate ◽  
Conjugate Pair ◽  
Complex Conjugate Pair

QUARK-GLUON VERTEX AND STRUCTURE OF MESONS AND DIQUARKS

1994 ◽  
Vol 09 (38) ◽  
pp. 3551-3563 ◽  
Author(s):  
S.J. STAINSBY ◽  
R.T. CAHILL
Keyword(s):  
Form Factors ◽  
Quark Propagator ◽  
The Other ◽  
Analytic Structure ◽  
Complex Conjugate ◽  
Branch Points ◽  
Conjugate Pair ◽  
Gluon Vertex ◽  
Scalar Diquark ◽  
Complex Conjugate Pair

Two quark propagators with different analytic structure are employed in Bethe-Salpeter type equations for the pion and scalar diquark form factors. One of the quark propagators has been calculated with the inclusion of a trivial (bare) quark-gluon vertex and, as a consequence, contains a complex conjugate pair of logarithmic branch points. The other quark propagator is obtained using a non-trivial (dressed) vertex ansatz and is entire, with an essential singularity at infinity. The effects of these different quark propagators on the BSE solutions are compared.

scholarly journals Coquaternionic Quantum Dynamics for Two-level Systems

10.14311/1394 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
D. C. Brody ◽  
E. M. Graefe
Keyword(s):  
State Space ◽  
Time Evolution ◽  
Quantum Dynamics ◽  
Complex Conjugate ◽  
Conjugate Pair ◽  
Open Orbit ◽  
Energy Eigenvalues ◽  
Complex Conjugate Pair ◽  
Two Level Systems ◽  
Different Characteristics

The dynamical aspects of a spin-1/2 particle in Hermitian coquaternionic quantum theory are investigated. It is shown that the time evolution exhibits three different characteristics, depending on the values of the parameters of the Hamiltonian. When energy eigenvalues are real, the evolution is either isomorphic to that of a complex Hermitian theory on a spherical state space, or else it remains unitary along an open orbit on a hyperbolic state space. When energy eigenvalues form a complex conjugate pair, the orbit of the time evolution closes again even though the state space is hyperbolic.

The macrodynamics of α-effect dynamos in rotating fluids

1975 ◽  
Vol 67 (3) ◽  
pp. 417-443 ◽  
Author(s):  
W. V. R. Maekus ◽  
M. R. E. Proctor
Keyword(s):  
Magnetic Fields ◽  
Large Scale ◽  
Magnetic Energy ◽  
Finite Amplitude ◽  
Small Scale ◽  
Past Study ◽  
Ohmic Loss ◽  
General Will ◽  
Rotating Systems ◽  
Stellar Magnetic Fields

Past study of the large-scale consequences of forced small-scale motions in electrically conducting fluids has led to the ‘α-effect’ dynamos. Various linear kinematic aspects of these dynamos have been explored, suggesting their value in the interpretation of observed planetary and stellar magnetic fields. However, large-scale magnetic fields with global boundary conditions can not be force free and in general will cause large-scale motions as they grow. I n this paper the finite amplitude behaviour of global magnetic fields and the large-scale flows induced by them in rotating systems is investigated. In general, viscous and ohmic dissipative mechanisms both play a role in determining the amplitude and structure of the flows and magnetic fields which evolve. In circumstances where ohmic loss is the principal dissipation, it is found that determination of a geo- strophic flow is an essential part of the solution of the basic stability problem. Nonlinear aspects of the theory include flow amplitudes which are independent of the rotation and a total magnetic energy which is directly proportional to the rotation. Constant a is the simplest example exhibiting the various dynamic balances of this stabilizing mechanism for planetary dynamos. A detailed analysis is made for this case to determine the initial equilibrium of fields and flows in a rotating sphere.

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

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