Subharmonic Perturbations Research Articles

Self-similarity and recurrence in stability spectra of near-extreme Stokes waves

We consider steady surface waves in an infinitely deep two-dimensional ideal fluid with potential flow, focusing on high-amplitude waves near the steepest wave with a 120 $^{\circ }$ corner at the crest. The stability of these solutions with respect to coperiodic and subharmonic perturbations is studied, using new matrix-free numerical methods. We provide evidence for a plethora of conjectures on the nature of the instabilities as the steepest wave is approached, especially with regards to the self-similar recurrence of the stability spectrum near the origin of the spectral plane.

Design and verification of q-axis perturbation based active islanding detection schemes for DG systems

The penetration of grid integrated distributed generation (DG) in the present decade, has benefited rural communities, the environment, and the power sector. These renewable power sources based DGs could eliminate the need of extensive transmission networks, especially in remote areas, reduce emissions and improve power supply reliability. A significant drawback of grid integrated DG systems is the islanding of DG units, which puts workers' safety at risk and raises the possibility of damaging electrical infrastructure. Therefore, islanding detection techniques are used to reduce the danger associated with islanded functioning of DG units. Fast detection, small non detection area and less power quality disturbance are the major requirements of any islanding detection method. To address this issue of islanding, researchers have proposed various islanding detection strategies. This paper compares various q-axis controller-based islanding identification approaches: sub-harmonic perturbation (SHP), complementary reactive power perturbation (CRPP), and even harmonic perturbation (EHP). In all three proposed methodologies, the perturbations introduced result in frequency deviations surpassing the predefined threshold values. But the time of islanding detection is least in the CRPP approach. CRPP can also drift the total harmonic distortion (THD) beyond the corresponding threshold in an appreciable way. The performance of these (Islanding detection methods) IDMs is evaluated through simulations using MATLAB-Simulink on a PV fed DG. The efficacy of the comparative analysis is ensured with necessary waveforms.

Rogue waves and instability arising from long-wave-short-wave resonance beyond the integrable regime.

We consider instability and localized patterns arising from the long-wave-short-wave resonance in the nonintegrable regime numerically. We study the stability and instability of elliptic-function periodic waves with respect to subharmonic perturbations, whose period is a multiple of the period of the elliptic waves. We thus find the modulational instability (MI) of the corresponding dnoidal waves. Upon varying parameters of dnoidal waves, spectrally unstable ones can be transformed into stable states via the Hamiltonian Hopf bifurcation. For snoidal waves, we find a transition of the dominant instability scenario between the MI and the instability with a bubblelike spectrum. For cnoidal waves, we produce three variants of the MI. Evolution of the unstable states is also considered, leading to formation of rogue waves on top of the elliptic-wave and continuous-wave backgrounds.

Stability of Hydroelastic Waves in Deep Water

Two-dimensional periodic travelling hydroelastic waves on water of infinite depth are investigated. A bifurcation branch is tracked that delineates a family of such solutions connecting small amplitude periodic waves to the large amplitude static state for which the wave is at rest and there is no fluid motion. The stability of these periodic waves is then examined using a surface-variable formulation in which a linearised eigenproblem is stated on the basis of Floquet theory and solved numerically. The eigenspectrum is discussed encompassing both superharmonic and subharmonic perturbations. In the former case, the onset of instability via a Tanaka-type collision of eigenvalues at zero is identified. The structure of the eigenvalue spectrum is elucidated as the travelling-wave branch is followed revealing a highly intricate structure.

Continuous-time balanced truncation for time-periodic fluid flows using frequential Gramians

Reduced-order models for flows that exhibit time-periodic behavior (e.g., flows in turbomachinery and wake flows) are critical for several tasks, including active control and optimization. One well-known procedure to obtain the desired reduced-order model in the proximity of a periodic solution of the governing equations is continuous-time balanced truncation. Within this framework, the periodic reachability and observability Gramians are usually estimated numerically via quadrature using the forward and adjoint post-transient response to impulses. However, this procedure can be computationally expensive, especially in the presence of slowly-decaying transients. Moreover, it can only be performed if the periodic orbit is stable in the sense of Floquet. In order to address these issues, we use the frequency-domain representation of the Gramians, which we henceforth refer to as frequential Gramians. First, these frequential Gramians are well-defined for both stable and unstable dynamics. In particular, we show that when the underlying system is unstable, these Gramians satisfy a pair of allied differential Lyapunov equations. Second, they can be estimated numerically by solving algebraic systems of equations that lend themselves to heavy computational parallelism and that deliver the desired post-transient response without having to follow physical transients. The computational gains that we can achieve by using the frequency domain are demonstrated on a simple three-dimensional toy model that exhibits time-periodic dynamics. We then demonstrate this method on a periodically-forced axisymmetric jet at Reynolds numbers Re=1250 and Re=1500. At the lower Reynolds number, the flow strongly amplifies subharmonic perturbations and exhibits vortex pairing about a Floquet-stable T-periodic solution. At the higher Reynolds number, the underlying T-periodic orbit is unstable and the flow naturally settles onto a 2T-periodic limit cycle characterized by pairing vortices. At both Reynolds numbers, we compute a reduced-order model and we use it to design a feedback controller and a state estimator capable of suppressing vortex pairing.

Stability of elliptic function solutions for the focusing modified KdV equation

We study the spectral and orbital stability of elliptic function solutions for the focusing modified Korteweg-de Vries (mKdV) equation and construct the corresponding breather solutions to exhibit stable or unstable dynamic behaviors. The elliptic function solutions of the mKdV equation and related fundamental solutions of the Lax pair are exactly represented by theta functions. Based on the ‘modified squared wavefunction’ (MSW) method, we construct all linear independent solutions of the linearized mKdV equation and then provide a necessary and sufficient condition of the spectral stability for elliptic function solutions with respect to subharmonic perturbations. In the case of spectrum stability, the orbital stability of elliptic function solutions is established in a suitable Hilbert space. Using Darboux-Bäcklund transformation, we construct breather solutions to exhibit unstable or stable dynamic behavior. Through analyzing the asymptotic behavior, we find that the breather solution under the cn-type solution background is equivalent to the elliptic function solution adding a small perturbation as t→±∞.

Spectral stability of elliptic solutions to the short-pulse equation

The short pulse (SP) equation describes the propagation of ultrashort optical pulses in nonlinear media and possesses a Lax pair of the Wadati–Konno–Ichikawa (WKI) type. In this paper, using integrability, we examine the spectral stability of elliptic solutions to the SP equation. Firstly, we analytically give an explicit description of the spectrum of the WKI-Lax operator to the SP equation for elliptic potentials. Then, by constructing the squared-eigenfunction connection between the non-standard linear stability problem (LZ=ΛZ′) and the Lax spectral problem, we prove that the elliptic solutions are spectrally stable with respect to subharmonic perturbations.

Periodic traveling waves in the ϕ^{4} model: Instability, stability, and localized structures.

We consider the instability and stability of periodic stationary solutions to the classical ϕ^{4} equationnumerically. In the superluminal regime, the model possesses dnoidal and cnoidal waves. The former are modulationally unstable and the spectrum forms a figureeight intersecting at the origin of the spectral plane. The latter can be modulationally stable, and the spectrum near the origin in that case is represented by vertical bands along the purely imaginary axis. The instability of the cnoidal states in that case stems from elliptical bands of complex eigenvalues far from the spectral plane origin. In the subluminal regime, there exist only snoidal waves which are modulationally unstable. Considering the subharmonic perturbations, we show that the snoidal waves in the subluminal regime are spectrally unstable with respect to all subharmonic perturbations, while for the dnoidal and cnoidal waves in the superluminal regime, the transition between the spectrally stable state and the spectrally unstable state occurs through a Hamiltonian Hopf bifurcation. The dynamical evolution of the unstable states is also considered, leading to some interesting localization events on the spatio-temporal backgrounds.

Sensitivity of wave merging and mixing to initial perturbations in Holmboe instabilities

Initial perturbations are commonly used in direct numerical simulations (DNS) to trigger the shear instability of stratified fluids. We investigate the effects of initial perturbations on the evolution of Holmboe instabilities with DNS. In particular, we model the interaction between a primary Holmboe wave and a subharmonic component that has a wavelength double that of the primary wave. We show that the phase difference and the amplitude of the primary and subharmonic components of the initial perturbation control the merging of Holmboe instabilities, which, in turn, influence diapycnal mixing in stratified flows. The amplitude difference has a more significant effect on the merging of Holmboe instabilities compared to the initial phase difference. For a given amplitude of the primary perturbation, a larger subharmonic perturbation results in an earlier merging event. In three-dimensional simulations, this preference of the subharmonic initial perturbation increased the amplitude of Holmboe waves by a factor of two. Although the subharmonic mode grows slower, it grows for longer producing more net mixing.

The orbital stability of the periodic traveling wave solutions to the defocusing complex modified Korteweg–de Vries equation

The stability of the elliptic solutions to the defocusing complex modified Korteweg–de Vries (cmKdV) equation is studied. Using the integrability of the defocusing cmKdV equation, we prove the spectral stability of the elliptic solutions. We show that one special linear combination of the first seven conserved quantities produces a Lyapunov functional, which implies that the elliptic solutions are orbitally stable with respect to the subharmonic perturbations.

Stability of elliptic solutions to the defocusing fourth order nonlinear Schrödinger equation

We study the stability of elliptic solutions to the integrable defocusing fourth order nonlinear Schrödinger (4NLS) equation with respect to subharmonic perturbations. With the help of the squared-eigenfunction connection, we prove the spectral stability by relating the Lax spectrum to the stability spectrum. We establish the orbital stability of elliptic solutions to the 4NLS equation by constructing a Lyapunov functional.

Analysis of the dynamics of subharmonic flow structures via the harmonic resolvent: Application to vortex pairing in an axisymmetric jet

We demonstrate the use of the harmonic resolvent to study the physics of subharmonic flow structures in the proximity of time-periodic solutions of the Navier-Stokes equation. As an illustrative example, we study the vortex pairing phenomenon in a periodically forced axisymmetric jet. While it is well-known that vortex pairing in jets and mixing layers arises from the interaction between subharmonic perturbations and the fundamental vortex shedding mode, here we provide additional insight. In particular, we uncover the nature of the strong cross-frequency interactions in the flow, and we compute the spatiotemporal mode that drives the pairing mechanism.

The instability of a helical vortex filament under a free surface

We perform a theoretical investigation of the instability of a helical vortex filament beneath a free surface in a semi-infinite ideal fluid. The focus is on the leading-order free-surface boundary effect upon the equilibrium form and instability of the vortex. This effect is characterised by the Froude number $F_r = U(gh^*)^{-{1}/{2}}$ where $g$ is gravity, and $U = \varGamma /(2{\rm \pi} b^*)$ with $\varGamma$ being the strength, $2{\rm \pi} b^*$ the pitch and $h^*$ the centre submergence of the helical vortex. In the case of $F_r \rightarrow 0$ corresponding to the presence of a rigid boundary, a new approximate equilibrium form is found if the vortex possesses a non-zero rotational velocity. Compared with the infinite fluid case (Widnall, J. Fluid Mech., vol. 54, no. 4, 1972, pp. 641–663), the vortex is destabilised (or stabilised) to relatively short- (or long-)wavelength sub-harmonic perturbations, but remains stable to super-harmonic perturbations. The wall-boundary effect becomes stronger for smaller helix angle and could dominate over the self-induced flow effect depending on the submergence. In the case of $F_r > 0$ , we obtain the surface wave solution induced by the vortex in the context of linearised potential-flow theory. The wave elevation is unbounded when the $m$ th wave mode becomes resonant as $F_r$ approaches the critical Froude numbers ${\mathcal {F}} (m) = (C_0^*/U)^{-1} (mh^*/b^*)^{-{1}/{2}}$ , $m=1, 2, \ldots,$ where $C_0^*$ is the induced wave speed. We find that the new approximate equilibrium of the vortex exists if and only if $F_r < {\mathcal {F}}(1)$ . Compared with the infinite fluid and $F_r \rightarrow 0$ cases, the wave effect causes the vortex to be destabilised to super-harmonic and long-wavelength sub-harmonic perturbations with generally faster growth rate for greater $F_r$ and smaller helix angle.

Stability of waves on fluid of infinite depth with constant vorticity

The stability of periodic travelling waves on fluid of infinite depth is examined in the presence of a constant background shear field. The effects of gravity and surface tension are ignored. The base waves are described by an exact solution that was discovered recently by Hur and Wheeler (J. Fluid Mech., vol. 896, 2020). Linear growth rates are calculated using both an asymptotic approach valid for small-amplitude waves and a numerical approach based on a collocation method. Both superharmonic and subharmonic perturbations are considered. Instability is shown to occur for any non-zero amplitude wave.

Transient linear stability of pulsating Poiseuille flow using optimally time-dependent modes

Time-dependent flows are notoriously challenging for classical linear stability analysis. Most progress in understanding the linear stability of these flows has been made for time-periodic flows via Floquet theory focusing on time-asymptotic stability. However, little attention has been given to the transient intracyclic linear stability of periodic flows since no general tools exist for its analysis. In this work, we explore the potential of using the recent framework of the optimally time-dependent (OTD) modes (Babaee & Sapsis, Proc. R. Soc. Lond. A, vol. 472, 2016, 20150779) to extract information about both the transient and the time-asymptotic linear stability of pulsating Poiseuille flow. The analysis of the instantaneous OTD modes in the limit cycle leads to the identification of the dominant instability mechanism of pulsating Poiseuille flow by comparing them with the spectrum and the eigenmodes of the Orr–Sommerfeld operator. In accordance with evidence from recent direct numerical simulations, it is found that structures akin to Tollmien–Schlichting waves are the dominant feature over a large range of pulsation amplitudes and frequencies but that for low pulsation frequencies these modes disappear during the damping phase of the pulsation cycle as the pulsation amplitude is increased beyond a threshold value. The maximum achievable non-normal growth rate during the limit cycle was found to be nearly identical to that in plane Poiseuille flow. The existence of subharmonic perturbation cycles compared with the base flow pulsation is documented for the first time in pulsating Poiseuille flow.

Stability of Elliptic Solutions to the sinh-Gordon Equation

Using the integrability of the sinh-Gordon equation, we demonstrate the spectral stability of its elliptic solutions. By constructing a Lyapunov functional using higher-order conserved quantities of the sinh-Gordon equation, we show that these elliptic solutions are orbitally stable with respect to subharmonic perturbations of arbitrary period.

Subharmonic dynamics of wave trains in reaction–diffusion systems

We investigate the stability and nonlinear local dynamics of spectrally stable wave trains in reaction–diffusion systems. For each N∈N, such T-periodic traveling waves are easily seen to be nonlinearly asymptotically stable (with asymptotic phase) with exponential rates of decay when subject to NT-periodic, i.e., subharmonic, perturbations. However, both the allowable size of perturbations and the exponential rates of decay depend on N, and, in particular, they tend to zero as N→∞, leading to a lack of uniformity in such subharmonic stability results. In this work, we build on recent work by the authors and introduce a methodology that allows us to achieve a stability result for subharmonic perturbations which is uniform in N. Our work is motivated by the dynamics of such waves when subject to perturbations which are localized (i.e. integrable on the line), which has recently received considerable attention by many authors.

Near-inertial parametric subharmonic instability of internal wave beams in a background mean flow

Abstract

Linear modulational and subharmonic dynamics of spectrally stable Lugiato-Lefever periodic waves

We study the linear dynamics of spectrally stable T-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. Such T-periodic solutions are nonlinearly stable to NT-periodic, i.e. subharmonic, perturbations for each N∈N with exponential decay rates of perturbations of the form e−δNt. However, both the exponential rates of decay δN and the allowable size of the initial perturbations tend to 0 as N→∞, so that this result is non-uniform in N and, in fact, empty in the limit N=∞. The primary goal of this paper is to introduce a methodology, in the context of the LLE, by which a uniform stability result for subharmonic perturbations may be achieved, at least at the linear level. The obtained uniform decay rates are shown to agree precisely with the polynomial decay rates of localized, i.e. integrable on the real line, perturbations of such spectrally stable periodic solutions of the LLE. This work both unifies and expands on several existing works in the literature concerning the stability and dynamics of such waves, and sets forth a general methodology for studying such problems in other contexts.

On the destabilization of a periodically driven three-dimensional torus

We report experimental evidence of the destabilization of a 3D torus obtained when a small subharmonic perturbation is added to a 2D torus characteristic of a driven relaxation oscillator. The Poincare sections indicate that the torus breakup is sensitive to the phase difference between the main driving frequency and its first subharmonic perturbing component. The observed transition confirms the Newhouse, Ruelle and Takens quasiperiodic transition to chaos on a 3D torus. Numerical results on a sinusoidally perturbed circle map mirror the experimental results and confirm the key role of the phase difference in the transition between distinct dynamical regimes.