Superharmonic Perturbations Research Articles

Superharmonic instability of stokes waves

AbstractA stability of nearly limiting Stokes waves to superharmonic perturbations is considered numerically in approximation of an infinite depth. Investigation of the stability properties can give one an insight into the evolution of the Stokes wave. The new, previously inaccessible branches of superharmonic instability were investigated. Our numerical simulations suggest that eigenvalues of linearized dynamical equations, corresponding to the unstable modes, appear as a result of a collision of a pair of purely imaginary eigenvalues at the origin, and a subsequent appearance of a pair of purely real eigenvalues: a positive and a negative one that are symmetric with respect to zero. Complex conjugate pairs of purely imaginary eigenvalues correspond to stable modes, and as the steepness of the underlying Stokes wave grows, the pairs move toward the origin along the imaginary axis. Moreover, when studying the eigenvalues of linearized dynamical equations we find that as the steepness of the Stokes wave grows, the real eigenvalues follow a universal scaling law, that can be approximated by a power law. The asymptotic power law behavior of this dependence for instability of Stokes waves close to the limiting one is proposed. Surface elevation profiles for several unstable eigenmodes are made available through http://stokeswave.org website.

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.

Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems

The problem of linear instability of a nonlinear travelling wave in a canonical Hamiltonian system with translational symmetry subject to superharmonic perturbations is discussed. It is shown that exchange of stability occurs when energy is stationary as a function of wave speed. This generalizes a result proved by Saffman (J. Fluid Mech., vol. 159, 1985, pp. 169–174) for travelling wave solutions exhibiting a wave profile with reflectional symmetry. The present argument remains true for any non-canonical Hamiltonian system that can be cast in Darboux form, i.e. a canonical Hamiltonian form on a submanifold defined by constraints, such as a two-dimensional surface wave on a constant shearing flow, revealing a general feature of Hamiltonian dynamics.

On the superharmonic instability of nonlinear three-dimensional interfacial waves of two infinite layers

Linear superharmonic stability analysis of nonlinear short-crested gravity interfacial waves between two semi-infinite fluids is numerically investigated. Like short-crested free surface waves, superharmonic instabilities correspond to the harmonic resonance phenomenon. The basic flow is calculated with a perturbation method and expansions are summed using Padé approximants. It is found that these instabilities develop near the poles due to harmonic resonance. The obtained results show that the instability of maximal growth rate develops for a density ratio μ=0.08 and occurs sporadically. In addition, for μ=0.34, the maximum growth rate is negligible and hence, for this value the short-crested interfacial waves are stable to superharmonic perturbations.

Linear stability of finite-amplitude capillary waves on water of infinite depth

AbstractWe study the linear stability of the exact deep-water capillary wave solution of Crapper (J. Fluid Mech., vol. 2, 1957, pp. 532–540) subject to two-dimensional perturbations (both subharmonic and superharmonic). By linearizing a set of exact one-dimensional non-local evolution equations, a stability analysis is performed with the aid of Floquet theory. To validate our results, the exact evolution equations are integrated numerically in time and the numerical solutions are compared with the time evolution of linear normal modes. For superharmonic perturbations, contrary to Hogan (J. Fluid Mech., vol. 190, 1988, pp. 165–177), who detected two bubbles of instability for intermediate amplitudes, our results indicate that Crapper’s capillary waves are linearly stable to superharmonic disturbances for all wave amplitudes. For subharmonic perturbations, it is found that Crapper’s capillary waves are unstable, and our results generalize to the highly nonlinear regime the analysis for small amplitudes presented by Chen & Saffman (Stud. Appl. Maths, vol. 72, 1985, pp. 125–147).

Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials

This work concerns superharmonic perturbations of a Gaussian measure given by a special class of positive weights in the complex plane of the form $w(z) = \exp(-|z|^2 + U^{\mu}(z))$, where $U^{\mu}(z)$ is the logarithmic potential of a compactly supported positive measure $\mu$. The equilibrium measure of the corresponding weighted energy problem is shown to be supported on subharmonic generalized quadrature domains for a large class of perturbing potentials $U^{\mu}(z)$. It is also shown that the $2\times 2$ matrix d-bar problem for orthogonal polynomials with respect to such weights is well-defined and has a unique solution given explicitly by Cauchy transforms. Numerical evidence is presented supporting a conjectured relation between the asymptotic distribution of the zeroes of the orthogonal polynomials in a semi-classical scaling limit and the Schwarz function of the curve bounding the support of the equilibrium measure, extending the previously studied case of harmonic polynomial perturbations with weights $w(z)$ supported on a compact domain.

Periodic solutions for a complex Hamiltonian system: New standing water-waves

The interaction of several (up to 39) two-dimensional, deep water, gravity, standing wave modes is studied to high order (up to 18) using a Hamiltonian formulation. Four new time-periodic solutions are found. The stability of periodic standing waves to superharmonic perturbations is studied. A new instability of the standing Stokes wave is found at moderate steepness (0.266). Two of the new waves are weakly unstable. Stability of the Stokes wave to subharmonic perturbation is also studied. The long time evolution of the perturbed waves is examined. The Hamiltonian method gives most accurate results. Recurrence of the unstable Stoke waves is revealed. The periodic solutions are useful for understanding the dynamical system of the standing wave modes.

On the stability of steep gravity waves

Previous calculations of the normal mode perturbations of steep gravity waves have suggested that the lowest superharmonic mode n = 2 becomes unstable at around ak = 0.436, where 2 a is the crest-to-trough height of the unperturbed wave and k is the wavenumber. This would correspond to the wave steepness at which the phase speed c is a maximum (considered as a function of ak ). However, numerical calculations at such high wave steepnesses can become inaccurate. The present paper studies analytically the conditions for the existence of a normal mode at zero limiting frequency. It is proved that for superharmonic perturbations such conditions will occur only for a pure phase-shift (corresponding to n = 1) or when the speed c is stationary with respect to the wave steepness, that is when d c = 0. Hence the limiting form of the instability found by Tanaka ( J. phys. Soc. Japan 52, 3047-3055 (1983)) near the value ak = 0.429 must be a pure phase-shift.

The dispersion of short wavelets in the presence of a dominant long wave

The characteristics are studied of short surface waves superimposed upon, and interacting with, a long, finite-amplitude dominant wave of frequency N . An asymptotic analysis allows the numerical investigation of Longuet-Higgins (1978) to be extended to higher superharmonic perturbations, and it is found that, although they are distorted by the underlying finite-amplitude wave, gravity wavelets continue to propagate freely provided the dominant wave does not break. Capillary waves can, however, be blocked by short, steep, non-breaking gravity waves, so that in a wind-wave tank at short fetch and high wind speed, freely travelling gravity-capillary waves can be erased by the successive dominant wave crests. A train or group of short gravity waves suffers modulations δ k in its local wave-number because of the straining of the long wave, and large modulations C δ k in its apparent frequency measured at a fixed point (where C is the long wave phase speed), largely because of the Doppler shifting produced by the dominant wave orbital velocity. The spectral signatures of a wave train are calculated by stationary phase and are found to have maxima at the upper wavenumber or frequency in the range. If an ensemble of short-wave groups is sampled at a given frequency f at a fixed point, the signal is derived from groups with a range of intrinsic frequencies δ, but is dominated by those at the long-wave crest for which f = δ + k.u 0 , where u 0 is the orbital velocity of the dominant wave. The apparent phase speed measured by a pair of such probes is the sum of the propagation speed c of the wavelet and the orbital velocity u 0 of the long wave. When f / N is large, the apparent phase speed approaches u 0 , independent of f . These results are consistent with measurements by Ramamonjiarisoa & Giovanangeli (1978) and others in which the apparent phase speed at high frequencies is found to be independent of the frequency — the measurements do not therefore imply a lack of dispersion of short gravity waves on the ocean surface.