Clustering of the resonant triads induced by vertical-shear instability in astrophysical discs

ABSTRACT Clustering of resonant triads that are induced by vertical-shear instability (VSI), driven by the combined effect of the vertical speed shear and small temperature gradients, is studied for vertically isothermal thin unmagnetized Keplerian discs. The authors’ recent study of isolated VSI resonant triads is extended to illustrate their clustering. The coupling conditions for two VSI resonant triads with one common mode are derived and generalized to higher dimension clustering. The clustering of two, three, and four triads connected via one common mode is numerically simulated. The numerical simulations demonstrate the chaotization of non-linear oscillations about the prototypes of the linearly stable modes with a growing cluster’s dimension that is accompanied by a decrease of the characteristic time of chaotization and an increase of the characteristic frequency of perturbations. The chaos associated with the VSI resonant clustering is believed to precede transition to sustainable turbulence in astrophysical discs.

Image Encryption Using Elliptic Curves and Rossby/Drift Wave Triads

We propose an image encryption scheme based on quasi-resonant Rossby/drift wave triads (related to elliptic surfaces) and Mordell elliptic curves (MECs). By defining a total order on quasi-resonant triads, at a first stage we construct quasi-resonant triads using auxiliary parameters of elliptic surfaces in order to generate pseudo-random numbers. At a second stage, we employ an MEC to construct a dynamic substitution box (S-box) for the plain image. The generated pseudo-random numbers and S-box are used to provide diffusion and confusion, respectively, in the tested image. We test the proposed scheme against well-known attacks by encrypting all gray images taken from the USC-SIPI image database. Our experimental results indicate the high security of the newly developed scheme. Finally, via extensive comparisons we show that the new scheme outperforms other popular schemes.

Linear and Weakly Nonlinear Energetics of Global Nonhydrostatic Normal Modes

Here the theory of global nonhydrostatic normal modes has been further developed with the analysis of both linear and weakly nonlinear energetics of inertia–acoustic (IA) and inertia–gravity (IG) modes. These energetics are analyzed in the context of a shallow global nonhydrostatic model governing finite-amplitude perturbations around a resting, hydrostatic, and isothermal background state. For the linear case, the energy as a function of the zonal wavenumber of the IA and IG modes is analyzed, and the nonhydrostatic effect of vertical acceleration on the IG waves is highlighted. For the nonlinear energetics analysis, the reduced equations of a single resonant wave triad interaction are obtained by using a pseudoenergy orthogonality relation. Integration of the triad equations for a resonance involving a short harmonic of an IG wave, a planetary-scale IA mode, and a short IA wave mode shows that an IG mode can allow two IA modes to exchange energy in specific resonant triads. These wave interactions can yield significant modulations in the dynamical fields associated with the physical-space solution with periods varying from a daily time scale to almost a month long.

Vertical shear-induced resonant triads in Keplerian discs

ABSTRACT The vertical-shear instability (VSI) is studied through weakly non-linear analysis of unmagnetized vertically isothermal thin Keplerian discs under small radial temperature gradients. Vertically global and radially local axisymmetric compressible perturbations are considered. The VSI excites three classes of quasi-resonant triads of non-linearly interacting modes characterized by distinct temporal evolution. Most of the triads belong to the two-mode regime of non-linear interaction. Such triads are comprised of one unstable non-linear mode that grows quasi-exponentially, and two other modes that practically decoupled from the former. The latter two modes perform non-linear oscillations around their either linear prototypes (class I) or respective initial values (class II). The rest of the resonant triads belong to class III where all three modes exhibit non-linear oscillations. The proposed model describes an intermediate non-linear stage of the VSI prior to its saturation.

Fast and slow resonant triads in the two-layer rotating shallow water equations

In this paper, we examine triad resonances in a rotating shallow water system when there are two free interfaces. This allows for an examination in a relatively simple model of the interplay between baroclinic and barotropic dynamics in a context where there is also a geostrophic mode. In contrast to the much-studied one-layer rotating shallow water system, we find that as well as the usual slow geostrophic mode, there are now two fast waves, a barotropic mode and a baroclinic mode. This feature permits triad resonances to occur between three fast waves, with a mixture of barotropic and baroclinic modes, an aspect that cannot occur in the one-layer system. There are now also two branches of the slow geostrophic mode, with a repeated branch of the dispersion relation. The consequences are explored in a derivation of the full set of triad interaction equations, using a multiscale asymptotic expansion based on a small-amplitude parameter. The derived nonlinear interaction coefficients are confirmed using energy and enstrophy conservation. These triad interaction equations are explored, with an emphasis on the parameter regime with small Rossby and Froude numbers.

Quantifying resonant and near-resonant interactions in rotating turbulence

Nonlinear triadic interactions are at the heart of our understanding of turbulence. In flows where waves are present, modes must not only be in a triad to interact, but their frequencies must also satisfy an extra condition: the interactions that dominate the energy transfer are expected to be resonant. We derive equations that allow direct measurement of the actual degree of resonance of each triad in a turbulent flow. We then apply the method to the case of rotating turbulence, where eddies coexist with inertial waves. We show that for a range of wavenumbers, resonant and near-resonant triads are dominant, the latter allowing a transfer of net energy towards two-dimensional modes that would be inaccessible otherwise. The results are in good agreement with approximations often done in theories of rotating turbulence, and with the mechanism of parametric instability proposed to explain the development of anisotropy in such flows. We also observe that, at least for the moderate Rossby numbers studied here, marginally near-resonant and non-resonant triads play a non-negligible role in the coupling of modes.

Critical and near-critical reflections of near-inertial waves off the sea surface at ocean fronts

In a balanced oceanic front, the possible directions of the group velocity vector for internal waves depart from the classic Saint Andrew’s cross as a consequence of sloping isopycnals and the associated thermal wind shear. However, for waves oscillating at the Coriolis frequency $f$, one of these directions remains horizontal, while the other direction allows for vertical propagation of energy. This implies the existence of critical reflections from the ocean surface, after which wave energy, having propagated from below, cannot propagate back down. This is similar to the reflection of internal waves, propagating in a quiescent medium, from a bottom that runs parallel to the group velocity vector. We first illustrate this phenomenon with a series of linear Boussinesq numerical experiments on waves with various frequencies, ${\it\omega}$, exploring critical (${\it\omega}=f$), forward (${\it\omega}>f$), and backward (${\it\omega}<f$) reflections. We then conduct the nonlinear equivalents of these simulations. In agreement with the classical case, backward reflection inhibits triadic resonances and does not exhibit prominent nonlinear effects, while forward reflection shows strong generation of harmonics that radiate energy away from the surface. Surprisingly though, critical reflections are associated with oscillatory motions that extend down from the surface. These motions are not freely propagating waves but instead take the form of a cluster of non-resonant triads which decays with depth through friction.