Resonant Triad Research Articles

S3T Stability of the Homogeneous State of Barotropic Beta-Plane Turbulence

Abstract Zonal jets and nonzonal large-scale flows are often present in forced–dissipative barotropic turbulence on a beta plane. The dynamics underlying the formation of both zonal and nonzonal coherent structures is investigated in this work within the statistical framework of stochastic structural stability theory (S3T). Previous S3T studies have shown that the homogeneous turbulent state undergoes a bifurcation at a critical parameter and becomes inhomogeneous with the emergence of zonal and/or large-scale nonzonal flows and that these statistical predictions of S3T are reflected in direct numerical simulations. In this paper, the dynamics underlying the S3T statistical instability of the homogeneous state as a function of parameters is studied. It is shown that, for weak planetary vorticity gradient β, both zonal jets and nonzonal large-scale structures form from upgradient momentum fluxes due to shearing of the eddies by the emerging infinitesimal flow. For large β, the dynamics of the S3T instability differs for zonal and nonzonal flows but in both the destabilizing vorticity fluxes decrease with increasing β. Shearing of the eddies by the mean flow continues to be the mechanism for the emergence of zonal jets while nonzonal large-scale flows emerge from resonant and near-resonant triad interactions between the large-scale flow and the stochastically forced eddies. The relation between the formation of large-scale structure through modulational instability and the S3T instability of the homogeneous state is also investigated and it is shown that the modulational instability results are subsumed by the S3T results.

Non-Gaussian interaction information: estimation, optimization and diagnostic application of triadic wave resonance

Abstract. Non-Gaussian multivariate probability distributions, derived from climate and geofluid statistics, allow for nonlinear correlations between linearly uncorrelated components, due to joint Shannon negentropies. Triadic statistical dependence under pair-wise (total or partial) independence is thus possible. Synergy or interaction information among triads is estimated. We formulate an optimization method of triads in the space of orthogonal rotations of normalized principal components, relying on the maximization of third-order cross-cumulants. Its application to a minimal one-dimensional, periodic, advective model leads to enhanced triads that occur between oscillating components of circular or locally confined wave trains satisfying the triadic wave resonance condition.

NONLINEAR DYNAMICS OF MAGNETOHYDRODYNAMIC ROSSBY WAVES AND THE CYCLIC NATURE OF SOLAR MAGNETIC ACTIVITY

The solar dynamo is known to be associated with several periodicities, with the nearly 11/22 yr cycle being the most pronounced one. Even though these quasiperiodic variations of solar activity have been attributed to the underlying dynamo action in the Sun's interior, a fundamental theoretical description of these cycles is still elusive. Here, we present a new possible direction in understanding the Sun's cycles based on resonant nonlinear interactions among magnetohydrodynamic (MHD) Rossby waves. The WKB theory for dispersive waves is applied to magnetohydrodynamic shallow-water equations describing the dynamics of the solar tachocline, and the reduced dynamics of a resonant triad composed of MHD Rossby waves embedded in constant toroidal magnetic field is analyzed. In the conservative case, the wave amplitudes evolve periodically in time, with periods on the order of the dominant solar activity timescale (~11 yr). In addition, the presence of linear forcings representative of either convection or instabilities of meridionally varying background states appears to be crucial in balancing dissipation and thus sustaining the periodic oscillations of wave amplitudes associated with resonant triad interactions. Examination of the linear theory of MHD Rossby waves embedded in a latitudinally varying mean flow demonstrates that MHD Rossby waves propagate toward the equator in a waveguide from –35° to 35° in latitude, showing a remarkable resemblance to the structure of the butterfly diagram of the solar activity. Therefore, we argue that resonant nonlinear magnetohydrodynamic Rossby wave interactions might significantly contribute to the observed cycles of magnetic solar activity.

A Lagrangian approach to the weakly nonlinear interaction of Kelvin waves and a symmetry-breaking bifurcation of a rotating flow

We develop a general framework of using the Lagrangian variables for calculating the energy of waves on a steady Euler flow and the mean flow induced by their nonlinear interaction. With the mean flow at hand we can determine, without ambiguity, all the coefficients of the amplitude equations to third order in amplitude for a rotating flow subject to a steady perturbation breaking the circular symmetry of the streamlines. Moreover, a resonant triad of waves is identified which brings in the secondary instability of the Moore–Saffman–Tsai–Widnall instability, and with the aid of the energetic viewpoint, resonant amplification of the waves without bound is numerically confirmed.

A numerical study on the energy transfer from surface waves to interfacial waves in a two-layer fluid system

AbstractInteraction between surface and interfacial waves with continuous energy spectra in a two-layer density stratified fluid system is investigated numerically. For an initial wave field which consists only of the surface waves all propagating in the same direction, it is confirmed that the spectra $S_{s}(k)$ of the surface waves and $S_{i}(k)$ of the interfacial waves change significantly due to the recently found class 3 triad resonance. When the bulk of the surface wave spectrum $S_{s}(k)$ is initially located well above the critical wavenumber $k_{crit}$, below which the class 3 triad resonance is prohibited, $S_{s}(k)$ downshifts gradually toward the lower wavenumber during the initial stage of evolution. However, this downshift halts when the peak of $S_{s}(k)$ reaches around $k_{crit}$, and after that a steep peak forms in $S_{s}(k)$ around $k_{crit}$. It is confirmed that the timescale of the spectral evolution is of $O(1/{\it\epsilon}^{2})$ (${\it\epsilon}$ is a characteristic non-dimensional wave amplitude) in most of the $k$ space, consistent with the prediction of the wave turbulence theory for a system with a decay-type dispersion relation. However, it is also found that the timescale of the formation and growth of the sharp peak in $S_{s}(k)$ around $k_{crit}$ is of $O(1/{\it\epsilon})$, i.e. the timescale of the deterministic three-wave resonance.

On the transition from the laminar to disordered flow in a precessing spherical-like cylinder

We investigate, through both asymptotic analysis and direct numerical simulation, precessionally driven flow of a homogeneous fluid confined in a fluid-filled circular cylinder that rotates rapidly about its symmetry axis and precesses about a different axis that is fixed in space. A particular emphasis is placed on a spherical-like cylinder whose diameter is nearly the same as its length. At this special aspect ratio, the strongest direct resonance occurs between the spatially simplest inertial mode and the precessional Poincaré forcing. An asymptotic analytical solution in closed form describing weakly precessing flow is derived in the mantle frame of reference for asymptotically small Ekman numbers. We also construct a nonlinear three-dimensional finite element model – which is validated against both the asymptotic solution and a constructed exact solution – for elucidating the nonlinear transition leading to disordered flow in the precessing spherical-like cylinder. Properties of both weakly and strongly precessing flows are investigated with the aid of a complete inertial-mode decomposition of the fully nonlinear solution. Despite a large effort being made, the well-known triadic resonance is not found in the precessing spherical-like cylinder. The energy contained in the precessionally forced inertial mode is primarily transferred, through nonlinear effects in the viscous boundary layers, to the geostrophic flow that becomes predominant when the precessional Poincaré force is sufficiently large. It is found that the nonlinear flow evolutes gradually and progressively from the laminar to disordered as the precessional force increases.

Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wavetrains

AbstractInternal gravity wavetrains in continuously stratified fluids are generally unstable as a result of resonant triad interactions which, in the inviscid limit, amplify short-scale perturbations with frequency equal to one half of that of the underlying wave. This so-called parametric subharmonic instability (PSI) has been studied extensively for spatially and temporally monochromatic waves. Here, an asymptotic analysis of PSI for time-harmonic plane waves with locally confined spatial profile is made, in an effort to understand how such wave beams differ, in regard to PSI, from monochromatic plane waves. The discussion centres upon a system of coupled evolution equations that govern the interaction of a small-amplitude wave beam with short-scale subharmonic wavepackets in a nearly inviscid uniformly stratified Boussinesq fluid. For beams with general localized profile, it is found that triad interactions are not strong enough to bring about instability in the limited time that subharmonic perturbations overlap with the beam. On the other hand, for quasi-monochromatic wave beams whose profile comprises a sinusoidal carrier modulated by a locally confined envelope, PSI is possible if the beam is wide enough. In this instance, a stability criterion is proposed which, under given flow conditions, provides the minimum number of carrier wavelengths a beam of small amplitude must comprise for instability to arise.

Wave motion in a heavy compressible fluid: Revisited

The effect of the compressibility of the ocean and its role in generating progressive compression-type waves (acoustic-gravity waves) are revisited. Originally, Longuet-Higgins (1950) obtained a solution for the nonlinear interaction of two opposite waves in a compressible fluid, which results in the generation of compression waves. In this paper we extend this solution to include the general interaction of waves of profoundly different wavelengths. We also fully solve the special triad resonance case obtained by Longuet-Higgins (1950).

Can tidal perturbations associated with sea level variations in the western Pacific Ocean be used to understand future effects of tidal evolution?

This study examines connections between mean sea level (MSL) variability and diurnal and semidiurnal tidal constituent variations at 17 open-ocean and 9 continental shelf tide gauges in the western tropical Pacific Ocean, a region showing anomalous rise in MSL over the last 20 years and strong interannual variability. Detrended MSL fluctuations are correlated with detrended tidal amplitude and phase fluctuations, defined as tidal anomaly trends (TATs), to quantify the response of tidal properties to MSL variation. About 20 significant amplitude and phase TATs are found for each of the two strongest tidal constituents, K1 (diurnal) and M2 (semidiurnal). Lesser constituents (O1 and S2) show trends at nearly half of all gauges. Fluctuations in MSL shift amplitudes and phases; both positive and negative responses occur. Changing overtides suggest that TATs are influenced by changing shallow water friction over the equatorial Western Pacific and the eastern coast of Australia (especially near the Great Barrier Reef). There is a strong connection between semidiurnal TATs at stations around the Solomon Islands and changes in thermocline depth, overtide generation, and the El Nino Southern Oscillation (ENSO). TATs for O1, K1, and M2 are related to each other in a manner that suggests transfer of energy from M2 to the two diurnals via resonant triad interactions; these cause major tidal variability on sub-decadal time scales, especially for M2. The response of tides to MSL variability is not only spatially complex, it is frequency dependent; therefore, short-term responses may not predict long-term behavior.

Experimental study of fluid flows in a precessing cylindrical annulus

The flow inside a precessing fluid cavity has been given particular attention since the end of the 19th century in geophysical and industrial contexts. The present study aims at shedding light on the underlying mechanism by which the flow inside a precessing cylindrical annulus transitions from laminar to multiple scale complex structures. We address this problem experimentally using ultrasonic Doppler velocimetry to diagnose the fluid velocity in a rotating and precessing cylindrical annulus. When precession is weak, the flow can be described as a superposition of forced inertial modes. Above a critical value of the precession rate, the forced flow couples with two free inertial modes satisfying triadic resonance conditions, leading to the classical growth and collapse. Using a Bayesian approach, we extract the wavenumber, frequency, growth rate, and amplitude of each mode involved in the instability. In some cases, we observe for the first time ever experimentally two pairs of free modes coexisting with the forced flow. At larger precession rates, we do not observe triadic resonance any more, instead we observe several harmonics whose frequencies are integer multiples of the rotation frequency.

An equivalent physical model of three acoustic waves interaction in nonlinear medium

Among the parametric effects, three waves interaction has been widely investigated in many fields of physics, such as nonlinear optics and plasma. The interaction of three acoustic waves in the medium with quadratic nonlinearity, which must satisfy conditions on their frequencies and wavenumbers, is essentially confined to energy exchange between three components. Based on Burgers equation, the propagation rules of three acoustic waves, which are called a resonant triad, are governed by a drastically simplified system of three coupled nonlinear ordinary differential equations. A mathematical equivalence between the equations for an acoustic triad and a simple parametric vibration system, the undamped elastic pendulum, is discussed in this paper by a multiple time-scale analysis. We study the dynamics of this system, drawing analogies between its behavior and that of the acoustic triad. Finally, it is certified that three acoustic waves interaction can be described by Mathieu type equations in case one acoustic wave is much stronger than the others in three waves. This means that experiments with an elastic pendulum can give us new insights into dynamics of mechanism in three acoustic waves interaction.

Wave turbulence in a rotating channel

AbstractThis paper describes wave-turbulence closure and its consequences for rapidly rotating (i.e. small Rossby number) turbulence confined by two infinite, parallel walls perpendicular to the rotation axis. Expressing the flow as a combination of inertial waveguide modes leads to a spectral matrix, whose diagonal elements express the distribution of energy over modes and whose off-diagonal elements represent correlations between modes of different orders. In preparation for wave-turbulence closure, the flow is decomposed into two-dimensional and wave components. The former is found to evolve as if it were a classical, two-dimensional, non-rotating flow, but with wall friction due to Ekman pumping by the boundary layers. Evolution equations for the wave-component elements of the spectral matrix are derived using a wave-turbulence approach. Detailed analysis of these equations shows that, surprisingly, the two-dimensional component has no effect on wave-component energetics. As expected for wave turbulence, energy transfer between wave modes is via resonant triads and takes place at times $O(\varepsilon ^{-2})$ multiples of the rotational period, where $\varepsilon $ is the Rossby number. Despite playing no role in wave-mode energetics, the two-dimensional component produces decay of the off-diagonal elements of the spectral matrix on a time scale that is small compared with $O(\varepsilon ^{-2})$ rotation periods. There are thus three asymptotically distinct stages in the evolution of the turbulence in the limit of small Rossby number: the two-dimensional flow begins to evolve at the usual large-eddy turnover time scale ($O(\varepsilon ^{-1})$ multiples of the rotation period) and continues to develop thereafter. This is followed by decorrelation of different wave orders and finally evolution of the wave energy spectra due to resonant interactions.

Resonant instability of the non-linearly saturated magnetorotational mode in thin Keplerian discs

The magneto-rotational decay instability (MRDI) of thin Keplerian discs threaded by poloidal magnetic fields is introduced and studied. The linear magnetohydrodynamic problem decouples into eigenvalue problems for in-plane slow- and fast- Alfv'een-Coriolis (AC), and vertical magnetosonic (MS) eigenmodes. The magnetorotational instability (MRI) is composed of a discrete number of unstable slow AC eigenmodes that is determined for each radius by the local beta. In the vicinity of the first beta threshold a parent MRI eigenmode together with a stable AC eigenmode (either slow or fast) and a stable MS eigenmode form a resonant triad. The three-wave MRDI relies on the nonlinear saturation of the parent MRI mode and the exponential growth of two daughter linearly stable waves, slow-AC and MS modes with an effective growth rate that is comparable to that of the parent MRI. If, however, the role of the AC daughter wave is played by a stable fast mode, all three modes remain bounded.

A Constructive Method for Computing Generalized Manley-Rowe Constants of Motion

AbstractThe Manley-Rowe constants of motion (MRC) are conservation laws written out for a dynamical system describing the time evolution of the amplitudes in resonant triad. In this paper we extend the concept of MRC to resonance clusters of any form yielding generalized Manley-Rowe constants (gMRC) and give a constructive method how to compute them. We also give details of a Mathematica implementation of this method. While MRC provide integrability of the underlying dynamical system, gMRC generally do not but may be used for qualitative and numerical study of dynamical systems describing generic resonance clusters.

Degradation of an internal wave beam by parametric subharmonic instability in an upper ocean pycnocline

Numerical simulations are performed to investigate the interaction of a semidiurnal internal wave (IW) beam with the nonuniform stratification of an upper ocean pycnocline. During the initial stage of the interaction, higher harmonics originate after reflection of the IW beam at the caustic and are trapped in the pycnocline. In cases where the pycnocline thickness is sufficiently large (approximately larger than twice the dominant wavelength in the beam), the incoming beam undergoes a parametric subharmonic instability (PSI) during refraction through the pycnocline at later time. Consequently, there is significant energy transfer to subharmonic motions that exhibit exponential growth with a rate of 2/3 day−1. The instability is demonstrated to be a resonant triad interaction by diagnostics in both wave frequency and wave number spaces. Smaller vertical scales originate during the PSI, lead to wave steepening, and produce convective overturns. This work identifies a potential mechanism that drives IW beam degradation during its propagation through the upper ocean.

An extended equatorial plane: linear spectrum and resonant triads

The usual equatorial -plane supports a range of wave types, including trapped baroclinic Rossby waves, a Yanai wave and a Kelvin wave, which are known to be important in equatorial wave dynamics, both linear and nonlinear. Here, we extend this model to the whole sphere, using a Mercator projection, in which the Coriolis parameter is retained in full, but the metric coefficients are approximated by their values at the equator. We find that while the trapped baroclinic waves are preserved in this model, the barotropic waves are also trapped, with consequences for nonlinear resonant triad interactions.

A Global Model for the Diapycnal Diffusivity Induced by Internal Gravity Waves

AbstractAn energetically consistent model for the diapycnal diffusivity induced by breaking of internal gravity waves is proposed and tested in local and global settings. The model [Internal Wave Dissipation, Energy and Mixing (IDEMIX)] is based on the spectral radiation balance of the wave field, reduced by integration over the wavenumber space, which yields a set of balances for energy density variables in physical space. A further simplification results in a single partial differential equation for the total energy density of the wave field. The flux of energy to high vertical wavenumbers is parameterized by a functional derived from the wave–wave scattering integral of resonant wave triad interactions, which also forms the basis for estimates of dissipation rates and related diffusivities of ADCP and hydrography fine-structure data. In the current version of IDEMIX, the wave energy is forced by wind-driven near-inertial motions and baroclinic tides, radiating waves from the respective boundary layers at the surface and the bottom into the ocean interior. The model predicts plausible magnitudes and three-dimensional structures of internal wave energy, dissipation rates, and diapycnal diffusivities in rough agreement to observational estimates. IDEMIX is ready for use as a mixing module in ocean circulation models and can be extended with more spectral components.

Nonlinear fate of internal wave attractors.

We present a laboratory study on the instability of internal wave attractors in a trapezoidal fluid domain filled with uniformly stratified fluid. Energy is injected into the system via standing-wave-type motion of a vertical wall. Attractors are found to be destroyed by parametric subharmonic instability via a triadic resonance which is shown to provide a very efficient energy pathway from long to short length scales. This Letter provides an explanation of why attractors may be difficult or impossible to observe in natural systems subject to large amplitude forcing.

The route to dissipation in strongly stratified and rotating flows

AbstractWe investigate the route to dissipation in strongly stratified and rotating systems through high-resolution numerical simulations of the Boussinesq equations (BQs) and the primitive equations (PEs) in a triply periodic domain forced at large scales. By applying geostrophic scaling to the BQs and using the same horizontal length scale in defining the Rossby and the Froude numbers, $\mathit{Ro}$ and $\mathit{Fr}$, we show that the PEs can be obtained from the BQs by taking the limit ${\mathit{Fr}}^{2} / {\mathit{Ro}}^{2} \rightarrow 0$. When ${\mathit{Fr}}^{2} / {\mathit{Ro}}^{2} $ is small the difference between the results from the BQ and the PE simulations is shown to be small. For large rotation rates, quasi-geostrophic dynamics are recovered with a forward enstrophy cascade and an inverse energy cascade. As the rotation rate is reduced, a fraction of the energy starts to cascade towards smaller scales, leading to a shallowing of the horizontal spectra from ${ k}_{h}^{- 3} $ to ${ k}_{h}^{- 5/ 3} $ at the small-scale end. The vertical spectra show a similar transition as the horizontal spectra and we find that Charney isotropy is approximately valid also at larger wavenumbers than the transition wavenumber. The high resolutions employed allow us to capture both ranges within the same simulation. At the transition scale, kinetic energy in the rotational and in the horizontally divergent modes attain comparable values. The divergent energy is several orders of magnitude larger than the quasi-geostrophic divergent energy given by the $\Omega $-equation. The amount of energy cascading downscale is mainly controlled by the rotation rate, with a weaker dependence on the stratification. A larger degree of stratification favours a downscale energy cascade. For intermediate degrees of rotation and stratification, a constant energy flux and a constant enstrophy flux coexist within the same range of scales. In this range, the enstrophy flux is a result of triad interactions involving three geostrophic modes, while the energy flux is a result of triad interactions involving at least one ageostrophic mode, with a dominant contribution from interactions involving two ageostrophic and one geostrophic mode. Dividing the ageostrophic motions into two classes depending on the sign of the linear wave frequency, we show that the energy transfer is for the largest part supported by interactions within the same class, ruling out the wave–wave–vortex resonant triad interaction as a mean of the downscale energy transfer. The role of inertia-gravity waves is studied through analyses of time-frequency spectra of single Fourier modes. At large scales, distinct peaks at frequencies predicted for linear waves are observed, whereas at small scales no clear wave activity is observed. Triad interactions show a behaviour which is consistent with turbulent dynamics, with a large exchange of energy in triads with one small and two large comparable wavenumbers. The exchange of energy is mainly between the modes with two comparable wavenumbers.

Nonlinear resonant interactions of interfacial waves in horizontal stratified channel flows

AbstractWe consider the problem of nonlinear resonant interactions of interfacial waves with the presence of a linear interfacial instability in an inviscid two-fluid stratified flow through a horizontal channel. The resonant triad consists of a (linearly) unstable wave and two stable waves, one of which has a wavelength that can be much longer than that of the unstable component. Of special interest is the development of the long wave by energy transfer from the base flow due to the coupled effect of nonlinear resonance and interfacial instability. By use of the method of multiple scales, we derive the interaction equations which govern the time evolution of the amplitudes of the interacting waves including the effect of interfacial instability. The solution of the evolution equations shows that depending on the flow conditions, the (stable) long wave can achieve a bi-exponential growth rate through the resonant interaction with the unstable wave. Moreover, the unstable wave can grow unboundedly even when the nonlinear self-interaction effect is included, as do the stable waves in the associated resonant triad. For the verification of the theoretical analysis and the practical application involving a broadbanded spectrum of waves, we develop an effective direct simulation method, based on a high-order pseudo-spectral approach, which accounts for nonlinear interactions of interfacial waves up to an arbitrary high order. The direct numerical simulations compare well with the theoretical analysis for all of the characteristic flows considered, and agree qualitatively with the experimental observation of slug development near the entrance of two-phase flow into a pipe.