Resonant Triad Research Articles

Nonlinear aspects of focusing internal waves

When a torus oscillates horizontally in a linearly stratified fluid, the wave rays form a double cone, one upward and one downward, with two focal points where the wave amplitude has a maximum due to wave focusing. Following a former study on linear aspects of wave focusing (Ermanyuk et al., J. Fluid Mech., vol. 813, 2017, pp. 695–715), we here consider experimental results on the nonlinear aspects that occur in the focal region below the torus for higher-amplitude forcing. A new non-dimensional number that is based on heuristic arguments for the wave amplitude in the focal area is presented. This focusing number is defined as $Fo=(A/a)\unicode[STIX]{x1D716}^{-1/2}f(\unicode[STIX]{x1D703})$, with oscillation amplitude $A$, $f(\unicode[STIX]{x1D703})$ a function for the variation of the wave amplitude with wave angle $\unicode[STIX]{x1D703}$, and $\unicode[STIX]{x1D716}^{1/2}=\sqrt{b/a}$ the increase in amplitude due to the focusing, with $a$ and $b$, respectively, the minor and major radius of the torus. Nonlinear effects occur for $Fo\geqslant 0.1$, with the shear stress giving rise to a mean flow which results in the focal region in a central upward motion partially surrounded by a downward motion. With increasing $Fo$, the Richardson number $Ri$ measured from the wave steepness monotonically decreases. Wave breaking occurs at $Fo\approx 0.23$, corresponding to $Ri=0.25$. In this regime, the focal region is unstable due to triadic wave resonance. For the different tori sizes under consideration, the triadic resonant instability in these three-dimensional flows resembles closely the resonance observed by Bourget et al. (J. Fluid Mech., vol. 723, 2013, pp. 1–20) for a two-dimensional flow, with only minor differences. Application to internal tidal waves in the ocean are discussed.

On resonant interactions of gravity‐capillary waves without energy exchange

AbstractWe consider resonant triad interactions of gravity‐capillary waves and investigate in detail special resonant triads that exchange no energy during their interactions so that the wave amplitudes remain constant in time. After writing the resonance conditions in terms of two parameters (or two angles of wave propagation), we first identify a region in the two‐dimensional parameter space, where resonant triads can be always found, and then describe the variations of resonant wavenumbers and wave frequencies over the resonance region. Using the amplitude equations recovered from a Hamiltonian formulation for water waves, it is shown that any resonant triad inside the resonance region can interact without energy exchange if the initial wave amplitudes and relative phase satisfy the two conditions for fixed point solutions of the amplitude equations. Furthermore, it is shown that the symmetric resonant triad exchanging no energy forms a transversely modulated traveling wave field, which can be considered a two‐dimensional generalization of Wilton ripples.

Effects of distributed roughness on crossflow instability through generalized resonance mechanisms

Experiments have shown that micron-sized distributed surface roughness can significantly promote transition in a three-dimensional boundary layer dominated by crossflow instability. This sensitive effect has not yet been fully explained physically and mathematically. Past studies focused on surface roughness exciting crossflow vortices and/or changing the local stability characteristics. The present paper seeks possible additional mechanisms by investigating the effects of distributed surface roughness on crossflow instability through resonant interactions with eigenmodes. A key observation is that the perturbation induced by roughness with specific wavenumbers can interact with two eigenmodes (travelling and stationary vortices) through triadic resonance, or interact with one eigenmode (stationary vortices) through Bragg scattering. Unlike the usual triadic resonance of neutral, or nearly neutral, eigenmodes, the present triadic resonance can take place among modes with $O(1)$ growth rates, provided that these are equal; unlike the usual Bragg scattering involving neutral waves, crossflow stationary vortices can also be unstable. For these amplifying waves, the generalized triadic resonance and Bragg scattering are put forward, and the resulting corrections to the growth rates are derived by a multiple-scale method. The analysis is extended to the case where up to four crossflow vortices interact with each other in the presence of suitable roughness components. The numerical results for Falkner–Skan–Cooke boundary layers show that roughness with a small height (a few percent of the local boundary-layer thickness) can change growth rates substantially (by a more-or-less $O(1)$ amount). This sensitive effect is attributed to two facts: (i) the resonant nature of the triadic interaction and Bragg scattering, which makes the correction to the growth rate proportional to the roughness height, and (ii) the wavenumbers of the roughness component required for the resonance are close to those of the neutral stationary crossflow modes, as a result of which a small roughness can generate a large response. Another important effect of roughness is that its presence renders the participating eigenmodes, which are otherwise independent, fully coupled. Our theoretical results suggest that micron-sized distributed surface roughness influences significantly both the amplification and spectral composition of crossflow vortices.

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.

Upgradient and Downgradient Potential Vorticity Fluxes Produced by Forced Rossby Waves. Part II: Parameter Sensitivity and Physical Interpretation

AbstractWe examine the potential vorticity (PV) flux produced by forced Rossby waves in a two-layer quasigeostrophic model, using a perturbation analysis. Rossby waves are excited by external forcing applied to the upper layer. The southward PV flux is produced in the lower layer by the higher-order Rossby waves that are excited by nonlinear wave–wave interactions, whereas the northward PV flux is produced in the upper layer. The direction of the PV flux is consistent with that obtained by an eddy-resolving model of the wind-driven circulation in previous studies. The southward PV flux is produced in a wide parameter range comparable to the eddy-resolving model. The basic features of the PV flux remain unchanged even in the limit of weak stratification. In this limit, stratification has nearly no effect on the flow, except that it isolates the lower layer from the direct effects of external forcing. The mechanism of the southward PV flux is explained using basic features of the barotropic Rossby waves and does not depend on details of the model. Furthermore, the resonant triad interaction of Rossby waves does not affect the PV flux. Stratification weakens or strengthens the PV flux depending on the horizontal scale of the external forcing.

On triadic resonance as an instability mechanism in precessing cylinder flow

Contained rotating flows subject to precessional forcing are well known to exhibit rapid and energetic transitions to disorder. Triadic resonance of inertial modes has been previously proposed as an instability mechanism in such flows, and that idea was developed into a successful model for predicting instability in a cylindrical container when departures from solid-body rotation are sufficiently small. Using direct numerical simulation and dynamic mode decomposition, we analyse instabilities of precessing cylinder flows whose three-dimensional basic states, steady in the gimbal frame of reference, may depart substantially from solid-body rotation. In the gimbal frame, the instability can be interpreted as resulting from a supercritical Hopf bifurcation that results in a limit-cycle flow. In the cylinder frame of reference, the basic state is a rotating wave with azimuthal wavenumber $m=1$, and the instability satisfies triadic-resonance conditions with the instability mode maintaining a fixed orientation with respect to the basic state. Thus, we are able to demonstrate the existence of two alternative but congruent explanations for the instability. Additionally, we show that basic states may depart substantially from solid-body rotation even with modest cylinder tilt angles, and growth rates for instabilities may be sufficiently large that nonlinear saturation to disordered states can occur within approximately ten cylinder revolutions, in agreement with experimental observations.

Precession effects on a liquid planetary core

Motivated by the desire to understand the rich dynamics of precessionally driven flow in a liquid planetary core, we investigate, through numerical simulations, the precessing fluid motion in a rotating cylindrical annulus, which simultaneously possesses slow precession. The same problemhas been studied extensively in cylinders, where the precessing flow is characterized by three key parameters: the Ekman number E, the Poincaré number Po and the radius-height aspect ratio Γ. While in an annulus, there is another parameter, the inner-radius-height aspect ratio ϒ, which also plays an important role in controlling the structure and evolution of the flow. By decomposing the nonlinear solution into a set of inertial modes, we demonstrate the properties of both weakly and moderately precessing flows. It is found that, when the precessional force is weak, the flow is stable with a constant amplitude of kinetic energy. As the precessional force increases, our simulation suggests that the nonlinear interaction between the boundary effects and the inertial modes can trigger more turbulence, introducing a transitional regime of rich dynamics to disordered flow. The inertial mode u111, followed by u113 or u112, always dominates the precessing flow when 0.001 ≤ Po ≤ 0.05, ranging from weak to moderate precession. Moreover, the precessing flow in an annulus shows more stability than in a cylinder which is likely to be caused by the effect of the inner boundary that restricts the growth of resonant and non-resonant inertial modes. Furthermore, the mechanism of triadic resonance is not found in the transitional regime from a laminar to disordered flow.

Rapidly rotating precessing cylinder flows: forced triadic resonances

Rapidly rotating cylinder flows subjected to low-amplitude precessional forcing are studied numerically over a range of cylinder and precessional rotation rates. For sufficiently small rotation rates, viscous effects lead to a forced overturning flow that is steady in the precession (table) frame of reference. Increasing the rotation rates, this forced flow loses stability in a Hopf bifurcation, which can be either supercritical or subcritical, and may preserve or break the symmetry of the system, depending on the parameter regime studied. Regardless of these details of the Hopf bifurcation, it is found that the Hopf instability is associated with a slightly detuned triadic resonance between the forced overturning flow and two free Kelvin modes (inviscid eigenmodes of the rotating cylinder). Further increases in rotation rates lead to a sequence of secondary instabilities which also follow a generic pattern irrespective of the parameter regime investigated. The relationship between this sequence of instabilities and the resultant nonlinear dynamics with the experimentally observed phenomenon of resonant collapse is discussed.

Silent inflow condition for turbulent boundary layers

A method is proposed to trigger inflow turbulence while minimizing spurious noise in order to study the noise radiated directly by a turbulent boundary layer. The strategy is based on a controlled transition by entering a resonant triad of unstable modes based on an inflectional velocity profile.

Inverse cascades and resonant triads in rotating and stratified turbulence

Kraichnan seminal ideas on inverse cascades yielded new tools to study common phenomena in geophysical turbulent flows. In the atmosphere and the oceans, rotation and stratification result in a flow that can be approximated as two-dimensional at very large scales, but which requires considering three-dimensional effects to fully describe turbulent transport processes and non-linear phenomena. Motions can thus be classified into two classes: fast modes consisting of inertia-gravity waves, and slow quasi-geostrophic modes for which the Coriolis force and horizontal pressure gradients are close to balance. In this paper we review previous results on the strength of the inverse cascade in rotating and stratified flows, and then present new results on the effect of varying the strength of rotation and stratification (measured by the ratio $N/f$ of the Brunt-V\"ais\"ala frequency to the Coriolis frequency) on the amplitude of the waves and on the flow quasi-geostrophic behavior. We show that the inverse cascade is more efficient in the range of $N/f$ for which resonant triads do not exist, $1/2 \le N/f \le 2$. We then use the spatio-temporal spectrum, and characterization of the flow temporal and spatial scales, to show that in this range slow modes dominate the dynamics, while the strength of the waves (and their relevance in the flow dynamics) is weaker.

Multiscale Atmosphere–Ocean Interactions and the Low-Frequency Variability in the Equatorial Region

Abstract In the present study a simplified multiscale atmosphere–ocean coupled model for the tropical interactions among synoptic, intraseasonal, and interannual scales is developed. Two nonlinear equatorial β-plane shallow-water equations are considered: one for the ocean and the other for the atmosphere. The nonlinear terms are the intrinsic advective nonlinearity and the air–sea coupling fluxes. To mimic the main differences between the fast atmosphere and the slow ocean, suitable anisotropic multispace/multitime scalings are applied, yielding a balanced synoptic–intraseasonal–interannual–El Niño (SInEN) regime. In this distinguished balanced regime, the synoptic scale is the fastest atmospheric time scale, the intraseasonal scale is the intermediate air–sea coupling time scale (common to both fluid flows), and El Niño refers to the slowest interannual ocean time scale. The asymptotic SInEN equations reveal that the slow wave amplitude evolution depends on both types of nonlinearities. Analytic solutions of the reduced SInEN equations for a single atmosphere–ocean resonant triad illustrate the potential of the model to understand slow-frequency variability in the tropics. The resonant nonlinear wind stress allows a mechanism for the synoptic-scale atmospheric waves to force intraseasonal variability in the ocean. The intraseasonal ocean temperature anomaly coupled with the atmosphere through evaporation forces synoptic and intraseasonal atmospheric variability. The wave–convection coupling provides another source for higher-order atmospheric variability. Nonlinear interactions of intraseasonal ocean perturbations may also force interannual oceanic variability. The constrains that determine the establishment of the atmosphere–ocean resonant coupling can be viewed as selection rules for the excitation of intraseasonal variability (MJO) or even slower interannual variability (El Niño).

Internal wave resonant triads in finite-depth non-uniform stratifications

We present a theoretical study of nonlinear effects that result from modal interactions in internal waves in a non-uniformly stratified finite-depth fluid with background rotation. A linear wave field containing modes $m$ and $n$ (of horizontal wavenumbers $k_{m}$ and $k_{n}$) at a fixed frequency $\unicode[STIX]{x1D714}$ results in two different terms in the steady-state weakly nonlinear solution: (i) a superharmonic wave of frequency $2\unicode[STIX]{x1D714}$, horizontal wavenumber $k_{m}+k_{n}$ and a vertical structure $\bar{h}_{mn}(z)$ and (ii) a time-independent term (Eulerian mean flow) with horizontal wavenumber $k_{m}-k_{n}$. For some $(m,n)$, $\bar{h}_{mn}(z)$ is infinitely large along specific curves on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane, where $N_{0}$ and $f$ are the deep ocean stratification and the Coriolis frequency, respectively; these curves are referred to as divergence curves in the rest of this paper. In uniform stratifications, a unique divergence curve occurs on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane for those $(m,n\neq m)$ that satisfy $(m/3)<n<(3m)$. In the presence of a pycnocline (whose strength is quantified by the maximum stratification $N_{max}$), divergence curves occur for several more modal interactions than those for a uniform stratification; furthermore, a given $(m,n)$ interaction can result in multiple divergence curves on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane for a fixed $N_{max}/N_{0}$. Nearby high-mode interactions in a uniform stratification and any modal interaction in a non-uniform stratification with a sufficiently strong pycnocline are shown to result in near-horizontal divergence curves around $f/\unicode[STIX]{x1D714}\approx 1$, thus implying that strong nonlinear effects often occur as a result of interaction within triads containing two different modes at the near-inertial frequency. Notably, self-interaction of certain modes in a non-uniform stratification results in one or more divergence curves on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane, thus suggesting that even arbitrarily small-amplitude individual modes cannot remain linear in a non-uniform stratification. We show that internal wave resonant triads containing modes $m$ and $n$ at frequency $\unicode[STIX]{x1D714}$ occur along the divergence curves, and their existence is guaranteed upon the satisfaction of two different criteria: (i) the horizontal component of the standard triadic resonance criterion $\boldsymbol{k}_{1}+\boldsymbol{k}_{2}+\boldsymbol{k}_{3}=0$ and (ii) a non-orthogonality criterion. For uniform stratifications, criterion (ii) reduces to the vertical component of the standard triadic resonance criterion. For non-uniform stratifications, criterion (ii) seems to be always satisfied whenever criterion (i) is satisfied, thus significantly increasing the number of modal interactions that result in strong nonlinear effects irrespective of the wave amplitudes. We then adapt our theoretical framework to identify resonant triads and hence provide insights into the generation of higher harmonics in two different oceanic scenarios: (i) low-mode internal tide propagating over small- or large-scale topography and (ii) an internal wave beam incident on a pycnocline in the upper ocean, for which our results are in qualitative agreement with the numerical study of Diamessis et al. (Dynam. Atmos. Oceans., vol. 66, 2014, pp. 110–137).

Triad resonant wave interactions in electrically charged jets

Nonlinear instability in electrically charged jets is studied using the governing electro-hydrodynamic equations describing stretching and thinning of a liquid jet. A jet flow system subject to both space and time evolving disturbances is considered. At the linear stage, the Rayleigh and conducting jet flow instability modes are uncovered. Nonlinear instability in the flow is explored via triad resonant waves which uncover fa- vorable operating modes not previously detected in the linear study of the problem. In particular, the jet radius is significantly reduced, and the electric field of the jet is properly oriented under the nonlinear study. It is found that taking into account the resonance triad modes provides a better mathematical description of a jet that stretches and thins due to tangential electric field effects. Both linear and nonlinear instability results in the jet flow system are presented and discussed.

The Arithmetic Geometry of Resonant Rossby Wave Triads

Linear wave solutions to the Charney--Hasegawa--Mima equation with periodic boundary conditions have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, the set of resonant triads may be described as the set of integer solutions to a particular homogeneous Diophantine equation, or as the set of rational points on a projective surface $X$. The set of all resonant triads was found by Bustamante and Hayat (2013) via mapping to quadratic forms. Our work independently finds all resonant triads via a rational parametrization of $X$. We provide a fiberwise description of $X$ as a rational singular elliptic surface, yielding many new results about the set of wavevectors belonging to resonant triads. In particular, we show there is an infinite number of resonant triads (with relatively prime wavevectors) containing a wavevector $(a,b)$ with $a/b=r$, where $r$ is any given rationa...

Tsunami mitigation by resonant triad interaction with acoustic-gravity waves.

Tsunamis have been responsible for the loss of almost a half million lives, widespread long lasting destruction, profound environmental effects, and global financial crisis, within the last two decades. The main tsunami properties that determine the size of impact at the shoreline are its wavelength and amplitude in the ocean. Here, we show that it is in principle possible to reduce the amplitude of a tsunami, and redistribute its energy over a larger space, through forcing it to interact with resonating acoustic–gravity waves. In practice, generating the appropriate acoustic–gravity modes introduces serious challenges due to the high energy required for an effective interaction. However, if the findings are extended to realistic tsunami properties and geometries, we might be able to mitigate tsunamis and so save lives and properties. Moreover, such a mitigation technique would allow for the harnessing of the tsunami's energy.

Internal wave attractors: different scenarios of instability

This paper presents an experimental study of different instability scenarios in a parallelogram-shaped internal wave attractor in a trapezoidal domain filled with a uniformly stratified fluid. Energy is injected into the system via the oscillatory motion of a vertical wall of the trapezoidal domain. Whole-field velocity measurements are performed with the conventional particle image velocimetry technique. In the linear regime, the total kinetic energy of the fluid system is used to quantify the strength of attractors as a function of coordinates in the parameter space defining their zone of existence, the so-called Arnold tongue. In the nonlinear regime, the choice of the operational point in the Arnold tongue is shown to have a significant impact on the scenario of the onset of triadic instability, most notably in terms of the influence of confinement on secondary waves. The onset of triadic resonance instability may occur as a spatially localized event similar to Scolan et al. (Phys. Rev. Lett., vol. 110, 2013, 234501) in the case of strong focusing or in form of growing normal modes as in McEwan (J. Fluid Mech., vol. 50, 1971, pp. 431–448) for the limiting case of rectangular domain. In the present paper, we describe also a new intermediate scenario for the case of weak focusing. We explore the long-term behaviour of cascades of triadic instabilities in wave attractors and show a persistent trend toward formation of standing-wave patterns corresponding to some discrete peaks of the frequency spectrum. At sufficiently high level of energy injection the system exhibits a ‘mixing box’ regime which has certain qualitatively universal properties regardless to the choice of the operating point in the Arnold tongue. In particular, for this regime, we observe a statistics of events with high horizontal vorticity, which serve as kinematic indicators of mixing.

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.

Nonlinear coupling of interfacial instabilities with resonant wave interactions in horizontal two-fluid plane Couette–Poiseuille flows: numerical and physical observations

We investigate mechanisms governing the initial growth and nonlinear evolution of interfacial waves in horizontal two-fluid plane Couette–Poiseuille flows. Nonlinear coupling of the Kelvin–Helmholtz interfacial instability with resonant wave interactions has been shown to be capable of rapidly generating long waves through the transfer of energy from linearly unstable short waves to stable long-wave components within the context of potential flow theory. The objective of this work is to determine whether that coupled mechanism persists in laminar and turbulent viscous flows. Utilizing both theoretical and computational methods, we analyse the initial Orr–Sommerfeld instability to quantify the frequencies and growth/decay rates of each wave mode for two-fluid laminar and turbulent channel flows. The obtained dispersion relation allows for the identification of resonant and/or near-resonant triads among (unstable and damped) wave components in an interfacial wave spectrum. We perform direct numerical simulations (DNS) of the two-phase Navier–Stokes equations with a fully nonlinear interface to formally establish the validity of our theoretical predictions for viscous flows. DNS results show the existence of a nonlinear energy cascade from unstable short- to damped long-wavelength waves due to resonant subharmonic and/or triadic interactions in both laminar Couette and turbulent Poiseuille flows. Spectral analysis of the interfacial evolution confirms that the combined instability–resonance mechanism persists in the presence of viscosity despite being derived under the assumption of potential flow theory. Finally, we perform a detailed examination of experimentally measured wave power spectra from Jurman et al. (J. Fluid Mech., vol. 238, 1992, pp. 187–219) and carry out a numerical sensitivity study of the flow conditions to demonstrate and verify the existence of the coupled instability–resonance mechanism in physical systems. Our analysis accurately predicts the initial instability and the resulting nonlinear energy cascade through subharmonic and triadic interfacial wave resonances.

Generation of long subharmonic internal waves by surface waves

A new set of Boussinesq equations is derived to study the nonlinear interactions between long waves in a two-layer fluid. The fluid layers are assumed to be homogeneous, inviscid, incompressible, and immiscible. Based on the Boussinesq equations, an analytical model is developed using a second-order perturbation theory and applied to examine the transient evolution of a resonant triad composed of a surface wave and two oblique subharmonic internal waves. Wave damping due to weak viscosity in both layers is considered. The Boussinesq equations and the analytical model are verified. In contrast to previous studies which focus on short internal waves, we examine long waves and investigate some previously unexplored characteristics of this class of triad interaction. In viscous fluids, surface wave amplitudes must be larger than a threshold to overcome viscous damping and trigger internal waves. The dependency of this critical amplitude as well as the growth and damping rates of internal waves on important parameters in a two-fluid system, namely the directional angle of the internal waves, depth, density, and viscosity ratio of the fluid layers, and surface wave amplitude and frequency is investigated.

Study of turbulence and interacting inertial modes in a differentially rotating spherical shell experiment

We present a study of inertial modes in a differentially rotating spherical shell (spherical Couette flow) experiment with a radius ratio of $\eta = 1/3$. Inertial modes are Coriolis-restored linear wave modes which often arise in rapidly rotating fluids. Recent experimental work has shown that inertial modes exist in a spherical Couette flow for $\Omega_{i}<\Omega_{o}$, where $\Omega_i$ and $\Omega_o$ is the inner and outer sphere rotation rate. A finite number of particular inertial modes has previously been found. By scanning the Rossby number from $-2.5 < Ro = (\Omega_{i}-\Omega_{o})/\Omega_{o} < 0$ at two fixed $\Omega_{o}$, we report the existence of similar inertial modes. However, the behavior of the flow described here differs much from previous spherical Couette experiments. We show that the kinetic energy of the dominant inertial mode dramatically increases with decreasing Rossby number that eventually leads to a wave-breaking and an increase of small-scale structures at a critical Rossby number. Such a transition in a spherical Couette flow has not been described before. The critical Rossby number scales with the Ekman number as0 $E^{1/5}$. Additionally, the increase of small-scale features beyond the transition transfers energy to a massively enhanced mean flow around the tangent cylinder. In this context, we discuss an interaction between the dominant inertial modes with a geostrophic Rossby mode exciting secondary modes whose frequencies match the triadic resonance condition.