scholarly journals Oceanic Internal-Wave Field: Theory of Scale-Invariant Spectra

2010 ◽  
Vol 40 (12) ◽  
pp. 2605-2623 ◽  
Author(s):  
Yuri V. Lvov ◽  
Kurt L. Polzin ◽  
Esteban G. Tabak ◽  
Naoto Yokoyama
Keyword(s):  
Kinetic Equation ◽  
Power Law ◽  
Spectral Power ◽  
Internal Gravity Waves ◽  
Collision Integral ◽  
Turbulence Theory ◽  
Coriolis Parameter ◽  
Scale Invariant ◽  
Weakly Nonlinear ◽  
Resonant Interactions

Abstract Steady scale-invariant solutions of a kinetic equation describing the statistics of oceanic internal gravity waves based on wave turbulence theory are investigated. It is shown in the nonrotating scale-invariant limit that the collision integral in the kinetic equation diverges for almost all spectral power-law exponents. These divergences come from resonant interactions with the smallest horizontal wavenumbers and/or the largest horizontal wavenumbers with extreme scale separations. A small domain is identified in which the scale-invariant collision integral converges and numerically find a convergent power-law solution. This numerical solution is close to the Garrett–Munk spectrum. Power-law exponents that potentially permit a balance between the infrared and ultraviolet divergences are investigated. The balanced exponents are generalizations of an exact solution of the scale-invariant kinetic equation, the Pelinovsky–Raevsky spectrum. A small but finite Coriolis parameter representing the effects of rotation is introduced into the kinetic equation to determine solutions over the divergent part of the domain using rigorous asymptotic arguments. This gives rise to the induced diffusion regime. The derivation of the kinetic equation is based on an assumption of weak nonlinearity. Dominance of the nonlocal interactions puts the self-consistency of the kinetic equation at risk. However, these weakly nonlinear stationary states are consistent with much of the observational evidence.

Numerical Investigation of Mechanisms Underlying Oceanic Internal Gravity Wave Power-Law Spectra

2020 ◽  
Vol 50 (9) ◽  
pp. 2713-2733
Author(s):  
Yulin Pan ◽  
Brian K. Arbic ◽  
Arin D. Nelson ◽  
Dimitris Menemenlis ◽  
W. R. Peltier ◽  
...  
Keyword(s):  
Power Law ◽  
Gravity Waves ◽  
High Frequency ◽  
Internal Gravity Wave ◽  
Numerical Data ◽  
Field Measurements ◽  
Internal Gravity Waves ◽  
Collision Integral ◽  
Turbulence Theory ◽  
Nonlinear Interactions

AbstractWe consider the power-law spectra of internal gravity waves in a rotating and stratified ocean. Field measurements have shown considerable variability of spectral slopes compared to the high-wavenumber, high-frequency portion of the Garrett–Munk (GM) spectrum. Theoretical explanations have been developed through wave turbulence theory (WTT), where different power-law solutions of the kinetic equation can be found depending on the mechanisms underlying the nonlinear interactions. Mathematically, these are reflected by the convergence properties of the so-called collision integral (CL) at low- and high-frequency limits. In this work, we study the mechanisms in the formation of the power-law spectra of internal gravity waves, utilizing numerical data from the high-resolution modeling of internal waves (HRMIW) in a region northwest of Hawaii. The model captures the power-law spectra in broad ranges of space and time scales, with scalings ω−2.05±0.2 in frequency and m−2.58±0.4 in vertical wavenumber. The latter clearly deviates from the GM76 spectrum but is closer to a family of induced-diffusion-dominated solutions predicted by WTT. Our analysis of nonlinear interactions is performed directly on these model outputs, which is fundamentally different from previous work assuming a GM76 spectrum. By applying a bicoherence analysis and evaluations of modal energy transfer, we show that the CL is dominated by nonlocal interactions between modes in the power-law range and low-frequency inertial motions. We further identify induced diffusion and the near-resonances at its spectral vicinity as dominating the formation of power-law spectrum.

Understanding discrete capillary-wave turbulence using a quasi-resonant kinetic equation

2017 ◽  
Vol 816 ◽  
Author(s):  
Yulin Pan ◽  
Dick K. P. Yue
Keyword(s):  
Kinetic Equation ◽  
Euler Equations ◽  
Power Law ◽  
Capillary Wave ◽  
Quantitative Measure ◽  
Energy Cascade ◽  
Turbulence Theory ◽  
Finite Domain ◽  
Spectral Slope ◽  
Infinite Domain

Experimental and numerical studies have shown that, with sufficient nonlinearity, the theoretical capillary-wave power-law spectrum derived from the kinetic equation (KE) of weak turbulence theory can be realized. This is despite the fact that the KE is derived assuming an infinite domain with continuous wavenumber, while experiments and numerical simulations are conducted in realistic finite domains with discrete wavenumbers for which the KE theoretically allows no energy transfer. To understand this, we first analyse results from direct simulations of the primitive Euler equations to elucidate the role of nonlinear resonance broadening (NRB) in discrete turbulence. We define a quantitative measure of the NRB, explaining its dependence on the nonlinearity level and its effect on the properties of the obtained stationary power-law spectra. This inspires us to develop a new quasi-resonant kinetic equation (QKE) for discrete turbulence, which incorporates the mechanism of NRB, governed by a single parameter $\unicode[STIX]{x1D705}$ expressing the ratio of NRB and wavenumber discreteness. At $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{0}\approx 0.02$, the QKE recovers simultaneously the spectral slope $\unicode[STIX]{x1D6FC}_{0}=-17/4$ and the Kolmogorov constant $C_{0}=6.97$ (corrected from the original derivation) of the theoretical continuous spectrum, which physically represents the upper bound of energy cascade capacity for the discrete turbulence. For $\unicode[STIX]{x1D705}<\unicode[STIX]{x1D705}_{0}$, the obtained spectra represent those corresponding to a finite domain with insufficient nonlinearity, resulting in a steeper spectral slope $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{0}$ and reduced capacity of energy cascade $C>C_{0}$. The physical insights from the QKE are corroborated by direct simulation results of the Euler equations.

scholarly journals Four-wave resonant interactions in the classical quadratic Boussinesq equations

2009 ◽  
Vol 618 ◽  
pp. 263-277 ◽  
Author(s):  
M. ONORATO ◽  
A. R. OSBORNE ◽  
P. A. E. M. JANSSEN ◽  
D. RESIO
Keyword(s):  
Energy Transfer ◽  
Kinetic Equation ◽  
Shallow Water ◽  
Water Waves ◽  
Wave Spectrum ◽  
Boussinesq Equations ◽  
Wave Model ◽  
Flat Bottom ◽  
Weakly Nonlinear ◽  
Resonant Interactions

We investigate theoretically the irreversibile energy transfer in flat bottom shallow water waves. Starting from the oldest weakly nonlinear dispersive wave model in shallow water, i.e. the original quadratic Boussinesq equations, and by developing a statistical theory (kinetic equation) of the aforementioned equations, we show that the four-wave resonant interactions are naturally part of the shallow water wave dynamics. These interactions are responsible for a constant flux of energy in the wave spectrum, i.e. an energy cascade towards high wavenumbers, leading to a power law in the wave spectrum of the form of k−3/4. The nonlinear time scale of the interaction is found to be of the order of (h/a)4 wave periods, with a the wave amplitude and h the water depth. We also compare the kinetic equation arising from the Boussinesq equations with the arbitrary-depth Hasselmann equation and show that, in the limit of shallow water, the two equations coincide. It is found that in the narrow band case, both in one-dimensional propagation and in the weakly two-dimensional case, there is no irreversible energy transfer because the coupling coefficient in the kinetic equation turns out to be identically zero on the resonant manifold.

Experimental internal gravity wave turbulence

2020 ◽  
Author(s):  
Géraldine Davis ◽  
Thierry Dauxois ◽  
Sylvain Joubaud ◽  
Timothée Jamin ◽  
Nicolas Mordant ◽  
...  
Keyword(s):  
Internal Wave ◽  
Gravity Waves ◽  
Closed Loop ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Wave Turbulence ◽  
Large Set ◽  
Weakly Nonlinear ◽  
Resonant Interactions ◽  
Set Up

&lt;p&gt;Stratified fluids may develop simultaneously turbulence and internal wave turbulence, the latter describing a set of a large number of dispersive and weakly nonlinear interacting waves. The description and understanding of this regime for internal gravity waves (IGW) is really an open subject, in particular due to their very unusual dispersion relation. In this presentation, I will show experimental signatures of a large set of weakly interacting IGW obtained in a 2D trapezoidal tank.&lt;/p&gt;&lt;p&gt;Due to the peculiar linear reflexion law of IGW on inclined slopes, this setup - for given excitation frequencies - focuses all the input energy on a closed loop called attractor. If the forcing is large enough, this attractor destabilizes and the system eventually achieves a nonlinear cascade in frequencies and wavevectors via triadic resonant interactions, which results at large forcing amplitudes in a k^-3 spatial energy spectrum. I will also show some results obtained in a much larger set-up -the Coriolis facility in Grenoble- with signature of 3D internal wave turbulence.&lt;/p&gt;

Resonant wave interactions in a stratified shear flow

1988 ◽  
Vol 190 ◽  
pp. 357-374 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Shear Flow ◽  
Gravity Waves ◽  
Critical Level ◽  
Internal Gravity Waves ◽  
Wave Interactions ◽  
Resonant Wave ◽  
Resonant Interactions ◽  
Stratified Shear Flow ◽  
Weak Resonance ◽  
Background Shear

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

Thermosonde Measurement of Temperature Structure Parameter and Temperature Spectral Power-Law Exponent Profile in the Lower Stratosphere

2014 ◽  
Vol 34 (5) ◽  
pp. 0501001
Author(s):  
吴晓庆 Wu Xiaoqing ◽  
黄宏华 Huang Honghua ◽  
钱仙妹 Qian Xianmei ◽  
汪平 Wang Ping ◽  
崔朝龙 Cui Chaolong
Keyword(s):  
Power Law ◽  
Lower Stratosphere ◽  
Spectral Power ◽  
Temperature Structure ◽  
Structure Parameter ◽  
Temperature Structure Parameter ◽  
Power Law Exponent

scholarly journals Generic Properties of Curvature Sensing through Vision and Touch

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Birgitta Dresp-Langley
Keyword(s):  
Power Law ◽  
Sensory Modality ◽  
Generic Properties ◽  
Scale Invariant ◽  
Curvature Sensing ◽  
Mathematical Properties ◽  
Universal Power Law ◽  
Sensory Modalities ◽  
Congenitally Blind ◽  
Blind Individuals

Generic properties of curvature representations formed on the basis of vision and touch were examined as a function of mathematical properties of curved objects. Virtual representations of the curves were shown on a computer screen for visual scaling by sighted observers (experiment 1). Their physical counterparts were placed in the two hands of blindfolded and congenitally blind observers for tactile scaling. The psychophysical data show that curvature representations in congenitally blind individuals, who never had any visual experience, and in sighted observers, who rely on vision most of the time, are statistically linked to the same mathematical properties of the curves. The perceived magnitude of object curvature, sensed through either vision or touch, is related by a mathematical power law, with similar exponents for the two sensory modalities, to the aspect ratio of the curves, a scale invariant geometric property. This finding supports biologically motivated models of sensory integration suggesting a universal power law for the adaptive brain control and balance of motor responses to environmental stimuli from any sensory modality.

scholarly journals SHORT-TIME EXISTENCE FOR SCALE-INVARIANT HAMILTONIAN WAVES

2006 ◽  
Vol 03 (02) ◽  
pp. 247-267 ◽  
Author(s):  
JOHN K. HUNTER
Keyword(s):  
Rayleigh Waves ◽  
Evolution Equations ◽  
Hyperbolic Surface ◽  
Smooth Solutions ◽  
Scale Invariant ◽  
Weakly Nonlinear ◽  
Hamiltonian Evolution ◽  
Short Time ◽  
Nonlinear Scale ◽  
Time Existence

We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations include ones that describe weakly nonlinear hyperbolic surface waves, such as nonlinear Rayleigh waves in elasticity.

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