scholarly journals Complete Hamiltonian formalism for inertial waves in rotating fluids

2017 ◽  
Vol 831 ◽  
pp. 128-150 ◽  
Author(s):  
A. A. Gelash ◽  
V. S. L’vov ◽  
V. E. Zakharov
Keyword(s):  
Hamiltonian Formalism ◽  
Wave Interaction ◽  
Wave Theory ◽  
Incompressible Fluids ◽  
Hamiltonian Approach ◽  
Inertial Waves ◽  
Rotating Fluids ◽  
Weakly Nonlinear ◽  
Interaction Processes ◽  
Inertial Wave

A complete Hamiltonian formalism is suggested for inertial waves in rotating incompressible fluids. Resonance three-wave interaction processes – decay instability and confluence of two waves – are shown to play a key role in the weakly nonlinear dynamics and statistics of inertial waves in the rapid rotation case. Future applications of the Hamiltonian approach to inertial wave theory are investigated and discussed.

Secondary instabilities in rapidly rotating fluids: inertial wave breakdown

1999 ◽  
Vol 382 ◽  
pp. 283-306 ◽  
Author(s):  
R. R. KERSWELL
Keyword(s):  
Three Dimensional ◽  
Finite Amplitude ◽  
Small Scale ◽  
Inertial Waves ◽  
Rotating Fluids ◽  
Initial Excitation ◽  
Geostrophic Flows ◽  
Inertial Wave ◽  
Secondary Instabilities ◽  
Wave Breakdown

Inertial waves are a ubiquitous feature of rapidly rotating fluids. Although much is known about their initial excitation, little is understood about their stability. Experiments indicate that they are generically unstable and in many cases catastrophically so, quickly causing the whole flow to collapse to small-scale disorder. The linear stability of two three-dimensional inertial waves observed to break down in the laboratory is considered here at experimentally small but finite Ekman numbers of [les ]10−4. Surprisingly small threshold amplitudes for instability are found. The results support the conjecture that triad resonances are the generic mechanism for secondary instability in rapidly rotating fluids but also highlight the ability of geostrophic flows to derive energy through a finite-amplitude inertial wave. This latter finding may go some way to explaining the significant mean circulations typically observed in inertial wave experiments.

scholarly journals Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence

2019 ◽  
Vol 879 ◽  
pp. 296-326 ◽  
Author(s):  
Thomas Le Reun ◽  
Benjamin Favier ◽  
Michael Le Bars
Keyword(s):  
Rotation Rate ◽  
Resonant Interaction ◽  
Wave Turbulence ◽  
Inertial Waves ◽  
Weakly Nonlinear ◽  
Elliptical Instability ◽  
Inertial Wave ◽  
Geostrophic Turbulence ◽  
Set Up ◽  
Saturation Flow

In this paper, we present an experimental investigation of the turbulent saturation of the flow driven by the parametric resonance of inertial waves in a rotating fluid. In our set-up, a half-metre wide ellipsoid filled with water is brought to solid-body rotation, and then undergoes sustained harmonic modulation of its rotation rate. This triggers the exponential growth of a pair of inertial waves via a mechanism called the libration-driven elliptical instability. Once the saturation of this instability is reached, we observe a turbulent state for which energy is injected into the resonant inertial waves only. Depending on the amplitude of the rotation rate modulation, two different saturation states are observed. At large forcing amplitudes, the saturation flow mainly consists of a steady, geostrophic anticyclone. Its amplitude vanishes as the forcing amplitude is decreased while remaining above the threshold of the elliptical instability. Below this secondary transition, the saturation flow is a superposition of inertial waves which are in weakly nonlinear resonant interaction, a state that could asymptotically lead to inertial wave turbulence. In addition to being a first experimental observation of a wave-dominated saturation in unstable rotating flows, the present study is also an experimental confirmation of the model of Le Reun et al. (Phys. Rev. Lett., vol. 119 (3), 2017, 034502) who introduced the possibility of these two turbulent regimes. The transition between these two regimes and their relevance to geophysical applications are finally discussed.

Free-surface breakdown in a rapidly rotating liquid

1978 ◽  
Vol 86 (3) ◽  
pp. 457-463 ◽  
Author(s):  
W. E. Scott
Keyword(s):  
Free Surface ◽  
Capillary Waves ◽  
Mode Frequency ◽  
Wave Form ◽  
Inertial Waves ◽  
Rotating Liquid ◽  
Surface Breakdown ◽  
Beat Phenomenon ◽  
Inertial Wave ◽  
Wave Modes

It is shown that the wavelets which appear on the inertial wave form of the inner free surface of a fully spun-up cylindrical mass of liquid contained in a vertical, rapidly rotating and gyrating gyrostat are capillary waves. It is further shown that the interaction between these capillary waves and the excited inertial waves is not the mechanism which effects an observed two-period collapse (‘breakdown’) and reappearance of the free-surface inertial wave form. Rather, the two-period breakdown can be explained by the conjecture that it is a beat phenomenon arising from the interaction of two differently structured inertial wave modes, which have the same frequency at small amplitudes of oscillation of the gyrostat but which, owing to the dependence of the inertial mode frequency on the amplitude of the gyrostatic motion, have slightly different frequencies at larger amplitudes of oscillation of the gyrostat.

scholarly journals Focusing of long waves with finite crest over constant depth

2013 ◽  
Vol 469 (2153) ◽  
pp. 20130015 ◽  
Author(s):  
Utku Kânoğlu ◽  
Vasily V. Titov ◽  
Baran Aydın ◽  
Christopher Moore ◽  
Themistoklis S. Stefanakis ◽  
...  
Keyword(s):  
Source Region ◽  
Wave Theory ◽  
Long Waves ◽  
Constant Depth ◽  
Large Wave ◽  
Weakly Nonlinear ◽  
Scaling Relationships ◽  
2011 Japan Tsunami ◽  
Wave Heights ◽  
Run Up

Tsunamis are long waves that evolve substantially, through spatial and temporal spreading from their source region. Here, we introduce a new analytical solution to study the propagation of a finite strip source over constant depth using linear shallow-water wave theory. This solution is not only exact, but also general and allows the use of realistic initial waveforms such as N -waves. We show the existence of focusing points for N -wave-type initial displacements, i.e. points where unexpectedly large wave heights may be observed. We explain the effect of focusing from a strip source analytically, and explore it numerically. We observe focusing points using linear non-dispersive and linear dispersive theories, analytically; and nonlinear non-dispersive and weakly nonlinear weakly dispersive theories, numerically. We discuss geophysical implications of our solutions using the 17 July 1998 Papua New Guinea and the 17 July 2006 Java tsunamis as examples. Our results may also help to explain high run-up values observed during the 11 March 2011 Japan tsunami, which are otherwise not consistent with existing scaling relationships. We conclude that N -waves generated by tectonic displacements feature focusing points, which may significantly amplify run-up beyond what is often assumed from widely used scaling relationships.

scholarly journals Energy intensity of inertial waves in a sphere

1983 ◽  
Vol 25 (2) ◽  
pp. 145-160
Author(s):  
W. W. Wood
Keyword(s):  
Energy Density ◽  
General Theory ◽  
Energy Intensity ◽  
Initial Disturbance ◽  
Inertial Waves ◽  
Inertial Wave

AbstractThe decay at large wavenumbers of the energy density in an inertial wave generated in a sphere by an arbitrary initial disturbance is determined as a first step to a comparison with the general theory of Phillips [17] for a statistically steady field of random inertial waves in an arbitrary cavity.

Stability and Three-Wave Interaction of Disturbances in a Supersonic Boundary Layer With Cooling

2010 ◽  
Vol 5 (3) ◽  
pp. 52-62
Author(s):  
Sergey A. Gaponov ◽  
Natalya M. Terekhova
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Considerable Change ◽  
Supersonic Boundary Layer ◽  
Surface Cooling ◽  
Weakly Nonlinear ◽  
Linear Evolution ◽  
Weakly Nonlinear Theory ◽  
Compressed Gas

In linear and nonlinear approach (weakly nonlinear theory of stability) interaction of disturbances on a boundary layer of compressed gas is considered at surface cooling. The regimes of moderate (Max number М = 2) and high (М = 5.35) are considered at supersonic speeds. It is established that the surface cooling leads to considerable change of linear evolution of disturbances: the vortical disturbances of the first mode are stabilised, and the acoustic disturbances of the second mode are destabilised, the change degree is defined by the degree of change of the temperature factor. The nonlinear interaction in three-wave systems on high (М = 5.35) supersonic regimes on a boundary layer of compressed gas is carried out between waves of the different nature (acoustic and vortical) in a regime of a parametrical resonance. As a rating wave the flat acoustic wave which raises three-dimensional subharmonic components of the vortical modes. However, the similar interactions for vortical waves at М = 2 considerably weaken. It is possible to expect that surface cooling will lead to delay of a laminar regime at М = 2 and to accelerate of turbulization at М = 5.35

scholarly journals Spherical vortices in rotating fluids

2018 ◽  
Vol 846 ◽  
Author(s):  
M. M. Scase ◽  
H. L. Terry
Keyword(s):  
Ideal Fluid ◽  
Rotation Rate ◽  
Wave Motion ◽  
Infinite Family ◽  
Vortex Rings ◽  
Rotating Fluids ◽  
Critical Rotation ◽  
Spherical Vortex ◽  
Inertial Wave ◽  
Special Case

A popular model for a generic fat-cored vortex ring or eddy is Hill’s spherical vortex (Phil. Trans. R. Soc. A, vol. 185, 1894, pp. 213–245). This well-known solution of the Euler equations may be considered a special case of the doubly infinite family of swirling spherical vortices identified by Moffatt (J. Fluid Mech., vol. 35 (1), 1969, pp. 117–129). Here we find exact solutions for such spherical vortices propagating steadily along the axis of a rotating ideal fluid. The boundary of the spherical vortex swirls in such a way as to exactly cancel out the background rotation of the system. The flow external to the spherical vortex exhibits fully nonlinear inertial wave motion. We show that above a critical rotation rate, closed streamlines may form in this outer fluid region and hence carry fluid along with the spherical vortex. As the rotation rate is further increased, further concentric ‘sibling’ vortex rings are formed.

Numerical Simulation of Non-Gaussian Wave Elevation and Kinematics Based on Two-Dimensional Fourier Transform

2006 ◽  
Author(s):  
Xiang Yuan Zheng ◽  
Torgeir Moan ◽  
Ser Tong Quek
Keyword(s):  
Fourier Transform ◽  
Wave Interaction ◽  
Wave Theory ◽  
Two Dimensions ◽  
Second Order ◽  
Interaction Matrix ◽  
Random Wave ◽  
Two Dimensional ◽  
Near Surface ◽  
Wave Elevation

The one-dimensional Fast Fourier Transform (FFT) has been extensively applied to efficiently simulate Gaussian wave elevation and water particle kinematics. The actual sea elevation/kinematics exhibit non-Gaussianities that mathematically can be represented by the second-order random wave theory. The elevation/kinematics formulation contains double-summation frequency sum and difference terms which in computation make the dynamic analysis of offshore structural response prohibitive. This study aims at a direct and efficient two-dimensional FFT algorithm for simulating the frequency sum terms. For the frequency difference terms, inverse FFT and FFT are respectively implemented across the two dimensions of the wave interaction matrix. Given specified wave conditions, not only the wave elevation but kinematics and associated Morison force are simulated. Favorable agreements are achieved when the statistics of elevation/kinematics are compared with not only the empirical fits but the analytical solutions developed based on modified eigenvalue/eigenvector approach, while the computation effort is very limited. In addition, the stochastic analyses in both time-and frequency domains show that the near-surface Morison force and induced linear oscillator response exhibits stronger non-Gaussianities by involving the second-order wave effects.

The reflexion of internal/inertial waves from bumpy surfaces. Part 2. Split reflexion and diffraction

1971 ◽  
Vol 49 (1) ◽  
pp. 113-131 ◽  
Author(s):  
P. G. Baines
Keyword(s):  
Wave Field ◽  
Logarithmic Singularity ◽  
Fluid Velocity ◽  
Wave Theory ◽  
Dispersive Wave ◽  
Inertial Waves ◽  
Incident Wavelength ◽  
Wave Fields ◽  
Reflected Wave ◽  
Square Root Singularity

This paper considers the linear inviscid reflexion of internal/inertial waves from smooth bumpy surfaces where a characteristic (or ray) is tangent to the surface at some point. There are two principal cases. When a characteristic associated with the incident wave is tangent to the surface we have diffraction; when the tangential characteristic is associated with a reflected wave we have split reflexion, a phenomenon which has no counterpart in classical non-dispersive wave theory. In both these cases the problem of determining the wave field may be reduced to a set of coupled integral equations with two unknown functions. These equations are solved for the simplest topography for each case, and the properties of the wave fields for more general topographies are discussed. For both split reflexion and diffraction, the fluid velocity has an inverse-square-root singularity on the tangential characteristic, and the energy density has a corresponding logarithmic singularity. The diffracted wave field penetrates into the shadow region a distance which is of the order of the incident wavelength. Possibilities for instability and mixing are discussed.

scholarly journals Response of a swirl flame to inertial waves

2017 ◽  
Vol 10 (4) ◽  
pp. 277-286 ◽  
Author(s):  
Alp Albayrak ◽  
Deniz A Bezgin ◽  
Wolfgang Polifke
Keyword(s):  
Steady State ◽  
Impulse Response ◽  
Finite Impulse Response ◽  
Reactive Flow ◽  
Inertial Waves ◽  
Mean Flow ◽  
Flow Equations ◽  
Flame Response ◽  
Inertial Wave ◽  
The Impact

Acoustic waves passing through a swirler generate inertial waves in rotating flow. In the present study, the response of a premixed flame to an inertial wave is scrutinized, with emphasis on the fundamental fluid-dynamic and flame-kinematic interaction mechanism. The analysis relies on linearized reactive flow equations, with a two-part solution strategy implemented in a finite element framework: Firstly, the steady state, low-Mach number, Navier–Stokes equations with Arrhenius type one-step reaction mechanism are solved by Newton’s method. The flame impulse response is then computed by transient solution of the analytically linearized reactive flow equations in the time domain, with mean flow quantities provided by the steady-state solution. The corresponding flame transfer function is retrieved by fitting a finite impulse response model. This approach is validated against experiments for a perfectly premixed, lean, methane-air Bunsen flame, and then applied to a laminar swirling flame. This academic case serves to investigate in a generic manner the impact of an inertial wave on the flame response. The structure of the inertial wave is characterized by modal decomposition. It is shown that axial and radial velocity fluctuations related to the eigenmodes of the inertial wave dominate the flame front modulations. The dispersive nature of the eigenmodes plays an important role in the flame response.

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