Excitation of superharmonics by internal modes in non-uniformly stratified fluid

2016 ◽  
Vol 793 ◽  
pp. 335-352 ◽  
Author(s):  
Bruce R. Sutherland
Keyword(s):  
Single Mode ◽  
Stratified Fluid ◽  
Buoyancy Frequency ◽  
Nonlinear Interactions ◽  
Initial State ◽  
Weakly Nonlinear ◽  
Resonant Wave ◽  
Vertical Scale ◽  
Scale Structures ◽  
Parametric Subharmonic Instability

Theory and numerical simulations show that the nonlinear self-interaction of internal modes in non-uniform stratification results in energy being transferred to superharmonic disturbances forced at twice the horizontal wavenumber and frequency of the parent mode. These disturbances are not in themselves a single mode, but a superposition of modes such that the disturbance amplitude is largest where the change in the background buoyancy frequency with depth is largest. Through weakly nonlinear interactions with the parent mode, the disturbances evolve to develop vertical-scale structures that distort and modulate the parent mode. Because pure resonant wave triads do not exist in non-uniformly stratified fluid, parametric subharmonic instability (PSI) is not evident even though noise is superimposed upon the initial state. The results suggest a new mechanism for energy transfer to dissipative scales (from large to small vertical scale and with frequencies larger and smaller than that of the parent mode) through forcing superharmonic rather than subharmonic disturbances.

Nonlinear interactions between the two wakes behind a pair of square cylinders

2014 ◽  
Vol 759 ◽  
pp. 295-320 ◽  
Author(s):  
J. Mizushima ◽  
G. Hatsuda
Keyword(s):  
Nonlinear Stability ◽  
Mixed Mode ◽  
Single Mode ◽  
Nonlinear Interactions ◽  
Symmetric Mode ◽  
Amplitude Equations ◽  
Equilibrium Solutions ◽  
Antisymmetric Mode ◽  
Weakly Nonlinear ◽  
Mode Solution

AbstractNonlinear interactions between the two wakes behind a pair of square cylinders, which are placed side by side in a uniform flow, are investigated by the linear and weakly nonlinear stability analyses and numerical simulations. It is known from the linear stability analysis that the flow past a pair of cylinders becomes unstable to a symmetric or an antisymmetric mode of disturbance, depending on the gap ratio, the ratio of the gap distance between the two cylinders to the cylinder diameter. The antisymmetric mode gives the critical condition for smaller gap ratios than a threshold value, and for larger gap ratios the symmetric mode becomes the most unstable. We focus on the flow pattern arising through the nonlinear interactions of the two modes of disturbance for gap ratios around the threshold value when both modes are growing. We derive a couple of amplitude equations for the two modes to properly describe the nonlinear interaction between them by applying the weakly nonlinear stability theory. The amplitude equations are shown to have three equilibrium solutions except the null solution such as a mixed-mode solution, symmetric and antisymmetric single-mode solutions. Examination of the stability of each equilibrium solution leads to a conclusion that the mixed-mode solution exchange its stability with both the symmetric and the antisymmetric single-mode solutions simultaneously. In the case where the mixed-mode solution is stable, both the symmetric and antisymmetric modes have finite amplitudes, and the resultant flow has an asymmetric flow pattern comprising of finite amplitudes of the two modes of disturbance superposed on the steady symmetric flow. While in the case where both the single-mode solutions are stable, either of the symmetric- and antisymmetric-mode solutions survives, overwhelming the other. Then, if the symmetric mode attains at an equilibrium finite amplitude and the antisymmetric mode vanishes, the resultant flow is symmetric, and if the antisymmetric mode survives and the symmetric mode decays out, the flow becomes asymmetric with the antisymmetric mode of disturbance superposed on the steady symmetric flow. Thus, the flow appearing due to instability differs depending on the initial condition, not uniquely determined, when both single-mode solutions are stable. We numerically delineated the region in the parameter space of the gap ratio and the Reynolds number where the mixed-mode solution is stable. The theoretical results obtained from the weakly nonlinear stability analyses are confirmed by numerical simulations. The conclusion derived from the stability analysis of the equilibrium solutions of the amplitude equations is widely applicable also to other double Hopf bifurcation problems.

scholarly journals Linear and Weakly Nonlinear Energetics of Global Nonhydrostatic Normal Modes

2019 ◽  
Vol 76 (12) ◽  
pp. 3831-3846 ◽  
Author(s):  
Carlos F. M. Raupp ◽  
André S. W. Teruya ◽  
Pedro L. Silva Dias
Keyword(s):  
Normal Modes ◽  
Physical Space ◽  
Wave Mode ◽  
Finite Amplitude ◽  
Vertical Acceleration ◽  
Wave Interactions ◽  
Weakly Nonlinear ◽  
Resonant Wave ◽  
Zonal Wavenumber ◽  
Resonant Triads

Abstract 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.

Dissipative effects on the resonant flow of a stratified fluid over topography

1988 ◽  
Vol 192 ◽  
pp. 287-312 ◽  
Author(s):  
N. F. Smyth
Keyword(s):  
Bottom Topography ◽  
Stratified Fluid ◽  
Flow Speed ◽  
Resonant Condition ◽  
Weakly Nonlinear ◽  
Dissipative Effects ◽  
Long Wave ◽  
Near Resonance ◽  
Wave Limit ◽  
Wave Modes

The effect of dissipation on the flow of a stratified fluid over topography is considered in the weakly nonlinear, long-wave limit for the case when the flow is near resonance, i.e. the basic flow speed is close to a linear long-wave speed for one of the long-wave modes. The two types of dissipation considered are the dissipation due to viscosity acting in boundary layers and/or interfaces and the dissipation due to viscosity acting in the fluid as a whole. The effect of changing bottom topography on the flow produced by a force moving at a resonant velocity is also considered. In this case, the resonant condition is that the force velocity is close to a linear long-wave velocity for one of the long-wave modes. It is found that in most cases, these extra effects result in the formation of a steady state, in contrast to the flow without these effects, which remains unsteady for all time. The flow resulting under the action of boundary-layer dissipation is compared with recent experimental results.

scholarly journals Dynamical Relationship between the Phase of North Atlantic Oscillations and the Meridional Excursion of a Preexisting Jet: An Analytical Study

2008 ◽  
Vol 65 (6) ◽  
pp. 1838-1858 ◽  
Author(s):  
Dehai Luo ◽  
Tingting Gong ◽  
Linhao Zhong
Keyword(s):  
North Atlantic ◽  
Stationary Wave ◽  
Negative Phase ◽  
Initial State ◽  
Weakly Nonlinear ◽  
The North ◽  
Induced Dipole ◽  
Dipole Component ◽  
Southward Shift ◽  
The Impact

Abstract In this paper, it is shown from an analytical solution that in the presence of a preexisting jet the interaction between the zonal jet and the topography of the land–sea contrast (LSC) in the Northern Hemisphere (NH) tends to induce a dipole component that depends crucially upon whether this zonal jet exhibits a north–south excursion. This phenomenon cannot be observed if the zonal jet has no north–south shift. When the preexisting jet is located more northward (southward), the induced dipole can have a low-over-high (high-over-low) structure and thus can make the center of the stationary wave anomaly shift southward (northward), which can be regarded as an initial state or embryo of a positive (negative) phase North Atlantic Oscillation (NAO). This dipole component can be amplified into a typical NAO event under the forcing of synoptic-scale eddies. To some extent, this result provides an explanation for why the positive (negative) phase of the NAO can be controlled by the northward (southward) shift of the zonal jet prior to the NAO. In addition, the impact of the jet shift on the occurrence of NAO is examined in a weakly nonlinear NAO model if the initial state of an NAO is prespecified. It is found that the northward (southward) shift of a zonal jet favors the occurrence of the subsequent positive (negative) phase NAO event and then results in a northward (southward)-intensified jet relative to the preexisting jet. In addition, during the decaying of the positive phase NAO, a strong blocking activity is easily observed over Europe as the jet is moved to the north.

A viscous internal wave in a stratified fluid whose buoyancy frequency varies with altitude

1975 ◽  
Vol 69 (3) ◽  
pp. 615-624 ◽  
Author(s):  
D. Gordon ◽  
U. R. Klement ◽  
T. N. Stevenson
Keyword(s):  
Internal Wave ◽  
Small Amplitude ◽  
Wave Amplitude ◽  
Stratified Fluid ◽  
Similarity Solutions ◽  
Buoyancy Frequency ◽  
Experimental Measurements ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Curved Paths

A viscous incompressible stably stratified fluid with a buoyancy frequency which varies slowly with altitude is considered. A simple harmonic localized disturbance generates an internal wave in which the energy propagates along curved paths. Small amplitude similarity solutions are obtained for two-dimensional and axisymmetric waves. It is found that under certain conditions the wave amplitude can increase with height. The two-dimensional theory compares quite well with experimental measurements.

Elliptic instability of a stratified fluid in a rotating cylinder

2010 ◽  
Vol 660 ◽  
pp. 240-257 ◽  
Author(s):  
D. GUIMBARD ◽  
S. LE DIZÈS ◽  
M. LE BARS ◽  
P. LE GAL ◽  
S. LEBLANC
Keyword(s):  
Stratified Fluid ◽  
Angular Frequency ◽  
Nonlinear Regime ◽  
Finite Cylinder ◽  
Linear Growth Rate ◽  
Small Eccentricity ◽  
Weakly Nonlinear ◽  
Instability Modes ◽  
Centrifugal Instability ◽  
New Type

In this paper, we analyse the characteristics of the elliptic instability in a finite cylinder in the presence of both background rotation and axial stratification. A general formula for the linear growth rate of the stationary sinuous modes is derived including viscous and detuning effects in the limit of small eccentricity. This formula is discussed and compared to experimental results which are obtained in a cylinder filled with salted water for two different eccentricities by varying the stratification, the background rotation and the cylinder rotation. A good agreement with the theory concerning the domain of instability of the sinuous modes is demonstrated. Other elliptic instability modes, oscillating at the cylinder angular frequency are also evidenced together with a new type of instability mode, which could be connected to a centrifugal instability occurring during the experimental phase of spin-up. The nonlinear regime of the elliptic instability is also documented. In contrast with the homogeneous case, no cycle involving growth, breakdown and re-laminarization is observed in the presence of strong stratification. The elliptic instability in a stratified fluid seems to yield either a persistent turbulent state or a weakly nonlinear regime.

The growth of a turbulent patch in a stratified fluid

1988 ◽  
Vol 190 ◽  
pp. 55-70 ◽  
Author(s):  
Harindra J. S. Fernando
Keyword(s):  
Energy Source ◽  
Mixed Layer ◽  
Turbulent Mixing ◽  
Interfacial Layer ◽  
Patch Size ◽  
Stratified Fluid ◽  
Buoyancy Frequency ◽  
External Energy ◽  
External Energy Source ◽  
Vertical Size

The behaviour of a turbulent region generated within a linearly-stratified fluid by an external energy source has been studied experimentally. A monoplanar grid that generated small-amplitude oscillations was used as the energy source. The results show that the mixed layer initially grows rapidly, as in an unstratified fluid, but when its physical vertical size becomes rf ∼ (K1/N)½, at a time tf ≈ 4.0 N−1, where N is the buoyancy frequency and K1 is the ‘action’ of the grid, the buoyancy forces become dominant and drastically reduce further vertical growth of the patch. While the patch size remains at rf, a well-defined density interfacial layer is formed at the entrainment interface. An important feature of the interfacial layer is the presence of internal waves, excited by the mixed-layer turbulence. If the grid oscillations are continuously maintained, the interfacial waves break and cause turbulent mixing, thereby increasing the size of the patch beyond rf at a very slow rate. Theoretical estimates are made for the growth characteristics and are compared with the experimental results.

scholarly journals Energy Transfer from High-Shear, Low-Frequency Internal Waves to High-Frequency Waves near Kaena Ridge, Hawaii

2012 ◽  
Vol 42 (9) ◽  
pp. 1524-1547 ◽  
Author(s):  
Oliver M. Sun ◽  
Robert Pinkel
Keyword(s):  
Internal Waves ◽  
High Frequency ◽  
Low Frequency ◽  
Buoyancy Frequency ◽  
Reynolds Stresses ◽  
Turbulent Dissipation ◽  
Transfer Rates ◽  
Energy Transfers ◽  
High Frequency Waves ◽  
Parametric Subharmonic Instability

Abstract Evidence is presented for the transfer of energy from low-frequency inertial–diurnal internal waves to high-frequency waves in the band between 6 cpd and the buoyancy frequency. This transfer links the most energetic waves in the spectrum, those receiving energy directly from the winds, barotropic tides, and parametric subharmonic instability, with those most directly involved in the breaking process. Transfer estimates are based on month-long records of ocean velocity and temperature obtained continuously over 80–800 m from the research platform (R/P) Floating Instrument Platform (FLIP) in the Hawaii Ocean Mixing Experiment (HOME) Nearfield (2002) and Farfield (2001) experiments, in Hawaiian waters. Triple correlations between low-frequency vertical shears and high-frequency Reynolds stresses, 〈uiw∂Ui/∂z〉, are used to estimate energy transfers. These are supported by bispectral analysis, which show significant energy transfers to pairs of waves with nearly identical frequency. Wavenumber bispectra indicate that the vertical scales of the high-frequency waves are unequal, with one wave of comparable scale to that of the low-frequency parent and the other of much longer scale. The scales of the high-frequency waves contrast with the classical pictures of induced diffusion and elastic scattering interactions and violates the scale-separation assumption of eikonal models of interaction. The possibility that the observed waves are Doppler shifted from intrinsic frequencies near f or N is explored. Peak transfer rates in the Nearfield, an energetic tidal conversion site, are on the order of 2 × 10−7 W kg−1 and are of similar magnitude to estimates of turbulent dissipation that were made near the ridge during HOME. Transfer rates in the Farfield are found to be about half the Nearfield values.

scholarly journals Study of Squeezing Properties in a Two-level System

1995 ◽  
Vol 48 (6) ◽  
pp. 907 ◽  
Author(s):  
Rui-hua Xie ◽  
Gong-ou Xu ◽  
Dun-huan Liu
Keyword(s):  
Single Mode ◽  
Nonlinear Interactions ◽  
Level System ◽  
Rotating Wave Approximation ◽  
Wave Approximation ◽  
Mode Field ◽  
Level Atom ◽  
Rotating Wave ◽  
Arbitrary Intensity ◽  
Intensity Dependent Coupling

We have studied the squeezing properties of a field and atom in a two-level system. The influence of nonlinear interactions (Le. the arbitrary intensity-dependent coupling of a single-mode field to a single two-level atom, the nonlinear interaction of the field with a nonlinear Kerr-like medium) on the squeezing is discussed in detail in the rotating wave approximation (RWA). We show numerically that the effect of the virtual-photon field suppresses dipole squeezing predicted in the RWA and leads to an increased squeeze revival period; the suppressed squeezing can be revived due to the presence of the nonlinear Kerr-like medium.

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