scholarly journals Unbalanced instabilities of rapidly rotating stratified shear flows

2007 ◽  
Vol 584 ◽  
pp. 373-396 ◽  
Author(s):  
J. VANNESTE ◽  
I. YAVNEH
Keyword(s):  
Couette Flow ◽  
Gravity Waves ◽  
Growth Rates ◽  
Maximum Growth ◽  
Kelvin Waves ◽  
Buoyancy Frequency ◽  
Kelvin Wave ◽  
Asymptotic Results ◽  
The Stability ◽  
Inertia Gravity Waves

The linear stability of a rotating stratified inviscid horizontal plane Couette flow in a channel is studied in the limit of strong rotation and stratification. Two dimensionless parameters characterize the flow: the Rossby number ε, defined as the ratio of the shear to the Coriolis frequency and assumed small, and the ratio s of the Coriolis frequency to the buoyancy frequency, assumed to satisfy s ≤ 1. An energy argument is used to show that unstable perturbations must have large, O(ε−1), wavenumbers. This motivates the use of a WKB-approach which, in the first instance, provides an approximation for the dispersion relation of the various waves that can propagate in the flow. These are Kelvin waves, trapped near the channel walls, and inertia–gravity waves with or without turning points.Although the waves have real phase speeds to all algebraic orders in ε, we establish that the flow is unconditionally unstable. This is the result of linear resonances between waves with oppositely signed wave momenta. Three modes of instabilities are identified, corresponding to the resonance between (i) a pair of Kelvin waves, (ii) a Kelvin wave and an inertia–gravity wave, and (iii) a pair of inertia–gravity waves. Whilst all three modes of instability are active when the Couette flow is anticyclonic, mode (iii) is the only possible instability mechanism when the flow is cyclonic.We derive asymptotic estimates for the instability growth rates. These are exponentially small in ε, i.e. of the form Im ω = a exp(-Ψ/ε) for some positive constants a and Ψ. For the Kelvin-wave instabilities (i), we obtain analytic expressions for a and Ψ; the maximum growth rate, in particular, corresponds to Ψ = 2. For the other types of instabilities, we make the simplifying assumption s ≪ 1 and find that the maximum growth rates correspond to Ψ=2.80 for (ii) and Ψ= π for (iii). The asymptotic results are confirmed by numerical computations. These reveal, in particular, that the instabilities (iii) have much smaller growth rates in cyclonic flows than in anticyclonic flows, even though Ψ = π in both cases.Our results highlight the limitations of the so-called balanced models, widely used in geophysical fluid dynamics, which filter out Kelvin and inertia–gravity waves and hence predict the stability of Couette flow. They are also relevant to the stability of Taylor–Couette flows and of astrophysical accretion disks.

scholarly journals Inertia–gravity-wave radiation by a sheared vortex

2008 ◽  
Vol 596 ◽  
pp. 169-189 ◽  
Author(s):  
E. I. ÓLAFSDÓTTIR ◽  
A. B. OLDE DAALHUIS ◽  
J. VANNESTE
Keyword(s):  
Gravity Wave ◽  
Couette Flow ◽  
Gravity Waves ◽  
Dimensional Structure ◽  
Analytic Description ◽  
Wave Radiation ◽  
Rossby Number ◽  
Spontaneous Radiation ◽  
Initial State ◽  
Inertia Gravity Waves

We consider the linear evolution of a localized vortex with Gaussian potential vorticity that is superposed on a horizontal Couette flow in a rapidly rotating strongly stratified fluid. The Rossby number, defined as the ratio of the shear of the Couette flow to the Coriolis frequency, is assumed small. Our focus is on the inertia–gravity waves that are generated spontaneously during the evolution of the vortex. These are exponentially small in the Rossby number and hence are neglected in balanced models such as the quasi-geostrophic model and its higher-order generalizations. We develop an exponential-asymptotic approach, based on an expansion in sheared modes, to give an analytic description of the three-dimensional structure of the inertia–gravity waves emitted by the vortex. This provides an explicit example of the spontaneous radiation of inertia–gravity waves by localized balanced motion in the small-Rossby-number regime.The inertia–gravity waves are emitted as a burst of four wavepackets propagating downstream of the vortex. The approach employed reduces the computation of inertia–gravity-wave fields to a single quadrature, carried out numerically, for each spatial location and each time. This makes it possible to unambiguously define an initial state that is entirely free of inertia–gravity waves, and circumvents the difficulties generally associated with the separation between balanced motion and inertia–gravity waves.

Non-normal stability analysis of a shear current under surface gravity waves

2008 ◽  
Vol 609 ◽  
pp. 49-58
Author(s):  
D. AMBROSI ◽  
M. ONORATO
Keyword(s):  
Growth Rate ◽  
Modal Analysis ◽  
Gravity Waves ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Surface Gravity ◽  
Horizontal Shear ◽  
Surface Gravity Waves ◽  
Shear Current ◽  
The Stability

The stability of a horizontal shear current under surface gravity waves is investigated on the basis of the Rayleigh equation. As the differential operator is non-normal, a standard modal analysis is not effective in capturing the transient growth of a perturbation. The representation of the stream function by a suitable basis of bi-orthogonal eigenfunctions allows one to determine the maximum growth rate of a perturbation. It turns out that, in the considered range of parameters, such a growth rate can be two orders of magnitude larger than the maximum eigenvalue obtained by standard modal analysis.

Inertia-gravity waves beyond the inertial latitude. Part 1. Inviscid singular focusing

2010 ◽  
Vol 664 ◽  
pp. 478-509 ◽  
Author(s):  
VICTOR I. SHRIRA ◽  
WILLIAM A. TOWNSEND
Keyword(s):  
Wave Field ◽  
Internal Waves ◽  
Gravity Waves ◽  
Deep Ocean ◽  
Vertical Shear ◽  
Buoyancy Frequency ◽  
Local Minima ◽  
Wave Evolution ◽  
Vertical Scales ◽  
Inertia Gravity Waves

The paper is concerned with analytical study of inertia-gravity waves in rotating density-stratified ideal fluid confined in a spherical shell. It primarily aims at clarifying the possible role of these motions in deep ocean mixing. Recently, it was found that on the ‘non-traditional’ β-plane inertia-gravity internal waves can propagate polewards beyond their inertial latitude, where the wave frequency equals the local Coriolis parameter, by turning into subinertial modes trapped in the narrowing waveguides around the local minima of buoyancy frequency N. The behaviour of characteristics was established: wave horizontal and vertical scales decrease as the wave advances polewards and tend to zero at a latitude corresponding to an attractor of characteristics. However, the basic questions about wave evolution, its quantitative description and the possibility of its reflection from the critical latitude remain open. The present work addresses these issues by studying the linear inviscid evolution of finite bandwidth wavepackets on the ‘non-traditional’ β-plane past the inertial latitude for generic oceanic stratification. Beyond the inertial latitude, the wave field is confined in narrowing waveguides of three distinct generic types around different local minima of the buoyancy frequency. In the oceanic context, the widest is adjacent to the flat bottom, the thinnest is the upper mixed layer, and the middle one is located between the seasonal and main thermocline. We find explicit asymptotic solutions describing the wave field in the WKB approximation. As a byproduct, the conservation of wave action principle is explicitly formulated for all types of internal waves on the ‘non-traditional’ β-plane. The wave velocities and vertical shear tend to infinity and become singular at the attractor latitude or its vicinity for both monochromatic and finite bandwidth packets. We call this phenomenon singular focusing. These WKB solutions are shown to remain valid up to singularity for the bottom and mid-ocean waveguides. The main conclusion is that even in the inviscid setting the wave evolution towards smaller and smaller horizontal and vertical scales is irreversible: there is no reflection. For situations typical of deep ocean, a simultaneous increase in wave amplitude and decrease of vertical scale causes a sharp increase of vertical shear, which may lead to wave breaking and increased mixing.

The effect of viscosity on the stability of a uniformly rotating liquid column in zero gravity

2007 ◽  
Vol 572 ◽  
pp. 261-286 ◽  
Author(s):  
J. P. KUBITSCHEK ◽  
P. D. WEIDMAN
Keyword(s):  
Growth Rates ◽  
Tricritical Point ◽  
Temporal Stability ◽  
Maximum Growth ◽  
Liquid Column ◽  
Azimuthal Mode ◽  
Wide Range ◽  
The Stability ◽  
Axisymmetric Disturbances ◽  
Stability Results

An investigation of the linear temporal stability of a uniformly rotating viscous liquid column in the absence of gravity is presented. The governing parameters are the rotational Reynolds number Re and the Hocking parameter L, defined as the ratio of surface tension to centrifugal forces. Though the viscosity-independent condition L≥(k2 + n2-1)−1 for stability to disturbances of axial wavenumber k and azimuthal mode number n has been known for some time, the preferred modes, growth rates and frequencies at onset of instability have not been reported. We compute these results over a wide range of L–Re space and determine certain asymptotic behaviours in the limits of L→0, L→∞ and Re→∞. The computations exhibit a continuous evolution toward known inviscid stability results in the large-Re limit and their ultimate transition to an n = 1 spiral mode at small Re. While viscosity is shown to reduce growth rates for axisymmetric disturbances, it also produces a destabilizing effect for n = 2 planar and n = 1 spiral disturbances in certain regions of parameter space. A special feature is the appearance of a tricritical point in L–Re space at which maximum growth rates of the axisymmetric, n = 1 spiral, and n = 2 planar disturbances are equal.

scholarly journals Short vertical-wavelength inertia-gravity waves generated by a jet–front system at Arctic latitudes – VHF radar, radiosondes and numerical modelling

2014 ◽  
Vol 14 (13) ◽  
pp. 6785-6799 ◽  
Author(s):  
A. Réchou ◽  
S. Kirkwood ◽  
J. Arnault ◽  
P. Dalin
Keyword(s):  
Gravity Waves ◽  
Potential Temperature ◽  
Wrf Model ◽  
Horizontal Wind ◽  
Buoyancy Frequency ◽  
Vertical Wavelength ◽  
Vhf Radar ◽  
The Waves ◽  
Inertia Gravity Waves ◽  
Good Agreement

Abstract. Inertia-gravity waves with very short vertical wavelength (λz≤1000 m) are a very common feature of the lowermost stratosphere as observed by the 52 MHz radar ESRAD (Esrange MST radar) in northern Scandinavia (67.88° N, 21.10° E). The waves are seen most clearly in radar-derived profiles of buoyancy frequency (N). Here, we present a case study of typical waves from 21 February to 22 February 2007. Good agreement between N2 derived from radiosondes and by radar shows the validity of the radar determination of N2. Large-amplitude wave signatures in N2 are clearly observed by the radar and the radiosondes in the lowermost stratosphere, from 9 km to 14–16 km height. Vertical profiles of horizontal wind components and potential temperature from the radiosondes show the same waves. Mesoscale simulations with the Weather Research and Forecasting (WRF) model are carried out to complement the analysis of the waves. Good agreement between the radar and radiosonde measurements and the model (except for the wave amplitude) shows that the model gives realistic results and that the waves are closely associated to the upper-level front in an upper-troposphere jet–front system. Hodographs of the wind fluctuations from the radiosondes and model data show that the waves propagate upward in the lower stratosphere confirming that the origin of the waves is in the troposphere. The observations and modelling all indicate vertical wavelengths of 700 ± 200 m. The radiosonde hodograms indicate horizontal wavelengths between 40 and 110 km and intrinsic periods between 6 and 9 h. The wave amplitudes indicated by the model are however an order of magnitude less than in the observations. Finally, we show that the profiles of N2 measured by the radar can be used to estimate wave amplitudes, horizontal wavelengths, intrinsic periods and momentum fluxes which are consistent with the estimates from the radiosondes.

scholarly journals Asymptotics of the Far Fields of Internal Gravity Waves Excited by a Source of Radial Symmetry

2021 ◽  
Vol 56 (5) ◽  
pp. 672-677
Author(s):  
V. V. Bulatov ◽  
Yu. V. Vladimirov
Keyword(s):  
Gravity Waves ◽  
Hankel Transform ◽  
Internal Gravity Waves ◽  
Radial Symmetry ◽  
Wave Mode ◽  
Buoyancy Frequency ◽  
Asymptotic Results ◽  
Model Distribution ◽  
Perturbation Source ◽  
Individual Wave

Abstract— The problem of the far field of internal gravity waves generated by a perturbation source of radial symmetry aroused at an initial instant of time is solved. The constant model distribution of the buoyancy frequency is considered and, using the Fourier–Hankel transform, an analytical solution to the problem is obtained in the form of the sum of wave modes. Asymptotics of the solutions that describe the spatial-temporal characteristics of elevation of the isopycnic lines and the vertical and horizontal velocity components far from the perturbation source are obtained. The asymptotics of the components of the wave field are expressed in terms of the square of the Airy function and its derivatives in the neighborhood of the wave fronts of an individual wave mode. The exact and asymptotic results are compared and it is shown that the asymptotic method makes it possible to calculate effectively the far wave fields at times of the order of ten and more of the Brunt–Väisälä periods.

Instability of Baroclinic Tidal Flow in a Stratified Fjord

2010 ◽  
Vol 40 (1) ◽  
pp. 139-154 ◽  
Author(s):  
Zhiyu Liu
Keyword(s):  
Growth Rates ◽  
Richardson Number ◽  
Tidal Flow ◽  
Maximum Amplitude ◽  
Buoyancy Frequency ◽  
Unstable Flow ◽  
Mode Structure ◽  
Turbulent Dissipation ◽  
Second Mode ◽  
The Stability

Abstract The Taylor–Goldstein equation is used to investigate the stability of a baroclinic tidal flow observed in a stratified fjord. The flow is analyzed at hourly intervals when turbulent dissipation measurements were made. The critical gradient Richardson number is often close to the Miles–Howard limit of 0.25, but sometimes it is substantially less. Although during 8 of the 24 periods examined the flow is marginally stable, it is either very stable or very unstable in others. For the unstable flow, the e-folding period of the fastest growing disturbances is 83–455 s, about 46% of the buoyancy period at the levels where the fastest growing disturbances have their maximum amplitude. These disturbances to the flows have wavelengths about 20%–72% of the water depth and have mostly a second-mode structure. Simultaneous measurements of the flow and turbulence allow for testing of the hypothesis that the growth rates of the most unstable disturbances are related to the turbulent dissipation rates. Dissipation is found to depend on the growth rates, but only to a power of about 1.2; there is a stronger (power 1.8) dependence on the buoyancy frequency.

Growth of inertia–gravity waves in sheared inertial currents

2008 ◽  
Vol 601 ◽  
pp. 85-100 ◽  
Author(s):  
K. B. WINTERS
Keyword(s):  
Gravity Waves ◽  
Rotation Frequency ◽  
Base Flow ◽  
Rotating Flows ◽  
Vertical Shear ◽  
Buoyancy Frequency ◽  
Rotating Frame ◽  
Structure Of Solutions ◽  
Unstable Solutions ◽  
Inertia Gravity Waves

The linear stability of inviscid non-diffusive density-stratified shear flow in a rotating frame is considered. A temporally periodic base flow, characterized by vertical shear S, buoyancy frequency N and rotation frequency f, is perturbed by infinitesimal inertia–gravity waves. The temporal evolution and stability characteristics of the disturbances are analysed using Floquet theory and the growth rates of unstable solutions are computed numerically. The global structure of solutions is addressed in the dimensionless parameter space (N/f, S/f, φ) where φ is the wavenumber inclination angle from the horizontal for the wave-like perturbations. Both weakly stratified rapidly rotating flows (N<f) and strongly stratified slowly rotating flows (N>f) are examined. Distinct families of unstable modes are found, each of which can be associated with nearby stable solutions of periodicity T or 2T where T is the inertial frequency 2π/f. Rotation is found to be a destabilizing factor in the sense that stable non-rotating shear flows with N2/S2>1/4 can be unstable in a rotating frame. Morever, instabilities by parametric resonance are found associated with free oscillations at half and integer multiples of the inertial frequency.

Sign in / Sign up

Export Citation Format

Share Document