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.

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.

scholarly journals Intraseasonal Modulation of the Surface Cooling in the Gulf of Guinea

2013 ◽  
Vol 43 (2) ◽  
pp. 382-401 ◽  
Author(s):  
Julien Jouanno ◽  
Frédéric Marin ◽  
Yves du Penhoat ◽  
Jean-Marc Molines
Keyword(s):  
Mixed Layer ◽  
Gravity Waves ◽  
Heat Budget ◽  
Vertical Shear ◽  
Upper Ocean ◽  
Equatorial Waves ◽  
Gulf Of Guinea ◽  
The Core ◽  
Mixed Layer Heat Budget ◽  
Inertia Gravity Waves

Abstract A regional numerical model of the tropical Atlantic Ocean and observations are analyzed to investigate the intraseasonal fluctuations of the sea surface temperature at the equator in the Gulf of Guinea. Results indicate that the seasonal cooling in this region is significantly shaped by short-duration cooling events caused by wind-forced equatorial waves: mixed Rossby–gravity waves within the 12–20-day period band, inertia–gravity waves with periods below 11 days, and equatorially trapped Kelvin waves with periods between 25 and 40 days. In these different ranges of frequencies, it is shown that the wave-induced horizontal oscillations of the northern front of the mean cold tongue dominate the variations of mixed layer temperature near the equator. But the model mixed layer heat budget also shows that the equatorial waves make a significant contribution to the mixed layer heat budget through modulation of the turbulent cooling, especially above the core of the Equatorial Undercurrent (EUC). The turbulent cooling variability is found to be mainly controlled by the intraseasonal modulation of the vertical shear in the upper ocean. This mechanism is maximum during periods of seasonal cooling, especially in boreal summer, when the surface South Equatorial Current is strongest and between 2°S and the equator, where the presence of the EUC provides a background vertical shear in the upper ocean. It applies for the three types of intraseasonal waves. Inertia–gravity waves also modulate the turbulent heat flux at the equator through vertical displacement of the core of the EUC in response to equatorial divergence and convergence.

scholarly journals Ageostrophic instability in rotating, stratified interior vertical shear flows

2014 ◽  
Vol 755 ◽  
pp. 397-428 ◽  
Author(s):  
Peng Wang ◽  
James C. McWilliams ◽  
Claire Ménesguen
Keyword(s):  
Energy Source ◽  
Kinetic Energy ◽  
Potential Energy ◽  
Gravity Waves ◽  
Shear Flows ◽  
Vertical Shear ◽  
Mean Flow ◽  
Efficient Energy Transfer ◽  
The Mean ◽  
Inertia Gravity Waves

AbstractThe linear instability of several rotating, stably stratified, interior vertical shear flows $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\overline{U}(z)$ is calculated in Boussinesq equations. Two types of baroclinic, ageostrophic instability, AI1 and AI2, are found in odd-symmetric $\overline{U}(z)$ for intermediate Rossby number ($\mathit{Ro}$). AI1 has zero frequency; it appears in a continuous transformation of the unstable mode properties between classic baroclinic instability (BCI) and centrifugal instability (CI). It begins to occur at intermediate $\mathit{Ro}$ values and horizontal wavenumbers ($k,l$) that are far from $l= 0$ or $k = 0$, where the growth rate of BCI or CI is the strongest. AI1 grows by drawing kinetic energy from the mean flow, and the perturbation converts kinetic energy to potential energy. The instability AI2 has inertia critical layers (ICL); hence it is associated with inertia-gravity waves. For an unstable AI2 mode, the coupling is either between an interior balanced shear wave and an inertia-gravity wave (BG), or between two inertia-gravity waves (GG). The main energy source for an unstable BG mode is the mean kinetic energy, while the main energy source for an unstable GG mode is the mean available potential energy. AI1 and BG type AI2 occur in the neighbourhood of $A-S= 0$ (a sign change in the difference between absolute vertical vorticity and horizontal strain rate in isentropic coordinates; see McWilliams et al., Phys. Fluids, vol. 10, 1998, pp. 3178–3184), while GG type AI2 arises beyond this condition. Both AI1 and AI2 are unbalanced instabilities; they serve as an initiation of a possible local route for the loss of balance in 3D interior flows, leading to an efficient energy transfer to small scales.

Inertia–Gravity Waves Generated within a Dipole Vortex

2007 ◽  
Vol 64 (12) ◽  
pp. 4417-4431 ◽  
Author(s):  
Chris Snyder ◽  
David J. Muraki ◽  
Riwal Plougonven ◽  
Fuqing Zhang
Keyword(s):  
Gravity Waves ◽  
Initial Conditions ◽  
Initial Period ◽  
Potential Temperature ◽  
Leading Edge ◽  
Vertical Shear ◽  
Coriolis Parameter ◽  
Dipole Vortex ◽  
The Waves ◽  
Inertia Gravity Waves

Abstract Vortex dipoles provide a simple representation of localized atmospheric jets. Numerical simulations of a synoptic-scale dipole in surface potential temperature are considered in a rotating, stratified fluid with approximately uniform potential vorticity. Following an initial period of adjustment, the dipole propagates along a slightly curved trajectory at a nearly steady rate and with a nearly fixed structure for more than 50 days. Downstream from the jet maximum, the flow also contains smaller-scale, upward-propagating inertia–gravity waves that are embedded within and stationary relative to the dipole. The waves form elongated bows along the leading edge of the dipole. Consistent with propagation in horizontal deformation and vertical shear, the waves’ horizontal scale shrinks and the vertical slope varies as they approach the leading stagnation point in the dipole’s flow. Because the waves persist for tens of days despite explicit dissipation in the numerical model that would otherwise damp the waves on a time scale of a few hours, they must be inherent features of the dipole itself, rather than remnants of imbalances in the initial conditions. The wave amplitude varies with the strength of the dipole, with waves becoming obvious once the maximum vertical vorticity in the dipole is roughly half the Coriolis parameter. Possible mechanisms for the wave generation are spontaneous wave emission and the instability of the underlying balanced dipole.

scholarly journals Inertia-gravity waves in the troposphere and lower stratosphere associated with a jet stream exit region

1999 ◽  
Vol 17 (1) ◽  
pp. 115-121 ◽  
Author(s):  
L. Thomas ◽  
R. M. Worthington ◽  
A. J. McDonald
Keyword(s):  
Gravity Waves ◽  
Lower Stratosphere ◽  
Middle Atmosphere ◽  
Vertical Shear ◽  
Jet Stream ◽  
Mesoscale Meteorology ◽  
Perturbation Velocity ◽  
Waves And Tides ◽  
Radar Measurements ◽  
Inertia Gravity Waves

Abstract. Radar measurements at Aberystwyth (52.4° N, 4.1° W) of winds at tropospheric and lower stratospheric heights are shown for 12-13 March 1994 in a region of highly curved flow, downstream of the jet maximum. The perturbations of horizontal velocity have comparable amplitudes in the troposphere and lower stratosphere with downward and upward phase propagation, respectively, in these two height regions. The sense of rotation with increasing height in hodographs of horizontal perturbation velocity derived for hourly intervals show downwards propagation of energy in the troposphere and upward propagation in the lower stratosphere with vertical wavelengths of 1.7 to 2.3 km. The results indicate inertia-gravity waves propagating in a direction similar to that of the jet stream but at smaller velocities. Some of the features observed contrast with those of previous observations of inertia-gravity waves propagating transverse to the jet stream. The interpretation of the hodographs to derive wave parameters has taken account of the vertical shear of the background wind transverse to the direction of wave propagation.Key words. Meteorology and atmospheric dynamics (mesoscale meteorology; middle atmosphere dynamics; waves and tides)

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.

On the effects of frontogenetic strain on symmetric instability and inertia–gravity waves

2012 ◽  
Vol 711 ◽  
pp. 620-640 ◽  
Author(s):  
Leif N. Thomas
Keyword(s):  
Gravity Waves ◽  
Richardson Number ◽  
Low Frequency ◽  
Deformation Field ◽  
Vertical Shear ◽  
Geostrophic Flow ◽  
Ageostrophic Circulation ◽  
The Waves ◽  
Symmetric Instability ◽  
Inertia Gravity Waves

AbstractThe dynamics of symmetric instability and two-dimensional inertia–gravity waves in a baroclinic geostrophic flow undergoing frontogenesis is analysed. A frontogenetic strain associated with a balanced deformation field drives an ageostrophic circulation and temporal variations in the basic state that significantly affect the properties of perturbations to the background flow. For stable stratification, perturbations to the basic state result in symmetric instability or inertia–gravity waves, depending on the sign of the Ertel potential vorticity and the magnitude of the Richardson number of the geostrophic flow. The kinetic energy (KE) of both types of motion is suppressed by frontogenetic strain due to the vertical shear in the ageostrophic circulation. This is because the perturbation streamlines tilt with the ageostrophic shear causing the disturbances to lose KE via shear production. The effect can completely dampen symmetric instability for sufficiently strong strain even though the source of KE for the instability (the vertical shear in the geostrophic flow) increases with time. Inertia–gravity waves in a baroclinic flow undergoing frontogenesis simultaneously lose KE and extract KE from the deformation field as they decay. This is because the horizontal velocity of the waves becomes rectilinear, resulting in a Reynolds stress that draws energy from the balanced flow. The process is most effective for waves of low frequency and for a geostrophic flow with low Richardson number. However, even in a background flow that is initially strongly stratified, frontogenesis leads to an exponentially fast reduction in the Richardson number, facilitating a rapid energy extraction by the waves. The KE transferred from the deformation field is ultimately lost to the unbalanced ageostrophic circulation through shear production, hence the inertia–gravity waves play a catalytic role in loss of balance. Given the large amount of KE in low-frequency inertia–gravity waves and the ubiquitous combination of strain and baroclinic geostrophic currents in the ocean, it is estimated that this mechanism could play a significant role in the removal of KE from both the internal wave and mesoscale eddy fields.

Balance in non-hydrostatic rotating stratified turbulence

2008 ◽  
Vol 596 ◽  
pp. 201-219 ◽  
Author(s):  
WILLIAM J. McKIVER ◽  
DAVID G. DRITSCHEL
Keyword(s):  
Gravity Waves ◽  
Potential Vorticity ◽  
Acoustic Waves ◽  
Buoyancy Frequency ◽  
Scaling Analysis ◽  
Stratified Turbulence ◽  
Dynamical Equations ◽  
Geostrophic Balance ◽  
Vorticity Balance ◽  
Inertia Gravity Waves

It is now well established that two distinct types of motion occur in geophysical turbulence: slow motions associated with potential vorticity advection and fast oscillations due to inertia–gravity waves (or acoustic waves). Many studies have theorized the existence of a flow for which the entire motion is controlled by the potential vorticity (or one ‘master variable’) – this is known as balance. In real geophysical flows, deviations from balance in the form of inertia–gravity waves or ‘imbalance’ have often been found to be small. Here we examine the extent to which balance holds in rotating stratified turbulence which is nearly balanced initially.Using the non-hydrostatic fluid dynamical equations under the Boussinesq approximation, we analyse properties of rotating stratified turbulence spanning a range of Rossby numbers (Ro≡|ζ|max/f) and the frequency ratios (c≡N/f) where ζ is the relative vertical vorticity, f is the Coriolis frequency and N is the buoyancy frequency. Using a recently introduced diagnostic procedure, called ‘optimal potential vorticity balance’, we extract the balanced part of the flow in the simulations and assess how the degree of imbalance varies with the above parameters.We also introduce a new and more efficient procedure, building upon a quasi-geostrophic scaling analysis of the complete non-hydrostatic equations. This ‘nonlinear quasi-geostrophic balance’ procedure expands the equations of motion to second order in Rossby number but retains the exact (unexpanded) definition of potential vorticity. This proves crucial for obtaining an accurate estimate of balanced motions. In the analysis of rotating stratified turbulence at Ro≲1 and N/f≫1, this procedure captures a significantly greater fraction of the underlying balance than standard (linear) quasi-geostrophic balance (which is based on the linearized equations about a state of rest). Nonlinear quasi-geostrophic balance also compares well with optimal potential vorticity balance, which captures the greatest fraction of the underlying balance overall.More fundamentally, the results of these analyses indicate that balance dominates in carefully initialized simulations of freely decaying rotating stratified turbulence up to O(1) Rossby numbers when N/f≫1. The fluid motion exhibits important quasi-geostrophic features with, in particular, typical height-to-width scale ratios remaining comparable to f/N.

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 Abyssal Mixing through Critical Reflection of Equatorially Trapped Waves off Smooth Topography

2019 ◽  
Vol 49 (2) ◽  
pp. 519-542 ◽  
Author(s):  
Bertrand L. Delorme ◽  
Leif N. Thomas
Keyword(s):  
Gravity Waves ◽  
Critical Reflection ◽  
Rossby Waves ◽  
Meridional Overturning Circulation ◽  
Vertical Shear ◽  
Trapped Waves ◽  
Wave Type ◽  
Meridional Overturning ◽  
Abyssal Mixing ◽  
Inertia Gravity Waves

AbstractThe inferred diapycnal upwelling in the abyssal meridional overturning circulation (AMOC) is intensified near the equator, but little is known as to why this is so. In this study, it is shown that the reflection of equatorially trapped waves (ETWs) off the bottom leads to seafloor-intensified mixing and substantial diapycnal upwelling near the equator when the full Coriolis force and the so-called nontraditional effects are taken into account. Using idealized simulations run with the MITgcm of downward-propagating ETWs of various types (i.e., inertia–gravity, Yanai, Kelvin, and Rossby waves) accounting for nontraditional effects, it is demonstrated that the reflection of ETWs off a flat seafloor generates beams of short inertia–gravity waves with strong vertical shear and low Richardson numbers that result in bottom-intensified, persistent, zonally invariant mixing at the inertial latitude of the ETW through the mechanism of critical reflection. The beams are more intense with weaker stratification and, for a given wave type, are stronger for waves with shorter periods and longer vertical wavelengths. The intensity of the beams also differs between wave types because their distinct meridional structures modulate the amount of energy fluxed to the bottom at the inertial latitude. As a result, equatorial inertia–gravity, Rossby, and eastward-propagating Yanai waves yield stronger mixing than Kelvin and westward-propagating Yanai waves in the simulations. It is estimated that this process can result in order 10 Sv (1 Sv ≡ 106 m3 s−1) of diapycnal upwelling per wavelength of ETW in the abyss and thus could play an important role in closing the AMOC.

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