Strato-rotational instability without resonance

2018 ◽  
Vol 846 ◽  
pp. 815-833 ◽  
Author(s):  
Chen Wang ◽  
Neil J. Balmforth
Keyword(s):  
Three Dimensional ◽  
Normal Modes ◽  
Phase Speed ◽  
Flow Speed ◽  
Rotational Instability ◽  
Buoyancy Frequency ◽  
Critical Levels ◽  
Mean Flow ◽  
Linear Eigenvalue Problem ◽  
Resonant Interactions

Strato-rotational instability (SRI) is normally interpreted as the resonant interactions between normal modes of the internal or Kelvin variety in three-dimensional settings in which the stratification and rotation are orthogonal to both the background flow and its shear. Using a combination of asymptotic analysis and numerical solution of the linear eigenvalue problem for plane Couette flow, it is shown that such resonant interactions can be destroyed by certain singular critical levels. These levels are not classical critical levels, where the phase speed $c$ of a normal mode matches the mean flow speed $U$, but are a different type of singularity where $(c-U)$ matches a characteristic gravity-wave speed $\pm N/k$, based on the buoyancy frequency $N$ and streamwise horizontal wavenumber $k$. Instead, it is shown that a variant of SRI can occur due to the coupling of a Kelvin or internal wave to such ‘baroclinic’ critical levels. Two characteristic situations are identified and explored, and the conservation law for pseudo-momentum is used to rationalize the physical mechanism of instability. The critical level coupling removes the requirement for resonance near specific wavenumbers $k$, resulting in an extensive continuous band of unstable modes.

Tilting at wave beams: a new perspective on the St. Andrew’s Cross

2017 ◽  
Vol 830 ◽  
pp. 660-680 ◽  
Author(s):  
T. Kataoka ◽  
S. J. Ghaemsaidi ◽  
N. Holzenberger ◽  
T. Peacock ◽  
T. R. Akylas
Keyword(s):  
Wave Field ◽  
Finite Length ◽  
Line Source ◽  
Cutoff Frequency ◽  
Three Dimensional ◽  
Resonance Phenomenon ◽  
Buoyancy Frequency ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Viscous Effects

The generation of internal gravity waves by a vertically oscillating cylinder that is tilted to the horizontal in a stratified Boussinesq fluid of constant buoyancy frequency, $N$, is investigated. This variant of the widely studied horizontal configuration – where a cylinder aligned with a plane of constant gravitational potential induces four wave beams that emanate from the cylinder, forming a cross pattern known as the ‘St. Andrew’s Cross’ – brings out certain unique features of radiated internal waves from a line source tilted to the horizontal. Specifically, simple kinematic considerations reveal that for a cylinder inclined by a given angle $\unicode[STIX]{x1D719}$ to the horizontal, there is a cutoff frequency, $N\sin \unicode[STIX]{x1D719}$, below which there is no longer a radiated wave field. Furthermore, three-dimensional effects due to the finite length of the cylinder, which are minor in the horizontal configuration, become a significant factor and eventually dominate the wave field as the cutoff frequency is approached; these results are confirmed by supporting laboratory experiments. The kinematic analysis, moreover, suggests a resonance phenomenon near the cutoff frequency as the group-velocity component perpendicular to the cylinder direction vanishes at cutoff; as a result, energy cannot be easily radiated away from the source, and nonlinear and viscous effects are likely to come into play. This scenario is examined by adapting the model for three-dimensional wave beams developed in Kataoka & Akylas (J. Fluid Mech., vol. 769, 2015, pp. 621–634) to the near-resonant wave field due to a tilted line source of large but finite length. According to this model, the combination of three-dimensional, nonlinear and viscous effects near cutoff triggers transfer of energy, through the action of Reynolds stresses, to a circulating horizontal mean flow. Experimental evidence of such an induced mean flow near cutoff is also presented.

Stability of internal gravity wave beams to three-dimensional modulations

2013 ◽  
Vol 736 ◽  
pp. 67-90 ◽  
Author(s):  
T. Kataoka ◽  
T. R. Akylas
Keyword(s):  
Modulational Instability ◽  
Three Dimensional ◽  
Harmonic Component ◽  
Internal Gravity Wave ◽  
Beam Width ◽  
Threshold Value ◽  
Resonant Interaction ◽  
Buoyancy Frequency ◽  
Mean Flow ◽  
Time Harmonic

AbstractThe linear stability of uniform, plane internal wave beams with locally confined spatial profile, in a stratified fluid of constant buoyancy frequency, is discussed. The associated eigenvalue problem is solved asymptotically, assuming perturbations of long wavelength relative to the beam width. In this limit, instability is found only for oblique disturbances which vary in the along-beam and the horizontal transverse directions. The mechanism of instability is a first-harmonic–mean resonant interaction between the underlying wave beam and three-dimensional perturbations that comprise a time-harmonic component, with the beam frequency, and a mean flow. Progressive beams which transport energy in one direction, in particular, are unstable if the beam steepness exceeds a certain threshold value, whereas purely standing beams are unstable even at infinitesimal steepness. A distinguishing feature of this three-dimensional modulational instability is the generation of circulating horizontal mean flows at large distances from the vicinity of the beam.

scholarly journals Marginal Instability?

2009 ◽  
Vol 39 (9) ◽  
pp. 2373-2381 ◽  
Author(s):  
S. A. Thorpe ◽  
Zhiyu Liu
Keyword(s):  
Richardson Number ◽  
Maximum Growth ◽  
Flow Speed ◽  
Maximum Growth Rate ◽  
Buoyancy Frequency ◽  
Mean Flow ◽  
Boundary Layer Flows ◽  
Naturally Occurring ◽  
Marginal State ◽  
The Mean

Abstract Some naturally occurring, continually forced, turbulent, stably stratified, mean shear flows are in a state close to that in which their stability changes, usually from being dynamically unstable to being stable: the time-averaged flows that are observed are in a state of marginal instability. By “marginal instability” the authors mean that a small fractional increase in the gradient Richardson number Ri of the mean flow produced by reducing the velocity and, hence, shear is sufficient to stabilize the flow: the increase makes Rimin, the minimum Ri in the flow, equal to Ric, the critical value of this minimum Richardson number. The value of Ric is determined by solving the Taylor–Goldstein equation using the observed buoyancy frequency and the modified velocity. Stability is quantified in terms of a factor, Φ, such that multiplying the flow speed by (1 + Φ) is just sufficient to stabilize it, or that Ric = Rimin/(1 + Φ)2. The hypothesis that stably stratified boundary layer flows are in a marginal state with Φ < 0 and with |Φ| small compared to unity is examined. Some dense water cascades are marginally unstable with small and negative Φ and with Ric substantially less than ¼. The mean flow in a mixed layer driven by wind stress on the water surface is, however, found to be relatively unstable, providing a counterexample that refutes the hypothesis. In several naturally occurring flows, the time for exponential growth of disturbances (the inverse of the maximum growth rate) is approximately equal to the average buoyancy period observed in the turbulent region.

scholarly journals Upwelling and isolation in oxygen-depleted anticyclonic modewater eddies and implications for nitrate cycling

2016 ◽  
Author(s):  
Johannes Karstensen ◽  
Florian Schütte ◽  
Alice Pietri ◽  
Gerd Krahmann ◽  
Björn Fiedler ◽  
...  
Keyword(s):  
High Resolution ◽  
Large Scale ◽  
Cold Water ◽  
Phase Speed ◽  
Ekman Transport ◽  
Buoyancy Frequency ◽  
Low Oxygen ◽  
Mean Flow ◽  
Mode Water ◽  
The Core

Abstract. The physical (temperature, salinity, velocity) and biogeochemical (oxygen, nitrate) structure of an oxygen depleted coherent, baroclinic, anticyclonic mode-water eddy (ACME) is investigated using high-resolution autonomous glider and ship data. A distinct core with a diameter of about 70 km is found in the eddy, extending from about 60 to 200 m depth and. The core is occupied by fresh and cold water with low oxygen and high nitrate concentrations, and bordered by local maxima in buoyancy frequency. Velocity and property gradient sections show vertical layering at the flanks and underneath the eddy characteristic for vertical propagation (to several hundred-meters depth) of near inertial internal waves (NIW) and confirmed by direct current measurements. A narrow region exists at the outer edge of the eddy where NIW can propagate downward. NIW phase speed and mean flow are of similar magnitude and critical layer formation is expected to occur. An asymmetry in the NIW pattern is seen that possible relates to the large-scale Ekman transport interacting with ACME dynamics. NIW/mean flow induced mixing occurs close to the euphotic zone/mixed layer and upward nutrient flux is expected and supported by the observations. Combing high resolution nitrate (NO3−) data with the apparent oxygen utilization (AOU) reveals AOU:NO3− ratios of 16 which are much higher than in the surrounding waters (8.1). A maximum NO3− deficit of 4 to 6 µmol kg−1 is estimated for the low oxygen core. Denitrification would be a possible explanation. This study provides evidence that the recycling of NO3−, extracted from the eddy core and replenished into the core via the particle export, may quantitatively be more important. In this case, the particulate phase is of keys importance in decoupling the nitrogen from the oxygen cycling.

scholarly journals Direct Estimate of Lateral Eddy Diffusivity Upstream of Drake Passage

2014 ◽  
Vol 44 (10) ◽  
pp. 2593-2616 ◽  
Author(s):  
Ross Tulloch ◽  
Raffaele Ferrari ◽  
Oliver Jahn ◽  
Andreas Klocker ◽  
Joseph LaCasce ◽  
...  
Keyword(s):  
Antarctic Circumpolar Current ◽  
Model Simulation ◽  
Phase Speed ◽  
Drake Passage ◽  
Eddy Diffusivity ◽  
Flow Speed ◽  
Mean Flow ◽  
Direct Estimate ◽  
Circumpolar Current ◽  
The Antarctic

Abstract The first direct estimate of the rate at which geostrophic turbulence mixes tracers across the Antarctic Circumpolar Current is presented. The estimate is computed from the spreading of a tracer released upstream of Drake Passage as part of the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES). The meridional eddy diffusivity, a measure of the rate at which the area of the tracer spreads along an isopycnal across the Antarctic Circumpolar Current, is 710 ± 260 m2 s−1 at 1500-m depth. The estimate is based on an extrapolation of the tracer-based diffusivity using output from numerical tracers released in a one-twentieth of a degree model simulation of the circulation and turbulence in the Drake Passage region. The model is shown to reproduce the observed spreading rate of the DIMES tracer and suggests that the meridional eddy diffusivity is weak in the upper kilometer of the water column with values below 500 m2 s−1 and peaks at the steering level, near 2 km, where the eddy phase speed is equal to the mean flow speed. These vertical variations are not captured by ocean models presently used for climate studies, but they significantly affect the ventilation of different water masses.

scholarly journals The Interaction of a Baroclinic Mean Flow with Long Rossby Waves

2005 ◽  
Vol 35 (5) ◽  
pp. 865-879 ◽  
Author(s):  
A. Colin de Verdière ◽  
R. Tailleux
Keyword(s):  
Doppler Shift ◽  
Rossby Waves ◽  
Numerical Solutions ◽  
Phase Speed ◽  
Standard Theory ◽  
Buoyancy Frequency ◽  
Implicit Assumption ◽  
Constant Sign ◽  
Mean Flow ◽  
The Mean

Abstract The effect of a baroclinic mean flow on long oceanic Rossby waves is studied using a combination of analytical and numerical solutions of the eigenvalue problem. The effect is summarized by the value of the nondimensional numberwhen the mean flow shear keeps a constant sign throughout the water column. Because previous studies have shown that no interaction occurs if the mean flow has the shape of the first unperturbed mode (the non–Doppler shift effect), an implicit assumption in the application of the present work to the real ocean is that the relative projections of the mean flow on the second and higher modes remain approximately constant. Because R2 is large at low latitudes between 10° and 30° (the southern branches of subtropical gyres or the regions of surface westward shear), the phase speed of the first mode is very slightly decreased from the no mean flow standard theory case. Between 30° and 40° latitudes (the northern branches of the subtropical gyres or the regions of surface eastward shear), R2 is O(10) and the westward phase speed can increase significantly (up to a factor of 2). At still higher latitudes when R2 is O(1) a critical transition occurs below which no discrete Rossby waves are found that might explain the absence of observations of zonal propagations at latitudes higher than 50°. Our case studies, chosen to represent the top-trapped and constant-sign shear profiles of observed mean flows, all show the importance of three main effects on the value of the first baroclinic mode Rossby wave speed: 1) the meridional gradient of the quantity N2/f (where N is the buoyancy frequency) rather than that of the potential vorticity fN2; 2) the curvature of the mean flow in the vertical direction, which appears particularly important to predict the sign of the phase speed correction to the no-mean-flow standard theory case: increase (decrease) of the westward phase speed when the surface-intensified mean flow is eastward (westward); and 3) a weighted vertical average of the mean flow velocity, acting as a Doppler-shift term, which is small in general but important to determine the precise value of the phase speed.

scholarly journals The Quasi-Nondispersive Regimes of Long Extratropical Baroclinic Rossby Waves over (Slowly Varying) Topography

2006 ◽  
Vol 36 (1) ◽  
pp. 104-121 ◽  
Author(s):  
Rémi Tailleux
Keyword(s):  
Rossby Waves ◽  
Normal Modes ◽  
Phase Speed ◽  
Dispersive Wave ◽  
Mean Flow ◽  
Coriolis Parameter ◽  
Zonal Wavenumber ◽  
Topographic Parameter ◽  
Ray Paths

Abstract Actual energy paths of long, extratropical baroclinic Rossby waves in the ocean are difficult to describe simply because they depend on the meridional-wavenumber-to-zonal-wavenumber ratio τ, a quantity that is difficult to estimate both observationally and theoretically. This paper shows, however, that this dependence is actually weak over any interval in which the zonal phase speed varies approximately linearly with τ, in which case the propagation becomes quasi-nondispersive (QND) and describable at leading order in terms of environmental conditions (i.e., topography and stratification) alone. As an example, the purely topographic case is shown to possess three main kinds of QND ray paths. The first is a topographic regime in which the rays follow approximately the contours f /hαc = a constant (αc is a near constant fixed by the strength of the stratification, f is the Coriolis parameter, and h is the ocean depth). The second and third are, respectively, “fast” and “slow” westward regimes little affected by topography and associated with the first and second bottom-pressure-compensated normal modes studied in previous work by Tailleux and McWilliams. Idealized examples show that actual rays can often be reproduced with reasonable accuracy by replacing the actual dispersion relation by its QND approximation. The topographic regime provides an upper bound (in general a large overestimate) of the maximum latitudinal excursions of actual rays. The method presented in this paper is interesting for enabling an optimal classification of purely azimuthally dispersive wave systems into simpler idealized QND wave regimes, which helps to rationalize previous empirical findings that the ray paths of long Rossby waves in the presence of mean flow and topography often seem to be independent of the wavenumber orientation. Two important side results are to establish that the baroclinic string function regime of Tyler and Käse is only valid over a tiny range of the topographic parameter and that long baroclinic Rossby waves propagating over topography do not obey any two-dimensional potential vorticity conservation principle. Given the importance of the latter principle in geophysical fluid dynamics, the lack of it in this case makes the concept of the QND regimes all the more important, for they are probably the only alternative to provide a simple and economical description of general purely azimuthally dispersive wave systems.

On the stratified Taylor column

1973 ◽  
Vol 58 (3) ◽  
pp. 517-537 ◽  
Author(s):  
Nelson G. Hogg
Keyword(s):  
Three Dimensional ◽  
Flow Speed ◽  
Bottom Current ◽  
Buoyancy Frequency ◽  
Coriolis Parameter ◽  
Height Scale ◽  
Topographic Parameter ◽  
Upstream Flow ◽  
Baroclinic Current ◽  
Taylor Column

We analyse the effects of small, circularly symmetric topography on the slow flow of an inviscid, incompressible, diffusionless, horizontally uniform, baroclinic current and show that the vertical influence depends primarily on three parameters: a stratification measureS(the square of the ratio of buoyancy frequency times height scale to Coriolis parameter times length scale), a topographic parameter β (ratio of scaled topographic height multiplied by scaled bottom current to Rossby number ε) and the scaled upstream shearu′0(z) (the dimensional upstream shear divided by the ratio of the r.m.s. upstream flow speed to height scale).Investigating a linear stratification model we find that the topographic effect is depth independent ifS[lsim ] ε and a Taylor column, as indicated by the appearance of closed streamlines above the bump, exists when β > 2. Moderate stratification (S∼ 1) causes the flow to be fully three-dimensional and the Taylor column to be a conical vortex whose height depends on βSandu′0). The results are compared with Davies's (1971, 1972) experiments.Our results tend to support the Taylor column theory of Jupiter's Great Red Spot but effects due to variations in the Coriolis parameter with latitude have been (unjustifiably) ne glected. Using typical values for the earths oceans we find that Taylor columns of significant height could be found there. Some pertinent observations from the ocean are discussed.

Resonant interactions between topographic planetary waves in a continuously stratified fluid

1978 ◽  
Vol 84 (4) ◽  
pp. 769-793 ◽  
Author(s):  
Lawrence A. Mysak
Keyword(s):  
Low Frequency ◽  
Phase Speed ◽  
Stratified Fluid ◽  
Planetary Waves ◽  
Buoyancy Frequency ◽  
Frequency Wave ◽  
Resonant Interactions ◽  
Frequency Correction ◽  
Resonant Triads ◽  
The Waves

The resonant interactions between topographic planetary waves in a continuously stratified fluid are investigated theoretically. The interacting waves form a resonant triad and travel along a channel with a uniformly sloping bottom. The basic state stratification in the channel is characterized by a constant buoyancy frequency. The existence of solutions to the quadratic resonance conditions is established graphically. Each wave by itself is a bottom-intensified oscillation of the type discovered by Rhines (1970) except for the addition of a small positive frequency correction. This correction must be included to satisfy higher-order terms in the bottom boundary condition. For strong stratification (r2[Gt ]L2, wherer= internal deformation radius andL= channel width), the waves are strongly bottom-trapped and this frequency correction is negligible. For weak stratification (r2[Lt ]L2) the waves are barotropic and the frequency correction isO(δ), where δ = fractional change in depth across the channel. In many oceanic contexts, δ lies in the range 0·1-0·4 and therefore this correction can produce a significant change in the phase speed. The amplitudes of the waves in the triad obey the classical gyroscopic equations usually encountered in quadratic resonance problems. In particular, the amplitudes evolve on the slow time scale\[ t=O(1/f_0\delta^2), \]which for our scaling assumptions is alsoO(1/f0Ro), whereRois the Rossby number.The results are applied to the Norwegian continental slope region. It is shown that, in this vicinity, there may exist resonant triads consisting of two short, high-frequency waves (periods around 3-4 days) and one long, low-frequency wave (period around 9 days).

Critical-level behaviour and wave amplification of a gravity wave incident upon a shear layer

1975 ◽  
Vol 72 (4) ◽  
pp. 661-671 ◽  
Author(s):  
I. A. Eltayeb ◽  
J. F. McKenzie
Keyword(s):  
Shear Layer ◽  
Gravity Wave ◽  
Critical Level ◽  
Phase Speed ◽  
Flow Speed ◽  
Wave Incident ◽  
Mean Flow ◽  
Wave Amplification ◽  
The Mean ◽  
Streaming Motion

The properties of reflexion, refraction and absorption of a gravity wave incident upon a shear layer are investigated. It is shown that one must expect these properties to be very different depending upon the parameters (such as the Richardson number Ri, the wavelength normalized by the length scale of the shear and the ratio of the flow speed to the phase speed of the wave) characterizing the interaction of a gravity wave with a shear layer. In particular, it is shown that for all Richardson numbers there is a discontinuity in the net wave-action flux across the critical level, i.e. at a height where the flow speed matches the horizontal phase speed of the wave. When Ri > ¼, this is accompanied by absorption of part of the energy of the incident wave into the mean flow. In addition it is shown that the phenomenon of wave amplification (over-reflexion) can arise provided that the ultimate shear flow speed exceeds the horizontal phase speed of the wave and Ri is less than a certain critical value Ric ≃ 0·1129, in which case the reflected wave extracts energy from the streaming motion. It is also pointed out that wave amplification can lead to instability if the boundary conditions are altered in such a way that the system can behave like an ‘amplifier’.

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