Mean flow generation by topographic Rossby waves

1979 ◽  
Vol 94 (1) ◽  
pp. 39-64 ◽  
Author(s):  
Alain Colin De Verdiere
Keyword(s):  
Rossby Waves ◽  
Phase Speed ◽  
Zonal Flow ◽  
Travelling Wave ◽  
General Tendency ◽  
Primary Wave ◽  
Mean Flow ◽  
Transfer Energy ◽  
Flow Strength ◽  
Topographic Rossby Waves

This paper makes use of the ease of modelling topographic Rossby waves in a laboratory context to investigate the ability of these waves to generate strong zonal mean flows when the geostrophic (f/H) contours are closed. A zonally travelling wave is forced in a narrow latitude band of a ‘polar beta plane’. Stronger signals occur when the motion of the driving is retrograde and at the phase speed of the gravest free modes. An important zonal westward mean flow occurs in the free interior while a compensating eastward jet is found at forced latitudes. The dependence of the mean flow strength upon the wave steepness indicates that genuine rectification processes are indeed taking place when the fluid is stirred by purely oscillating devices.This general tendency for topographic Rossby waves to transfer energy to zonal components is first analysed theoretically by investigating a side-band instability mechanism within an unforced fluid. Among the products of the interactions between a primary wave of wavenumberkand its side bands of wavenumberk± δk, the zonal flow is prominent. Wave steepnesses of order (|δk|/|k|)½only are required for zonal energy to grow whereas non-zonal components of scale longer or shorter than the primary wave need huge steepnesses [of order (|δk|/|k|−3/2] for amplification. This supplements the earlier notion that ‘nearly zonal’ waves may be generated by weak resonant interaction.For gentle driving certain classical aspects of Rossby wave propagation can be checked against the experiments. The linear theory provides also a convenient framework to discuss the meridional structure of the wave-induced Reynolds stress. For more energetic driving, a test of the potential vorticity mixing theory can be carried out and sheds further light upon the rectification mechanisms.

One-dimensional models for topographic Rossby waves in elongated basins on the f-plane

1986 ◽  
Vol 170 ◽  
pp. 435-459 ◽  
Author(s):  
Thomas Stocker ◽  
Kolumban Hutter
Keyword(s):  
Stream Function ◽  
Depth Profile ◽  
Rossby Waves ◽  
Boundary Value ◽  
Phase Speed ◽  
Approximate Model ◽  
Convergence Properties ◽  
Cross Sectional ◽  
Weighted Residuals ◽  
Topographic Rossby Waves

Topographic Rossby waves in elongated basins on the f-plane are studied by transforming the linear boundary-value problem for the mass transport stream function into a class of two-point boundary-value problems of which the independent spatial variable is the (curved) basin axis. The procedure for deriving the substitute problems is the Method of Weighted Residuals. What emerges is a vector differential equation and associated boundary conditions, its dimension indicating the order of the approximate model. It is shown that each substitute problem in the class entails the qualitative features typical of topographic waves, and increasing the order of the model corresponds to higher-order approximations. Equations are explicitly presented for cross-sectional distributions of the lake topography which has a power-law representation and permits the analysis of weak and strong topographies.Straight channels in which the depth profile does not change with position along the axis are studied in detail. The dispersion relation is discussed and dispersion curves are shown for the three lowest-order models. Convergence properties are thereby uncovered and phase speed and group velocity properties are found as they depend on wavenumber and topography. Further, for the lowest two modes, cross-channel stream-function distributions are presented. Apart from further convergence properties these distributions show that for U-shaped channels wave activity is nearer to the shore than for V-shaped channels, important information in the design of mooring systems.An analysis of topographic Rossby wave reflection follows, which emphasizes the importance of the depth profile in the reflecting zone. Based on these results some lake solutions are presented.

The propagation of atmospheric Rossby gravity waves in latitudinally sheared zonal flows

1977 ◽  
Vol 287 (1340) ◽  
pp. 115-143 ◽  
Keyword(s):  
Gravity Waves ◽  
Rossby Waves ◽  
Zonal Flow ◽  
Flow Speed ◽  
Wave Number ◽  
Mean Flow ◽  
Zonal Flows ◽  
Planetary Scale ◽  
The Mean ◽  
Critical Latitude

Using the B-plane approximation we formulate the equations which govern small perturbations in a rotating atmosphere and describe a wide class of possible wave motions, in the presence of a background zonal flow, ranging from ‘moderately high’ frequency acoustic-gravity-inertial waves to ‘low’ frequency planetary-scale (Rossby) waves. The discussion concentrates mainly on the propagation properties of Rossby waves in various types of latitudinally sheared zonal flows which occur at different heights and seasons in the earth’s atmosphere. However, it is first shown that gravity waves in a latitudinally sheared zonal flow exhibit critical latitude behaviour where the ‘intrinsic ’ wave frequency matches the Brunt-Vaisala frequency (in contrast to the case of gravity waves in a vertically sheared flow where a critical layer exists where the horizontal wave phase speed equals the flow speed) and that the wave behaviour near such a latitude is similar to that of Rossby waves in the vicinity of their critical latitudes which occur where the ‘intrinsic’ wave frequency approaches zero. In the absence of zonal flow in the atmosphere the geometry of the planetary wave dispersion equation (which is described by a highly elongated ellipsoid in wave-number vector space) implies that energy propagates almost parallel to the /--planes. This feature may provide a reason why there seems to be so little coupling between planetary scale motions in the lower and upper atmosphere. Planetary waves can be made to propagate eastward, as well as westward, if they are evanescent in the vertical direction. The W.K.B. approximation, which provides an approximate description of wave propagation in slowly varying zonal wind shears, shows that the distortion of the wave-number surface caused by the zonal flow controls the dependence of the wave amplitude on the zonal flow speed. In particular it follows that Rossby waves propagating into regions of strengthening westerlies are intensified in amplitude whereas those waves propagating into strengthening easterlies are diminished in amplitude. A classification of the various types of ray trajectories that arise in zonal flow profiles occurring in the Earth’s atmosphere, such as jet-like variations of westerly or easterly zonal flow or a belt of westerlies bounded by a belt of easterlies, is given, and provides the conditions giving rise to such phenomena as critical latitude behaviour and wave trapping. In a westerly flow there is a tendency for the combined effects on wave propagation of jet-like variations of B and zonal flow speed to counteract each other, whereas in an easterly flow such variations tend to reinforce each other. An examination of the reflexion and refraction of Rossby waves at a sharp jump in the zonal flow speed shows that under certain conditions wave amplification, or over-reflexion, can arise with the implication that the reflected wave can extract energy from the background streaming motion. On the other hand the wave behaviour near critical latitudes, which can be described in terms of a discontinuous jump in the ‘wave invariant’, shows that such latitudes can act as either wave absorbers (in which case the mean flow is accelerated there) or wave emitters (in which case the mean flow is decelerated there).

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.

Troposphere-Stratosphere Coupling and the Role of Critical Layer Nonlinearity

2020 ◽  
Author(s):  
Imogen Dell
Keyword(s):  
Rossby Waves ◽  
Phase Speed ◽  
Nonlinear Effects ◽  
Critical Layer ◽  
Coupling Mechanism ◽  
Mathematical Framework ◽  
Mean Flow ◽  
Weather And Climate ◽  
Dynamical Coupling

<p>There exists a coupling mechanism between the troposphere and the stratosphere, which plays a fundamental role in weather and climate. The coupling is highly complex and rests upon radiative and chemical feedbacks, as well as dynamical coupling by Rossby waves. The troposphere acts as a source of Rossby waves which propagate upwards in to the stratosphere, affecting the zonal mean flow. Rossby waves are also likely to play a significant role in downward communication of information via reflection from the stratosphere in to the troposphere. A mechanism for this reflection could be from a so-called critical layer. A shear flow exhibits a critical layer where the phase speed equals the flow velocity, where viscous and nonlinear effects become important. A wave incident upon a critical layer may be absorbed, reflected or overreflected, whereby the amplitude of the reflected wave is larger than that of the incident wave. In the case of troposphere-stratosphere coupling, the concept of critical layer overreflection is key to understanding atmospheric instability.</p><p>Motivated by this, a mathematical framework for understanding the coupling will be presented together with an investigation in to the role of nonlinearity versus viscosity inside the critical layer.</p>

Propagation of Topographic Rossby Waves in the Deep Basin of the South China Sea Based on Abyssal Current Observations

2021 ◽  
Author(s):  
Hua Zheng ◽  
Xiao-Hua Zhu ◽  
Chuanzheng Zhang ◽  
Ruixiang Zhao ◽  
Ze-Nan Zhu ◽  
...  
Keyword(s):  
Energy Source ◽  
South China Sea ◽  
South China ◽  
Rossby Waves ◽  
Phase Speed ◽  
The South China Sea ◽  
The South ◽  
Deep Basin ◽  
China Sea ◽  
Topographic Rossby Waves

AbstractTopographic Rossby waves (TRWs) are oscillations generated on sloping topography when water columns travel across isobaths under potential vorticity conservation. Based on our large-scale observations from 2016 to 2019, near 65-day TRWs were first observed in the deep basin of the South China Sea (SCS). The TRWs propagated westward with a larger wavelength (235 km) and phase speed (3.6 km/day) in the north of the array and a smaller wavelength (80 km) and phase speed (1.2 km/day) toward the southwest of the array. The ray-tracing model was used to identify the energy source and propagation features of the TRWs. The paths of the near 65-day TRWs mainly followed the isobaths with a slightly downslope propagation. The possible energy source of the TRWs was the variance of surface eddies southwest of Taiwan. The near 65-day energy propagated from the southwest of Taiwan to the northeast and southwest of the array over ~100–120 and ~105 days, respectively, corresponding to a group velocity of 4.2–5.0 and 10.5 km/day, respectively. This suggests that TRWs play an important role in deep-ocean dynamics and deep current variation, and upper ocean variance may adjust the intraseasonal variability in the deep SCS.

Planetary (Rossby), Inertia-Gravity (Poincaré) and Kelvin waves on the f-plane and β-plane in the presence of a uniform zonal flow

2021 ◽  
Author(s):  
Yair De-Leon ◽  
Chaim I. Garfinkel ◽  
Nathan Paldor
Keyword(s):  
Numerical Solutions ◽  
Bottom Topography ◽  
Phase Speed ◽  
Zonal Flow ◽  
Dispersion Curves ◽  
Kelvin Waves ◽  
Linear Wave ◽  
Hermite Function ◽  
Mean Flow ◽  
Zonal Wavenumber

<p>A linear wave theory of the Rotating Shallow Water Equations (RSWE) is developed in a channel on either the mid-latitude f-plane/β-plane or on the equatorial β-plane in the presence of a uniform mean zonal flow that is balanced geostrophically by a meridional gradient of the fluid surface height. We show that this surface height gradient is a potential vorticity (PV) source that generates Rossby waves even on the f-plane similar to the generation of these waves by PV sources such as the β–effect, shear of the mean flow and bottom topography. Numerical solutions of the RSWE show that the resulting planetary (Rossby), Inertia-Gravity (Poincaré) and Kelvin-like waves differ from their counterparts without mean flow in both their phase speeds and meridional structures. Doppler shifting of the “no mean-flow” phase speeds does not account for the difference in phase speeds, and the meridional structure does not often oscillate across the channel but is trapped near one the channel's boundaries in mid latitudes or behaves as Hermite function in the case of an equatorial channel. The phase speed of Kelvin-like waves is modified by the presence of a mean flow compared to the classical gravity wave speed but their meridional velocity does not vanish. The gaps between the dispersion curves of adjacent Poincaré modes are not uniform but change with the zonal wavenumber, and the convexity of the dispersion curves also changes with the zonal wavenumber. In some cases, the Kelvin-like dispersion curve crosses those of Poincaré modes, but it is not an evidence for the existence of instability since the Kelvin waves are not part of the solutions of an eigenvalue problem. </p>

scholarly journals Streamwise-travelling waves of spanwise wall velocity for turbulent drag reduction

2009 ◽  
Vol 627 ◽  
pp. 161-178 ◽  
Author(s):  
MAURIZIO QUADRIO ◽  
PIERRE RICCO ◽  
CLAUDIO VIOTTI
Keyword(s):  
Drag Reduction ◽  
Travelling Waves ◽  
Phase Speed ◽  
Travelling Wave ◽  
Wall Turbulence ◽  
Energetic Efficiency ◽  
Friction Drag ◽  
Mean Flow ◽  
The Waves ◽  
Spanwise Velocity

Waves of spanwise velocity imposed at the walls of a plane turbulent channel flow are studied by direct numerical simulations. We consider sinusoidal waves of spanwise velocity which vary in time and are modulated in space along the streamwise direction. The phase speed may be null, positive or negative, so that the waves may be either stationary or travelling forward or backward in the direction of the mean flow. Such a forcing includes as particular cases two known techniques for reducing friction drag: the oscillating wall technique (a travelling wave with infinite phase speed) and the recently proposed steady distribution of spanwise velocity (a wave with zero phase speed). The travelling waves alter the friction drag significantly. Waves which slowly travel forward produce a large reduction of drag that can relaminarize the flow at low values of the Reynolds number. Faster waves yield a totally different outcome, i.e. drag increase (DI). Even faster waves produce a drag reduction (DR) effect again. Backward-travelling waves instead lead to DR at any speed. The travelling waves, when they reduce drag, operate in similar fashion to the oscillating wall, with an improved energetic efficiency. DI is observed when the waves travel at a speed comparable with that of the convecting near-wall turbulence structures. A diagram illustrating the different flow behaviours is presented.

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.

Coupled Interannual Rossby Waves in a Quasigeostrophic Ocean–Atmosphere Model

2007 ◽  
Vol 37 (5) ◽  
pp. 1192-1214 ◽  
Author(s):  
Riccardo Farneti
Keyword(s):  
Wave Propagation ◽  
Rossby Wave ◽  
Rossby Waves ◽  
Phase Relationship ◽  
Phase Speed ◽  
Middle Latitude ◽  
Mean Flow ◽  
Ocean Atmosphere ◽  
Rossby Wave Propagation ◽  
Atmosphere Model

Abstract Rossby wave propagation is investigated in the framework of an idealized middle-latitude quasigeostrophic coupled ocean–atmosphere model. The Rossby waves are observed to propagate faster than both the classical linear theory (unperturbed solution) and the phase speed estimates when the effect of the zonal mean flow is added (perturbed solution). Moreover, using statistical eigentechniques, a clear coupled Rossby wave mode is identified between a baroclinic oceanic Rossby wave and an equivalent barotropic atmospheric wave. The spatial phase relationship of the coupled wave is similar to the one predicted by Goodman and Marshall, suggesting a positive ocean–atmosphere feedback. It is argued that oceanic Rossby waves can be efficiently coupled to the overlying atmosphere and that the atmospheric coupling is capable of adding an extra speedup to the wave; in fact, when the ocean is simply forced, the Rossby wave propagation speed approaches the perturbed solution.

Reynolds stresses and mean fields generated by pure waves: applications to shear flows and convection in a rotating shell

2008 ◽  
Vol 602 ◽  
pp. 303-326 ◽  
Author(s):  
E. PLAUT ◽  
Y. LEBRANCHU ◽  
R. SIMITEV ◽  
F. H. BUSSE
Keyword(s):  
Shear Flows ◽  
Rossby Waves ◽  
Zonal Flow ◽  
Quantitative Agreement ◽  
Model Systems ◽  
Geometrical Factor ◽  
Wave Flow ◽  
Nonlinear Frequency ◽  
Two Dimensional ◽  
Reynolds Stresses

A general reformulation of the Reynolds stresses created by two-dimensional waves breaking a translational or a rotational invariance is described. This reformulation emphasizes the importance of a geometrical factor: the slope of the separatrices of the wave flow. Its physical relevance is illustrated by two model systems: waves destabilizing open shear flows; and thermal Rossby waves in spherical shell convection with rotation. In the case of shear-flow waves, a new expression of the Reynolds–Orr amplification mechanism is obtained, and a good understanding of the form of the mean pressure and velocity fields created by weakly nonlinear waves is gained. In the case of thermal Rossby waves, results of a three-dimensional code using no-slip boundary conditions are presented in the nonlinear regime, and compared with those of a two-dimensional quasi-geostrophic model. A semi-quantitative agreement is obtained on the flow amplitudes, but discrepancies are observed concerning the nonlinear frequency shifts. With the quasi-geostrophic model we also revisit a geometrical formula proposed by Zhang to interpret the form of the zonal flow created by the waves, and explore the very low Ekman-number regime. A change in the nature of the wave bifurcation, from supercritical to subcritical, is found.

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