scholarly journals Linear Waves in Midlatitudes on the Rotating Spherical Earth

2009 ◽  
Vol 39 (12) ◽  
pp. 3204-3215 ◽  
Author(s):  
Yair De Leon ◽  
Nathan Paldor
Keyword(s):  
Rossby Waves ◽  
Eigenvalue Problems ◽  
Phase Speed ◽  
Channel Width ◽  
Small Radius ◽  
Harmonic Waves ◽  
Kelvin Wave ◽  
Linear Waves ◽  
Spherical Earth ◽  
Poincare Waves

Abstract The linear waves of the shallow water equations in a zonal channel in midlatitudes on the rotating spherical earth are investigated analytically and numerically by solving several relevant eigenvalue problems. For baroclinic deformation radii in the ocean, the phase speed of long Rossby waves in a sufficiently wide channel on the sphere can be 5 times that of their harmonic β-plane counterparts. The difference between the two phase speeds increases with the channel width and decreases with 1) the latitude of the equatorward wall, 2) the radius of deformation, and 3) the mode number. For Poincaré (inertia–gravity) waves, the phase speed on the sphere is slightly lower than that of harmonic waves on the β plane. The meridionally dependent amplitude of the meridional velocity is identical for both waves and is trapped near the equatorward wall—that is, its amplitude is maximal within a few deformation radii from this wall. The phase speeds of the Kelvin and anti-Kelvin waves on a sphere are determined by the latitudes of the equatorward and poleward walls, respectively, where they attain their maximal height amplitude. Accordingly, the phase speed of the anti-Kelvin wave is larger than that of the westward-propagating Poincaré waves in a certain wavenumber range, whereas the phase speed of eastward-propagating Poincaré waves does not approach that of the Kelvin wave even at large wavenumbers. Analytical expressions for the phase speed of trapped Poincaré and Rossby waves are obtained for small deformation radii in wide channels by approximating the meridional velocity’s eigenfunction by an Airy function that decays with distance from the equatorward wall. The exact latitude of the poleward wall does not affect the solution, provided it is several deformation radii away from the equatorward boundary and the exact channel width increases with the radius of deformation. For a sufficiently small radius of deformation, such as that observed in the ocean, the solution is trapped, even for very narrow channels, and the phase speed is only slightly larger than that of harmonic waves.

Rossby-Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves

1989 ◽  
Vol 202 ◽  
pp. 149-176 ◽  
Author(s):  
Satoshi Sakai
Keyword(s):  
Gravity Waves ◽  
Rossby Waves ◽  
Baroclinic Instability ◽  
Phase Speed ◽  
Kelvin Wave ◽  
Layer System ◽  
Geostrophic Balance ◽  
New Type ◽  
Frontal Instability ◽  
Zero Phase

An ageostrophic version of Phillips’ model is studied. All instabilities found are systematically interpreted in terms of resonance of wave components. The instability occurs if there is a pair of wave components which propagate in the opposite direction to the basic flow and these wave components have almost the same Doppler-shifted frequency. A new instability, identified as a resonance between the Kelvin wave and the Rossby waves, is found at Froude number F ≈ 0.7. The Rossby waves are almost completely in geostrophic balance while the ageostrophic Kelvin wave is the same as in a one-layer system. Doppler shifting matches frequencies which would otherwise be very different. This instability is presumably the mechanism of the frontal instability observed by Griffiths & Linden (1982) in a laboratory experiment. Ageostrophic, baroclinic instability with non-zero phase speed is also observed in the numerical calculation. This instability is caused by resonance between different geostrophic modes.

Planetary (Rossby) waves and inertia–gravity (Poincaré) waves in a barotropic ocean over a sphere

2013 ◽  
Vol 726 ◽  
pp. 123-136 ◽  
Author(s):  
Nathan Paldor ◽  
Yair De-Leon ◽  
Ofer Shamir
Keyword(s):  
Gravity Waves ◽  
Rossby Waves ◽  
Phase Speed ◽  
Mode Number ◽  
Rotating Sphere ◽  
Zonal Wavenumber ◽  
Speed Of Gravity ◽  
Meridional Mode ◽  
Poincare Waves ◽  
The Waves

AbstractThe construction of approximate Schrödinger eigenvalue equations for planetary (Rossby) waves and for inertia–gravity (Poincaré) waves on an ocean-covered rotating sphere yields highly accurate estimates of the phase speeds and meridional variation of these waves. The results are applicable to fast rotating spheres such as Earth where the speed of barotropic gravity waves is smaller than twice the tangential speed on the equator of the rotating sphere. The implication of these new results is that the phase speed of Rossby waves in a barotropic ocean that covers an Earth-like planet is independent of the speed of gravity waves for sufficiently large zonal wavenumber and (meridional) mode number. For Poincaré waves our results demonstrate that the dispersion relation is linear, (so the waves are non-dispersive and the phase speed is independent of the wavenumber), except when the zonal wavenumber and the (meridional) mode number are both near 1.

The Effects of Wind Forcing and Background Mean Currents on the Latitudinal Structure of Equatorial Rossby Waves

2005 ◽  
Vol 35 (5) ◽  
pp. 666-682 ◽  
Author(s):  
Renellys C. Perez ◽  
Dudley B. Chelton ◽  
Robert N. Miller
Keyword(s):  
Rossby Wave ◽  
Rossby Waves ◽  
Current System ◽  
Phase Speed ◽  
Wave Structure ◽  
Wind Forcing ◽  
Thermocline Depth ◽  
Linear Waves ◽  
Mean Current ◽  
Equatorial Rossby Waves

Abstract The latitudinal structure of annual equatorial Rossby waves in the tropical Pacific Ocean based on sea surface height (SSH) and thermocline depth observations is equatorially asymmetric, which differs from the structure of the linear waves of classical theory that are often presumed to dominate the variability. The nature of this asymmetry is such that the northern SSH maximum (along 5.5°N) is roughly 2 times that of the southern maximum (along 6.5°S). In addition, the observed westward phase speeds are roughly 0.5 times the predicted speed of 90 cm s−1 and are also asymmetric with the northern phase speeds, about 25% faster than the southern phase speeds. One hypothesized mechanism for the observed annual equatorial Rossby wave amplitude asymmetry is modification of the meridional structure by the asymmetric meridional shears associated with the equatorial current system. Another hypothesis is the asymmetry of the annually varying wind forcing, which is stronger north of the equator. A reduced-gravity, nonlinear, β-plane model with rectangular basin geometry forced by idealized Quick Scatterometer (QuikSCAT) wind stress is used to test these two mechanisms. The model with an asymmetric background mean current system perturbed with symmetric annually varying winds consistently produces asymmetric Rossby waves with a northern maximum (4.7°N) that is 1.6 times the southern maximum (5.2°S) and westward phase speeds of approximately 53 ± 13 cm s−1 along both latitudes. Simulations with a symmetric background mean current system perturbed by asymmetric annually varying winds fail to produce the observed Rossby wave structure unless the perturbation winds become strong enough for nonlinear interactions to produce asymmetry in the background mean current system. The observed latitudinal asymmetry of the phase speed is found to be critically dependent on the inclusion of realistic coastline boundaries.

scholarly journals A harmonic projection and least–squares method for quantifying Kelvin wave activity

2016 ◽  
Author(s):  
Andrew Delman ◽  
Janet Sprintall ◽  
Julie McClean ◽  
Lynne Talley
Keyword(s):  
Indian Ocean ◽  
Least Squares ◽  
Sea Level ◽  
Rossby Waves ◽  
Phase Speed ◽  
Wave Activity ◽  
Kelvin Waves ◽  
Three Dimensions ◽  
Kelvin Wave ◽  
Sea Level Anomaly

Abstract. A new method for isolating the equatorial and coastal Kelvin wave signal from alongtrack satellite altimetry data is presented and applied to sea level anomaly (SLA) observations in the tropical Indian Ocean. The method consists of sequential projections onto the SLA data, starting with meridional or cross-shore Kelvin wave profiles derived from shallow water theory (y-projections). Next, Fourier basis functions in x-t (along-waveguide distance and time respectively) space with the phase speed ranges of Kelvin and Rossby waves are projected onto the y-projections. After projections in all three dimensions have been carried out, least-squares methods are applied to optimize the non-orthogonal basis function coefficients and minimize the misfit of their along-waveguide forcing and dissipation. Lastly, the westward-propagating (Rossby wave-related) signals are removed, generating a Kelvin wave coefficient K that represents Kelvin wave activity. Along the Indian Ocean equatorial-coastal waveguide, Hovmöller diagrams of K show reduced high-wavenumber noise compared to analogous diagrams of pre-processed sea level anomaly. Results from a Monte Carlo simulation demonstrate that Kelvin wave signals generated a priori can be effectively recovered even when superimposed with strong Rossby waves; the signs of all but the weakest waves are diagnosed correctly in over 90 % of cases. When the method is applied to 21 years of satellite observations and the SLA signal associated with K is removed, the large residual in the equatorial SLA signal has a spatial distribution consistent with wind-forced Rossby waves. The equatorial SLA variability in the western part of the basin is poorly correlated with the SLA field associated with K, as the superimposed SLA profile of Rossby waves can distort the true origin locations of Kelvin waves in the raw SLA field. Therefore, this method offers improved tracking of Kelvin waves compared to the raw SLA dataset, and may provide the opportunity to study weakly nonlinear aspects of these waves by comparison with linear models.

Getting Rid of Fast Waves: Slow Dynamics

2018 ◽  
Author(s):  
Vladimir Zeitlin
Keyword(s):  
Rossby Waves ◽  
Baroclinic Instability ◽  
Nonlinear Effects ◽  
Kelvin Wave ◽  
Beta Plane ◽  
Equatorial Wave ◽  
Flow Over Topography ◽  
Wave Guides ◽  
Rotating Shallow Water ◽  
Fast Waves

After analysis of general properties of horizontal motion in primitive equations and introduction of principal parameters, the key notion of geostrophic equilibrium is introduced. Quasi-geostrophic reductions of one- and two-layer rotating shallow-water models are obtained by a direct filtering of fast inertia–gravity waves through a choice of the time scale of motions of interest, and by asymptotic expansions in Rossby number. Properties of quasi-geostrophic models are established. It is shown that in the beta-plane approximations the models describe Rossby waves. The first idea of the classical baroclinic instability is given, and its relation to Rossby waves is explained. Modifications of quasi-geostrophic dynamics in the presence of coastal, topographic, and equatorial wave-guides are analysed. Emission of mountain Rossby waves by a flow over topography is demonstrated. The phenomena of Kelvin wave breaking, and of soliton formation by long equatorial and topographic Rossby waves due to nonlinear effects are explained.

Waves, jets and vortices: Regime transitions in shallow water turbulence

2021 ◽  
Author(s):  
Nikos Bakas
Keyword(s):  
Shallow Water ◽  
Large Scale ◽  
Rossby Waves ◽  
Phase Speed ◽  
Critical Value ◽  
Turbulent Regime ◽  
Wave Regime ◽  
Frequency Spectra ◽  
Spectra Analysis ◽  
Zonal Jets

<p>Forced-dissipative beta-plane turbulence in a single-layer shallow-water fluid has been widely considered as a simplified model of planetary turbulence as it exhibits turbulence self-organization into large-scale structures such as robust zonal jets and strong vortices. In this study we perform a series of numerical simulations to analyze the characteristics of the emerging structures as a function of the planetary vorticity gradient and the deformation radius. We report four regimes that appear as the energy input rate ε of the random stirring that supports turbulence in the flow increases. A homogeneous turbulent regime for low values of ε, a regime in which large scale Rossby waves form abruptly when ε passes a critical value, a regime in which robust zonal jets coexist with weaker Rossby waves when ε passes a second critical value and a regime of strong materially coherent propagating vortices for large values of ε. The wave regime which is not predicted by standard cascade theories of turbulence anisotropization and the vortex regime are studied thoroughly. Wavenumber-frequency spectra analysis shows that the Rossby waves in the second regime remain phase coherent over long times. The coherent vortices are identified using the Lagrangian Averaged Deviation (LAVD) method. The statistics of the vortices (lifetime, radius, strength and speed) are reported as a function of the large scale parameters. We find that the strong vortices propagate zonally with a phase speed that is equal or larger than the long Rossby wave speed and advect the background turbulence leading to a non-dispersive line in the wavenumber-frequency spectra.</p>

scholarly journals Instability of combined gravity-inertial-Rossby waves in atmospheres and oceans

2011 ◽  
Vol 29 (6) ◽  
pp. 997-1003 ◽  
Author(s):  
J. F. McKenzie
Keyword(s):  
Energy Exchange ◽  
Wave Speed ◽  
Rossby Waves ◽  
Low Frequency ◽  
Phase Speed ◽  
Ocean Dynamics ◽  
Stratified Medium ◽  
Instability Condition ◽  
Long Wavelength ◽  
Critical Latitude

Abstract. The properties of the instability of combined gravity-inertial-Rossby waves on a β-plane are investigated. The wave-energy exchange equation shows that there is an exchange of energy with the background stratified medium. The energy source driving the instability lies in the background enthalpy released by the gravitational buoyancy force. It is shown that if the phase speed of the westward propagating low frequency-long wavelength Rossby wave exceeds the Poincaré-Kelvin (or "equivalent" shallow water) wave speed, instability arises from the merging of Rossby and Poincaré modes. There are two key parameters in this instability condition; namely, the equatorial/rotational Mach (or Froude) number M and the latitude θ0 of the β-plane. In general waves equatorward of a critical latitude for given M can be driven unstable, with corresponding growth rates of the order of a day or so. Although these conclusions may only be safely drawn for short wavelengths corresponding to a JWKB wave packet propagating internally and located far from boundaries, nevertheless such a local instability may play a significant role in atmosphere-ocean dynamics.

scholarly journals Long-Wave Dynamics of Sea Level Variations during Indian Ocean Dipole Events

2009 ◽  
Vol 39 (5) ◽  
pp. 1115-1132 ◽  
Author(s):  
Dongliang Yuan ◽  
Hailong Liu
Keyword(s):  
Indian Ocean ◽  
Sea Level ◽  
Indian Ocean Dipole ◽  
Rossby Waves ◽  
Western Boundary ◽  
Kelvin Wave ◽  
Wave Pulse ◽  
Wave Dynamics ◽  
Long Wave ◽  
Equatorial Rossby Waves

Abstract Long-wave dynamics of the interannual variations of the equatorial Indian Ocean circulation are studied using an ocean general circulation model forced by the assimilated surface winds and heat flux of the European Centre for Medium-Range Weather Forecasts. The simulation has reproduced the sea level anomalies of the Ocean Topography Experiment (TOPEX)/Poseidon altimeter observations well. The equatorial Kelvin and Rossby waves decomposed from the model simulation show that western boundary reflections provide important negative feedbacks to the evolution of the upwelling currents off the Java coast during Indian Ocean dipole (IOD) events. Two downwelling Kelvin wave pulses are generated at the western boundary during IOD events: the first is reflected from the equatorial Rossby waves and the second from the off-equatorial Rossby waves in the southern Indian Ocean. The upwelling in the eastern basin during the 1997–98 IOD event is weakened by the first Kelvin wave pulse and terminated by the second. In comparison, the upwelling during the 1994 IOD event is terminated by the first Kelvin wave pulse because the southeasterly winds off the Java coast are weak at the end of 1994. The atmospheric intraseasonal forcing, which plays an important role in inducing Java upwelling during the early stage of an IOD event, is found to play a minor role in terminating the upwelling off the Java coast because the intraseasonal winds are either weak or absent during the IOD mature phase. The equatorial wave analyses suggest that the upwelling off the Java coast during IOD events is terminated primarily by western boundary reflections.

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.

scholarly journals Phase-speed Spectra of Eddy Tracer Fluxes Linked to Isentropic Stirring and Mixing in the Upper Troposphere and Lower Stratosphere

2016 ◽  
Vol 73 (12) ◽  
pp. 4711-4730 ◽  
Author(s):  
Marta Abalos ◽  
William J. Randel ◽  
Thomas Birner
Keyword(s):  
Rossby Waves ◽  
Lower Stratosphere ◽  
Southern Oscillation ◽  
Phase Speed ◽  
Transient Eddy ◽  
Tracer Transport ◽  
Quasi Biennial Oscillation ◽  
Upper Troposphere ◽  
Eddy Fluxes ◽  
Subtropical Jets

Abstract The regions around the subtropical jets in the upper troposphere and lower stratosphere (UTLS) are characterized by strong isentropic stirring and mixing. In this work, the wave spectrum of the associated eddy tracer fluxes is examined using an artificial passive tracer advected on isentropes by the two-dimensional flow. The eddy diffusivity computed from the flux–gradient relation captures the main features of the mixing structure. Eddy transport in the UTLS is strongest in the summer hemisphere, and weak eddy fluxes are found at the core and poleward of the subtropical jets, especially in the winter hemisphere. There is an important contribution of stationary planetary equatorial Rossby waves in the tropical upper troposphere. The transient eddy tracer transport is primarily linked to medium-scale waves (wavenumbers ~4–7) breaking in the regions of weak westerlies around the subtropical jets and to planetary-scale waves at high latitudes. Phase-speed spectra for transient eddy fluxes show a close relationship of waves to the background zonal wind. In the deep tropics, traveling equatorial and Rossby waves of extratropical origin lead to cross-equatorial tracer transport throughout the upper troposphere. Interannual changes show that eddy tracer fluxes closely follow the shifts in the zonal winds associated with El Niño–Southern Oscillation and the quasi-biennial oscillation.

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