scholarly journals The Short-Wave Limit of Linear Equatorial Kelvin Waves in a Shear Flow

2005 ◽  
Vol 35 (6) ◽  
pp. 1138-1142 ◽  
Author(s):  
John P. Boyd
Keyword(s):  
Numerical Solutions ◽  
Phase Speed ◽  
Nonlinear Behavior ◽  
Short Wave ◽  
Kelvin Waves ◽  
Jet Flows ◽  
Strong Nonlinearity ◽  
Kelvin Wave ◽  
Zonal Wavenumber ◽  
The Mean

Abstract The effects of latitudinal shear on equatorial Kelvin waves in the one-and-one-half-layer model are examined through a mixture of perturbation theory and numerical solutions. For waves proportional to exp(ikx), where k is the zonal wavenumber and x is longitude, earlier perturbation theories predicted arbitrarily large distortions in the limit k → ∞. In reality, the distortions are always finite but are very different depending on the sign of the equatorial jet. When the mean jet is westward, the Kelvin wave becomes very, very narrow. When the mean jet flows eastward, the Kelvin wave splits in two with peaks well off the equator and exponentially small amplitude at the equator itself. The phase speed is always a bounded function of k, asymptotically approaching a constant. This condition has important implications for the nonlinear behavior of Kelvin waves. Strong nonlinearity cannot be balanced by contracting longitudinal scale, as in the author’s earlier Korteweg–deVries theory for equatorial solitons: for sufficiently large amplitude, the Kelvin wave must always evolve to a front.

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 Equatorial Waves in Opposite QBO Phases

2011 ◽  
Vol 68 (4) ◽  
pp. 839-862 ◽  
Author(s):  
Gui-Ying Yang ◽  
Brian J. Hoskins ◽  
Julia M. Slingo
Keyword(s):  
Group Velocity ◽  
Lower Stratosphere ◽  
Phase Speed ◽  
Kelvin Waves ◽  
Western Hemisphere ◽  
Equatorial Waves ◽  
Vertical Wavelength ◽  
Quasi Biennial Oscillation ◽  
Zonal Wavenumber ◽  
The Waves

Abstract A methodology for identifying equatorial waves is used to analyze the multilevel 40-yr ECMWF Re-Analysis (ERA-40) data for two different years (1992 and 1993) to investigate the behavior of the equatorial waves under opposite phases of the quasi-biennial oscillation (QBO). A comprehensive view of 3D structures and of zonal and vertical propagation of equatorial Kelvin, westward-moving mixed Rossby–gravity (WMRG), and n = 1 Rossby (R1) waves in different QBO phases is presented. Consistent with expectation based on theory, upward-propagating Kelvin waves occur more frequently during the easterly QBO phase than during the westerly QBO phase. However, the westward-moving WMRG and R1 waves show the opposite behavior. The presence of vertically propagating equatorial waves in the stratosphere also depends on the upper tropospheric winds and tropospheric forcing. Typical propagation parameters such as the zonal wavenumber, zonal phase speed, period, vertical wavelength, and vertical group velocity are found. In general, waves in the lower stratosphere have a smaller zonal wavenumber, shorter period, faster phase speed, and shorter vertical wavelength than those in the upper troposphere. All of the waves in the lower stratosphere show an upward group velocity and downward phase speed. When the phase of the QBO is not favorable for waves to propagate, their phase speed in the lower stratosphere is larger and their period is shorter than in the favorable phase, suggesting Doppler shifting by the ambient flow and a filtering of the slow waves. Tropospheric WMRG and R1 waves in the Western Hemisphere also show upward phase speed and downward group velocity, with an indication of their forcing from middle latitudes. Although the waves observed in the lower stratosphere are dominated by “free” waves, there is evidence of some connection with previous tropical convection in the favorable year for the Kelvin waves in the warm water hemisphere and WMRG and R1 waves in the Western Hemisphere, which is suggestive of the importance of convective forcing for the existence of propagating coupled Kelvin waves and midlatitude forcing for the existence of coupled WMRG and R1 waves.

scholarly journals The Spectrum of Convectively Coupled Kelvin Waves and the Madden–Julian Oscillation in Regions of Low-Level Easterly and Westerly Background Flow

2012 ◽  
Vol 69 (7) ◽  
pp. 2107-2111 ◽  
Author(s):  
Paul E. Roundy
Keyword(s):  
Longwave Radiation ◽  
Wave Dispersion ◽  
Warm Pool ◽  
Kelvin Waves ◽  
Madden Julian Oscillation ◽  
Westerly Wind ◽  
Kelvin Wave ◽  
Low Level ◽  
Geographical Regions ◽  
Zonal Wavenumber

Abstract The zonal wavenumber–frequency power spectrum of outgoing longwave radiation in the global tropics suggests that power in convectively coupled Kelvin waves and the Madden–Julian oscillation (MJO) is organized into two distinct spectral peaks with a minimum in power in between. This work demonstrates that integration of wavelet power in the wavenumber–frequency domain over geographical regions of moderate trade winds yields a similar pronounced spectral gap between these peaks. In contrast, integration over regions of background low-level westerly wind yields a continuum of power with no gap between the MJO and Kelvin bands. Results further show that signals in tropical convection are redder in frequency in these low-level westerly wind zones, confirming that Kelvin waves tend to propagate more slowly eastward over the warm pool than other parts of the world. Results are consistent with the perspective that portions of disturbances labeled as Kelvin waves and the MJO that are proximate to Kelvin wave dispersion curves exist as a continuum over warm pool regions.

The Influence of Amazon Rainfall on the Atlantic ITCZ through Convectively Coupled Kelvin Waves

2007 ◽  
Vol 20 (7) ◽  
pp. 1188-1201 ◽  
Author(s):  
Hui Wang ◽  
Rong Fu
Keyword(s):  
South America ◽  
Surface Wind ◽  
Phase Speed ◽  
Kelvin Waves ◽  
Tropical Atlantic ◽  
Kelvin Wave ◽  
Reanalysis Dataset ◽  
Large Variability ◽  
Boreal Spring ◽  
Atlantic Itcz

Abstract Using outgoing longwave radiation (OLR) and Tropical Rainfall Measuring Mission (TRMM) daily rain-rate data, systematic changes in intensity and location of the Atlantic intertropical convergence zone (ITCZ) were detected along the equator during boreal spring. It is found that the changes in convection over the tropical Atlantic may be induced by deep convection in equatorial South America. Lagged regression analyses demonstrate that the anomalies of convection developed over the land propagate eastward across the Atlantic and then into Africa. The eastward-propagating disturbances appear to be convectively coupled Kelvin waves with a period of 6–7.5 days and a phase speed of around 15 m s−1. These waves modulate the intensity and location of the convection in the tropical Atlantic and result in a zonal variation of the Atlantic ITCZ on synoptic time scales. The convectively coupled Kelvin wave has substantial signals in both the lower and upper troposphere. Both a reanalysis dataset and the Quick Scatterometer (QuikSCAT) ocean surface wind are used to characterize the Kelvin wave. This study suggests that synoptic-scale variation of the Atlantic ITCZ may be linked to precipitation anomalies in South America through the convectively coupled Kelvin wave. The results imply that the changes of Amazon convection could contribute to the large variability of the tropical Atlantic ITCZ observed during boreal spring.

scholarly journals Observed Relationships between Oceanic Kelvin Waves and Atmospheric Forcing

2006 ◽  
Vol 19 (20) ◽  
pp. 5253-5272 ◽  
Author(s):  
Paul E. Roundy ◽  
George N. Kiladis
Keyword(s):  
Wind Stress ◽  
El Niño ◽  
El Nino ◽  
Phase Speed ◽  
Kelvin Waves ◽  
Apparent Difference ◽  
Kelvin Wave ◽  
Central Pacific ◽  
Basin Scale ◽  
The Waves

Abstract The Madden–Julian oscillation (MJO) has been implicated as a major source of the wind stress variability that generates basin-scale Kelvin waves in the equatorial Pacific. One source of debate concerning this relationship is the apparent difference in the frequencies of the two processes. This work utilizes data from the Tropical Atmosphere Ocean (TAO) array of moored buoys along with outgoing longwave radiation data to show by means of a multiple linear regression model and case studies that the frequency discrepancy is due to a systematic decrease in the phase speeds of the Kelvin waves and an increase in the period of the waves toward the east as conditions adjust toward El Niño. Among the potential contributing factors to this phase speed decrease is an apparent air–sea interaction that enhances the wind forcing of some of the Kelvin waves, allowing them to continue to amplify because the propagating wind stress anomaly decelerates to the speed of the developing Kelvin wave instead of the significantly faster speed of the typical MJO. Kelvin waves appear to be most effectively amplified during periods when the temperature gradient above the thermocline across the equatorial central Pacific is strong, the thermocline shoals steeply toward the east in the central Pacific, and/or when the phase speed of the propagating wind stress forcing is closest to that of the Kelvin wave. These conditions tend to occur as the ocean adjusts toward El Niño. Since Kelvin waves are instrumental to the development of El Niño events, isolating the detailed relationship between the waves and the MJO will lead to a better understanding of interannual ocean–atmosphere interactions.

scholarly journals Convectively Coupled Kelvin Waves and Tropical Cyclogenesis in a Semi-Lagrangian Framework

2016 ◽  
Vol 144 (11) ◽  
pp. 4131-4139 ◽  
Author(s):  
Carl J. Schreck
Keyword(s):  
North Atlantic ◽  
Phase Speed ◽  
Kelvin Waves ◽  
Tropical Cyclogenesis ◽  
Kelvin Wave ◽  
Easterly Waves ◽  
The North ◽  
Lagrangian Framework ◽  
Easterly Wave ◽  
Vertically Aligned

Abstract This study examines how convectively coupled Kelvin waves interact with the semi-Lagrangian circulation of easterly waves to modulate tropical cyclogenesis. Recent studies have shown that fewer tropical cyclones form in the three days before passage of the Kelvin wave’s peak convection and more develop in the three days thereafter. Separately, other studies have identified the recirculation of moisture and vorticity within easterly waves using a semi-Lagrangian frame of reference. That framework is achieved by subtracting the easterly wave phase speed from the earth-relative winds. This study combines these recent findings by testing whether the equatorial westerlies from Kelvin waves can help close the semi-Lagrangian circulation. Past studies have shown that Kelvin waves tilt westward with height in the troposphere such that equatorial westerlies build upward from the surface in the days following the convective peak. This study shows that the easterly wave’s semi-Lagrangian closed circulation grows upward as it intersects the Kelvin wave’s westward tilt. The Kelvin wave’s westerly anomalies reach 500 hPa about three days after the convection has passed, which establishes the deep, vertically aligned easterly wave vortex necessary for tropical cyclogenesis. This study focuses on the eastern Pacific, but similar results are found for the North Atlantic. In other basins, the Kelvin wave accentuates the westerlies from the Madden–Julian oscillation and/or the monsoon trough. Given that Kelvin waves often last weeks and circumnavigate the globe, these results may advance long-range tropical cyclogenesis forecasting.

scholarly journals Low-frequency scattering of Kelvin waves by continuous topography

1993 ◽  
Vol 248 ◽  
pp. 173-201 ◽  
Author(s):  
E. R. Johnson
Keyword(s):  
Direct Method ◽  
Low Frequency ◽  
Short Wave ◽  
Kelvin Waves ◽  
Order Effects ◽  
Kelvin Wave ◽  
Inviscid Flows ◽  
Explicit Formulae ◽  
Weak Stratification ◽  
Higher Order Effects

This paper continues the analysis of Johnson (1990, hereinafter referred to as I) of the scattering of Kelvin waves by collections of ridges and valleys. General results, flow patterns and explicit solutions follow by restricting attention to waves whose period is long compared to the inertial period but without the additional further simplification introduced in I of approximating general features by stepped topography. A simple direct method is presented giving explicit formulae for the amplitude of the transmitted Kelvin wave and the scattered topographic long waves. A simple but accurate approximation to the solution is also given. The accuracy and usefulness of the apparently crude method of I are confirmed and a superior method presented for choosing the positions of steps in the approximation of general topography. Inviscid flows, the effects of weak dissipation and weak stratification, the form and relevance of the short-wave field over downslopes, the partition of mass and energy flux between the long-wave and short-wave fields and the size and form of higher-order effects are also discussed.

On the stability of Kelvin waves

1989 ◽  
Vol 206 ◽  
pp. 1-23 ◽  
Author(s):  
W. K. Melville ◽  
G. G. Tomasson ◽  
D. P. Renouard
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Resonance ◽  
Approximate Solutions ◽  
Kelvin Waves ◽  
Kelvin Wave ◽  
Governing Equations ◽  
Weakly Nonlinear ◽  
Approximate Analytical Solutions ◽  
The Stability ◽  
Good Agreement

We consider the evolution of weakly nonlinear dispersive long waves in a rotating channel. The governing equations are derived and approximate solutions obtained for the initial data corresponding to a Kelvin wave. In consequence of the small nonlinear speed correction it is shown that weakly nonlinear Kelvin waves are unstable to a direct nonlinear resonance with the linear Poincaré modes of the channel. Numerical solutions of the governing equations are computed and found to give good agreement with the approximate analytical solutions. It is shown that the curvature of the wavefront and the decay of the leading wave amplitude along the channel are attributable to the Poincaré waves generated by the resonance. These results appear to give a qualitative explanation of the experimental results of Maxworthy (1983), and Renouard, Chabert d'Hières & Zhang (1987).

scholarly journals A Toy Model of the Instability in the Equatorially Trapped Convectively Coupled Waves on the Equatorial Beta Plane

2008 ◽  
Vol 65 (12) ◽  
pp. 3736-3757 ◽  
Author(s):  
Joseph Allan Andersen ◽  
Zhiming Kuang
Keyword(s):  
Growth Rates ◽  
Wave Spectrum ◽  
Phase Speed ◽  
Wave Theory ◽  
Linear Instability ◽  
Kelvin Wave ◽  
Convectively Coupled Waves ◽  
Beta Plane ◽  
Zonal Wavenumber ◽  
Wave Modes

Abstract The equatorial atmospheric variability shows a spectrum of significant peaks in the wavenumber–frequency domain. These peaks have been identified with the equatorially trapped wave modes of rotating shallow water wave theory. This paper addresses the observation that the various wave types (e.g., Kelvin, Rossby, etc.) and wavenumbers show differing signal strength relative to a red background. It is hypothesized that this may be due to variations in the linear stability of the atmosphere in response to the various wave types depending on both the specific wave type and the wavenumber. A simple model of the convectively coupled waves on the equatorial beta plane is constructed to identify processes that contribute to this dependence. The linear instability spectrum of the resulting coupled system is evaluated by eigenvalue analysis. This analysis shows unstable waves with phase speeds, growth rates, and structures (vertical and horizontal) that are broadly consistent with the results from observations. The linear system, with an idealized single intertropical convergence zone (ITCZ) as a mean state, shows peak unstable Kelvin waves around zonal wavenumber 7 with peak growth rates of ∼0.08 day−1 (e-folding time of ∼13 days). The system also shows unstable mixed Rossby–gravity (MRG) and inertio-gravity waves with significant growth in the zonal wavenumber range from −15 (negative indicates westward phase speed) to +10 (positive indicates eastward phase speed). The peak MRG n = 0 eastward inertio-gravity wave (EIG) growth rate is around one-third that of the Kelvin wave and occurs at zonal wavenumber 3. The Rossby waves in this system are stable, and the Madden–Julian oscillation is not observed. Within this model, it is shown that in addition to the effect of the ITCZ configuration, the differing instabilities of the different wave modes are also related to their different efficiency in converting input energy into divergent flow. This energy conversion efficiency difference is suggested as an additional factor that helps to shape the observed wave spectrum.

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.

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