scholarly journals Uniform Asymptotics for the Linear Kelvin Wave in Spherical Geometry

2008 ◽  
Vol 65 (2) ◽  
pp. 655-660 ◽  
Author(s):  
John P. Boyd ◽  
Cheng Zhou
Keyword(s):  
Asymptotic Approximation ◽  
Rossby Waves ◽  
Spherical Geometry ◽  
Uniform Asymptotics ◽  
Kelvin Wave ◽  
Beta Plane ◽  
Zonal Wavenumber ◽  
Equatorial Wave ◽  
Wave Limit ◽  
Two Parameters

Abstract The Kelvin wave is the gravest eigenmode of Laplace’s tidal equation. It is widely observed in both the ocean and the atmosphere. In the absence of mean currents, the Kelvin wave depends on two parameters: the zonal wavenumber s (always an integer) and Lamb’s parameter ε. An asymptotic approximation valid in the limit s2 + ε ≫ 1 is derived that generalizes the usual “equatorial wave” limit that ε → ∞ for fixed s. Just as shown for Rossby waves two decades ago, the width of the Kelvin wave is (ε + s2)−1/4 rather than ε−1/4 as in the classical equatorial beta-plane approximation.

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.

Dynamic Core of the MJO

2020 ◽  
Author(s):  
Chidong Zhang ◽  
Ji-Eun Kim
Keyword(s):  
Harmonic Oscillator ◽  
Kelvin Wave ◽  
Stochastic Forcing ◽  
Continuous Transition ◽  
Damped Harmonic Oscillator ◽  
Beta Plane ◽  
Zonal Wavenumber

<p>This presentation introduces a theory in which the dynamic core of the MJO is described in terms of a harmonic oscillator that can be excited by stochastic forcing. The mechanism for selecting MJO scales comes from momentum damping. The resonant solution to the equation for a damped harmonic oscillator on an equatorial beta plane represents the equatorial Kelvin wave for small damping and large zonal wavenumbers and the MJO for large (3 – 5 days) damping and zonal wavenumber one. This theory demonstrates the distinction between the Kelvin wave and MJO and their continuous transition. In contrast to most other MJO theories that compete against each other, this theory embraces most other theories in that they provide possible sources of energy (forcing) to the dynamic core of the MJO.</p>

scholarly journals Effect of Overturning Circulation on Long Equatorial Waves: A Low-Frequency Cutoff

2018 ◽  
Vol 75 (5) ◽  
pp. 1721-1739 ◽  
Author(s):  
Amanda Back ◽  
Joseph A. Biello
Keyword(s):  
Large Scale ◽  
Rossby Waves ◽  
Meridional Circulation ◽  
Low Frequency ◽  
Hadley Cell ◽  
Equatorial Waves ◽  
Kelvin Wave ◽  
Background Flow ◽  
Overturning Circulation ◽  
Equatorial Wave

Zonally long tropical waves in the presence of a large-scale meridional and vertical overturning circulation are studied in an idealized model based on the intraseasonal multiscale moist dynamics (IMMD) theory. The model consists of a system of shallow-water equations describing barotropic and first baroclinic vertical modes coupled to one another by the zonally symmetric, time-independent background circulation. To isolate the effects of the meridional circulation alone, an idealized background flow is chosen to mimic the meridional and vertical components of the flow of the Hadley cell; the background flow meridionally converges and rises at the equator. The resulting linear eigenvalue problem is a generalization of the long-wave-scaled version of Matsuno’s equatorial wave problem with the addition of meridional and vertical advection. The results demonstrate that the meridional circulation couples equatorially trapped baroclinic Rossby waves to planetary, barotropic free Rossby waves. The meridional circulation also causes the Kelvin wave to develop an equatorially trapped barotropic component, imparting a westward-tilted vertical structure to the wave. The total energy of the linear system is positive definite, so all waves are shown to be neutrally stable. A critical layer exists at latitudes where the meridional background flow vanishes, resulting in a minimum frequency cutoff for physically feasible waves. Therefore, linear Matsuno waves with periods longer than the vertical transport time of the meridional circulation do not exist in the equatorial waveguide. This implies a low-frequency cutoff for long equatorial waves.

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 A Reduced-Order Representation of the Madden–Julian Oscillation Based on Reanalyzed Normal Mode Coherences

2019 ◽  
Vol 76 (8) ◽  
pp. 2463-2480 ◽  
Author(s):  
Vassili Kitsios ◽  
Terence J. O’Kane ◽  
Nedjeljka Žagar
Keyword(s):  
Normal Mode ◽  
Gravity Waves ◽  
Rossby Waves ◽  
Spatial Scales ◽  
Longwave Radiation ◽  
Kelvin Waves ◽  
Madden Julian Oscillation ◽  
Kelvin Wave ◽  
Zonal Wavenumber ◽  
Vertical Scales

Abstract The Madden–Julian oscillation (MJO) is presented as a series of interacting Rossby and inertial gravity waves of varying vertical scales and meridional extents. These components are isolated by decomposing reanalysis fields into a set of normal mode functions (NMF), which are orthogonal eigenvectors of the linearized primitive equations on a sphere. The NMFs that demonstrate spatial properties compatible with the MJO are inertial gravity waves of zonal wavenumber k = 1 and the lowest meridional index n = 0, and Rossby waves with (k, n) = (1, 1). For these horizontal scales, there are multiple small vertical-scale baroclinic modes that have temporal properties indicative of the MJO. On the basis of one such eastward-propagating inertial gravity wave (i.e., a Kelvin wave), composite averages of the Japanese 55-year Reanalysis demonstrate an eastward propagation of the velocity potential, and oscillation of outgoing longwave radiation and precipitation fields over the Maritime Continent, with an MJO-appropriate temporal period. A cross-spectral analysis indicates that only the MJO time scale is coherent between this Kelvin wave and the more energetic modes. Four mode clusters are identified: Kelvin waves of correct phase period and direction, Rossby waves of correct phase period, energetic Kelvin waves of larger vertical scales and meridional extents extending into the extratropics, and energetic Rossby waves of spatial scales similar to that of the energetic Kelvin waves. We propose that within this normal mode framework, nonlinear interactions between the aforementioned mode groups are required to produce an energetic MJO propagating eastward with an intraseasonal phase period. By virtue of the selected mode groups, this theory encompasses both multiscale and tropical–extratropical interactions.

Atmospheric Subseasonal Variability and Circulation Regimes: Spectra, Trends, and Uncertainties

2020 ◽  
Vol 33 (21) ◽  
pp. 9375-9390
Author(s):  
Nedjeljka Žagar ◽  
Žiga Zaplotnik ◽  
Khalil Karami
Keyword(s):  
Gravity Waves ◽  
Large Scale ◽  
Rossby Waves ◽  
Wave Activity ◽  
Subseasonal Variability ◽  
Zonal Wavenumber ◽  
Equatorial Wave ◽  
Wave Patterns ◽  
Meridional Mode ◽  
The One

AbstractThe globally integrated subseasonal variability associated with the two main atmospheric circulation regimes, the balanced (or Rossby) and unbalanced (or inertia–gravity) regimes, is evaluated for the four reanalysis datasets: ERA-Interim, JRA-55, MERRA, and ERA5. The results quantify amplitudes and trends in midlatitude traveling and quasi-stationary Rossby wave patterns as well as in the equatorial wave activity across scales. A statistically significant reduction of subseasonal variability is found in Rossby waves with zonal wavenumber k = 6 along with an increase in variability in wavenumbers k = 3–5 in the summer seasons of both hemispheres. The four reanalyses also agree regarding increased variability in the large-scale Kelvin waves, mixed Rossby–gravity waves, and westward-propagating inertio-gravity waves with the lowest meridional mode. The amplitude and sign of trends in inertia–gravity modes with smaller zonal scales and greater meridional modes differ between the ERA-Interim and JRA-55 datasets on the one hand and the ERA5 and MERRA data on the other. An increased variability in the ERA-Interim and JRA-55 accounts for positive trends in their total subseasonal variability.

scholarly journals Cloud Masks Derived from the Mars Daily Global Maps and an Application to the Tropical Cloud Belt on Mars

Geosciences ◽  
2021 ◽  
Vol 11 (8) ◽  
pp. 324
Author(s):  
Huiqun Wang ◽  
Gonzalo González Abad
Keyword(s):  
Processing Technique ◽  
Oscillatory Behavior ◽  
Early Summer ◽  
Dust Storms ◽  
Transitional Period ◽  
Image Processing Technique ◽  
Kelvin Wave ◽  
Blue Channel ◽  
Zonal Wavenumber ◽  
Shift Pattern

An image processing technique is used to derive cloud masks from the color Mars Daily Global Maps (MDGMs) that are composed from the Mars Reconnaissance Orbiter (MRO) Mars Color Imager (MARCI) wide-angle image swaths. The blue channel of each MDGM is used to select cloud candidates and the blue-to-red ratio map is compared with a reference ratio map to filter out false positives. Quality control is performed manually. The derived cloud masks cover 1 Mars year from the summer of Mars year (MY) 28 to the summer of MY 29. The product has a 0.1° longitude by 0.1° latitude resolution and is available each day. This makes it possible to characterize the evolution of the tropical cloud belt from several new perspectives. The tropical cloud belt steadily builds up during northern spring and early summer, peaks near the early- to mid-summer transitional period, and rapidly declines afterward. From the perspective of cloud occurrence frequency and time mean, the cloud belt appears meandrous and zonally discontinuous, with minima in the Amazonis Planitia and Arabia Terra longitudinal sectors. A pronounced cloud branch diverges from the main cloud belt and extends from the Valles Marineris towards the Noachis and Hellas region. The cloud belt exhibits noticeable oscillatory behavior whereby cloud brightening alternates between the western and eastern hemispheres near the equator with a periodicity between 20 and 30 sols. The cloud belt oscillation occurred each Mars year around Ls = 140°, except for the Mars years when intense dust storms made disruptions. The phenomenon appears to be associated with an eastward propagating equatorial Kelvin wave with zonal wavenumber 1. This wave has a much longer wave period than the diurnal and semidiurnal Kelvin waves discussed in most of the previous studies and may be an important factor for the intra-seasonal variability of the tropical cloud belt. The convolution of clouds’ local time variation with MRO’s orbit shift pattern results in a seemingly highly regular 5-day traveling wave in Hovmöller diagrams of cloud masks.

scholarly journals Convectively Coupled Equatorial Waves. Part I: Horizontal and Vertical Structures

2007 ◽  
Vol 64 (10) ◽  
pp. 3406-3423 ◽  
Author(s):  
Gui-Ying Yang ◽  
Brian Hoskins ◽  
Julia Slingo
Keyword(s):  
Composite Structures ◽  
Wave Structure ◽  
Wave Theory ◽  
Western Hemisphere ◽  
Equatorial Waves ◽  
Kelvin Wave ◽  
Ambient Flow ◽  
Convectively Coupled Equatorial Waves ◽  
Eastern Hemisphere ◽  
Equatorial Wave

Abstract Multilevel 15-yr ECMWF Re-Analysis (ERA-15) and satellite-observed brightness temperature (Tb) data for the period May–October 1992 are used to examine the horizontal and vertical structures of convectively coupled equatorial waves. Dynamical waves are isolated using a methodology developed previously. Composite structures of convectively coupled equatorial waves are obtained using linear regression/correlation between convection (Tb) and dynamical structures. It is found that the relationship depends on the ambient flow and the nature of the convective coupling, and varies between off-equatorial- and equatorial-centered convection, different hemispheres, and seasons. The Kelvin wave structure in the Western Hemisphere is generally consistent with classic equatorial wave theory and has its convection located in the region of low-level convergence. In the Eastern Hemisphere the Kelvin wave tends to have convection in the region of enhanced lower-tropospheric westerlies and a tilted vertical structure. The Kelvin wave also tends to have a third peak in zonal wind amplitude at 500 hPa and exhibits upward propagation into the lower stratosphere. Lower-tropospheric westward-moving mixed Rossby–gravity (WMRG) and n = 1 Rossby (R1) wave structures and their relationship with convection are consistent with classic equatorial wave theory and the implied lower-tropospheric convergences. In the Eastern Hemisphere the WMRG and R1 waves have first baroclinic mode structures in the vertical. However, in the Western Hemisphere, the R1 wave has a barotropic structure. In the Eastern Hemisphere the R1 wave, like the Kelvin wave, tends to have equatorial convection in the region of enhanced lower-level westerlies, suggesting that enhanced surface energy fluxes associated with these waves may play an important organizing role for equatorial convection in this warm-water hemisphere. In the upper troposphere, eastward-moving Rossby–gravity (EMRG) and n = 1 gravity waves are found in the Eastern Hemisphere, and eastward-moving WMRG and R1 waves are found in the Western Hemisphere, suggestive of Doppler shifting of waves by the ambient flow.

scholarly journals Planetary Scale Selection of the Madden–Julian Oscillation*

2009 ◽  
Vol 66 (8) ◽  
pp. 2429-2443 ◽  
Author(s):  
Tim Li ◽  
Chunhua Zhou
Keyword(s):  
Boundary Layer ◽  
Rossby Wave ◽  
Initial Perturbation ◽  
Madden Julian Oscillation ◽  
Kelvin Wave ◽  
Scale Selection ◽  
Planetary Scale ◽  
Zonal Wavenumber ◽  
Nonlinear Heating ◽  
Selection Of

Abstract Numerical experiments with a 2.5-layer and a 2-level model are conducted to examine the mechanism for the planetary scale selection of the Madden–Julian oscillation (MJO). The strategy here is to examine the evolution of an initial perturbation that has a form of the equatorial Kelvin wave at zonal wavenumbers of 1 to 15. In the presence of a frictional boundary layer, the most unstable mode prefers a short wavelength under a linear heating; but with a nonlinear heating, the zonal wavenumber 1 grows fastest. This differs significantly from a model without the boundary layer, in which neither linear nor nonlinear heating leads to the long wave selection. Thus, the numerical simulations point out the crucial importance of the combined effect of the nonlinear heating and the frictional boundary layer in the MJO planetary scale selection. The cause of this scale selection under the nonlinear heating is attributed to the distinctive phase speeds between the dry Kelvin wave and the wet Kelvin–Rossby wave couplet. The faster dry Kelvin wave triggered by a convective branch may catch up and suppress another convective branch, which travels ahead of it at the phase speed of the wet Kelvin–Rossby wave couplet if the distance between the two neighboring convective branches is smaller than a critical distance (about 16 000 km). The interference between the dry Kelvin wave and the wet Kelvin–Rossby wave couplet eventually dissipates and “filters out” shorter wavelength perturbations, leading to a longwave selection. The boundary layer plays an important role in destabilizing the MJO through frictional moisture convergences and in retaining the in-phase zonal wind–pressure structure.

scholarly journals Propagation of Rossby waves in stratified shear flows

1989 ◽  
Vol 12 (3) ◽  
pp. 547-557
Author(s):  
Palani G. Kandaswamy ◽  
B. Tamil Selvi ◽  
Lokenath Debnath
Keyword(s):  
Wave Equation ◽  
Relative Frequency ◽  
Shear Flows ◽  
Rossby Waves ◽  
Zonal Flow ◽  
Single Wave ◽  
Beta Plane ◽  
Valve Effect ◽  
The Waves ◽  
Isothermal Atmosphere

A study is made of the propagation of Rossby waves in a stably stratified shear flows. The wave equation for the Rossby waves is derived in an isothermal atmosphere on a beta plane in the presence of a latitudinally sheared zonal flow. It is shown that the wave equation is singular at five critical levels, but the wave absorption takes place only at the two levels where the local relative frequency equals in magnitude to the Brunt Vaisala frequency. This analysis also reveals that these two levels exhibit valve effect by allowing the waves to penetrate them from one side only. The absorption coefficient exp(2πμ)is determined at these levels. Both the group velocity approach and single wave treatment are employed for the investigation of the problem.

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