Dispersion and asymmetry of chiral gravitational waves in gyroscopic mechanical systems. Part 1: Discrete lattice strips

Detailed precipitation characteristics coupled with equatorial Rossby and Kelvin waves are investigated. We prepare a rainfall event dataset using the Tropical Rainfall Measurement Mission (TRMM) Precipitation Radar (PR) level 2 data. Utilizing three indices, area size, maximum echo-top height, and stratiform precipitation ratio, rainfall events are classified into mesoscale convective system (MCS), deep, congestus, and shallow convective events, and “other” type. We perform composite analyses based on the wave phase defined in Part I. Precipitation amount in Rossby waves is in phase with column water vapor (CWV) anomalies and is mainly contributed by MCSs, which are simultaneously activated with deep convection. The large CWV can support deep development and organization of convection. Shallow and congestus convective events indicate their peaks just before the active phase on 10°N or in the later part of the convectively suppressed phase on the equator. A five-step evolution is shown in Kelvin waves. In the first stage, shallow convective events are triggered by high SST, followed by a dominance of congestus convective events. Then, in the developing stage, deep convective events become dominant. In the mature stage, heavy precipitation is contributed by MCSs, and mostly stratiform rain is maximized at later stages. Kelvin waves indicate relatively weak connection to CWV fluctuation. Although contrasted evolutions are indicated, large contributions by MCSs to precipitation amount are common among the two coupled waves. This is considered to result in the commonality of the equivalent depths with their top-heavy heating. Significance Statement A coupling mechanism between equatorial waves and convective activity is a key issue in the tropical meteorology. While many previous idealized studies suggested some instability mechanisms, detailed precipitation characteristics is not enough investigated. We prepare rainfall-event dataset observed from a spaceborne Precipitation Radar on board the TRMM satellite to quantify detailed precipitation characteristics statistically and compare equatorial Rossby waves and equatorial Kelvin waves. We found that organized convective systems and deep convection are simultaneously activated in the Rossby waves and a clear transition of convective activity is shown in the Kelvin waves. These characteristics highly correspond to waves’ synoptic-scale structures. Our observational results of detailed evolutions of rainfall events will improve understanding of coupling processes.

In this paper, the role of oceanic Rossby waves in climate variability is reviewed, as well as their dynamics in tropical oceans and at mid-latitudes. For tropical oceans, both the interactions between equatorial Rossby and Kelvin waves, and off-equatorial Rossby waves are privileged. The difference in the size of the basins induces disparities both in the forcing modes and in the dynamics of the tropical waves, which form a single quasi-stationary wave system. For Rossby waves at mid-latitudes, a wide range of periods is considered, varying from a few days to several million years when very-long-period Rossby waves winding around the subtropical gyres are hypothesized. This review focuses on the resonant forcing of Rossby waves that seems ubiquitous: the quasi-geostrophic adjustment of the oceans favors natural periods close to the forcing period, while those far from it are damped because of friction. Prospective work concentrates on the resonant forcing of dynamical systems in subharmonic modes. According to this new concept, the development of ENSO depends on its date of occurrence. Opportunities arise to shed new light on open issues such as the Middle Pleistocene transition.

Equatorial Rossby (ER) waves are identified in an 8‐year data set of National Meteorological Center operational analyses. The westward moving waves have symmetric circulations about the equator and eastward energy dispersion, with maximum zonal wind perturbations along the equator and meridional wind maxima roughly 10°–15° off the equator. They have maximum amplitude in the lower troposphere and are associated with convective signals at roughly the mean latitude of the tropical convergence zones. The circulations thus possess many of the features of the equatorially trapped n = 1 Rossby modes derived by Matsuno [1966]. They are generally around zonal planetary wavenumber 6 scale and have a deep, nearly equivalent barotropic structure up to 100 mbar. This is in contrast with the Madden‐Julian Oscillation and mixed Rossby gravity waves, which generally have a first baroclinic mode vertical structure. During the intensive observing period (IOP) of the Tropical Ocean Global Atmosphere (TOGA) Coupled Ocean‐Atmosphere Response Experiment (COARE) and in other years, westward propagating ER disturbances along the equator are observed by using 6‐ to 30‐day filtered circulation data at 850 mbar. Zonal wind anomalies at 850 mbar are preferentially associated with like‐signed anomalies in a deep layer at this frequency, as was observed during COARE. The ER waves are most common during northern winter in the central Pacific, although they can be detected sporadically over the other ocean basins and in other seasons as well. One source for the equatorial Rossby waves appears to be upper level extratropical Rossby wave activity propagating into the eastern Pacific, although the equatorial waves are sometimes present during the northern summer when the extratropical source is absent. We cite evidence that propagation of ER wave activity from the central into the western Pacific is related to occurrences of westerly wind bursts.

This study investigates precipitation amounts and apparent heat sources, which are coupled with equatorial Kelvin waves and equatorial Rossby waves, using TRMM PR level 2 data products. The synoptic structures of wave disturbances are also studied using the ERA5 dataset. We define the wave phase of equatorial waves based on FFT-filtered brightness temperature and conduct composite analyses. Rossby waves show a vertically upright structure and their upright vortices induce large-amplitude column water vapor (CWV) anomalies. Precipitation activity is almost in phase with CWV, and thus is consistent with a moisture mode. Kelvin waves, on the other hand, indicate a nearly quadrature phase relationship between temperature and vertical velocity, like gravity wave structure. Specific humidity develops from near the surface to the middle troposphere as the Kelvin wave progresses. A clear negative CWV anomaly also does not exist despite the existence of negative precipitation anomalies. Convective activity corresponds well with its tilting structure of moisture and modulates the phase relationship between temperature and vertical motion. For both wave cases, apparent heat sources can amplify available potential energy despite the difference of coupling mechanisms of these two waves; precipitation is driven by CWV fluctuation for the Rossby wave case, and by buoyancy-based fluctuations for the Kelvin wave case. These can be observational evidence of actual coupling processes that is comparable to previous idealized studies. Significance Statement A coupling mechanism between equatorial waves and convective activity is a significant issue in tropical meteorology. While many previous idealized studies suggested some instability mechanisms, their true roles are not yet clear because detailed precipitation characteristics are not well investigated. We aim to quantify precipitation and synoptic-scale wave disturbances, and compare equatorial Rossby waves and equatorial Kelvin waves, which should have different instability coupling modes between each other, in order to shed light on a convectively coupling mechanism. We found that precipitation is actually driven by column moisture in Rossby waves and by dynamical fluctuation in Kelvin waves. Despite these competing mechanisms, similar top-heavy heating can maintain convectively coupled disturbances. Our observational results will support and improve theoretical studies.

Equatorially trapped waves, such as Kelvin Waves, Equatorial Rossby Waves and Westward‐moving Mixed Rossby–Gravity (WMRG) Waves, play a major role in organising tropical convection on synoptic to sub‐seasonal time‐scales. These waves have the potential to provide an important source of predictability for high‐impact weather in Southeast (SE) Asia and the Tropics more widely. To aid understanding of the role played in high‐impact weather by such waves, the observed statistical relationship between identified equatorial waves and heavy rainfall in SE Asia is examined for the period 1998–2016. Increases in the amount of precipitation and the likelihood of extreme precipitation in SE Asia are linked to all three types of waves that are included in analysis; Kelvin, equatorial Rossby and WMRG waves. There is both increased mean rainfall and increased probability of occurrence of heavy rainfall on days when high‐amplitude waves propagate over SE Asia. In particular, heavy precipitation can be up to three times more likely in regions of SE Asia during equatorial waves, including Malaysia, Indonesia and the Philippines. Kelvin waves have a large influence on heavy rainfall over Indonesia, WMRG and Kelvin waves impact Malaysia rainfall, and equatorial Rossby and WMRG waves are linked to increased rainfall over the Philippines. Based on this study it can be concluded that the probability of extreme precipitation in this region is dependent on equatorial wave activity. Therefore, the skill in probabilistic prediction of extreme precipitation in SE Asia would be expected to be conditional on the skill in equatorial wave prediction, and the modelled relationship between equatorial waves and convection.

An analysis of the energy equation of the three‐dimensional normal modes of the atmospheric circulation provides evidence on the behaviour of the most important equatorial waves which compose the Madden–Julian Oscillation. They are the equatorial Rossby waves of zonal wavenumber 1 and 3 and the Kelvin waves of zonal wavenumber 1. In the mean, the equatorial Rossby waves gain energy by convective heating and atmospheric column stretching and transfer it to the circulation field, whereas the Kelvin waves draw energy from the circulation field and dissipate it by surface friction. The convergence associated with the Kelvin waves contributes to the moisture fuelling of the convection centre. Therefore, the coupling between equatorial Rossby and Kelvin waves is mediated by convective heating/column stretching excitation of Rossby waves and moisture convergence by Kelvin waves.

Abstract. Atmospheric convection over the subtropical western North Pacific (SWNP) during boreal summer varies on timescales of around 10 d, with significant effects on both local and remote circulation. One of less understood effects of this variability is its coupling with equatorial wave dynamics. This paper quantifies equatorial wave perturbations and their evolution throughout the lifecycle of SWNP convection using wave-space regression between outgoing longwave radiation over the SWNP region and spectral expansion coefficients of global tropospheric circulation from ERA5 reanalyses. The regression distinguishes between convection-coupled linear Rossby and Kelvin waves, and mixed Rossby-gravity (MRG) and inertia-gravity (IG) waves. The former two correspond to the Gill-type tropical circulation response to asymmetric heating. The results show that MRG and IG waves exhibit amplitudes comparable to those of the Gill-response component in the upper troposphere. In particular, MRG and IG waves dominate the cross-equatorial northerly flow over the Maritime Continent, with MRG waves playing the larger role. These findings suggest caution when applying the Gill solution to interpret circulation responses to asymmetric heating in model simulations. As SWNP convection intensifies, the MRG-wave northerly winds across the equator strengthen, while IG waves represent enhanced upper-tropospheric outflow over the SWNP region. By contrast, the combined effect of Kelvin and Rossby waves reinforces circulation on the equatorward flank of the anticyclone over the SWNP region, with Rossby wave easterlies being about three times stronger than those associated with Kelvin waves. The Rossby wave signal resembles the n=1 Rossby wave, with its Southern Hemisphere (SH) subtropical anticyclonic gyre forming over the southern Indian ocean during the decay phase of the SWNP convection. This gyre, together with the SH IG meridional flow provides a dynamical bridge linking SWNP convection and extratropical circulation during austral winter.

Using shallow water equations on an equatorial beta plane, the nonlinear dynamics of the equatorial waves is investigated. A general mathematical procedure to study the nonlinear dynamics of these waves is developed using the asymptotic method of multiple scales. On faster temporal and spatial scales the equations describe the equatorial wavesviz, the Rossby waves, Rossby gravity waves, the inertia gravity waves and the Kelvin waves. Assuming that the amplitude of these waves are functions of slower time and space scales, it is shown that the evolution of the amplitude of these waves is governed by the nonlinear Schrodinger equation. It is then shown that for the dispersive waves like Rossby waves and Rossby-gravity waves, the envelope of the amplitude of the waves has a ‘soliton’ structure.

Equatorward-propagating wave trains in the upper troposphere are observed to be associated with deep convection over the eastern tropical Pacific on the submonthly timescale during northern winter. The convection occurs in the regions of ascent and reduced static stability ahead of cyclonic anomalies in the wave train. In this study an atmospheric primitive equation model is used to examine the roles of the dry wave dynamics and the diabatic heating associated with the convection. Many features of a dry integration initialized with a localized wave train in the African‐Asian jet on a threedimensional climatological basic state quantitatively agree with the observations, including the zonal wavenumber 6‐7 scale of the waves, the time period of approximately 12 days, and the cross-equatorial Rossby wave propagation over the eastern Pacific. There is ascent and reduced static stability ahead of the cyclonic anomalies, consistent with the interpretation that the waves force the convection. The spatial scale of the waves appears to be set by the basic state; baroclinic growth upstream in the Asian jet favors waves with zonal wavenumber 6. On reaching the Pacific sector, lower-wavenumber components of the wave train are not refracted so strongly equatorward, while higher-wavenumber components are advected quickly along the Pacific jet before they can propagate equatorward. Once over the Pacific, the wave train approximately obeys barotropic Rossby wave dynamics. The observed lower-tropospheric anomalies include an equatorial Rossby wave that propagates westward from the region of cross-equatorial wave propagation and tropical convection. However, this equatorial Rossby wave is not forced directly by the dry equatorward-propagating wave train but appears in a separate integration as a forced response to the observed diabatic heating associated with the tropical convection.

Equatorial Kelvin waves can be affected by subtropical Rossby wave dynamics. Previous research has demonstrated the Kelvin wave growth in response to subtropical forcing and the resonant growth due to eddy momentum flux convergence. However, the relative importance of the wave–mean flow and wave–wave interactions for the Kelvin wave growth compared to the direct wave excitation by the external forcing has not been made clear. This study demonstrates the resonant Kelvin wave excitation by interactions of subtropical Rossby waves and the mean flow using a spherical shallow-water model. The use of Hough harmonics as basis functions makes Rossby and Kelvin waves prognostic variables of the model and allows the quantification of terms contributing to their tendencies in physical and wave spaces. The simulations show that Kelvin waves are resonantly excited by interactions of Rossby waves and the balanced zonal mean flow in the subtropics, provided the Rossby and Kelvin wave frequencies, which are modified by the mean flow, match. The resonance mechanism is substantiated by analytical expressions. The Kelvin wave tendencies are caused by velocity and depth tendencies: The velocity tendencies due to the meridional advection of the zonal mean velocity can be outweighed by the zonal advection of the Rossby wave velocity or by the depth tendencies due to Rossby wave divergence. Identifying the resonant excitation mechanism in data should contribute to the quantification of Kelvin wave variability originating in the subtropics. Significance Statement This study seeks to understand how Kelvin waves, which are eastward-propagating disturbances in the tropical atmosphere, are connected to Rossby wave dynamics in the subtropics. Using idealized simulations, the Kelvin wave excitation is explained as a resonance effect due to interactions of Rossby waves and the zonal mean flow. The mechanism contributes to the understanding of atmospheric wave interactions and extratropical effects on the tropics. Further work searching for evidence of the new mechanism in atmospheric data may shed a new light on subseasonal variability in the tropics, such as the Madden–Julian oscillation.

For mid-latitude Rossby waves (RWs) in the atmosphere, the expression for the energy flux for use in a model diagnosis, and without relying on a Fourier analysis or a ray theory, has previously been derived using quasi-geostrophic equations and is singular at the equator. By investigating the analytical solution of both equatorial and mid-latitude waves, the authors derive an exact universal expression for the energy flux which is able to indicate the direction of the group velocity at all latitudes for linear shallow water waves. This is achieved by introducing a streamfunction as given by the inversion equation of Ertel’s potential vorticity, a novel aspect for considering the energy flux. For ease of diagnosis from a model, an approximate version of the universal expression is explored and illustrated for a forced/dissipative equatorial basin mode simulated by a single-layer oceanic model that includes both mid-latitude RWs and equatorial waves. Equatorial Kelvin Waves (KWs) propagate eastward along the equator, are partially redirected poleward as coastal KWs at the eastern boundary of the basin, and then shed mid-latitude RWs that propagate westward into the basin interior. The connection of the equatorial and coastal waveguides has been successfully illustrated by the approximate expression of the group-velocity-based energy flux of the present study. This will allow for tropical-extratropical interactions in oceanic and atmospheric model outputs to be diagnosed in terms of an energy cycle in a future study.

The mean of the sea level deviation data derived from the TOPEX/Poseidon altimeter in the equatorial Pacific, between 10°S and 10°N, and between 120°E and 78°W, from cycles 2 to 136 (3 October 1992–2 June 1996), are extracted using a maximum–minimum average method. Then, two-dimensional (2D) sea level deviation time series are developed to visualize the dynamics of equatorial waves. The complex singular value decomposition (CSVD) method is applied to decompose these 2D time series into empirical orthogonal modes. Using this method, zonal and meridional structures, propagation directions, periods, and propagation speeds of these empirical modes are obtained. The first empirical mode is propagating westward, and its structure is asymmetric to the equator. It has an average phase speed c = −0.6 m s−1 within 4°–6°N and c = −0.4 m s−1 within 6°–8°S, respectively, and a period of 15 months, which is associated with an interannual Rossby wave. The second empirical mode is propagating eastward along the equator and has a phase speed of 2.5 m s−1 and a period of 7 months, which is associated with an equatorial Kelvin wave. The asymmetric feature of the empirical Rossby wave, which is also observed in the equatorial Pacific, may suggest that the background currents and wind fields in the equatorial Pacific Ocean affect its propagation. The amplitude of the empirical Kelvin mode increases as it propagates eastward. This is associated with an eastward shoaling of the thermocline depth along the equatorial Pacific Ocean. The results of both empirical modes are consistent with those predicted by the theory of Kelvin and Rossby waves and closely represent the actual features of both waves observed in the equatorial Pacific Ocean. Therefore, the CSVD is a suitable method for revealing the dynamics of equatorial waves.

Summary The effect of the Coriolis force is demonstrated for chiral continuum models describing waves in the equatorial region and the polar regions on a rotating sphere. Novel asymptotic features of equatorial waves are presented in this paper. We show that the shape of a ridge of a polar vortex can be approximated by the governing equations of a gyropendulum. Theoretical deductions are accompanied by illustrative examples.

It is shown that resonant coupling between ultra long equatorial Rossby waves and packets of either short Rossby or short westward-traveling gravity waves is possible. Simple analytic formulas give the discrete value of the packet wave number k, for which the group velocity of the packet of meridional mode number n matches the group velocity of a nondispersive long Rossby wave of odd mode number m. The equations that describe the coupling are derived via the method of multiple scale and tables of the interaction coefficients are numerically calculated. For realistic parameter values, it appears this coupling could be important in the tropical ocean. The principal physics of the coupled equations is threefold: 1) modulational or “side band” instability of plane waves, 2) instability of a short wave packet with respect to growing long waves if no long waves are initially present, and 3) solitary waves which consist of an envelope soliton of short waves of Nonlinear Schrodinger type traveling in con...

Aims. The influence of a toroidal magnetic field on the dynamics of Rossby waves in a thin layer of ideal conductive fluid on a rotating sphere is studied in the “shallow water” magnetohydrodynamic approximation for the first time. Methods. Dispersion relations for magnetic Rossby waves are derived analytically in Cartesian and spherical coordinates. Results. It is shown that the magnetic field causes the splitting of low order (long wavelength) Rossby waves into two different modes, here denoted fast and slow magnetic Rossby waves. The high frequency mode (the fast magnetic Rossby mode) corresponds to an ordinary hydrodynamic Rossby wave slightly modified by the magnetic field, while the low frequency mode (the slow magnetic Rossby mode) has new and interesting properties since its frequency is significantly smaller than that of the same harmonics of pure Rossby and Alfven waves.