Inertia-gravity Waves Research Articles

Waves in a Cloudy Vortex

Abstract This paper derives a system of equations that approximately govern small-amplitude perturbations in a nonprecipitating cloudy vortex. The cloud coverage can be partial or complete. The model is used to examine moist vortex Rossby wave dynamics analytically and computationally. One example shows that clouds can slow the growth of phase-locked counter-propagating vortex Rossby waves in the eyewall of a hurricane-like vortex. Another example shows that clouds can (indirectly) damp discrete vortex Rossby waves that would otherwise grow and excite spiral inertia–gravity wave radiation from a monotonic cyclone at high Rossby number.

The high-resolution vertical structure of internal tides and near-inertial waves measured with an ADCP over the continental slope in the Bay of Biscay

ADCP measurements of the velocity structure in the permanent thermocline at two locations over the continental slope in the Bay of Biscay are presented. The vertical variation of the contribution of the inertia-gravity waveband to the kinetic energy, vertical motion, and current shear are analysed. The semi-diurnal tides together with near-inertial waves appear to provide over 70% of the high-frequency kinetic energy (>1/3 cpd). Over the vertical range of the ADCP observations the phase of the harmonic M 2 tide changes up to 155°, while the kinetic energy varies in the vertical by a factor of 3.8, showing the importance of the contribution of internal waves to the observed tidal motion. Both semi-diurnal internal tidal waves and near-inertial waves have a vertically restricted distribution of the variance of the horizontal and vertical velocity, as in internal wave beams. The short-term 14-day averaged amplitude and phase lag of the M 2 tide shows large temporal changes, with a characteristic 40–45 day time scale. These changes are probably related to variations in generation sites and propagation paths of the internal tide, because of changes in the temperature and salinity stratification due to the presence of meso-scale eddies. The relatively large shear in the inertia-gravity wave band, mainly at near-inertial frequencies, supports low-gradient Richardson numbers that are well below 1 for nearly half of the time. This implies that the large shear may support turbulent mixing for a large part of the time.

Time-dependent rotating stratified shear flow: Exact solution and stability analysis

A solution of the Euler equations with Boussinesq approximation is derived by considering unbounded flows subjected to spatially uniform density stratification and shear rate that are time dependent [S(t)= partial differentialU3/partial differentialx2]. In addition to vertical stratification with constant strength N(v)2, this base flow includes an additional, horizontal, density gradient characterized by N(h)2(t). The stability of this flow is then analyzed: When the vertical stratification is stabilizing, there is a simple harmonic motion of the horizontal stratification N(h)2(t) and of the shear rate S(t), but this flow is unstable to certain disturbances, which are amplified by a Floquet mechanism. This analysis may involve an additional Coriolis effect with Coriolis parameter f, so that governing dimensionless parameters are a modified Richardson number, R=[S(0)2+N(h)4(0)/N(v)2]1/2, and f(v)=f/N(v), as well as the initial phase of the periodic shear rate. Parametric resonance between the inertia-gravity waves and the oscillating shear is demonstrated from the dispersion relation in the limit R-->0. The parametric instability has connection with both baroclinic and elliptical flow instabilities, but can develop from a very different base flow.

Characteristics of inertia-gravity waves around jet stream from radiosonde observations in Wuhan (30.5°N, 114.4°E)

The 5 years’ radiosonde data obtained from January 2000 to December 2004 in Wuhan (30.5°N, 114.4°E) have been used for studying the behaviors of inertia-gravity waves in the vicinity of the jet stream. It is observed that the wave intensity has a similar seasonal variation with the jet stream intensity with a strong winter maximum and a summer minimum. Moreover, a similar inter-annual trend for both the wave intensity and jet stream intensity is also found. These results suggest that the jet stream may be the predominant source of the inertia-gravity waves in the troposphere and lower stratosphere over Wuhan in the period of the 5 years. It is noticed from 28 radiosonde profiles during wintertime that the energy of inertia-gravity waves exhibits upward and downward propagation respectively above and below the jet stream. This indicates that the source of the inertia-gravity waves is within the jet stream. In these cases, the twin waves below and above the jet stream usually hold similar amplitudes. The horizontal propagation of the twin waves also shows some interesting relationship.

The persistence of balance in geophysical flows

Rotating stably stratified geophysical flows can exhibit a near ‘balanced’ evolution controlled by the conservative advection of a single scalar quantity, the potential vorticity (PV). This occurs frequently in the Earth's atmosphere and oceans where motions tend to be weak compared with the background planetary rotation and where stratification greatly inhibits vertical motion. Under these circumstances, both high-frequency acoustic waves and lower-frequency inertia–gravity waves (IGWs) contribute little to the flow evolution compared with the even-lower-frequency advection of PV. Moreover, this ‘slow’ PV-controlled balanced evolution appears unable to excite these higher-frequency waves in any significant way – i.e. balance persists.The present work pushes the limits of balance by systematically exploring the evolution of a range of highly nonlinear flows in which motions are comparable with the background rotation. These flows do not possess a frequency separation between PV advection and IGWs. Nonetheless, the flows exhibit a remarkable persistence of balance. Even when flows are not initialized to minimize the amount of IGWs initially present, and indeed even when flows are deliberately seeded with significant IGW amplitudes, the flow evolution – over many inertial periods (days) – remains strongly controlled by PV advection.

Equatorial wave attractors and inertial oscillations

Three different approximations to the axisymmetric small-disturbance dynamics of a uniformly rotating thin spherical shell are studied for the equatorial region assuming time-harmonic motion. The first is the standard β-plane model. The second is Stern's (Tellus, vol. 15, 1963, p. 246) homogeneous, equatorial β-plane model of inertial waves (that includes all Coriolis terms). The third is a version of Stern's equation extended to include uniform stratification. It is recalled that the boundary value problem (BVP) that governs the streamfunction of zonally symmetric waves in the meridional plane becomes separable only for special geometries. These separable BVPs allow us to make a connection between the streamfunction field and the underlying geometry of characteristics of the governing equation. In these cases characteristics are each seen to trace a purely periodic path. For most geometries, however, the BVP is non-separable and characteristics and therefore wave energy converge towards a limit cycle, referred to as an equatorial wave attractor. For Stern's model we compute exact solutions for wave attractor regimes. These solutions show that wave attractors correspond to singularities in the velocity field, indicating an infinite magnification of kinetic energy density along the attractor. The instability that arises occurs without the necessity of any ambient shear flow and is referred to as geometric instability.For application to ocean and atmosphere, Stern's model is extended to include uniform stratification. Owing to the stratification, characteristics are trapped near the equator by turning surfaces. Characteristics approach either equatorial wave attractors, or point attractors situated at the intersections of turning surfaces and the bottom. At these locations, trapped inertia–gravity waves are perceived as near-inertial oscillations. It is shown that trapping of inertia–gravity waves occurs for any monochromatic frequency within the allowed range, while equatorial wave attractors exist in a denumerable, infinite set of finite-sized continuous frequency intervals. It is also shown that the separable Stern equation, obtained as an approximate equation for waves in a homogeneous fluid confined to the equatorial part of a spherical shell, gives an exact description for buoyancy waves in uniformly but radially stratified fluids in such shells.

The Primary Nonlinear Dynamics of Modal and Nonmodal Perturbations of Monochromatic Inertia–Gravity Waves

Abstract The breaking of an inertia–gravity wave (IGW), initiated by its leading normal modes (NMs) or singular vectors (SVs), and the resulting small-scale eddies are investigated by means of direct numerical simulations of a Boussinesq fluid characterizing the upper mesosphere. The focus is on the primary nonlinear dynamics, neglecting the effect of secondary instabilities. It is found that the structures with the strongest impact on the IGW and also the largest turbulence amplitudes are the NM (for a statically unstable IGW) or short-term SV (statically and dynamically stable IGW) propagating horizontally transversely with respect to the IGW, possibly in agreement with observations of airglow ripples in conjunction with statically unstable IGWs. In both cases these leading structures reduce the IGW amplitude well below the static and dynamic instability thresholds. The resulting turbulent dissipation rates are within the range of available estimates from rocket soundings, even for IGWs at amplitudes low enough precluding NM instabilities. Thus SVs can help explain turbulence occurring under conditions not amenable for the classic interpretation via static and dynamic instability. Because of the important role of the statically enhanced roll mechanism in the energy exchange between IGW and eddies, the turbulent velocity fields are often conspicuously anisotropic. The spatial turbulence distribution is determined to a large degree by the elliptically polarized horizontal velocity field of the IGW.

Filtering inertia-gravity waves from the initial conditions of the linear shallow water equations

A method for filtering inertia-gravity waves from elevation and depth-averaged velocity is described. This filtering scheme is derived from the linear shallow water equations for constant depth and constant Coriolis frequency. The filtered solution is obtained by retaining only the eigenvectors corresponding to the geostrophic equilibrium and by discarding explicitly the eigenvectors corresponding to the fast moving inertia-gravity waves. An alternative formulation is derived using a variational approach. Both filtering methods are tested numerically for a periodic domain with constant depth and the variational approach is implemented for a closed domain with large topographic variations. The filtering methods significantly reduce the amplitudes of the inertia-gravity waves while preserving the mean flow. The variational method is compared to the Incremental Analysis Update technique and the benefits of the variational filter are presented.

A Consistent Theory for Linear Waves of the Shallow-Water Equations on a Rotating Plane in Midlatitudes

Abstract The present study provides a consistent and unified theory for the three types of linear waves of the shallow-water equations (SWE) in a zonal channel on the β plane: Kelvin, inertia–gravity (Poincaré), and planetary (Rossby). The new theory is formulated from the linearized SWE as an eigenvalue problem that is a variant of the classical Schrödinger equation. The results of the new theory show that Kelvin waves exist on the β plane with vanishing meridional velocity, as is the case on the f plane, without any change in the dispersion relation, while the meridional structure of their height amplitude is trivially modified from exponential on the f plane to a one-sided Gaussian on the β plane. Similarly, inertia–gravity waves are only slightly modified in the new theory in comparison with their characteristics on the f plane. For planetary waves (which exist only on the β plane) the new theory yields a similar dispersion relation to the classical theory only for large gravity wave phase speed, such as those encountered in a barotropic ocean or an equivalent barotropic atmosphere. In contrast, for low gravity wave phase speed, for example, those in an equivalent barotropic ocean where the relative density jump at the interface is 10−3, the phase speed of planetary waves in the new theory is 2 times those of the classical theory. The ratio between the phase speeds in the two theories increases with channel width. This faster phase propagation is consistent with recent observation of the westward propagation of crests and troughs of sea surface height made by the altimeter aboard the Ocean Topography Experiment (TOPEX)/Poseidon satellite. The new theory also admits inertial waves, that is, waves that oscillate at the local inertial frequency, as a genuine solution of the eigenvalue problem.

Gravity-wave breaking: Linear and primary nonlinear dynamics

In a summary of recent work on the breaking of gravity waves (GWs) and the corresponding onset of turbulence the question of critical thresholds is given a systematic discussion. Contrary to a widespread belief static and dynamic stability do not indicate a GW’s stability against breaking. Low-frequency inertia–gravity waves can be destabilized by singular vectors even if their amplitude is too weak to produce local Richardson numbers less than a quarter. High-frequency gravity waves have horizontal gradients strong enough so that even at statically and dynamically stable amplitudes they are destabilized by normal modes. The dynamics of the processes is discussed within a framework which takes a careful path from linear stability analyses to direct numerical simulations. The latter show that subcritical GW breaking can lead to turbulence of a strength as observed in the middle atmosphere. In many cases the GW amplitude is reduced way below the conventional thresholds of static or dynamic instability.

Radiosonde observations of vertical wave number spectra for gravity waves in the lower atmosphere over Central China

Abstract. Vertical wave number spectra of inertial gravity waves in the troposphere and lower stratosphere over six stations at latitudes from 20° N to 40° N were statistically studied by using the data from Radiosonde observation on a twice daily basis at 08:00 and 20:00 LT. Statistically, the spectral characteristics seem to be independent of the local observation time, and show considerable conformity between the spectral of zonal and meridional kinetic energy densities. Compared with the spectra of the kinetic energy density, the spectra of the potential energy density are steeper. in addition the characteristic wave numbers of the spectra also show considerable consistency among the observations at different stations. As for the spectral slopes, they are systematically smaller (in magnitude) than the canon value of –3, and exhibit slight height, seasonal and latitudinal variability. In addition to these universal characteristics, the spectral structures also exhibit departures and variations, and most of the departures and variations are related to the strong tropospheric jets. Generally, in the case of strong shear due to the tropospheric jet, there usually occur larger characteristic wave numbers and smaller spectral slopes. These departures seem to be persistent and climatological rather than transitory, indicating the significant impacts of the sheared background winds on the spectral structures of gravity waves.

Simulation of Inertia–Gravity Waves in a Poleward-Breaking Rossby Wave

Poleward-breaking Rossby waves often induce an upper-level jet streak over northern Europe. Dominant inertia–gravity wave packets are observed downstream of this jet. The physical processes of their generation and propagation, in such a configuration, are investigated with a mesoscale model. The study is focused on an observational campaign from 17 to 19 December 1999 over northern Germany. Different simulations with the fifth-generation Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5) have been performed. For a high-resolution process study, three domains were set up that encompass the evolution of Rossby waves and that of inertia–gravity waves. To minimize the impact of model damping, the horizontal and vertical resolution has been adjusted appropriately. With a novel statistical approach, the properties of inertia–gravity wave packets have been estimated. This method uses the horizontal divergence field and takes into account the spatial extension of a wave packet. It avoids the explicit treatment of the background field and works for arbitrary wavelength. Two classes of inertia–gravity waves were found: subsynoptic waves with a horizontal wavelength of about 500 km and mesoscale waves with a horizontal wavelength of about 200 km. The subsynoptic structures were also detected in radiosonde observations during this campaign. The similarity between simulated and observed wavelengths and amplitudes suggests that the simulations can be considered as near realistic. Spontaneous radiation from unbalanced flow is an important process of inertia–gravity wave generation. Synoptic-scale imbalances in the exit region of the upper-tropospheric jet streak were identified with the smoothed cross-stream Lagrangian Rossby number. In a number of simulations with different physics, it was found that the inertia–gravity wave activity was related to the tropospheric jet, orography, and moist convection. The upward propagation of inertia–gravity waves was favored during this event of a poleward-breaking Rossby wave. The presence of the polar vortex induced background winds exceeding the critical line. Consequently, the activity of inertia–gravity waves in the lower stratosphere increased by an order of magnitude during the case study. The successful simulation of the complex processes of generation and propagation showed the important role of poleward Rossby wave breaking for the appearance of inertia–gravity waves in the midlatitudes.

Gravity waves above the Andes detected from GPS radio occultation temperature profiles: Jet mechanism?

A significant wave activity (WA) in the upper troposphere and lower stratosphere, mainly during winter, was detected at midlatitudes in the southern hemisphere (30–40S) above the Andes Range, from an analysis of Global Positioning System Radio Occultation (GPS RO) temperature profiles retrieved by CHAMP (CHAllenging Minisatellite Payload) and SAC‐C (Satélite de Aplicaciones Científicas‐C) Low Earth Orbit (LEO) satellites, between May 2001 and February 2006. The possible main gravity wave sources in this region are: i) orographic forcing, ii) geostrophic adjustment and iii) deep convection. The available vertical resolution of GPS RO soundings does not rule out any of these alternatives. Based on satellite imaginary, the WA enhancements cannot be attributed to deep convection events. Inertia‐gravity waves (IGWs) could be generated after a geostrophic adjustment process, following a perturbation of the zonal jet situated above the Andes Mountains by mountain waves (MWs). The monthly WA intensity follows the zonal wind velocity strength according to its seasonal variability at jet altitudes. As the GPS‐LEO lines of sight are roughly meridionally aligned and the morphology of the Andes at middle latitudes is predominantly north‐south, it was possible to detect MWs as well as IGWs from GPS RO temperature profiles. This characteristic does not apply for other mountain range alignments. From the analysis of a numerical simulation at the time and location of a single RO event with very strong WA, two main modes of oscillation with horizontal wavelength around 40 and 200 km were identified. The first one is attributed to a MW and the second one to an IGW.

An Improved Calculation of Coriolis Terms on the C Grid

Abstract The central differencing grid with fully staggered velocity components (C grid) is widely used in primitive equation oceanographic models despite potential problems in simulating baroclinic inertia–gravity and Rossby waves that can arise due to the averaging of velocity components in the Coriolis terms. This note proposes a new averaging of the velocity components in order to calculate the Coriolis terms on the C grid. The averaging weights are calculated from the minimum of a suitably defined cost function that optimally minimizes the error in the inertial part of frequencies of inertia–gravity waves and maintains the second-order accuracy of the computations. The theoretical analysis of wave frequency diagrams shows that the new scheme results in more accurate frequencies of long inertia–gravity and Rossby waves, especially when the Rossby radius of deformation is not well resolved by the grid resolution.

Lower-Stratospheric and Upper-Tropospheric Disturbances Observed by Radiosondes over Thailand during January 2000

Lower-stratospheric and upper-tropospheric disturbances over Thailand during 12–21 January 2000 were studied using the Global Energy and Water Cycle Experiment (GEWEX) Asian Monsoon Experiment-Tropics (GAME-T) intensive rawinsonde observations with fine temporal sampling intervals of 3 h. Analysis was focused on the wind disturbances with a period shorter than about 10 days. Frequency spectra showed three distinct peaks: a 1-day period above a height of 20 km, a near-inertial period around 19 and 27 km, and periods of 2.5–9 days (or longer) in the height range of 12–17 km. The wave with a 1-day period was interpreted as a diurnal tide. A comparison with the migrating tide in the global-scale wave model showed that the observational results had larger amplitude and shorter vertical wavelength than the model. The difference between the observation and the model may be caused by the superposition of the nonmigrating tide. The wave with the near-inertial period was interpreted as an internal inertial gravity wave. A hodograph analysis was performed in order to investigate the wave properties. It was found that the wave, which appeared at a height around 19 km (just above the tropopause level), propagated southwestward with a ground-based group velocity of about 1.4 m s−1. The longer period disturbances, which appeared at heights of 12–17 km, had layered structures with the vertical scales of 2–4 km. They were considered to be due to inertial instability, based on the fact that the potential vorticity of the background atmosphere was nearly zero and that their phase structures were consistent with theory. It was shown by a backward trajectory analysis that the air parcel with negative potential vorticity had its origin in equatorial Indonesia. It was also shown by a forward trajectory analysis that the air parcel was transported to the Pacific south of Japan. This is consistent with the existence of similar layered disturbances that are shown using rawinsonde data at a station there.

Inertia gravity waves in the upper troposphere during the MaCWAVE winter campaign – Part II: Radar investigations and modelling studies

Abstract. We present an experimental and modelling study of a strong gravity wave event in the upper troposphere/lower stratosphere near the Scandinavian mountain ridge. Continuous VHF radar measurements during the MaCWAVE rocket and ground-based measurement campaign were performed at the Norwegian Andoya Rocket Range (ARR) near Andenes (69.3° N, 16° E) in January 2003. Detailed gravity wave investigations based on PSU/NCAR Fifth-Generation Mesoscale Model (MM5) data have been used for comparison with experimentally obtained results. The model data show the presence of a mountain wave and of an inertia gravity wave generated by a jet streak near the tropopause region. Temporal and spatial dependencies of jet induced inertia gravity waves with dominant observed periods of about 13 h and vertical wavelengths of ~4.5–5 km are investigated with wavelet transform applied on radar measurements and model data. The jet induced wave packet is observed to move upstream and downward in the upper troposphere. The model data agree with the experimentally obtained results fairly well. Possible reasons for the observed differences, e.g. in the time of maximum of the wave activity, are discussed. Finally, the vertical fluxes of horizontal momentum are estimated with different methods and provide similar amplitudes. We found indications that the derived positive vertical flux of the horizontal momentum corresponds to the obtained parameters of the jet-induced inertia gravity wave, but only at the periods and heights of the strongest wave activity.

Inertia gravity waves in the upper troposphere during the MaCWAVE winter campaign – Part I: Observations with collocated radars

Abstract. During the {MaCWAVE} campaign, combined rocket, radiosonde and ground-based measurements have been performed at the Norwegian Andøya Rocket Range (ARR) near Andenes and the Swedish Rocket Range (ESRANGE) near Kiruna in January 2003 to study gravity waves in the vicinity of the Scandinavian mountain ridge. The investigations presented here are mainly based on the evaluation of continuous radar measurements with the ALWIN VHF radar in the upper troposphere/ lower stratosphere at Andenes (69.3° N, 16.0° E) and the ESRAD VHF radar near Kiruna (67.9° N, 21.9° E). Both radars are separated by about 260 km. Based on wavelet transformations of both data sets, the strongest activity of inertia gravity waves in the upper troposphere has been detected during the first period from 24–26 January 2003 with dominant vertical wavelengths of about 4–5 km as well as with dominant observed periods of about 13–14 h for the altitude range between 5 and 8 km under the additional influence of mountain waves. The results show the appearance of dominating inertia gravity waves with characteristic horizontal wavelengths of ~200 km moving in the opposite direction than the mean background wind. The results show the appearance of dominating inertia gravity waves with intrinsic periods in the order of ~5 h and with horizontal wavelengths of 200 km, moving in the opposite direction than the mean background wind. From the derived downward energy propagation it is supposed, that these waves are likely generated by a jet streak in the upper troposphere. The parameters of the jet-induced gravity waves have been estimated at both sites separately. The identified gravity waves are coherent at both locations and show higher amplitudes on the east-side of the Scandinavian mountain ridge, as expected by the influence of mountains.

Observations of intermediate‐scale diurnal waves in the equatorial mesosphere and lower thermosphere

The purpose of this study is to seek observational evidence for diurnal, vertically propagating inertia‐gravity waves (IGWs) in the mesosphere and lower thermosphere (MLT). Numerical modeling studies indicate that vertically propagating IGWs excited by tropical heating provide a significant source of momentum for the semiannual oscillation (SAO) in the equatorial zonal mean winds. The power spectrum of these waves has a strong diurnal component. We analyze global patterns of ascending and descending node differences in MLT satellite temperatures, which are assumed in this study to be proxies for waves of diurnal period. Juxtaposed upon the planetary‐scale features are localized variations with longitudinal wavelengths ranging from 25° to 50°, and with vertical wavelengths between 13 and 25 km. These “intermediate‐scale” variations are spatially coherent, and persist for several weeks. Their amplitudes generally increase with altitude, while their phase structures suggest both eastward and westward propagation. The variance of wave numbers 9–17 in the MLT is examined in relation to the underlying stratospheric zonal mean zonal gradient winds. Stronger variances generally coincide with periods where the underlying zonal mean winds are relatively light, or unidirectional. The weak inverse association between variance strength and zonal wind magnitude is suggestive of a wave filtering mechanism.

Vortex–wave interaction in a rotating stratified fluid: WKB simulations

In this paper we present ray-tracing results on the interaction of inertia–gravity waves with the velocity field of a vortex in a rotating stratified fluid. We consider rays that interact with a Rankine-type vortex with a Gaussian vertical distribution of vertical vorticity. The rays are traced, solving the WKB equations in cylindrical coordinates for vortices with different aspect ratios. The interactions are governed by the value of , for the incident waves. In a sequel paper, we compare WKB simulations with experimental results.

Upscale effects in simulations of tropical convection on an equatorial beta-plane

A cloud-resolving model is configured to span the full meridional extent of the tropical atmosphere and have sufficient zonal extent to permit the representation of tropical cloud super-clusters. This is made computationally feasible by the use of anisotropic horizontal grids where one horizontal coordinate direction has over an order of magnitude finer resolution than the other direction. Typically, the meridional direction is chosen to have the coarser resolution (40 km grid spacing) and the zonal direction has enough resolution to ‘permit’ crude convective squall line ascent (1 km grid spacing). The aim was to run in cloud-resolving model (CRM) mode yet still have sufficient meridional resolution and extent to capture the equatorial trapped waves and the Hadley circulation. The large-scale circulation is driven by imposed uniform tropospheric cooling in conjunction with a fixed sea surface temperature distribution. At quasi-equilibrium the flow is characterized by sub-tropical jetstreams, tropical squall line systems that form eastward-propagating super-clusters, tropical depressions and even hurricanes. Two scientific issues are briefly addressed by the simulations: what forces the Hadley circulation and the nature of stratospheric waves appearing in the simulation. It is found that the presence of a meridional sea surface temperature gradient is not sufficient on its own to force a realistic Hadley circulation even though convection communicates the underlying temperature gradient to the atmosphere. It is shown in a simulation that accounts for the observed time and zonal-mean momentum forcing effect of large-scale eddies (originating in middle latitudes) that the heaviest precipitation is concentrated near the equator in association with moisture flux convergence driven by the Trade winds. A spectral analysis of the stratospheric waves found on the equator using the dispersion relation for equatorially-trapped waves provides strong evidence for the existence of a domain-scale Kelvin wave together with eastward and westward propagating inertia-gravity waves. The eastward-propagating stratospheric waves appear to be part of a convectively coupled wave system travelling at about 15 ms −1.