wavelength, longer period MRG wave basic states, with the vertical mode number increasing as the square of the MRG wave period.An appendix deals with the case of zonally long and intermediate wavelength MRG waves, where a weak instability regime causes a moderate adjustment involving resonant triad interactions without leading to jet formation. For eastward phase propagating waves, adjustment does not lead to significant angular momentum redistribution.

The dynamics of internal gravity waves is modelled using Wentzel–Kramer–Brillouin (WKB) theory in position–wave number phase space. A transport equation for the phase‐space wave‐action density is derived for describing one‐dimensional wave fields in a background with height‐dependent stratification and height‐ and time‐dependent horizontal‐mean horizontal wind, where the mean wind is coupled to the waves through the divergence of the mean vertical flux of horizontal momentum associated with the waves. The phase‐space approach bypasses the caustics problem that occurs in WKB ray‐tracing models when the wave number becomes a multivalued function of position, such as in the case of a wave packet encountering a reflecting jet or in the presence of a time‐dependent background flow. Two numerical models were developed to solve the coupled equations for the wave‐action density and horizontal mean wind: an Eulerian model using a finite‐volume method and a Lagrangian ‘phase‐space ray tracer’ that transports wave‐action density along phase‐space paths determined by the classical WKB ray equations for position and wave number. The models are used to simulate the upward propagation of a Gaussian wave packet through a variable stratification, a wind jet and the mean flow induced by the waves. Results from the WKB models are in good agreement with simulations using a weakly nonlinear wave‐resolving model, as well as with a fully nonlinear large‐eddy‐simulation model. The work is a step toward more realistic parametrizations of atmospheric gravity waves in weather and climate models.

A finite-volume model of the classic differentially heated rotating annulus experiment is used to study the spontaneous emission of gravity waves (GWs) from jet stream imbalances, which may be an important source of these waves in the atmosphere and for which no satisfactory parameterisation exists. Experiments were performed using a classic laboratory configuration as well as using a much wider and shallower annulus with a much larger temperature difference between the inner and outer cylinder walls. The latter configuration is more atmosphere-like, in particular since the Brunt-Väisälä frequency is larger than the inertial frequency, resulting in more realistic GW dispersion properties. In both experiments, the model is initialised with a baroclinically unstable axisymmetric state established using a two-dimensional version of the code, and a low-azimuthal-mode baroclinic wave featuring a meandering jet is allowed to develop. Possible regions of GW activity are identified by the horizontal velocity divergence and a modal decomposition of the small-scale structures of the flow. Results indicate GW activity in both annulus configurations close to the inner cylinder wall and within the baroclinic wave. The former is attributable to boundary layer instabilities, while the latter possibly originates in part from spontaneous GW emission from the baroclinic wave.

The available meridional sections of zonal velocity with high vertical and meridional resolution reveal tall eastward jets at 2N and 2S, named the extra-equatorial jets (EEJ), straddling the stacked eastward and westward jets of smaller vertical scales right at the equator, the so-called equatorial deep jets (EDJ). In contrast to the semi-annual to interannual fluctuations in the zonal velocity component, the measured meridional velocity component is dominated by intraseasonal period. We argue here that the formation mechanism for both types of jets is linked to the intraseasonal variability in meridional velocity and the associated wave motions. A process study is complemented by high resolution primitive equation simulations based on a realistic background stratification and an oscillating forcing inside the western boundary layer. The forcing confined to the upper 2500 m excites a spectrum of waves, including a baroclinic short Mixed Rossby-Gravity (MRG), whose instability leads to the formation of the EDJ and short barotropic Rossby waves, whose instability gives rise to the EEJ. The modeled EEJ and EDJ response is confined to the same depth range as the forcing. Potential vorticity is homogenized within specific depth ranges of westward EDJ and is found to be latitudinally confined between 2N and 2S by the EEJ. The combined EDJ and EEJ increase lateral mixing at the equator but also act as barriers at ±2 degrees of latitude.

Meridionally confined zonally propagating wave solutions to the linear hydrostatic Boussinesq equations on a generalized equatorial b plane that includes the ''nontraditional'' Coriolis force terms associated with the poleward component of planetary rotation are calculated. Kelvin, Rossby, inertia-gravity, and mixed Rossby-gravity modes generalize from the traditional model with the dispersion relation unchanged. The effects of the nontraditional terms on all waves are the curving upward with latitude of the surfaces of constant phase and the equatorial trapping width of the solutions (the equatorial radius of deformation) increasing by order (V/N) 2 compared to the traditional case, where V is the planetary rotation rate and N the buoyancy frequency. In addition, for the Rossby, inertia-gravity, and mixed Rossby-gravity modes, there is a phase shift of O(V/N) in the zonal and vertical velocity components relative to the meridional component, and their spatial structures are further modified by differences of O(V/N) 2 . For the Rossby and inertia-gravity waves, the modifications depend also on the phase speed of the wave. In the limit N ) V, the traditional approximation is justified and lines of constant phase in the y-z plane become horizontal, whereas for N ( V phase lines become everywhere almost parallel to the planetary rotation vector. In both limits, the phase lines are perpendicular to the dominant restoring force-respectively, gravity and the centrifugal force associated with the solid-body rotation of the atmosphere at rest in the rotating frame. * Current affiliation: Institü t fü r Atmosphä re und Umwelt,

A systematic approach to the direct numerical simulation (DNS) of breaking upper mesospheric inertia‐gravity waves of amplitude close to or above the threshold for static instability is presented. Normal mode or singular vector analysis applied in a frame of reference moving with the phase velocity of the wave (in which the wave is a steady solution) is used to determine the most likely scale and structure of the primary instability and to initialize nonlinear “2.5‐D” simulations (with three‐dimensional velocity and vorticity fields but depending only on two spatial coordinates). Singular vector analysis is then applied to the time‐dependent 2.5‐D solution to predict the transition of the breaking event to three‐dimensional turbulence and to initialize three‐dimensional DNS. The careful choice of the computational domain and the relatively low Reynolds numbers, on the order of 25,000, relevant to breaking waves in the upper mesosphere, makes the three‐dimensional DNS tractable with present‐day computing clusters. Three test cases are presented: a statically unstable low‐frequency inertia‐gravity wave, a statically and dynamically stable inertia‐gravity wave, and a statically unstable high‐frequency gravity wave. The three‐dimensional DNS are compared to ensembles of 2.5‐D simulations. In general, the decay of the wave and generation of turbulence is faster in three dimensions, but the results are otherwise qualitatively and quantitatively similar, suggesting that results of 2.5‐D simulations are meaningful if the domain and initial condition are chosen properly.

We have performed fully resolved three-dimensional numerical simulations of a statically unstable monochromatic inertia–gravity wave using the Boussinesq equations on an $f$-plane with constant stratification. The chosen parameters represent a gravity wave with almost vertical direction of propagation and a wavelength of 3 km breaking in the middle atmosphere. We initialized the simulation with a statically unstable gravity wave perturbed by its leading transverse normal mode and the leading instability modes of the time-dependent wave breaking in a two-dimensional space. The wave was simulated for approximately 16 h, which is twice the wave period. After the first breaking triggered by the imposed perturbation, two secondary breaking events are observed. Similarities and differences between the three-dimensional and previous two-dimensional solutions of the problem and effects of domain size and initial perturbations are discussed.

Depth-dependent barotropic instability of short mixed Rossby-gravity (MRG) waves is proposed as a mechanism for the formation of equatorial zonal jets. High-resolution primitive equation simulations show that a single MRG wave of very short zonal wavelength and small to moderate amplitude is unstable and leads to the development of a largely barotropic, zonally symmetric flow, featuring a westward jet at the equator and extra-equatorial jets alternating in direction with latitude. At higher but still moderate amplitude, westward flow still prevails at the equator at depths of maximum horizontal velocity amplitude in the initial wave, but the long-term equilibrated state can also feature eastward ''superrotating'' jets at the equator near the depths of zero horizontal velocity in the initial wave. The formation of the superrotating jets in the simulations is found to be sensitive to the inclusion of the nontraditional Coriolis force in the equations of motion. A linear theory is used to demonstrate the existence of exponentially growing horizontally nondivergent perturbations with a dominant zonally symmetric zonal velocity component. An argument for the sense and positioning of the jets relative to the equator is given in terms of inertial instability and the meridional mixing of planetary vorticity by the small zonal-scale components of the linearly unstable modes. In the long time evolution of the flow, if the amplitude of the westward equatorial jet becomes too great, zonally symmetric inertial instability limits the growth of the jets, and inertial adjustment leads to the homogenization of potential vorticity in latitude and depth around the equator.