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We discuss applications of a recently developed method for model reduction based on linear response theory of weakly coupled dynamical systems. We apply the weak coupling method to simple stochastic differential equations with slow and fast degrees of freedom. The weak coupling model reduction method results in general in a non-Markovian system; we therefore discuss the Markovianization of the system to allow for straightforward numerical integration. We compare the applied method to the equations obtained through homogenization in the limit of large timescale separation between slow and fast degrees of freedom. We numerically compare the ensemble spread from a fixed initial condition, correlation functions and exit times from a domain. The weak coupling method gives more accurate results in all test cases, albeit with a higher numerical cost.

Motions on planetary spatial scales in the atmosphere are governed by the planetary geostrophic equations. However, little attention has been paid to the interaction between the baroclinic and barotropic flows within the planetary geostrophic scaling. This is the focus of the present study, which utilizes planetary geostrophic equations for a Boussinesq fluid supplemented by a novel evolution equation for the barotropic flow. The latter is affected by meridional momentum flux due to baroclinic flow and drag by the surface wind. The barotropic wind, on the other hand, affects the baroclinic flow through buoyancy advection. Via a relaxation towards a prescribed buoyancy profile the model produces realistic major features of the zonally symmetric wind and temperature fields. We show that there is considerable cancellation between the barotropic and the baroclinic surface zonal mean zonal winds. Linear and nonlinear model responses to steady diabatic zonally asymmetric forcing are investigated, and the arising stationary waves are interpreted in terms of analytical solutions. We also study the problem of baroclinic instability on the sphere within the present model.

A general circulation model is used to study the interaction between parameterized gravity waves (GWs) and large-scale Kelvin waves in the tropical stratosphere. The simulation shows that Kelvin waves with substantial amplitudes (∼10 m s−1) can significantly affect the distribution of GW drag by modulating the local shear. Furthermore, this effect is localized to regions above strong convective organizations that generate large-amplitude GWs, so that at a given altitude it occurs selectively in a certain phase of Kelvin waves. Accordingly, this effect also contributes to the zonal-mean GW drag, which is large in the middle stratosphere during the phase transition of the quasi-biennial oscillation (QBO). Furthermore, we detect an enhancement of Kelvin-wave momentum flux due to GW drag modulated by Kelvin waves. The result implies an importance of GW dynamics coupled to Kelvin waves in the QBO progression.
Plain Language Summary: The variability of the tropical atmosphere at altitudes of about 18–40 km is dominated by a large-amplitude long-term oscillation of wind, the quasi-biennial oscillation, which has a broad impact on the climate and seasonal forecasting. This oscillation is known to be driven by various types of atmospheric waves with multiple spatial scales. Using a numerical model, this study reports a process of interaction between those waves on different scales, which has not been illuminated before. The result implies a potential importance of this process in the progression of the quasi-biennial oscillation. Proper model representations of these multiscale waves and tropical convection are required to simulate this process.

Motivated by the question of whether and how wave–wave interactions should be implemented into atmospheric gravity-wave parametrizations, the modulation of triadic gravity-wave interactions by a slowly varying and vertically sheared mean flow is considered for a non-rotating Boussinesq fluid with constant stratification. An analysis using a multiple-scale WKBJ (Wentzel–Kramers–Brillouin–Jeffreys) expansion identifies two distinct scaling regimes, a linear off-resonance regime, and a nonlinear near-resonance regime. Simplifying the near-resonance interaction equations allows for the construction of a parametrization for the triadic energy exchange which has been implemented into a one-dimensional WKBJ ray-tracing code. Theory and numerical implementation are validated for test cases where two wave trains generate a third wave train while spectrally passing through resonance. In various settings, of interacting vertical wavenumbers, mean-flow shear, and initial wave amplitudes, the WKBJ simulations are generally in good agreement with wave-resolving simulations. Both stronger mean-flow shear and smaller wave amplitudes suppress the energy exchange among a resonantly interacting triad. Experiments with mean-flow shear as strong as in the vicinity of atmospheric jets suggest that internal gravity-wave dynamics are dominated in such regions by wave modulation. However, triadic gravity-wave interactions are likely to be relevant in weakly sheared regions of the atmosphere.