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Abstract Two key challenges in the development of data‐driven gravity‐wave parameterizations are generalization, how to ensure that a data‐driven scheme trained on the present‐day climate will continue to work in a new climate regime, and calibration, how to account for biases in the “host” climate model. Both problems depend fundamentally on the response to out‐of‐sample inputs compared with the training dataset, and are often conflicting. The ability to generalize to new climate regimes often goes hand in hand with sensitivity to model biases. To probe these challenges, we employ a one‐dimensional (1D) quasibiennial oscillation (QBO) model with a stochastic source term that represents convectively generated gravity waves in the Tropics with randomly varying strengths and spectra. We employ an array of machine‐learning models consisting of a fully connected feed‐forward neural network, a dilated convolutional neural network, an encoder–decoder, a boosted forest, and a support‐vector regression model. Our results demonstrate that data‐driven schemes trained on “observations” can be critically sensitive to model biases in the wave sources. While able to emulate accurately the stochastic source term on which they were trained, all of our schemes fail to simulate fully the expected QBO period or amplitude, even with the slightest perturbation to the wave sources. The main takeaway is that some measures will always be required to ensure the proper response to climate change and to account for model biases. We examine one approach based on the ideas of optimal transport, where the wave sources in the model are first remapped to the observed one before applying the data‐driven scheme. This approach is agnostic to the data‐driven method and guarantees that the model adheres to the observational constraints, making sure the model yields the right results for the right reasons. 

We extend the Matsuno–Gill model, originally developed on the equatorial$$\beta$$-plane, to the surface of the sphere. While on the$$\beta$$-plane the non-dimensional model contains a single parameter, the damping rate$$\gamma$$, on a sphere the model contains a second parameter, the rotation rate$$\epsilon ^{1/2}$$(Lamb number). By considering the different combinations of damping and rotation, we are able to characterize the solutions over the$$(\gamma, \epsilon ^{1/2})$$plane. We find that the$$\beta$$-plane approximation is accurate only for fast rotation rates, where gravity waves traverse a fraction of the sphere's diameter in one rotation period. The particular solutions studied by Matsuno and Gill are accurate only for fast rotation and moderate damping rates, where the relaxation time is comparable to the time on which gravity waves traverse the sphere's diameter. Other regions of the parameter space can be described by different approximations, including radiative relaxation, geostrophic, weak temperature gradient and non-rotating approximations. The effect of the additional parameter introduced by the sphere is to alter the eigenmodes of the free system. Thus, unlike the solutions obtained by Matsuno and Gill, where the long-term response to a symmetric forcing consists solely of Kelvin and Rossby waves, the response on the sphere includes other waves as well, depending on the combination of$$\gamma$$and$$\epsilon ^{1/2}$$. The particular solutions studied by Matsuno and Gill apply to Earth's oceans, while the more general$$\beta$$-plane solutions are only somewhat relevant to Earth's troposphere. In Earth's stratosphere, Venus and Titan, only the spherical solutions apply. 

Abstract An intermediate complexity moist general circulation model is used to investigate the sensitivity of the quasi‐biennial oscillation (QBO) to resolution, diffusion, tropical tropospheric waves, and parameterized gravity waves. Finer horizontal resolution is shown to lead to a shorter period, while finer vertical resolution is shown to lead to a longer period and to a larger amplitude in the lowermost stratosphere. More scale‐selective diffusion leads to a faster and stronger QBO, while enhancing the sources of tropospheric stationary wave activity leads to a weaker QBO. In terms of parameterized gravity waves, broadening the spectral width of the source function leads to a longer period and a stronger amplitude although the amplitude effect saturates in the mid‐stratosphere when the half‐width exceedsm/s. A stronger gravity wave source stress leads to a faster and stronger QBO, and a higher gravity wave launch level leads to a stronger QBO. All of these sensitivities are shown to result from their impact on the resultant wave‐driven momentum torque in the tropical stratosphere. Atmospheric models have struggled to accurately represent the QBO, particularly at moderate resolutions ideal for long climate integrations. In particular, capturing the amplitude and penetration of QBO anomalies into the lower stratosphere (which has been shown to be critical for the tropospheric impacts) has proven a challenge. The results provide a recipe to generate and/or improve the simulation of the QBO in an atmospheric model.