Authors:

Ray Chew, Mark Schlutow, and Rupert Klein

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Abstract:

The traditional approximation neglects the cosine components of the Coriolis acceleration, and this approximation has been widely used in the study of geophysical phenomena. However, the justification of the traditional approximation is questionable under a few circumstances. In particular, dynamics with substantial vertical velocities or geophysical phenomena in the tropics have non-negligible cosine Coriolis terms. Such cases warrant investigations with the non-traditional setting, i.e. the full Coriolis acceleration. In this manuscript, we study the effect of the non-traditional setting on an isothermal, hydrostatic and compressible atmosphere assuming a meridionally homogeneous flow.

Employing linear stability analysis, we show that, given appropriate boundary conditions, i.e. a bottom boundary condition that allows for a vertical energy flux and non-reflecting boundary at the top, the atmosphere at rest becomes prone to a novel unstable mode. The validity of assuming a meridionally homogeneous flow is investigated via scale analysis. Numerical experiments were conducted, and Rayleigh damping was used as a numerical approximation for the non-reflecting top boundary. Our three main results are as follows: (i) experiments involving the full Coriolis terms exhibit an exponentially growing instability, yet experiments subjected to the traditional approximation remain stable; (ii) the experimental instability growth rate is close to the theoretical value; (iii) a perturbed version of the unstable mode arises even under sub-optimal bottom boundary conditions. Finally, we conclude our study by discussing the limitations, implications and remaining open questions. Specifically, the influence on numerical deep-atmosphere models and possible physical interpretations of the unstable mode are discussed.

Plain Language Summary:

A simplification of the effects of the Earth’s rotation is often used in the study of geophysical phenomena. However, this simplification may not be valid under certain conditions. We demonstrate via mathematical analysis that if the simplification assumptions are removed, and the full effect of the Earth’s rotation is considered, the Earth’s atmosphere exhibits a hitherto unknown instability under the condition that there is energy transfer from the Earth’s surface into the atmosphere. We conducted idealised computational experiments that corroborate this analysis. The existence of this novel instability has implications for real-world weather prediction and may explain certain physical observations.

Key Points:

1. We demonstrated via linear stability analysis that, given appropriate boundary conditions, the atmosphere at rest becomes prone to a novel unstable mode.
2. Numerical simulations persistently exhibit the unstable mode across varying experimental setups and for boundary conditions that deviate from those studied in theory.
3. This novel unstable mode may have implications for numerical weather prediction employing the non-traditional setting and may explain some physical phenomena.