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This study derives a complete set of equatorially confined wave solutionsfrom an anelastic equation set with the complete Coriolis terms, which includeboth the vertical and meridional planetary vorticity. The propagation mechanismcan change with the effective static stability. When the effective staticstability reduces to neutral, buoyancy ceases, but the role of buoyancy as aneastward-propagation mechanism is replaced by the compressional beta effect,i.e., vertical density-weighted advection of the meridional planetaryvorticity. For example, the Kelvin mode becomes a compressional Rossby mode.Compressional Rossby waves are meridional vorticity disturbances that propagateeastward owing to the compressional beta effect. The compressional Rossby wavesolutions can serve as a benchmark to validate the implementation of thenontraditional Coriolis terms (NCTs) in numerical models; with an effectivelyneutral condition and initial large-scale disturbances given a half verticalwavelength spanning the troposphere on Earth, compressional Rossby waves areexpected to propagate eastward at a phase speed of 0.24 m s${}^{-1}$. The phasespeed increases with the planetary rotation rate and the vertical wavelengthand also changes with the density scale height. Besides, the compressional betaeffect and the meridional vorticity tendency are reconstructed using reanalysisdata and regressed upon tropical precipitation filtered for the Madden$-$Julianoscillation (MJO). The results suggest that the compressional beta effectcontributes 10.8% of the meridional vorticity tendency associated with the MJOin terms of the ratio of the minimum values.

The compressional beta effect: Analytical solution, numerical benchmark, and data analysis