The mixed Rossby-gravity wave on the spherical Earth

Area of Science:

• Atmospheric dynamics
• Geophysical fluid dynamics
• Wave theory

Background:

• The mixed Rossby-gravity (MRG) wave is a key phenomenon in atmospheric and oceanic dynamics.
• Existing theories often rely on approximations like the equatorial β-plane, limiting their applicability.

Purpose of the Study:

• To revisit and refine the theory of the MRG wave on a sphere.
• To develop an analytic solution valid for a wider range of parameters.
• To compare spherical and equatorial β-plane approximations.

Main Methods:

• Derivation of an ad hoc analytic solution for the MRG wave.
• Analysis of westward-propagating waves and their properties.
• Approximation of governing equations using Schrödinger eigenvalue equations.
• Numerical solutions of shallow-water equations for validation.

Main Results:

• A novel analytic solution accurately describes MRG waves for moderate Lamb parameters.
• Westward-propagating waves with specific phase speeds exist on a sphere, unlike the equatorial β-plane.
• MRG waves are identified as ground-state solutions connecting Rossby and zonally symmetric waves.
• Longuet-Higgins' asymptotic solutions are unsuitable for high Lamb parameters; Matsuno's relation is accurate for smaller Lamb parameters than expected.

Conclusions:

• The derived analytic solution provides a more robust description of MRG waves on a sphere.
• The study highlights limitations of the equatorial β-plane approximation for MRG waves.
• This work advances the theoretical understanding of atmospheric and oceanic wave dynamics.