The energy flux of three-dimensional waves in the atmosphere: Exact expression for a basic model diagnosis with no equatorial gap

A model diagnosis for the energy flux of off-equatorial Rossby waves in the atmosphere has previously been done using quasi-geostrophic equations and is singular at the equator. The energy flux of equatorial waves has been separately investigated in previous studies using a space-time spectral analysis or a ray theory. A recent analytical study has derived an exact universal expression for the energy flux which can indicate the direction of the group velocity for linear shallow water waves at all latitudes. This analytical result is extended in the present study to a height-dependent framework for three-dimensional waves in the atmosphere. This is achieved by investigating the classical analytical solution of both equatorial and off-equatorial waves in a Boussinesq fluid. For the horizontal component of the energy flux, the same expression has been obtained between equatorial waves and off-equatorial waves in the height-dependent framework, which is linked to a scalar quantity inverted from the isentropic perturbation of Ertel’s potential vorticity. The expression of the vertical component of the energy flux requires computation of another scalar quantity that may be obtained from the meridional integral of geopotential anomaly in a wavenumber-frequency space. The exact version of the universal expression is explored and illustrated for three-dimensional waves induced by an idealized Madden-Julian Oscillation forcing in a basic model experiment. The zonal and vertical fluxes manifest the energy transfer of both equatorial Kelvin waves and off-equatorial Rossby waves with a smooth transition at around 10°S and around 10°N. The meridional flux of wave energy represents connection between off-equatorial divergence regions and equatorial convergence regions.