A Formulation of Unified Three-Dimensional Wave Activity Flux of Inertia–Gravity Waves and Rossby Waves

Abstract

A companion paper formulates the three-dimensional wave activity flux (3D-flux-M) whose divergence corresponds to the wave forcing on the primitive equations. However, unlike the two-dimensional wave activity flux, 3D-flux-M does not accurately describe the magnitude and direction of wave propagation. In this study, the authors formulate a modification of 3D-flux-M (3D-flux-W) to describe this propagation using small-amplitude theory for a slowly varying time-mean flow. A unified dispersion relation for inertia–gravity waves and Rossby waves is also derived and used to relate 3D-flux-W to the group velocity. It is shown that 3D-flux-W and the modified wave activity density agree with those for inertia–gravity waves under the constant Coriolis parameter assumption and those for Rossby waves under the small Rossby number assumption. To compare 3D-flux-M with 3D-flux-W, an analysis of the European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Re-Analysis (ERA-Interim) data is performed focusing on wave disturbances in the storm tracks during April. While the divergence of 3D-flux-M is in good agreement with the meridional component of the 3D residual mean flow associated with disturbances, the 3D-flux-W divergence shows slight differences in the upstream and downstream regions of the storm tracks. Further, the 3D-flux-W magnitude and direction are in good agreement with those derived by R. A. Plumb, who describes Rossby wave propagation. However, 3D-flux-M is different from Plumb’s flux in the vicinity of the storm tracks. These results suggest that different fluxes (both 3D-flux-W and 3D-flux-M) are needed to describe wave propagation and wave–mean flow interaction in the 3D formulation.

The 11-year solar cycle in current reanalyses: a (non)linear attribution study of the middle atmosphere

“…Hence, it would be interesting to extend the discussion of dynamical effects for other relevant characteristics, for example, for the analysis of wave propagation and wave-mean flow interaction using the 3-D formulation (Kinoshita and Sato, 2013). This paper is fully focused on the SC influence, i.e.…”

A Formulation of Three Dimensional Wave Activity Flux Describing Wave Propagation on the Mass-Weighted Isentropic Time Mean Equation

“…However, if the unified dispersion relation applicable to both inertia-gravity waves and Rossby waves is used to calculate the group velocity, the wave activity density needs to be modified (Kinoshita and Sato 2013b).…”

Generalized Transformed Eulerian Mean (GTEM) Description for Boussinesq Fluids

“…Since the EP tensor E is closely related to the wave energy equations (N2010; (C1) in Appendix C), the direction of the EP flux deviates from the group velocity for the Rossby wave (Longuet-Higgins 1964;Noda 1986). This aspect has been discussed in detail by Kinoshita and Sato (2013b) by using an explicit waveform.…”

“…However, it can be shown after manipulation that ( 38′ ) is equivalent to (38) with f 0 replaced by f under the WKB approximation for the perturbation equations used by Kinoshita et al (2010) and Kinoshita and Sato (2013a). On the other hand, Kinoshita and Sato (2013b) showed the difference between ( 38′ ) and (38) when the β effect of the earthʼs rotation was explicitly considered. They also showed that in the quasi-geostrophic approximation −fc KS in ( 38′ ) becomes twice of the geostrophic eddy kinetic energy.…”