Wave ray tracing method

1. Wave ray equations based on horizontally nonuniform basic flow

The dispersion relation describing the propagation characteristics of perturbations can be derived from the linearized barotropic non-divergent vorticity equation on a time-mean slowly varying basic state with the WKB approximation (eg. Karoly 1983, Li and Nathan 1997, Li and Li 2012, Li et al. 2015, Zhao et al. 2015) as

                                                                                                  ,                                   (1) 

 where  are the zonal and meridional component of the basic flow under Mercator projection, ,are the gradient of the basic state absolute vorticity along the longitude and latitude, are the zonal wavenumber, meridional wavenumber, and the angular frequency, respectively. For stationary waves, the wavenumber is defined as

                                                                                                      .                                                (2)

The zonal and meridional components of group velocity take the form

                                                                                                   ,                         (3)

                                                                                                    .                        (4)

The ray path is a trajectory locally tangent to the group velocity vector (Lighthill 1978). Hence, we can detect the energy dispersion by calculating the ray trajectories. Due to the longitudinal and latitudinal variation of the basic state, both l and k change along the ray paths. Their evolution is determined by kinematic wave theory (Whitham 1960) as follows:

                                                                                                       ,     (5)

                                                                                                        ,        (6)

where  denotes the Lagrangian variation moving at the group velocity. Therefore, the ray trajectory can be integrated through equations (3)-(6) after the basic state and the initial state given.

References

Karoly, D. J., 1983: Rossby wave propagation in a barotropic atmosphere. Dyn. Atmos. Oceans, 7, 111-125.

Li, L., and T. R. Nathan, 1997: Effects of low-frequency tropical forcing on intraseasonal tropical-extratropical interactions. J. Atmos. Sci., 54, 332-346.

Li Y., and J. Li, 2012: Propagation of planetary waves in the horizontal non-uniform basic flow (in Chinese). Chinese J. Geophys., 55, 361-371.

Li, Y., J. Li, F. Jin, S. Zhao. 2015: Interhemispheric propagation of the stationary Rossby waves in a horizontally non-uniform basic flow.J. Atmos. Sci., 72, 3233-3256, doi:10.1175/JAS-D-14-0239.1.

Lighthill, J., 1978: Waves in fluids. Cambridge University Press, 504.

Whitham, G., 1960: A note on group velocity. J. Fluid Mech., 9, 347-352.

Zhao, S., J. Li, and Y. Li (2015), Dynamics of an interhemispheric teleconnection across the critical latitude through a southerly duct during boreal winter, J. Clim., 28, 7437-7456, doi:10.1175/JCLI-D-14-00425.1.

2. Availability to the code

The code is written in Fortran 90 language. For those finding of interest in using the wave ray tracing method as a verification of the wave energy dispersion pathways, please ask Prof. Jianping Li (ljp@ouc.edu.cn) to get the code. It is appreciated if you'd like to introduce your plan about the use of this code. We have used it to examine the energy dispersion trajectory in several published papers. 

3. Publications using this wave ray tracing code

Sun, C., J. Li*, S. Zhao, 2015: Remote influence of Atlantic multidecadal variability on Siberian warm season precipitation. Sci. Rep., online, doi: 10.1038/srep16853.

Zhao, S., J. Li*, Y. J. Li, 2015: Dynamics of an Interhemispheric Teleconnection across the Critical Latitude through a Southerly Duct during Boreal Winter. J. Climate., 28, 7437-7456, doi:10.1175/JCLI-D-14-00425.1.

Li, Y. J., J. Li*, F. F. Jin, and S. Zhao, 2015: Interhemispheric Propagation of Stationary Rossby Waves in a Horizontally Nonuniform Background Flow. J. Atmos. Sci., 72, 3233–3256, doi: 10.1175/JAS-D-14-0239.1

Xu, H. L., J. Li*, J. Feng, J. Y. Mao, 2013: The asymmetric relationship bewteen the winter NAO and the precipitation in southwest China. Acta Meteorol. Sin., 70, 1276-1291. 

Li Y., and J. Li, 2012: Propagation of planetary waves in the horizontal non-uniform basic flow (in Chinese). Chinese J. Geophys., 55, 361-371.