On the Energy Dispersion of Magnetic Rossby Waves

Abstract

The energy dispersion of magnetic Rossby waves has been investigated by applying the two-dimensional incompressible magnetohydrodynamic (MHD) equations in both uniform basic flow and basic magnetic field. The dispersion relation suggests that the magnetic Rossby waves can be divided into fast- and slow-propagating modes, respectively. The fast-propagating mode propagates eastward and is similar to the fast Alfvén waves. The energy dispersion speed is faster than the phase speed, which means the perturbation energy can lead the perturbations themselves to arrive downstream. The slow-propagating waves with smaller (larger) horizontal scales are similar to those of the slow Alfvén waves (Rossby waves). The zonal group velocity is slower than the zonal phase speed for the slow-propagating magnetic Rossby waves. For the slow-propagating waves that propagate eastward, this means that the perturbation energy may trigger new perturbations that are located upstream of the perturbations themselves. The group velocity vector is basically same as (opposite of) the wavevector for the fast-propagating (slow-propagating) magnetic Rossby waves that propagate eastward. The endpoints of the group velocity vectors and the wavevector multiplying a factor are located on a cycle in the wavenumber space. Due to the uniform basic flow and the uniform basic magnetic field, the energy dispersion paths (called rays) are straight lines. Along the straight-line rays, the wave action, wave energy, and amplitude keep their initial values, and the wave neither develops nor decays.

The energy dispersion of magnetic Rossby waves in zonally nonuniform basic states

“…w ˆ, 2 w ˆand 3 w ˆare three unequal real roots, representing three types of neutral magnetic Rossby waves. I note that there are only two modes of magnetic Rossby waves if the basic states are zonally uniform as in the previous study (Li 2022)…”

“…where t denotes time and t d denotes a short time interval; S d denotes the unit area that is perpendicular to the local group velocity vector. Based on the method, the variations in wave energy and amplitude along a moving ray are extensively discussed for Rossby waves on the Earth's atmosphere (Li et al 2021b, 2022.…”

“…Wave ray theory can be applied in interpreting teleconnection phenomena on the Earth's atmosphere and thereby has been extensively investigated (Karoly and Hoskins 1982, Karoly 1983, Hoskins and Ambrizzi 1993, Li and Nathan 1994, Yang and Hoskins 1996, Li and Nathan 1997, Li et al 2015. Recently, Li et al (2021a) derived a formula to solve the divergence of group velocity and discussed the variations in wave energy and amplitude (Li et al 2021a, Li et al 2021b, 2022.…”

“…And if the basic magnetic field and motion field are uniform, the group velocity components are constant values and the rays propagate in straight lines (Acheson and Hide 1973). Recently, Li (2022) discussed the upstream and downstream effects for magnetic Rossby waves and pointed out the endpoints of group velocity vector and wave vector are on a cycle on wavenumber space, analogical to the ones for HD Rossby waves. However, the energy dispersion paths (called rays) are straight lines due to uniform basic states.…”

The Dynamics of Magnetic Rossby Waves in the Quasigeostrophic Shallow Water Magnetohydrodynamic Theory

“…The QG equations can also be derived from the rotational compressible MHD flow with well-prepared initial data (Kwon et al 2018) and from separating the baroclinic modes of the three-dimensional Boussinesq fluid (Balk 2022). Linearization of the QG SWMHD system over the rest state with a constant background magnetic field results in the dispersion relation for magnetic Rossby waves (Zeitlin 2013), which embodies the influence of the divergence of a free surface, and hence slightly different from the one in the two-dimensional incompressible MHD system (e.g., Hide 1966;Zaqarashvili et al 2007;Li 2022).…”