Study of Propagation Properties of Rossby Waves in the Atmosphere and Relationship Between the Phase Velocity and the Group Velocity

Using Rossby wave equations, the dispersion equation is developed. The wave normal diagram for Rossby waves on a beta plane is a circle in wave number (k, l) space whose center is displaced along the negative X axis, and whose radius is less than this displacement, which means that phase propagation is entirely westward. The phase velocity diagram is a circle whose center is displaced along the negative X axis, the group velocity diagram is an ellipse whose center is displaced westward and whose major and minor axes give the maximum all the directions group speeds as function of the frequency and parameter Q.


Introduction
The wave type that is of most importance for large-scale meteorological processes is the Rossby wave, or planetary wave. In an inviscid barotropic fluid of constant depth (where the divergence of the horizontal velocity must vanish), the Rossby wave is an absolute vorticity-conserving motion that owes its existence to the variation of the Coriolis parameter with latitude, the so-called β-effect. More generally, in baroclinic atmosphere, the Rossby wave is a potential vorticity conserving motion that owes its existence to the isentropic gradient of potential vorticity [1]. Rossby waves play a central role in geophysical fluid dynamics and dynamical meteorology, particularly in the dynamics of quasi-geostrophic flow [2]. Their reflection properties at coast lines help to explain certain features of western boundary currents [3]. The anisotropic and dispersive propagation properties of Rossby waves (in particular the "backward" property in which phase and group velocities are opposite in the N-S direction) have been invoked, together with equatorial heating, to explain dipole-like formation of equatorial easterly jets accompanied, at higher latitudes, by westerly jets [4]. In addition, an inverse turbulent cascade together with Rossby waves can lead to the formation of zonal flows [2]. These ideas have been further developed recently [5] in models of an eddy-driven jet. Rossby wave propagation can be understood in a qualitative fashion by considering a closed chain of fluid parcels initially aligned along a circle of latitude [1].
The major type of planetary scale waves can be classified into several cases, based on energy source or horizontal and vertical structure. Based on energy source, of planetary scale waves can be divided into free and forced modes. The free modes are the normal oscillations in the atmosphere. They can be excited by small random forcing and are limited in amplitude by dissipation. The free planetary (rossby) modes are global extent and with periods of about 2, 5, 10 and 16 days [6,7]. The forced Rossby modes can be excited by flow over topography or by land-sea heating contrasts. Based on horizontal structure classification, planetary waves can be divided into two, global modes and equatorial modes. The global modes are the planetary waves can propagate both meridionally and zonally. Rossby waves are the most important global modes. The equatorial modes are trapped in the equatorial wave-guide and propagate zonally along the equator. Kelvin and Rossby-gravity waves are the most important equatorial modes. In addition to the above two classification, planetary waves can be divided into external and internal modes from their vertical structure. External modes are vertically trapped modes and the energy density of such modes decays exponentially in the vertical. Internal modes are vertically propagating (phase surfaces tilt with height) modes and they can transfer momentum and energy vertically over many scale heights. The forced Rossby waves, Kelvin waves, Rossby-gravity waves and Gravity waves can propagate vertically and transfer energy and momentum between the lower and upper atmosphere under suitable background conditions [8].
Rossby waves play a vital role in the dynamics of the middle atmosphere. They primarily responsible for the asymmetries in the polar vortex, sudden warming in the stratosphere, forcing of the Quasi-biennial Oscillation (OBO), Semi-Annual Oscillation (SAO). Meridionally propagating Rossby waves carry angular momentum between the tropics and middle latitudes. Planetary waves can reach large amplitudes if they propagate upwards to the MLT region. Vertical propagation is determined by wave/mean flow interactions [9]. Most traveling waves are westward propagating, and so they can propagate through the eastward winds of the winter stratosphere, but not the westward winds of the summer stratosphere. The continental configuration and topography of the northern hemisphere produces more planetary wave activity than in the southern hemisphere. This planetary wave activity acts to cause sudden stratospheric warming and break down the polar vortex at the end of winter in the northern hemisphere. In contrast, the southern polar vortex is more stable and there has only been one major sudden stratospheric warming in the southern hemisphere, which occurred in 2002. [7] 2. The − β β β β Plane We saw in the derivation of the barotropic vorticity equation the potential importance of the fact that the Coriolis parameter varies with latitude, a consequence of spherical geometry. However, dealing with spherical geometry is (a little) more complicated than with planar geometry, so it is common to represent a strip of the sphere-limited in latitude but going all the way around the world in longitude-as a plane, as in Figure 1.
Assuming that the width of the strip is small enough, we can approximate ( ) f ϕ as Taylor series about the central latitude: . Substituting for y , we get

Purpose of Study
Study of relationship between the phase velocity and the group velocity of Rossby waves in the atmosphere.

The Dispersion Equation
To get the dispersion equation for Rossby waves lets depend on the quasi-geostrophic equations: is the geostrophic stream function, p′ is the perturbation in pressure.
Hydrostatic balance can be written as: Subtraction (7) from (6) and using the mass-continuity equation (3) to obtain: is the Z component of the absolute vorticity associated with the geostrophic flow. Using equations (4) and(5) the vertical velocity can be expressed as: Substitution of this expression into the vorticity equation (8) and further careful manipulation lead finally to the quasigeostrophic potential vorticity equation: Where Γ is the elliptic operator.
In addition, the quasi-geostrophic potential vorticity equation linearizes to: Whereψ is a complex amplitude, , , k l m are wavenumbers on X, Y, Z. We obtain the dispersion relation for Rossby waves:

The Phase Velocity
The phase velocity p v K ω = can be written:

The Group Velocity
The group velocity g v K ω ∂ = ∂ can be written It is now straightforward to eliminate k in favour of gx v from equation (21), which on substitution into the square of equation (22) gives directly the group velocity curve in the form The southward group velocity is simply a reflection of the northward group velocity in the x-axis. Equation (24) can be written The group velocity diagram is a ellipse whose origin is displaced westward by represented in Figure 4. The maximum eastward and westward group speeds are given by and the maximum northward (southward) group speed also follows as Examples of the behavior of external group velocities are shown in Figure 5. The ellipse collapses to the origin as 2 Which represented in Figure 6. It is of some interest to emphasize that Rossby waves are backward in the sense that the latitudinal components of their phase and group velocities are always in opposite directions. This property can be invoked to describe the formation of a dipole pair of jets in the following way: Northward (away from the equator) wave energy flux is associated with southward, towards the equator, wave momentum flux; and the opposite in the case of southward directed energy flux, away from the equator, corresponds to northward (towards the equator) momentum flux. In other words, a pole ward energy flux from the equator is associated with an equator ward flux of momentum. Hence, Rossby wave dynamics implies that localized equatorial heating gives rise to equatorial easterly zonal jets. This "convergence" of equatorial momentum implies a deficit at higher latitudes such that a westerly jet must necessarily form there [12].

Conclusion
the phase speed is negative, so the phase of rossby waves always propagates westward. Since ω is a nonlinear function of k , Rossby waves are dispersive. The magnitude of the group velocity is, typically, greater for the westward propagating long waves than for the eastward propagating short waves. The latitudinal components for Rossby waves of their phase and group velocities are always in opposite direction.