Following Hoskins and Karoly (1981), the barotropic vorticity equation can be written as the Mercator projection of a sphere,
where $t$ is time; $x$ is the $x$-axis in a Mercator projection; $\psi $ is the horizontal streamfunction perturbation; ${\bar u_{\rm{M}}}$ and ${\beta _{\rm{M}}}$ are the zonal wind and meridional gradient of absolute vorticity in a Mercator projection, respectively. These two terms are expressed as
where $\bar \Omega $ is the Earth’s angular velocity of rotation, ${\rm{a}}$ is the radius of the Earth and $y$ represents the $y$-axis in a Mercator projection.
The dispersion relationship for the wave-form solution $\exp [i\left({kx + ly - \omega t} \right)]$ of Eq. (1) is
where k, l, and ω are the zonal wavenumber, meridional wavenumber, and circular frequency, respectively. According to Eq. (3), the meridional propagation of Rossby waves requires
where $c = \omega /k$ is the zonal phase velocity. When ${l^2} < 0$, the meridional propagation is trapped. The latitude where ${l^2} = 0$ is called the turning latitude; a wave ray shifts direction from north to south (or from south to north) when arriving at the turning latitude, which reflects a wave ray similar to how a mirror reflects lights. Therefore, the turning latitude is a natural boundary of a wave ray.
When ${l^2} = 0$, Eq. (4) becomes
This means that ${\bar u_{\rm{M}}}$ and ${\,\beta _{\rm{M}}}$ should satisfy Eq. (5) at the turning latitude(s) when the zonal wavenumber and the circular frequency are fixed. South of the jet center, ${\bar u_{\rm{M}}}$ is generally monotonically increasing. If ${\,\beta _{\rm{M}}}$ is also monotonic, Eq. (5) determines at most one turning latitude; if ${\,\beta _{\rm{M}}}$ is double-valued south of the jet center, Eq. (5) determines at most two turning latitudes. North of the jet center, ${\bar u_{\rm{M}}}$ is generally monotonically decreasing, and the situation is the same as that south of the jet center. To conclude, if ${\,\beta _{\rm{M}}}$ is monotonic in the jet, there will be at most one turning latitude; if ${\,\beta _{\rm{M}}}$ is double-valued, there will be at most two turning latitudes; if ${\,\beta _{\rm{M}}}$ is triple-valued, there will be at most three turning latitudes; and so on. Therefore, it is obvious that the number of turning latitudes is determined mainly by ${\,\beta _{\rm{M}}}$, which consists of two parts: the meridional gradient of the planetary vorticity and the relative vorticity. The former depends on the rotation of the Earth and monotonically decreases with latitude, and the latter is determined mainly by the second derivative of ${\bar u_{\rm{M}}}$. Consequently, the variation in ${\,\beta _{\rm{M}}}$ is determined mainly by the second derivative of ${\bar u_{\rm{M}}}$. In any range where the second derivative of ${\bar u_{\rm{M}}}$ increases from a negative to a positive value and the corresponding slope value is larger than that of the absolute meridional gradient of the planetary vorticity, whose slope is negative, ${\,\beta _{\rm{M}}}$ would be a double-valued function of the latitude in that range. This range could appear south of the jet center, north of the jet center, or near the jet center depending on the specific distribution of ${\bar u_{\rm{M}}}$. In any range not far from the jet center, the second derivative of ${\bar u_{\rm{M}}}$ decreases to a negative minimum value at the jet center and then increases. Therefore, if the second derivative of ${\bar u_{\rm{M}}}$ in the range is larger than the absolute meridional gradient of the planetary vorticity (corresponding to a sharper jet), ${\,\beta _{\rm{M}}}$ would be a double-valued function.
On the other hand, the latitude where ${\bar u_{\rm{M}}} = c$ (and hence ${l^2} \to \infty $) defines another boundary of the same wave ray along which the ray would infinitely tend toward the latitude but never approach it. It is a critical line and near which the wave is trapped and the Wentzel−Kramers−Brillouin method is invalid. The phase velocity is a specific value when the zonal wavenumber and the circular frequency are fixed. Since ${\bar u_{\rm{M}}}$ is generally a double-valued function of latitude, it is easy to deduce that there would be two critical lines situated north and south of the jet center, respectively.
In this case, it is clear that the wave energy dispersion path, represented by the wave ray, would not propagate across the entire sphere but would be restricted to a limited region surrounded by critical lines, by turning latitudes, or by both a critical line and a turning latitude. Based on different combinations of critical lines and turning latitudes, we conclude that there are three types of wave energy dispersion regions. The first type is surrounded by two critical lines; the second type is surrounded by a critical line and a turning latitude; and the third type is surrounded by two turning latitudes. It should be noted that Yang and Hoskins (1996) already mentioned these three types of propagation regions. However, they did not carefully analyze the variations in the wave energy and the amplitudes along the wave rays in these types of propagation regions, which will be highlighted and discussed explicitly in this paper.
Now let us examine another important latitude where ${\,\beta _{\rm{M}}} = 0$, which is the necessary condition for barotropic instability. ${\,\beta _{\rm{M}}} = 0$ does not necessarily appear. If ${\,\beta _{\rm{M}}} > 0$ in a westerly jet, Rossby waves would be stable. If ${\,\beta _{\rm{M}}} < 0$ within a certain region in a westerly jet, Rossby waves would have a chance to become unstable. According to Eq. (4), ${l^2} = 0 - {k^2} < 0$ when ${\,\beta _{\rm{M}}} = 0$. This means that Rossby waves cannot propagate across the latitude. Therefore, even though the existence of the latitude is a necessary condition, this latitude would never be located along the path of wave propagation.
According to Eq. (3), the group velocity can be derived as
where ${K^2} = {k^2} + {l^2}$ is the square of the total wavenumber. Three additional useful relations are
where ${{{D_g}}}/{{Dt}} = {\partial }/{{\partial t}} + {{{c}}_g} \cdot \nabla$ represents the rate of change along the wave ray and cg is the group velocity. The first equation in Eq. (7) states that the circular frequency is conserved along the wave ray because the frequency is independent of time in the dispersion relationship of Eq. (3). The second equation in Eq. (7) predicts the invariance of k along the wave ray since the dispersion relationship in Eq. (3) is invariable in the x direction. Equation (6) and the third equation in Eq. (7) constitute a complete system of equations with three variables ($y,k,l$) and three equations. The wave ray routes can be obtained by applying the Runge−Kutta method to integrate the system numerically. It should be noted that, along the wave ray, l could also be directly solved according to Eq. (3) since $\omega $ and k do not change along the wave ray. This is equivalent to solving the third equation in Eq. (7).
According to Bretherton and Garrett (1969), the wave action conservation equation is
where $F = E{\rm{/}}\omega '$ is the wave action density, $E = {K^2}{A^2}/{4}$ is the wave energy density (wave energy averaged over a period), A is the wave amplitude, and $\omega ' = \omega - {\bar u_{\rm{M}}}k$ is the intrinsic frequency. Wave action density is a combined variable to fulfill the formal conservation. It is not convenient to discuss the variations of the wave energy and the amplitude. Therefore, Eq. (8) can be easily rewritten as the wave energy equation
Following the group velocity, the individual variability of the wave energy is
Equation (10) demonstrates that the individual variability of the wave energy along a ray is determined by two factors. The first one, represented by the divergence of the group velocity, denotes the concentration or dispersion of the energy. If the wave energy is concentrated along its dispersion path, the wave energy will increase. According to Liu and Liu (2011),
Therefore, the second one, represented by the product of the wave scale and the gradient of the basic flow, denotes the barotropic energy budget of the wave. If it is larger than zero, the wave extracts barotropic energy from the basic flow by eddy activities, which leads to an increase in the wave energy.
Equation (10) provides a method to calculate the variations in the wave energy and hence the amplitude along a ray and distinguishes the effects of the energy concentration or dispersion and the wave energy budget from the basic flow on the wave energy. It has explicit physical meaning, although it is non-conservative. Therefore, it is more suitable to apply the wave energy equation. However, Eq. (10) cannot be directly integrated along a ray by utilizing the group velocity expression in Eq. (6) because the group velocity values neighboring the wave ray are unknown (Lighthill, 1978). Therefore, it is hard to calculate $\nabla \cdot {{{c}}_g}$ along a ray. Inspired by Lighthill (1978) and Karoly and Hoskins (1982), a new method is proposed here. Since $\nabla \cdot {{{c}}_g} = 0$ means that ${{{c}}_g}$ is a solenoidal vector field (Lighthill, 1978), in terms of the cross-sectional area ${{\delta S}}$ of a thin ray tube (tubular surface made up of rays, to which of course ${{{c}}_g}$ is universally tangential), $\nabla \cdot {{{c}}_g} = 0$ can be expressed as ${{{c}}_g} \cdot \delta {{S}} = {\rm{constant}}$ along a ray. If above derivation works, $\nabla \cdot {{{c}}_g} \ne 0$ means that ${{{c}}_g} \cdot \delta {{S}}$ varies along a ray. According to Karoly and Hoskins (1982), ${{{c}}_g} \cdot \delta {{S}} = {c_{g,x}}\delta x + {c_{g,y}}\delta y$, where $\delta x$ and $\delta y$ are the section areas in x- and y- axis, respectively. Therefore,
where the subscript t denotes time and $\delta t$ represents a short time interval. According to Eq. (12), $\nabla \cdot {{{c}}_g}$ along a wave ray can be easily solved by applying the values of the group velocity along the ray. Then, the wave energy Eq. (10) can be easily solved.
A developing Rossby wave may be accompanied by increasing wave energy or amplitude, or both. Then, how to define a developing Rossby wave becomes a real question. Here, we adopt the definition by Lu and Zeng (1981), who pointed out that a perturbation develops only if both its energy and amplitude increase. Compared with the criteria that highlight either the wave energy or the amplitude, it is a strict one. They made such a definition based on two reasons. One is that, although the wave energy is increasing, it is dispatched into larger areas due to larger horizontal scale (e.g., longer wavelength). This leads to a decrease in the amplitude. In such a case, the wave is not significant and even decays a little. The other one is that, although the amplitude is increasing, the wave energy is decreasing. Of course, as they suggested, the concentration of the wave energy in a small local region can also make significant transient local synoptic phenomena. This is of interest too. Since a developing Rossby wave is defined by both an increase in the wave energy and the amplitude, a decaying wave is defined by both a decrease in the wave energy and the amplitude.