Theoretical basis for stationary Rossby wave propagation

1 Classic theory on zonal basic flow $\bar{u} = \bar{u}(y), \bar{v} = 0$

The classical Rossby wave propagation theory (Hoskins and Karoly 1981) begin with a nondivergent barotropic vorticity equation linearized about $\bar{u} = \bar{u}(y)$ on a Mercator projection of a sphere takes the form

$$ \left( \frac{\partial}{\partial t} + \bar{u}_m \frac{\partial}{\partial x} \right) \nabla_M^2 \psi' + \bar{\beta}_m \frac{\partial \psi' }{\partial x} = 0 $$

where $\bar{u}_m = \frac{u}{\cos \varphi}$ is the ambient zonal wind and

$$ \bar{\beta}_m = \frac{2 \Omega \cos^2 \varphi}{a} - \frac{\cos\varphi}{a^2} \frac{\partial}{\partial \varphi} \left[ \frac{1}{\cos\varphi} \frac{\partial \bar{u} \cos\varphi }{\partial \varphi} \right] $$

is the meridional gradient of the absolute vorticity in the Mercator projection.

The dispersion relation is

$$ \omega = \bar{u}_m k - \frac{ \bar{\beta}_m k }{k^2+l^2} $$

where $\omega$ is the frequency of the disturbances; $k$ and $l$ are zonal and meridional wavenumbers, respectively. The latitudinal distribution of the meridional wavenumber $l$ can be calculated for a given zonal wavenumber $k$ for the stationary wave ($\omega =0$), yielding $$ l^2(\varphi) = K_s^2 - k^2$$ where $K_s$ is the stationary wavenumber, defined as $K_s = \sqrt{{\bar{\beta}_m}/{\bar{u}_m}}$.

The stationary waves can propagate if the flow is westerly ($\bar{u}_m$ positive) with positive meridional absolute vorticity gradient ($\bar{\beta}_m$ positive). The case with easterly flow and negative $\bar{\beta}_m$ is usually not considered since it is rare in the real atmosphere.

Please refer to and Hoskins and Ambrizzi (1993) for the details.

Hoskins and Ambrizzi (1993) suggested that the classic theory for zonally averaged flow can be applied to a realistic longitudinally varying flow by considering the local latitudinal zonal wind profile; i.e., by replacing $\bar{\beta}_m(\varphi)$ and $\bar{u}_m(\varphi)$ in the expression of stationary wavenumber with the local flow state $\bar{\beta}_m(\lambda, \varphi)$ and $\bar{u}_m(\lambda, \varphi)$ separately to obtain the local stationary wavenumber $K_s(\lambda, \varphi)$. This extension is based on the assumptions that (1) meridional flow and the zonal absolute vorticity gradient may be neglected to a first order approximation compared with zonal flow and the meridional absolute vorticity gradient, respectively; and (2) the meridional wavelength is greater than or equal to the zonal wavelength.

Many interhemispheric teleconnections cannot be understood via the classic stationary Rossby wave theory (see discussions in Zhao et al. 2015).

2 Rossby wave theory on a horizontally non-uniform flow $\bar{u} = \bar{u}(x, y), \bar{v} = \bar{v}(x, y)$

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 (Karoly 1983, Li and Nathan 1997, Li et al. 2015, Zhao et al. 2015) as

$$ \omega = \bar{u}_m k+\bar{v}_m l + \frac{ \bar{q}_x l - \bar{q}_y k }{k^2+l^2} $$

where $\bar{u}_m = \frac{u}{\cos \varphi}, \bar{v}_m = \frac{v}{\cos \varphi}$ are the zonal and meridional component of the basic flow under Mercator projection, $\bar{q}_x$ and $\bar{q}_y$ are the gradient of the basic state absolute vorticity $\bar{q}=2\Omega \sin⁡ \varphi + \nabla^2 \bar{\psi}$ along the longitude and latitude, $k, l, \omega$ are the zonal wavenumber, meridional wavenumber, and the angular frequency, respectively.

Using the dispersion relation and kinematic wave theory (Whitham 1960; Bühler 2009) gives the following ray tracing equations $$\frac{D_g k}{Dt}=-k\frac{∂\bar{u}_m}{∂x}-l\frac{∂\bar{v}_m}{∂x}-\frac{l \frac{∂^2\bar{q}}{∂x^2}-k\frac{∂^2\bar{q}}{∂x∂y}}{K^2}$$ $$\frac{D_g l}{Dt}=-k\frac{∂\bar{u}_m}{∂y}-l\frac{∂\bar{v}_m}{∂y}-\frac{l \frac{∂^2\bar{q}}{∂x∂y}-k\frac{∂^2\bar{q}}{∂y^2}}{K^2}$$ $$\frac{D_g x}{Dt}=u_g=\bar{u}_m+\frac{(k^2-l^2)\frac{∂\bar{q}}{∂y}-2kl\frac{∂\bar{q}}{∂x}}{K^4}$$ $$\frac{D_g y}{Dt}=v_g=\bar{v}_m+\frac{2kl\frac{∂\bar{q}}{∂y}+(k^2-l^2)\frac{∂\bar{q}}{∂x}}{K^4}$$ where the local group velocity vector ${c}_g=\left(u_g,v_g\right)$ and $\frac{D_g}{Dt}=\partial/\partial t+{c}_g\bullet\nabla$ represents the derivative along group velocity rays.

See Li et al. (2015) and Zhao et al. (2015) for details of Rossby wave propagation behaviors on a horizontally non-uniform flow.