Ray paths of stationary Rossby waves emanating from a local midlatitude source are usually refracted equatorward. However, this general tendency for equatorward propagation is mitigated by the presence of a midlatitude jet that acts as a zonal waveguide. This opens up the possibility of circum-global teleconnections and quasi-resonance, which suggests that the ability to guide a wave in the zonal direction is an important jet property. This paper investigates waveguidability of idealized midlatitude jets in a barotropic model on the sphere. A forced-dissipative model configuration with a local source for Rossby waves is used in order to quantify waveguidability by diagnosing the latitudinal distribution of waviness in a longitudinal sector far downstream of the forcing. Systematic sensitivity experiments show that waveguidability increases smoothly with increasing jet amplitude and with decreasing jet width. This result is contrasted with the predictions from two idealized theoretical concepts based on (1) ray tracing as derived from Wentzel–Kramers–Brillouin (WKB) theory and (2) a sharp jet with a zonally oriented front of potential vorticity. The existence of two so-called turning latitudes, which is the key diagnostic for a zonal waveguide according to ray tracing theory, turns out to be a poor predictor for the dependence of waveguidability on jet amplitude and jet width obtained in the numerical simulations. By contrast, the meridional gradient of potential vorticity correlates fairly well with the diagnosed waveguidability. The poor predictions from ray tracing are not surprising, because the underlying WKB assumptions are not satisfied in the current context. The failure of WKB is traced back to the properties of the underlying equations, and a heuristic argument is presented to elucidate the potential of the potential vorticity (PV) gradient to act as a proxy for waveguidability.

Rossby waves are a ubiquitous feature of the atmospheric flow in the upper troposphere

An important aspect in connection with midlatitude Rossby waves is the extent to which they are ducted in the zonal direction. As is well known, there is a general tendency for Rossby waves to be refracted equatorward owing to the sphericity of the Earth

To the extent that the midlatitude background flow represents an efficient waveguide, this may lead to circumglobal Rossby waves, which in turn can result in circumglobal teleconnections

A key argument in the work of previous authors to support the theory of quasi-resonance draws on the ideas of ray tracing, which in turn are based on the Wentzel–Kramers–Brillouin (WKB) theory

The basic tenet of WKB theory is that the scale

This state of affairs motivates the current work, in which we consider the waveguidability of midlatitude jets in an idealized modeling framework and investigate the validity of ray tracing arguments. Our approach is partly based on the work of

The paper is organized as follows. First in Sect.

We consider non-divergent barotropic flow on a sphere in a forced-dissipative configuration. In our analysis we focus on the stationary part of the solution, which is obtained through temporal averaging. The key model variable is absolute vorticity

The forcing is implemented as pseudo-orographic forcing with a rather local orography, i.e.,

The dissipative term in Eq. (

Any deviation from purely zonal flow will be considered to be an eddy or, more specifically, a wave. Correspondingly, we will use maps of the meridional wind

The flow is initialized with the zonally symmetric background state. The latter is defined by specifying a zonally symmetric zonal flow field

Numerical solutions are obtained using a standard pseudo-spectral scheme. To ensure numerical stability, we add a hyperdiffusion term

For some choices of the background wind our numerical solutions turn out to be highly transient. This transience is due to barotropic instability in the case of a strong narrow jet. In all our experiments these unstable modes are characterized by a large eastward phase velocity. This is illustrated in Fig.

Development of a barotropically unstable initial state for a model setup with a strong narrow jet at 45

For illustration we consider two very different background flows, one with pure solid-body rotation (corresponding to

Latitudinal profiles of the background zonal wind

Illustration of the numerical solution for an initial state with

Apparently, for pure solid-body rotation (Fig.

By contrast, the solution for the strong jet case (Fig.

Before we do so, however, in this section we review some well-known concepts for the analysis of stationary Rossby waves and their propagation in a spherical domain. This will help us to interpret our results in the later parts of the paper.

First, we linearize the model equation (Eq.

For further progress, it turns out it is convenient to perform a coordinate transformation corresponding to a Mercator projection of the sphere onto a plane (see

For later reference we note that for a background flow with pure solid-body rotation as in Eq. (

Diagnosing the zonally symmetric background state for three different cases: solid-body rotation

If the factors

Ray tracing arguments consider the propagation of Rossby waves in those regions which allow wave propagation (e.g.,

Turning latitudes can easily be diagnosed from profiles of

The jet scenario is particularly interesting in the present context. Whenever there are two turning latitudes (

Let us return to the case of a background flow with pure solid-body rotation according to Eq. (

Another school of thought associates a midlatitude zonal waveguide with the existence of a sharp meridional gradient of PV

The background flow associated with a PV discontinuity is a westerly jet with a cusp-like peak at the latitude of the discontinuity. Solutions of the linearized equations can be found which are wavelike in the zonal direction and evanescent in the meridional direction like

Based on the theoretical concepts sketched out in the previous section, we now define waveguidability in the framework of our numerical model, explore it systematically for various background flows, and compare the results with predictions from ray tracing theory.

As we argued before, background solid-body rotation can be considered a reference scenario in which the waves propagate along great circles from one hemisphere to the other. As a consequence, wave activity emanating from a local northern hemispheric Rossby wave source can be expected to be located almost entirely in the Southern Hemisphere some 180

These considerations motivated us to use the following method to quantify waveguidability. Introducing wave enstrophy of the stationary part of the solution as

Probability density function

The probability density function

Our method to quantify waveguidability is based on, but also extends, the ideas from

We now systematically vary the amplitude

Dependence of waveguidability

We tested the sensitivity of our results with respect to the particular metric (Eq.

The numerical results from Fig.

Zonal Fourier spectrum of

One could argue that the above interpretation of ray tracing theory is too brute force because it does not account for the existence of an entire spectrum of zonal wavenumbers (see Fig.

It is enlightening to focus on the behavior in Fig.

How does the other theoretical concept, in which the meridional gradient of PV plays the key role for waveguidability, fare? To the extent that the PV gradient is a relevant proxy for waveguidability, we expect a close correlation between

Same as Fig.

Results from the parameter sweep involving

Overall, a comparison between Figs.

We now aim to obtain a deeper understanding for why WKB-based ray tracing theory provides partly misleading results regarding waveguidability. For this purpose we linearize the model equation (Eq.

We, first, investigate whether or not our numerical solutions are close to linear. For this purpose we repeated the simulations with the forcing amplitude

Numerical solution of the stationary part

We now write the perturbation PV in terms of the perturbation streamfunction,

The other question we want to discuss here is why the PV gradient within the jet is possibly a more appropriate proxy for its waveguidability. Let us, for the moment, consider the unforced, undamped version of Eq. (

The remaining question in this line of argument is why the

In this paper we investigated the waveguidability of idealized midlatitude jets using a barotropic model for flow on a sphere. Our focus was on stationary waves in a forced-dissipative model configuration with a local source for Rossby waves. This setup allowed us to quantify waveguidability (in percent) by diagnosing the latitudinal distribution of the explicitly simulated wave enstrophy in a longitudinal sector far downstream of the forcing. We carried out a systematic sensitivity study by varying the amplitude and the width of the jet.

Our numerical solutions indicate that waveguidability increases smoothly with increasing jet amplitude and with decreasing jet width. This result is in contrast to the prediction from WKB-based ray tracing theory, where the waveguide is either perfect or nonexistent, depending on the existence or nonexistence of two turning latitudes (one on either side of the jet). Another weakness of ray tracing theory is that the profile of the all-important stationary wavenumber saturates at some point for fairly narrow and strong jets; this theory would, therefore, not predict any further increase in waveguidability as the jet's amplitude is further increased, which is in conflict with the simulated behavior. To be sure, the overall poor predictive power of ray tracing theory in the current application is not surprising because the underlying WKB assumptions are not satisfied. We conclude that a literal application of ray tracing theory can be misleading; in particular, the existence of two turning latitudes is by no means a binary event at which the waveguidability suddenly changes in a fundamental way.

In addition, we found that waveguidability in our simulations correlates much better with the meridional gradient of background PV at jet latitude than with the predictions from ray tracing theory: both the meridional PV gradient and waveguidability vary smoothly as a function of jet amplitude and jet width. This result is consistent with the idealized concept of Rossby waves on a zonally oriented PV front suggesting that sharp meridional gradients of PV are conducive to zonal Rossby wave propagation. The good correlation with the background PV gradient was argued to be due to the forced-dissipative character of our model setup. A heuristic argument was presented based on the idea that the reach of a Rossby wave packet is determined by a balance between the dissipation rate and the speed of propagation of the Rossby wave packet; to the extent that the zonal speed of propagation is proportional to the meridional PV gradient, waveguidability in the zonal direction should, indeed, be related to the PV gradient.

Analysis of the linearized model equation (but without the WKB approximation) indicates that the amplitude of the forced stationary solution is governed by an inhomogeneous one-dimensional Helmholtz equation. This implies that the solution at a given grid point depends on the spatial distribution of the coefficients in a nonlocal fashion. Furthermore, a wave-like solution can tunnel through small regions of evanescent behavior, which means that the existence of two turning latitudes is not necessarily equivalent to a perfect waveguide. By contrast, the ray path solution only depends on the local values of the coefficients which prevents the possibility of tunneling. This argument helps to understand why the literal interpretation of ray tracing theory can be misleading and why the existence of two turning latitudes is often associated with values of waveguidability well below 100 %.

Overall, our analysis is not very affirmative regarding the utility of ray tracing for diagnosing waveguidability of Rossby waves in midlatitude jets. This seems to be in conflict with results from earlier publications such as

Despite these caveats, an analysis that focuses on the existence of two turning latitudes in the background flow may still have some merit. As suggested by our Fig.

On the more constructive side, our work suggests that the meridional PV gradient might be a suitable proxy for waveguidability. Obviously, this hypothesis is currently lacking any underlying closed theory. In addition, it should be tested in more realistic modeling frameworks, because our barotropic model entirely lacks the partitioning of PV into a contribution due to vorticity and a contribution due to static stability. By contrast, Ertel PV in a primitive equation model would be able to capture the sharp gradients of PV on isentropes at the boundary between the upper troposphere and the lower stratosphere

The computer code is available from the author upon request.

The author declares that he has no conflict of interest.

The author wants to thank Olivia Martius and Tim Woollings for constructive criticism on the previous version of this paper, helping to improve the presentation of the material.

Part of the research leading to these results has been carried out within the subproject “Upscale impact of diabatic processes
from convective to near-hemispheric scale” of the Transregional Collaborative Research Center SFB/TRR 165 “Waves to Weather” (

This paper was edited by Juerg Schmidli and reviewed by Tim Woollings and Olivia Romppainen-Martius.