Diaz, M., and A. Aiyyer, 2013: Energy Dispersion in African Easterly Waves. *J. Atmos. Sci.*, **70**, 130–

This study sets out to understand why African easterly waves (AEW) have a westward phase speed, but an eastward group velocity. Previous studies of AEW energetics haven’t really considered this aspect of AEWs, but this study presents a convincing case that techniques used to understand midlatitude baroclinic waves can be very useful to understand AEW dynamics.

First, group velocity can be unfamiliar to some, so let’s lay out a definition.

Group Velocity: The propagation speed of the shape of the waves’ amplitudes, or envelope of the wave packet. This can be physically interpreted as the velocity at which energy or information is conveyed along a wave.

I find this whole idea of a wave packet group velocity difficult to visualize, unless I think of how it looks in a Hovmoller plot. Below is a composite wave packet from this paper showing waves having westward phase velocity. The eastward group velocity occurs when the peak amplitude of the subsequent waves moves to the east with time.

Viewing this in a composite sense makes me wonder if the compositing method might mask large variability in the group velocity for individual events. Maybe a better question is: if there was large variability in the group velocity, how would this smear out the composite?

In the case of mid-latitude wave packets, the phase and group velocity are generally in the same direction. The figure below shows a lag regression of mid-latitude baroclinic waves withan eastward phase and group velocity from Chang (1993). In this case, waves are amplified downstream, at the expense of the upstream waves.

The main tool of this study is the *eddy kinetic energy* budget, shown in (1). This budget requires us to define the time or space scales of the eddies that we are interested in. For a very detailed derivation of the eddy kinetic energy budget, see here.

The first term on the LHS of (1) is the “pressure work” term. A series of previous papers starting with Orlansky and Katzfey (1991) have used this type of budget analysis to analyze the energy dispersion in midlatitude baroclinic waves.

Expanding the pressure work term in (2) we get the *geopotential flux convergence* (GFC) and the baroclinic conversion of eddy available potential energy to eddy kinetic energy. The GFC is useful for diagnosing energy dispersion and group velocity. However, the GFC is a scalar quantity, and we would like to know the direction of the flux. The geostrophic wind is *non-convergent*, so it naturally doesn’t add anything to the GFC. Therefore we need the *ageostrophic* wind to calculate a geopotential flux vector that is relevant for explaining a pattern of GFC.

The geopotential flux convergence and vectors can be seen in the panel c below, and it’s clear that this is a major contribution to the total EKE tendency in d.

Another interesting aspect of this study is the spatial variability of the group velocity. Although the AEW packets have a preferential eastward group velocity, parts of the wave can have actually have a *westward* group velocity.

“The negative PV gradient associated with the anticyclonic shear side of the AEJ is surrounded on all sides by a positive PV gradient. The barotropic Rossby wave dispersion relation predicts westward phase ve- locity and eastward group velocity within a positive meridional PV gradient and the reverse within a nega- tive gradient. Since AEWs extend into both gradients, they could theoretically have both eastward and west- ward group velocity.”

The figure below shows the composite ageostrophic geopotential flux averaged over a full wavelength, along with the mean PV gradient in contours. The area of strongest negative PV gradient around 15N, which is also the level where the African easterly jet maximizes and AEWs have their strongest amplitude, indicates a *westward* group velocity at 650hPa. North and south of this area the flux indicates an eastward group velocity.

“This pattern implies that AEWs at 650 hPa have both eastward and westward group velocity depending on their latitude and is consistent with the view that barotropic instability involves the mutual interaction of two counterpropagating Rossby edge waves in the horizontal plane (e.g., Hoskins et al. 1985).”

At lower levels around 15N the flux is eastward, so the AEW group velocity changes direction with height. However, in the vertical integral, these signals cancel out giving no net group velocity at ~15N. South and North of this region the vertically integrated flux is eastward.