Dynamics in the Mesosphere: Interplay of Rossby Waves and Gravity Waves

For a long time, the mesosphere was also called the "ignoresphere" because observation is difficult there and the height region is beyond the reach of atmospheric general circulation models.

By the early 1980s, it was understood that a weak wind layer exists at a height of approximately 90 km, which is the top of the mesosphere; that this layer is maintained by the momentum carried by gravity waves; and that there is a general circulation from the summer polar region to the winter polar region, as described in the previous section "General Circulation in the Middle Atmosphere."

However, the detailed physics in the mesosphere were not fully elucidated.

We have been studying the detailed structure and physics in the mesosphere by performing simulations using an atmospheric general circulation model that resolves gravity waves and simulates to the top of the mesopause. As a result, we have found that the mesosphere is a very dynamically active region.

There are two main types of waves in the atmosphere: large horizontal-scale waves (2,000 km to 40,000 km) called Rossby waves, and small horizontal-scale waves (tens of km to 2,000 km) called gravity waves.

The numbers in parentheses are approximate horizontal scales, although there are both larger-scale gravity waves and smaller-scale Rossby waves. The difference between them is the restoring force. The restoring force for gravity waves is the buoyancy, whereas the restoring force for Rossby waves is the horizontal gradient of absolute vorticity (when the mean wind is zero, this is simply the latitudinal gradient of the Coriolis force, which is always positive). In addition, Rossby waves are geostrophically balanced, while gravity waves are not. Thus, these two waves are completely different.

The main sources of gravity waves are large-scale winds over mountains, cumulus convection, jet streams, and cyclones. Except for cumulus convection, gravity waves are caused by the geostrophic wind balance, so in a broad sense, gravity waves are generated by Rossby waves.

In the mesosphere, however, gravity waves are the cause of Rossby wave generation, which we will now discuss in detail.

Figure 1 shows the distribution of potential vorticity when Rossby waves are generated in the mesosphere in January, as reproduced by our general circulation model. Although in the usual case the potential vorticity is larger in the north, it has a maximum at 45 degrees north. On the polar side of this peak, the potential vorticity is lower in the north. This anomalous latitudinal gradient is an unstable state for Rossby waves (baroclinic and/or barotropic instability).

**Figure 1** Distribution of potential vorticity (red represents large potential vorticity). Contours represent height (km). The vertical axis is the potential temperature.

This instability is caused by gravity wave forcing.

Figure 2 shows the wave forcing by Rossby waves and gravity waves. In the usual case, gravity wave forcing is strongly negative (a westward acceleration) above the eastward jet in the upper mesosphere.

**Figure 2** Rossby wave (top) & gravity wave (bottom) activity fluxes (arrows point in the direction of the group velocity, while their length indicates wave amplitude). Colors represent respective wave forcing. Contours represent mean zonal winds. During normal conditions (left) and during Rossby wave generation (right).

However, when Rossby waves are excited in the mesosphere, the wave forcing and jet are modified as follows (see also Fig. 2 and Fig. 3):

Strong Rossby waves propagate into the stratosphere from the troposphere, causing strong negative Rossby wave forcing (with westward acceleration) at heights of 50–60 km.
The eastward jet, normally located at 45 degrees north (see left panel of Fig. 2), is decelerated and shifts poleward and downward (see right panel of Fig. 2).
The negative gravity wave forcing above the jet also shifts poleward and downward along with the jet.
Upwelling occurs on the low-latitude side of this gravity wave forcing. The upwelling cause a temperature decrease through adiabatic expansion. As a result, the potential vorticity (PV) at mid-latitude increases and a maximum forms.

**Figure. 3** Formation of potential vorticity (PV) maximum at mid-latitude through the interplay of Rossby waves (PW) and gravity waves (GW).

Thus, Rossby waves from the troposphere trigger a change in the gravity wave forcing, which cause the formation of potential vorticity maxima at mid-latitudes.

Here is where it gets interesting. As previously mentioned, the negative latitudinal gradient of potential vorticity is an unstable field for Rossby waves, which then emits Rossby waves. This can be seen from the structure of the Rossby wave forcing in the upper right panel of Fig. 2, which is positive in the north and negative in the south above 65 km.

Figure 4 shows the relationship between this wave forcing and the meridional flux of potential vorticity. Positive wave forcing carries potential vorticity northward, while negative wave forcing carries potential vorticity southward. Thus, the negative and positive Rossby wave forcings reduce the potential vorticity maximum in the mid-latitudes. In other words, the Rossby waves are generated to eliminate the field that is unstable for them.

Furthermore, Rossby waves on the north and south sides, which transport potential vorticity northward and southward, have different characteristics. The Rossby waves on the south side have long periods of more than 10 days, while the Rossby waves on the north side have short periods of 2–4 days. In other words, the Rossby waves are generated in pairs to reduce the potential vorticity maximum induced by gravity waves.

**Figure. 4** Latitudinal profile of potential vorticity in the mesosphere. The equation in the red box is the theoretical relationship between Rossby wave forcing and potential vorticity flux .

Moreover, this process also changes the structure of the large-scale jet stream in the stratosphere and mesosphere, the meridional circulation, and the large-scale temperature structure. Thus, the mesosphere is a world in which gravity waves and Rossby waves work together.

Adapted from Sato and Nomoto (2015).